Bertrand competition

Based on Wikipedia: Bertrand competition

Here's a puzzle that has haunted economists for over a century: if two gas stations sit across the street from each other, selling identical fuel, why don't they just keep undercutting each other until neither one makes any money at all?

The unsettling answer, according to a French mathematician named Joseph Bertrand, is that they should. And in a perfectly competitive world, they would.

The Accidental Revolutionary

In 1883, Bertrand was reviewing a fifty-year-old economics book when he stumbled onto something that would reshape how we think about market competition. The book was Antoine Augustin Cournot's Researches into the Mathematical Principles of the Theory of Wealth, published in 1838—a foundational text arguing that competing firms should each pick a quantity to produce, then let the market determine the price.

Cournot's model was elegant and reassuring to businesses. It predicted that even with competition, firms could charge more than their costs and earn healthy profits. The math worked out. The conclusions felt right.

Bertrand wasn't buying it.

His objection was devastatingly simple: firms don't compete by choosing quantities. They compete by choosing prices. And once you flip that assumption, everything changes.

The Race to the Bottom

Imagine two companies selling an identical product—let's call them Firm A and Firm B. They have the same costs, the same production capabilities, and customers who care about nothing except getting the lowest price. There's no brand loyalty, no switching costs, no reason whatsoever to pay more than necessary.

Now suppose both firms are charging ten dollars, and it costs them each five dollars to make the product. Both are earning five dollars profit per unit. Life is good.

But here's the thing: Firm A realizes that if it drops its price to nine dollars and ninety-nine cents, every single customer will switch over. Why pay ten dollars when you can pay a penny less? Firm A would capture the entire market and make slightly less per unit but far more in total.

Of course, Firm B sees this coming. So Firm B drops to nine dollars and ninety-eight cents.

Then A goes to nine ninety-seven.

Then B goes to nine ninety-six.

Where does this end?

Bertrand's answer: it ends at the cost of production. The price drops all the way down to five dollars—the marginal cost—at which point neither firm makes any profit at all. Any lower and they'd be losing money on every sale. Any higher and the other firm could undercut them.

This is the Bertrand paradox: two firms, competing head-to-head, should theoretically drive each other's profits to zero.

Why This Matters (And Why It Should Worry You)

The Bertrand model produces what economists call a Nash equilibrium—a situation where neither player can improve their position by changing their strategy alone. If both firms price at cost, neither has any incentive to change. Raising your price means losing all your customers. Lowering it means losing money on every sale.

But it's a peculiar kind of equilibrium. Economists call it a weak Nash equilibrium because the firms don't actually lose anything by deviating—they just can't gain anything either. It's equilibrium through mutual exhaustion rather than mutual benefit.

The implications are stark. In a market with homogeneous goods—products that are completely interchangeable—price competition alone should destroy all profits. Two competitors are just as ruthlessly efficient as a hundred competitors, or a thousand. The number doesn't matter. What matters is that someone, somewhere, is willing to undercut you.

This should give pause to anyone who thinks that "setting the price" gives sellers meaningful power. In Bertrand's world, the seller who posts the price is trapped by it. They're not controlling the market; they're being controlled by the relentless logic of competition.

The Empty Set Problem

There's a mathematical wrinkle in Bertrand competition that reveals something deep about the nature of competitive pressure.

Suppose your competitor prices somewhere between cost and the monopoly price—the price they'd charge if they had no competition at all. What should you do?

The intuitive answer is: undercut them by just a tiny amount. If they're charging eight dollars, you charge seven ninety-nine. You steal all their customers while sacrificing almost nothing on margin.

But here's the problem: no matter how small an undercut you choose, you could always choose a smaller one. Undercut by a penny? Why not half a penny? Why not a tenth of a penny? In the continuous world of mathematical economics, there's always a smaller number.

This means that when your competitor prices above cost but below the monopoly price, your "best response" is technically the empty set—there is no single best price you can choose. Every candidate price is dominated by something slightly lower.

This isn't just mathematical pedantry. It captures something real about competitive dynamics: the pressure to undercut is infinite and insatiable. There's no natural stopping point, no comfortable margin where both firms can rest easy. The only stable prices are the extremes—the monopoly price (when you're alone) or the cost price (when you're not).

Why Real Markets Don't Work This Way

If Bertrand competition accurately described reality, most businesses would be bankrupt. Clearly, something is missing from the model.

Several things, actually.

First, products are rarely truly homogeneous. Even gas stations differentiate themselves—through location, convenience, cleanliness, snack selection, or just the vague sense that one brand is "better" than another. These small differences create breathing room. If customers aren't perfectly indifferent between sellers, the race to the bottom slows down.

Second, real firms have capacity constraints. A company can't instantly produce unlimited quantities to meet all market demand. If Firm A drops its price and suddenly everyone wants to buy from Firm A, it might not be able to serve them all. Some customers end up at Firm B anyway, which means Firm B doesn't lose everything by keeping prices higher.

Third, information isn't perfect. Customers don't instantly know every price from every seller. Search takes time and effort. This friction allows price differences to persist.

Fourth, and perhaps most importantly, business isn't a one-shot game. Firms compete against each other day after day, year after year. In repeated interactions, there's room for tacit cooperation—an unspoken agreement to keep prices reasonable rather than trigger a mutually destructive price war. This isn't collusion in the illegal sense, just the recognition that scorched-earth competition hurts everyone.

Bertrand Versus Cournot

The contrast between Bertrand and Cournot competition reveals how much depends on your assumptions about what firms actually control.

In Cournot's model, firms choose quantities. They decide how much to produce, and the market price emerges from the total supply meeting demand. This tends to produce prices above cost and positive profits, even with competition.

In Bertrand's model, firms choose prices. Customers then decide how much to buy from each seller based on who's cheapest. This tends to drive prices down to cost and eliminate profits entirely.

Which model is right? It depends on the industry.

Some markets genuinely are more "Cournot-like"—industries where production decisions are made far in advance, where capacity is expensive and hard to adjust, where firms commit to quantities before they know exactly what prices will clear the market. Oil production, semiconductor manufacturing, and large-scale agriculture all have this flavor.

Other markets are more "Bertrand-like"—retail, services, anything where firms can easily adjust to meet whatever demand materializes at their chosen price. If you're selling software downloads or consulting hours, you're not really constrained by quantity. You're competing on price.

The genius of Bertrand's critique was recognizing that the choice of strategic variable—price versus quantity—isn't neutral. It shapes everything that follows.

The Edgeworth Extension

Bertrand himself never formalized his insight into a proper mathematical model. That work fell to Francis Ysidro Edgeworth, an Irish economist who picked up Bertrand's ideas in 1889 and worked out the technical details.

Edgeworth also introduced an important complication: what happens when firms do have capacity constraints? When neither firm can serve the entire market?

The answer, it turns out, is chaos. With capacity limits, there may be no stable equilibrium at all. Prices cycle endlessly—rising when both firms are near capacity, crashing when someone sees an opportunity to grab market share. This "Edgeworth cycle" shows up in real markets, particularly gasoline, where prices often follow sawtooth patterns of gradual increases followed by sudden drops.

The Deeper Lesson

Bertrand competition is a theoretical extreme, a pure case that rarely exists in the wild. But theoretical extremes are useful precisely because they reveal underlying forces.

The underlying force here is substitutability. When customers genuinely don't care which seller they buy from—when products are perfect substitutes—sellers lose all their bargaining power. Every penny of profit becomes vulnerable to undercutting. Competition compresses margins toward zero.

This helps explain why businesses work so hard to differentiate themselves. Branding, customer service, loyalty programs, convenience features, product variations—all of these serve, among other purposes, to weaken the force of Bertrand competition. They create reasons for customers to prefer one seller over another even when prices aren't identical.

It also explains why monopolies are so profitable and so persistent. The moment competition appears, margins come under pressure. The more competitive the market, the more pressure. Bertrand's model is just the limiting case: perfect competition, zero profits.

Who Really Sets the Price?

There's a folk wisdom that says whoever posts the price has power over it. Sellers put stickers on shelves, so sellers must control prices. Right?

Bertrand competition suggests otherwise. Yes, the seller physically writes the number. But what number can they write? In a competitive market, they're constrained by what their rivals might write. They're trapped between their own costs and the threat of being undercut.

The price isn't really set by the seller at all. It's set by the structure of competition itself.

This doesn't mean sellers are powerless—they can differentiate their products, build brand loyalty, create switching costs, or find ways to implicitly coordinate with competitors. But all of these strategies are, in a sense, attempts to escape Bertrand competition, to create conditions where the pure logic of undercutting doesn't apply.

The firms that succeed in escaping are the ones that earn profits. The firms that don't escape compete their margins away. And Bertrand's insight, more than 140 years old now, still explains why.