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This article surveys the development of nonparametric models and methods for estimation of choice models with nonlinear budget sets. The discussion focuses on the budget set regression, that is, the conditional expectation of a choice variable given the budget set. Utility maximization in a nonparametric model with general heterogeneity reduces the curse of dimensionality in this regression. Empirical results using this regression are different from maximum likelihood and give informative inference. The article also considers the information provided by kink probabilities for nonparametric utility with general heterogeneity. Instrumental variable estimation and the evidence it provides of heterogeneity in preferences are also discussed. 

The elasticity of taxable income with respect to the net of tax rate is a key parameter for predicting the effect of tax reform or designing an income tax. Bunching at kinks and notches in a single budget set has been used to estimate the taxable income elasticity. We show that when the distribution of preferences is unrestricted the amount of bunching at a kink or a notch is not informative about the size of the taxable income elasticity, and neither is the entire distribution of taxable income for a convex budget set. Kinks do provide information about the size of the elasticity when a priori restrictions are placed on the preference distribution. They can identify the elasticity when the preference distribution is completely and correctly specified across the kink and provide bounds under restrictions on the preference distribution. We find wide estimated bounds in an empirical example using data like Saez (2010) based on upper and lower bounds for the preference density.