A Unified Approach to Measuring Poverty and Inequality
8. 9.
10. 11.
12. 13.
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For example, if the elements of two distributions are converted from Indian rupees to U.S. dollars, then the direction of inequality between any two distributions should not change if the inequality measure satisfies unit consistency. An inequality measure that satisfies scale invariance also satisfies unit consistency, but the converse is not necessarily true. A class of decomposable inequality measures satisfying unit consistency has been developed by Zheng (2007a). In this book, however, we focus on relative inequality measures satisfying the scale invariance. For a more in-depth theoretical discussion of the transfer sensitivity property, see Shorrocks and Foster (1987). A geographical interpretation of the residual term can be found in Lambert and Aronson (1993), where the residual term is shown to be an effect of the re-ranking effect. The inequality of a distribution is computed in three steps: (a) within-group inequalities are computed in each subgroup; (b) the groups are ranked by their mean incomes and a concentration curve representing between-group inequalities is constructed; and (c) the Lorenz curve is constructed. The difference between the Lorenz curve of the distribution in the third step and the concentration curve from the second step is known as the residual term. The Lorenz curve was developed by Max Lorenz (1905). Interested readers, who may desire to have further theoretical understanding of the properties and their interrelationship, should see Zheng (1997) and Chakravarty (2009). A related but weaker property has been developed by Zheng (2007b). See note 8. This axiom is also known in the literature as strong transfer (see Zheng 2000). However, to keep the terminologies comparable across sections, we prefer to use the term transfer principle. A weaker version of this property exists that is known in the literature as weak transfer (see Chakravarty 1983), which can be stated as follows: if distribution x' is obtained from distribution x by a regressive transfer between two poor people while the poverty line is fixed at z and the number of poor does not change, then P(x'; z) > P(x; z). If distribution x" is obtained from another distribution x by a progressive transfer between two poor people while the poverty line is fixed at z and the number of poor does not change, then PS(x"; z) < P(x; z). Note that this property is different from the weak transfer principle that we define in this book.