Optimal taxation with multiple incomes and types

We analyze the optimal nonlinear income tax schedule for taxpayers with multiple incomes and multiple unobserved characteristics. We identify smoothness assumptions and extensions of the single crossing conditions that enable the characterization of the optimum through variational calculus. Both the tax perturbation and mechanism design approaches yield identical results when the number of incomes equals the number of unobserved characteristics. Notably, the mechanism design approach requires slightly less stringent assumptions than the tax perturbation approach. Additionally, we introduce a numerical method to determine the optimal tax schedule. Applied to couples, the optimal isotax curves are nearly linear and parallel. Additional contributions include a Pareto efficiency test and a condition on primitives ensuring the sufficiency of the government's necessary conditions, thereby guaranteeing the uniqueness of the solution.