Article Politics,Philosophy&Economics 12(4)375–391 Exploitation, altruism, ªTheAuthor(s)2013 Reprintsandpermissions: and social welfare: sagepub.co.uk/journalsPermissions.nav DOI:10.1177/1470594X13496066 An economic exploration ppe.sagepub.com Matthias Doepke Northwestern University, USA Abstract Child labor is often condemned as a form of exploitation. I explore how the notion of exploitation, as used in everyday language, can be made precise in economic models of childlabor.Exploitationisdefinedrelativetoaspecificsocialwelfarefunction.Ifirstshow that under the standard dynastic social welfare function, which is commonly applied to intergenerationalmodels,childlaborisneverexploitative.Incontrast,underaninclusive welfarefunction,whichplacesadditionalweightonthewelfareofchildren,childlaboris alwaysexploitative.Neitherwelfarefunctioncapturesthemoregradualdistinctionsthat common usage of the term exploitation allows. I resolve this conflict by introducing a welfare function with minimum altruism, in which child labor in a given family is judged relativetoa specific social standard. Under thiscriterion,child labor is exploitativeonly infamilieswheretheparent(orguardian)displaysinsufficientaltruismtowardsthechild. I argue that this welfarefunction bestcaptures the conventional conceptofexploitation andhasusefulpropertiesforinformingpoliticalchoicesregardingchildlabor. Keywords Child labor, exploitation,social welfarefunction, altruism Introduction In both popular and scientific writing, child labor is frequently portrayed as a form of exploitationofchildren.1Indeed,searchingfor‘‘childlabor’’and‘‘exploitation’’onGoo- gle Scholar yields about 22,000 hits in June 2013.2 Yet within the context of economic Correspondingauthor: Matthias Doepke, Department of Economics, Northwestern University, 2001 Sheridan Road, Evanston, IL60208,USA. Email:[email protected] 376 Politics, Philosophy & Economics 12(4) modelsofchildlabor,thereisusuallynoclearlydefinednotionofexploitation.Tobesure, theliteraturehasidentifiedanumberofconditionsunderwhichsomerestrictionsofchild labor may be desirable. However, these arguments are based on standard efficiency grounds(generatedby,forexample,humancapitalexternalities)andarenotlinkedtothe conceptofexploitation.3 Inthispaper,Iexplorehowexploitationcanbedefinedinthecontextofaneconomic modelinawaythatrhymeswiththeconventionalunderstandingoftheterm.Thatis,I am taking seriously the moral intuitions that lie behind the usage of ‘‘exploitation’’ in commonlanguage,andaimtotranslatetheseintuitionsintothelanguageofwelfareeco- nomics.Centraltothisundertakingistodevelopawelfarecriterionthatcanbeusedto decide which forms ofchild laboramount toexploitation,andwhich do not. To see why this is not a trivial undertaking, consider the standard welfare concept usedindynamiceconomicmodels.SincetheseminalworkofBarro(1974),Beckerand Barro(1988),andBarroandBecker(1989),ithasbeenstandardpracticetotreataltruis- ticallylinkeddynastiesinawaythatisanalogoustoasingleagentwhohasaninfinite lifespan.4Inthecaseofasingleagentwholivesforever,thenaturalwelfareconceptis theinitiallifetimeutilityoftheagent.Iftherearemultiplesuchagents,thewelfarecri- terionisa(possiblyweighted)sumoflifetimeutilities.Applyingthesameconcepttoa world of dynasties, a welfare criterion emerges that evaluates the welfare of the entire dynastythroughtheutilityoftheheadofthedynasty.Futuremembersofadynastyonly enter consideration insofar as the initial member of the dynasty is altruistic towards them, i.e. theirutility enters theutility ofthe dynastyhead. Defining a dynastic utility criterion in this way has practical advantages. Standard resultsfromdynamiceconomicscarryovertothedynasticsetting,implying,forexam- ple,that,intheabsenceofexternalitiesandwithcompletemarkets,competitiveequili- briaare efficient. However, using the standard dynastic welfare criterion also makes it impossible to second-guesswhetherparentsmakethe‘‘right’’choicesfortheirchildren.Theimplicit socialwelfarefunctionacceptstherevealedpreferenceofparentsfortheirchildren,and doesnotattachanyimportancetothechildrenotherthanthroughthealtruismthattheir parentsfeelforthem.Thislimitationofstandarddynasticwelfareexplainswhyitisdif- ficulttospeakofexploitationofchildreninthisclassofeconomicmodels.Whenwefeel that children are being exploited, implicitly we are saying that their parents (or, more generally,guardians)arenottakingsufficientcareofthem.Yetsuchajudgementcannot beexpressedwhenawelfarecriterionisusedthatassumesfromtheoutsetthatthepar- entscarejusttherightamount.Hence,thestandarddynasticsocialwelfarefunctionren- ders theconcept ofexploitation vacuous. Theaimofthispaperistoaddressthemismatchbetweentherhetoricofexploitation andthewelfarecriteriathatareusuallyappliedtoeconomicmodels.Todothis,Ipropose alternativewelfarecriteriathatcanbeusedtogivethetermexploitationamoreprecise meaning within an economic model of child labor, and then connect this discussion to arguments for and against regulation of child labor. Based on the common language understanding of exploitation, for the purposes of this paper we will say that people arebeingexploitediftheyareunfairlyusedforthebenefitofothers.5Inthecaseofchild labor, the ‘‘being used for the benefit of others’’ condition will usually be satisfied Doepke 377 because it is generally parents (or other guardians) who make decisions on child labor andstandtobenefitfromtheincome.Thequestion,then,iswhetherthisuseofthechild is unfair. Anumberofdifferentnotionsoffairnesshavebeenproposedbywritersinethicsand politicalphilosophy.Myobjectivehereistostayascloseaspossibletoconventionalcon- ceptsofsocialwelfareineconomicmodels,whilealsoarrivingataformulationthatgives meaningtothetermexploitation.Thenotionoffairnesswillthereforebebasedonsocial welfarefunctionsthatareultimatelybasedonindividualpreferences.6Inaddition,acriter- ionisdesiredunderwhichthequestionofwhetherchildlaboramountstoexploitationdoes nothaveanabsoluteanswer,butratherdependsonspecificcircumstances.Thiswillrecon- cile, through the lens of the formal criterion, the common intuition that, for example, extremeformsofchildlaborsuchasprostitutionconstituteexploitation,whereasactivities suchashelpingoutonthefamilyfarmaremorebenign.Inevaluatingdifferentsocialwel- farefunctions,Iwillbelookingforcriteriawheresuchgradualjudgementsarepossible. Tofindanotionoffairnessthatmeetstheseobjectives,itisessentialtoallowadis- tinctionbetweenparentalaltruismandthe‘‘social’’altruismthatisusedtodefineexploi- tation. In particular, the basic idea of the welfare criteria proposed here is to allow for socialwelfarefunctionsthatmaygivemore(orless)weighttochildrenthantheweight implicit in their parents’ altruism. Whether child labor amounts to exploitation then dependsonthedegreeofparentalaltruism.Indeed,ifaparenthadnoregardwhatsoever forthewelfareofthechild,childlaborwouldbeakintoslavery,andpresumablyslavery shouldamounttoexploitationunderanydefinition.Conversely,ifaparenthadcomplete disregardforhisorherownutilityandwassolelymotivatedbymaximizingthechild’s happiness,childlaborwouldonlyoccurtobenefitthechild,andthusshouldnotconsti- tute exploitation. Unlike the standard dynastic social welfare function, the preferred social welfarefunction developedhere leads tothe same conclusions. Inthenextsection,Ioutlineabasicmodelofchildlaborthatwillprovidetheframe- workfortheanalysis.Thefollowingsectioncontainsthecoreoftheanalysisandoutlines how exploitation does or does not arise from the perspective of alternative criteria for judgingsocialwelfare. Thelast section concludes. A simple model of child labor and altruism Whiletheargumentsbelowshouldbeapplicabletoawideclassofeconomicmodels,it isusefultodevelopthemainideasinaspecificsetting.Thekeyfeaturesoftheenviron- ment considered here are that there are parents and children; parents decide on child labor;andparentsarealtruistictowardstheirchildren.Tosimplifytheanalysis,themain ideasaredevelopedinastaticsettingwithasinglegenerationofparentsandchildren.7A dynamicextension isconsidered below. Economic environment Weenvisionaneconomypopulatedbyafinitenumberofadults,indexedbyi,whoare characterized by their human capital h >0 and a preference parameter z, where i i 0(cid:2)z (cid:2)1. Each adult has exactly one child. The preference parameter z represents i i 378 Politics, Philosophy & Economics 12(4) altruism,i.e.therelativeweightofthechild’sutilityintheutilityfunctionoftheparent. Theremaybevariationinthedegreeofaltruismamongtheparents.Theparent’slifetime utility V asafunctionof humancapital h is givenby P;i i V ðhÞ¼ð1(cid:3)zÞuðc ðh;lÞÞþzV ðh Þ: ð1Þ P;i i i P i i i C C Herec ðh;lÞistheparent’sconsumption(whichdependsonhumancapitalandthe P i i supply of child labor) and V is the child’s utility (as an adult). The parent works full C time, and chooses the child’s labor supply l. The remaining time 1(cid:3)l serves to raise i i the child’shumancapital. The child’shumancapital is givenby h ¼eð1(cid:3)lÞ: ð2Þ C i We can think of e as a production function for human capital. One interpretation is that1(cid:3)l istimethatchildrenspendinschoolandthatthereforeraisestheirhumancap- i ital. However, wider interpretations are also possible; for example, one component of human capital is health, and e might represent the detrimental effects of child labor onfuture health. Thechild’spreferences are givenby V ðhÞ¼uðc ðhÞÞ: ð3Þ C C Ofcourse,inadynamicdynasticsettingthechild’sutilitywouldhavethesamefunc- tionalformasthatoftheparent,withanadditionaltermfortheutilityofthegrandchildren. Forillustration,Iwilladoptparticularfunctionalforms.Parentalconsumptionisgivenby c ¼hþl: ð4Þ P That is, the wage is equal to one per unit of human capital, and children’s human capital is normalized to one. Alternatively, we can abstract from labor markets alto- gether and interpret (4) as a linear production function that is available to each household. Theproductionfunction foreducation is assumed tobelinear eð1(cid:3)lÞ¼1(cid:3)l: ð5Þ Given thatthere are nograndchildren, the children’sconsumption is givenby c ¼h ¼1(cid:3)l: ð6Þ C C We alsoassume thatthe utilityfunction is logarithmic uðcÞ¼lnðcÞ: ð7Þ i Unconstrained equilibrium Wecannowcharacterizetheoutcomeinthiseconomyifparentsareunconstrainedinthe choice oftheir child’slaborsupply. Proposition 1 (Unconstrained Equilibrium) In the unconstrained equilibrium, parentschoose the followingchild laborsupplyl i l ¼maxf1(cid:3)zð1þhÞ;0g: i i i Doepke 379 Proof: Theparent’s optimization problemis maxfð1(cid:3)zÞlnðh þlÞþz lnð1(cid:3)lÞg: i i i i i 0(cid:2)li(cid:2)1 Thefirst-order conditionfor aninterior optimumis 1(cid:3)z z i ¼ i ; h þl 1(cid:3)l i i i which yields l ¼1(cid:3)zð1þhÞ i i i Thiscondition,togetherwiththenon-negativityconstraintonligivestheresult.c As intuition suggests, a parent that is entirely selfish (z ¼0) will always set child i labor to the maximum level, l ¼1. Altruistic parents will leave at least some time i for their child’s education, l <1, and the more so the higher their own human capital i h is. i Exploitation and social welfare Defining exploitation relative to a social welfare function Havingdescribedtheunconstrainedoutcomeintheeconomy,Inowwouldliketocon- siderethicalcriteriaforevaluatingtheequilibriumallocation.Isitappropriatetosaythat childrenarebeingexploitedinthisworld?Forthepurposesofthediscussion,wewillsay thatpeoplearebeingexploitediftheyareunfairlyusedforthebenefitofothers.Inthe specificcaseofalaborrelationship,exploitationhappensifonepersonforcesanotherto workwithout adequate compensation. Atfirstsight,itmaythenappearthatinourmodelchildlaboramountstoexploita- tionbydefinition,giventhatoneeconomicagent(theparent)imposesaworkrequire- ment on the other (the child), and no formal compensation is provided. But this does not rhyme with common sense: the parent does care about the child after all, and if zisclosetoone,theparentwouldbewillingtoenduregreatsufferingforthewelfare ofthechild.Itwouldbedifficulttoarguethatasmallamountofchildlaborthathelps toprovidesubsistenceincometotheparentrepresentsexploitation.Itistruethatinthe modelthechilddoesnotreceiveformalcompensationforthelabor.However,itisalso truethattheparentpresumablyincurredexpensesandeffortinraisingthechild(which isimplicitintheadultbudgetconstraint),andchildlaborcouldbeviewedascompen- sation for such efforts. Giventhesearguments,itappearsthatthequestionofwhetherchildlaborisexploi- tation is not an either–or question; rather, the answer should depend on the degree of altruismz.Arelationshipwhereoneagentcontrolsthelaborsupplyoftheothermaystill not represent exploitation as long as the controlling agent cares sufficiently about the controlled agent. Of course, what ‘‘sufficiently’’ means here depends on the meaning of‘‘fairness’’inthe definitionofexploitation. 380 Politics, Philosophy & Economics 12(4) Asoutlinedintheintroduction,thepositionadoptedhereisthat‘‘exploitation’’needs tobejudgedrelativetosomesocialwelfarefunction.Foragivensocialwelfarefunction, Idefine exploitation asfollows. Definition1(Exploitation)Foragivensocialwelfarefunction,childlaboriscon- sideredexploitation if thefollowing conditions aresatisfied: 1. Achildisrequiredtoworkmorethanwhatthesocialwelfarefunctionprescribes. 2. Theadditionalwork benefits the parentrelative tothe socialoptimum. Ofcourse,thecontentofthedefinitiondependsonthesocialwelfarefunction.Rather thantryingtogiveonespecificmeaningtothetermexploitation,thepurposeofthedef- initionisthereverse:Iwouldliketocomparedifferentpossiblesocialwelfarefunctions intermsoftheirimplicationsforthemeaningofexploitation,andthenuseourintuitive understanding of exploitation to determine which social welfare function most closely matches this understanding. Thus, the analysis will help determine what kind of social welfare function may usefully represent the moral intuitions behind the common- language useofexploitation. The dynastic social welfare function We limit attention to social welfare functions that are utilitarian in the sense of being functions ofthe utilitiesof themembers ofthe society (cid:4)(cid:2) (cid:3)(cid:5) SWF¼f V ;V : P;i C;i Thesocialoptimumistheallocationthatmaximizesthesocialwelfarefunctionsub- ject toresourceconstraints. Definition2(SocialOptimum)Thesocialoptimumgivenasocialwelfarefunc- tionf is the allocation thatsolves the followingproblem (cid:2) (cid:2) (cid:3)(cid:3) max f V ;V P;i C;i subject to(1)to(7) above. We will start with the social welfare function that is most commonly employed in dynasticeconomicmodelsofthekindusedhere,namelywhatItermthedynasticsocial welfarefunction.Thisfunctionisgivenbythesumoftheutilitiesoftheparents,without directlyaccountingforthechildren(giventhattheyarealreadyincludedintheirparents’ utility).This socialwelfare functionis asfollows.8 Definition 3 (Dynastic Social Welfare Function) The dynastic social welfare functionis given by X SWF ¼ V : Dynastic P;i i Doepke 381 Whilethissocialwelfarefunctioniscommonlyapplied,itimpliesthataslongasmar- ketsarecompleteandtherearenoexternalities,thedecentralizedequilibriumcoincides withthesocialoptimum.Thus,parentstreattheirchildrenexactlyasthesocialwelfare functionprescribes, andinthissense there is noexploitation. Proposition2(NoExploitationunderDynasticSocialWelfareFunction)Under thedynasticsocialwelfarefunction,thesocialoptimumcoincideswiththeuncon- strainedequilibrium. Proof:Thesocialwelfarefunctionisasumoftheindividualobjectivefunctionsthat underlytheunconstrainedequilibrium,theconstraintsetisthesame,andthereisno interaction between different dynasties, implying that the optimum is the same as well.c ThisfindingreflectsthefamiliarFirstWelfareTheorem,namelythat,intheabsence ofexternalitiesandmarketincompleteness,equilibriumoutcomesareParetooptimal.Of course,Paretooptimalisdefinedhererelativetoonlytheparents’utility,butthesame judgement is includedinthe dynasticsocial welfarefunction. We could generate a different result with the same welfare function by introducing market failures. To illustrate, consider an extension of the model with a humancapital externality. All dynasties are identical with the same altruism factor z and same initial humancapitalh.Theutilityfunctionsandtheproductionfunctionforhumancapitalare asabove.However,Inowassumethatproductioninthesecondperiod(whenthechil- dren areadult) involvesa humancapital externality c ¼pffihffiffiffiffiffipffih(cid:2)ffiffiffiffiffi: ð8Þ C C C Thatis,thechildren’sconsumptiondependsnotjustontheirownhumancapitalh , C butalsoontheaveragehumancapitalh(cid:2) inthepopulation.Householdsarenotcompen- C sated for their contribution to average human capital in the economy; thus, a classic externality is now present in the model. The externality leads to a wedge between the equilibriumallocation oflaborandthe socially optimalallocation. Proposition3(EquilibriumandSocialOptimumwithHumanCapitalExtern- ality)Inthemodelwithahumancapitalexternalityasin(8),theequilibriumlabor allocation is (cid:9) (cid:7) (cid:7) (cid:8)(cid:8) (cid:10) 2 h l¼max 1(cid:3)z 1þ ;0 ; 2(cid:3)z 2 whereas thesocially optimal allocation is l¼maxf1(cid:3)zð1þhÞ;0g: Thus,allchildrenworkmoreintheequilibriumallocationcomparedtothesocial optimum (aslong asequilibrium child laboris positive). 382 Politics, Philosophy & Economics 12(4) Proof:Theparent’soptimizationproblemis(afterpluggingallconstraintsintothe objective andomittingconstants) (cid:9) (cid:10) 1 max ð1(cid:3)zÞ lnðhþlÞþ z lnð1(cid:3)lÞ : 0(cid:2)l(cid:2)1 2 The first-orderconditionfor aninterior optimum is 1(cid:3)z z ¼ ; hþl 2ð1(cid:3)lÞ which yields (cid:7) (cid:7) (cid:8)(cid:8) 2 z 2 h l¼ (cid:3) ð2þhÞ¼ 1(cid:3)z 1þ : 2(cid:3)z 2(cid:3)z 2(cid:3)z 2 This conditiontogether with the non-negativityconstraintonl givesthe result. Forthesocialoptimum,theplannerrecognizesthath ¼h(cid:2) inequilibrium,imply- C C ingthatchildren’sconsumption is stillgivenby c ¼h C C asin(8),sothatthesocialoptimumcoincideswiththeequilibriumallocationchar- acterized inProposition 1. c Thus,inthemodelwiththehumancapitalexternality,childlaborisinefficientlyhigh intheequilibriumallocation.However,eventhoughchildrenworktoomuch,themodel stilldoesnotfulfillourdefinitionofexploitation.Thereasonisthattheincreasedlabor supply of the children in the equilibrium allocation does not actually make the parents betteroffrelativetothesocialoptimum,whichwasthesecondrequirementinthedef- initionofexploitation.Infact,theparentsareworseoffintheequilibrium:alleconomic agents, parentsandchildren,are negativelyaffectedbythe presenceoftheexternality. Thefollowing corollarysummarizes the result. Corollary1(NoExploitationdespiteHumanCapitalExternality)Inthemodel withahumancapitalexternality,childrenworkmorethaninthesocialoptimum, but nevertheless child labor does not constitute exploitation in the sense of Definition1. Thisexample should make clear that it is difficult to satisfy the definition of exploi- tation when the usual dynastic social welfare function is used. If all parents are identi- cal, the social optimum under this welfare function delivers the highest possible utility to the parents given the constraints of the economy, so that they cannot be made better off by exploiting their children through child labor in some way. A parallel argument would apply to other forms of market failures.9 We therefore move on to alternative welfare functions where the tight link between social optima and parental welfare no longer applies. Doepke 383 A social welfare function with direct weight on children Onecanarguethatthebasicdefectofthedynasticsocialwelfarefunctionisthatitplaces direct weightonly on the parents,while ignoring the children. Thus,let usconsider an alternative welfare function where the utility of children enters directly, in addition to theindirectweightingthroughparentalaltruism.Amongothers,thissocialwelfarefunc- tion is advocated by Farhi and Werning (2007).10 We term this welfare function the inclusive socialwelfare function. Definition 4 (Inclusive Social Welfare Function) The inclusive social welfare functionis given by X X SWF ¼ V þg V : Inclusive P;i C;i i i Hereg>0istheadditional,directweightattachedtotheutilityofthechildren. Theinclusivewelfarefunctionleads totheoppositeconclusionofthedynasticwel- farefunction:aslongaschildrenareworkingatall,thisalwaysrepresentsexploitation, regardless of the degree of altruism z. To establish this result, I revert to the original modelwithoutahumancapitalexternality(allowingforexternalitieswouldnotalterthe results). Proposition4(SocialOptimumunderInclusiveWelfareFunction)Underthe inclusivesocialwelfarefunction,thesociallyoptimalchild laborsupplyisgiven by: (cid:9) (cid:10) 1 l ¼max ð1(cid:3)zð1þhÞ(cid:3)ghÞ;0 : i 1þg i Thus, if child labor is positive in equilibrium, the socially optimal child labor supplyislowerthanequilibriumlaborsupply(whichischaracterizedinProposi- tion1). Proof: Given that there is no interaction across dynasties, we can solve for the socially optimal child labor supply for each dynasty individually. For dynasty i, the problemofmaximizing socialwelfare is givenby maxfð1(cid:3)zÞ lnðh þlÞþðz þgÞlnð1(cid:3)lÞg: i i i i i 0(cid:2)li(cid:2)1 Thefirst-order conditionfor aninterior optimumis 1(cid:3)z z þg i ¼ i ; h þl 1(cid:3)l i i i which yields 1 l ¼ ð1(cid:3)zð1þhÞ(cid:3)ghÞ: i 1þg i 384 Politics, Philosophy & Economics 12(4) Thisconditiontogetherwiththenon-negativityconstraintonligivestheresult.c Corollary2(ExploitationunderInclusiveWelfareFunction)Undertheinclu- sivesocialwelfarefunction,ifchildlaborispositiveintheunconstrainedequili- brium, itconstitutes exploitation inthesense ofDefinition 1. Proof: Proposition 4 shows that if child labor is positive in equilibrium, it has to be lower in the social optimum under the inclusive welfare criterion, satis- fying the first requirement of the definition. In addition, the equilibrium allo- cation yields the highest feasible utility for the parents, implying that parents are better off in equilibrium compared to the social optimum, which satisfies the second requirement of the definition. c Theresultundertheinclusivesocialwelfarefunctionisequallyunsatisfactoryasour findingsunderthedynasticsocialwelfarefunction.Anychildlaborisnowconsideredto amounttoexploitation.Nomatterhowmuchtheparentcaresaboutthechild,fromthe perspectiveofthesocialwelfarefunctiontheparentdoesnotcareenough.Thetechnical reason is thatthe socialaltruism is inadditiontoparental altruism. More generally, the welfare criterion implies that parents give too little to their chil- dren in any dimension, be it protection from child labor, consumption, human capital investment, orbequests.Whereas under the dynastic socialwelfare function parents can do no wrong, under the inclusive social welfare function they can do no right. Such a blanket condemnation appears equally unattractive as the blanket approval of parental actions that is implied by the dynastic social welfare function. The next proposed wel- fare criterion tries to find a middle ground between these two extremes. A social welfare function with universal altruism Atheart,asocialwelfarefunctiondescribeshowwefeelaboutotherpeople’schoices.In ourspecificsetting,whatisatissueishowpeoplefeelabouttheappropriatedistributionof resourcesbetweenparentsandchildreninotherfamilies.Itwouldseemplausiblethatin making this judgement,people apply the same altruism factorz that they apply intheir i ownfamily.Thus,eachindividualwouldpossessasocialwelfarefunctionwherethesame weightz isattachedtothewelfareofallchildren,regardlessofthepreferencesofthepar- i entsofthesechildren.Socialjudgementswouldthendependonwhichfamily’ssocialwel- farefunctionisused.Asimplepossibilityistosumallfamilies’welfarefunctions,inthe samewaythatthedynasticwelfarefunctionsumstheutilityofalldynastyheads.Weterm thiswelfarefunctiontheuniversallyaltruisticwelfarefunction.Thesocialdiscountfactor z(cid:2)thatisbeingappliedtoallchildrenistheaverageofallindividualaltruismfactors 1X z(cid:2)¼ z: i N i Here N is the totalnumber ofdynasties. Definition 5 (Social Welfare Function with Universal Altruism) The univer- sally altruisticsocial welfarefunction is givenby