By H. A. Freeman, Milton Friedman, Frederick Mosteller, Allen W. Wallis (eds.)
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1991) Rasche et al. 1 SOME CONCEPTS FROM ECONOMICS 29 of the Ortega et al. (1991) and Chotikapanich (1993) as well as the Rao –Tam (1987) and Chotikapanich (1993) models, they concluded that the additive models perform distinctly better than either constituent model and moreover yield a satisfactory fit over the entire range of income. Among the many further proposals, we should mention the work of Maddala and Singh (1977b) who suggested expressing the Lorenz curve as a sum of powers of u and 1 À u.
Briefly, the argument runs as follows. L X2 is equivalent to (Arnold, 1987) ðx 0 F~ 1 (t) dt ! ðx 0 F~ 2 (t) dt for all x [ Rþ (2:69) 48 GENERAL PRINCIPLES and also to ð1 F~ 1 (t) dt ! f. of the mean-scaled random variable Xi =E(Xi ), i ¼ 1, 2. If F i [ RV1 (Àai ), ai . 0, i ¼ 1, 2, it follows using property RV 3 under appropriate regularity conditions that the integrated upper tails are in RV1 (Àai þ 1). 70) Ð1 F~ 1 (t) dt ! 1, 1 ~ F (t) dt g(x) :¼ Ðx x for all x [ Rþ , () a1 a2 : 2 This shows that if size distributions with regularly varying tails are ordered in the Lorenz sense, then the more unequal distribution necessarily exhibits heavier tails (namely, a smaller a).
A further well-known functional form is the one proposed by Rasche et al. (1980) who suggested L(u) ¼ [1 À (1 À u)a ]1=b , 0 , u , 1, (2:17) where 0 , a, b 1. 7) obtained for b ¼ 1 and a , 1. 1), as pointed out by Anstis (1978). In order to overcome the drawback of many previously considered functional forms, namely, a lack of fit over the entire range of income, several authors have proposed generalizations or combinations of the previously considered functions. Quite recently, Sarabia, Castillo, and Slottje (1999) have suggested a family of parametric Lorenz curves that synthetizes and unifies some of the previously considered functions.