Defining poverty lines as a fraction of central tendency.

• Author: Muller, Christophe

1. Introduction

   For many years the relative notions of poverty have been important. These notions account for the evolution of perceptions of basic needs evolving in society (Sen 1983; Foster 1998). Being poor among a population of poor people can be considered very differently from being poor in a wealthy environment. This concern is often met by updating the poverty line across time in relation to the distribution of living standards. In these conditions, are the evolution patterns of poverty measures a real economic phenomena or only hidden consequences of methodological choices? (1) This paper addresses this question.

   The literature on poverty lines is extensive and varied (van Praag, Goedhart, and Kapteyn 1978; Hagenaars and van Praag 1985; Callan and Nolan 1991; Citro and Michael 1995; Short 1998; Ravallion 1998; Madden 2000). In particular, fractions of the median or the mean of the living standard distribution have been used to update poverty lines, notably for dynamic poverty analyses by national and international administrations (Fuchs 1969; Plotnick and Skidmore 1975; Fiegehen, Lansley, and Smith 1977; O'Higgins and Jenkins 1990; Central Statistical Authority 1997; Chambaz and Maurin 1998; Oxley 1998; Stewart 1998. See Zheng 2001 for other references). An example of a major country where administrations use a fraction of the median of income as a component of poverty threshold is the United Kingdom (Oxley 1998). The United States will probably use this approach in the future as it is recommended in Citro and Michael (1995). (2)

   Other updating procedures exist, such as poverty lines anchored on the mean living standard of households whose living standards are close to the desired poverty line (Ravallion 1998), or poverty lines relying on subjective perceptions of poverty by individuals (van Praag, Goedhart, and Kapteyn 1978; Hagenaars and van Praag 1985; Pradhan and Ravallion 1998). This paper does not cover these procedures.

   An index of poverty is a real valued function P, which, given a poverty line z, associates to each income profile y [member of] [R.sup.n.sub.+], a value P(y, z) indicating its associated level of poverty. For example, using a household consumption survey, an estimation of a poverty measure provides an indicator of the amount of poverty in the country. The results can be used to guide economic and social policies. We consider in this paper a large class of poverty measures under lognormality of the living standard distribution. This class covers all the poverty measures used in applied work. However, we also stress two major poverty measures for which we have explicit parametric results: (i) the Watts measure (Watts 1968; Zheng 1993), one of the most popular axiomatically sound poverty measures; and (ii) the head-count index, which is the most used poverty measure.

   The aim of the paper is to show that using a fraction of a central tendency as the poverty line restricts the evolution of poverty statistics to be stable when the inequality is stable. This situation may occur in particular for proportional taxation, uniform value-added tax (VAT), and fixed-rate sharecropping arrangements. Therefore, for null or low levels of inequality changes--the usual case--using such popular updating procedures leads to confusing the evolution of poverty over years with the evolution of inequality described by using the Gini coefficient. This is important for policy because this procedure is frequently implemented in poverty studies, which generates pictures of limited changes in poverty. Browning (1989) shows that it is crucial for government policy to distinguish inequality and poverty. While helping the truly needed is favored, extending that role to permit redistribution is often counterproductive. Section 2 describes the properties of poverty measures when poverty lines are updated by a fraction of central tendency. The consequences of using different relative poverty lines are also compared. Section 3 concludes our research.

2. Poverty Lines and Poverty Change

   Setting

   The results are largely based on the assumption of lognormality of the distribution of living standards. The lognormal approximation has often been used in applied analysis of living standards (Slesnick 1993; Alaiz and Victoria-Feser 1996). (3) Although it has sometimes been found statistically consistent with income data (e.g., van Praag, Hagenaars, and van Eck 1983), other distribution models for living standards or incomes may be statistically closer to the data. Using U.S. data, Cramer (1980) finds the lognormal distribution is no longer dominated by other distribution models if measurement errors are incorporated.

   What is wanted in this paper is (i) to obtain simplifications in calculus while simultaneously considering the three major central tendencies of a distribution (mean, median, and mode); and (ii) to simultaneously obtain a simple parametric expression of the Watts measure, the head-count index, and the Gini coefficient of inequality. This is generally not possible with nonlognormal distributions. Then, the goodness-of-fit of the distribution model is of rather secondary interest. The lognormal model is used as a simple way of illustrating a general argument that could be extended to more flexible specifications of the income distribution. In this paper, a more statistically adequate distribution model would not allow us to present the point more clearly. However, much of the qualitative intuition of the results should work with other usual income distributions.

   The variance of the logarithms, denoted [[sigma].sup.2], is a well-known inequality measure, not always consistent with the Lorenz ordering (Foster and Ok 1999). This is not the case under lognormality. Then, under lognormality, the Gini coefficient is

   G = 2[PHI]([sigma]/[square root of 2]) - 1, (1)

   where [PHI] is the cumulative distribution function (cdf) of the standard normal law and the Theil coefficient is [sigma]/2. [sigma] corresponds one-to-one with the Gini coefficient and Theil coefficient. This paper only mentions one of these inequality measures in the qualitative statements.

   When updating the poverty line, by defining it as a fraction of the median (mean or mode), measured aggregate poverty is conserved under lognormality when [sigma] is constant. Let us recall that the median of a lognormal distribution LN(m, [[sigma].sup.2]) is [e.sup.m], the mode is [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and the mean is [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Then, for example, a poverty line defined as a fraction of the median has a formula: z = [Ke.sup.m], with K a given number between 0 and 1. In practice, parameters m and [sigma] are not perfectly known, but are estimated instead. To avoid mixing too many questions, we do not discuss estimation errors in this paper. However, there are sampling confidence intervals for poverty indicators in the application. And now, in the theoretical part, it can be assumed that m and [sigma] are known.

   The first part starts with a very general class of additive poverty measures of the form

   P = [[integral].sup.2.sub.0] k(y, z) d[mu](y), (2)

   where y is the income variable, [mu] is the cdf LN(m, [[sigma].sup.2]), and z is the poverty line. P can be rewritten after a change in the variable

   [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (3)

   where [phi] is the pdf of the standard normal law. Therefore, P only depends on parameters Z([equivalent to] (In z - m)/[sigma]), [sigma], and m. Note that the level of m cannot be described as merely the scale of the incomes. In particular, when m rises with a given [sigma], the variance of the incomes also rises. Now, if the poverty measure can be written as

   P =...