The typical method of solving for the optimal nonlinear tax schedule relies on deriving optimal incentive-compatible allocations. While this “primal approach” is mathematically rigorous, it lacks intuitive appeal. This paper considers a different method that relies on directly solving for the optimal tax system. This “dual approach” is much closer to actual tax policy as it centers around the welfare effects of tax reforms. I show that it can easily incorporate preference heterogeneity, as well as individual utility misoptimization. Beyond solving for the optimal tax system, the dual approach allows one to obtain insights into the welfare effects of small nonlinear tax reforms outside the optimum.