تسلط لورنز برای توزیع درآمد تبدیل شده: اثبات یک فرمول ساده

This article provides a simple proof of the Lorenz dominance criterion for two non-decreasing income transformations. The criterion is extended to the most general case, with only very mild restrictions on the form of initial income distribution or the properties of the income transformations.

To investigate the causes and consequences of inequality and also to assess government policy targeting disparity, we need to measure and compare inequality associated with different income transformations. The most popular ordering of inequality is Lorenz dominance introduced in Atkinson (1970). Atkinson demonstrated that if the Lorenz curve (which shows the proportion of total income received by the poorest t% of the population) for one distribution lies below the Lorenz curve associated with another, then inequality in the first case is higher for a wide class of inequality measures. However, it is somewhat cumbersome to construct the Lorenz curve every time we need to compare the inequality resulting from different income transformations. In some circumstances, for example when a new tax code is designed, we may not even know the true income distribution. In such a situation we need to know when one income transformation leads to higher inequality than another for any initial income distribution. The Lorenz Dominance criterion helps us to answer this question. In particular, it states that for two non-decreasing strictly positive income transformations the necessary and sufficient condition for Lorenz dominance is that the ratio of two transformations is a monotonic function. Jakobsson (1976) and Kakwani (1977) first formulated and proved the Lorenz dominance criterion for income transformations. However, the formulations provided in Jakobsson (1976) and Kakwani (1977) were rather complicated and the criterion was proved only for continuous and differentiable income distribution functions and for differentiable income transformations. Keen et al. (2000) have proved a similar criterion for discrete income distributions. Le Breton et al. (1996) provided the proof for discrete distributions. Although this proof, without loss of generality, could be extended to the case of general income distributions, this work has not been completed. Moreover, the proof is likely to be rather complicated. The present paper provides a much simpler proof and extends the class of income transformations. In particular, we do not restrict the domain of income to be finite or positive and we do not require the income transformations to be non-negative. The aim of this article is to provide a simple proof of the Lorenz dominance criterion for two income transformations in the case of the most general form of initial income distribution and non-decreasing income transformations.

This paper provides a simple proof of the Lorenz dominance criterion for income transformations for a given general form of income distribution function. I believe this work provides a useful reference for the researcher investigating the impact of disparity on different economic outcomes.