Designing a proper taxation system exemplifies the trade-off between equity and  efficiency. Governments often face the dilemma of how to generate sufficient  revenues without undermining the incomes, savings, and innovation of people,  particularly the poor. Tax collection can be likened to a leaky bucket, as Okun  (1975) put it, with the leak representing the administrative costs and social  welfare loss associated with taxes. A good taxation system, therefore, minimizes tax  administration and compliance costs while maximizing social welfare. This paper concerns the problem of social welfare loss induced by a tax system. It provides a methodology to estimates magnitudes of welfare loss from taxation  using alternative social welfare functions.

This paper builds on the pioneering and innovative contributions to the measurement  of tax progressivity (regressively) by Pigou (1949) and Musgrave and Thin (1948).  Kakwani in 1977 developed an index to measure the progressivity (or regressivity) of  tax systems. This index, widely known as Kakwani index, has been extensively used to  analyze equity in taxation and government expenditures, as well as equity in access  to health, education and essential services. In particular, the index has become  popular in analyzing equity in finance and delivery of health care.

The progressivity of a tax system is defined based on the average tax rates along the  income scale. A tax system is said to be progressive (regressive) when the average  tax rate rises (falls) with income. It is proportional when the average tax rate is constant throughout the  income scale. Kakwani (1977) developed his index by measuring the extent of the  overall deviation of a tax system from proportionality. This deviation is related to  the concept of tax elasticity, which can be shown to be one at all income levels when  the tax system is proportional (Kakwani, 1977). A suitable measure of tax  progressivity should depend on the overall deviation of tax elasticity from 1.

The overall measure of the deviation of the tax elasticity from 1 is the Kakwani index  given by K= C – G, where C is the concentration index of taxes and G is the Gini index  of the pre-tax income distribution. The Kakwani index measures the extent of  deviation of a tax system from proportionality and therefore, has seemingly no  welfare implications. Suits (1977) also proposed a measure of tax progressivity  defined as one minus twice the area under the relative concentration curve. This index  also measures the overall deviation of a tax system from proportionality. However,  since these two indices of progressivity are widely applied to taxation policy, they  should not be used merely as a statistical device to measure tax progressivity.  Instead, the indices should incorporate normative judgments implicit in a social  welfare function. There is now much consensus that for every measure of inequality, there is some  underlying notion of social welfare. Thus, every inequality measure must be derived  from a social welfare function (Dalton 1920, Atkinson 1970, Kolm 1976). Similar to  inequality measures, a tax progressivity measure can be derived from a social welfare  function.

This paper develops a framework of social welfare to derive tax progressivity  measures from alternative social welfare functions proposed in the literature. Like  inequality measures, every progressivity measure has an implicit social welfare function that incorporates a  society's distributional judgments. The paper derives the Kakwnai index of tax  progressivity from Sen's social welfare function. The main contribution of the paper,  however, is that it derives a new class of progressivity measures that incorporate a  distribution judgment parameter capturing inequality aversion. 

Inequality aversion parameter means that social welfare is more sensitive to a shift  in the tax function in favor of poor persons than to the same shift affecting rich individuals. As the inequality aversion parameter rises, more weight accrues to tax  rates at the lower end of the income distribution than at the top. This class of  progresssivity measures is derived using Atkinson's social welfare functions.

The paper also draws social welfare implications of Suits' measure of tax  progressivity. Finally, the paper proposes a new measure of tax progressivity derived  from Bonferroni social welfare function (Bonferroni 1930).

The methodology developed in the paper is used to illustrate an empirical analysis of  the Australian taxation data. 

This paper is structured as follows. Section 2 presents the social welfare framework  for tax progressivity. Section 3 is devoted to the redistributive impacts of taxation  system on inequality. Section 4 shows how the Kakwani index can be derived from the  Gini social welfare function. Section 5 derives a class of progressivity measures  from Atkinson's social welfare functions. While section 6 develops the social welfare implications of the Suits measure of  progressivity, section 7 derives a new measure of tax progressivity using the  Bonferroni social welfare function. Finally, section 8 illustrates the paper's  methodology by applying to the Australian tax system. Section 9 concludes the paper.