Cournot Equilibrium with Unequal Costs

Why Cost Differences Matter in Cournot Competition

In Cournot competition, firms choose quantities at the same time, knowing that the market price depends on their combined output. When firms have different marginal costs, each firm’s output choice reflects not only the demand facing the market but also its own cost. A lower-cost firm generally produces more; the higher-cost firm may produce less or, under some conditions, nothing.

The analysis here uses a two-firm model. The earlier material on Cournot best responses establishes the strategic setting; the key task is to solve the two firms’ choices when their costs differ. The result depends on whether the proposed equilibrium quantities are both positive, so it is important to check that condition rather than apply an interior formula automatically.

Model and assumptions: Two firms sell a homogeneous good and choose nonnegative quantities \(q_1\) and \(q_2\) simultaneously. Inverse market demand is \(P=a-bQ\), where \(Q=q_1+q_2\), \(a>0\), and \(b>0\). Firm 1 and firm 2 have constant marginal costs \(c_1\) and \(c_2\), respectively, with no fixed costs. Each firm takes the other firm’s quantity as given when choosing its own. The linear demand applies over the relevant output range; if the model permits prices below zero, the equilibrium must still be economically meaningful for the market being studied.

Best Responses with Different Costs

Given firm 2’s output \(q_2\), firm 1’s profit is revenue less variable cost. Firm 2’s profit has the analogous form. Since price depends on total output, each firm’s choice affects the price earned on all units it sells.

\pi_1(q_1,q_2)=[a-b(q_1+q_2)-c_1]q_1,\qquad \pi_2(q_1,q_2)=[a-b(q_1+q_2)-c_2]q_2.

For a positive output choice, differentiating each profit with respect to its own quantity gives the first-order condition. Each profit is a concave quadratic in its own quantity because its second derivative is \(-2b<0\). Thus the first-order condition gives the profit-maximizing quantity when that quantity is positive; when the unconstrained choice is negative, the nonnegativity constraint makes the best response zero.

\frac{\partial \pi_1}{\partial q_1}=a-c_1-2bq_1-bq_2,\qquad \frac{\partial \pi_2}{\partial q_2}=a-c_2-bq_1-2bq_2.

Best responses: With nonnegative output, the firms’ best-response functions are

q_1=\max\left\{0,\frac{a-c_1-bq_2}{2b}\right\},\qquad q_2=\max\left\{0,\frac{a-c_2-bq_1}{2b}\right\}.

The maximum with zero matters: a negative unconstrained output is not feasible and means that the firm chooses not to produce.

Interior Equilibrium: Both Firms Produce

An interior equilibrium is one in which both firms choose positive quantities. In that case, both first-order conditions hold with equality and the maximum-with-zero portions of the best responses do not bind. Solving the two equations gives quantities that depend separately on the two firms’ costs.

2bq_1+bq_2=a-c_1,\qquad bq_1+2bq_2=a-c_2.

To solve, subtract the second equation from twice the first. This eliminates \(q_2\) and gives \(3bq_1=a-2c_1+c_2\). Subtracting the first equation from twice the second similarly gives \(3bq_2=a-2c_2+c_1\). Therefore, the interior equilibrium quantities are:

q_1^*=\frac{a-2c_1+c_2}{3b},\qquad q_2^*=\frac{a-2c_2+c_1}{3b}.

These quantities describe an interior equilibrium only when both numerators are positive. If either numerator is zero or negative, the corresponding interior formula cannot be used as a feasible positive-output choice; the relevant best response must instead be evaluated at zero.

Interior-equilibrium condition: Both firms produce positive quantities when

a-2c_1+c_2>0 \quad\text{and}\quad a-2c_2+c_1>0.

Equivalently, each firm’s cost is sufficiently low relative to the demand intercept and the other firm’s cost.

Adding the two equilibrium quantities gives total output. Substitution into inverse demand gives the market price. The price and profits take especially useful forms:

Q^*=\frac{2a-c_1-c_2}{3b},\qquad P^*=\frac{a+c_1+c_2}{3}.

Because each firm’s first-order condition implies \(P^*-c_i=bq_i^*\), its operating profit can also be written as \(b(q_i^*)^2\). This expression is valid here because there are no fixed costs.

\pi_i^*=(P^*-c_i)q_i^*=b(q_i^*)^2,\qquad i=1,2.

Deriving the Price and Profit Results

The price formula follows directly from total output. The sum of the two interior quantities is \((2a-c_1-c_2)/(3b)\). Substituting this sum into \(P=a-bQ\) yields the stated price. For firm 1, its first-order condition is \(a-c_1-2bq_1^*-bq_2^*=0\). Rearranging gives:

$$ a-b(q_1^*+q_2^*)-c_1=bq_1^*. $$

The left side is \(P^*-c_1\), the difference between market price and firm 1’s marginal cost. Multiplying both sides by \(q_1^*\) proves \(\pi_1^*=(P^*-c_1)q_1^*=b(q_1^*)^2\). Applying the same reasoning to firm 2 proves its profit formula. This also explains why a positive equilibrium quantity implies a positive price-cost margin in this model.

Worked Example: Unequal Costs and Interior Equilibrium

Suppose inverse demand is \(P=120-2Q\), so \(a=120\) and \(b=2\). Firm 1 has marginal cost \(c_1=20\), and firm 2 has marginal cost \(c_2=40\). Find each firm’s equilibrium quantity, the market price, and both profits. The method is to solve the two interior first-order conditions, then verify positivity and each firm’s best response.

Firm 1’s quantity. Substitute into the interior formula: \(q_1^*=(120-2(20)+40)/(3(2))=(120-40+40)/6=120/6=20\).

Firm 2’s quantity. \(q_2^*=(120-2(40)+20)/6=(120-80+20)/6=60/6=10\). Both quantities are positive, so the interior calculation is valid.

Total output and price. \(Q^*=20+10=30\), so \(P^*=120-2(30)=120-60=60\). The price formula gives the same result: \((120+20+40)/3=180/3=60\).

Profits. Firm 1 earns \((60-20)(20)=40(20)=800\). Firm 2 earns \((60-40)(10)=20(10)=200\).

Check. Firm 1’s first-order condition evaluates to \(120-20-2(2)(20)-2(10)=100-80-20=0\). Firm 2’s evaluates to \(120-40-2(2)(10)-2(20)=80-40-40=0\). Both conditions hold, and the calculated profits are positive.

Worked Example: How a Cost Difference Changes Output and Price

Let inverse demand be \(P=120-Q\). Firm 1 has marginal cost \(30\), while firm 2 has marginal cost \(45\). Calculate the equilibrium quantities, price, and profits, then compare them with the equal-cost outcome in which both firms have marginal cost \(30\). The method is to solve each case separately; replacing two different costs with an average would hide the allocation between firms.

Unequal-cost quantities. Here \(a=120\), \(b=1\), \(c_1=30\), and \(c_2=45\). Thus \(q_1^*=(120-60+45)/3=105/3=35\), and \(q_2^*=(120-90+30)/3=60/3=20\). Both firms produce positive quantities.

Unequal-cost price and profits. Total output is \(35+20=55\), so price is \(120-55=65\). Firm 1’s profit is \((65-30)(35)=35(35)=1{,}225\). Firm 2’s profit is \((65-45)(20)=20(20)=400\).

Equal-cost comparison. If both costs are \(30\), each quantity is \((120-30)/3=30\). Total output is \(60\), price is \(120-60=60\), and each firm’s profit is \((60-30)(30)=900\).

Interpretation and check. Relative to the equal-cost case, the lower-cost firm produces 35 rather than 30, while the higher-cost firm produces 20 rather than 30. Total output falls from 60 to 55 and price rises from 60 to 65. The first-order conditions in the unequal-cost case are \(120-30-2(35)-20=0\) for firm 1 and \(120-45-35-2(20)=0\) for firm 2. The resulting margins, 35 and 20, equal the respective quantities, as \(b=1\) requires.

When the Higher-Cost Firm Produces Nothing

The interior formulas do not cover every cost combination. Suppose firm 1 is the lower-cost firm and produces alone. Its best response to \(q_2=0\) is \(q_1=(a-c_1)/(2b)\), provided \(a>c_1\). To check whether firm 2 wants to enter, evaluate its marginal profit from a small increase in output at zero. At \(q_2=0\), that marginal profit is \(a-c_2-bq_1\). Firm 2 optimally remains at zero if this value is nonpositive.

Substituting firm 1’s single-firm quantity into that condition gives:

a-c_2-b\frac{a-c_1}{2b}\leq 0 \quad\Longleftrightarrow\quad c_2\geq\frac{a+c_1}{2}.

Thus, when \(a>c_1\) and \(c_2\) is at least \((a+c_1)/2\), the lower-cost firm produces \((a-c_1)/(2b)\), the higher-cost firm produces zero, and price is \(a-bq_1=(a+c_1)/2\). At equality, the high-cost firm’s marginal profit at zero is exactly zero, and zero remains a best response. If both firms’ costs are at least \(a\), neither can earn positive operating profit by producing, and the zero-output outcome applies.

Worked Example: Checking for a Zero-Output Firm

Suppose inverse demand is \(P=100-Q\), firm 1’s marginal cost is \(20\), and firm 2’s marginal cost is \(65\). Determine whether both firms produce, and calculate the equilibrium price and profits. The method is to test the interior condition and, if it fails, check firm 2’s incentive to produce when firm 1 is alone.

Test the interior candidate. The formula gives \(q_1=(100-40+65)/3=125/3\) and \(q_2=(100-130+20)/3=-110/3\). A negative quantity is infeasible, so this is not an interior equilibrium.

Find firm 1’s output if firm 2 produces zero. Its best response is \(q_1=(100-20)/(2(1))=80/2=40\). The resulting price is \(100-40=60\).

Check firm 2’s incentive. The entry threshold is \((a+c_1)/2=(100+20)/2=60\). Since \(c_2=65\geq60\), firm 2’s best response is zero. Directly, its marginal profit at zero output is \(100-65-40=-5\), which is negative.

Profits and check. Firm 1 earns \((60-20)(40)=1{,}600\); firm 2 earns zero. Firm 1’s first-order condition is \(100-20-2(40)=0\), and firm 2 has no incentive to enter. The corner equilibrium is therefore consistent with both firms’ choices.

Common Exam Trap: Averaging the Costs

A tempting shortcut is to average the costs and then apply a symmetric Cournot formula. That shortcut can produce the correct total output or price in some numerical cases, but it does not correctly determine which firm produces how much or what each firm earns.

For the first worked example, the average cost is \((20+40)/2=30\). Treating both firms as if their cost were 30 gives the symmetric formula \(q=(120-30)/(3(2))=15\) for each firm. That allocation is wrong: the unequal-cost equilibrium is \(q_1=20\) and \(q_2=10\). The average-cost shortcut happens to give total output 30 and price 60 in this example, but it assigns the wrong quantities and would give the wrong firm-level profits. Use each firm’s own marginal cost in its own first-order condition, then check the quantity constraints.

Key takeaway: With two firms and linear inverse demand, an interior Cournot equilibrium gives each firm a quantity determined by both costs, but with twice the weight on its own cost. Verify that both quantities are positive. If the high-cost firm’s interior quantity is nonpositive, check whether the lower-cost firm’s single-firm output makes entry unprofitable.

Check Your Understanding

For each problem, calculate or verify the requested equilibrium using the firms’ own costs and the nonnegative-quantity condition.

Inverse demand is \(P=84-2Q\). Marginal costs are \(c_1=12\) and \(c_2=24\). Find both equilibrium quantities, the price, and each firm’s profit. (Answer: \(q_1=14,\ q_2=8,\ P=40,\ \pi_1=392,\ \pi_2=128\).)
Inverse demand is \(P=150-3Q\), with marginal costs \(c_1=30\) and \(c_2=45\). Find the interior quantities and profits, and verify the equilibrium price from total output. (Answer: \(q_1=15,\ q_2=10,\ Q=25,\ P=75,\ \pi_1=675,\ \pi_2=300\).)
Inverse demand is \(P=100-2Q\), with \(c_1=20\) and \(c_2=60\). Check the interior candidate, then determine the equilibrium quantities, price, and profits. (Answer: The interior candidate has \(q_2=0\) at the boundary; equilibrium \(q_1=20,\ q_2=0,\ P=60,\ \pi_1=800,\ \pi_2=0\).)
Inverse demand is \(P=120-Q\), with \(c_1=30\) and \(c_2=80\). Use the entry condition to verify that firm 2 produces nothing, then find firm 1’s quantity, price, and profit. (Answer: The threshold is \((120+30)/2=75\), so \(c_2=80\) is high enough; \(q_1=45,\ q_2=0,\ P=75,\ \pi_1=2{,}025,\ \pi_2=0\).)
Inverse demand is \(P=90-Q\), with \(c_1=50\) and \(c_2=55\). Determine whether either firm produces in equilibrium. (Answer: Both costs are below \(a=90\), and the interior quantities are \(q_1=(90-100+55)/3=15,\ q_2=(90-110+50)/3=10\); both produce.)

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