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Competitive equilibria and the grand coalition
Available online at www.sciencedirect.com
Journal of Mathematical Economics 44 (2008) 697–706
Competitive equilibria and the grand coalition Carlos Herv´es-Beloso a,∗ , Emma Moreno-Garc´ıa b b
a RGEA, Facultad de Econ´ omicas, Universidad de Vigo, Spain Facultad de Econom´ıa y Empresa, Universidad de Salamanca, Spain
Received 17 January 2006; received in revised form 7 November 2006; accepted 16 November 2006 Available online 8 December 2006
Abstract This paper provides a new characterization of competitive equilibrium allocations based on the veto mechanism. The main theorem shows that, in pure exchange economies with a continuum of non-atomic agents, the competitive equilibria can be characterized by strengthening the veto power of the grand coalition, formed by all the agents in the economy. The welfare theorems are obtained as easy corollaries of our main result. Furthermore, in the case of finite economies, we show that the characterizations of the Walrasian equilibria based on the veto power of the grand coalition are particular cases of our main theorem. © 2007 Elsevier B.V. All rights reserved. JEL Classification: D51 Keywords: Competitive equilibrium; Continuum economies; Core–Walras equivalence; Dominated allocations; Grand coalition
1. Introduction We consider pure exchange economies with a continuum of non-atomic agents and finitely many commodities Aumann (1964, 1966). In these economies, competitive allocations can be characterized by means of the veto mechanism. Precisely, Aumann (1964) showed that the set of competitive allocations coincides with the core of the economy (Core–Walras equivalence). That is, competitive allocations are those feasible allocations that are not blocked by any coalition of agents. Actually, for the equivalence is essential the consideration of the veto power of infinitely many coalitions. In fact, in the case of a finite economy, where only a finite number of coalitions can be formed, the characterization of the Walrasian allocations using the veto mechanism is only asymptotic (Debreu and Scarf, 1963). The main result in this paper is a characterization of the competitive equilibrium allocations in terms of the veto power of the grand coalition. That is, instead of infinitely many coalitions, we only consider the blocking power of the grand coalition formed by all the agents, but exercised in a family of economies obtained by perturbing the agents’ initial endowments. Thus, this is a kind of Core–Walras equivalence theorem in which we do not consider the veto power of infinitely many coalitions, but the veto power of a single coalition in infinitely many economies. We define non-dominated allocations in an economy as those allocations (feasible or not) that are not blocked by the grand coalition. We remark that if an allocation is competitive, then it is efficient and therefore it is not dominated in any economy in which the endowments are redistributions of the total resources of the initial economy. We prove ∗
Corresponding author. Tel.: +34 986225126; fax: +34 986812401. E-mail addresses: [email protected] (C. Herv´es-Beloso), [email protected] (E. Moreno-Garc´ıa).
0304-4068/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.jmateco.2006.11.002
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that competitive allocations are robustly efficient.1 That is, a competitive allocation f is a non-dominated allocation not only in the initial economy but in all economies obtained by modifying, in the direction of f, the initial endowments of any coalition of agents. Reciprocally, robustly efficient allocations are competitive. Indeed, the results of Schmeidler (1972) and Vind (1972), which show that non-competitive allocations are those blocked by coalitions of a given measure, allow us to give a sharper interpretation. Actually we show that in continuum economies the competitive allocations are characterized as allocations that are non-dominated in any economy obtained by a slight perturbation of the real initial endowments of the agents belonging to either arbitrarily small coalitions, arbitrarily large coalitions, or coalitions of a given measure. This characterization of competitive allocations by the veto power of the grand coalition leads us to obtain the first and second welfare theorems as easy corollaries. Moreover, applying our theorem to a continuum economy with n different types of agents we obtain, as a particular case, the characterization of the Walrasian equilibrium allocations for finite pure exchange economies showed by Herv´es-Beloso et al. (2005a). Finally, we explore the case of infinitely many commodities. We prove that, under the hypothesis which guarantee the Core–Walras equivalence and the extension of Vind’s (1972) result, our main theorem also holds for economies where the commodity space is the space of bounded sequences ∞ . The rest of the paper is as follows. In Section 2 we set the model and state and comment the assumptions. In Section 3 the main result is presented and proved. Section 4 addresses the particular case of a pure exchange economy with n agents, recovering previous characterizations of Walrasian equilibrium allocations. Finally in Section 5 we consider economies with infinitely many commodities. 2. The model Consider a pure exchange economy E with a continuum of agents. The commodity space is the Euclidean space R . The space of agents is represented by the measure space (I, A, μ), where I is the real interval [0, 1], A is the Lebesgue σ-algebra of subsets of I and μ is the Lebesgue measure. Each agent t ∈ I is characterized by her consumption set R+ , her initial endowment ω(t) ∈ R+ , and her preference relation t . As in Aumann (1964), we state the following assumptions on endowments and preference relations: (H.1). The mapping ω : I → R+ , that associates to each agent her initial endowment, is integrable and dμ(t) 0.
I
ω(t)
(H.2). For every consumer t ∈ I, the preference relation t is continuous and strictly monotone. (H.3). The preferences are measurable ({t ∈ I|xt y} is a measurable set of agents for all bundles x, y ∈ R+ ). (H.1) is an assumption about boundedness of endowments which asserts that each of the commodities is actually present in the market. A preference relation is strictly monotone if x ≥ y and x = y implies that x y. Note that (H.2) states the usual continuity assumption on preferences and also requires a desirability condition that says that each trader always wants more of every commodity. Strict monotonicity on preferences is required by Schmeidler (1972) and Vind (1972) to show characterizations of the core in atomless economies that we will use to give a sharper interpretation of our main theorem. (H.3) is a measurability assumption which is only of technical significance and constitutes no real economic restriction. This measurability assumption is the same used by Vind (1972) and it is slightly different than the one required by Aumann (1964) which allows for the comparison of arbitrary allocations and not simply for the comparison of constant allocations. As it was remarked by Ostroy and Zame (1994) both measurability requirements are equivalent in several scenarios including the case of finite dimensional commodity spaces. These hypothesis are required to show Aumann’s (1964) core equivalence result. Actually, as in Aumann’s (1964) work, the preferences are not assumed to be complete and nor even transitive. 1
See Definition 3.2.
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Then, the economy we consider is an atomless economy which defined by E ≡ ((I, A, μ), R+ , ω(t), t , t ∈ I) and satisfies standard assumptions regarding preferences (which may be neither complete nor transitive) and initial endowments. An allocation is a μ-integrable function f : I → R+ . An allocation f is feasible if I f (t) dμ(t) ≤ I ω(t) dμ(t). A price system is an element of Δ, where Δ denotes the ( − 1)-dimensional simplex of R+ , that is, = {p ∈ R+ such that h=1 ph = 1}. A competitive equilibrium for the economy E is a pair (p, f ), where p is a price system and f is a feasible allocation such that, for almost every agent t, the bundle f (t) maximizes the preference relation t in the budget set Bt (p) = {y ∈ R+ such thatp · y ≤ p · ω(t)}. A coalition of agents is any positive measure subset of the set of individuals. A feasible allocation belongs to the core of the economy if it is not blocked by any coalition of agents. A coalition S blocks an allocation f via another allocation g in the economy E if: (i) S g(t) dμ(t) ≤ S ω(t) dμ(t) and (ii) g(t)t f (t) for almost all t ∈ S. Let us denote by Core(E) the set of allocations which belong to the core of the economy E. It is easy to show that any Walrasian allocation belongs to the core. Moreover, Aumann (1964) showed that, under the hypothesis (H.1)–(H.3), the core coincides with the set of competitive equilibrium allocations in continuum economies with a finite dimensional commodity space. This Core–Walras equivalence will be used in the proof of our main result. 3. The main result In this section we state our main result which provides a characterization of Walrasian equilibria in terms of the veto power of the grand coalition. For this, we set the definition of robustly efficient allocation and we adapt to continuum economies the notion of dominated allocations stated for finite economies in Herv´es-Beloso et al. (2005a). Definition 3.1. An allocation h (feasible or not) is dominated (or blocked by the grand coalition) in the economy E if there exists a feasible allocation g such that g(t)t h(t) for almost all t ∈ I. Given a coalition S, a feasible allocation f, and a real number α, with 0 ≤ α ≤ 1, let E(S, f, α) be a continuum exchange economy which coincides with E except for the initial endowment allocation that is given by ω(t) if t ∈ I \ S ω(S, f, α)(t) = (1 − α)ω(t) + αf (t) if t ∈ S That is, the continuum economy E(S, f, α) is the same as E when α = 0 and otherwise, E(S, f, α) coincides with E except for the initial endowment of the agents in the coalition S that is given by the convex combination defined by α of f and ω. Then, the economy E(S, f, α) is defined by E(f, S, α) ≡ ((I, A, μ), R+ , ω(S, f, α)(t), t , t ∈ I). Note that I ω(S, f, α)(t) dμ(t) = I ω(t) dμ(t) − α S (ω(t) − f (t)) dμ(t). Therefore, when the size of the coalition S is either arbitrarily small or arbitrarily big, the total resources of the economy E(S, f, α) are arbitrarily close to those of the former economy E. Moreover, if αn is a sequence converging to zero, then ω(S, f, αn )(t) converges to ω(t) for n every t ∈ I and I ω(S, f, α )(t) dμ(t) converges to I ω(t) dμ(t). Definition 3.2. A feasible allocation f is robustly efficient in the economy E if f is a non-dominated allocation in every economy E(S, f, α). We can interpret the perturbed economies E(S, f, α) as different ways to get f starting from the initial allocation ω. Then, an allocation f is robustly efficient if it is not blocked by the grand coalition in any economy which is a path from ω to f.
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Furthermore,let f be a dominated allocation in the economy E(S, f, α), with α < 1. Consider the measure ν defined as ν(A) = (μ(A), A (ω(t) − f (t)) dμ(t)), for every A ⊂ S. Applying Liapunov’s convexity theorem to the vector measure ν, restricted to S, we obtain that there exists S ⊂ S with μ(S ) = αμ(S) and S (ω(t) − f (t)) dμ(t) = α S (ω(t) − f (t)) dμ(t). That is, if f is dominated in the economy E(S, f, α), with α < 1, then there exists S ⊂ S such that f is also dominated in the economy E(S , f, 1). The proof of our main theorem requires the following Lemma. Lemma 3.1. Let f be an allocation that is blocked by the coalition S via g. Then, given any α ∈ (0, 1), there exists an allocation h : S → R+ such that h(t)t f (t) for every t ∈ S and S h(t) dμ(t) = S (αg(t) + (1 − α)f (t)) dμ(t). This result is used by Vind (1972) who refers the reader to a previous paper on preferences (Vind (1964)). Here, for the sake of completeness, we state a direct proof. Proof. Consider the vectorial measure ν(A) = (μ(A), A (g(t) − f (t)) dμ(t)) restricted to measurable subsets of the coalition S. Let anyα ∈ (0, 1). Applying Liapunov’s convexity theorem we obtain that there exits A ⊂ S such that μ(A) = αμ(S) and A (g(t) − f (t))dμ(t) = α S (g(t) − f (t)) dμ(t). By continuity and measurability of preferences, there exist g˜ : A → R+ and δ > 0 such that A g˜ (t) dμ(t) = ˜ (t)t f (t) for every t ∈ A. (See Vind (1972)). A g(t)f (t) dμ(t) − δ and g Let the allocation h : S → R+ defined as follows: ⎧ g˜ (t) if t ∈ A ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ h(t) = ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ δ ⎪ ⎩ f (t) + if t ∈ S \ A μ(S \ A) By monotonicity of preferences h(t)t f (t) for every t ∈ S. Moreover,
δ f (t) + dμ(t) h(t) dμ(t) = g˜ (t) dμ(t) + μ(S \ A) S A S\A g(t) dμ(t) − δ + f (t) dμ(t) − f (t) dμ(t) + δ = A
=α =
S
S
S
(g(t) − f (t)) dμ(t) +
A
S
f (t) dμ(t)
(αg(t) + (1 − α)f (t)) dμ(t).
We remark that, applying the previous Lemma, we can conclude that if an allocation f is dominated in the economy E(S, f, α), then f is also dominated in any economy E(S, f, β), for every β ≤ α. Theorem 3.1. Let us consider a continuum economy E satisfying assumptions (H.1)–(H.3). An allocation f is a competitive equilibrium allocation in E if and only if f is robustly efficient. Proof. Let (p, f ) be a competitive equilibrium for the economy E. Suppose that there exist a coalition S and a number α ∈ (0, 1] such that f is dominated in the economy E(S, f, α). Then, there exists g : I → R+ such that (i) I g(t) dμ(t) ≤ I ω(S, f, α)(t) dμ(t), and (ii) g(t)t f (t) for almost all agent t ∈ I. Since f is a competitive equilibrium allocation in the economy E, we have that p · f (t) ≤ p · ω(t) for almost all agent t ∈ I and from condition (ii) we deduce that p · g(t) > p · ω(t) ≥ p · f (t), for almost all agent t ∈ I.
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Multiplying the above inequalities by (1 − α) and by α, respectively, we obtain that the following inequalities hold for almost all t ∈ S: p · (1 − α)g(t) > p · (1 − α)ω(t)
and p · αg(t) > p · αf (t)
Thus, p · g(t) > p · ((1 − α)ω(t) + αf (t)) for almost all agent t ∈ S. Therefore, we have that p · g(t) dμ(t) > p · ω(t) dμ(t) + p · ((1 − α)ω(t) + αf (t)) dμ(t) = p · ω(S, f, α)(t) dμ(t) I
I\S
S
I
which is a contradiction with (i), that is, a contradiction with the feasibility of g in the economy E(S, f, α). Let f be a feasible allocation in E and assume that f is a non-dominated allocation for every economy E(A, f, α). In particular f is Pareto optimal. Assume that f is not a competitive equilibrium allocation for the economy E. Then, applying the Core–Walras equivalence, we have that f does not belong to the core of the continuum economy E. That is, there exists a coalition S ⊂ I, μ(S) < μ(I), and there exists g : S → R+ , such that (i) S g(t) dμ(t) ≤ S ω(t) dμ(t) and (ii) g(t)t f (t) for every t ∈ S. Then, by continuity and measurability of preferences, we can take g such that S g(t) dμ(t) ≤ S ω(t) dμ(t) − δ, with δ > 0. By the previous Lemma, given any α ∈ (0, 1), there exists an allocation h : S → R+ such that S h(t) dμ(t) = S (αg(t) + (1 − α)f (t)) dμ(t) and h(t)t f (t) for every t ∈ S. Let us consider the allocation z : I → R+ given by ⎧ if t ∈ S ⎨ h(t) α z(t) = δ if t ∈ I \ S ⎩ f (t) + μ(I \ S) Note that μ(I \ S) > 0 and that by monotonicity of preferences z(t)t f (t) for almost every t ∈ I. By construction, we obtain (z(t) − ω(I \ S, f, α)(t)) dμ(t) = ((1 − α)f (t) + αg(t)) dμ(t) − ω(t) dμ(t) I
+
S
α δ μ(I \ S)
I\S
S
S
dμ(t) −
(1 − α)(f (t) − ω(t)) dμ(t) +
−
S
f (t) +
I\S
=
I\S
(ω(I \ S, f, α)(t)) dμ(t)
α(g(t) − ω(t)) dμ(t) +
((1 − α)ω(t) + αf (t)) dμ(t) ≤ (1 − α)
= (1 − α)
S
I\S
f (t) dμ(t) + αδ − (1 − α)
(f (t) − ω(t)) dμ(t) + (1 − α)
= (1 − α)
I
f (t) dμ(t) + αδ
(f (t) − ω(t)) dμ(t)
− αδ + (1 − α)
S
I\S
I\S
I\S
ω(t) dμ(t)
(f (t) − ω(t)) dμ(t)
(f (t) − ω(t)) dμ(t) ≤ 0
Then, z is a feasible allocation in the economy E(I \ S, f, α). Therefore, the grand coalition blocks f via z in the economy E(I \ S, f, α), which is a contradiction with the fact that f is a non-dominated allocation for every economy E(A, f, α).
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Remark 3.1. Note that, in the statement of the above theorem, the parameter α can be chosen arbitrarily close to zero. On the other hand, Schmeidler (1972) showed that any allocation that is not blocked by “small” coalitions is in the core. Then, in Theorem 3.1, the set of perturbed economies E(S, f, α) can be restricted to those where the endowments are only modified on coalitions which have arbitrarily small measure. Furthermore, by Vind’s (1972) result, if an allocation f does not belong to the core of the economy E, then for any ε ∈ (0, 1) there exists a coalition S ⊂ I, with μ(S) = ε, that blocks f. That is, in the statement of our main result, the set of coalitions can be restricted to those coalitions S such that μ(S) = ε for any fixed ε > 0. Therefore, we have actually shown that the competitive allocations are those that are not dominated in any economy obtained by perturbing slightly the initial endowments of the agents belonging to either arbitrarily small coalitions, arbitrarily large coalitions, or coalitions of a fixed measure. Note also that the characterization can be stated in the following way. Let α¯ ∈ (0, 1) and let f be a feasible allocation in the economy E satisfying assumptions (H.1)–(H.3). Then, f is a competitive allocation in E if and only if f is a ¯ non-dominated allocation for every economy E(S, f, α). Furthermore, Theorem 3.1 can be rewritten as follows. A feasible allocation f is not competitive in the economy E if and only if there exists a sequence of coalitions Sk and a sequence of real numbers αk , with both μ(Sk ) and αk converging to zero and Sk+1 ⊂ Sk , such that f is a dominated allocation in every economy E(Sk , f, αk ). Remark 3.2 (The welfare theorems). The first welfare theorem is an immediate consequence of Theorem 3.1. In fact, if f is a Walrasian equilibrium allocation, then f is a Pareto optimal allocation not only in the economy E but also in any economy E(S, f, α) where f is feasible. On the other hand, observe that if f is a Pareto optimal allocation in E, then f is also a Pareto optimal allocation in the ˆ f, α) are equal to Eˆ Then, if economy Eˆ in which the initial endowment allocation is f. Note that all the economies E(S, f holds the assumption (H.1), we can apply Theorem 3.1 to the economy Eˆ and we obtain, exactly, the second welfare theorem. Therefore, both welfare theorems are particular cases of Theorem 3.1. 4. Economies with n agents Let us consider an economy En , with n consumers and commodities, where each consumer i is characterized by her preference relation i and her initial endowments ωi ∈ R+ . Given En , we define an associated continuum economy Ec with n different types of agents. The set of agents is represented by the real interval [0, 1], with the Lebesgue measure μ. We write I = [0, 1] = ni=1 Ii , where Ii = [i − 1/n, i/n), if i = n, and In = [n − 1/n, 1]. Each consumer t ∈ Ii is characterized by her consumption set R+ , her preference relation t = i and her initial endowment ω(t) = ωi ∈ R+ . We will refer to Ii as the set of agents of type i in the atomless economy Ec with a finite number of different types is
economy Ec . Then, the associated continuum given by Ec = R+ , I = ni=1 Ii , ωi , i , i = 1, . . . , n In this section, we assume that the economy Ec is in the hypothesis (H.1)–(H.3) and also (H.4). For every i = 1, . . . , n the preference relation i is convex (i.e., if xi y and zi y, then λx + (1 − λ)zi y for every λ ∈ (0, 1)). (H.5). For every i, if yi x and y ≤ z then zi x. As in this section we address finite economies, the convexity assumption (H.4) is standard. We remark that we deal with preferences that are not required to be neither complete nor transitive. The hypothesis (H.5), that Grodal (1972, p. 582) calls weak monotonicity, actually implies some kind of transitivity. Note that if preferences were transitive then (H.5) would be redundant. Observe that an allocation x in En can be interpreted as an allocation f in Ec , where f is the step function given by f (t) = xi , if t∈ Ii . Reciprocally, an allocation f in Ec can be interpreted as an allocation x = (x1 , . . . , xn ) in En , where xi = 1/μ(Ii ) Ii f (t) dμ(t). Observe also that (x, p) is an equilibrium for the economy En if and only if (f, p) is an equilibrium for the continuum economy Ec , where f (t) = xi if t ∈ Ii . The aim of this section is to apply Theorem 3.1 to the n-types continuum economy Ec in order to characterize Walrasian equilibrium allocations in economies with n agents obtaining an extension of the characterization result in Herv´es-Beloso et al. (2005a). The following lemma will be essential in the proof of the main result of this section.
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Lemma 4.1. Let be a continuous and convex and weakly monotone ((H.5)) preference relation and let g be an allocation such that g(t) x for every t ∈ S, with μ(S) > 0. Then 1/μ(S) S g(t) dμ(t) x. Proof. As before, by continuity there exist g˜ and δ(t) > 0such that g(t) = g˜ (t) − δ(t) and g˜ (t) x for every t ∈ S. Then y = 1/μ(S) S g(t) dμ(t) = δ + b, where b = 1/μ(S) S g˜ (t) dμ(t) and δ = 1/μ(S) S δ(t) dμ(t). Observe that b belongs to the closed convex hull of g˜ (S) (see, for example, Diestel and Uhl, 1997). That is, b is the limit of a sequence bn in the convex hull of g˜ (S) which, by convexity of preferences, is contained in the set {z|z x} and therefore bn x for all n. Moreover, bn< b + δ for all n large enough.2 Finally, we have 1/μ(S) S g(t) dμ(t) = y = b + δ > bn x and, by (H.5), 1/μ(S) S g(t) dμ(t) x. Observe that the proof of the previous lemma requires continuity, convexity and “weak monotonicity” of preferences. With the same assumptions it is not difficult to see that the result holds if the commodity space is ∞ and we consider the Bochner integral. For the case in which the preference relation is a complete pre-order represented by a concave utility function and the commodity space is finite dimensional, Lemma 4.1 is a consequence of the Jensen’s inequality. For the infinite dimensional setting and convex and complete pre-orders the same result is proved in Garc´ıa-Cutr´ın and Herv´es-Beloso (1993). In order to set our next result we need some notation. Given an allocation x and a vector a = (a1 , . . . , an ), with 0 ≤ ai ≤ 1, let E(a, x) be a pure exchange economy which coincides with En except for the initial endowments that are given by ω(a, x) = aω + (1 − a)x. That is, E(a, x) ≡ (R+ , (i , ωi (ai , xi ) = ai ωi + (1 − ai )xi ), i = 1, . . . , n) Theorem 4.1. A feasible allocation x is a Walrasian allocation in the economy En if and only if x is a non-dominated allocation for every economy E(a, x). Proof. Let x be a Walrasian allocation for the economy En . Then the step function f is a Walrasian allocation for the n-types continuum economy Ec , where f (t) = xi if t ∈ Ii . By Theorem 3.1 f is a non-dominated allocation for every economy E(S, f, α). It is straightforward to see that this implies that x is a non-dominated allocation for every economy E(a, x). Reciprocally, let x be a non-dominated allocation for every economy E(a, x). Assume that x is not a Walrasian allocation for the economy En . Then, the step allocation f given by x is not a Walrasian allocation for the continuum economy with n different types of agents. By Theorem 3.1, there exist a coalition S and a real number such that f is dominated in the economy E(S, f, α). That is, there exists an allocation g : [0, 1] → R+ such that (i) I g(t) dμ(t) ≤ I\S ω(t) dμ(t) + S ((1 − α)ω(t) + αf (t)) dμ(t), and (ii) g(t)t f (t) for almost all agent t ∈ I. Now, in the finite economy En , let us consider the allocation (g1 , . . . , gn ), where gi = 1/μ(Ii ) Ii g(t) dμ(t). By Lemma 4.1, gi i xi , for , n}. every agent i ∈ {1, . . . By (i) we have that ni=1 gi ≤ ni=1 si ωi + ni=1 (1 − si )((1 − α)ωi + αxi ), where si = nμ((I \ S) Ii ). Therefore, x is a dominated allocation in the economy E(a, x), with ai = α(1 − si ), which is a contradiction. Observe that the assumptions in Theorem 4.1(i.e, strictly positivity of total initial endowments, continuity, monotonicity and convexity of preferences) differ from the assumptions in Herv´es-Beloso et al. (2005a). Here the preferences are not required to be neither complete nor transitive and only the total resources and not the individual initial endowments are assumed to be strictly positive. In contrast, the monotonicity assumption is here stronger. Next we set a result on replicated economies which is not a direct consequence of the Theorem 4.1. For each positive integer r, the r-fold replica economy of En is a new exchange economy with rn agents, indexed by ij, i = 1, . . . , n; j = 1, . . . , r such that the consumer ij is characterized by the initial endowment ωij = ωi and the preference relation ij = i . Every allocation x = (x1 , . . . , xn ) of En can be considered as an allocation rx = In fact, if we can choose δ 0 then the order interval (b − δ, b + δ) is a neighborhood of b and the conclusion is obvious. Otherwise, for some commodity h, we have gh (t) = g˜ h (t) = 0 for almost every t ∈ S. Then bnh = 0 for all n and the conclusion follows as well. 2
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(x11 , . . . , x1r , . . . , xn1 , . . . , xnr ) in the r-fold replica economy rE, where xij = xi , for j = 1, . . . , r and i = 1, . . . , n. The allocation rx is called equal treatment allocation. It is known that, under convexity, a feasible allocation x is a Walrasian allocation in the economy En if and only if rx is a Walrasian allocation in the replicated economy rE and, by Theorem 4.1, if and only if rx is a non-dominated allocation for every economy rE(a, rx). In which follows we show that, if we enlarge the economy a given economy En , Walrasian equilibrium allocations are characterized as non-dominated allocations in economies obtained by perturbing the initial endowments of coalitions with relatively small size. As we have remarked, we can not obtain such a characterization as an immediate consequence of Theorem 4.1. Given an allocation x and a vector a = (a1 , . . . , an ), with 0 ≤ ai ≤ 1, let ER (a, x) be a pure exchange economy which coincides with the replicated economy rEn except for the initial endowments of at most one agent ij of each type i, that are given by ωij (a, x) = ai ωi + (1 − ai )xi . Theorem 4.2. A feasible allocation x is a Walrasian allocation in the economy En if and only if rx is a non-dominated allocation for every economy ER (a, x). Proof. Let x be a Walrasian allocation in the economy En . Assume that rx is a dominated allocation in the economy ER (a, x). Then, there exists an allocation g such that rx is blocked the whole coalition in the economy ER (a, x). By convexity of preferences (Lemma n n 4.1), g can be taken to be an equal treatment allocation. Hence, there exists n g such n rg ≤ (r − 1)ω + a ω + (1 − a )x and g x for every i = 1, . . . , n. This implies that i i i i i i i i i i=1 i=1 i=1 i=1 gi ≤ n (1 − (1 − a /r))ω + (1 − a /r)x . Therefore, x is a dominated allocation in the economy E(a, x), and applying i i i i i=1 the Theorem 4.1. this is in contradiction with the fact that x is a Walrasian allocation in the economy En . Reciprocally, assume that x is not a Walrasian allocation in the economy En . Then the corresponding step function f, given by f (t) = xi if t ∈ Ii , is not a competitive allocation in the associated continuum economy Ec . Let α = 1/r. Applying Theorem 3.1, there exists a coalition S such that f is dominated in the economy E(S, f, 1/r). That is, the grand coalition blocks f via an allocation g in the economy E(S, f, 1/r). As before, by nLemma 4.1, n g can be taken n to be an equal treatment allocation. This implies that there exists g such that g x and g ≤ s ω + i i i i i i=1 (1 − si )[(1 − i=1 i i=1 n n 1/r)ω + (1/r)x ], where s = nμ(S I ) for each type of agents i = 1, . . . , n. Hence g i i i i=1 i ≤ i=1 (r − 1/r)ωi + n i 1/r[s ω + (1 − s )x ]. Therefore, we conclude that rx is a dominated allocation in the economy ER (s, x). i i i i i=1 5. An approach to the infinite dimensional setting The aim of this section is to explore Theorem 3.1 in economies with infinitely many commodities. In this context, the relation between Walrasian and non-dominated allocations becomes even more interesting since the diversity in agents’ preferences and endowments is now potentially higher and, therefore, blocking may become more difficult. To establish Theorem 3.1 we have required Aumann’s (1964) Core–Walras equivalence. It is known that this equivalence result may fail in an infinite dimensional setting. However, under monotonicity and Mackey continuity of preferences and a boundedness condition on initial endowments, Bewley (1973) showed the core equivalence of equilibria for continuum economies with ∞ (the space of bounded sequences) as commodity space. Mackey continuity of preferences was shown by Araujo (1985) to be a necessary condition for existence of individually rational Pareto allocations and, therefore, for existence of Walrasian equilibria. Mackey upper semi-continuous preferences are known to be upper myopic, in the sense that gains in the distant future are negligible (see Brown and Lewis, 1981; Araujo, 1985). Mackey lower semi-continuity of preferences corresponds to lower myopia: losses in the distant future are negligible (in the terminology of Mas-Colell and Zame (1991), Example 4.1). Roughly speaking, lower myopia allows us to drop tails from a blocking allocation, and, subsequently, use a finite-dimensional approach. Thus, let us consider a pure exchange economy E∞ with a continuum of agents, represented by the space (I, A, μ), and ∞ as the commodity space. Each agent t ∈ I = [0, 1] is characterized by her consumption set ∞ +, ∞ and her preference relation given by a complete pre-order on her initial endowments ω(t) = (ωj (t))∞ ∈ t + j=1 ∞ ∞ + × + .
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Theorem 3.1 as well as the already cited results of Schmeidler (1972) and Vind (1972) rely on Liapunov’s convexity theorem, which does not hold in an infinite dimensional setup. This implies that an immediate extension of these results to economies with an infinite dimensional commodity space is not obvious. However, Herv´es-Beloso et al. (2000) provided an extension of the theorems stated in the papers by Schmeidler (1972); Grodal (1972) and Vind (1972) (three notes in same issue of Econometrica) to continuum economies, whose commodity space is ∞ , under the following hypothesis3 : (A.1). The map ω : I → ∞ + , which associates to each agent her initial endowment, is Bochner integrable. Bochner integrability of the endowments allocation was identified by Gretsky and Ostroy (1991) with physical thickness of markets. (A.2). For every measurable set A ⊂ I, with μ(A) > 0, there exists a real number a = a(A) > 0, such that ω (t) dμ(t) ≥ a for all j. A j Note that the natural extension of the strict positivity condition on the endowments in finite economies to the continuum case is ωj (t) ≥ a(t) > 0, for all j. However, we remark that for an extension of this model to the case of infinitely many commodities this assumption is stronger than (A.2). (A.3). Preference relations are complete pre-orders which are Mackey continuous and monotone for almost all t ∈ I. ∞ ∞ Let C∗ (∞ + ) denote the space of Mackey continuous functions on (+ ). Note that, since , endowed with the Mackey topology, is separable, (A.3) implies that preference relations are representable by Mackey continuous utility functions.
(A.4). The mapping U : I → C∗ (∞ + ), which associates to each agent t an utility function U(t) = Ut representing her preference relation, is measurable. n n ∞ n ∞ Given x = (xh )∞ h=1 ∈ and n ∈ N, we denote by x the element of defined by xh = xh if 1 ≤ h ≤ n and xh = 0 if h > n. Under the above assumptions (A.1)–(A.4), the following property holds Herv´es-Beloso et al. (2000):
If a feasible allocation f is blocked in the economyE∞ by a coalition S via an allocation g, then there exists a positive integer N = N(f ) such that the coalition S blocks f via the allocation gN , where ghN = gh if 1 ≤ h ≤ N and ghN = 0 if h > N. As a consequence of this property, Theorem 3.1 and the interpretation stated in a Remark in Section 3 can be extended to this infinite dimensional framework. Moreover, this result can be extended to the case of differential information by following the same argument and applying the results in Herv´es-Beloso et al. (2005b). More work is needed to verify whether the result still holds for other commodity spaces. Of particular interest is the space of measures, where the Walrasian allocations are known to resist manipulation by arbitrarily small coalitions (see Ostroy and Zame (1994) and Herv´es-Beloso et al. (1999)) and may, therefore, be regarded as true perfectly competitive outcomes. Acknowledgements This work is partially supported by Research Grants BEC2003-09067-C04-01 (Ministerio de Ciencia y Tecnolog´ıa and FEDER), PGIDT04XIC30001PN (Xunta de Galicia) and SA070A05 (Junta de Castilla y Le´on). The authors are grateful to an anonymous referee for his/her helpful suggestions. 3
See Herv´es-Beloso et al. (2000) for the comments on the hypothesis.
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References Araujo, A., 1985. Lack of Pareto optimal allocations in economies with infinitely many commodities: the need for impatience. Econometrica 53, 455–461. Aumann, R.J., 1964. Markets with a continuum of traders. Econometrica 32, 39–50. Aumann, R.J., 1966. Existence of competitive equilibria in markets with a continuum of traders. Econometrica 34, 1–17. Bewley, T., 1973. The equality of the core and the set of equilibria in economies with infinitely many commodities and a continuum of agents. International Economic Review 14, 383–393. Brown, D., Lewis, L., 1981. Myopic economic agents. Econometrica 49, 359–368. Debreu, G., Scarf, H., 1963. A limit theorem on the core of an economy. International Economic Review 4, 235–246. Diestel, J., Uhl, J.J., 1997. Vector Measures. American Mathematical Society, Providence, Rhode Island. Garc´ıa-Cutr´ın, J., Herv´es-Beloso, C., 1993. A discrete approach to continuum economies. Economic Theory 3, 577–584. Gretsky, N., Ostroy, J., 1991. Thick and thin market non-atomic exchange economies. In: Aliprantis, C.D., Burkinshaw, O., Rothman, N.J. (Eds.), Advances in Equilibrium Theory, Lecture Notes in Economics and Mathematical Systems. Springer-Verlag, Berlin, p. 224. Grodal, B., 1972. A second remark on the core of an atomless economy. Econometrica 40, 581–583. Herv´es-Beloso, C., Moreno-Garc´ıa, E., P´ascoa, M.R., 1999. Manipulation-proof equilibrium in atomless economies with commodity differentiation. Economic Theory 14, 545–563. Herv´es-Beloso, C., Moreno-Garc´ıa, E., N´un˜ ez-Sanz, C., P´ascoa, M., 2000. Blocking efficacy of small coalitions in myopic economies. Journal of Economic Theory 93, 72–86. Herv´es-Beloso, C., Moreno-Garc´ıa, E., Yannelis, N.C., 2005a. An equivalence theorem for a differential information economy. Journal of Mathematical Economics 41, 844–856. Herv´es-Beloso, C., Moreno-Garc´ıa, E., Yannelis, N.C., 2005b. Characterization and incentive compatibility of Walrasian expectations equilibrium in infinite dimensional commodity spaces. Economic Theory 26, 361–381. Mas-Colell, A., Zame, W.R., 1991. Equilibrium theory in infinite dimensional spaces. In: Hildenbrand, W., Sonnenschein, H. (Eds.), In: Handbook of Mathematical Economics, vol. IV. North-Holland, Amsterdam, pp. 1835–1898 (Chapter 34). Ostroy, J.M., Zame, W.R., 1994. Nonatomic economies and the boundaries of perfect competition. Econometrica 62, 593–633. Schmeidler, D., 1972. A remark on the core of an atomless economy. Econometrica 40, 579–580. Vind, K., 1964. Edgeworth allocations in an exchange economy with many traders. International Economic Review 5, 165–177. Vind, K., 1972. A third remark on the core of an atomless economy. Econometrica 40, 585–586.
Journal of Mathematical Economics 44 (2008) 697–706
Competitive equilibria and the grand coalition Carlos Herv´es-Beloso a,∗ , Emma Moreno-Garc´ıa b b
a RGEA, Facultad de Econ´ omicas, Universidad de Vigo, Spain Facultad de Econom´ıa y Empresa, Universidad de Salamanca, Spain
Received 17 January 2006; received in revised form 7 November 2006; accepted 16 November 2006 Available online 8 December 2006
Abstract This paper provides a new characterization of competitive equilibrium allocations based on the veto mechanism. The main theorem shows that, in pure exchange economies with a continuum of non-atomic agents, the competitive equilibria can be characterized by strengthening the veto power of the grand coalition, formed by all the agents in the economy. The welfare theorems are obtained as easy corollaries of our main result. Furthermore, in the case of finite economies, we show that the characterizations of the Walrasian equilibria based on the veto power of the grand coalition are particular cases of our main theorem. © 2007 Elsevier B.V. All rights reserved. JEL Classification: D51 Keywords: Competitive equilibrium; Continuum economies; Core–Walras equivalence; Dominated allocations; Grand coalition
1. Introduction We consider pure exchange economies with a continuum of non-atomic agents and finitely many commodities Aumann (1964, 1966). In these economies, competitive allocations can be characterized by means of the veto mechanism. Precisely, Aumann (1964) showed that the set of competitive allocations coincides with the core of the economy (Core–Walras equivalence). That is, competitive allocations are those feasible allocations that are not blocked by any coalition of agents. Actually, for the equivalence is essential the consideration of the veto power of infinitely many coalitions. In fact, in the case of a finite economy, where only a finite number of coalitions can be formed, the characterization of the Walrasian allocations using the veto mechanism is only asymptotic (Debreu and Scarf, 1963). The main result in this paper is a characterization of the competitive equilibrium allocations in terms of the veto power of the grand coalition. That is, instead of infinitely many coalitions, we only consider the blocking power of the grand coalition formed by all the agents, but exercised in a family of economies obtained by perturbing the agents’ initial endowments. Thus, this is a kind of Core–Walras equivalence theorem in which we do not consider the veto power of infinitely many coalitions, but the veto power of a single coalition in infinitely many economies. We define non-dominated allocations in an economy as those allocations (feasible or not) that are not blocked by the grand coalition. We remark that if an allocation is competitive, then it is efficient and therefore it is not dominated in any economy in which the endowments are redistributions of the total resources of the initial economy. We prove ∗
Corresponding author. Tel.: +34 986225126; fax: +34 986812401. E-mail addresses: [email protected] (C. Herv´es-Beloso), [email protected] (E. Moreno-Garc´ıa).
0304-4068/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.jmateco.2006.11.002
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that competitive allocations are robustly efficient.1 That is, a competitive allocation f is a non-dominated allocation not only in the initial economy but in all economies obtained by modifying, in the direction of f, the initial endowments of any coalition of agents. Reciprocally, robustly efficient allocations are competitive. Indeed, the results of Schmeidler (1972) and Vind (1972), which show that non-competitive allocations are those blocked by coalitions of a given measure, allow us to give a sharper interpretation. Actually we show that in continuum economies the competitive allocations are characterized as allocations that are non-dominated in any economy obtained by a slight perturbation of the real initial endowments of the agents belonging to either arbitrarily small coalitions, arbitrarily large coalitions, or coalitions of a given measure. This characterization of competitive allocations by the veto power of the grand coalition leads us to obtain the first and second welfare theorems as easy corollaries. Moreover, applying our theorem to a continuum economy with n different types of agents we obtain, as a particular case, the characterization of the Walrasian equilibrium allocations for finite pure exchange economies showed by Herv´es-Beloso et al. (2005a). Finally, we explore the case of infinitely many commodities. We prove that, under the hypothesis which guarantee the Core–Walras equivalence and the extension of Vind’s (1972) result, our main theorem also holds for economies where the commodity space is the space of bounded sequences ∞ . The rest of the paper is as follows. In Section 2 we set the model and state and comment the assumptions. In Section 3 the main result is presented and proved. Section 4 addresses the particular case of a pure exchange economy with n agents, recovering previous characterizations of Walrasian equilibrium allocations. Finally in Section 5 we consider economies with infinitely many commodities. 2. The model Consider a pure exchange economy E with a continuum of agents. The commodity space is the Euclidean space R . The space of agents is represented by the measure space (I, A, μ), where I is the real interval [0, 1], A is the Lebesgue σ-algebra of subsets of I and μ is the Lebesgue measure. Each agent t ∈ I is characterized by her consumption set R+ , her initial endowment ω(t) ∈ R+ , and her preference relation t . As in Aumann (1964), we state the following assumptions on endowments and preference relations: (H.1). The mapping ω : I → R+ , that associates to each agent her initial endowment, is integrable and dμ(t) 0.
I
ω(t)
(H.2). For every consumer t ∈ I, the preference relation t is continuous and strictly monotone. (H.3). The preferences are measurable ({t ∈ I|xt y} is a measurable set of agents for all bundles x, y ∈ R+ ). (H.1) is an assumption about boundedness of endowments which asserts that each of the commodities is actually present in the market. A preference relation is strictly monotone if x ≥ y and x = y implies that x y. Note that (H.2) states the usual continuity assumption on preferences and also requires a desirability condition that says that each trader always wants more of every commodity. Strict monotonicity on preferences is required by Schmeidler (1972) and Vind (1972) to show characterizations of the core in atomless economies that we will use to give a sharper interpretation of our main theorem. (H.3) is a measurability assumption which is only of technical significance and constitutes no real economic restriction. This measurability assumption is the same used by Vind (1972) and it is slightly different than the one required by Aumann (1964) which allows for the comparison of arbitrary allocations and not simply for the comparison of constant allocations. As it was remarked by Ostroy and Zame (1994) both measurability requirements are equivalent in several scenarios including the case of finite dimensional commodity spaces. These hypothesis are required to show Aumann’s (1964) core equivalence result. Actually, as in Aumann’s (1964) work, the preferences are not assumed to be complete and nor even transitive. 1
See Definition 3.2.
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Then, the economy we consider is an atomless economy which defined by E ≡ ((I, A, μ), R+ , ω(t), t , t ∈ I) and satisfies standard assumptions regarding preferences (which may be neither complete nor transitive) and initial endowments. An allocation is a μ-integrable function f : I → R+ . An allocation f is feasible if I f (t) dμ(t) ≤ I ω(t) dμ(t). A price system is an element of Δ, where Δ denotes the ( − 1)-dimensional simplex of R+ , that is, = {p ∈ R+ such that h=1 ph = 1}. A competitive equilibrium for the economy E is a pair (p, f ), where p is a price system and f is a feasible allocation such that, for almost every agent t, the bundle f (t) maximizes the preference relation t in the budget set Bt (p) = {y ∈ R+ such thatp · y ≤ p · ω(t)}. A coalition of agents is any positive measure subset of the set of individuals. A feasible allocation belongs to the core of the economy if it is not blocked by any coalition of agents. A coalition S blocks an allocation f via another allocation g in the economy E if: (i) S g(t) dμ(t) ≤ S ω(t) dμ(t) and (ii) g(t)t f (t) for almost all t ∈ S. Let us denote by Core(E) the set of allocations which belong to the core of the economy E. It is easy to show that any Walrasian allocation belongs to the core. Moreover, Aumann (1964) showed that, under the hypothesis (H.1)–(H.3), the core coincides with the set of competitive equilibrium allocations in continuum economies with a finite dimensional commodity space. This Core–Walras equivalence will be used in the proof of our main result. 3. The main result In this section we state our main result which provides a characterization of Walrasian equilibria in terms of the veto power of the grand coalition. For this, we set the definition of robustly efficient allocation and we adapt to continuum economies the notion of dominated allocations stated for finite economies in Herv´es-Beloso et al. (2005a). Definition 3.1. An allocation h (feasible or not) is dominated (or blocked by the grand coalition) in the economy E if there exists a feasible allocation g such that g(t)t h(t) for almost all t ∈ I. Given a coalition S, a feasible allocation f, and a real number α, with 0 ≤ α ≤ 1, let E(S, f, α) be a continuum exchange economy which coincides with E except for the initial endowment allocation that is given by ω(t) if t ∈ I \ S ω(S, f, α)(t) = (1 − α)ω(t) + αf (t) if t ∈ S That is, the continuum economy E(S, f, α) is the same as E when α = 0 and otherwise, E(S, f, α) coincides with E except for the initial endowment of the agents in the coalition S that is given by the convex combination defined by α of f and ω. Then, the economy E(S, f, α) is defined by E(f, S, α) ≡ ((I, A, μ), R+ , ω(S, f, α)(t), t , t ∈ I). Note that I ω(S, f, α)(t) dμ(t) = I ω(t) dμ(t) − α S (ω(t) − f (t)) dμ(t). Therefore, when the size of the coalition S is either arbitrarily small or arbitrarily big, the total resources of the economy E(S, f, α) are arbitrarily close to those of the former economy E. Moreover, if αn is a sequence converging to zero, then ω(S, f, αn )(t) converges to ω(t) for n every t ∈ I and I ω(S, f, α )(t) dμ(t) converges to I ω(t) dμ(t). Definition 3.2. A feasible allocation f is robustly efficient in the economy E if f is a non-dominated allocation in every economy E(S, f, α). We can interpret the perturbed economies E(S, f, α) as different ways to get f starting from the initial allocation ω. Then, an allocation f is robustly efficient if it is not blocked by the grand coalition in any economy which is a path from ω to f.
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Furthermore,let f be a dominated allocation in the economy E(S, f, α), with α < 1. Consider the measure ν defined as ν(A) = (μ(A), A (ω(t) − f (t)) dμ(t)), for every A ⊂ S. Applying Liapunov’s convexity theorem to the vector measure ν, restricted to S, we obtain that there exists S ⊂ S with μ(S ) = αμ(S) and S (ω(t) − f (t)) dμ(t) = α S (ω(t) − f (t)) dμ(t). That is, if f is dominated in the economy E(S, f, α), with α < 1, then there exists S ⊂ S such that f is also dominated in the economy E(S , f, 1). The proof of our main theorem requires the following Lemma. Lemma 3.1. Let f be an allocation that is blocked by the coalition S via g. Then, given any α ∈ (0, 1), there exists an allocation h : S → R+ such that h(t)t f (t) for every t ∈ S and S h(t) dμ(t) = S (αg(t) + (1 − α)f (t)) dμ(t). This result is used by Vind (1972) who refers the reader to a previous paper on preferences (Vind (1964)). Here, for the sake of completeness, we state a direct proof. Proof. Consider the vectorial measure ν(A) = (μ(A), A (g(t) − f (t)) dμ(t)) restricted to measurable subsets of the coalition S. Let anyα ∈ (0, 1). Applying Liapunov’s convexity theorem we obtain that there exits A ⊂ S such that μ(A) = αμ(S) and A (g(t) − f (t))dμ(t) = α S (g(t) − f (t)) dμ(t). By continuity and measurability of preferences, there exist g˜ : A → R+ and δ > 0 such that A g˜ (t) dμ(t) = ˜ (t)t f (t) for every t ∈ A. (See Vind (1972)). A g(t)f (t) dμ(t) − δ and g Let the allocation h : S → R+ defined as follows: ⎧ g˜ (t) if t ∈ A ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ h(t) = ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ δ ⎪ ⎩ f (t) + if t ∈ S \ A μ(S \ A) By monotonicity of preferences h(t)t f (t) for every t ∈ S. Moreover,
δ f (t) + dμ(t) h(t) dμ(t) = g˜ (t) dμ(t) + μ(S \ A) S A S\A g(t) dμ(t) − δ + f (t) dμ(t) − f (t) dμ(t) + δ = A
=α =
S
S
S
(g(t) − f (t)) dμ(t) +
A
S
f (t) dμ(t)
(αg(t) + (1 − α)f (t)) dμ(t).
We remark that, applying the previous Lemma, we can conclude that if an allocation f is dominated in the economy E(S, f, α), then f is also dominated in any economy E(S, f, β), for every β ≤ α. Theorem 3.1. Let us consider a continuum economy E satisfying assumptions (H.1)–(H.3). An allocation f is a competitive equilibrium allocation in E if and only if f is robustly efficient. Proof. Let (p, f ) be a competitive equilibrium for the economy E. Suppose that there exist a coalition S and a number α ∈ (0, 1] such that f is dominated in the economy E(S, f, α). Then, there exists g : I → R+ such that (i) I g(t) dμ(t) ≤ I ω(S, f, α)(t) dμ(t), and (ii) g(t)t f (t) for almost all agent t ∈ I. Since f is a competitive equilibrium allocation in the economy E, we have that p · f (t) ≤ p · ω(t) for almost all agent t ∈ I and from condition (ii) we deduce that p · g(t) > p · ω(t) ≥ p · f (t), for almost all agent t ∈ I.
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Multiplying the above inequalities by (1 − α) and by α, respectively, we obtain that the following inequalities hold for almost all t ∈ S: p · (1 − α)g(t) > p · (1 − α)ω(t)
and p · αg(t) > p · αf (t)
Thus, p · g(t) > p · ((1 − α)ω(t) + αf (t)) for almost all agent t ∈ S. Therefore, we have that p · g(t) dμ(t) > p · ω(t) dμ(t) + p · ((1 − α)ω(t) + αf (t)) dμ(t) = p · ω(S, f, α)(t) dμ(t) I
I\S
S
I
which is a contradiction with (i), that is, a contradiction with the feasibility of g in the economy E(S, f, α). Let f be a feasible allocation in E and assume that f is a non-dominated allocation for every economy E(A, f, α). In particular f is Pareto optimal. Assume that f is not a competitive equilibrium allocation for the economy E. Then, applying the Core–Walras equivalence, we have that f does not belong to the core of the continuum economy E. That is, there exists a coalition S ⊂ I, μ(S) < μ(I), and there exists g : S → R+ , such that (i) S g(t) dμ(t) ≤ S ω(t) dμ(t) and (ii) g(t)t f (t) for every t ∈ S. Then, by continuity and measurability of preferences, we can take g such that S g(t) dμ(t) ≤ S ω(t) dμ(t) − δ, with δ > 0. By the previous Lemma, given any α ∈ (0, 1), there exists an allocation h : S → R+ such that S h(t) dμ(t) = S (αg(t) + (1 − α)f (t)) dμ(t) and h(t)t f (t) for every t ∈ S. Let us consider the allocation z : I → R+ given by ⎧ if t ∈ S ⎨ h(t) α z(t) = δ if t ∈ I \ S ⎩ f (t) + μ(I \ S) Note that μ(I \ S) > 0 and that by monotonicity of preferences z(t)t f (t) for almost every t ∈ I. By construction, we obtain (z(t) − ω(I \ S, f, α)(t)) dμ(t) = ((1 − α)f (t) + αg(t)) dμ(t) − ω(t) dμ(t) I
+
S
α δ μ(I \ S)
I\S
S
S
dμ(t) −
(1 − α)(f (t) − ω(t)) dμ(t) +
−
S
f (t) +
I\S
=
I\S
(ω(I \ S, f, α)(t)) dμ(t)
α(g(t) − ω(t)) dμ(t) +
((1 − α)ω(t) + αf (t)) dμ(t) ≤ (1 − α)
= (1 − α)
S
I\S
f (t) dμ(t) + αδ − (1 − α)
(f (t) − ω(t)) dμ(t) + (1 − α)
= (1 − α)
I
f (t) dμ(t) + αδ
(f (t) − ω(t)) dμ(t)
− αδ + (1 − α)
S
I\S
I\S
I\S
ω(t) dμ(t)
(f (t) − ω(t)) dμ(t)
(f (t) − ω(t)) dμ(t) ≤ 0
Then, z is a feasible allocation in the economy E(I \ S, f, α). Therefore, the grand coalition blocks f via z in the economy E(I \ S, f, α), which is a contradiction with the fact that f is a non-dominated allocation for every economy E(A, f, α).
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Remark 3.1. Note that, in the statement of the above theorem, the parameter α can be chosen arbitrarily close to zero. On the other hand, Schmeidler (1972) showed that any allocation that is not blocked by “small” coalitions is in the core. Then, in Theorem 3.1, the set of perturbed economies E(S, f, α) can be restricted to those where the endowments are only modified on coalitions which have arbitrarily small measure. Furthermore, by Vind’s (1972) result, if an allocation f does not belong to the core of the economy E, then for any ε ∈ (0, 1) there exists a coalition S ⊂ I, with μ(S) = ε, that blocks f. That is, in the statement of our main result, the set of coalitions can be restricted to those coalitions S such that μ(S) = ε for any fixed ε > 0. Therefore, we have actually shown that the competitive allocations are those that are not dominated in any economy obtained by perturbing slightly the initial endowments of the agents belonging to either arbitrarily small coalitions, arbitrarily large coalitions, or coalitions of a fixed measure. Note also that the characterization can be stated in the following way. Let α¯ ∈ (0, 1) and let f be a feasible allocation in the economy E satisfying assumptions (H.1)–(H.3). Then, f is a competitive allocation in E if and only if f is a ¯ non-dominated allocation for every economy E(S, f, α). Furthermore, Theorem 3.1 can be rewritten as follows. A feasible allocation f is not competitive in the economy E if and only if there exists a sequence of coalitions Sk and a sequence of real numbers αk , with both μ(Sk ) and αk converging to zero and Sk+1 ⊂ Sk , such that f is a dominated allocation in every economy E(Sk , f, αk ). Remark 3.2 (The welfare theorems). The first welfare theorem is an immediate consequence of Theorem 3.1. In fact, if f is a Walrasian equilibrium allocation, then f is a Pareto optimal allocation not only in the economy E but also in any economy E(S, f, α) where f is feasible. On the other hand, observe that if f is a Pareto optimal allocation in E, then f is also a Pareto optimal allocation in the ˆ f, α) are equal to Eˆ Then, if economy Eˆ in which the initial endowment allocation is f. Note that all the economies E(S, f holds the assumption (H.1), we can apply Theorem 3.1 to the economy Eˆ and we obtain, exactly, the second welfare theorem. Therefore, both welfare theorems are particular cases of Theorem 3.1. 4. Economies with n agents Let us consider an economy En , with n consumers and commodities, where each consumer i is characterized by her preference relation i and her initial endowments ωi ∈ R+ . Given En , we define an associated continuum economy Ec with n different types of agents. The set of agents is represented by the real interval [0, 1], with the Lebesgue measure μ. We write I = [0, 1] = ni=1 Ii , where Ii = [i − 1/n, i/n), if i = n, and In = [n − 1/n, 1]. Each consumer t ∈ Ii is characterized by her consumption set R+ , her preference relation t = i and her initial endowment ω(t) = ωi ∈ R+ . We will refer to Ii as the set of agents of type i in the atomless economy Ec with a finite number of different types is
economy Ec . Then, the associated continuum given by Ec = R+ , I = ni=1 Ii , ωi , i , i = 1, . . . , n In this section, we assume that the economy Ec is in the hypothesis (H.1)–(H.3) and also (H.4). For every i = 1, . . . , n the preference relation i is convex (i.e., if xi y and zi y, then λx + (1 − λ)zi y for every λ ∈ (0, 1)). (H.5). For every i, if yi x and y ≤ z then zi x. As in this section we address finite economies, the convexity assumption (H.4) is standard. We remark that we deal with preferences that are not required to be neither complete nor transitive. The hypothesis (H.5), that Grodal (1972, p. 582) calls weak monotonicity, actually implies some kind of transitivity. Note that if preferences were transitive then (H.5) would be redundant. Observe that an allocation x in En can be interpreted as an allocation f in Ec , where f is the step function given by f (t) = xi , if t∈ Ii . Reciprocally, an allocation f in Ec can be interpreted as an allocation x = (x1 , . . . , xn ) in En , where xi = 1/μ(Ii ) Ii f (t) dμ(t). Observe also that (x, p) is an equilibrium for the economy En if and only if (f, p) is an equilibrium for the continuum economy Ec , where f (t) = xi if t ∈ Ii . The aim of this section is to apply Theorem 3.1 to the n-types continuum economy Ec in order to characterize Walrasian equilibrium allocations in economies with n agents obtaining an extension of the characterization result in Herv´es-Beloso et al. (2005a). The following lemma will be essential in the proof of the main result of this section.
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Lemma 4.1. Let be a continuous and convex and weakly monotone ((H.5)) preference relation and let g be an allocation such that g(t) x for every t ∈ S, with μ(S) > 0. Then 1/μ(S) S g(t) dμ(t) x. Proof. As before, by continuity there exist g˜ and δ(t) > 0such that g(t) = g˜ (t) − δ(t) and g˜ (t) x for every t ∈ S. Then y = 1/μ(S) S g(t) dμ(t) = δ + b, where b = 1/μ(S) S g˜ (t) dμ(t) and δ = 1/μ(S) S δ(t) dμ(t). Observe that b belongs to the closed convex hull of g˜ (S) (see, for example, Diestel and Uhl, 1997). That is, b is the limit of a sequence bn in the convex hull of g˜ (S) which, by convexity of preferences, is contained in the set {z|z x} and therefore bn x for all n. Moreover, bn< b + δ for all n large enough.2 Finally, we have 1/μ(S) S g(t) dμ(t) = y = b + δ > bn x and, by (H.5), 1/μ(S) S g(t) dμ(t) x. Observe that the proof of the previous lemma requires continuity, convexity and “weak monotonicity” of preferences. With the same assumptions it is not difficult to see that the result holds if the commodity space is ∞ and we consider the Bochner integral. For the case in which the preference relation is a complete pre-order represented by a concave utility function and the commodity space is finite dimensional, Lemma 4.1 is a consequence of the Jensen’s inequality. For the infinite dimensional setting and convex and complete pre-orders the same result is proved in Garc´ıa-Cutr´ın and Herv´es-Beloso (1993). In order to set our next result we need some notation. Given an allocation x and a vector a = (a1 , . . . , an ), with 0 ≤ ai ≤ 1, let E(a, x) be a pure exchange economy which coincides with En except for the initial endowments that are given by ω(a, x) = aω + (1 − a)x. That is, E(a, x) ≡ (R+ , (i , ωi (ai , xi ) = ai ωi + (1 − ai )xi ), i = 1, . . . , n) Theorem 4.1. A feasible allocation x is a Walrasian allocation in the economy En if and only if x is a non-dominated allocation for every economy E(a, x). Proof. Let x be a Walrasian allocation for the economy En . Then the step function f is a Walrasian allocation for the n-types continuum economy Ec , where f (t) = xi if t ∈ Ii . By Theorem 3.1 f is a non-dominated allocation for every economy E(S, f, α). It is straightforward to see that this implies that x is a non-dominated allocation for every economy E(a, x). Reciprocally, let x be a non-dominated allocation for every economy E(a, x). Assume that x is not a Walrasian allocation for the economy En . Then, the step allocation f given by x is not a Walrasian allocation for the continuum economy with n different types of agents. By Theorem 3.1, there exist a coalition S and a real number such that f is dominated in the economy E(S, f, α). That is, there exists an allocation g : [0, 1] → R+ such that (i) I g(t) dμ(t) ≤ I\S ω(t) dμ(t) + S ((1 − α)ω(t) + αf (t)) dμ(t), and (ii) g(t)t f (t) for almost all agent t ∈ I. Now, in the finite economy En , let us consider the allocation (g1 , . . . , gn ), where gi = 1/μ(Ii ) Ii g(t) dμ(t). By Lemma 4.1, gi i xi , for , n}. every agent i ∈ {1, . . . By (i) we have that ni=1 gi ≤ ni=1 si ωi + ni=1 (1 − si )((1 − α)ωi + αxi ), where si = nμ((I \ S) Ii ). Therefore, x is a dominated allocation in the economy E(a, x), with ai = α(1 − si ), which is a contradiction. Observe that the assumptions in Theorem 4.1(i.e, strictly positivity of total initial endowments, continuity, monotonicity and convexity of preferences) differ from the assumptions in Herv´es-Beloso et al. (2005a). Here the preferences are not required to be neither complete nor transitive and only the total resources and not the individual initial endowments are assumed to be strictly positive. In contrast, the monotonicity assumption is here stronger. Next we set a result on replicated economies which is not a direct consequence of the Theorem 4.1. For each positive integer r, the r-fold replica economy of En is a new exchange economy with rn agents, indexed by ij, i = 1, . . . , n; j = 1, . . . , r such that the consumer ij is characterized by the initial endowment ωij = ωi and the preference relation ij = i . Every allocation x = (x1 , . . . , xn ) of En can be considered as an allocation rx = In fact, if we can choose δ 0 then the order interval (b − δ, b + δ) is a neighborhood of b and the conclusion is obvious. Otherwise, for some commodity h, we have gh (t) = g˜ h (t) = 0 for almost every t ∈ S. Then bnh = 0 for all n and the conclusion follows as well. 2
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(x11 , . . . , x1r , . . . , xn1 , . . . , xnr ) in the r-fold replica economy rE, where xij = xi , for j = 1, . . . , r and i = 1, . . . , n. The allocation rx is called equal treatment allocation. It is known that, under convexity, a feasible allocation x is a Walrasian allocation in the economy En if and only if rx is a Walrasian allocation in the replicated economy rE and, by Theorem 4.1, if and only if rx is a non-dominated allocation for every economy rE(a, rx). In which follows we show that, if we enlarge the economy a given economy En , Walrasian equilibrium allocations are characterized as non-dominated allocations in economies obtained by perturbing the initial endowments of coalitions with relatively small size. As we have remarked, we can not obtain such a characterization as an immediate consequence of Theorem 4.1. Given an allocation x and a vector a = (a1 , . . . , an ), with 0 ≤ ai ≤ 1, let ER (a, x) be a pure exchange economy which coincides with the replicated economy rEn except for the initial endowments of at most one agent ij of each type i, that are given by ωij (a, x) = ai ωi + (1 − ai )xi . Theorem 4.2. A feasible allocation x is a Walrasian allocation in the economy En if and only if rx is a non-dominated allocation for every economy ER (a, x). Proof. Let x be a Walrasian allocation in the economy En . Assume that rx is a dominated allocation in the economy ER (a, x). Then, there exists an allocation g such that rx is blocked the whole coalition in the economy ER (a, x). By convexity of preferences (Lemma n n 4.1), g can be taken to be an equal treatment allocation. Hence, there exists n g such n rg ≤ (r − 1)ω + a ω + (1 − a )x and g x for every i = 1, . . . , n. This implies that i i i i i i i i i i=1 i=1 i=1 i=1 gi ≤ n (1 − (1 − a /r))ω + (1 − a /r)x . Therefore, x is a dominated allocation in the economy E(a, x), and applying i i i i i=1 the Theorem 4.1. this is in contradiction with the fact that x is a Walrasian allocation in the economy En . Reciprocally, assume that x is not a Walrasian allocation in the economy En . Then the corresponding step function f, given by f (t) = xi if t ∈ Ii , is not a competitive allocation in the associated continuum economy Ec . Let α = 1/r. Applying Theorem 3.1, there exists a coalition S such that f is dominated in the economy E(S, f, 1/r). That is, the grand coalition blocks f via an allocation g in the economy E(S, f, 1/r). As before, by nLemma 4.1, n g can be taken n to be an equal treatment allocation. This implies that there exists g such that g x and g ≤ s ω + i i i i i i=1 (1 − si )[(1 − i=1 i i=1 n n 1/r)ω + (1/r)x ], where s = nμ(S I ) for each type of agents i = 1, . . . , n. Hence g i i i i=1 i ≤ i=1 (r − 1/r)ωi + n i 1/r[s ω + (1 − s )x ]. Therefore, we conclude that rx is a dominated allocation in the economy ER (s, x). i i i i i=1 5. An approach to the infinite dimensional setting The aim of this section is to explore Theorem 3.1 in economies with infinitely many commodities. In this context, the relation between Walrasian and non-dominated allocations becomes even more interesting since the diversity in agents’ preferences and endowments is now potentially higher and, therefore, blocking may become more difficult. To establish Theorem 3.1 we have required Aumann’s (1964) Core–Walras equivalence. It is known that this equivalence result may fail in an infinite dimensional setting. However, under monotonicity and Mackey continuity of preferences and a boundedness condition on initial endowments, Bewley (1973) showed the core equivalence of equilibria for continuum economies with ∞ (the space of bounded sequences) as commodity space. Mackey continuity of preferences was shown by Araujo (1985) to be a necessary condition for existence of individually rational Pareto allocations and, therefore, for existence of Walrasian equilibria. Mackey upper semi-continuous preferences are known to be upper myopic, in the sense that gains in the distant future are negligible (see Brown and Lewis, 1981; Araujo, 1985). Mackey lower semi-continuity of preferences corresponds to lower myopia: losses in the distant future are negligible (in the terminology of Mas-Colell and Zame (1991), Example 4.1). Roughly speaking, lower myopia allows us to drop tails from a blocking allocation, and, subsequently, use a finite-dimensional approach. Thus, let us consider a pure exchange economy E∞ with a continuum of agents, represented by the space (I, A, μ), and ∞ as the commodity space. Each agent t ∈ I = [0, 1] is characterized by her consumption set ∞ +, ∞ and her preference relation given by a complete pre-order on her initial endowments ω(t) = (ωj (t))∞ ∈ t + j=1 ∞ ∞ + × + .
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Theorem 3.1 as well as the already cited results of Schmeidler (1972) and Vind (1972) rely on Liapunov’s convexity theorem, which does not hold in an infinite dimensional setup. This implies that an immediate extension of these results to economies with an infinite dimensional commodity space is not obvious. However, Herv´es-Beloso et al. (2000) provided an extension of the theorems stated in the papers by Schmeidler (1972); Grodal (1972) and Vind (1972) (three notes in same issue of Econometrica) to continuum economies, whose commodity space is ∞ , under the following hypothesis3 : (A.1). The map ω : I → ∞ + , which associates to each agent her initial endowment, is Bochner integrable. Bochner integrability of the endowments allocation was identified by Gretsky and Ostroy (1991) with physical thickness of markets. (A.2). For every measurable set A ⊂ I, with μ(A) > 0, there exists a real number a = a(A) > 0, such that ω (t) dμ(t) ≥ a for all j. A j Note that the natural extension of the strict positivity condition on the endowments in finite economies to the continuum case is ωj (t) ≥ a(t) > 0, for all j. However, we remark that for an extension of this model to the case of infinitely many commodities this assumption is stronger than (A.2). (A.3). Preference relations are complete pre-orders which are Mackey continuous and monotone for almost all t ∈ I. ∞ ∞ Let C∗ (∞ + ) denote the space of Mackey continuous functions on (+ ). Note that, since , endowed with the Mackey topology, is separable, (A.3) implies that preference relations are representable by Mackey continuous utility functions.
(A.4). The mapping U : I → C∗ (∞ + ), which associates to each agent t an utility function U(t) = Ut representing her preference relation, is measurable. n n ∞ n ∞ Given x = (xh )∞ h=1 ∈ and n ∈ N, we denote by x the element of defined by xh = xh if 1 ≤ h ≤ n and xh = 0 if h > n. Under the above assumptions (A.1)–(A.4), the following property holds Herv´es-Beloso et al. (2000):
If a feasible allocation f is blocked in the economyE∞ by a coalition S via an allocation g, then there exists a positive integer N = N(f ) such that the coalition S blocks f via the allocation gN , where ghN = gh if 1 ≤ h ≤ N and ghN = 0 if h > N. As a consequence of this property, Theorem 3.1 and the interpretation stated in a Remark in Section 3 can be extended to this infinite dimensional framework. Moreover, this result can be extended to the case of differential information by following the same argument and applying the results in Herv´es-Beloso et al. (2005b). More work is needed to verify whether the result still holds for other commodity spaces. Of particular interest is the space of measures, where the Walrasian allocations are known to resist manipulation by arbitrarily small coalitions (see Ostroy and Zame (1994) and Herv´es-Beloso et al. (1999)) and may, therefore, be regarded as true perfectly competitive outcomes. Acknowledgements This work is partially supported by Research Grants BEC2003-09067-C04-01 (Ministerio de Ciencia y Tecnolog´ıa and FEDER), PGIDT04XIC30001PN (Xunta de Galicia) and SA070A05 (Junta de Castilla y Le´on). The authors are grateful to an anonymous referee for his/her helpful suggestions. 3
See Herv´es-Beloso et al. (2000) for the comments on the hypothesis.
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