The Limits of Price Discrimination When the Market Is Unknown
Irfan Tekdir
AI 速览：在仅知可能价值集合、不知具体价值分布的情况下，刻画市场与市场分割所能产生的消费者剩余与生产者剩余组合。
Bergemann, Brooks, and Morris (2015) characterize the consumer and producer surplus pairs that segmentation can produce when the distribution of values is known. I fix only the set of possible values and characterize the pairs that some market and some segmentation can produce. They form a convex polygon. Its outer edge is the largest efficient surplus a market can have when its uniform-price profit is capped, net of that profit, and truncated equal-revenue markets attain it. At the values no segmentation is needed; strictly between two values, reaching the frontier requires splitting a market into two extremal markets. With a continuum of values up to $\bar{v}$, consumer surplus is at most $π\ln(\bar{v}/π)$ when producer surplus (the seller's profit) is $π$, this bound is reached without segmentation, and no market gives consumers more than $\bar{v}/e$.
In a duopoly with captive and contested consumers, I characterize every pair of consumer and producer surplus that can arise under some market composition and some public segmentation, and construct a market that produces each pair. The set is lens-shaped. At any level of industry profit, consumers do worst when captive and contested consumers are perfectly separated and best in an unsegmented market in which the two firms have equally many captives. A market with unequal captive bases and some contested consumers cannot reach this upper frontier under any segmentation. Between Bertrand competition and monopoly, the frontier itself lies strictly below what consumers would get if everyone paid a single deterministic price yielding the same profit. Firms must randomize, and with downward-sloping demand the resulting price dispersion lowers consumer surplus.