Discussion Papers In Economics
And Business

An Axiomatic Characterization of the Price-Money
Message Mechanism
Ken Urai
Hiromi Murakami
Discussion Paper 15-31-Rev.

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANAn Axiomatic Characterization of the Price-Money
Message Mechanism
Ken Urai
Hiromi Murakami
Discussion Paper 15-31-Rev.

March 2016

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANAn Axiomatic Characterization of the Price-Money
Message Mechanism ∗Ken Urai†
Hiromi Murakami‡

March 3, 2016

Abstract
The axiomatic characterization of price or market equilibrium is one of the most important problems
in the general equilibrium theory. There does not seem to exist, however, so many papers on theaxiomatic characterization problem of monetary equilibrium. The overlapping-generations model witha double infinity of commodities and agents is one of the most fundamental frameworks for introducing
money into an economic model, although a simple game-theoretic or welfare characterization on therole of money under competitive mechanism is widely known to be diﬃcult. In this paper, we showthat the informational eﬃciency axiomatic characterization as in Hurwicz (1960), Mount and Reiter(1974) and Sonnenschein (1974) is possible for the price-money competitive mechanism for overlappinggenerations economies among the class of all allocation mechanisms with messages. In particular,
the category theoretic universal mapping characterization in Sonnenschein (1974) is generalized andapplied to the overlapping-generations framework through our monetary version of Debreu-Scarf’s
core limit theorem of Urai and Murakami (2015). Our argument is also closely related to the replica
characterization approaches of Walrasian social choice mechanism like Thomson (1988) and Nagahisa(1994), and provides a comprehensive perspective on them.
Keywords: Axiomatic Characterization, Resource Allocation Mechanism, Informational Eﬃciency,
Monetary Equilibrium, Overlapping-Generations Model, Replica Core Equivalence, Universal Mapping
PropertyJEL Classification: C60, C71, D51, D82, E00∗ A preliminary version of this paper was presented in Poster Session at Niigata University (May 23, 2015, Spring Meetingof Japanese Economic Association). We would like to thank Professor Ivan Brenes (Osaka University) for his kind help. A
part of this research was supported by the Grant-in-Aid for Scientific Research from the Japan Society for the Promotion of
Science in 2013 (No.25380227) and 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

The axiomatic characterization of price or market equilibrium is one of the most important problems in
the general equilibrium theory. There does not seem to exist, however, so many papers on the axiomaticcharacterization problem of monetary equilibrium (an equilibrium with fiat money or the non-negative
wealth transfer).
In order to describe money in the static economic model, it is widely known that an ordinary finite
economic setting is insuﬃcient and the double infinity of agent-commodity structure, e.g., the overlappinggenerations framework, is necessary. The overlapping-generations model was firstly proposed by Samuelson
in 1958 (Samuelson 1958), and has generated much discussion and many papers. An outstanding characteristic of this model is that competitive equilibria may not necessarily be Pareto-optimal and that the outsidemoney has the potential to improve on the welfare of the economy. Although Samuelson’s argument was
not entirely accurate, it is known that the characterization of money in an overlapping-generations economy is not so straightforward from the welfare and game theoretic viewpoint (see Hayashi 1976, Okuno
and Zilcha 1980 and Esteban 1986) except for the elementary welfare relation between the monetary
equilibrium and the weakly Pareto-optimal allocation (Balasko and Shell 1980 and Esteban 1986).
In this paper, to obtain an alternative explanation for the importance of money, we provide the axiomaticcharacterization of the price-money competitive mechanism in overlapping-generations economies among
the class of allocation mechanisms with messages together with several axioms and category theoretic
treatments in Sonnenschein (1974). In his seminal paper (1974 Propositions 1, 2 and 7), Sonnenscheincharacterizes the price mechanism as the unique resource allocation mechanism that can be referenced(i) uniquely under many-to-one configurations by (ii) many allocation mechanisms satisfying the several
important axioms, where (i) means a certain kind of eﬃciency and (ii) means a universality for the
price message mechanism. His result uses Debreu-Scarf’s replica core equivalence theorem and a similar
characterization under the social choice framework is given as a replica stability condition in characterizing
Walrasian mechanism like Thomson (1988) and Nagahisa (1994). Also, his result can be related to theinformational eﬃciency problem like Hurwicz (1960), Mount and Reiter (1974), Osana (1978) and Jordan(1982).1
In the double infinity economies including overlapping-generations ones, the non-negative wealth transfer
or outside money plays an essential role for equilibrium states, so it is important to characterize thecompetitive market mechanism as the allocation mechanism with a message space not only of prices
but also of wealth transfers, the price-money message mechanism. We use in this paper the method ofSonnenschein (1974) since his categorical treatment is appropriate for our infinite dimensional and partially
economy-dependent (monetary) message space settings. It should also be noted that in the approach of
Sonnenschein, the Cobb-Douglas dependent argument is not necessary. Recently the authors have shown acore limit theorem for monetary overlapping-generations economies (Urai and Murakami 2015). Our limit
theorem together with Sonnenschein’s axioms and category theoretic framework is suﬃcient to characterize
the eﬃciency and universality properties of the price-money message mechanism.

2

Economies
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:
1 See Sonnenschein (1974, Propositions 3–6).

1

(E.1) {It }∞t=1 : a countable family of mutually disjoint finite subsets of N such that∪∞

t=1 It ⊂ N ,I1 ̸= ∅ and for each t ∈ N , if It = ∅, then It+1 = ∅. 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)}∪∞such that k(t) ∈ N , ℓ(t) ∈ N , t=1 Kt = N , k(t) < k(t + 1) ≦ k(t) + ℓ(t) for all t ∈ N . Kt is theindex set of commodities available to generation t.
(E.3) {(≿i , ω i )}i∈∪t∈N It : countably many agents, where ≿i is a continuous, strictly monotonic,
strictly convex and rational weak preference relation on the commodity space for each generation,
Kt
i
t
RK
+ , for each i ∈ It . The initial endowment of i, ω , is an element of R++ = {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∞as the set of all p in RN(with+ 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).
t∪
For each E = ({It }, {Kt }, {(≿i , ω i )}) ∈ Econ, sequence (xi ∈ RK+ )i∈ t∈N It is called an allocation fort∪
E. Allocation (xi ∈ RK+ )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∗ : t=1 It → R+ , and feasible allocationit∪(xi∗ ∈ RK+ )i∈ t∈N It , is called a monetary Walras allocation for E, if for each t ∈ N and i ∈ It , x∗ is a

∗
i
∗
i
∗
t
≿i -greatest element in set {xi ∈ RK
+ | p · x ≦ p · ω + ME (i)}. Since the non-negative wealth transfer isan abstraction of the money supply in perfect-foresight overlapping-generations economies, we denote theset of all monetary Walras 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 bundles∑
y i for all i ∈ S such that i∈S (y i − ω i ) = 0 and y i ≿i xi for all i ∈ S, and y i ≻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.

2

3

Replica Core Equivalence Theorem

The replica core equivalence theorem for overlapping-generations economy was proved in Aliprantis andBurkinshaw (1990). Their result, however, does not include the case with monetary (non-negative wealthtransfer) equilibrium. Our previous paper (Urai and Murakami 2015) treats this problem through thefollowing replica finite core concept.
t∪
Given economy E = ({It }, {Kt }, {(≿i , ω i )}) ∈ Econ and its feasible allocation x = (xi ∈ RK+ )i∈ t∈N It
we denote by E(x) the economy whose initial endowment allocation, ω = (ω i ), is replaced by x = (xi ).2
Hence, we have E = E(ω). Let us consider the replica economy

Em (x) ⊕ En (ω),(3)

consisting of all members of the m-fold replica economy of E(x) and all members of the n-fold replica
economy of E(ω) for each m and n in N . Note that when the number of agents of E is finite, Core(Em+n )is equal to Fcore(Em+n ) and for each feasible allocation x of E, if its (m+n)-fold replica allocation belongs
to Core(Em+n ), then the replica allocation also belongs to Fcore(Em (x) ⊕ En (ω)).3 Moreover, as we see
below, every (m + n)-fold replica allocation of x belongs to Fcore(Em (x) ⊕ En (ω)) if and only if x is amonetary Walras allocation. Therefore the standard replica core limit theorem (Debreu and Scarf 1963)
can be restated as follows (because there is no diﬀerence between our monetary Walras allocation and theWalras allocation for finite economies).
A feasible allocation x for E is a Walras allocation if and only if its (m+n)-fold replica allocation
belongs to Fcore(Em (x) ⊕ En (ω)) for all m ∈ N and n ∈ N suﬃciently large.
In general, the number of agents in E is not finite, and Fcore(Em (x) ⊕ En (ω)) gives us a completecharacterization of monetary Walras allocations. That is, an element of MWalras(E) has a certain kindof stability-under-replication property with respect to the form Em (x) and En (ω) for the finite core solution
concept. The next theorem is proved in Urai and Murakami (2015) showing the above replica finite corestability condition for the element of MWalras(E). Thus a replica stability condition like Thomson (1988)
and Nagahisa (1994) is satisfied automatically in our price-money message mechanism.
Theorem 1 (Replica Core Equivalence Theorem) : A feasible allocation x for E is a monetary Walras
allocation if and only if its (m + n)-fold replica allocation belongs to Fcore(Em (x) ⊕ En (ω)) for everym ∈ N and n ∈ N .
Let Cm⊕n (E) be the set of allocations x for E such that the (m+n)-fold replica allocation of x belongs to′′
Fcore(Em (x)⊕En (ω)). Then Cm⊕n (E) ⊃ Cm ⊕n (E) ⊃ · · · where m′ ≧ m and n′ ≧ n, and MWalras(E) =
∩∞ ∩∞
m⊕n
(E). Thus we have obtained an extension of Debreu-Scarf replica core limit theorem to
n=1
m=1 C
monetary Walras allocations in economies including overlapping-generations settings.

∪

2 In the following, the subscript, i ∈t∈N It , for an allocation is sometimes omitted as long as there is no risk of confusion.
3 Assume that an (m + n)-fold replica allocation xm+n for Em (x) ⊕ En (ω) does not belong to Fcore(Em (x) ⊕ En (ω)). If

the number of agents of E is finite, then allocation x is feasible for the finite set of all agents of E. It is possible, therefore, toextend every finite coalition blocking xm+n in Fcore(Em (x) ⊕ En (ω)) to a finite coalition including all members of m-fold
replica economy of E, i.e., to a finite coalition blocking xm+n in Em+n (ω).

3

4

An Axiomatic Characterization of the Price-Money Message
Mechanism
The replica finite core limit theorem in the above section, Theorem 1, enables us to provide an axiomaticcharacterization of the price-money message mechanism through the universal mapping problem as inSonnenschein (1974). Our Theorem 2 on the price-money dictionary is corresponding to Sonnenschein’s
Proposition 1 and Theorem 3 on the isomorphism property is to his Proposition 7.
A social choice correspondence g : Econ → (R∞ )N is said to be compatible with a finite-core andweakly Pareto-optimal allocation if and only if g(E) ⊂ Fcore(E) and g(E) ⊂ {x| x ∈ Fcore(E(x))} for all
E ∈ Econ. An abstract message mechanism (an allocation mechanism with messages under the frameworkof Sonnenschein 1974) based on social choice correspondence g is a triple (A, µ, f ), where A is a set thatis called a message domain, µ is a correspondence that indicates for each economy E the set µ(E) ⊂ A ofequilibrium messages for E, and f is a function that defines for each agent, i, and each message, a, theresponse, f i (E, a), of the agent in E to the message, satisfying that g(E) = {(f i (E, a))∞i=1 | a ∈ µ(E)}.
The monetary Walrasian social choice correspondence associates with each economy the monetary Walras
allocations of the economy. In this section, we treat allocations that are compatible with finite-core andweakly Pareto-optimal only. For the standard message mechanism, let A be the product of RN+ and{M | M : Econ → RN+ }, µ(E) be the set of equilibrium prices with the non-negative wealth transfers of E,
and f be the excess demand function of each consumer relative to price-money messages (more precisely,
see below). We use the following idempotency axiom.
Axiom I (Idempotency): For each economy E ∈ Econ and message a ∈ A, f (E(f (E, a)), a) = f (E, a).
In the above, we use the notation E(x) in section 3. Moreover, let us consider the following axiom ofSonnenschein (1974).
Axiom S (Sonnenschein): For each finite list of agents and economies, (i1 , E1 ), (i2 , E2 ), . . . , (im , Em ),each message a ∈ A and each list of responses (f is (Es , a))ms=1 , there exists an economy E∗ including{i1 , i2 , . . . , im } such that a is an equilibrium message for E∗ for which the equilibrium list (f i (E∗ , a))∞i=1
is an extension of (f is (Es , a))ms=1 .
It follows that we do not treat messages that cannot constitute an equilibrium state for any suﬃciently
large extension of a certain list of agents and economies. The above axiom together with the replica
equivalence theorem (Theorem 1) is closely related to the replication stability axiom of Thomson (1988).
We assume in the following that the commodity structure {Kt }∞t=1 is fixed. The set of all economies∗∞
with the commodity structure {Kt }t=1 is denoted by Econ . Denote by I(t) the set of all agents in∪tgenerations from 1 to t, i.e., I(t) = s=1 Is , and by K(t) the set of all commodities that are available∪tfor agents in I(t), i.e., K(t) = s=1 Ks . For each t, denote by ∆K(t) the unit simplex in RK(t) and byK(t)K(t′ )

K(t)∆++ its relative interior, R++ ∩ ∆K(t) . Let us consider the projective system (∆++ , ϱt′ t )t′ ,t∈N andpr′ pK(t′ )
K(t)K(t′ )
projective limit ∆++ = ←−lim (∆++ , ϱt′ t ), where ϱt′ t : ∆++ → ∆++ is defined as ϱt′ t (p) = ∥ prK(t ′) p∥ .4
K(t )∞We note that ∆++ can be recognized as a subset of R∞++ by identifying the equivalence class [(xt )t=1 ]
∏
pr
p
K(t)
∞
∞
K(t)
of (xt )∞t=1 ∈t=1 ∆++ with the element p ∈ R++ such that prK(1) p = x1 and ∥ prp∥ = xt forK(t)4 The projective limit is non-empty since t ∈ N is countable and every ϱ ′ is surjective (see Bourbaki 1939, p.198,
t t
Proposition 5).

4

all t = 2, 3, · · ·.5

We can take the price and non-negative wealth transfer domain as P × M = {p ∈
pr

p

∗
K(t)
R∞ | ∃[(xt )∞t=1 ] ∈ ∆++ , prK(1) p = x1 , ∥ prK(t) p∥ = xt , for each t = 1, 2, · · ·} × {M | M : Econ ∋ E 7→NME ∈ R+}. The excess demand function ei for the i-th consumer (≿i , ωi ) in E ∈ Econ∗ is defined asiiei : P × R N+ ∋ (p, ME ) 7→ e (p, ME ) ∈ R∞ , where e (p, ME ) is identified with the points of the consumption
Ktiit
set RK| p · xi ≦ p · ω i + ME (i)} for each i ∈ It
+ , such that e (p, ME ) is the ≿i -greatest point in {x ∈ R
and each t ∈ N .

Define e : Econ∗ × (P × M) → (R∞ )N by e(E, p, M ) = (ei (p, ME ))i∈∪t∈N It .6 If π(E) denotes the set of
all monetary Walrasian equilibrium messages (the set of market equilibrium price and non-negative wealthtransfer messages) for each E ∈ Econ∗ , then (P × M, π, e) is a message mechanism based on social choicecorrespondence MWalras(E). This is called the price-money message mechanism.
Theorem 2 (Price-Money Dictionary Theorem) : If (A, µ, f ) is a message mechanism based on social
choice correspondence g that is compatible with the finite-core and weakly Pareto-optimal allocations,and if (A, µ, f ) satisfies Axioms I and S, then (i) there exists a unique function ϕ : A → P × M such thatthe following triangle commutes, and (ii) on ϕ(A) ⊂ P × M, the price-money message mechanism satisfies
axioms I and S.
(R∞ )N eEcon∗ × (P × M)
6
1Econ∗ × ϕ

@
I f
@
@

Econ∗ × A

Proof : (i) Assume that (A, µ, f ) is a message mechanism based on social choice correspondence g satisfying Axioms I and S, and let a be an element of A. Define for each t ∈ N , h(t) (x, ≿i ) for eachconsumption x ∈ RKs for agent i ∈ Is ⊂ I(t) of an economy E ∈ Econ∗ as h(t) (x, ≿i ) = {p ∈ ∆K(t) | y ≻i
x implies p·y ≧ p·x}, where every RKs is canonically identified with a subspace of K(t). We first show that
∩ (t) ih (f (E, a), ≿i ) is non-empty for each t ∈ N , where the intersection is over all consumers and economies
in Econ∗ , and f i (E, a) is a response of i ∈ I(t) in E to message a in (A, µ, f ). Because ∆K(t) is compact,
and because each of the sets in the collection of which we are forming the intersection is closed, it is suf∩mficient to show that s=1 h(t) (f is (Es , a), ≿is ) is non-empty for any [(i1 , E1 ), (i2 , E2 ), . . . , (im , Em )]. Given
the list [(i1 , E1 ), (i2 , E2 ), . . . , (im , Em )] of agents in I(t) and economies, by Axiom S there exists E∗ ∈ Econ∗
containing {i1 , i2 , . . . , im } and a ∈ µ(E∗ ), such that the equilibrium list, (f i (E∗ , a))∞i=1 , is an extension of
s
is
m
i
∞
(f (E , a))s=1 . Because (f (E∗ , a))i=1 is weakly Pareto-optimal, by Balasko and Shell (1980) and Este∩mban (1986), it is supported by a price as a price-wealth equilibrium, and thus s=1 h(t) (f is (Es , a), ≿is ) is∩∞
non-empty. Moreover, because for some economy and its agents, i=1 h(t) (f i (E, a), ≿i ) is singleton and is∩K(t)an element of ∆++ , it follows that h(t) (f i (E, a), ≿i ) is composed of a single point p(t).
By definition of h(t) , for all t′ ≦ t, p(t′ ) = ϱt′ t (p(t)), and we obtain a unique element p ∈ P by identifying∩m
it with the unique element of the projective limit ←−lim s=1 h(t) (f is (Es , a), ≿is ) ⊂ ∆++ . Let us denote that
5 For example, if K(1) = {1, 2}, K(2) = {1, 2, 3}, K(3) = {1, 2, 3, 4}, · · ·, then the element ((1/2, 1/2), (1/3, 1/3, 1/3),(1/4, 1/4, 1/4, 1/4), · · ·) of ∆++ can be identified with (1/2, 1/2, 1/2, 1/2, 1/2, · · ·) of R∞++ .
6 For the argument in the following, we use the property that for each price-money message (p, M ) there is an economy inEcon∗ such that (p, M ) is the unique price-money equilibrium message supporting the equilibrium allocation. For this, thecondition that Econ∗ includes the class of economies consisting of agents with Cobb-Douglas utility functions, is suﬃcient
but not necessary.

5

1ipoint, p, by ϕ1 (a), and define ϕ2 (a) = M , M : Econ∗ ∋ E 7→ ME ∈ RN
+ , where ME (i) = ϕ (a) · (f (E, a) −ω i ), which we prove as follows that it is non-negative under Axioms S, I and Theorem 1. Let ϕ(a) beN(ϕ1 (a), ϕ2 (a)) ∈ P × {M | M : Econ∗ ∋ E 7→ ME ∈ R+
} = P × M. To establish the theorem, it is suﬃcient∗i∞
to show that for each economy E∗ ∈ Econ and each a ∈ A, allocation y∗ = (y∗i )∞i=1 = (f (E∗ , a))i=1 issuch that, for each i, y∗i = f i (E∗ , a) can be extended (through Axiom S) to a monetary Walras allocation

under the message a for a large economy including i. To see this, by using Axiom S, let E∗∗ be anieconomy including i such that a is an equilibrium message for E∗∗ and response y∗∗= f i (E∗∗ , a) is equal
to y∗i = f i (E∗ , a). We show that the list of responses, y∗∗ = f (E∗∗ , a), is a monetary Walras allocation.
Assume that it is not. Then, by Theorem 1, there are m and n such that (m + n)-fold replica allocationj ∞nof y∗∗ = (y∗∗)j=1 is not an element of Fcore(Em∗∗ (y∗∗ ) ⊕ E∗∗ (ω)). Then there are (k1 + k2 ) members
n
m
of Em
∗∗ (y∗∗ ) ⊕ E∗∗ (ω) blocking y∗∗ , where k1 members belong to E∗∗ (y∗∗ ) and k2 members belong toEn∗∗ (ω). Under Axiom I, by applying Axiom S again, we obtain an economy E∗∗∗ including the (k1 + k2 )nmembers of Em∗∗ (y∗∗ ) ⊕ E∗∗ (ω) such that a is a solution message for E∗∗∗ satisfying that the equilibriummlist, (f i (E∗∗∗ , a))∞i=1 , a finite core allocation, is an extension of those of the k1 members of E∗∗ (y∗∗ ) (thatare by Axiom I, f i (E∗∗∗ , a) = f i (E∗∗ (y∗∗ ), a) = f i (E∗∗ , a), equal to their original responses) and the k2

members of En∗∗ (ω) under a ∈ A. However, this is impossible, because the (k1 + k2 ) members can improveupon it.
(ii) For the latter assertion, one can observe in the above argument, y∗∗ = f (E∗∗ , a) = e(E∗∗ , ϕ(a)) is amonetary Walras allocation, which proves that axiom S is satisfied on ϕ(A). Moreover, it is straightforwardthat the commutativity of the diagram with axiom I for (A, µ, f ) means that axiom I is satisfied on ϕ(A)since e(E(e(E, ϕ(a))), ϕ(a)) = e(E(f (E, a)), ϕ(a)) = f (E(f (E, a)), a) = f (E, a) = e(E, ϕ(a)).

■

The above theorem shows that the price-money message mechanism represents all the message mechanisms based on social choice correspondences compatible with finite-core and weakly Pareto-optimal
allocations under axioms I and S. The theorem, therefore, can be interpreted as a price-money message
mechanism representation theorem. We call ϕ the price-money dictionary.
A non-negative price-wealth transfer message, (p, M ), has a huge scope. In particular, the domain of
M is Econ∗ , and it must specify all non-negative wealth transfers for all economies. Such a message may
become too complicated to be called “simple” in the sense of Sonnenschein (1974) and one can easily
verify that the price-money message mechanism may not satisfy axioms I and S.7 However, if we restrict
the domain of non-negative price-wealth transfer messages to the class satisfying Axioms I and S, PM∗ ={(p, M ) ∈ P × M| (p, M ) is an image of ϕ for some (A, µ, f ) in Theorem 2 satisfying axioms I and S.},
we have the following isomorphism theorem. For this assertion, we use the next axiom on the dependency
of monetary messages on economic structures.
∞∪
Axiom D (Dependency on the Economic Structure) : If E = ({It }∞t=1 , {Kt }t=1 , {(≿i , ωi )i∈ t∈N It }) and
′
′ ∞
′
∞
′ ∞
∞
′ ∞
∪
E′ = ({It′ }∞
t=1 , {Kt }t=1 , {(≿i , ωi )i∈ t∈N It }) are such that {It }t=1 = {It }t=1 , {Kt }t=1 = {Kt }t=1 and∪ωi = ωi′ for all i ∈ t∈N It , then ME = ME′ for all M ∈ M.
7 Axiom I will be satisfied if we restrict our attention to monetary mesages that intend to realize a certain desirable levelof income for each person independently from their initial endowments for each economy E, but in general, it will not besatisfied. To obtain a counter example for axiom S, assume that {Kt }∞t=1 is the two period one commodity overlappinggeneration structure. Consider M such that amount p · ω i + M E(i) > 0 for each E, i ∈ I(1), is greater than the amounts of thetotal value of their initial endowments under (1, 1, 1, · · ·) ∈ R∞
+ that is greater than all equilibrium prices for commodities inK(1). (Note that a price (p∗1 , p∗2 , p∗3 , · · ·) is assumed to satisfy p∗1 + p∗2 = 1. See definition of P.) Thus, we cannot expect theabove message to be an equilibrium of any E ∈ Econ. In Sonnenschein (1974), the mnemonic S is used because the condition
states that “any finite number of individuals can be swamped” and it “restricts the messages of a message mechanism to be
simple.”6

Theorem 3 (Isomorphism Theorem) : Consider the restriction of price-monetary message mechanism(PM∗ , π, e). Let (P ′ , π ′ , e′ ) be a message mechanism based on social choice correspondence g on Econ∗compatible with the finite core and weakly Pareto-optimal allocations. If (P ′ , π ′ , e′ ) satisfies axioms I andS, and if, for every message mechanism (A, µ, f ) satisfying axioms I and S, there exists a unique mappingϕ′ : A → P ′ such that f (E, a) = e′ ◦ [1Econ∗ × ϕ′ ](E, a), then (i) there exists an isomorphism (bijection) h′such that h′ : PM∗ → P ′ and e = e′ ◦ [1Econ∗ × h′ ]. (ii) Moreover, assume that monetary messages satisfy
Axiom D. If we can restrict the problem on spaces with topological (resp. on each component space ofan inverse system with diﬀerentiable) structures and continuous mappings (resp. diﬀerentiable coordinate
mappings ), then the isomorphism can be taken as the homeomorphism (resp. diﬀeomorphism for eachcomponent space).8
Proof : Because (PM∗ , π, e) is now assumed to be a message mechanism based on social choice correspondence g satisfying axioms I and S, we have the next diagram by assumption.
(R∞ )N e′Econ∗ × P ′6
1Econ∗ × ϕ′

@
I e
@
@

Econ∗ × PM∗
Moreover, because (P ′ , π ′ , e′ ) is also a message mechanism based on social choice correspondence g, theprevious theorem shows that we have the next diagram.
(R∞ )N eEcon∗ × PM∗
6
1Econ∗ × ϕ

@
I e′
@
@

Econ∗ × P ′Since the identity mapping is the unique mapping for P ′ to P ′ satisfying e′ = e′ ◦ id and PM∗ to PM∗
satisfying e = e ◦ id, we have ϕ′ ◦ ϕ = id and ϕ ◦ ϕ′ = id, which means that ϕ and ϕ′ are bijectives. Let us
define h′ as h′ = ϕ′ , then we have the first assertion.
For the second assertion, for each (p, M ) ∈ PM∗ , each economy E ∈ Econ∗ and each generation t,
consider two agents is and js , s ∈ {1, . . . , t} such that e = (· · · , ei1 , · · · , ej1 , · · · , ei2 , · · · , ej2 , · · ·) on P × M
is one to one, continuous and/or diﬀerentiable.9■
8 In this paper, the price-money message space can be identified with an inverse limit of the finite dimensional domainsof each coordinate function ei for e and the diﬀerentiable structure can be discussed under such finite dimensional domains.
K(t)
I(t)
For example, we can take the domain of ei for i ∈ I(t) as ∆++ × R+ and identify it with a subset of PM∗ under thecanonical identification R∞ ⊂ R∞ . Then, we can also obtain under h′ , the image of ∆++ × R+ in P ′ , the domain of all
e′i , i ∈ I(t), and the diﬀeomorphism between them.
9 If generation s ∈ {1, . . . , t} of E consists of a single member, alternatively consider an economy Ê including all membersof generations 1, · · · , t of economy E such that every generation s of Ê consists of at least two members and (p, M ) is anequilibrium message of it by Axiom S. To obtain a concrete example for such a one-to-one, continuous and/or diﬀerentiablemapping, take a pair of Cobb-Douglas and Leontief utility agents for each generation. Note that the Leontief type utility is
not diﬀerentiable, but the demand function induced from it can be diﬀerentiable.
K(t)

7

I(t)

By Theorem 3, we see that an allocation mechanism with messages that can play the role of the dictionary
property described in Theorem 2 is essentially unique. The result can also be restated through theuniversal mapping theorem (Bourbaki 1939, p.284) that assures the existence of such an object unique up
to isomorphism as a solution to the universal mapping problem.

5

Conclusion

1. We have seen that for every private representation (A, µ, f ) of a social choice correspondence gcompatible with the finite core and weakly Pareto-optimal allocations satisfying Axioms I and S, there
exists a dictionary function, ϕ : A → P × M uniquely (Theorem 2). In other words, the result of sucha private representation (an allocation mechanism with message) can be universally realized through theprice-money message mechanism (P × M, π, e) uniquely (the universality and the eﬃciency property).
2. If we restrict the domain of an abstract message mechanism to those satisfying axioms I and S, theprice-money message mechanism has the unique (up to isomorphism) minimum size of the information
space satisfying the dictionary property (Theorem 3).
3. Based on the diﬀerentiable structure, to develop the alternative informational eﬃciency argument toour framework is also desirable. By considering the diﬀerentiable structure on the message and commodity
spaces, we will be able to obtain the same results as Propositions 3–6 in Sonnenschein (1974) that will
provide an informational eﬃciency theorem like Hurwicz (1960), Mount and Reiter (1974) and Osana(1978), as well as a uniqueness theorem of the competitive message mechanism like Jordan (1982). These
will be, however, the subjects of further investigations.
4. It should also be noted that our monetary equilibrium concept (a competitive equilibrium with nonnegative wealth transfer) is related to the concept of “dividend equilibrium” or “equilibrium with slack”(see, e.g., Aumann and Drèze 1986) and the role of our replica finite core concept can also be identified
with that of “rejective core” in Konovalov (2005) on the limit core characterization for such equilibria.
Therefore, if we generalize the preference satiation structure for the double infinity economy (see our Uraiand Murakami 2015, footnote 15), our results are possible to be identified with an another replica corecharacterization for such dividend equilibria.
5. Thomson (1988) and Nagahisa (1994) use the replication stability axiom like Axiom S of Sonnenschein(1974) to characterize Walrasian correspondences in a direct way for the standard social choice setting.
It will be interesting to apply their arguments for our double infinity monetary economic situation, sincevarious social choice axioms and frameworks would be desirable for describing the institution like money.
6. The method of Sonnenschein is appropriate for our setting of (i) the infinite dimensional commodity
and generation structure and (ii) the partially economy-dependent property of messages. Moreover, his
strongly structured response function setting has an advantage over the message-process argument in
Hurwicz (1960), Mount and Reiter (1974), Osana (1978) and Jordan (1982) since it does not necessarily
depend on (iii) the Cobb-Douglas utility assumption of the agents.

8

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