Cournot Competition with Differentiated Products

Why Differentiated Products Change Cournot Competition

In Cournot competition, each firm chooses a quantity while treating its rival’s quantity as fixed. With differentiated products, each firm’s price depends both on its own output and on the rival’s output. A larger rival quantity may reduce the price a firm can charge, but generally does not eliminate its customers: buyers may regard the products as imperfect substitutes rather than identical goods.

The analysis follows the same basic quantity-choice logic as other Cournot models, but now each firm has its own inverse demand equation. The cross-quantity effect determines how strongly a rival’s output affects a firm’s price—and therefore its best response. This tutorial is aligned with the SOA VEE Economics (Validation by Educational Experience) syllabus.

Definition: A two-firm Cournot model with differentiated products specifies each firm’s inverse demand as a function of both firms’ quantities. A standard linear form is $p_1=a_1-b_1q_1-\gamma q_2$ and $p_2=a_2-b_2q_2-\gamma q_1$, where $q_i$ and $p_i$ are firm $i$’s quantity and price, $a_i$ is a demand intercept, $b_i>0$ is the own-quantity effect, and $\gamma\geq 0$ is the cross-quantity effect. The firms choose quantities simultaneously.

The inverse-demand form is convenient for Cournot calculations: it gives each firm’s price at any pair of quantities. In direct-demand form, the corresponding economic idea is that a product’s demand typically falls when its own price rises and rises when a substitute’s price rises. The inverse-demand cross-quantity term captures the effect of additional rival output on a firm’s price. The linear model above assumes that this effect is constant over the relevant range.

Assume each firm has constant marginal cost $c_i\geq 0$, and, if included, a fixed cost $F_i\geq 0$. Operating profit is $(p_i-c_i)q_i$; accounting for fixed cost gives total profit $\pi_i=(p_i-c_i)q_i-F_i$. Fixed costs do not affect the quantity choice when they do not vary with output. The interior-equilibrium results below apply when both calculated quantities are positive and the firms’ profit functions are strictly concave in their own quantities. If a calculated quantity is nonpositive, the relevant firm’s nonnegative-output constraint must instead be considered.

Best Responses and the Interior Equilibrium

Firm 1 chooses $q_1$ to maximize profit, taking $q_2$ as given. Substituting its inverse demand into operating profit gives $\pi_1=(a_1-b_1q_1-\gamma q_2-c_1)q_1-F_1$. Differentiating with respect to $q_1$ yields its first-order condition. Firm 2’s condition follows in the same way.

\frac{\partial \pi_1}{\partial q_1}=a_1-c_1-2b_1q_1-\gamma q_2=0, \qquad \frac{\partial \pi_2}{\partial q_2}=a_2-c_2-2b_2q_2-\gamma q_1=0.

Each second derivative is negative: $\partial^2\pi_1/\partial q_1^2=-2b_1<0$ and $\partial^2\pi_2/\partial q_2^2=-2b_2<0$. Thus, for a given rival quantity, the first-order condition identifies the profit-maximizing quantity, provided it is feasible. Rearranging gives the firms’ best-response functions:

q_1=\frac{a_1-c_1-\gamma q_2}{2b_1}, \qquad q_2=\frac{a_2-c_2-\gamma q_1}{2b_2}.

These equations show strategic interdependence directly: each firm chooses its best quantity in response to the quantity it expects from its rival. Because the slope of either best response is negative when $\gamma>0$, a larger rival quantity reduces the firm’s best-response quantity. In this model, quantities are strategic substitutes.

Theorem: For an interior equilibrium, the two first-order conditions form a linear system. If $4b_1b_2-\gamma^2>0$, the system has a unique solution:

q_1^*= \frac{2b_2(a_1-c_1)-\gamma(a_2-c_2)} {4b_1b_2-\gamma^2}, \qquad q_2^*= \frac{2b_1(a_2-c_2)-\gamma(a_1-c_1)} {4b_1b_2-\gamma^2}.

For these expressions to describe an interior Cournot equilibrium, both quantities must be positive. The determinant condition ensures the linear system is nonsingular; it does not by itself guarantee positive quantities. Once the quantities are found, substitute both into the inverse-demand equations to find prices, then calculate profits using the stated cost assumptions.

Why the Equilibrium Formula Holds

The two first-order conditions can be written as a simultaneous system. Solving that system explains both the denominator and the way each firm’s costs and demand intercept enter the answer.

Write the two equations.
Set $A_1=a_1-c_1$ and $A_2=a_2-c_2$. The first-order conditions become $2b_1q_1+\gamma q_2=A_1$ and $\gamma q_1+2b_2q_2=A_2$.

Eliminate firm 2’s quantity.
Multiply the first equation by $2b_2$ and the second by $\gamma$, then subtract. The $q_2$ terms cancel, leaving $(4b_1b_2-\gamma^2)q_1=2b_2A_1-\gamma A_2$.

Solve and obtain the other quantity.
Divide by the nonzero determinant to get the formula for $q_1^*$. Eliminating $q_1$ instead gives $(4b_1b_2-\gamma^2)q_2=2b_1A_2-\gamma A_1$, which yields the formula for $q_2^*$.

The determinant condition also has a useful interpretation. A very large cross-effect relative to the own-quantity effects can undermine the standard interior solution. In the symmetric case, the condition is $\gamma<2b$. Even when it holds, always check the resulting quantities rather than assuming the algebra produces a feasible equilibrium.

Worked Equilibria

Worked Example: Unequal Demand Intercepts and Costs

Suppose two firms sell imperfect substitutes with inverse demands $p_1=100-2q_1-q_2$ and $p_2=80-2q_2-q_1$. Their constant marginal costs are $c_1=\$20$ and $c_2=\$10$, and there are no fixed costs. Find equilibrium quantities, prices, and profits. This method uses the two first-order conditions as a system; it avoids treating either firm’s price as independent of its rival’s output.

Step 1: Form the first-order conditions. For firm 1, $a_1=100$, $b_1=2$, and $\gamma=1$, so $4q_1+q_2=100-20=80$. For firm 2, $a_2=80$, $b_2=2$, and $c_2=10$, so $q_1+4q_2=80-10=70$.

Step 2: Solve the system. Multiply the first equation by 4 to obtain $16q_1+4q_2=320$. Subtract $q_1+4q_2=70$, giving $15q_1=250$, so $q_1=50/3$. Substituting into $4q_1+q_2=80$ gives $q_2=80-200/3=40/3$.

Step 3: Calculate prices. $p_1=100-2(50/3)-40/3=160/3=\$53.33$ and $p_2=80-2(40/3)-50/3=110/3=\$36.67$, rounded to the nearest cent.

Step 4: Calculate and check profits. Firm 1’s margin is $160/3-20=100/3$, so its profit is $(100/3)(50/3)=5000/9=\$555.56$. Firm 2’s margin is $110/3-10=80/3$, so its profit is $(80/3)(40/3)=3200/9=\$355.56$. Each quantity is positive, and substitution verifies $4(50/3)+40/3=80$ and $50/3+4(40/3)=70$. Final dollar amounts are rounded to two decimals.

Worked Example: Stronger Substitutability and Best Responses

Consider symmetric inverse demands $p_1=100-2q_1-\gamma q_2$ and $p_2=100-2q_2-\gamma q_1$, with marginal cost $\$20$ for each firm and no fixed costs. Compare the equilibrium when $\gamma=0.5$ with the equilibrium when $\gamma=1.5$. In both cases, the own-quantity coefficient and intercept are held fixed.

Step 1: Use symmetry. The firms have identical demand and cost parameters, so an interior symmetric equilibrium has $q_1=q_2=q$. The first-order condition is $4q+\gamma q=100-20=80$, or $q=80/(4+\gamma)$.

Step 2: Calculate equilibrium quantities. When $\gamma=0.5$, $q=80/4.5=160/9=17.7778$ units per firm. When $\gamma=1.5$, $q=80/5.5=160/11=14.5455$ units per firm. Quantities are rounded to four decimal places.

Step 3: Compare prices and profits. At $\gamma=0.5$, each price is $100-2.5(160/9)=500/9=\$55.56$, and each profit is $(500/9-20)(160/9)=51200/81=\$632.10$. At $\gamma=1.5$, each price is $100-3.5(160/11)=540/11=\$49.09$, and each profit is $(540/11-20)(160/11)=51200/121=\$423.14$. Prices and profits are rounded to two decimals.

Interpretation and check. The best-response slope is $-\gamma/(2b)=-\gamma/4$, so it changes from $-0.125$ to $-0.375$. A larger cross-effect makes each firm’s best response more sensitive to the rival’s quantity. The positive quantities satisfy the interior condition $\gamma<4$, and substituting either symmetric quantity into its first-order condition gives $(4+\gamma)q=80$. Stronger substitutability lowers equilibrium quantity, price, and profit here. This comparison holds the other parameters fixed; it does not assert that every change in market conditions has the same effect.

Worked Example: Asymmetric Own-Quantity Effects and Fixed Costs

A hypothetical market has inverse demands $p_1=120-3q_1-2q_2$ and $p_2=100-2q_2-2q_1$. Marginal costs are $c_1=\$30$ and $c_2=\$20$; fixed costs are $F_1=\$50$ and $F_2=\$100$. Find the interior equilibrium and each firm’s profit after fixed costs.

Step 1: Construct the first-order conditions. Firm 1’s condition is $6q_1+2q_2=120-30=90$. Firm 2’s condition is $2q_1+4q_2=100-20=80$. The determinant is $6(4)-2(2)=20>0$.

Step 2: Solve. Multiply the first equation by 2 to get $12q_1+4q_2=180$. Subtract the second equation to obtain $10q_1=100$, so $q_1=10$. Substitution into $6q_1+2q_2=90$ gives $60+2q_2=90$, hence $q_2=15$.

Step 3: Find prices and profits. $p_1=120-3(10)-2(15)=60$, and $p_2=100-2(15)-2(10)=50$. Firm 1’s profit after fixed costs is $(60-30)(10)-50=\$250$. Firm 2’s is $(50-20)(15)-100=\$350$.

Check. The first-order conditions give $6(10)+2(15)=90$ and $2(10)+4(15)=80$, and both equilibrium quantities are positive. Fixed costs are subtracted when calculating profit but do not enter these first-order conditions because they are constant with respect to quantity.

Reading the Cross-Effect and Avoiding Traps

In the inverse-demand equations, a larger $\gamma$ means that an increase in one firm’s quantity reduces the rival’s price more strongly, holding the rival’s own quantity fixed. When comparing models, keep the other parameters fixed if the goal is to isolate this effect. Under symmetric parameters and an interior equilibrium, the best-response slope becomes more negative as $\gamma$ rises, and equilibrium quantity per firm is $(a-c)/(2b+\gamma)$. In asymmetric models, calculate both outputs from the system; the effect on each firm depends on the parameters of both firms.

A frequent exam error is forgetting that the own-quantity term is differentiated twice in revenue. For firm 1, revenue is $p_1q_1=(a_1-b_1q_1-\gamma q_2)q_1$. Its derivative with respect to $q_1$ is $a_1-2b_1q_1-\gamma q_2$, not $a_1-b_1q_1-\gamma q_2$. The latter incorrectly treats price as fixed when the firm expands output. Another check is to substitute computed quantities back into both first-order conditions before calculating prices and profits.

Key takeaway: With differentiated products, each firm’s Cournot best response depends on its rival’s quantity through the cross-effect in inverse demand. Derive both first-order conditions, solve them together, check feasibility, then calculate prices and profits. In the standard linear model with a positive cross-effect, quantities are strategic substitutes.

Check Your Understanding

For each problem, assume firms choose nonnegative quantities simultaneously and have constant marginal costs. Check that any proposed interior quantities are positive.

Inverse demands are $p_1=70-2q_1-q_2$ and $p_2=70-2q_2-q_1$. Both firms have marginal cost $\$10$ and no fixed costs. Find the symmetric equilibrium quantity per firm, price, and profit. (Answer: $q_1=q_2=12$, price $\$34$, profit $\$288$ per firm.)
Inverse demands are $p_1=90-3q_1-q_2$ and $p_2=75-2q_2-q_1$; marginal costs are $\$15$ and $\$10$, respectively. Find the interior equilibrium quantities. (Answer: $q_1=235/23\approx10.2174$, $q_2=315/23\approx13.6957$.)
In a symmetric market, $p_1=120-2q_1-1.2q_2$ and $p_2=120-2q_2-1.2q_1$, and both marginal costs are $\$24$. Find each firm’s equilibrium quantity and price. (Answer: $q_1=q_2=240/13\approx18.4615$, price $=120-3.2(240/13)\approx\$60.92$.)
Inverse demands are $p_1=110-2q_1-1.5q_2$ and $p_2=90-2q_2-1.5q_1$, with marginal costs $\$20$ and $\$10$, respectively. Find both equilibrium quantities and verify the two first-order conditions. (Answer: $q_1=192/11\approx17.4545$, $q_2=148/11\approx13.4545$; the conditions are $4q_1+1.5q_2=90$ and $1.5q_1+4q_2=80$.)

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