An Index of Inequality: With Applications to Horizontal Equity and Social Mobility

Abstract

Horizontal equity and social mobility are discussed in terms of a non-utilitarian social welfare function which takes account of the process by which an ex ante distribution is mapped into an ex post distribution (by the tax system, for example). An index of inequality is proposed which decomposes into two components, corresponding to vertical and horizontal equity respectively. A functional form for the social welfare function is derived for the purposes of empirical work, and an application to data for 5895 UK households is presented. The paper contains a theoretical application of the index to a model of optimal taxation. IT IS CONVENTIONAL to assess the merits of alternative public policies in terms of a trade-off between equity and efficiency. In practice, however, a change in, say, the tax system involves three effects. First, it may have incentive or disincentive effects leading to efficiency gains or losses. Secondly, it may alter the distribution of welfare levels. Thirdly, it may alter the ranking of individuals (or households) within the distribution. These three effects correspond to efficiency, vertical equity, and, we shall argue, certain aspects of horizontal equity, respectively, and any assessment of a tax change must take into account all three. The principal assumption of this paper is that government is concerned about the trade-off between these three effects. A strict utilitarian is concerned only with the consequences of an action, and in the evaluation of a particular reform a utilitarian measure of social welfare is defined over the vector of ex post utilities. No account is taken of the process by which the vector of ex ante utilities is mapped into the ex post vector, and no ethical status is awarded to the ex ante distribution. One does not have to adopt an entitlement theory of justice to believe that the utilitarian approach may ignore some relevant considerations, such as the fairness of the redistributive process. A striking example of this arises in the model used by Mirrlees [11] to examine optimal income taxation. In that model individuals have identical preferences (defined over consumption and leisure) and differ only in respect of their potential wage rates or ability levels. Clearly, in the absence of taxation individual utility is an increasing function of ability. Suppose the government uses redistributive lump-sum taxes to achieve the first-best optimum. Then as Mirrlees [11, 12] shows, the first-best optimum for a utilitarian social welfare