Augeraud-Véron, Emmanuelle

d'Albis, Hippolyte

English

Research Papers in Economics

Hippolyte d￿ Albis ￿ Emmanuelle Augeraud-VØron
Balanced cycles in an OLG model with a continuum of
￿nitely-lived individuals
Received: / Accepted:
Abstract This paper studies the intertemporal equilibrium of a barter econ-
omy populated with a continuum of ￿nitely-lived overlapping generations.
Assuming isoelastic preferences and zero endowments at the beginning and
the end of the individuals￿life-span, it proves the existence of an Hopf bifur-
cation and provides su¢ cient conditions on parameters for its occurrence.
Keywords: Overlapping-generations Models, Mixed-type Functional Di⁄er-
ential Equations, Endogenous Fluctuations, Hopf Bifurcation.
JEL Classi￿cation Numbers: C6, D9.
￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿
The authors would like to thank an anonymous referee, Alain Venditti and Francesco
Ricci for helpful comments and suggestions. The usual disclaimer applies.
￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿
Hippolyte d￿ Albis
GREMAQ, University of Toulouse I, 31000 Toulouse, FRANCE
E-mail: hippolyte.d-albis@univ-tlse1.fr
Emmanuelle Augeraud-VØron
LMA, University of La Rochelle, 17042 La Rochelle, FRANCE
E-mail: eaugerau@univ-lr.fr
11 Introduction
This paper investigates the long-run ￿ uctuations that may emerge in station-
ary OLG economies. To this end, we follow Demichelis and Polemarchakis
[5] who study a continuous-time model with a continuum of ￿nitely-lived
individuals1. The intertemporal equilibrium is shown to be the solution of
a functional di⁄erential equation of mixed-type (MFDE). By extending this
model to isoelastic preferences, the MFDE is nonlinear. The proof of the ex-
istence of long-run ￿ uctuations then relies on Rustichini [10] and Benhabib
and Rustichini [2]: it uses the linearized MFDE that characterizes the local
motion of the economy and looks for solutions which have Hopf bifurcation
values. Assuming zero discounting and a speci￿c endowment distribution,
we show that there exist some sets of parameters such that a barter economy
exhibits a cycle on the neighborhood of a steady state. An elasticity of in-
tertemporal substitution lower than one and zero endowments at beginning
and end of the individuals￿life-span are su¢ cient conditions to obtain this
result. This paper consequently shows that an increase in the frequency of
trade during the individual￿life-span does not eliminate the possibility of
endogenous ￿ uctuations in OLG economies2.
The sketch of the paper is as follows. Section 2 presents an overlapping-
generations model with continuous trading and ￿nitely-lived individuals. Sec-
1The pioneer OLG model of this kind is due to Cass and Yaari [4].
2Note that this issue is usually debated in discrete time environment: see notably
Aiyagari [1] and Ghiglino and Tvede [6] in a framework with many generations and [8],
[7] and [9] in models with two generations.
2tion 3 characterizes the intertemporal equilibrium of a barter economy and
analyses the existence and uniqueness properties of the steady state. Section
4 studies the spectral decomposition of the linearized dynamics in the neigh-
borhood of a steady state and gives su¢ cient conditions for the existence of
an Hopf bifurcation.
2 The model
Time is continuous and has a ￿nite starting point; let t ￿ 0 denote the
time index. Individuals live for an interval of time of length 1. They only
derive utility from consumption and have isoelastic preferences and no time
discount. Let c(s;t) ￿ 0 denotes the consumption of an individual who born
at time s as of time t. Hence, the intertemporal utility of an individual who
born at time s ￿ 0, denoted as u(s), is:
u(s) =
Z s+1
s
c(s;t)
1￿ 1
￿
1 ￿ 1
￿
dt (1)
where ￿ > 0 stands for the elasticity of intertemporal substitution. During
a lifetime, an age-dependent endowment is received; it is denoted w(t ￿ s)
and satis￿es:
w(t ￿ s) =
￿
w(t) if t 2 [s + ￿;s + ￿]
0 otherwise (2)
with 0 ￿ ￿ < ￿ ￿ 1. Individuals have access to a competitive asset mar-
ket that yields the interest rate r(t). Let a(s;t) denotes the real wealth of
an individual who born at time s as of time t. The instantaneous budget
3constraint is therefore:
@a(s;t)
@t
= r(t)a(s;t) + w(t ￿ s) ￿ c(s;t) (3)
Individuals are born with no ￿nancial assets and cannot die indebted. There-
fore, initial and terminal conditions write:
a(s;s) = 0 (4)
a(s;s + 1) ￿ 0 (5)
It is assumed that a(s;t) and c(s;t) are C1 ￿
R2
+
￿
and that r(t) and w(t ￿ s)
are continuous for all t 2 [s;s + 1]. The individual program is to maximize
(1) subject to (3), (4) and (5).
Lemma 1 The optimal consumption pro￿le satis￿es:
c(s;t) =
R s+￿
s+￿ w(z)e￿
R z
s r(u)dudz
R s+1
s e￿(1￿￿)
R z
s r(u)dudz
e
￿
R t
s r(u)du (6)
Proof: The ￿rst order conditions are:
@c(s;t)
@t
= ￿r(t)c(s;t) (7)
and a(s;s + 1) = 0. Integrating forward condition (3), and using the second
optimal condition yields:
Z s+1
s
c(s;z)e
￿
R z
s r(u)dudz =
Z s+￿
s+￿
w(z)e
￿
R z
s r(u)dudz (8)
Replacing (7) in (8) gives c(s;s) and consequently (6). ￿
4The demographic structure is in overlapping generations. Each individual
belongs to a cohort whose size is normalized to 1. There is no population
growth and, at each point of time, a new cohort enters the economy since the
oldest one leaves it. Hence, the aggregate counterpart, x(t), of any individual
variable, x(s;t); is obtained by integrating over the birth date, such that:
x(t) =
Z t
t￿1
x(s;t)ds (9)
Assume there exists a single non storable good and that the aggregate endow-
ment equals the size of the population; then:
R t
t￿1 w(t ￿ s)ds = 1. Replacing
the endowment distribution rule given by (2) yields: w(t) = 1=(￿ ￿ ￿). Us-
ing (6), the aggregate consumption, denoted c(t), hence satis￿es:
c(t) =
1
(￿ ￿ ￿)
Z t
t￿1
R s+￿
s+￿ e￿
R z
s r(u)dudz
R s+1
s e￿(1￿￿)
R z
s r(u)dudz
e
￿
R t
s r(u)duds (10)
There is no money in this economy and assets are constituted by consumption
loans. The aggregate wealth is denoted a(t); using (3), (4) and the optimal
condition: a(s;s + 1) = 0, its dynamics writes:
da(t)
dt
= r(t)a(t) + 1 ￿ c(t) (11)
3 Equilibrium and steady states
De￿nition 1 An equilibrium with perfect foresight is a function F (t) =
(c(t);a(t);r(t)), F : R+ ! R2
+ ￿ R, C1 (R+) such that (i) individuals max-
imize their utility subject to the budget constraint, (ii) the aggregate con-
sumption equals the aggregate endowment (i.e. c(t) = 1), and (iii) the
aggregate asset is zero (i.e. a(t) = 0).
5Using (3), it can be inferred from de￿nition 1, that an equilibrium reduces
to a function r(t).
De￿nition 2 A steady state equilibrium is a triplet (c;a;r) that satis￿es:
(i) c = ￿(r) and a = (c ￿ 1)=r, (ii) c = 1 and a = 0; where function
￿ : R ! R++ is such that:
￿(r) =
R 1
0 e￿rzdz
(￿ ￿ ￿)
R ￿
￿ e￿rzdz
R 1
0 e￿(1￿￿)rzdz
(12)
Property 1 A steady state interest rate is a r that satis￿es ￿(r) = 1 if
r 6= 0 and ￿
0 (r) = 0 if r = 0.
Proof: For r 6= 0, the property is an immediate implication of de￿nition 2;
for r = 0; just observe that ￿(0) = 1 and use l￿ H￿pital￿ s Rule. ￿
Lemma 2 r = 0 is a steady state interest rate if and only if ￿ + ￿ = 1.
Proof: Replace r = 0 in ￿
0 (r) and use property 1 to conclude. ￿
Corollary 1 The individual consumption is age-independant if and only if
￿ + ￿ = 1.
Lemma 3 There exist (￿;￿;￿) such that there is no steady-state equilibrium.
Proof: It can be easily shown that there is no solution r 6= 0 such that
￿(r) = 1 if ￿ = 0 and ￿ 2 (0;1 ￿ ￿] or if ￿ = 1 and ￿ 2 [￿;1). ￿
Lemma 4 There exist (￿;￿;￿) such that there are multiple steady-state equi-
libria.
6Proof: Suppose ￿ = 1 ￿ ￿ and recall with lemma 2 that r = 0 is a steady
state; it can be easily shown that there exist at least two other solutions
r 6= 0 such that ￿(r) = 1 if ￿ 2 (￿;2￿(1 ￿ ￿)). ￿
4 Endogenous cycles
The equilibrium satis￿es the following non linear dynamics:
1
(￿ ￿ ￿)
Z t
t￿1
R s+￿
s+￿ e￿
R z
s r(u)dudz
R s+1
s e￿(1￿￿)
R z
s r(u)dudz
e
￿
R t
s r(u)duds = 1 (13)
In this section, it is the local dynamics around steady-state r￿ which is stud-
ied; it is the one of x(t) de￿ned such that r(t) = r￿ + ￿x(t).
Property 2 The characteristic function of x(t) is denoted Q(￿) and satis￿es:
Q(￿) = ￿
R ￿
￿ e(￿￿r￿)sdz
R ￿
￿ e￿r￿zdz
+ (1 ￿ ￿)
R 1
0 e(￿￿(1￿￿)r￿)zdz
R 1
0 e￿(1￿￿)r￿zdz
+ ￿
R 1
0 e￿r￿zdz
R 1
0 e(￿r￿￿￿)zdz
(14)
Proof: Replace r(t) = r￿ + ￿x(t) in (13). Then do a Taylor expansion in
the neighborhood of ￿ = 0 and rearrange using ￿(r) = 1. De￿ne X(t) =
R t
0 x(u)du. It yields:
X(t) =
R t
t￿1 e￿r￿(t￿s)
￿R s+￿
s+￿ e￿r￿(z￿s)X(z)dz
(￿￿￿) ￿
(1￿￿)
R s+1
s e￿(1￿￿)r￿(z￿s)X(z)dz
R 1
0 e￿r￿zdz
￿
ds
￿
R 1
0 e￿(1￿￿)r￿zdz
(15)
Finally, Q(￿) is obtained by the following change of variable: x(t) = e￿t and
rearranging using ￿(r) = 1. ￿
Lemma 5 The characteristic function Q(￿) has an in￿nity of complex roots
with negative real parts and an in￿nity of complex roots with positive real
parts.
7Proof: Roots of Q(￿) are asymptotic to those of the following equations3:
P1 (￿) = ￿￿
 
￿￿
2
Z 1
0
e
￿r￿zdz +
(1 ￿ ￿)e￿￿(1￿￿)r￿
R 1
0 e￿(1￿￿)r￿zdz
!
(16)
P2 (￿) = ￿￿
 
￿￿
2
Z 1
0
e
￿r￿zdz +
(1 ￿ ￿)e￿￿+￿r￿
R 1
0 e￿(1￿￿)r￿zdz
!
(17)
Asymptotically, roots of P1 (￿) have a positive real part while those of P2 (￿)
have a negative real part. ￿
Corollary 2 The dynamics is generically characterized by oscillations that
decrease in magnitude and eventually disappear.
Consider now the case ￿ = 1￿￿ and focus on the neighborhood of the steady
state r￿ = 0. The characteristic function (14) rewrites:
Q(￿) = ￿
R 1￿￿
￿ e￿zdz
(1 ￿ 2￿)
+ (1 ￿ ￿)
￿Z 1
0
e
￿zdz
￿
+ ￿
1
￿R 1
0 e￿￿zdz
￿ (18)
Lemma 6 There exist (￿;￿;￿) such that there are pure imaginary roots
which are Hopf bifurcation values.
Proof: Let ￿ = p+iq. The proof proceeds in two steps: (i) it supposes ￿ = iq
and proves that for ￿ < 2￿(1 ￿ ￿), there exists a q > 0 such that Q(iq) = 0;
it hence de￿nes (￿0;￿0 (￿0)) the pair of parameters for which this root does
exist; (ii) it uses ￿ as a bifurcation parameter and shows that there exists a
neighborhood of ￿0 such that dRe(Q(￿))=d￿ is not equal to zero.
(i) Replace ￿ = iq in (18) and de￿ne ~ Q(q) such that:
~ Q(q) =
cos(￿q) ￿ cos((1 ￿ ￿)q)
1 ￿ 2￿
+ (1 ￿ ￿)[cos(q) ￿ 1] ￿
￿q2
2
(19)
3See Bellman and Cooke (1963) p. 410.
8A Taylor expansion in the neighborhood of q = 0 yields:
~ Q(q) =
q4
12
￿
￿(1 ￿ ￿) ￿
￿
2
￿
+ O
￿
q
6￿
> 0 (20)
Moreover, limq!+1 ~ Q(q) = ￿1. Therefore, there exists a q > 0 such that
~ Q(q) = 0.
(ii) From (18), it yields:
dRe(Q(￿))
d￿
￿ ￿ ￿ ￿
￿0
=
sin(q)
q
￿
q2
2(1 ￿ cos(q))
￿ 1
￿
(21)
The roots of this latter function are those of sin(q) and cos(q) = 1 ￿ q2=2.
They are the f2￿+k￿;k 2 Zg[fq1;q2;q3;q4g, where q1;q2;q3 and q4 are the
4 roots of the following polynomial:
￿
1 ￿
q2
2
￿2
+
￿
1 ￿
(q ￿ ￿
2)2
2
￿2
= 1 (22)
Since elements of f2￿ + k￿;k 2 Zg [ fq1;q2;q3;q4g are independent with
respect to a0, there exists a0, such that a0 = 2 f2￿+k￿;k 2 Zg[fq1;q2;q3;q4g.
￿
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10

A.: Vintage capital, investment and growth.

Can there be short-period deterministic cycles when people are long lived?

Debt, liquidity and dynamics.

Di⁄erential di⁄erence equations.

Dynamics in OG economies.

Endogenous growth cycles in an overlapping generations model with investment gestation lags.

Frequency of trade and the determinacy of equilibrium paths: logarithmic economies of overlapping generations under certainty. Economic Theory (forthcoming)

Hopf bifurcation for functional di⁄erential equations of the mixed type.

Optimal growth and cycles in overlapping generations models.

Balanced Cycles in an OLG Model with a Continuum of Finitely-Lived Individuals.

Balanced Cycles in an OLG Model with a Continuum of Finitely-Lived Individuals.

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