Bertrand Model | Econ Academy

Two coffee shops sit on opposite corners of the same block. They serve identical americanos and have the same costs. On Monday, Shop A charges 5 dollars a cup and Shop B charges 4.50. Within an hour Shop A is empty — identical coffee, 50 cents cheaper across the street, no reason to stay. So Shop A drops to 4 dollars. Shop B drops to 3.50. The race only ends at the price floor neither shop can profitably go below: marginal cost.

That’s the Bertrand model (1883). Two firms, identical product, simultaneous price-setting. The Nash equilibrium has both firms pricing at $M C$ and earning zero economic profit — the perfectly competitive outcome from just two firms. Economists call this the Bertrand paradox.

Key Term

Why P = MC is the only stable price

Walk through every other possibility and watch them collapse:

Both above MC, same price ( $P_{1} = P_{2} > M C$ ). Half the market each, both profitable. Either firm can shave one cent off and take the whole market — instant profit jump. Not stable.
Both above MC, different prices ( $P_{1} > P_{2} > M C$ ). The cheaper firm wins everything, the dearer firm sells nothing. The losing firm will undercut on its next move. Not stable.
Anyone below MC. Every cup sold loses money. The losing firm will raise back to at least $M C$ . Not stable.
Both at MC ( $P_{1} = P_{2} = M C$ ). Each firm earns zero, but neither can profitably deviate: going below loses money, going above loses all customers. Stable.

The Nash equilibrium is the only configuration with no arrow pointing away from it.

Bertrand equilibrium with identical products: price equals marginal cost, zero profit.

Interactive — Two coffee shops in a price war

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Two identical coffee shops, four states. Click Next to walk through the undercutting race down to P = MC.

The widget shows the unraveling explicitly: at every state except the last, one firm has an arrow pointing away — a profitable deviation. Only at $P = M C$ does that arrow vanish for both. The pull toward $M C$ comes entirely from one-sided undercutting incentives, not from many firms entering the market. Just two firms are enough.

But real-world duopolies — Coca-Cola and Pepsi, Boeing and Airbus, Visa and Mastercard — earn fat profits, not zero. Identical-goods Bertrand pushes too hard. Four tweaks bring the model back to earth and let firms charge above $M C$ :

Product differentiation. Buyers prefer some brands and won’t all defect for a one-cent price cut.
Capacity constraints. A firm physically can’t serve the whole market at $M C$ , so the rival doesn’t actually lose all its customers when undercut.
Switching costs. Customers stick with what they know.
Repeated interaction. Future punishment (price wars, retaliation) deters today’s price cut.

Check yourself · no marks

How does the Bertrand model’s prediction compare to Cournot’s, and why do they differ so much?

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Practice · 1 / 4

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The Bertrand model:

In the Bertrand model with homogeneous products, the Nash Equilibrium price is:

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