decentralizing equality of opportunity

INTERNATIONAL ECONOMIC REVIEWVol. 50, No. 1, February 2009

DECENTRALIZING EQUALITY OF OPPORTUNITY∗

BY CATERINA CALSAMIGLIA1

Universitat Autonoma de Barcelona, Spain

In a global justice problem, equality of opportunity is satisfied if individualwell-being is independent of exogenous irrelevant characteristics. Policymakers,however, address questions involving local justice problems. We interpret a col-lection of local justice problems as the decentralized global justice problem. Weshow that controlling for effort locally, which is not required by the global justiceobjective, is sufficient for decentralizing equality of opportunity. Moreover, undersome conditions, equalizing rewards to effort is not only sufficient but necessary.This implies in particular that most affirmative action policies may not contributeto providing equality of opportunity.

1. INTRODUCTION

Although global and local distributive justice problems seem to be related, acritical gap exists between them. In the first chapter of his book Equity, PeytonYoung (1994) describes this disconnect: “Theories of justice in the large do nottell us how to solve concrete, everyday distributive problems such as who shouldget into medical school or how much to charge for a subway ride.” He arguesthat the reason for this is that “[i]ssues of local justice tend to be compartmental-ized. Societies make no effort to coordinate distributive decisions across differentdomains.” For example, kidney agencies do not give precedence to patients whofailed to get into university or a worker’s wage at a given job is not increasedbecause the government located a waste dump near his home. He concludes bysaying that “there is no mechanism comparable to the invisible hand of the mar-ket for coordinating distributive justice at the micro into just outcomes at themacro level.” He then argues that local distributive justice should be derived byinduction, that is, formulating general rules from what we observe is being donein different local distributive justice problems.

∗ Manuscript received December 2006; revised May 2007.1 I thank Stephen Morris, John E. Roemer, Hanming Fang, and Donald J. Brown for valuable

advice, and very specially Antoni Calvo-Armengol, who has influenced me deeply and whom Iwill sorely miss. I am also grateful to two referees and the editor for very useful comments thathave helped improve the article substantially. I also thank Xavier Calsamiglia, Ariel Rubinstein, andseminar participants for helpful comments and encouragement. The work is partially supported bythe Generalitat de Catalunya through grant 2005SGR00454 and by the Spanish Ministry of Educa-tion and Science through grant SEJ2005-01481/ECON and FEDER, the Barcelona Economics Pro-gram (XREA), and the CONSOLIDER-INGENIO 2010 (CSD2006-00016) grant. Please addresscorrespondence to: Caterina Calsamiglia, Departament d’Economia i Historia Economica,Universitat Autonoma de Barcelona, Edifici B- Campus de Bellaterra, 08193 Barcelona, Spain.E-mail: [email protected].

273C© (2009) by the Economics Department of the University of Pennsylvania and the Osaka UniversityInstitute of Social and Economic Research Association

274 CALSAMIGLIA

In this article, we interpret a collection of local distributive justice problemsas essentially the decentralized global distributive justice problem. Then, a localjustice rule can be defined as the rule that decentralizes a given distributive justicecriterion when information is disperse and decision making decentralized. Wethen ask whether we can solve the global distributive justice problem by solving aseries of local distributive justice problems. In particular, we characterize the setof local distributive justice rules that decentralize equality of opportunity.

There is a wide literature dealing with the issue of compensation and responsibil-ity.2 The results in this literature are mainly axiomatic characterizations clarifyingthe conflicts and interdependencies of different ethical principles and their differ-ent axiomatic representations. But the context is that of a global justice problem.To analyze the point in this article we focus on a very simple formalization ofequality of opportunity, initiated by Rawls (1971) and formalized and discussedin Roemer (1998). Equality of opportunity requires that an individual’s successin life be independent of irrelevant characteristics, that is, of characteristics thatthe individual should not be responsible for. Such irrelevant characteristics mayinclude, for example, race, gender, or parents’ income.3 A large part of the socialdebate concerns precisely what the set of irrelevant characteristics should be, butthis article abstracts from this debate and takes the set of irrelevant characteristicsas given.

In the literature the difference between global and local problems has beenlargely ignored. Specifically, local equality of opportunity has been interpretedas local outcomes or local utilities being independent of irrelevant characteristicsaffecting the local outcome. This is the analogous definition of equality of oppor-tunity in the global problem for the local problem.4 We show that such notionof local equality of opportunity may not lead to equality of opportunity glob-ally. Specifically, local independence of outcomes from irrelevant characteristicswill not aggregate into independence of total welfare from irrelevant character-istics. This article shows that a collection of local mechanisms will decentralizethe provision of equality of opportunity if it provides local equality of rewards toeffort, where effort is a productive individual choice variable, for example, timedevoted to a given activity.5 Moreover, it shows that if the mechanisms and theenvironment are rich, a notion that will be made precise later in the article, globalequality of opportunity will be decentralized only if the collection of mechanismsprovides local equality of rewards to effort. Although equalizing opportunitiesdoes not require explicitly controlling for effort in the global problem, it does for

2 See Fleurbaey and Maniquet (2004) for an excellent review of this literature.3 Whether welfare or outcomes should be equalized has been part of the debate, but this article does

not directly address this issue, although indirectly it relates to it, since the global objective is welfarebut locally the planner focuses on outcomes. See for example Dworkin (1981a, 1981b) or Roemer(1986) for a discussion on this issue.

4 Some examples are Benabou and Ok (2001), Betts and Roemer (2002), Llavador and Roemer(2001), or Roemer et al. (2003).

5 Roemer (1998) proposes an algorithm that equalizes outcomes for individuals who exert the samedegree of effort. This differs substantially from equalizing rewards to effort as shown in Section 2.

DECENTRALIZING EQUALITY OF OPPORTUNITY 275

the decentralized problem. Effort, an endogenous variable, needs to be controlledfor in the local problems in order to achieve equality of opportunity.

When decentralizing equality of opportunity, a problem that arises is that ofproperly accounting for the cost of local effort, because it generally depends oninformation that the local policymaker does not have.6 This is the main reasonwhy providing equality of opportunity in a decentralized manner requires thattwo individuals differing in irrelevant characteristics receive the same outcomewhen they exert the same effort. This suggests that when designing policies oneshould analyze the effect of the policy on the rewards to effort and not only onthe outcome it will generate. Implementing this may be hard and may requiremechanism design techniques because effort is usually unobservable. Thereforesome may interpret this as a critique to the equal opportunities enterprise since wepresent a more difficult problem. Others, however, may interpret this result as astep toward understanding what the provision of global equality of opportunitiesin society entails and view the local rule as a useful tool for policy design.

This article does not analyze how this moral objective interacts with other objec-tives that policymakers may have. It does not address how efficient or profitableit may be for the policymaker to equalize rewards to effort. So clearly futureresearch should embed this criterion in a more general welfare function that in-cludes other desirable properties or objectives that the policymaker may care for.A general approach to this issue would involve an axiomatic analysis similar tothat of Fleurbaey and Maniquet (2005, 2008). In more concrete settings it willbe important to analyze how equalizing rewards to effort limits the policymaker.This would allow us address important issues like which is the most efficient wayof providing equality of rewards to effort or how restrictive equalizing rewardsto effort is in other respects. However, in competitive environments like auctions,tournaments, or contests it has been shown that asymmetries between the playersreduces their incentives to compete and that reducing these asymmetries may bebeneficial to the administrator.7 How these asymmetries are resolved, however,has a big impact for the administrator. In particular, equalizing chances of win-ning of asymmetric players through a lump sum bonus may decrease the effortsexerted by all individuals. But equalizing rewards to effort enhances competitionand increases the efforts exerted by all individuals.8 In this case, therefore, prop-erly providing local equality of opportunity maximizes the performance of thecompetition.

The structure of the remainder of the article is as follows. Section 2 presents anexample to illustrate the main ideas underlying the results. Section 3 presents the

6 For example, colleges may be interested in equalizing the probability of admission into collegefor students who only differ in their neighborhood characteristics. But if different individuals haveinvested a different number of hours of study, colleges want to compensate for the individual’s costof studying the extra hours by admitting them with a higher probability. But this opportunity costdepends on how much effort or time the individual is spending on the other activities available to him,for example, training basketball or working in a paid job.

7 See Maskin and Riley (1995), Myerson (1981), McAfee and McMillan (1989), and Corns andSchotter (1996) for details.

8 See Franke (2008) and Calsamiglia et al. (2007).

276 CALSAMIGLIA

benchmark model and derives the main result. Section 4 presents some concludingremarks.

2. EDUCATION AND BASKETBALL: AN EXAMPLE

Paul and Richard are intelligent, highly motivated, and hardworking teenagers,denoted by i ∈ {P, R}. Both would like to become professional basketball playersand get a college degree. Paul and Richard are identical except that Richard’sneighborhood has better public schools.9 Unlike education, Paul and Richardhave identical resources to improve their basketball skills, that is, they have thesame number of courts and teams of similar qualities in which to train.

Paul and Richard have to decide how much time to spend on homework, de-noted by eS, and on playing basketball, denoted by eB. Their ultimate objective isto get into college and the NBA.

In this example we assume that individuals are characterized by a vector ofrelevant characteristics r = (r S, r B), which correspond to education and basket-ball innate ability, and a vector of irrelevant characteristics, b = (bS, bB), rep-resenting neighborhood school resources and basketball facilities. In particularlet (bP

S , bPB) = ( 1

2 , 1) and (bRS , bR

B) = ( 710 , 1), and (r P

S , r PB ) = (r R

S , r RB) = ( 1

3 , 15 ). Note

that Paul and Richard only differ in bS.College admissions decide who gets admitted into college and observe

education-specific characteristics, r S and bS, and NBA recruiters decide who getsadmitted into the NBA and observe basketball-specific characteristics, r B and bB.

Out of the characteristics and effort, Paul and Richard get a performance ineducation and basketball that can be measured by the outcome functions “nor-malized SAT score” and “normalized NBA test score.” The normalized SAT score(percentile in the distribution of SAT scores in the population) can be representedby

aS = aS(bS, rS, eS) = bSrS(eS)12 .

College admissions therefore know that aS = 13 bS(eS)

12 . The normalized NBA test

score (percentile in the distribution of NBA scores in the population) can berepresented by

aB = aB(bB, rB, eB) = bBrB(eB)12 .

NBA recruiters therefore know that aB = 15 (eB)

12 . The cost of training and studying

to Paul and Richard is represented by

c(eS, eB) = 1500

(eS + eB)2.

9 Paul (P) comes from the Poor neighborhood and Richard (R) from the Rich neighborhood.


eS

pSR

pSPceS

P

cR

eS

eB

c eB

P

pB

P, R

*PeS*ReS

*ReBeB

*P

pS

*R

pS*P

pB

*R

pB*P ce

B

R

FIGURE 1

CHOICES AND OUTCOMES WHEN COLLEGE AND NBA RECRUITERS FOLLOW THE MARKET POLICY

College admissions officers and NBA recruiters, with education-specific andbasketball-specific information, respectively, decide the probability of individualsbeing admitted into college and into the NBA, that is pS(bS, r S, eS) and pB(bB, r B,eB).10 Paul and Richard’s utility is determined by the probability of them beingadmitted into college or the NBA and by the cost of devoting time to either of thetwo activities:

U = pS(bS, rS, eS) + pB(bB, rB, eB) − c(eS, eB).

We say that college and NBA recruiters follow a market policy if they admitindividuals with probability pj = a j .

The marginal cost in one activity depends on the effort being spent at the otheractivity. Specifically, the more time one spends in one activity, the higher themarginal cost of effort for the other activity. As seen in Figure 1, when maximizinghis utility, Richard devotes relatively more time to school than Paul given hishigher productivity of studying. This leaves Richard with a higher marginal costof playing basketball, leading him to train fewer hours than Paul. In particular,Paul will choose (eP

S , ePB) = (4.2, 6) and get pP

S = 0.34, pPB = 0.5, and U P = 0.62; and

Richard will choose (eRS , eR

B) = (6.6, 4.8) and get pRS = 0.6, pR

B = 0.44, and U R =0.77. Notice that equality of opportunity is violated because Richard has a higherwelfare than Paul even though they only differ in irrelevant characteristics.

Equality of opportunity is a concern when irrelevant characteristics affectingindividual’s achievement differ. Since irrelevant characteristics affecting basket-ball are the same for Paul and Richard, NBA recruiters need not worry about

10 In this example we are not modelling an admission policy that is allocating a fixed number of seats,but assuming that the admissions policy for Paul can be changed without Richard’s being affected,which is somewhat unrealistic, but greatly simplifies presenting the point of the example.

278 CALSAMIGLIA

violating equality of opportunity. For them the market policy, for example, will bein accordance with equality of opportunity.11

Suppose that college admissions had as a first priority the contribution to theprovision of equality of opportunity through providing local equality of opportu-nity or equality of educational opportunity. It is not clear, a priori, what equalityof educational opportunity would require. One possibility is that a consultant,having read some political philosophy, Roemer’s book on equality of opportunity,or the literature on compensation and responsibility tells them that since Paul andRichard are identical except for their irrelevant characteristic, the compensationprinciple suggests equalizing the chances that they get admitted into college. Thiswill lead to college admissions designing an admissions policy that ensures Pauland Richard being admitted with equal probability. There will generally be a largeset of policies satisfying this property. The following examples point out two poli-cies that are often discussed and even applied in the political debate that are inthis set but do not lead to global equality of opportunity.

EXAMPLE 1: Equalizing local outcomes through Affirmative Action. Collegesset the admissions probability according to a modified SAT score, pS = aS + T(bS),to evaluate the different applicants. The transfers T(bS) are such that Paul andRichard get the same modified score. In particular it gives a positive transfer ofT(bP

S ) = aRS − aP

S = 0.26 to Paul and a zero transfer to Richard.12

As seen in Figure 2, p′PS corresponds to pP

S shifted up until p′∗PS = p∗R

S . Givencollege admissions policy Paul and Richard will not change their hours devotedto education and basketball because of the lump sum nature of the transfer, i.e.,the rewards to time into either activity are left unchanged. We observe that Paulis better off than Richard under this policy (U P = 0.88 and U R = 0.77).

College admissions accept both Paul and Richard with equal probabilities. NBArecruiters accept Paul with higher probability than Richard because his perfor-mance is better—he chooses to devote more time to basketball.13

This intervention gives Paul admission to college at a lower “price” in terms ofeffort than to Richard. Richard devotes more time to it and is equally rewardedwith the same chances of being admitted. Paul can then spend more time trainingbasketball at a lower marginal cost. Equality of opportunity is violated since Paul’sutility is higher than Richard’s.

11 In the literature an important principle of justice that authors have imposed is the principle ofnatural reward, introduced by Fleurbaey (1995), which requires respecting inequalities resulting fromaspects other than irrelevant characteristics. In particular then it would suggest that if individuals areidentical with respect to irrelevant characteristics there would be no reason to add any modificationto the policy. See Fleurbaey and Maniquet (2004) for further discussion.

12 Affirmative Action policies have been of this form in some universities and have been formalizedin this manner in the literature. Disadvantaged minorities were given a fixed number of extra pointsin their evaluation procedure. See for example Schotter and Weigelt (1992).

13 Notice that we did not consider Paul sophisticated enough to know that the college admissionswere equalizing outcomes. If that were the case Paul would have spent NO time in school becausehe’d know that would not affect his education degree relatively to Richard’s.


eS

pSR

pSP

c eS

P

c Re

S

eB

c eB

P

c Re

B

pB

P, R

*PeS*R*ReS

eB eB*P

pS

*Rp’S

*P

pB

*R

pB*P

p’SP

ceS

P

FIGURE 2

EFFECT OF AFFIRMATIVE ACTION CONSISTING OF ADDING A CONSTANT NUMBER OF POINTS TO SAT SCORES

eS

pS

Rc ReS

eB

eB

c’Pc R

eB pB

P, R

*PeS*R*ReS

eBe’B

*P

pS

*Rp’S

*P

pB

*R

p’B*P

p’S

P

e’S*P

c Pe

B

FIGURE 3

EFFECT OF POLICY THAT FORCES PAUL TO STAY IN SCHOOL LONG ENOUGH TO GET THE SAME EDUCATIONAL

OUTCOME AS RICHARD

EXAMPLE 2: Equalizing local outcomes “paternalistically.” College admissionspaternalistically force Paul to study long enough (through having the teacher forcehim to stay more hours in school) so that he gets the same SAT score as Richard.That is, force Paul to study a certain amount of hours, e′P

S , so that aS( 13 , e′P

S , 0.5) =aS( 1

3 , eRS , 0.7). If the market policy is now implemented the probability of being

admitted is equalized.

Figure 3 shows that Paul is now forced to study long enough so that he gets thesame SAT score as Richard. Paul has to study longer hours, e′P

S = 13, and therefore

280 CALSAMIGLIA

faces a larger marginal cost for training basketball that leads him to choose to trainless hours, e′P

B = 2.6. This lowers his utility to U P = 0.44. Paul and Richard areequally likely to get into college, but Richard has higher chances of entering theNBA. Richard gets a utility of U R = 0.77. Paul is now not only worse than Richard,but worse than he was under no equalizing educational opportunity policy!

These two examples illustrate that independence of local outcomes to irrelevantcharacteristics does not lead to equality of opportunity. In particular a policymakertrying to contribute to the provision of equality of opportunity by making localoutcomes independent of irrelevant characteristics may end up generating newand stronger inequalities. In the first example Richard, originally advantaged,ends up with a lower utility than Paul. In the second example, Paul, originallydisadvantaged, is made worse off than before the intervention.

This article shows that the rule that decentralizes equality of opportunity is onethat equalizes rewards to effort or time, that is, that it requires the same “price” interms of effort for each unit of local outcome. In this case, the goal is that for everyextra hour of studying, Paul and Richard get the same increase in their probabilityof being admitted. There will generally be a large set of policies that satisfy thisproperty.14 The following is just an example of a policy that would accomplishsuch goal.

EXAMPLE 3: Equalizing rewards to effort. College admissions correct not onlyfor the bias in outcomes but also in rewards to effort and bases his admissionsdecisions on the following modified SAT score: a′

S(bS) = T(bS) · aS and sets pS =a′

S. This would imply increasing the probability of being admitted proportionallyto the SAT achieved. In this case TR (bR

S , bPS ) = 1 and TP(bR

S , bPS ) = bP

S

bRS

= 75 would

make the marginal increase in the probability of being admitted for one more hourstudied equal for Paul and Richard. As Figure 4 shows, under this policy Paul andRichard’s choice sets are equalized, that is, the probabilities of entering collegeand the NBA for any chosen level of effort are independent of neighborhoodeducation.

The problems Paul and Richard face, the trade-offs and payoffs from studyingand training, are exactly the same, and therefore their welfare will be equal. Pauland Richard choose to study the same number of hours, train the same numberof hours, get the same chances to enter college and the NBA, and and get equalutilities.

3. THE MODEL

This section shows that the results derived from Section 2 are not specific to theexample, but can be extended to general environments. The model only includestwo activities, but the results can be easily extended to n activities.

14 The policymaker will generally have other objectives, for example, efficiency, that will help himdecide which policy to implement from the set satisfying equality of rewards to effort. We do notaddress this issue in this example but comment on it in the introduction of the article.


eS

pS

P

eS

c’ P, R

eB

pB

P, R

*P,ReS

pS*P,R

p’S

P, R

p’B* P,R

c’eB

P, R

*P,ReB

aS

Pp’S

P-

FIGURE 4

EFFECT OF AFFIRMATIVE ACTION CONSISTING OF ADDING A PROPORTION OF EXTRA POINTS TO SAT SCORES

There is a continuum of individuals in an interval I and two utility-providingactivities, j ∈ {1, 2}.15

3.1. Individual Characteristics and Preferences. Each individual i ∈ I is char-acterized by two vectors of exogenous characteristics: irrelevant characteristics,b = (b1, b2) ∈ IR2, which represent the characteristics that society thinks shouldnot affect individual well-being, and relevant characteristics, r = (r1, r2) ∈ IR2,which represent the characteristics that society thinks may affect individual well-being.16 Individual characteristics are described by the joint distribution F : IR2 ×IR2 → IR. Let � be the set of conceivable joint distributions F.

Individuals devote effort or time to each of the activities, e = (e1, e2) ∈ IR2.17

An individual with characteristics (b, r) will exert an effort e = ϕ(b, r), where ϕ :IR2 × IR2 → IR2. Let � be the set of conceivable such functions.

Individual utility depends on the outcome consumed for each activity and theeffort exerted in each activity:

15 Having a continuum of individuals is necessary in the Proof of Theorem 2, but is not needed forTheorem 1. See footnote 23 for exposition.

16 There may be activities for which society believes that no characteristic is irrelevant. This maycorrespond to spheres in life for which societies believe individuals are fully responsible and thereforeno government intervention is required. By having all characteristics in an activity be relevant thismodel is perfectly compatible with there being a private sphere, which has been defended by politicalphilosophers and economists.

17 The analysis contains only one dimensional efforts, one for each activity. This may well correspondto similar underlying efforts. For instance, eating healthy food will improve your performance inall activities (some foods will help you more for some activities, but clearly restraining yourself ofeating some foods will improve most activities). This could be included in the model by choosing theappropriate cost function. Another aspect of importance would be the efforts being multidimensional.In that case the sufficiency results presented in this section would clearly hold. As for the necessitypart, some modifications in the modeling would need to be included, which would not incorporate anyadditional intuitions but would complicate the analysis.

282 CALSAMIGLIA

U(x, e) = u(x) − c(e),

where u is a strictly increasing function representing the utility out of the outcomevector x = (x1, x2) and c is a nonseparable function in e1 and e2 that representsthe cost of the effort vector, e = (e1, e2).18

3.2. Production. Outcome produced in each activity depends only on thecharacteristics of the specific activity. That is, outcome in activity j will depend onthe marginal distribution of characteristics (bj , r j ), F j , and the efforts exerted,e j ∈ ϕ j (bj , r j ), where ϕ j (bj , r j ) = {e j | ϕ(bj , b− j , r j , r− j ) = (e j , e− j )}. There areno externalities in production between these two activities.

The outcome production function of individuals with characteristics and effort(b, r, e) can be represented by a vector of strictly increasing functions:

a(b, r, e; F, ϕ) = (a1(b1, r1, e1; F1, ϕ1), a2(b2, r2, e2; F2, ϕ2)) ∈ IR2,

where a j : IR3 × � j × � j → IR represents the outcome produced by individ-ual with characteristics and effort (bj , r j , e j ) in activity j when living in societywith characteristics described by F j and exerting efforts according to ϕ j . There-fore bj and r j are relevant and irrelevant characteristics potentially affecting theproduction in activity j.

DEFINITION 1 (SOCIAL MECHANISM): A social mechanism µ is a mapping thatassigns for every distribution F ∈ � and effort function ϕ ∈ � an outcome vectorto each individual:

µ : IR3 × IR3 × � × � → IR × IR

µ ≡ (µ1, µ2) : (b, r, e; F, ϕ) → (x1, x2)

s.t.∫

IR2×IR2µ j (b, r, e; F, ϕ) dF ≤

∫IR2×IR2

a j (bj , r j , e j ; Fj , ϕ j ) dF j for j = 1, 2.

That is, x j is the outcome that the individual with characteristics (b, r) and efforte gets out of activity j when the distribution of individual characteristics is F andthe efforts exerted are described by ϕ. For example, the social mechanism couldassign to each individual what he produces, in which case µ(b, r , e; F , ϕ) = a(b, r ,e; F , ϕ).

18 The analysis can be extended to the case where relevant characteristics in the preferences areincluded, that is, for the case where U(t , x, e) = u(t , x) − c(t , e), where t is a relevant characteristic.The model can also be extended to the case where the asymmetry in preferences is an irrelevantcharacteristic but linked to a specific activity. For example, if some individuals get less utility out ofeducation because of their parent’s background and influence than others. The results in this articleapply equivalently to these extended frameworks. More general asymmetries in preferences couldbe included but in some cases defining what policymaker is responsible for compensating it in thedecentralized case will be less natural.


3.3. Individual Choice and Equilibrium. Given a social mechanism, an indi-vidual’s effort vector will maximize his utility, given the efforts of others:

e∗(b, r ; F, ϕ) ≡ arg maxe∈IR2

u(µ(b, r, e; F, ϕ)) − c(e),

where ϕ∗ is an equilibrium if for all individuals (b, r) ∈ supp F , ϕ∗(b, r) = e∗(b, r ;F , ϕ∗). Every individual is behaving optimally given what society is doing.

Individual utility in equilibrium is denoted by

u∗(b, r ; F, ϕ∗) = u(µ(b, r, ϕ∗(b, r); F, ϕ∗)) − c(ϕ∗(b, r)).

3.4. The Global Objective. We now define the objective of a social plannerwanting to provide equality of opportunity. As mentioned in the introduction,this is a simple formalization of equality of opportunity, inspired by the Rawlsiantradition in political philosophy and formalized and discussed in Roemer (1998).

DEFINITION 2 (EQUALITY OF OPPORTUNITY): Let (bi , r i , ei ) denote the character-istics of individual i ∈ I. A social mechanism µ provides equality of opportunity iffor all F ∈ � and ϕ∗ ∈ �

∀i, k ∈ I, r i = rk ⇒ u∗(bi , r i ; F, ϕ∗) = u∗(bk, rk; F, ϕ∗).

A social mechanism provides equality of opportunity if the welfare achieved byeach individual varies with his relevant characteristics but not with his irrelevantcharacteristics. Here, the planner is concerned about indirect utility, and thereforeshould not be concerned about the effort exerted by individuals, but only aboutfinal welfare. This is consistent with the fact that effort as a choice variable hasnot been emphasized in the political philosophy literature.

If one policymaker had all information he could act as a social planner and designthe social mechanism that guarantees equality of opportunity. But if informationand decision making is dispersed the provision of equality of opportunity willbe decentralized. In what follows we describe how the information and decisionmaking is distributed in this model.

3.5. The Local Problem. There is one policymaker responsible for assigningoutcomes for each activity j, who observes individual characteristics and effortexerted concerning activity j, and does not observe characteristics affecting theother activity. This implies that the local planner for activity j only knows themarginal distribution F j , and ϕ j . Given this information he chooses how to allo-cate outcomes for activity j.

DEFINITION 3 (LOCAL MECHANISM): A local mechanism is a mapping φ j thatassigns to every marginal distribution F j and ϕ j a local outcome vector to eachindividual:

284 CALSAMIGLIA

φ j : IR3 × � j × � j → IR

φ j : (bj , r j , e j ; Fj , ϕ j ) → xj

s.t.∫

IR×IRφ j (bj , r j , e j ; Fj , ϕ j ) dF j ≤

∫IR×IR

a j (bj , r j , e j ; Fj , ϕ j ) dF j .

A local mechanism assigns to every individual an outcome for a particularactivity conditional on his characteristics and effort choices specific to that activity,and potentially on other individuals’ characteristics and choices.

A pair of local mechanisms generates a social mechanism such that µ = (µ1,µ2)(b, r , e; F , ϕ) = (φ1(b1, r1, e1; F1, ϕ1), φ2(b2, r2, e2; F2, ϕ2)). Instead of themapping µ assigning an outcome vector to each individual we have (φ1, φ2), withφ j assigning the outcome for activity j. Therefore, a pair of local mechanisms, {φ1,φ2}, provides equality of opportunity if the induced social mechanism µ = (φ1, φ2)provides equality of opportunity. Equivalently, we say that a pair of local mech-anisms decentralizes equality of opportunity if the resulting social mechanismprovides equality of opportunity.

We now define the rule that decentralizes the provision of equality of opportu-nity. In contrast to the equality of opportunity requirement, it introduces effort asan important variable that needs to be explicitly controlled for.

DEFINITION 4 (LOCAL EQUALITY OF REWARDS TO EFFORT): Let suppF j be the sup-port of F j . And let Imgϕ j = {e j | ∃(bj , r j ) ∈ F j s.t. ϕ j (bj, r j) = ej}, that is, theimage of ϕ j . A local mechanism provides local equality of rewards to effort if forall F j and ϕ∗

j

∀e j ∈ Imgϕ∗j ∀i, k ∈ I

r ij = rk

j = r j and eij = ek

j = e j ⇒ φ j(bi

j , r j , e j ; Fj , ϕ∗j

) = φ j(bk

j , r j , e j ; Fj , ϕ∗j

).

Equivalently, if there exists w j such that for all F j and ϕ∗j ,

∀e j ∈ Imgϕ∗j ∀bj , r j ∈ suppFj φ j

(bj , r j , e j ; Fj , ϕ

∗j

) = w j(r j , e j ; Fj , ϕ

∗j

).

A local mechanism providing local equality of rewards to effort allows rewardsto effort only to depend on relevant characteristics. In other words, two individ-uals with different irrelevant characteristics but the same relevant characteristicsshould get the same if they choose to do the same effort.19

THEOREM 1. If the pair of local mechanisms (φ1, φ2) provides local equality ofrewards to effort, then it decentralizes equality of opportunity.

19 The rewards have to be the same only for the levels of effort that can be potentially chosenby some individual (e j ∈ Imgϕ∗

j ). Whatever happens for levels of effort that are never chosen is notimportant. But for policy design the distinction is small and economically not important (since thedifference is on levels of effort that are never chosen) so it is simpler and practically equivalent tofocus on making rewards to effort independent of irrelevant characteristics for all levels of effort.


Next we introduce conditions under which equalizing rewards to effort is notonly sufficient but also necessary. We introduce notation that allows us to describethe set of efforts that the policymaker in activity j can expect an individual withcharacteristics (bj , r j ) to choose in a given equilibrium. Given a pair of localmechanisms φ1, φ2 and the marginal distribution of characteristics in activity jand efforts in equilibrium, ϕ∗

j , let Ej (bj , r j ; F j , ϕ∗j ) be the set of effort levels in

activity j that an individual with characteristics (bj , r j ) chooses for some (b− j ,r− j ) ∈ supp F−j under some distribution of characteristics and efforts in the otheractivity, F− j ∈ �− j and ϕ∗

− j ∈ �− j .Formally,

E1(b1, r1; F1, ϕ

∗1

) = {e1

∣∣ ∃F2 ∈ �2, ϕ∗2 ∈ �2 and (b2, r2) ∈ suppF2

s.t. e1 = e∗1

(b1, b2, r1, r2; F1, F2, ϕ

∗1 , ϕ∗

2

)}E2

(b2, r2; F2, ϕ

∗2

) = {e2 | ∃F1 ∈ �1, ϕ

∗1 ∈ �1 and (b1, r1) ∈ suppF1

s.t. e2 = e∗2

(b1, b2, r1, r2; F1, F2, ϕ

∗1 , ϕ∗

2

)}.

The following condition on the mechanisms and the environment will force thatpolicymakers equalize rewards to effort in order to provide equality of opportunityin the aggregate.

DEFINITION 5 (RICHNESS): The mechanisms and the environment are rich if forall F j ∈ � j , Imgϕ∗

j ⊆ Ej (bj , r j ; F j , ϕ∗j ) for all (bj , r j ) ∈ supp Fj.

The mechanisms and the environment are rich whenever the set of efforts ob-served by the planner in equilibrium, e∗

j , can be chosen by all individuals with (bj ,r j ) ∈ supp Fj under some circumstance in the other activity −j. For example, ifschools observe that individuals put from 1 to 10 hours of study, then there mustbe individuals of any characteristic in school willing to choose that effort for somecircumstance outside of school. Therefore, whether you are in a good or bad schoolthere should be circumstances outside of school that make you interested in any ofthe 1 to 10 hours of study. This does not mean that individuals with different char-acteristics have the same probability of choosing a given effort, but that there is aninstance where every individual, as observed by the policymaker, is interested inchoosing any of the efforts in equilibrium. This means that the policymaker cannotnot rule out the possibility of any individual exerting a certain level of effort.20

This is a condition on both the environment and the mechanisms since it requiresenough heterogeneity in individual’s characteristics and the outputs they get withthose characteristics in each activity.

If richness is not satisfied, then equalizing rewards to effort may not be a neces-sary condition for equality of opportunity to be decentralized. The following two

20 One circumstance where this property would never hold, independently of the space of individualcharacteristics or distributions, is in the case where the utility and the cost function are separable inactivity 1 and 2, since then the choice e j is just a function of (bj , r j ). In that case it is clearly not necessaryto equalize rewards to effort: Equalizing partial utilities would be sufficient. But this assumes awayany interaction between the cost of time devoted to different activities.

286 CALSAMIGLIA

examples illustrate how the violation of richness may allow for decentralizationwithout equalizing rewards to effort. Example 1 shows a case where the mecha-nisms forces richness to be violated and example 2 one in which the environmentdoes so.

EXAMPLE 1. Let u(x1, x2, e) = x1 + x2 − (e1 + e2)2. Suppose there are norelevant characteristics and let a1(b1, e1; F1, ϕ1) = e1b1, a2(b2, e2; F2, ϕ2) = K +e2. Suppose policymaker 2 imposes a local mechanism such that ϕ2(b2, e2; F2,ϕ2) = K. Let policymaker 1 set φ1(b1, e1; F1, ϕ1) = e1b1 + T(b), where T(b) =∫ b2

4 dF2 − b2

4 . Individuals will choose e1 = b12 and e2 = 0. Final welfare is constant

for all individuals, u∗ = K + ∫ b2

4 dF2.

Richness is violated in this example because if bi1 �= bk

1, then e∗1(bi

1) �= e∗1(bk

1)independently of what happens in the other activity. An individual with bk

1 willnever choose e∗

1(bi1). Policymaker 2 is not equalizing rewards to effort but equality

of opportunity is provided.

EXAMPLE 2. Let u(x1, x2, e) = x1 + x2 − (e1 + e2)2. Let b1, b2 ∈ (0, 1). Supposethere is perfect correlation between b1 and b2, say b1 = b2. Let a1(b1, e1; F1, ϕ1) =K + e1 and a2(b2, e2; F2, ϕ2) = K + e2. If each policymaker sets respectively φ1 =K + e1 if b1 ≤ b and φ1 = 0 otherwise, and φ2 = K + e2 if b2 > b and φ2 = 0otherwise. Each individual will specialize, if b1 ≤ b in activity 1 and in activity 2otherwise. Final welfare will be u∗(b) = K + 1

4 . Here, the environment is forcingrichness to be violated (if b1 > b the individual will never exert effort in activity1) and decentralizing equality of opportunity does not require equalizing rewardsto effort.

Overall, richness requires that there be little correlation between informationin the different sectors in order to limit the capacity of a local planner to behaveas a social planner.

We now present the second result of the paper. It states that equalizing re-wards to effort is not only sufficient but necessary if richness of the mechanismsand the environment holds. We also impose differentiability on the mechanisms,which simplifies the proof of the theorem. This result shows that the objective ofindependent policymakers wanting to contribute to the provision of equality ofopportunity should be equalizing rewards to effort.

THEOREM 2. If the pair of mechanisms and the environment are rich, and if thepair of local mechanisms (φ1, φ2) is differentiable, then it decentralizes equality ofopportunity only if it provides local equality of rewards to effort.

4. CONCLUDING REMARKS

The allocation of rights and resources in a society is done in a decentralizedmanner, with each policymaker deciding how to allocate resources based on partialinformation. This article interprets local distributive justice as decentralized globaldistributive justice. Thinking of local justice as decentralized global justice allows


us to evaluate local interventions on the basis of their success in achieving globaljustice.

We analyze what local justice requires if we want to achieve the global justicegoal of providing equality of opportunity. We show that under some conditions acollection of local mechanisms decentralizes the provision of equality of opportu-nity if and only if each mechanism provides local equality of rewards to effort. Thisimplies that if a society is concerned about providing equality of opportunity, thenits institutions and policymakers should be concerned about equalizing rewardsto effort or equivalently giving the same output to individuals who exert the sameamount of effort. In other words, the incentives individuals are given should notdepend on irrelevant characteristics.

The result of the article may indirectly provide a rationale for the difference be-tween the distributive justice notions that political philosophers propose, i.e., nor-mative distributive justice, and those that people express when voting for welfareprograms or when stating their opinions in surveys or experiments, i.e., positivenotions of distributive justice. Political philosophers overlook the role of effortbecause it is endogenous and ultimately a function of exogenous characteristics.Their debate is centered on what exogenous characteristics should be irrelevant.21

Individuals, when asked in surveys or through experiments, also differ in whichcharacteristics they consider to be irrelevant. However, they mostly agree thateffort exerted of those on assistance programs is crucial to evaluate the fairnessof a given allocation. For example, Bowles and Gintis (1998) report that “the wel-fare state is in trouble not because selfishness is rampant (it is not), but becausemany egalitarian programs no longer evoke, and sometimes offend, deeply heldnotions of fairness. . . . In a 1995 CBS/NYT survey, for example, 89% supportedand mandated work requirement for those on welfare.”22 Yet, Konow (2003) in asurvey on positive distributive justice states “[a] common view is that differencesowing to birth, luck and choice are all unfair and that only differences attributableto effort are fair. A common finding (and claim) among social scientists is thatindividual effort affects the perceived fairness of allocations.” We believe thisdifference between political philosophers and individuals in society may be theresult of philosophers examining global distributive justice whereas individualsalways being asked in the frame of a local distributive justice. If global equality ofopportunity is desirable, then, even though we can ignore effort—or more gener-ally endogenous choice variables—in the global distributive justice problem, wemust control it in the local distributive justice problem. The difference between

21 An example of a discussion on rewarding effort and justice is in Sen’s article “Merit and Justice,” inMeritocracy and Economic Inequality (Arrow et al., 2000), where he states that “an incentive argumentis entirely ‘instrumental’ and does not lead to any notion of intrinsic ‘desert.’ If paying a person moreinduces him or her to produce more desirable results, then an incentive argument may exist for thatperson’s pay being greater. This is an instrumental and contingent justification—it does not assert thatthe person intrinsically ‘deserves’ to get more.” Actions are rewarded for what they help to bringabout, but the rewarding is not valued in itself.

22 Bowles and Gintis (1998) argue that the reason people care about the effort recipients of welfareexert is due to strong reciprocity. “Homo reciprocans is not committed to the abstract goal of equaloutcomes but to a rough balancing out of burdens and rewards.”

288 CALSAMIGLIA

an individual in a survey and a philosopher then is the framework in which theyexpress their opinions. There is not necessarily an inconsistency between what thetwo express when talking about justice.

APPENDIX

PROOF OF THEOREM 1. Every individual maximizes his utility given the socialmechanism generated by the pair of local mechanisms and given the other indi-viduals’ choices. Since the local mechanisms provide local equality of rewards toeffort we know that the social mechanism and therefore the choice problem facedby the individual has the following form:

maxe∈IR2

u((

w1(r1, e1; F1, ϕ

∗1

), w2

(r2, e2; F2, ϕ

∗2

))) − c(e).

The effort level chosen will not depend on b since the utility function does notdepend on b. The problem that two identical individuals face in equilibrium isidentical. Even if the effort chosen is different the utility is still the same sincein equilibrium for every effort level they get the same payoff. If one obtaineda higher payoff that would contradict individuals behaving optimally. Individualfinal welfare is then a function only of the relevant characteristics r:

u∗(b, r) = v(r) = u((

w1(r1, e∗

1(r)), w2

(r2, e∗

2(r))) − c

(e∗

1(r), e∗2(r)

)�

PROOF OF THEOREM 2. After choosing the optimal level of effort the individualwill have a utility of

u∗(b, r) = u(φ1

(b1, r1, e∗

1; F1, ϕ∗1

), φ2

(b2, r2, e∗

2; F2, ϕ∗2

)) − c(e∗

1, e∗2

).

If the pair of local mechanisms provides equality of opportunity, then the indi-rect utility needs to remain unchanged with respect to b, which implies that thefollowing is true:

∀F ∈ � ∀b, r ∈ suppF∂u∗(b, r)

∂b1= ∂u∗(b, r)

∂b2= 0.

Note that since each individual has zero mass, his actions will not change the dis-tribution of efforts in equilibrium.23 Therefore, differentiating the indirect utilitywith respect to bj we obtain the following expressions:

23 Continuity simplifies doing comparative statics on welfare when one individual’s characteristicschange. We analyze the effect on an individual’s welfare of changing his characteristics within the sameequilibrium. If the model had a discrete number of individuals, doing this comparative statics wouldinvolve understanding how the equilibrium correspondence changes with a change in one individual’scharacteristics, which could unnecessarily complicate the model and the analysis.


∂u(φ∗

1 , φ∗2

)∂φ1

(∂φ∗

1

∂b1+ ∂φ∗

1

∂e1

∂e∗1

∂b1

)

+ ∂u(φ∗

1 , φ∗2

)∂φ2

∂φ∗2

∂e2

∂e∗2

∂b1− ∂c(e∗)

∂e1

∂e∗1

∂b1− ∂c(e∗)

∂e2

∂e∗2

∂b1= 0.

Rewriting we obtain,

∂u(φ∗

1 , φ∗2

)∂φ1

∂φ∗1

∂b1+

(∂u

(φ∗

1 , φ∗2

)∂φ1

∂φ∗1

∂e1− ∂c(e∗)

∂e1

)∂e∗

1

∂b1

+(

∂u(φ∗

1 , φ∗2

)∂φ2

∂φ∗2

∂e2− ∂c(e∗)

∂e2

)∂e∗

2

∂b1= 0.

Using the first order conditions of the individual we know that ∂u(φ∗1 ,φ∗

2 )∂φ j

∂φ∗j

∂e j−

∂c(e∗)∂e j

= 0 for j = 1, 2, which implies that ∂u(φ∗1 ,φ∗

2 )∂φ1

∂φ∗1

∂b1= 0. Similarly we can show

that ∂u(φ∗1 ,φ∗

2 )∂φ2

∂φ∗2

∂b2= 0.

Therefore, since u is strictly increasing we have that

∂u(φ∗

1 , φ∗2

)∂φ j

∂φ∗j

∂bj= 0 ⇒ ∂φ j

∂bj(bj , r j , e∗

j (b, r ; F, ϕ∗)) = 0.(A.1)

Take e j ∈ Imgϕ∗j . Let (bj , r j ) ∈ supp Fj be an individual optimally choosing e j .

By (1) we know that for any (bj , r j ) ∈ supp Fj in an open ball around (bj , r j ) thelocal outcome given when effort is e remains unchanged. But we want this to betrue for all (bj , r j ) ∈ supp Fj . By richness we know that for all (bj , r j ) ∈ supp Fj

Imgϕ∗ ⊆ Ej (bj , r j ; F j , ϕ∗j ), which implies that for all (bj , r j ) ∈ supp Fj there exists

F− j and (b− j , r− j ) ∈ supp F− j such that e∗j (bj , r j , b− j , r− j ; Fj , F− j , ϕ

∗j , ϕ

∗j ) = e j .

Therefore local outcome assigned to effort e j should remain unchanged around(bj , r j ) for every bj , and therefore local outcome should be equal when bj

changes. This shows that the local mechanisms provides local equality of rewards toeffort. �

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decentralizing equality of opportunity

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