Discussion Papers In Economics
And Business

Replica Core Equivalence Theorem: An Extension of
the Debreu-Scarf Limit Theorem to Double Infinity
Monetary Economies
Ken Urai
Hiromi Murakami
Discussion Paper 14-35-Rev.2

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANReplica Core Equivalence Theorem: An Extension of
the Debreu-Scarf Limit Theorem to Double Infinity
Monetary Economies
Ken Urai
Hiromi Murakami
Discussion Paper 14-35-Rev.2

May 2015

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANReplica Core Equivalence Theorem: An Extension of
the Debreu-Scarf Limit Theorem to Double Infinity
Monetary Economies∗
Ken Urai†
Hiromi Murakami‡

Abstract
An overlapping generations model with the double infinity of commodities and agents is the mostfundamental framework to introduce outside money into a static economic model. In this model,
competitive equilibria may not necessarily be Pareto-optimal. Although Samuelson (1958) emphasized
the role of fiat money as a certain kind of social contract, we cannot characterize it as a cooperativegame-theoretic solution like a core. In this paper, we obtained a finite replica core characterization ofWalrasian equilibrium allocations under non-negative wealth transfer and a core-limit characterizationof Samuelson’s social contrivance of money. Preferences are not necessarily assumed to be ordered.
Keywords: Monetary Equilibrium, Overlapping Generations Model, Core Equivalence, Replica Economy, Non-Ordered Preference
JEL Classification: C62, C71, D51, E00∗ We express special gratitude to professor Takashi Hayashi (Glasgow University) for his important comments and advice.
We are also grateful for the comments by professors Atsushi Kajii (Kyoto University), Tomoyuki Kamo (Kyoto Sangyo
University), and Tomoki Inoue (Meiji University) as well as the participants at a poster session at Seinan Gakuin University
on October 11, 2014 at the autumn meeting of the Japanese Economic Association. We also appreciated several commentsfrom an anonymous referee of the Journal of Economic Theory and many valuable suggestions from the chief editor of the
Journal of Mathematical Economics. Part of this research was supported by a Grant-in-Aid for Scientific Research from theJapan Society for the Promotion of Science in 2013 (No.25380227) and the Osaka University Cross-Boundary Innovation
Program.
† Graduate School of Economics, Osaka University, E-mail: urai@econ.osaka-u.ac.jp‡ Graduate School of Economics, Osaka University, E-mail: murakami@math.econ.osaka-u.ac.jp

1

Introduction

In this paper, we generalize the Debreu-Scarf core limit theorem to a case with a double infinity economy
that includes such typical examples as Samuelson consumption-loan models with money.
For standard finite general equilibrium settings, the Walrasian equilibrium is Pareto-optimal (the first
fundamental theorem of welfare economics), and every Pareto-optimal allocation is an equilibrium allocation relative to a price-wealth system (the second fundamental theorem of welfare economics). If westrengthen the concept of Pareto-optimal allocation to a replica core allocation (an allocation whose replica
is a core allocation for each replica economy), we obtain an equivalence result where every replica core
allocation is a competitive allocation and every competitive allocation is a replica core allocation (Debreu
and Scarf 1963, Theorems 1 and 3). This equivalence theorem between the replica core and the competitive
equilibria is commonly known as the Debreu-Scarf core limit theorem.
In a double infinity economy, competitive equilibrium (with or without money) is not necessarily Paretooptimal (Samuelson 1958), but it is known to be weakly Pareto-optimal (Esteban 1986 and Balasko andShell 1980). It is also known that every weakly Pareto-optimal allocation is an equilibrium allocation relative to a price-wealth system (Balasko and Shell 1980). Chae (1987), Aliprantis and Burkinshaw (1990)
and Chae and Esteban (1993) treat the core equivalence problem for Walrasian equilibrium allocations
in overlapping generations models. Their approaches, however, fail to treat competitive equilibrium allocations with money.1 Of course, an equilibrium with money (non-negative wealth transfers from thegovernment) is one critical issue that the overlapping generations model tries to describe.
In this paper, we show that if we strengthen the concept of weak Pareto-optimal allocation to replica
finite core allocation (an allocation whose replica is a certain kind of finite core allocation for each replica
economy 2 ), we obtain an equivalence result analogous to Aliprantis and Burkinshaw (1990) where a replica
finite core allocation is a competitive allocation with non-negative wealth transfers and every competitive
allocation with non-negative wealth transfers is a replica finite core allocation.3
Our replica core equivalence approach (as well as that of Aliprantis and Burkinshaw 1990) has threeimportant advantages: by concentrating on the equivalence argument without using the equal treatment
property, (i) we can show a weak core theoretic equivalence result merely based on weak optimality
conditions, (ii) we obtain a limit theorem of the core instead of a theorem in the limit measure space like
Chae and Esteban, and (iii) we can allow for an argument based on non-ordered preferences, and hence
our result may also be considered a non-ordered extension of the Debreu-Scarf core equivalence theorem.4

1 Their concepts, such as the short-term core (Aliprantis and Burkinshaw 1990) and the short-run core (Chae 1987),
exclude equilibrium allocations with non-zero fiat money in two-period overlapping generations economies. For the shortterm core argument, as a simple one-good per period economy example pointed out in Esteban (1986), such a monetary
equilibrium is blocked by a coalition of all agents after a certain period without changing all but the first finite members’
allocations. In the short-run core, we can also easily construct an example under which a typical Samuelson-type monetary
equilibrium allocation is always blocked by the t-generation for each t-economy for all t = 1, 2, · · ·.
2 More precisely, this is an allocation, x, whose replica is a finite core allocation for each replica economy even when theendowments of some members are replaced by the allocation, x, itself.
3 In this paper, we use “wealth transfer” instead of “monetary transfer” because these two concepts are diﬀerent unless
we use the perfect-foresight assumption on the expectation for dynamics. In the sense of Esteban and Millán (1990), weconcentrate on the set of all monetary equilibrium allocations and competitive equilibrium allocations without money.
4 Aliprantis and Burkinshaw (1990), however, do not successfully treat non-ordered preference cases. In our model, astrong sense of the local non-satiation in (E.3) plays an essential role in the proof of Theorem 1.

1

2

The Model

Let N be the set of all positive integers and R be the set of real numbers. A pure exchange overlapping
generations economy, or more simply, an economy, E, is comprised of the following list:∪∞(E.1) {It }∞t=1 : a countable family of mutually disjoint finite subsets of N such that
t=1 It ⊂ N ,and if It = ∅, It+1 = ∅ for each t ∈ N . It is the index set of agents in generation t.
(E.2) {Kt }∞t=1 : a countable family of non-empty finite intervals, Kt = {k(t), k(t) + 1, · · · , k(t) + ℓ(t)}∪∞where k(t) and ℓ(t) are elements of N such that t=1 Kt = N , k(t) < k(t + 1) ≦ k(t) + ℓ(t) for allt ∈ N , and {t | n ∈ Kt } is finite for each n ∈ N . Kt is the index set of commodities available togeneration t.
(E.3) {(≿i , ω i )}i∈∪t∈N It : countably many agents, where ≿i is a reflexive binary relation on comt
modity space for each generation RK+ , representing a preference of i ∈ It . We write xi ∼i yi iﬀ
xi ≿i yi and yi ≿ xi , and xi ≻i yi iﬀ xi ≿i yi and xi ̸∼ yi . Strict preference ≻i is continuous (having
Ktt
an open graph in RK+ × R+ ), strictly monotonic (xi ≧ yi and xi ̸= yi implies xi ≻i yi ), and has aconvex better set ({yi | yi ≻i xi } is convex) at every xi such that ωi ⊁i xi . The closure of the graphof ≻i in RKt × RKt is the graph of ≿i (a strong sense of local non-satiation). The initial endowmenttof i, ω i , is an element of RK++ = {x | x : Kt → R++ } for each i ∈ It .
Nt
It is convenient to identify the commodity space for each generation RK+ with a subset of R , which ist
the set of all functions from N to R, by considering x ∈ RK
+ a function that takes value 0 on N \ Kt .
Kt
Then we can define the total commodity space for economy ⊕∞t=1 R+ as the set of all finite sums amongKt
the points in the commodity spaces of the generations. Clearly, ⊕∞t=1 R+ can be identified with a subset
of direct sum R∞ , the set of all finite real sequences, which is a subspace of the set of all real sequences,

R∞ ≈ RN with pointwise convergence topology.
i∞∪Given an economy, E = ({It }∞t=1 , {Kt }t=1 , {(≿i , ω )}i∈ t∈N It ), the price space for E, P(E), is defined∞(withas the set of all p in RN+ such that under the duality between R∞ (with relative topology) and R
pointwise convergence topology), p positively evaluates all the agents’ initial endowments:(1)i
P(E) = {p ∈ RN
+ | p · ω > 0 for all i ∈ It , for all t ∈ N }.

Nt
Since for all i ∈ It , ω i belongs to RK++ for all t ∈ N , the price space of E always includes R++ for all E
in Econ, where Econ denotes the set of all economies satisfying conditions (E.1), (E.2) and (E.3).

For each E = ({It }, {Kt }, {(≿i , ω i )}) ∈ Econ, sequence (xi ∈ RKt )i∈∪t∈N It is called an allocation for
E. Allocation (xi ∈ RKt )i∈∪t∈N It is said to be feasible if
(2)

∑∑

xi ≦

t∈N i∈It∑∑ωi ,t∈N i∈It

where the summability in RN of both sides of the inequality is assured by (E.2). The list of price∪∞vector p∗ ∈ P(E), non-negative wealth transfer function ME∗ : N = t=1 It → R+ , and feasible allocation(xi∗ ∈ RKt )i∈∪t∈N It , is called a non-negative wealth transfer Walrasian equilibrium for E, if for each t ∈ N
and i ∈ It , xi∗ is a ≻i -maximal element in set {xi ∈ RKt | p∗ · xi ≦ p∗ · ω i + ME∗ (i)}. Since the non-negative
wealth transfer is an abstraction of the money supply in perfect-foresight overlapping generations settings,
we denote the set of all non-negative wealth transfer Walrasian equilibrium allocations by MWalras(E).
∪∞A coalition in economy E = ({It }, {Kt }, {(≿i , ω i )}) ∈ Econ is a set of consumers S ⊂ t=1 It .
Allocation x for economy E is said to be blocked by coalition S if it is possible to find commodity bundles2

x̂i for all i ∈ S such that∑i∈S (x̂i− ω i ) = 0 and x̂i ≿i xi for all i ∈ S, and x̂i ≻i xi for at least one i ∈ S.

For each E = ({It }, {Kt }, {(≿i , ω i )}) ∈ Econ, the set of all feasible allocations that cannot be blocked
by any coalition is said to be the core of economy E and is denoted by Core(E). Element x ∈ Core(E) iscalled a core allocation. The set of all feasible allocations that cannot be blocked by any finite coalition iscalled the finite core of economy E and is denoted by Fcore(E). Element x ∈ Fcore(E) is called a finite
core allocation for E.

3

Replica Core Equivalence Theorem

For each feasible allocation x = (xi ∈ RKt )i∈∪t∈N It for E = ({It }, {Kt }, {(≿i , ω i )}) ∈ Econ, we denoteby E(x) an economy where initial endowment allocation ω = (ω i ) is replaced by x = (xi ).5 Hence, wehave E = E(ω).
Consider the following replica economy,
Em (x) ⊕ En (ω),(3)

which consists of all the members of the m-fold replica economy of E(x) and the n-fold replica economyof E(ω) for each m ∈ N and n ∈ N . Let us denote by Cm⊕n (E) the set of allocations x for E such thatthe (m + n)-fold replica allocation of x belongs to Fcore(Em (x) ⊕ En (ω)).6 Moreover, let us denote byCn (E) the set of allocations x for E such that the n-fold replica allocation of x belongs to Core(En ). It
is easy to check that if x is a feasible allocation of E = E(ω) such that (m + n)-fold replica allocationof x does not belong to Fcore(Em (x) ⊕ En (ω)), the replica allocation does not belong to Core(Em+n ).7
Therefore, we can write Cm⊕n (E) ⊃ Cm+n (E) for each m ∈ N and n ∈ N . It is also easy to check that′′Cm⊕n (E) ⊃ Cm ⊕n (E) ⊃ · · · where m′ ≧ m, n′ ≧ n.8 For finite economy E, the Debreu-Scarf limit theorem∩∞ ∩∞∩∞can be restated as m+n=2 Cm+n (E) = Walras(E). We see below (Theorem 1), n=1 m=1 Cm⊕n (E) =MWalras(E). Hence, the restriction of Theorem 1 to the case with finite economy E provides the following
extension of the replica core version of the Debreu-Scarf limit theorem because there is no diﬀerence inour settings between Walras(E) and MWalras(E).
For finite economy E, feasible allocation x for E is a competitive equilibrium allocation iﬀ its(m + n)-fold replica allocation belongs to Fcore(Em (x) ⊕ En (ω)) for every suﬃciently large∩∞ ∩∞m ∈ N and n ∈ N . That is, Walras(E) = n=1 m=1 Cm⊕n (E).
As above, concept Fcore gives a unified replica core equivalence characterization for all non-negative
wealth transfer Walrasian equilibrium allocations. Note that allocation x such that x ∈ Fcore(E(x))
is the weak Pareto-optimal allocations in Balasko and Shell (1980). It is easy to check that the n-fold
replica allocation of x∗ ∈ MWalras(E) belongs to Fcore(En (x∗ )) for all n ∈ N and Fcore(En (ω)) for all

5 In the following, we sometimes omit the subscript i ∈∪t∈N It of an allocation for an economy as long as there is no riskof confusion.
6 The (m + n)-fold replica allocation of x is the allocation for Em (x) ⊕ En (ω) such that for each replica agent i in E(x) orE(ω), we assign the same allocation under x in economy E.
7 Clearly, the replica allocation, xm+n , is feasible for Em (x) ⊕ En (ω) and Em+n = Em+n (ω). If xm+n ∈/ Fcore(Em (x) ⊕En (ω)), then there exists a finite coalition S in Em (x) ⊕ En (ω) that blocks allocation xm+n . We can write S = S1 ∪ S2 ,where S1 (resp. S2 ) consists of members in Em (x) (resp. En (ω)). Then, coalition S ∗ consisting of all members of Em (x) andS2 also blocks xm+n ∈ Em (x) ⊕ En (ω). Therefore xm+n ∈/ Core(Em+n ).
8 Note that the equal treatment property is not necessary for ensuring the above inclusion relations. Here we are following
the replica core equivalence approach in Aliprantis and Burkinshaw (1990).

3

n ∈ N .9

But we have an example where the n-fold replica of allocation x, which is not an element ofMWalras(E), belongs to Fcore(En (x)) ∩ Fcore(En (ω)) for all n ∈ N .10
Theorem 1: Feasible allocation x for E is a non-negative wealth transfer Walrasian equilibrium allocation
iﬀ its (m + n)-fold replica allocation belongs to Fcore(Em (x) ⊕ En (ω)) for every m ∈ N and n ∈ N . That∩∞ ∩∞is, MWalras(E) = n=1 m=1 Cm⊕n (E).
Proof : [Suﬃciency] Let x∗ = (xi∗ ) be an element of MWalras(E) under price p∗ and non-negative
wealth transfer function ME∗ . Assume that S = S1 ∪ S2 is a finite coalition of Em (x∗ ) ⊕ En (ω) forsome m and n in N blocking the (m + n)-fold replica allocation of x∗ = (xi∗ ), where S1 is a coalition
in Em (x∗ ) and S2 is a coalition in En (ω). Then, under (E.3), especially by using the strong sense of
∑
∑
∑
i
i
i
local non-satiation, an allocation (xi )i∈S for S exists such thati∈S x =i∈S1 x∗ +i∈S2 ω andxi ≻i xi∗ for all i ∈ S. Then we have p∗ · xi > p∗ · xi∗ for all i ∈ S. This implies, however, that
∑
∑
∑
∑
∑
∑
p∗ · ( i∈S1 xi + i∈S2 xi ) > p∗ · i∈S1 xi∗ + p∗ · i∈S2 xi∗ ≧ p∗ · ( i∈S1 xi∗ + i∈S2 ω i ), a contradiction to
∑
∑
∑
i
i
i
i∈S x =i∈S1 x∗ +i∈S2 ω .

i∞∪[Necessity] Let x̄ = (x̄i ) be an allocation for E = ({It }∞t=1 , {Kt }t=1 , (≿i , ω )i∈ t∈N It ) such that every(m + n)-fold replica allocation of x̄ belongs to Fcore(Em (x̄) ⊕ En (ω)) for all m and n in N . In this∪tproof, we denote by I(t) the set of all agents in generations from 1 to t, s=1 Is , and by K(t) the set∪tof all commodities that are available for agents in I(t), s=1 Ks . Define for each i ∈ It , t ∈ N , Γi as
Γi = {β i z1i +(1−β i )z2i | ω i +z1i ≻i x̄i , x̄i +z2i ≻i x̄i , 0 ≦ β i ≦ 1} ⊂ RKt . Then, take the convex hull Γ(t) of∪t∪finite union i∈I(t) Γi ⊂ s=1 RKs ≈ RK(t) for each t ∈ N . Since, for every i, Γi is non-empty and convex,∑
non-empty convex set Γ(t) consists of all vectors z that can be written as i∈I(t) αi (β i z1i + (1 − β i )z2i ),∑with αi ≧ 0, i∈I(t) αi = 1, where z1i + ω i ≻i x̄i and z2i + x̄i ≻i x̄i for each i.
We will show in the similar way as in the proof of Debreu and Scarf (1963, Theorem 3) that Γ(t) does
not have 0 as its element for each t ∈ N . Let us suppose that 0 belongs to Γ(t). Then, one can write
∑
∑
i
i
i
i
i
i
i
i
i i
i
i i
i∈I(t) α = 1, and z1 + ω ≻i x̄ and z2 + x̄ ≻i x̄ for eachi∈I(t) α (β z1 + (1 − β )z2 ) = 0, with α ≧ 0,iii i
i. For each k suﬃciently large, let a1k and a2k be the smallest integers greater than kα β and kαi (1 − β i )
i

i

β i
i
respectively. Also, let I be the set of i ∈ I(t) for which αi > 0. For each i ∈ I, we define z1k
as kα
z1
ai
i

1k

i

) i
i
i
i
and z2k
as kα a(1−β+ ω i belongs to the segment [ω i , z1i + ω i ], z2k+ x̄i belongs toz2 . Observe that z1k
i2k

the segment [x̄i , z2i + x̄i ].
I
I1(αi > 0, 0 ≦ β i ≦ 1)

(β i ̸= 0)
βi = 01 − βi = 0

I2(1 − β i ̸= 0)

Figure 1: The union of I1 and I2 is equals to I.
Let I1 be the set of i ∈ I such that β i ̸= 0, and I2 be the set of i ∈ I such that 1 − β i ̸= 0. Note thatii
I1 ∪ I2 = I (see Figure 1). For i ∈ I1 , z1k+ ω i tends to z1i + ω i , and for i ∈ I2 , z2k+ x̄i tends to z2i + x̄i
9 To see this, in the proof of Theorem 1, [Suﬃciency], let S or S be an empty set.
1
2
10 See Appendix.

4

i
as k tends to infinity. The continuity assumption on preferences implies that z1k+ ω i ≻i x̄i for all i ∈ I1
iand z2k+ x̄i ≻i x̄i for all i ∈ I2 for all k suﬃciently large. Select one of such k. Then we have

(4)

0 =

k

∑

αi (β i z1i + (1 − β i )z2i ) =

i∈I

∑

=

kαi β i z1i +

=

i∈I

∑

i∈I

kα (β i z1i + (1 − β i )z2i ) +ii∈I1 ∩I2

i∈I1 \I2

∑

∑
∑
i
i
(ai1k z1k
+ ai2k z2k)=
kαi (β i z1i + (1 − β i )z2i )i
ai1k z1k+∑ii(ai1k z1k
+ ai2k z2k)+i∈I1 ∩I2

i∈I1 \I2∑kαi (1 − β i )z2i

i∈I2 \I1∑iai2k z2ki∈I2 \I1()
Let us consider the (maxi∈I ai1k ) + (maxi∈I ai2k ) -fold replica economy of E. Take the coalition composedi, and ai2k replica membersof ai1k replica members of i for each i ∈ I1 to each one of whom we assign ω i +z1ki2
of i for each i ∈ I2 to each one of whom we assign x̄i2 + z2k. This coalition blocks the allocation (x̄i ) asiiequation (4) and the fact that z1k+ ω i ≻i x̄i for all i ∈ I1 and z2k+ x̄i ≻i x̄i for all i ∈ I2 show. Thisis a contradiction to the definition of Fcore(Em (x̄) ⊕ En (ω)). Hence, we have established that 0 does notbelong to the convex set Γ(t) for each t ∈ N .
Let π(t) ⊂ R∞ be the set of prices such that p · z ≧ 0 for all z ∈ Γ(t) ⊂ RK(t) ⊂ R∞ , which is non-empty
by the separating hyperplane theorem. π(t) is closed in RK(t) × R × · · · = R∞ . Moreover, under theK(t)resource-related structure assured by (E.2) and ω i ≫ 0 for all i, π(t) is a subset of R++ × R × · · · ⊂∩R∞ .11 Next, we will obtain p∗ ∈ t∈N π(t). From the definition of each Γ(t) ⊂ RK(t) , we haveΓ(1) ⊂ Γ(2) ⊂ Γ(3) ⊂ · · · in R∞ . Hence, we have π(1) ⊃ π(2) ⊃ π(3) ⊃ · · · in R∞ . Thus we see∩t
s=1 π(s) = π(t). For finite economy, we have π(t) = π(t + 1) = · · · for all suﬃciently large t, hence theresult is obvious. If the number of agents is infinite, for each t ∈ N , choose price p(t) = (p1 (t), p2 (t), · · ·)K(t)in π(t) ⊂ R++ × R × · · · ⊂ R∞ (see Figure 2).

p1 (1)
p1 (2)
p2 (1) · · ·RK(1)p2 (2)

p3 (2)

p1 (s − 1) p2 (s − 1) · · ·
p1 (s)|
p4 (2) · · ·
RK(2)



p(1)
s

π(1)



q
q
q

q
q
q



RK(s−1)p2 (s) · · ·π(s − 1)

⇓
p∗1

}|

p(s − 1)s π(s)
RK(s)

{z

p(2)
s

π(2)



{z

⇓

}

|

{z

⇓

p∗2

}



p(s)
s
q
q
q

p∗s

Figure 2: How to construct the limit price p∗.

11 Indeed, π(t) is the set of supporting price vectors for the better set at x̄i under the strictly monotonic preference foreach i ∈ I(t), where every x̄i is necessarily evaluated positively (at least as great as the value of ω i ).

5Let us define for each t ∈ N , compact set ∆K(t) = {qt | qt ∈ RK(t) , ∥qt ∥ = 1}. Moreover, for each s, t ∈ N ,prps ≦ t, define mapping hst : π(t) → ∆K(s) as hst (p) = ∥ prK(s) p∥ for each p ∈ π(t), where pr denotes theK(s)projection.12 Since ∆K(1) is compact, if t → ∞, h1t (p(t)) converges to a limit, p̂∗1 ∈ ∆K(1) ∩ h11 (π(1)).
Then we take a subsequence, {p(t)}t∈N(1) , of {p(t)}t∈N , where N (1) is a cofinal subset of N , such thath1t (p(t)) converges to p̂∗1 . Define p∗1 ∈ RK(1) as p∗1 = p̂∗1 . Next, since ∆K(2) is compact, when t → ∞, bytaking a subsequence, {p(t)}t∈N(2) , of {p(t)}t∈N(1) , where N (2) is a cofinal subset of N (1) constructedby elements greater than or equal to 2, h2t (p(t)) also has a limit p̂∗2 ∈ ∆K(2) ∩ h22 (π(2)). We definep∗2 ∈ RK(2)\K(1) as ∥ pr 1 p̂∗ ∥ prK(2)\K(1) p̂∗2 . Generally, from the compactness of ∆K(s) for each s ∈ N , if
K(1) 2t → ∞, by taking a subsequence, {p(t)}t∈N(s) , of {p(t)}t∈N(s−1) , where N (s) is a cofinal subset of N (s−1)
constructed by elements greater than equal to s, hst (p(t)) has a limit p̂∗s ∈ ∆K(s) ∩ hss (π(s)). Hence we
can define p∗s ∈ RK(s)\K(s−1) as p∗s = ∥ pr 1 p̂∗ ∥ prK(s)\K(s−1) p̂∗s . By repeating the above procedure, we
K(1)s(p∗ ,···,p∗ )obtain p∗ = (p∗1 , p∗2 , · · ·). Since for each s ∈ N , ∥(p∗1 ,···,ps∗ )∥ = p̂∗s is an element of hss (π(s)), p∗ belongs to
s
1
∩
∗
∗
h−1
ss (p̂s ) = π(s), so we have p ∈t∈N π(t).
Since xi ≻i x̄i means that both xi −ω i and xi − x̄i belong to Γi , we have p∗ ·xi ≧ p∗ ·ω i and p∗ ·xi ≧ p∗ · x̄i .
By taking xi arbitrarily near to x̄i (from the local non-satiation property), we can see that p∗ · x̄i ≧ p∗ · ω i .
Define ME∗ (i) ≧ 0 as ME∗ (i) = p∗ · x̄i − p∗ · ω i for all i. Then, we have p∗ · x̄i = p∗ · ω i + ME∗ . In addition,
the condition of initial endowments ω i ≫ 0 for all i, implies that p∗ · ω i > 0. Since xi ≻i x̄i means thatp∗ · xi ≧ p∗ · x̄i , the continuity of preference together with p∗ · ω i + ME∗ (i) > 0 implies that for every i, x̄i
is an individual maxima under p∗ and ME∗ .
■′′As we mentioned before, Cm⊕n (E) ⊃ Cm ⊕n (E) ⊃ · · ·, where m′ ≧ m, n′ ≧ n, and MWalras(E) =
∩∞ ∩∞
m⊕n
(E). Thus we obtained a replica finite-core limit equivalence theorem to the non-negative
n=1
m=1 C
wealth transfer Walrasian equilibrium allocations, especially, the perfect-foresight monetary Walrasianequilibrium allocations for overlapping generation economies. Note that in the above proof, we do notassume preferences ≻i s to be ordered. Our equivalence theorem can also be utilized to axiomaticallycharacterize the price-wealth message mechanisms as Sonnenschein (1974), where the Debreu-Scarf limit
theorem plays an essential role in showing the category theoretic main result (see, Urai and Murakami2015).

REFERENCES
Aliprantis, C. D. and Burkinshaw, O. (1990): “An overlapping generations model core equivalence theorem,” Journal of Economic Theory 50, 362–380.
Balasko, Y. and Shell, K. (1980): “The overlapping-generations model. I. The case of pure exchange
without money,” Journal of Economic Theory 23, 281–306.
Bourbaki, N. (1966): General Topology. Hermann, Paris. English Translation: Springer-Verlag, 1995.
Chae, S. (1987): “Short run core equivalence in an overlapping generations model,” Journal of Economic
Theory 43, 170–183.
12 Let ∆K(t) ∩ h (π(t)) be X for each t ∈ N . Then the family (X )tttt t∈N and the family of mappings (hst ), s ≦ t,s, t ∈ N form an inverse system of compact spaces. By Bourbaki (1966, Chapter I, §9, no.6, Proposition 8), the inverse limit
X = ←−
lim Xt is non-empty. And price p∗ that we seek in the following can be identified as an element of X.

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Chae, S. and Esteban, J. (1993): “Core equivalence in an overlapping generations model,” Journal of
Economic Theory 59, 417–425.
Debreu, G. and Scarf, H. (1963): “A limit theorem on the core of an economy,” International Economic
Review 4, 235–246. Reprinted in G. Debreu, Mathematical Economics, pp. 151-162. Cambridge
University Press, Cambridge, 1983.
Esteban, J. (1986): “A characterization of core in overlapping-generations economies,” Journal of Economic Theory 39, 439–456.
Esteban, J. and Millán, T. (1990): “Competitive equilibria and the core of overlapping generations
economies,” Journal of Economic Theory 50, 155–174.
Samuelson, P. A. (1958): “An exact consumption loans model of interest with or without social contrivanceof money,” Journal of Political Economy 66(6), 467–482.
Sonnenschein, H. (1974): “An axiomatic characterization of the price mechanism,” Econometrica 42(3),425–433.

Appendix
Let us consider overlapping generations economy E such that every agent lives for young and old two
periods, where there is one consumption good for each period, and every t generation consists of two
agents, it and i′t , for t = 1, 2, · · ·. Each agent has initial endowment (2 + 2ϵ , 2ϵ ), where ϵ > 0 will be definedin the following as suﬃciently small.
′′
Consider allocation x = (xi1 , xi1 , xi2 , xi2 , · · ·) such that xit = (0.1 + ϵ, 0.1 + ϵ) for all t = 1, 2, · · ·,′′xi1 = (3.9, 0.9) and xit = (2.9, 0.9) for all t = 2, 3, · · ·. Clearly, x is feasible. Assuming that all the agents’marginal rate of substitution at x is 1 (see Figure 3 for generations t ≧ 2), we have p = (1, 1, · · ·) as the
supporting price for allocation x, which means that the replica allocation of x is weakly Pareto-optimal
for all replica economies: the n-fold replica allocation of x is an element of Fcore(En (x)) for all n ∈ N .
Assume further that all agents’ preferences are of the Cobb-Douglas type; their utility becomes arbitrarilya small level when their consumption at one of their life time periods is near to 0. Then we can check thatthe replica allocation of x is a finite core allocation for all replica economies: the n-fold replica allocationof x is an element of Fcore(En (ω)) for all n ∈ N .

Old

i3

0.1

↗
s

0

0.1

i3′

Old

p
s

1

(1,0.1)
s
2

0.9
0.8
Young

0

↗
s
s
2

2.9

p
(3.9,0.8)s
3.9

Figure 3: MRS for each agent is 1. Parameter ϵ > 0 for each allocation is neglected.

7

Young

′

For example, let uit (y1 , y2 ) = y10.5 y20.5 for each t = 1, 2, · · ·, ui1 (y1 , y2 ) = (y1 − 3)0.5 y20.5 when y1 ≧ 3,

i′1

′

i′t

u (y1 , y2 ) = y1 − 3 when 0 ≦ y1 ≦ 3, u (y1 , y2 ) = (y1 − 2)0.5 y20.5 when y1 ≧ 2 and uit (y1 , y2 ) = y1 − 2when 0 ≦ y1 ≦ 2 for each t = 2, 3, · · ·. Then, no finite coalition in En (ω) can improve upon the n-fold
replica allocation of x. Any finite coalition among members in the first generation fails to improve upon
(0.1 + ϵ, 0.1 + ϵ) as long as ϵ < 10−3 . Indeed, utility level of at least one of such coalition members, i∗ ,∗should be less than or equal to ui (2 + 2ϵ , 2ϵ ) = (2 + 2ϵ )0.5 2ϵ 0.5 under the maximality for utility allocation

with Cobb-Douglas type utility functions of homogeneity of degree 1. When ϵ < 10−3 , such coalition fails
to block any utility allocations greater than or equals to those under (0.1 + ϵ, 0.1 + ϵ), hence never improveupon those under the n-times replica allocation of x. Suppose that the n-times replica allocation of x
cannot be improved upon by any finite coalition among members of generations from 1 to k − 1. We show
in the following that any finite coalition, S, among members from 1 to k also fails to block the n-times
replica allocation of x. Let us denote S by S1 ∪ S2 , where S1 is the set of members in generations from 1 tok − 1, and S2 is the set of members in generation k. Note that between S1 and S2 , we have only to consider
two cases that there is a non-negative transfer of endowment commodity in period k from S2 to S1 or that
there is a positive transfer of it from S1 to S2 . For the first case, under the same discussion in the previous
paragraph, it is impossible to make utility levels of members of S2 greater than or equal to those under(0.1 + ϵ, 0.1 + ϵ). For the second case, it would be possible to keep all utility levels of members of S1 as
good as those under x, but if so, by not doing such a positive endowment transfer, S1 can improve upon
the replica allocation of x, which contradicts the assumption. It follows that, by mathematical induction,
the n-fold replica allocation of x is an element of Fcore(En (ω)).
Allocation x is not a non-negative wealth transfer Walrasian equilibrium under p. (The wealth transfer
for type it agents should be negative.) The two-fold replica allocation of x does not belong to Fcore(E1 (x)⊕E1 (ω)) since, for example, i3 in E1 (ω) and i′3 in E1 (x) block the replica allocation of x with (1 + 2ϵ , 0.1 + 2ϵ )for i3 and (3.9, 0.8) for i′3 .

8