Double limit analysis of optimal personal income taxation

A double limit analysis is developed to determine the optimal personal income tax based on the principle that moving a marginal dollar of tax revenue from one income interval to another should not raise social welfare. The first order condition can be differentiated with respect to an interval boundary to yield a second order differential equation for the optimal income tax that can be solved to yield specific solutions. Application of an optimal income tax to a broader income range in general reduces tax revenues and requires greater subsidies at low-income levels. Optimal solutions are provided assuming Cobb-Douglas or Constant Elasticity of Substitution (CES) utility and lognormal or Pareto productivity distributions.