On longitudinal analysis of poverty conceptualised as a fuzzy state

When poverty is viewed as a matter of degree, i.e. as a fuzzy measure, two additional aspects are introduced into the analysis compared with the conventional poor/non-poor dichotomous approach: (i) the choice of membership functions i.e. quantitative specification of individuals' or households' degrees of poverty and deprivation; and (ii) the choice of rules for the manipulation of the resulting fuzzy sets, rules defining their complements, intersections, union and averaging. Specifically, for longitudinal analysis of poverty using the fuzzy set approach, we need joint membership functions covering more than one time period, which have to be constructed on the basis of the series of cross-sectional membership functions over those time periods. In this paper we propose a general rule for the construction of fuzzy set intersections, that is, rules for the construction of longitudinal poverty measures from a sequence of cross-sectional measures. On the basis of the results obtained, various fuzzy poverty measures over time can be constructed as consistent generalisations of the corresponding conventional (dichotomous) measures. Examples are rates of any-time, persistent and continuous poverty, distribution of persons and poverty spells according to duration, rates of exit and re-entry into the state of poverty, etc. The proposed rule has been developed in a logical, step-by-step, manner, satisfying the required marginal constraints. This is important since there are reasons to believe that, hitherto, the rules of fuzzy set operations in the context of multi-dimensional and longitudinal poverty analysis have not been well or widely understood. In an annex to this paper, we also present some numerical illustrations using survey data from the Italian European Community Household Panel, 1994-2001, with breakdown by Macro-region in Italy. The main objective, however, is to provide quantitative comparison between the conventional and fuzzy approaches. Noteworthy from a methodological point is the difference in the performance of the approaches concerning persistence of poverty. Movements in and out of poverty may be somewhat over-estimated (and hence the persistent or continuous poverty rates under-estimated) with the conventional approach, perhaps because it gives too much weight even to small movements across the poverty line.