Revisiting Bertrand Competition with Captive Customers
Myatt & Ronayne: "Bertrand competition and captive customers" Economics Letters, Volume 257 (December 2025)
Textbook treatments of Bertrand competition suggest a simple outcome: rivals ferociously undercut each other to grab price-sensitive customers until prices hit marginal cost. This leads to the classic prediction of the “law of one price” so that cost-pricing is ubiquitous.
When marginal costs are strictly asymmetric, the standard intuition (that the two most efficient firms compete down to the second-lowest cost and the lowest-cost firm scoops all sales) faces a theoretical challenge because this resulting price is weakly dominated for the second-lowest-cost firm (that firm makes no sales, yet charges a price such that if it did make a sale, it would make nothing on it). In addition, there are very many (infinitely many) predictions for all but the lowest-cost firm’s price. Addressing those issues requires harnessing insights beyond the benchmark model.
Recent research by David P. Myatt (London Business School) and David Ronayne (ESMT Berlin, Project A04), explores how pricing predictions can be refined when firms with asymmetric costs have small masses of “captive” customers. In this context, where the masses of captive buyers approach zero, the analysis picks out a unique prediction. The efficient firm serves all customers at a price exactly equal to the second-lowest marginal cost, netting a positive profit, while its competitors earn nothing, just as in the benchmark model. However, all other prices are uniquely pinned down and markedly different from the classroom intuition of at-cost pricing. All inefficient firms exclusively use above-cost (undominated) prices, and each sets the monopoly price with strictly positive probability. This outcome suggests that rather than conforming to a strict “law of one price,” the equilibrium reflects a “law of monopoly pricing,” where most competitors sit far above cost, while a single efficient firm anchors the transaction price at cost.
In contrast to duopolies where a single high-cost firm disciplines the price setter, their oligopoly analysis reveals that it is not necessarily a single competitor that anchors the efficient firm’s pricing. The unique equilibrium can readily involve the participation of multiple inefficient firms in mixed-strategy pricing. Specifically, the strategies are defined by a sequence of pairwise ‘‘dances’’ between the efficient firm and its rivals, where partner swaps occur as prices rise. They show that the order in which the masses of captive buyers vanish can matter for determining the prescribed mixed strategies for each individual inefficient firm, potentially involving all inefficient firms in these competitive “dances”. This result highlights how the structure of the non-participating inefficient firms influences the pricing dynamics, reinforcing that competition in this limit is rarely confined to just the two most efficient rivals.
Link: Bertrand competition and captive customers
