Linear Demand Systems are Consistent with Discrete Choice

Aggregate linear demand systems can be micro-founded by a population of heterogeneous consumers making discrete choices. They can also be extended to incorporate income effects. I prove these results and show how to structurally estimate this class of models using only market-level data. The distribution of consumer preferences is recovered by disciplining the corner solutions with aggregate terms only. This approach allows for the same richness of Berry, Levinsohn, and Pakes (1995) [BLP] while retaining the analytical and computational advantages of linear demand: it does not require exogenously fixing market size, allows for heterogeneity in choice sets, and results in a flexible substitution matrix which varies endogenously as different consumers enter/exit the market. The discrete-choice microfoundation facilitates welfare analysis and the flexible measurement of markups an cost pass-throughs, among a wide range of other applications.