The well-known Atkinson and Stiglitz (1976, section 5) theorem on the superiority of nonlinear directtaxation over nonlinear indirect taxation is proved to hold under the second order approach, even on in-tervals where the monotonicity constraint is binding. this confirms for the nonlinear case previous find-ings by Kaplow (2006) and Laroque (2005), who considered a joint combination of linear indirect taxesand, possibly suboptimal, nonlinear income taxation. the intuition behind is that, when indirect taxationis not restricted to linearity, its elimination is only justified if the income taxation scheme becomes opti-mally chosen, as happens in Atkinson and Stiglitz (1976).In addition, our analysis allows for a more general class of sub-utility functions than the one in the Atkin-son and Stiglitz theorem, which includes work done or income further to consumption goods and servic-es. Finally, the possibility that optimal allocations fail to be differentiable, or continuous, is also contem-plated unlike in the standard literature.