Both buyers and sellers of goods and services may bene.t from letting their economic transactions go unrecorded for tax purposes. The supplier reduces his tax burden by underreporting income, whereas the consumer gains from buying a non-taxed lower-priced product. The distributional implications...

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...