  We develop new robust discrete choice tools to learn about the average
willingness to pay for a price subsidy and its effects on demand given
exogenous, discrete variation in prices. Our starting point is a nonparametric,
nonseparable model of choice. We exploit the insight that our welfare
parameters in this model can be expressed as functions of demand for the
different alternatives. However, while the variation in the data reveals the
value of demand at the observed prices, the parameters generally depend on its
values beyond these prices. We show how to sharply characterize what we can
learn when demand is specified to be entirely nonparametric or to be
parameterized in a flexible manner, both of which imply that the parameters are
not necessarily point identified. We use our tools to analyze the welfare
effects of price subsidies provided by school vouchers in the DC Opportunity
Scholarship Program. We robustly find that the provision of the status quo
voucher and a wide range of counterfactual vouchers of different amounts have
positive benefits net of costs. This positive effect can be explained by the
popularity of low-tuition schools in the program; removing them from the
program can result in a negative net benefit. Relative to our bounds, we also
find that comparable logit estimates potentially understate the benefits for
certain voucher amounts, and provide a misleading sense of robustness for
alternative amounts.
