Discussion Papers In Economics
And Business

Universality and Efficiency of Price Mechanism basedon an Expansion Possibility of Economies
Kohzo Shiraishi
Ken Urai
Hiromi Murakami
Discussion Paper 16-10

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANUniversality and Efficiency of Price Mechanism basedon an Expansion Possibility of Economies
Kohzo Shiraishi
Ken Urai
Hiromi Murakami
Discussion Paper 16-10

May 2016

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANUniversality and Eﬃciency of Price Mechanism basedon an Expansion Possibility of Economies∗
Kohzo Shiraishi†
Ken Urai‡
Hiromi Murakami§

Abstract
In this paper, we axiomatically characterize the universality and eﬃciency of price mechanism basedon an expansion possibility of economic environments like trading opportunities, immigrants, agents’consumption sets through the cultural globalization, and so on. Together with some generalized
settings like economy dependent message-response structure, we base our argument on the frameworkof Sonnenschein (1974). In Sonnenschein (1974), the price mechanism is characterized in more specific
way than that of Hurwicz (1960), Mount and Reiter (1974), etc. for the informational eﬃciency
problem. His model, however, has an advantage to describe the universality and eﬃciency of the pricemechanism from the category-theoretic viewpoint through the basic economic tool of the excess demand
function. His approach also enables us to characterize the price mechanism through cooperative gametheoretic settings like the core quivalence that is closely related to the replica stability axiom of social
choice settings like Thomson (1988) and Nagahisa (1994).

Keywords: Price Mechanism, Axiomatic Characterization, Informational Eﬃciency, Universality, Message Mechanism, Cultural Globalization
JEL classification: D50, D71∗ Part of this research was supported by JSPS KAKENHI Grant Numbers 25380227 and 15J01034.
† Faculty of Economics, Kobegakuin University, kporco@eb.kobegakuin.ac.jp‡ 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

An axiomatic characterization of the price mechanism as an eﬃcient allocation mechanism provides
one of the most important progresses in the general equilibrium theory (see, e.g., Hurwicz 1960, Mount
and Reiter 1974, Sonnenschein 1974, Osana 1978, Jordan 1982, etc.). Such arguments are concernedwith problems on the desirability of the price mechanism, which gives a path-breaking framework for thedevelopment in the 1970s and 80s of the field called the mechanism design.
In this paper, we base our argument on the framework of Sonnenschein (1974), and axiomaticallycharacterize the universality and eﬃciency of price mechanism from the category-theoretic viewpoint
together with an expansion possibility of economies. Our arguments include the problems concerning not
only the simple expansion of trading opportunities, expansion of members like immigrants, the increaseof women’s labor force and so forth, but also expansions of agents’ consumption sets caused by cultural
enlightenments and/or globalization.
Although the message mechanism of Sonnenschein (1974) is characterized in more specific way thanthat of Hurwicz (1960) or Mount and Reiter (1974), his approach has advantages such that (i) it describes
the category-theoretic universality and eﬃciency of the price mechanism through the elementary economic
tool of the excess demand function, (ii) it allows for the message-response structure partially depending on
the economies, and (iii) it enables us to characterize the price mechanism by cooperative game theoretic
setting of the core equivalence that is closely related to the replica stability axiom of social choice settings
like Thomson (1988) and Nagahisa (1994).
It has generally been said that the market or price mechanism has a close relation to the progress ofglobalization. There seems, however, few articles in the economic theory that is successfully abstract theidea under the purely mathematical general equilibrium framework. In this paper, by using Sonnenschein’scategory theoretical axiomatic characterization of the price mechanism, we show that the price mechanismis the unique message mechanism that are compatible with the nature of the globalization.
To show our result, we have to generalize the Debreu and Scarf (1963) core equivalence theorem so as toinclude the cases in which agents’ consumption sets may not have full dimensions in the commodity space.
We overcome this problem by adjusting the feasibility and preference conditions as well as generalizing
the Sonnenschein’s axiom S and message dependency on the economic structure. In the above sense,
this paper is a generalization of the model and results of Sonnenschein (1974) to the cases incorporating
partially economy-dependent messages for his response function, and figure out the relationship between
the price mechanism and the expansion possibility of an economy.
In this paper, we identify the extension of an economy in the axiom of Sonnenschein (1974; AxiomS) with the real expansion of the size of an economy including an expansion of consumption set for eachconsumer. We can interpret such situation, especially, the dependency of agents’ responses for messages on
the totality or size of an economy, as the cultural globalization. In other words, the opportunity for tradesof participants in an economy is generally expanded if there is a possibility for a society to commodify orcommoditize something that was not previously considered as a commodity.
To allow for the possibility that the function of messages is partially dependent on the economy as wellas on the individual characteristics, is quite natural in treating such a cultural problem together withthe expansion of the economy. Moreover by doing so, we can discuss both the planned and decentralized
economic systems in the universal class or domain of message spaces to characterize the market pricemechanism.
Our conclusion is as follows. Assume that the messages are restricted to those defining core allocations

1and having the property descrived like the axiom of Sonnenschein together with some minor natural
conditions for a universal class of economy dependent message spaces.1 Then, the only mechanism thatsatisfies these conditions is the market price mechanism.

2

The Model

In this paper, we use I, I ′ , · · · etc. as finite index sets of agents. Note that such an index will be used
independently with the notations for economies like E, E ′ , · · ·, so we sometimes use I as a set of agents inboth E and E ′ . Let I be the set of agents in economy E. Economy E consists of the feasible consumption
set, the preference preordering and the initial endowment for i ∈ I, denoted respectively by XiE , ≾Ei and
ωiE . For each i ∈ I, we define XiE as a subset of Rℓ such that there exists a finite set of coordinates,
Ki ⊂ {1, 2, · · · , ℓ} satisfying xi ∈ XiE only if t-th coordinate of xi is 0 for all t ∈/ Ki . The preferenceE
EE
preordering of i in economy E, ≾i , is a subset of Xi × Xi and the initial endowment of i in E, ωiE , isKi 2ℓan element of R+
such that PrRKi ωiE ∈ R++.
We can write an economy, E, as E = (I, ≾E , ω E ), where ≾E and ω E are identified with functions on I,

i.e., for each i ∈ I, ≾E (i) = ≾iE and ω E (i) = ωiE . In this paper, we suppose that the preference, ≾Ei , isE
E
E
represented by a utility function of each individual, ui : Xi → R, and each ui satisfies continuity, strict
monotonicity and strict quasi-concavity (strict convexity in the sense of Debreu 1959).3 Moreover, werequest each economy to satisfy the next resource relatedness or irreducibility condition.
(Irreducibility): In an economy, for each agent i and j, there exists a chain of agents, i0 =i, i1 , · · · , im = j, such that Kit ∩ Kit+1 ̸= ∅ for all t = 0, 1, · · · , m − 1 (see Clark 1979; Lemma2,
Irreducibility).
For each E = (I, ≾E , ω E ), sequence (xi ∈ XiE )i∈I is called an allocation for E. Allocation (xi ∈ XiE )i∈I
is said to be feasible if

∑

xi ≦

i∈I

∑

ωi ,

(1)

i∈I

where the ordering ≦ on Rℓ is defined as x ≦ y iﬀ for each k = 1, 2, · · · , ℓ, k-th coordinate of x is less thanor equal to the k-th coordinate of y. A coalition in economy E = (I, {(≿i , ωi )}i∈I ) is a set of agents S ⊂ I.
Feasible allocation x is said to be the core allocation if there is no coalition S, and no y = (yi )i∈S satisfies∑∑(a) i∈S yi ≦ i∈S ωi , and (b) yi ≿ xi for all i ∈ S and yi ≻ xi for at least one i ∈ S. We call the set of
all core allocations the core of economy E and denote it by Core(E). Allocation x is said to be blocked bycoalition S if conditions (a) and (b) hold.
Now, let us define the concept of expansion of economy E. In our setting, the expansion of an economy
may cause the change of each agent’s characteristics and the change will be described by the extension of
the feasible consumption set and initial endowments.

1 Sonnenshcein’s axiom, Axiom S, is concerning about the simplicity of messages such that the small part of an economy
cannot have a big influence on the entire economy.
2 In this paper, we canonically identify RKi with the subset of Rℓ such that {x = (x , · · · , x ) ∈ Rℓ | x = 0 for all t ∈/ Ki }.
t
1ℓ
Ki
The range of the projection, PrRKi , from Rℓ to RKi and the set R++= {(xj )j∈Ki | xj > 0 for all j ∈ Ki .} are also identified
with subsets of Rℓ .
3 For each x ∈ X E , and for each v ∈ RKi \ {0}, we assume x + v ≻ x. To assure the resource relatedness among agentsii+iℓ , we use thein economy E where each agent i ∈ I does not necessarily have full-dimensional consumption set, XiE ⊂ R+
strict monotonicity condition for preferences. The condition can be dropped if we assume a certain kind of insatiability forthe society (see Murakami and Urai 2016).

2

We write the set of economies as Econ. In the following, we define a message mechanism on an economy.
Let A be a set. Given a message, a ∈ A, we assume that for each economy E = (I, ≾E , ω E ), allocation∏f (a, E) = (f i (a, E))i∈I ∈ i∈I XiE is defined. We call f on A × Econ such that (a, E) 7→ f (a, E) ∈∏E
i∈I Xi a response function. In addition, we consider an equilibrium correspondence µ : Econ ∋ E 7→µ(E) ⊂ A. As in Sonnenschein (1974), given a correspondence, g, that defines for each economy E =(I, ≾E , ω E ) a subset of its feasible allocations, we call the triple (A, µ, f ) as an abstract message mechanism(a resource allocation mechanism with messages) based on social choice correspondence g, if
g(E) = {(f i (a, E))i∈I | a ∈ µ(E)}.
′

(2)

′

For economies, E = (I, ≾E , ω E ) and E ′ = (I ′ , ≾E , ω E ), we write E ,→ E ′ to mean that (i) I ⊂ I ′ , (ii)
E′
E
E′
E
E
I′
I′
I
I′
E
E
for ≾Ei ⊂ Xi × Xi and ≾i ⊂ Xi × Xi , we have Xi ⊂ Xi and ≾i = Xi × Xi ∩ ≾i for each i ∈ I,′′
and (iii) for each i ∈ I, ωiE ≦ ωiE . We simply write E ⊂ E ′ if E ,→ E ′ and XiE = XiE for all i ∈ I.
Let us consider the following axioms for f and µ.
(C1 ) Responses are invariant for the expansion of the economy. (Mechanism is decentralized.) That is, ∀a ∈ A, ∀E ∈ Econ, ∀E ′ ∈ Econ, E ⊂ E ′ ,f (a, E) is a restriction of f (a, E ′ ) on members of E.

(3)(C2 ) Equilibrium responses are core compatible. That is, ∀a ∈ A, E ∈ Econ,a ∈ µ(E) =⇒ f (a, E) ∈ g(E) ⊂ Core(E).

(4)(C3 ) Mechanism satisfies Sonnenschein’s Axiom S. That is, for each economy E and eachmessage a ∈ A, there exists an economy E ′ ⊃ E such that a is an equilibrium message for E ′ .
ℓDefine the set of price vectors, P , as P = {(p1 , · · · , pℓ ) ∈ R+|∑ℓ

k=1 pk= 1}. The

price mechanism is an allocation mechanism with messages, (P, π, e), where for each E =(I, ≾E , ω E ) ∈ Econ, π(E) ⊂ P denotes the set of all competitive equilibrium prices for E and∏for each p ∈ P , e(p) = (ei (p))i∈I ∈ i∈I XiE is the list of the value of each agent’s excess
demand function.4
R∞



e

@ f
I
@
@

C×P
6
1C × hC×A

Figure 1: Commutative Diagram for the Universal Mapping Problem in Sonnenschein (1974)

4 The excess demand function exists since each agent’s utility function is strictly quasi-concave.

3The commutative diagram in Figure 1 with respect to the class of agents’ characteristics, C,
information sets, excess demand structure and any equilibrium structures satisfying the axioms(C1 ), (C2 ) and (C3 ) was proved in Sonnenschein (1974) as the next proposition.

PROPOSITION (Sonnenschein 1974; Propositions 1 and 7) : If (A, µ, f ) is amessage mechanism based on a social choice correspondence g, and if (A, µ, f ) satisfies axioms(C1 ), (C2 ) and (C3 ), then there exists a unique function h : A → P such that the triangle in
Figure 1 commutes (Dictionary Property). Moreover, the mechanism that can play the above
dictionary property is unique up to isomorphism (Universal Mapping Property).

3

Examples, Lemma and Theorem

Now, we consider a more general conditions to characterize the price mechanism under the
globalization or the economic expansion framework. In the following, the axioms, (C1 ), (C2 )and (C3 ), are reformulated by making allowance for the expansion concept of economies. The
private representation axiom, (C1 ), will be dissolved and absorbed into our new axiom (C3′ ), an
analogue of Sonnenschein’s axiom S incorporating the economy depending response functions
and the meanings of economic expansion as the enlightenment under cultural globalization.
Let us consider the following condition.
(C3′ ) For each message a ∈ A, for each finite list of economies E 1 , E 2 , · · · , E m and the∗∗list of responses f (E 1 , a), f (E 2 , a), · · · , f (E m , a), there exists economy E ∗ = (I ∗ , ≾E , ω E ),E 1 ,→ E ∗ , · · · , E m ,→ E ∗ , including all members of E 1 , E 2 , · · · , E m as diﬀerent agents, suchthat a is an equilibrium message for E ∗ under which the equilibrium list (fi (E ∗ , a))i∈I ∗ is anextension of f (E 1 , a), f (E 2 , a), · · · , f (E m , a).
Instead of the conditions (C1 ) and (C3 ) in the previous section, we use the condition (C3′ )saying that for any message a and for any m-sequence of agents and economies with various
stages of enlightenments, there is a suﬃciently large economy E ∗ such that E ∗ includes all mmembers and a is an equilibrium message for E ∗ . This is a natural extension of Sonnenschein’s
axiom S or (C3 ) and his setting (C1 ) to our situation including economic expansion and canbe interpreted as the feature of messages such that “any finite individuals can be swamped”
or as “a restriction on messages to be simple in the sense that they do not carry too muchinformation about the membership of an economy” (see Sonnenschein 1974).
In the previous section, (C1 ) indicates that the responses of agents do not depend on
the scale of the economy. The above extended axiom (C3′ ) of (C1 ) and (C3 ) generalizes theprivate representation (decentralization) settings in Sonnenschein (1974), enabling us to discussproblems such as cultural globalization under the changes of consumption sets like Xi ⊂ Xi′
through an expansion of the economy. We give some examples in the following.
Example 1: (Dietary Culture) Under the expansion of the economy, if a certain commodity that were not thought to be edible becomes a commodity for food in the new culture,
we can treat such a situation by an expansion of feasible consumption set Xi of each consumer,4

e.g., sushi and sashimi culture for Western countries, the dietary culture of milk and beef forJapanese in Meiji era, and so on.
Example 2: (Female labor force)

We can handle the situation that the woman labor

force can be utilized for newly possible purposes in the new culture after the expansion of an
economy. By considering the labor as negative consumption good, the extension of a dimensionof the consumption set will appropriately describe the situation.
Example 3: (Immigrants) It can be said that the inflow of immigrants is an expansionof the economy. If the labor force of immigrants is appreciated in the expanded economy morethan the original one, we can describe it as an extension of feasible consumption set. In thiscase, we consider many kinds of labor forces of immigrants as negative consumption goods, e.g.,
immigration from the society where the baby-sitting, window cleaning, etc., are not recognized
as a wage labour, emigration to the country where a person with a high IT ability is more
appreciated, and so on.
Example 4: (Free Trade Zone)

If we ignore the problem of monopolistic market powerof global multinational corporations, we can interpret the extension of feasible consumption
set like here, as treating the merit of the normal free trade argument under the pure exchange
framework (without considering the production). In this case, we can treat the expansion in
the positive direction of the production set as the expansion in the negative direction of theconsumption set.
Example 5: (Multinational Corporation) Our model does not include the production.
However, if we identify the negative consumption (e.g., the supply of the labor force) as inputs
for a certain production, we can treat the problem such that after the expansion of an economy adomestic worker has a possibility to be hired by a multinational cooperation in another country.
The situation is also possible to be considered an expansion of his consumption set.
To our result, it is necessary to generalize the Debreu-Scarf (1963) core equivalence theorem
to our framework, so that the consumption sets of agents may not have full dimensions in thecommodity space Rℓ .

LEMMA: (Debreu-Scarf Limit Theorem without Full Dimensional Consumption Sets) In our settings of section 2, for every economy E = (I, ≾E , ω E ) ∈ Econ, feasible
allocation x for E is a competitive equilibrium allocation if its N -fold replica allocation belongs
to Core(E N ) for every positive integer N , where E N denotes the N -fold replica economy of E.
■

PROOF: See Appendix A.

To show the extension theorem of Sonnenschein (1974; Propositions 1 and 7), it is necessary
to describe the commutative diagram in Figure 1 by considering the dependence of responses
on economy E.

5

R∞



e

@ f
I
@
@

Econ × P
6
1Econ × hEcon × A

Figure 2: A Commutative Diagram for Economy Dependent Message Mechanisms

THEOREM: If (A, µ, f ) is a message mechanism based on social choice correspondenceg, and if (A, µ, f ) satisfies axioms (C2 ) and (C3′ ), then there exists a unique function h : A → P
such that the triangle in Figure 2 commutes (Dictionary Property). Moreover the mechanismthat can play the above dictionary property is unique up to isomorphism (Universal Mapping
Property).
■

PROOF: See Appendix B.

REFERENCES
Clark, S. A. (1979): “Pareto-Optimality in the Pure Distribution Economy with an Infinite
Number of Consumers and Commodities,” Journal of Economic Theory 21, 336–347.
Debreu, G. (1959): Theory of Value. Yale University Press, New Haven, CT.
Debreu, G. and Scarf, H. (1963): “A limit theorem on the core of an economy,” International
Economic Review 4, 235–246. Reprinted in G. Debreu, Mathematical Economics, pp.
151-162. Cambridge University Press, Cambridge, 1983.
Hurwicz, L. (1960): “Optimality and Informational Eﬃciency in Resource Allocation Processes,” in Mathematical Methods in the Social Sciences 1959, (Arrow, K. J., Karlin, S.,
and Suppes, P. ed) , pp. 27–46, Stanford University Press, Stanford. Also in Readings in
Welfare Economics, edited by K.J.Arrow and T.Scitovsky. Irwin, New York, 1969.
Jordan, J. S. (1982):“The Competitive Allocation Process is Informationally Eﬃcient

Uniquely,” Journal of Economic Theory 28, 1–18.
Mount, K. and Reiter, S. (1974): “The Informational Size of Message Spaces,” Journal of
Economic Theory 8, 161–192.
Murakami, H. and Urai, K. (2016): “Replica Core Limit Theorem for Economy with Satiation,”
Discussion Paper No. 16-09, Faculty of Economics and Osaka School of International
Public Policy, Osaka University.
Nagahisa, R. (1994): “A Necessary and Suﬃcient Condition for Walrasian Social Choice,”
Journal of Economic Theory 62, 186–208.
Osana, H. (1978): “On the Informational Size of Message Spaces for Resource Allocation
Processes,” Journal of Economic Theory 17, 66–78.
Schaefer, H. H. (1971): Topological Vector Spaces. Springer-Verlag, New York/Berlin.

6

Sonnenschein, H. (1974): “An axiomatic characterization of the price mechanism,” Econometrica 42(3), 425–433.
Thomson, W. (1988): “A Study of Choice Correspondences in Economies with a Variable
Number of Agents,” Journal of Economic Theory 46, 237–254.

Appendix A
Proof of LEMMA
This lemma can be proved in the similar way to the proof of Debreu and Scarf (1963;
Theorem 3) except for the continuity argument for suﬃciently large k concerning the formationof the blocking coalition (in our proof, see the limit argument around equation (5) usingKi
projection on R+
and the weak feasibility condition) and positive price argument based on
the minimum wealth condition (in our proof, see the paragraph that contains footnote 6). For
the sake of completeness, we shall give a simple sketch of the total proof.

Let x̄ = (x̄i )i∈I be a feasible allocation for economy E = (I, ≾E , ω E ) such that every N fold replica allocation of x̄ belongs to Core(E N ) for all N ≧ 1. Define for each i ∈ I, Γi as
Γi = {zi | zi + ωiE ≻i x̄i } ⊂ RKi where Ki denotes the set of commodities such that xi ∈ XiEiﬀ t-th coordinate of xi is 0 for all t ∈/ Ki . Then, take the convex hull Γ of finite union∪ℓi∈I Γi ⊂ R . Since Γi is convex and non-empty for every i, Γ becomes a non-empty convex
set.
∑
We will show 0 ∈/ Γ. Let us suppose the contrary. Then, one can write i∈I αi zi = 0, with∑αi ≧ 0 and i∈I αi = 1, zi + ωiE ≻i x̄i for each i ∈ I. For suﬃciently large k, let aki be thesmallest integer greater than kαi . Also, let J be the set of all i ∈ I for which αi > 0. For eachii ∈ J, we define zik as kαz . Observe that zik + ωiE belongs to the segment [ωiE , zi + ωiE ].
ak i
i

For i ∈ J, zik +ωiE tends to zi +ωiE as k tends to infinity. Hence, we have PrRKi (zik +ωiE ) also
tends to zi +ωiE ∈ RKi , where the range of projection PrRKi , RKi , is canonically identified withthe subset of Rℓ . The continuity assumption on preferences implies that PrRKi (zik + ωiE ) ≻i x̄ifor all i ∈ J, for all k suﬃciently large. Select one of such k. Then we have∑i∈J

aki zik = k∑

αi zi = 0.

(5)

i∈J

Let us consider the replica economy E N with N = maxi∈J aki . Take the coalition, S,
composed of aki replica members of i for each i ∈ J to each one of whom we assign zik + ωiE .
This coalition blocks the allocation (x̄i ) by equation (5) and the fact that PrRKi (zik +ωiE ) ≻i x̄i∑∑for each i ∈ J, since i∈S (PrRKi (zik + ωiE )) ≦ i∈S (zik + ωiE ) under the non negativity ofzik (t) + ωiE (t) ∈ [0 = zi (t) + ωiE (t), ωiE (t)] for each t ∈/ Ki . This is a contradiction to theNdefinition of Core(E ) and our definition of the feasibility under ≦. Hence, we have established
0∈
/ Γ.

∪
Let K = i∈I Ki and π be the set of prices such that π = {p ∈ RK ∩ ∆ | p · z ≧ 0 for all z ∈Γ}, where ∆ represents the standard (♯K − 1)-dimensional simplex of RK , i.e., ∆ = {p | p =∑KKK(p1 , p2 , · · · , pK ) ∈ R+,
k=1 pk = 1}. Set π is closed in R+ and is non-empty since there

7exists p ∈ RK \ {0} by the separating hyperplane theorem.5
Ki
From p ∈ π and ωi ∈ R++
together with the irreducibility condition, it is a routine task
to show that p · ωi > 0 for all i ∈ I. Then, if a price of some commodity k ∈ K, pk , is zero,
we have a contradiction as follows. From the strict monotonicity condition for preferences andthe irreducibility, there exist some agent who demands the commodity k at x̄. We call one
such agent as i. Consider first the case that p · x̄i = 0. Then, since p · ωi > 0, let δ ∈ R++ besuﬃciently small value such that p · ωi > pk δ. A vector x̄i + (0, · · · , 0, +δ, 0, · · · , 0) − ωi suchthat x̄i + (0, · · · , 0, +δ, 0, · · · , 0) is strictly preferred to x̄i , where +δ > 0 is the k-th coordinateof a commodity, will not be non-negatively supported by p. This is a contradiction to thedefinition of Γ. Secondly, if p · x̄i > 0, we have pk = 0 and there exist a commodity k ′ ̸= k suchthat pk′ > 0 and x̄ik′ > 0. Then, a vector x̄i + (0, · · · , 0, +ϵ, 0, · · · , 0, −η, 0, · · · , 0) such thatx̄i + (0, · · · , 0, +ϵ, 0, · · · , 0, −η, 0, · · · , 0) is strictly preferred to x̄i ,6 where +ϵ > 0 is the k-th
coordinate of a commodity and −η < 0 is the k ′ -th coordinate of a commodity, will not beK
non-negatively supported by p. This is a contradiction to the definition of Γ. Hence, p ∈ R++
holds for each p ∈ π. Let us choose one of such p arbitrarily and denote it by p∗ .
For each i ∈ I, since xi ≻i x̄i means that xi − ωi belongs to Γi , we have p∗ · xi ≧ p∗ · ωi .
Moreover, for each i ∈ I, since p∗ is non-negative and the strict monotonicity holds on thispoint x̄i , we can take xi arbitrarily near to x̄i . Then we have p∗ · x̄i ≧ p∗ · ωi . Feasibility,∑∑i∈I ωi , means that x̄i satisfies the budget constraint and is an individual maximai∈I xi ≦under price p∗ . Hence, allocation x̄ is a competitive equilibrium allocation.

■

Appendix B
Proof of THEOREM
The first assertion: Dictionary Property
1

1

Let a ∈ A be a message and E 1 = (I 1 , ≾E , ω E ) ∈ Econ be an arbitrary economy. Under∗∗(C3′ ), there exists E ∗ = (I ∗ , ≾E , ω E ) such that E 1 ,→ E ∗ and a ∈ µ(E ∗ ). Without loss ofgenerality, we can assume that there is at least one individual in E ∗ who has Cobb-Douglas
utility function. Since by (C2 ), f (a, E ∗ ) ∈ Core(E ∗ ), we can take pa,E 1 ,E ∗ for supporting pricevector on both f (a, E ∗ ) and f (a, E 1 ), which is unique due to the existence of a Cobb-Douglas
utility agent in E ∗ .
Next, we show that under pa,E 1 ,E ∗ , fi (a, E ∗ ) satisfies the budget constraint for all i ∈ I ∗ .
Assume not. Then, the allocation f (a, E ∗ ) is not a Walras allocation. By our limit lemma, there
exist a positive n such that the n-fold replica allocation of f (a, E ∗ ) cannot be in the core of then-fold replica economy of E ∗ , (E ∗ )n . Hence, we have a set G of the individuals in (E ∗ )n who
can block the n-fold replica allocation of f (a, E ∗ ). By applying (C3′ ) for (E ∗ )n , by identifying
it with E ∗ ,→ E ∗ ,→ · · · ,→ E ∗ , there exists E ∗∗ such that a ∈ µ(E ∗∗ ) and E ∗ ,→ E ∗∗ .Moreoverby (C2 ), f (a, E ∗∗ ) is an element of Core(E ∗∗ ) satisfying that fi (a, E ∗ ) = fi (a, E ∗∗ ) for all i of
5 For example, consider any element z ∈ Γ. For element z ∈ Γ, in the non-negative direction of every coordinate, there
exist ωi + zi + ek that is preferred to ωi + zi by some agent, and there also exists z + ek ∈ Γ from the monotonicity condition.
Note that ek is a unit vector ek = (0, · · · , 0, 1, 0, · · · , 0) of RK where the k-th coordinate is 1. Hence, from the convexity of Γ,Γ has interior points. Concerning the separating hyperplane theorem, see, for example, Schaefer (1971; p.46, Theorem 3.1).
6 From the monotonicity and irreducibility conditions, we have (x̄ + (0, · · · , 0, +ϵ, 0, · · · , 0)) ≻ x̄ for each +ϵ > 0. Then,ii ifrom the continuity of preferences, we also have x̄i + (0, · · · , 0, +ϵ, 0, · · · , 0, −η, 0, · · · , 0) ≻i x̄i for suﬃciently small η > 0.

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(E ∗ )n . But this is impossible because G blocks the utility allocation under f (a, E ∗∗ ). Therefore
pa,E 1 ,E ∗ · (fi (a, E ∗ ) − ωi∗ ) = 0 for all i ∈ I ∗ .
Finally we check that the choice of pa,E 1 ,E ∗ does not depend on E 1 and E ∗ . Let us define
E 1′ and E ∗′ as E 1′ ,→ E ∗′ and a ∈ µ(E ∗′ ), and take the supporting price pa,E 1′ ,E ∗′ in exactly
the same way with the previous paragraph. Let us consider an economy E 2 that consistsof all members of E ∗ and E ∗′ . By applying (C3′ ) on E ∗ ,→ E 2 , we obtain E ∗∗∗1 such thatE ∗ ,→ E 2 ,→ E ∗∗∗1 and a ∈ µ(E ∗∗∗1 ). Note that f (a, E ∗∗∗1 ) is an extension of f (a, E ∗ ) and is acore allocation. Therefore the unique supporting price pa,E 1 ,E ∗ on f (a, E ∗ ) and the supporting
price on f (a, E ∗∗∗1 ) are equal. Moreover, by applying (C3′ ) again on E ∗′ ,→ E 2 ,→ E ∗∗∗1 , weobtain E ∗∗∗2 such that E ∗′ ,→ E 2 ,→ E ∗∗∗1 ,→ E ∗∗∗2 and a ∈ µ(E ∗∗∗2 ). Note that f (a, E ∗∗∗2 )is an extension of f (a, E ∗′ ) and f (a, E ∗∗∗1 ), and is a core allocation. Therefore, under the
existence of Cobb-Douglas utility agent in both E ∗ and E ∗′ , the unique supporting price on
allocation f (a, E ∗∗∗2 ), pa,E 1′ ,E ∗′ , and pa,E 1 ,E ∗ are equal.

The second assertion: Universal Mapping Property
This property is a direct result of fundamental mathematical theorem on the universal■

mapping. See, e.g., Bourbaki(1966).

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