Discussion Papers In Economics
And Business
Existence of Monetary Equilibrium forOverlapping Generations Economies with
Satiated AgentsKEN URAIHIROMI MURAKAMI
TAKAHIRO MORIYA

Discussion Paper 22-03

August 2022

Graduate School of Economics
Osaka University, Toyonaka, Osaka 560-0043, JAPANExistence of Monetary Equilibrium forOverlapping Generations Economies with
Satiated Agents∗
Ken Urai†
Hiromi Murakami‡
Takahiro Moriya§

Abstract
This paper treats a general equilibrium existence problem on overlapping generations(OLG) economies with satiated agents and ﬁat money. The setting is important because it allows the analysis to incorporate Pareto optimality problems of the typewhere the value functions do not necessarily converge. For example, this is thecase for optimal allocations such as a basic income under a budget deﬁcit, which
can only be realized with policies that require non-negative wealth transfers for all
agents. Because of the existence of satiated agents, our argument can be identiﬁed
with a dividend equilibrium (Aumann and Dreze 1986) or an equilibrium with slack(Mas-Colell 1992). For OLG settings, however, some conditions (like arguments forstrictly positive endowments, resource relatedness and minimum wealth conditions
with negative prices, and limit arguments for dividends) must be reconsidered. We
note that by considering the OLG framework together with satiated agents, not only
can the existence of a monetary equilibrium be assumed in advance, but the existence of a monetary equilibrium based on an arbitrary money supply is guaranteed.
Taking into account that our setting includes production (without discount factor),
the model here provides a basis for a wide range of diﬃcult problems, including theplayer’s (equilibrium-dependent) survival problem, multi-sectoral capital accumulation, bequests, ﬁrm formation, and so on.
Keywords: Dividend Equilibrium, Monetary Equilibrium, Overlapping Generations
Economy, Satiation, Basic Income.
JEL Classiﬁcation: C62, D51, E40∗ A preliminary version. Do not circulate without permission of the authors.

† Graduate School of Economics, Osaka University, E-mail: urai@econ.osaka-u.ac.jp‡ Faculty of Economics, Otemon Gakuin University, E-mail: hi-murakami@otemon.ac.jp§ Department of Economics, Stony Brook University, E-mail: takahiro.moriya@stonybrook.edu

1Introduction
In this paper, we treat the general equilibrium existence problem on overlapping gen-erations (OLG) economies with possibly satiated agents and ﬁat money. The setting isimportant to incorporate monetary equilibrium allocations that are possible only after
non-negative wealth transfers are introduced.
Because of the existence of satiated agents, the equilibrium concept that we are concerned with can be identiﬁed with a dividend equilibrium or an equilibrium with slack (seeAumann and Drèze 1986 or Mas-Colell 1992). Equilibrium existence theorems on dividend equilibria for such ﬁnite economies are treated in Mas-Colell (1992), Kajii (1996),
Allouch and Florenzano (2013) and so forth. For the OLG framework, however, further
considerations and arrangements for conditions are necessary.
To investigate the existence of equilibrium with slack or money, the approaches by MasColell (1992) and Kajii (1996) were conﬁned to ﬁnite time horizon cases. Their approaches
also take the minimum wealth condition as given or as trivial under a certain initialendowment interiority condition. For the OLG framework, however, we must reconsider
several conditions like strictly positive initial endowments and several limit arguments forequilibrium states including negative prices. In this paper, we use the condition of social
non-satiatedness (SNS) of preferences and extend the method in Balasko and Shell (1980)
and Balasko et al. (1980) to assure the existence of an equilibrium for the OLG economy.
In section 2, rigorous deﬁnitions for the economies treated in this paper will be given.
The main theorem of the existence of equilibrium is given in Section 3, where several
discussions and remarks are also provided. The proof of the theorem is given in Section 4.

2

Economies
Let N be the set of all positive integers and R be the set of real numbers. For ﬁnite

set A, denote by ♯A the number of elements of A. We denote by RA , the ♯A dimensionalA
A
and R++, the non-negative and strictly positive orthants
Euclidean space R♯A and by R+
of RA , respectively. If A is a subset of B, RA is canonically identiﬁed with the subset ofRB .
An 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 non-empty ﬁnite subsets of∪∞
N such that t=1 It = I = N . It is the index set of agents in generation t for eacht ∈ N.
{Kt }∞
t=1 : a countable family of non-empty ﬁnite integer intervals, Kt =∪∞{k(t), k(t)+1, · · · , k(t)+ℓ(t)} such that t=1 Kt = K = N , k(t) < k(t+1) ≦ k(t)+ℓ(t)(E.2)

for all t ∈ N . Kt is the index set of commodities available to generation t for eacht ∈ N . We suppose that ℓ = maxt∈N ℓ(t) exists.
K(t)(E.3) {Yt }∞t=1 : a sequence of technologies. Yt is a compact convex subset of R∪thaving 0 as its element, where K(t) denotes the set s=1 Ks . We suppose that foreach t, there exists t̄ such that for all s ≧ t̄, Ys ∩ RK(t) = {0}. Denote by Y (t) the1

summation∑t

s=1 Ys .

We suppose that technology Yt is owned only by agents in∪tI(t + ℓ) for a positive integer ℓ, where I(t) denotes the set s=1 Is .
(E.4) {(≿i , ωi )}i∈I : countably many agents, where ≿i is continuous and convex inKt
the sense of Debreu (1959) on the commodity space of each generation, R+, for
Kteach i ∈ It and t ∈ N . The initial endowment of i, ωi , is an element of R+\ {0} foreach i ∈ It and t ∈ N .
t(E.5) {{mti }i∈I }∞t=1 : mi (yt , p) is a non-negative continuous proﬁt distribution func-tion for agent i with respect to technology Yt , where we suppose that yt ∈ Yt , p ∈ RN ,∑and i∈I(t+ℓ) mti (yt , p) = p · yt .1 We assume that for each i ∈ It , msi (ys , p) = 0 for
all s > t + ℓ.
We note that in (E.4), no monotonicity conditions are assumed, so that preferences areallowed to be satiated.
In the following, it would be convenient to identify the commodity space for each genKt
eration R+
with a subset of RK = RN , which is the set of all functions from K = N to
Kt
R, by considering x ∈ R+
a function that takes value 0 on N \ Kt . Then we can deﬁneKt
the total commodity space for economy ⊕∞t=1 R+ as the set of all ﬁnite sums of points
Kt
in commodity spaces of 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∞ ≈ RN under the pointwise convergence topology.
∞∞t∞Given an economy, E = ({It }∞t=1 , {Kt }t=1 , {Yt }t=1 , {(≿i , ωi )}i∈I , {{mi }i∈I }t=1 ), theKtprice space for E, P, is deﬁned as the set of all p in RN . A sequence, (xi ∈ R+
)i∈I ,
Kt
is called an allocation for E. Allocation (xi ∈ R+
)i∈I is said to be feasible if there exists

a sequence {yt ∈ Yt }∞t=1 such that∑∑t∈N i∈It

xi =

∑
t∈N

yt +

∑∑

ωi ,

(1)

t∈N i∈It

where the summability in RK = RN of both sides of the equality is assured by (E.2)
and (E.3). In equation (1), the no free disposability is assumed since we do not use any
monotonicity condition for each agent. Under the attainability condition (1) and thecondition on Ys and RK(t) in (E.3), we can obtain for each i ∈ It a compact convex set
Ktsuch that every feasible allocation for i is a relative interior point of Xi , so thatX i ⊂ R+for individual maximization problem there is no loss of generality (under the convexity of
Kt 2.
preferences) to restrict i’s consumption set to Xi instead of R+

The list of price vector p∗ ∈ P, non-negative slack variables (Mi∗ ∈ R+ )i∈I , and an
Kt)i∈I feasible under (yt∗ ∈ Yt )t∈N , is called a dividend equilibrium state
allocation (x∗i ∈ R+

for E, if for each t ∈ N , p∗ ·yt∗ ≧ p∗ ·yt for all yt ∈ Yt , and for each i ∈ It , x∗i is a ≿i -greatest
1 In this paper, we use the inner product notation even for two inﬁnite dimensional vectors in RN ≈ R∞

as long as one of which can be identiﬁed with a ﬁnite vector (through the identiﬁcation like RKt ⊂ R∞ ⊂R∞ explained in the next following paragraphs), and the summability for the coordinate products isclearly warranted. For the continuity of msi , we take the topology of RN ≈ R∞ as the product (pointwise
convergence) topology.
2 In this paper, for the sake of simplicity, we assume that each Y , t = 1, 2, . . ., is compact. It followstthat we can take such compact Xi for each i naturally from the feasibility condition (1). For more generalcases, however, we have to ensure the compactness of attainable set by using conditions on asymptotic
cones like Debreu (1959). The compactness of Yt and Xi should be derived from such conditions.

2element in the set
Kt{xi ∈ Xi ⊂ R+
| p∗ · x i ≦ p∗ · ω i +t+ℓ∑msi (ys∗ , p∗ ) + Mi∗ }.

(2)

s=1

Since the non-negative wealth transfer is possible to be identiﬁed with the money supply in
perfect-foresight economies, a dividend equilibrium allocation can also be called a monetary
Wlaras allocation for E.
In order to assure the existence of equilibrium for OLG economies, a certain kind ofresource relatedness condition is necessary. For this purpose, instead of the monotonicity
of preferences, we assume the following condition.
(SNS: Socially Non-Satiated preference conﬁguration)
For all generation t, for each commodity k, s ∈ Kt and feasible allocationx = (xi )i∈I , there exists at least one agent i ∈ It having a positive endowmentof commodity s and non-empty open interval (0, ϵ) ⊂ R+ satisfying one of
the following two conditions: (1) for every δ ∈ (0, ϵ), xi + (0, · · · , 0, δ, 0, · · ·) is
strictly preferred to xi , or (2) for every δ ∈ (0, ϵ), xi + (0, · · · , 0, −δ, 0, · · ·) is
strictly preferred to xi , where the non-zero entry of the additional vector, δ or−δ, is the k-th coordinate.
The condition for a ﬁnite satiation economy with non-negative price settings was introduced in Murakami and Urai (2017) and Murakami and Urai (2019) as one of the simplest
method to assure the resource relatedness for the entire economy. Condition SNS is automatically satisﬁed when for each generation t, there is at least one agent i ∈ It whose
Kt
preference is strictly monotonic and ωi ∈ R++. On the other hand, the condition does notsatisﬁed if there exists a commodity k such that under a certain feasible allocation (xi )i∈I ,all agents possible to consume commodity k are satiated with k. We also note that SNS∑
Ktmeans that for all t ∈ N , i∈It ωi ∈ R++.

3

Existence of Dividend Equilibrium
For each T ∈ N , denote by E(T ) a ﬁnite truncated economy that consists of all agents∪T
s=1 Is . The proof of the existence theorem is essentially equivalent to thein I(T ) =usual diagonal argument (see, Balasko-Call-Shell 1980, etc.), which derives the equilibriumof the total economy E as a limit of a sequence of equilibria for truncated economies.
There are some diﬃculties, however, associated with the necessity to include preference
saturation and the possibility of negative prices, together with the resource relatedness(SNS) condition speciﬁc in this paper. In particular, we use the Mas-Colell (1992) theorem
to prove the existence of an equilibrium in economy E(T ), but problems speciﬁc to OLG
models (especially, in guaranteeing minimum wealth conditions for agents whose initialendowments cannot be strictly positive) prevent the theorem from a direct application(see, for example, equation (5) in section 4.2). We have the following theorem.
Theorem 1 . Under (E.1)-(E.5) and SNS conditions, an overlapping generations economy
E has a dividend equilibrium.
3■

Proof : See Section 4.

Discussion 1 (Basic Income Policy)
Nowadays, government intervention for the purpose of redistribution to a certain entity
has become more and more diﬃcult and is not working practically. For example, resistances from corporations and rich people would be strong, and from the public-ﬁnance
perspective, tax distortions are not desirable. This study focuses on government basic
income-like spending policies as a possible new redistribution. Above all, this has a lumpsum tax aspect and does not cause distortion (i.e., non-negative transfers could be accepted with lower resistances by all economic agents). By using the above theorem, we
can say that there is an equilibrium and that the weak Pareto optimality in the sense ofBalasko and Shell (1980) is always assured (the allocation cannot be improved by any ﬁnite agents). Of course, we have many Pareto optimal examples in OLG economy that are
possible only after positive money supply (negative budgetary deﬁcit of the government)
is introduces. Under certain criteria, the equilibrium with BI is better. (This is where weneed to extend the model to say that our new redistribution policy is properly supported
by microeconomics and we need to expand the model.)Discussion 2 (Capital Accumulation)
We also note that by considering the OLG framework together with the existence of satiated agents, we can easily introduce bequests and capital accumulations into the model.
Indeed, if the existence of satiated agents and their consumptions are followed by appropriate non-negative wealth transfers to others, the situation cannot be distinguished fromthe existence of bequests as long as the determination problem of the heirs are exogenously
given. It is also possible to understand that such bequests are bought by the government(so satiated agents are convinced in that their bequests are saved as government-bankdeposits), lended to the next-generation technology, transformed to commodities in the
future (including capital goods), and the capital accumulation (by using the ﬁat moneyor the ﬁscal deﬁcits) takes place. For such circumstances, it is easy to construct Pareto
optimal monetary equilibrium examples under which ﬁscal deﬁcits (government bonds) do
not necessarily be redeemed. (To obtain such Pareto optimal allocations, the ﬁscal deﬁcits
for eternal time periods are necessary.)

Discussion 3 (Survival Problem)
The framework of preference saturation and government dividends provides an interesting open question regarding the so-called general equilibrium with survival problem.
Here, the survival problem is a general equilibrium existence problem in which, undersome price conditions, the survival conditions of some of the consumers in the economy
are not satisﬁed (i.e., the budget constraint set is empty under some price). Suppose thata group of people in the economy are able to saturate everyone under their initial holdings

4and technology, and that the surplus of output under this saturated situation is suﬃcient
to survive all the rest of the people in the economy. Can the government’s dividend policy then guarantee the existence of dividend equilibrium in the economy with a survival
problem? We also note that

Discussion 4 (General Dynamic Game)The ﬁrst thing that characterizes our model from the viewpoint of inﬁnite-horizon dynamic games is that it can handle problem setups and equilibrium paths in which theobjective function and value function are not necessarily ﬁnite. Needless to say, thismeans that solution paths that cannot be captured by dynamic programming based on
the Bellman equation are also subject to analysis. Second, while the above is well known ineconomic models with overlapping generations for consumption, in this paper it is treatedin a general way (without the use of discount factors) by including production. Although
individual production activities are directly depicted as closed within a ﬁnite period, they
overlap generationally, so that the model introduces technology that spans an inﬁnite
number of periods as a whole. Thus, our model can serve as a basis for asking moregeneral questions from a microeconomic standpoint toward capital accumulation, multisector growth issues, and in some cases, ﬁrm formation and industrial structure. Third,
the government is introduced as the money-issuing entity, but its role is approximately
minimal in an economic context, including the fact that the value of money may be zero(its value is determined by equilibrium). However, as a quid pro quo for introducing thisspecial player, it is possible to deal with the survival problem as discussed in Discussion 3.
This means that the setting can include situations where the players’ choice set is empty,
depending on how the equilibrium is (such as choices under prices), and allows for a morediverse treatment of hard-to-handle issues related to the game structure itself (such as
bankruptcy and default).

4

Proof

4.1

Setting Truncated Economy till generation T

Let E(T ) = ({It }Tt=1 , {Kt }Tt=1 , {Yt }Tt=1 , {{(≿i , ωi )}i∈I(T ) , {mti }}i∈I(T ) }Tt=1 ) be an economy truncated at a generation T . Let h be an index for a good, and let ph be a pricefor the good h. We denote by H(T ) the total number of goods till generation T , i.e.,H(T ) = ♯K(T ) = k(T ) + ℓ(T ). Price vectors for E(T ) will be denoted like p(T ) or p∗ (T ),and we deﬁne sets of prices, P(T ), P∗ (T ) and P∞ (T ) as follows:• p(T ) = (p1 , p2 , · · · , ph , · · · , pH(T ) ),• p∗ (T ) = (p∗1 , p∗2 , · · · , p∗h , · · · , p∗H(T ) ),• P(T ) = {p(T ) ∈ RH(T ) |||p(T )|| ≤ 1},• P∗ (T ) = {p∗ (T ) ∈ RH(T ) \{0}|||p∗ (T )|| ≤ 1},• P∞ (T ) = {(p∗1 , p∗2 , · · · , p∗H(T ) , pH(T )+1 , · · ·) ∈ P∗ (T ) × R × R × R · · · | ||p∗ (T )|| ≤ 1}.
5

Here, each production set is assumed to be compact.3 So, naturally, each consumption
set, Xi , in the economy can also be assumed to be compact. This is because the production
plan is ﬁnite, and, therefore, consumers can consume at most ﬁnite amounts in total.
In the following proof, we ﬁrst restrict our argument to the truncated economy E(T ).4

4.2

Budget constraint modiﬁcation

To show the existence of an equilibrium for E(T ), we have to modify the budget constraint in equation (2). Let us introduce a modiﬁed budget set for individual i ∈ It ⊂ I(T )for price p(T ) ∈ P(T ), as follows:• If ||p(T )|| ̸= 0, the modiﬁed budget set is described as
Kt{xi ∈ Xi ⊂ R+
| p(T ) · xi ≦ p(T ) · ωi +T∑

msi (ys , p(T )) +s=1

1 − ||p(T )||}.
||p(T )||

(3)

• Otherwise, it isKt| p(T ) · xi ≦ p(T ) · ωi +{xi ∈ Xi ⊂ R+

T∑

msi (ys , p(T )) + A},

(4)

s=1

where A is taken to be a large constant such that it covers the whole consumption set Xi
for all i ∈ I(T ). Since Xi is compact and I(T ) is ﬁnite, an easy calculation shows that we
can take A as a real number greater than the diameter of set Xi for all i ∈ I(T ).5)||
Adjustment term 1−||p(T||p(T )|| is a device to ensure the minimum wealth condition (Mas-Colell 1992, condition V) at every price whose norm is less than 1 (including the point,||p(T )|| = 0). For prices such that ||p(T )|| = 1, however, we need a further elaboration
since by the OLG framework, it is not appropriate to assume that an initial endowment
for i is strongly positive.6 So in order to ensure the minimum wealth condition for alli ∈ I(T ), some income redistribution is necessary. Since every ωi is semi-positive, theminimum wealth condition is always satisﬁed as long as p(T ) is strictly positive. The
minimum wealth condition is also satisﬁed for all agent as long as p(T ) has at least one
negative coordinate. Therefore, the problem occurs only for points near to price p(T ) suchthat ∥p(T )∥ = 1 and p(T ) has at least one coordinate equal to 0, the relative boundary ofK(T ){p(T ) ∈ P(T )| ∥p(T )∥ = 1} ∩ R+

in the surface of P(T ).

Here, we can utilize condition (SNS). As we shall see in subsection 4.6, for economy
E(T ), there is a positive lower bound bl and upper bound bu such that |pk |/|ps | ∈ [bl , bu ]for all k, s ∈ K(T ) as long as p(T ) = (p1 , p2 , . . . , pH(T ) ) ∈ P(T ) such that ∥p(T )∥ = 1 is anequilibrium price for E(T ). This necessary condition for equilibrium prices also provides
a minimum absolute value ϵs > 0 for each coordinate, s ∈ K(T ). Indeed, if |ps | < ϵ, there
3 We do not have to assume that the production set is compact. Assumptions on production sets basedon their asymptotic cones like Debreu (1959) would be suﬃcient.
4 We ﬁrst focus on the truncated economy till time T . However, Balasko and Shell (1980) consider an
economy with an inﬁnite horizon setting directly. See, Balasko et al. (1980) for a more detailed treatment.
5 In our setting, we have to consider the cases that ||p(T )|| = 0 because we have to consider negative
prices (the price space, P(T ), is taken to be a ball shape containing {0}).
6 Only assumed in (E.4) is that for each i ∈ I , ω ∈ RKt \ {0} though we know by (SNS) that
t
i
+
∑
Kt
ω
∈
R
.
i
i∈It
++

6

must exist at least one k ∈ K(T ) \ {s} such that |pk | is greater than (1 − ϵ)/(H(T ) − 1).7
Then, by taking ϵ > 0 to be suﬃciently small, we have |pk |/|ps | > bu , a contradiction.
Deﬁne ϵ > 0 as the minimum of {ϵs | s ∈ K(T )}. Denote by B the relative boundary of set
K(T ){p(T ) ∈ P(T )| ∥p(T )∥ = 1} ∩ R+

in the surface of the closed unit ball, P(T ). Let Uϵ be

the open ϵ neighborhood of 0 and denote by U (B) the open ϵ neighborhood, Uϵ + ∂B, of
B. Note that every p(T ) ∈ U (B) ∩ P(T ) cannot be an equilibrium price for E(T ). Since
P(T ) is closed, set P(T ) \ U (B) is closed, so for each p(T ) ∈ U (B), the distance betweenp(T ) and P(T )\U (B), d(p(T )) = dist(p(T ), P(T )\U (B)) is strictly positive (and less than∑K(T )or equal to ϵ). Let ω(T ) ∈ R++ (by SNS condition) be the vector ( i∈I(T ) ωi )/♯I(T ).
Now, based on the modiﬁed budget set (3) and (4), we further modify budget set forTi ∈ It ⊂ I(T ) under production {ys }s=1
, by deﬁning income function mi ({ys }Ts=1 , p(T ))as follows:() ∑Tϵ − d(p(T ))d(p(T ))p(T ) ·ωi +ω(T ) +msi (ys , p(T )) = mi ({ys }Ts=1 , p(T )),
ϵ
ϵ
s=1

(5)

where mi can be identiﬁed with the same notation in Mas-Colell (1992). So, we candirectly apply the way employed in Mas-Colell (1992) for the existence of equilibrium
prices for E(T ). Note that if p(T ) belongs to P(T ) \ U , the left hand side of equation (5)∑Tis equivalent to p(T ) · ωi + ( s=1 msi (ys , p(T )) in equations(3) and (4). For p(T ) in U , the)∑∑d(p(T ))
ϵ−d(p(T ))ωi +ω(T ) is equal to p(T ) · i∈I(T ) ωi , so misummation i∈I(T ) p(T ) ·ϵϵ
is nothing but a continuous income redistribution that is strictly positive for all agents(satisfying the minimum wealth condition V of Mas-Colell 1992).

4.3

Existence of Equilibrium for E(T )For economy E(T )=({It }Tt=1 ,{Kt }Tt=1 , {Yt }Tt=1 , {{(≿i , ωi )}i∈I(T ) , {mti }i∈I(T ) }Tt=1 ), a
dividend equilibrium till generalization T is deﬁned as a triple, (p∗ (T ), {{x∗i }i∈It }Tt=1 ,{yt∗ }Tt=1 ) which satisﬁes the following conditions: (6) feasibility, (7) proﬁt maximizationand (8) utility maximization.
∑∑

x∗i =

t∈T i∈It

∑
t∈T

yt∗ +

∑∑

ωi ,

(6)

t∈T i∈Ityt∗ ∈ Yt , and p∗t yt ≤ p∗t yt∗ for all yt ∈ Yt , for all t = 1, . . . , T.

(7)

Kt
under the budget,
For all i ∈ It and t = 1, . . . , T , x∗i is a ≿i -greatest element in Xi ⊂ R+p∗ (T ) · xi ≤ p∗ (T ) · ωi +T∑

msi (ys∗ , p∗ (T )) +s=1

1 − ||p∗ (T )||.
||p∗ (T )||(8)

The proof is directly from Mas-Colell (1992) by using the modiﬁcation (equation (5)) of∗(T )||
consumer budgets. The last term 1−||p||p∗ (T )|| , can be identiﬁed with an equilibrium mone-tary (non-negative wealth) transfer to consumer i ∈ I(T ), Mi∗ (T ), which is automaticallyequally to all consumers in I(T ) in our present setting. As will be discussed in subsection
4.8, the distribution ratio of money for each agent, i ∈ I(T ), can be set freely, by using∗∗(T )||
1−||p (T )||parameter Zi ∈ R++ and considering term Zi 1−||p||p∗ (T )|| instead of ||p∗ (T )|| for budget

constraints in (3) and (8).8
7 If not, the norm of p(T ) cannot be greater than or equal to 1.
8 Since the term has the role to ensure the minimum wealth condition (so that the continuity of budget

7

4.4

Eliminate the case ∃h, p∗h = 0 for economy E(T )If p∗h , h ≤ H(T ) is 0, from SNS condition, ∃ i ∈ It ⊂ I(T ), h ∈ Kt such that i has apositive initial endowment of s ∈ Kt and ph = 0 can never be a supporting price for i
at every attainable allocation. It follows that till generation T , a competitive equilibrium(CE) price, p∗ (T ) for E(T ) cannot contain a coordinate 0. Especially, CE price space till
generation T , P∗ (T ), does not contain 0.
P∗ (T ) = {p∗ (T ) ∈ RH(T ) \{0}|||p∗ (T )|| ≤ 1}.

4.5

(9)

Bounds using price ratio

We need to derive an upper and lower bound of the competitive price for accommodating
inﬁnite price sequence.9 Let me deﬁne xi + = x∗i +(· · · , δk , · · ·) or xi − = x∗i +(· · · , −δk , · · ·),where δk , ∈ R++ .
From SNS condition, ∃i ∈ It in every generation t, for his consumption bundle xi ∈H(T )−R+ , we can take strictly preferred x+
i or xi by taking δk on any good k. Moreover,
′
′
H(T )
| d(xi , x+continuity of preference ensures that, on open ball Br+k (xi ) (x+i ) = {xi ∈ R+
i )<
′
H(T )
−
−
−
′
rk(xi ) } or Brk (xi ) (xi ) = {xi ∈ R+
| d(xi , xi ) < rk(xi ) }. Note that rk(xi ) ∈ R++ is+−dependent on xi and xi or xi . When rk(xi ) is considerably small, within its open ball,
′

′

′′

′′

we can pick up any xi that satisﬁes xi ≻i xi . Note that xi is any arbitrary point within′′′′
the open ball Bq (xi ) = {xi ∈ RH(T ) | d(xi , x∗i ) < q}.
The bounds of CE prices are decided so that the hyperplane through Bq (xi ) made by−−
the CE price vector does not cross Br+k (xi ) (x+i ) or Brk (xi ) (xi ). That is, for the individual,

at x∗i ∈ R+H(T ), he can pick up more desired consumption point. This is because from SNS

condition under any δk ∈ R++ , xi + = x∗i + (· · · , δk , · · ·) or xi − = x∗i + (· · · , −δk , · · ·) would
be also in his feasible set. It is contradiction to the property that x∗i is a maximum point.
As follows, we prove formally for any goods pair s and k, the CE price rate is bounded.
NOTE THAT the proof is incomplete and we will check it later.

4.5.1

x∗is is inner or corner of consumption set

NOTE THAT we need to check this subsection later.
This is the case where x∗is is inner of consumption set
Under CE price till time T , p∗ (T ) = (p∗1 , p∗2 , · · · , p∗T ), for i ∈ I(T ) who falls into individualx∗x∗with SNS condition, p∗ (T ), should be at least on the region where hyperplane hi i , hi i =
x∗

x∗

N
i
{xi ∈ R+| p∗ (T )(x∗i − xi ) = 0} satisﬁes hi i ∩ Brk (x∗i ) (x+(x−i ) = ϕ or hi ∩ Brk (x∗i ) = ϕ.
i)Note that this creates ”biggest” upper bound. (Here we use property of SNS condition
that when x∗ik = 0 better set is located at right side for sure. Otherwise just applying SNS
condition does not help restrict the CE price space.

correspondence) for each agent at p(T ) = 0, we cannot set Zi to be equal to 0, as long as we use themethod of Mas-Colell (1992)
9 Only eliminating the case ∃h p∗ = 0 is not enough because we need closeness. Intuitively, eliminatingh∃h p∗h = 0 leads the space to be open.

8

If x∗is = 0 in this case the same argument, that is, the hyperplane through x∗i does nothelp because x∗is = 0 (in this case better set is not ball but hemisphere). Then, we use the
SNS condition saying initial endowment of goods s is strictly positive. So, instead of using
the hyperplane through x∗i , we use the hyperplane through individual i’s initial endowment.
Always individual i can consume his initial endowment. Therefore, at least under the CEN∗iprice the hyperplane through initial endowment hω
i = {xi ∈ R+ | p (T )(ωi − xi ) = 0}

does not intersect with his better set. (NOTE THAT this is also ”biggest” upper bound.
∑t+l)||This means there is term s msi (y s , p(T )) + 1−||p(T||p(T )|| . So, possible hyperplane is steeper
also.
Particularly, We mention how to get the bound of arbitrary price ratio. Focusing on
arbitrary two goods, let them goods s and k without loss of generality. Imagine exchangeof the two goods through market. From SNS condition, ∀t ∈ T there is at least one non−
∗
∗
∗
∗
∗
∗
satiated individual. He strictly prefer x+i or xi to xi = (xi1 , xi2 , · · · , xik , xis · · · , xiT ). Of

course He can move −p∗k /p∗s slope, exchanging unites of goods k instead of unites of s
through market. We can regard this price rate as a kind of hyperplane through x∗i or ωi(this is corner solution case). If x∗i is CE allocation, that |p∗k /p∗s | holding CE is restrictedx∗ito where the hyperplane passing through x∗i (or ωi ), hi i (or hωi ), must not interact−−
with Br+k (x∗ ) (x+i ) or Brk (x∗ ) (xi ). Otherwise that individual can achieve strictly preferrediiconsumption, which is contradict to the condition of CE allocation.

4.5.2

Details

NOTE THAT we need to check this subsection later.
There are three cases (1). x∗i is located on the inner of his consumption set, (2). x∗i islocated corner, while x∗is > 0 and (3). x∗i is located corner, while x∗is = 0.
• (1). x∗i is located on the inner and (2). x∗i is located corner, while x∗is > 0
We can use the discussion we mentioned above, open ball argument. So, we omit detailed
explanation. See the above discussion.
• (3). x∗i is located corner, while x∗is = 0.
In this case, we need careful attention. As follows, we describe in detail. Suppose x∗i islocated corner while x∗is = 0 (case (3)), the possible GE prices are the 4 cases. Moreover,
we have to take into account the better set (left-sided or right-sided). In sum, supposep∗ (T ) and (x∗i , yt∗ ) are CE price and consumption for individual i with SNS condition,
there are 8 possible cases.
• a. p∗k > 0 p∗s > 0 and better set is right-sided• a’. p∗k > 0 p∗s > 0 and better set is left-sided• b. p∗k > 0 p∗s < 0 and better set is right-sided• b’. p∗k > 0 p∗s < 0 and better set is left-sided
9• c. p∗k < 0 p∗s > 0 and better set is right-sided• c’. p∗k < 0 p∗s > 0 and better set is left-sided• d. p∗k < 0 p∗s < 0 and better set is right-sided• d’. p∗k < 0 p∗s < 0 and better set is left-sided
We check whether we can get upper bound one by one in this subsection. Note thatin every case it is enough to check the normal vector of −pk /ps slope passing throughTTωi = (· · · , ωis > 0, · · ·) ∈ R+
and (· · · , x∗ki + ϵ − r, 0, · · ·) ∈ R+
. This is because, even if∑t+l s s
1−||p(T )||there is additional term s mi (y , p(T )) + ||p(T )|| in his budget constraint, the normalT
vector through ωi = (· · · , ωis > 0, · · ·) ∈ R+
must be steeper.

That is why in the following we only focus on the normal vector through ωi .
Case a.: From strictly positiveness of ωis , the |p∗k /p∗s | is restricted as the a normal vectorof ωi does not intersect with the better set Br+k (x∗ ) (x+i ). Otherwise, individual i with SNSicondition have incentive to change his GE consumption x∗i . This is contradiction.

Case a’.: In this case the better set Br−k (x∗ ) (x−i ) is located within his feasible set atiﬁrst, because p∗k > 0 p∗s > 0 means feasible set is left side of the normal vector. So, it is
impossible that x∗i is maximum point at ﬁrst.

Case b.: From strictly positiveness of ωis , the |p∗k /p∗s | is restricted as the a normal vectorof ωi does not intersect with the better set Br+k (x∗ ) (x+i ). Otherwise, individual i with SNSicondition have incentive to change his GE consumption x∗i . This is contradiction.

Case b’.: In this case the better set Br−k (x∗ ) (x−i ) is located within his feasible set atiﬁrst, because p∗k > 0 p∗s < 0 means feasible set is left side of the normal vector. So, it is
impossible that x∗i is maximum point at ﬁrst.

Case c.: In this case the better set Br+k (x∗ ) (x+i ) is located within his feasible set at ﬁrst,
ibecause p∗k < 0 p∗s > 0 means feasible set is right side of the normal vector. So, it is
impossible that x∗i is maximum point at ﬁrst.

Case c’: From strictly positiveness of ωis , the |p∗k /p∗s | is restricted as the a normal vectorof ωi does not intersect with the better set Br−k (x∗ ) (x−i ). Otherwise, individual i with SNSicondition have incentive to change his GE consumption x∗i . This is contradiction.

Case d: In this case the better set Br+k (x∗ ) (x+i ) is located within his feasible set at ﬁrst,
ibecause p∗k < 0 p∗s < 0 means feasible set is right side of the normal vector. So, it is
impossible that x∗i is maximum point at ﬁrst.

Case d’: From strictly positiveness of ωis , the |p∗k /p∗s | is restricted as the a normal vectorof ωi does not intersect with the better set Br−k (x∗ ) (x−i ). Otherwise, individual i with SNSi10condition have incentive to change his GE consumption x∗i . This is contradiction.
So, suppose p∗ (T ) and (x∗i , yt∗ ) are CE price and consumption for individual i with SNS
condition, only possible cases are a, b, c’ or d’. In each case we can derive upper bound,as we will show in the ”examples of such bounds” section.

4.5.3

Examples of such bounds

NOTE THAT we need to check this subsection later.
(Just to be sure, here we give a few examples of such bounds)• Case.1 : x∗is is inner of consumption set and better set is rightT
Let x∗i = (· · · , x∗ik , x∗is , · · ·) ∈ R+
without loss of generality. When we focusing on goods′

Tk and s, we can always, from SNS condition, take better set Br+k (x∗ ) (x+i ) = {xi ∈ R+ |
i

′

∗
∗
∗
} ,here x+d(xi , x+i ) < rk(x∗i = xi = (· · · , xik + ϵ, xis , · · ·). So, at CE the bounds of pricei)rate p∗k and p∗s is decided in the following way (from elementary geometry)22∗ ∗||(· · · , 0, p∗k , p∗s , 0, · · ·) + x∗i − x+
i || = ϵ − r ∧ ||(· · · , 0, pk , ps , 0, · · ·)|| = r.

So, given x∗i and ϵ, r,r|p∗k /p∗s | = √ϵ 1−ϵ• Case.2 : x∗is = 0 and better set is rightLet ωi = (· · · , ωis > 0, · · ·). In this situation, the ratio of p∗k and p∗s is decided as follows,(· · · , 0, p∗k , p∗s , 0, · · ·)·((· · · , x∗ik + ϵ − r, 0, · · ·) − ωi ) = 0.
So, given x∗i and ϵ, r,|p∗k /p∗s | =x∗ik + ϵ − r − ωikωisSimilarly, we can derive CE price rate of other cases like better set is left side.
So we omit the rest.

4.6

Relationship between generation T and T + 1

s,k
Let bs,kl (T ) and bu (T ) be lower and upper bounds of the price ratio between goodss ∈ K(T ) and k ∈ K(T ) within which every CE price ratio till generation T must belong.
s,kNext, for T + 1, if we obtain the same two bounds, bs,kl (T + 1) and bu (T + 1), we cans,kreplace them by bs,kl (T ) and bu (T ), since if “pk /ps is not a CE price ratio (supporting

all agents’ optimal consumption vectors) till generation T ,” then it is clear that “pk /ps is
not a CE price ratio (supporting all agents’ optimal consumption vectors) till generations,k
T + 1,” so the upper and lower bounds, bs,kl (T ) and bu (T ), also deﬁnes the interval that

CE price ratio pk /ps till generation T + 1 must belong.
11

Hence, for each two commodities s and k, by considering the ﬁrst T such that s ∈ K(T )and k ∈ K(T ), we obtain the lower and upper bounds of the price ratio between goods ss,ks,kand k, bs,k= bs,kll (T ) and bu = bu (T ), within which every CE price ratio (pk /ps ) must
belong.
Let us summarize the above by using a set-theoretic form.
For each s and k,
Rs,k (T + 1) ⊆ Rs,k (T ),where Rs,k (T ) is deﬁned as
Rs,k (T ) = {|p∗k /p∗s || p∗k /p∗s is a CE price ratio till generation T },(10)

Rs,k (T + 1) = {|p∗k /p∗s || p∗k /p∗s is a CE price ratio till generation T + 1}.

(11)

and

As is mentioned in section 4.2, we can use such lower and upper bounds for eachcommodity-price ratio to deﬁne a compact range for each commodity where general equilibrium prices (if such exist) necessarily belong. One of the simplest ways is to normalize
the norm of the ﬁrst coordinate of a price, p∗1 , which cannot be 0 as stated in section 4.4,
to be 1. Now, we can deﬁne for each commodity k = 1, 2, ..., a compact interval,[ak , bk ] ⊂ R, 0 < ak ≦ bk , k = 1, 2, . . . ,(12)such that the norm (the absolute value) of the k-th coordinate |p∗k | must belong to [ak , bk ]as long as p∗ = (p∗1 , p∗2 , . . .) is a candidate for an equilibrium price. (Of course we candeﬁne a1 and b1 as a1 = b1 = 1.)

4.7

Limits of x∗i (T ), yt∗ (T ) and Mi∗ (T )About the limits of competitive prices when T → ∞, from the above discussion and by
Tychonoﬀ’s theorem, we can obtain a compact set for CE prices (normalized their ﬁrst
coordinates as norm 1) as
P∗ = {p = (p1 , p2 , . . .) ∈ R∞ | |pk | ∈ [ak , bk ] for all k = 1, 2, . . .}.

(13)

For each T = 1, 2, ..., and T -generation economy E(T ), we have an equilibrium pricep∗ (T ) in P∗ (T ) identiﬁed with a subset of R∞ through P∞ (T ) ⊂ R∞ . So, we can take a∗convergent subsequence of {p∗ (T )}∞T =1 in its compact competitive price set P when T →∏∞∞.10 Note that the topology of P∗ = k=1 ([−bk , −ak ] ∪ [ak , bk ]) assured by Tychonoﬀ’s

theorem for compactness is the product topology that is equivalent to the relativized
topology to P∗ of R∞ . In other words, without loss of generality (since the choice ofﬁnite truncation economies is arbitrary), we may identify {p∗ (T )}∞T =1 with a pointwise(coordinatewise) convergent sequence in R∞ to a certain p∗ = (p∗1 , p∗2 , . . .) ∈ P∗ .
With the convergent sequence, {p∗ (T )}∞T =1 , is associated three kinds of sequences of∗∞T -period equilibrium states, {x∗i (T )}∞T =1 for each i ∈ I, {yt (T )}T =1 for each t = 1, 2, . . .,10 The argument here might be better to use the ordinary coordinatewise convergence together with the
diagonal method instead of using Tychonoﬀ’s theorem. See Murakami and Urai (2017) for a detailed priceconvergence argument in OLG economy under SNS condition.

12

∗
∗
and {Mi∗ (T )}∞T =1 for each i ∈ I. Since the convergence of p (T ) to p is coordinatewise,

and since for each t = 1, 2, . . ., yt∗ (T ) is a solution of the ordinary proﬁt maximization
problem in RK(t) , we may conclude that (by using the Berge’s maximum theorem andtaking a subsequence if necessary) that yt∗ (T ) converges to a point yt∗ such that yt∗ is a
proﬁt maximization point in the compact Yt under p∗ . Of course, the convergence of yt∗ (T )to yt∗ together with p∗ (T ) to p∗ means the convergence of mti (yt∗ (T ), p∗ (T )) to the amountmti (yt∗ , p∗ ) for each i and t since every mti is continuous. It is also clear (since every Xi
is compact) that we may suppose that (by taking subsequences repeatedly) {x∗i (T )}∞T =1
converges to x∗i ∈ Xi for each i ∈ I. As limits satisfying condition (6), it is also clear thatthe feasibility condition (1) is satisﬁed. Therefore, all we have to check is to assure eachconsumer’s utility maximization under (2), although it should be noted that the value Mi∗
for each i is still not ﬁxed. Here, note that for each i ∈ It , sequence {Mi∗ (T )}∞T =1 (after
following the normalization of prices, p∗ (T ), T = 1, 2, ...), has an obvious upper bound,∑H(t)At k=1 |bk |, where At is the budget modiﬁcation constant in E(t), since for equilibrium
prices of E(t) normalized to the Euclidean norm of 1, the norm of another normalization∑H(t)based on the ﬁrst coordinate cannot exceed the sum, k=1 |bk | (the value of maximum
variational norm). It follows that, again by taking subsequences repeatedly (if necessary),∗
we can assume for all i, sequence {Mi∗ (T )}∞T =1 converges to Mi ∈ R+ . Now it would be aroutine task for every i to check that x∗i is a utility maximization point under (2) since theargument is nothing but to check (again by using Berge’s maximum theorem) the uppersemicontinuity of Marshallian demand correspondence.

4.8

)||
is diﬀerent from person
Extension to the case where 1−||p(T||p(T )||
to person

Since we use the method in Mas-Collel (1982), we have to take Mi∗ (T ) as Mi∗ (T ) =1−||p(T )||||p(T )|| , i.e., the monetary transfer is the same for everyone in the economy till generation

T . We can extend this situation to where each person’s dividend is diﬀerent.
Let Zi ∈ R++ be a dividend operator, which allows diﬀerence between the amounts ofdividend an individual gets. With Zi , we modify the budget constraint as follows,p(T )xi ≤ p(T )ωi +T∑

msi (ys , p(T )) + Zi

s=1

1 − ||p(T )||.
||p(T )||

(14)

As long as Zi ∈ R++ , the minimum wealth condition still holds. So, the the rest ofargument will not change. Hence, we can prove there is CE in the same way as before.

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Prices,” Econometrica 53, 1271–1318.

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Balasko, Y., Cass, D., and Shell, K. (1980): “Existence of Competitive Equilibrium in a
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Kajii, A. (1996): “How to discard non-satiation and free-disposal with paper money,”
Journal of Mathematical Economics 25, 75–84.
Mas-Colell, A. (1992): “Equilibrium theory with possibly satiated preferences,” in Equilibrium and dynamics: Essays in honor of David Gale., (Majumdar, M. ed) , Chapter 9, pp. 201–213, Macmillan, London.
Murakami, H. and Urai, K. (2017): “Replica core limit theorem for economies with satiation,” Economic Theory Bulletin 5(2), 259–270.
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Message Mechanism for Economies with Satiation,” Journal of Mathematical Economics 82, 264–271.

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