The properties of a class of tax policies

We consider a class of tax policies, which are constrained to yield the same amount in tax revenue. Every policy generates a transformed income distribution, which is stochastically dominated by the initial pre-tax income distribution, and a transformed Lorenz curve and transformed Gini coefficient. The extremes of the transformed Lorenz curves and the ranges of the transformed coefficients indicate the broadness of the class. Every Gini coefficient within the obtained ranges and every point within the closed region, limited by the extreme Lorenz curves, are attainable by a member of the class. One main result is that continuity is a necessary condition if one demands that the income inequality should remain or be reduced. In our previous studies of tax policies, the assumption was that the transformations were differentiable and satisfy a derivative condition. In this study, we show that it is possible to reduce this assumption to a continuity condition. We present the necessary and sufficient condition that a given Lorenz curve within that closed region is attainable by a tax policy belonging to the given class.