Bertrand Competition with Unequal Marginal Costs

Why Cost Differences Change Bertrand Pricing

In homogeneous-product Bertrand competition, consumers buy from the firm offering the lower price. When firms have the same marginal cost, undercutting can drive price toward cost. With unequal costs, however, the lower-cost firm may be able to earn a positive margin while the higher-cost firm cannot profitably match or undercut it. The key comparison is between the low-cost firm’s unconstrained monopoly price and the rival’s marginal cost.

The result depends on the model’s assumptions. Each firm chooses one price simultaneously; the products are identical; consumers buy from the lowest-price firm; and, at a tie, the firms split demand equally. Firms have no capacity constraints or fixed costs, and each has constant marginal cost. Demand is positive at the prices being considered. These assumptions matter: capacity limits and differentiated products change the pricing incentives.

Definition: Let $c_L$ and $c_H$ be the constant marginal costs of the low-cost and high-cost firms, with $c_L<c_H$. Let $D(p)$ be market demand at price $p$. A firm charging $p_i$ receives all demand if it charges less than its rival, half if the firms tie, and none if it charges more. Its profit is its unit margin times the quantity it sells.

If firm $L$ is the sole seller at price $p$, its profit is $\pi_L(p)=(p-c_L)D(p)$. At a tie, each firm sells half of $D(p)$, so firm $i$’s profit is $(p_i-c_i)D(p_i)/2$. A firm that makes no sales earns zero, regardless of its posted price, provided it has no fixed costs.

The Low-Cost Firm’s Monopoly Price

Define $p_M$ as the price that maximizes the low-cost firm’s profit when it serves the whole market. This is its unconstrained monopoly price: it does not yet account for the possibility that its rival might undercut. For linear demand $D(p)=a-bp$, with $a>0$, $b>0$, and positive demand at the optimum, the profit-maximizing price follows directly from the profit function.

\pi_L(p)=(p-c_L)(a-bp)

Expanding and differentiating gives the first-order condition. The second derivative is negative, so the stationary point is the unique profit maximum over the relevant range.

\frac{d\pi_L}{dp}=a+bc_L-2bp=0 \qquad\Longrightarrow\qquad p_M=\frac{a+bc_L}{2b}, \qquad \frac{d^2\pi_L}{dp^2}=-2b<0

The maximum is useful for equilibrium analysis, but it is not automatically the price the low-cost firm can charge in equilibrium. If it sets a price above $c_H$, the high-cost firm may be able to undercut it while still charging more than its own marginal cost.

Worked Example: A Monopoly Price Below the Rival’s Cost

Suppose market demand is $D(p)=240-4p$, with $c_L=20$ and $c_H=50$. Find the low-cost firm’s monopoly price and profit, then test whether $p_L=40,\ p_H=55$ is an equilibrium.

Method. First maximize the low-cost firm’s profit as a sole seller. Then compare each firm’s possible deviations with its profit at the proposed price pair.

For the low-cost firm, $a=240$, $b=4$, and $c_L=20$. Therefore, $p_M=(240+4(20))/(2(4))=320/8=40$. Demand at this price is $D(40)=240-4(40)=80$. Its monopoly profit is $(40-20)(80)=20(80)=\$1{,}600$.

At the proposed prices, the low-cost firm is the sole seller and earns the same $\$1{,}600$. Since $40$ maximizes its sole-seller profit, no other price below $55$ gives it more. A price above $55$ would leave it with no sales. Thus it has no profitable deviation.

The high-cost firm sells nothing at $55$ and earns zero. If it undercuts the low firm’s price, it must charge less than $40$, which is below its marginal cost of $50$; selling at such a price would generate a loss. Matching $40$ would also mean selling at a loss, while charging more than $40$ leaves it with no sales. The price pair is therefore a pure-strategy Nash equilibrium. The high-cost firm’s no-sales responses are not limited to prices above its cost: any price strictly above $40$ yields zero sales and zero profit in this outcome.

Check. At the proposed prices, the low-cost firm earns $\$1{,}600$ and the high-cost firm earns zero; neither has a profitable deviation under the stated assumptions. This method avoids assuming that both firms must sell in equilibrium.

When the Monopoly Price Is Sustainable

If $p_M<c_H$, the low-cost firm can charge its monopoly price while keeping the high-cost firm from making a profitable sale by undercutting. At $p_L=p_M$, any $p_H>p_M$ leaves the high-cost firm with no sales. If it undercuts $p_M$, it must charge less than $p_M<c_H$, so its margin is negative. Matching $p_M$ also gives it a negative margin at a tie. The low-cost firm, meanwhile, is already maximizing its profit as sole seller.

The same equilibrium description applies when $p_M=c_H$, provided the high-cost firm sets a price strictly above $p_M$. At a tie at $p_M=c_H$, the high-cost firm’s profit would be zero, but the low-cost firm would prefer to undercut slightly and capture nearly all demand. Thus the strict inequality in the high-cost firm’s price matters.

Theorem: Under the stated assumptions, if $p_M\leq c_H$ and demand is positive at $p_M$, the pure-strategy equilibria have $p_L=p_M$ and $p_H>p_M$. The low-cost firm earns its maximum sole-seller profit, and the high-cost firm earns zero. The high-cost firm’s price is not uniquely determined; it may choose any price strictly above $p_M$.

The reason the high-cost firm’s equilibrium price is not pinned down is that it earns zero whenever it sells nothing. Raising its price while remaining above $p_L$ does not change its sales or profit. Nor can the low-cost firm necessarily raise its price: at $p_M$, any higher sole-seller price lowers its profit, and a price above the rival’s price would leave it with no sales.

Worked Example: A High-Cost Firm’s Viable Price Choices

Let demand be $D(p)=180-2p$, with $c_L=15$ and $c_H=45$. Determine the equilibrium low-cost price and evaluate two possible high-cost prices, $46$ and $40$.

Method. Compare the low-cost firm’s monopoly price with the high-cost firm’s marginal cost, then check whether each proposed high-cost price leaves it above the low-cost firm’s price.

Here $a=180$, $b=2$, so $p_M=(180+2(15))/(2(2))=210/4=52.5$. Since $52.5>45$, the theorem for $p_M\leq c_H$ does not apply. In particular, $p_L=52.5,\ p_H=60$ cannot be an equilibrium: firm $H$ could undercut to $52$, above its cost of $45$. At $52$, demand is $180-104=76$, so its profit as sole seller would be $(52-45)(76)=7(76)=\$532$, rather than zero.

For the proposed prices $p_L=40$, $p_H=46$, firm $H$ sells nothing. If instead $p_H=40$, the firms tie below $c_H$, so firm $H$ sells half of $D(40)=100$ at a loss: $(40-45)(100/2)=-\$250$. Thus $46$ is a better response than $40$ to $p_L=40$, but that alone does not establish equilibrium; firm $L$ must also be best responding. It is not: at $p_L=40$, firm $L$ earns $(40-15)(100)=\$2{,}500$, while at $p_L=52.5<p_H=60$ it earns $(52.5-15)(75)=37.5(75)=\$2{,}812.50$.

Check. The high-cost firm’s deviation from $60$ to $52$ is profitable when the low firm charges $52.5$; the low firm’s deviation from $40$ to $52.5$ is profitable when the rival charges $60$. This method avoids treating “the high-cost firm cannot sell below cost” as proof that a candidate price pair is an equilibrium.

When the Low-Cost Firm’s Monopoly Price Exceeds Rival Cost

If $p_M>c_H$, there is generally no pure-strategy equilibrium in this continuous-price model with positive demand near $c_H$. The low-cost firm would like to charge more than $c_H$ when it is the sole seller, but at a price above $c_H$ the high-cost firm can undercut it at a price that still covers its own marginal cost. If the low-cost firm instead charges below $c_H$, it may be protected from a profitable undercut, but it has an incentive to raise its price if the rival’s price leaves room to do so.

The boundary price $c_H$ does not generally solve the problem. If the high-cost firm prices above $c_H$, the low-cost firm can raise a price below $c_H$ when doing so improves its profit. If both firms price at $c_H$, the low-cost firm can undercut slightly, capture almost all demand, and earn more than its tied profit. If the low-cost firm prices above $c_H$, the high-cost firm can undercut profitably. These incentives prevent a pure-price equilibrium when the low-cost firm’s monopoly price is strictly above $c_H$.

This conclusion relies on continuous prices: a firm can choose a price arbitrarily close to a rival’s price without having to match it exactly. There is no smallest positive undercut. A minimum price increment or other pricing restrictions can change the result; the continuous-price conclusion should not be applied automatically to such settings.

Find $p_M$.
Maximize $(p-c_L)D(p)$, using the linear-demand formula when its assumptions hold.

Compare $p_M$ with $c_H$.
If $p_M\leq c_H$, the low-cost firm can charge $p_M$ and the high-cost firm can avoid sales by setting $p_H>p_M$.

If $p_M>c_H$, test both firms’ incentives.
A price above $c_H$ invites a potentially profitable undercut by the high-cost firm; a lower price may give the low-cost firm an incentive to raise its price. Do not assert a pure equilibrium without checking deviations.

Common Mistake: Checking Only One Firm

A proposed price pair is a Nash equilibrium only if neither firm can increase its profit by changing its own price while the rival’s price stays fixed. Showing that the high-cost firm cannot undercut profitably is not enough; the low-cost firm might still prefer a different price.

For example, let $D(p)=300-3p$, $c_L=30$, and $c_H=80$, with proposed prices $p_L=70$ and $p_H=90$. At $70$, demand is $300-3(70)=90$, so the low-cost firm earns $(70-30)(90)=40(90)=\$3{,}600$. Its monopoly price is $p_M=(300+3(30))/(2(3))=390/6=65$; at $65$, demand is $300-195=105$, and profit is $(65-30)(105)=35(105)=\$3{,}675$. It therefore gains $\$75$ by deviating from $70$ to $65$. The high-cost firm cannot profitably undercut $70$, because doing so requires a price below its cost of $80$, but that does not prevent the low-cost firm’s profitable deviation. The proposed pair is not an equilibrium.

Key takeaway: Compare the low-cost firm’s monopoly price with the high-cost firm’s marginal cost, then check both firms’ incentives. When $p_M\leq c_H$, the low-cost firm can charge $p_M$ while the high-cost firm earns zero at any strictly higher price. When $p_M>c_H$, a pure equilibrium generally fails under continuous pricing and positive demand near $c_H$.

Check Your Understanding

For each problem, use the continuous-price, homogeneous-product assumptions stated above. Check both firms’ incentives where a candidate price pair is given.

Demand is $D(p)=160-2p$, with $c_L=12$ and $c_H=38$. Find $p_M$, compare it with $c_H$, and state whether the model has a pure-strategy equilibrium. (Answer: $p_M=(160+24)/4=46>38$; there is no pure-strategy equilibrium under the stated assumptions.)
Demand is $D(p)=210-3p$, with $c_L=20$ and $c_H=55$. Find the equilibrium low-cost price and profit, and describe the high-cost firm’s equilibrium prices. (Answer: $p_M=(210+60)/6=45$, $D(45)=75$, and low-cost profit is $(45-20)(75)=\$1{,}875$; $p_L=45$ and any $p_H>45$ is an equilibrium response.)
Demand is $D(p)=250-5p$, with $c_L=15$ and $c_H=40$. Determine whether the monopoly price is sustainable and calculate the low-cost firm’s equilibrium profit if it is. (Answer: $p_M=(250+75)/10=32.5<40$, $D(32.5)=87.5$, and profit is $(32.5-15)(87.5)=\$1{,}531.25$; the high-cost firm must charge more than $32.5$.)
Demand is $D(p)=140-2p$, with $c_L=10$ and $c_H=30$. A candidate pair is $p_L=28,\ p_H=40$. Calculate the low-cost firm’s profit and its profit at $p_L=29$, holding the rival price fixed. Is the pair an equilibrium? (Answer: At $28$, demand is $84$ and profit is $18(84)=\$1{,}512$. At $29$, demand is $82$ and profit is $19(82)=\$1{,}558$, so the low-cost firm gains $\$46$; the pair is not an equilibrium.)
Demand is $D(p)=360-4p$, with $c_L=25$ and $c_H=65$. Find the equilibrium low-cost price and profit. Is $p_H=60$ a viable high-cost price in that equilibrium? (Answer: $p_M=(360+100)/8=57.5<65$, $D(57.5)=130$, and low-cost profit is $32.5(130)=\$4{,}225$; yes, $p_H=60>57.5$ leaves the high-cost firm with zero sales even though its price is below its marginal cost.)

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