Discussion Papers In Economics
And Business

Local Independence, Monotonicity, Incentive
Compatibility and Axiomatic Characterization ofPrice-Money Message Mechanism
Ken Urai
Hiromi Murakami
Discussion Paper 17-08

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANLocal Independence, Monotonicity, Incentive
Compatibility and Axiomatic Characterization ofPrice-Money Message Mechanism
Ken Urai
Hiromi Murakami
Discussion Paper 17-08

April 2017

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

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 abundant literature, 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 and local independence or the monotonicity, or the incentive compatibility conditions of social choice correspondence among the allocation mechanisms with messages under
the category theoretic approach in Sonnenschein (1974). We utilize Sonnenschein’s market extension
axiom for swamped economies that is closely related to the replica stability axiom of Thomson (1988).
We can see how these conditions characterize the price-money message mechanism universally amonga wide class of mechanisms, and eﬃciently in the sense that it has the minimal message spaces (pricemoney dictionary theorems). Moreover, by using the category theoretic framework, we can obtain the
up-to-isomorphism uniqueness for such a dictionary object (isomorphism theorems).
Keywords : Resource Allocation Mechanism, Social Choice Correspondence, Overlapping-Generations
Economy, Monetary Walras Allocation, Local Independence, Monotonicity, Incentive Compatibility,
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 Shigehiro Serizawa (Osaka University), Tomoyuki Kamo (Kyoto
Sangyo University) and Kazuhiko Hashimoto (Osaka University of Economics), and to participants in the Joint Seminarof 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 from the 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‡ Faculty of Commerce, Doshisha University, E-mail: murakami@math.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 axiomatic characterization 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 thesocial choice and mechanism design settings. In this paper, we treat this problem through the settingof the allocation 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 formessage 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 the social choiceframework. 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, the monotonicity and the incentive compatibility, through the category
theoretic approach of Sonnenschein (1974).
By treating the number of agents as a variable and allowing for an arbitrarily general class of message
spaces including the class of finite dimensional topological manifolds as in Jordan (1982), Sonnenschein’scategory theoretic framework together with his axiom S enables us to obtain a stronger type of an axiomaticcharacterization, which we call the universal implementability, than the simple uniqueness and/or eﬃciency
theorems provided in the traditional informational eﬃciency arguments like Hurwicz (1960), Mount andReiter (1974), Osana (1978) and Jordan (1982).
Both Sonnenschein (1974) and the traditional informational eﬃciency arguments treat an allocation
mechanism with messages, although Sonnenschein describes an allocation as a response to (a function of)a message for each economy instead of describing a general relation of the message process among thesevariables, an allocation, an economy, and a message. Under Axiom S on the concept of an equilibrium
relative to an increase in the number of economic agents, Sonnenschein (1974) shows that among an
arbitrarily general class of messages, every message together with a response to it (a process using themessage) can uniquely be realized through a certain price message together with the optimal demand
1 See Urai and Murakami (2016a) and Urai and Murakami (2016b).

1

response to the price (the demand process of agents using the price). Let us call this property as thedictionary property of the price mechanism. Then, the price mechanism will be characterized as theunique mechanism having such a dictionary property. The uniqueness and the minimality (eﬃciency)
results on the message space of prices are also given simultaneously as the unique up to isomorphismfeature of the dictionary object obtained through an immediate mathematical property on the response
commutative diagrams among processes.
Mathematically, the above characterization is known as a universal mapping property and the unique
object defined under the isomorphism feature is called the solution of the universal mapping problem. An
object defined like a solution of a universal mapping problem is quite important since such an object iscompletely characterized by relations (commutativity of several mappings) and can be grasped through afinite simple algebra among such relations (mappings) without referring anything to the concrete specificobject itself. To obtain and/or define a concept in a simple, general and mathematically accurate way asmuch as possible, the setting like the universal mapping problem under the category theoretic framework
is known to be one of the most appropriate and desirable manners from the viewpoint of the scientificuniversal language.
If we identify each economic agent’s consumption set with its strategy set and a message space for an
economy with a strategy set of an additional imaginary agent, the above message process (by considering
the response functions as constraint correspondences) defines a generalized game for the economy in exactly
the same way as defining the abstract economy for the competitive market. It would be natural, therefore,
to call the above message process in the Sonnenschein’s response function form as a mechanism (in ageneralized sense) or a message mechanism.
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. Theorems
4 and 5 (6 and 7, resp.) show that the price-money dictionary theorem (Theorem 2) and the isomorphism theorem (Theorem 3) can also be obtained through the monotonicity (incentive compatibility, resp.)
instead of the local independency.

2

The Model
As in our previous paper (Urai and Murakami 2016b), 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
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

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
where It ̸= ∅ for each t ∈ N . It is the index set of agents in generation t.

∪∞

t=1 It = N ,(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 x = (xi )i∈∪t∈N It is said to be feasible if
(2)

∑∑

∑∑

xi =

t∈N i∈Itωi ,t∈N i∈It

where the summability in RN of both sides of the equation is assured by (OG2).3 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(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 = (xi )i∈∪t∈N It , for economy E = ({It }, {Kt }, {(≿i , ωi )}) ∈ Econ is said to be∑∑
weakly Pareto-optimal (WPO), if there is no allocation y = (yi ) with the propertyt∈Ni∈It yi =∑∑t∈Ni∈It xi , yi = xi except for a finite number of i, and yi ≿i xi with at least one strict preference
3 In the following, we treat a sequence, (x

i ∈ R

Kt

)i∈∪

t∈N It(yi ) by y and so forth.

3, as a vector and denote it by a single letter, e.g., (xi ) by x,∪≻i for i ∈ t∈N It .4 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).5 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).6 It is possible

to consider a weak version of monotonicity by replacing the condition for finite agents with for a singele
agent. We also define an incentive compatibility condition for social choice correspondence g as follows:
if E is ({It }, {Kt }, {(≿i , ωi )}), E′ is ({It }, {Kt }, {(≿′i , ωi )}), ≿′i = ≿i except for a single agent, i∗ ,x = (xi ) ∈ g(E) and x′ = (x′i ) ∈ g(E′ ), then we have xi∗ ≿i∗ x′i∗ .

3

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) : (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 in RK(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 Walrasiansocial 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.7
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 preferenceof i to ≻′i so that ≻′i satisfies ωi ≻′i xi and all assumptions in (OG3), and p remains to be a supporting
4 See Balasko and Shell (1980).
5 The definition was originally given and used in Nagahisa (1991) for finite and diﬀerentiable economies.
6 Note that the above monotonicity is weaker than the local independence condition as long as we ignore the condition of
the existence of a unique supporting hyperplane at allocation x. The local independence condition also enables us to obtaina simple characterization of monetary Walrasian allocation in the next section (Theorem 1).
7 To obtain an analogous result based on the monotonicity and/or incentive compatibility, we need Axiom S in the next
section.

4

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,8
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 →RN+ }, 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 .
8 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.

5The above condition is closely related to the replica stability axiom of Thomson (1988) (see Urai and
Murakami 2016b). 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.9 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 LI (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).10
Axiom (Monotonicity) : For each economy E and message a, if for some s ∈ N , 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 of i in E′ for eachi ∈ I(s), then we have f (E, a) = f (E′ , a).
For our result, the following single-agent formed weak version of the monotonicity condition is suﬃcient.
Axiom M (Single Agent Monotonicity) : For each economy E and message a, if E′ is an economy having
the same indices and endowments of agents in E together with the same preferences except for an agent,i, such that the better set of i at fi (E, a) in E includes the better set of i at the same point in E′ , then we
have fi (E, a) = fi (E′ , a).
An economic interpretation of Axiom LI is that for a fixed message a, agents’ responses do not change aslong as their local evaluations on the relative importance of each commodity do not change. In the sameway, the monotonicity axioms are interpreted that for message a, agents do not change their responses aslong as they think that the importance of their choices do not decrease.11
In relation to the above monotonicity conditions, we also use the next incentive compatibility or no
incentive to misrepresent condition to obtain an axiomatic characterization of the price mechanism.
Axiom IC (Incentive Compatibility) : For each economy E and message a, if E′ is an economy having
the same indices and endowments of agents in E together with the same preferences except for an agent,i, then we have fi (E, a) ≿i fi (E′ , a), where ≿i denotes the preference relation of i in E.

9 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.
10 It is possible to weaken Axiom LI 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 LI and M as
conditions not on equilibria but merely on responses.
11 The importance of their choices do not decrease in the sense that the better sets at their choice points do not increase.
In view of an axiom for the message mechanism, it can be said that we request messages to be self-convincing for agents
having such preference changes.

6

From the viewpoint of an axiom for a message mechanism, the condition says that messages are suchthat for agents who mistake their own preferences, their responses fails to be better ones.12
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 , withthe 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 ∞
be recognized as a subset of R++ by identifying the equivalence class [(x )t=1 ] of (xt )∞t=1 ∈t=1 ∆++
pr

p

K(t)
1
t
with the element p ∈ R∞++ such that prK(1) p = x and ∥ prK(t) p∥ = x for all t = 2, 3, . . .. We takethe price and non-negative wealth transfer domain as P × M = {p ∈ R∞ | ∃[(xt )∞t=1 ] ∈ ∆++ , prK(1) p =

pr

p

N
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.13
Theorem 2 (Price-Money Dictionary Theorem under Axiom LI) : If (A, µ, f ) is a message mechanism
based on a WPO-IR compatible social choice correspondence, and if (A, µ, f ) satisfies Axioms S and LI,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 LI.
R∞



e

Econ∗ × (P × M)
6
1Econ∗ × ϕ

@
I f
@
@

Econ∗ × A

Proof : (i) Assume that (A, µ, f ) is a message mechanism based on a social choice correspondence satisfying Axioms S and LI, 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 h(t) (fi (E, a), ≿i ) is non-empty for each t ∈ N , where the intersection is over all consumers andeconomies 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
12 In social choice theory, it is known that if we restrict the domain of preferences to the class of linear orderings, theincentive compatibility necessarily means the monotonicity (see Mas-Colell et al. 1995, p.811, Proof of Theorem 21.E.2).
For a general equilibrium setting, we cannot restrict preferences to be an element of such a narrow domain. For a message
mechanism argument, however, since we can fix a certain message a, Axiom S enable us to use Axiom IC in the same way
with Axiom M (see the proof of Theorem 6, below).
13 For Axiom LI, see (2) of Theorem 1. For Axiom S, see footnote 7 of Urai and Murakami (2016b).

7

compact, and because each of the sets in the collection from which we are forming the intersection is closed,∩m
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 agents in I(t) and economies, by Axiom S there existsE∗ ∈ Econ∗ containing {i1 , i2 , . . . , im } and a ∈ µ(E∗ ), such that the equilibrium list, (fi (E∗ , a))∞i=1 , is anm∞extension of (fis (Es , a))s=1 . Because (fi (E∗ , a))i=1 is an element of g(E), the allocation is weakly Paretooptimal, so by Balasko and Shell (1980) and Esteban (1986), it is supported by a price as a price-wealth∩mequilibrium, and thus s=1 h(t) (fis (Es , a), ≿is ) is non-empty. Moreover, because for some economy and its
agents in generations in 1 to t, their intersection of h(t) (fi (E, a), ≿i ) is a singleton and its unique element
∩
K(t)
K(t)
belongs to ∆++ , 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, LI and Theorem
1. Let ϕ(a) be (ϕ1 (a), ϕ2 (a)) ∈ P × M. In order to establish the theorem, it is suﬃcient to show thatfor each economy E∗ ∈ Econ∗ and a ∈ A, an allocation y ∗ = (yi∗ ) = (fi (E∗ , a)) is such that for each i,yi∗ = fi (E∗ , a) satisfies ME∗ (i) = ϕ1 (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.14 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 LI (see footnote 10). 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 thecommutativity of the diagram with Axiom LI for (A, µ, f ) means that Axiom LI is satisfied on ϕ(A).

■By Theorem 2, from every message mechanism (A, µ, f ) based on a WPO-IR compatible social choicecorrespondence satisfying Axiom S and LI , there exists a unique price dictionary function, ϕ : A → P×M.
In other words, the results of such message mechanisms can be realized universally and eﬃciently throughthe price-money message mechanism (P × M, π, e). Thus we have obtained the price-money dictionary
theorem as in our previous paper (Theorem 2 of Urai and Murakami 2016b: based on a WPO and finite
core compatible social choice correspondence).
We can also obtain an isomorphism theorem for the price-money message mechanism (Theorem 3 ofUrai and Murakami 2016b) as follows. Denote by PM∗LI 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 LI. The following axiom on the dependenceof monetary messages on the economic structure is necessary to show the second assertion.
∞∪
Axiom D (Dependency of Money Supply on E) : 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.

14 It is always possible to add finite agents in constructing economy E in Axiom S.
∗

8Theorem 3 (Isomorphism Theorem under Axiom LI) : Consider the restriction of price-money message mechanism (PM∗LI , π, e). Let (P ′ , π ′ , e′ ) be a message mechanism based on a WPO-IR compatiblesocial choice correspondence on Econ∗ . If (P ′ , π ′ , e′ ) satisfies Axioms S and LI, and if, for every message mechanism (A, µ, f ) satisfying Axioms S and LI, there exists a unique mapping ϕ′ : A → P ′ suchthat f (E, a) = e′ ◦ [1Econ∗ × ϕ′ ](E, a), then (i) there exists an isomorphism (bijection) h′ such thath′ : PM∗LI → P ′ and e = e′ ◦ [1Econ∗ × h′ ]. (ii) Moreover, if monetary messages satisfy Axiom D, we canrestrict the problem on spaces with topological (resp. diﬀerentiable on each inverse-system component)
structures and continuous mappings (resp. diﬀerentiable coordinate mappings) to obtain continuous (resp.
diﬀerentiable on each component space) unique ϕ in the previous theorem, so that isomorphism h′ can betaken as a homeomorphism (resp. diﬀeomorphism for each component space).15
Proof : Because (PM∗LI , π, e) is now assumed to be a message mechanism based on a WPO-IR compatiblesocial choice correspondence satisfying Axioms S and LI, we have the next diagram by assumption.
(R∞ )N e′Econ∗ × P ′

@
I e
@
@

6
1Econ∗ × ϕ′Econ∗ × PM∗LI

Moreover, because (P ′ , π ′ , e′ ) is also a message mechanism based on a social choice correspondence
satisfying Axiom S and LI, the previous theorem shows that we have the next diagram.
(R∞ )N eEcon∗ × PM∗LI
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∗LI toPM∗LI 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, by considering on PM∗LI , the relativised topology of the product topology
∏
∞
of R+
× E∈Econ (R♯E ), where ♯E denotes the cardinal number of the set of agents in E, we can easily
check the continuity (resp. diﬀerentiability on each component space) of the mapping ϕ in Theorem 2
through the local homeomorphism (resp. diﬀeomorphism) property at each point of the domain of the
15 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∗LI . 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∗LI 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 .

9excess demand function that is appropriately chosen for each E by using Axiom D. Then the proof of thefirst assertion is suﬃcient for assuring the isomorphism poroperty of h′ .
■Hence, Theorem 3 asserts that if we restrict the domain of the price-money messages to where Axioms
S and LI 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 LI).
Theorem 4 (Price-Money Dictionary Theorem under Axiom M) : If (A, µ, f ) is a message mechanism
based on a WPO-IR compatible social choice correspondence, 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)
6
1Econ∗ × ϕ

@
I f
@
@

Econ∗ × A
Proof : (i) We can repeat the argument in the first paragraph in the proof of Theorem 2 and obtain the∩K(t)single point p(t) in h(t) (fi (E, a), ≿i ) ⊂ ∆++ , where the intersection is over all consumers in generations 1 to t and economies in Econ∗ . Also by definition of h(t) , for all t′ ≦ t, we have p(t′ ) = ϱt′ t (p(t)),
and obtain a unique element p ∈ P by identifying it with the unique element of the projective limit∩m
lim s=1 h(t) (fis (Es , a), ≿is ) ⊂ ∆++ . Let us define ϕ1 (a) as this unique element, p, 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 inthe 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 eacheconomy E∗ ∈ Econ∗ and a ∈ A, an allocation y ∗ = (yi∗ ) = (fi (E∗ , a)) 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 ≿i of i to ≿′ithat is obtained through the homothetic transformation of the surface at yi∗ under ≿i (see Figure 1).16≿

16 Let U (y ∗ ) be the indiﬀerent surface at y ∗ of i under∗∗∗iii . By IR, yi ̸= 0. By the strict monotonicity, αU (yi ), α > 0
Ks
Ksi
covers the R+ . 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).

10ωis
ϕ1 (a)ssyi∗
0≿i in E∗∗ · · · ≿′i in E′∗∗s

Figure 1: Homothetic Transformation 1
Consider the economy E′∗∗ such that the preference of i in E∗∗ is replaced with ≿′i . By Axiom M,
ˆ ′ satisfies ωi ≿
ˆ ′ y∗
ˆ ′ satisfying condition (OG3) such that ≿fi (E′∗∗ , a) = yi∗ . Change the preference of i to ≿
i
i
i i
′
′
ˆ , and ϕ1 (a) remains̸ ˆ i ωi , the better set of i at yi∗ under ≿′i is included by that of i at yi∗ under ≿and yi∗ ≿ito be a supporting hyperplane of the better set (see Figure 2).

ωis
ϕ1 (a)
s fi (E′′ , a)
∗∗

s
yi∗
0

≿i in E∗∗ · · · ≿′i in E′∗∗
ˆ ′ · · · ≿′′ in E′′
≿
∗∗
i
i

s

Figure 2: Homothetic Transformation 2
ˆ ′ , of i by using the indiﬀerence
Moreover, consider the homothetic transformation of the preference, ≿isurface at yi∗ , and call it ≿′′i . Let us denote by E′′∗∗ the economy where we replace the preference ≿′i of iwith ≿′′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 sets of i and j at fi (E′′∗∗ , a) and fj (E∗∗ , a), respectively, by
the WPO property of f (E∗∗∗ , a), it follows that fi (E′′∗∗ , a) must be diﬀerent from yi∗ by the IR property
11

of f (E∗∗∗ , a) under ≿′′i . The point, fi (E′′∗∗ , a), should, however, be equal to some point zi at which theindiﬀerence surface under ≿′′i of the point is supported by ϕ1 (a). That is, fi (E′′∗∗ , a) = zi ̸= yi∗ = fi (E′∗∗ , a).
This is a contradiction since fi (E′∗∗ , a) is equal to fi (E′′∗∗ , a) under Axiom M.
(ii) Repeat the arguments in the proof (ii) of Theorem 2 (replace Axiom LI 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 a WPO-IR compatiblesocial choice correspondence 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 ′ suchthat f (E, a) = e′ ◦ [1Econ∗ × ϕ′ ](E, a), then (i) there exists an isomorphism (bijection) h′ such thath′ : PM∗M → P ′ and e = e′ ◦ [1Econ∗ × h′ ]. (ii) Moreover, if monetary messages satisfy Axiom D, we canrestrict the problem on spaces with topological (resp. diﬀerentiable on each inverse-system component)
structures and continuous mappings (resp. diﬀerentiable coordinate mappings) to obtain continuous (resp.
diﬀerentiable on each component space) unique ϕ in the previous theorem, so that isomorphism h′ can betaken as a homeomorphism (resp. diﬀeomorphism for each component space).
Proof : Repeat the proof of Theorem 3 (replace Axiom LI and PM∗LI respectively with Axiom M andPM∗M ).
■
We can also obtain the dictionary and the isomorphism theorems under Axiom IC instead of M. See thefollowing Theorems 6 and 7.
Theorem 6 (Price-Money Dictionary Theorem under Axiom IC) : If (A, µ, f ) is a message mechanism
based on a WPO-IR compatible social choice correspondence, and if (A, µ, f ) satisfies Axioms S and IC,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 IC.
R∞



e

Econ∗ × (P × M)

@
I f
@
@

6
1Econ∗ × ϕEcon∗ × A

Proof : (i) Follow the arguments in the first four paragraphs of the proof of Theorem 4. By appropriatelychanging the reference to Axiom M with Axiom IC, obtain the homothetically transformed preference, ≿′i ,of the indiﬀerent surface at yi∗ of ≿i .
Consider the economy E′∗∗ such that the preference of i in E∗∗ is replaced with ≿′i (see Figure XX).
Let yi∗∗ be fi (E′∗∗ , a). Consider an economy E∗∗∗ including i ∈ E′∗∗ and i ∈ E∗∗ as diﬀerent agents suchthat message a is an equilibrium message for E∗∗∗ under responses yi∗∗ and yi∗ through Axiom S. By the
WPO property of equilibrium allocations, yi∗∗ cannot belong to the indiﬀerence surface of yi∗ in E∗∗ that12

is necessarily equal to the indiﬀerence surface of yi∗ in E′∗∗ except for the case that yi∗ = yi∗∗ . Indeed,y ∗ +y ∗∗if yi∗ ̸= yi∗∗ , allocation among these two agents can be improved by replacing yi∗ and yi∗∗ with i 2 i inE∗∗∗ without changing any other agents’ allocations. Hence, if yi∗ ̸= yi∗∗ , there are only two cases: (i-i)
yi∗∗ = fi (E′∗∗ , a) is preferred to yi∗ for i in both E∗∗ and E′∗∗ , and (i-ii) yi∗ is preferred to yi∗∗ = fi (E′∗∗ , a) fori in both E∗∗ and E′∗∗ . In Case (i-i), for i in E∗∗ , there is an incentive to misrepresent his/her preference

as the one in E′∗∗ , and in Case (i-ii), for i in E′∗∗ , there is an incentive to misrepresent his/her preference
as the one in E∗∗ . It follows that we must have yi∗∗ = fi (E′∗∗ , a) = yi∗ under Axiom IC.
ˆ ′ satisfying (OG3) such that ≿ˆ ′ satisfies ωi ≿
ˆ ′ y ∗ and y ∗ ̸ ≿ˆ ′ ωi ,Now, change the preference of i to ≿
i
i
i
i i
i
ˆ ′ , and ϕ1 (a) remains to be a
the better set of i at y ∗ under ≿′ is included by that of i at y ∗ under ≿
i

i

i

i

supporting hyperplane of the better set (see Fig. xxx). Moreover, consider the homothetic transformation
ˆ ′ , of i by using the indiﬀerence surface at y ∗ , and call it ≿′′ . Let us denote by E′′ theof the preference, ≿ieconomy where we replace the preference ≿′i of i with ≿′′i .

i

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∗∗∗ . Point fi (E′′∗∗ , a) must be diﬀerent from and strictly preferred to yi∗ under ≿′′i by the
IR property of f (E∗∗∗ , a). That is, fi (E′′∗∗ , a) ≻′′i fi (E′∗∗ , a) = yi∗ , which also means that fi (E′′∗∗ , a) ≻′i yi∗ .

Indeed, fi (E′′∗∗ , a) that is known to be diﬀerent from yi∗ = yi∗∗ = fi (E′∗∗ , a), must be a part of weakly
Pareto optimal allocation in E∗∗∗ and have ϕ1 (a) as a supporting price for the ≻′′i -better set. Since ≻′′i

is homothetic, point fi (E′′∗∗ , a) must belong to the closed half line from 0 containing yi∗ satisfying thecondition “≿′′i -preferred to yi∗ .” By the monotonicity of preferences, the subset of closed half line from 0
containing yi∗ satisfying the condition “≿′′i -preferred to yi∗ ” is equal to the subset of closed half line from 0

containing yi∗ satisfying the condition “≿′i -preferred to yi∗ .” It follows that fi (E′′∗∗ , a) ≻′i yi∗ . This, however,
implies that there is an incentive for agent i in E′∗∗ to misrepresent his/her preference as the one in E′′∗∗ ,a contradiction under Axiom IC.
(ii) Repeat the arguments in the proof (ii) of Theorem 2 (replace Axiom LI with Axiom IC).

■

Denote by PM∗IC the set of all (p, M ) ∈ P × M which is an image of ϕ for some (A, µ, f ) in Theorem 6
satisfying Axioms S and IC.
Theorem 7 (Isomorphism Theorem under Axiom IC) : Consider the restriction of price-money message mechanism (PM∗IC , π, e). Let (P ′ , π ′ , e′ ) be a message mechanism based on a WPO-IR compatiblesocial choice correspondence on Econ∗ . If (P ′ , π ′ , e′ ) satisfies Axioms S and IC, and if, for every message mechanism (A, µ, f ) satisfying Axioms S and IC, there exists a unique mapping ϕ′ : A → P ′ suchthat f (E, a) = e′ ◦ [1Econ∗ × ϕ′ ](E, a), then (i) there exists an isomorphism (bijection) h′ such thath′ : PM∗M → P ′ and e = e′ ◦ [1Econ∗ × h′ ]. (ii) Moreover, if monetary messages satisfy Axiom D, we canrestrict the problem on spaces with topological (resp. diﬀerentiable on each inverse-system component)
structures and continuous mappings (resp. diﬀerentiable coordinate mappings) to obtain continuous (resp.
diﬀerentiable on each component space) unique ϕ in the previous theorem, so that isomorphism h′ can betaken as a homeomorphism (resp. diﬀeomorphism for each component space).
Proof : Repeat the proof of Theorem 3 (replace Axiom LI and PM∗LI respectively with Axiom IC andPM∗IC ).
■

13

5

Conclusion

We have thus obtained three kinds of price-money dictionary theorems (Theorem 2, 4 and 6) and theisomorphism theorems (Theorem 3, 5 and 7). It can be said that the price-money dictionary theorem,
which asserts that the price-money message mechanism can be referenced uniquely and universally among
the category 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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15