1. Utilitarianism for Agents Who Like Equity, but Dislike Decreases in Income Peter P. Wakker (& Veronika Köbberling)

2. {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

3.  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 ),

4. 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

5. (0,,0,…, ) s Six individuals {1,…,6}. Suppose (0,,0,…, ) 5 Example of our technique: c 6 190100 610500 300400 300400 ~ ~ Then [610;500] ~ [190;100] * We often write  instead of [  ;  ]

6. 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 ~ iyix ~ iy c c c c s ss s rank-dependent sign 777 If comonotonic cosigned

7.  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(  ).

8. 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

9. (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).

10. 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)