A Worked Symmetric Cournot Model

Introduction

In Cournot competition, firms choose quantities at the same time, knowing that the market price depends on their combined output. Each firm must therefore account for the quantity it expects its rival to produce. This tutorial works through the symmetric two-firm case: the firms face the same linear inverse demand and have the same constant marginal cost.

The central calculation is not to apply a monopoly formula twice. Instead, each firm chooses its own best response to the other firm’s quantity, and equilibrium occurs where both firms’ choices are best responses to one another. The methods here are aligned with the SOA VEE Economics (Validation by Educational Experience) syllabus.

Model assumptions: Two firms sell a homogeneous good and choose nonnegative quantities simultaneously. Total quantity is $Q=q_1+q_2$, and inverse demand is $P(Q)=a-bQ$, where $a>0$ and $b>0$. Each firm has the same constant marginal cost $c$, with $0\leq c<a$, and no fixed costs. There are no capacity constraints. Each firm takes its rival’s quantity as given when choosing its own quantity.

Here, $q_i$ is firm $i$’s output and $\pi_i$ is its profit. In the interior case, where both firms produce positive quantities, profit equals price minus marginal cost, multiplied by the firm’s own quantity:

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

The inverse-demand equation describes the market price for total output. A firm does not set a separate price: it chooses its quantity, and the resulting market price is determined by $Q$. The assumption $c<a$ ensures that positive production can be profitable when the rival produces zero.

Deriving the Symmetric Equilibrium

To find a firm’s best response, hold the rival’s quantity $q_j$ fixed and maximize firm $i$’s profit with respect to $q_i$. The profit function is a quadratic in $q_i$:

\pi_i=(a-c-bq_j)q_i-bq_i^2

Its first derivative with respect to $q_i$ is $a-c-bq_j-2bq_i$, and its second derivative is $-2b<0$. Thus, when the optimum is positive, setting the first derivative equal to zero gives the firm’s profit-maximizing quantity. Including the possibility of a zero quantity, the best response is:

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

In a symmetric interior equilibrium, both firms produce the same positive quantity, so $q_1=q_2=q$. Substituting the rival’s quantity $q$ into the best-response equation gives the equilibrium condition. Solving it produces each firm’s quantity:

q=\frac{a-c-bq}{2b} \qquad\Longrightarrow\qquad 3bq=a-c \qquad\Longrightarrow\qquad q=\frac{a-c}{3b}

The model’s parameter restrictions make this solution positive. Total output is twice each firm’s output. Substituting total output into inverse demand gives the equilibrium price. Since each firm sells $q$ units at that price and has marginal cost $c$, its profit is $(P-c)q$.

Symmetric two-firm Cournot result: When $a>c\geq 0$, $b>0$, and the assumptions above hold, each firm produces $q=(a-c)/(3b)$. Total output is $Q=2(a-c)/(3b)$, the market price is $P=(a+2c)/3$, and each firm’s profit is $\pi=(a-c)^2/(9b)$.

Why the Equilibrium Formulas Hold

The best-response condition follows directly from profit maximization: for a fixed rival quantity, the profit increase from raising output is the first derivative of profit with respect to own output. At an interior optimum, that derivative must be zero; the negative second derivative confirms that profit is strictly concave in own output, so this stationary point is a maximum.

The equilibrium quantity follows because symmetry requires each firm to choose the other firm’s quantity as its own best response. The price and profit formulas then follow from total output, inverse demand, and the definition of profit. In detail, substituting $Q=2(a-c)/(3b)$ into demand gives the price:

P=a-b\left(\frac{2(a-c)}{3b}\right) =a-\frac{2(a-c)}{3} =\frac{a+2c}{3}

The difference between price and marginal cost is $(a-c)/3$. Multiplying that margin by each firm’s quantity gives:

\pi=(P-c)q =\left(\frac{a-c}{3}\right)\left(\frac{a-c}{3b}\right) =\frac{(a-c)^2}{9b}

This result is for two firms with identical costs in the specified linear-demand model. It should not be carried over unchanged to firms with different costs, more than two firms, or a different demand specification.

Worked Example: Calculate and Check the Base Equilibrium

Two firms sell a product with inverse demand $P=90-Q$. Each firm has constant marginal cost $c=30$ dollars per unit and no fixed costs. Find each firm’s equilibrium quantity, total output, price, and profit. Then check firm 1’s best response when firm 2 produces its equilibrium quantity.

Method. Match the demand equation to $P=a-bQ$: $a=90$ and $b=1$. The common marginal cost is $c=30$, so $a-c=90-30=60>0$, and the interior symmetric solution applies.

Each firm produces:

q=\frac{a-c}{3b} =\frac{90-30}{3(1)} =\frac{60}{3} =20

Total output is $Q=20+20=40$. The market price is $P=90-40=50$ dollars per unit. Each firm’s profit is its margin times its own output: $(50-30)(20)=20(20)=400$ dollars.

Check. If firm 2 produces $20$, firm 1’s profit as a function of its own quantity is $[90-(q_1+20)-30]q_1=(40-q_1)q_1$. Its derivative is $40-2q_1$, which equals zero at $q_1=20$; its second derivative is $-2<0$. Thus firm 1’s chosen quantity is its best response. By symmetry, firm 2’s quantity is also a best response.

Common mistake avoided: Treating the combined output of 40 as each firm’s output would give the wrong profit calculation. Each firm sells 20 units, while the price depends on the total of 40 units.

Worked Example: Calculate Outcomes with a Different Demand Slope

Suppose inverse demand is $P=120-2Q$, and both firms have constant marginal cost $c=24$ dollars per unit, with no fixed costs. Calculate each firm’s equilibrium quantity, total output, price, and profit. Verify the result using the best-response equation.

Method. Here $a=120$, $b=2$, and $a-c=120-24=96>0$. Each firm’s equilibrium quantity is:

q=\frac{120-24}{3(2)} =\frac{96}{6} =16

Total output is $Q=2(16)=32$. Substitution into inverse demand gives $P=120-2(32)=120-64=56$ dollars per unit. Each firm’s profit is $(56-24)(16)=32(16)=512$ dollars.

To check, firm $i$’s best response to the rival’s output is $(a-c-bq_j)/(2b)=(96-2q_j)/4$, when this is positive. At $q_j=16$, it is $(96-32)/4=64/4=16$, as required. The result is internally consistent.

Common mistake avoided: The slope $b=2$ affects both the best response and the equilibrium quantity. Using $b=1$ in the denominator would incorrectly give $q=32$ for each firm.

How a Marginal-Cost Change Affects the Calculation

A parameter change can be analyzed by calculating the equilibrium once with the original parameter and again with the changed value, keeping the other assumptions and parameters fixed. In this model, a change in $c$ affects the net demand intercept $a-c$. The example below shows how that change carries through to each firm’s quantity, total output, price, and profit.

Worked Example: Recalculate Equilibrium After a Cost Increase

Two identical firms face inverse demand $P=100-2Q$. Initially, each has constant marginal cost $c=28$ dollars per unit. The marginal cost then rises to $c=34$ dollars per unit for both firms. Assume no fixed costs before or after the change. Calculate and compare the symmetric equilibria.

Method. First use $a=100$, $b=2$, and $c=28$. Each firm initially produces $(100-28)/(3(2))=72/6=12$ units, so total output is $Q=24$. Price is $100-2(24)=52$ dollars per unit, and each firm’s profit is $(52-28)(12)=24(12)=288$ dollars.

After the cost increase, use $c=34$, keeping $a=100$ and $b=2$. Each firm then produces $(100-34)/(3(2))=66/6=11$ units, so total output is $Q=22$. Price is $100-2(22)=56$ dollars per unit, and each firm’s profit is $(56-34)(11)=22(11)=242$ dollars.

Comparison. Each firm’s output falls by $12-11=1$ unit, total output falls by $24-22=2$ units, and price rises by $56-52=4$ dollars per unit. Each firm’s profit falls by $288-242=46$ dollars. The higher market price does not mean that profit rises: the margin increases from $52-28=24$ to $56-34=22$ dollars per unit, while output also falls. Multiplying the margin by quantity gives the stated profits.

Common mistake avoided: Reusing the old equilibrium price after changing marginal cost would ignore the firms’ new best responses and the resulting change in total output. Recompute quantities first, then use demand to find price.

Common Mistake: Using the Monopoly Quantity for Each Firm

For a single monopolist facing the same linear inverse demand and constant marginal cost, the monopoly quantity is found by maximizing the monopolist’s profit over total market output. That is a different choice problem from the one faced by either firm in Cournot competition. The monopoly quantity is not the quantity for each firm in a two-firm Cournot equilibrium.

For example, with $P=90-Q$ and $c=30$, applying the linear-demand monopoly quantity formula gives $Q=(90-30)/(2(1))=30$ for the monopolist. A mistaken approach might assign that quantity to each Cournot firm, giving total output $Q=60$ and price $P=90-60=30$, equal to marginal cost. The resulting profit for each firm would be $(30-30)(30)=0$. But at a rival output of 30, a firm’s best response is $(90-30-30)/(2)=15$, not 30. The proposed quantities are therefore not mutual best responses. The correct Cournot quantities are 20 for each firm, as the first worked example verifies.

A reliable calculation sequence is: identify $a$, $b$, and $c$; calculate each firm’s quantity; add the quantities to obtain $Q$; use inverse demand to find $P$; and calculate each firm’s profit using its own quantity. Keep the distinction between each firm’s output and market output explicit at every step.

Key takeaway: In the symmetric two-firm Cournot model with linear inverse demand $P=a-bQ$ and common constant marginal cost $c<a$, each firm’s positive equilibrium quantity is $(a-c)/(3b)$. Total output is the sum of the two firm quantities; price comes from market demand, and each firm’s profit uses its own output.

Check Your Understanding

For each problem, calculate the requested outcomes from the stated two-firm model. Assume firms choose quantities simultaneously, have identical constant marginal costs, and have no fixed costs.

Inverse demand is $P=84-Q$, and marginal cost is $c=24$ dollars per unit. Find each firm’s equilibrium quantity, total output, market price, and profit per firm. (Answer: Each firm produces 20 units; total output is 40, price is $44, and profit per firm is $400.)
Inverse demand is $P=150-3Q$, and each firm’s marginal cost is $c=30$ dollars per unit. Find each firm’s quantity, total output, price, and profit per firm. (Answer: Each firm produces $40/3$ units; total output is $80/3$, price is $70, and profit per firm is $1600/3$ dollars, or $533.33 to the nearest cent.)
Inverse demand is $P=96-2Q$, and common marginal cost is $c=24$ dollars per unit. Calculate equilibrium price and profit per firm, then verify one firm’s best response to its rival’s equilibrium quantity. (Answer: Each firm produces 12 units, total output is 24, price is $48, and profit per firm is $288; the best response to 12 rival units is 12 units.)
Inverse demand shifts from $P=110-2Q$ to $P=134-2Q$, while both firms’ marginal cost remains $c=26$ dollars per unit. Calculate each firm’s quantity, price, and profit before and after the shift. (Answer: Before, each produces 14 units, price is $54, and profit per firm is $392. After, each produces 18 units, price is $62, and profit per firm is $648.)
Inverse demand is $P=140-2Q$, and both firms’ marginal cost rises from $c=20$ to $c=32$ dollars per unit. Calculate each firm’s quantity, market price, and profit per firm under both cost levels. (Answer: At $c=20$, each produces 20 units, price is $60, and profit per firm is $800. At $c=32$, each produces 18 units, price is $68, and profit per firm is $648.)

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