We provide a method to estimate resource shares—the fraction of total household expenditure allocated to each household member—using linear (e.g., ordinary least squares) estimation of Engel curves. The method is a linear reframing of the 2013 nonlinear model of Dunbar, Lewbel, and Pendakur, extended to allow single-parent and other complex households, scale economies in assignable goods, and complementarities between nonassignable goods and supplemented with a linear identification test. We apply the model to data from 12 countries and investigate resource shares, gender gaps, and poverty at the individual level. We reject equal sharing and find large gender gaps in resource shares, and consequently in poverty rates, in some countries.

We prove a new identification theorem showing nonparametric identification of the joint distribution of random coefficients in general nonlinear and additive models. This differs from existing random coefficients models by not imposing a linear index structure for the regressors. We then model unobserved preference heterogeneity in consumer demand as utility functions with random Barten scales. These Barten scales appear as random coefficients in nonlinear demand equations. Using Canadian data, we compare estimated energy demand functions with and without random Barten scales. We find that unobserved preference heterogeneity substantially affects the estimated consumer surplus costs of an energy tax.