How optimal nonlinear income taxes change when the distribution of the population changes

The impacts of changing the number of individuals of a particular skill level on the solutions to two versions of the finite population optimal nonlinear income tax problem are investigated. In one version, preferences are quasilinear in leisure. For this version, it is shown that it is possible to sign the directions of change in everyone's optimal consumptions and optimal marginal tax rates. In the other version, preferences are quasilinear in consumption. For this version, it is shown that it is possible to sign the directions of change in everyone's optimal before-tax incomes and optimal marginal tax rates. Moreover, the directions of change in the optimal marginal tax rates are the same for the two specifications of preferences.