Atkinson and Stiglitz show that with weakly separability, differential commodity taxes are unnecessary given an optimal nonlinear income tax. Deaton showed that with an optimal linear progressive income tax, commodity taxes are superfluous under weakly separable and linear Engel curves. Using the latter case as an example, we derive two main results. If the income tax is less progressive than optimal, necessities should bear a lower tax rate than luxuries. If low-income households are income-constrained so cannot afford luxuries, it may be optimal to tax necessities at higher rates than luxuries, depending whether labor varies along the intensive or extensive margin.
optimal income tax Atkinson-Stiglitz Theorem indirect taxes