Bertrand Competition with Differentiated Products

Why Product Differentiation Changes Price Competition

With identical products, a buyer can switch entirely to the lower-priced seller, making even a small price difference important. With differentiated products, customers may still prefer one firm’s product when its price is somewhat higher. Each firm therefore faces a downward-sloping demand for its own product and must account for how its price affects its sales. This changes the price-setting problem: a firm’s best price depends on the rival’s price.

The model below uses linear demand to make that dependence explicit. Its conclusions apply to an interior outcome in which both firms sell positive quantities. They are not a claim that every linear demand equation gives sensible quantities at every possible price.

Definition: In a two-firm differentiated-product Bertrand model, firms choose prices $p_1$ and $p_2$ simultaneously. A firm’s price best response is the price that maximizes its profit for a specified price of the rival. An interior price equilibrium is a pair of prices at which each firm’s price is a best response to the other firm’s price and both firms have positive demand.

Linear Demand and the Price Best Responses

Let the two firms’ demands be

q_1=A_1-B_1p_1+G_1p_2,\qquad q_2=A_2-B_2p_2+G_2p_1.

Here $q_i$ is firm $i$’s quantity sold; $A_i$ is a demand intercept; $B_i>0$ measures the effect of the firm’s own price on its demand; and $G_i\geq 0$ measures the effect of the rival’s price on its demand. A positive cross-price coefficient represents products that are substitutes: if the rival raises its price, some customers shift toward firm $i$. The products are differentiated because a firm retains some customers when its own price exceeds the rival’s price.

Assume each firm has constant marginal cost $c_i$, chooses its price to maximize operating profit, and has no capacity constraint. For the interior calculation, take demand as linear in the region being analyzed. The candidate equilibrium must have $q_1>0$ and $q_2>0$; if a calculated demand is negative, the interior solution is invalid. A model that truncates demand at zero would need to consider that boundary separately.

Firm 1’s profit is $\pi_1=(p_1-c_1)q_1$. Holding $p_2$ fixed, differentiate with respect to $p_1$:

\frac{\partial\pi_1}{\partial p_1} =q_1+(p_1-c_1)(-B_1) =A_1-B_1p_1+G_1p_2-B_1(p_1-c_1).

Setting this derivative to zero gives firm 1’s best response. The same method applies to firm 2.

p_1=\frac{A_1+B_1c_1+G_1p_2}{2B_1}, \qquad p_2=\frac{A_2+B_2c_2+G_2p_1}{2B_2}.

These first-order conditions identify profit-maximizing prices in the interior: each firm’s second derivative of profit with respect to its own price is $-2B_i<0$, so profit is strictly concave in that price. The formulas are not by themselves proof of an equilibrium with positive sales; the resulting quantities must still be checked.

Key result: At an interior best response, the first-order condition can also be written $q_i=B_i(p_i-c_i)$. Thus, if firm $i$ has positive demand, its price exceeds marginal cost. This markup result relies on the specified linear demand and interior profit maximization; it does not mean that every price in every differentiated-product model must exceed cost.

Solving for an Interior Equilibrium

An interior equilibrium is found by solving the two best-response equations together, since each price depends on the rival’s price. Define $R_1=A_1+B_1c_1$ and $R_2=A_2+B_2c_2$. The simultaneous equations are

2B_1p_1-G_1p_2=R_1,\qquad -G_2p_1+2B_2p_2=R_2.

If $D=4B_1B_2-G_1G_2>0$, these equations have the unique solution

p_1=\frac{2B_2R_1+G_1R_2}{D}, \qquad p_2=\frac{G_2R_1+2B_1R_2}{D}.

The condition $D>0$ ensures that the two linear equations have a nonzero determinant and a unique intersection. It does not guarantee that the associated quantities are positive. After solving, substitute both prices into the original demand equations and check the interior condition.

For a symmetric version, suppose $A_1=A_2=A$, $B_1=B_2=B$, $G_1=G_2=G$, and $c_1=c_2=c$. Equal prices $p$ satisfy both firms’ best responses:

p=\frac{A+Bc+Gp}{2B} \quad\Longrightarrow\quad p=\frac{A+Bc}{2B-G},

provided $2B-G\neq 0$. The symmetric candidate’s quantity is $q=A-(B-G)p$, which must be positive for the interior result to apply.

Worked Example: Finding a Symmetric Price Equilibrium

Two firms sell differentiated products with demands $q_1=100-2p_1+0.5p_2$ and $q_2=100-2p_2+0.5p_1$. Each has constant marginal cost $c=10$. Find the symmetric interior equilibrium prices, quantities, and each firm’s profit.

Method. Use the symmetric best-response formula with $A=100$, $B=2$, $G=0.5$, and $c=10$:

p=\frac{100+(2)(10)}{2(2)-0.5} =\frac{120}{3.5} =\frac{240}{7} \approx 34.2857.

Both prices are $p_1=p_2=240/7$. Substitute into either demand equation:

q=100-2\left(\frac{240}{7}\right)+0.5\left(\frac{240}{7}\right) =100-\frac{480}{7}+\frac{120}{7} =\frac{340}{7} \approx 48.5714.

The quantity is positive, so the interior condition holds. Each firm’s margin is $240/7-10=170/7$, and its profit is

\pi=\left(\frac{170}{7}\right)\left(\frac{340}{7}\right) =\frac{57{,}800}{49} \approx \$1{,}179.59.

Check. The best-response condition requires $q=B(p-c)$. Here $2(170/7)=340/7=q$, as required. Profit is rounded to the nearest cent; prices and quantities are shown to four decimal places. The common mistake this method avoids is treating the rival’s price as irrelevant: the $0.5p_j$ cross-price term affects the firm’s optimal price.

Why Positive Demand Supports a Markup

The markup result follows directly from the firm’s marginal trade-off. Raising price increases the margin on every unit still sold, but reduces quantity by $B_i$ units per unit increase in its own price. At an interior optimum, the gain and loss balance:

q_i-B_i(p_i-c_i)=0 \quad\Longrightarrow\quad p_i-c_i=\frac{q_i}{B_i}.

Since $B_i>0$, positive demand $q_i>0$ implies $p_i-c_i>0$. Differentiation matters because customers do not all abandon a firm when its price rises relative to the rival’s; the firm’s remaining demand makes a positive margin possible in this model.

In the symmetric case, the cross-price coefficient $G$ measures how strongly the rival’s price affects a firm’s demand. For fixed positive $A$, $B$, and $c$, and with $2B-G>0$, the equilibrium price is $p=(A+Bc)/(2B-G)$. Its change as $G$ increases is

\frac{dp}{dG}=\frac{A+Bc}{(2B-G)^2}>0

when $A+Bc>0$. Under these stated conditions, a lower cross-price effect (weaker substitution) lowers the equilibrium price relative to a stronger cross-price effect. It can nevertheless leave the price above marginal cost if both firms continue to sell positive quantities. The comparison concerns this linear model with the other parameters fixed; it is not a universal prediction for every market.

Worked Example: Comparing Two Degrees of Substitution

Two otherwise identical firms have demand $q_i=120-3p_i+Gp_j$, where $G$ is either $0.5$ or $1.5$. Each firm’s marginal cost is $10$. For each value of $G$, find the symmetric equilibrium price, quantity, and markup. Then interpret the comparison.

Method. Use $p=(A+Bc)/(2B-G)$, with $A=120$, $B=3$, and $c=10$. For weaker substitution, $G=0.5$:

p=\frac{120+(3)(10)}{6-0.5} =\frac{150}{5.5} =\frac{300}{11} \approx 27.2727, \qquad q=120-3\left(\frac{300}{11}\right)+0.5\left(\frac{300}{11}\right) =\frac{570}{11} \approx 51.8182.

The markup is $300/11-10=190/11\approx17.2727$, which is positive. For stronger substitution, $G=1.5$:

p=\frac{150}{6-1.5} =\frac{100}{3} \approx 33.3333, \qquad q=120-3\left(\frac{100}{3}\right)+1.5\left(\frac{100}{3}\right) =70.

The markup in this case is $100/3-10=70/3\approx23.3333$. Both quantities are positive, and both equilibrium prices exceed marginal cost. With the other parameters held fixed, the weaker-substitution case has the lower price, but still has a positive markup.

Check. In the first case, $3(p-c)=3(190/11)=570/11=q$. In the second, $3(p-c)=3(70/3)=70=q$. These checks confirm the first-order condition in both cases. Prices and quantities are rounded to four decimal places where needed.

Asymmetric Firms and a Common Exam Trap

The best-response method also works when firms have different demand intercepts, own-price effects, cross-price effects, or marginal costs. In that case, do not impose equal prices. Write each firm’s best response using its own parameters, solve the two equations simultaneously, and verify both demands. The following example illustrates that procedure.

Worked Example: Solving with Different Costs and Demand Slopes

Suppose demands are $q_1=80-2p_1+0.4p_2$ and $q_2=70-1.5p_2+0.3p_1$. Marginal costs are $c_1=8$ and $c_2=12$. Find the interior equilibrium prices and quantities.

Method. For firm 1, $A_1=80$, $B_1=2$, and $G_1=0.4$. Its best response is

p_1=\frac{80+(2)(8)+0.4p_2}{4} =24+0.1p_2.

For firm 2, $A_2=70$, $B_2=1.5$, and $G_2=0.3$. Its best response is

p_2=\frac{70+(1.5)(12)+0.3p_1}{3} =\frac{88+0.3p_1}{3} =\frac{88}{3}+0.1p_1.

Substitute firm 2’s expression into firm 1’s:

p_1=24+0.1\left(\frac{88}{3}+0.1p_1\right) =\frac{404}{15}+0.01p_1, \qquad 0.99p_1=\frac{404}{15}, \qquad p_1=\frac{8080}{297}\approx27.2054.

Then

p_2=\frac{88}{3}+0.1\left(\frac{8080}{297}\right) =\frac{9520}{297} \approx32.0539.

Substituting these prices into demand gives

q_1=80-2\left(\frac{8080}{297}\right)+0.4\left(\frac{9520}{297}\right) =\frac{11408}{297} \approx38.4108, $$ $$ q_2=70-1.5\left(\frac{9520}{297}\right)+0.3\left(\frac{8080}{297}\right) =\frac{8934}{297} \approx30.0808.

Both quantities are positive, so the interior solution is valid. As a further check, $q_1=2(p_1-c_1)=2(19.2054)\approx38.4108$, and $q_2=1.5(p_2-c_2)=1.5(20.0539)\approx30.0808$. Prices and quantities are rounded to four decimal places.

A frequent error is to calculate each firm’s price using only its own demand intercept and cost, then stop. For example, in the first worked example, omitting the rival-price term gives $p=(100+2(10))/(2(2))=30$. That is not an equilibrium price: at $p_2=30$, firm 1’s correct best response is $(100+20+0.5(30))/4=33.75$, not $30$. The correction is to include the cross-price term in each best response and solve the equations together.

Key takeaway: In a linear differentiated-product price model, each firm’s best response reflects its own price effect, marginal cost, and the rival-price effect. Solve the best responses jointly, then check that both quantities are positive. With positive interior demand, the first-order condition implies price above marginal cost.

Check Your Understanding

For each problem, use the stated linear-demand model and check the calculated quantities when an interior equilibrium is requested.

Two firms have symmetric demands $q_i=60-2p_i+0.5p_j$ and marginal cost $5$ each. Derive the symmetric equilibrium price and find each firm’s quantity. (Answer: $p_1=p_2=20$, $q_1=q_2=30$.)
Firm 1 has demand $q_1=50-2p_1+0.5p_2$, marginal cost $4$, and the rival’s price is fixed at $20$. Find firm 1’s best-response price, demand, and profit. (Answer: $p_1=17$, $q_1=26$, $\pi_1=\$338$.)
Demands are $q_1=90-2p_1+0.5p_2$ and $q_2=75-1.5p_2+0.25p_1$, with marginal costs $c_1=5$ and $c_2=7$. Solve the two best responses and find both equilibrium quantities. (Answer: $p_1\approx28.86$, $p_2\approx30.91$, $q_1\approx47.73$, $q_2\approx35.86$.)
For symmetric demand $q_i=100-2p_i+Gp_j$, marginal cost is $8$. Find the symmetric equilibrium price when $G=0.4$ and when $G=1.2$, and state which case has the higher price. (Answer: $p\approx32.22$ when $G=0.4$, $p\approx41.43$ when $G=1.2$; the higher cross-price effect gives the higher price.)
Two firms have symmetric demand $q_i=40-2p_i+0.5p_j$ and marginal cost $12$. Find the symmetric equilibrium price and quantity, and verify that the price exceeds marginal cost. (Answer: $p_1=p_2\approx18.29$, $q_1=q_2\approx12.57$; the markup is about $6.29$.)

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