During professor Peter Lambert's visit in the spring 1994 in Helsinki we started a joint project. Later docent Markus Jäntti joined our group. Our main problem was the optimal strategy for taxation and redistribution of incomes. The optimality is attained if the inequality reduction is maximal. We started this study from results presented by Fellman first in a preliminary report and later at the Econometric Society European Meeting 1993 in Uppsala (Fellman, 1993). Our theoretical result is that we have given a general and a very simple proof of a result of Fei (Equity oriented fiscal programs. Econometrica 1981:49:869-888). Furthermore, we have developed a yardstick for measuring to what extent real tax and transfer policies reduce the income inequality. The obtained results have been applied to Finnish data for the period 1975-1990. These studies have resulted in two papers (Fellman et al., 1996, 1999). We consider classes of tax and transfer policies which are constrained to keep the internal order of incomes intact. In addition, under the assumption that the initial incomes are not affected, the tax policies yield the same amount in tax revenue. Satisfying only these restrictions, these classes are very general and hence an adaptable tool for inequality and welfare studies. Every policy generates a transformed income distribution and consequently, a transformed Lorenz curve and transformed Gini and welfare indices. The properties of this class are studied and some mathematical results are established. We measure the broadness of the class by determining the extremes of the transformed Lorenz curves and the ranges of the transformed indices. The obtained bounds indicate the attainable redistributive effects of the tax policies. In general, every Gini index within the obtained range is attainable by a member of the class of policies. In addition, every point within the closed region, limited by the extreme Lorenz curves, is also attainable (Fellman, 1995, 2000). Furthermore, we have studied the conditions that a given Lorenz curve within that closed region corresponds to a policy belonging to the given class, and the obtained results are formulated in terms of stochastic dominance (Fellman, 2002). Similar results for transfer policies are developed (Fellman, 2003). Results from these mathematical studes have also been presented at scientific conferences. Our studies continue and the goal is to generalize earlier results. We have replaced the differentiability condition and the derivative restriction for the class of tax policies with a continuity condition. Furthermore, we have considered discontinuous transfer policies.