Revealed statistical consumer theory

We provide a microfoundation for using aggregated data (e.g. mean purchases) when evaluating consumer choice data. We present a model of statistical consumer theory where the individual maximizes their utility with respect to a distribution of bundles that is constrained by a statistic of the distribution (e.g. mean expenditure). We show that this behavior is observationally equivalent to an individual whose preferences depend only on the statistic of the distribution. This means that despite working with distributions, the empirical content of the model only depends on a finite-dimensional statistic. Statistical consumer theory neither nests nor is nested in the random utility approach. We show this approach generalizes quasilinear utility with random preferences and income, mean-variance preferences, and preferences that depend on arbitrary moments.