A Universal Implementability of Price-Money
Mechanism for Overlapping-Generations Economies
AbstractAxioms

We axiomatically characterize price-money mechanism for◮ Axiom I (Idempotency)
For each E ∈ Econ and a ∈ A, f (E (f (E , a)), a) = f (E , a).

overlapping-generations economies, using the core limit theoremof Urai and Murakami (2016). The category theoretic argument
in Sonnenschein (1974) is generalized and reinterpreted as a-------------------------------◮ Axiom S (Sonnenschein)

For each (i1, E 1 ), · · · ,(im, E m ), there exists an economy E∗ including

universal implementability of price-money mechanism.

{i1, · · · , im} such that a is an solution message for E∗ for which theequilibrium list (f i (E∗, a))i∞=1 is an extension of (f is (E s, a))ms=1.

Price-money mechanism is shown to be uniquely and efficiently
implement all of other allocation mechanisms satisfying several
basic axioms. We also show an isomorphism theorem, which says

Universal Implementability Theorem

that we obtain a minimum information space having theuniversal implementability property.

If (A, µ, f ) is a message mechanism based on a social choice correspondence that is compatible with Fcore and WPO allocations,ModelI1
I2
I3

k (1)

and if (A, µ, f ) satisfies Axioms I and S, then there exists a uniquek (1) + 1
k (2)k(1) + ℓ(1)k(2) + ℓ(2)k (3)function φ : A → P × M such that the next triangle commutes.

ℓ(1) = 4ℓ(2) = 2ℓ(3) = 2( R∞

k(3) + ℓ(3)e←− Econ × (P × M)
Price-Money↑տf1Econ × φ DictionaryN)

Econ × A∗ (%i , ωi) represents agent i ∈ It

◮ An economy E is identified with the list, ({It}, {Kt}, {(%i, ωi)}),

Isomorphism Theoremand Econ denotes the set of all economies.

◮ MWalras(E ) is the set of all monetary equilibrium alloca-

Let (P ×M, π, e) satisfy Axioms I and S. Then there exists
isomorphism h ′ : P × M → P ′ and e = e ′ ◦ [1Econ × h ′ ] for any

tions: the list of p∗ ∈ P (E ), feasible allocation x∗ = (xi∗ ∈ RKt )i∈St∈N It

S∞∗
and money supply function ME : N = t=1 It → R+, where xi∗ is%i-greatest element in {xi ∈ RKt | p∗ · xi ≦ p∗ · ωi + M∗E (i)}.

′Fcore-WPO compatible (P , π ′, e ′ ) satisfying Axioms I and S.

(R∞◮ Allocation x is weakly Pareto-optimal (WPO) if there is no ywith ∑t∈N ∑i∈It yi = ∑t∈N ∑i∈It xi, such that yi = xi except for finitee′←−′↑1×φտeEcon)Nagents, yi %i xi for all i and yi ≻i xi for at least one i.

Econ × P ′(R∞e←− Econ × (P × M)տ e ′ ↑ 1Econ × φ)N

Econ × P ′Econ × (P × M)

◮ Finite core of an economy E , Fcore(E ), is the set of feasible

If the problem is restricted on spaces with topological (differentia-allocation that cannot be blocked by any finite coalition.

ble) structures and continuous (differentiable) mappings, then h′can be taken as a homeomorphism (diffeomorphism).

Message Mechanism
Conclusion◮ A social choice correspondence is g : Econ ∋ E 7→ g(E ) ⊂ (R∞)N.
◮ A message mechanism based on g is a triple (A, µ, f ), where

Unique up to Isomorphism

A is message space, µ : E 7→ µ(E ) ⊂ A assigns equilibrium

Price-Money Message

messages, f : Econ × A → (R∞ )N defines responses to messagea ∈ A for each agent in E , satisfying g(E ) = {(fi (E , a))i∞=1| a ∈ µ(E )}.

φ↑ Unique Function exists Universally. ↑φ′◮ Price-money mechanism is defined by (P × M, π, e), wherePrK(t) p∞t∞
1
P × M = { p ∈ R | ∃[(x )t=1] ∈ ∆++, PrK(1) p = x , k Pr pk = xt}×K (t){M | M : Econ ∋ E 7→ ME ∈ RN+ }, π (E ) is the set of all price-money

Aequilibrium messages, e is the excess demand function such thatKei(p, ME ) is the %i-greatest point in {xi ∈ R t | p · xi ≦ p · ωi + ME (i)}.

′AP

VariousMessage
Classes′P◮ The universality and the efficiency property ofprice-money message mechanism is proved.

Replica Core Equivalence Theorem

References[1] Sonnenschein (1974) “An axiomatic characterization of the price mechanism,” Econometrica.

A feasible allocation x for E is a monetary competitive equilibrium[2] Urai and Murakami (2016) “Replica core equivalence theorem: An extension of the Debreu–Scarf

allocation if and only if its (m + n)-fold replica allocation belongs tolimit theorem to double infinity monetary economies,” JMathE. [3] Bourbaki (1939) “Theory of Sets,”

Fcore(E m(x) ⊕ E n(ω )) for every m ∈ N and n ∈ N . [Urai & Murakami 2016]

Hiromi Murakami (Osaka University, JSPS Fellows DC2)
murakami@math.econ.osaka-u.ac.jp
JSME Methodology Subcommittee-Spring 2017 MeetingSpringer. [4] Hurwicz (1960) “Optimality and Informational Efficiency in Resource Allocation Processes,” Mathematical Methods in the Social Sciences,” Stanford University Press.

with Ken Urai (Osaka University)
March 30, 2017