Replica core equivalence theorem: An extension of the Debreu–Scarf limit theorem to double infinity monetary economies

Journal of Mathematical Economics 66 (2016) 83–88

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Journal of Mathematical Economics journal homepage: www.elsevier.com/locate/jmateco

Replica core equivalence theorem: An extension of the Debreu–Scarf limit theorem to double infinity monetary economies✩ Ken Urai, Hiromi Murakami ∗ Graduate School of Economics, Osaka University, Japan

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Article history: Received 4 November 2014 Received in revised form 10 August 2016 Accepted 19 August 2016 Available online 27 August 2016 Keywords: Monetary equilibrium Overlapping generations model Core equivalence Replica economy Non-ordered preference

abstract An overlapping generations model with the double infinity of commodities and agents is the most fundamental framework to introduce outside money into a static economic model. In this model, competitive equilibria may not necessarily be Pareto-optimal. Although Samuelson (1958) emphasized the role of fiat money as a certain kind of social contract, we cannot characterize it as a cooperative game-theoretic solution like the core. In this paper, we obtained a finite replica core characterization of Walrasian equilibrium allocations under non-negative wealth transfer and a core-limit characterization of Samuelson’s social contrivance of money. Preferences are not necessarily assumed to be ordered. © 2016 Elsevier B.V. All rights reserved.

1. Introduction In this paper, we generalize the Debreu–Scarf core limit theorem to a case with a double infinity economy that includes such typical examples as Samuelson consumption-loan models with money. For standard finite general equilibrium settings, the Walrasian equilibrium is Pareto-optimal (the first fundamental theorem of welfare economics), and every Pareto-optimal allocation is an equilibrium allocation relative to a price-wealth system (the second fundamental theorem of welfare economics). If we strengthen the concept of Pareto-optimal allocation to a replica core allocation (an allocation whose replica is a core allocation for each replica economy), we obtain an equivalence result where

✩ We express special gratitude to professor Takashi Hayashi (Glasgow University) for his important comments and advice. We are also grateful for the comments by professors Atsushi Kajii (Kyoto University), Tomoyuki Kamo (Kyoto Sangyo University), and Tomoki Inoue (Meiji University) as well as the participants at a poster session at Seinan Gakuin University on October 11, 2014 at the autumn meeting of the Japanese Economic Association. We also appreciate many insightful and accurate comments from anonymous referees of the Journal of Mathematical Economics and Journal of Economic Theory together with valuable suggestions from the chief editor of Journal of Mathematical Economics. We would also like to thank Professor Ivan Brenes (Osaka University) for his kind help. Part of this research was supported by JSPS KAKENHI Grant Numbers JP25380227 and JP15J01034. ∗ Corresponding author. E-mail addresses: [email protected] (K. Urai), [email protected] (H. Murakami).

http://dx.doi.org/10.1016/j.jmateco.2016.08.003 0304-4068/© 2016 Elsevier B.V. All rights reserved.

every replica core allocation is a competitive allocation and every competitive allocation is a replica core allocation (Debreu and Scarf, 1963, Theorems 1 and 3). This equivalence theorem between the replica core and the competitive equilibria is commonly known as the Debreu–Scarf core limit theorem. In a double infinity economy, competitive equilibrium (with or without money) is not necessarily Pareto-optimal (Samuelson, 1958). The notion of weak Pareto-optimality (see Balasko and Shell, 1980), which is weak and different from the usual Paretooptimality, however, is possible to characterize such competitive equilibrium allocations. The competitive equilibrium (with or without money) is known to be weakly Pareto-optimal (Esteban, 1986; Balasko and Shell, 1980). It is also known that every weakly Pareto-optimal allocation is an equilibrium allocation relative to a price-wealth system (Balasko and Shell, 1980). Chae (1987), Aliprantis and Burkinshaw (1990) and Chae and Esteban (1993) treat the core equivalence problem for Walrasian equilibrium allocations in overlapping generations models. Their approaches, however, fail to treat competitive equilibrium allocations with money.1 Of course, an equilibrium with money (non-negative

1 Their concepts, such as the short-term core (Aliprantis and Burkinshaw, 1990) and the short-run core (Chae, 1987), exclude equilibrium allocations with non-zero fiat money in two-period overlapping generations economies. For the short-term core argument, as a simple one-good per period economy example pointed out in Esteban (1986), such a monetary equilibrium is blocked by a coalition of all agents after a certain period without changing all but the first finite members’

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K. Urai, H. Murakami / Journal of Mathematical Economics 66 (2016) 83–88

wealth transfers from the government) is one critical issue that the overlapping generations model tries to describe. In this paper, we show that if we strengthen the concept of weakly Pareto-optimal allocation to replica finite core allocation (an allocation whose replica is a certain kind of finite core allocation for each replica economy),2 we obtain an equivalence result analogous to Aliprantis and Burkinshaw (1990) where a replica finite core allocation is a competitive allocation with nonnegative wealth transfers and every competitive allocation with non-negative wealth transfers is a replica finite core allocation.3 Our replica core equivalence approach (as well as that of Aliprantis and Burkinshaw, 1990) has three important advantages: by concentrating on the equivalence argument without using the equal treatment property, (i) we can show a weak core theoretic equivalence result merely based on weak optimality conditions, (ii) we obtain a limit theorem of the core instead of a theorem in the limit measure space like Chae and Esteban, and (iii) we can allow for an argument based on non-ordered preferences, and hence our result may also be considered an extension of the Debreu–Scarf core equivalence theorem incorporating nonordered preferences.4 We should also note that our monetary equilibrium concept (a competitive equilibrium with non-negative wealth transfer) is closely related to the concept of ‘‘dividend equilibrium’’ or ‘‘equilibrium with slack’’ (see, e.g. Aumann and Drèze, 1986). The dividend equilibrium is a general equilibrium concept for economies with possibly satiated consumers. For a finite economy with satiated consumers, a competitive equilibrium might fail to exist, and ‘‘the existence of an equilibrium can be restored if we give consumers appropriate extra amounts of income to spend’’ (Mas-Colell, 1992). In such case, some consumers’ budget constraints under positive extra wealth transfers must be held with strict inequality. Otherwise, the resource feasibility does not hold by Walras’ law. For economies with infinitely many agents and commodities, however, the lack of Walras’ law under the double infinity structure allows us to use the resource allocation mechanism with non-negative wealth transfers without satiated consumers, as a monetary Walrasian equilibrium is compatible with the resource feasibility. Konovalov (2005) defines ‘‘rejective core’’ and proves that the rejective core coincides with the set of dividend equilibrium allocations in a continuum exchange economy. Though our model is not for economy with satiation but for a double infinity economy with or without money, our replica core concept F core(Em (x) ⊕ En (ω)) in Section 3 is similar to his rejective core, and our theorem presents an analogue of core convergence argument relative to his core equivalence result.5 2. The model Let N be the set of all positive integers and R , R+ and R++ be the sets of real numbers, non-negative real numbers and positive

real numbers, respectively. A pure exchange overlapping generations economy, or more simply, an economy, E, is comprised of the following list: (E.1) {It }∞ t =1 : a countable ∞ family of mutually disjoint finite subsets of N such that t =1 It ⊂ N , I1 ̸= ∅ and for each t ∈ N , if It = ∅, then It +1 = ∅. It is the index set of agents in generation t. (E.2) {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 , k(t ) < k(t + 1) 5 k(t ) + ℓ(t ) for all t ∈ N . Kt is the index set of commodities available to generation t. (E.3) {(%i , ωi )}i∈t ∈N It : countably many agents, where %i is a reflexive binary relation on commodity space for each K generation R+t = {x | x : Kt → R+ }, representing a preference of i ∈ It . We write xi ∼i yi iff xi %i yi and yi %i xi , and xi ≻i yi iff xi %i yi and xi i yi . Strict preference ≻i is continuous K K (having an open graph in R+t × R+t ), strictly monotonic i i i i i i (x = y and x ̸= y implies x ≻i y ), and has a convex better set ({yi | yi ≻i xi } is convex) at every xi such that ωi ̸≻i xi . The K K closure of {yi | yi ≻i xi } in R+t × R+t is {yi | yi %i xi } for K

each xi ∈ R+t (a strong sense of local non-satiation). The initial Kt endowment of i, ωi , is an element of R++ = {x | x : Kt → R++ } for each i ∈ It . It is convenient to identify the commodity space for each K generation R+t with a subset of R N , which is the set of all functions K

from N to R, by considering x ∈ R+t a function that takes value 0 on N \ Kt . Then we can define the total commodity space for Kt economy ⊕∞ t =1 R+ as the set of all finite sums among the points K

t in the commodity spaces of the generations. Clearly, ⊕∞ t =1 R+ can be identified with a subset of direct sum R∞ , the set of all finite real sequences, which is a subspace of the set of all real sequences, R ∞ ≈ R N with pointwise convergence topology. ∞  i Given an economy, E = ({It }∞ t =1 , {Kt }t =1 , {(%i , ω )}i∈ t ∈N It ),

N the price space for E, P(E), is defined as the set of all p in R+ such that under the duality between R∞ (with relative topology) and R ∞ (with pointwise convergence topology), p positively evaluates all the agents’ initial endowments: N P(E) = {p ∈ R+ | p · ωi > 0 for all i ∈ It , for all t ∈ N }.

Since for all i ∈ It , ωi belongs to R++ for all t ∈ N , the price space N of E always includes R++ for all E in Econ, where Econ denotes the set of all economies satisfying conditions (E.1), (E.2) and (E.3). For each E = ({It }, {Kt }, {(%i , ωi )}) ∈ Econ, sequence i (x )i∈t ∈N It with xi ∈ R+Kt for t ∈ N and i ∈ It , is called an allocation for E. Allocation (xi )i∈t ∈N It is said to be feasible if

 t ∈N i∈It

allocations. In the short-run core, we can also easily construct an example under which a typical Samuelson-type monetary equilibrium allocation is always blocked by the t-generation for each t-economy for all t = 1, 2, . . . . 2 More precisely, this is an allocation, x, whose replica is a finite core allocation for each replica economy even when the endowments of some members are replaced by the allocation, x, itself. 3 In this paper, we use ‘‘wealth transfer’’ instead of ‘‘monetary transfer’’ because these two concepts are different unless we use the perfect-foresight assumption on the expectation for dynamics. In the sense of Esteban and Millán (1990), we concentrate on the set of all monetary equilibrium allocations and competitive equilibrium allocations without money. 4 Aliprantis and Burkinshaw (1990), however, do not successfully treat nonordered preference cases. In our model, a strong sense of the local non-satiation in (E.3) plays an essential role in the proof of Theorem 1 (see footnote 12). 5 Concerning the above argument, we thank to the suggestion of a referee.

(1)

Kt

xi =



ωi ,

(2)

t ∈N i∈It

where the summability in R N of both sides of the equality is ∗ assured by (E.2). The list of price ∞ vector p ∈ P(E), non-negative ∗ wealth transfer function ME : t =1 It → R+ , and feasible allocation (xi∗ )i∈t ∈N It , is called a non-negative wealth transfer Walrasian equilibrium for E, if for each t ∈ N and i ∈ It , xi∗ is a ≻i -maximal

element in set {xi ∈ R+t | p∗ · xi 5 p∗ · ωi + ME∗ (i)}. Since the non-negative wealth transfer is an abstraction of the money supply in perfect-foresight overlapping generations settings, we denote the set of all non-negative wealth transfer Walrasian equilibrium allocations by MW alras(E). We also denote by Walras(E) the subset of MW alras(E) for allocations associated with wealth transfer ME∗ (i) = 0 for each i. K

K. Urai, H. Murakami / Journal of Mathematical Economics 66 (2016) 83–88

A coalition in economy E = ({It }, {Kt }, {(%i , ωi )}) ∈ Econ ∞ is a set of consumers S ⊂ t =1 It . Allocation x for economy E is said to be blocked by coalition S if it to find commodity is possible bundles xˆ i for all i ∈ S such that xi − ωi ) = 0 and xˆ i %i xi i∈S (ˆ for all i ∈ S, and xˆ i ≻i xi for at least one i ∈ S. For each E = ({It }, {Kt }, {(%i , ωi )}) ∈ Econ, the set of all feasible allocations that cannot be blocked by any coalition is said to be the core of economy E and is denoted by Core(E). Element x ∈ Core(E) is called a core allocation. The set of all feasible allocations that cannot be blocked by any finite coalition is called the finite core of economy E and is denoted by F core(E). Element x ∈ F core(E) is called a finite core allocation for E. 3. Replica core equivalence theorem For each feasible allocation x = (xi )i∈t ∈N It for E = ({It }, {Kt },

{(%i , ωi )}) ∈ Econ, we denote by E(x) an economy where initial endowment allocation ω = (ωi ) is replaced by x = (xi ).6 Hence, we have E = E(ω). Consider the following replica economy, E (x) ⊕ En (ω), m

(3)

which consists of all the members of the m-fold replica economy of E(x) and the n-fold replica economy of E(ω) for each m ∈ N and n ∈ N . Let us denote by Cm⊕n (E) the set of allocations x for E such that the (m + n)-fold replica allocation of x belongs to F core(Em (x) ⊕ En (ω)).7 Moreover, let us denote by Cn (E) the set of allocations x for E such that the n-fold replica allocation of x belongs to Core(En ). It is easy to check that if x is a feasible allocation of E = E(ω) such that (m + n)-fold replica allocation of x does not belong to F core(Em (x) ⊕ En (ω)), the replica allocation does not belong to Core(Em+n ).8 Therefore, we can write Cm⊕n (E) ⊃ Cm+n (E) for each m ∈ N and n ∈ N . It is also easy ′ ′ to check that Cm⊕n (E) ⊃ Cm ⊕n (E) where m′ = m, n′ = n.9 For finite economy E with It = ∅ for some t ∈ N , the Debreu–Scarf ∞ limit theorem can be restated as m+n=2 Cm+n (E) = Walras(E). ∞ ∞ We see below (Theorem 1), n=1 m=1 Cm⊕n (E) = MW alras(E). Hence, the restriction of Theorem 1 to the case with finite economy E provides the following extension of the replica core version of the Debreu–Scarf limit theorem because there is no difference in our settings between Walras(E) and MW alras(E). For finite economy E, feasible allocation x for E is a competitive equilibrium allocation iff its (m + n)-fold replica allocation n belongs to F core(Em ( x) ⊕ E (ω)) for every m ∈ N and n ∈ N . ∞ ∞ That is, Walras(E) = n=1 m=1 Cm⊕n (E). As above, concept F core gives a unified replica core equivalence characterization for all non-negative wealth transfer Walrasian equilibrium allocations. Note that allocation x such that x ∈ F core(E(x)) is a weakly Pareto-optimal allocation in the sense of Balasko and Shell (1980). It is easy to check that the n-fold replica allocation of x∗ ∈ MW alras(E) belongs to F core(En (x∗ )) for all n ∈ N and F core(En (ω)) for all n ∈ N .10 But we have an example

6 In the following, we sometimes omit the subscript i ∈  t ∈N It of an allocation for an economy as long as there is no risk of confusion. 7 The (m + n)-fold replica allocation of x is the allocation for Em (x) ⊕ En (ω) such

that we assign the same commodity bundle under x in economy E to each replica agent in E(x) or E(ω). 8 Clearly, the replica allocation, xm+n , is feasible for Em (x) ⊕ En (ω) and Em+n =

Em+n (ω). If xm+n ̸∈ F core(Em (x) ⊕ En (ω)), then there exists a finite coalition S in Em (x) ⊕ En (ω) that blocks allocation xm+n . We can write S = S1 ∪ S2 , where S1 (resp. S2 ) consists of members in Em (x) (resp. En (ω)). Then, coalition S ∗ consisting of all members of Em (x) and S2 also blocks xm+n in replica economy Em (x) ⊕ En (ω). Therefore xm+n ̸∈ Core(Em+n ). 9 Note that the equal treatment property is not necessary for ensuring the above

inclusion relations. Here we are following the replica core equivalence approach in Aliprantis and Burkinshaw (1990). 10 To see this, in the proof of Theorem 1, (Sufficiency), let S or S be an empty set. 1

2

85

where the n-fold replica of allocation x, which is not an element of MW alras(E), belongs to F core(En (x)) ∩ F core(En (ω)) for all n ∈ N .11 Theorem 1. Feasible allocation x for E is a non-negative wealth transfer Walrasian equilibrium allocation iff its (m + n)-fold replica n allocation belongs to F core(Em ( x) ⊕ E (ω)) for every m ∈ N and ∞ ∞ n ∈ N . That is, MW alras(E) = n=1 m=1 Cm⊕n (E). Proof (Sufficiency). Let x∗ = (xi∗ ) be an element of MW alras(E) under price p∗ and non-negative wealth transfer function ME∗ . Assume that S = S1 ∪ S2 is a finite coalition of Em (x∗ ) ⊕ En (ω) for some m and n in N blocking the (m + n)-fold replica allocation of x∗ = (xi∗ ), where S1 is a coalition in Em (x∗ ) and S2 is a coalition in En (ω). Then, by an allocation (xi )i∈S such  definition, there is i i i i i that i∈S x = i∈S1 x∗ + i∈S2 ω , x %i x∗ for all i ∈ S and j

xj ≻j x∗ for some j ∈ S. Note that the equilibrium price p∗ is strictly positive under the strict monotonicity of preferences. The equilibrium condition means that xi ≻i xi∗ implies p∗ · xi > p∗ · xi∗ . Moreover, under (E.3), especially by using the strong sense of local non-satiation, we have xi %i xi∗ implies p∗ · xi = p∗ · xi∗ .12 It    follows that we have p∗ · ( i∈S1 xi + i∈S2 xi ) > p∗ · i∈S1 xi∗ +

p∗ ·





i∈S

i∈S2

xi =

xi

= p∗ · (

  i i i∈S1 x∗ + i∈S2 ω ), a contradiction to  i i i∈S1 x∗ + i∈S2 ω .

∗

∞ (Necessity). Let x¯ = (¯xi ) be an allocation for E = ({It }∞ t =1 , {Kt }t =1 ,  i (%i , ω )i∈ t ∈N It ) such that every (m + n)-fold replica allocation of x¯ belongs to F core(Em (¯x) ⊕ En (ω)) for all m and n in N . In this proof, we t denote by I (t ) the set of all agents in generations from 1 to t , s=1 Is , and by K (t ) the set of all commodities that are t available for agents in I (t ), s=1 Ks . Define for each i ∈ It , t ∈ N , Γi as Γi = {β i z1i + (1 − β i )z2i | z1i + x¯ i ≻i x¯ i , z2i + ωi ≻i x¯ i , 0 5 i β R Kt . Then, take the convex hull Γ (t ) of finite union  5 1} ⊂  t Ks ⊂ R K (t ) for each t ∈ N . Since, for every i, Γi i∈I (t ) Γi ⊂ s=1 R is non-empty and convex, non-empty Γ (t ) consists of convexi set i i i i all vectors z that can be written as i∈I (t ) α (β z1 + (1 − β )z2 ),

with α i = 0, i∈I (t ) α i = 1, where z1i + x¯ i ≻i x¯ i and z2i + ωi ≻i x¯ i for each i. We will show in the similar way as in the proof of Debreu and Scarf (1963, Theorem 3) that Γ (t ) does not have 0 as its element for each t ∈ N . Let us suppose that 0 belongs to Γ (t ) for some t ∈ N .  Then, one can write i∈I (t ) α i (β i z1i + (1 − β i )z2i ) = 0, with α i = 0,





i∈I (t )

α i = 1, and z1i + x¯ i ≻i x¯ i and z2i + ωi ≻i x¯ i for each i. For each

κ sufficiently large, let ai1κ and ai2κ be the smallest integers greater than κα i β i and κα i (1 − β i ) respectively. Also, let I be the set of κα i β i i ∈ I (t ) for which α i > 0. For each i ∈ I, we define z1i κ as i z1i a and z2i κ as

κα i (1−β i ) i ai2κ

1κ

z2 . Observe that z1i κ + x¯ i belongs to the segment

[¯xi , z1i + x¯ i ] and z2i κ + ωi belongs to the segment [ωi , z2i + ωi ]. Let J1 be the set of i ∈ I such that β i ̸= 0, and J2 be the set of i ∈ I such that 1 − β i ̸= 0. Note that J1 ∪ J2 = I (see Fig. 1). For i ∈ J1 , z1i κ + x¯ i tends to z1i + x¯ i , and for i ∈ J2 , z2i κ +ωi tends to z2i +ωi as κ tends to infinity. The continuity assumption on preferences

11 See Appendix. 12 Since under the strong sense of local non-satiation, such xi is a limit point of the better set as long as the better set is not empty that is assured by the strict monotonicity. Under the minimum wealth condition, p∗ · xi < p∗ · xi∗ implies that we have a point yi in the better set such that p∗ · yi < p∗ · xi∗ , a contradiction. Note that the standard concept of local non-satiation fails to guarantee the above argument without using the transitivity of preferences.

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K. Urai, H. Murakami / Journal of Mathematical Economics 66 (2016) 83–88

Fig. 1. The union of J1 and J2 is equal to I.

implies that z1i κ + x¯ i ≻i x¯ i for all i ∈ J1 and z2i κ + ωi ≻i x¯ i for all i ∈ J2 for all κ sufficiently large. Select one of such κ . Then we have 0 = κ



α i (β i z1i + (1 − β i )z2i ) =

i∈I

=



κα i (β i z1i + (1 − β i )z2i )

i∈I

κα i β i z1i +



κα i (β i z1i + (1 − β i )z2i )

i∈J1 ∩J2

i∈J1 \J2

+





κα (1 − β i )z2i i

i∈J2 \J1

=

 i∈J1 \J2

ai1κ z1i κ +



(ai1κ z1i κ + ai2κ z2i κ ) +

i∈J1 ∩J2



ai2κ z2i κ .

(4)

i∈J2 \J1

Let us consider the replica economy Em (x) ⊕ En (ω) with m = maxi∈I ai1κ and n = maxi∈I ai2κ . Take the coalition composed of ai1κ replica members of i for each i ∈ J1 to each one of whom we assign z1i κ + x¯ i , and ai2κ replica members of i for each i ∈ J2 to each one of whom we assign z2i κ + ωi . This coalition blocks the allocation (¯xi ) as Eq. (4) and the fact that z1i κ + x¯ i ≻i x¯ i for all i ∈ J1 and z2i κ + ωi ≻i x¯ i for all i ∈ J2 show. This is a contradiction to the definition of F core(Em (¯x) ⊕ En (ω)). Hence, we have established that 0 does not belong to the convex set Γ (t ) for each t ∈ N . For each t = 1, 2, 3, . . . , let π (t ) ⊂ R ∞ be the set of prices such that π (t ) = {p ∈ R K (t ) × R × · · · | p · z = 0 for all z ∈ Γ (t )}.13 Set π (t ) is closed in R K (t ) × R × · · · = R ∞ and by the separating hyperplane theorem, there exists p ∈ π (t ) such that   p ∈ R K (t ) \ {0} × R × · · · . It is easy to check that every p ∈ π (t ) K (t )

satisfies that prK (t ) p ∈ R+ since a negative price cannot separate 0 from any strictly monotone better set at x¯ i of each i ∈ I (t ).14 Moreover, we can also prove that every p ∈ π (t ) such that K (t ) prK (t ) p ̸= 0 satisfies prK (t ) p ∈ R++ as long as I1 , I2 , . . . , It are nonempty sets. Indeed, prK (t ) p is a supporting price vector for both of the better set at x¯ i and the direction of the better set from ωi for each i ∈ I (t ). Hence, if pr{k} p = 0 for some commodity k ∈ Ks ⊂ K (t ), as long as there exists an agent i ∈ Is whose wealth under p is strictly positive (i.e., p · ωi > 0), we have a contradiction as follows. Consider first the case that p · x¯ i = 0. Then, from the strict monotonicity of preferences, a vector x¯ i + (0, . . . , 0, +δ, 0, . . .) − ωi such that x¯ i + (0, . . . , 0, +δ, 0, . . .) is strictly preferred to x¯ i , where +δ > 0 is the k-th coordinate, will not be nonnegatively supported by p. Secondly, if p · x¯ i > 0, the strict monotonicity and continuity of preferences also means that for a strictly preferred point x¯ i + (0, . . . , 0, +ϵ, 0, . . . , 0, −δ, 0, . . .) to x¯ i of i, vector (0, . . . , 0, +ϵ, 0, . . . , 0, −δ, 0, . . .) will not be nonnegatively supported by p for sufficiently small δ , where +ϵ > 0 is the k-th coordinate and the coordinate of −δ < 0, say ℓ, is equal to the coordinate of x¯ i that is evaluated positively under p as (pr{ℓ} p)(pr{ℓ} x¯ i ) > 0.15 The non-negativity of the wealth under

13 Note that π(t ) is defined independently of the price space P(E). 14 In the above, pr denotes the projection on the subspace defined by the A coordinates in A ⊂ N . 15 Such coordinate ℓ necessarily exists under condition p · x¯ i > 0 and should be different from k since pr{k} p = 0.

p, however, is assured as follows. There exists at least one agent j ∈ Is′ ⊂ I (t ) such that j’s wealth under p is strictly positive.16 So, every price for commodities in Ks′ can be proved to be positive by using the above argument. Then, the wealth of all agents of generation s′′ with Ks′ ∩ Ks′′ ̸= ∅ are positive by the initial endowment condition in (E.3). By repeating these processes we can show that all agents in I (t ) have positive wealth under p.17 It follows that we have K (t ) π (t ) ⊂ (R++ ∪ {0}) × R × · · · ⊂ R∞. Next, we will obtain p∗ ∈ t ∈N (π (t ) ∩ (R K (t ) \ {0}) × R × · · ·). From the definition of each Γ (t ) ⊂ R K (t ) , we have Γ (1) ⊂ Γ (2) ⊂ Γ (3) ⊂ · · · in R∞ . Hence, we have π (1) ⊃ π (2) ⊃ π (3) ⊃ · · · in t R ∞ . Thus we see s=1 π (s) = π (t ). For finite economy, we have t such that It ̸= ∅, It +1 , It +2 , . . . are empty, and π (t ) = π (t + 1) = · · ·. Hence by the non-emptiness of π (t ) ∩ (R K (t ) \ {0}) × R × · · ·, the result is obvious. If the number of agents is infinite, for each t ∈ N , we choose price p(t ) = (p1 (t ), p2 (t ), . . .) in π (t ) ∩ (R K (t ) \ K (t ) {0}) × R × · · · ⊂ R++ × R × · · · ⊂ R ∞ (see Fig. 2). Define for each t ∈ N , compact set ∆K (t ) = {qt | qt ∈ K (t ) R , ∥qt ∥ = 1}.18 Moreover, for each s, t ∈ N , s 5 t, define mapping hst : π (t ) ∩ (R K (t ) \ {0}) × R × · · · ∋ p → hst (p) ∈

∆K (s) ∩ prK (s) π (s) as hst (p) =

prK (s) p

∥prK (s) p∥

for each p. For simplicity of

notation, we denote by D(t ) the domain, π (t ) ∩ (R K (t ) \ {0}) × R × · · ·, of hst . Note that ∆K (s) ∩ prK (s) π (s) is a compact subset of K (t )

R++ . In addition, we can prove that the image of D(t ) under hst is also compact. Indeed, if it is not, there is a point y∗ ∈ ∆K (s) ∩ prK (s) π (s), y∗ ̸∈ hst (D(t )), and a sequence {yν = hst (qν )} in hst (D(t )) such that yν converges to y∗ . Since π (t ) is a cone we can normalize such qν as ∥qν ∥ = 1, so by taking a subsequence, we obtain q∗ = lim qν that is obviously an element of D(t ). This together with the continuity of hst shows that hst (q∗ ) = y∗ , a contradiction.19 Let {p(t )}t ∈N be a sequence such that p(t ) ∈ D(t ) for every t ∈ N . Since ∆K (1) ∩ prK (1) π (1) is compact, when t → ∞, by taking a subsequence, {p(t )}t ∈N (1) , of {p(t )}t ∈N , where N (1) is a cofinal subset of N , h1t (p(t )) converges to a limit, pˆ ∗1 ∈ K (1)

h11 (D(1)) ⊂ (∆K (1) ∩ prK (1) π (1)). Define p∗1 ∈ R++ as p∗1 = pˆ ∗1 . Next, since ∆K (2) ∩ prK (2) π (2) is compact, when t → ∞, by taking a subsequence, {p(t )}t ∈N (2) , of {p(t )}t ∈N (1) , where N (2) is a cofinal subset of N (1) constructed by elements greater than or equal to 2, h2t (p(t )) also has a limit pˆ ∗2 ∈ h22 (D(2)) ⊂ (∆K (2) ∩ prK (2) π (2)). K (2)\K (1)

as ∥pr 1 pˆ ∗ ∥ prK (2)\K (1) pˆ ∗2 . Generally, K (1) 2 from the compactness of ∆K (s) ∩ prK (s) π (s) for each s ∈ N , if t → ∞, by taking a subsequence, {p(t )}t ∈N (s) , of {p(t )}t ∈N (s−1) , where N (s) is a cofinal subset of N (s − 1) constructed by elements greater than or equal to s, hst (p(t )) has a limit pˆ ∗s ∈ hss (D(s)) ⊂

We define p∗2

∈ R++

K (s)\K (s−1)

(∆K (s) ∩ prK (s) π (s)). Hence we can define p∗s ∈ R++ as p∗s = ∥pr 1 pˆ ∗ ∥ prK (s)\K (s−1) pˆ ∗s . By repeating the above procedure, K (1) s

(p ,...,p ) we obtain p∗ = (p∗1 , p∗2 , . . .). Since for each s ∈ N , ∥(p∗1 ,...,p∗s )∥ = pˆ ∗s s 1 ∗

∗

K (t )

16 Vector p is an element of R + \ {0} and for each commodity in K (t ), there is at least one agent who has positive endowment of it in I (t ). 17 The above strong preference-endowment structure assured by (E.2) and (E.3) would be replaced with more general resource-relatedness conditions for standard overlapping generations economies and their equilibrium existence arguments. See, e.g., Balasko and Shell (1980, p. 290) and Arrow and Hahn (1971, p. 117). 18 In the following, the notation ∥ · ∥ will be used to represent the Euclidean norm for the subspace.   19 Let X be h (D(t )) ⊂ ∆K (t ) ∩ pr t tt K (t ) π(t ) for each t ∈ N . Then the family (Xt )t ∈N and the family of mappings (hst ), s 5 t, s, t ∈ N form an inverse system of compact spaces. By Bourbaki (1966, Chapter I, section 9, no. 6, Proposition 8), the inverse limit X = lim Xt is non-empty. And price p∗ that we seek in the following ←−

can be identified as an element of X .

K. Urai, H. Murakami / Journal of Mathematical Economics 66 (2016) 83–88

87

Fig. 2. How to construct the limit price p∗ . 1 ∗ is an element of hss (D(s)) , p∗ belongs to h− ps ) ⊂ D(s), so we ss (ˆ   ∗ K (t ) have p ∈ t ∈N D(t ) = t ∈N (π (t ) ∩ (R \ {0}) × R × · · ·). Since xi ≻i x¯ i means that both xi − ωi and xi − x¯ i belong to Γi , we have p∗ · xi = p∗ · ωi and p∗ · xi = p∗ · x¯ i . By taking xi arbitrarily near to x¯ i (from the strict monotonicity), we can see that p∗ · x¯ i = p∗ · ωi . Define ME∗ (i) = 0 as ME∗ (i) = p∗ · x¯ i − p∗ · ωi for all i. Then, we have p∗ · x¯ i = p∗ · ωi + ME∗ (i). In addition, the condition of initial

t endowments, ωi ∈ R++ for all i such that i ∈ It for all t ∈ N , implies that p∗ · ωi > 0. Since xi ≻i x¯ i means that p∗ · xi = p∗ · x¯ i , the continuity of preference together with p∗ · ωi + ME∗ (i) > 0 implies that for every i, x¯ i is an individual maxima under p∗ and ME∗ . 

K

Note that in the above proof, we do not assume each preference,

≻i , to be ordered. Our equivalence theorem can also be utilized to axiomatically characterize the price-money message mechanisms as Sonnenschein (1974), where the Debreu–Scarf limit theorem plays an essential role in showing the category theoretic main result (see, Urai and Murakami, 2016). Appendix The purpose of this Appendix is to provide an example of allocation x such that the n-fold replica of allocation x belongs to F core(En (x))∩F core(En (ω)) for all n ∈ N , but is not an element of MW alras(E). Since our example does not depend on the shape of the indifference curves for sufficiently smallutility levels (less than the level obtained through allocation 3ϵ , 3ϵ for each consumer in the following example), we use the Cobb–Douglas utility functions, although they are not strictly monotone on the boundary. Let us consider overlapping generations economy E such that every agent lives for young and old two periods, where there is one consumption good for each period, and every t generation consists of two agents, it and i′t , for t = 1, 2, . . . . Each agent has initial endowment (2 + 2ϵ , 2ϵ ), where ϵ > 0 will be defined in the following as sufficiently small. ′ ′ Consider allocation x = (xi1 , xi1 , xi2 , xi2 , . . .) such that xit = ′ (0.1 + ϵ, 0.1 + ϵ) for all t = 1, 2, . . . , xi1 = (3.9, 0.9) and i′t x = (2.9, 0.9) for all t = 2, 3, . . . . Clearly, x is feasible. Assuming that all the agents’ marginal rate of substitution at x is 1 (see Fig. 3 for generations t = 2), we have p = (1, 1, . . .) as the supporting price for allocation x, which means that the replica allocation of x is weakly Pareto-optimal in the sense of Balasko and Shell (1980) for all replica economies: the n-fold replica allocation of x is an element of F core(En (x)) for all n ∈ N . Assume further

that all agents’ preferences are of the Cobb–Douglas type; their utility becomes arbitrarily a small level when their consumption at one of their life time periods is near to 0. Then we can check that the replica allocation of x is a finite core allocation for all replica economies: the n-fold replica allocation of x is an element of F core(En (ω)) for all n ∈ N . For example, let uit (y1 , y2 ) = y01.5 y02.5 for each t = 1, 2, . . . , ′

′

ui1 (y1 , y2 ) = (y1 − 3)0.5 y02.5 when y1 = 3, ui1 (y1 , y2 ) = y1 − 3 0.5 0.5 y2

i′t

when 0 5 y1 5 3, u (y1 , y2 ) = (y1 − 2) ′

when y1 = 2 and

uit (y1 , y2 ) = y1 − 2 when 0 5 y1 5 2 for each t = 2, 3, . . . . Then, no finite coalition in En (ω) can improve upon the n-fold replica allocation of x. Any finite coalition among members in the first generation fails to improve upon (0.1 + ϵ, 0.1 + ϵ) as long as ϵ < 10−3 . Indeed, utility level of at least one of such coalition members, i∗ , should be

 0.5

less than or equal to ui (2 + 2ϵ , 2ϵ ) = (2 + 2ϵ )0.5 2ϵ under the maximality for utility allocation with Cobb–Douglas type utility functions of homogeneity of degree 1. When ϵ < 10−3 , such coalition fails to block any utility allocations greater than or equals to those under (0.1 + ϵ, 0.1 + ϵ), hence never improve upon those under the n-fold replica allocation of x. Suppose that the n-fold replica allocation of x cannot be improved upon by any finite coalition among members of generations from 1 to k − 1. We show in the following that any finite coalition, S, among members from 1 to k also fails to block the n-fold replica allocation of x. Let us denote S by S1 ∪ S2 , where S1 is the set of members in generations from 1 to k − 1, and S2 is the set of members in generation k. Note that between S1 and S2 , we have only to consider two cases that there is a non-negative transfer of endowment commodity in period k from S2 to S1 or that there is a positive transfer of it from S1 to S2 . For the first case, under the same discussion in the previous paragraph, it is impossible to make utility levels of members of S2 greater than or equal to those under (0.1 + ϵ, 0.1 + ϵ). For the second case, it would be possible to keep all utility levels of members of S1 as good as those under x, but if so, by not doing such a positive endowment transfer, S1 can improve upon the replica allocation of x, which contradicts the assumption. It follows that, by mathematical induction, the n-fold replica allocation of x is an element of F core(En (ω)). Allocation x is not a non-negative wealth transfer Walrasian equilibrium under p. (The wealth transfer for type it agents should be negative.) The two-fold replica allocation of x does not belong to F core(E1 (x) ⊕ E1 (ω)) since, for example, i3 in E1 (ω) and i′3 in E1 (x) block the replica allocation of x with (1 + 2ϵ , 0.1 + 2ϵ ) for i3 and (3.9, 0.8) for i′3 . ∗

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Fig. 3. MRS for each agent is 1. Parameter ϵ > 0 for each allocation is neglected.

References Aliprantis, C.D., Burkinshaw, O., 1990. An overlapping generations model core equivalence theorem. J. Econom. Theory 50, 362–380. Arrow, K., Hahn, F., 1971. General Competitive Analysis. Holden-Day, San Francisco. Aumann, R.J., Drèze, J.H., 1986. Values of markets with satiation or fixed prices. Econometrica 53, 1271–1318. Balasko, Y., Shell, K., 1980. The overlapping-generations model. I. The case of pure exchange without money. J. Econom. Theory 23, 281–306. Bourbaki, N., 1966. General Topology. Hermann, Paris, English Translation: Springer-Verlag, 1995. Chae, S., 1987. Short run core equivalence in an overlapping generations model. J. Econom. Theory 43, 170–183. Chae, S., Esteban, J., 1993. Core equivalence in an overlapping generations model. J. Econom. Theory 59, 417–425. Debreu, G., Scarf, H., 1963. A limit theorem on the core of an economy. Internat. Econom. Rev. 4, 235–246. Reprinted in G. Debreu, Mathematical Economics, Cambridge University Press, Cambridge, 1983, pp. 151–162.

Esteban, J., 1986. A characterization of core in overlapping-generations economies. J. Econom. Theory 39, 439–456. Esteban, J., Millán, T., 1990. Competitive equilibria and the core of overlapping generations economies. J. Econom. Theory 50, 155–174. Konovalov, A., 2005. The core of an economy with satiation. Econom. Theory 25, 711–719. Mas-Colell, A., 1992. Equilibrium theory with possibly satiated preferences. In: Majumdar, M. (Ed.), Equilibrium and Dynamics: Essays in Honor of David Gale. Macmillan, London, pp. 201–213. (Chapter 9). Samuelson, P.A., 1958. An exact consumption loans model of interest with or without social contrivance of money. J. Polit. Econ. 66 (6), 467–482. Sonnenschein, H., 1974. An axiomatic characterization of the price mechanism. Econometrica 42 (3), 425–433. Urai, K., Murakami, H., 2016. An axiomatic characterization of the pricemoney message mechanism, Discussion Paper No. 15-31-Rev., Faculty of Economics and Osaka School of International Public Policy, Osaka University.