Discussion Papers In Economics
And Business

Local Independence, Monotonicity and Axiomatic
Characterization of Price-Money Message
Mechanism
Ken Urai
Hiromi Murakami
Discussion Paper 16-08

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANLocal Independence, Monotonicity and Axiomatic
Characterization of Price-Money Message
Mechanism
Ken Urai
Hiromi Murakami
Discussion Paper 16-08

April 2016

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

April 22, 2016

Abstract
To characterize money in a static economic model, it is known to be important to consider the agentcommodity double-infinity settings, i.e., the overlapping-generations framework. There does not seemto exist any papers, however, treating the axiomatic characterization problems for such monetary
Walras allocations under the social choice and/or mechanism design settings. We show that themonetary Walras allocation for the economy with double infinities is characterized by weak Paretooptimality, individual rationality, local independence or the monotonicity conditions of social choicecorrespondence among the allocation mechanisms with messages under the category theoretic approachin Sonnenschein (1974). We utilize Sonnenschein’s market extension axiom for swamped economies thatis closely related to the replica stability axiom of Thomson (1988). We can see how these conditions
characterize the price-money message mechanism universally among a wide class of mechanisms, andeﬃciently in the sense that it has the minimal message spaces (price-money dictionary theorems).
Moreover, by using the category theoretic framework, we can obtain the up-to-isomorphism uniquenessfor such a dictionary object (isomorphism theorems).
Keywords : Resource Allocation Mechanism, Social Choice Correspondence, Overlapping-Generations
Economy, Monetary Walras Allocation, Local Independence, Monotonicity, Informational Eﬃciency,
Universal Mapping Property
JEL Classification: C60, D50, D51, D71, E00∗ We wish to express special gratitude to professor Ryoichi Nagahisa (Kansai University) for his important comments andadvice. We are also grateful for comments by professors Tomoyuki Kamo (Kyoto Sangyo University) and Kazuhiko Hashimoto(Osaka University), and to participants in the Joint Seminar of Mathematical Economics at Osaka University (August 7,2015), Poster Session at Sophia University (Octover 10, 2015, Autumn Meeting of Japanese Economic Association) and the
21st Decentralization Conference at Keio University (Octover 12, 2015). We would also 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 fromthe Japan Society for the Promotion of Science in 2013 (No.25380227) and in 2015 (No.15J01034).
† Graduate School of Economics, Osaka University, E-mail: urai@econ.osaka-u.ac.jp‡ Graduate School of Economics, Osaka University, E-mail: pge027mh@student.econ.osaka-u.ac.jp

1

Introduction

To introduce money in a static general equilibrium model, the overlapping-generations model with the
double infinity of commodities and agents is known to be the most fundamental framework. The model was
firstly introduced by Samuelson in 1958 (Samuelson 1958). It generated various discussions because of its
outstanding feature that competitive equilibria may not necessarily be Pareto-optimal. Although Samuelson
characterized the role of fiat money as a certain kind of social contract that leads to Pareto improvement,
his argument was not accurate enough. In the 1970s and the 1980s, many papers on monetary generalequilibria make certain that the existence of fiat money may not necessarily cause Pareto improvements
nor necessarily assure the existence of monetary equilibria that may or may not be Pareto-optimal (see,for example, Shell 1971, Hayashi 1976 and Okuno and Zilcha 1980). Moreover, a monetary equilibriumis unstable from the cooperative game theoretic viewpoint (Esteban 1986) as well as from the viewpointof equilibrium dynamics (Gale 1973). The only aﬃrmative result that we have for a characterizationof equilibrium in overlapping-generations economies with money is the relation between weakly Paretooptimality and valuation equilibrium (see Balasko and Shell 1980 and Esteban 1986).
Recently, the authors presented a replica finite core equivalence and characterization for the monetary Walrasian correspondence under the overlapping-generations framework and provided an axiomaticcharacterization for it.1 Except for our works, however, there do not seem to exist papers treating suchaxiomatic characterization problems for the double-infinity monetary equilibrium allocations under the social choice and mechanism design settings. In this paper, we treat this problem through the setting of theallocation mechanism with messages like Sonnenschein (1974), Hurwicz (1960), Mount and Reiter (1974),
Osana (1978) and Jordan (1982). In order to facilitate a typical infinite dimensional treatment for message for message spaces on this problem, we especially follow the strongly structured response-function
approach in Sonnenschein (1974) and Sonnenschein’s market extension axiom for swamped economies
which is closely related to the replica stability axiom of Thomson (1988) and Nagahisa (1994) for thesocial choice framework. We show that the monetary equilibrium allocation and the price-money message
mechanism are possibly characterized axiomatically by using well-known social choice theoretic normative
criteria, especially the local independency and the monotonicity, through the category theoretic approachof Sonnenschein (1974).
Theorem 1 shows in the domain of double infinity exchange economies, suﬃcient and/or necessary relations between the social choice correspondence satisfying the property of weak Pareto-optimality (WPO),
individual rationality (IR), the local independence (LI) and the correspondence that allocate for each economy its monetary Walrasian equilibria, the monetary Walrasian correspondence. Based on this knowledge, Theorem 2 asserts that every allocation mechanism with messages whose equilibrium results arecompatible with the three conditions, WPO, IR and LI, can universally and uniquely be identified witha part of monetary Walrasian (price-money) message mechanism. We can also obtain Theorem 3 thatassures the uniqueness of such a message space as a solution to the universal mapping problem. Theorem 4
and 5 show that the price-money dictionary theorem (Theorem 2) and the isomorphism theorem (Theorem3) can also be obtained through the monotonicity instead of the local independency.

1 See Urai and Murakami (2015a) and Urai and Murakami (2015b).

1

2

The Model

As in our previous paper (Urai and Murakami 2015b), we define the general overlapping-generations
settings under the duality between R∞ and R∞ . Since we are concerned with one shot (perfect foresight)
equilibrium states, this kind of overlapping-generations model (one good for each period and (ℓ(t) + 1)periods lifetime-span for each generation t) is suﬃciently general to include all types with ℓ-goods andn-periods lifetime for each generation t.2
We denote by N the set of all positive integers and by R the set of real numbers. An overlappinggenerations economy, or an economy, E, is a list of:∪∞(OG1) {It }∞t=1 ; a countable family of mutually disjoint finite subsets of N such that
t=1 It = N ,where It ̸= ∅ for each t ∈ N . It is the index set of agents in generation t.
(OG2) {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 and k(t) ≦ k(t + 1) ≦ k(t) + ℓ(t) for
all t ∈ N , and {t | n ∈ Kt } is finite for each n ∈ N . Kt is the index set of commodities available togeneration t.
(OG3) {(≿i , ωi )}i∈∪t∈N It ; countably many agents, where ≿i is a rational weak preference on thecommodity space, RKt , of i ∈ It for t ∈ N . Every preference, ≿i , can be represented by a continuous
utility function, ui : RKt → R, that is strictly quasi-concave and strictly monotonic. The initialtendowment of i, ωi , is an element of RK++ = {x | x : Kt → R++ } for each i ∈ It .

Nt
The commodity space for each generation, RK+ , can be recognized as the subset of R , the set of alltfunctions from N to R, by identifying x ∈ RK
+ with the function that takes value 0 on N \Kt . The totalcommodity space for an economy is, therefore, the set of all finite sums in RN among points in commodity
Kt
Kt
N
∞
spaces of some generations, ⊕∞t=1 R+ ⊂ R . Clearly, ⊕t=1 R+ can also be identified with a subset of thedirect sum, R∞ , the set of all finite real sequences, which is a subspace of the set of all real sequences,
R∞ ≈ RN .
∞∪Given an economy, E = ({It }∞t=1 , {Kt }t=1 , {(≿i , ωi )}i∈ t∈N It ), the price space for E, P(E), is defined∞as the set of all p in RN+ such that under the duality between R∞ and R , p evaluates all agents’ initialendowments positively, i.e.,(1)

P(E) = {p ∈ RN
+ | p · ωi > 0 for all i ∈ It , for all t ∈ N }.

Nt
Since for all i ∈ It , ωi ∈ RK++ , for all t ∈ N , the price space of E always includes R++ for all E in Econ,
the set of all economies satisfying conditions (OG1), (OG2) and (OG3).

For each E = ({It }, {Kt }, {(≿i , ωi )}) ∈ Econ, sequence (xi ∈ RKt )i∈∪t∈N It is called an allocation for
E. An 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 equation is assured by (OG2). The list of a price vector∪∞p∗ ∈ P(E), a non-negative wealth transfer function ME∗ : N = t=1 It → R+ , and a feasible allocation
2 With respect to the topological structure for this kind of double infinity economies, we also use the simplest case of
the duality between the direct limit of finite dimensional commodity spaces and the inverse limit of finite dimensional pricespaces (see Aliprantis et al. 1989, Urai 1990 and Urai 1994), which is equivalent to treating the duality between R∞ (⊂ R∞ )and R∞ under the ordinary product topology.

2

(x∗i ∈ RKt )i∈∪t∈N It is called a monetary Walras allocation for E, if for each t ∈ N and i ∈ It , x∗i is a
≿i -greatest element in the set {xi ∈ RKt | p∗ · xi ≦ p∗ · ωi + ME∗ (i)} (see, e.g., Balasko and Shell 1981 andEsteban and Millán 1990). We denote the set of all monetary Walras allocations by MWalras(E).
An allocation, x, for economy E = ({It }, {Kt }, {(≿i , ωi )}) ∈ Econ is said to be weakly Pareto-optimal∑∑∑∑(WPO), if there is no y with the property t∈N i∈It yi = t∈N i∈It xi , yi = xi except for a finite∪
number of i, and yi ≿i xi with at least one strict preference ≻i for i ∈ t∈N It .3 Moreover, we say that∪
allocation x is individually rational (IR) if xi ≿i ωi for all i ∈ t∈N It .
Social choice correspondence g on domain Econ is a correspondence that assigns some allocations for∪s∪seach E, g : Econ ∋ E 7→ g(E) ⊂ RN . Let K(s) = t=1 Kt and I(s) = t=1 It for each s ∈ N . We saythat social choice correspondence g satisfies the condition of local independence (LI) if x ∈ g(E) impliesx ∈ g(E′ ) whenever E is ({It }, {Kt }, {(≿i , ωi )}), E′ is ({It }, {Kt }, {(≿′i , ωi )}), ≿′i = ≿i except for finite
agents, and if there exists s ∈ N such that I(s) includes all such finite agents and a unique supporting
hyperplane Hs ⊂ RK(s) at x ∈ g(E) of every better set {yi | yi ≻i xi } of i ∈ I(s) also supports every{yi | yi ≻′i xi } of i ∈ I(s).4 Social choice correspondence g is said to satisfy the condition of monotonicity
if x ∈ g(E) implies x ∈ g(E′ ) whenever E is ({It }, {Kt }, {(≿i , ωi )}), E′ is ({It }, {Kt }, {(≿′i , ωi )}),≿′i = ≿i except for finite agents, and there exists s ∈ N such that I(s) includes all such finite agents andevery better set {yi | yi ≻i xi } at x of i ∈ I(s) includes {yi | yi ≻′i xi } of each i ∈ I(s).53

A Preliminary Theorem

Social choice correspondence g : Econ → RN is said to be monetary Walrasian if g(E) = MWalras(E).
We have the next theorem that characterizes the monetary Walrasian social choice correspondence throughthe LI condition.
Theorem 1 (LI Characterization Theorem)6 : (i) Assume that social choice correspondence g is WPO
and IR. If g satisfies LI and all better sets of i ∈ I(s) at x ∈ g(E) are supported by a unique price p inRK(s) , then for all i ∈ I(s), their initial endowments are evaluated less than or equal to the value of xi
under p. In particular, if g satisfies LI and all better sets of agents at x ∈ g(E) is supported by a unique
price, x is a monetary Walras allocation (g is monetary Walrasian). (ii) On the other hand, monetary
Walrasian social choice correspondence g is WPO and IR valued, and if the non-negative wealth transfer
ME does not depend on the preferences of agents, it satisfies the LI condition.
Proof : Assume that an allocation x ∈ g(E) is WPO and IR, and social choice correspondence g satisfies∪∞
the LI. By the weak Pareto-optimality, we have a price, p ∈ R∞ , such that for each i ∈ t=1 It , x′i ≻i xi
implies that p · x′i > p · xi (Balasko and Shell 1980). If p · xi ≧ p · ωi for all i, by defining ME (i) as
p · xi − p · ωi ∈ R+ , we can identify x as a monetary Walras allocation. Hence, the latter part of theassertion (i) follows from the former. Suppose that for some i ∈ I(s), p · xi < p · ωi and all better sets ofi ∈ I(s) at x are supported by a unique price p ∈ RK(s) . Then, it is possible to change the preference
3 See Balasko and Shell (1980).
4 See Nagahisa (1991).
5 It is obvious that the above monotonicity is weaker than the local independence condition as long as we ignore theprobably inessential condition of the existence of a unique supporting hyperplane at allocation x. The local independence
condition, however, enables us to obtain a simple characterization of monetary Walrasian allocation in the next section(Theorem 1).
6 To obtain an analogous result based on the monotonicity, we need Axiom S in the next section.

3

of i to ≻′i so that ≻′i satisfies ωi ≻′i xi and all assumptions in (OG3), and p remains to be a supporting
hyperplane of the better set of i under ≻′i at x. Under the LI, x should be in the value of the social choice
for such an economy, which is impossible, however, since ωi ≻′i xi contradicts the IR.
On the other hand, monetary Walras allocation is obviously IR and is well known to be WPO (seeBalasko and Shell 1980 and Esteban 1986). Moreover, for each allocation x ∈ g(E) having a uniquesupporting hyperplane Hs ∈ RK(s) for better sets of agents in I(s) for some s ∈ N , we can confirm
that the monetary Walrasian social choice correspondence g satisfies the LI. Indeed, as long as each nonnegative wealth transfer does not depend on preferences, any preference changes from economy E to E′
to check the LI condition do not aﬀect the property of allocation x to be agents’ individual price-wealthmaximands.
■
Note that in the definition of LI, the uniqueness property of the supporting hyperplane for better setsof agents at allocation x of g is important. The LI condition does not say anything for allocations that do
not have this uniqueness property. For diﬀerentiable class of economies (treated in section 4 Theorem 3),
the uniqueness property is satisfied at every allocations.

4

Axiomatic Characterization of the Price-Money Message Mechanism

Three conditions in Theorem 1, the weak Pareto-optimality (WPO), the individual rationality (IR),and the local independence (LI), enable us to provide an axiomatic characterization of the monetary
Walrasian (price-money) message mechanism through the category theoretic framework as in Sonnenschein(1974). We formulate, especially, the local independence condition and, interchangeably, the monotonicity
condition as axioms for such allocation mechanisms.
At first, we reformulate the concepts in Sonnenschein (1974) into the social choice settings. A WPO-IR
compatible social choice correspondence associates with each economy E a set of allocations which are
WPO and IR allocations for E. An allocation mechanism with messages in our model or an (abstract)message mechanism based on such a social choice correspondence, g, is a triple, (A, µ, f ): the set A is amessage domain, µ is a correspondence which indicates for each economy E the set µ(E) ⊂ A of equilibrium
messages for E, and f is a function which defines for each agent, i, and each message, a, the response,7
f i (E, a), of the agent to the message, satisfying that g(E) = {(f i (E, a))∞The list of
i=1 | a ∈ µ(E)}.
equilibrium responses associated with a ∈ µ(E) assigns to each agent in E his response to the message a.
The monetary Walrasian social choice correspondence associates with each economy the monetary Walras
allocations of the economy. The standard message mechanism is such that A = RN
+ × {M | M : Econ →NR+ }, µ(E) is the set of equilibrium prices with non-negative wealth transfer of E, and f gives the excess
demand function of each consumer relative to price-money messages. Let us consider the following axioms.
Axiom S (Sonnenschein) : For each finite list of the economies and the members, (i1 , E1 ), (i2 , E2 ), . . . ,(im , Em ), each message a ∈ A and each list of responses (fis (Es , a))ms=1 , there exists an economy E∗
including {i1 , i2 , . . . , im } such that a is an equilibrium message for E∗ satisfying that the equilibrium list,m(fi (E∗ , a))∞i=1 , is an extension of (fis (Es , a))s=1 .
7 For this concept, Sonnenschein (1974) uses the word private representation. The word “private,” however, is not appropriate for our setting, since the responses of agents to the messages are partly dependent on the economy.

4The above condition is closely related to the replica stability axiom of Thomson (1988) (see Urai and
Murakami 2015b). Note that since the non-negative wealth transfer may be diﬀerent among agents having
the same individual characteristics, it would be desirable to treat general messages that are partly economydependent.8 Hence the finite agents in the previous axioms should be listed with the economies to which
they belong.
The following axioms redefine the local independency and the monotonicity conditions in the previous
section through the terms in the allocation mechanism with messages.
Axiom L (Local Independency) : For each economy E and message a, if there exist generation s ∈ N
and a unique hyperplane Hs ⊂ RK(s) that supports the better set at fi (E, a) of every i ∈ I(s), then foreach economy E′ having the same indices of agents and commodities, endowments and possibly diﬀerent
preferences of agents in I(s) of economy E, such that Hs is also a supporting hyperplane of the better set
at fi (E, a) of every i ∈ I(s), we have f (E, a) = f (E′ , a).9
Axiom M (Monotonicity) : For each economy E and message a, if s ∈ N and E′ is an economy having
the same indices and endowments of agents in E together with the same preferences except for agents inI(s) such that every better set at fi (E, a) in E includes the better set at the same point in E′ for eachi ∈ I(s), then we have f (E, a) = f (E′ , a).
Consumer i is a pair (≿i , ωi ), where ≿i and ωi satisfy the conditions in (OG3). We assume in the∗following the commodity structure, {Kt }∞t=1 , is fixed, and identify the set of all economies, Econ , with∞the set of those in Econ with the commodity structure {Kt }t=1 . 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 for∪tK(t)agents in I(t), i.e., K(t) = s=1 Ks . For each t, by ∆K(t) , we denote the unit simplex in RK(t) and by ∆++
′
K(t
)
K(t)
its relative interior, R++ ∩∆K(t) . Let us consider projective system (∆++ , ϱt′ t )t′ ,t∈N and projective limit
K(t′ )

K(t)K(t′ )pr′p∆++ = ←−lim (∆++ , ϱt′ t ), where ϱt′ t : ∆++ → ∆++ is defined as ϱt′ t (p) = ∥ prK(t ′) p∥ . Note that ∆++ can
K(t )
∏∞
K(t)
t ∞
t ∞
be recognized as a subset of R∞++ by identifying the equivalence class [(x )t=1 ] of (x )t=1 ∈t=1 ∆++
prK(t) p
1twith the element p ∈ R∞++ such that prK(1) p = x and ∥ prK(t) p∥ = x for all t = 2, 3, . . .. We take∞the price and non-negative wealth transfer domain as P × M = {p ∈ R | ∃[(xt )∞t=1 ] ∈ ∆++ , prK(1) p =prpN
x1 , ∥ prK(t) p∥ = xt , for each t = 1, 2, . . .}×{M | M : Econ∗ ∋ E 7→ ME ∈ R+}. The excess demand functionK(t)∗of the i-th consumer, (≿i , ωi ), in E ∈ Econ is defined as ei : P × M ∋ (p, M ) 7→ ei (p, ME ) ∈ R∞ , whereei (p, ME ) is the ≿i -greatest point in {xi ∈ RKt | p · xi ≦ p · ωi + ME (i)}, for each i ∈ It and t ∈ N .
Define e : Econ∗ × (P × M) → R∞ by e(E, ≿i , ωi , p, M ) = (ei (p, ME )i∈It )t∈N . If for each E ∈ Econ∗ ,π(E) denotes the set of all price-money equilibrium messages, then (P × M, π, e) is a message mechanism
based on WPO-IR compatible social choice correspondence MWalras(E). It is called the price-money
message mechanism. Note that (P × M, π, e) does not satisfy Axioms S and L.10

8 In the sense that the message is related not only to each agent’s characteristics, i.e., the initial endowment and thepreference, but also to their places in the economy to which they belong.
9 It is possible to weaken Axiom L by restricting the condition to the messages satisfying a ∈ µ(E) as long as we use
the axiom with Axiom S. From the viewpoint of independency among axioms, it is desirable to define axioms L and M as
conditions not on equilibria but merely on responses. Note that if Axiom L is satisfied, the sub-correspondence of a social
choice correspondence g that is defined by the set of allocations under g of E relating to a certain message a ∈ µ(E) as its
responses, also satisfies the LI condition.
10 For Axiom L, see (2) of Theorem 1. For Axiom S, see footnote 7 of Urai and Murakami (2015b).

5

Theorem 2 (Price-Money Dictionary Theorem under Axiom L) : If (A, µ, f ) is a message mechanism
based on WPO-IR compatible social choice correspondence g, and if (A, µ, f ) satisfies Axioms S and L,then (i) there exists a unique function ϕ : A → P × M, such that the following triangle commutes, and(ii) on ϕ(A) ⊂ P × M, the price-money message mechanism satisfies Axioms S and L.
R∞



e

Econ∗ × (P × M)

@
I f
@
@

6
1Econ∗ × ϕEcon∗ × A

Proof : (i) Assume that (A, µ, f ) is a message mechanism based on g satisfying Axioms S and L, and let abe an element of A. Define for each t ∈ N , h(t) (x, ≿i ) for each consumption 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 h(t) (fi (E, a), ≿i ) is non-empty for eacht ∈ N , where the intersection is over all consumers and economies in Econ∗ , and fi (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∩mfrom which we are forming the intersection is closed, it is suﬃcient to show that s=1 h(t) (fis (Es , a), ≿is ) is
non-empty for any [(i1 , E1 ), (i2 , E2 ), . . . , (im , Em )]. Given the list [(i1 , E1 ), (i2 , E2 ), . . . , (im , Em )] of agentsin I(t) and economies, by Axiom S there exists E∗ ∈ Econ∗ containing {i1 , i2 , . . . , im } and a ∈ µ(E∗ ),∞msuch that the equilibrium list, (fi (E∗ , a))∞i=1 , is an extension of (fis (Es , a))s=1 . Because (fi (E∗ , a))i=1 isan element of g(E), the allocation is weakly Pareto-optimal, so by Balasko and Shell (1980) and Esteban∩m(1986), it is supported by a price as a price-wealth equilibrium, and thus s=1 h(t) (fis (Es , a), ≿is ) is non∩∞empty. Moreover, because for some economy and its agents, t=1 h(t) (fi (E, a), ≿i ) is singleton and is an∩K(t)element of ∆++ , it follows that h(t) (fi (E, a), ≿i ) is composed of a single point p(t).
By definition of h(t) , we have p(t′ ) = ϱt′ t (p(t)) for all t′ ≦ t, and obtain a unique element p ∈ P by∩midentifying it with the unique element of the projective limit ←−lim s=1 h(t) (fis (Es , a), ≿is ) ⊂ ∆++ . Let
us denote that point, p, by ϕ1 (a), and define ϕ2 (a) = M , M : Econ∗ ∋ E 7→ ME ∈ RN , as ME (i) =ϕ1 (a) · (fi (E, a) − ωi ), which will be proved as non-negative in the following by Axioms S, L and Theorem
1. Let ϕ(a) be (ϕ1 (a), ϕ2 (a)) ∈ P × M. In order to establish the theorem, it is suﬃcient to show that for∞each economy E∗ ∈ Econ∗ and a ∈ A, an allocation y ∗ = (yi∗ )∞
i=1 = (fi (E∗ , a))i=1 is such that for each∗
1i, yi = fi (E∗ , a) satisfies ME∗ (i) = ϕ (a) · (fi (E∗ , a) − ωi ) ≧ 0. Fix a member i of E∗ . By using Axiom
S, let E∗∗ be an economy including i such that a is an equilibrium message for E∗∗ and the responseyi∗∗ = fi (E∗∗ , a) is equal to yi∗ = fi (E∗ , a). Without loss of generality we can assume that E∗∗ has, atleast to the generation s of member i, one consumer in each generation whose supporting hyperplane forany better set at their individually rational point is unique.11 We show that response y ∗∗ = f (E∗∗ , a) is
a monetary Walras allocation. This allocation is IR and WPO. Moreover the social choice rule definedby a and its responses for all economies in Econ∗ having the same indices of agents in E satisfies the LI
condition by Axiom L (see footnote 9). So by (i) of Theorem 1, the allocation belongs to MWalras(E∗∗ ).
(ii) One can observe in the above argument, y ∗∗ = f (E∗∗ , a) = e(E∗∗ , ϕ(a)) is a monetary Walras
allocation, which proves that Axiom S is satisfied on ϕ(A). Moreover, it is straightforward that the
11 It is always possible to add finite agents in constructing economy E in Axiom S.
∗

6

commutativity of the diagram with Axiom L for (A, µ, f ) means that Axiom L is satisfied on ϕ(A).

■By Theorem 2, from every message mechanism (A, µ, f ) based on WPO-IR compatible g satisfying
Axiom S and L , there exists a unique price dictionary function, ϕ : A → P × M. In other words, theresults of such message mechanisms can be realized universally and eﬃciently through the price-money
message mechanism (P × M, π, e). Thus we have obtained the price-money dictionary theorem as in ourprevious paper (Theorem 2 of Urai and Murakami 2015b: based on WPO and finite core compatible g).
We can also obtain an isomorphism theorem for the price-money message mechanism (Theorem 3 ofUrai and Murakami 2015b) as follows. Denote by PM∗L the set of all (p, M ) ∈ P × M which is an imageof ϕ for some (A, µ, f ) in Theorem 2 satisfying Axioms S and L. The following axiom on the dependenceof monetary messages on the economic structure is necessary to show the second assertion.
∞∪
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.

Theorem 3 (Isomorphism Theorem under Axiom L) : Consider the restriction of price-money message
mechanism (PM∗L , π, e). Let (P ′ , π ′ , e′ ) be a message mechanism based on WPO-IR compatible social
choice correspondence g on Econ∗ . If (P ′ , π ′ , e′ ) satisfies Axioms S and L, and if, for every message
mechanism (A, µ, f ) satisfying Axioms S and L, there exists a unique mapping ϕ′ : A → P ′ such thatf (E, a) = e′ ◦ [1Econ∗ × ϕ′ ](E, a), then (i) there exists an isomorphism (bijection) h′ such that h′ : PM∗L →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 inverse-system component space with
diﬀerentiable) structures and continuous mappings (resp. diﬀerentiable coordinate mappings), then theisomorphism can be taken as the homeomorphism (resp. diﬀeomorphism for each component space).12
Proof : Because (PM∗L , π, e) is now assumed to be a message mechanism based on WPO-IR compatiblesocial choice correspondence g satisfying Axioms S and L, we have the next diagram by assumption.
(R∞ )N e′Econ∗ × P ′6
1Econ∗ × ϕ′

@
I e
@
@

Econ∗ × PM∗L
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.
12 In this paper, the price-money message space has been treated as an inverse limit of finite dimensional domains ofcoordinate functions of e. The diﬀerentiability for e and a diﬀerentiable structure on its domain, however, is appropriate to
be treated on each of its coordinate functions, ei , whose domain is always possible to be identified with a finite dimensional′′ ∞subspace of P × M ⊃ PM∗L . More precisely, under the definitions of e = (ei )∞i=1 and e = (ei )i=1 with Axiom D, the bijection′′
h gives an algebraic isomorphism between the domain of ei and ei for each i, to which the diﬀeomorphism argument canbe applied. We can construct (as a subspace of PM∗L under the identification of R∞ ⊂ R∞ ) a direct limit of the finite
dimensional projection, Pt , of the domain of (ei )i∈I(t) for each t ∈ N, so that the bijection h′ gives an algebraic isomorphism
between the domains Pt of (ei )i∈I(t) and Pt′ of (e′i )i∈I(t) for each t. In this sense, each restriction of h′ gives a diﬀeomorphism′ ∞between the direct systems, (Pt )∞t=1 and (Pt )t=1 .

7

(R∞ )N eEcon∗ × PM∗L

@
I e′
@
@

6
1Econ∗ × ϕEcon∗ × P ′Since the identity mapping is the unique mapping for P ′ to P ′ satisfying e′ = e′ ◦ id and PM∗L to PM∗L
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∗L , for each economy E ∈ Econ∗ and for each generationt, 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.13 Then, the continuity (resp. diﬀerentiability) at (p, M )of h′ will be ensured by the continuity (resp. diﬀerentiability) of e and e′ .

■Hence, Theorem 3 asserts that if we restrict the domain of the price-money messages to where Axioms
S and L are satisfied, the price-wealth formed monetary message mechanism is essentially the only object
having the above universality and eﬃciency as a solution to the universal mapping problem.
The above price-money dictionary theorem (Theorem 2) and the isomorphism theorem (Theorem 3) can
also be obtained through the monotonicity axiom (Axiom M) instead of the local independency axiom(Axiom L).
Theorem 4 (Price-Money Dictionary Theorem under Axiom M) : If (A, µ, f ) is a message mechanism
based on WPO-IR compatible social choice correspondence g, and if (A, µ, f ) satisfies Axioms S and M,
then (i) there exists a unique function ϕ : A → P × M, such that the following triangle commutes, and(ii) on ϕ(A) ⊂ P × M, the price-money message mechanism satisfies Axioms S and M.
R∞



e

Econ∗ × (P × M)

@
I f
@
@

6
1Econ∗ × ϕEcon∗ × A

Proof : (i) We can repeat the argument in the first paragraph in the proof of Theorem 2 and obtain∩∞K(t)the single point p(t) in 1=1 h(t) (fi (E, a), ≿i ) ⊂ ∆++ . Also by definition of h(t) , for all t′ ≦ t, we havep(t′ ) = ϱt′ t (p(t)), and obtain a unique element p ∈ P by identifying it with the unique element of the∩mprojective limit ←−lim s=1 h(t) (fis (Es , a), ≿is ) ⊂ ∆++ . Let us define ϕ1 (a) as this unique element, p, and
13 If generation s ∈ {1, . . . , t} of E consists of a single member, alternatively consider an economy Ê including all members ofgenerations 1, · · · , t of economy E such that every generation s ∈ {1, . . . , t} consists of at least two members and (p, M ) is theequilibrium message of Ê 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.

8define ϕ2 (a) = M , M : Econ∗ ∋ E 7→ ME ∈ RN , as ME (i) = ϕ1 (a) · (fi (E, a) − ωi ), which will be proved
as non-negative in the following by Axioms S and M.
Let ϕ(a) be (ϕ1 (a), ϕ2 (a)) ∈ P × M. To establish the theorem, it is suﬃcient to show that for each∞economy E∗ ∈ Econ∗ and a ∈ A, an allocation y ∗ = (yi∗ )∞
i=1 = (fi (E∗ , a))i=1 is such that for each i,yi∗ = fi (E∗ , a) satisfies ME∗ (i) = ϕ1 (a) · (fi (E∗ , a) − ωi ) ≧ 0.
Assume the contrary, that is, that there is a member i of E∗ such that at yi∗ = fi (E∗ , a) we have
ϕ1 (a) · (fi (E∗ , a) − ωi ) < 0. By using Axiom S, let E∗∗ be an economy including i such that a is anequilibrium message for E∗∗ and the response yi∗∗ = fi (E∗∗ , a) is equal to yi∗ = fi (E∗ , a), and there is atleast one agent j ̸= i such that i and j are in the same generation s and the supporting hyperplane at
fj (E∗∗ , a) for the better set of j is unique (that is necessarily equal to ϕ1 (a)).
Fix the indiﬀerence surface of i in E∗∗ at yi∗ = yi∗∗ = fi (E∗∗ , a) and change the preference of i to what
is obtained through the homothetical transformation of the surface at yi∗ , ≾i′ .14

Consider the economy E′∗∗ such that the preference of i in E∗∗ is replaced with ≾′i . By Axiom M,
ˆ ′ ωi and allˆ ′ so that ≾ˆ ′ satisfies ωi ≾
ˆ ′ y ∗ and y ∗ ̸ ≾fi (E′∗∗ , a) = yi∗ . Change the preference of i to ≾
i
i
i
i
i i
ˆ′
assumptions in (OG3), and ϕ1 (a) remains to be a supporting hyperplane of the better set of i under ≾i

at yi∗ . Moreover, consider the homothetical transformation of the preference of i by using the indiﬀerenceˆ ′ , and call it ≾′′ . Let us denote by E′′ the economy where we replace the preference ≾′surface at yi∗ of ≾
∗∗
i
i
i

of i with ≾′′i .
By Axiom S, we have an economy E∗∗∗ including i in E′′∗∗ and j in E∗∗ , and a is an equilibrium message
for E∗∗∗ . Since ϕ1 (a) must support the better set of j at fj (E∗∗ , a) by WPO, fi (E′′∗∗ , a) must be diﬀerent

from yi∗ by the condition IR, but must be equal to some point zi at which the indiﬀerence surface of thepoint is supported by ϕ1 (a). That is, fi (E′′∗∗ , a) = zi ̸= yi∗ = fi (E′∗∗ , a). This is a contradiction sincefi (E′∗∗ , a) is equal to fi (E′′∗∗ , a) under Axiom M.

(ii) Repeat the arguments in the proof (ii) of Theorem 2 (replace Axiom L with Axiom M).

■

Denote by PM∗M the set of all (p, M ) ∈ P × M which is an image of ϕ for some (A, µ, f ) in Theorem 4
satisfying Axioms S and M.
Theorem 5 (Isomorphism Theorem under Axiom M) : Consider the restriction of price-money message
mechanism (PM∗M , π, e). Let (P ′ , π ′ , e′ ) be a message mechanism based on WPO-IR compatible social
choice correspondence g on Econ∗ . If (P ′ , π ′ , e′ ) satisfies Axioms S and M, and if, for every message
mechanism (A, µ, f ) satisfying Axioms S and M, there exists a unique mapping ϕ′ : A → P ′ such thatf (E, a) = e′ ◦[1Econ∗ ×ϕ′ ](E, a), then (i) there exists an isomorphism (bijection) h′ such that h′ : PM∗M →P ′ and e = e′ ◦[1Econ∗ ×h′ ]. Moreover, (ii) assume that monetary messages satisfy Axiom D and the set of
all economies, Econ∗ , consists of those having commodity structure {Kt }∞t=1 and include at least two agentsfor each generation. If we can restrict the problem on spaces with topological (resp. on each inverse-systemcomponent space with diﬀerentiable) structures and continuous mappings (resp. diﬀerentiable coordinate
mappings), then the isomorphism can be taken as the homeomorphism (resp. diﬀeomorphism for eachcomponent space).

14 Let U (y ∗ ) be the indiﬀerent surface at y ∗ of i. Under the IR, y ∗∗ ̸= 0. By the strict monotonicity, αU (y ∗ ), α > 0 covers
i
i
i
i
Ks
i
s
the RK+ . Moreover for each x ∈ R+ , there exist α(x) such that x ∈ α(x)U (y∗ ) and such α(x) is unique under the strong′

monotonicity. Thus by defining u(x) as u(x) = α(x), we have the preference, ≾i , satisfying all the conditions in (OG3).

9

Proof : Repeat the proof of Theorem 3 (replace Axiom L and PM∗L respectively with Axiom M andPM∗M ).
■

5

Conclusion

We have thus obtained two kinds of price-money dictionary theorems (Theorem 2 and 4) and theisomorphism theorems (Theorem 3 and 5). It can be said that the price-money dictionary theorem, whichasserts that the price-money message mechanism can be referenced uniquely and universally among thecategory satisfying the axioms in question (a property for itself ), together with a restriction of pricemoney messages to the place where all such axioms are satisfied (a property on itself ), enables us toshow the isomorphism theorem which says that the price-money message mechanism can be characterized
as an essentially unique mechanism (up to isomorphism between the message spaces) in the category of
allocation mechanisms with messages satisfying those axioms.

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