! " '() # ! % $ & * + ) & &, ! ! #$%%& ' ' ! " # ' '(# The Walrasian Distribution of Opportunity Sets: An Axiomatic Characterization Koichi Tadenuma Department of Economics Hitotsubashi University Kunitachi, Tokyo 186-8601, Japan Email: koichi.t[at]econ.hit-u.ac.jp Yongsheng Xu Department of Economics Andrew Young School of Policy Studies Georgia State University, Atlanta, GA, 30303, U.S.A. Email: yxu3[at]gsu.edu December 2010 Abstract Economic systems generate various distributions of opportunity sets for individuals to choose consumption bundles. This paper presents an axiomatic analysis on distributions of opportunity sets. We introduce several reasonable properties of distributions of opportunity sets, and characterize the distributions of opportunity sets in the market economy by these properties. Keywords: distribution of opportunity sets, market economy, Walrasian distribution, axiomatic characterization. JEL Classification Numbers: D50, D60, D70, P00 1 Introduction Economic and social systems are assessed on various criteria. For example, the market system has been evaluated on such criteria as efficiency, equity, and freedom. On the efficiency ground, modern economic theory has established the fundamental theorems of welfare economics (Arrow (1951) and Debreu (1951, 1954)), which state that the market system, if perfectly operated, generates a socially efficient outcome. Though the efficiency aspect of the market system is very appealing, many writers have also argued that a most compelling advantage of the market system lies in its promotion of individual freedom in terms of opportunities to achieve as well as its guarantee of individual liberty in terms of autonomy of decisions.1 Sen (1988) stresses the importance of the opportunity aspect of freedom: It seems reasonable to argue that if we really do attach importance to the actual opportunity that each person has, subject to feasibility, to lead the life that he or she would choose, then the opportunity aspect of freedom must be quite central to social evaluation (Sen, 1988, p.527). Following Sen (1993), we consider that the degree of freedom of choice enjoyed by an individual in a social and economic system is reflected in the set of opportunities available to the individual. For example, in the market system, a consumer’s opportunities can be described by her budget set, namely, the set of consumption bundles that are available and affordable for her. Indeed, standard textbooks of microeconomics start with postulating that each consumer chooses her most preferred bundle in her budget set. In contrast, the central planning system assigns to each individual a specific bundle, giving her no freedom of choice. Somewhere in between, distortions and/or regulations in the market will change opportunities available to individuals. For instance, if the government subsidizes low-income individuals to purchase necessities, their opportunities are different from those of highincome individuals; if, on the other hand, the government sets a regulation on the quantity of consumption of gasoline for each individual, then each individual’s budget set is truncated by the limit. Such examples are abundant and all illustrate the following: each social and economic system gives 1 We refer to, for instance, Hayek (1976), Nozick (1974), Buchanan (1986), Friedman (1962), Kornai (1988), and Lindbeck (1988) for further discussions on these and related issues. 1 various degrees of freedom for individuals; these various degrees of freedom for individuals are reflected by their opportunity sets. In order to better understand and compare the opportunity aspect of individual freedom in various systems, it is essential to study properties of distributions of opportunity sets generated by the systems. To the best of our knowledge, however, there exist few studies on general properties of distributions of opportunity sets.2 In this paper, we formulate several reasonable properties of distributions of opportunity sets, and then, by using these properties, we characterize the distribution of opportunity sets generated by the most prominent economic system in the modern age, the market system. We will show that, a distribution of opportunity sets satisfies three properties, Feasibility, No Coordination Surplus, and Minimum Attainability (see below for informal descriptions of these properties) if and only if it is a distribution of opportunity sets generated by the market system, which we call a Walrasian distribution. In what follows, let us explain more about these properties that characterize Walrasian distributions. Concerning our first property, Feasibility, we note first that, for most economic systems, opportunity sets of individuals are inter-dependent. This point has been observed by Basu (1987) and Pattanaik (1994). In fact, on a moment’s reflection of an Edgeworth box diagram, one will see that whether a consumption bundle in a budget set is attainable or not for an individual depends on the choices of the other individuals. For example, suppose that there are two individuals, i and j, each of whom is endowed with 5 units of bread and 3 units of meat, and that the price of each good is unity. Then, individual i’s choice of 6 units of bread and 2 units of meat is not compatible with individual j’s choice of 7 units of bread and 1 unit of meat, while it is compatible with j’s choice of 4 units of bread and 4 units of meat. Hence, to formulate a very basic notion of feasibility of a distribution of opportunity sets, a careful consideration is necessary. Certainly, it is too strong to require that any combination of choices of individuals from their respective opportunity sets constitute a feasible allocation. In fact, under a very mild assumption, the only distribution of opportunity sets satisfying this requirement is the central planning system which assigns to each individual a specific bundle or less. 2 There is a related literature on ranking opportunity sets. But its focus is on ranking opportunity sets by a single individual, and do not consider distributions of opportunity sets in a social setting. See, for instance, Barbera, Bossert and Pattanaik (2004), and Pattanaik and Xu (1990, 2000). 2 To consider various types of distributions of opportunity sets, therefore, we need a weaker notion of feasibility. Here we propose to define a feasible distribution of opportunity sets as one in which each individual’s choice from the boundary of her opportunity set constitutes a feasible allocation with some choices of the other individuals from the boundaries of their respective opportunity sets. Our axiom of Feasibility requires that a distribution of opportunity sets be feasible in this weak sense. The second property, No Coordination Surplus, says that no “coordination” of choices by individuals over their respective opportunity sets gives these individuals a larger total amount of commodities. More formally, it requires that, for any group M of individuals, and any vector of consumption bundles (xi )i∈M of the members of M such that each xi belongs to the boundary of i’s opportunity set, there exists no vector of consumption ∑ ∑bundles (yi )i∈M in their respective opportunity sets such that i∈M yi > i∈M xi . This property is essential for a decentralized decision by each individual to be incentive compatible. If it were violated, a group of individuals would want to increase each member’s consumption by coordinating its decisions. The third property, Minimum Attainability, states that, for any feasible allocation in the economy, at least one consumption bundle in the allocation should be contained in the opportunity set of the corresponding individual. This condition excludes the case in which no individual can attain his consumption bundle at some allocation in the given economic system even though it is a feasible allocation. Such a case arises, for instance, in the central planning system. Each of our three properties is a fairly weak and natural requirement, and yet these properties together are enough to characterize the class of distributions of opportunity sets generated by the market system. We show that a distribution of opportunity sets satisfies Feasibility, No Coordination Surplus, and Minimum Attainability if and only if it is a Walrasian distribution with positive prices. The rest of the paper is organized as follows. In Section 2, we present notation and basic definitions. We also formally define a Walrasian distribution of opportunity sets. Section 3 introduces and discusses properties of distributions of opportunity sets. Section 4 presents our characterization of the Walrasian distributions of opportunity sets, and Section 5 concludes. The proof of the main result is contained in the appendix. 3 2 Basic Definitions and Notation There are n individuals and k goods. Let N := {1, . . . , n} be the set of individuals, and K := {1, . . . , k} the set of goods. An allocation is a vector k x = (x1 , . . . , xn ) ∈ Rnk + where for each i ∈ N , xi = (xi1 , . . . , xik ) ∈ R+ is 3 a consumption bundle of individual i. There exists some fixed amount of social endowments of goods, which are∑represented by the vector ω ∈ Rk++ . n 4 An allocation x ∈ Rnk Let Z be the set of + is feasible if i=1 xi ≤ ω. all feasible allocations. For each x ∈ Rnk , and each i ∈ N , we denote + x−i := (x1 , . . . , xi−1 , xi+1 , . . . , xn ). For each i ∈ N , an opportunity set for individual i is a set in Rk+ that is non-empty, compact, and comprehensive. Recollect that a set S ∈ Rk+ is comprehensive if for all x, y ∈ Rk+ , x ∈ S and y ≤ x imply y ∈ S. Comprehensiveness is a reasonable assumption on opportunity sets under free disposal of goods. Let S := {S ⊂ Rk+ | S is non-empty, compact, and comprehensive}. A distribution of opportunity sets is an n-tuple (S1 , . . . , Sn ) ∈ S n . Given a price vector p ∈ Rk+ and an income mi ∈ R+ , define B(p, mi ) := {xi ∈ Rk+ | p · xi ≤ mi } and B ∗ (p, mi ) := {xi ∈ Rk+ | p · xi = mi }. Let X(ω) := {xi ∈ Rk+ | xi ≤ ω}. Walrasian distribution: A distribution of opportunity sets (S1 , . . . , Sn ) ∈ n k n S ∑ is Walrasian if there exist p ∈ R+ and (m1 , . . . , mn ) ∈ R+ such that i∈N mi = p·ω and Si = B(p, mi )∩X(ω) for all i ∈ N . If, in addition, p ≫ 0, then we call the distribution of opportunity sets a Walrasian distribution with positive prices. Notice that our definition of a Walrasian distribution of opportunity sets slightly departs from the standard definition of a distribution of budget sets because each budget set is bounded by the resource constraint. This notion of constrained budget sets was introduced by Hurwicz, Maskin and Postlewaite (1982, 1995), and plays an important role in implementation theory. In particular, Hurwicz, Maskin and Postlewaite (1982, 1995) showed that the unconstrained Walrasian correspondence is not implementable in Nash equilibria, but the constrained Walrasian correspondence ∑ is. n For all (S1 , . . . , Sn ) ∈ S , and all M ⊆ N , define j∈M Sj := {x ∈ Rk+ | 3 As usual, R+ is the set of all non-negative real numbers, and R++ is the set of all positive real numbers. 4 Vector inequalities are defined as follows: For all x, y ∈ Rk+ , x ≥ y ⇔ (x − y) ∈ Rk+ ; x > y ⇔ [x ≥ y and x 6= y]; and x ≫ y ⇔ (x − y) ∈ Rk++ . 4 ∃(xj )j∈M ∈ Πj∈M Sj such that x = ∑ j∈M xj }. For all S ⊆ Rk+ , define ∂ + S := {x ∈ S |6 ∃y ∈ S such that y > x}. That is, ∂ + S is the “undominated boundary of S”. For all x ∈ Rk+ , and all ε ∈ R++ , let D(x, ε) := {y ∈ Rk+ | ||y − x|| < ε}. For all S ⊆ Rk+ , define intS := {x ∈ S | ∃ε ∈ R++ , D(x, ε) ⊆ S}. 3 Properties of Distributions of Opportunity Sets An opportunity set of an individual prescribes the range of alternatives from which she can choose. In social situations, however, choices made by individuals involved are often interdependent. Thus, it is too strong to require that,∑ for all x = (x1 , . . . , xn ) ∈ Rnk + , if xi ∈ Si for all i ∈ N , then x ∈ Z, i.e., i∈N xi ≤ ω. This requirement says that every individual can attain any consumption bundle in her opportunity set regardless of the choices of the other individuals. A distribution of opportunity sets (S1 , . . . , Sn ) satisfies this condition if and only if, there exists x̄ ∈ Rnk + such that x̄ ∈ Z, and, for all k i ∈ N , Si ⊆ {x ∈ R+ | x ≤ x̄}. Hence, we need to weaken the requirement in order to consider and compare reasonable distributions of opportunity sets. Our proposal is the following: Feasibility: For every i ∈ N and every xi ∈ ∂ + Si , there exists x−i ∈ Πj6=i ∂ + Sj such that (x1 , . . . , xn ) ∈ Z. Feasibility states that, for each individual, any choice from the boundary of her opportunity set is attainable for some choices of the other individuals from the boundaries of their respective opportunity sets. It is a very weak, reasonable requirement. Were it violated, some alternative ai of individual i from the boundary of her opportunity set is compatible with no combination of choices of the other individuals from the boundaries of their respective opportunity sets. Since we consider the case where objects of individual choice are goods (not bads), each individual would choose from the boundary of her opportunity set, given freedom of choice. Then, individual i can never attain the alternative ai . Our feasibility condition excludes such cases. As we will show, Walrasian distributions of opportunity sets satisfy this condition, and so do many other kinds of distributions. Figure 1 depicts 5 02 02 ❅ ❅ S1 ❅ S2 ❅ ❅ ❅ S2 S1 01 01 (a) (b) Figure 1: Examples of feasible distributions of opportunity sets several examples where each corresponding distribution of opportunity sets satisfies Feasibility. There are distributions of opportunity sets that do not satisfy Feasibility. Consider the following example. Let k = 2, n = 2 and ω = (2, 2). Let S1 = {z ∈ R2+ | z ≤ (1, 2)} = S2 . Then, (S1 , S2 ) does not satisfy Feasibility because ∂ + S1 = {(1, 2)} and ∂ + S2 = {(1, 2)}, and (1, 2) + (1, 2) = (2, 4) > ω. Given a feasible distribution, it may still leave rooms for improving opportunities of all individuals. Consider the following example. Let k = 2, n = 2 and ω = (4, 4). Let S1 = {z ∈ R2+ | z ≤ (2, 2)} = S2 . Then, (S1 , S2 ) satisfies Feasibility. However, we can improve both individuals’ opportunities. For example, consider S1′ = S1 ∪ {z ∈ R2+ | z ≤ (1, 3)} and S2′ = S2 ∪ {z ∈ R2+ | z ≤ (3, 1)}. Because ∂ + S1′ = {(2, 2), (1, 3)} and ∂ + S2′ = {(2, 2), (3, 1)}, the distribution (S1′ , S2′ ) is feasible, and it may be argued that S1′ offers more opportunities to individual 1 than S1 , and S2′ offers more opportunities to individual 2 than S2 . To understand our next property, let xi ∈ Si and xj ∈ Sj be the chosen consumption bundles of individuals i and j in their opportunity sets, respectively. Suppose that both xi and xj are on the boundaries of their opportunity sets respectively, and hence each individual cannot obtain more 6 goods by herself. Suppose, however, that there exist other alternatives yi ∈ Si and yj ∈ Sj such that yi +yj > xi +xj . Then, the two individuals will coordinate their actions, and do not choose xi and xj but choose yi and yj in order to obtain a larger total amount of goods. By appropriately redistributing the total among them, each individual can receive a greater amount of goods than xi or xj . Thus, coordination among individuals is beneficial in this case. Our next property says that such cases never arise, and hence coordination of choices among any individuals generates no surplus for them. No Coordination Surplus: For every M ⊆ N ,∑if xi ∈ ∂ + S∑ i for all i ∈ M , then there exists no (yi )i∈M ∈ Πi∈M Si such that i∈M yi > i∈M xi . We will show that Walrasian distributions of opportunity sets satisfy No Coordination Surplus later. There are many other distributions of opportunity sets that satisfy this property. For examle, let z̄ ∈ R2+ be given, and define Si = {z ∈ R2+ | z ≤ z̄} for every i ∈ N . Then, the distribution (S1 , . . . , Sn ) satisfies No Coordination Surplus. There are also distributions of opportunity sets that do not satisfy this property. Let M = {1, 2} ⊆ N , and define S1 = {z ∈ R2+ | z12 + z22 ≤ 4} = S2 . Let x1 = (0, 2) and x2 = (2, 0). Then, xi ∈ ∂ + Si for every i ∈ M , and x1 + x2 = (2, 2). Let y1 = (1.4, 1.4) ∈ S1 and y2 = (1.4, 1.4) ∈ S2 . Then, y1 + y2 = (2.8, 2.8) ≫ x1 + x2 . The last property requires that given any feasible allocation, among the n consumption bundles specified in the allocation, there must be at least one bundle belonging to the opportunity set of the corresponding individual. In other words, any feasible allocation must be “attainable” for some individual by his choice from his opportunity set. Minimum Attainability: x i ∈ Si . For every x ∈ Z, there exists i ∈ N such that Walrasian distributions of opportunity sets satisfy Minimum Attainability as we show later. There are many other distributions that satisfy this property as well as those that violate it. In Figure 1, case (b) satisfies Minimum Attainability whereas case (a) viloates it. 7 4 Characterizations of the Walrasian Distribution of Opportunity Sets We have introduced three reasonable properties of distributions of opportunity sets. Each of these properties is fairly weak, and there are many kinds of distributions of opportunity sets that satisfy it. However, requiring the three properties together singles out one particular type of distributions of opportunity sets, namely, Walrasian distributions with positive prices. First, we show that Walrasian distributions of opportunity sets with positive prices satisfy all these properties. Proposition 1 Every Walrasian distribution of opportunity set with positive prices satisfies Feasibility, No Coordination Surplus, and Minimum Attainability. Proof. Let (S1 , . . . , Sn ) ∈ S n be a Walrasian distribution of opportunity n sets with positive prices. Let p ∈ Rk++ and (m1 , . . . , mn ) ∈ R ∑+ be such that k Si = {xi ∈ R+ | p·xi ≤ mi and xi ≤ ω} for every i ∈ N , and i∈N mi = p·ω. Feasibility: Let i ∈ N and xi ∈ ∂ + Si be given. Then, p · xi = mi . Define v := ω − xi . Notice that v ≥ 0. If v = 0, then Sj = {0} for all j 6= i, and Si = X(ω). Clearly, this distribution of opportunity sets is feasible. Consider v > 0 next. For every j ∈ N with j 6= i, let αj ∈ R+ be such that p · (αj v) = mj , and define xj := αj v. Then, ∑ xj ∈ ∂ + Sj for every On the other ∑ j ∈ N. ∑ hand, we have ∑p·v = p·(ω −xi ) = h∈N mh −mi = j6=i mj = ∑j6=i p·(αj v). Hence, (1 −∑ j6=i αj )p · v =∑0. Since p · v > ∑0, it follows that j6=i αj = 1. Therefore, h∈N xh = xi + j6=i xj = xi + j6=i αj v = xi + v = ω. Thus, we have proven that (S1 , . . . , Sn ) is feasible. ∑ No Coordination Surplus: Let M ⊆ N be given. For every y ∈ j∈M Sj , ∑ there exists (xj )∑ that y = j∈M xj , and hence j∈M ∈ Πj∈M Sj such∑ ∑ ∑p · y = + Obviously, z ∈ j∈M Sj . j∈M p · xj ≤ j∈M mj . Let z ∈ j∈M ∂ Sj . ∑ + There exists (w ) ∈ Π ∂ S such that z = j j∈M j∈M j j∈M wj , and ∑ ∑ ∑thus p · z = j∈M p · wj = j∈M mj . Since ∑ p ≫ 0, there cannot exist y ∈ ∑ j∈M S+j with + y >∑z. Therefore, z ∈ ∂ ( j∈M Sj ). We have shown that j∈M ∂ Sj ⊆ ∂ + ( j∈M Sj ). Minimum Attainability: Suppose, on the contrary, that there exists a feasible allocation x ∈ Z such that ∈ N , xi ∈ / Si . Then, for ∑ for every i∑ every ∑ i ∈ N , p · xi > mi . Hence, i∈N p · xi ∑ > i∈N mi = p · ω. However, since i∈N xi ≤ ω and p ≫ 0, we must have i∈N p · xi ≤ p · ω, which is a 8 contradiction. Thus, for every x ∈ Z, there exists i ∈ N such that xi ∈ Si . Our main theorem is a characterization of the class of Walrasian distributions of opportunity sets with positive prices. Theorem 1 A distribution of opportunity sets (S1 , . . . , Sn ) ∈ S n satisfies Feasibility, No Coordination Surplus, and Minimum Attainability if and only if it is Walrasian distribution of opportunity sets with positive prices. The proof of Theorem 1 is long, and is relegated in the appendix. It should be noted that the properties in the above theorem are logically independent: (i) case (a) of Figure 1 satisfies Feasibility and No Coordination Surplus, but violates Minimum Attainability; (ii) case (b) of Figure 1 satisfies Feasibility and Minimum Attainability, but violates No Coordination Surplus; (iii) the distribution of the opportunity sets defined by setting Si = X(ω) for all i ∈ N satisfies No Coordination Surplus and Minimum Attainability, but violates Feasibility. 5 Conclusion In this paper, we have provided a new approach to economic systems by focusing on distributions of opportunity-freedom for individuals generated in the systems. In particular, we have axiomatically characterized the distributions of opportunity sets in the market system with three properties: Feasibility, No Coordination Surplus, and Minimum Attainability. Each property captures some characteristic of the market system. Feasibility reflects and captures the inter-dependence of opportunity sets of different individuals in a social setting. No Coordination Surplus is a necessary condition for decentralized decisions to be incentive compatible. Were it violated, a group of individuals would be able to “improve” each member’s situation by coordinating its decisions. Minimum Attainability requires the non-existence of a feasible allocation that is “unreachable” for all individuals in the following sense: for every feasible allocation, at least one individual has his consumption bundle specified by the feasible allocation belonging to his opportunity set. 9 For each of these three properties, there are many other distributions of opportunity sets that meet the other two properties. However, the only distributions that satisfy all the three properties are the distributions of opportunity sets generated by the market system. Two final remarks are in order. First, one may think that convexity of opportunity sets is a desirable and natural property. Convexity of opportunity sets is essential for the optimal choice correspondence to be continuous with respect to changes in opportunity sets. It is also intuitively natural if opportunity sets are regarded as the set of available consumption bundles during some period of time such as a week or a month: if two consumption plans are attainable in a month, then the average of the two plans should also be attainable. It is then interesting to note that this important and natural property, convexity, of opportunity sets is not imposed a priori, but is implied by the three properties in our theorem. (See Lemma 3 in the appendix.) Thanks to this result, we can fully utilize the separating hyperplane theorem to establish our main theorem. Second, there is no equity-type property (in any sense) imposed on a distribution of opportunity-freedom in our characterization theorem. In fact, the extreme distribution such that S1 = X(ω) and Si = {0} for all i 6= 1 satisfies all the three properties and it is Walrasian. This aspect reflects the very nature of the market system. No mechanism to realize an equitable distribution of opportunity sets is embedded in the market system itself. In order to consider equity in distributions of opportunity-freedom, we need to introduce a measure of equity. As a further and future study, it would be interesting to formally define notions of equity in distributions of opportunity sets and investigate their implications. Our approach to analyze economic systems based on distributions of opportunity sets has opened a new avenue for investigating and making comparisons of various economic systems. The present study has axiomatically characterized the distribution of opportunity sets generated by the market system. It would be interesting to investigate other economic systems. Acknowledgments We would like to thank Marc Fleurbaey for helpful comments and discussions. Financial support from the Ministry of Education, Culture, Sports, Science and Technology, Japan, through Grant-in-Aid for Scientific Research (B) 10 No. 21330044 and (S) No. 20223001 and the grant for the Global Center of Excellence Program is gratefully acknowledged. References Arrow, K. (1951) An extension of the basic theorems of classical welfare economics. In J. Neyman (ed.), Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability. Berkeley: University of California Press. Barberà, S., W. Bossert and P.K. Pattanaik (2004) Ranking sets of objects. In S. Barberà, P.J. Hammond and C. Seidl (eds.), Handbook of Utility Theory, Volume 2: Extensions. Dordrecht: Kluwer Academic Publishers. Basu, K. (1987) Achievements, capabilities and the concept of well-being. Social Choice and Welfare 4: 69-76. Buchanan, J.M. (1986) Liberty, Market and the State. New York: Wheatsheaf. Debreu, G. (1951) The coefficient of resource utilization, Econometrica 19: 273-292. Debreu, G. (1954) Valuation equilibrium and Pareto optimum. Proceedings of the National Academy of Sciences of the United States of America 40: 588-592. Friedman, M. (1962) Capitalism and Freedom. Chicago Press. Chicago: University of Hayek, F.A. (1976) Law, Legislation,and Liberty. Chicago: University of Chicago Press. Hurwicz, L., E. Maskin and A. Postlewaite (1982) Feasible implementation of social choice correspondences by Nash equilibria. Unpublished mimeo. 11 Hurwicz, L., E. Maskin and A. Postlewaite (1995) Feasible Nash implementation of social choice rules when the designer does not know endowments or production sets. In J. O. Ledyard (ed.), The Economics of Informational Decentralization: Complexity, Efficiency and Stability. Amsterdam: Kluwer Academic Publishers. Kornai, J. (1988) Individual freedom and the reform of socialist economy. European Economic Review 32: 233-267. Lindbeck, A. (1988) Individual freedom and welfare state policy. European Economic Review 32: 295-318. Nozick, R. (1974) Anarchy, State and Utopia. Oxford: Blackwell. Pattanaik, P. K. (1994) Rights and freedom in welfare economics. European Economic Review 38: 731-738. Pattanaik, P. K. and Y. Xu (1990) On ranking opportunity sets in terms of freedom of choice. Recherches Economiques de Louvain 56: 383-390. Pattanaik, P. K. and Y. Xu (2000) On ranking opportunity sets in economic environments. Journal of Economic Theory 93: 48-71. Sen, A.K. (1988) Freedom of choice: concept and content. European Economic Review 32: 269-294. Sen, A.K. (1993) Markets and freedoms: achievements and limitations of the market mechanism in promoting individual freedoms. Oxford Economic Papers 45: 519-541. Appendix To prove Theorem 1, it is useful to introduce the following property: Tightness: A distribution of opportunity sets (S1 , . . . , Sn ) ∈ S n is tight if for every i ∈ N , and every xi ∈ ∂Si , there exists x−i ∈ Πj6=i ∂Sj such that ∑ h∈N xh = ω. Notice that Tightness implies Feasibility. The proof of Theorems 1 relies on the following sequence of lemmas. 12 Lemma 1 If S := (S1 , . . . , Sn ) ∈ S n satisfies Feasibility and Minimum Attainability, then it satisfies Tightness. Proof. Suppose that S = (S1 , . . . , Sn ) ∈ S n satisfies Feasibility and Minimum Attainability, but violates Tightness. Then, ∑ there exist i ∈ N and xi ∈ ∂ + Si such that for all x−i ∈ Πj6=i ∂ + Sj , xi + j6=i xj 6= ω. By Feasibility, ∑ however, there exists y−i ∈ Πj6=i ∂ + Sj with xi + j6=i yj < ω. Define z ∈ Rnk + ∑ ∑ by zi := xi + n1 (ω − xi − j6=i yj ) > xi and zj := yj + n1 (ω − xi − j6=i yj ) > yj . Then, z is a feasible allocation, but for all h ∈ N , zh ∈ / Sh . This contradicts the supposition that S satisfies Minimum Attainability. For each xi ∈ Rk+ and each ε > 0, define V (xi , ε) := {yi ∈ Rk+ | |yiℓ −xiℓ | < ε for every ℓ ∈ K}. Recollect that X(ω) = {a ∈ Rk+ | a ≤ ω}. Lemma 2 Suppose that S := (S1 , . . . , Sn ) ∈ S n satisfies Feasibility and Minimum Attainability. For every i ∈ N and every xi ∈ Si , if V (xi , ε) ∩ (Rk+ \ Si ) ∩ X(ω) 6= ∅ for every ε > 0, then xi ∈ ∂ + Si . Proof. Suppose that S = (S1 , . . . , Sn ) ∈ S n satisfies Feasibility and Minimum Attainability. Since S is feasible, Si ⊆ X(ω) for every i ∈ N . By Lemma 1, S satisfies Tightness. Suppose, on the contrary, that there exist i ∈ N and xi ∈ Si such that V (xi , ε) ∩ (Rk+ \ Si ) ∩ X(ω) 6= ∅, for every ε > 0, + and yet xi ∈ / ∂ + Si . Then, there exists z∑ i ∈ ∂ Si with zi > xi . By Tightness, there is z−i ∈ Πj6=i ∂ + Sj such that zi + j6=i zj = ω. 1 Define the allocation t ∈ Rnk + by ti := xi and tj := zj + n−1 (zi − xi ) > zj for every j 6= i. Then, t is a feasible allocation and tj ∈ / Sj for every j 6= i. Let K(xi ) := {ℓ ∈ K | ωℓ > xiℓ }. If K(xi ) = ∅, then xiℓ = ωℓ for every ℓ ∈ K, and comprehensiveness of Si implies Si = X(ω). Hence, we have (Rk+ \ Si ) ∩ X(ω) = ∅, which is a contradiction. Thus, K(xi ) 6= ∅. For each ε > 0, define the consumption bundle xi (ε) ∈ Rk+ by { xiℓ + ε if ℓ ∈ K(xi ) xiℓ (ε) = xiℓ otherwise. Claim: xi (ε) ∈ / Si for every ε > 0. Suppose, on the contrary, that xi (ε) ∈ Si for some ε > 0. Let yi ∈ V (xi , ε). If yiℓ > ωℓ for some ℓ ∈ K, then yi ∈ / X(ω). Otherwise, yiℓ ≤ ωℓ for 13 all ℓ ∈ K. Then, yi ≤ xi (ε) ∈ Si . It follows from comprehensiveness of Si that yi ∈ Si and hence yi ∈ / (Rk+ \ Si ). Therefore, we have V (xi , ε) ∩ (Rk+ \ Si ) ∩ X(ω) = ∅, which is a contradiction. Thus, the claim has been proven. For each ℓ ∈ K(xi ), define J(ℓ) := {j ∈ N ∑ \ {i} | tjℓ > 0}. Since ℓ ∈ K(xi ), we have tiℓ = xiℓ < ωℓ . Together with h∈N thℓ = ωℓ , it follows that J(ℓ) 6= ∅. Let εℓ > 0 be such that tjℓ − εℓ ≥ 0 for every j ∈ J(ℓ). For every j ∈ N \ {i}, since Sj is closed and tj ∈ / Sj , it follows that there exists εj > 0 such that V (tj , εj ) ∩ Sj = ∅. Let ε∗ := min{ min εℓ , min εj }. ℓ∈K(xi ) j∈N \{i} Define the allocation v ∈ Rnk + by { ∗ xiℓ + |J(ℓ)| ε2 if ℓ ∈ K(xi ) viℓ = xiℓ otherwise. and for every j 6= i, { ∗ tjℓ − ε2 if ℓ ∈ K(xi ) and j ∈ J(ℓ) vjℓ = tjℓ otherwise. Then, v is a feasible allocation. By the above claim, we have vi ∈ / Si . For ε∗ ∗ every j ∈ N \ {i}, because 2 < ε ≤ εj and V (tj , εj ) ∩ Sj = ∅, it follows that vj ∈ / Sj . This contradicts the supposition that S satisfies Minimum Attainability. Lemma 3 If S := (S1 , . . . , Sn ) ∈ S n satisfies Feasibility, No Coordination Surplus and Minimum Attainability, then Si is convex for all i ∈ N . Proof. Suppose that S = (S1 , . . . , Sn ) ∈ S n satisfies Feasibility, No Coordination Surplus and Minimum Attainability. By Lemma 1, S satisfies Tightness. Suppose, on the contrary, that Si is not convex for some i ∈ N . Then, there exist xi , yi ∈ ∂ + Si such that zi := 21 (xi +yi ) ∈ / Si . Since xi , yi ∈ Si and S is feasible, it follows that xi , yi ∈ X(ω) and hence zi ∈ X(ω) as well. Let α := max{α′ ∈ R | α′ zi ∈ Si }. Because Si is compact and comprehensive, such α exists and 0 ≤ α < 1. Let vi := αzi . It is clear that for every ε > 0, V (vi , ε) ∩ (Rk+ \ Si ) ∩ X(ω) 6= ∅. By Lemma 2, vi ∈ ∂ + Si . It follows from Tightness that there exist x−i , y−i , v−i ∈ Πj6=i ∂ + Sj such that ∑ xi + xj = ω j6=i 14 yi + ∑ yj = ω vi + ∑ vj = ω. j6=i j6=i Since vi < 21 (xi + yi ), we have j6=i vj = ω − vi > ω − 12 (xi + yi ). Then, we have ∑ ∑ 1 vi + xj < (xi + yi ) + xj 2 j6=i j6=i ∑ 1 1 = xi + xj − xi + yi 2 2 j6=i ∑ yi + ∑ vj j6=i 1 1 = ω − xi + yi 2 2 1 > yi + ω − (xi + yi ) 2 1 1 = ω − xi + yi 2 2 Hence, yi + ∑ vj > vi + j6=i + ∑ xj . j6=i ∑ Therefore, (vi , x−i ) ∈ / ∂ ( h∈N Sh ). But this means that S does not satisfy No Coordination Surplus, which is a contradiction. Lemma 4 If S := (S1 , . . . , Sn ) ∈ S n satisfies Feasibility, No Coordination Surplus and Minimum Attainability, then for all i ∈ N , there exist pi ∈ Rk+ , pi 6= 0 and mi ∈ R+ such that ∂ + Si ⊆ B ∗ (pi , mi ) ∩ X(ω). Proof. Assume that S = (S1 , . . . , Sn ) ∈ S n satisfies Feasibility, No Coordination Surplus and Minimum Attainability. By Lemma 1, S satisfies Tightness. It follows from Lemma 3 that Si is convex for ∑every i ∈ N . Let i ∈ N . By Feasibility, ∂ + Si ⊂ X(ω). Define Ti := j∈N \{i} Sj and k Vi := {ω} − Ti := {xi ∈ R | ∃zi ∈ Ti , xi = ω − zi }. Since Sj is convex for all j ∈ N \ {i}, Ti is convex, and so is (−Ti ). Hence, Vi is convex. By No Coordination Surplus, ∑ ∂ + Sj ⊆ ∂ + Ti . (1) j∈N \{i} 15 It follows from Tightness and (1) that (A) for all yi ∈ ∂ + Si , there exists zi ∈ ∂ + Ti such that yi + zi = ω. In order to show that Vi ∩ intSi = ∅, (2) suppose, to the contrary, that there exists xi ∈ Vi ∩ intSi . Since xi ∈ intSi , there exists x′i ∈ ∂ + Si with x′i > xi . From (A), there exists zi ∈ ∂ + Ti such that x′i + zi = ω. On the other hand, since xi ∈ Vi , there exists zi′ ∈ Ti with xi = ω − zi′ . Together we have zi′ = ω − xi > ω − x′i = zi , which contradicts zi ∈ ∂ + Ti . Thus, (2) must hold true. By the separating hyperplane theorem, there exist pi ∈ Rk , pi 6= 0 and mi ∈ R such that Si ⊆ {xi ∈ Rk+ | pi · xi ≤ mi } and Vi ⊆ {xi ∈ Rk+ | pi · xi ≥ mi }. Since Si is comprehensive, it follows that pi > 0. To complete the proof, we need to show that ∂ + Si ⊆ {xi ∈ Rk+ | pi · xi = mi } (3) Suppose, to the contrary, that there exists si ∈ ∂ + Si such that pi · si < mi . From (A) above, there exists ti ∈ ∂ + Ti with si + ti = ω. Then, p · (ω − ti ) = p · si < mi . But since ω − ti ∈ Vi , we have p · (ω − ti ) ≥ mi . This is a contradiction. Thus, the relation (3) must hold true. Lemma 5 If S := (S1 , . . . , Sn ) ∈ S n satisfies Feasibility, No Coordination Surplus and Minimum Attainability, then, for all i ∈ N , there exist pi ∈ Rk++ and mi ∈ R+ such that Si = B(pi , mi ) ∩ X(ω). Proof. Suppose that S = (S1 , . . . , Sn ) ∈ S n satisfies Feasibility, No Coordination Surplus and Minimum Attainability. By Lemma 1, S satisfies Tightness. Let N 0 := {i ∈ N | Si = {0}} and N ∗ := N \ N 0 . By Tightness, N ∗ 6= ∅. If i ∈ N 0 , then it is obvious that Si = B(pi , 0) ∩ X(ω) for any pi ∈ Rk++ . If N ∗ = {i}, then Tightness implies that Si = X(ω). Then, it is clear that for any pi ∈ Rk++ , Si = B(pi , mi ) ∩ X(ω) with mi = pi · ω. In the rest of the proof, we assume that |N ∗ | ≥ 2. Let i ∈ N ∗ . By Lemma 4, there exists pi ∈ Rk+ and mi ∈ R+ such that ∂ + Si ⊆ B ∗ (pi , mi ) ∩ X(ω). Step 1: mi > 0. Suppose, to∑the contrary, that mi = 0. Let M := N ∗ \ {i} 6= ∅. We will show that j∈M Sj ⊇ X(ω). Suppose, to the contrary, that there exists 16 ∑ ∑ y0 ∈ Rk+ such that y0 ≤ ω, but y0 ∈ / j∈M Sj . Since j∈M Sj is closed, there ∑ exists z0 ∈ Rk+ with z0 ≪ ω and z0 ∈ / j∈M Sj . For each j ∈ M , define λj ∈ R+ by λj := max{λ ∈ R+ | λz0 ∈ Sj }. ∑ ∑ ∑ ∑ By definition, ∑ j∈M λj z0 = ( j∈M λj )z0 ∈ j∈M Sj . Since z0 ∈ / j∈M Sj , we must have j∈M λj < 1. Define ∑ 1 − j∈M λj > 0. ε := |N ∗ | − 1 Define an allocation z ∈∑ Rnk + by zj := (λj + ε)z0 for each j ∈ M , and zi := ω − z0 ≫ 0. Then, j∈N zj = ω. However, from the definition of λj , we have zj ∈ / Sj for all j ∈ M . Because Si ⊆ B(p, 0) and pi > 0, it follows that zi ∈ / Si . This is a contradiction with Minimum Attainability of S. ∑ Thus, we have shown that S ⊇ X(ω). By No Coordination Surplus, j∈M j ∑ ∑ + + + ∂ j ⊇ j∈M S∑ j∈M ∂ Sj . Therefore, there exists no (yj )j∈M ∈ Πj∈M ∂ Sj such that / N 0 , there exists j∈M yj < ω. On the other hand, since i ∈ + xi ∈ ∂ Si with∑xi > 0. By Tightness, ∑ there exists (yj )j∈M ∈ Πj∈M ∂ + Sj such that xi + j∈M yj = ω, and hence j∈M yj = ω − xi < ω. This is a contradiction. Step 2: Si = B(pi , mi ) ∩ X(ω). In view of Lemma 4, it is enough to show that Si ⊇ B(pi , mi ) ∩ {xi ∈ Rk+ | xi ≤ ω}. Suppose, to the contrary, that there exists xi ∈ Rk+ such that xi ∈ B(pi , mi ) ∩ X(ω) but xi ∈ / Si . Since Si is comprehensive, for all x′i ∈ Rk+ with x′i ≥ xi , x′i ∈ / Si . From Step 1, mi > 0. Hence one can find x′i ∈ B(pi , mi ) ∩ X(ω) such that x′i ∈ / Si and pi · x′i > 0. Because Si is closed k ∗ k in R+ , there exists xi ∈ R+ such that x∗i ∈ / Si and pi · x∗i < pi · x′i ≤ mi . (4) ∑ Next we will show that (ω − x∗i ) ∈ / j∈M Sj . Suppose, to the contrary, that ∑ ∑ (ω−x∗i ) ∈ j∈M Sj . Since {ω}− j∈M Sj ⊆ {x0 ∈ Rk+ | pi ·x0 ≥ mi } from the proof of Lemma 4, we have pi · [ω − (ω − x∗i )] ≥ mi . Hence, pi · x∗i ≥ m ∑i , which is in contradiction with (4). Thus, it must be true that (ω − x∗i ) ∈ / j∈M Sj . Then, by a similar argument to the proof of Step 1, we can find a feasible allocation z ∈ Rnk / Sh for all h ∈ N . This is a contradiction + such that zh ∈ with Minimum Attainability of S. 17 Step 3: pi ≫ 0. Suppose that for some good ℓ ∈ {1, . . . , k}, piℓ = 0. Then, for all xi ∈ ∂ + Si , xiℓ = ωℓ . By Tightness, (B) for all j ∈ N ∗ with j 6= i, and all xj ∈ ∂ + Sj , xjℓ = 0. On the other hand, by applying Steps 1 and 2 to each j ∈ N ∗ with j 6= i, we obtain that for all j ∈ N ∗ with j 6= i, there exist pj ∈ Rk+ and mj > 0 such that Sj = B(pj , mj ) ∩ X(ω). Since mj > 0, there exists xj ≫ 0 such that pj · xj ≤ mj , and hence xj ∈ Sj . But then, there exists x′j ∈ ∂ + Sj with x′j ≥ xj ≫ 0. This contradicts (B). Thus, it must be true that pi ≫ 0. This completes the proof. Lemma 6 If S := (S1 , . . . , Sn ) ∈ S n satisfies Feasibility, No Coordination Surplus and Minimum Attainability, then there exist p ∈ Rk++ and n (m ∑ 1 , . . . , mn ) ∈ R+ such that Si = B(p, mi ) ∩ X(ω) for all i ∈ N , and i∈N mi = p · ω. Proof. Assume that S = (S1 , . . . , Sn ) ∈ S n satisfies Feasibility, No Coordination Surplus and Minimum Attainability. By Lemma 1, S satisfies Tightness. Let N 0 := {i ∈ N | Si = {0}} and N ∗ := N \ N 0 . By Tightness, N ∗ 6= ∅. It follows from Lemma 5 that for each i ∈ N ∗ , there exist pi ∈ Rk++ and mi ∈ R++ such that Si = B(pi , mi ) ∩ X(ω). Claim: for all i, j ∈ N ∗ , and all s, t ∈ K, pis pjs = . pit pjt Suppose, on the contrary, that pis pit K. Without loss of generality, let pjs pjt 6= pis pit for some i, j ∈ N ∗ and some s, t ∈ > pjs . pjt Since pi , pj ≫ 0, mi , mj > 0, and pi · ω > mi , there exist ε > 0, ai ∈ ∂ + Si and bj ∈ ∂ + Sj such that i a′i := (ai1 , . . . , ais − ε, . . . , ait + ppsi ε, . . . , ain ) ∈ ∂ + Si and b′j := (bj1 , . . . , bjs + ε, . . . , bjt − pjs ε, . . . , bjn ) pjt t + ∈ ∂ Sj . Since S satisfies No Coordination Surplus, it follows that ai + bj ∈ ∂ + (Si + Sj ). On the other hand, because a′i ∈ Si and b′j ∈ Sj , we have a′i + b′j ∈ (Si + Sj ). Observe that a′ir + b′jr = air + bjr i for all r 6= t, and a′it + b′jt = ait + bjt + [ ppsi − t 18 pjs )]ε pjt > ait + bjt . Therefore, ai + bj 6∈ ∂ + (Si + Sj ), which contradicts ai + bj ∈ ∂ + (Si + Sj ). Thus, it must j i be true that ppsi = ppsj for all i, j ∈ N ∗ , and all s, t ∈ K. t t pi i 1 1 Choose i ∈ N ∗ , and define p := (1, p2i , . . . , ppni ) ∈ Rk++ . Then, it follows from the above claim that pj = pj1 p for all j ∈ N ∗ . For each j ∈ N ∗ , define m′i := mi /pj1 . Then, we have Sj = B(p, m′i ) ∩ X(ω) for all j ∈ N ∗ . For all j ∈ N 0 , let m′j = 0. Then, it is clear that for all j ∈ N 0 , Sj = {0} = B(p, m′j ) ∩ X(ω). Let i ∈ N ∗ , and let ∑ xi ∈ ∂ + Sj . By Lemma 1, there exist (xj )j∈N \{i} ∈ + Πj∈N \{i} ∂Sj such that h∈N xh = ω. For all h ∈ N , because ∂ Sh = B ∗ (p, m′h ) ∩ X(ω), it follows that p · xh = m′h . Then, ∑ ∑ p · xh m′h = h∈N h∈N = p· ∑ xh h∈N = p · ω. This completes the proof. Proof of Theorem 1: By Proposition 1, every Walrasian distribution of opportunity sets with positive prices satisfies Feasibility, No Coordination Surplus and Minimum Attainability. It follows from Lemma 6 that if a distribution of opportunity sets (S1 , . . . , Sn ) ∈ S n satisfies Feasibility, No Coordination Surplus and Minimum Attainability, then it is Walrasian with positive prices. 19