notions of equal, or equivalent, opportunities

Soc Choice Welfare (1994) 11:137-156

Social Choice dWelfare

© Springer-Verlag 1994

Notions of equal, or equivalent, opportunities*

William Thomson

Department of Economics, University of Rochester, Harkness, Rochester, NY 14627, USA

Received October 23, 1989/Accepted August 12, 1993

Abstract. We formulate and study three concepts of equity designed to capture certain notions of "equal", or "equivalent", opportunities. The central concept is that of a family of choice sets. Given such a family ~ , a feasible allocation z is alternatively required to be such that (i) there is B e ~ such that each agent i maximizes his satisfaction in B at zi, (ii) there is B e G / s u c h that each agent i is indifferent between z i and the maximizer of his satisfaction in B, (iii) for each agent i there is B i e ~ such that z i maximizes agent i ' s satisfaction in the union of the Bj and zi is in B i . Most of the standard concepts of equity can be obtained as particular cases of these general definitions by appropriately choosing ~ . We identify conditions on ~ guaranteeing that the resulting allocations be efficient. We apply the definitions to economies with only private goods, and to economies with public goods.

1. Introduction

In this paper, we formulate three notions of "equal", or "equivalent", opportunities and apply them to private good economies and to public good economies. We also examine their relationship to other notions of equity that are commonly considered.

A solution to the problem of fair allocation often proposed is the equal-income Walrasian solution. Although most writers are not very explicit about their reasons for being in favor it, the fact that such allocations are obtained by everyone

* This is a revised version of an earlier paper circulated under the title of "Notions of equal opportunities". Early drafts were presented at the Conference on "Economic Models and Distributive Justice", held in Bruxelles and Namur, January 1987, and at the Public Choice Society meeting in Tucson, Arizona, March 1987. The author thanks the participants, in particular T. Schwartz, for their comments, and NSF for its support, under grant No. 85 11136 and 88 09822. The suggestions of two anonymous referees, D. Diamantaras, L. Gevers, and H. Konishi are also gratefully acknowledged.

138 W. Thomson

choosing in a common choice set seems to be an important underlying motivation: when all incomes are equal, the Walrasian mechanism can be described as giving agents equal opportunities.

However, there is no reason a priori why concepts of "equal opportunities as equal choice sets" should have to involve Walrasian notions. Our purpose here is to show that concepts of "equal opportunities as equal, or equivalent, choice sets" can indeed be developed independently of, although in a manner compatible with, Walrasian notions. We propose and study three def'mitions.

These definitions are motivated by practical as well as theoretical considerations. First, we note that even in economies where resources are supposed to be allocated via the Walrasian mechanism, i.e. with all agents facing parallel straight line budget sets, a variety of distortions typically occur that create substantial non-linearities : quantity discounts, minimal or maximal purchase requirements, non-linear tax rates, and welfare payments are important examples. Hausman (1985) gives an extensive list of such distortions and describes the effect they have on the shape of budget sets: non-convexities and non-differentiabilities are not rare. A theory of economic justice that can accommodate these departures from the pure Walrasian model would be quite desirable. It is often thought that non- linearities in choice sets necessarily generate inefficiencies; it is one of our ob- jectives to identify the extent to which this is true. Linear choice sets do have the merit of simplicity, however, in addition to underlying the Walrasian idealization of competitive markets, and we will often use them as examples. In addition, they will come up quite naturally on several occasions. When a choice set is interpreted as the set of bundles that can be obtained by operating some technology, they correspond to the case of linear technologies. For such technologies, the activity of one agent has no impact on the options open to the others. This absence of "external" effects makes them a particularly useful benchmark.

There are also good theoretical reasons to be interested in a general, non- Walrasian, theory of equity based on equal choice sets. An extension of revealed preference and demand theories from Walrasian budget sets to arbitrary choice sets has already been successfully carried out (see Richter, 1979, for an abstract treatment). We believe that a much better understanding of the power, but also of the limitations, of the Walrasian paradigm can be obtained by identifying which of its properties hold for such generalizations.

Our point of departure then is a class of choice sets, assumed to be given. At first, we impose no restrictions on the class. This will permit a great generality in our definitions. The emphasis of our study however is on the second step, where for each of our three concepts we characterize the restriction placed on the family of choice sets by requiring that the resulting equitable allocations satisfy certain desirable properties. As we will show, several of these restrictions, in turn, imply that the set of equitable allocations necessarily contains, and in some cases is equal to, the set of equal-income Walrasian allocations.

We now briefly and informally describe the three concepts. First, we require of an allocation that it be obtained by having all agents choose from a common choice set. Second, we require the allocation to be equivalent to one obtained by having all agents choose from a common choice set. Third, we require the allocation to be obtained by having each agent choose from some individualized choice set, and to be such that no agent would prefer choosing from someone else's choice set. Slightly more formally, and given a class of choice sets, first we say that an allocation is an equal opportunity allocation relative to that class if

Notions of equal, or equivalent, opportunities 139

there is a member of the class such that, for each agent, his component of the allocation maximizes his satisfaction in that choice set. Next, we declare an allocation equal opportunity equivalent relative to the class if there is a member of the class such that each agent finds his component of the allocation indifferent to the maximizer(s) of his satisfaction in that choice set. Finally, we say that an allocation exhibits no envy o f opportunities relative to the class if, for each agent, there is a member of the class in which he maximizes his satisfaction at his component of the allocation, and such that he prefers his component of the allocation to any point in anyone else's choice set.

We apply our definitions to standard domains, economies with private goods only, and economies with both private goods and public goods. We establish several existence and non-existence results and we clarify the relationship of these concepts not only to the Walrasian solution but also to some other notions that are commonly discussed in the theory of fair allocation.

Our generalizations of the Walrasian concepts are indeed also inspired by the concept that has arguably met with the greatest success in the analysis of distri- butional questions, namely that of an envy-free allocation (an allocation such that no agent would prefer switching bundles with anyone else), and to the concept of an egalitarian-equivalent allocation (an allocation such that all agents find their bundles indifferent to a common reference bundle). The latter concept has played a fundamental role in some recent literature.

By taking the notion of a family of choice sets as a starting point, we are therefore able to unify much of the existing literature. Moreover, we will see that several new concepts, recently introduced and given axiomatic characterizations, fall out as particular cases of our general definitions.

2. Preliminaries

First, we introduce notation, we define the concepts that have been central to the recent literature on equity, and we briefly state their main properties in private good economies and in public good economies. 1

We start with private good economies. There are g commodities and n agents. The set of agents is denoted by N, with generic element i. Each agent i ~ N is equipped with a continuous preference relation (a complete, reflexive and tran- sitive preorder) defined over IRe_, denoted by R i. Let Pi denote the strict preference relation associated with R i and Ii the indifference relation. Given zi~lRe+, L(Ri ,z i)~_IRe+ is the set of consumptions to which agent i, with preference relation R~, weakly prefers z~. We take as given the aggregate endowment f2 ~ IR~, and the aggregate production possibilities Y___ IRe, assuming away differences in productivities across agents. Therefore, an economy can simply be described by a list of preference relations R = (R 1 .... , Rn). For simplicity, we only consider domains of preferences that are strictly monotone in IR~ ÷ (i.e. for all zi, z[ ~ IR~+ +, z i ~ z[. = z i P iz[ ).2 We denote by ~ c the domain of classical preferences, that is, economies where in addition, preferences are convex, and by

1 More detailed reviews and references to the relevant papers are given in Thomson and Varian (1985) and Thomson (1993). 2 Vector inequalities: x ~ y means x k >_ y~ for all k; x _> y means x __> y and x ~a y; x > y means x~ > yx for all k.

140 W. Thomson

L the domain of linear preferences, that is, preferences that can be given linear numerical representations.

Let Z be the set of feasible allocations : for exchange economies, (i.e. if Y= { 0}), Z =- {z ~ IR~+n 127z i = f2 }; for production economies, Z - { z ~ IR ~_n ] Zz i _ g2 e Y}. Let Zi be the projection of Z onto agent i 's consumption space. We denote by P ( R ) the set of Pareto-efficient allocations o f R: P ( R ) = { z ~ Z[ there does not exist z" ~ Z such that z; R~ z i for all i e N, strict preference holding for at least one i ~ N}. An equal division allocation is one where all agents consume the same bundle; this bundle may have been obtained by operating the technology Y. For exchange economies, there is a unique such allocation: it is a5 = (f2/n,. . . , O/n). In a production economy, there are many such allocations, of the form ((g2 +y) /n ... . . (£2 +y) /n) where y ~ Y. The set of allocations that are individually rational from equal division for R is I (R ) = { z e Z [ z~ R~ (Zz~ /n) V i }. Note that this is not the standard definition.

We will also consider economies with public goods, using the following standard model: one of the g goods is a private good and the remaining g - 1 goods are public goods. Initially, an amount ~2 E IR+ of the private good is available and there is none of the public goods. Here, the feasible set is Z - {(xl .... , xn ' y) ~ IR%+ e- l [ (27x i _ g2, y) ~ Y}, where x i E IR+ denote agent i 's consumption of the private good and y ~ IR~-1 denotes the vector of public good levels.

We denote by A e-1 the ( g - 1)-dimensional simplex. In private good economies, the following solution will play a central role:

Definition. The allocation z = (z I . . . . . z~) ~ Z is an equal-income Walrasian allocation for R if there exists p ~ A e- 1 such that,

(i) for all i ~ N , and for all z; tire+ satisfying pz; <=pg2/n+A/n, zgRiz;, where (ii) A = m a x { p y [ y ~ Y}.

Let W(R) be the set of these allocations. We refer to W as the equal-income Walrasian correspondence.

In public good economies, we will often consider the following solution:

Definition. The allocation z = (x 1 .... , x~, y) ~ Z is a Lindahl allocation for R relative to the profile of endowments co = (o~1,..., a~) ~ IR~_ " and profile of profit shares 0 = (01 .. . . . 0 n) ~ A " - 1 if there exists zc = (n 1,..., ztn) e A (e-l)" such that

(i) for all i ~ N , and for all (x; ,y ' )~IR~+ satisfying x;+zr~y'<=a~i+OgA , (x i, y) R i (x: , y ' ) where (ii) A = m a x { x ' +(271re)y' [(x',y')~ Y}. The following three concepts are central to the literature on fair allocation:

Definition. The allocation z e Z satifies no-domination if for all i , j ~ N , Z i ~ Zj.

Definition (Foley 1967). The allocation z ~ Z is envy-free for R if for all i , j ~ N, z~ R~ zj. Let F(R) be the set of these allocations.

Definition (Pazner and Schmeidler 1978). The allocation z ~ Z is egalitarian- equivalent for R if there exists some reference bundle z 0 ~ IR e such that for all i ~ N, zl I~ z o. Let E~- (R) be the set of these allocations.

\


z* R1

O~ 0 2 Fig. 1

The intersection of two correspondences is denoted by juxtaposing the symbols representing each of them: to illustrate, given preferences R, we write FP(R)= F(R)nP(R).

Assume that preferences are strictly monotone in IR~_. Then, no-envy implies no-domination, and in public good economies with a single private good, no- envy is equivalent to no-domination.

In exchange economies, envy-free and efficient allocations exist under standard assumptions on preferences :3 any equal-income Walrasian allocation enjoys both properties. In economies with production however, there may be no such allocations when agents have different productivities (for that reason, we have ex- cluded that case from consideration). In exchange economies, an allocation may be envy-free and efficient without Pareto-dominating equal division. In two- person classical exchange economies all allocations Pareto-dominating equal division are envy-free. Egalitarian-equivalent and efficient allocations exist very generally.

The situation with public goods is quite different. We will illustrate some of the differences in the case of one private good and one public good. Assume that the public good is produced according to a linear technology, and choose the units of measurement of the goods so that one unit of the input yields one unit of the output. Using the Kolm triangle representation, we find that the set of envy-free allocations is the vertical segment through the top vertex. In general, this segment intersects the efficient set at a finite number of points. In Fig. 1, this intersection is a singleton denoted z*. Any one of the points of the segment is an equal bundle allocation. Since it is based on individualized prices, by operating the Lindahl correspondence, (which from a number of viewpoints, is the natural counterpart for public good economies of the Walrasian correspon-

3 The most general existence results are due to Varian (1974), Svensson (1983), and Diamantaras (1992). None of these authors requires convexity of preferences. Instead, they impose assumptions on the shape of the set of efficient allocations.

142 W. Thomson

dence), from an arbitrary equal bundle allocation, 4 it is clear that we do not in general reach an envy-free allocation: in Fig. 1, o) is an equal bundle allocation, but z, obtained by operating the Lindahl correspondence from co, is not envy- free.

It is also worth noting that the Lindahl correspondence does not necessarily treat identical agents identically, in contrast with the Walrasian correspondence. This was pointed out by Champsaur (1976) who showed that nevertheless, under standard assumptions, there always are Lindahl allocations at which identical agents are treated identically. Finally, given an arbitrary point of equal division co, the set of envy-free and efficient allocations almost never contains the set of efficient allocations Pareto-dominating co. In Fig. 1, this would happen only for CO = Z * . 5

Although the concept of egalitarian-equivalence can be criticized for its failure to meet the no-domination requirement (see Thomson 1993, for a discussion of this point), it is nevertheless an extremely useful one. Selections from the egalitarian-equivalent correspondence were recently found to respond well to changes in resources, technologies, or number of participants. In particular, for economies with pun ic goods, a selection from the egalitarian-equivalent and efficient correspondence, first identified by Mas-Colell (1980b), was shown by Moulin (1987b) to be the only solution to satisfy a condition of this type (together with some other but minor conditions). Other selections are found by Thomson (1987b) to satisfy other appealing monotonicity requirements.

3. Equal opportunities

Our first notion, that of equal opportunities, is briefly discussed in Thomson and Varian (1985). It simply and directly says that all agents should face the same choice set, as suggested by Kolm (1972) and probably many others; whatever differences exist between final bundles can then be entirely attributed to differences in tastes. It is noted in Thomson and Varian that this set cannot be specified once and for all since the various choices made from it by the agents typically will not be compatible. Let us imagine instead that we have access to a whole family of choice sets, 2 , where each B ~ 2 is a non-empty subset of lR+. If 3 is rich enough, then for each economy, compatibility of choices will hold for some B ~ , ~ .

Definition. The allocation z ~ Z is an equal opportunity allocation relative to the family ~ for R if there exists B e ~ such that for each i ~ N, z i maximizes R i in B. Let O ~ (R) be the set of these allocations.

4 In the constant-returns-to-scale case under examination, there is no need to specify profit shares since at equilibrium profits are zero. s Recently, Sato (1987) proposed an alternative definition of equity for economies with public goods. According to this definition, the results given above for exchange economies do have counterparts in which the Lindahl correspondence plays the role of the Walrasian correspondence. However, Sato's concept may not be as natural as Foley's original one. Sato (1985) also proposed a definition of "fairness in terms of consumer surplus" for economies with a single private good. This definition coincides with egalitarian-equivalence (see below) of net trades when the reference bundle is required to be proportional to the unit vector relative to the private good.


It follows directly from this definition that O ~ is a subsolution of the no- envy solution. Also, in exchange economies, if (2In ~ B for all B ~ ~¢2, then O ~ is a subsolution of the solution that associates with each economy its set of allocations that Pareto-dominate the point of equal division.

These observations give us a first indication of the possible usefulness of our approach: our first concept allows us to perform selections from the no-envy solution. 6 As simple examples show, the set of envy-free and efficient allocations is sometimes very large. In such cases, one would like to be able to recommend selections from it. If Pareto-domination of equal division is found desirable, and indeed, a number of writers have argued in favour of this requirement, then the criterion proposed here will help.

A natural condition to impose on ~ is that for each admissible economy R, the set O~, (R) be non-empty and that moreover, each of its points be an efficient allocation. Although one may conjecture on the basis of our experience with Walrasian budget sets when distorted by taxes, quantity constraints, etc., as described in the introduction, that this may be difficult to achieve, investigating the extent to which efficiency is indeed possible is worth undertaking. The following paragraphs provide some answers. Let us formally introduce the requirement:

(c0 for each R ~ ~ " , 0 ~ O ~ (R)c_ P (R) .7

Let ~ be the family of "equal-income budget sets", hereafter called the Wal- rasian family" Given p e A e-1, let y ~ IR~_ be a maximizer of py' in y ' ~ Y, Wp - {Zo e IR ~ I pZo = p (f~ + y)/n}, and 7 f = { Wp [p E A e- 1}. Under standard assumptions on preferences and technologies, ~ satifies (e). Recall that the notation W designates the correspondence that associates with each economy R the set of equal-income Walrasian allocations for R. Obviously we have O ~ = W, an equality that solves the question of existence of families satisfying (e) on classical domains. Are there other such families? The answer is yes:

Example. Consider the domain of 2-person classical economies. Let ~ .~ - ~ u { K } , where K is the choice set depicted in Fig. 2: K is piece-wise linear and K n {z 0 ~ IR ~ [ z o < O} is symmetric with respect to equal division.

If two points z 1 and z 2 of K add up to g?, and are maximizers over K of R 1 and R2 respectively, then z -= (zl, z2) e P (R) (whether or not preferences are convex). For all R, O~(R)~_ W(R), and for some economies, such as the one depicted in Fig. 2, the containment is strict. (A possible interpretation of the shape of K: by having the price of the good that is measured on the vertical axis be relatively higher for small and for large quantities, one encourages intermediate consumptions. This may be a desirable social objective for some goods). The same kind of example could be constructed for an arbitrary number of commodities. Further generalizations are described later on.

Are there other, perhaps less artificial, examples of families 2 satisfying (c~)? Is it possible to characterize all families satisfying (00 or at least all the allocations obtainable in this way on some useful domains? These seem worthwhile questions to ask. Although we do not have complete answers, we show next that under fairly natural additional assumptions on the richness of the family of choice sets,

6 For another approach to the selection problem, see Diamantaras and Thomson (1990). 7 A weaker requirement is that for each R ~ ~n, Os ~ P(R)~g

144 W. Thomson

R2 Z2 f~

• R1 Fig. 2

the equal-income Walrasian allocations necessarily result as equal opportunity allocations. First, we note that the concept proposed in this section (this will also be true of the one proposed in Sect. 5), satisfies an interesting property that is far from being always satisfied, and this first piece of information will be quite useful in helping us understand the implications of our proposal.

Definition. The solution ~0: ~n - - - ,Z is weakly monotonic if for all R, R ' ~ ~ n , for all z ~ o ( R ) , i f L ( R ; , z i ) ~ _ L ( R i , z i ) for all i ~ N , then z ~ o ( R ' ) .

This say that if z is "~o-optimal" for some economy and preferences are then changed in such a way that for all i ~ N, zi does not fall in agent i ' s estimation relative to any other consumption, then z remains (p-optimal.

Weak monotonicity is a variant introduced by Gevers (1985) and also used by Nagahisa (1990, 1991) of a condition whose importance for the theory of implementation was discovered by Maskin (1977). Maskin 's monotonicity differs only in that the inclusion L ( R / , z i ) ~ L ( R i , ze) is replaced by the inclusion L ( R ; , z i ) n Z i D _ L ( R i , z i ) m Z i . 8

There is a close relation between weakly monotonic correspondences and the Walrasian correspondence, which, as easily checked, is weakly monotonic. This relationship is described in the following Theorem, which should be compared to a result due to Hurwicz (1979). We omit its straightforward proof. 9

Theorem 1. Let ~o : ~ Z be such that

( i) ~ . ~ L , (ii) ~o is weakly monotonic,

( iii ) i f R ~ ~ Ln and z ~ Z satisfies z i 1 i Zo for all i ~ N and for some z o e IRe+ such that (z o . . . . . zo) ~ P ( R ), then z e ¢ ( R ). Then ~o ~ W.

8 Alternatively, we could have worked on the domain of preferences that are continuous, convex, and strictly monotone in ~,% +, and such that any consumption with a zero coordinate is indifferent to the zero consumption. Then, under the assumption that each choice set B ~ is convex and contains an interior point, the correspondence O~ would be Maskin-monotonic. Allowing indifference curves that are transversal to the axes, or non-convex preferences, or non-convex B, prevents Maskin-monotonicity of O~. 9 It appears in this form, but with monotonicity in the sense of Maskin and specialized to the case of exchange economies, in Thomson (1987a). See also Gevers (1985).


We apply this result to ~0 - O~,. Condition (i) is a natural domain assumption. We claimed earlier that O ~ satifies (ii) and this is easy to check. Finally, it is very natural to require that if an equal bundle allocation is efficient, then it, as well as any feasible allocation that is Pareto-indifferent to it, be declared equitable. This property is satisfied by virtually all existing preference-based solutions to the problem of fair allocation. Requirement (iii) is even weaker since its application is restricted to the very special case in which agents have linear preferences. Nevertheless, together with the other two conditions stated above it has strong implications for the richness of the family O ~ . Indeed, Theorem 1 states that O ~ contains the equal-income Walrasian correspondence.

The same inclusion can be obtained on the basis of other considerations, involving the requirement that each point of each choice set be part of a list of choices defining a feasible allocation, together with some regularity assumptions on the shape of the choice set] °

Theorem 1 identifies circumstances in which the equal-income Walrasian allocations cannot be avoided, but they do not tell us all that can be accomplished, and this remains an open question. To show that the notion of equal opportunities studied here can indeed lead to other allocations, even under the assumptions of Theorem 1, we exhibit a family ~ such that, for n = 2, O ~ is in fact equal to the individually rational from equal division and efficient solution. This family generalizes the example K seen earlier.

Lemma 1. On the domain of 2-person classical economies there exists a family 32 such that O ~ = IP. 11

Proof For each p , p ' e A e - 1 , and for all d e ] R + , let B ( p , p ' , d ) { ze lRe+ Iz<=f2, pz=pf2 /2 , p ' f 2 / 2 - d < p 'z < p' f2/2 + d } u { z e lRe+ [z<=Y2, p ' z = p ' f 2 / 2 + d , pz<_pf2/2}u{zelRe+ [z<_f2, p ' z = p ' t ' 2 / 2 - d , pz>=p£2/2}. Let ~.~={B(p,p' ,d)-(p,p' e A e- 1, d e 1R +}.

It is clear that O ~ _~ I since each choice set in ~ contains t2/2. Also, O ~ __. P by the argument made in the discussion of K. Conversely, given z ~ IP(R), let f i e A e-1 be a supporting price. IflSz 1 =fit2/2, set p = p ' =/7 and let d be arbitrary. If /~z 1 > /7 f2 /2 (the case/~zl < 15f2/2 would be treated analogously), let p" = p and p be such that for all z~ e 1Re__ with z~ R2(t2/2), pz~ >=pt2/2. This is possible because z 2 R 2 0 / 2 . Then, let d be such that p ' z~ = p ' Q / 2 + d. One can easily check that R 1 and R 2 a r e maximized on B(p, p', d) at zl and z 2 respectively. Q.E.D.

We note that Mas-Colell's (1992) generalization of the Walrasian solution designed to accommodate economies with possibly satiated preferences, when applied to the problem of fair division, can be seen as an instance of the general notion developed here. In that application, the budget sets have a straight-line boundary but they do not necessarily pass through the point of equal division. However, all agents are required to maximize over a common choice set.

We now turn to public good economies. Things are much less satisfactory here, as indicated by the following negative result, which is proved by way of a very simple example with only 2 goods and 2 agents, and a linear technology.

lo We omit the formal statement and the proof of this claim, the regularity conditions being somewhat technical. The early version of this paper contains the details. 11 This result does not extend to n > 2 since in that case it is not true that I~Fand yet, as we saw earlier, we necessarily have O~ _~F.

146 W. Thomson

Theorem 2. Let g = 2 and n = 2, and suppose that the technology is linear. There & no family ~ such that for all R e 5~ c", 0--/: O ~ (R)c_P(R).

Proof Let R e 5~ c~ be an economy such that FP(R) contains a single point z = (Xo, x o, y) and the two indifference curves through z 0 = (x 0, y) have unique but not identical lines of support at that point. Since O ~ c_F, requiring 0--/: O ~ (R)c_ P(R) implies that z e O ~ (R). Let B be a member of ~ ration- alizing z as an element of O ~ (R). Since both agents maximize their preferences on B at z 0, B has a kink there. Since z e P (R), the marginal rates of substitution at z 0 add up to the marginal rate of transformation at the corresponding production point (the Samuelson condition). Then, let R ' =(R2,R2). Clearly, z e O ~ (R ' ) (the same B e ~ can serve to rationalize z as a member of O ~ (R')) , but z¢ P(R ' ) since the Samuelson condition does not hold any- more. Q.E.D.

4. Equal-opportunity equivalence

Our next definition can be seen as generalizing the definition of the previous section but it is also inspired by the notion of egalitarian-equivalence and the idea, underlying this notion, of evaluating an allocation by comparing it to allocations that are not necessarily feasible (indeed the list (z 0 .. . . . zo), where z o is the reference consumption in the definition of egalitarian-equivalence, is not in general a feasible allocation). This fundamental idea can be applied in other ways. For instance, consider the following definition:

Definition (Pazner 1977). The allocation z e Z is envy-free equivalent for R if there is z ' = ( z ~ , . . . , z ' ) e IR~- n such that z; R~zj for all i , j e N and ziIiz" for all i ff N. (Here, z ' is not required to be in Z.) Let F -~ (R) be the set of these allocations.

Our proposal here is to combine the idea of equal opportunities with that of egalitarian-equivalence.

Def'mition. The allocation z e Z is equal-opportunity equivalent relative to the family ~ for R if there exists B ~ ~ such that for each i e AT, zl is indifferent for agent i to the maximizer of Ri on B. Let O ) ( R ) be the set of these allocations.

Note that for all families of choice sets ~ , F = _ O ) _~ 0 ~ . The correspondence O ) is not in general weakly monotonic, but it has that property for some families 2 . 1 2 Several of the standard equity concepts can be derived from the above definition by appropriately choosing ~ . For instance, if ~ = {{Zo} I z o ~ IRe_ }, then O ~ = E ~-, and if ~ -- {{Zl,..., z=} I (zl , . . . , z,) e IR?}, then O)=F~-. Also, we have:

Lemma 2. If ~ = ~ , then O ) = W.

Proof Let z e O ~ ( R ) be given: there is p e a e-1 such that for all i e N , z e/, zg* where zg* maximizes R i in { z" e IRe+ [pz/<= p (g2 + y)/n with p y ~ py" for all y ' e Y}. We claim that pz~ =pz* =p (~2 + y)/n for all i e N. First, note the obvious fact that pz~ >__pz* for all i e N. Supposing by contradiction that for

12 If ~ = ~" (see Lemma 2) and under the assumptions detailed in footnote 10, it is in fact monotonic.

Notions of equal, or equivalent, opportunities

L z3

R~

z 0

R3

o ~ / 3

2 R2

147

a b

Fig. 3

some i ~ N, pz i > pz*, we obtain pXzi : X p z i > Xpz* = p ( f 2 + y ) , which con- tradicts the feasibility of z. Therefore O ~ c_ W. The inclusion W _ O ~ is straightforward. Q.E.D.

Let 2 be the family of linear choice sets: L ~ 2 if and only if there is p e A e- 1 a n d a ~ I R + s.t. L={xelRe+ I P x = ~ } .

Lemma 3. I f n = 2 , O ~ = E - - . 13 I f n > 2, there is no necessary containment between 07~ P and E'- P, even on the domain of classical exchange economies.

Proof The first statement follows from the fact that z = (zl, z2) e E-- (R) if and only if the two indifference surfaces through zt and z 2 intersect, and this is equivalent to saying that they have a common hyperplane of support; the intersection of this hyperplane with IR + can serve as choice set L e 2 to show that z ~ O~(R) . The second statement is established by the examples of exchange economies of Fig. 3. In Fig. 3a, z e O ~ P ( R ) (in fact, z e W ( R ) ) but z¢ E--(R) since the three indifference curves through the three consumptions Zl, z 2, and z 3 have no point in common. In Fig. 3b, z e E - - P ( R ) but z ¢ O ~ ( R ) since the only common line of support to agents 1 and 2's indifference curves through z 1 and z 2 is not a line of support to agent 3's indifference curve through z 3. Q.E.D.

I f Y is convex and we are interested only in efficient allocations, we can limit ourselves to choice sets B satisfying the following condition:

(i) There exists z0 e B that is not in the relative interior of ( Y + {f2})/n. To see this, suppose that B _ r e l . int. {(Y+{f2})/n}. Let z e Z be such that

for all i ~ N, z e/,. z* where z* maximizes R~ on B. For all e > 0 and for all i e N , ( z * + e ( 1 .... ,1))P~z~ and by convexity of Y, for e small enough, Z ' z * + n e ( 1 .... , 1 ) ~ Y+{O}: z is Pareto-dominated by the feasible allocation ( z * + e (1,.. . , 1 n ))i=1. Therefore, zCP(R) .

13 Note that if n = 2, the inclusion O ) _ E = holds for any family ~ ' of convex choice sets.

148 W. Thomson

Also, if ~ ~ ~ (this time, whether or not Y is convex), and if only efficient allocations are desired, we should impose on each L e ~ the requirement that (ii) There exists Yo e (Y+{g2})/nc~iR~+ that is on or above L.

Indeed, suppose that every Y0 e (Y+ {f~})/n is below L. Then, designating by p e A e - I the normal to L, we find that if z e Z is such that for each i e N , z f l iz* where z* maximizes R i in L, then pzi>=pz* >PYo for all Yo e (Y+{f2})/n. Therefore, p Z z i =Xpzi > npy o for all Yo e (Y+{f2}) /n or pY, z~>py for all y e Y+{O}, so that zCZ.

Let ~ ' be the subfamily of 2 satisfying (i) and (ii). It is easy to construct examples showing that the no-domination condition

can be violated by allocations in O ) ( R ) , even if 3 = 2 ' . The next Theorem gives a necessary and sufficient condition on ~ for this not to happen on the class of economies with convex technologies: once again, the only allocations that remain admissible are the equal-income Walrasian allocations!

Theorem 4. Suppose Y is convex. A necessary and sufficient condition for a subfamily ~ of 2 ' to be such that for all R e ~ c , and for all z e O ) (R ), z satisfies no-domination, is that ~ c g f . (Then, in fact, O ) c W)).

Proof Let L e 2 ' \ g f be given. Let p e A e- 1 be a normal vector to L. Let zoeL \ re l , int{(Y+{~?})/n }. By conditions (i) defining 2 ' , such z 0 exists. Let y be a maximizer of {py' lY' e (Y+{~}) /n} . By conditions (ii) defining 2 ' , p y > = p z o. If p y = p z o, L e g/Y and we are done. Otherwise, there exists p" E A e- 1 such that p ' z o > p 'y and y ' e ( Y + {f~})/n) n IR + + such that p ' z o > p 'y ' . Then, for e > 0 but small enough, the points z 1 ( e ) = y ' - ( 1 .... ,1 )e

and z2 (e )=y ' +(1 ... . . 1)e lie above L and p'zo>=p'z2(e ). Now, let R e ~ c , be an economy such that (a) each agent has an indifference surface tangent to L at Zo; (b) agents 1 and 2's indifference surfaces through Zo pass through z 1 (e) and Zz(e ) respectively; (c) for all k > 2, agent k's indifference surface through z o passes through y; (d) finally, these indifference surfaces have p" as a supporting price at z 1 (e), z2(e ), and y respectively. 14 Note that z e O~) (R) and yet since z 2 (e) > z 1 (e), z violates no-domination.

Conversely, if ~ c 7f , then O ) ~_ W and since W satisfies no-domination, we are done. Q.E.D.

Here is a useful example of a subfamily of 2 whose associated equity notion is non-empty: the subfamily of choice sets normal to a given price vector. Formally, given p e int{A e- 1}, let 2 (p) be the subfamily of c o of all linear choice sets normal to p.

Theorem 5. The solution O~.(p) P is well-defined for all p ~ int {A ~- 1}.

Proof is Index the family 2 (p) by "incomes": 2 (p) = {L (p, I) [I e IR + }, where L ( p , I ) - { Z o e IRe+ ipZo<=i}" For each i t N, let u~ : IRe ~ I R be a continuous numerical representation of R~. For each l e IR +, let v; (I)-= max{u~ (zi) I z; e L (p, I)}. As I varies from 0 to o0, the vector v (I) = (v~ (I) . . . . . v, (I)) traces out

a continuous monotone path in utility space. Let 6 be the intersection of the path with the boundary of the compact set u (Z)-= {x ~ IR = I~ z e Z s.t. V i ~ N,

~4 We omit the tedious analytical expressions of utility functions achieving this. 15 The proof is essentially the same as the standard proof that egalitarian-equivalent and efficient allocations exist for a reference consumption proportional to a given bundle (Pazner and Schmeidler 1978).


ui(zi)=xi}, the image of the set of feasible allocations Z under the list of representations u=(u~)i~u. This intersection exists uniquely. Finally, let ~o(R) - { zeZ[u( z )=5} . It is clear that 0¢~0 =O~(p)P. Q.E.D.

We should note that each solution O ~ (p) has the advantage of being "essentially single-valued" (that is, if z, z ' ~ O ~ (p) (R), then zi I i z / for all i ~ N). Theorem 5, (as well as Theorem 6 below), can be generalized by observing that the only relevant properties of the family ~ (p) that matter in its proof are shared by any family ~.~ ={B(2 ) [2 e IR+} such that

(i) B(0) ={0} (ii) for all r > 0, there exists 2 such that B(2)___{ze IRe I ]]z[I ~r}

(iii) for all 2 , 2 ' e ]R + if 2 ~ 2 ", then B (2) ___ B (2 ' ) (iv) B( . ) is a continuous correspondence.

Depending upon the assumptions made on preferences and the technology, the non-emptiness of O ) can be achieved in other ways, in particular by considering families obtained by first subjecting the existing technology to simple transformations, which can be interpreted as reflecting a lower or a greater productivity of some factor, or the introduction of a fixed cost, or of a fixed "bonus". Here are a few examples that may prove useful.

Given Y___IRe+ and 2 > 0, let

T~ ( Y,2 )=-{y ~ IRe]S y" e Ys.t. y~= y/~ if y~__<O and y~ = 2y~ otherwise} ,

T2(Y, 2)--~{yelRel3y' e Ys.t. yk=y~/2 i f y f =<0

and Yk = Y[ otherwise} ,

T3(Y, 2)=-{yelRe[3y" e Ys.t. y = 2 y ' } .

Given 2 ~ IR, and the choice of a good k, let

T4(Y, 2 ,k )=-{ye lRe[qy ' e Ys.t. y=y" +2e~, where ek is the k th unit vector} .

The technology T~ ( Y, 2) is obtained from Yby making each input combination 2-times more productive. The technology T 2 (Y, 2) results when output combination can be produced by 1/2-times any input combination that produced it in Y. The technology T 3 (Y, 2) is simply a radial expansion of Y of factor 2. To obtain T 4 (IT, 2, k) when 2 < 0, think of a fixed cost of 2 units of the k th good being added to each input-output combination; if the k th good was an input this operation means that more of that input will be required, in conjunction with the same quantities of the other inputs, to produce the same output vector; if it was an output, the operation means that less of that output will be obtained from the same input vector, or that the k th good will become an input. The case 2 > 0 can be interpreted in a symmetric way.

Let us consider the family associated with the first transformation T~. Again, for each i ff N, let u/: IR~_ ~ I R be a continuous numerical representation of R i.

150 W. Thomson

\

R1

Z

I'

i I

0 )

Fig. 4 01 ~' 0 2

Given X > 0, let v, (X) = argmax { u; (x,) ] x, ~ T~ ( I1, X) + { ~/n}}. Under natural assumptions on Y and preferences that we will not explore in detail, the point v(2)=-(Vl(~.) .... ,v,(A)) moves continuously along a monotone path; also, v(O) e u(Z) and for A large enough, v(,t)¢u(Z). Therefore, by the same argument as in the proof of Theorem5, we obtain that O~P(R)q=O for 2 - { T~ ( Y, ,t) + { ~/n} I 2 e IR + }. The same reasoning applies to the other examples of transformations although the assumptions under which non-emptiness of O ~ (R) is guaranteed will differ from one to the other. It is important to emphasize here that the families so defined involve the technology Y actually available in the economy.

We now turn to public good economies. The notion of equal opportunity equivalence is perfectly well-defined in such economies, and for natural families 2 , O):/=0. Before stating existence results however, we first explain how to identify the set of egalitarian-equivalent and efficient allocations in the Kolm triangle. As for private good economies, this concept will fall out as a particular case. Let z e P(R) be given. To see whether z ~ E-- (R), take the symmetric image of I, agent l 's indifference curve through zl, with respect to the equal division line, (the vertical line through the top vertex of the triangle). This is the dashed curve labeled g (I) in Fig. 4. If rc (I) and agent 2's indifference curve through z 2 have a point in common, as is the case in Fig. 4 (the point %), then z ~ E--(R). Note that in that case, these two curves have a common tangency line (the line L) which can serve to rationalize z as an element of O~(R). Conversely, if z ~ O ) (R), that is, if the two curves have a common tangency line, then they intersect and any point of intersection can serve as reference point to rationalize z as an element of E-~ (R). (This observation actually provides the proof of the first statement of Lemma 4 below.)

The set E~-P(R) is connected. Its end-points are obtained by finding two indifference curves such that the symmetric image of one is contained in, and tangent to, the other.

As for private good economies, the correspondences E--- and F-- can be obtained by appropriately choosing ~ . We also have the following counterpart of Lemma 3.

Notions of equal, or equivalent, opportunities

z~

z~

z~

z 1

R1

a

Fig. 5

151

z~ Zo

1

Lemma 4. I f n = 2, O~,= E-~. 16 I f n > 2, there is no necessary containment between O ) P (R ) and E--- P ( R ) even i f the technology is a constant-returns-to-scale technology and R e ~,~ c~

Proof The proof of the first statement is similar to that of the first statement of Lemma 5 (also see the discussion of Fig. 4). The proof of the second statement is given by the examples of Fig. 5. In each case, let Y be a constant-returns-to- scale technology with a marginal rate of transformation equal to the sum of the three agents' marginal rates of substitution at zt, z 2, and z3, and let s9 e 1R+ be such that z be feasible for the resulting economy. Then, z e P(R) . In Fig. 5a, z e O ) P (R) but z ¢ E-- (R) since the three indifference curves through zl, zz, and z 3 do not intersect at a common point. In Fig. 5b, z e E-- P ( R ) but z¢ O ~ ( R ) since the unique common line of support to agent 1 and 3's indifference curves through zl and z 3 is not a line of support to agent 2's indifference curve through z 2. Q.E.D.

On the other hand, the equality O ~ = W that we obtained earlier for exchange economies has no counterpart: although a Lindahl allocation from some point of equal division may be in O~,(R), as illustrated by point z of Fig. 4, (z can be obtained by operating the Lindahl mechanism from a)), not all Lindahl allocations from some point of equal division need be in the set. This is illustrated by the point z ' which is reached by operating the Lindahl mechanism from oJ'. Indeed, the symmetric image zr ( I ' ) with respect to the equal division line of I ' , agent l 's indifference curve through z~, does not intersect agent 2's indifference curve through z~.

We do have, however, a counterpart of the existence theorem for a class of economies often considered, namely economies in which indifference curves are asymptotic to the axes. Given the family ~ ( p ) of linear choice sets normal to the price p e A e- 2, an argument similar to that made in the proof of Theorem 5 shows that O~(p)P ~ 0. Of course, the allocations in the set will accidentally be envy-free.

16 Again, this holds for any O) when ~ is a convex family.

152 W. Thomson

Theorem 6. Consider the class o f economies with strictly monotone preferences in ~e+ + and such that for all i e N and for all z i ~ IRe+, ziIiO if zie. = 0 for some e ' , and Y is convex. Then, O~,(p) P--/: 0 for all p ~ A e- 1

The proof is like that of Theorem 5. The restrictions on preferences are im- posed to ensure that the image of the feasible set in utility space is strictly comprehensive (i.e. if d e u (Z) , and u(O)<_d'<_d, then d ' ~ i n t { u ( Z ) } ) . There- fore, the intersection of the path with the boundary of the feasible set in utility space is the image of an efficient allocation.

Theorem 6 can be generalized in the same manner Theorem 5 was. We conclude this section by illustrating with the help of one more example

the compatibility of our approach with other approaches. Mas-Colell (1980a) and Moulin (1987a) propose to select allocations z satisfying the following property: there is a constant-returns-to-scale technology such that each agent i ~ N is indifferent between zi and the best consumption that he could achieve if he had access to the technology, given his endowment. This constant-returns-to-scale equivalent solution is also considered by Moulin and Roemer (1989) and Roemer and Silvestre (1987). Here, we assume that agents have collective ownership of the economy's resources, instead of each being endowed with some vector of goods, but otherwise, this notion coincides with the notion of equal opportunity equivalence relative to the family of linear choice sets passing through a given point (here g2/n). I t is of interest that Moulin arrived at this solution via the axiomatic route, whereas our proposals are based on intuitive considerations of equity.

5. No envy of opportunities

We close with the formation of another concept, which generalizes a definition proposed by Varian (1976) 17 and further studied by Archibald and Donaldson (1979). Here, agents may have different choice sets but no agent would rather have access to the choice set of any other agent.

Definition. The allocation z ~ Z exhibits no envy of opportunities relative to the family ~ for R if for each i e N, there is B i ~ ~ such that z i maximizes R i on B i and for no i , j ~ N, agent i prefers any point of B: to z z. Let G~, (R) be the set of these allocations.

Suppose that z ~ Z is such that for each i ~ N, agent i maximizes R i in some Bi ~ ~ but for a pair {i , j} , BF_B :. This situation, which is considered by Ar- chibald and Donaldson when the choice sets are the budget sets defined by the supporting hyperplanes to the agents' indifference surfaces at their respective consumptions, is called by them one of "strong inequality". Then, (if in fact, B F _ int{B:},) the existence of envy of opportunities can be inferred simply on the basis of monotonicity of preferences.

~7 Varian's definition applies only to allocations in P (R) and iniplicitly assumes the Walrasian correspondence to be operated. Given an allocation z ~ P(R), imagine that each agent i ~ N is able to trade from z i at the prices supporting z. The set of allocations so obtained is defined to be his opportunity set. Say that an allocation z ~ P(R) is opportunity fair if no agent i ~ N prefers any point of the opportunity set of any other agent to his own consumption z i.


R 2

z 2

a

R1

z; R 2

L2

Z1

Fig. 6

It is clear that for all ~ ' , O ~ c_ G~, __. F, and that for all ~ , G ~ is a weakly monotonic correspondence. Also, we have the following elementary facts: I f ~ - - ~ , and all agents have smooth preferences in ]R~_+, then G~P(R)nint{Z}=W(R)nint{Z}. And if ~={{Zo}[Zo~lRe+}, then G ~ =F .

The correspondence G ~ is not a subsolution of the pareto solution, as illustrated in Fig. 6a for an exchange economy R ~ ~ cn. Also, the correspondence G~,P is not a subcorrespondence of the equal-income Walrasian correspondence, even if agents have smooth preferences in IR~_ +. This is illustrated in Fig. 6b, also for an exchange economy R ~ 2 cn. There, z is a boundary allocation.

In the Edgeworth box, G~(R) is a subset of the set of points that are "beyond" both offer curves drawn from equal division. Indeed, if z e Z is such that for some i ~ N the line of support to agent i's indifference curve at z i goes above f2/2, then agent j ~ N, if given access to this choice set, will necessarily be able to do better than zj.

In the public good case, G~(R) is empty for some R ~ 2 cn. In the Kolm triangle, for the allocation z ~ Z to be in this set, it should first of all be a point of the vertical segment through the top vertex since G~_F. In addition, the agents' indifference curves at z should have lines of support that are symmetric of each other with respect to the segment. There is no guarantee that such allocation exists and even if it does, it will be efficient only if these lines of support are vertical, an atypical situation.

Weakenings of the above definition can be formulated that allow existence in general situations. Two possible formulations are discussed in Appendix B. They take us as far as the line of investigation followed in this paper seems to be able to take us.

6. Concluding comments

We have proposed three notions of equal, or equivalent, opportunities. These notions generalize and unite the concepts that have played the main role in the literature, including no-envy and egalitarian-equivalence. Also, we discovered

154 W. Thomson

that for the simplest specification of the families of choice sets on which these definitions are based, the resulting equity notions often lead to the equal-income Walrasian allocations. A more detailed investigation of the solutions that would obtain for more general, non linear, families might produce new solutions of interest. We leave this question for future research.

The following diagram summarizes the logical relations between our concepts.

F*___ O)~_ Ose

F ~ F 2 ~ O ~

We should note that the concepts presented here can be adapted in a straightforward way to the problem of evaluating the equity of a trade vector instead of that of an allocation. Instead of considering sets of possible consumptions, simply consider sets of possible trades. Given a family ,ff of such sets, replace 2 by S in our three main definitions.

Finally, we would like to emphasize that we have analyzed only a few of the possible meanings of the phrases "equal opportunities" and "equivalent opportunities". "Equal opportunities" is sometimes understood in other ways and in order to differentiate our analysis from some related literature we conclude by briefly describing some of these other meanings.

Taking the no-envy concept as a point of departure, note first that an outcome at which one agent envies another agent may well be judged equitable if it is the result of a process in which all agents have had equal opportunities. For instance, consider the problem of allocating some indivisible object and suppose that there exists no means of effecting (monetary) compensations across agents. Then, the probabilistic mechanism assigning all agents equal chances of winning the object might be judged ex-ante equitable, although it will generate allocations with envy.

"Equal opportunities" may also mean that the "transition" mechanism that takes agents from the initial position to the final position is fair; for instance disparities of incomes may be found acceptable in societies where it is nevertheless thought to be a fundamental principle of fairness that the educational process (the transition mechanism) give all children equal opportunities to realize their potential.

Finally, if agreement exists on the transition mechanism, equal opportunities may mean equal or "equivalent" initial resources. This idea was pursued in Thomson (1983), where certain invariance properties of final allocations with respect to exchanges of endowments were formulated and studied.

Appendix

Here we propose alternatives to the main definition of Sect. 5. The first one is motivated by the following variant, which appears in Thomson

(1982) and Baumol (1986), of the notion of an envy-free allocation. It says that no agent should prefer the average of what the other agents have received to what he has received. (See these references for motivation and analyses of this concept.)

Definition. The allocation z ~ Z is average-envy-free for R if for all i ~ N,

ziRi(~iZj/(n--1) ) •


Then, we propose to evaluate a feasible allocation obtained by each agent choosing in his own choice set, by comparing his consumption to the best consumption he could achieve if given access to the "average" choice set of the others.

Definition. The allocation z ~ Z exhibits no envy of average opportunities relative to the family ~ for R if for each i ~ N, there is B i ~ ~ such that zi maximizes R i in Bi and for no i ~ N, agent i prefers any point of ~. B j / ( n - l )

=-- z/~lRe+l z ; = zj / ( n - 1 ) w h e r e f o r e a c h j ~ N , j ~ i , z j ~ B g J . i

It can be seen by means o f simple examples that if agent i does not envy the opportunities of any of the other agents, he may nevertheless envy their average opportunities, and conversely.

Although this definition may be useful in some contexts, it will not help us solve our existence problem in the public good case of Sect. 5 (the existence of allocations that are equal opportunity equivalent and efficient) since this problem occurs already when n = 2 and in that case the definition above coincides with that earlier definition.

Here is another possible weakening of the main definition of Sect. 5. It says that an allocation is acceptable if it is equivalent (in the sense of Pareto-indifference) to a list of consumptions obtained by giving agents perhaps different opportunities, but such that no one envies the opportunities of anyone else.

Definition. The allocation z ~ Z is no-envy of opportunities equivalent relative to the family ~ for R if for each i ~ N, there is B+ ~ ~ such that ziI i z*, where z* is the maximizer of Ri on B~ and for no i, j ~ N, agent i prefers any point of Bj to z*. Let G ) ( R ) be the set of these allocations.

I t is clear that for all ~@, G ) contains both G ~ and O ) . However, since the latter correspondence is non-empty for natural choices of ~ , even in public good economies (Theorems 5 and 6), one could argue that this last definition is too much of a generalization. So we will not pursue its analysis.

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notions of equal, or equivalent, opportunities

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