Discussion Papers In Economics
And BusinessStability and Universal Implementability of the Price
Mechanism
Kohzo Shiraishi
Ken Urai
Hiromi Murakami
Discussion Paper 18-16

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANStability and Universal Implementability of the Price
Mechanism
Kohzo Shiraishi
Ken Urai
Hiromi Murakami
Discussion Paper 18-16

May 2018

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANStability and Universal Implementability of the Price
Mechanism∗
Kohzo Shiraishi†
Ken Urai‡
Hiromi Murakami§

Abstract
An axiomatic characterization of price or market mechanisms is one of the most important problems
in general equilibrium theory. Based on the general equilibrium framework and the characterization of
the price mechanism, this paper provides a new perspective or a unified viewpoint on some axioms insocial choice theory and a setting for the informational eﬃciency problem of the allocation mechanisms.
Our arguments focus on a contemporary reconsideration and generalization of the category theoretic
method in Sonnenschein (1974) and the replica stability arguments in social choice theory like Thomson (1988) and Nagahisa (1994). Sonnenschein’s axiomatic characterization of the price mechanism isextended to an economy-dependent welfare form of a universal implementability theorem. The framework provides new methods and general settings in treating mechanisms with messages or information,and many social choice axioms. Our result also has an important economic interpretation that theprice mechanism can be characterized as a universal rule that is stable in assuring suﬃciently highutility levels for each member of a small economy relative to its large expansions.
Keywords: Price Mechanism, Axiomatic Characterization, Category Theory, Informational Eﬃciency, Universal Implementability, Message Mechanism
JEL classification: C60, D50, D71∗ Part of this research was supported by JSPS KAKENHI Grant Numbers 25380227 and 15J01034.
† Faculty of Economics, Kobegakuin University, E-mail: kporco@eb.kobegakuin.ac.jp‡ Graduate School of Economics, Osaka University, E-mail: urai@econ.osaka-u.ac.jp§ School of Business Administration, Kwansei Gakuin University, E-mail: murakami@math.econ.osaka-u.ac.jp

1

Introduction

An axiomatic characterization of price or market mechanisms is one of the most important problems
in general equilibrium theory. Based on the general equilibrium framework and the characterization of
the price mechanism, this paper provides a new perspective or a unified viewpoint on some axioms insocial choice theory and a setting for the informational eﬃciency problem on allocation mechanisms withmessages like Hurwicz (1960), Mount and Reiter (1974), Sonnenschein (1974), and so forth. Our arguments are particularly concerned with a contemporary reconsideration and generalization of the category
theoretical axiomatic method in Sonnenschein (1974), its stability axiom for messages with respect to economic expansion, and the replica stability axiom in such social choice arguments as Thomson (1988) andNagahisa (1994). These stability axioms and characterizations of the price mechanism have closely related
theoretical structures. Such a unified view enables us to obtain a generalization of Sonnenschein’s axiomatic characterization of the price mechanism to extend his dictionary property theorem based directlyon agent-characteristics to a universal implementability theorem based on economies and utility levels or
welfare conditions.
In this paper, we characterize the price mechanism as a unique mechanism that implements the resource
allocation process of all other message mechanisms and satisfies several important axioms: core property,
replica stability of responses, Sonnenschein’s expansion possibility, etc.1 The replica stability social choiceaxiom is reformulated into an axiom on equilibrium responses to an allocation mechanism with messages.
Sonnenschein’s axiom is generalized to incorporate economy-dependent responses and arguments based on
utility or welfare levels.
Based on a standard general equilibrium setting, our framework provides a new method to treat therelation between arguments on such informational eﬃciency problems as Hurwicz and axiomatic characterizations through fundamental social choice axioms like individual rationality, Pareto-optimality, local
independency, monotonicity, and incentive compatibility.2 Moreover, it provides a contemporary gametheoretic framework of a mechanism to treat a general class of messages, allowing for the number of agentsand an equilibrium concept in an economy as variables.3 Sonnenschein’s arguments are reconstructed
and formulated here as a universal implementability theorem of the price system, insisting that the pricemechanism is a unique mechanism that can implement all other message mechanisms having equilibrium
concepts compatible with the core (Axiom C2 ) and satisfying a certain stability condition on equilibrium
messages with respect to economic expansion (Axiom C3′ ).4
Our result also characterizes an interesting feature of the price mechanism. By rigorously treating therole of Sonnenschein’s expansion possibility axiom while separating from it the replica stability propertyof responses, we determined that the condition that uniquely and universally characterizes the pricemechanism or a price message is nothing but the following feature: a message that assures a suﬃciently
1 A ground-breaking work on this axiomatic characterization problem was given in Sonnenschein (1974) as a corollary to
the Debreu-Scarf core limit theorem. His approach is important to relate the classical informational eﬃciency problem on
allocation mechanisms like Hurwicz (1960) and Mount and Reiter (1974) to a game-theoretic treatment of the mechanism.
Sonnenschein’s category theoretic characterization enables us to treat a more general class of messages and contemporarymodel settings than classical informational eﬃciency arguments. Unfortunately, the meaning and significance of Sonnenschein’s approach are not fully understood.
2 See Concluding Remark 2. Hence, the approach is meaningful to relate the classical informational eﬃciency problem on
allocation mechanisms with many social choice arguments.
3 A contemporary game theoretic interpretation of a mechanism (like Mas-Colell et al. 1995; Chapter 23) for our approach
will be treated in the Appendix.
4 See Concluding Remarks 1 and 3. We also note that the category theoretic argument or the universal implementability
is diﬀerent from the simple uniqueness about an informational size or isomorphism argument in classical studies like Osana(1978), Jordan (1982), etc. Universal implementability is a method to define a certain concept as a special object that canbe uniquely referenced in a wide class of objects that is axiomatically determined.

1

satisfactory allocation possibility for members of each small economy relative to its large expansions. Moreprecisely, price can be characterized as a message that assures each agent in small economy E a suﬃciently
good allocation in equilibrium in the following sense: with respect to the message, if it is an equilibrium
message for E under a certain (e.g., a game theoretic) equilibrium concept, one has to anticipate thepossibility that the utility level of each member for the equilibrium of a large expansion of E is at most thelevel that she obtained in E.5 Although it is true that the price mechanism is a universal rule for a global
economy, the property that ensures its universality is its stability feature as a good message for each small
economy.

2

The Basic ModelIn this paper, E, E′ , · · · and I, I ′ , · · · are used to denote economies and index sets of agents, respectively.
An economy E with index set of finite agents I consists of the preference preordering and the initialendowment for i ∈ I, denoted respectively by ≾i and ωi . For each i ∈ I, we define feasible consumptionℓset Xi as Xi = R+
. The preference preordering of i in economy E, ≾i , is a subset of Xi × Xi and theℓinitial endowment of i in E, ωi , is an element of R++.
We can write economy E as E = (≾i , ωi )i∈I . Suppose that the preference ≾i is represented by autility 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). Moreover, for the sake ofsimplicity, we restrict the class of all economies to those including at least one agent whose utility function
is diﬀerentiable.6 We write the set of economies as Econ.
For each E = (≾i , ωi )i∈I in Econ, sequence (xi ∈ Xi )i∈I is called an allocation for E. Allocationx = (xi ∈ Xi )i∈I is said to be feasible if
∑
∑
xi =
ωi .
(1)
i∈I

i∈I

A coalition in economy E = (≿i , ωi )i∈I is a non-empty set of agents S ⊂ I. Feasible allocation x is said to∑∑be a core allocation if there are no coalition S and no y = (yi )i∈S , satisfying (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 coreof economy E and denote it by Core(E). Allocation x is said to be blocked by coalition S if conditions (a)and (b) hold.
Next, 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 , ωi )i∈I , allocation f (a, E) = (fi (a, E))i∈I ∈ i∈I Xi is defined. We
call f on A × Econ a response function. It is also possible to consider the response function of each agent
i, fi , as a composition of an constraint correspondence in Xi for individual choices under message a anda profit maximization process of agent i. In this sense, our economy-dependent form of Sonnenschein’s
response function can be identified with a generalized form of the contemporary game theoretic mechanism
concept (see Concluding Remark 1 and Appendix). As in Sonnenschein (1974), we consider an equilibrium
correspondence, µ : Econ ∋ E 7→ µ(E) ⊂ A, and a correspondence, g, that defines for each economy
E = (≿i , ωi )i∈I a subset of its feasible allocations. We call the triple, (A, µ, f ), as a message mechanism

5 See Axiom (C ′ ) in Section 3. Note that in our framework, the concept of equilibrium, µ, is not confined to any restricted
3
senses. Concept µ is treated axiomatically in the relation among other concepts like message space A, response function f ,and so forth.
6 For our result, the conditions that imply the Debreu-Scarf core limit theorem will be suﬃcient. We use the diﬀerentiability
condition, however, since it simplifies many arguments and the proof of the main theorem.

2(an resource allocation mechanism with messages) based on a social choice correspondence, g, if
g(E) = {(fi (a, E))i∈I | a ∈ µ(E)}.

(2)

For economies, E = (≿i , ωi )i∈I and E′ = (≿′i , ωi′ )i∈I ′ , we write E ⊂ E′ to mean that (i) I ⊂ I ′ , (ii) for

each i ∈ I, the preference of i, ≾i is equal to ≾′i , and (iii) for each i ∈ I, ωi = ωi′ .

Now we reconstruct the category theoretic argument in Sonnenschein (1974) while explicitly treating his
several implicit settings and assumptions as independent axioms. Let us consider the following conditions
on f and µ.
Axiom (C 1 ): Responses are invariant for the extension of the economy. (Messages are not economydependent.) 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)

Axiom (C 2 ): Equilibrium responses are core compatible. That is, ∀a ∈ A, E ∈ Econ,a ∈ µ(E) =⇒ f (a, E) ∈ Core(E).

(4)

Axiom (C 3 ): Mechanism satisfies Sonnenschein’s Axiom S. That is, for each economy E and each messagea ∈ A, there exists an economy E′ ⊃ E such that a is an equilibrium message for E′ .
∑ℓℓDefine the set of price vectors, P , as P = {(p1 , · · · , pℓ ) ∈ R+
| k=1 pk = 1}. The price mechanism is
an allocation mechanism with messages, (P, π, e), where for each E = (≾i , ωi )i∈I ∈ Econ, π(E) ⊂ P is the∏set of all competitive equilibrium prices for E, and for each p ∈ P and E, e(p, E) = (ξi (p))i∈I ∈ i∈I Xi
is the list of the value of each agent’s demand function.7

Rℓ



ξ

@ f
I
@
@

C×P
6
1C × hC×A

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

The commutative diagram in Figure 1 with respect to the class of agents’ characteristics C, information
sets, demand structure and any equilibrium structures satisfying Axioms (C1 ), (C2 ) and (C3 ) was provedin Sonnenschein (1974). His results can be restated as follows.
Proposition (Sonnenschein 1974; Propositions 1 and 7) : If (A, µ, f ) is a message mechanism
based on a social choice correspondence, and if (A, µ, f ) satisfies Axioms (C1 ), (C2 ) and (C3 ), then we
have the following characterization of the price mechanism:
7 The demand function exists since each agent’s utility function is strictly quasi-concave.

3

(i) There exists a unique function h : A → P such that the triangle in Figure 1 commutes (Dictionary
Property).
(ii) Function h can be taken as continuous (or diﬀerentiable) when the domain of message spaces are
restricted on topological spaces (resp. diﬀerentiable manifolds) with continuous (resp. diﬀerentiable)
response functions.
(iii) Moreover, the mechanism that can play the above dictionary property isunique up to isomorphism (Universal Mapping Property).

3

Generalized Axioms and Theorems
Now, we consider more general conditions to characterize the price mechanism. In the following, Axioms(C1 ), (C2 ) and (C3 ) are reformulated and weakened. The economy-independent axiom of messages, (C1 ),will be replaced with the following replica stability axiom for responses, (C1′ ).8 Moreover, Sonnenschein’s
Axiom S is generalized by incorporating an economy-dependent response function together with a utility
form characterization as the following, (C3′ ).
Axiom (C ′1 ) : ∀E ∈ Econ, ∀a ∈ µ(E), ∀n, f (a, En ) = (f (a, E))n .9
Axiom (C ′3 ) : For each message a ∈ A, there exists E∗ ∈ Econ such that a ∈ µ(E∗ ). Moreover, for eachE ∈ Econ and a ∈ µ(E), and for each E′ ∈ Econ such that E ⊂ E′ , there exists E∗ = (≾∗i , ωi∗ )i∈I ∗ , E′ ⊂ E∗ ,such that a is an equilibrium message for E∗ and ui (fi (a, E)) ≧ ui (fi (a, E∗ )) for each agent i in E.
In the previous section, Axiom (C1 ) indicates that the responses of agents do not depend on the scale ofthe economy. The above extended axioms, (C1′ ) and (C3′ ), generalize the settings in Sonnenschein (1974)
in the sense that responses are restricted to equilibrium messages and allowed to be economy-dependent
except for the replica extension. Note also that Axiom (C3′ ) is given in the utility form. The framework
would enable us to discuss problems such as relation between the stability of price mechanism and an
expansion possibility of economies.
To obtain an extension theorem of Sonnenschein (1974; Propositions 1 and 7), we use an alternative of
the commutative diagram in Figure 1 by considering the dependence of responses on economy E. Now wehave a main theorem that is an extension of the proposition in the previous section. Moreover, for thispurpose, we focus our attention on equilibrium messages.10
Main Theorem: If a message mechanism based on a social choice correspondence, (A, µ, f ), satisfies
Axioms (C1′ ), (C2 ) and (C3′ ), we have the following characterization of the price mechanism:(i) There exists a unique function h : A → P such that the triangle in Figure 2 commutes for eachE ∈ Econ and a ∈ µ(E) (Universal Implementability).
8 See Thomson (1988), Nagahisa (1994), etc.

In those frameworks the replica stability concept was applied to a social
choice result or an equilibrium. Here, we use the same concept on the structure of agents’ responses to the messages.
9 For each allocation x in economy E, we denote by xn the n-fold replica allocation in the n-fold replica economy, En , forn = 1, 2, , · · ·.
10 Our approach does not restrict the domain of messages, A, when for every message a ∈ A, there exists at least one
economy E such that a ∈ µ(E) under Axiom (C3′ ). The convention therefore is suﬃcient for our purpose to characterize theuniversal implementability of the price mechanism since condition (C3′ ) as well as (C3 ) request that every message will be
an equilibrium message.

4

(Rℓ )∞ 

e

@ f
I
@
@

Econ × P
6
1Econ × hEcon × A

Figure 2: A Commutative Diagram for Economy Dependent Message Mechanisms(ii) Function h can be taken as continuous (or diﬀerentiable) when the domain of message spaces are
restricted on topological spaces (resp. diﬀerentiable manifolds) with continuous (resp. diﬀerentiable)
response functions.
(iii) Moreover, the mechanism that can play the above universal implementability property is unique
up to isomorphism (Universal Mapping Property) among the class of mechanisms satisfying Axioms(C1′ ), (C2 ) and (C3′ ).
Proof : [The first and second assertions: Universal Implementability] Let a and E be an
arbitrary pair of a message and an economy satisfying E = (≾i , ωi )i∈I ∈ Econ and a ∈ µ(E). Since a isan equilibrium message for E, f (a, E) is Pareto-optimal. By considering that at least one member of E is
diﬀerentiable, we obtain a unique supporting price p for allocation f (a, E).
Next, we show that under p, fi (a, E) satisfies the budget constraint for all i ∈ I. Assume not. Then, theallocation f (a, E) is not a Walras allocation. By Debreu-Scarf core limit theorem (Debreu and Scarf 1963;
Theorem 3), there exist a positive n such that the n-fold replica allocation of f (a, E) cannot be in the coreof the n-fold replica economy of E, En . Hence, we have a set G of the agents in En who can block then-fold replica allocation of f (a, E). Under (C1′ ), we have (f (a, E))n = f (a, En ). Furthermore, by applying(C3′ ) on En , there exists E∗ such that a ∈ µ(E∗ ), En ⊂ E∗ and ui (fi (a, E∗ )) ≦ ui (fi (a, En )) for all i of En .
It follows that, by (C2 ), f (a, E∗ ) is an element of Core(E∗ ) satisfying ui (fi (a, E∗ )) ≦ ui (fi (a, En )) for all iof En . But this is impossible because G in En blocks the utility allocation under (f (a, E))n . Therefore,p · (fi (a, E) − ωi ) = 0 for all i ∈ I. Define h as h(a) = p, then p · (fi (a, E) − ωi ) = 0 and the strict convexityof preferences necessarily implies that fi (a, E) = ξi (h(a)).
Function h is necessarily continuous (or diﬀerentiable) when we restrict the class of message spaces on
topological spaces (resp. diﬀerentiable manifolds) with continuous (resp. diﬀerentiable) response functions.
Indeed, for each a ∈ A, it is possible (e.g., by considering an economy with Cobb-Douglas utility agents,)
to choose an economy such that near h(a), a coordinate of the list of demand functions, ξi , provides
a local homeomorphism (resp. diﬀeomorphism). The composition of such local homeomorphism (resp.
diﬀeomorphism) and the coordinate response function fi of f is locally equal to h. Therefore h is continuous(resp. diﬀerentiable).
[The third assertion: Universal Mapping Property] Though it would be possible to obtain
this result as a corollary to the fundamental mathematical theorem on the universal mapping (see, e.g.,
Bourbaki 1970; Chapter IV, Section 3), we present a direct proof since (i) the completeness of this paper
is also desirable and (ii) the restriction of our commutative diagram argument to equilibrium messages
requires some special treatments for a direct application of the universal mapping theorem.

5Note that Axiom (C3′ ) ensures that every message a ∈ A can be an equilibrium message for at least one
economy in Econ. Suppose that (P, π, e) and (P ′ , π ′ , e′ ) are two message mechanisms satisfying Axioms(C1′ ), (C2 ) and (C3′ ) and having the above universal implementability property. Then, by the first assertionof this theorem, we obtain two mappings h : P ′ → P and h′ : P → P ′ such that the diagram in Figure
2 commutes for each equilibrium messages. Moreover, the uniqueness of such mappings also implies thatboth h ◦ h′ : P → P and h′ ◦ h : P ′ → P ′ are identity mappings. It follows that h and h′ must be bijections
such that h−1 = h′ . The requests on h and h′ to be continuous (or diﬀerentiable) imply that both h andh′ are homeomorphisms (resp. diﬀeomorphisms). Bijection (homeomorphism, diﬀeomorphism, resp.) halso gives the bijection (homeomorphism, diﬀeomorphism, resp.) between the values π(E) and π ′ (E) for
each E as well as the restrictions of graphs of e(·, E) on the graph of π and e′ (·, E) on the graph of π ′ foreach E. Hence, the mechanism satisfying the universal implementability (restricted on the graph of its
equilibrium correspondence) is unique up to isomorphism.
■

4

Concluding Remarks

1. We have thus obtained a generalization of Sonnenschein (1974) theorem on the price mechanism in an
economy-dependent manner based on each agent’s utility level. The condition has a meaning that the pricemechanism can be characterized as a representative mechanism that can implement all economy-dependent
message mechanisms ensuring core property and suﬃciently high utility levels or stable actions for membersof each small economy. The concept of mechanism in our paper can be interpreted in a contemporary
game theoretic context that can treat a wide class of economies with messages by identifying its message
space with a component of the strategy sets (see Appendix).
2. The setting in this paper suggests a new method or a perspective to treat arguments on the allocation
mechanism with messages and axiomatic characterizations through fundamental social choice axioms like
individual rationality, Pareto-optimality, local independency, monotonicity, incentive compatibility and soforth. Based on our papers on replica core limit arguments (Urai and Murakami 2016a and Murakami
and Urai 2017a), we are preparing other papers on overlapping-generations price-money mechanism, pricedividend mechanism for satiation economies, and characterization based on monotonicity and incentive
compatibility, etc. (Urai and Murakami 2016b, Murakami and Urai 2017b and Urai and Murakami 2017).
For each cases, the utility form (welfare) characterization of this paper will provide other extension possibilities and interesting arguments.
3. In this paper, Sonnenschein’s arguments are reconstructed as a pure category theoretic universal
implementability theorem. By treating the number of agents as a variable, we can incorporate various
topological or dimensional structures of message spaces. For this purpose, the equilibrium correspondenceµ : Econ ∋ E 7→ µ(E) ⊂ A is used in the triple (A, µ, f ) of the message mechanism. The idea toutilize the equilibrium correspondence µ to characterize the role of messages in economies consisting of
diﬀerent number of participants has an advantage in applying the same replication method to various social
choice and information problems. As mentioned in the remark 2, the local independency, monotonicity,
incentive compatibility, etc., can also be reconsidered from this general viewpoint to characterize the pricemechanism.

6

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Debreu, G. (1959): Theory of Value. Yale University Press, New Haven, CT.
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Mount, K. and Reiter, S. (1974): “The Informational Size of Message Spaces,” Journal of Economic Theory
8, 161–192.
Murakami, H. and Urai, K. (2017a): “Replica core limit theorem for economies with satiation,” Economic
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Murakami, H. and Urai, K. (2017b): “A Universal Implementability of the Price Mechanism for Economies
with Satiation,” mimeo, Osaka University.
Nagahisa, R. (1994): “A Necessary and Suﬃcient Condition for Walrasian Social Choice,” Journal of
Economic Theory 62, 186–208.
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Thomson, W. (1988): “A Study of Choice Correspondences in Economies with a Variable Number ofAgents,” Journal of Economic Theory 46, 237–254.
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Appendix
A Contemporary Game Theoretic Interpretation of the Message Mechanism
We can identify our economy-dependent response function form of the message mechanism with ageneralized sense of the contemporary game theoretic mechanism as follows. (For convenience sake, in thefollowing, the number of agents of an economy will be fixed as n.)
Let S0 , S1 , . . ., Sn be n + 1 strategy sets of n + 1 players. The 0-th player will be considered a special
participant, and its strategic space S0 will be identified with the space of messages. The spaces S1 , · · · , Snwill be strategic spaces of consumer players.
With each strategy profile, (s0 , s1 , . . . , sn ) ∈ S = S0 × S1 × · · · × Sn , is associated an outcome, g(s0 , s1 ,. . . , sn ) ∈ X, where function g is called an outcome function and X is called an outcome space. The
list (S0 , S1 , . . . , Sn , g) is called a mechanism (Mas-Colell et al. 1995; Chapter 23, p. 866). If we add a
constraint correspondence K : S → S and a condition that a strategy profile s = (s0 , s1 , . . . , sn ) is anelement of K(s), we obtain a generalized sense of mechanism, (S0 , S1 , . . . , Sn , K, g).
Now we see that an economy-dependent message mechanism (A, µ, f ) with respect to economies with nagents in Econ can be identified with a generalized sense of mechanism.
Consider a space of messages A as S0 , and feasible consumption sets X1 , . . . , Xn of n agents as S1 , . . . , Sn ,respectively. For constraint correspondence K, we consider a special case that value K(a, x1 , . . . , xn ) of(a, x1 , . . . , xn ) ∈ A × X1 × · · · × Xn does not depend on x1 , . . ., xn . The domain and the range of outcome
function g are fixed to be A × X1 × · · · × Xn and X1 × · · · × Xn , respectively. Moreover, g is defined as
g = prX1 × · · · × prXn where prXi is the projection on Xi for each i = 1, . . . , n. We call such a mechanism,(A, X1 , . . . , Xn , K, g), a message mechanism. A special case that A is the price space and K is the budget
constraint correspondence, is called the price mechanism.
When an economy E ∈ Econ consisting of n agents is given, the utility functions and initial endowmentsof agents in E on X1 , . . ., Xn are specified and we obtain a generalized game setting on the above message
mechanism, (A, X1 , . . . , Xn , K, g). The list of best response functions of agents, 1, . . ., n, will be identified
with our economy-dependent form of the response function.

8