Partial Identification of Heterogeneity in Preference Orderings Over Discrete Choices

We study a variant of a random utility model that takes a probability distribution over preference relations as its primitive. We do not model products using a space of observed characteristics. The distribution of preferences is only partially identified using cross-sectional data on varying budget sets. Imposing monotonicity in product characteristics does not restore full identification. Using a linear programming approach to partial identification, we show how to obtain bounds on probabilities of any ordering relation. We also do constructively point identify the proportion of consumers who prefer one budget set over one or two others. This result is useful for welfare. Panel data and special regressors are two ways to gain full point identification.