Nonparametric Welfare Analysis for Discrete Choice

We consider empirical measurement of equivalent/compensating variation resulting from price-change of a discrete good using individual-level data, when there is unobserved heterogeneity in preferences. We show that for binary and unordered multinomial choice, the marginal distributions of EV/CV can be expressed as simple closed-form functionals of conditional choice-probabilities under essentially unrestricted preference-distributions. These results hold even when the distribution/dimension of unobserved heterogeneity are neither known nor identified and utilities are neither quasi-linear nor parametrically specified. The welfare distributions take simple forms which are easy to compute in applications. In particular, average EV for a price-rise equals the change in average Marshallian consumer-surplus and is smaller than average CV for a normal good. These nonparametric point-identification results fail for ordered choice if the unit-price is identical for all alternatives, thereby providing a connection to Hausman-Newey's (2014) partial identification results for the limiting case of continuous choice