Optimality Conditions for Distributive Justice

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1 Optmalty Codtos for Dstrbutve Justce J. N. Hooer Carege Mello Uversty Abstract We aalyze utltara ad Rawlsa crtera for dstrbuto of lmted resources by dervg optmalty codtos for approprate optmzato problems. We assume that some dvduals are more productve tha others, so that a equtable dstrbuto of resources creates greater overall utlty. We derve codtos uder whch a dstrbuto of wealth a maxmzes utlty, b maxmzes a utlty fucto that accouts for the socal cost of equalty, ad c satsfes a lexmax crtero that reflects the Rawlsa dfferece prcple. We show that a utltara soluto a ca dstrbute resources equally oly whe all dvduals have the same margal productvty. Equalty s possble uder b a dverse populato whe the cost of equalty s suffcetly large. Equalty s possble uder the Rawlsa opto c whe o segmet of socety has a much greater average productvty tha the rest. Equalty s more lely to be cosstet wth Rawlsa justce whe there are rapdly decreasg returs to greater vestmet productvty, whe the most productve dvduals are ot much more productve tha the average, ad, rocally, whe people are more terested gettg rch. 1 Itroducto Utltarasm ad the Rawlsa dfferece prcple mply dfferet crtera for dstrbutve justce, but both ca be vewed as mathematcal optmzato problems. Utltarasm maxmzes a socal utlty fucto whose argumets represet wealth dstrbuted to dvduals. The Rawlsa dfferece prcple calls for a lexcographc maxmum of the utltes allotted to dvduals. Ths suggests that the theory of optmzato mght provde some sght to the codtos uder whch a dstrbuto of wealth satsfes a utltara or a Rawlsa crtero. I partcular, we use classcal optmalty codtos to aalyze dstrbutos over odetcal dvduals. Ths s a departure from most axomatc treatmets of dstrbutve justce, whch assume that dvduals are dstgushable Blacorby, Bossert, & Doaldso Ths capablty allows us to study oe of the pereal ssues of dstrbutve justce the extet to whch a effcet dstrbuto of wealth requres equalty. It s sometmes argued that more utlty s created whe greater shares of wealth are alloted to The author retas the copyrght for ths artcle. dvduals who are more taleted, more productve, or wor harder. We use the modelg devce of assgg to each dvdual a productvty fucto u α that measures the total utlty evetually created whe dvdual s tally alloted wealth α. We the fd the dstrbuto of tally avalable wealth that ultmately results the greatest total utlty. We vestgate the degree of equalty that s requred to maxmze utlty, ad well as codtos uder whch a completely egaltara dstrbuto maxmzes utlty. We perform a smlar aalyss whe the calculato of utlty accouts for the fact that excessve equalty may dsrupt socal harmoy ad ultmately reduce total utlty. I partcular, we determe whe the cost of equalty s hgh eough so that a egaltara dstrbuto maxmzes utlty. The Rawlsa dfferece prcple states roughly that equalty should be tolerated oly whe t s ecessary ad suffcet to result greater utlty for everyoe. We follow the commo practce of terpretg ths as a mperatve to fd a lexmax dstrbuto. To do so we suppose that dvduals have a commo utlty fucto vα that measures the persoal utlty that results whe a dvdual s alloted wealth α. We further suppose that the fracto of the total utlty that s evetually ejoyed by a dvdual s proportoal to the utlty of that dvdual s tal wealth allocato. Thus we vew the tal allocato of resources to dvduals as assgg socal status ad prvlege. We derve codtos uder whch a dstrbuto of wealth satsfes the lexmax crtero, as well as codtos uder whch the lexmax dstrbuto s completely egaltara. 2 Utltara Dstrbuto We frst formulate the utltara problem. Let the utlty geerated by perso from wealth x be u x. If the total resource budget s 1, the problem of dstrbutg wealth to maxmze utlty s max u x a x 0, all b c 1

2 If we assocate Lagrage multpler λ wth the costrat 1b, ay optmal soluto of 1 whch each x > 0 must satsfy x λ =0,,..., Elmatg λ yelds 1x 1 = = x 2 Thus a wealth dstrbuto s optmal oly whe the margal productvty of wealth s the same for everyoe. Assume that dvduals 1,...,are dexed by creasg margal productvty: +1 α u α for all α 0 ad =1,..., 1 3 I ths case, 2 s satsfed oly f x 1 x. Thus the less productve dvduals receve less wealth, as oe mght expect. Furthermore, a utltara dstrbuto s completely egaltara x 1 = = x =1/ oly whe the margal productvtes are equal: 11/ = = 1/ To obta some dea of how sewed the wealth dstrbuto mght be, t s helpful to assume a specfc form u x =c x p 4 for the utlty fuctos, where p 0 ad each c 0. Here c dcates the productvty of perso. Whe p =1, perso produces utlty proporto to the wealth receved. Whe 0 <p<1, greater wealth has decreasg margal utlty, ad p =0dcates ablty to use wealth to create utlty. If dvduals are dexed order of margal productvty, we have that c 1 c. Sce a optmal soluto of 1 whch each x > 0 must satsfy 1b ad 2, t s 1 x = c 1 1 p c 1 1 p j 5 j=1 whe 0 p<1. Whe p 1, t s clear o specto that a optmal soluto sets x =1ad x =0for = 1,..., 1. The the optmal dstrbuto s completely uequal whe utlty geerated s proportoal to wealth p = 1. The most productve member of socety receves all the wealth. The dstrbuto becomes creasgly egaltara as p approaches zero, reachg the lmt a dstrbuto whch each perso receves wealth proporto to c. Thus the most egaltara dstrbuto that s possble ths utltara model s oe whch people receve wealth proporto to ther productvty. Moreover, ths occurs oly the lmtg case whe the utlty geerated s depedet of the wealth receved p = 0. Whe 0 < p 1, a utltara dstrbuto ca be completely egaltara x 1 = = x oly whe c 1 = = c. Whe p>1, oe dvdual must receve all the wealth eve whe c 1 = = c. Usg ths model, more egaltara dstrbutos are less effcet. I a optmal dstrbuto wth 0 p<1, the total utlty s 1 p c 1 1 p 6 I a completely egaltara dstrbuto, each x j =1/, ad the total utlty s p 1 c 7 The rato 7/6 dcates the utlty cost of a egaltara dstrbuto. 3 Cost of Iequalty The rudmetary utltara model above mples that a utltara soluto ca result cosderable equty whe dvduals have dfferet abltes. A classcal defese of utltarasm, however, s that excessve equty geerates dsutlty by cotrbutg to socal dsharmoy. The model 1 does ot accout for ay such cost of equalty. A more adequate model may result utltara wealth dstrbutos that are more equtable. A smple way to try to capture the cost of equty s to model t as a proportoal to the total rage of comes. The model 1 becomes max u x β max{x } m{x } x 0, all Presumably, a postve cost factor β could result utltara solutos that dstrbute wealth more equally. It s also terestg to derve how large β must be to result a completely egaltara dstrbuto. The aalyss s easer f we learze problem 8 usg the followg lemma. We aga assume that dvduals are dexed by creasg margal productvty, as 3. Lemma 1 If the utlty fuctos u satsfy 3, ad 8 has a optmal soluto, the the followg problem has the same optmal value as 8: max u x βx x 1 a x x +1,,..., 1 x 0, all b c d Proof. Let x be a optmal soluto of 8 wth optmal value U.Ifx j >x for some j, wth j<, the create a ew soluto x 1 defed by x 1 j = x, x1 = x j, ad x1 = x for j,. IfU 1 s the objectve fucto value of soluto x 1 8, the U 1 = U + u j x u j x j+u x j u x 10 But due to 3, u x j u x u j x j u j x 8 9

3 because j<. Ths ad 10 mply that U 1 U. Now f x 1 j >x1 for some j, wth j<, create a ew soluto x 2 the same maer, ad observe aga that the objectve fucto of 8 does ot decrease. Cotue wth the sequece x 1,...,x t utl x t 1 x t. The x t s feasble the problem max u x β max{x } m{x } x x +1,,..., 1 x 0, all 11 ad has a objectve fucto optmal value o less tha U. But 11 has a optmal value o greater tha U because t s more hghly costraed tha 8. Thus 8 ad 11 have the same optmal value. But 11 s obvously equvalet to 9, whch mples that 8 ad 9 have the same optmal value, as clamed. To characterze optmal solutos of 9, we assocate Lagrage multpler λ wth 9b ad multplers µ 1,...,µ 1 wth the costrats 9c. The Karush-Kuh-Tucer KKT optmalty codtos mply that x s optmal 9 oly f there are a value of λ ad oegatve values of µ 1,...,µ 1 such that 1x 1 +β λ µ 1 =0 x λ + µ 1 µ =0,=2,..., 1 x β λ + µ 1 =0 12 where µ =0f x <x +1 the soluto. We frst exame the case whch each dvdual receves a dfferet wealth allotmet x. I ths case each µ =0, ad we ca elmate λ from 12 to obta 2x 2 = = 1x 1 1x 1 = 2x 2 β x = 2x 2 +β Thus all dvduals who are ot at the extremes of the dstrbuto have equal margal productvty a utltara dstrbuto, just as they do the soluto of the orgal model 1. The dvdual at the bottom of the dstrbuto, however, has margal productvty that s β smaller tha that of those the mddle, whle the dvdual at the top has margal productvty that s β larger tha that of those the mddle. Ths teds to result somewhat larger allotmet for the dvdual at the bottom, ad a smaller allotmet for the oe at the top. Sce the remag dvduals are forced to le betwee these extremes, the et result s a dstrbuto that s less sewed tha the orgal model. 4 Equalty a Utltara Dstrbuto We ca also determe what value of β results a completely egaltara model. I ths case the multplers µ ca be ozero. Aga elmatg λ from the KKT codtos 12, we get 2µ 1 µ 2 = d 1 µ 1 + µ µ +1 = d,=2, µ 1 + µ 1 = d 1 where d = 1 x 1 +1 x +1+β, =1,..., 1 d 1 = 1 x 1 x 14 +2β It ca be checed that the followg solves 13 µ = 1 d 1 1 d 15 = for =1,..., 1. Substtutg 14 to 15, we get µ = β + 1 x m x 16 for =1,..., 1. We ow cosder a egaltara soluto, whch each /. Sce each µ 0 a optmal soluto, we obta the followg from 16. Theorem 2 Suppose that dvduals are dexed order of creasg margal productvty. The a utltara dstrbuto the model 8 s egaltara x 1 = = x oly f β m 1/ 1 1/ 17 for =1,..., 1. Ths may be easer to terpret for the specfc productvty fuctos defed earler. Corollary 3 If the productvty fucto u are gve by 4, a utltara dstrbuto the model 8 s egaltara oly f β p p 1 c 1 c for =1,..., 1. Thus to determe the mmum β requred to esure equalty, we exame each group of smallest coeffcets c 1,...,c. The value of β depeds o the dfferece betwee the average of these coeffcets ad the average of the remag coeffcets. Thus f there s a group of dvduals who are much less productve o the average tha the remag dvduals, relatve to the overall rage of productvtes, a larger β s requred to esure equalty. Ths could occur a two-class socety wth a relatvely homogeeous uderclass ad relatvely homogeous eltes, for example.

4 5 Rawlsa Dstrbuto A lexmax lexcographc maxmum model ca be used to represet a wealth dstrbuto that satsfes the Rawlsa dfferece prcple. As before we let u x be the socal utlty geerated by a perso who tally receves wealth x. We also suppose that the fracto of total utlty receved by perso s proporto to the persoal utlty vx of perso s tal wealth allocato. Thus everyoe has the same persoal utlty fucto, eve though dfferet people may have dfferet productvty fuctos. If y s the utlty ejoyed by perso, ay soluto of the followg problem s a Rawlsa dstrbuto: lexmax y a y = vx y 1 vx 1,=2,..., b y = u x c 18 x 0,,..., d where y = y 1,...,y. By defto, y solves 18 f ad oly f y solves problems L 1,...,L, where L s the problem max m {y,...,y } y 1,...,y 1 =y1,...,y b 18e The lexmax soluto s frequetly defed wth respect to a partcular orderg y 1,...,y of the varables, whch case L 1 maxmzes y rather tha m{y,...,y }. Ths s approprate for the Rawlsa problem because we do ot ow advace how the soluto values y wll ra sze. Suppose, however, that persos 1,..., are dexed by creasg margal productvty as 3. The we ca assume wthout loss of geeralty that persos wth less margal productvty are earer the bottom of the dstrbuto. Lemma 4 Suppose that 3 holds ad that vα s mootoe odecreasg for α 0. The f 18 has a soluto, t has a soluto whch y 1 y. Proof. Sce v s mootoe, t suffces to show that 18 has a soluto x, ȳ whch x 1 x. For ths t suffces to exhbt a soluto x, ȳ that solves L for = 1,...,ad for whch x 1 x. Let x,y be a soluto of 18, ad let x 0,y 0 = x,y. If x 0 1 x0 for =2,...,, the x0 solves L 1 ad we let x 1 = x 0. Otherwse we suppose x 0 = m {x 0 } ad defe x 1 by x 1 1 = x 0, x1 = x0 1, ad x 1 = x 0 for 1,. We defe y 1 to satsfy 18b-18c. We ca see as follows that x 1,y 1 solves L 1. If U 0 = u x s the total utlty for soluto x 0,y 0, the the total utlty for soluto x 1,y 1 s U 1 = U 0 + u x 0 1 u x 0 +u 1 x 0 u 1 x 0 1 e But we have from 3 that u x 0 1 u 1 x 0 u 1 x 0 1 u 1 x 0 Thus U 1 U 0, ad x 1 geerates o less total utlty tha x 0. Sce utlty s allotted to the y 1 s proporto to vx 1, ad v s mootoe ocreasg, we get y1 1 y1. 0 Thus x 1,y 1 solves L 1. Now f x 1 1 x1 for =2,...,, the x1,y 1 solves L 1,L 2 ad we let x 2,y 2 =x 1,y 1. Otherwse we suppose x 1 = m 2{x 1 } ad defe x2 by x 2 1 = x 1, x2 = x1 1, ad x 2 = x1 for >2ad. We ca show as above that x 2,y 2 solves L 1,L 2. I ths fasho we costruct the sequece x 1,y 1,...,x,y ad let x, ȳ =x,y. By costructo, x 1 x. Sce x, ȳ solves L 1,...,L, t solves 18. To aalyze solutos of 18, t s coveet to elmate the varables y from each L. Usg costrats 18b 18c, we get u x y = vx,,..., vx Usg Lemma 4, L ca be wrtte u x max vx vx a x 1,...,x 1 =x 1,...,x 1 b c x x x 0 d e 20 where x 1,...,x 1 are prevously computed solutos of L 1,...,L 1, respectvely. We focus frst o L 1. Assocatg Lagrage multplers µ 1,...,µ 1 wth the costrats 20d, the KKT optmalty codtos mply that a soluto x wth each x > 0 s optmal 20 oly f there are oegatve values of µ 1,...,µ 1 such that v x 1 Σu + vx 1 u 1x 1 v x 1 Σu 2 λ µ 1 =0 vx 1 u x v x Σu 2 λ + µ 1 µ =0, =2,..., 1 vx 1 u x v x Σu 2 λ + µ 1 =0 21 where Σu = c u x, = vx

5 ad where µ =0f x <x +1 the soluto. We beg by examg the case whch each dvdual receves a dfferet allotmet x. Here each µ =0, ad 21 mples v x 1 vx 1 + u 1 x 1 Σu v 1 x 1 = u x Σu v x for =1,..., 1, assumg vx 1 > 0. Ths says that the margal dfferece betwee productvty ad persoal utlty s the same for everyoe except the lowest raed dvdual, for whom the dfferece s somewhat less. Ths teds to crease the allotmet to the lowest dvdual, reducg the gap betwee ths perso ad the others. The optmalty codtos for L 2 are smlar ad lewse move the secod closest dvdual closer to those who are more hghly raed. Thus geeral, the lexmax soluto results a dstrbuto that s more egaltara tha oe whch the margal dfferece betwee productvty ad persoal utlty s the same for everyoe. 6 Equalty a Rawlsa Dstrbuto We ow exame codtos uder whch a Rawlsa dstrbuto ca be egaltara. We foud earler that a utltara dstrbuto wth utlty fuctos u x = c x p, vx =x q caot be egaltara uless dvduals are detcal ther productvty. We wll show that a Rawlsa dstrbuto ca, uder certa codtos, be egaltara a more dverse populato. I a egaltara dstrbuto ay µ ca be ozero. We elmate λ from the optmalty codtos 21 for L 1 to obta v x 1 vx 1 + u 1x 1 Σu v x 1 1 vx 1 Σu µ 1 = u x Σu v x vx 1 Σu µ 1 µ for =2,..., 1, ad v x 1 vx 1 + u 1 x 1 Σu v x 1 1 vx 1 Σu µ 1 = u x v x Σu vx 1 Σu µ 1 Ths yelds the followg. Theorem 5 Suppose the productvty fuctos are gve by u α =c α p ad the utlty fucto by vα =α q. The L 1 has a egaltara soluto x 1 = = x oly f 1 c 1 c q p c 24 or equvaletly, 1 1+ q p for =1,..., 1. c 1 1 q p 25 c Proof. The equatos ca be wrtte as 13 where d = vx 1 Σu v x 1 vx 1 u +1x +1 1x 1 Σu + v x +1 v x 1 for =1,..., 1. Substtutg x 1 = = x =1/ ad the fuctos u, v as gve above, we obta d = q p j=1 c j p p c +1 c 1 26 Sce 15 solves 13, we ca substtute 26 to 15 ad get q µ = p 1+p c + 1 c 1 c p for =1,..., 1. The KKT codtos mply that x = = x =1/ ca be a optmal soluto oly f µ 0 for =1,..., 1, whch mples 24. A egaltara soluto x 1 = = x solves L 1 f ad oly f t solves the lexmax problem 18. If t solves L 1, the a lexmax soluto must have x 1 =1/, whch mples by 18d that x 2 = = x = 1/. If a egaltara soluto does ot solve L 1, the some dstrbuto wth x 1 < 1/ solves L 1, whch mples that x 1 < 1/ ay lexmax soluto. Thus we have Corollary 6 If the productvty fuctos are gve by u α =c α p ad the utlty fucto by vα =α q, the a lexmax dstrbuto s egaltara x 1 = = x oly f 24 ad 25 hold. Thus a Rawlsa dstrbuto s completely egaltara whe the gap betwee the average productvty of the least productve dvduals ad that of the remag populato s ot too great for ay. The maxmum gap s proportoal to q/p ad /. Ths meas that a smaller gap s requred whe the margal utlty of wealth decreases rapdly wth the level of wealth q s small, ad whe the opposte s true of margal productvty p s large. Thus a egaltara dstrbuto s more lely whe dvduals do ot care very much about gettg rch ad are satsfed wth a moderate level of prosperty. Iequalty s also more lely whe allocatg greater advatages to taleted or dustrous people reaps cosstetly greater rewards. A egaltara dstrbuto also requres a smaller productvty gap betwee the hghest class ad the remag populato.e., whe = 1 tha betwee the lowest class ad the remag populato =1. Thus f the dstrbuto of talets ad dustry has a log tal at the upper ed, as s commoly supposed, the codto for equalty could be hard to meet. 7 Cocluso We fd that a utltara dstrbuto of wealth ca result substatal equalty whe some dvduals are more productve tha others. The dstrbuto s completely egaltara oly whe every dvdual has the same margal

6 productvty. Whe margal productvtes are uequal, the most egaltara dstrbuto that s possble s oe whch dvduals are allocated wealth proporto to ther margal productvty, ad ths occurs oly whe there are rapdly decreasg margal returs for greater allocatos of wealth. A more egaltara dstrbuto results whe the utlty fucto cludes a pealty to accout for socal dysfucto that equalty may cause. I partcular, f the pealty s proportoal to the gap betwee the rchest ad poorest dvduals, we ca calculate a costat of proportoalty that results a completely egaltara dstrbuto. Ths costat teds to be larger whe there s large gap average productvty betwee two segmets of socety. That s, there a group of dvduals that have a much smaller average margal productvty tha the remag dvduals, relatve to the overal rage of productvtes. Ths may occur, for example, whe eltes ad commo people form farly homogeous groups separated by a large gap average productvty. Fally, the Rawlsa dfferece prcple ca result a completely egaltara dstrbuto whe o two segmets of socety have a large gap average productvty. Equalty s more lely to occur whe there are decreasg returs for placg greater vestmet taleted ad dustrous people. Somewhat surprsgly, equalty s also more lely whe people are early as cocered about gettg rch as about lvg a mmally comfortable lfestyle. Whe people wat rches more, a prvleged class s less lely to be cosstet wth Rawlsa justce. Fally, equalty s more lely whe the most taleted ad dustrous dvduals are ot much more productve tha the average perso, eve though the least productve dvduals may fall far below the mea. Refereces Blacorby, C.; Bossert, W.; ad Doaldso, D Utltarasm ad the theory of justce. I Arrow, K.; Se, A.; ad Suzumura, K., eds., Hadboo of Socal Choce ad Welfare, Vol. 1, volume 19 of Hadboos Ecoomcs. Amsterdam: Elsever Bouveret, S., ad Lematre Fdg lexm-optmal solutos usg costrat programmg: New algorthms ad ther applcato to combatoral auctos. I Edrss, U., ad Lag, J., eds., 1st Iteratoal Worshop o Computatoal Socal Choce. Iserma, H Lear lexcographc optmzato. OR Spetrum 123: