Oligopoly and monopolistic competition

Perfect competition and monopoly are the two ends of a range, and almost every market anyone actually buys from sits between them.

What makes the middle hard is that a firm with a few rivals must forecast what those rivals will do, and their best move depends on what it does. There is no way to choose without solving for everyone at once. This is the point at which microeconomics becomes game theory, and the equilibrium concept used throughout this lesson is the one from that subject: a set of choices such that no firm can do better by changing its own, holding the others fixed.

Competing in quantities

Antoine Augustin Cournot set out the first solution in 1838, using two owners of mineral springs. Each chooses a quantity, the total determines the price through market demand, and each takes the other's quantity as given.

Take inverse demand $p = 120 - Q$ with $Q = q_{1} + q_{2}$ , and constant marginal cost £30 for both. Firm 1's profit is

π_{1} = (120 - q_{1} - q_{2}) q_{1} - 30 q_{1}

Differentiating with respect to $q_{1}$ and setting to zero gives $120 - 2 q_{1} - q_{2} - 30 = 0$ , so

q_{1} = \frac{90 - q_{2}}{2}

That is firm 1's best response function: the profit-maximising output for each possible choice by the rival. Firm 2's is the mirror image. Equilibrium is where both hold simultaneously, and by symmetry $q_{1} = q_{2} = q$ , so $2 q = 90 - q$ and $q = 30$ each.

Example. Complete that calculation: find the total quantity, the price and each firm's profit, and compare with monopoly and with perfect competition.

Total output is 60 and the price is $120 - 60 = £ 60$ . Each firm earns $(60 - 30) \times 30 = £ 900$ , so the industry earns £1800. A monopolist would set $MR = 120 - 2 Q = 30$ , producing 45 at a price of £75 for a profit of £2025. A competitive industry would produce where $p = MC$ , giving $Q = 90$ at £30 and no profit. So the duopoly sits between the two on every measure: more output than monopoly, less than competition; lower price than monopoly, higher than competition; and less total profit than a monopolist would earn, which is the first hint that the two firms are hurting each other.

Now you. Three firms face $p = 100 - Q$ with marginal cost £10 each. Find each firm's output, the price and each firm's profit.

Answer

Firm $i$ 's first-order condition is $100 - 2 q_{i} - Q_{- i} - 10 = 0$ . With symmetry, $q_{i} = q$ and $Q_{- i} = 2 q$ , so $90 = 4 q$ and $q = 22.5$ . Total output is 67.5 and the price is £32.50. Each firm earns $(32.5 - 10) \times 22.5 = £ 506.25$ .

More firms means more competition

The general solution for $n$ identical firms facing $p = a - b Q$ with marginal cost $c$ is

q_{i} = \frac{a - c}{b (n + 1)}, p = c + \frac{a - c}{n + 1}

The markup over marginal cost falls as $1 / (n + 1)$ . On the first example, one firm gives a price of £75, two give £60, three give £52.50, ten give £38.18, and the limit as $n$ grows is £30, the competitive price. Cournot competition therefore reproduces monopoly at one firm and perfect competition in the limit, with a continuum between, which is why it remains the workhorse model of oligopoly nearly two centuries later.

Competing in prices, and a paradox

Joseph Bertrand reviewed Cournot in 1883 and objected that firms set prices, not quantities. Redo the problem that way and the answer changes completely.

Two firms sell an identical product with the same constant marginal cost, and buyers go to whoever is cheaper. If firm 2 prices above marginal cost, firm 1 can undercut by a penny and take the whole market, so any price above marginal cost invites undercutting. The only equilibrium is $p_{1} = p_{2} = MC$ , with zero profit. Two firms are enough for the competitive outcome. That is the Bertrand paradox, and it is a paradox because real duopolies plainly do not price at marginal cost.

Three things resolve it, and each is a real feature of real markets.

Capacity. If neither firm can serve the whole market, undercutting no longer captures it, so the incentive weakens. Francis Edgeworth made this point in 1897, and Kreps and Scheinkman showed in 1983 that firms choosing capacity first and prices second reach the Cournot outcome. That result is the modern justification for using quantity competition to model industries where capacity is decided long before price.

Repetition. A one-shot undercutting gain has to be weighed against losing a profitable future, which the collusion section below makes precise.

Differentiation. If the products are not identical, cutting price does not capture the whole market, because some buyers prefer the other brand at any small price gap. This is the most important resolution in practice, since almost nothing is truly homogeneous.

Example. Two differentiated firms face $q_{1} = 80 - 2 p_{1} + p_{2}$ and the mirror image, with marginal cost £20. Find the equilibrium prices, quantities and profits.

Firm 1 maximises $π_{1} = (p_{1} - 20) (80 - 2 p_{1} + p_{2})$ . Differentiating, $80 - 4 p_{1} + p_{2} + 40 = 0$ , so the best response is $p_{1} = (120 + p_{2}) / 4$ . By symmetry $4 p = 120 + p$ , so $p = £ 40$ . Each sells $80 - 80 + 40 = 40$ units and earns $(40 - 20) \times 40 = £ 800$ . Price sits comfortably above marginal cost with only two firms, which is what differentiation buys.

Now you. The same two firms, but marginal cost rises to £26. Find the prices, quantities and profits.

Answer

The best response becomes $p_{1} = (80 + 52 + p_{2}) / 4 = (132 + p_{2}) / 4$ , so $4 p = 132 + p$ and $p = £ 44$ . Each sells $80 - 88 + 44 = 36$ units and earns $(44 - 26) \times 36 = £ 648$ . A £6 cost increase raised price by £4, so two thirds passed through, higher than the half a monopolist with linear demand would pass through.

Why cartels form and why they break

Both firms would be better off agreeing to the monopoly output and splitting it. The problem is that the agreement is not a stable point of the game.

Example. In the first Cournot market, $p = 120 - Q$ with marginal cost £30, the firms agree to split the monopoly output of 45, taking 22.5 each. Compute each firm's profit under the agreement, then compute what firm 1 earns by cheating.

Under the agreement the price is £75 and each earns $(75 - 30) \times 22.5 = £ 1012.50$ , better than the £900 of Cournot competition. Now let firm 1 cheat while firm 2 keeps its word. Firm 1's best response to $q_{2} = 22.5$ is $q_{1} = (90 - 22.5) / 2 = 33.75$ . Total output becomes 56.25 and the price falls to £63.75, so firm 1 earns $33.75 \times 33.75 = £ 1139.06$ and firm 2 earns only $33.75 \times 22.5 = £ 759.38$ . Cheating pays £126.56, and it pays whatever the other firm does, so both cheat and both end up at the Cournot outcome of £900. The cartel has the structure of a prisoner's dilemma.

Now you. Two firms face $p = 200 - 2 Q$ with marginal cost £20 and agree to split the monopoly output. Find the agreed profit each, and the profit from cheating.

Answer

Monopoly output solves $200 - 4 Q = 20$ , giving $Q = 45$ at £110, so each takes 22.5 and earns $(110 - 20) \times 22.5 = £ 2025$ . Firm 1's best response to 22.5 is $q_{1} = (180 - 45) / 4 = 33.75$ , making $Q = 56.25$ and $p = £ 87.50$ , so firm 1 earns $67.5 \times 33.75 = £ 2278.13$ and firm 2 earns £1518.75. Cheating gains £253.13.

Repetition is what makes collusion possible at all. If the game recurs indefinitely and each firm threatens to revert to Cournot competition forever after any cheating, then cheating is worth it only if the immediate gain outweighs the discounted stream of lost future profit. On the first example, collusion holds when the discount factor satisfies

δ \geq \frac{1139.06 - 1012.50}{1139.06 - 900} = 0.53

Firms that value next period at more than about 53 per cent of this one can sustain the cartel. That is the flavour of the folk theorem, worked out properly in the Game Theory course, and it predicts what is observed: collusion survives where the participants are few, meet repeatedly, can see each other's prices quickly, and expect the industry to continue.

The record bears it out and shows the fragility too. OPEC has repeatedly agreed quotas and repeatedly seen members exceed them. The lysine cartel run by Archer Daniels Midland and three Asian producers from 1992 raised the price by roughly 70 per cent within months, and collapsed when an insider turned informant for the FBI. The vitamins cartel of the 1990s ran for nearly a decade before attracting record fines on both sides of the Atlantic. Detection, not restraint, is what usually ends them.

Monopolistic competition

Edward Chamberlin and Joan Robinson both published in 1933 on the case that fits the largest number of real firms: many sellers, differentiated products, and free entry. Restaurants, hairdressers, plumbers and corner shops are all in it.

Each firm has a little market power, because its product is not identical to any other, so it faces a downward-sloping demand curve of its own and prices above marginal cost. But free entry means any profit attracts a new entrant, whose arrival takes customers and pushes each incumbent's demand curve inward. Entry stops when profit reaches zero, which happens when the firm's demand curve is tangent to its average cost curve.

Take a firm with $C (q) = 800 + 10 q + 0.5 q^{2}$ , so average total cost is minimised at $q = 40$ with a value of £50. After entry has run its course, suppose its own demand curve is $p = 90 - 1.5 q$ . Marginal revenue is $90 - 3 q$ and marginal cost is $10 + q$ , which meet at $q = 20$ , where the price is £60 and average total cost is $40 + 10 + 10 = £ 60$ too. Profit is exactly zero, as free entry requires, and the tangency has landed the firm at half its efficient scale, paying £60 for what could be made for £50.

That is the excess capacity theorem: monopolistic competition delivers firms operating below minimum average cost, with unused capacity, permanently. The empty tables at eight restaurants rather than full tables at four is not a market failure to be corrected but the price of variety, and whether the variety is worth it is exactly the question the surplus arithmetic cannot answer, since it has no way to value having a choice of eight.

Measuring concentration

Since structure sits on a continuum, it has to be measured rather than classified. The standard tool is the Herfindahl-Hirschman index, the sum of the squared market shares in percentage points, so a monopoly scores 10,000 and ten equal firms score 1,000. Squaring is what makes it informative: it weights large firms far more heavily than a simple count of firms does.

United States merger guidelines have used it since 1982, though the thresholds have moved: the 2010 guidelines called anything below 1,500 unconcentrated and anything above 2,500 highly concentrated, while the 2023 guidelines lowered the second line to 1,800. In every version the change in the index caused by a merger matters as much as the level. The index has an obvious weakness, which is that it depends entirely on how the market is defined, and market definition is where most competition cases are actually decided. That is the SSNIP test from the elasticity lesson, and it is why cross-price elasticities end up in courtrooms.

Everything so far, from perfect competition through monopoly to the middle ground, has assumed that the costs and benefits of a trade fall on the people trading. Dropping that assumption is the next lesson, and it breaks the efficiency result even when competition is perfect.