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scottish institute for research in economics SIRE DISCUSSION PAPER SIRE-DP-2008-13 Which Inequality?The Inequality of Endowments Versus theInequality of RewardsEd HopkinsTatiana KornienkoUniversity of Edinburgh www.sire.ac.uk Which Inequality?The Inequality of Endowments Versus theInequality of Rewards ∗ Ed Hopkins † EconomicsUniversity of EdinburghEdinburgh EH8 9JY, UKTatiana Kornienko ‡ EconomicsUniversity of EdinburghEdinburgh EH8 9JY, UKMarch, 2008 Abstract Society often allocates valuable resources - such as prestigious positions, salaries,or marriage partners - via tournament-like institutions. In such situations, in-equality a ff ects incentives to compete and hence has a direct e ff ect on equilibriumchoices and hence material outcomes. We introduce a new distinction betweeninequality in initial endowments (e.g. ability, inherited wealth) and inequality of what one can obtain as rewards (e.g. prestigious positions, money). We showthat these two types of inequality have opposing e ff ects on equilibrium behaviorand wellbeing. Greater inequality of rewards tends to hurt most people — boththe middle class and the poor, — who are forced into greater e ff ort. In contrast,greater inequality of endowments tends to bene fi t the middle class. Thus, whichtype of inequality is considered hugely a ff ects the correctness of our intuitionsabout the implications of inequality. Keywords: inequality, endowments, rewards, relative position, ordinal rank, games,tournaments, dispersive order, star order. JEL codes: C72, D63, D62, D31. ∗ We thank Helmut Bester, Simon Clark, Kai Konrad, Benny Moldovanu, Andrew Oswald, PrasantaPattanaik, Mike Peters, József Sákovics and participants at the Edinburgh social economics workshopand the Public Economic Theory conference, Marseille for helpful discussions. Ed Hopkins thanks theEconomic and Social Research Council, Research Fellowship Scheme award reference RES-000-27-0065,and a Leverhulme Trust Study Abroad Fellowship for support. †
[email protected], http://homepages.ed.ac.uk/hopkinse ‡
[email protected], http://homepages.ed.ac.uk/tkornie2 1 Introduction Perhaps there is no other economic debate older than that over inequality. As Sen(1980) points out, while most have agreed that some form of equality is desirable, therehas been less consensus on what should be equalized. There is even debate over whatis meant by equality and inequality (see Sen (1980), Phelps Brown (1988), Roemer(1996), Lamont (2003) and many others). Despite this diversity of opinion, typicallyequality is treated as a moral question. There may be some distributions of incomeor of wealth or some methods of distribution of endowments which are simply unfair.In contrast, in economics, the second fundamental welfare theorem seemed to separatethese moral issues from the mainstream of economic analysis, the study of e ffi ciency.Only recently it has been suggested that people have “social preferences”, so that theycare directly about what others receive as well as their own income or consumption.Di ff ering formulations have been proposed by Frank (1985), Fehr and Schmidt (1999)and Charness and Rabin (2002) amongst many.Here, we take a purely economic approach and examine a model where individualscare only about their own consumption, yet inequality has a material e ff ect on marketoutcomes. We assume a society where individuals di ff er in terms of initial endowments,whether it is innate ability, education received or inherited wealth. Second, the rewardsthat individuals receive as a result of their achievements also vary. A fi xed set of rewards, that could represent cash prizes, places at a prestigious university, attractive jobs, desirable spouses, social esteem, monopoly rents or any combination of these, isassigned by a tournament. Individuals, whose endowments are private information,make a simultaneous decision over e ff ort or performance. Then each individual is givena reward according to his rank in the distribution of performance: fi rst prize is givento fi rst place, second prize to second place, and so on.Such a tournament creates important positional externalities, to obtain a top rewardone must occupy a top position, and by doing so one excludes others from that positionand hence that reward. This induces competitors to behave as though they had adesire for high rank or status, an observation fi rst due to Cole, Mailath and Postlewaite(1992), discussed further by Postlewaite (1998). In turn, this leads to equilibrium e ff ortbeing ine ffi ciently high and equilibrium utility being ine ffi ciently low. Crucially, theseexternalities also imply that the equilibrium choice of e ff ort and equilibrium utilitydepend on both the initial distribution of endowments and the distribution of rewards.Therefore, there is no need to appeal to any notion of justice for equality to matter. Itmatters because what others have a ff ects the job one gets, the wage one is paid and theamount of leisure one takes.Thus, both the distribution of endowments and the distribution of rewards a ff ectindividual choices and equilibrium utility. However, we fi nd that changes in the inequal-ity of endowments have the opposite e ff ect to changes in the inequality of rewards. Anincrease in the equality of competitors’ endowments raises the return to e ff ort as it iseasier to overtake one’s rivals. This leads to higher e ff ort for low and middle ranking1 agents. Furthermore, equilibrium utility falls at middle and high ranks and even thosewith higher endowments can be worse o ff in the more equal and hence more competitivedistribution. However, an increase in the equality of rewards implies there is less di ff er-ence between a high prize and a low one. This leads to a reduction in incentives and adecrease in equilibrium e ff ort for low and middle ranking competitors, and an increasein their equilibrium utility. Simply put, greater equality of rewards will typically bene fi tmost of society, greater equality of endowments can harm the majority.In our analysis, we assume formal equality of opportunity. That is, “there is nolegal bar to access to education, to all positions and jobs, and that all hiring is merito-cratic” (Roemer, 1996, p. 163) and where individuals are rewarded according to theirproductivity (Lamont, 2003). However, the tournament mechanism of resource alloca-tion allows one to separate the other two forms of inequality — that of endowments andof rewards. Talent could vary widely, but the most talented could receive a monetaryreward only slightly greater than the least talented. Alternatively, small di ff erences intalent could lead to big di ff erences in outcomes. This distinction between inequality of endowments and inequality of rewards we believe to be novel.Why should we assume that success is driven by relative performance? An importantreason is empirical. Tournament-like mechanisms are used in practice to determine uni-versity admissions, entry into certain professions and promotions and pay within fi rms.More generally, there is now a signi fi cant body of research that suggests that indicatorsof wellbeing such as job satisfaction (Brown et al., (2004)), health (Marmot et al. (1991),Marmot (2004)) and overall happiness (Easterlin (1974)) are strongly determined byrelative position. That is, a highly ranked individual in a poor country can have greaterhealth and happiness than a low ranked individual in a richer country, even though thelatter has greater material prosperity. However, this existing literature on relative po-sition and relative concerns has tended to assume that if relative position matters, thisimplies a distaste for inequality. In particular, Frank (1999, 2000) has argued forcefullythat greater inequality exacerbates wasteful social competition. Hopkins and Kornienko(2004) showed that this argument may be problematic in that, in a model with relativeconcerns, an increase in equality of endowments can make everyone worse o ff . Here, weprovide partial support for Frank’s argument but we emphasize that it is an increase inthe inequality of rewards, rather than greater inequality of endowments, that worsenswasteful competition.We observe that the inequality of rewards has a much better fi t with common beliefsabout the e ff ects of inequality. An increase in this type of inequality bene fi ts the richin endowments and hurts the middle-class and poor, who are forced into greater e ff ort.Under some regularity conditions, even stronger welfare e ff ects are possible - namely,that greater inequality of rewards can make all worse o ff . In contrast, greater inequalityof endowments has an opposing e ff ect and can bene fi t the middle-classes. Of course,this latter fi nding is in con fl ict with common intuitions about the role of inequality.What these results suggest in practical terms is that policies to promote equality couldhave very di ff erent e ff ects depending on the form of the intervention. For example, an2 attempt to equalize educational achievement, in the sense that it equalizes endowmentsof those entering the labor market, would have a quite di ff erent e ff ect than a policyto equalize incomes, the rewards from labor market activity. The fi rst policy wouldincrease competitive pressure on those in the middle of the range of ability, while thesecond would reduce it. Thus, once the impact on work e ff ort is included, the fi rstpolicy surprisingly could have a negative e ff ect on those it was aimed to help, in a sensethat it would create pressures on those in the middle to perform more, at the expense of their happiness. In contrast, the second policy would reduce these pressures to perform,increasing wellbeing for those in the middle at the expense of lower performance.What is crucial in the tournament model we consider is that the shape and the rangeof the distributions of endowments and rewards themselves determine the marginalreturn to e ff ort. Consequently, even policy interventions such as lump-sum taxes andtransfers can have an impact on incentives if they change either the distributions of endowments or rewards. In fact, there are two distinct e ff ects from any changes in thelevel of inequality. The fi rst, which we call the direct e ff ect, is simply that under amore equal distribution of endowments or rewards lower ranked individuals will havegreater endowments or rewards respectively. However, in either case, there is also thesecond e ff ect, which we call the incentive or social competitiveness e ff ect. Crucially, theincentive e ff ect of an increase in equality of endowments is positive and opposite of thatof an increase in the equality of rewards, which decreases incentives. Note that thisincentive e ff ect is created by the competitive externalities present in our tournamentmodel. So, in their absence such as in more conventional neoclassical models, there areonly the direct e ff ects so that reward and endowment inequality would appear to havesimilar results. This may be why the distinction between rewards and endowments hasnot been made before.One important assumption of our tournament model is that there is a fi xed dis-tribution of indivisible rewards. The justi fi cation for this is that in reality there aremany desirable things, jobs, places at university, marriage opportunities, that do di ff erin quality and which are not divisible. A subtle criticism is that even if rewards are in-divisible, there might be the possibility of side payments. This possibility is analyzed ina di ff erent literature where workers are matched to (indivisible) jobs by an endogenouswage schedule. For example, Costrell and Loury (2004) and Suen (2007) have consid-ered changes in the distribution of ability of workers and in the quality of jobs. There isno incentive e ff ect as there is no choice of e ff ort by workers and all outcomes are Paretoe ffi cient, in distinct contrast to the situation we model. Nonetheless, the shape of thedistributions of ability and of jobs a ff ects the distribution of wages. That is, inequalitycan have a material e ff ect on outcomes even in the presence of side payments. Thissuggests that what is crucial is simply that rewards in society vary and that some areindivisible and thus positional externalities are present. 1 1 More technically, inequality of endowments and inequality of rewards will have substantial butopposing e ff ects whether matching between competitors and jobs or rewards is done under transferableutility or non-transferable utility. 3
