By Mark McGillivray (eds.)
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K -land k = 2, 3, ... , n, and upon simplifying, we obtain SO k-l k-l L [xk- x;] :S L i=l SO (c) that X ~ASF ~ADP <; ~L y. [Yk- Y;], V k = 2, 3, ... 24) ~ (n- h + 1) Yh- [Yh + Yh+l + · · · + Yn] for all h = 1, 2, ... , n- 1. The proof then proceeds in (n- 1) successive steps. h = 1. 25) that x1 ~ Yl· STEP z: h = 2. 28) + Y2· h: h =h. 31) hence x1 + Xz + · · · + Xn-h+I 2: YI + Yz + · · · + Yn-h+l· STEP n- 1: h = n- 1. 34) hencex1 +xz+···+Xn-I '=:YI +yz+···+Yn-I· We have shown that xi + xz + · · · + xh ::: YI + Y2 + · · · + Yh1 for all h = 1121...
It might be helpful to begin with a benchmark result that constitutes a first step towards a more general solution. By construction, distributions x 2 , x 3 , x 4 and x 5 in Example 1 obtain from x 1 by means of a single rank-preserving progressive transfer of one income unit. Close inspection reveals that the only case where the resulting distribution dominates the original distribution according to the differentials quasi-ordering is when the progressive transfer involves the richest and the poorest individual.
These include the number-dampened utilitarian principles (Ng 1986) other than classical utilitarianism and their generalized counterparts (see Blackorby, Bossert and Donaldson 2003). Critical-level generalized utilitarianism If strong Pareto is satisfied, a critical level represents a minimally acceptable level of utility such that the ceteris paribus addition of a single individual with a greater lifetime utility is a social improvement. Because no one in the existing population is affected, it is natural to choose a constant critical-level function.