Patrick Moyes

An axiomatic characterization of Yitzhaki’s index of individual deprivation

Abstract This note presents and discusses a number of properties a measure of individual deprivation should satisfy. The axioms we propose are necessary and sufficient to characterize Yitzhaki’s index of deprivation making the structure of the index more transparent.

Consistent Income Tax Structures When Households Are Heterogeneous

The paper examines the redistributive impact of income taxation for heterogeneous populations. An equivalent income function is introduced in order to adjust the incomes of different types of households and the redistributive performance of the tax system is evaluated by comparing the Lorenz curves of the distributions of before and after tax equivalent incomes. Inequality reduction after tax imposes mild consistency conditions which are met by most current tax systems. If the inequality reducing effect of taxation is required to be independent of the reference household type, then the structure of the equivalent income function is fully determined by the inequality concept. Journal of Economic Literature Classification Numbers: D31, D63, H24.

The impossibility of a progressive tax structure

Abstract When households are identical in all respects other than income, a non-decreasing average tax rate is both necessary and sufficient for post-tax incomes to be always at least as equal as pre-tax incomes. This paper investigates whether this well known result can be extended to households with different needs. A tax structure which never increases inequality (in the Lorenz sense) is shown to be unable to discriminate between household types. If, in addition, the tax structure never reduces the relative income of the poorest member of the neediest group of households, then the same proportional tax must be applied to all households, and redistribution is impossible either within or between groups of households with differing needs. A similar conclusion is reached using conventional equivalence scale arguments, and also applies when the parameters of the tax schedules are determined endogenously. We discuss the significance of this impossibility result and its implications for tax design.

Robust international comparisons of distributions of disposable income and regional public goods

The paper provides robust normative comparisons of 12 OECD countries based on their distributions of disposable income and access to two regional public goods: infant mortality and pupil-teacher ratios at public schools. Comparisons are performed using two and three-dimensional dominance criteria that coincide with the unanimity of utilitarian judgments taken over specific classes of utility functions. The criteria succeed in ranking conclusively about 30% of all possible comparisons in the two-dimensional case, compared with 67% for one-dimensional income-based comparisons and 6% for three-dimensional ones. Introducing local public goods seems to worsen the relative standing of Anglo-Saxon countries.

Redistributive effects of minimal equal sacrifice taxation

We examine the circumstances under which minimal equal sacrifice taxation reduces income inequality. Focusing on absolute sacrifice and relative inequality, we show that after tax incomes are more equally distributed than before tax incomes and that after tax inequality decreases with the revenue to be raised if and only if the elasticity of the marginal utility of income is decreasing. However, other things equal, after tax inequality is shown to decline as the utility function becomes more concave. We investigate next how these conclusions extend when one takes the absolute inequality view and considers that it is the relative loss in utility that has to be equalized across individuals.

Deprivation, welfare and inequality

We provide a characterization of the generalised satisfaction—in our terminology non-deprivation—quasi-ordering introduced by S.R. Chakravarty (Keio Econ Stud 34:17–32, (1997)) for making welfare comparisons. The non-deprivation quasi-ordering obeys a weaker version of the principle of transfers: welfare improves only for specific combinations of progressive transfers, which impose that the same amount be taken from richer individuals and allocated to one arbitrary poorer individual. We identify the extended Gini social welfare functions that are consistent with this principle and we show that the unanimity of value judgements among this class is identical to the ranking of distributions implied by the non-deprivation quasi-ordering. We extend the approach to the measurement of inequality by considering the corresponding relative and absolute ethical inequality indices.

A Non-welfarist Approach to Inequality Measurement

Following Kolm (1969) and Atkinson (1970) there is a wide agreement in the literature to appeal to the Lorenz curve for measuring inequality. A distribution of income is typically considered as being no more unequal than another distribution if its Lorenz curve lies nowhere below that of the latter distribution. Besides its simple graphical representation, much of the popularity of the so-called Lorenz criterion originates in its relationship with the notion of progressive transfers. It is traditionally assumed that inequality is reduced by a progressive transfer; that is, when income is transferred from a richer to a poorer individual without affecting their relative positions on the ordinal income scale. The principle of tranfers, which captures this judgement, is closely associated with the Lorenz quasi-ordering of distributions of equal means. Indeed half a century ago, Hardy et al. (1952) demonstrated that if a distribution Lorenz dominates another distribution, then the former can be obtained from the latter by means of a finite sequence of progressive transfers, and conversely.1 This relationship between progressive transfers and the Lorenz quasi-ordering constitutes the cornerstone of the modern theory of welfare and inequality measurement. As a consequence, the literature has concentrated on Lorenz consistent inequality measures; that is, indices such that a progressive transfer is always recorded as reducing inequality or increasing welfare.