A General Theory of Inverse Welfare Functions

Optimal taxation problems typically involve finding a tax schedule to maximize a welfare function. This paper considers the reverse problem of finding an inverse welfare function that rationalizes a given tax schedule as optimal. Inverse welfare functions encode the implicit interpersonal comparisons a society must make in order to justify a tax schedule. We develop a general theory to recover the inverse social welfare function not only for income tax schedules, but also for substantially more complex tax systems that incorporate many different forms of taxation and multidimensional agent heterogeneity. The key insight is that even in complex tax environments, the (Gateaux) derivative of government revenue with respect to the tax schedule is the key empirical object required to construct the inverse welfare function. Additionally, our framework allows us to characterize Pareto efficient schedules in complex environments and extend the Atkinson-Stiglitz result. Our framework can also be augmented to construct inverse welfare functions when there are general equilibrium effects of taxation and when agents make optimization errors. We provide a number of example inverse welfare function constructions related to the taxation of couples, income taxation with labor demand and endogenous wages, piecewise linear income taxation, and joint taxation of income and housing rent.