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Utilitarianism for Agents Who Like Equity, but Dislike Decreases in Income

Utilitarianism for Agents Who Like Equity, but Dislike Decreases in Income Peter P. Wakker (& Veronika Köbberling). 2. {1,…,n}: population of n persons. X: outcome set; general set x = (x 1 ,…,x n ): allocation , outcome x j for person j, j = 1,…,n

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Utilitarianism for Agents Who Like Equity, but Dislike Decreases in Income

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

  2. 2 {1,…,n}:populationof npersons X:outcomeset; general set x = (x1,…,xn):allocation, outcome xj forpersonj, j = 1,…,n Xn: set of allocations  on Xn:preference relnof policy maker

  3. 3 Notation: ix is (x with xi replaced by ), nx = (x1,..,xn-1,) We assume also  on X given; Pareto optimality: ix  ix; e.g. 1x = (,x2,..,xn),

  4. & Tversky & Kahneman ‘92 (x1,...,xn) wjU(xj) n j=1 Schmeidler (1989) 6 5 6 4 sign very recent (Weymark ’81) rank-dependent Traditional approach:weighted utilitarianism c s U:X: utility of representative agent; wj: importance weightof person j. wj0; wj = 1. Harsanyi (1955) characterized through lotteries and expected utility; nowadays: c s c: dependence on comonotonic class s: dependence on sign-profile c s c s better not commit to EU.

  5. * Then [610;500] ~ [190;100] 6 5 Example of our technique: Six individuals {1,…,6}. Suppose (0,,0,…, ) (0, ,0,…, ) 190 300 ~ 100 400 ~ (0,,0,…, ) (0, ,0,…, ) 610 300 500 400 c s We often write  instead of [;]

  6. c c c c s s s s c c c c s s s s cosigned comonotonic c s c s c s c s then~  * rank-dependent Lemma.Under weighted utilitarianism, ~ U() - U()= U() - U(). * 7 7 7 6 If there exist nonnull i, x, y with: ix~iy wiU() + jiwjU(xj) = wiU() + jiwjU(yj) andix ~ iy wiU() + jiwjU(xj) = wiU() + jiwjU(yj) thenwiU() -wiU() = wiU()-wiU() c s U() -U() = U()-U() sign c s

  7. comonotonic ix ~iy ~’ / ix ~ iy cosig- ned cosig- ned ’jf ~jg jf ~ jg como-notonic como-notonic 8 8 7 (1) ~*  U() - U()= U() - U(). c s (2) ’~* U(’) - U()= U() - U(). c s  ’~ .. U(’) = U() 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 s c s sign- In preferences: There are no i,x,y,j.f,g with: (2) (1)

  8. rank-dependent comonotonic 9 10 8 Theorem. Assume solvability. The following two statements are equivalent: (i) weighted utilitarianism. (ii) four conditions: (a) weak ordering; (b) Pareto optimality; (c) Archimedeanity; (d) tradeoff consistency. sign sign-

  9. comonotonic! * [13;10]~ [35;30] OK! person 1poorest person 1richest * [12;10]~ [35;30] No! Equity caused a change in importance of persons not comonotonic! 4 9 Example. Two persons {1,2}, (writing [;] instead of ). (40,13)~(46,10) (40,35)~(46,30) c (12,20)~(10,24) (35,20)~(30,24) Violates theory! Say the policy maker likes equity. Ebert (2001), Zank (2001).

  10. * [-1;-4] ~ [6;2] * [-2;-4] ~ [6;2] echt 4 !!!!! echt 4 !!!!! 4 10 Example. Two persons {1,2}. All allocations below are comonotonic. (10,-1) ~ (15,-4) (10, 6) ~ (15, 2) c No. Not cosigned. (-2,-9) ~ (-4,-6) (6,-9) ~ (2,-6) c No. Not cosigned. Zank (2001)

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