Multidimensional Product Design

Abstract

Products like radio shows and insurance contracts are designed by firms to sort for the most valuable users. An empirically relevant model of this process requires users to be heterogeneous along multiple dimensions of their preferences and their values, but existing models do not allow for multidimensional heterogeneity or require restrictive assumptions to be tractable. We show that a simple price theoretic analysis is possible when user heterogeneity is of high dimension relative to the firm's design instruments. We obtain necessary conditions for profit and welfare maximization in terms of moments of the distribution of user heterogeneity, where the power of an instrument to sort for valuable users is proportional to the covariance, within the set of marginal users, between the value of users and their marginal utility for the instrument. Our model allows for non-transferable utility, consumption externalities, cream-skimming distortions, adverse/advantageous selection, non-linear pricing, third-degree discrimination and imperfect competition. We discuss applications to broadcast media, the credit card industry and imperfect competition in insurance provision.