FUZZY MEASUREMENT OF INCOME INEQUALITY - SOME POSSIBILITY RESULTS ON THE FUZZIFICATION OF THE LORENZ ORDERING

This paper starts from the premise that the concept of income inequali ty is ill-defined, and hence, it studies the measurement of income ine quality from a fuzzy set theoretical point of view. It is argued that the standard (fuzzy) transitivity concepts are not compatible with fuz zy inequality orderings which respect Lorenz ordering. For instance, w e show that there does not exist a max-min transitive fuzzy relation o n a given income distribution space which ranks distributions unambigu ously according to the Lorenz criterion whenever they can actually be ranked by it. Weakening the imposed transitivity concept, it is possib le to escape from the noted impossibility theorems. We introduce some alternative transitivity concepts for fuzzy relations, and subsequentl y, construct a class of fuzzy orderings which preserve Lorenz ordering and satisfy these alternative transitivities. It is also shown that f uzzy measurement can be used to construct confidence intervals for the crisp conclusions of inequality indices.