Homogeneous Product Oligopoly

Imagine a small town with only two petrol stations, both selling an identical fuel. How do they decide how much to sell, or what price to charge? They know the other is watching, and any move they make will provoke a reaction. This chapter is about that kind of strategic dance—when a handful of firms sell exactly the same product and must figure out the best course of action, knowing their rivals are just as clever.

The Big Picture#

When products are identical, competition looks very different depending on whether firms choose quantities (how much to produce) or prices (what to charge). If they pick quantities, we get a middle-ground result between monopoly and perfect competition. If they pick prices instead, the outcome can swing all the way to zero profit—a paradox that surprises our intuition. Real‑world markets often fall somewhere in between because firms can take steps, like building a plant of a certain size, that turn a price‑setting game into a quantity‑setting one. Understanding this interplay between quantity, price, and capacity lets us see why some industries are so fiercely competitive while others enjoy comfy margins.

Cournot Quantity Competition#

In 1838, the French economist Augustin Cournot asked a simple question: what happens if two or more firms decide how much to produce at the same time, and the market price is then set by total supply? This model—the Cournot oligopoly—is the foundation of modern competition economics.

The residual demand idea#

Suppose every firm sells the exact same product. Buyers only care about the total quantity available; the price that clears the market comes from the overall demand curve. But from a single firm’s perspective, it does not face the whole market demand—it faces what is left after everyone else has put their output on the shelves. That left‑over part is called residual demand.

Residual demand: The demand left for a firm after subtracting what its rivals have already produced: , where is the total output of all other firms.

In other words, Firm 1 cannot control the total quantity; it can only choose its own , and it must accept that the market price will be whatever clears the remaining supply given .

A concrete linear example#

To make this tangible, let the market inverse demand be linear:

where . For simplicity, suppose there are two firms (duopoly), each with a constant marginal cost , and no fixed costs. Firm 1’s profit is

Firm 1 maximises this by choosing , treating as a given number—this is the essence of Nash equilibrium: everyone does the best they can, taking others’ actions as fixed.

Setting the derivative to zero:

This is Firm 1’s best‑response function (or reaction function). It says: “If my rival increases output by one unit, I should cut my own output by half a unit.” The more the rival produces, the less profitable it is for me to flood the market.

For Firm 2, symmetry gives

Solving these two equations together yields the Cournot equilibrium outputs:

Total output is , and the market price is

Compare this with monopoly (, ) and perfect competition (, ). Cournot duopoly sits right in the middle—less output than perfect competition, more than monopoly, and a price that is above marginal cost but below the monopoly price.

Strategic substitutes#

Notice the slope of the best‑response function: . When one firm increases its quantity, the other’s best reply is to decrease its quantity. This property is called strategic substitutes. In any strategic setting, an action is a strategic substitute if a more aggressive move by your rival makes you want to be less aggressive. With identical products, quantities are strategic substitutes—extra output from a competitor shrinks the residual demand, so you respond by pulling back.

Strategic substitutes: A setting where, if your rival raises its output, your profit from producing one more unit falls, so your best reply is to produce less. Reaction functions slope downward.

Extending the logic to firms, the symmetric Cournot output per firm becomes , and the price falls towards as grows. So more firms push the market closer to the competitive ideal.

📝 Section Recap: When identical‑product firms pick quantities simultaneously, each faces a residual demand left by rivals; their best‑response functions slope downward (strategic substitutes), leading to a price above marginal cost that gradually approaches the competitive level as the number of firms rises.

The Bertrand Paradox#

In 1883, Joseph Bertrand published a strong criticism of Cournot’s work. He argued that if firms really sell identical goods and set prices instead of quantities, the logic leads to a dramatically different result. Suppose two firms choose prices and at the same time, with constant marginal cost . Buyers flock to the cheaper seller. If the prices are equal, we can assume they split the market.

Now ask: can a pair of prices where either firm charges more than form an equilibrium? No. Here is why.

The undercutting logic#

Assume both firms initially set the same price . Each sells half the market and earns a positive profit. But look at the incentives. Firm 1 could drop its price by just a pinch—one cent—and steal the entire market. Its profit would jump from to approximately , which is almost double. Firm 2 faces the same temptation. So the only stable situation is one where no one can gain by a further price cut. That happens when price is driven all the way down to marginal cost: , and profits are zero.

Bertrand paradox (homogeneous goods): Even with only two firms, price competition pushes the price down to marginal cost, just like perfect competition. The only stable outcome is zero profit.

Why it is called a paradox#

It feels shocking—surely a duopoly should have some market power? The paradox relies on the fact that the goods are perfect substitutes and there are no capacity limits. If Firm 1 tries to raise its price slightly above marginal cost, it loses all its customers instantly because the rival’s product is identical and every buyer is perfectly informed. Real markets rarely, if ever, satisfy all these sharp conditions, which is why firms often find ways to avoid the Bertrand trap: product differentiation, capacity constraints, or repeated interaction.

In this scenario, a smooth reaction function does not exist. When , Firm 1’s best response is to set (just below ). If , the best response is . That is a discontinuous “jump” rather than a gently sloping line. We cannot talk about strategic substitutes or complements in the usual way, because the undercutting dynamic is so fierce that the only outcome is the perfectly competitive one.

📝 Section Recap: With identical products and no capacity limits, price competition is cut‑throat to the extreme—undercutting drives price to marginal cost and eliminates all profit. This “Bertrand paradox” illustrates how dramatically the choice of strategic variable (quantity vs. price) can alter market outcomes.

Capacity Precommitment and Cournot Equivalence#

The sharp contrast between Cournot and Bertrand has puzzled economists for a long time. Think about a real industry: an airline that sets its ticket prices does not ignore how many seats it has. A concrete manufacturer cannot sell more cement than its kiln capacity allows. This suggests a bridge between the two models: if firms first install some capacity (a production limit) and then compete on prices, the outcome may look much more like Cournot than like the Bertrand paradox.

A two‑stage game#

Imagine a two‑stage interaction:

1. Stage 1 (capacity choice): Each firm independently decides a capacity , incurring a cost of per unit of capacity. This capacity is the maximum it can produce and sell later.
2. Stage 2 (price competition): With capacities fixed, the firms simultaneously set prices . Consumers buy from the cheapest seller up to its capacity; any excess demand spills over to the higher‑priced firm if it still has spare capacity.

What happens in the price subgame? If total capacity is fairly large relative to demand at marginal cost, the equilibrium will be a market‑clearing price where demand equals . The intuition: no firm can undercut to grab extra sales beyond its capacity, so the incentive to slash prices is blunted. Instead, both firms set a price that clears the market, and each sells exactly its capacity. If the capacities are modest (so is less than the competitive output), that market‑clearing price will be above the marginal production cost , yielding profits.

How capacity makes price reactions upward‑sloping#

With capacity limits, a firm’s residual demand becomes kinked. Suppose Firm 2 prices at and has capacity . If Firm 1 sets , it captures the whole market but can sell only up to . Any extra demand beyond goes to Firm 2, which then earns on its capacity. If Firm 1 prices above , it becomes the high‑price supplier, getting only the residual demand after Firm 2’s capacity is exhausted.

The best‑response function for price now turns out to be upward‑sloping over a certain range: if Firm 2 raises its price, Firm 1 can also raise its own price without losing (much) additional business, because the extra demand that would have come to Firm 1 is capped by its capacity anyway. This property is the opposite of the Cournot case—prices become strategic complements.

Strategic complements and substitutes—a broader lens#

• Strategic substitutes (Cournot quantities): ; reaction functions slope down. Doing more of the action makes the other firm want to do less.
• Strategic complements (prices with capacity): ; reaction functions slope up. Doing more (raising price) makes the rival want to raise its price too.

Strategic complements: When your rival does more of something (like raising its price), it becomes more profitable for you to do more of it too. Your best responses move in the same direction.

In the two‑stage capacity‑then‑price game, the equilibrium in the subgame (given ) forces the price to equal the level at which total demand just absorbs . Therefore, the overall outcome matches the Cournot equilibrium that would arise if the firms had simply chosen output levels to begin with. This is often called Cournot equivalence or the Kreps–Scheinkman result: capacity precommitment turns a price‑setting game into a quantity‑setting game in disguise.

So, when it is costly to expand capacity quickly, the market outcome may be Cournot‑like even though firms quote prices every day. This insight helps explain why many manufacturing industries display intermediate levels of concentration and mark‑ups, rather than the devastating price wars of the pure Bertrand model.

📝 Section Recap: If firms first commit to capacity and then compete on prices, the need to avoid under‑pricing beyond their limits softens price competition. Prices become strategic complements in the price subgame, and the final outcome replicates the Cournot quantity‑setting benchmark—capacity acts as a commitment device that avoids the Bertrand paradox.

Profit Maximization in Oligopoly: From Monopoly to Perfect Competition#

Having seen the three main models—Cournot, Bertrand, and the capacity‑precommitment story—we can now step back and look at how an oligopolist maximises profit and where the market lands on the spectrum between monopoly and perfect competition.

No matter the strategic variable, a firm’s optimal choice always boils down to balancing two forces:

• The direct price‑cutting effect: Selling one more unit brings in the current price, but also pushes the price down a little for all the units you already sell.
• The strategic effect: Your choice influences what your rivals will do. In Cournot, if you expand output, rivals contract, which partly offsets the price decline; in Bertrand with identical goods, the strategic effect is catastrophic—a small price cut steals the whole market until the floor is hit.

A useful way to capture the degree of market power is the Lerner index, , which measures the mark‑up as a share of price. In our linear Cournot duopoly, (positive). With firms, as grows, tending to zero. In Bertrand with homogeneous goods, immediately. In the capacity‑precommitment equilibrium, the mark‑up is the same as Cournot.

So a manager maximising profit in an oligopoly must consider not only demand and cost but also the type of competitive rivalry. If capacities are flexible and products are indistinguishable, the market may behave like Bertrand—low margins, high volume. If capacity is rigid or output is hard to adjust, Cournot logic applies, offering more comfortable margins. Real‑world firms often try to shape these conditions: building a large plant (capacity commitment), making their product different, or building customer loyalty are all ways to escape the Bertrand trap and enjoy higher mark‑ups.

📝 Section Recap: Oligopoly profit maximisation always involves balancing the marginal gain from extra output against the depressing effect on price, compounded by rivals’ reactions. Where the market falls between the Bertrand zero‑profit and the Cournot positive‑profit extremes depends on whether firms compete primarily on price or on quantity—and whether capacity commitments transform the game.

Summary#

We have journeyed through the three main models of competition when firms sell identical products. Cournot’s quantity‑setting world gives us a calm, predictable oligopoly where each firm’s best response is to pull back when a rival expands, leading to prices above cost. Bertrand’s price‑setting world, by contrast, is a race to the bottom—with no limits, undercutting destroys all profit, a paradox that helps us think about real markets. The capacity‑precommitment bridge shows how firms can soften price competition by building a credible limit on what they can sell, effectively turning a price game into a quantity game and restoring Cournot‑like outcomes. Understanding these forces helps you see why some industries are dominated by a few comfortable giants while others lose profit despite a handful of players.