The Atkinson-Stiglitz Theorem shows that with weakly separable preferences, differential commod- ity taxes are not needed if an optimal nonlinear income tax is imposed. Redistributive objectives can be achieved with the income tax alone even if goods differ considerably in their income elasticities of demand. Deaton showed that if the government is restricted to a linear progressive income tax along with commodity taxes, the latter are superfluous if preferences are not only weakly separable but also yield linear Engel curves whose slopes are common to all households. These have potentially strong policy implications since they suggest that the common practice of giving preferential commodity tax treatment to necessities is not warranted. Using the linear progressive income tax as an example and assuming the Deaton conditions are satisfied, we derive two contradictory results. First, if the income tax is less progressive than optimal, necessities should be taxed preferentially relative to luxuries. Second, if low-income households are income-constrained so are unable to afford any luxury goods, it may be optimal to tax necessity goods at higher rates than luxuries, depending on whether labor supply varies along the intensive or extensive margin. The analysis is extended to allow for nonlinear taxes and endogenous demand for a public good.