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theorem on the commodity space ca(K). The ﬁrst main topic of this paper is to
explore this issue. In particular, we prove the existence of competitive equilibria for
LIEs with the commodity spaces ℓ∞ and ca(K).
We believe that our choice of the commodity spaces, ℓ∞ and ca(K), is justiﬁedin that these spaces have clear and deﬁnitive interpretations. Moreover, the modelsof these spaces have common mathematical characters. In particular, those authors
who studied the ℓ∞ or ca(K) spaces all worked with the weak* topology on thespace; hence, the resource condition in terms of the Gelfand integral. In addition,
they set the price space as the predual of the commodity space. This is in contrast
with [35, 46] in which the price space is simply taken as the norm dual of thecommodity space.
Aumann’s fundamental observation is that the convexity of the preferences is
unnecessary for ﬁnite dimensional models. The ﬁrst problem to generalize his result to inﬁnite dimensional settings is that we need the convexity in the inﬁnite
dimensional settings. Indeed, all of the aforementioned authors assumed the convexity of preferences or similar assumptions1. The convexity assumptions obviously
weaken the impact of Aumann’s classical result and reveal the “convexifying effect” of large numbers of the economic agents that is a result of Fatou’s lemmaor Liapunov’s convexity theorem, which are known to fail on the inﬁnite dimensional spaces. Therefore, we must assume the convexity directly or determine someconditions that recover the missing convexity.
Rustichini and Yannelis [55] was probably the ﬁrst paper to address the equilibrium existence problem for a model on an inﬁnite dimensional commodity space
without the convexity of preferences. They concluded that to obtain any Fatou-type
theorem (lemma), one must have “many more agents than commodities.” According to Mertens [45, p.189], this “many more agents than commodities” thesis isseemingly at ﬁrst addressed by Aumann himself in the context of the core equivalence theorem2; hence we call it Aumann’s thesis. Aumann’s thesis in the sense ofRustichini–Yannelis is stated as follows. Let AE = {A ∩ E| A ∈ A} be the sub-σ
algebra of A restricted to E ∈ A. Recall that for any (real) vector space, an algebraic Hamel basis exists. The cardinality of any Hamel basis of a vector space L isthe same, and we denote it as dim(L). Rustichini and Yannelis proposed the next
condition:(RY): For any E ∈ A with λ(E) > 0, dim(LE∞ (λ)) > dim(L),where LE∞ (λ) is the space of essentially bounded functions on E and L is the commodity space. Note that this condition involves both consumers and commodities
spaces. Podczeck criticized this condition that “one may wish to interpret an atomless measure spaces as an idealization of a large but ﬁnite number of them. From
this point of view, it is preferable to keep a measure space of agents ‘small’ ([50],p.386).”

1Noguchi assumed that a commodity vector does not belong to the convex hull of its preferred

set.
2Aumann was indeed correct, see [51, 65].

INDIVIDUALIZED AND DISTRIBUTIONALIZED ECONOMIES

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In the present paper, we propose that Aumann’s thesis is manifestly represented
when we set a saturated (or super-atomless3) measure space of consumers. A measurespace (A, A, λ) is saturated if its subalgebra AE are uncountable generated for
any E ∈ A with λ(E) > 0 modulo null sets. Aumann’s thesis is now embodied
intrinsically within the measure space of consumers rather than a condition imposedon it from outside. It is realized naturally in our models.
As Podczeck [52] indicated, the saturated measure space itself does not necessarily
have an extraordinarily large cardinality. This can be explained by the nontrivial
Loeb measure spaces (Keisler and Sun [21]) that are important examples of thesaturated measure spaces. It is possible for some nontrivial Loeb measure spaces tohave the cardinality of the continuum; hence, they can be identiﬁed with the unit
interval on the real line.
On the saturated measure space, we can prove Fatou’s lemma for the Gelfand
integrable maps [14, 30]. This is the key result for proving the existence of equilibria
in the spaces ℓ∞ and ca(K) without the convexity of preferences. Moreover, thesaturation is known to be equivalent with the convexity of the Gelfand integral ofcorrespondences [52, 57]. We use it in our proofs of core equivalence theorems for
those spaces.
There is another formalism of large economies that we call the large distributionalized economy (LDE). The LDE was invented by Hart, Hildenbrand, and Kohlberg[16] in which the economy was deﬁned as a probability measure µ on the space ofagents’ characteristics P × Ω. The allocation of this economy is also deﬁned as aprobability measure ν on X × P × Ω, where X is a consumption set that is assumed
to be identical among all consumers. The marginal distribution of ν on P × Ω isrequired to coincide with µ for consistency. We call the competitive equilibrium of
the LDE the distributive equilibrium. Their purpose of these notions of economy andequilibria is to introduce collective or macroeconomic perspectives into the generalequilibrium theory:
If the set A of agents is small (e.g., #A = 3), then the description ofan economy as an assignment a 7→ (≿a , ea ) of agents to characteristics seems to be the appropriate one. However, when the set A islarge (e.g., #A = 1000), one is tempted to give up this individualistic description and to replace it by a more collective view, namely toconsider the distribution of the mapping E [16, pp. 159–160, italics
by Hart–Hildebrand–Kohlberg].
They also recognized that this equilibrium concept is not suitable for core equilibrium:... [I]f an economy is described by a distribution of agents’ characteristics only, then it is not clear how the core of that economy
should be deﬁned. The concepts of ‘coalition’ and ‘to improve upon’require the individualistic description of an economy as a mapping
which assigns to every individual agent his characteristics [ibid.].

3Super-atomless was coined by Podczeck [52].

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The LDE formalism was successfully applied to the model of the space ca(K) by
Mas-Colell [41] and Jones [19]. They could prove the existence of distributive equilibria without assuming the convexity of preferences because neither Fatou’s lemma
nor Liapunov’s theorem is used in the proof for the existence of these equilibria. Mas-Colell also proved the core equivalence by invoking the (individualistic
or mapping) representation for distributive equilibrium, where a pair of measurable mappings ξ : A → X and E : A → P × Ω is called a representation4 of ν if
ν = (ξ, E)∗ λ ≡ λ ◦ (ξ, E)−1 . Their observations were further conﬁrmed for the modelof ℓ∞ by [59].
The saturated measure space also casts light on the relationship between the
allocative and distributive equilibria. When X × P × Ω is a complete and separable
metric space, the LDE µ and its distributive equilibrium ν have a representationof E and (ξ, E ′ ), respectively. We must notice that E and E ′ are not necessarilyequal; hence, the important question is the following. Let an economy µ and its
equilibrium ν be given. For each representation E of µ, do we have an equilibrium
allocation map ξ such that (ξ, E) represents ν? Generally, the answer is no, assuggested by Mas-Colell;
While given an economy µ, we can prove the existence of an equilibrium distribution ν (and therefore of an equilibrium allocation
ξ : A → X for some representation of µ, E : A → P × Ω), it isunlikely that given a representation E : A → P × Ω, there is anallocation ξ : A → X that is an equilibrium with respect to E [41,p. 273 with notations changed].
However, we obtain an aﬃrmative answer when the measure space is saturated(Theorem 4.4). Hence, the distributive equilibrium ν does not lose any individual
information when one works with the saturated economies. In addition, the allocative and the distributive equilibria are equivalent in a strong sense for economies
with the saturated measure space of consumers. We discuss more the realization of
the distributive equilibria in Section 4.
The paper is organized as follows. Section 2 is devoted to mathematical preliminaries. Its aim is twofold. The ﬁrst is to ﬁx our mathematical symbols andnotations. The second is to collect all mathematical results used in the text for ourreaders’ convenience and enable the paper to be self-contained. In particular, thedeﬁnition of saturated measure spaces and a Fatou’s lemma for the Gelfand integral
on them that plays a key role in the proof of the existence theorems are presented.
However, many of the results are standard. Hence, experts can skip this section.
In Section 3, we present our basic model that includes exchange economies withthe commodity spaces ℓ∞ and ca(K). The consumption set of the basic modelis assumed to contain a ﬁnite dimensional and unbounded part for relaxing theinteriority assumptions on the initial endowments that are commonly assumed in
the literature. Indeed, [6, 35, 46, 47, 55, 60, 61] all assumed that individual initialendowments belong to the (norm) interior of the consumption set. Obviously, such
4Similarly, a mapping E : A → P × Ω is a representation of µ, or E represents µ if µ = E λ ≡λ ◦ E −1 . Notice that representations are not generally unique.

∗INDIVIDUALIZED AND DISTRIBUTIONALIZED ECONOMIES

983

an assumption is very strong in the economies with a continuum of traders 5. We
dispense with the interiority assumption by making use of the monotonicity ofpreferences. As is well known, these remarks do not apply for models on ca(K) in
which all consumers are assumed to have the nonnegative orthant of ca(K) as theircommon consumption set. See also Section 5.
In Section 4, we discuss the relationships between large individualized and distribuitionalized economies. The ﬁrst issue is the realization of distributive equilibria
explained previously. A distributive equilibrium of a distributionalized economy µis said to be realizable if for every individualized economy E that represents µ, an
equilibrium allocation ξ of E represents ν or ν = (ξ, E)∗ λ. As stated, Theorem 4.4
shows that distributive equilibria are realizable if the underlying measure space issaturated. This result has been already obtained and applied in speciﬁc models[31, 62]. We present and discuss it in a general framework. The second issue to be
explored is the symmetricity of equilibria that was pioneered in the noncooperative
game theory [32, 42]. Intuitively, a distributive equilibrium is called symmetric if
the consumers with the same characteristics consume the same commodity bundle.
We discuss the symmetric equilibria and related results.
In Section 5, we study the existence and core equivalence of allocative equilibria
for economies with commodity spaces of ℓ∞ and ca(K). First, the existence andthe core equivalence theorems [59] for LDEs are reformulated and translated into
LIEs. Next, we show similar results for LIE with the commodity space ca(K). Our
fundamental observation is the following. As is well known, the basic problem concerning space ca(K) is that the positive orthant has an empty norm interior. Hence,
one needs some conditions to remedy this mathematical diﬃculty. Jones [19, 20]
introduced the condition of a bounded marginal rate of substitution ((US), Assumption 5.5). This is a local condition in the sense that it restricts the substitutability
between two commodities locally or commodities being suﬃciently similar, and heproved the existence of equilibria (hence core) under the assumption (US). Unfortunately the local substitutability is not suﬃcient for the core equivalence. The
references [15, 48] for instance worked with the bounded rate of substitution called(BRS) in the global sense that generates an open cone of the commodity space.
This condition was originally invented by Mas-Colell [43] as the properness. Ostroy
and Zame [48] assumed the local6 and the global substitutability at once for bothof the existence and core equivalence. Greinecker and Podczeck [15] discarded themetrizability of K and weakened conditions on the bounds for local and global substitutablilities. But they kept the basic stance of [48] and did not ask the question ofexistence. We clariﬁed the roles of local and global substitutability for existence andcore equivalence, respectively, namely that the local substitutability is required forexistence but not for core equivalence and core equivalence can be proved without
local substitutability. Speciﬁcally speaking, we will prove in Theorem 5.7 existenceof an equilibrium assuming (US) without any global substitutability conditions and5Precisely, Khan–Yannelis, Noguchi, and Rustichini–Yannelis assumed that there exists z ∈ Xsuch that ω − z belongs to the (norm) interior of X.
6They proposed an another version of the local substitutability called (PLD) which is weaker
than (US). They proved two versions of core equivalence theorems corresponding to (US) and(PLD) respectively. The condition (BRS) is maintained throughout their paper.

984

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in Theorem 5.11 a core equivalence by utilizing a version of global substitutability((CD), Assumption 5.10) proposed by Rustichini and Yannelis [56] without assuming local substitutability. Finally, all the proofs are given in Section 6.
2. Mathematical preliminaries
2.1. Some Measure Space Theories. For a topological (or Banach) space Y ,we denote the Borel σ-algebra of Y which is deﬁned as the σ-algebra generated byopen subsets of Y by B(Y ) and the set of Borel probability measures by M(Y ).
We begin with the next well-known fact ([49, Theorem I.2.1]).
Fact 2.1. Let X be a separable metric space. Then, for every measurable set B of
measure space (X, B(X), λ), there exists a closed set CB ⊂ B satisfying λ(CB ) =λ(B) and for any closed set C ⊂ B with λ(C) = λ(B), CB ⊂ C.
The closed set CB in Fact 2.1 is called the support of B and denoted by support(B).
Then, the next fact is also well known ([49, Theorem I.3.9]).
Fact 2.2. Let Y1 and Y2 be a complete separable metric space and B1 ∈ B(Y1 ) andB2 ⊂ Y2 . Let f : B1 → Y2 be a measurable and injective (one-to-one) map withf (B1 ) = B2 . Then, B2 ∈ B(Y2 ).
Let a measurable space (A, A) and a set E be given. For a map f : E → A, deﬁneσ(f ) = {f −1 (B)| B ∈ A}; the smallest σ-algebra on E that makes f measurable.
One has ([1, Theorem 4.41])
Fact 2.3. Let f : E → A and g : E → Y , where Y is complete and separable7.
Then, g is σ(f )-measurable if and only if there exists an A-measurable map h : A →Y such that g = h◦f .
Given any probability space (A, A, λ), a function on A is called almost one-toone if it is one-to-one on A except for some λ-null set of A. We have ([7, Theorem
9.6.3]),
Fact 2.4. Let Y be a complete separable metric space and m ∈ ca(Y ). If(Y, B(Y ), m) is an atomless measure space, there is an almost one-to-one map from([0, 1], B([0, 1]), dx) to Y , where dx is the Lebesgue measure on the Borel σ-algebraB([0, 1])8.
Let f be a Borel measurable map from (A, A, λ) to a topological space X. The
direct image measure λ ◦ f −1 is denoted by f∗ λ. The operator ∗ λ is a map from theset of all measurable maps of A to X that is denoted by L0 (A, X) to M(X). We
then have ([22, Lemma 2.1])
Fact 2.5. For an atomless measure space (A, A, λ), the map ∗ λ : L0 (A, X) → M(X)is surjective.
7We can replace the range space R in the cited result by the Polish space X; see [25, footnote

26].
∫

8In this notation, the Lebesgue integral is understood to be denoted as ∫ f (x)dx rather thanf (x)d(dx).

INDIVIDUALIZED AND DISTRIBUTIONALIZED ECONOMIES

985

A measure algebra is a pair (A, λ) where A is a Boolean σ-algebra with binary
operations ∧ and ∨, a unary operation c and λ is a real valued function satisfying the following conditions:(i) λ(B) = 0 if and only if B = ∅, where ∅ = Ac
and A = ∅c P
are the smallest and the largest elements in A, respectively; (ii)
∞
λ(∨∞
E
)
=
n
n=1
n=1 λ(En ) for every sequence {En } in A with En ∩Em = ∅ wheneverm 6= n. A map Φ : A → B between measure algebras (A, λ) and (B, µ) is calledhomomorphism if it is one-to-one, Φ(Ac ) = Φ(A)c , Φ(A ∨ B) = Φ(A) ∨ Φ(B) andλ(A) = µ(Φ(A)). Measure algebras (A, λ) and (B, µ) are isomorphic if there exists
a homomorphism which is onto.
A subalgebra of A is a subset of A, which contains A and is closed under the
Boolean operation ∧, ∨, and c . The order ≤ on A is given by B ≤ C if and only if
B = B∧C. A subalgebra U of A is order-closed with respect to ≤ if any nonempty
upwards directed subsets of U with their supremum in A have the supremum in U .
A subset U ⊂ A completely generates A if the smallest order-closed subalgebra in
A containing U is A itself. The Maharam type of (A, λ) is the smallest cardinal ofany subset U , which completely generates A.
Let (A, A, λ) be a ﬁnite measure space. We deﬁne an equivalence relation on
A by E ∼ F if and only if λ(E∆F ) = 0, where E∆F = (E ∧ F c ) ∨ (E c ∧ F ).
The quotient space is denoted by Â = A/ ∼. The equivalence class representedby E ∈ A is denoted Ê. Then, the lattice operation and the unary operation c\\is deﬁned naturally on Â, Ê ∨ F̂ = E∪ F (union), Ê ∧ F̂ = E∩ F (intersection),c[Ê = A\E (complement). The pair (Â, λ̂) is a measure algebra associated with
(A, A, λ), where λ̂(Ê) = λ(E). Moreover, (Â, λ̂) becomes a complete metric spaceby the metric ρ(E, F ) = λ(E∆F ) (see [1], Lemma 13.13). The measure algebra(Â, λ̂) is separable if it is a separable metric space. The Maharam type of (A, A, λ)is deﬁned to be that of (Â, λ̂).
Let AE = {A ∩ E| A ∈ A} the sub-σ algebra of A restricted to E ∈ A. We
denote the restriction of λ to AE by λE , or λE (B) = λ(B) for every B ∈ AE . A
ﬁnite measure space (A, A, λ) is (Maharam type) homogeneous if for every E ∈ A
with λ(E) > 0, the Maharam type of (E, AE , λE ) is equal to (A, A, λ). It is easy tosee (e.g., [52, p.838])
Fact 2.6. A ﬁnite measure space (A, A, λ) is atomless if and only if for every E ∈ A
with λ(E) > 0, the Maharam type of (E, AE , λE ) is inﬁnite.
This fact motivates the next deﬁnition.
Deﬁnition 2.7. A ﬁnite measure space (A, A, λ) is saturated (or super-atomless) if
for every E ∈ A with λ(E) > 0, the Maharam type of (E, AE , λE ) is uncountable.
Lebesgue space is homogeneous, and its Maharam type is countable ([13, 331X,
p.130]). Note that the saturated measure spaces are not necessarily Maharam homogeneous. Typical examples of the (homogeneous) saturated measure spaces are
the atomless Loeb space [36], product spaces of the form [0, 1]c and {0, 1}c , where cis an uncountable cardinal, [0, 1] equipped with the Lebesgue measure, and {0, 1}
the “half-half” measure. Measure algebras of [0, 1]c and {0, 1}c are homogeneouswith their Maharam type c and are isomorphic whenever c is inﬁnite cardinal (see[13, Theorems 331I and 331K]), and they are separable if and only if c is countable.

986

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Let (A, A, λ) be a ﬁnite measure space, X and Y be complete separable metricspaces. The next deﬁnition reveals a crucial property of the saturated spaces.
Deﬁnition 2.8. A ﬁnite measure space (A, A, λ) is said to satisfy the saturation
property for a measure µ ∈ M(X×Y ) if for every measurable function f on A withf∗ λ = µX , there exists a measurable function g on Y that satisﬁes (f, g)∗ λ = µ,where µX means the marginal distribution of µ on X.
Let L1 (λ) be the set of all λ-integrable functions on A,Z1|f (a)|dλ < +∞ .
L (λ) = f : A → R
A
1
Denote by LE (λ) be the vector subspace of L1 (λ) whose element is a restrictionof each function in L1 (λ) to E. It is well known that (Â, λ̂) is separable if and only if

L1 (λ) is separable ([1, Lemma 13.14]). The next characterization of the saturation
is well known (see [13, 22]) and would be sometimes useful.
Fact 2.9. Let (A, A, λ) be a ﬁnite measure space and X and Y be complete separable metric spaces. Then, the following conditions are equivalent.
(a) (A, A, λ) is saturated,(b) (A, A, λ) is atomless and satisﬁes the saturation property for every µ ∈ M(X×Y ),(c) L1E (λ) is nonseparable for every E ∈ A with λ(E) > 0.
Let ([0, 1], L([0, 1]), dx) be the usual Lebesgue space, i.e., the completion of themeasure space ([0, 1], B([0, 1]), dx). Keisler–Sun [22] proved
Fact 2.10. Let (A, A, λ) be a ﬁnite measure space, X and Y be complete separable
metric spaces and f : A → X, g : A → Y be measurable maps, and assume thatf∗ λ is atomless. Suppose that (A, A, λ) has the saturation property for (f, g)∗ λ,but the Lebesgue space ([0, 1], L([0, 1]), dx) does not. Then, (A, A, λ) is saturated.
The next result is known as Egorov’s Theorem ([11, p. 97]).
Fact 2.11. Let (X, dX ) be a metric space and (A, A, λ) be a ﬁnite measure space.
If (fn ) is a sequence of measurable maps on A to X that converges a.e to a map
f : A → X, then:(i) f is measurable,(ii) for every measurable set E ∈ A and every ϵ > 0, there exists a set F ∈ A withF ⊂ E and λ(E\F ) < ϵ such that (fn ) converges uniformly f on F .
Let {K n } be an increasing sequence of closed subsets of a compact metric space
K converging to K in the topology of closed convergence. If qn : K n → R t 7→
qn (t) is continuous, we write (K n , qn ) → (K, q) if q = q(t) ∈ C(K) and for every
subsequence nk and tnk ∈ K nk with tnk → t, qnk (tnk ) → q(t). We have the following:
Fact 2.12. Let {mn } be a bounded sequence in ca(K) with support(mn ) ⊂ K n
and mn → m in the weak* topology, and (K n , qn ) → (K, q). Then, qn mn → qm
(Mas-Colell [41]).
Let (K n , qn ) be a sequence as above. We say that it is equicontinuous if for all
ϵ > 0 there is a δ > 0 such that for all t, s ∈ K n with d(t, s) ≤ δ, |qn (t) − qn (s)| ≤ ϵ.
Mas-Colell [41] also proved the following:INDIVIDUALIZED AND DISTRIBUTIONALIZED ECONOMIES

987

Fact 2.13. Let {K n } be a sequence of closed sets of a compact metric space K with
K n ⊂ K n+1 ⊂ · · · → K in the topology of closed convergence and {qn } a sequencein C(K) with kqn k ≤ 1. If (K n , qn ) is equicontinuous, then there is a subsequence
nk and q ∈ C(K) with (K nk , qnk ) → (K, q).
2.2. Topological Vector Lattices. Let L and M be Banach spaces. The bilinear
form for the duality pair hL, M i is denoted by hz, ri or zr for short, where z ∈ Land r ∈ L∗ . However, rz denoted by the reverse order is more convenient foreconomic applications, and we follow this notation in the following. We summarize
the basic properties of the spaces used in the text.
The space of all bounded sequencesℓ∞ = x = (xt )| supt≥1 |xt | < +∞ ,is a nonseparable Banach space with respect to the norm kxk = supt≥1 |xt | forx ∈ ℓ∞ with the dual spaceba = q : 2N → R supE⊂N |q(E)| < +∞, q(E ∪ F ) = q(E) + q(F )whenever E ∩ F = ∅which is the spaceof bounded and ﬁnitely additive set functions on N under theR∞duality qz = zdq with z ∈ ℓ∞ and q ∈ ba. Let ℓ∞+ = {z ∈ ℓ | z ≥ 0} be∞∞
the nonnegative orthant of ℓ . The topology on ℓ deﬁned by the point-wise
convergence on ba is called weak topology and denoted as σ(ℓ∞ , ba).
It is well known that the dual space of the space of all summable sequences,
(
)
∞
X
1
t
t
ℓ = p = (p )|p | < +∞ ,t=1

Ptis ℓ∞ , and ℓ1 is a separable Banach space with the norm kpk = ∞t=1 |p |. The
1∞
1
nonnegative orthant ℓ+ is deﬁned similarly as ℓ+ . Thus, the space ℓ is isomorphic
to the subspace ca of ba,


∞
X
∞
ca = p ∈ ba p(∪n=1 En ) =p(En ) whenever Ei ∩ Ej = ∅ (i 6= j) ,n=1

which is the space of the bounded and countably additive set functions on N. The
topology on ℓ∞ deﬁned by the point-wise convergence on ℓ1 is called weak* topology
and denoted as σ(ℓ∞ , ℓ1 ).
The set function q ∈ ba is called purely ﬁnitely additive if q ′ = 0 whenever q ′ ∈ ca
and 0 ≤ q ′ ≤ q. The relation between the ba and ca is made clear by the next
fundamental theorem,
Fact 2.14. If q ∈ ba and q ≥ 0, then there exist set functions qc ≥ 0 and qp ≥ 0 inba such that qc is countably additive and qp is purely ﬁnitely additive and satisfyq = qc + qp . This decomposition is unique ([68]).

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Let (K, dK ) be a compact metric space. The space ca(K) is the set of bounded
countably additive set functions (signed measures) on K,
∞X
ca(K) = m : B(K) → R supE⊂K |m(E)| < +∞, m(∪∞E)=
m(Ei )
i=1 i


i=1

whenever Ei ∩ Ej = ∅ (i 6= j) .
Then, ca(K) is a Banach space with respect to the norm)
( nX
kmk = sup|m(Ei )| Ei ∩ Ej = ∅ for i 6= j, n ∈ N for m ∈ ca(K).
i=1

Let C(K) be the set of all continuous functions on K. C(K) is also a separable
Banach space with respect to the norm kq(t)k = sup{|q(t)|| t ∈ K} for q = q(t) ∈C(K). The Riesz representation theorem ([53, p. 357]) asserts that
R the dual spaceof C(K) is ca(K), or C(K)∗ = ca(K) under the duality hm, qi = q(t)dm(t) withm ∈ ca(K) and q = q(t) ∈ C(K) (see [1, Theorem 14.15]). The topology on ca(K)deﬁned by the point-wise convergence on C(K) is called the weak* topology anddenoted by σ(ca(K), C(K)). Bounded subsets of ℓ∞ and ca(K) are σ(ℓ∞ , ℓ1 ) andσ(ca(K), C(K))-weakly compact, respectively, i.e., that the weak* closure of thesets are weak* compact by Banach–Alaoglu’s theorem.
Fact 2.15. If L is a Banach space, then the unit ball of L∗ , B = {r ∈ L∗ | krk ≤ 1}
is compact in the σ(L∗ , L)-topology. Moreover, if L is a separable Banach space,then norm-bounded subset of L∗ is a compact metric space ([54, pp. 68-70]).
We use the following notations for the vector space orderings. For x = (xt ) ∈ Rk
or ℓ∞ or ℓ1 , x ≥ 0 means that xt ≥ 0 for all t and x > 0 means that x ≥ 0 and
x 6= 0. x  0 means that xt > 0 for all t. Finally, for x = (xt ) ∈ ℓ∞ , we denote byx ≫ 0 if and only if there exists an ϵ > 0 such that xt ≥ ϵ for all t. For m ∈ ca(K),m ≥ 0 means that m(B) ≥ 0 for every B ∈ B(K). m > 0 means that m ≥ 0 and
m 6= 0. The nonnegative orthant of ca(K) or ca+ (K) = {m ∈ ca(K)| m ≥ 0} is
nothing but the set of Borel measures M(K) on K. Because we have assumed K
to be a compact metric space, M(K) is a complete and separable metric space inthe weak* topology; see[66].
For t ∈ K, the Dirac measure δt is deﬁned by δt (E) = 1 when t ∈ E, δt (E) = 0
when t ∈/ E. Because (K, d) is a compact metric space, it is separable. Hence, there
exists a countable dense subset {t1 , t2 . . . } of K. Let LS(t1 . . . tn ) be the linear spacespanned by {δt1 . . . δtn }. It is well known that the set ∪∞n=1 LS(t1 . . . tn ) is dense inca(K) in the weak* topology.
These spaces are examples of locally convex topological vector spaces, that is, real
vector spaces equipped with a locally convex Hausdorﬀ topology (i.e., topology having a neighborhood base at 0 consisting of convex sets) such that the addition andthe scalar multiplication are jointly continuous. Moreover, they are also topological
vector lattices. In general, an ordered vector space is a vector space L endowed witha reﬂexive, transitive and antisymmetric relation ≤ that satisﬁes (i) if x ≤ y anda ∈ R+ , then ax ≤ ay, (ii) if x ≤ y, then x + z ≤ y + z for each z ∈ L. Let L be

INDIVIDUALIZED AND DISTRIBUTIONALIZED ECONOMIES

989

an ordered vector space and S ⊂ L. An element y ∈ L is an upper bound of S if
and only if x ≤ y for every x ∈ S. An element sup S is called the supremum (least
upper bound) if it is an upper bound of S and sup S ≤ y for every upper boundy of S. Similarly, z is a lower bound of S if z ≤ x for every x ∈ S. inf S is theinﬁmum (greatest lower bound) if it is a lower bound of S and z ≤ inf S for every
lower bound z of S. We usually write x ∨ y rather than sup{x, y} and x ∧ y rather
than inf{x, y}. If every pair x, y of an ordered vector space L has the supremumx ∨ y and the inﬁmum x ∧ y, then we call L a vector lattice or Riesz space. We
write x+ = x ∨ 0 and x− = (−x) ∨ 0 and call the positive part and the negative
part of x, respectively. Then, x = x+ − x− , and we write |x| = x+ + x− and callthe absolute value of x. A topological vector lattice L is called a Banach lattice if it
is a Banach space and satisﬁes kxk ≤ kyk whenever |x| ≤ |y|. The following facts
together with Fact 2.15 are also included as the “essential mathematical structures([44, p. 1838])”.
Fact 2.16. Let L be a locally convex topological vector space and L′ its topologicaldual. If A and B are disjoint convex sets such that one of which has an interior
point, then there is a nonzero continuous linear functional r ∈ L′ such that rx ≤ ry
for each x ∈ A and each y ∈ B (Hahn–Banach separation theorem, [1, Theorem
5.67]).
P
Fact 2.17. Let L be a vector lattice and letPx1 . . . xn , z ∈ L+ satisfy z ≤ ni=1 xi .
Then, there are z1 . . . zn ∈ L+ that satisfy ni=1 zi = z and zi ≤ xi (Riesz decomposition theorem, [1, p. 319]).
2.3. Integrations of Vector-Valued Maps. Let (A, A, λ) be a ﬁnite measure∗space, L a Banach space and M =PnL its norm dual. A map f : A → L is called
simple if it is of the form f (a) = i=1 xi 1Ai (a), where xi ∈ L, Ai are measurable
partitions of A with ∪ni=1 Ai = A and 1Ai (a) is the characteristic function of Ai ,or 1Ai (a) = 1 when a ∈ Ai and 1Ai (a) = 0 when a ∈/ Ai . A map f : A → L is
said to be strongly measurable if there exists a sequence (fn ) of simple functions
with kfn (a) − f (a)k R→ 0 a.e. A strongly measurable function f isR said to be
Bochner Rintegrable if A kfn (a)R − f (a)kdλ → 0. In this case, we denote A f (a)dλ =limn→∞ A fn (a)dλ and call A f (a)dλ the Bochner integral. It is well known ([10,Theorem
R 2, p. 45]) that a strongly measurable map f is Bochner integrable if andonly if A kf (a)kdλ < ∞.
A map f : A → L is said to be weakly measurable if for each r ∈ L∗ , rf (a) ismeasurable. A weakly measurable map f (a) is said to be Pettis
integrable if thereR
exists an elementRξf ∈ L such that for each r ∈ L∗ , rξf = A rf (a)dλ. The vector
ξf is denoted by A f (a)dλ and called Pettis integral of f .
A map f : A → L∗ is said to be weak* measurable if for each z ∈ L, zf (a) ismeasurable. A weak* measurable map f (a) is said to be Gelfand
integrable if thereR∗
exists an element Rπf ∈ L such that for each z ∈ L, zπf = A zf (a)dλ. The vector
πf is denoted by A f (a)dλ and called Gelfand integral of f .
In general, let L be a locally convex topological vector space and L′ be its topological dual. A map f : A → L is said to be weakly measurable if for each r ∈ L′ ,rf (a) is measurable. A weakly measurable map f (a) is said to be RPettis integrable
if there exists an element ξf ∈ L such that for each r ∈ L′ , ξf r = A rf (a)dλ. The

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R
vector ξf is denoted by A f (a)dλ and called Pettis integral of f . Those integral
concepts can be generalized to correspondences.
Fact 2.18. Let L be a Banach space and L∗ its norm dual space. If f : A → L∗ is
weak* measurable and zf (a) is integrable function for all z ∈ L, then f is Gelfand
integrable ([10, pp. 53–54]).
Fact 2.19. Let {fn } be a sequence of Gelfand integrable functions Rfrom A to L∗
which
converges a.e. to f in the weak* topology. Then, it follows that A fn (a)dλ →
R
f
(a)dλ
in the weak* topology.
A
For instance, let L = ℓ1 . Then M = L∗ = ℓ∞ . A map f : A → ℓ∞ is weak*measurable if for each p ∈ ℓ1 , pf (a) is measurable. The Gelfand
integrable of fR
is an element xf ∈ ℓ∞ such that for each p ∈ ℓ1 , pxf = A pf (a)dλ. Similarly,
a map f : A → ca(K) is weak* measurable if for each q ∈ C(K), qf (a) is ameasurable
if there
R function on (A, A, λ), and it is Gelfand integrable
R
R exists anelement A f (a)dλ ∈ ca(K) such that for each q ∈ C(K), q A f (a)dλ =R A qf (a)dλ.
In particular, forR every Borel set RB ∈ B(K), the value of the measure A f (a)dλ at
B is deﬁned by A f (a)dλ(B) ≡ A f (a)(B)dλ.
The integral of a correspondences can be deﬁned as in the ﬁnite dimensional
cases. For instance let ϕ : A → L∗ be a correspondence and L (ϕ) = {f : A →L∗ | f is Gelfand integrable, f (a) ∈ ϕ(a) a.e} the set of Gelfand integrable selectionsof ϕ. We then deﬁneZZϕ(a)dλ =f (a)dλ f ∈ L (ϕ) .
AA

Let X be a topological space and Fn a sequence of subsets of X. The topologicallimes superior Ls(Fn ) is deﬁned by the following:x ∈ Ls(Fn ) if and only if there exists a subsequence Fnk with xnk ∈ Fnk for all k
and xnk → x (k → ∞).
The next fact is also well known (Fatou’s lemma in ℓ dimensions, [18, p. 69]).
Fact 2.20. Let (fn )n∈N be a sequence
R of integrable functions of a measure space(A, A, λ) to Rℓ+ . Suppose that limn A fn (a)dλ exists. Then, there exists an integrable function f : (A, A, λ) → Rℓ+ such that(a) Rf (a) ∈ Ls(fn (a)) a.e.R in A,(b) A f (a)dλ ≤ limn→∞ A fn (a)dλ.
Consider a sequence of maps ϕn : A → X. For each a ∈ A, if Ls(ϕn (a)) 6= ∅, we
can deﬁne a correspondence Ls(ϕn (·)) : A → X, a 7→ Ls(ϕn (a)). The next theorem
is attributed to Khan–Sagara–Suzuki [30], which is an inﬁnite dimensional versionof the Fatou’s lemma and plays a crucial role in our proofs (see also [14]).
Fact 2.21. Let (A, A, λ) be a complete and ﬁnite measure space that is saturatedand L∗ be the dual space of a separable Banach space L. Let fn : A → L∗ be asequence of Gelfand integrable mappings from A to L∗ such that there exists an
integrable function g(a) with supn kfn (a)k ≤ g(a) a.e.
Then, there exists a Gelfand integrable map f : A → L∗ withZZ
f (a)dλ ∈ Ls
fn (a)dλ and f (a) ∈ Ls(fn (a)) a.e.
AA

INDIVIDUALIZED AND DISTRIBUTIONALIZED ECONOMIES

991

Podczeck–Sun–Yannelis ([52, 57]) proved the following:
Fact 2.22. Let (A, A, λ) be a ﬁnite measure space and L∗ be the dual space of aseparable
Banach space L. Then, the following conditions are equivalent.
R(i) A ϕ(a)dλ is convex for every correspondence ϕ : A → L∗ ,(ii) the measure space (A, A, λ) is saturated.
Remark 2.23. Recall that a topological space is a Suslin space if and only if it is
the image of a continuous map from a complete and separable metric space (Polishspace). A separable Banach space L is a Suslin space, and its dual space L∗ endowedwith the weak* topology is also a Suslin space (see [64, p. 67]). Hence, ℓ∞ with the
weak* topology σ(ℓ∞ , ℓ1 ) and ca(K) with the weak* topology σ(ca(K), C(K)) for
a compact metric space K are examples of Suslin spaces, because ℓ1 and C(K) are
separable Banach spaces when K is compact ([12, p. 437]). A map from a measurespace (A, A, λ) to a locally convex Suslin space is Borel measurable if and only if it
is weakly measurable ([64, Theorem 1]). Because σ(ℓ∞ , ℓ1 ) and σ(ca(K), C(K)) arelocally convex, maps from (A, A, λ) to ℓ∞ or ca(K) are weak* measurable if andonly if they are Borel measurable (with respect to the weak*topologies).
3. Basic model
3.1. Setup and Assumptions. Let L be a Banach lattice with its order relation
denoted by ≥ that is a dual space of the separable Banach lattice M , or L = M ∗ .
The order relation on M is also denoted by ≥ if there is no danger of confusion.
As usual, we deﬁne ξ > 0 if and only if ξ ≥ 0 and ξ 6= 0 and L+ = {ξ ∈ L| ξ ≥0} (L− ≡ −L+ ), and similarly for M . Let Z be a convex and weak* closed subset of
L+ . Throughout the paper, the commodity space is assumed to be Q = Rk ×L, andthe (common) consumption set is X = Rk+ ×Z. Typical elements of X are called theconsumption vectors and denoted by ξ = (x, x), ζ = (z, z), ϕ = (u, m), ψ = (y, n)and so on, where u, x, y, z ∈ Rk+ and m, n, x, z ∈ Z. Accordingly, we assume theprice space to be P = Rk ×M , and its elements are called price vectors that areoften denoted as π = (p, p) or ρ = (q, q), p, q ∈ Rk+ and p, q ∈ M+ .
As usual, a preference ≿⊂ X×X is a complete, transitive, and reﬂexive binary
relation on X. We denote (ξ, ζ) ∈≿ by ξ ≿ ζ. ξ ≺ ζ means that (ξ, ζ) ∈≿.
/Let P
be the set of allowed preference relations. The assumptions for the preferences are
as follows:
Assumption 3.1. (PR) (i) For every ≿∈ P, ≿⊂ X×X is complete, transitive,
and reﬂexive, which is closed in X×X in the weak* topology,(ii) (Monotonicity). For every ≿∈ P, ξ ≺ ζ whenever ξ, ζ ∈ X and ξ < ζ.
As stated in Section 1, we do not need the convexity for the preferences, comparedwith [6, 35, 40, 46, 47, 48, 61]. Because P ⊂ F (X×X) by (i), we can endow P withthe topology of closed convergence on F(X × X) ([18, pp. 15–19]).
An endowment vector is an element of X. We denote the set of all endowment
vectors by Ω and assume it to be weak* compact subset of X. Throughout the paper,
we consider the measurable space of consumers’ characteristics (P × Ω, B(P × Ω)).

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3.2. Equilibria and Core. Let (A, A, λ) be a complete probability space of theconsumers. The deﬁnition of the economy is standard.
Deﬁnition 3.2. A LIE E is a Borel measurable mapping E : A → P × Ω deﬁned by
a 7→ (≿a , ω(a)). The economy is called saturated (or super-atomless) if the measurespace (A, A, λ) is saturated.
An allocation Ris a Gelfand Rintegrable map ξ : A → X a.e. An allocation
is saidRtobefeasibleifξ(a)dν
≤ω(a)dν.
It
iscalledexactly
feasibleifξ(a)dν=AA
A
R
A ω(a)dν.
We now state a deﬁnition of a competitive equilibrium.
Deﬁnition 3.3. A pair (π, ξ) of a price vector π ∈ P+ with π 6= 0 and an allocation
ξ : A → X is called an allocative equilibrium of the economy E if the following
conditions hold:(E-1) πξ(a)≤ πω(a)R
R and ξ(a) ≿a ζ whenever πζ ≤ πω(a) a.e.,(E-2) A ξ(a)dλ = A ω(a)dλ.
A nonnull measurable subset S of the measure space of consumers (A, A, λ) is
called a coalition. We furnish the conventional deﬁnition of the core.
Deﬁnition 3.4. A coalition S ⊂ A is said to block an allocation ξ if there exists anallocation ζ such thatRR(i) S ζ(a)dλ ≤ S ω(a)dλ;(ii) ξ(a) ≺a ζ(a) on S.
A feasible allocation ξ : A → X belongs to the core of an economy E if there exist
no coalitions S ⊂ A which block ξ.
4. Realization of distributive equilibria
4.1. LIE Representations for LDEs. In this section, we assume that X is acomplete and separable metric space (Polish space) in the weak* topology and that
P is also a Polish space in the closed convergence topology ([18, p. 19]). Theseassumptions on X and P are satisﬁed for the models in Section 5. The distributive
economy is deﬁned as follows.
Deﬁnition 4.1. A LDE is a probability measure µ on the measurable space(P×Ω, B(P × Ω)). A distributionalized economy µ is called atomless if it is anatomless probability measure.
A probability measure ν on X ×P×Ω is called an allocation
R distribution
R if νP×Ω =µ. An allocation distribution is called (exactly) feasible if X ι(ξ)dνX = Ω ι(ω)dµΩ ,whereι is theR inclusion map. Because ι(ξ) = ξ for all ξ, we hereafter denoteRι(ξ)dν
X = X ξdνX , and so on. The distributive equilibrium is deﬁned as follows.
X
Deﬁnition 4.2. A pair (π, ν) of a price vector π ∈ P+ with π 6= 0 and an allocation
distribution ν on X × P×Ω is called a distributive equilibrium of the economy µ if
the following conditions hold,(D-1) ν({(ξ, ≿, ω) ∈ X × P×Ω| πξ ≤ πω and ξ ≿ ηwhenever πη ≤ πω}) = 1,
RR
(D-2) X ξdνX = Ω ωdµΩ ,INDIVIDUALIZED AND DISTRIBUTIONALIZED ECONOMIES

993(D-3) νP×Ω = µ,where the marginals of µ are denoted by subscripts, e.g., µP denotes the marginal
on P, and so on.
Let (A, A, λ) be an atomless probability measure space for a measurable map
f : A → P×Ω.
Deﬁnition 4.3. For an economy µ, a measurable map E : A → P×Ω such thatµ = E∗ λ is called a representation of µ. The representation is called saturated if themeasure space (A, A, λ) is saturated.
Note that a representation is not unique even if it exists. Because P × Ω is a
Polish space, the representations of µ exists by Fact 2.5 in Section 2.1. Moreover,because the saturated measure spaces are atomless, the saturated representations
also exist. Similarly, for every allocation distribution ν, a measurable map (ξ, E) :A → X × P×Ω, which satisﬁes ν = (ξ, E)∗ λ is the representation of ν. The map
ξ : A → X is only simply an allocation. The representations for ν also exist by thesame reason for µ. We may call the existence of the representations for ν the weakequivalence of the individual and distributive equilibria ([25]).
A fundamental problem is the realization of the distributive equilibrium or thestrong equivalence of the two equilibria; given an equilibrium ν of an economy µ andan individual economy E, which represents µ, can we obtain an allocation ξ suchthat (ξ, E) represents ν? The answer is generally negative for atomless measurespaces of the consumers. However, the positive answer is the rule for the saturated
measure spaces.
Theorem 4.4. Let (A, A, λ) be a saturated probability space. Let distributive economy µ and its equilibrium ν be given. For every individual economy E : A → P×Ω
which represents µ, there exists an equilibrium allocation ξ : A → X such that
ν = (ξ, E)∗ λ.
4.2. Symmetric Equilibria. The concept of the symmetric equilibria in the next
deﬁnition is attributed to [42] (see also [32]).
Deﬁnition 4.5. The equilibrium ν is called symmetric if there exists a measurable
map θ : P × Ω → X such that ν(Graph(θ)) = 1, where Graph(θ) = {(ξ, ≿, ω) ∈X × P × Ω| ξ = θ(≿, ω)}.
If an equilibrium is symmetric, then the consumers with the identical characteristics consume the identical consumption vector.
Deﬁnition 4.6. Let a distributionalized economy µ and its equilibrium ν be given.
A probability space (A, A, λ) realizes ν, or (A, A, λ) is a realization of ν, if every
individual economy E : A → P×Ω which represents µ has a measurable map ξ :A → X such that ν = (ξ, E)∗ λ.
Let E : (A, A, λ) → P × Ω be an individual representation that is deﬁned on
the measure space of consumers of the distributionalized economy µ and deﬁne
σ(E) = {E −1 (B)| B ∈ B(P × Ω)}; the smallest σ-algebra on A which makes E
measurable. The following results are already known for the large atomless games
[25].

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Theorem 4.7. Assume that X is a complete separable metric space and let (π, ν)be a distributive equilibrium of µ and E a representation of µ deﬁned on (A, A, λ).
Then, ν is symmetric if and only if ν = (ξ, E)∗ λ for a σ(E)-measurable equilibrium
allocation ξ of E.
Given any probability space (A, A, λ), recall that a function on A is called almost one-to-one if it is one-to-one on A except for some λ-null set of A. If λ is
atomless, we can show by Fact 2.4 in Section 2.1 that there exists an almost one-toone Lebesgue representation, i.e., a representation deﬁned on the Lebesgue space([0, 1], B([0, 1]), dx).
Theorem 4.8. Let E be a representation of µ deﬁned on ([0, 1], B([0, 1]), dx). Assume that E is almost one-to-one.
(a) If ξ : [0, 1] → X is a competitive equilibrium allocation of E that is measurablein B([0, 1]), then ν = (ξ, E)∗ λ is a symmetric equilibrium distribution of µ.
(b) Let ξ : [0, 1] → X be any Gelfand integrable function that is measurable inB([0, 1]), and let ν = (ξ, E)∗ dx. If ν is an equilibrium distribution of µ, then ξ isan equilibrium allocation of E and ν is symmetric.
Theorem 4.9. Let an atomless distributionalized economy µ and its equilibrium ν
be given. Then, the following conditions are equivalent.
(a) ν is symmetric,(b) every atomless probability space is a realization of ν,(c) every atomless nonsaturated probability space is a realization of ν,(d) the measure space ([0, 1], B([0, 1]), dx) is a realization of ν.
Corollary 4.10. An atomless probability space (A, A, λ) realizes a nonsymmetricequilibrium of an atomless distributionalized economy µ if and only if it is saturated.
5. Existence and core equivalence of allocative equilibria
5.1. A Market with Inﬁnite Time Horizon. In this section, we set L = ℓ∞and M = ℓ1 ; then, the commodity space is Q = Rk × ℓ∞ . We assume k ≥ 1. The
commodity vectors are written by ξ = (x, x), ζ = (z, z), where x = (xi ), z = (z i ) ∈Rk and x = (xt ), z = (z t ) ∈ ℓ∞ .
The price vector is assumed to be a vector π = (p, p) ∈ P+ = Rk+ × ℓ1+ , whereξ = (x, x) ∈ Q
p = (pi ) ∈ Rk+ and p = (pt ) ∈ ℓ1+ . The value of a commodity P
t tevaluated by a price vector π = (p, p) ∈ P+ is given by πξ = px + ∞t=1 p x .
Let (A, A, λ) be a complete probability space of the consumers. We assume
that the consumption set X∞ that is identical among all consumers is deﬁned by
X∞ = Rk+ ×Z∞ , where Z∞ is a cube in ℓ∞ ,Z∞ = {x ∈ L| 0 ≤ x ≤ x̂1},
where 1 = (1, 1 . . . ) ∈ ℓ∞ and x̂ is a positive constant. Of course, the x̂ > 0 is
intended to be a very large number. By Fact 2.15 in Section 2.2, Z∞ is a compact
metric subspace of L+ with respect to the weak* topology that is obviously convex.
As usual, a preference ≿ is a complete, transitive, and reﬂexive binary relation
on X∞ that is close in X∞ ×X∞ in the weak* topology. Recall that P is the set of

INDIVIDUALIZED AND DISTRIBUTIONALIZED ECONOMIES

995

allowed preferences. Because X∞ is a locally compact, complete separable metricspace, we can show as in Hildenbrand [18, Lemma, p. 98] that P is a Borel set.
An endowment vector is an element of Q+ . An endowment vector is usually
written as ω = (e, e) ∈ Q+ , e = (ei ) ∈ Rk+ and e = (et ) ∈ ℓ∞
+ . We denote the set of
all endowment vectors by Ω and assume that it is of the formΩ∞ = {ω = (e, e) ∈ Q| 0 ≤ ω ≤ ω̂1},for some ω̂ > 0. The number ω̂ is intended to be far smaller than x̂. The set Ω∞ isalso a compact metric space in the weak* topology. The same caveat for X∞ also
applies to Ω∞ . In what follows, we often write for simplicity X∞ and Ω∞ as X andΩ, respectively.
The endowment map a 7→ ω(a) in the economy E : A → P × Ω is Gelfand
integrable by Fact 2.18 in Section 2.3. The next assumption saying that the total
endowment belongs to the norm interior of the consumption set is standard.
Assumption
5.1. (PE) (Positive endowments). (i) e(a) > 0 a.e,R(ii) A ω(a)dλ ≫ 0.
We can now state
Theorem 5.2. Let E : A → P × Ω be an economy that is saturated. Then, there
exists a competitive equilibrium (π, ξ) for E.
Theorem 5.3. Let E : A → P × Ω be an economy that satisﬁes the conditions
assumed in Theorem 5.2. A feasible allocation ξ : A → X belongs to the core of
the economy E if and only if there exists a price vector π ∈ P such that (π, ξ) is acompetitive equilibrium for E.
Remark 5.4. In order to prove Theorem 5.3, the saturation Ris not required (cf.
[51] or [56]). Since the weak* closure of the Gelfand integral A Ψ(a)dλ is weak*compact and convex (see [23] or [67]), the Hahn-Banach separation theorem canbe still applied with minor modiﬁcations. However, if we drop the saturation, nonemptiness of the equilibrium hence the core can be no longer guaranteed, therefore
relevance of such a generalization is questionable. Moreover, under the assumptionof saturation we can invoke Fact 2.22 which makes the proof simple and straightforward. By these reasons we present Theorem 5.3 assuming the saturation. The
same remark applies to Theorem 5.11.
5.2. A Market with Diﬀerentiated Commodities. Let (K, dK ) be a compact
metric space. In this section, we set L = ca(K) and M = C(K); hence, Q =Rk ×ca(K) and P = Rk ×C(K) (k ≥ 0). A typical element of Q is denoted by
ϕ = (u, m) or ψ = (y, n), u, y ∈ Rk and m, n ∈ ca(K). The price vector is written
as ρ = (q, q), q = (q i ) ∈ Rk and q = q(t) ∈ C(K).
Following [19, 41], the economic interpretation of K is a space of the commodity
characteristics. Hence, each t ∈ K represents the complete list of characteristics
that describes the commodity. A (diﬀerentiated) commodity bundle m is deﬁned
as a signed measure on K; hence, an element of ca(K). In particular, the Diracmeasure δt is the (one unit of) commodity bundle that contains characteristicst ∈ K. The consumption set XM is XM = Rk+ ×ZM , ZM = ca(K)+ . We assume996

TAKASHI SUZUKI

k ≥ 0; hence, we do not require the nondiﬀerentiated (or homogeneous) goods. We
need an additional assumption for the preferences:
Assumption 5.5. (US) For all y ∈ Rk+ , for all ≿∈ P and for every α > 1, there
exists ε > 0 such that if diameter (J) < ε and m, n ∈ ZM satisfy m(J) ≤ n(J),then(y, m|J + m|K\J ) ≺ (y, α(n|J + m|K\J )),where diameter (J) = sup{d(s, t) | s, t ∈ J} and m|J is the restriction of m to J.
The assumption US is called by Ostroy and Zame [48] by uniform substitutability, meaning that for nearby commodities, consumers prefer any feasible trade inwhich the “terms” (namely α) are strictly greater than one. Let PU S be the set ofpreferences that satisfy Assumptions PR and US.
An initial endowment is assumed to be a nonnegative vector ϖ of XM . We often
denote ϖ = (f, f ), f = (f i ) ∈ Rk+ and f ∈ ca(K)+ . Let ΩM ⊂ XM be the set of
initial endowments. We assume it is of the form
ΩM = {ϖ = (f, f ) ∈ Q| 0 ≤ f i , f (K) ≤ ω̂, i = 1 . . . k}, ω̂ > 0.
As before the suﬃx M is often omitted from XM and ΩM .
Let (A, A, λ) be a complete probability space of consumers. An endowment
assignment is a Gelfand integrable map ϖ : A → ΩM , a 7→ ϖ(a) = (f (a), f (a)).
The assumption on the endowments for this economy isR
AssumptionR 5.6. (AE)
(Adequate
endowments).
(i)
A f (a)dλ  0,(ii) support A f (a)dλ = K (see Fact 2.1).
The assumption AE simply says that all commodity characteristics are available
in the market. Let EP be the composition of the map E and the projection of P × Ω
to P, a 7→≿a . The existence of equilibria for the model with the diﬀerentiatedcommodities is established by the following:
Theorem 5.7. Let E be an economy that is saturated and satisﬁes EP (A) ⊂ PU S andAssumption AE. Then, there exists a competitive equilibrium (ρ, ϕ) with ρ ∈ P+ \{0}for E.
We say that τ (6= 0) ∈ X is an extremely desirable commodity if there exists an
weak* open neighborhood U of 0 such that for each ϕ ∈ X we have ϕ ≺ ϕ + ατ − ψ
whenever α > 0, ατ ≥ ψ, and ψ ∈ αU . This notion is interpretable geometricallyas follows. Let τ (6= 0) ∈ X, U be an weak* open neighborhood U of 0 and deﬁnean open cone C by
C = {ατ − ψ| ψ ∈ Q, ψ ∈ αU, α > 0}.
Then, the commodity bundle τ is extremely desirable if for each ϕ ∈ X we haveϕ ≺ χ whenever χ ∈ (C + ϕ) ∩ X. Note that this implies that τ is an extremely
desirable commodity if for each ϕ ∈ X it follows that ((−C + ϕ) ∩ X) ∩ {χ ∈ X| ϕ ≺χ} or equivalently (−C) ∩ {χ − ϕ ∈ X| ϕ ≺ χ} = ∅. Rustichini and Yannelis [56]strengthened the concept of the extremely desirable commodity.
Deﬁnition 5.8. A pair of commodities (χ, υ) ∈ X×X is said to be a desirable
commodity pair if for every ψ ∈ X we have ψ ≺a ψ + χ − υ whenever ψ + χ ≥ υ foreach a ∈ A.

INDIVIDUALIZED AND DISTRIBUTIONALIZED ECONOMIES

997

Deﬁnition 5.9. A pair (χ, υ) ∈ Q×Q is said to P
have the splitting property if form
any m-tuple (ψ1 . . . ψm ) ∈ Q × · · · ×Q such thatP i=1
ψi = (χ − υ)− there existsman m-tuple (γ1 . . . γm ) ∈ Q × · · · × Q such that i=1 γi = (χ − υ)+ and the pair(ψi , γi ) is a desirable commodity pair.
The next condition was proposed by [56].
Assumption 5.10. (CD) (Commodity pair desirability). There exists a commodity bundle τ (6= 0) ∈ X and an weak* open neighborhood U of 0 such that any
commodity bundle pair (χ, υ) of the form χ = ατ , α > 0, and υ ∈ αU has the
splitting property.
Let PCD be the set of preferences that satisfy the conditions PR and CD. Let
P ′ ⊂ PCD be a Borel set of allowed preferences. Note that in [19, 41], the sets ofallowed preferences are assumed to be compact. Our requirement of measurability
is much weaker. The core equivalence theorem for the economy E on ca(K) is now
stated as follows.
Theorem 5.11. Let E be an economy that is saturated and satisﬁes EP (A) ⊂ P ′and Assumption AE. A feasible allocation ϕ : A → X belongs to the core of theeconomy E if and only if there exists a price vector ρ ∈ P+ \{0} such that (ρ, ϕ) is
a competitive equilibrium for E.
6. Proofs
6.1. Proof of Theorem 4.4. First, we show
Lemma 6.1. Let (A, A, λ) be an atomless measure space, E : A → P × Ω be a
representation of µ and ξ : A → X a measurable mapping. Deﬁne ν = (ξ, E)∗ λ.
Then, ξ is an equilibrium allocation of E if and only if ν is an equilibrium distributionof µ.
Proof. Suppose that ξ is an equilibrium allocationof E.
R
R Then, there exists a pricevector π(6= 0) ∈ ℓ∞
+ with λ(E) = 1 and A ξ(a)dλ = A ω(a)dλ, where E = {a ∈A| πξ(a) = πω(a) and ξ(a) ≿a ζ whenever πζ ≤ πω(a)}. Let
F = {(ξ, ≿, ω) ∈ X × P × Ω| πξ = πω and ξ ≿ ζ whenever πζ ≤ πω}.
Then, (ξ, E)−1 (F ) = E, hence ν(F ) = (ξ, E)∗ λ(F ) = λ(E) = 1, which proves thecondition (D-1). Becauseξ∗ λ R= νX and ω∗ λ = νΩ = µΩ , we have from the change ofRvariable formula X xdνX = Ω ωdµΩ . Hence, the condition (D-2) is met. Finally,
the condition (D-3) follows from νP×Ω = E∗ λ = µ. The converse is also proved in asimilar way.
□
We now prove Theorem 4.4. Because E∗ λ = µ = νP×Ω , we have from Fact 2.5
in Section 2.1 a measurable map ξ with ν = (ξ, E)∗ λ. Then, ξ is an equilibrium
allocation by Lemma 6.1.
□
6.2. Proof of Theorem 4.7. To prove Theorem 4.7, we shall prove the following:
Lemma 6.2. Let (A, A, λ) be an atomless measure space, E : A → P × Ω bea representation of µ and ν be a symmetric equilibrium of µ, i.e., there exists a
measurable mapping θ : P × Ω → X such that ν(Graph(θ)) = 1. Deﬁne ξ : A → X
by ξ(a) = θ(E(a)). Then, ν = (ξ, E)∗ λ and ξ is an equilibrium allocation of E.

998

TAKASHI SUZUKI

Proof. Let B ∈ X × P × Ω. Then, we haveν(B ∩ Graph(θ)) = ν({(θ(≿, ω), ≿, ω)| (θ(≿, ω), ≿, ω) ∈ B})= µ({(≿, ω)| (θ(≿, ω), ≿, ω) ∈ B})= λ({a ∈ A| (ξ(a), E(a)) ∈ B}) = (ξ, E)∗ λ(B),which implies that ν = (ξ, E)∗ λ and ξ is an equilibrium allocation of E from Lemma
6.1.
□
Now we turn to prove Theorem 4.7. Let ν be a symmetric equilibrium of theeconomy µ with a measurable map θ : P × Ω → X such that ν(Graph(θ)) = 1. Set
ξ = θ ◦ E. It is easy to verify that ξ is σ(E)-measurable. It follows from Lemma 6.2that ν = (ξ, E)∗ λ and ξ is an equilibrium allocation of E.
Next, suppose that ν = (ξ, E)∗ λ for a σ(E)-measurable equilibrium allocation of
E. By Lemma 6.1, ν is an equilibrium distribution of µ. It follows from Fact 2.3 in
Section 2.1 that there exists a measurable map θ : P × Ω → X such that ξ = θ ◦ E.
It is suﬃcient to show that ν(Graph(h)) = 1, yielding the following:ν(Graph(θ)) = (ξ, E)∗ λ(Graph(θ)),= λ({a ∈ A| (ξ(a), E(a)) ∈ Graph(θ)}),= λ({a ∈ A| (θ ◦ E(a), E(a)) ∈ Graph(θ)}),= λ({a ∈ A| E(a) ∈ P × Ω}) = 1,
which proves Theorem 4.7.

□

6.3. Proof of Theorem 4.8. (a): Let I ′ be a Borel subset of I = [0, 1] such thatdx(I ′ ) = 1 and E is one-to one on I ′ . We can assume without loss of generalitythat both E and ξ are constant on I\I ′ . Lemma 6.1 shows that for a competitive
equilibrium allocation ξ : [0, 1] → X of E that is measurable in B([0, 1]), ν =(ξ, E)∗ dx is an equilibrium distribution of µ. It remains to show that ν is symmetric.
Let B ∈ P × Ω and C = ξ −1 (B)∩I ′ . Because ξ is measurable, C ∈ B([0, 1]). By
Fact 2.2 in Section 2.1, E(C) ∈ B(P × Ω); hence, C ∈ σ(E). Theorem 4.7 implies
that ν is symmetric.
(b): Let ξ : [0, 1] → X be a Gelfand integrable function which is measurable inB([0, 1]), and set ν = (ξ, E)∗ dx. By Lemma 6.1, ξ is a competitive equilibrium
allocation of E. The proof of the part (a) shows that ξ is σ(E) measurable; hence,ν is symmetric.
6.4. Proofs of Theorem 4.9 and Corollary 4.10. To prove that (a) implies(b) in Theorem 4.9, let ν be a symmetric equilibrium of an atomless economy µ,and (A, A, λ) an atomless probability space. Then, there exists a measurable map
θ : P × Ω → X such that ν(Graph(θ)) = 1. Suppose that E : A → P × Ω be a
representation of µ. Set ξ = θ◦E. Then, it follows from Lemma 6.2 that ν = (ξ, E)∗ λand ξ is an equilibrium allocation. Therefore, (a) implies (b). Obviously, (b) ⇒ (c)⇒ (d).
Now, suppose that (d) holds. Because µ is atomless, it follows from Fact 2.4
that there exists a representation E of µ on a measure space ([0, 1], B([0, 1]), dx)
which is almost one-to-one. Because ([0, 1], B([0, 1]), dx) realizes ν, there exists an

INDIVIDUALIZED AND DISTRIBUTIONALIZED ECONOMIES

999

equilibrium allocation ξ of E such that ν = (ξ, E)∗ dx. Then, it follows from Theorem
4.8 that ν is symmetric.
□
Finally, we deduce the Corollary. Let (A, A, λ) be saturated. Then, it realizes any distributive equilibrium, i.e., a symmetric equilibrium. Conversely, let ν
be nonsymmetric distributive equilibrium of an economy µ. We can easily showthat ([0, 1], B([0, 1]), dx) is a realization of ν if and only if the Lebesgue space([0, 1], L([0, 1]), dx) realizes ν. Therefore, by Theorem 4.9, neither ([0, 1], B([0, 1]), dx)
nor ([0, 1], L([0, 1]), dx) realizes ν. Because (A, A, λ) realizes ν, it follows from Fact
2.10 in Section 2.1 that (A, A, λ) is saturated.
□
6.5. Proof of Theorem 5.2. Step 1: Let E : A → P×Ω be the economy. For eachn ∈ N, let Ln be the canonical projection of ℓ∞ to Rn , Ln = {ξ = (ξ t ) ∈ ℓ∞ | ξ =(ξ 1 , ξ 2 . . . ξ n , 0, 0 . . . )}. We then deﬁne
X n = X ∩ Ln , ≿na =≿a ∩(X n ×X n ), P n = P ∩ 2L ×L , Ωn = Ω∩Ln .
nnSimilarly, we denote the canonical projection of ω = (ω 1 , ω 2 . . . ω n , ω n+1 . . . ) ∈ Ω
as ωn = (ω 1 , ω 2 . . . ω n , 0, 0 . . . ) ∈ Ωn . Obviously, ωn → ω in the weak* topology.
They induce ﬁnite dimensional economies E n : A → P n × Ωn deﬁned by E n (a) =(≿na , ωn (a)), n = 1, 2 . . . . Then, we have the following:
Lemma 6.3. For each n ≥ k, there exists a quasicompetitive equilibrium for the
economy E n , or a price-allocation pair (πn , ξn (a)) that satisﬁes(Q-1n) πn ξn (a) ≤ πn ω(a) and ξn (a) ≿a ζ whenever πn ζ ≤ πn ωn (a) and πn ωn (a) >0 a.e., R
R(Q-2n)A ξn (a)dλ ≤ A ωn (a)dλ.
Proof. See Khan and Yamazaki [34], Proposition 2.

□

Step 2: Because ωn (a) = (en (a), en (a)) → ω(a) = (e(a), e(a)) a.e., we haveZZωn (a)dλ →ω(a)dλAA

byPnFactt 2.19 in Section 2.3. Withoutt loss of generality, we can assume that πn 1 =t=1 pn = 1 for all n, where πn = (pn ) and 1 = (1, 1 . . . ). Here, we have identiﬁed
πn ∈ Rn+ with a vector in ℓ1+ that is also denoted by πn as πn = (πn , 0, 0 . . . ).
We denote ξn (a) = (xn (a), xn (a)) ∈ X n . The ﬁnite dimensional Fatou’s lemma(Fact 2.20 in Section 2.3) implies that there exists
R a measurable mapR x : A →Rk+ Rsuch that x(a) ∈ Ls(xn (a)) a.e. in A and A x(a)dλ ≤ limn→∞ A xn (a)dλ(≤ A e(a)dλ). By Fact 2.21 in Section 2.3, we have aR Gelfand integrable
functionR
x : A → X such that x(a) ∈ Ls(xn (a)) a.e., and A x(a)dλ ≤ A e(a)dλ. Let
ξ(a) = (x(a), x(a)). Then we have obtained thatZZ
ω(a)dλ.
ξ(a)dλ ≤AASince the set ∆ = {π ∈ ba + | kπk = π1 = 1} is weak* compact by the Alaoglu’s theorem (Fact 2.15 in Section 2.2), we have a subnet (πn(κ) , ξn(κ) (a)) with(πn(κ) , ξn(κ) (a)) → (π̂, ξ(a)) a.e. in the weak* topology, where π̂ ∈ ba + with π̂1 = 1.

1000

TAKASHI SUZUKI

1
Step
R 3: Deﬁne the set P =R {a ∈ A| π̂ω(a)R > 0}. Because πn ∈ ℓ for all n, we
have A πn(κ) ω(a)dλ = πn(κ) A ω(a)dλ → π̂ A ω(a)dλ > 0 by Assumption PE (ii)and π̂1 = 1; hence, we obtain λ(P ) > 0. The essence of the proof is contained in
the next Lemma.

Lemma 6.4. ξ(a) ≺a ζ implies that π̂ω(a) < π̂ζ a.e. on P .
Proof. If the lemma was false, there exists ζ(a) = (ζ t (a)) ∈ X such that π̂ζ(a) ≤π̂ω(a) and ξ(a) ≺a ζ(a) on a subset of P with λ-positive measure. Let Qn ={a ∈ A| a does not satisfy (Q-1n)} and R = {a ∈ A| ξ(a) ∈/ Ls(ξn (a))}. Set Q =
∪∞
Q
.
Since
Q
∪
R
is
of
measure
0,
P
\(Q
∪
R)
is
nonempty.
Let a ∈ P \(Q ∪ R).
n=1 n
We can assume without loss of generality that π̂ζ(a) < π̂ω(a) and ξ(a) ≺a ζ(a).
Let ζn (a) = (ζ 1 (a) . . . ζ n (a), 0, 0 . . . ) be the projection of ζ(a) to X n . Because
ζn (a) → ζ(a), we have for suﬃciently large n0 that π̂ζn0 (a) ≤ π̂ζ(a) < π̂ω(a) andξ(a) ≺a ζn0 (a).
Because (πn(κ) , ξn(κ) ) → (π̂, ξ(a)), there is a κ1 with n(κ1 ) ≡ n1 ≥ n0 such that0 ≤ πn1 ζn0 (a) < πn1 ω(a) = πn1 ωn1 (a), and ξn1 (a) ≺a ζn0 (a), or ξn1 (a) ≺na 1 ζn0 (a).
This contradicts the fact that (πn1 , ξn1 (a)) is a quasi-equilibrium for E n1 .
□By Lemma 6.4 and Assumption PR (ii) we obtain p  0; hence, it follows
from Assumption PE (i) that λ(P ) = 1. Let π̂ = π + πp be the Yosida–Hewitt
decomposition where π ∈ ℓ1+ is the countably additive part and πp is purely theﬁnitely additive part. Lemma 6.4 combined with Assumption PR (ii) imply thatπξ(a) ≥ πω(a) a.e. Then, it follows from the resource feasibility condition that wehave the budget conditions πξ(a) = πω(a) for almost all a ∈ A. The condition(E-1) follows
R immediately
Rfromt Lemma 6.4.
tBecause A x (a)dλ ≤ A e (a)dλ ≤ ω̂ < x̂ for each t, there exists a positive
amount of consumers with xt (a) < x̂. Then, by the monotonicity
PR
R (ii), one
Robtains that pt > 0 for all t. Because p  0, it follows that A ξ(a)dλ = A ω(a)dλ,or the condition (E-2) is met. This completes the proof of Theorem 5.2.
□
6.6. Proof of Theorem 5.3. The proof that a competitive equilibrium allocation
is a core allocation is standard. Therefore, it is skipped.
Step 1: We shall show C (E) ⊂ W (E). Let ϕ(·) ∈ C (E) and deﬁne P : A → ℓ∞by P (a) = {ξ ∈ ℓ∞ | ϕ(a) ≺a (ξ + ω(a))} and Ψ : A → ℓ∞ by Ψ(a) = P (a) ∪ {0},respectively. It follows from Assumption PR (ii) and unboundedness of the ﬁrst k
goods that P (a) 6= ∅, a.e.
Lemma 6.5. Graph(Ψ) ∈ B(A × ℓ∞ ).
Proof. By the measurability of P and the continuity of preferences, the set
G = {(≺, ξ, η) ∈ P × ℓ∞ × ℓ∞ | η ≺ ξ}
is a Borel set of P × ℓ∞ × ℓ∞ . Deﬁning a map ψ : A × ℓ∞ → P × ℓ∞ × ℓ∞ byψ(a, ξ) = (≺a , ξ +ω(a), ϕ(a)), it follows that Graph(P ) = ψ −1 (G). Because the mapψ is measurable by Remark 2.23 of Section 2.3, the graph of Ψ is measurable. □Because Ψ(a) isRweak* measurable by Lemma
6.5 and Remark 2.23, we canRdeﬁne the integral A Ψ(a)dλ. Because
R 0 ∈ A Ψ(a)dλ, it is nonempty and convex
by Fact 2.22. We now show that A Ψ(a)dλ ∩ ℓ∞
− = {0}. Suppose not. Then,INDIVIDUALIZED AND DISTRIBUTIONALIZED ECONOMIES

1001

R
there exists a function h : A → ℓ∞ such that h(a) ∈ Ψ(a) a.e., and A h(a)dλ < 0.
Set C = {a ∈ A| h(a) 6= 0}. Obviously, λ(C) > 0. Deﬁne an integrable function
g : A → ℓ∞ byR
h(a)dλg(a) = h(a) + ω(a) − A.
λ(C)We can write g(a) = (g̃(a), g(a)) = ((g̃ i (a)), (g t (a))) ∈ Rk × ℓ∞ , and let ĝ(a) =(ĝ t (a)) by ĝ t (a) = inf{g t (a), x̂}. Settingĝ(a) =R
R (g̃(a), ĝ(a)), it can be easily seenthat ϕ(a) ≺a ĝ(a) a.e. in C and C ĝ(a)dλ = C ω(a)dλ,R contradicting ϕ ∈ C (E).
We can apply Fact 2.16 to the two disjoint convex sets A Ψ(a)dλ and ℓ∞− \{0} andobtain a vector π̂ ∈ ba with π̂ > 0 such thatZ
0 ≤ π̂ζ whenever ζ ∈Ψ(a)dλ.
A

Step 2: We shall show thatπ̂η ≥ π̂ω(a) whenever ϕ(a) ≺a η a.e.
To do this, we ﬁrst verify π̂ϕ(a) = π̂ω(a) a.e. in A. Let C ⊂ A be a measurable
set with λ(C) > 0 and take an ϵ > 0 and d ∈ Rk+ \0. We deﬁne j : A → X by thefollowing(
ϕ(a) − ω(a) + ϵ(d, 0) for a ∈ C,
j(a) =
0for a ∈
/ C.
Then, it follows from Assumption PR (ii) that j(a) ∈ Ψ(a); hence,
Z

Z
π̂
ϕ(a) + ϵλ(C)(d, 0) −ω(a)dλ ≥ 0,
C

R

C

R

and
rearrangingR this, we obtain C π̂ϕ(a)dλ ≥ C π̂ω(a)dλ − ϵλ(C)π̂(d, 0); hence,R
ϵ > 0R is arbitrary. We thenC π̂ϕ(a)dλ ≥ C π̂ω(a)dλ for any C ⊂ A, because
R
conclude π̂ϕ(a) ≥ π̂ω(a) a.e. It follows from A ϕ(a)dλ ≤ A ω(a)dλ that π̂ϕ(a) =π̂ω(a) a.e., as desired. By
in the deﬁnition of j(a) by η with ϕ(a) ≺a η
R replacing ϕ(a)R
for a ∈ C, we obtain C π̂ηdλ ≥ C π̂ω(a)dλ. Because C ⊂ A is arbitrary, we
conclude π̂η ≥ π̂ω(a) for every η such that ϕ(a) ≺a η a.e. in A.
The remaining part of the proof proceeds in the same way as in the last part ofthe proof of Theorem 5.2. Let π̂ = πc + πp be the Yosida–Hewitt decomposition
where πc ≡ (p, q) ∈ Rk+ × ℓ1+ is the countably additive part and πp is purely ﬁnitely
additive part. The budget conditions πc ϕ(a) = πc ω(a) a.e. can be shown as usual.
The above condition and Assumption PR (ii) imply p  0, hence πc ω(a) > 0 a.e.
by Assumption PE. We then conclude ϕ(a) ≺a η implies πc η > πc ωa , a.e. This
completes the proof.
□
6.7. Proof of Theorem 5.7. Step 1: Let ϵn be a sequence of positive numbers
decreasing to zero. As in [19, 41], we can construct a sequence of ﬁnite subsets
K n = {tn1 . . . tnmn } of K and a sequence of pairwise disjoint open sets Bin with

1002

TAKASHI SUZUKInntni ∈ Bin for i = 1 . . . mn such that denoting B n = ∪mi=1 Bi ,dK (tni , t) ≤ ϵn for every t ∈ Bin and for all n, i = 1 . . . mn ,ZZnϖ(a)dλ(B ) =ϖ(a)dλ(K),
A
n

A

K ⊂K

n+1

for all n, and K n → K in the topology of closed convergence.

For each n, let Ln = LS(tn1 . . . tnmn ) ⊂ ca(K) be the linear space spanned by{δtn1 . . . δtnmn } and set X n = Rk × (Z∩Ln ). We then deﬁne P n = P ∩ (X n ×X n )and let rn be a map from P to P n deﬁned byrn (≿) =≿n =≿ ∩(X n × X n ).
Jones [19, Lemma 6] proved that the map rn (·) is continuous. It is obvious thatrn (≿) →≿ as n → +∞.
For each n, let Λn : P × X → P n × X n be a map deﬁned byΛn (≿, ϖ) = (≿n , ϖn ), ≿n = rn (≿), fn = f, fn =mnXf (Bin )δtni .

i=1

It is obvious from deﬁnition that ϖn (K) = ϖ(K). Because rn is continuous and
Bin are open, the map Λn is measurable. Set E n = Λn ◦ E : A → P n × X n ,n
E n (a) = (≿na , ϖn (a)).
R Then,E is an economy with ﬁnite number of commodities.
Because support A f (a)dλ = K, we have by constructionZϖn (a)dλ  0,Anwhere we have identiﬁed X n with Rk+m
. Then, we have the following+

Lemma 6.6. The economy E n has a competitive equilibrium, or there exists a pricenwith ρn 6= 0 and an allocation ϕn : A → X n that satisfy
vector ρn ∈ Rk+m+
(E-1n) ρR n ϕn (a) ≤ ρn ϖRn (a) and ϕn (a) ≿a ψ whenever ρn ψ ≤ ρn ϖn (a) a.e. in A,(E-2n) A ϕn (a)dλ = A ϖn (a)dλ.
R
Proof. Because A ϖn (a)dλ  0 and the preferences are monotone, the assumptions
for Theorem 2 in Hildenbrand [18, p. 151] are satisﬁed.
□nStep 2: We denote ρn = (qn , qn ) = ((qni ), (qn (t))) ∈ Rk+ ×Rm+ . Without loss ofgenerality, we can assume kρn k = max{qni , qn (t)| i = 1 . . . k, t ∈ K n } = 1 for everyn. In the next lemma, the assumption of bounded marginal rate of substitution US
plays an essential role.

Lemma 6.7. Let (ρn , ϕn ) be the equilibrium obtained by Lemma 6.6. Then, (K n , qn )are equicontinuous.
Proof. Suppose that (K n , qn ) are not equicontinuous. Then, we can assume that
there exist sequences (tn ), (sn ) such thatd(tn , sn ) → 0 and lim

qn (tn )n→∞ qn (sn )> 1.

INDIVIDUALIZED AND DISTRIBUTIONALIZED ECONOMIESFor tn with mn ({tn }) > 0, deﬁnemn(tn ,sn ) = mn − mn ({tn })δtn +BecauseR

A mn (a)dµ =Rqn (tn )qn (sn )1003


mn ({tn })δsn .

A en (a)dµ  0, we haveµ ({a ∈ A| mn (a)(tn ) > 0}) > 0.
If mn (a)(tn ) > 0, then by Assumption US, we have for n suﬃciently large that(un (a), mn (a)) ≺a (un (a), mn(tn ,sn ) (a)).
(t ,s )

This is a contradiction because qn mn n n (a) = qn mn (a).

□We can assume without loss of generality that qn → q ∈ Rk+ , and it follows
from Fact 2.13 in Section 2.1 that we can assume that (K n , qn ) → (K, q) for someq ∈ C(K). Let ρn = (qn , qn ) and ρ = (q, q). Clearly kρk = 1.
Lemma 6.8. Suppose that (K n , qn ) → (K, q) and ϖn → ϖ ∈ Ω with ρϖ > 0. Letnϕn (a) = (un (a), mn (a)) ∈ Rk+ ×Rm+ and suppose un (a) → u(a), and for each n,
ρn ϕn (a) ≤ ρn ϖn and ψ ≿na ϕn (a) whenever ρn ψ ≤ ρn ϖn . Then, if q(t∗ ) = 0 forsome t∗ ∈ K, then mn (a)(K) → +∞.
Proof. Suppose not. Then, by taking subsequence if necessary, we can assume thatmn (a) → m̂ for some m̂ ∈ Z. We now claim that ρϕ(a) ≤ ρϖ(a) and ϕ(a) ≿a ψ
whenever ρψ ≤ ρϖ(a), where ϕ(a) = (u(a), m̂) and ψ = (y, n). To see this, supposethat there exists ψ ∈ X with ρψ ≤ ρϖ(a) and ϕ(a) ≺a ψ. Because ρϖ(a) > 0 we
can assume without loss ofthat ρψ < ρϖ(a) and ϕ(a) ≺a ψ. Setting
P generalitynn )δ n , we have ψ → ψ; hence, ρ ψ < ρ ϖ (a)n(Bψn = (y, nn ), where nn = m
ti
n
n n
n n
i
i=1
and ϕn (a) ≺na ψn for n large enough, contradicting the assumption. This cannot be
the case, however, because q(m + δt∗ ) = qm and (u(a), m̂) ≺a (u(a), m̂ + δt∗ ). □Step 3: We now claim that there exists an ϵ > 0 such that qn (t) ≥ ϵ > 0 for all t ∈K n , n = 1, 2 . . . . If this is not the case, there exists a sequence {tn } ⊂ K n such thattn → t∗ for some t∗ ∈ K with qn (tn ) → 0. Obviously, this implies that q(t∗ ) = 0 as
well. Choose s ∈ K such that q(s) = 1/2 and take an open neighborhood U of s suchthat q(t) > 1/2 for t ∈ U . Let B = {a ∈ A| ϖ(a)(U ) > 0}. Clearly, B is measurable,
and by Assumption AE, λ(B) > 0. Because qn fn (a) → qf (a) > 0 a.e. on B by
Fact 2.12 in Section 2.2,
R we have ρn ϖnR(a) → ρϖ(a) > 0 a.e.
R on B, and it follows
from Lemma 6.8 that A fn (a)dλ(K) = A mn (a)dλ(K) ≥ B mn (a)dλ(K) → +∞.
However, this contradicts thatZZ
f (a)dλ(K) < +∞.
fn (a)dλ(K) →A

A

Because qn ≥ ϵ and kϖ(a)k ≤ ω̂, it follows that0 ≤ mn (a)(K) ≤ω̂
for all n large enough, a.e.
ϵ

1004

TAKASHI SUZUKI

Then, by applying Fact 2.21 in Section 2.3, we obtain that there exists a measurable
map m(a) such that
ZZ
m(a)dλ ∈ Ls
mn (a)dλ and m(a) ∈ Ls(xn (a)) a.e.
A

It follows from
lim

A

Z

n→∞ A

Z
un (a)dλ ≤ lim

n→∞ A

Z
fn (a)dλ = limn→∞ Af (a)dλ < +∞and Fact 2.20 that thereR exists a measurableRmap u : A → Rk+ such that u(a) ∈Ls(un (a)) a.e. in A and A u(a)dλ ≤ limn→∞ A un (a)dλ. Let ϕ(a) = (u(a), m(a)).
We show that
R (ρ, ϕ(·)) is an
R equilibrium for E.
Because A ϖn (a)dλ = A ϕn (a)dλ, ϕn (a) → ϕ(a) a.e. and ϖn (a) → ϖ(a) a.e., itfollows from Fact 2.19 in Section 2.3 thatZ
Z
Z
Z
ϖ(a)dλ = lim
ϖn (a)dλ = lim
ϕn (a)dλ =ϕ(a)dλ,
A

n

A

n

A

A

and one obtains from ρn → ρ and Fact 2.12 in Section 2.2 that
ρϕ(a) ≤ ρϖ(a) a.e.
Hence, the condition (E-2) and the budget conditions are met.
Suppose that there exists ψ = (y, n) ∈ X such that ρψ ≤ ρϖ(a) and ϕ(a) ≺a ψ.
If ρϖ(a) > 0, then because the preferences are continuous, we canPassume withoutnnnloss of generality that ρψ < ρϖ(a) and ϕ(a) ≺a ψ. Let ψn = (y, mi=1 n(Bi )δti ).
Because ρn ϖn (a) → ρϖ(a) and ρn ψn → ρψ, it follows from (E-1n) and (6.2) thatρn ψn < ρn ϖn (a) and ϕn (a) ≺a ψn for n large enough, a contradiction. If ρϖ(a) = 0,
we can easily show from Assumptions PR (ii) and AE that q  0. Then, because
q(t) ≥ ϵ for all t ∈ K, the budget set is a singleton, or {ϕ ∈ X| ρϕ ≤ 0} = {0};hence, ϕ(a) = 0 is trivially a maximal element in the budget set. Therefore, thecondition (E-1) is also met and we complete the proof.
□
6.8. Proof of Theorem 5.11. Step 1: In the following proof, we omit the prime of
P ′ and denote it simply as P. The proof that a competitive equilibrium allocationis also a core allocation is standard. Therefore, it is skipped. For the opposite
direction, let ϕ : A → X be a core allocation, and we need the following auxiliary
result.
Lemma 6.9. Deﬁne the upper contour set of ϕ(a) byΦϕ(a) = {γ ∈ X | ϕ(a) ≺a γ}.
Then, the correspondence a 7→ Φϕ(a) has a measurable graph.
Proof. By the measurability of P and the continuity of preferences, the set G ={(≿, ψ, γ) ∈ P×X×X | ψ ≺ γ} belongs to B(P×X×X). Deﬁne Γ : A×X →P×X×X by Γ(a, γ) = (≿a , ϕ(a), γ). Obviously, Γ is a measurable map by theremark 2.23; hence,
Graph (Φϕ(a) ) = Γ−1 (G) ∈ B(A×X).

□

INDIVIDUALIZED AND DISTRIBUTIONALIZED ECONOMIES

1005

We show that there exists a price vector ρ ∈ P such that (ρ, ϕ) is a competitive
equilibrium for E. Deﬁne the correspondence Ψ : A → 2X by
Ψ(a) = Φϕ(a) ∪ {ϖ(a)}.
Step 2: Let C = ∪α>0 α(τ −U ), where τ (6= 0) ∈ X and U are given in Assumption
CD. We show that
ZZ
Ψ(a)dλ − ϖ(a)dλ ∩ −C = ∅.
A

Because −C is weak* open, it Ris suﬃcient to
R show that for any Gelfand integrable
function ψ : A → X such that A ψ(a)dλ ∈ A Ψ(a)dλ, and there exists a sequencesof Gelfand integrable maps ψn : A → X and ϖn : A →R X such that Rψn (a) → ψ(a)
and
ϖ
(a)
→
ϖ(a)
a.e.
in
the
weak*
topology,hencen
A ψn (a)dλ → A ψ(a)dλ andRR
A ϖn (a)dλ → A ϖ(a)dλ andZZψn (a) −ϖn (a)dλ ∈
/ −C
A

A

for a large enough n. Let S = {a ∈ A|
If λ(S) = 0, then ψ(a) = ϖ(a)R ϕ(a) ≺a ψ(a)}.
R
a.e. by deﬁnition of Ψ(a); hence, A ψ(a)dλ − A ϖ(a)dλ ∈/ −C. Hence, we may
assume that λ(S) > 0.
For each n ∈ N, deﬁne Sn = {a ∈ S| kψ(a)k(= ψ(a)(K)) ≤ n}. Obviously,S1 ⊂ S2 ⊂ . . . are measurable and ∪∞n=1 Sn = S, λ(Sn ) → λ(S) as n → ∞. Hence,
we can take a measurable subset SN of S such that λ(SN ) is arbitrarily close to λ(S)and ψ(a) is norm bounded on SN . If we restrict our discussion on SN , then we canassume without loss of generality that {ψn (a)} are included in a bounded subset
X̂ = {ϕ ∈ X| kϕk ≤ N + 1}. On X̂, the weak* topology is metrizable by Alaoglu’s
theorem (Fact 2.15 in Section 2.1); hence, in the same way as [59, Lemma 7], wehave a sequence of simple functions ψ̂n : S → X with ψ̂n (a) → ψ(a) a.e. in Ŝn in
the weak* topology. Then, by Egorov’s Theorem (Fact 2.11 in Section 2.1), there
exists a measurable set S ′ ⊂ SN of positive measure with λ(S ′ ) arbitrarily close toλ(SN ) such that ψ̂n (a) → ψ(a) uniformly on S ′ . By the continuity of preferences,
for suﬃciently large n, we have ϕ(a) ≺a ψ̂n (a) a.e. in S ′ .
We can writemnXψ̂n (a) =ψ̂ni 1Sni (a),
i=1

where
ϕ(a) ≺a ψ̂ni for all a ∈ Sni and all i = 1 . . . mn ,inand 1Sni (·) is the indicator function of the set Sni with S ′ = ∪mi=1 Sn . Without
loss of generality, we maythat λ(Sn1 ) = · · · = λ(Snmn ) = λ̂. Let ϖn (a) = assume
R
Pmn
P
mn
i
−1
i .
i =
i ϖ(a)dλ 1Sni=1 λ̂
i=1 ϖn 1Sn
SnRR
We now show that S ′ ψ̂n (a) − S ′ ϖn (a)dλ ∈/ −C. Suppose not. Then, for some
α > 0 one has
mn
mn
X
X
ψ̂ni λ̂ −ϖni λ̂ ∈ −α(τ + U )
i=1

i=1

1006

TAKASHI SUZUKI

and thereforemnXψ̂ni + χ − υ =

i=1

mn
X

ϖni ,

i=1

P n i
where χ = (α/λ̂)τ and υ ∈ (α/λ̂)U . Because m
i=1 ϖn ≥ 0, it follows from thepresented equation thatmnXψ̂ni .
(χ − υ)− ≤i=1

Then, by the Riesz decomposition theorem (Fact 2.17 in Section 2.2), we can ﬁnd
P n i−
0 ≤ ψni ∈ Q with 0 ≤ ψni ≤ ψ̂ni , i = 1 . . . mn and mi=1 ψn = (χ − υ) . It follows
1mfrom
CD that there exist mn commodity bundles γn . . . γn n such that
Pmn Assumptioni+
i=1 γn = (χ − υ) andψ̂ni ≺a ψ̂ni + γni − ψni for all a ∈ Sni and all i = 1 . . . mn .
Set ψ̃ni = ψ̂ni + γni − ψni . We have ϕ(a) ≺a ψ̃ni from ϕ(a) ≺a ψ̂ni for all a ∈ Sni andP n i
all i and the transitivity of ≺a . Deﬁne ψ̃n (a) = mi (a). Then,i=1 ψ̃n 1Sn
Z
Z
mn
mn
X
X
ψ̃n (a)dλ =ψ̃ni λ̂ =ϖni λ̂ =ϖ(a)dλ.
S′

i=1

i=1

S′

We have found an allocation ψ̃n (a) that is feasible for the coalition S ′ and preferred to ϕ(a) a.e. in S ′ , which contradicts the assumption that ϕ(a) is a core
allocation.
Deﬁne ψn : A → X by(ψ̂n (a) for a ∈ S ′ ,ψn (a) =ψ(a) for a ∈/ S′.
Similarly, we deﬁne ϖn : A → X by(
ϖn (a) =ϖn (a) for a ∈ S ′ ,ϖ(a) for a ∈/ S′.

Because S ′ ⊂ RS and λ(SR′ ) is arbitrarily close to λ(S) and C is weak* open, we
conclude that A ψn (a) − A ϖn (a)dλ ∈/ −C; hence,
Z
Z
ϕ(a)dλ − ϖ(a)dλ ∩ −C = ∅.
A

R
Step 3: By Fact 2.22 in Section 2.3, A Ψ(a)dλ is convex. By the Hahn–Banachseparation theorem (Fact 2.16 in Section 2.2), there exists ρ ∈ P such thatZ
ϖ(a)dλρψ ≥ ρRA

for every ψ ∈ A Ψ(a)dλ. The monotonicity PR (ii) implies that ρ ≥ 0.
We will show that for a.e.
ψ a ϕ(a) implies that ρψ ≥ ρϖ(a).

INDIVIDUALIZED AND DISTRIBUTIONALIZED ECONOMIES

1007

To do this, ﬁrst we now show that ρϕ(a) = ρϖ(a) a.e. Let C ⊂ A, λ(C) > 0 and
ϵ > 0. Deﬁne ĵ : A → X by(R
ϕ(a) + ϵ A ϖ(a)dλ for a ∈ C,ĵ(a) =ϖ(a)for a ∈
/ C.
Then, we haveZZZ
ρϕ(a)dλ + ϵλ(C)ϖ(a)dλ +
C

A

!
ϖ(a)
A\C

Z
≥ρ

ϖ(a)dλ.
A

R
Given
that
ϵ
>
0
is
arbitrary,
werearrange
thisinequality
to
have
that
C
R
R ρϕ(a)dλ ≥ρϖ(a)dλ
for
any
C
⊂
A.
It
follows
that
ρϕ(a)
≥
ρϖ(a)
a.e.
BecauseA ϕ(a)dλ =RCA ϖ(a)dλ, we have ρϕ(a) =R ρϖ(a) a.e.R Replacing in the deﬁnition of ĵ(a) byψ ∈ Ψ(a) for a ∈ C, we have C ρψdλ ≥ C ρϖ(a)dλ. Because C ⊂ A is arbitrary,
we conclude thatψ a ϕ(a) implies that ρψ ≥ ρϖ(a) a.e.
The following argument is standard. For a ∈ A with ρϖ(a) > 0, we can showthat ρψ > ρϖ(a) whenever ϕ(a) ≺a ψ. By Assumption PR (ii), we can easily showthat ρ is strictly positive. Then, for a ∈ A such that ρϖ(a) = 0, the budget set
is a singleton {0}, meaning that ϕ(a) = 0 is utility maximizing under the budget
constraint; hence, condition (E-1) of Deﬁnition 5.8 holds. This shows that (ρ, ϕ) is
a competitive equilibrium.
□
Acknowledgements
This work has been accomplished under collaboration with Professors M. Ali
Khan and Nobusumi Sagara. The author expresses his sincere gratitude to themand his regret that they did not think it enough to accept coauthorship. He also
thanks to an anonymous referee of the journal for spotting errors and ambiguities.
Of course, he is solely responsible for remaining errors and shortcomings.
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Manuscript received December 30 2020
revised September 5 2020

Takashi Suzuki
Department of Economics, Meiji-Gakuin University, 1-2-37 Shirokanedai, Minato-ku, Tokyo 108–8636, Japan
E-mail address: takashisuz@jcom.home.ne.jp