Inverse Optimal Taxation in Closed Form

For any policy question with distributional implications, the planner's social welfare function plays an important role in shaping the optimal policy. In the context of an optimal taxation problem in which individuals differ ex ante with respect to labor productivity, the planner's social welfare function defines the relative Pareto weights placed on individuals with different productivities. In this paper, we show that if one assumes that the observed income tax schedule has been chosen optimally, then it is sometimes possible to derive a closed-form expression for the social welfare function. This social welfare function involves parameters defining the shape of the observed tax and transfer system, as well as parameters defining preference elasticities and the shape of the underlying cross-sectional productivity distribution. The key result is that inverse optimum problem (finding a social welfare function that justifies a given tax function) is tractable in quantitatively relevant cases where the optimum problem (solving for the optimal tax function for a given social welfare function) is not. In particular, the inverse optimum problem is tractable given a standard utility specification with curvature over both consumption and labor effort. This tractability makes it easier to understand how optimal taxation works.