Poverty: fuzzy measurement and crisp ordering

Two other new directions are multidimensional poverty measurement and partial poverty orderings, respectively. Multidimensional poverty measurement is based on the argument that an individual’s welfare is essentially multidimensional and hence an individual’s poverty status must also be evaluated from a multidimensional perspective. Contributions to this research include Tsui (2002), Bourguignon and Chakravarty (2003), Duclos et al. (2006), and Alkire and Foster (2011). The research on partial poverty orderings accepts the use of a single poverty line but the exact location of the line is uncertain and may vary within a range. The research seeks conditions under which poverty rankings remain valid for a range of poverty lines. Contributions to this area of research include Atkinson (1987), Foster and Shorrocks (1988), Jenkins and Lambert (1997), and Zheng (1999).

These two conditions ensure that a poverty measure satisfies the monotonicity axiom (poverty is reduced if the income of a poor individual is increased) and the transfer axiom (poverty is reduced by a progressive transfer of income among the poor). For surveys on poverty axioms, poverty measures and poverty orderings, see Zheng (1997, 2000), Lambert (2001) and, more recently, Chakravarty (2009).

In some elections such as the US presidential election, this axiom is violated as voters in different states carry different weights. In the US gubernatorial and other local elections, however, this axiom is satisfied.

An example of \(\rho (z)\) iswhere \(\gamma >0\) and \(\theta >0\) satisfywith \(w=\frac{1}{2}(c^{2}-d^{2})-cz_{l}+dz_{u}>0\). It is easy to verify that such a \(\rho (z)\) is a “density function” for a poverty membership function. Note that the definition excludes a small neighborhood at \(z_{l}\) and \(z_{u}\) to enable \(\rho (z_{l})=\rho (z_{u})=0\).

Note that the dominance conditions derived for crisp decomposable measures often are only sufficient for rank-based poverty measures such as the Sen measure—see Zheng (2000) for a detailed exposition.

One could also consider a more general intermediate poverty measure, \(\tilde{ p}(\frac{z-x}{z^{\eta }})\), which contains both relative and absolute measures as special cases (\(\eta =1\) and \(0\), respectively). For a general treatment of intermediate poverty measures, see Zheng (2007).

We do not illustrate Propositions 4.2 and 4.3 since they require the estimation of poverty membership functions which is not feasible with the CPS data.

The author of this paper is currently engaging in collaborations in estimating the poverty membership functions using the World Bank’s LSMS data and EU’s SILC data.