Bunching in rank-dependent optimal income tax schedules

Considering optimal non-linear income tax problems when the social welfare function only depends on ranks as in Yaari (Econometrica 55(1):95–115, 1987) and weights agreeing with the Lorenz quasi-ordering, we extend the analysis of Simula and Trannoy (Am Econ J Econ Policy, 2021) in two directions. First, we establish conditions under which bunching does not occur in the social optimum. We find a sufficient condition on individual preferences, which appears as a reinforcement of the Spence-Mirrlees condition. In particular, the marginal dis-utility of gross income should be convex, but less convex the higher the productivity. We also show that, for all productivity distributions with a log-concave survival function, bunching is precluded under the maximin, Gini, and "illfare-ranked single-series Ginis". Second, we turn to a discrete population setting, and provide an "ABC" formula for optimal marginal tax rates, which is related to those for a continuum of types found in Simula and Trannoy (2021), but remain essentially distinct.