Using distance functions to derive optimal progressive earnings tax and commodity tax structures

Much of the research program in optimal taxation rests on the Atkinson-Stiglitz theorem (1976) | in the presence of optimal nonlinear earnings taxation, if leisure is weakly separable from goods, there is no role for differential commodity taxation. The nonlinear earnings tax in the theorem is one where, conditional on reported earnings, the government can choose tax paid and the marginal tax rate (mtr). The relationship between the average tax rate (atr) and mtr is unrestricted. Most governments operate progressive nonlinear tax systems in which, for each person paying taxes, mtr is not less than atr. I build on Deaton's work on distance functions and taxation to show that the AS theorem fails in the presence of optimal progressive earnings taxation. Conditional on mtr atr, the search for optimal earnings tax structures cannot be undertaken without simultaneously studying optimal commodity taxation whether or not leisure is weakly separable from goods. The formal theory in the paper assumes two types. I also discuss a nite-type example of an optimal progressive earnings, and commodity, tax structure and present numerical examples with four types.