Discussion Papers In Economics
And BusinessStability and Universal Implementability of the Price
Mechanism
Kohzo Shiraishi
Ken Urai
Hiromi Murakami
Discussion Paper 17-09

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 17-09

April 2017

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
This paper provides a unified viewpoint on some axioms in social choice theory and a setting forthe allocation mechanism with messages in the informational eﬃciency problem. In particular, ourarguments are concerned with the category theoretical axiomatic method in Sonnenschein (1974) andthe replica stability axiom in the social choice arguments like Thomson (1988) and Nagahisa (1994).
The unified view enables us to obtain an extension of Sonnenschein’s axiomatic characterization of
the price mechanism as an agent-characteristics form dictionary property to a utility form economydependent universal implementability theorem.
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, kporco@eb.kobegakuin.ac.jp‡ 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

This paper provides a new perspective or a unified viewpoint on some axioms in social choice theory anda setting for the informational eﬃciency problem on allocation mechanisms with messages like Hurwicz(1960), Mount and Reiter (1974), Sonnenschein (1974), and so forth. Particularly, our arguments areconcerned with the category theoretical axiomatic method in Sonnenschein (1974) and the replica stability
axiom in the social choice arguments as in Thomson (1988) and Nagahisa (1994). Sonnenschein’s axiomaticargument through his expansion possibility axiom (Axiom S in Sonnenschein 1974) and the social choiceargument through the replica stability axiom (see Thomson 1988) have closely related theoretical structures
as a characterization of the price mechanism. The unified view enables us to obtain an extension ofSonnenschein’s axiomatic characterization of the price mechanism as an agent-characteristics form ofdictionary property to an economy-dependent utility form of universal implementability theorem.
In this paper, we see that the price mechanism can be characterized as the unique mechanism that
implements resource allocations of all other message mechanisms satisfying several important axioms (coreproperty, replica stability of responses, Sonnenschein’s expansion possibility, etc.). The replica stabilitysocial choice axiom is reformulated into an axiom not on equilibria but on responses for an allocation
mechanism with messages. The Sonnenschein’s axiom, on the other hand, is weakened so as to incorporate
economy-dependent utility form representation.1
Our result as an extension of Sonnenschein’s theorem (Propositions 1 and 7 in Sonnenschein 1974)
characterize an interesting feature of the price mechanism. By rigorously treating the role of Sonnenschein’s
expansion possibility axiom while separating from it the replica stability property of responses, we canfind out that the condition that uniquely and universally characterize the price mechanism is nothing but
the feature that there exists a possibility that no agents can be better oﬀ when they admit an expansionof the economy. In other words, the price can be characterized as a message that prepares for each small
economy, E, a suﬃciently good allocation at least in the following sense: under the message, we haveto expect a possibility that the utility level of every member is not improved for a certain kind of large
extension of E.2 Although it is true that the price mechanism is a universal rule for the global economy,
the property ensuring its universality (universal implementability) is rather its stability feature as a goodmessage for each small local economy.

2

The Basic ModelIn this paper, I, I ′ , · · · etc. represent finite index sets of agents. Note that such an index will be used
independently with the notations for economies like E, E′ , · · ·, so I is sometimes used 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 Xi , ≾i and ωi .
ℓ
For each i ∈ I, we define Xi as Xi = R+
. The preference preordering of i in economy E, ≾i , is a subsetℓof Xi × Xi and the initial endowment of i in E, ωi , is an element of R++.
1 The setting in this paper also provides a general perspective on the relation between the arguments on the informationaleﬃciency problem for the allocation mechanism and axiomatic characterization through fundamental social choice axioms
like individual rationality, Pareto-optimality, local independency, monotonicity, incentive compatibility and so on. Based on
our papers, Urai and Murakami (2016a) and Murakami and Urai (2017a), on the replica core limit theorem, we are preparing
for axiomatic characterization results as Urai and Murakami (2016b) of the price-money mechanisms and Murakami and
Urai (2017b) of the price-dividends mechanism. On the monotonicity and the incentive compatibility, we are also preparing
a paper, Urai and Murakami (2017).
2 For every economy, E, an extension Ê of E exists such that the message is an equilibrium for Ê and the utility levels of
all members of E are not improved under the equilibrium of Ê (see Axiom (C3′ ) in section 3).

1

We can write an 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 andstrict quasi-concavity (strict convexity in the sense of Debreu 1959). We write the set of economies as
Econ.
For each E = (≾i , ωi )i∈I , sequence (xi ∈ Xi )i∈I is called an allocation for E. Allocation (xi ∈ Xi )i∈I issaid to be feasible if

∑

xi =

i∈I

∑

ωi .

(1)

i∈I

A coalition in economy E = (≿i , ωi )i∈I is a set of agents S ⊂ I. Feasible allocation x is said to be the∑∑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 core of
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 such that (a, E) 7→ f (a, 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 , ωi )i∈I a subset of its feasible allocations, wecall the triple (A, µ, f ) as a message mechanism (an resource allocation mechanism with messages) basedon 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 restate the category theoretic argument in Sonnenschein (1974) while reconstructing it by
explicitly treating several implicit settings and assumptions as independent axioms. Let us consider thefollowing conditions on f and µ.
Axiom (C 1 ) : Responses are invariant for the extension of the economy. (Messages are not economy
dependent.) 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′ .
ℓDefine the set of price vectors, P , as P = {(p1 , · · · , pℓ ) ∈ R+|∑ℓ

k=1 pk = 1}. The price mechanism is anallocation mechanism with messages, (P, π, e), where for each E = (≾i , ωi )i∈I ∈ Econ, π(E) ⊂ P denotes

2

R∞



e

C×P

@ f
I
@
@

6
1C × hC×A

Figure 1: Commutative Diagram for the Universal Mapping Problem in Sonnenschein (1974)
the set of all competitive equilibrium prices for E and for each p ∈ P , e(p) = (ei (p))i∈I ∈list of the value of each agent’s excess demand function.3∏

i∈I Xi is the

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 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 a message mechanism
based on a social choice correspondence, and if (A, µ, f ) satisfies Axioms (C1 ), (C2 ) and (C3 ), thenthere exists a unique function h : A → P such that the triangle in Figure 1 commutes (Dictionary
Property). 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. Moreover, the mechanism that can play the above dictionary property is unique up toisomorphism (Universal Mapping Property).

3

Generalized Axioms and Theorems on Universal Implementability

Now, we consider more general conditions to characterize the price mechanism under the globalizationor the economic expansion framework. In the following, Axioms (C1 ), (C2 ) and (C3 ) are reformulated bymaking allowance for an expansion concept of economies. The economy independent axiom of messages,(C1 ), will be replaced with the following replica stability axiom for responses, (C1′ ).4 Moreover, Sonnenschein’s Axiom S is generalized by incorporating an economy dependent response function together witha utility form characterization as the following, (C3′ ).
Axiom (C ′1 ) : ∀E ∈ Econ, ∀a ∈ µ(E), ∀n, f (a, En ) = (f (a, E))n .5
Axiom (C ′3 ) : For each E ∈ Econ and message a ∈ A, there exists economy 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.

3 The excess demand function exists since each agent’s utility function is strictly quasi-concave.
4 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.
5 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, , · · ·.

3

In the previous section, (C1 ) indicates that the responses of agents do not depend on the scale of theeconomy. The above extended axioms, (C1′ ) and (C3′ ), generalize the settings in Sonnenschein (1974)
in the sense that responses are allowed to be economy-dependent except for the replica extension withequilibrium messages. Note that condition (C3′ ) is given in the utility form. The framework enables us todiscuss problems such as relation between the stability of price mechanism and an expansion possibilityof economies like the globalization .
To obtain an extension theorem of Sonnenschein (1974; Propositions 1 and 7), we describe the commutative diagram in Figure 1 by considering the dependence of responses on economy E. To this purpose, we
focus our attention on equilibrium messages.6 Moreover, for the sake of simplicity, we restrict the classof all economies Econ as the set of economies in which there is at least one agent whose utility function is
diﬀerentiable.
R∞



e

@ f
I
@
@

Econ × P
6
1Econ × hEcon × A

Figure 2: A Commutative Diagram for Economy Dependent Message Mechanisms

Theorem 1: If a message mechanism based on a social choice correspondence, (A, µ, f ), satisfies Axioms(C1′ ), (C2 ) and (C3′ ), then there exists a unique function h : A → P such that the triangle in Figure
2 commutes for each E ∈ Econ and a ∈ µ(E) (Universal Implementability). 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. 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 assertion: Universal Implementability] Let a and E1 be an arbitrary pair of amessage and an economy satisfying E1 = (≾1i , ωi1 )i∈I 1 ∈ Econ and a ∈ µ(E1 ). Since a is an equilibrium
message for E1 , f (a, E1 ) is Pareto-optimal. By considering that at least one member of E1 is diﬀerentiable,
we obtain a unique supporting price p for allocation f (a, E1 ).
Next, we show that under p, fi (a, E1 ) satisfies the budget constraint for all i ∈ I 1 . Assume not. Then,the allocation f (a, E1 ) 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, E1 ) cannot
be in the core of the n-fold replica economy of E1 , (E1 )n . Hence, we have a set G of the individualsin (E1 )n who can block the n-fold replica allocation of f (a, E1 ). Under (C1′ ), we have (f (a, E1 ))n =f (a, (E1 )n ). Furthermore, by applying (C3′ ) on (E1 )n , there exists E∗ such that a ∈ µ(E∗ ), (E1 )n ⊂ E∗
and ui (fi (a, E∗ )) ≧ ui (fi (a, (E1 )n )) for all i of (E1 )n . It follows that, by (C2 ), f (a, E∗ ) is an element ofCore(E∗ ) satisfying ui (fi (a, E∗ )) ≧ ui (fi (a, (E1 )n )) for all i of (E1 )n . But this is impossible because G in()
(E1 )n blocks the utility allocation under (f (a, E1 ))n . Therefore, p · fi (a, E1 ) − ωi1 = 0 for all i ∈ I 1 .

6 Our approach does not restrict the domain of messages as long as for every message a ∈ A, there exists at least one
economy E such that a ∈ µ(E). Such convention is also suﬃcient for our purpose to characterize the universal implementabilityof the price mechanism for all of other message mechanisms and equilibrium allocations.

4It is easy to check that such 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 quite easy (e.g., by considering an economy with CobbDouglas utility agents,) to choose an economy such that near h(a), a coordinate of the list of excess
demand functions, e, provides a local homeomorphism (resp. diﬀeomorphism). The composition of such
local homeomorphism (resp. diﬀeomorphism) and a coordinate response function of f gives an anotherlocal representation of h that is necessarily continuous (resp. diﬀerentiable).
[The second 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 1971), 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.
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).
■

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