  Welfare effects of price changes are often estimated with cross-sections;
these do not identify demand with heterogeneous consumers. We develop a
theoretical method addressing this, utilizing uncompensated demand moments to
construct local approximations for compensated demand moments, robust to
unobserved preference heterogeneity. Our methodological contribution offers
robust approximations for average and distributional welfare estimates,
extending to price indices, taxable income elasticities, and general
equilibrium welfare. Our methods apply to any cross-section; we demonstrate
them via UK household budget survey data. We uncover an insight: simple
non-parametric representative agent models might be less biased than complex
parametric models accounting for heterogeneity.
