The present paper provides a simple and complete proof of the existence of a stationary monetary equilibrium for a stochastic overlapping generations model with a finite state space. Differently from previous studies, we show that all the prices are positive without the Frobenius theorem.Stochastic overlapping generations model, Stationary monetary equilibrium, Existence, Fixed-point theorem, Frobenius theorem.

Eisei Ohtaki

English

Research Papers in Economics

     
 
 
  
  
Volume 31, Issue 1 
  
A note on the existence of monetary equilibrium in a stochastic OLG model 
with a finite state space 
  
 
 
Eisei Ohtaki  
Graduate School of Economics, Keio University 
Abstract 
The present paper provides a simple and complete proof of the existence of a stationary monetary equilibrium for a 
stochastic overlapping generations model with a finite state space. Differently from previous studies, we show that all 
the prices are positive without the Frobenius theorem.
The author thanks the editor and an anonymous referee for helpful and valuable comments; Professor Yasuo Maeda (Fac. of Economics, Keio 
University) for his supervision; and Professor Hiroyuki Ozaki (Fac. of Economics, Keio University) for his encouragement. The author is solely 
responsible for any remaining errors in this paper. 
Citation: Eisei Ohtaki, (2011) ''A note on the existence of monetary equilibrium in a stochastic OLG model with a finite state space'', 
Economics Bulletin, Vol. 31 no.1 pp. 485-492. 
Submitted: Jan 01 2011.   Published: February 01, 2011. 
 
     1. Introduction
It is well-known that the class of overlapping generations (henceforth, OLG) models is
a useful class of models for exploring properties of equilibrium over time. The aim of this
paper is to provide a proof of the existence of a stationary monetary equilibrium (henceforth,
SME) for a stochastic OLG model with a ﬁnite state space, which was studied by Magill
and Quinzii (2003, henceforth, MQ). MQ, of course, showed the existence of a SME. They,
however, referred to the paper by Gottardi (1996) for the ﬁxed point argument and did
not provide a complete proof of the existence of a SME. Since Gottardi dealt with a more
complicated OLG model with heterogenous agents and several securities, it is worthwhile
to provide a simple and complete proof of the existence that is tailored to the simple, but
rather canonical, model of monetary OLG economy studied by MQ.
This paper shows the existence of a SME by applying the Brouwer ﬁxed-point theo-
rem.1 While Gottardi and MQ applied the Frobenius theorem to show that all the prices
are positive, this paper shows it without the Frobenius theorem.2 Moreover, this paper
demonstrates that we can ﬁnd a SME for a broader range of OLG models than MQ’s.
2. The Model
2.1. Fundamentals
We consider the same model as that of Magill and Quinzii (2003), that is, a one-good,
pure-exhange overlapping generations economy in which agents live for two periods and
have random endowments. The uncertainty is modeled by the realization of one of ﬁnite
number of the shocks at the beginning of each period, s 2 S = f1,...,Sg, s0 being the
initial shock. Let st = (s0,...,st) 2 S
1+t denote the history of the shocks occurred from
period 0 to period t and let Σt := S
1+t denote the set of all such histories up to date t.
Note that Σ0 = S and Σ1 = S
1. Also let Σ := [1
t=0Σt denote the collection of all such
histories for all periods. s = (s0,...,st,...) 2 Σ is denoting a typical path of the event-tree
Σ. We assume that shocks follow a ﬁrst-order Markov process and denote by P the induced
probability on the event-tree Σ.
In each period t ¸ 0, a new generation enter the economy after the resolution of the
period t uncertainty. Therefore, it is impossible for the young and the old to enter into risk
sharing contracts. We assume that there is only one agent per generation and that all agents
born in and after the initial period, called “newly born agents,” share the same endowment
and preference structures. At date-event st 2 Σ, a newly born agent is endowed with the
random endowment stream ω(st) = (ω1(st),(ω2(st,s0))s02S) 2 R+ £ RS
+. We assume that
the endowment stream depends only on the shock st realized when the agent is young, i.e,
ω(st) ´ ω(st) = (ω1
st,(ω2
sts0)s02S) for st = (s0,...,st).
All newly born agents maximize the expected utility of their lifetime consumption
streams, with the same utility indices. A newly born agent born at date-event st 2 Σ
ranks the possible consumption streams ct(st) = (c1(st),(c2(st,s0))s02S) 2 R+ £ RS
+ ac-
cording to a utility function Ust : R
1+S
+ ! R.
To close the model, we introduce the old generation present in the initial period, which
we call the “initial old.” This generation contains one old agent, whom is endowed with
1See also Manuelli (1990). He showed the existence of a stationary monetary equilibrium for a stochastic
OLG model with a continuous state space by applying the Shauder ﬁxed-point theorem.
2See Debreu and Herstein (1953) and Takayama (1974) for more details on the Frobenius theorem.
1ω0(s0) units of the consumption good at date-event s0. At date-event s0 2 Σ0, he/she ranks
the possible consumption plan c2
0(s0) 2 R+ according to a utility function U0
s0 : R+ ! R.
2.2. Assumptions on Fundamentals
Following Magill and Quinzii (2003), we impose the following system of assumptions on
the model.
Assumption 1 (Markov Structure). There exists a Markov transition matrix ρ =
[ρss0]s,s02S with ρss0 > 0 for all s,s0 2 S such that P(st+1 = s0jst = s) = ρss0 for all
s,s0 2 S.
Assumption 2 (Positive Endowment).
1. ω(s) = (ω1
s,(ω2
ss0)s02S) 2 R
1+S
++ for all s 2 S;
2. ω2
0(s0) 2 R++ for all s0 2 S.
Assumption 3 (Preferences).
1. There exist increasing, concave, and diﬀerentiable functions u1,u2 : R+ ! R with
limc#0 u0
i(c) = 1 for i = 1,2 such that
Ust(c(st)) = u1(c
1(st)) +
∑
s02S
u2(c
2(st,s
0))ρss0
for all st 2 Σ and all c(st) = (c1(st),(c2(st,s0))s02S) 2 R+ £ RS
+;
2. ¡cu00
2(c)/u0
2(c) · 1 for all c 2 R++;
3. U0
s0(c2
0(s0)) = c2
0(s0).
2.3. Equilibrium Price Processes (Vectors)
To deﬁne equilibria as consequences of intergenerational trade, we assume that there is
an inﬁnitely-lived (outside) asset available in positive supply, normalized to 1, which yields
no dividends (usually called ﬁat money). The asset is initially held by the initial old and
is then exchanged (unless prices are zero) at each date between the old and the young. Let
q(st) 2 R+ denote the price of the asset at date-event st 2 Σ. A newly born agent at
date-event st chooses z(st) 2 R to maximize Ust(ct(st)) subject to budget constraints
c
1
t(st) = ω
1(st) ¡ q(st)z(st),
(8s
0 2 S) c
2
t+1(st) = ω
2(st,s
0) + q(st,s
0)z(st).
Under Assumption 3.1, the necessary and suﬃcient condition for this choice problem is
provided by
(8st 2 Σ) q(st)u
0
1(ω
1
st ¡ q(st)z(st)) =
∑
s02S
q(st,s
0)u
0
2(ω
2
sts0 + q(st,s
0)z(st))ρsts0
Since, in any equilibrium, agents choice must be consistent with the market clearing condi-
tion, z(st) = 1 for all st 2 Σ, we are then led to the deﬁnition of an equilibrium.
2Deﬁnition 1. (q(st))st2Σ with q(st) 2 R+ for all st 2 Σ is an equilibrium price process if
(8st 2 Σ) q(st)u
0
1(ω
1
st ¡ q(st)) =
∑
s02S
q(st,s
0)u
0
2(ω
2
sts0 + q(st,s
0))ρsts0. (E)
We concentrate our attention on “stationary equilibrium” rather than equilibrium pro-
cesses. The next deﬁnition deﬁnes stationary equilibrium price vectors:
Deﬁnition 2. (q¤
s)s2S 2 RS
+ is a (strongly) stationary equilibrium price vector if
(8s 2 S) q
¤
su
0
1(ω
1
s ¡ q
¤
s) =
∑
s02S
q
¤
s0u
0
2(ω
2
ss0 + q
¤
s0)ρss0. (E¤)
3. Results
One can easily ﬁnd a trivial stationary equilibrium price vector.
Proposition 0. ¯ q ´ 0 is a stationary price vector.
Proof. ¯ q ´ 0, i.e., ¯ qs = 0 for all s 2 S, obviously satisﬁes Eq.(E¤), so that it is a stationary
equilibrium price vector. Q.E.D.
We add an assumption to present a MQ’s result. Deﬁne the S £ S matrix Π0 by
Π
0 = [π
0
ss0]s,s02S :=
[
u0
2(ω2
ss0)ρss0
u0
1(ω1
s)
]
s,s02S
.
Since, under Assumption 2, Π0 is a matrix with positive coeﬃcients, it follows from the
Frobenius theorem that Π0 has a unique positive eigenvalue associated with a positive
eigenvector. Let λf(Π0) denote this eigenvalue.
Assumption 4. λf(Π0) > 1.
Proposition 1 (Magill and Quinzii, 2003). Under Assumptions 1,2,3, and 4, a unique
positive stationary equilibrium price vector exists.
Proof. See the proof of Proposition 1 in Magill and Quinzii (2003). One can ﬁnd that
the proof of the result that all the prices are positive crucially depends on the Frobenius
theorem. Q.E.D.
Notice that in the proof by Magill and Quinzii (2003), Assumption 2, i.e., the hypothesis
that ω2 À 0, plays an important role to apply the Frobenius theorem. However, if ω2
ss0 = 0
for some pair of s and s0, then Π0 is no longer well-deﬁned, so that we can no longer ensure
that all the prices are positive by the Frobenius theorem. Here, we provide an existence
result that includes such cases.
We now describe our main result. Instead of Assumptions 2 and 4 of Magill and Quinzii
(2003), we impose the following assumptions.
3Assumption 20. ω1
s 2 R++ for all s 2 S.
Assumption 40. ω2
tt0 = 0 for some t,t0 2 S or mins2S
∑
s02S[u0
2(ω2
ss0)/u0
1(ω1
s0)]ρss0 > 1.
Note that the model has presumed that ω2
ss0 is nonnegative for all s,s0 2 S. Hence,
ω2
ss0 must be positive for all s,s0 2 S and, under Assumption 40, it must follow that
mins2S
∑
s02S[u0
2(ω2
ss0)/u0
1(ω1
s0)]ρss0 > 1 whenever it is false that ω2
tt0 = 0 for some t,t0 2 S.
We then show that a nontrivial stationary equilibrium price vector exists and it is positive
without the Frobenius theorem.
Proposition 10. Under Assumptions 1,2 0,3, and 4 0, a unique positive stationary equilib-
rium price vector exists.
Proof. (Existence) Our proof strategy is constructed from four steps:
1. Deﬁne some continuous mapping Φ of RS
++ to RS
++;
2. Find a compact and convex set A such that A ½ RS
++ and Φ maps A into itself;
3. Applying the Brouwer ﬁxed-point theorem to Φ on A, we show the existence of a ﬁxed
point a¤ 2 A;3
4. Verify that we can construct a positive stationary equilibrium price vector from a¤.
1. Our ﬁrst task is to deﬁne the continuous mapping Φ of RS
++ to RS
++. For each s 2 S,
let Qs := [0,ω1
s] and its interior be denoted by int.Qs. For all s 2 S, deﬁne the function
fs : Qs£R+ ! R by fs(q,a) := qu0
1(ω1
s ¡q)¡a for all (q,a) 2 Qs£R+. Notice that, for all
s 2 S and all a 2 R+, fs(¢,a) is continuous and increasing on int.Qs by Assumption 3.1.
Also notice that, for all s 2 S and all a 2 R++, fs(0,a) = ¡a < 0 and limq"ω1
s fs(q,a) =
1 > 0. Hence, for all all s 2 S and a 2 R++, the intermediate value theorem ensures the
existence and the uniqueness of ˆ qs(a) 2 int.Qs such that fs(ˆ qs(a),a) = 0. Moreover, since
∂fs/∂q = u0
1 ¡ qu00
1 > 0 and ∂fs/∂a = ¡1 < 0, the implicit function theorem implies that,
for all s 2 S, ˆ qs(¢) is a continuous and increasing function on R++.
Deﬁne the mapping φ : RS
++ ! RS
++ by φ(a) := (ˆ qs(as))s2S for all a = (as)s2S 2 RS
++.
Also deﬁne the mapping ψ : RS
++ ! RS
++ by ψ(q) := (
∑
s02S qs0u0
2(ω2
ss0 + qs0)ρss0)s2S for
all q = (qs)s2S 2 RS
++. It is easy to verify that both φ and ψ are continuous on RS
++.
Moreover, φ is increasing and ψ is nondecreasing on RS
++. It is obvious that φ is increasing,
since ˆ qs(¢) is increasing for all s 2 S. On the other hand, to see that ψ is nondecreasing,
notice that, for all s,s0 2 S, ∂[qu0
2(ω2
ss0 +q)]/∂q = u0
2 +qu00
2 ¸ u0
2 ¢(1+(q +ω2
ss0)u00
2/u0
2) ¸ 0,
where the ﬁrst inequality follows from concavity of u2 and the fact that ω2
ss0 ¸ 0, and the
second inequality follows from Assumption 3.2. Hence, qu0
2(ω2
ss0 + q) is nondecreasing in q
for all s,s0 2 S. Therefore, it follows that ψ(q0) · ψ(q1) for all q0,q1 2 RS
++ such that
q0 · q1.
Now, deﬁne the mapping Φ : RS
++ ! RS
++ by Φ(a) := (ψ ±φ)(a) for all RS
++. It follows
from continuity and monotonicity of φ and ψ that Φ is continuous and nondecreasing on
RS
++.
2. Next, we ﬁnd a compact and convex set A such that A ½ RS
++ and Φ maps A
into itself. Let As := [0,ω1
su0
1(ω1
s)] for all s 2 S and A :=
∩
s2S As. The interiors of As
3See, for example, Takayama (1974) for details on the Brouwer ﬁxed-point theorem.
4and A are denoted by int.As and int.A, respectively. For all s,s0 2 S, deﬁne the function
zs(s0,¢) : As0 ! R++ by
(8a 2 As0) zs(s
0,a) :=
u0
2(ω2
ss0 + a/u0
1(ω1
s0))
u0
1(ω1
s0 ¡ a/u0
1(ω1
s0))
.
Then, it follows from Assumption 3.1 that zs(s0, ¢) is continuous and decreasing on int.As0.
Notice that it follows from the deﬁnition of ˆ qs(¢) and concavity of u1 that ˆ qs(a) = a/u0
1(ω1
s¡
ˆ qs(a)) · a/u0
1(ω1
s) for all s 2 S and all a 2 R++. Thus, it follows from concavity of u1 and
u2 that
(8s,s
0 2 S)(8a 2 int.As0) zs(s
0,a) ·
u0
2(ω2
ss0 + ˆ qs0(a))
u0
1(ω1
s0 ¡ ˆ qs0(a))
. (1)
Deﬁne the function ˆ z : A ! R++ by ˆ z(a) := mins2S
∑
s02S zs(s0,a)ρss0 for all a 2 A.
By properties of zs(s0, ¢), ˆ z is continuous and decreasing on int.A. Let ˆ z(0) := lima#0 ˆ z(a).
If ω2
ss0 = 0 for some s,s0 2 S, then it follows from Assumption 3.1 that ˆ z(0) = 1 > 1.
On the other hand, if ω2
ss0 > 0 for all s,s0 2 S, then Assumption 40 ensures that ˆ z(0) > 1.
Hence, in all cases, we have ˆ z(0) > 1. Since ˆ z is continuous and decreasing on int.A, there
exists at least one a 2 int.A such that ˆ z(0) > ˆ z(a) ¸ 1. Also let a := ω1u0
2(ω2 + q), where
ω1 := maxs2S ω1
s, ω2 := mins,s02S ω2
ss0, and q := mins2S ˆ qs(a) > 0. Notice that ω1 ¸ ω1
s0,
ω2 · ω2
ss0, and q · ˆ qs0(a) for all s,s0 2 S and all a ¸ a. Therefore, it follows from concavity
of u2 that
∑
s02S
ω
1
s0u
0
2(ω
2
ss0 + ˆ qs0(as))ρss0 · a (2)
for all s 2 S and all (as0)s02S such that as0 ¸ a for all s0 2 S.
We claim that a < a. To see this, notice that
(8s
0 2 S) a < ω
1
s0u
0
1(ω
1
s0) · ω
1u
0
1(ω
1
s0) = a
u0
1(ω1
s0)
u0
2(ω2 + q)
, (3)
where the ﬁrst (strict) inequality follows from the fact that a 2 int.A, the second inequality
follows from the fact that ω1
s0 · ω1, and the last equality follows from the deﬁnition of ω1.
Then, it follows that
(8s,s
0 2 S) a > a
u0
2(ω2 + q)
u0
1(ω1
s0)
¸ a
u0
2(ω2
ss0 + ˆ q0
s(a))
u0
1(ω1
s0)
¸ a
u0
2(ω2
ss0 + ˆ qs(a))
u0
1(ω1
s0 ¡ ˆ qs(a))
¸ azs(s
0,a), (4)
where the ﬁrst (strict) inequality follows from Eq.(3), the second follows from concavity
of u2 and the facts that ω2 · ω2
ss0 and ˆ qs(a) ¸ q for all s,s0 2 S, the third follows from
concavity of u1 and the fact that ˆ qs(a) > 0, and the last inequality follows from Eq.(1).
Hence we have
(8s 2 S) a = a
∑
s02S
ρss0 > a
∑
s02S
zs(s
0,a)ρss0 ¸ aˆ z(a) ¸ a,
where the ﬁrst equality follows from the deﬁnition of ρ, the second (strict) inequality follows
from Eq.(4), the third follows from the deﬁnition of ˆ z, and the last follows from the fact
that ˆ z(a) ¸ 1. This establishes the claim.
5We also claim that a · Φ(a) ¿ a for all a 2 RS
++ with a · a · a, where a :=
(a,...,a),a := (a,...,a) 2 RS
++. Notice that, for all a 2 RS
++ with a ¸ a and all s 2 S, ∑
s02S ˆ qs0(as0)u0
2(ω2
ss0 + ˆ qs0(as0))ρss0 <
∑
s02S ω1
s0u0
2(ω2
ss0 + ˆ qs0(as0))ρss0 · a, where the ﬁrst
(strict) inequality follows from the fact that ˆ qs(a) < ω1
s for all positive a, and the second
inequality follows from Eq.(2). Hence, Φ(a) ¿ a for all a 2 RS
++ with a ¸ a. On the
other hand, notice that
∑
s02S
ˆ qs0(a)u
0
2(ω
2
ss0 + ˆ qs0(a))ρss0
= a
∑
s02S
u0
2(ω2
ss0 + ˆ qs0(a))
u0
1(ω1
s0 ¡ ˆ qs0(a))
ρss0
¸ a
∑
s02S
u0
2(ω2
ss0 + a/u0
1(ω1
s0))
u0
1(ωs0 ¡ a/u0
1(ω1
s0))
ρss0
¸ aˆ z(a)
¸ a,
where the ﬁrst equality follows from the deﬁnition of ˆ qs0(¢), the second inequality follows
from Eq.(1), the third follows from the deﬁnition of ˆ z, and the last follows from the fact
that ˆ z(a) ¸ 1. This implies that Φ(a) ¸ a. Since Φ is nondecreasing, it follows that
Φ(a) ¸ Φ(a) ¸ a for all a 2 RS
++ with a ¸ a. This establishes the claim.
Let A := [a,a]S. A is obviously compact and convex and it is a subset of RS
++. Moreover,
we have veriﬁed that Φ is continuous on A and Φ(A) ½ A.
3. We apply the Brouwer ﬁxed-point theorem to Φ on A.4 Since A is compact and
convex, Φ is continuous, and Φ(A) ½ A, it follows from the Brouwer ﬁxed-point theorem
that Φ has a ﬁxed point a¤ = (a¤
s)s2S in A, i.e., there exists some a¤ 2 A such that
Φ(a¤) = a¤. Since a¤ 2 RS
++, it follows that ˆ qs(a¤
s) 2 int.Qs for all s 2 S.
4. We omit the proof of Step 4, since it is easy to verify that q¤ := (ˆ qs(a¤
s))s2S is a
positive stationary equilibrium price vector. This establishes the proof of “existence” part
of Proposition 10.
(Uniqueness) Deﬁne the mapping Π : RS
++ ! R
S£S
++ by Π(q) := [πss0(q)]s,s02S, where
(8s,s
0 2 S)(8q 2 R
S
++) πss0(q) :=
u0
2(ω2
ss0 + qs0)ρss0
u0
1(ω1
s ¡ qs)
.
We have veriﬁed that at least one positive stationary equilibrium vector exists in the above
argument. Notice that, for arbitrary positive stationary equilibrium price vector q¤ 2 RS
++,
we have the Frobenius root λf(Π(q¤)) = 1, since Π(q¤)q¤ = q¤ (See the deﬁnition of
equilibrium price vector). Also notice that q,q0 2 RS
++ with q > q0, we have πss0(q) ·
πss0(q0) with at least one strict inequality. This follows immediately from concavity of ui
for i = 1,2. The rest of the proof of the “uniqueness” part is the almost same as that of
Magill and Quinzii (2003), so that see their work for more details. Q.E.D.
4One can apply the Tarski ﬁxed-point theorem instead of the Brouwer ﬁxed-point theorem, since A is a
complete partial order set and Φ is nondecreasing. In this case, we no longer need to require continuity of
Φ, i.e., continuity of φ and ψ. See, for example, Ok (2007) for the details of the Tarski ﬁxed-point theorem.
64. Concluding Remarks
The proof of the existence of a stationary monetary equilibrium (SME) for a complicated
overlapping generations (OLG) model with heterogeneous agents and several securities has
been already provided. However, a simple and complete proof of the existence of SME for a
simple, but rather canonical, model of monetary OLG economy presented in this paper has
not provided yet. The main contribution of this paper is to provide a simple and complete
proof of the existence of a SME for such a simple and canonical OLG model.
While previous studies applied the Frobenius theorem to show that all the prices are
positive, this paper has shown it without the Frobenius theorem. This indicates that a
simple OLG framework does not necessarily require the Frobenius theorem to ensure that
all the prices are positive.
Also remark that we can no longer show that all the prices are positive by the Frobenius
theorem when the amount of the second-period endowment is zero in some state, i.e., when
ω2
ss0 = 0 for some s,s0 2 S, since the matrix Π0 is no longer well-deﬁned under Assumption
3.1. In such cases, therefore, we can no longer verify that prices of money are positive by
a similar procedure to the one adopted by previous studies. On the other hand, this paper
provided the existence result that includes such cases. This indicates that this paper has
shown the existence of a SME for a broader range of models than that of previous studies
such as Magill and Quinzii (2003).
References
Debreu, G. and I.N. Herstein (1953) “Nonnegative square matrix,” Econometrica 21, 597–
607.
Gottardi, P. (1996) “Stationary monetary equilibria in overlapping generations models with
incomplete markets,” Journal of Economic Theory 71, 75–89.
Magill, M. and M. Quinzii (2003) “Indeterminacy of equilibrium in stochastic OLG models,”
Economic Theory 21, 435–454.
Manuelli, R. (1990) “Existence and optimality of currency equilibrium in stochastic over-
lapping generations models: The pure endowment case,” Journal of Economic Theory
51, 268–294.
Ok, E.A. (2007) Real Analysis with Economic Applications. Princeton University Press:
Princeton.
Takayama, A. (1974) Mathematical Economics. The Dryden Press: Hinsdale. IL.
7

Existence and optimality of currency equilibrium in stochastic overlapping generations models: The pure endowment case,”

Nonnegative square matrix,”

Real Analysis with Economic Applications.

Stationary monetary equilibria in overlapping generations models with incomplete markets,”

A note on the existence of monetary equilibrium in a stochastic OLG model with a finite state space

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A note on the existence of monetary equilibrium in a stochastic OLG model with a finite state space

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