The paradox of endogenous poverty line in the poverty indices

When measuring poverty, the poverty line is considered to be relatively determined as a percentage of the median income or as a percentage of the average income so as to allow international comparisons. The poverty line relative to the median income may be described as endogenous in contrast with the poverty line relative to the average income which may be described as exogenous because it is independent of the normalized distribution of incomes. As the existing literature is focused on the properties of the indices but not on the respective interest of the endogenous and exogenous poverty lines, this paper compares the two definitions of the relative poverty. The Relative Cost of Poverty Reduction is the distance from the Lorenz curve to the poverty line, that is, the income that must be spent to make all poor non-poor or the sum that must be spent to eliminate poverty completely, determined as a proportion of total revenue; this indicator is important because it shows how much a country or an international organization must pay to solve the poverty problem, as a proportion of its total income. A significant paradox is reported here: when the poverty line is relative to the median, a higher poverty index (with the Sen or Sen-Shorrocks-Thon family of indices but also the Foster-Greer-Thorbecke family), associated with successive dominated Lorenz curves, is compatible in many cases with a lower Relative Cost of Poverty Reduction. However the paradox vanishes if the poverty line is relative to the average income. This has been demonstrated algebraically or numerically (when the equations could not be solved for mathematical reasons), by considering four typical families of concentration curves so as to generate increasing poverty levels and comparing the cost of reducing poverty for the same population. These families of Lorenz curves are such that the curves are non-intersecting. One has directly used normalized continuous non-intersecting Lorenz curves generated from various mathematical functions (power function, polynomial function, elliptic function and exponential function); or one has generated Lorenz curves from the Pareto distribution. And one has demonstrated analytically that the Relative Cost of Poverty Reduction must be decreasing sooner or later when the Lorenz curve tends toward its limit. The conclusion is that the poverty line should be relative to the average income rather than to the median income.