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A refinement and extension of the no-envy concept
217
Economics Letters 33 (1990) 217-222 North-Holland
A REFINEMENT
AND EXTENSION
OF THE NO-ENVY
CONCEPT
*
Dimitrios DIAMANTARAS Temple
University, Philadelphia,
PA 19122, USA
William THOMSON University of Rochester,
Rochester,
NY 14627, USA
Received 23 February 1988 Accepted 20 October 1989
We propose a new criterion of equity in the spirit of Foley’s (1967) no-envy concept. We prove the existence of efficient allocations satisfying this criterion for a class of economies containing many for which envy-free and efficient allocations do not exist; when a large set of the latter exists, our criterion selects a small subset of it.
1. Introduction The concept that has recently played the most important role in the economic analysis of equity issues is probably that of no-envy [Foley (1967)]: An allocation z is envy-free is no agent would prefer someone else’s consumption bundle to his own. Of course, the introduction of this concept did not solve all distributional issues. This is in part because the set of envy-free and efficient allocations may be quite large or it may be empty. In the first case, there arises the issue of deciding which of the envy-free and efficient allocations are preferable from the viewpoint of distribution. In the second case, the concept offers no recommendation at all. Unfortunately, each of these two problems is quite frequent. For instance, in a two-person exchange economy, the set of envy-free and efficient allocations usually strictly contains the set of efficient allocations that Pareto dominate equal division of resources (fig. 1). Conversely, in a production economy in which the agents are differently productive, there may be no envy-free efficient allocations [Pazner and Schmeidler (1974)]. Strengthening the no-envy concept would help solving the multiplicity problem but would make matters worse so far as existence is concerned; weakening the concept would have the opposite effect. In this note, we propose a new concept of equity which simultaneously solves the two problems. This concept, which is still very much in the spirit of Foley’s original concept of no-envy, can be seen as maximally strengthening it to solve the multiplicity problem and minimally weakening it to recover existence. * D. Diamantaras thanks the College of Arts and Science of the University of Rochester for support under an R.N. Ball dissertation fellowship. W. Thomson thanks the NSF for its support under grant SES 85-11136. We also wish to acknowledge the comments of A. Chaudhuri. This paper has some resemblance to the one wrongly published in Vol. 30, no. 3 (pp. 103-107). There are, however, substantial differences. 0165-1765/90/$3.50
0 1990 - Elsevier Science Publishers B.V. (North-Holland)
Q. Diamantaras,
W Thornron / R&zing
the no-enuy concept
Fig. 1
2. Notation, definitions, result There are k 2 1 private goods, m 2 0 public goods, n 2 1 agents, and n types of ‘time’, one for each agent. Time of type i can either be consumed by agent i as leisure or contributed to the production process as labor. (We could have more agent-specific inputs, or none at all, with no difficulty.) We denote the set of agents by N. Agent i, for each i E N, is characterized by: (i) a closed and convex consumption set 2, c IR:+~+“’ containing the origin, and (ii) a preference relation defined on Zi and assumed to admit a continuous and strictly increasing numerical representation ui : 2, + IF8+. Let U = (u,, . . . , u,). [Strictly increasing is taken to mean ii > zi implies u,(ii) > ui(z,) ‘.I Let 9=(L, X)ER”=,k be the vector of initial endowments. L,, for i in N, is the time endowment of agent i; Xq, for 4 = 1,. . _, k, is the aggregate endowment of the qth private good. Note that X is not indexed by agents; all agents are collectively entitled to all goods, initially available or produced, except time, which cannot be redistributed. There is a production set Y c R’“+k+m, which is closed, contains - lRn+k+m, and is such that [Y + Q] I? R T’k+m is bounded. The allocation space s&‘= Wt+nk+“! An allocation (l, x, y) E.EZ is feasible for (Y, s2, Y) if for all i, (l;, xi, y) E Zi, and (I - L, xx, - X, y) E Y. Let A(&?, Y) Cd be the set of feasible allocations of (U, D, Y), and P(U, 52, Y) c A(ti, Y) its set of Pareto-efficient allocations. An equity criterion E associates with each economy (U, ,(2, Y) a set E(U, 9, Y) c A(f2, Y) of allocations at which agents are thought to be equitably sharing in the society’s resources. The intersection of the equity criterion E with the Pareto criterion P is denoted by EP. The best known equity criterion is that of an envy-free allocation:
’ Vector inequalities:
23, i,
2.
D. Diamantaras, W. Thomson / Refining the no-envy concept
219
z = (1, x, y) E A(O, Y) is envy-free for (U, D, Y) if there is no pair 1. An allocation i. j E N such that ui(Zr, xi, y) < u,(/,, x,, y). Let F( U, a, Y) designate the set of these allocations.
Definition
As noted earlier, FP(U, 9, Y) may be very large, but it also may be empty. We now turn to our concept, for which we draw upon ideas recently developed by Chaudhuri (1986). This author proposes evaluating the extent to which an allocation may fail to be envy-free by comparing each agent’s consumption to the other agent’s consumptions subjected to appropriate radial contractions. More precisely, suppose that agent i envies agent j, i.e., u,(z,) > uI( z,). Chaudhuri suggests taking the value of l/X - 1 for which ui(Xzj) = ui(zi) as a measure of how much agent i envies agent j. He ‘then defines an aggregate measure of envy by summing up these individual measures over all pairs i, j such that ui( z,) > u,(z,). Here we would like to evaluate the extent to which an allocation may fail to meet the no-envy requirement as well as the extent to which an allocation may exceed it. To that purpose we consider radial contractions and expansions of consumption bundles. Then, by taking the maximal A for which u,(z,) 2 u,(Xz,) for all i, j as a measure of envy or distance from envy at z and then maximizing the smallest such h over the set of Pareto-efficient allocations we identify which allocations minimally violate the no-envy criterion when the criterion cannot be met by an efficient allocation, or maximally satisfy it when it is met by multiple efficient allocations. Our approach enables us to perform selections from the set of envy-free and efficient allocations when it is not Chaudhuri’s aggregate empty; Chaudhuri’s approach is not sufficient for this. 2 Also, minimizing measure of envy can lead to allocations at which the distribution of envy is very skewed. We feel that our criterion, which is designed to equate envy, or distance from envy, across agents, is more in keeping with the spirit that has guided the search for equity criteria. Definition 2. Given X E R +, the allocation z = (I, x, y) E A(J2, Y) is A-envy-free for (17, S;r, Y) if there is no pair i, j, i Zj, such that u,(Z,, xi, y) < u,(XI,, Xx,, Xy). Let F’(U, L’, Y) be the set of these allocations. Note that F’(U, 0, Y) = F(U, Q, Y), and F”(U, c F”\U, 9, Y). Given (U, 0, Y), let X(U, 9, Y) be the maximal example shows that F’(U, 9, Y) may be non-empty may not exist.
9, Y) =A(&?, Y). Also, if A > X’, F”(U,
9, Y)
X such that F’P(U, 9, Y) = ,@. The following for all h E R +, and, consequently, X(U, 52, Y)
Example 1. There are no labor inputs, and k = 2, m = 0, n = 2. Let ui(z,) = zii, u2(z2) = z22. Also, Y = {zO E lR$ lza I (1, 1)). In this (exchange) economy, P(U, s2, Y) = (((1, 0), (0, 1))) = {z* }, and z*~F’(U,ti, Y)forall h~lR+. Although we feel that it would be meaningful to extend the domain of definition of the function X by allowing it to take on the value + 00 in economies such as the one in the example, we will find it more convenient to make an assumption 3 on the economy guaranteeing that given any (I, x, y) E A(S2, Y) there is maximal h such that for no pair i, j u,(1,, x,, y) < u,(hl,-, Xx,, hy). ’ Chaudhuri
also advocates using radial contractions of bundles only if preferences are homothetic. We have not imposed this restriction. 3 The same assumption is used in Mas-Cole11 (1985, p.300) to obtain the boundedness of the set of envy-free and efficient allocations in a large economy.
220
Assumption compact.
D. Diamantaras, W. Thomson / Refining the no-envy concept
A.
For all i, and for all c E range uI, the set {(I,,
x,,
y)
E
Z, 1c
2 ~;(l;,
Xi,
y)}
is
Under Assumption A our main definition, given next, is non-vacuous, as we will show. Definition 3. Given (U, 9, Y) such that A(U, 9, Y), the maximal exists, let G(U, Q, Y) = Fx(u~Q5y)P(U, 52, Y).
X such that F”P(U,
s2, Y) # 0
Remark. Alternatively, we could have required that the maximum h(U, Q, Y) be sought among the set of X which satisfy: For all i, jhz, is feasible for i if i has access to Y, s2. An examination of the following proofs shows that such a definition would lead to existence without the need for Assumption A. The definition is illustrated in fig. 1, where it is put to use to perform a selection from FP(U, 52, Y). In fig. 1, FP(U, fi, Y) is the curvilinear segment [z’, x2], where zi is such that u,(z~) = I, and z2 satisfies u,(z,‘) = u(zz). There is a unique allocation z* E G(U, 9, Y). It is such that for some X > 1, ui(z:) = ui(Xz~) and u2(z2) = u2(XxT). For all z E A(O, Y), for all i, j, let
IXz,EZ,,
P&)‘{-~+ A,,(z)=max{XER+ Lemma 1.
u;(z,)
2u,(Xzj)}
and
lXEPjj(z)}.
Under Assumption
A, each Xii: A(S2, Y) + R + is a continuous function.
Proof. For all z E A(S2, Y), Pi,(z) f 8, because 0 E pii( plj( z ) is also closed, by the closedness of Z, and the continuity of u,, and bounded above, by Assumption A. Therefore, for all z E A(O, Y), h,,(z) #,f3. Clearly, X,j( z ) 1s . a singleton, thus h ij is a well-defined function. To show continuity, let f : Iw+X A(O, Y) --* [w+ be defined by f (A, z) = h. Obviously, f is a continuous function. We have
We claim that pij is a compact-valued and continuous correspondence. We have already shown compact-valuedness of pij; we now show that pij is upper hemi-continuous and lower hemi-continuous, thus continuous. Let F c [w+ be a closed set (relative to Iw+). Then, if a = min F, we have S= {z~A(ti, =A(fi,
Y)I/?,j(z)nF#fl}
Y)\{zEA(O,
Y) Iui(z,)
or azjEZj},
and S is clearly a closed set. Thus pij is upper her&continuous, by Hildenbrand Proposition 1 (iii)]. Let now G c R + be an open set (relative to lR +)_ Then if b = inf G’# 0, we have T=
{zEA(O,
= {zEA(SZ,
Y)I/3;j(z)nG=fl) Y)Iui(zi)
or
bzj4int
Z,},
[1974, p. 22,
D. Diamantaras, W. Thomson / Refining the no-enuy concept
221
and T is clearly a closed set; if b = 0, then T = ,@, and again it is closed. Thus pij is lower hen&continuous, by Hildenbrand [1974, p. 27, Proposition 7 (iii)]. We have now proven that pij is a continuous correspondence. By a corollary to the maximum theorem [Hildenbrand (1974, p. 30)], A,, is an upper hen-n-continuous correspondence, hence it is 0 continuous, since it is a function [Hildenbrand (1974, p. 21, Ex. l)]. Theorem 1. Proof
Under Assumption
Let z E A(O,
A, G(U,
f2, Y) # 0.
Y) be given. Let Xi, be defined as above, and let
Ai(z)-min{h,(z)]j#i},
and
h(z)=min{h,(z)]i~N}.
By Lemma 1, the function X : A(O, Y) + R + is G(U, Sz, Y) = argmax{X(z) 1z E P(U, 52, Y)}. A(L2, Y) is a compact set. P(U, 9, Y) c A( 52, Y), continuity of the ui and the closedness and boundedness be adapted to our framework]. Therefore, P(U, 9, Y) is
well defined
and continuous.
We have
and P(U, 52, Y) is a closed set, by the of Y [Mas-Cole11 (1985, p. 154) can easily a compact set, and thus G(U, Q, Y) # 0.
[7
We discussed earlier the main advantages of our concept: It is an ordinal concept, very much in the spirit of the no-envy concept, it is minimal as an extension of this concept, and maximal as a refinement. We now pursue in greater detail the comparison of the two concepts and we mention a few open questions. If n = 2, and z E G(U, 52, Y), then for each i EN, u;(z,) = u,(X(U, 9, Y)z,). This is an appealing result which says that envy (or lack of envy) has truly been equalized across agents. However, we do not know whether the following extension of this result to n > 2 is true: If z E G(U, 9, Y), then, for all i E N there exists j E N such that ui(z,) = u,(X(U, Q, Y)zj). Similarly, we do not have an answer to the related question of essential single-valuedness of G, i.e. whether, if z, z’ E G(U, 52, Y), then for all i E N, uj(ii) = u,(z,‘).
Fig.
222
D. Diamantaras, W. Thomson / Refining the no-envy concept
The two concepts share the following share the following limitations. (i) If z E G(U, 52, Y), then z need not Pareto-dominate equal division. (ii) Pareto-indifference is violated: It is possible to have z E G(U, s2, Y), z’ E P(lJ 9, Y) with ai = Us for all i E N, and yet z’6C G(U, 9, Y). (iii) Finally, the following somewhat puzzling phenomenon may occur [Thomson (1987)]: In the Edgeworth box, of all the allocations that are Pareto-indifferent to z E G(U, 52, Y), z is the farthest away from equal division (fig. 2).
References Chaudhuri, A., 1986, Some implications of an intensity measure of envy, Social Choice, and Welfare 3, 255-270. Foley, D., 1967, Resource allocation and the public sector, Yale Economic Essays 7, 45-98. Hildenbrand, W., 1974, Core and equilibria of a large economy (Princeton University Press, Princeton, NJ). Mas-Colell, A., 1985, The theory of general economic equilibrium; a differentiable approach, Econometric Society Monograph no. 9 (Cambridge University Press, Cambridge). Pazner E. and Schmeidler, D., 1974, A difficulty with the concept of fairness, Review of Economic Studies 41, 441-443. Thomson, W., 1987, Lecture notes on equity (University of Rochester, Rochester, NY) Mimeo.
Economics Letters 33 (1990) 217-222 North-Holland
A REFINEMENT
AND EXTENSION
OF THE NO-ENVY
CONCEPT
*
Dimitrios DIAMANTARAS Temple
University, Philadelphia,
PA 19122, USA
William THOMSON University of Rochester,
Rochester,
NY 14627, USA
Received 23 February 1988 Accepted 20 October 1989
We propose a new criterion of equity in the spirit of Foley’s (1967) no-envy concept. We prove the existence of efficient allocations satisfying this criterion for a class of economies containing many for which envy-free and efficient allocations do not exist; when a large set of the latter exists, our criterion selects a small subset of it.
1. Introduction The concept that has recently played the most important role in the economic analysis of equity issues is probably that of no-envy [Foley (1967)]: An allocation z is envy-free is no agent would prefer someone else’s consumption bundle to his own. Of course, the introduction of this concept did not solve all distributional issues. This is in part because the set of envy-free and efficient allocations may be quite large or it may be empty. In the first case, there arises the issue of deciding which of the envy-free and efficient allocations are preferable from the viewpoint of distribution. In the second case, the concept offers no recommendation at all. Unfortunately, each of these two problems is quite frequent. For instance, in a two-person exchange economy, the set of envy-free and efficient allocations usually strictly contains the set of efficient allocations that Pareto dominate equal division of resources (fig. 1). Conversely, in a production economy in which the agents are differently productive, there may be no envy-free efficient allocations [Pazner and Schmeidler (1974)]. Strengthening the no-envy concept would help solving the multiplicity problem but would make matters worse so far as existence is concerned; weakening the concept would have the opposite effect. In this note, we propose a new concept of equity which simultaneously solves the two problems. This concept, which is still very much in the spirit of Foley’s original concept of no-envy, can be seen as maximally strengthening it to solve the multiplicity problem and minimally weakening it to recover existence. * D. Diamantaras thanks the College of Arts and Science of the University of Rochester for support under an R.N. Ball dissertation fellowship. W. Thomson thanks the NSF for its support under grant SES 85-11136. We also wish to acknowledge the comments of A. Chaudhuri. This paper has some resemblance to the one wrongly published in Vol. 30, no. 3 (pp. 103-107). There are, however, substantial differences. 0165-1765/90/$3.50
0 1990 - Elsevier Science Publishers B.V. (North-Holland)
Q. Diamantaras,
W Thornron / R&zing
the no-enuy concept
Fig. 1
2. Notation, definitions, result There are k 2 1 private goods, m 2 0 public goods, n 2 1 agents, and n types of ‘time’, one for each agent. Time of type i can either be consumed by agent i as leisure or contributed to the production process as labor. (We could have more agent-specific inputs, or none at all, with no difficulty.) We denote the set of agents by N. Agent i, for each i E N, is characterized by: (i) a closed and convex consumption set 2, c IR:+~+“’ containing the origin, and (ii) a preference relation defined on Zi and assumed to admit a continuous and strictly increasing numerical representation ui : 2, + IF8+. Let U = (u,, . . . , u,). [Strictly increasing is taken to mean ii > zi implies u,(ii) > ui(z,) ‘.I Let 9=(L, X)ER”=,k be the vector of initial endowments. L,, for i in N, is the time endowment of agent i; Xq, for 4 = 1,. . _, k, is the aggregate endowment of the qth private good. Note that X is not indexed by agents; all agents are collectively entitled to all goods, initially available or produced, except time, which cannot be redistributed. There is a production set Y c R’“+k+m, which is closed, contains - lRn+k+m, and is such that [Y + Q] I? R T’k+m is bounded. The allocation space s&‘= Wt+nk+“! An allocation (l, x, y) E.EZ is feasible for (Y, s2, Y) if for all i, (l;, xi, y) E Zi, and (I - L, xx, - X, y) E Y. Let A(&?, Y) Cd be the set of feasible allocations of (U, D, Y), and P(U, 52, Y) c A(ti, Y) its set of Pareto-efficient allocations. An equity criterion E associates with each economy (U, ,(2, Y) a set E(U, 9, Y) c A(f2, Y) of allocations at which agents are thought to be equitably sharing in the society’s resources. The intersection of the equity criterion E with the Pareto criterion P is denoted by EP. The best known equity criterion is that of an envy-free allocation:
’ Vector inequalities:
23, i,
2.
D. Diamantaras, W. Thomson / Refining the no-envy concept
219
z = (1, x, y) E A(O, Y) is envy-free for (U, D, Y) if there is no pair 1. An allocation i. j E N such that ui(Zr, xi, y) < u,(/,, x,, y). Let F( U, a, Y) designate the set of these allocations.
Definition
As noted earlier, FP(U, 9, Y) may be very large, but it also may be empty. We now turn to our concept, for which we draw upon ideas recently developed by Chaudhuri (1986). This author proposes evaluating the extent to which an allocation may fail to be envy-free by comparing each agent’s consumption to the other agent’s consumptions subjected to appropriate radial contractions. More precisely, suppose that agent i envies agent j, i.e., u,(z,) > uI( z,). Chaudhuri suggests taking the value of l/X - 1 for which ui(Xzj) = ui(zi) as a measure of how much agent i envies agent j. He ‘then defines an aggregate measure of envy by summing up these individual measures over all pairs i, j such that ui( z,) > u,(z,). Here we would like to evaluate the extent to which an allocation may fail to meet the no-envy requirement as well as the extent to which an allocation may exceed it. To that purpose we consider radial contractions and expansions of consumption bundles. Then, by taking the maximal A for which u,(z,) 2 u,(Xz,) for all i, j as a measure of envy or distance from envy at z and then maximizing the smallest such h over the set of Pareto-efficient allocations we identify which allocations minimally violate the no-envy criterion when the criterion cannot be met by an efficient allocation, or maximally satisfy it when it is met by multiple efficient allocations. Our approach enables us to perform selections from the set of envy-free and efficient allocations when it is not Chaudhuri’s aggregate empty; Chaudhuri’s approach is not sufficient for this. 2 Also, minimizing measure of envy can lead to allocations at which the distribution of envy is very skewed. We feel that our criterion, which is designed to equate envy, or distance from envy, across agents, is more in keeping with the spirit that has guided the search for equity criteria. Definition 2. Given X E R +, the allocation z = (I, x, y) E A(J2, Y) is A-envy-free for (17, S;r, Y) if there is no pair i, j, i Zj, such that u,(Z,, xi, y) < u,(XI,, Xx,, Xy). Let F’(U, L’, Y) be the set of these allocations. Note that F’(U, 0, Y) = F(U, Q, Y), and F”(U, c F”\U, 9, Y). Given (U, 0, Y), let X(U, 9, Y) be the maximal example shows that F’(U, 9, Y) may be non-empty may not exist.
9, Y) =A(&?, Y). Also, if A > X’, F”(U,
9, Y)
X such that F’P(U, 9, Y) = ,@. The following for all h E R +, and, consequently, X(U, 52, Y)
Example 1. There are no labor inputs, and k = 2, m = 0, n = 2. Let ui(z,) = zii, u2(z2) = z22. Also, Y = {zO E lR$ lza I (1, 1)). In this (exchange) economy, P(U, s2, Y) = (((1, 0), (0, 1))) = {z* }, and z*~F’(U,ti, Y)forall h~lR+. Although we feel that it would be meaningful to extend the domain of definition of the function X by allowing it to take on the value + 00 in economies such as the one in the example, we will find it more convenient to make an assumption 3 on the economy guaranteeing that given any (I, x, y) E A(S2, Y) there is maximal h such that for no pair i, j u,(1,, x,, y) < u,(hl,-, Xx,, hy). ’ Chaudhuri
also advocates using radial contractions of bundles only if preferences are homothetic. We have not imposed this restriction. 3 The same assumption is used in Mas-Cole11 (1985, p.300) to obtain the boundedness of the set of envy-free and efficient allocations in a large economy.
220
Assumption compact.
D. Diamantaras, W. Thomson / Refining the no-envy concept
A.
For all i, and for all c E range uI, the set {(I,,
x,,
y)
E
Z, 1c
2 ~;(l;,
Xi,
y)}
is
Under Assumption A our main definition, given next, is non-vacuous, as we will show. Definition 3. Given (U, 9, Y) such that A(U, 9, Y), the maximal exists, let G(U, Q, Y) = Fx(u~Q5y)P(U, 52, Y).
X such that F”P(U,
s2, Y) # 0
Remark. Alternatively, we could have required that the maximum h(U, Q, Y) be sought among the set of X which satisfy: For all i, jhz, is feasible for i if i has access to Y, s2. An examination of the following proofs shows that such a definition would lead to existence without the need for Assumption A. The definition is illustrated in fig. 1, where it is put to use to perform a selection from FP(U, 52, Y). In fig. 1, FP(U, fi, Y) is the curvilinear segment [z’, x2], where zi is such that u,(z~) = I, and z2 satisfies u,(z,‘) = u(zz). There is a unique allocation z* E G(U, 9, Y). It is such that for some X > 1, ui(z:) = ui(Xz~) and u2(z2) = u2(XxT). For all z E A(O, Y), for all i, j, let
IXz,EZ,,
P&)‘{-~+ A,,(z)=max{XER+ Lemma 1.
u;(z,)
2u,(Xzj)}
and
lXEPjj(z)}.
Under Assumption
A, each Xii: A(S2, Y) + R + is a continuous function.
Proof. For all z E A(S2, Y), Pi,(z) f 8, because 0 E pii( plj( z ) is also closed, by the closedness of Z, and the continuity of u,, and bounded above, by Assumption A. Therefore, for all z E A(O, Y), h,,(z) #,f3. Clearly, X,j( z ) 1s . a singleton, thus h ij is a well-defined function. To show continuity, let f : Iw+X A(O, Y) --* [w+ be defined by f (A, z) = h. Obviously, f is a continuous function. We have
We claim that pij is a compact-valued and continuous correspondence. We have already shown compact-valuedness of pij; we now show that pij is upper hemi-continuous and lower hemi-continuous, thus continuous. Let F c [w+ be a closed set (relative to Iw+). Then, if a = min F, we have S= {z~A(ti, =A(fi,
Y)I/?,j(z)nF#fl}
Y)\{zEA(O,
Y) Iui(z,)
or azjEZj},
and S is clearly a closed set. Thus pij is upper her&continuous, by Hildenbrand Proposition 1 (iii)]. Let now G c R + be an open set (relative to lR +)_ Then if b = inf G’# 0, we have T=
{zEA(O,
= {zEA(SZ,
Y)I/3;j(z)nG=fl) Y)Iui(zi)
or
bzj4int
Z,},
[1974, p. 22,
D. Diamantaras, W. Thomson / Refining the no-enuy concept
221
and T is clearly a closed set; if b = 0, then T = ,@, and again it is closed. Thus pij is lower hen&continuous, by Hildenbrand [1974, p. 27, Proposition 7 (iii)]. We have now proven that pij is a continuous correspondence. By a corollary to the maximum theorem [Hildenbrand (1974, p. 30)], A,, is an upper hen-n-continuous correspondence, hence it is 0 continuous, since it is a function [Hildenbrand (1974, p. 21, Ex. l)]. Theorem 1. Proof
Under Assumption
Let z E A(O,
A, G(U,
f2, Y) # 0.
Y) be given. Let Xi, be defined as above, and let
Ai(z)-min{h,(z)]j#i},
and
h(z)=min{h,(z)]i~N}.
By Lemma 1, the function X : A(O, Y) + R + is G(U, Sz, Y) = argmax{X(z) 1z E P(U, 52, Y)}. A(L2, Y) is a compact set. P(U, 9, Y) c A( 52, Y), continuity of the ui and the closedness and boundedness be adapted to our framework]. Therefore, P(U, 9, Y) is
well defined
and continuous.
We have
and P(U, 52, Y) is a closed set, by the of Y [Mas-Cole11 (1985, p. 154) can easily a compact set, and thus G(U, Q, Y) # 0.
[7
We discussed earlier the main advantages of our concept: It is an ordinal concept, very much in the spirit of the no-envy concept, it is minimal as an extension of this concept, and maximal as a refinement. We now pursue in greater detail the comparison of the two concepts and we mention a few open questions. If n = 2, and z E G(U, 52, Y), then for each i EN, u;(z,) = u,(X(U, 9, Y)z,). This is an appealing result which says that envy (or lack of envy) has truly been equalized across agents. However, we do not know whether the following extension of this result to n > 2 is true: If z E G(U, 9, Y), then, for all i E N there exists j E N such that ui(z,) = u,(X(U, Q, Y)zj). Similarly, we do not have an answer to the related question of essential single-valuedness of G, i.e. whether, if z, z’ E G(U, 52, Y), then for all i E N, uj(ii) = u,(z,‘).
Fig.
222
D. Diamantaras, W. Thomson / Refining the no-envy concept
The two concepts share the following share the following limitations. (i) If z E G(U, 52, Y), then z need not Pareto-dominate equal division. (ii) Pareto-indifference is violated: It is possible to have z E G(U, s2, Y), z’ E P(lJ 9, Y) with ai = Us for all i E N, and yet z’6C G(U, 9, Y). (iii) Finally, the following somewhat puzzling phenomenon may occur [Thomson (1987)]: In the Edgeworth box, of all the allocations that are Pareto-indifferent to z E G(U, 52, Y), z is the farthest away from equal division (fig. 2).
References Chaudhuri, A., 1986, Some implications of an intensity measure of envy, Social Choice, and Welfare 3, 255-270. Foley, D., 1967, Resource allocation and the public sector, Yale Economic Essays 7, 45-98. Hildenbrand, W., 1974, Core and equilibria of a large economy (Princeton University Press, Princeton, NJ). Mas-Colell, A., 1985, The theory of general economic equilibrium; a differentiable approach, Econometric Society Monograph no. 9 (Cambridge University Press, Cambridge). Pazner E. and Schmeidler, D., 1974, A difficulty with the concept of fairness, Review of Economic Studies 41, 441-443. Thomson, W., 1987, Lecture notes on equity (University of Rochester, Rochester, NY) Mimeo.