On the qualitative properties of the optimal income tax

We explore the precise requirements for the qualitative results on optimum income taxation to hold, with the aim of extending their application to a larger space of solutions than that of continuous, piecewise differentiable functions assumed in the literature. In particular, properties (R1)-(R8) in Ebert (1992) are shown to hold when the endogenous variables of the problem are defined by non-smooth or even discontinuous functions, provided consumption is supposed to be normal and leisure non-inferior. Moreover, the referred properties continue to hold, without assuming the normality of consumption, if it is supposed that the function descriptive of gross income becomes absolutely continuous. In addition, a characterization of the set of potential solutions stemming from Lebesgue's Decomposition Theorem has been used to analyze the relevance of properties (R1)-(R8), vis-à-vis other possible features of optimal tax schedules. The conclusion is that, even assuming the normality of consumption, the case for regressivity should be viewed, on the lines suggested by Kaneko (1982) within a somehow different model, as an exceptional outcome versus other income tax structures that may arise at the optimum.