Cournot and Bertrand: why the same two firms give two different answers

CB2 · Business Economics · 11 min read

What makes oligopoly different

In perfect competition a firm is too small to matter, so it treats the price as given. In monopoly there is nobody else, so there is nothing to react to. Oligopoly is the case in between, and it is harder for one reason: what you should do depends on what your rival does, and what they should do depends on what you do.

That circularity has no answer until you say precisely what each firm believes the other will hold fixed. Change that assumption and you change the result, not by a little but completely. Two assumptions are standard, and they sit at opposite ends.

Cournot. Each firm chooses a quantity, believing the rival's quantity is fixed. The price then comes off the market demand curve for the combined output.

Bertrand. Each firm chooses a price, believing the rival's price is fixed. The lower-priced firm takes the market.

Neither is the "right" one. Which one describes a real market depends on what the firm can actually change quickly — and that is a question about capacity, not about theory. A cement plant commits to an output and lets the price find its level; a software seller posts a price and serves whoever comes.

The Cournot mechanism: best responses that meet

Take market demand and your rival's output as given. Subtract their output from the market and what is left is a demand curve for you alone — your residual demand. You are then a monopolist over that residue, so you do what a monopolist does: produce where your marginal revenue equals your marginal cost.

Do that for every possible rival output and you have a reaction function: your best output as a function of theirs. It slopes down, and that is the mechanism in one sentence — the more your rival makes, the less of the market is left for you, and the less you want to make.

The rival has a reaction function too, by the same argument. Equilibrium is where the two are simultaneously satisfied: a pair of outputs such that neither firm, looking at what the other has actually done, wants to change. Nothing about it requires the firms to cooperate, communicate or even like each other.

The result sits between the two extremes, and comparing the three is the fastest way to see what oligopoly is.

And the number of firms moves you along that line. With linear demand and identical costs, each of n firms produces (a − c)/(n + 1) of the quantity intercept, so total output is n(a − c)/(n + 1). At n = 1 that is half the competitive output, which is the monopoly answer; as n grows it climbs towards the whole of it. Cournot is a dial between monopoly and competition, and the number of firms is the dial.

The Bertrand mechanism: undercutting with nothing to stop it

Now let the firms set prices instead, with a genuinely identical product and no capacity limit.

Suppose both charge a price above marginal cost. Either firm can cut its price by one paisa, lose almost nothing per unit, and take the entire market, because no buyer has any reason to prefer the dearer of two identical things. So any price above marginal cost invites an undercut. The undercutting has only one stopping point: marginal cost itself, below which a sale loses money.

The conclusion is startling and it is the point of the model. Two firms are enough to produce the competitive outcome. Price equals marginal cost, output is the competitive output, and profit is nil — with a market of two. That is the Bertrand paradox, and it is a paradox only because the Cournot answer with the same two firms is so comfortable.

What the paradox really shows is how much work two innocent-looking assumptions were doing: a perfectly identical product, and no limit on how much either firm can supply. Loosen either and the collapse stops.

Differentiate the product — a brand, a location, a service record, a claims-handling reputation — and undercutting no longer takes the whole market, because some buyers stay. Each firm keeps a downward-sloping demand curve of its own and a price above marginal cost.

Add a capacity limit and undercutting cannot take the whole market either, because the cheaper firm physically cannot serve it. The residue goes to the dearer firm at its own price, which makes undercutting much less attractive.

Both repairs point the same way, and so does the empirical observation that two-firm markets are rarely ruinous: the Bertrand result is a warning about the assumptions, not a prediction about duopolies.

Why a cartel looks attractive and then falls apart

If the Cournot firms could agree, they would. Together they would produce the monopoly quantity, charge the monopoly price and split a bigger profit than the Cournot pair earns. The arithmetic below shows the gain.

The trouble is in the same reaction function that produced the equilibrium. At the collusive output, your rival is holding back. Look at your own best response to that restraint and it is to produce more than your share of the agreement — because the price is high and the residual demand left to you is large. Cheating pays. And it pays for both of you, which means the agreement is not a resting point at all: it is a point from which both firms want to move.

That is the structure of a prisoners' dilemma, and it explains two things that otherwise look contradictory. Cartels keep forming, because the prize is real. And cartels keep collapsing into price wars, because each member's private interest is to break them. What holds a cartel together, when anything does, is the prospect of being punished in future rounds — which is why collusion is more stable in markets where firms meet each other repeatedly, see each other's prices quickly, and have the capacity to retaliate.

Where marks are lost

Treating the rival's output as a variable in your own first-order condition. The whole content of the Cournot assumption is that you hold it fixed while you optimise. Differentiate with respect to your own quantity only, and impose symmetry afterwards. Imposing it too early collapses the model into something else.

Forgetting that price comes from total output. Your profit per unit depends on q1 + q2, not on your own output. Writing the demand curve with only your own quantity in it is the commonest wrong start.

Quoting the Bertrand result as though it describes duopolies. It describes duopolies selling a perfectly identical product with unlimited capacity. Say so. An answer that states the assumptions and then the conclusion is worth more than one that states only the conclusion, and it is also the honest one.

Confusing "no profit" with "no normal profit". Price equal to marginal cost means no profit above the normal return. The firms are not going bankrupt; they are earning exactly what their capital could earn elsewhere.

Questions students ask

Which model should I use if a question does not say?

Read what the firms are described as choosing. If the question gives quantities, capacities or output decisions, it is Cournot. If it gives prices and an identical product, it is Bertrand. If it gives a differentiated product and prices, it is Bertrand with differentiation, and the collapse to marginal cost does not happen. The giveaway is almost always in the first two lines.

Where does the kinked demand curve fit in?

It is a different answer to the same circularity, built on an asymmetric belief: that rivals will match a price cut but ignore a price rise. If that is true, demand facing the firm is elastic above the going price (a rise loses customers to rivals who stayed put) and inelastic below it (a cut gains little, because rivals follow). Marginal revenue then has a gap in it at the going price, and marginal cost can move within that gap without the price changing at all. It is a model of why oligopoly prices are sticky, and that is the thing it is for.

Is Cournot realistic if nobody actually behaves like that?

Firms do not literally assume a frozen rival. The defence is that the Cournot outcome is where a market settles when firms commit to capacity before competing on price — the quantity is the commitment, and the price-setting that follows delivers the Cournot price. Read that way it is not a claim about beliefs at all; it is a claim about the order in which decisions are made, which is observable.

A worked example

An invented market, built for this page. Two companies supply 20-litre drinking-water jars in Visakhapatnam. The product is identical, so a buyer has no reason to prefer one. Market demand is

with P the price in rupees per jar and Q the total quantity in lakh jars a month. Both firms have a constant marginal cost of ₹40 per jar and no fixed cost. The figures are illustrative and chosen for clean arithmetic.

Benchmark 1: one firm, as a monopolist

Revenue is (100 − Q)Q, so marginal revenue is 100 − 2Q — falling twice as steeply as demand, because selling one more jar means repricing all the others. Set it equal to marginal cost:

Benchmark 2: perfect competition

Price is driven to marginal cost.

Cournot: build the reaction function

Firm 1's profit, with firm 2's output held fixed:

Differentiate with respect to q1 only and set to zero:

That is the reaction function, and it says the sensible thing: every extra jar the rival makes costs you half a jar of your own best output. Firm 2's is the mirror image.

Cournot: find where the two meet

By symmetry q1 = q2 = q:

Check it is really a resting point. Put q2 = 20 into firm 1's reaction function: (60 − 20)/2 = 20. Firm 1's best answer to 20 is 20, so neither firm wants to move. That is what equilibrium means here, and it is a one-line check worth doing every time.

Why the monopoly split is not an equilibrium

Suppose firm 2 produced the whole monopoly quantity of 30. Firm 1's best response is (60 − 30)/2 = 15, giving Q = 45 and P = ₹55. Firm 1 earns 15 × 15 = ₹225 lakh and firm 2 earns 30 × 15 = ₹450 lakh. Firm 2 is now unhappy: its best response to 15 is (60 − 15)/2 = 22.5, not 30. The pair keeps adjusting, and 20/20 is the only place the adjusting stops.

Bertrand: the same two firms, choosing prices

At any common price above ₹40, either firm can shave a paisa off and take the whole market, since the jars are identical and neither firm faces a capacity limit. There is no price above ₹40 that survives the next undercut, so

Identical to perfect competition, with two firms rather than many.

The four outcomes together

The three-firm row uses the formula: each firm makes (100 − 40)/(3 + 1) = 15, so Q = 45, P = ₹55, and each earns 15 × 15 = ₹225 lakh. Check the formula against the rows above it: at n = 1 it gives 30, the monopoly answer, and at n = 2 it gives 2 × 60/3 = 40, the Cournot answer. As n grows the total climbs towards 60, the competitive answer, without ever quite reaching it. Monopoly and competition are the two ends of one Cournot family.

Notice how far apart the two duopoly rows are. Same two firms, same product, same costs, same demand — and the industry earns ₹800 lakh under one assumption and nothing at all under the other. The assumption is not a technical detail; it is the answer.

The cartel, and why it will not hold

Suppose the two agree to split the monopoly output: 15 lakh jars each.

Better than the ₹400 lakh each earned at Cournot — so the agreement is worth making. Now let firm 1 cheat while firm 2 keeps to 15. Firm 1's reaction function gives its best response:

Cheating lifts firm 1 from ₹450 to ₹506.25 lakh, a gain of ₹56.25, and costs the loyal firm ₹112.50. Firm 2 faces the identical temptation, and if both cheat they end back at 20 each and ₹400 each — worse than the ₹450 the agreement offered.

Check the whole set of outcomes once: ₹450 each if both keep the deal, ₹400 each if both break it, and ₹506.25 against ₹337.50 if one breaks it alone. Whatever the other does, breaking it pays — 506.25 beats 450, and 400 beats 337.50. Both therefore break it, and both end up worse off than if neither had. The arithmetic of the cartel's collapse is in those four numbers.

Where this fits in CB2

This page explains one idea from CB2 Business Economics. The full CB2 syllabus, with what each part is examined on, sits on the course page, and past-paper style mock exams with full solutions are where you find out whether the idea has stuck under time pressure.

Other CB2 topics explained

• Price elasticity of demand, and what it decides about a price rise
• Income and substitution effects: splitting one price change in two
• Profit maximisation and the shutdown decision, from one cost table
• Adverse selection and moral hazard, and what an insurer does about them
• National income and the multiplier: where the rest of the spending comes from
• The IS-LM model and crowding out, solved with numbers
• How monetary policy reaches the economy: the transmission mechanism

Stuck on a term rather than the idea? The actuarial glossary defines the notation used above. If something here is wrong or unclear, tell us — these pages get corrected.