توزیع درآمد و اقدامات واگرایی غیرساخت مجموع

Inequality indices evaluate the divergence between the income distribution and the hypothetical situation where all individuals receive the mean income, and are unambiguously reduced by a Pigou–Dalton progressive transfer. This paper proposes a new approach to evaluate the divergence between any two income distributions, where the latter can be a reference distribution for the former. In the case where the reference distribution is perfectly egalitarian – and uniquely in this case – we assume that any progressive transfer reduces the divergence, and that the divergence can be additively separated into inequality and efficiency loss. We characterize the unique class of decomposable divergence measures consistent with these views. We derive the associated relative and absolute subclasses, and we illustrate the applicability of our results. This approach extends the generalized entropy studied in inequality measurement.

When individuals are identical in every aspect other than their respective incomes, theories of justice agree that an egalitarian distribution might be the best outcome for society as a whole. In this context, there is a consensus in the literature to use inequality indices, or dominance criteria such as the Lorenz quasi-ordering, for making normative judgments about the fairness of the income distribution. In practice however, individuals differ in many respects and an equally distributed income is no longer a social norm. As an immediate consequence conventional inequality indices become meaningless, unless other evaluation tools are introduced. The aim of this paper is to provide a unified framework for the economic analysis of income distributions, when objectives other than the strict equality of incomes are considered. We build upon previous works on inequality measurement, by rethinking or extending some usual normative views. We then identify, through an axiomatic characterization, a large class of measures. The conditions we impose are fairly reasonable, and not very demanding. More importantly, we claim that such an approach may shed new light on important issues in inequality measurement. The cornerstone of the inequality measurement theory is the Pigou–Dalton principle of transfers. This principle states that any progressive transfer from an individual to one poorer than him – transfer that does not modify the respective positions on the income scale – always reduces the inequality. Even if this principle is well established, it is not immune to criticism, and indeed not universally approved [5]. The principle of transfers actually encapsulates two normative views. Other things being equal, it first defines a path which characterizes an unambiguous improvement of the social welfare. Then, it describes a strictly egalitarian distribution as a social objective, since the equalizing process is completed when all individuals have the mean income. These two dimensions have been separately investigated and criticized in the literature. First, whereas the income inequality is unambiguously reduced among the individuals involved in a progressive transfer, it is not so obvious that the inequality is also decreased in society as a whole [15]. Some combinations of progressive transfers can modify the distribution in a questionable direction, resulting for example in an increase in polarization [22] and [38]. Then, a strictly egalitarian distribution does not necessarily appear as a reference point for the social planner. Some income inequality, for example stemming from differences in personal responsibility – such as effort – may be viewed as fair, and might not be compensated [6], [10], [24] and [35]. In this paper we assume as a first normative requirement that, for any given income distribution (denoted by x), there exists a representative, reference or objective distribution (denoted by y). This view significantly weakens the second feature of the principle of transfers. Depending on the situation in which the measure is applied, the reference distribution can be, for example, fair according to the ethics of responsibility. We do not characterize this reference. We only assume its existence. Hence, other literature has to be invoked to complement our approach. There are now several possibilities to define what exactly is meant by improvement, holding total income constant, to get closer to the reference distribution. The approach we embrace in this paper is not really innovative, even if it slightly weakens the standard view of the principle of transfers – according to which a progressive transfer is always a suitable transformation. We assume that a progressive transfer is always an admissible path if, in the reference distribution, every individual has the mean income of the initial distribution. But we also assume that the effect of such a transfer may be ambiguous if the reference distribution is not fully egalitarian. This property is simply called principle of transfers, even if our definition is weaker than the standard one. The second normative requirement we impose, relatively new in the literature, involves a situation where the mean income of the actual distribution x and the reference distribution y may be different. We assume that, if the reference distribution is egalitarian – any income c for all – the measures can be additively decomposed into two components. The first component evaluates the inequality within the distribution x, or equivalently, the divergence between x and the hypothetical situation where all individuals have the mean. The second component evaluates the efficiency loss, due to the divergence between the hypothetical situation where all individuals have the mean and the reference distribution, characterized by the income c for all. This property, called judgment separability, is undoubtedly the key element of our approach. Finally, if the actual distribution corresponds exactly to the distribution the social planner wants to achieve, it seems fair to recognize that only the status quo is acceptable. This is a consequence of the two first requirements. Adding some reasonable conditions to the normative judgments described above, we characterize a large class of decomposable measures which evaluate the divergence between any two income distributions, where the second one can be a reference for the first. As explained later, we focus on divergence measures, as opposed to distance measures. What follows is a brief overview of the measures we obtain View the MathML source Turn MathJax on where N denotes the considered population, consisting of n individuals. ϕ is a twice differentiable and strictly convex function. Traditional inequality indices can be replaced in this context. As already mentioned, an inequality index implicitly evaluates the divergence between the income distribution under consideration and the hypothetical situation where all individuals have the mean income. We show that our divergence measures extend the usual decomposable families of inequality indices: Under some restrictive conditions, or normalizations, they boil down to the relative and absolute versions of the generalized entropy initiated by [16], [18], [36] and [37]. The evaluation of the divergence between any two income distributions is not really new in the literature. Cowell [17] characterizes a large class of divergence measures, called measures of distributional change. Nevertheless our approach is conceptually different, on two main features, namely, (i) the properties required for the measures, and (ii) the measures obtained. (i) Since divergence measures generalize inequality indices, Cowell [17] proposes to generalize the principle of transfers to a property called monotonicity in distance. Whereas it represents an appropriate extension of the principle of transfers in a more general framework, this property is quite demanding. The property we impose, called simply principle of transfers, is weaker. Moreover, the measures identified by [17] are not consistent with the other main property we assume, called judgment separability. (ii) The divergence measures obtained by [17] and the divergence measures characterized in the present paper are different. Surprisingly enough, both classes of measures are widely used in information theory and information geometry. The relative measures identified by [17] are Csiszár f-divergences, independently introduced by [19] and [2]. The measures we obtain are Bregman divergences[12]. A well-known result in information theory is that the classes are distinct, but coincide in one specific case: This divergence – which is a single measure – is called Kullback–Leibler divergence[29]. Also, Cowell [17] characterized a class of absolute measures. There is only one divergence at the intersection of this class and our class, namely the squared Euclidean distance, a generalization of the variance. This paper is only a first step towards the new direction we propose. Its distinctive feature lies in the fact that we use one distribution as a reference. Although the characteristics of this distribution have to be specified, this is not really a weakness. Indeed, it leaves the application of the measures open to a wide number of fields where income distributions are involved. The real question is to preserve the standard principle of transfers, as accepted in this paper, when the income is equally distributed in the reference distribution. Accordingly, some modifications of the distributions, such as polarization, cannot be tracked. That seems however to be a necessary prelude. Based on the results derived here, many extensions can be undertaken, by redefining the path towards the reference distribution. The following section sets out the notation and illustrates our framework in the context of inequality measurement. We introduce in Section 3 the main conditions, namely smoothness, judgment separability, anonymity and the principle of transfers, which will be imposed on all divergence measures. We isolate a class of measures compatible with them. The class is characterized by a single evaluation function. The consistency with the principle of transfers, combined with the anonymity requirement, is captured by the strict Schur-convexity of this function. We then focus on decomposable measures, consistent with a non-negativity requirement. The implication is an additive structure. The entire class, compatible with all the properties, is finally identified. Section 4 hints at some directions to refine the general class by way of normalizations. We also discuss the relationship with the related literature. Section 5 concludes the paper and illustrates how the measures can be applied. We focus on the analysis of households with differing needs, the ethics of responsibility, and mobility measurement.