Do pure indivisibilities prevent core equivalence? Core equivalence theorem in an atomless economy with purely indivisible commodities only Tomoki Inoue∗ Graduate School of Economics, Keio University, Tokyo, Japan This Version: January 2, 2005

Abstract We consider a pure exchange economy with a continuum of agents and finitely many indivisible commodities. Every commodity can be consumed only in integer amounts. Thus, agents’ preferences are locally satiated and no commodity bundle has necessarily local cheaper points. We introduce a core which is an intermediate concept between the strong core and the weak core. In our economy, this core is the most natural concept in the sense that it coincides with the set of all exactly feasible Walras allocations. JEL classification: C71; D51 Keywords: Indivisible commodities; Core equivalence; Atomless exchange economy; Separation theorem

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Introduction The purpose of this paper is to establish a core equivalence theorem in an atomless

exchange economy with indivisible commodities only. Some commodities, such as televisions and cars, are physically indivisible. Wine is physically divisible, but its trading ∗

E-mail address: [email protected] (T. Inoue).

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unit is usually a bottle. Thus, wine is institutionally indivisible. Even water and gas for home usage are charged according to standard units. In this paper, we analyze atomless economies where every commodity can be consumed only in integer amounts. Therefore, the commodity space of our economy is given by the products of the set of integers. Not only the countability, but also the discreteness, of the commodity space plays an important role. Our argument cannot apply to any commodity space with countable elements. Therefore, to distinguish them from other types of indivisible commodities, we refer to commodities that can be consumed only in integer amounts as purely indivisible. In our economy, the size of the core depends heavily on the notion of improvement used in defining the core. The strong improvement by a coalition, as used by Aumann (1964) and Hildenbrand (1968) in atomless economies with divisible commodities, turns out to be too strong as it makes the set of core allocations “too big” for the purpose of characterizing the set of Walras allocations. In fact, core allocations may not be envy-free in net trades, although Walras allocations are. (See Example 2.10 in Section 2.) For this reason, in this paper, we refer to the standard notion of the core defined by the strong improvement as the weak core instead of simply the core. On the other hand, in finite economies, it is standard practice to define the core using the notion of the weak improvement.1 The weak improvement requires only a partial strong improvement by a coalition. We call this concept of the core the strong core. The strong core is “too small” to characterize the set of Walras allocations in our setup. (See Example 2.12 in Section 2.) Thus, in our search for an appropriate improvement notion by a coalition, we are naturally led to define the core using a notion of improvement situated between the strong and the weak improvement. In this paper, the core is defined by the notion of improvement used by Wako (1999) and Florig (2001), although in a very different specification of a model from ours. The main result of this paper is that the core, defined by the above notion of improvement, completely characterizes the set of all exactly feasible Walras allocations if agents’ preference relations are nonsatiated in positive directions and if the initial endowment mapping is essentially bounded. In our economy, because of the indivisibility of commodities, Walras allocations are not always exactly feasible, i.e., a free disposal 1

The weak improvement was used in defining the core by, e.g., Debreu and Scarf (1963).

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Walras equilibrium may exist, even though agents’ preference relations satisfy a discrete version of monotonicity. (See Example 2.5 in Section 2.) On the other hand, the improvement used in defining the core guarantees the exact feasibility of core allocations if agents’ preference relations are nonsatiated in positive directions. Our theorem tells us that the core coincides with the set of Walras allocations that are exactly feasible. In an atomless economy with divisible commodities, Aumann (1964) showed that the weak core coincides with the set of Walras allocations if agents’ preference relations are strongly monotone. Hildenbrand (1968) succeeded in weakening the monotonicity assumption to local nonsatiation. It can be easily shown that Walras allocations are in the weak core without any assumptions. To prove the converse inclusion, Aumann (1964) and Hildenbrand (1968) took two steps. First, they showed that for each weak core allocation, there exists a price vector such that, for each agent, if another consumption bundle is strictly preferred to his allocated bundle, then this preferred bundle is more expensive than or equal to his wealth level under that price vector. This implies that, under that price vector, each agent’s allocated bundle satisfies his budget constraint because weak core allocations are resource-feasible by definition and agents’ preference relations are locally nonsatiated by assumption. Therefore, in the first step of their proofs, Aumann and Hildenbrand showed that each weak core allocation could be supported as a pseudoequilibrium without explicitly considering agents’ budget constraints. In other words, it was shown that there exists a price vector that satisfies the following conditions: (1) under that price vector, each agent’s allocated bundle satisfies his budget constraint; (2) a commodity bundle that is in an agent’s budget set, but is not on his budget surface, is not strictly preferred to his allocated bundle. However, some commodity bundles on an agent’s budget surface may be strictly preferred to his allocated bundle. In the second step of their proofs, Aumann and Hildenbrand showed that such a pseudo-equilibrium is actually a Walras equilibrium. That is, they showed that (3) under the price vector obtained in the first step, any commodity bundle on any agent’s budget surface is not strictly preferred to his bundle allocated by the given weak core allocation. An indispensable assumption in the second step is that, for every agent, each consumption bundle on his budget surface has local cheaper points, in the sense that there is another consumption bundle in its neighborhood that is strictly less expensive than the agent’s wealth level 3

with respect to the corresponding price vector. In our economy, due to the discreteness of the commodity space, agents’ preference relations are necessarily locally satiated and no commodity bundle has local cheaper points. Thus, the method of proof of Aumann (1964) or Hildenbrand (1968) does not help in our economy. Indeed, not all pseudo-equilibria are Walrasian. (See Examples 4.1 and 4.2 in Section 4.) Our strategy is to show directly that core allocations are Walrasian. Thus, the goal is to find a price vector that satisfies conditions (1)-(3) simultaneously. Our new separation theorem (Theorem 5.2 in Section 5) guarantees the existence of such a price vector. We will describe the idea of the proof of our core equivalence theorem briefly. Given a core allocation, let E be taken to represent net trades that give rise to commodity bundles strictly preferred by some agents with a positive measure to bundles allocated by the given core allocation. Let H be the linear subspace spanned by agents’ excess demand vectors. The improvement used in defining the core makes disjoint two sets, H and the convex hull of E. Our separation theorem guarantees the existence of a vector which strongly separates those two sets. Thus, there exists a separating hyperplane determined by that vector such that the distances from each of the set H and the convex hull of E to the hyperplane are strictly positive. The vector obtained by our separation theorem is actually an equilibrium price vector. Since H is a linear subspace, that vector is an element of the orthogonal complement of H. Thus, each agent’s bundle allocated by the given core allocation satisfies his budget constraint. Moreover, the strong separation implies that each agent’s allocated bundle maximizes his utility under the budget constraint. Therefore, that vector is an equilibrium price vector. An intuitive reason why the convex hull of E and the linear subspace H can be strongly separated is that, due to the discreteness of our commodity space, the infimum of the distance from an agent’s allocated bundle to its strict upper contour set is strictly positive. By contrast, in an economy with local nonsatiated agents, this infimum is zero. The assumption of essential boundedness of the initial endowment mapping is stronger than that of Aumann (1964) or Hildenbrand (1968). In our economy, the assumption implies that the number of types of initial endowment vectors is finite. This assumption cannot be dispensed with in our economy. (See Example 3.3 in Section 3.) This paper is organized as follows. The precise description of our model is given 4

in the next section. Section 3 provides our core equivalence theorem. Our method for proving the core equivalence theorem is explained in Section 4. Section 5 provides a new separation theorem and its proof. Section 6 gives the proof of our core equivalence theorem. Finally, in Section 7, the relationship of our paper to other literature is studied.

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The model We begin with some notation. Let R, Z, and Q be the set of real numbers, integers,

and rational numbers, respectively. For x = (x(1) , . . . , x(m) ) and y = (y (1) , . . . , y (m) ) in Rm , we write x ≥ y if x(j) ≥ y (j) for all j ∈ {1, . . . , m}; x > y if x ≥ y and x 6= y; x À y if x(j) > y (j) for all j ∈ {1, . . . , m}. The symbol 0 denotes the origin in Rm , m m m as well as the real number zero. Rm + = {x ∈ R | x ≥ 0}; R++ = {x ∈ R | x À 0}; ∑m (j) (j) m m m Zm of x + = {x ∈ Z | z ≥ 0}; Q+ = {x ∈ Q | x ≥ 0}. The inner product j=1 x y

and y in Rm is denoted by x · y. For a topological space T , B(T ) stands for the σ-algebra of the Borel subsets in T . The integral of a vector function is to be taken as the vector of the integral of the components.

2.1

The economy We consider a pure exchange economy with a continuum of agents and finitely many

indivisible commodities. As usual, a commodity is characterized by its physical properties, the date at which it will be available, and the location at which it will be available. There exist L commodities in our economy, where L is a natural number and every commodity can be consumed only in integer amounts. Therefore, the commodity space is given by ZL . We follow the formulation of Hildenbrand (1974), except for the discrete commodity space.2 Let ¯b be a given positive number. A subset X of ZL+ belongs to the universal class X of consumption sets if the set X ∩{x ∈ RL | x ≤ (¯b, . . . , ¯b)} is nonempty.3 The universal 2

For the interpretation of the restriction of consumption sets on the universal class, see Hildenbrand

(1974, pp. 84-86). For a method of introducing the topology to the universal class and the space of preference relations, see Hildenbrand (1974, pp. 96-98). 3 The lower bound of consumption sets in the universal class is zero for simplicity.

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class X endowed with the topology of closed convergence is compact and metrizable. A pair (X, -) of a consumption set X ∈ X and a reflexive, transitive, and complete binary relation - on X is called a preference relation.4 We denote the set of all preference relations by P. A preference relation (X, -) can be identified with the subset - of ZL × ZL and the set {- | (X, -) ∈ P}, endowed with the topology of closed convergence, is separable and completely metrizable, so is P. Given a preference relation (X, -) ∈ P, we define binary relations  and ∼ on X as follows: x  y if and only if not (x - y); x ∼ y if and only if x - y and y - x. Let (A, A, ν) be a probability space of economic agents. We assume that the probability space (A, A, ν) is atomless to describe perfect competition.5 An economic agent a ∈ A is characterized by his preference relation (X(a), -a ) ∈ P and his initial endowment e(a) ∈ ZL+ . Let E : (A, A, ν) → P × ZL+ be a mapping that assigns agents’ characteristics. We write E(a) = (X(a), -a , e(a)). A mapping E : A → P × ZL+ is called an economy if E is A/B(P × ZL+ )-measurable, e : A → ZL+ is ν-integrable, and the set ∫ ∫ ∫ ({ A e(a)dν(a)} − RL+ ) ∩ A X(a)dν(a) is nonempty, where A X(a)dν(a) is the integral of the correspondence X of A into ZL+ . Given an economy E and a B ∈ A, a mapping f : B → ZL+ is called an allocation within B for E if f is ν-integrable and f (a) ∈ X(a) for ν-a.e. a ∈ B. An allocation within A is simply called an allocation. An allocation ∫ ∫ f : A → ZL+ for E is said to be feasible if the inequality A f (a)dν(a) ≤ A e(a)dν(a) holds. Note that, by the definition of an economy, every economy has a feasible allocation. If an ∫ ∫ allocation f satisfies the equality A f (a)dν(a) = A e(a)dν(a), the allocation f is said to be exactly feasible. In our analysis, we assume that every commodity is desirable. We believe that the desirability of commodities is consistent with the weak resource feasibility that allows free disposal of commodities.6 The following desirability condition is weaker than a discrete version of monotonicity, but it is sufficient for our core equivalence theorem, as will be seen in Section 3. Definition 2.1. A preference relation (X, -) is said to be nonsatiated in positive direc4 5 6

By an abuse of the language, - is sometimes called a preference relation. For a more detailed discussion, see the seminal paper of Aumann (1964). See Remark 6.4 in Section 6.

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tions if, for every x ∈ X and every infinite subset Y of {x}+ZL+ , there exists a commodity bundle y ∈ X ∩ co(Y ) with y  x, where co(Y ) denotes the convex hull of Y . Note that nonsatiation in positive directions does not require that consumption set X satisfies co(X) ∩ ZL = X, but X must satisfy X ∩ co(Y ) 6= ∅ for every x ∈ X and every infinite subset Y of {x} + ZL+ . We will give a heuristic example of a nonsatiated preference relation in positive directions. Example 2.2. Let L = 1 and X = 2 Z+ = {0, 2, 4, 6, . . .}. A binary relation - on X is given by the utility function:

  0 if x is even, u(2x) =  x if x is odd,

for x ∈ Z+ . Note that co(X) ∩ Z ) X. Note also that if Y = {4, 8, 12, . . .}, then 2  y holds for each element y of Y , while 6 ∈ X ∩ co(Y ) and 6  2. This preference relation (X, -) is nonsatiated in positive directions. A preference relation (X, -) is said to be discretely monotone if X + ZL+ ⊂ X and y  x whenever x, y ∈ X and y > x. Clearly, discretely monotone preference relations are nonsatiated in positive directions. Note that the set {X ∈ X | X + ZL+ ⊂ X} is countable, but even the set {(X, -) ∈ P | X = ZL+ and (X, -) is discretely monotone.} is uncountable. Therefore, the commodity space of our economy is a countable set, but agents’ preference relations may be full of uncountable variety.

2.2

Walras equilibria Walras equilibrium is a solution concept based on a market mechanism. Its precise

definition is as follows: Definition 2.3. Let E : A → P × ZL+ be an economy. A pair (p, f ) of a price vector p ∈ RL+ and a feasible allocation f for the economy E is called a Walras equilibrium for E if f (a) is maximal for -a in the set {x ∈ X(a) | p · x ≤ p · e(a)} for ν-a.e. a ∈ A, i.e., there exists a B ∈ A with ν(B) = 1 such that, for every a ∈ B, f (a) ∈ X(a) and p · f (a) ≤ p · e(a), and for every a ∈ B, if x ∈ X(a) and p · x ≤ p · e(a), then x -a f (a). An 7

allocation f for E is called a Walras allocation for E if there exists a price vector p ∈ RL+ such that (p, f ) is a Walras equilibrium for E. The set of all Walras allocations for E is denoted by W (E) and the set of all exactly feasible Walras allocations for E is denoted by W ∗ (E). Note that if (p, f ) is a Walras equilibrium, then (αp, f ) is also a Walras equilibrium for every positive number α. In our economy, due to the indivisibility of commodities, Walras equilibria may not exist. (See Example 3.2 in Section 3.) It should be emphasized that we adopt RL as the space of price vectors, not ZL or QL . Following Debreu (1959), it is assumed that the economy works without the help of a good serving as a medium of exchange. Thus, as is shown in the following example, the pro forma exchange rate of a particular pair of commodities is not always rational. Therefore, we adopt RL as the space of price systems. Example 2.4. Let (]0, 1], B(]0, 1]), λ) be a space of economic agents where λ is the one-dimensional Lebesgue measure. Let L = 4 and X(a) = Z4+ for all a ∈]0, 1]. The initial endowment mapping e :]0, 1] → Z4+ is given by e(a) = (n, n, n, n) ∈ Z4+ for agent a ∈](n + 1)−2 , n−2 ] (n = 1, 2, . . .). Note that e is λ-integrable but not λ-essentially bounded. Every agent has the same preference relation, represented by the following utility function: u(x(1) , x(2) , x(3) , x(4) ) = x(1) + x(2) +

√

2 x(3) +

√

2 x(4)

for (x(1) , x(2) , x(3) , x(4) ) ∈ Z4+ . It should be noted that this preference relation can be represented by an integer-valued utility function. In fact, since the set {x ∈ Z4+ | u(x) ≤ m} is finite for all m ∈ R+ , we can arrange all commodity bundles in nondecreasing order of utility level. An allocation f :]0, 1] → Z4+ is defined by   (n − 1, n + 1, n − 1, n + 1) if a ∈]2−1 ((n + 1)−2 + n−2 ), n−2 ] (n = 1, 2, . . .), f (a) =  (n + 1, n − 1, n + 1, n − 1) if a ∈](n + 1)−2 , 2−1 ((n + 1)−2 + n−2 )] (n = 1, 2, . . .). Clearly, if the allocation f is realized in the market, all commodities must be traded. It √ √ can be shown that f is Walrasian under the price vector p∗ = (1, 1, 2, 2). 8

We will show that f is not Walrasian under p = (1, p(2) , p(3) , p(4) ) with p 6= p∗ . Suppose that f is Walrasian under p. From the form of the utility function, it follows that 1 = p(2) and p(3) = p(4) > 0. Thus, we can write p = (1, 1, q, q) for some q > 0. Note that a slight change of prices causes a change of demand vectors for some agents since the √ initial endowment mapping e is not essentially bounded. If q < 2, then either commodity √ 3 or 4 has excess demand. On the other hand, if q > 2, then either commodity 1 or 2 has excess demand. Therefore, p∗ is a unique normalized price vector that makes the allocation f Walrasian. In our core equivalence theorem (Theorem 3.1 in Section 3), contrary to the above example, the initial endowment mapping is assumed to be essentially bounded. In this case, with the assumption of nonsatiation in positive directions of agents’ preference relations, for each core allocation f , we can choose a price vector from ZL under which f is Walrasian. Note that an equilibrium price vector is nonnegative by definition. The reason for this restriction is that free disposal of commodities is assumed. The zero price is excluded in a Walras equilibrium if some agents with a positive measure have nonsatiated preference relations in the usual sense. It should be noted that not all Walras allocations are exactly feasible even if agents’ preference relations are discretely monotone, as is shown in the next example. In our economy, agents’ preference relations are locally satiated, and therefore agents do not always use up their wealth. Hence, there may exist a Walras equilibrium with free disposal. Example 2.5. Let (A, A, ν) be an atomless probability space of economic agents and {A1 , A2 } be a measurable partition of A such that ν(A1 ) = ν(A2 ) = 1/2. Let L = 2 and X(a) = Z2+ for every agent a ∈ A. Agent a ∈ A1 has a preference relation represented by the following utility function: ua (x(1) , x(2) ) =

 

2x(1) + x(2)

if x(1) ∈ {0, 1},

 1 (x(1) + 2x(2) + 3) if x(1) ≥ 2. 2

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(See Figure 1.) Agent a ∈ A2 has a preference relation represented by the following utility function: ua (x(1) , x(2) ) = x(1) + x(2) . (See Figure 2.) Thus, every agent has a discretely monotone preference relation. The initial endowment mapping e : A → Z2+ is given by   (3, 1) if a ∈ A , 1 e(a) =  (1, 3) if a ∈ A . 2 We call this economy E1 .7 Let d : A → Z2+ be defined by   (1, 2) if a ∈ A , 1 d(a) =  (2, 2) if a ∈ A . 2 Then,

∫

d(a)dν(a) = (3/2, 2) < (2, 2) = A

∫ A

e(a)dν(a). One can check that a pair (p, d) is

a Walras equilibrium for the price vector p = (1, p(2) ) with 1 < p(2) < 4/3. Thus, d is a Walras allocation that is not exactly feasible. Therefore, W ∗ (E1 ) ( W (E1 ).

2.3

The cores Our purpose is to characterize the set of all exactly feasible Walras allocations using

the solution concept of cooperative game theory. First, we give the precise definition of the core that completely characterizes the set of exactly feasible Walras allocations, as is shown later. Definition 2.6. Let f : A → ZL+ be an allocation for economy E : (A, A, ν) → P × ZL+ . A coalition S ∈ A can improve upon the allocation f if there exists an allocation g for E such that ν({a ∈ S | f (a), g(a) ∈ X(a) and g(a) Âa f (a)}) > 0,8 g(a) = f (a) for ν-a.e. ∫ ∫ a ∈ S \ {b ∈ S | f (b), g(b) ∈ X(b) and g(b) Âb f (b)}, and S g(a)dν(a) ≤ S e(a)dν(a). The set of all feasible allocations for the economy E that cannot be improved upon by any coalition is called the core of E and is denoted by C(E). 7

This economy will be dealt with in Examples 2.10, 2.11, 2.12, 3.4, and 4.1. In these examples,

notation of specific allocations is unified. 8 The measurability of the set {a ∈ S | f (a), g(a) ∈ X(a) and g(a) Âa f (a)} can be shown by an argument similar to Claim 5 in the proof of Lemma 6.1 in Section 6.

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Wako (1999) introduced this concept of the core in the Shapley-Scarf economy. Florig (2001) used the same improvement as the core to define some sort of optimality. We will compare this core with other familiar core concepts: the weak core and the strong core. The definitions are as follows: Definition 2.7. Let f : A → ZL+ be an allocation for economy E : (A, A, ν) → P × ZL+ . A coalition S ∈ A with ν(S) > 0 can strongly improve upon the allocation f if there exists an ∫ ∫ allocation g for E such that g(a) Âa f (a) for ν-a.e. a ∈ S and S g(a)dν(a) ≤ S e(a)dν(a). The set of all feasible allocations for the economy E that cannot be strongly improved upon by any coalition S ∈ A with ν(S) > 0 is called the weak core of E and is denoted by Cw (E). Definition 2.8. Let f : A → ZL+ be an allocation for economy E : (A, A, ν) → P × ZL+ . A coalition S ∈ A can weakly improve upon the allocation f if there exists an allocation g for E such that ν({a ∈ S | f (a), g(a) ∈ X(a) and g(a) Âa f (a)}) > 0, f (a) -a g(a) for ∫ ∫ ν-a.e. a ∈ S, and S g(a)dν(a) ≤ S e(a)dν(a). The set of all feasible allocations for the economy E that cannot be weakly improved upon by any coalition is called the strong core of E and is denoted by Cs (E). Schmeidler (1972) proved that in economies with divisible commodities, for every ε ∈ ]0, 1], the weak core coincides with the set of feasible allocations that are not strongly improved upon by any coalition with measure less than ε. His argument is valid in our economy. Namely, for every ε ∈ ]0, 1], the core, the weak core, and the strong core of our economy do not expand if we restrict possible coalitions to those with measure less than ε. By definition, Cs (E) ⊂ C(E) ⊂ Cw (E) for every economy E. In an economy where all commodities are perfectly divisible, and agents have strongly monotone preference relations, the three core concepts above coincide. Actually, if a coalition S can weakly improve upon a feasible allocation f via an allocation g, then by transferring small amounts of commodities from agents in S who prefer g to f to other agents in S, we can construct another allocation which is feasible within S and which is preferred to f for almost every agent in S. Therefore, in such an economy, the strong core coincides with the weak core. 11

Moreover, these cores are equal to the set of Walras allocations. (See Aumann (1964).) In our economy, however, a small amount of commodities cannot be transferred, so these three cores may differ. In contrast to an economy with divisible commodities, the weak core in our economy may be much larger than the set of Walras allocations. Actually, Walras allocations are always envy-free in net trades, but not all weak core allocations are. The precise definition of envy-freeness is as follows: Definition 2.9. A feasible allocation f : A → ZL+ for economy E : (A, A, ν) → P × ZL+ is said to be envy-free in net trades if there exists a B ∈ A with ν(B) = 1 such that for every a and every b in B, if (f (b) − e(b)) + e(a) ∈ X(a), then (f (b) − e(b)) + e(a) -a f (a). The set of all envy-free allocations in net trades for E is denoted by EF (E). It is easy to show that W (E) ⊂ Cw (E) ∩ EF (E) for every economy E. The next example illustrates that not all weak core allocations are envy-free in net trades. Example 2.10. We consider the economy E1 from Example 2.5. Let {A21 , A22 } be a measurable partition of A2 with ν(A21 ) = ν(A22 ) = 1/4. We define an allocation h : A → Z2+ by

   (3, 1) if a ∈ A1 ,   h(a) =

(0, 4) if a ∈ A21 ,     (2, 2) if a ∈ A . 22

It is easy to show that h ∈ Cw (E1 ). If a ∈ A1 and b ∈ A21 , then (h(b) − e(b)) + e(a) = (2, 2) Âa (3, 1) = h(a). Thus, we have h ∈ / EF (E1 ). Hence, Cw (E1 ) ∩ EF (E1 ) ( Cw (E1 ). Even the intersection of the weak core and the set of envy-free allocations may be strictly larger than the set of Walras allocations. The next example illustrates this point. Example 2.11. We consider the economy E1 from Example 2.5. We define an allocation / W (E1 ) and g : A → Z2+ by g(a) = (2, 2) for all a ∈ A. One can easily check that g ∈ g ∈ Cw (E1 ) ∩ EF (E1 ). Thus, W (E1 ) ( Cw (E1 ) ∩ EF (E1 ). 12

From the examples above, it follows that the strong improvement used in defining the weak core is too strong to characterize the set of all Walras allocations. On the other hand, the weak improvement used in defining the strong core is too weak to characterize the set of all exactly feasible Walras allocations. As a direct consequence of Theorem 3.1 in the next section, under mild assumptions, Cs (E) ⊂ W ∗ (E) holds for every economy E. However, this inclusion may be strict. The next example illustrates this point. Example 2.12. We consider the economy E1 from Example 2.5. To show the strict inclusion Cs (E1 ) ( W ∗ (E1 ), by virtue of Theorem 3.1 in the next section, it suffices to say that there exists an exactly feasible Walras allocation that is not a strong core allocation. We will consider the initial allocation e. It can be easily shown that for every price vector p = (1, p(2) ) with 1 < p(2) < 4/3, a pair (p, e) is a Walras equilibrium. Thus, e is an exactly feasible Walras allocation, i.e., e ∈ W ∗ (E). However, e is not a strong core allocation. Indeed, the allocation g : A → Z2+ defined by g(a) = (2, 2) for all a ∈ A is feasible, g(a) Âa e(a) for all a ∈ A1 , and g(a) ∼a e(a) for all a ∈ A2 . Hence, e is not a strong core allocation.9 Therefore, we have Cs (E1 ) ( W ∗ (E1 ).

3

Core equivalence theorem The main result of this paper is that the set of all exactly feasible Walras allocations

is completely characterized by the core in our economy with purely indivisible commodities only. The precise statement is as follows: Theorem 3.1. Let E : (A, A, ν) → P × ZL+ be an economy which satisfies the following conditions. (i) For ν-a.e. a ∈ A, the preference relation (X(a), -a ) is nonsatiated in positive directions. 9

From Example 2.11, g is a weak core allocation but is not a core allocation. A Walras allocation e

is a core allocation by Theorem 3.1 but is weakly improved upon via g.

13

(ii) The initial endowment mapping e : A → ZL+ is ν-essentially bounded. Then, the set C(E) of core allocations coincides with the set W ∗ (E) of exactly feasible Walras allocations. Hence, Cs (E) ⊂ C(E) = W ∗ (E) ⊂ W (E) ⊂ Cw (E) ∩ EF (E) ⊂ Cw (E). We will explain the method of proof of Theorem 3.1 in the next section and give the formal proof in Section 6. It should be noted that, in our economy, a Walras equilibrium may not exist. In such a case, the equivalence still holds in the sense that both C(E) and W ∗ (E) are empty. A sufficient condition for the existence of an exactly feasible Walras allocation or, equivalently, the nonemptiness of the core is not known yet, but we believe that this theorem is meaningful because of the following two facts. (1) Some economies have exactly feasible Walras allocations. (See economy E3 of Example 3.2 and economy E1 of Example 3.4.) (2) This equivalence theorem provides a new light on the equilibrium existence problem. In fact, we may explore a sufficient condition for nonemptiness of the core instead of considering the existence problem of an exactly feasible Walras allocation. In the next example, we find all exactly feasible Walras allocations and all core allocations and show that the equivalence holds both in the case where an exactly feasible Walras allocation exists and in the case where it does not exist. Example 3.2. Let (A, A, ν) be an atomless probability space of economic agents and {A1 , A2 } be a measurable partition of A. Let L = 2 and X(a) = Z2+ for all a ∈ A. Agents’ preference relations are represented by the utility function:   x(1) + 2x(2) if a ∈ A , 1 (1) (2) ua (x , x ) =  2x(1) + x(2) if a ∈ A . 2

(See Figures 3 and 4.) The initial endowment mapping e : A → Z2+ is given by   (1, 2) if a ∈ A , 1 e(a) =  (2, 1) if a ∈ A . 2 We will consider two economies, E2 and E3 . The difference between E2 and E3 lies only in the proportion of A1 to A2 . In economy E2 , ν(A1 ) = 1/3 and ν(A2 ) = 2/3 hold. On the other hand, in economy E3 , ν(A1 ) = ν(A2 ) = 1/2 holds. 14

We begin with economy E2 . In economy E2 , both the core C(E2 ) and the set W ∗ (E2 ) of exactly feasible Walras allocations are empty. Claim 1. W ∗ (E2 ) = ∅. Proof. Since agents’ preference relations are discretely monotone, each equilibrium price vector p = (p(1) , p(2) ) is strictly positive if it exists. If p(2) /p(1) > 1 (resp. p(2) /p(1) < 1), then commodity 1 (resp. commodity 2) has excess demand. Let p(1) = p(2) > 0. In this case, the demand vector of agent a ∈ A1 is (0, 3) and that of agent a ∈ A2 is (3, 0). Thus, the aggregate demand is (2, 1), whereas the aggregate initial endowment is (5/3, 4/3). Therefore, p(1) = p(2) > 0 cannot be an equilibrium price vector. Hence, W ∗ (E2 ) = ∅. This completes the proof of Claim 1. Claim 2. C(E2 ) = ∅. Proof. Let f be a core allocation of economy E2 . From individual rationality of core ∑ allocations, we have kf (a)k1 ≥ 3 for ν-a.e. a ∈ A, where kxk1 = Li=1 |x(i) | for x ∈ RL . If ν({a ∈ A | kf (a)k1 > 3}) > 0, then, due to resource feasibility, we have ν({a ∈ A | kf (a)k1 < 3}) > 0. Therefore, kf (a)k1 = 3 for ν-a.e. a ∈ A. Again, due to individual rationality, f (a) ∈ {(0, 3), (1, 2)} for ν-a.e. a ∈ A1 and f (a) ∈ {(2, 1), (3, 0)} for ν-a.e. a ∈ A2 . If ν({a ∈ A1 | f (a) = (1, 2)}) > 0 and ν({a ∈ A2 | f (a) = (2, 1)}) > 0, then f can be strongly improved upon. Thus, we have ν({a ∈ A1 | f (a) = (1, 2)})×ν({a ∈ A2 | f (a) = (2, 1)}) = 0. Due to resource feasibility, we have ν({a ∈ A1 , | f (a) = (0, 3)}) = 1/3, ν({a ∈ A2 , | f (a) = (2, 1)}) = 1/3, and ν({a ∈ A2 | f (a) = (3, 0)}) = 1/3. The coalition A1 ∪ {a ∈ A2 | f (a) = (2, 1)} can improve upon f . This contradicts f ∈ C(E2 ). This completes the proof of Claim 2. Next, we will consider economy E3 . Let g : A → Z2+ be an allocation defined by   (0, 3) if a ∈ A , 1 g(a) =  (3, 0) if a ∈ A . 2 We show that W ∗ (E3 ) = [ g ] = C(E3 ), where [ g ] = {h : A → Z2+ | h is an allocation and h = g a.e.}. 15

Claim 3. W ∗ (E3 ) = [ g ]. Proof. One can easily check that g is Walrasian under the price vector (1, 1). Thus, [ g ] ⊂ W ∗ (E3 ). From an argument similar to the proof of Claim 1, any vector p = (p(1) , p(2) ) À 0 with p(1) 6= p(2) is not an equilibrium price vector. Therefore, W ∗ (E3 ) = [ g ]. This completes the proof of Claim 3. Claim 4. C(E3 ) = [ g ]. Proof. One can easily show that g ∈ C(E3 ). Let h ∈ C(E3 ). From an argument similar to the proof of Claim 2, we have h(a) ∈ {(0, 3), (1, 2)} for ν-a.e. a ∈ A1 , h(a) ∈ {(2, 1), (3, 0)} for ν-a.e. a ∈ A2 , and ν({a ∈ A1 | h(a) = (1, 2)}) × ν({a ∈ A2 | h(a) = (2, 1)}) = 0. Thus, due to resource feasibility, h(a) = (0, 3) for ν-a.e. a ∈ A1 and h(a) = (3, 0) for ν-a.e. a ∈ A2 . Therefore, h = g a.e. Hence, C(E3 ) = [ g ]. This completes the proof of Claim 4.

Condition (ii) of Theorem 3.1 implies that the number of types of initial endowment vectors is essentially finite. If the initial endowment mapping is not essentially bounded, core allocations may not be Walrasian. The next example illustrates this point. Example 3.3. Let (]0, 1], B(]0, 1]), λ) be a space of economic agents where λ is the one-dimensional Lebesgue measure. Let L = 2 and X(a) = Z2+ for all a ∈ ]0, 1]. The initial endowment mapping e : ]0, 1] → Z2+ is given by e(a) = (1, n) ∈ Z2 for agent a ∈ ](n + 1)−2 , n−2 ] (n = 1, 2, . . .). Note that the allocation e is λ-integrable but not λessentially bounded. Every agent has the same lexicographic preference -, as follows: for (x(1) , x(2) ) and (y (1) , y (2) ) in Z2+ , (x(1) , x(2) ) - (y (1) , y (2) ) if and only if x(1) < y (1) , or x(1) = y (1) and x(2) ≤ y (2) . Notice that in this economy, it is not possible for two distinct elements of the consumption set to be indifferent, so the core coincides with the strong core. It is easy to show that e is a core allocation. We will show that e cannot be achieved as a Walras equilibrium. Let p = (p(1) , p(2) ) ∈ R2+ be a price vector. If p(1) = 0 or p(2) = 0, then (p, e) is clearly not a Walras equilibrium. Hence, we consider the case where p(1) > 0 and p(2) > 0. There exists a natural number n0 such that p(1) /p(2) ≤ n for all n ≥ n0 . For

16

n ≥ n0 , we have p · (2, 0) = 2 p(1) ≤ p(1) + n p(2) = p · (1, n). Since (2, 0) Â (1, n) holds for every agent, e(a) is not maximal in the set {x ∈ X(a) | p·x ≤ ∪ p · e(a)} for agent a ∈ n≥n0 ](n + 1)−2 , n−2 ]. This means that (p, e) cannot be a Walras equilibrium for any price vector p À 0. Therefore, e is not a Walras equilibrium allocation. As is shown in the next example, for some economy E, every inclusion Cs (E) ⊂ C(E) = W ∗ (E) ⊂ W (E) ⊂ Cw (E) ∩ EF (E) ⊂ Cw (E) may be strict. Example 3.4. We consider the economy E1 from Example 2.5. Note that the strong core of economy E1 is nonempty. Actually, one can show that an allocation f : A → Z2+ for E1 , defined by

  (1, 3) if a ∈ A , 1 f (a) =  (3, 1) if a ∈ A , 2

is a strong core allocation, i.e., f ∈ Cs (E1 ). By summing up all the results from Theorem 3.1 and Examples 2.5, 2.10, 2.11, and 2.12, we have ∅ 6= Cs (E1 ) ( C(E1 ) = W ∗ (E1 ) ( W (E1 ) ( Cw (E1 ) ∩ EF (E1 ) ( Cw (E1 ).

4

Discussion of the method for proving the core equivalence theorem In this section, we first state the reason why a minor modification of the proof of

Aumann (1964) or Hildenbrand (1968) does not help our core equivalence theorem. Next, we state the method of our proof. In an economy with some divisible commodities,10 Aumann (1964) and Hildenbrand (1968) first showed that, for each weak core allocation f , there exists a price vector p ∈ RL+ \ {0} such that (i) for ν-a.e. a ∈ A, if x ∈ X(a) and x Âa f (a), then p · x ≥ p · e(a). Under the assumption of local nonsatiation of agents’ preference relations, from (i) and the feasibility of the weak core allocation f , we have (ii) p · f (a) = p · e(a) for ν-a.e. 10

In this paragraph, the consumption set X(a) of agent a is assumed to be a subset of RL with a

continuum of elements.

17

a ∈ A. Therefore, the pair (p, f ) is a pseudo-equilibrium.11 In an economy with some divisible commodities, the concept of a pseudo-equilibrium, which is a weaker concept than a Walras equilibrium, is useful because a sufficient condition for a pseudo-equilibrium to be a Walras equilibrium is known. From the standard argument,12 one can show that a pseudo-equilibrium (p, f ) is actually a Walras equilibrium if, for almost every agent a ∈ A, each commodity bundle x ∈ X(a) with p · x = p · e(a) has local cheaper points, that is, there exists a sequence (xn )n of X(a) such that xn → x and p · xn < p · x for all n. In our economy, however, the commodity space ZL is discrete, so any commodity bundle has no local cheaper points. Actually, a pseudo-equilibrium is not an effective concept in our economy. The following examples illustrate this point. Example 4.1. We consider the economy E1 from Example 2.5. The allocation g : A → Z2+ , g(a) = (2, 2) for all a ∈ A, is a pseudo-equilibrium allocation under the price vector (1, 1), but g is not Walrasian. (See Example 2.11.) Example 4.2. We consider the economy from Example 3.3. The initial endowment allocation e is a pseudo-equilibrium allocation under the price vector (1, 0), but e is not Walrasian. In our proof, we directly show that core allocations are Walrasian without going through pseudo-equilibria. The proof of Theorem 3.1 consists of three lemmas: Lemmas 6.1, 6.2, and 6.3. In Lemma 6.1, we show that, if agents’ preference relations are nonsatiated in positive directions, then every core allocation is exactly feasible. As we saw before, not all Walras allocations are exactly feasible. (See Example 2.5.) Since the weak core includes the set of all Walras allocations, weak core allocations may not be exactly feasible. Lemma 6.1 tells us that the strong improvement used in defining the weak core is too strong to guarantee the exact feasibility, but the improvement used in defining the core is sufficiently weak. In Lemma 6.2, we show that core allocations are essentially bounded if agents’ prefer11

A pair (p, f ) of a price vector p ∈ RL + \ {0} and a feasible allocation f is called a pseudo-equilibrium

if it satisfies conditions (i) and (ii) above. 12 The idea is the same as Debreu (1959, (1), 4.9).

18

ence relations are nonsatiated in positive directions and if the initial endowment mapping is essentially bounded. Our commodity space is ZL , so the essential boundedness of the initial endowments means that the number of types of agents’ initial endowment vectors is essentially finite. Lemma 6.2 tells us that the number of types of agents’ consumption bundles allocated by a core allocation is also finite, although agents’ preference relations may have uncountable variety. In Lemma 6.3, we show that, if the initial endowments are essentially bounded, then every exactly feasible essentially bounded core allocation can be decentralized by a suitable price system. Let f be an exactly feasible essentially bounded core allocation. Since f − e is essentially bounded, the range of f − e may be regarded as a finite set. Let H = span{f (a) − e(a) | a ∈ A} and ϕ(a) = {x ∈ X(a) | x Âa f (a)} − {e(a)} for a ∈ A. (∪ ) ( ) If there exists an A∗ ∈ A with ν(A∗ ) = 1 and co a∈A∗ ϕ(a) ∩ H − RL+ = ∅, then, from Theorem 5.2 in the next section, there exists a vector p ∈ H ⊥ ∩ ZL+ and a posi(∪ ) tive real number ε such that p · z ≥ ε for all z ∈ co a∈A∗ ϕ(a) ,13 that is, the vector (∪ ) p strongly separates two sets of co a∈A∗ ϕ(a) and H − RL+ . (The strong separation requires the existence of a separating hyperplane that has strictly positive distances from each of the separated sets.) The vector p is actually an equilibrium price vector. Indeed, p · f (a) = p · e(a) for all a ∈ A∗ since p ∈ H ⊥ . Thus, almost every agent satisfies his own budget constraint under p. Moreover, if a ∈ A∗ , x ∈ X(a), and x Âa f (a), then p · (x − e(a)) ≥ ε > 0. Thus, (p, f ) is a Walras equilibrium. Therefore, the majority of the proof of Lemma 6.3 is to show that the improvement used in defining the core is weak (∪ ) enough to guarantee that co a∈A∗ ϕ(a) and H − RL+ are disjoint for some A∗ ∈ A with ν(A∗ ) = 1. An intuitive reason why the two sets

∪ a∈A∗

ϕ(a) and H − RL+ can be strongly sep-

arated is that, due to the discreteness of the commodity space, inf{kx − f (a)k∞ | x ∈ X(a) and x Âa f (a)} ≥ 1 holds, where kxk∞ = max{|x(i) | | i = 1, . . . , L}. Thus, in contrast to an economy with divisible commodities, the discreteness of the commodity space is useful rather than obstructive.

13

H ⊥ means the orthogonal complement of H.

19

5

Separation theorem In this section, we explain our separation theorem, which plays a central role in the

proof of the core equivalence theorem. First, we prove the following lemma, which was originally shown by Gordan (1873).14 For completeness, we provide a proof. Lemma 5.1. Let v in RL and E be a nonempty subset of {x ∈ ZL | x ≥ v}. We define a subset E 0 of E as follows: x ∈ E 0 if and only if x ∈ E and there exists no y ∈ E such that y < x. Then, the set E 0 is nonempty and finite. Moreover, E ⊂ E 0 + ZL+ . Proof. Clearly, the set E 0 is nonempty and E ⊂ E 0 + ZL+ . It suffices to show that the set E 0 is finite. We will show this fact by induction on the dimension of RL . Clearly, this fact holds for L = 1. Suppose that this fact holds for L = k. We will prove this fact when L = k + 1. Since the set E is nonempty, we can choose a point x∗ = (x∗ (1) , . . . , x∗ (k+1) ) ∈ E. For j ∈ {1, . . . , k + 1} and n ∈ Z with v (j) ≤ n ≤ x∗ (j) , we define a subset Ej,n of E as follows: x = (x(1) , . . . , x(k+1) ) ∈ Ej,n if and only if x ∈ E, x(j) = n, and there exists no y = (y (1) , . . . , y (k+1) ) ∈ E such that y (j) = n and y < x. By the induction hypothesis, the set Ej,n is finite or empty. Therefore, the set E 0 is finite if we can show that ∪ ∪ (1) (k+1) ) ∈ E 0 \ {x∗ }. Then, there E 0 ⊂ {x∗ } ∪ k+1 j=1 v (j) ≤n≤x∗ (j) Ej,n . Let x = (x , . . . , x exists j ∈ {1, . . . , k + 1} such that x(j) < x∗ (j) . Hence, we have x ∈ Ej,x(j) . This implies ∪ ∪ 0 that E 0 ⊂ {x∗ } ∪ k+1 j=1 v (j) ≤n≤x∗ (j) Ej,n . Therefore, the set E is finite. This completes the proof of Lemma 5.1. Note that this lemma relies on the additivity of the set of integers. Now, we are ready to give our separation theorem. Theorem 5.2. Let v in RL and E be a nonempty subset of {x ∈ ZL | x ≥ v}. Let H be a linear subspace of RL . If the set co(E) ∩ (H − RL+ ) is empty, then we have the followings: (i) There exists a nonempty subset U of H ⊥ ∩ RL+ and a positive number ε such that U is open in H ⊥ ∩ RL+ and p · z ≥ ε for every p ∈ U and every z ∈ co(E), where H ⊥ is the orthogonal complement of H and co(E) is the closed convex hull of E.

14

I owe this reference to Kazuo Murota.

20

(ii) If H is a linear subspace spanned by some elements of ZL , then there exists a vector p ∈ H ⊥ ∩ ZL+ and a positive number ε such that p · z ≥ ε for every z ∈ co(E). In this theorem, the inherent natures of Z are essential. If E = {(x(1) , x(2) ) ∈ R2 | x(1) ≥ −5, x(2) ≥ 0, x(1) + x(2) > 0} and H = {(x(1) , x(2) ) ∈ R2 | x(1) + x(2) = 0}, then co(E) ∩ (H − R2+ ) = ∅. However, two sets of co(E) and H − R2+ cannot be strongly separated. Proof. (i) We define a subset E 0 of E as follows: x ∈ E 0 if and only if x ∈ E and there exists no y ∈ E such that y < x. By Lemma 5.1, the set E 0 is nonempty and finite. Moreover, E ⊂ E 0 + ZL+ . It follows that co(E) ⊂ co(E 0 + ZL+ ) = co(E 0 ) + RL+ . Since the set E 0 is finite, the convex hull co(E 0 ) of E 0 is compact. From co(E)∩(H −RL+ ) = ∅, it follows that co(E 0 ) ∩ (H − RL+ ) = ∅. Since the set co(E 0 ) is compact and the set H − RL+ is closed and convex, by the well-known separation theorem, there exists a vector p∗ ∈ RL \ {0} and two real numbers α and β such that p∗ · z ≥ α > β ≥ p∗ · y for every z ∈ co(E 0 ) and every y ∈ H − RL+ . From this, it follows that p∗ ∈ RL+ and α>β≥

sup p∗ · y ≥ sup p∗ · y = 0. y∈H−RL +

y∈−RL +

Moreover, since H is a linear subspace, we have p∗ ∈ H ⊥ . Summing up these facts, there exists a vector p∗ ∈ H ⊥ ∩ RL+ and an α ∈ R++ such that p∗ · z ≥ α for every z ∈ co(E 0 ). Since the set co(E 0 ) is a compact set, by Berge’s theorem, the mapping p 7→ min{p · z | z ∈ co(E 0 )} is continuous. Thus, there exists an open subset V of RL with p∗ ∈ V such that p · z > α/2 for every p ∈ V and every z ∈ co(E 0 ). For p ∈ RL+ , we have inf{p · z | z ∈ co(E 0 + ZL+ )} = inf{p · z | z ∈ co(E 0 ) + RL+ } = min{p · z | z ∈ co(E 0 )}. Let U = V ∩ H ⊥ ∩ RL+ . Since p∗ ∈ U , the set U is nonempty. In addition, the set U is open in H ⊥ ∩ RL+ . It follows that p · z > α/2 for every p ∈ U and every z ∈ co(E 0 + ZL+ ). 21

Since co(E 0 + ZL ) = co(E 0 ) + RL+ is closed, we have co(E) ⊂ co(E 0 + ZL+ ). Therefore, we have established (i). (ii) Let H be a linear subspace spanned by some elements of ZL . From the orthonormalization of Schmidt, the orthogonal complement H ⊥ of H is also spanned by some elements of ZL . Thus, the set H ⊥ ∩ QL is dense in H ⊥ . From (i), there exists a vector p ∈ H ⊥ ∩ QL+ and an ε ∈ R++ such that p · z ≥ ε for every z ∈ co(E). Since mp ∈ H ⊥ ∩ ZL+ for a sufficiently large natural number m, it follows that (mp)·z ≥ mε for every z ∈ co(E). This completes the proof of (ii). From the proof of Theorem 5.2, we have the following: Remark 5.3. Let E be a nonempty and bounded-from-below subset of ZL . Let H be a linear subspace of RL . Then, min{p · z | z ∈ co(E)} > 0} is well-defined for p ∈ RL+ , and a subset U ∗ of RL defined by } { U ∗ = p ∈ H ⊥ ∩ RL+ | min{p · z | z ∈ co(E)} > 0 is open in H ⊥ ∩ RL+ . Moreover, if the set co(E) ∩ (H − RL+ ) is empty, then the set U ∗ is nonempty.

6

Proof of the core equivalence theorem In this section, we give a proof of Theorem 3.1. Without any assumptions, one can

easily show that W ∗ (E) ⊂ C(E) for every economy E. Thus, our purpose is to show the converse inclusion. To this end, we take three lemmas. In each lemma, we try to clarify the role of assumptions. Hence, the assumptions of Theorem 3.1 are not all used in Lemmas 6.1 and 6.3. Lemma 6.1. Let E : (A, A, ν) → P × ZL+ be an economy that satisfies condition (i) of Theorem 3.1. Let B ∈ A with ν(B) > 0 and let f : B → ZL+ be an allocation within B such ∫ ∫ that B f (a)dν(a) < B e(a)dν(a). Then, there exists an allocation g : B → ZL+ within B ∫ ∫ such that B g(a)dν(a) = B e(a)dν(a), ν({a ∈ B | f (a), g(a) ∈ X(a) and g(a) Âa f (a)}) > 0, and g(a) = f (a) for ν-a.e. a ∈ B \ {b ∈ B | f (b), g(b) ∈ X(b) and g(b) Âb f (b)}. In 22

particular, every core allocation is exactly feasible. Proof. For simplicity, we assume that for every a ∈ B, f (a) ∈ X(a), and (X(a), -a ) is nonsatiated in positive directions. By replacing the coordinates, if necessary, we may ∫ ∫ assume B f (1) (a)dν(a) < B e(1) (a)dν(a). Let N be a natural number such that ∫ ∫ (1) N ν(B) > e (a)dν(a) − f (1) (a)dν(a). B

B

Let Y = {2N, 2N + 1, . . .} × {0}L−1 . Then, Y is an infinite subset of ZL+ . Note that co({f (a)} + Y ) ∩ ZL = {f (a)} + co(Y ) ∩ ZL = {f (a)} + Y . Since agents’ preference relations are nonsatiated in positive directions, for every a ∈ B, there exists a v ∈ Y such that f (a) + v ∈ X(a) and f (a) + v Âa f (a). For j ∈ Z+ , let Bj = {a ∈ B | f (a), f (a) + (2N + j, 0 . . . , 0) ∈ X(a) and f (a) + (2N + j, 0, . . . , 0) Âa f (a)}. Note that B =

∪∞ j=0

Bj .

Claim 5. Bj ∈ A for every j ∈ Z+ . Proof of Claim 5. Let j ∈ Z+ and let D = {(X, -, x, y) ∈ P × ZL × ZL | x, y ∈ X and not (x - y)}. From an argument similar to that of Hildenbrand (1974, Corollary 2, p.98), we have D ∈ B(P × ZL × ZL ) = B(P) ⊗ B(ZL ) ⊗ B(ZL ). The last equality relies on the spaces P and ZL being separable and metrizable. We define a mapping r : B → P × ZL × ZL by r(a) = (X(a), -a , f (a) + (2N + j, 0, . . . , 0), f (a)). Since the mapping r is A/B(P) ⊗ B(ZL ) ⊗ B(ZL )-measurable, we have Bj = r−1 (D) ∈ A. This completes the proof of Claim 5. ∪j−1

Bi for j ≥ 1. Then, the sequence (Cn )n≥0 of measurable ∪ sets is pairwise disjoint and B = ∞ j=0 Cj . Therefore, there exists a natural number k Let C0 = B0 and Cj = Bj \

i=0

such that

We define a mapping h :

k ∑

∪k j=0

ν(Cj ) >

j=0

1 ν(B). 2

Cj → ZL+ by

h(a) =

k ∑

(2N + j, 0, . . . , 0) ICj (a),

j=0

23

where ICj is the indicator function of Cj . From the definition of each Cj , it follows that ∪ f (a) + h(a) ∈ X(a) and f (a) + h(a) Âa f (a) for every a ∈ kj=0 Cj . In addition, we have ∫ h(1) (a)dν(a) =

Sk

j=0

Cj

k ∑ (2N + j) ν(Cj ) j=0

≥ 2N

k ∑

ν(Cj )

j=0

> N ν(B) ∫ ∫ (1) f (1) (a)dν(a). e (a)dν(a) − > B

{∫

Since the set

B

¯ } k ¯ ∪ ¯ Cj h(a)dν(a) ∈ RL ¯ S ∈ A, S ⊂ ¯ S j=0

is convex by Lyapunov’s theorem, there exists an S ∈ A such that S ⊂ (∫

∫

∫ e (a)dν(a) − (1)

h(a)dν(a) = B

S

f

(1)

∪k j=0

Cj , and

) (a)dν(a), 0, . . . , 0 .

B

Clearly, ν(S) > 0. We define a mapping g : B → ZL+ by g(a) = f (a) + h(a) IS (a). We have ∫

∫

∫

f (a)dν(a) + h(a)dν(a) ≤ S ∫ ∫ g (1) (a)dν(a) = e(1) (a)dν(a),

∫

g(a)dν(a) =

B

B

B (i)

B

for every 2 ≤ i ≤ L and every a ∈ B,

(i)

g (a) = f (a) g(a) Âa f (a) g(a) = f (a) If the inequality

∫ B

e(a)dν(a), B

for every a ∈ S, and for every a ∈ B \ S.

g(a)dν(a) <

∫ B

e(a)dν(a) still holds, by taking the same steps as

above, we construct a required allocation which is exactly feasible within B. This ends the proof of Lemma 6.1. Lemma 6.2. Let E : (A, A, ν) → P × ZL+ be an economy that satisfies conditions (i) and (ii) of Theorem 3.1. Then, every core allocation for E is ν-essentially bounded. Proof. Assume to the contrary that a core allocation f : A → ZL+ is not ν-essentially 24

bounded. For simplicity, we assume that, for every a ∈ A, f (a) ∈ X(a) holds and (X(a), -a ) is nonsatiated in positive directions. Since e : A → ZL+ is ν-essentially bounded by assumption, the set {x ∈ ZL+ | ν({a ∈ A | e(a) = x}) > 0} is nonempty and finite. We denote this set by {x1 , . . . , xK }. For k = 1, . . . , K, we define Ak = {a ∈ A | e(a) = xk } ∈ ∑ A. Clearly, ν(Ak ) > 0 for every k ∈ {1, . . . , K} and K k=1 ν(Ak ) = ν(A) = 1. From the fact that f is not ν-essentially bounded, it follows that there exists a k ∈ {1, . . . , K} such that the set Fk = {z ∈ ZL+ | ν({a ∈ Ak | f (a) = z}) > 0} is an infinite set. We define a subset Fk0 of Fk as follows: z ∈ Fk0 if and only if z ∈ Fk and there exists no y ∈ Fk such that y < z. From Lemma 5.1, the set Fk0 is nonempty and finite, and Fk ⊂ Fk0 + ZL+ . Therefore, there exists a z ∗ ∈ Fk0 such that Fk ∩ ({z ∗ } + ZL+ ) is an infinite ∗

set. Let Y = (Fk ∩ ({z ∗ } + ZL+ )) \ {z ∗ } and Azk = {a ∈ Ak | f (a) = z ∗ } ∈ A. By ∗

the definition of the set Fk , we have ν(Azk ) > 0. Since agents’ preference relations are ∗

nonsatiated in positive directions, for each a ∈ Azk , there exists a y ∈ X(a) ∩ co(Y ) such that y Âa z ∗ = f (a). For y ∈ ZL+ , let ∗

By = {a ∈ Azk | y ∈ X(a) and y Âa z ∗ }. ∗

Then, By ∈ A for every y ∈ ZL+ and Azk =

∪ co(Y )∩ZL +

∑

∗

0 < ν(Azk ) ≤

y∈co(Y

By . Since ν(By ),

)∩ZL +

there exists a y ∗ ∈ co(Y ) ∩ ZL+ such that ν(By∗ ) > 0. From y ∗ ∈ co(Y ), it follows that there exists a finite subset {y1 , . . . , yJ } of Y and positive numbers {α1 , . . . , αJ } such that ∑J ∑J yj ∗ j=1 αj = 1 and y = j=1 αj yj . For j ∈ {1, . . . , J}, let Ak = {a ∈ Ak | f (a) = ∗

y

yj } ∈ A. Then, ν(Akj ) > 0 since yj ∈ Fk . Note that the family {Azk , Ayk1 , . . . , AykJ } is pairwise disjoint. Let δ = min{ν(Ayk1 ), . . . , ν(AykJ )} > 0. By Lyapunov’s theorem, for y

every j ∈ {1, . . . , J}, there exists a Cj ∈ A such that Cj ⊂ Akj and ν(Cj ) = αj δ. Let ∪ C = Jj=1 Cj . Then, J J ∑ ∑ ν(C) = ν(Cj ) = αj δ = δ. j=1

We also have

∫ f (a)dν(a) = C

J ∑

j=1

yj ν(Cj ) = δ

j=1

J ∑ j=1

25

αj yj = ν(C) y ∗ .

Again, by Lyapunov’s theorem, there are D and E in A such that D∩E = ∅, D∪E ⊂ By∗ , ∗

∗

ν(D) = ν(C) ν(By∗ ), and ν(E) = ν(Azk ) ν(By∗ ). (Note that ν(C) + ν(Azk ) ≤ 1.) For each j ∈ {1, . . . , J}, the set ¯ {(∫ ) } ∫ ¯ yj 2L ¯ f (a)dν(a), e(a)dν(a) ∈ R ¯ F ∈ A, F ⊂ Ak \ Cj F

F

is convex. Therefore, for each j ∈ {1, . . . , J}, there exists an Fj ∈ A such that Fj ⊂ y

Akj \ Cj , ∫

∫ f (a)dν(a) = ν(By∗ ) Fj

∫

∫

e(a)dν(a) = ν(By∗ ) Fj

The set {(∫

)

∫ f (a)dν(a),

G

f (a)dν(a), and

y

Akj \Cj

e(a)dν(a)

∈ R2L

G

e(a)dν(a).

y

Akj \Cj

¯ (J )} ¯ ∪ y ¯ j z∗ Ak ∪ Ak ¯ G ∈ A, G ⊂ Ak \ ¯ j=1

∪ ∗ y is also convex. Thus, there exists a Gk ∈ A such that Gk ⊂ Ak \ ( Jj=1 Akj ∪ Azk ), ∫ ∫ f (a)dν(a) = ν(By∗ ) f (a)dν(a), and S y Ak \(

Gk

∫

∫

e(a)dν(a) = ν(By∗ ) Gk

J j=1

∗

Akj ∪Azk )

S yj z∗ Ak \( J j=1 Ak ∪Ak )

e(a)dν(a).

For l ∈ {1, . . . , K} \ {k}, the set ¯ ) {(∫ } ∫ ¯ 2L ¯ f (a)dν(a), e(a)dν(a) ∈ R ¯ G ∈ A, G ⊂ Al G

G

is convex. Thus, for each l ∈ {1, . . . , K} \ {k}, there exists a Gl ∈ A such that Gl ⊂ Al , ∫ ∫ f (a)dν(a) = ν(By∗ ) f (a)dν(a), and Gl Al ∫ ∫ e(a)dν(a) = ν(By∗ ) e(a)dν(a). Gl

Let S = D ∪ E ∪

∪J

j=1 Fj ∪ Gk ∪

Al

∪ l6=k

Gl . Note that {D, E, F1 . . . , FJ , G1 , . . . , GK } is

pairwise disjoint. We define g : S → ZL+ by   y ∗ if a ∈ D, g(a) =  f (a) if a ∈ S \ D. 26

∗

From D ⊂ By∗ ⊂ Azk , it follows that y ∗ ∈ X(a) and y ∗ Âa z ∗ = f (a) for every a ∈ D. Therefore, ν({a ∈ S | f (a), g(a) ∈ X(a) and g(a) Âa f (a)}) = ν(D) = ν(C) ν(By∗ ) > 0. We also have ∫ g(a)dν(a) S

∫

∗

= y ν(D) +

J ∫ ∑

f (a)dν(a) + E

{

= ν(By∗ ) y ν(C) + z

∗

∗ ν(Azk )

+

Ak \(

= ν(By∗ )

SJ

j=1

Akj ∪Azk )

∫

f (a)dν(a) +

∗

∫ + Ak \(

{∫ = ν(By∗ )

SJ

j=1

f (a)dν(a) +

∗

y

∑∫

Ak

}

∑∫

f (a)dν(a)

l6=k

Al

j=1

Akj \Cj

l6=k

Al

J ∫ ∑

f (a)dν(a)

y

∑∫

Akj ∪Azk )

f (a)dν(a) +

= ν(By∗ )

f (a)dν(a) +

Azk

C

∫

f (a)dν(a) +

∗

y

l6=k

f (a)dν(a)

y

Akj \Cj

j=1

+

f (a)dν(a) +

}

}

f (a)dν(a)

f (a)dν(a)

Al

l6=k

f (a)dν(a) ∫A

= ν(By∗ )

e(a)dν(a) {∫ A

= ν(By∗ )

∫ e(a)dν(a) + ∫

+ Ak \(

SJ

= xk ν(C) ν(By∗ ) +

j=1

y

∗

∗ xk ν(Azk ) ν(By∗ )

e(a)dν(a) +

+ Gk

=

e(a)dν(a) +

J ∫ ∑ y

+

j=1

∑∫ l6=k

J ∫ ∑

e(a)dν(a)

j=1

Akj \Cj

l6=k

Al

∑∫

Akj ∪Azk )

∫ ∫

e(a)dν(a) +

∗

Azk

C

∑∫

Gk

J ∫ ∑

∫ {∫

f (a)dν(a) +

Fj

j=1

∗

∫

}

e(a)dν(a)

e(a)dν(a) Fj

e(a)dν(a)

Gl

e(a)dν(a). S

This contradicts f ∈ C(E). This completes the proof of Lemma 6.2. 27

Gl

f (a)dν(a)

Lemma 6.3. Let E : (A, A, ν) → P × ZL+ be an economy that satisfies condition (ii) of Theorem 3.1. If f : A → ZL+ is a ν-essentially bounded exactly feasible core allocation for E, then f is a Walras allocation, i.e., f ∈ W ∗ (E). Proof. Let f : A → ZL+ be a ν-essentially bounded exactly feasible core allocation for E. For simplicity, we assume that f (a) ∈ X(a) for every a ∈ A. Since both mappings e and f are ν-essentially bounded, the mapping f − e : A → ZL is also ν-essentially bounded. Thus, the set {z ∈ ZL | ν({a ∈ A | f (a) − e(a) = z}) > 0} is nonempty and finite. We denote this set by {z1 , . . . , zM }. Let Am = {a ∈ A | f (a) − e(a) = zm } for ∑ m = 1, . . . , M . Clearly, ν(Am ) > 0 for every m ∈ {1, . . . , M } and M m=1 ν(Am ) = 1. Let H = span{z1 , . . . , zM }. Claim 6. 0 ∈ ri(co{z1 , . . . , zM }), where ri(C) is the relative interior of the set C. Proof of Claim 6. Since the mapping f is exactly feasible, we have ∫ 0 = (f (a) − e(a))dν(a) A

=

M ∑

zm ν(Am )

m=1

∈ ri(co{z1 , . . . , zM }) This completes the proof of Claim 6. We define a relation ϕ of

∪M m=1

Am into ZL by

ϕ(a) = {z ∈ ZL | z + e(a) ∈ X(a) and z + e(a) Âa f (a)} = {x ∈ X(a) | x Âa f (a)} − {e(a)}. For m ∈ {1, . . . , M } and z ∈ ZL , let Azm = {a ∈ Am | z + e(a) ∈ X(a) and z + e(a) Âa f (a)} ∈ A. ∗ = Am \ Let Am

∪ z∈ZL ν(Azm )=0

Azm ∈ A for m ∈ {1, . . . , M } and let A∗ =

ν(A∗m ) = ν(Am ) for every m ∈ {1, . . . , M }, we have ν(A∗ ) =

M ∑

ν(Am ) = 1.

m=1

28

∪M m=1

A∗m ∈ A. Since

∪

ϕ(a) = ∅,15 then f is a Walras allocation with respect to the price system 0. ∪ Therefore, in the rest of this proof, we assume a∈A∗ ϕ(a) 6= ∅. If

a∈A∗

( Claim 7. co

∪

) ϕ(a)

∩ (H − RL+ ) = ∅.

a∈A∗

∪ Proof of Claim 7. Suppose that co( a∈A∗ ϕ(a)) ∩ (H − RL+ ) 6= ∅. Then, there exists a ∪ point z ∈ co( a∈A∗ ϕ(a)) and a point y ∈ H such that z ≤ y. Following Aumann (1964, p.45) (see also Hildenbrand (1982, pp.843-844)), one can show that there exists an S ∈ A and a (vector-valued) simple function g : S → ZL such that 0 < ν(S) < min{ν(A1 ), . . . , ν(AM )}, g(a) ∈ ϕ(a) for all a ∈ S, and ∫ ν(S) z = g(a)dν(a). S

From Claim 6, there exists a point y0 ∈ co{z1 , . . . , zM } and a real number t ∈ ]0, 1[ such that t y + (1 − t) y0 = 0. Since y0 ∈ co{z1 , . . . , zM }, there are nonnegative numbers {α1 , . . . , αM } such that M ∑

αm = 1

and

m=1

y0 =

M ∑

α m zm .

m=1

Hence, we have 0 = t ν(S) y + (1 − t) ν(S) y0 M ∑ ≥ t ν(S) z + (1 − t) ν(S) αm zm ∫

m=1

∫ αm = t g(a)dν(a) + (1 − t) ν(S) (f (a) − e(a))dν(a) ν(Am \ S) Am \S S m=1 ∫ ∫ M ∑ αm f (a)dν(a) = t (g(a) + e(a))dν(a) + (1 − t) ν(S) ν(A \ S) m A \S S m m=1 ∫ ∫ M ∑ αm −t e(a)dν(a) − (1 − t) ν(S) e(a)dν(a). ν(A \ S) m S A \S m m=1 M ∑

15

Recall that in this lemma we do not make any desirability assumptions about preference relations. ∪ Thus, the set a∈A∗ ϕ(a) may be empty.

29

Since ν(Am \ S) > 0 for every m ∈ {1, . . . , M }, we can choose a real number δ such that ν(Am \ S) > δ > 0 for every m ∈ {1, . . . , M }. Then, 0 < tδ < 1, 0 ≤ (1 − αm δ < 1 for every m ∈ {1, . . . , M }, and t)ν(S) ν(Am \ S) ∫

∫ αm δ t δ (g(a) + e(a))dν(a) + (1 − t) ν(S) f (a)dν(a) ν(A \ S) m S A \S m m=1 ∫ ∫ M ∑ αm δ ≤ t δ e(a)dν(a) + (1 − t) ν(S) e(a)dν(a). ν(A \ S) m S A \S m m=1 M ∑

By Lyapunov’s theorem, the set {(∫ ) ∫ (g(a) + e(a))dν(a), e(a)dν(a), ν(T ) ∈ R2L+1 T

T

¯ } ¯ ¯ T ∈ A, T ⊂ S ¯

is convex. Then, there exists a T0 ∈ A such that T0 ⊂ S, ν(T0 ) = t δ ν(S), ∫ ∫ (g(a) + e(a))dν(a) = t δ (g(a) + e(a))dν(a), and T0 S ∫ ∫ e(a)dν(a) = t δ e(a)dν(a). T0

S

For m ∈ {1, . . . , M }, the set ) {(∫ ∫ f (a)dν(a), e(a)dν(a) ∈ R2L T

T

¯ } ¯ ¯ T ∈ A, T ⊂ Am \ S ¯

is convex. Thus, for every m ∈ {1, . . . , M }, there exits a Tm ∈ A such that Tm ⊂ Am \ S, ∫ ∫ αm δ f (a)dν(a) = (1 − t) ν(S) f (a)dν(a), and ν(Am \ S) Am \S Tm ∫ ∫ αm δ e(a)dν(a). e(a)dν(a) = (1 − t) ν(S) ν(Am \ S) Am \S Tm Let T =

∪M m=0

Tm ∈ A. We define a mapping h : T → ZL by   g(a) + e(a) if a ∈ T , 0 h(a) = ∪  f (a) if a ∈ M m=1 Tm .

For a ∈ T0 , we have h(a) ∈ X(a) and h(a) = g(a) + e(a) Âa f (a) since g(a) ∈ ϕ(a). Then, ν({a ∈ T | f (a), h(a) ∈ X(a) and h(a) Âa f (a)}) = ν(T0 ) = t δ ν(S) > 0. 30

Moreover, we have ∫ h(a)dν(a) T

∫

=

(g(a) + e(a))dν(a) + T0

M ∫ ∑ m=1

∫

f (a)dν(a)

Tm

∫ αm δ f (a)dν(a) = t δ (g(a) + e(a))dν(a) + (1 − t) ν(S) ν(A \ S) m A \S S m m=1 ∫ ∫ M ∑ αm δ ≤ t δ e(a)dν(a) + (1 − t) ν(S) e(a)dν(a) ν(A \ S) m S A \S m m=1 ∫ ∫ M ∑ e(a)dν(a) + e(a)dν(a) = ∫

T0

=

M ∑

m=1

Tm

e(a)dν(a). T

This contradicts f ∈ C(E). This completes the proof of Claim 7. Since the mapping e = (e(1) , . . . , e(L) ) is ν-essentially bounded, there exists a vector v ∈ RL+ such that |e(i) (a)| ≤ v (i) for ν-a.e. a ∈ A and all i ∈ {1, . . . , L}. Therefore, ϕ(a) ⊂ X(a) − {e(a)} ⊂ ZL+ − {e(a)} ⊂ ZL+ − {v}

for ν-a.e. a ∈ A∗ .

∪

ϕ(a) ⊂ ZL+ − {v}. Then, from Theorem 5.2, there ∪ exists a vector p ∈ H ⊥ ∩ZL+ and an ε ∈ R++ such that p·z ≥ ε for every z ∈ co( a∈A∗ ϕ(a)). For simplicity, we assume that

a∈A∗

For agent a ∈ A∗m , we have p · (f (a) − e(a)) = p · zm = 0, since p ∈ H ⊥ . Therefore, p·f (a) = p·e(a) for every a ∈ A∗ . If a ∈ A∗ , x ∈ X(a), and x Âa f (a), then x − e(a) ∈ ϕ(a). Thus, we have p · (x − e(a)) ≥ ε > 0. Since ν(A∗ ) = 1, a pair (p, f ) is a Walras equilibrium. Hence, f ∈ W ∗ (E). This completes the proof of Lemma 6.3. Remark 6.4. In Lemma 6.3, there is no desirability assumption, but free disposal of improving allocations is essential. (See the proof of Claim 7.) Of course, if agents’ preference relations are nonsatiated in positive directions, from Lemma 6.1, core allocations are exactly feasible. Moreover, non-core allocations can be improved upon via an allocation 31

that is exactly feasible within the corresponding coalition. This fact tells us that our model does not cover economies with bads. In an economy with bads, the resource-feasible constraint should be restricted to exact feasibility; Walras allocations and core allocations should be exactly feasible.16 Moreover, an improving allocation should be exactly feasible within the corresponding coalition. If our economy has bads, and all resource feasibility constraints are restricted to exact ones, then core allocations may not be Walrasian. Example 6.5 below illustrates this point. On the other hand, in an atomless economy with divisible commodities, the core equivalence theorem holds even if some commodities are bads and resource feasibility constraints are exact. (See Hildenbrand (1968).) This result, however, should be considered carefully. First, in such an economy, Walras equilibria may not exist, as was pointed out by Hara (2004). Therefore, in some atomless economies with bads, the equivalence holds in the sense that both the weak core and the set of Walras allocations are empty. Second, for a sequence of finite economies with bads, there may exist a sequence of weak core allocations such that the competitive gap measure given by Anderson (1978) does not converge in measure to zero, although the weak core coincides with the set of Walras allocations in the limit.17 This discrepancy between a continuum economy with bads and a large finite economy with bads comes from implicit assumptions in the measure space formulation of a continuum economy. (See Anderson (1992).) Example 6.5. Let (A, A, ν) be an atomless probability space of economic agents. Let L = 2 and X(a) = Z2+ for every a ∈ A. The agents’ type is unique, that is, every agent has the same initial endowment vector e0 = (1, 1) and the same preference relation -. The preference relation - is given by the following ordering: for (x(1) , x(2) ), (y (1) , y (2) ) in Z2+ , (x(1) , x(2) ) - (y (1) , y (2) ) if and only if x(1) < y (1) , or x(1) = y (1) and x(2) ≥ y (2) . Thus, commodity 1 is a good and commodity 2 is a bad. Note that the initial endowment mapping e : A → ZL+ , e(a) = e0 for all a ∈ A, is a core allocation. However, any vector p ∈ R2 cannot strongly separate the initial endowment vector e0 from its strict upper contour set {x ∈ Z2+ | x  e0 }. Thus, e is not Walrasian. 16 17

In an economy with bads, equilibrium price vectors need not be positive. This fact was pointed out by Hara (2004).

32

7

Discussion of the literature As was shown in Example 3.2, in our economy, Walras equilibria may not exist. The

reason for the nonexistence of Walras equilibria is that, due to the discreteness of the commodity space in our economy, agents’ demand correspondences may not be upper hemicontinuous. Since a Walras equilibrium may not exist, Dierker (1971) introduced an approximate equilibrium concept in a finite economy with the commodity space ZL and considered its existence. To guarantee the existence of an “exact” Walras equilibrium, the economy must have some divisibility. Inoue (2003) considered the equilibrium existence problem in a two-period economy, without uncertainty, where all commodities are purely indivisible, but with a divisible security that makes wealth transfer between two periods possible. The divisible security is not directly included in agents’ utility functions. Thus, the commodity space is the same as ours. Inoue showed that an equilibrium exists if the distribution of agents’ security holdings is dispersed, in the sense that the set of agents with the same security holdings is measure zero. Because of such dispersion, the set of agents whose demand correspondences are not upper hemicontinuous at a particular price vector is negligible relative to the size of the economy. This is true for any price vector. Therefore, the aggregate demand correspondence recovers the upper hemicontinuity. This is the key to Inoue’s (2003) equilibrium existence theorem. However, an equilibrium whose existence is guaranteed by this theorem is not exactly feasible because almost all agents cannot use up their wealth. Inoue’s (2003) argument followed the equilibrium existence proof of Mas-Colell (1977) and Yamazaki (1981). Mas-Colell (1977) considered an atomless economy where every × R+ . Thus, just one commodity is perfectly agent has the same consumption set ZL−1 + divisible. He showed that, if the distribution of agents’ initial endowments is dispersed, then a Walras equilibrium exists, although each agent’s demand correspondence is not upper hemicontinuous at some price systems due to the indivisibility. Yamazaki (1981) showed that if the distribution of agents’ initial endowments is dispersed, a Walras equilibrium still exists even though different agents have different and nonconvex consumption

33

sets. A convergence version of our core equivalence theorem was considered by Inoue (2004). He showed that, in a type sequence of economies, if the number of agents is sufficiently large but finite, the strong core is included in the set of Walras allocations. In contrast to a continuum economy, even if the number of economic agents is arbitrarily large but finite, not all core allocations are Walrasian. Inoue (2004) gave an example of an increasing sequence of economies such that, in each finite economy, there exists a core allocation that is not Walrasian, and in the limit atomless economy the core coincides with the set of exactly feasible Walras allocations. The feature that every strong core allocation is decentralized by a suitable price system with only finitely many agents is not seen in an economy with divisible commodities. (See Debreu and Scarf (1963).) Shapley and Scarf (1974) analyzed a different indivisible commodity market from the one we analyze. In their economy, there are finitely many agents. Each agent has only one indivisible commodity (e.g., a house), which is differentiated. It is assumed that every agent consumes, at most, one unit of one commodity. Shapley and Scarf showed that Walras equilibria exist by using the top-trading-cycle method of David Gale. Wako (1999) introduced the concept of the core in the Shapley-Scarf economy and proved that the core coincides with the set of Walras allocations, thus obtaining the nonemptiness of the core. In the Shapley-Scarf model, a resource-feasible allocation can be represented by a permutation of the initial allocation. Wako’s (1999) method of proof depends on this special structure, so it cannot be applied to our core equivalence theorem. Aumann (1964) and Vind (1964) formulated atomless economies and proved core equivalence theorems. In Aumann’s (1964) model, individual agents are the primitive concept, whereas in Vind’s (1964) model, coalitions are the primitive concept. In this paper, we followed Aumann’s formulation. Aumann (1964) proved that the weak core coincides with the set of exactly feasible Walras allocations in the framework of an exchange economy under the assumptions that every agent has the same consumption set RL+ and has a strongly monotone preference relation. Hildenbrand (1968) obtained a core equivalence theorem in the framework of a production economy and he weakened the strong monotonicity to local nonsatiation of preference relations. The methods of proof of the authors differ most strongly in terms of the stage when the separation theorem for 34

convex sets is applied. Aumann (1964) applied the separation theorem for the convex hull of the union of agents’ strict upper contour sets of their bundles allocated by a given weak core allocation, whereas Hildenbrand (1968) applies it for the integral of a correspondence that assigns each agent to his strict upper contour set. Our method of proof followed Aumann’s approach. Khan and Yamazaki (1981) investigated a core equivalence in Mas-Colell’s (1977) model. They considered an atomless economy where every agent has the same consumption set, ZL−1 × R+ , and showed that, under some assumptions, the weak core coincides + with the set of pseudo-equilibrium allocations where the corresponding price of the divisible commodity is positive. In addition, they showed that those two sets were nonempty. Their method of proof relied on Hildenbrand (1968). Their model as well as ours states that, due to the presence of indivisible commodities, not all weak core allocations are envy-free in net trades. As was explained in Section 4, pseudo-equilibrium (p, f ) is Walrasian if, for almost every a ∈ A, every x ∈ X(a) with p · x = p · e(a) has local cheaper points. In the Khan-Yamazaki (1981) economy, if the price of the divisible commodity is positive, i.e., p(L) > 0, then the set of consumption bundles that do not have local cheaper points is the countable set ZL−1 × {0}. Hence, from their equivalence theorem, we immediately obtain + the fact that the weak core coincides with the set of Walras allocations if the distribution of the initial endowments is dispersed, in the sense that ν({a ∈ A | e(L) (a) = w}) = 0 for all w ∈ R. Acknowledgements I am indebted to Chiaki Hara and Akira Yamazaki for helpful discussions. An earlier version of this paper was presented at the meeting of the Japanese Economic Association held at Meiji University and at the First Asian Workshop on General Equilibrium Theory (GETA 2004) held at Keio University. I am grateful to Atsushi Kajii, Tomoyuki Kamo, and Sinsuke Nakamura for valuable comments. I am also grateful to a guest editor and an anonymous referee for helping me to improve the paper.

35

References Anderson, R.M., 1978, An elementary core equivalence theorem, Econometrica 46, 14831487. Anderson, R.M., 1992, The core in perfectly competitive economies, in: R.J. Aumann and S. Hart, eds., Handbook of game theory, Vol.1 (North-Holland, Amsterdam) 413-457. Aumann, R.J., 1964, Markets with a continuum of traders, Econometrica 32, 39-50. Debreu, G., 1959, Theory of value (John Wiley & Sons, New York). Debreu, G., 1962, New concepts and techniques for equilibrium analysis, International Economic Review 3, 257-273. Debreu, G. and H. Scarf, 1963, A limit theorem on the core of an economy, International Economic Review 4, 235-246. Dierker, E., 1971, Equilibrium analysis of exchange economies with indivisible commodities, Econometrica 39, 997-1008. Florig, M., 2001, Hierarchic competitive equilibria, Journal of Mathematical Economics 35, 515-546. Gordan, P., 1873, Ueber die aufl¨osung linearer gleichungen mit reellen coefficienten, Mathematische Annalen 6, 23-28. Hara, C., 2004, Existence of equilibria and core convergence in economies with bads, Cambridge Working Paper Series in Economics 0413. Hildenbrand, W., 1968, The core of an economy with a measure space of economic agents, Review of Economic Studies 35, 443-452. Hildenbrand, W., 1974, Core and equilibria of a large economy (Princeton University Press, Princeton). Hildenbrand, W., 1982, Core of an economy, in: K.J. Arrow and M.D. Intriligator, eds., Handbook of mathematical economics, Vol. 2 (North-Holland, Amsterdam) 831877.

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Inoue, T., 2003, Existence of equilibrium in economies with indivisible commodities and savings, mimeo. Inoue, T., 2004, Indivisible commodities and decentralization of strong core allocations, mimeo. Khan, M.A. and A. Yamazaki, 1981, On the cores of economies with indivisible commodities and a continuum of traders, Journal of Economic Theory 24, 218-225. Mas-Colell, A., 1977, Indivisible commodities and general equilibrium theory, Journal of Economic Theory 16, 443-456. Schmeidler, D., 1972, A remark on the core of an atomless economy, Econometrica 40, 579-580. Shapley, L. and H. Scarf, 1974, On cores and indivisibility, Journal of Mathematical Economics 1, 23-37. Vind, K., 1964, Edgeworth-allocations in an exchange economy with many traders, International Economic Review 5, 165-177. Wako, J., 1999, Coalition-proofness of the competitive allocations in an indivisible goods market, in: M. H. Wooders, ed., Topics in mathematical economics and game theory: essays in honor of Robert J. Aumann, Fields Institute Communications 23 (American Mathematical Society, Providence) 277-283. Yamazaki, A., 1981, Diversified consumption characteristics and conditionally dispersed endowment distribution: regularizing effect and existence of equilibria, Econometrica 49, 639-654.

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commodity 2

e(a) commodity 1

0

Figure 1: endowment and indifference curves of agent a ∈ A1

commodity 2

e(a)

0

commodity 1 Figure 2: endowment and indifference curves of agent a ∈ A2

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commodity 2

e(a)

commodity 1

0

Figure 3: endowment and indifference curves of agent a ∈ A1

commodity 2

e(a) 0

commodity 1 Figure 4: endowment and indifference curves of agent a ∈ A2

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