Bertrand Competition

Introduction: why firms race on price 📉

students, imagine two snack shops across the street from each other. They sell the same candy bar, and customers can easily walk to either one. If one shop charges more than the other, many buyers will switch. This is the big idea behind Bertrand competition: firms compete by setting prices instead of quantities.

In this lesson, you will learn how to:

set up a Bertrand competition model,
explain why price competition can become extremely intense,
compare price competition with quantity competition.

Bertrand competition is important because it helps explain why some markets have very low prices, especially when products are very similar. It also shows a famous result in economics: even a small amount of price cutting can push profits down a lot when consumers can easily compare prices.

The basic Bertrand model

The classic Bertrand model has a few key assumptions. Two firms sell a product that customers view as identical. Each firm chooses a price at the same time. Buyers always purchase from the firm with the lower price, because there is no difference in the products. If both firms set the same price, they split the market.

Let the two firms set prices $p_1$ and $p_2$. Let marginal cost be $c$ for both firms. A simple demand setup is:

if $p_1 < p_2$, firm 1 gets all the demand,
if $p_2 < p_1$, firm 2 gets all the demand,
if $p_1 = p_2$, the firms split demand.

This can be written as a payoff rule. If total market demand at price $p$ is $Q(p)$, then firm 1’s profit is

$\pi_1$ = (p_1 - c)Q(p_1) \quad \text{if } p_1 < p_2,

and similarly for firm 2. If the prices are equal, each firm gets half:

$\pi_1$ = $\frac{1}{2}$(p_1 - c)Q(p_1), \quad $\pi_2$ = $\frac{1}{2}$(p_2 - c)Q(p_2).

This setup creates a powerful incentive to undercut the rival by just a little. If firm 2 sets a price of $p_2$, firm 1 can steal the whole market by setting $p_1 = p_2 - \varepsilon$, where $\varepsilon$ is a tiny positive number. That small move can greatly increase sales, as long as price stays above cost.

Why price competition becomes intense 🔥

Bertrand competition is famous because it often leads to very low prices. The reason is simple: when products are identical and customers can switch easily, each firm wants to be a little cheaper than the other.

Suppose both firms have the same marginal cost $c$. If both set a price above cost, say $p_1 = p_2 = c + 2$, then each earns some profit. But if firm 1 slightly lowers its price to $p_1 = c + 1.99$, it may capture the entire market. As long as the new price is still above cost, the extra sales can be worth it.

This creates a chain reaction:

if one firm lowers price, the other responds,
each firm tries to stay just below the rival,
the process pushes prices downward.

In the standard model with identical products, equal costs, and no capacity limits, the Nash equilibrium is

$p_1 = p_2 = c.$

At this price, each firm earns zero economic profit:

$\pi_1 = \pi_2 = 0.$

Why is this surprising? Because the firms are not colluding. They are each trying to make the best individual choice. But the logic of competition is so strong that any price above cost invites undercutting.

A real-world example

Think about selling gasoline on the same road 🚗. If drivers can easily compare prices, and the gas is nearly identical, a station charging even a little more may lose many customers. That pressure can make prices very close across stations.

Another example is online retail. If two websites offer the same phone charger, shoppers can compare prices instantly. That makes it hard for either seller to keep prices high unless there is some difference in service, shipping speed, or trust.

Solving the Bertrand equilibrium

students, let’s see the logic step by step. Assume:

two firms, 1 and 2,
identical products,
constant marginal cost $c$,
consumers buy from the lowest-priced firm.

Now ask: could both firms set a price above $c$ in equilibrium? Suppose both choose $p > c$. Then either firm can slightly lower price to $p - \varepsilon$, where $\varepsilon > 0$ is very small. That firm then gets the whole market and earns

(p - \varepsilon - c)Q(p - \varepsilon),

which is positive if $p - \varepsilon > c$.

So a price above cost cannot be stable.

Could the price be below cost? If $p < c$, each sale would lose money. A firm could raise price a bit and reduce losses, so pricing below cost is not stable either.

That leaves $p = c$ as the equilibrium price. At this price, a firm cannot profitably undercut, because pricing below cost would create losses. It also cannot profitably raise price, because it would lose all customers.

This is the Bertrand paradox: with only two firms, price competition can be as tough as perfect competition, even though there are only a few sellers.

What changes the result? Differentiation and capacity limits 🧩

The basic Bertrand model is very strong, but real markets are not always perfectly identical. When products differ, price pressure becomes weaker.

For example, two burger restaurants may both sell burgers, but one may have better service, a nicer atmosphere, or a more convenient location. Then customers are not perfectly willing to switch. In that case, a firm may be able to charge a price above marginal cost because some buyers prefer it.

Another important change is capacity constraints. In the basic model, each firm can serve everyone if its price is lowest. If a firm can only make a limited number of units, then undercutting may not guarantee it can take the whole market. That can soften price competition and allow prices above cost.

When the assumptions of the simple model are relaxed, Bertrand competition still matters, but the outcome is less extreme.

Price competition versus quantity competition 📊

Bertrand competition is about choosing price. Another famous model, Cournot competition, is about choosing quantity.

Here is the key difference:

In Bertrand competition, firms ask, “What price should I charge?”
In Cournot competition, firms ask, “How much should I produce?”

The outcomes are often very different.

In the simple Bertrand model with identical products, the equilibrium price equals marginal cost:

$p = c.$

This means consumers get a low price, and firms earn zero economic profit.

In a Cournot model, firms restrict output strategically. Because each firm chooses quantity instead of price, the market price usually ends up above marginal cost, so firms earn positive profits.

Why does the choice variable matter so much? Because in price competition, lowering price steals customers directly. In quantity competition, a firm cannot directly take customers away by changing output; instead, total industry supply changes the market price.

Simple comparison example

Imagine two identical firms in a market with strong consumer switching.

Under price competition, if one firm sets $p = 10$ and the other sets $p = 9.99$, the cheaper firm may get nearly all sales.
Under quantity competition, if one firm produces more, the market price falls, but both firms still share the market according to total supply.

This is why Bertrand competition can be much harsher than Cournot competition. It is one of the main reasons economists study both models side by side.

Why Bertrand competition matters in real markets

Bertrand competition helps explain many market outcomes in everyday life. It is especially useful when products are close substitutes and buyers can easily compare prices.

Examples include:

airline tickets on the same route ✈️,
generic medicines,
online electronics,
fuel stations near each other,
digital goods with nearly identical features.

In these settings, firms may not be able to keep prices far above cost unless there is some brand loyalty, product difference, or limited capacity.

The model also helps policymakers understand why markets with few firms do not always have high prices. If firms compete aggressively on price, prices may stay low. But if firms coordinate, differentiate their products, or face barriers that limit switching, prices may rise.

Conclusion

Bertrand competition shows how price-setting rivalry can be extremely intense when products are identical and consumers can switch easily. In the simplest version of the model, the equilibrium price equals marginal cost, so firms earn zero economic profit. This is a striking result because it shows that even a small number of firms can behave as if the market were perfectly competitive.

At the same time, the model teaches an important lesson: real markets depend on details. Product differences, brand loyalty, and capacity limits can all reduce price pressure. Compared with Cournot competition, Bertrand competition usually leads to lower prices because firms compete directly through pricing.

Study Notes

Bertrand competition is a model where firms choose prices at the same time.
The classic assumptions are identical products, equal marginal cost $c$, and consumers buying from the lowest-priced firm.
If one firm sets a slightly lower price than the other, it can often capture the whole market.
In the standard model, the Nash equilibrium is $p_1 = p_2 = c$.
At $p = c$, firms earn zero economic profit: $\pi_1 = \pi_2 = 0$.
Price competition can be intense because undercutting even by a tiny amount can win all customers.
This result is called the Bertrand paradox.
The model changes if products are differentiated or if firms have capacity limits.
Bertrand competition usually produces lower prices than Cournot competition.
Cournot competition is quantity competition, while Bertrand competition is price competition.