Oligopoly pricing: kinked demand, Cournot, Nash and Stackelberg

How do a few interdependent firms set price and output?

An oligopoly has only a few firms, so each one’s decision depends on how it expects the others to react. The profit rule is unchanged: produce where marginal revenue (MR) equals marginal cost (MC) and charge what the demand curve allows. What changes is that the firm’s demand curve depends on its rivals, so there is no single oligopoly supply curve. Different assumptions about rivals give different models.

Oligopoly pricing modelsfew firms, interdependent decisions

• Kinked demand Memorize rivals match a price cut, ignore a rise: sticky prices
• Cournot Memorize simultaneous; each firm takes rivals' output as fixed
• Nash equilibrium Memorize each strategy is a best response; no one gains by changing alone
• Stackelberg Memorize sequential; the leader commits first, the follower reacts
• Dominant firm Use it a low-cost leader (often 40%+ share) sets the price, others follow

What does the kinked demand curve explain?

The kinked demand curve model assumes rivals will match a price cut but ignore a price rise. Above the current price, demand is elastic: raise the price and many customers leave for rivals. Below it, demand is much less elastic: cut the price and the rivals follow, so little is gained.

That shape puts a vertical gap in the marginal revenue curve at the current price. Marginal cost can rise or fall inside the gap without changing the profit-maximizing price or output. The model therefore explains sticky prices. Its weakness is that it cannot explain how the prevailing price was set in the first place.

Exam tip Match cuts, ignore rises: that is the kink. Costs moving inside the MR gap leave price unchanged.

What is the Cournot assumption?

Under the Cournot assumption each firm picks the output that maximizes its profit, taking its rivals’ output as fixed, and all firms decide at the same time. In equilibrium no firm wants to change its output given what the others produce.

With linear demand and equal, constant marginal cost, n identical Cournot firms produce a total of n/(n + 1) times the competitive output.

$$Q = \frac{n}{n + 1} \cdot \frac{a - MC}{b}, \qquad P = a - bQ$$

Firms (n)	Total output	Price
1 (monopoly)	45	55
2	60	40
3	67.5	32.5
4	72	28
9	81	19
Perfect competition (P = MC)	90	10

The duopoly sits between monopoly and competition, and each extra firm pushes price towards marginal cost. That is the logic behind competition law.

$$P_{monopoly} > P_{Cournot} > P_{competition} = MC, \qquad Q_{monopoly} < Q_{Cournot} < Q_{competition}$$

What is a Nash equilibrium?

Memorize

Game theory is the study of decisions when each participant’s best move depends on what the others do. A Nash equilibrium is a set of strategies in which each firm’s strategy is its best response to the others’, so no firm can raise its profit by changing strategy alone. The firms do not cooperate. Each maximizes its own profit.

To find one in a payoff grid, mark your best payoff against each rival choice, do the same for the rival, and look for a cell where both are marked.

That gap between 150 and 220 is why collusion is tempting: both firms would be better off Holding, but each is better off Expanding if the other Holds.

Go deeper A Nash equilibrium without a dominant strategy

Two airlines, A and B, each pick Route 1 or Route 2. Payoffs (A, B): both on Route 1: 3, 3; A on 1 and B on 2: 8, 6; A on 2 and B on 1: 6, 8; both on Route 2: 3, 3. A’s best response to B on Route 1 is Route 2 (6 > 3), and to B on Route 2 is Route 1 (8 > 3). The same holds for B. Neither has a dominant strategy, yet there are two Nash equilibria: the cells where the airlines take different routes. The best-response method finds both.

What are Stackelberg, price leadership and collusion?

In the Stackelberg model decisions are sequential. The leader commits to its output first, and the follower chooses after observing it. The leader earns more than it would under Cournot and the follower earns less: a first-mover advantage. The leader can even overproduce to squeeze the follower.

Under price leadership one dominant firm, often with a share of 40% or more and usually the lowest-cost producer, sets the price. Smaller rivals follow, because undercutting a lower-cost leader risks a price war they would lose. Over time the leader’s share tends to erode as profits attract entrants.

When firms agree openly and formally to restrict output or fix prices, the group is a cartel . Collusion is tempting because it raises profit, reduces uncertainty and can help build barriers to entry. It is illegal in most countries and fragile, because each member gains by cheating.

Collusion is more likely when...

• Few firms, or one dominant firm Memorize similar market shares make it harder
• Homogeneous products Memorize
• Similar cost structures Memorize
• Orders are small and frequent Memorize cheating is caught quickly and pays little
• Retaliation is credible and severe Memorize
• Little competition from outside the group Memorize

Long-run economic profits are possible in an oligopoly, but they tend to erode as rivals enter or members cheat.

Go deeper Cournot reaction functions

With demand $P = a - b(q_1 + q_2)$ and constant cost $MC$, firm 1 sets $MR_1 = a - 2bq_1 - bq_2 = MC$, so its best response is $q_1 = (a - MC - bq_2) \div 2b$. Firm 2’s is symmetric. Solving both together gives $q_1 = q_2 = (a - MC) \div 3b$, so total output is two-thirds of the competitive output. For n firms the same steps give n/(n + 1).

Go deeper Signaling and price wars

Firms in an oligopoly rarely sign agreements. Instead they signal through announced price changes and capacity moves, and punish defectors with price cuts. Price wars hurt everyone, so they are usually short and tend to end with a return to the old price, which is why the kinked demand outcome is so persistent.

Go deeper When an oligopoly is not really an oligopoly

If entry is easy, a handful of firms will price close to cost, because any profit invites a newcomer. Count the barriers, not only the firms. The concentration measures in the next lesson do not capture this.