# Integrability and Identification in Multinomial Choice Models

## TL;DR

This paper establishes shape-restrictions for multinomial choice models that enable nonparametric identification of preferences and unobserved heterogeneity, extending the theoretical foundation of random utility models with practical implications.

## Contribution

It provides new shape-restrictions that characterize when multinomial choice probabilities can be rationalized by random utility models with nonparametric heterogeneity, including equivalence to additive models.

## Key findings

- Shape-restrictions characterize rationalizability of choice probabilities.
- Nonparametric identification of preference distributions is achieved without infinite identification assumptions.
- Slutsky-symmetry is equivalent to absence of income-effects.

## Abstract

McFadden's random-utility model of multinomial choice has long been the workhorse of applied research. We establish shape-restrictions under which multinomial choice-probability functions can be rationalized via random-utility models with nonparametric unobserved heterogeneity and general income-effects. When combined with an additional restriction, the above conditions are equivalent to the canonical Additive Random Utility Model. The sufficiency-proof is constructive, and facilitates nonparametric identification of preference-distributions without requiring identification-at-infinity type arguments. A corollary shows that Slutsky-symmetry, a key condition for previous rationalizability results, is equivalent to absence of income-effects. Our results imply theory-consistent nonparametric bounds for choice-probabilities on counterfactual budget-sets. They also apply to widely used random-coefficient models, upon conditioning on observable choice characteristics. The theory of partial differential equations plays a key role in our analysis.