Utilitarianism for Agents Who Like Equity, but Dislike Decreases in Income Peter P. Wakker (& Veronika Köbberling)
{1,…,n}: population of n persons 2 X: outcome set; general set x = (x 1,…,x n ): allocation, outcome x j for person j, j = 1,…,n X n : set of allocations on X n : preference rel n of policy maker
n x = (x 1,..,x n 1, ) We assume also on X given; Pareto optimality: i x i x; Notation: i x is ( x with x i replaced by ), 3 e.g. 1 x = ( ,x 2,..,x n ),
U: X : utility of representative agent; w j : importance weight of person j. w j 0; w j = 1. Harsanyi (1955) characterized through lotteries and expected utility; nowadays: s (x 1,...,x n ) w j U(x j ) n j=1 c c s c 4 Traditional approach: weighted utilitarianism better not commit to EU. recent (Weymark ’81) rank-dependent c: dependence on comonotonic class s c s very sign & Tversky & Kahneman ‘92 s: dependence on sign-profile Schmeidler (1989) 566
(0,,0,…, ) s Six individuals {1,…,6}. Suppose (0,,0,…, ) 5 Example of our technique: c ~ ~ Then [610;500] ~ [190;100] * We often write instead of [ ; ]
c Lemma. Under weighted utilitarianism, ~ U( ) U( ) = U( ) U( ). * U( ) U( ) = U( ) U( ) s w i U( ) + j i w j U(x j ) then w i U( ) w i U( ) = w i U( ) w i U( ) c c c c and i x ~ i y w i U( ) + j i w j U(y j ) = s s s s w i U( ) + j i w j U(y j )w i U( ) + j i w j U(x j ) c c c c s s s s = then ~ * sc 6 there exist nonnull i, x, y with: ix ~ iyix ~ iy c c c c s ss s rank-dependent sign 777 If comonotonic cosigned
U( ) U( ) = U( ) U( ). c c If any one of the four outcomes in a relation ~* is replaced by a nonequi-valent outcome, then the ~* relation does not hold any more: tradeoff consistency. c (2) ’ ~* c s s s s 7 U( ’) = U( ) ~ ’ ~ ’ / i x ~ i y i x ~ i y j f ~ j g ’ j f ~ j g In preferences: There are no i,x,y,j.f,g with: (1) (2) como- notonic comonotonic sign- como- notonic cosig- ned 88 ’ ~ .. (1) ~* U( ’) U( ) = U( ) U( ).
The following two statements are equivalent: (i) weighted utilitarianism. (ii) four conditions: (a) weak ordering; (b) Pareto optimality; (c) Archimedeanity; (d) tradeoff consistency. 8 rank-dependent sign comonotonic sign- Theorem. Assume solvability. 910
(writing [ ; ] instead of ). Example. Two persons {1,2}, (12,20) ~ (10,24) (35,20) ~ (30,24) 9 [13;10] ~ [35;30] * (40,13) ~ (46,10) (40,35) ~ (46,30) Say the policy maker likes equity. Violates theory! [12;10] ~ [35;30] * No! Equity caused a change in importance of persons person 1 poorest person 1 richest not comonotonic! comonotonic! c OK! 4 Ebert (2001), Zank (2001).
No. Not cosigned. Example. Two persons {1,2}. 10 ( 2, 9) ~ ( 4, 6) (6, 9) ~ (2, 6) (10, 1) ~ (15, 4) (10, 6) ~ (15, 2) [ 1; 4] ~ [6;2] * [ 2; 4] ~ [6;2] * All allocations below are comonotonic. c c No. Not cosigned. 4echt 4 !!!!! Zank (2001)