Directory UMM :Data Elmu:jurnal:J-a:Journal Of Economic Dynamics And Control:Vol25.Issue5.Apr2001:

Journal of Economic Dynamics & Control
25 (2001) 765}787

Intersectoral external e!ects,
multiplicities & indeterminaciesq
Jean-Pierre Drugeon!, Alain Venditti",*
!CNRS, EUREQUA, Maison des Sciences Economiques, 106-112 Bd de l+Ho( pital, 75647 Paris,
Cedex 13, France
"CNRS-GREQAM, Centre de la Vieille Charite 2, rue de la Charite, 13002 Marseille, France
Received 1 May 1997; accepted 1 May 1999

Abstract
This contribution focuses on the scope for indeterminacies that originate from global
capital stock externalities in a reference two-sector growth model. A set of su$cient
conditions for local indeterminacies and oscillations is established and builds upon a new
class of intersectoral dependency in competitive economies. The uniqueness of the steady
state is also questioned and conditions for global indeterminacies are delimited. The
underlying features of preferences and sectoral production technologies are assessed in
this paper. It is shown that the principal attribute of a two-sector environment, i.e.,
a non-linear production possibility frontier, directly underlies indeterminacies. It is the
in#uence of external e!ects on the relative price of the investment good that leads to these

phenomena, a key role being detected in this perspective for external e!ects in the
consumption good sector. ( 2001 Elsevier Science B.V. All rights reserved.
JEL classixcation: E12; E32; O41
q
The material used in this contribution is based upon the "rst part of a contribution by the same
authors entitled &On Externalities, Indeterminacies & Homothetic Growth Paths in a Canonical
Model of Economic Growth' (GREQAM Working Paper 96A40) which was presented at the EEA
96 Congress, Istanbul, &Nonlinear Dynamics' conferences, Paris, May 1996 and Marseille, May 1997,
Esem'97, Toulouse, Femes 97, Hong Kong.

* Corresponding author. Tel.: #33-04-91-14-07-42; fax: #33-04-91-90-02-27; The authors
would like to thank an anonymous referee for useful comments and suggestions as well as Grace
Meagher for a careful and detailed reading. They remain entirely to blame for any imprecisions or
mistakes.
E-mail address: venditti@ehess.cnrs-mrs.fr (A. Venditti).

0165-1889/01/$ - see front matter ( 2001 Elsevier Science B.V. All rights reserved.
PII: S 0 1 6 5 - 1 8 8 9 ( 9 9 ) 0 0 0 4 6 - 9

766 J.-P. Drugeon, A. Venditti / Journal of Economic Dynamics & Control 25 (2001) 765}787

Keywords: Intersectoral external e!ects; Multiple steady states; Endogenous #uctuations; Indeterminacies

1. Introduction
This article focuses on the scope for indeterminacies that originate from
capital stock externalities in a reference two-sector growth model. A given
con"guration will be refered to as indeterminate as soon as, starting from a given
initial value for the capital stock, a multiplicity of distinct equilibrium paths
comes into existence. Building upon a standard multisector growth model and
an argument that involves intersectoral external e!ects, the central purpose of
the current contribution is "rst to characterise the area for local and global
indeterminacies and then to identify the underlying mechanisms of intertemporal preferences and sectoral production technologies. From a methodological
standpoint, the aim is to reach a perception of multiplicity phenomena which is
in the line with the standards of optimal growth literature.
Dating from the in#uential contribution of Benhabib and Farmer (1994), the
parameter relevance of indeterminacies has been thoroughly investigated. However, various attempts have been made to focus on a more fundamental insight.
Within a discrete time environment and for a one-sector technology, Boldrin
and Rustichini (1994) were the "rst to establish a canonical determinacy result
based upon positive external e!ects.1 In a two-sector framework, both of these
authors and Venditti (1998) emphasise the theoretical possibility of locally

indeterminate steady states in the presence of positive external e!ects. These
conclusions remain incomplete in (i) conveying no clearly identi"ed phenomena
on intertemporal preferences and sectoral production technologies and (ii) neglecting the other major facet of indeterminacy issues, i.e., the conceivability of
multiple steady-state positions from a given initial value of the capital stock. It
should be mentioned that Benhabib and Farmer (1996) have independently
developed a related framework2 that relies on sector-specixc externalities.
The properties of their environment are largely complementary to the
ones associated with global external e!ects and intersectoral dependencies. Indeed, and although the completion of a theoretical de"nition of
a competitive equilibrium becomes a much more intricate task, the equilibrium
values of the external e!ects and the resulting dynamics are by far simpler to
circumscribe.

1 The case of negative externalities leaves room for indeterminacies, a computed example of which
was provided by Kehoe (1991).
2 It has just been extended by Benhabib and Nishimura (1998).

J.-P. Drugeon, A. Venditti / Journal of Economic Dynamics & Control 25 (2001) 765}787 767

Contemplating a broader perception of the indeterminacy issue, the present
contribution will focus on a continuous time version of the canonical multisector growth model with one capital good.3 The main points are as follows. In

comparison with a standard one-sector setting, a two-sector framework is
speci"c in bringing independently de"ned notions of increasing returns to scale
and decreasing investment costs: it reveals that their combination directly feeds
the indeterminacy outcome. More precisely, a univocal link emerges between
the indeterminacy of the steady state and the in#uence of the external e!ect on
the relative price of the investment good. Such an explanation directly relates to
intersectoral arbitrages: it is not relevant when an environment with a unique
homogeneous good is considered.
An articulation between indeterminacy and earlier conclusions of endogenous
#uctuations literature, i.e., endogenous cycles and multiple steady states, is
further developed and put in evidence. As an alternative to the myopy or multiple
consumption goods argument developed by Benhabib and Nishimura (1979a,b)
in convex environments, external e!ects may induce both types of phenomena.4
In particular and even for a benchmark con"guration with a zero discount rate,
conditions for uniqueness may be violated. Finally, the ways in which multiplicities or endogenous #uctuations relate to or even underlie local indeterminacies
are assessed.
These results are stated in terms of requesites on an indirect utility function
that is de"ned from the instantaneous utility function and the production
possibility frontier. In line with these concerns, a "nal examination completes
a characterisation of more meaningful direct requesites on preferences and

sectoral production technologies. A related articulation between the characteristics of the production possibility frontier and the properties of the sectoral
production function is available from the fundamental insights of Hirota and
Kuga (1971) and Kuga (1972). This optimal growth apparatus is currently
extended to the study of competitive equilibria with externalities. It is shown
that the nature of external e!ects in the consumption good sector has direct
implications on the determinacy properties of the steady state as well as on the
scope for multiplicity.5

3 Benhabib and Rustichini (1994) also introduced numerous techniques in the direction of
a general understanding of multisector dynamic competitive equilibria with externalities. Their
approach, being limited in scope, however leaves unanswered numerous issues as yet unexplored.
4 Their argument requires the introduction of heterogeneous capital, so the investigation for cycles
would be out of purpose in an optimal growth model with a unique capital good. Vide the argument
of Lemma A.1 in Appendix A.3.
5 These conclusions have been con"rmed in a non-stationary context by Drugeon and Venditti
(1996).

768 J.-P. Drugeon, A. Venditti / Journal of Economic Dynamics & Control 25 (2001) 765}787

A computed example con"rms that all the aforementioned theoretical results

correspond to actual possibilities: local indeterminacies, multiple stationary
states and in fact limit cycles emerge under given parameter con"gurations.6
The basic framework is analysed in Section 2. Section 3 focuses on the
uniqueness issue. Section 4 is concerned with the scope for local indeterminacies
and endogenous #uctuations. Section 5 characterises the implications of these
results on the fundamentals. A fully characterised illustration is provided in
Section 6. The main proofs are gathered together in the "nal appendix.

2. The model
Times is continuous. The economy is populated by a continuum [0, 1] of
in"nitely lived agents with identical preferences. These preferences are described
by an intertemporal utility functional U( ) ) de"ned over a consumption path
C assigning a consumption #ow c(t) at t3R :
`

P

U(C)"

`=

u[c(t)] exp (!ot) dt, o3R
`
t/0

(1)

for u( ) ) an instantaneous utility function such that:7
Assumption P.1. u3C2(RH , R) and satis"es u@'0, uA(0 for any c'0, u(0)"0,
`
u(R)"R, lim
u@"R, lim
u@"0.
c?0
c?=
Any consumer i3[0, 1] is endowed with a unitary amount of labour that he
allocates between a consumption good sector, or j"0, and a capital good
sector, or j"1. Any individual holds a "rm in any of the sectors and is endowed
with an initial capital stock x (0)"x . Both capital and labour are freely
i
0

shiftable from one sector to the other.
The production technology of a given "rm Fj in any of the sectors will depend
on privately held capital and labour, or xj(t) and lj(t), as well as on the aggregate
value of the capital stock X(t) at the same date:
Assumption T.1. Fj3C2(RH ]RH ]R , R ), j"0, 1, satis"es, for a given X3R :
`
` `
`
`
(i) Fj( ) , ) , X) is increasing, concave and homogeneous of degree one;
(ii) Fj ( ) , lj, X)(0 for any lj3]0, 1].
11
6 Precise requirements for limit cycles are outlined in detail. This analysis builds on sophisticated
material from the theory of dynamical systems, a detailed account of which is available in Drugeon
and Venditti (1996).
7 Assumptions concerning preferences will be denoted as P, restrictions involving technology will
be referred to as T and joint restrictions will be designated as PT.

J.-P. Drugeon, A. Venditti / Journal of Economic Dynamics & Control 25 (2001) 765}787 769

Let y(t) denote the production of the capital good sector and d3R feature
`
the depreciation rate of the capital stock. The investment resource constraint is
given by x5 (t)"y(t)!dx(t) at date t3R . The maximum sustainable value of the
`
capital stock states as
Assumption PT.1. For a given X3R , there exists x6 such that F1(x, 1, X)'dx
`
for any x(x6 , and F1(x, 1, X)(dx for any x'x6 . In addition, F1(0, l1, X)"0
for any l13[0, 1].
The de"nition of the production possibility frontier then results from solving
the following problem:
c"¹(x, y, X) "
: Max F0(x0, l0, X)
M
xj, ljN
s.t. y4F1 (x1, l1, X),
x0#x14x,

S(X)

l0#l141,
l0, l1, x0, x150.
¸et y"f (x, X) denote the solution of ¹(x, y X)"0 for any given X. The
analysis is narrowed down to the set of feasible capital stocks X:
X"
: M(x(t), x5 (t))3R ]RD04x(t)4x6 and x5 (t)4f [x(t), X(t)]!dx(t)
`
for any given X(t)3R N,
(2)
`
for X a convex set with Xs O0 as long as f ( ) , ) ) is not identical to zero over
[0, x6 ]]R . It is convenient at this stage to introduce an indirect utility function:
`
;[x(t), x5 (t), X(t)] "
: u[¹(x(t), x5 (t)#dx(t), X(t))].
(3)
Under the concavity Assumptions P.1 and T.1 on preferences and sectoral
production technologies, standard arguments8 ensure that, for any given
X3R , ;( ) , ) , X) is also a concave function. However, a strict concavity prop`

erty requires an extra restriction:9
Assumption PT.2. ;( ) , ) , ) ) is such that ;3C2(R ]R]R , R), D2;( ) , ) , X) is
`
`
negative de"nite over X for any given X3R .
`

8 E.g., available in Madden (1986).
9 Negative-de"niteness of the Hessian matrix is obtained from the imposition of strong concavity
assumptions on instantaneous utility and the production technology of the consumption good,
a recent and detailed account of these issues is available in Venditti (1997).

770 J.-P. Drugeon, A. Venditti / Journal of Economic Dynamics & Control 25 (2001) 765}787

Competitive market capital-paths s assuming a time-t value of x(t) solve, for
a given external-path s6, the following problem:10
Max := ;(x(t), x5 (t), X(t))exp(!ot) dt
t/0
M N
x
s.t. (x(t), x5 (t))3X, ∀t3R ,
`
P
=
x(0)"x , x(0) given,
0
X(t)3R , ∀t3R , o3R ,
`
`
`
plus a "xed-point argument between x(t) and X(t), i.e., x(s6, t)"X(t) for all
t3R .
`
The set of admissible externality paths is restrained to those which preserve
a "nite value for the integral of welfare:
Assumption PT.3. := exp(!ot);[x(t), x5 (t), X(t)](#R.
t/0
Also, assuming that an interior solution to P exists, the latter is character=
ised by satisfying11 of the Euler}Lagrange equation
; (x, x5 , x)!(d/dt)[; (x, x5 , x)]#o; (x, x5 , x)"0,
(4)
1
2
2
plus a transversality condition on the boundary, i.e., lim
M!; [x(t),
t?=
2
x5 (t), x(t)] ) x(t)exp(!ot)N"0.

3. Uniqueness versus multiplicities
As a benchmark:12
Dexnition 1. For a given o3R , an interior steady state is a pair (xH, 0 )3Xs , for
`
xH"XH, such that Z (xH)"; (xH, 0, xH)#o; (xH, 0, xH)"0.
o
1
2
The existence of such a position is ensured by extra minor restrictions on the
production technology of the capital good:
Assumption PT.4. Let x1(xH, dxH, xH) and l1(xH, dxH, xH) denote the steady input
demand functions. The productivity of the investment good
F1[x1(xH, dxH, xH), l1(xH, dxH, xH), xH] is such that lim H F1'd#o and
1
x ?0 1
lim H F1(d#o.
x ?= 1
10 See Kehoe et al. (1991) and Mitra (1995) for a detailed assessment in the discrete time case.
11 For recent and enlightening discussions on this topic, see Blot and Cartigny (1995).
12 In order to lighten the presentation, the dependency with respect to the discount rate will be
omitted when it is not explicitly considered.

J.-P. Drugeon, A. Venditti / Journal of Economic Dynamics & Control 25 (2001) 765}787 771

Fig. 1. (a) Local indeterminacy; (b) Local determinacy but global indeterminacy.

Lemma 1. Under Assumptions P.1, T.1, PT.1}4, there exists at least one interior
steady-state position.
Proof. See Appendix A.1. h
The implications of external e!ects may dominate the concavity properties of
the environment summarised in Assumptions P.1, T.1 and PT.2. This in turn can
have an e!ect upon the uniqueness properties and thus result in dynamical
complexities related to global concerns. For instance, as this is illustrated in
Fig. 1 within a con"guration with complex roots, stable manifolds of distinct
steady states may overlap for a given interval of values for the capital stock and
then a!ord multiple long-run positions from a given set of initial conditions:

772 J.-P. Drugeon, A. Venditti / Journal of Economic Dynamics & Control 25 (2001) 765}787

The subsequent de"nition will provide a useful benchmark for these multiplicity phenomena:
Dexnition 2. Consider an equilibrium trajectory s converging to a steady-state
(xH, 0, xH) for a given x(0)"x 3[0, x6 ]. Globally indeterminate equilibrium
0
dynamics emerge if there exists another path s@ which assumes ((xH)@, 0, (xH)@) as
its asymptotic position and satis"es x@(0)"x(0)"X(0) but (xH)@OxH.
As a preliminary step, a straightforward application13 of Brock's14 (1973,
Theorem 1) uniqueness argument remains available for the current environment:
Proposition 1. Under Assumptions P.1, T.1, PT.1}4, assume also that for o"0,
there exists a unique stationary state (xH, 0, xH). Then if [(; #o; )#
11
21
(; #o; )](0 is satisxed over [0, o6[, there exists a unique stationary state for
13
23
any o3[0, o6[.
From a broader perspective, a new area for indeterminacies emerges:
Proposition 2. Under Assumptions P.1, T.1, PT.1}4, assume that there exists values
of o3R such that (¹ #¹ )!o(¹ #¹ )50, multiple solutions can no
`
11
13
21
23
longer be discarded.
Proof. See Appendix A.2. h
Remark 1. From Propositions 1 and 2, a necessary condition for the violation of
uniqueness is summarised as a positive sign for the determinant. Section 4.3 is
concerned with a PoincareH }Hopf bifurcation where purely imaginary eigenvalues
give su$cient conditions to violate uniqueness. Such an outcome could also have
resulted from another phenomenon associated with a unique null eigenvalue, or
a saddle-node type bifurcation. See Guckenheimer and Holmes (1986, pp.
146}150) for a detailed expose& and Cazavillan et al. (1998) for a recent example of
a related phenomenon within an environment with heterogenous agents.
4. Local indeterminacies and oscillations
4.1. The saddlepoint property
As outlined in Lemma A.1 in Appendix A.3, the linearisation of the
Euler}Lagrange equation around a steady state (xH, 0, xH) does not rule out the
13 It is understood that Z (xH)"0 is parameterised by a given value XH"xH for the externality.
o
14 Following a topological approach based on the Hopf Lemma and an extension of Milnor (1965,
p. 36), Benhabib and Nishimura (1979b) have provided a more general understanding of the
uniqueness issue in a multisector model.

J.-P. Drugeon, A. Venditti / Journal of Economic Dynamics & Control 25 (2001) 765}787 773

possibility of complex eigenvalues and in this regard contradicts the predictions
of the optimal growth literature for a one capital good model. More precisely,15
for ; " ; (xH, 0, xH), i, j"1, 2, 3 and R the spectrum of the characteristic
ij
ij
polynomial:
Lemma 2. Under Assumptions P.1, T.1, PT.1}4, let k3R, there then exists a k@3R
such that R(k@)"!R(k)#o!(; /; ).
23 22
Proof. See Appendix A.3. h
This lemma is informative by the generalisation it provides of a well-known
rule of the optimal growth literature, see Levhari and Liviatan (1972), which
assesses a pair root structure for the spectrum of the Jacobian matrix. It is worth
noticing that the original formation of their rule, which associates !o#k to
any root k3R, remains valid in the present setting for external e!ects such that
; O0, ; O0 but ; "0. At this stage, a rough sketch of the indeterminacy
3
13
23
outcome is available. Considering a negative real eigenvalue k, the above
modi"ed rule suggests that there is an actual potential for a remaining negative
eigenvalue when o!(; /; )(0.
23 22
A benchmark structure with a dimension for the stable manifold equal to the
number of state variables is stated as follows:
Dexnition 3. Let s denote an equilibrium trajectory for a given x(0)"x and
0
consider a steady state (xH, 0, xH), for xH"XH and with R"Mk, k@N. It is
a regular saddlepoint if kk@(0.
Proposition 3. Under Assumptions P.1, T.1, PT.1}4, necessary and suzcient
conditions for (xH, 0, xH), to be a regular saddlepoint are summarised by the
fulxlment of (; #o; )#(; #o; )(0.
11
21
13
23
Proof. See Appendix A.3. h
Hence, the saddlepoint condition of an optimal growth environment, i.e.,
; #o; (0, appears as a mere part of the condition that pertains to
11
21
a competitive equilibrium with externalities. Let us now turn to the way this
relates to the uniqueness concerns mentioned earlier in this paper:
Corollary 1. Under Assumptions P.1, T.1, PT.1}4, assume also that for o"0, there
exists a unique stationary state (xH, 0, xH). If it satisxes the saddlepoint property for
any o3[0, oN [, then it is unique for any o within this interval.

15 It is understood that ; "L2;/Li Lj, i, j"1, 2, 3, is evaluated at (xH, 0, xH).
ij

774 J.-P. Drugeon, A. Venditti / Journal of Economic Dynamics & Control 25 (2001) 765}787

In spite of the introduction of external e!ects and though this now involve
extra terms, there still exists a direct link between the saddlepoint property and
uniqueness.
4.2. Local indeterminacies
Attention is now turned to the local acceptation of an indeterminate steadystate position:
Dexnition 4. Let s denote an equilibrium trajectory for a given
x(0)"x , x 3[0, x6 ] and with a steady-state position of (xH, 0, xH), xH"XH. It
0 0
is locally indeterminate if the associated stable manifold is of dimension 2.
Proposition 4. Under Assumptions P.1, T.1, PT.1}4, necessary and suzcient
conditions for indeterminate equilibrium dynamics are
(i) ; (0 holds,
23
(ii) ; /; 'o holds,
23 22
(iii) o; #; 'D; #o; D.
21
13
11
23
Proof. See Appendix A.4. h
The scope for indeterminacies is related to the occurrence of ; (0. On the
23
contrary, any sign for ; may a priori leave room for indeterminacies.16
13
A more detailed examination of the properties on preferences and sectoral
production technologies which underlie the signs of ; and ; is given in
13
23
Section 5.
Remark 2. Benhabib and Farmer (1996) and Benhabib and Nishimura (1998)
consider sector-specixc externalities. On a technical basis, internal factors of the
production technologies are duplicated by external ones as Fj(xj, li, x6 j, lM j)"
(xj)aj (lj)bj (x6 j)aj (lM j)bj, with a #b #a #b "1. The constant returns property
j
j
j
j
at a social level together with the absence of supplementary intersectoral
dimension enables one to build on standard arguments adaptated from optimal
growth theory material. On economic grounds, the associated equilibrium
e!ects di!er in being intrinsically linked to the properties of sector-speci"c
demands whereas the current approach focuses on intersectoral spillover e!ects
which arise from global external stock e!ects.

16 The requesite (iii) in Proposition 4 is "lled when D (0.
o

J.-P. Drugeon, A. Venditti / Journal of Economic Dynamics & Control 25 (2001) 765}787 775

4.3. Oscillations
The purpose of this section is to examine the area for the type of sustained
oscillations which were "rst pointed out by Benhabib and Nishimura (1979a) in
a multisector environment. A consideration of their potential articulation with
the preceding local indeterminacy concerns is made.
Assessing "rst the existence of a critical value for the discount rate:
Assumption PT.5. The dynamical system is C5 parameterised by o, there exists
a unique oH such that o![; (o)/; (o)]~0 for o~oH and the eigenvalues
23
22
remain complex in an open neighbourhood de"ned around oH.
It is observed that such an assumption directly builds upon the presence of
external e!ects in the characteristic polynomial: (i) without external e!ects, the
term that features the trace of the associated Jacobian matrix summarises to
o'0, (ii) without external e!ects and as is clear from Appendix A.3, under
Assumption PT.2, both eigenvalues are real. As this is made clear by the
subsequent formal statement, from an application of the PoincareH -Hopf bifurcation Theorem for #ows17 in R2, a new type of attractor, i.e., closed curves around
the steady state, appears:
Proposition 5. Under Assumptions P.1, T.1, PT.1}5, a Poincare& }Hopf bifurcation
occurs at o"oH and there exists a family of closed orbits in one side of an open
neighbourhood ]oH!ι, oH#ι[, ι3RH , dexned around oH. If dR[k(o)]/doD H
`
o/o
O0, each closed orbit is locally unique.
It is noted that a related conclusion was reached by Benhabib and Nishimura
(1979a), but their conclusions require the introduction of heterogeneous capital
goods. Their endogenous cycles result is based upon a two-capital goods environment and an asymmetric structure for the matrix ; , which is out of
12
purpose when ; is a scalar. In opposition to this, the current argument hinges
12
on an environment augmented by external e!ects but with a unique capital
good.
Parallel to this, an invariant closed curve around ]xH, 0, xH[ and an indeterminate steady state coincide only when the PoincareH }Hopf bifurcation is subcritical, i.e., when the closed curve appears when the complex eigenvalues have
negative real parts. This states that the only class of closed curves that "ts the
dynamical indeterminacies of De"nition 3 are locally repulsive. An analytical
treatment of this global facet of the indeterminacy issue is provided through the
comprehensive examination of an example in Section 6.

17 See Guckenheimer and Holmes (1986, p. 151) for an exposition.

776 J.-P. Drugeon, A. Venditti / Journal of Economic Dynamics & Control 25 (2001) 765}787

5. Looking for the fundamentals
5.1. A basic argument
As interesting as they are, the actual meaning of Propositions 4 and 5 remains
obscured by their phrasing in terms of the second-order derivatives of an
indirect utility function, the latter being itself de"ned, through (3), from an
immediate utility function u( ) ) and a production possibility frontier ¹( ) , ) , X).
This section intends to achieve a dearer understanding of the mechanisms on
preferences and sectoral production technologies which underlie the previous
indeterminacy conclusions.
Examining the contents of Propositions 2 and 4, it becomes clear that the
signs of ; and ; are decisive in the area for indeterminacies. Immediate
13
23
computations deliver, for d"0:
; "uA¹ ¹ #u@¹ ,
(5a)
13
3 1
13
; "uA¹ ¹ #u@¹ .
(5b)
23
3 2
23
A thorough understanding of the indeterminacy issue is obtained by directly
building on the properties of the externalities-augmented production possibility
frontier. Hirota and Kuga (1971) and Kuga (1972) completed a fundamental
argument by outlining in detail how one could recover properties of sectoral
production functions starting from the derivatives of the production possibility
frontier of an optimal growth framework. Extending their approach to the
current suboptimal equilibrium, Appendix A.5 unveils a close link between the
two derivatives at the core of the indeterminacy issue:
¹ "F0 (Lx0/LX)#F0 (Ll0/LX)#F0 ,
13
12
11
13
!¹ F1#[F1 (Lx1/LX)#F1 (Ll1/LX)#F1 ]F0
13 1 ,
12
11
13 1
¹ "
23
(F1)2
1

(6a)
(6b)

for
(Lx0/LX)"!(Lx1/LX),

(7a)

(Ll0/LX)"!(Ll1/LX),

(7b)

(Lx0/LX)"[F1!F1(Ll0/LX)/F1].
1
2
3
A noticeable clari"cation emerges:

(7c)

Lemma 3. For o3RH and d"0, the indeterminacy condition (iii) of Proposition 4
`
no longer depends on ¹ .
13
Proof. See Appendix A.5. h

J.-P. Drugeon, A. Venditti / Journal of Economic Dynamics & Control 25 (2001) 765}787 777

Thus there emerges a direct link between the indeterminacy of the steady-state
and the in#uence of the external e!ect on the relative price of the investment
good, i.e., ; . Such an explanation is directly related to intersectoral arbitrages:
23
it is therefore not surprising that it was out of purpose when a standard
environment with a unique homogeneous good was considered.
Further details are conceivable when external e!ects are further narrowed to
assume positive values. This requires the sectoral production technologies to
further satisfy (i) Fj (xj, lj, X)50, (ii) Fj (xj, lj, X)50 and (iii) Fj (xj, lj, X)50
23
13
3
in sectors j"0, 1. Then and from the role of ; (0 in the indeterminacy
23
outcome of Proposition 4, the expression of ; and the concavity of u( ) ), it
23
becomes clear that the occurrence of ; (0 requires the parallel holding of
23
¹ (0.
23
5.2. The homogeneous case
Building on the general case analysed in Lemma 3, the following statement
focuses more speci"cally on homogeneous sectoral production technologies:
Proposition 6. Let Fj( ) , ) , ) ), j"0, 1, be homogeneous of degree a '1. The derivaj
tive ¹ is strictly negative at the steady-state if the following conditions are
23
satisxed:
(i) Ll0/LX(0.
(ii) F0 !F1 (F0/F1)'0,
13 1 1
13
(iii) F0 !F1 (F0/F1)#(F1/F1)[F0 #F1 (F0/F1)]50.
11 1 1
3 1 11
13 1 1
13
Proof. See Appendix A.6. h
Condition (ii) assesses the need for a relatively greater in#uence of the
external e!ect on the return on capital in the consumption good sector. Condition (iii) requires this external return of capital to overcome the stabilising
in#uence of the private return to capital. It can be concluded from this that
positive external e!ects in the consumption good sector clearly favour indeterminacy whereas similar e!ects in the investment good sector have opposite
implications.

6. Global indeterminacies: An illustration
This section will dispense with a comprehensive analytical understanding of
an example that hinges on a parameterised representation inspired by Sutherland

778 J.-P. Drugeon, A. Venditti / Journal of Economic Dynamics & Control 25 (2001) 765}787

(1970).18 In order to clarify the exposition, the more technical details have been
relegated to the working paper version available as Drugeon and Venditti
(1996).
Consider the following speci"cation for the indirect utility function of the
Sutherland (1970) type:
;(x, x5 , X)"!7x2!12xx5 !38x5 2#6xX!x5 X
#(18/3)x3!(9/4)x4#(g/4)x5 4,

(8)

with g a parameter whose sign is discussed later on. It satis"es, at least locally,
the whole range of properties under which the previous analysis has been
performed. It is proved19 that ; '0, ; (0, ; (0, ; (0, ; (0 hold
1
2
11
22
12
in a neighbourhood of a steady position for which x5 "0.
Consider now the implications of the externality. First, note that
; "6x!x5 . The focus is on positive externalities around stationary states.
3
Also, note that ; "6'0 and ; "!1(0: from Proposition 4(i), indeter13
23
minacies cannot a priori be ruled out. Letting y"x5 , the dynamical system is
derived as
x5 "y ;

(9a)

1
[(!8!13o#18x!9x2)x!(76o!goy2!1)y].
y5 "
3gy2!76

(9b)

Hence a "rst stationary position xH located at the origin. The set of interior
steady-states is given by the values of x'0 which solve a quadratic form, or
x2!2x#((8#13o)/9)"0. This leads to a pair of candidate solutions for the
capital stock de"ned from xHH,HHH"1$[1!(8#13o)/9]1@2.20 Thus there is
an actual possibility of multiple stationary positions for the dynamical system.
Consider the Jacobian matrix in the neighbourhood of the aforementioned
steady-state position, it is shown that the trace is invariant and given by
!1/76#o, for o3RH . From Proposition 5, a PoincareH }Hopf bifurcation
`
occurs at o "1/76. However two con"gurations are to be distinguished
HH
according to the value of g. For the limit con"guration with g"0, the local
bifurcation is of a degenerated type21 and the dynamical system is conservative

18 See Liviatan and Samuelson (1969) for a related argument in an environment with a unique
homogeneous good and wealth e!ects in the utility function.
19 See Drugeon and Venditti (1996).
20 The de"nition of the stationary values of the captial stock remain unaltered for any g.
21 From a formal point of view, the conclusion obtained signi"es that it is necessary to establish
the nullity of all the coe$cients associated with terms of order greater than or equal to 2 in the
normal form of the dynamical system. See Guckenheimer and Holmes (1986) and Chenciner (1981).

J.-P. Drugeon, A. Venditti / Journal of Economic Dynamics & Control 25 (2001) 765}787 779

or hamiltonian, a detailed examination of such a system is available in Matsuyama (1991). The subsequent argument assumes the opposite and focuses on
the stability properties of closed orbits in the generic case.
Consider the non-linear dynamical system (9) for o "1/76. As assessed in
HH
Proposition 5, the essential di$culty in the analysis of the stability of a periodic
orbit is that one is faced with the existence of a center manifold at the steadystate that implies the need for an explicit consideration of the non-linearities of
the dynamical system. The subsequent coordinate transformation will thus
simplify the analytic expression of the vector "eld on the center manifold. After
lengthy computations, see Drugeon and Venditti (1996), a normal form of the
dynamical system emerges as
z5 "0 ) 22375iz!z2z6 [0.0000032499g#(0.198468!0.000018422g)i]
#O(DzD4)

(10)

with z as the complexixcation of the expression of the variables x and y in the
coordinates of the eigenspace. The stability properties of the periodic orbit result
from the sign of the real part of the coe$cient associated with the cubic term.
Here, the cubic component is summarised by z2z6 and this coe$cient is such that
its real part is Re"!0.0000032499g. From Guckenheimer and Holmes (1986,
pp. 151}152), the PoincareH }Hopf bifurcation is supercritical * the periodic
orbit is a limit cycle * for g'0 and subcritical for g(0. In a particular case,
for g"0, one recovers the degenerated example * a critical bifurcation * previously mentioned.
All these results are con"rmed by numerical simulations completed from the
fully non-linear dynamical system and for g"1000.22 Firstly, considering the
stability properties of the steady state xHH, from Fig. 2a, it reveals that the steady
o
state is strongly attractive for o"1/100. In opposition to this, for the critical
bifurcation value of o "1/76, attraction towards the steady state become
HH
much weaker in Fig. 2b.23 In this latter con"guration, a supercritical PoincareH }Hopf bifurcation emerges.
Beyond this bifurcation value, an assessment of the stability properties of the
periodic orbit is in order. Firstly, in Fig. 2c and for o"1/73, the initial
conditions lie outside the periodic orbit but the paths converge towards the

22 Note that, had lower values of g been considered, the real part of the coe$cient would have
been signi"cantly weaker and the orbit would have only been weakly attracting. In order to reach
a clear graphical illustration of the preceding theoretical assessement, an admittedly high value of
g has been retained. For more reasonable values of g, similar conclusion hold, but the convergence
speed towards the periodic orbit is limited while its basin of attraction becomes smaller.
23 The stability properties of the steady state are entirely derived from the non-linear component
of the system, xHH being a vague attractor if g'0.

780 J.-P. Drugeon, A. Venditti / Journal of Economic Dynamics & Control 25 (2001) 765}787

Fig. 2. (a) Stable steady state (Discount factor: o"0.01; Initial conditions: x(0)"0.96 and
y(0)"0.001); (b) Stable steady state (&weak attractor') (Discount factor: o"oH"1/76; Initial
conditions: x(0)"1.2 and y(0)"0.001); (c) Stable periodic orbit (&from the outside') (Discount factor:
o"1/73; Initial conditions: x(0)"1.295 and y(0)"0.001); (d) Stable periodic orbit (&from the
inside') (Discount factor: o"1/73; Initial conditions: x(0)"0.696 and y(0)"0.0005); (e) Stable
periodic orbit with initial conditions outside its basin of attraction (Discount factor: o"1/68; Initial
conditions: x(0)"1.26 and y(0)"0.001).

latter. Secondly, in Fig. 2d, the initial conditions lie within the closed curve. As
expected, xHH is locally repulsive where the periodic orbit is attractive.
A further interesting phenomenon occurs for (o"1/68) and a steady-state of
xHH"0.700218 when the initial position is located just outside of the basin of
attraction of the periodic orbit. As this appears in Fig. 2e, at a given moment in
time, the orbit integrates the basin of attraction of the saddle point

J.-P. Drugeon, A. Venditti / Journal of Economic Dynamics & Control 25 (2001) 765}787 781

Fig. 2. Continued.

782 J.-P. Drugeon, A. Venditti / Journal of Economic Dynamics & Control 25 (2001) 765}787

xHHH"1.29978 and is thus attracted by the latter. The interest of such a global
phenomenon lies in that it explicitly builds from multiple steady-state positions
and that non-linearities play a decisive role in its occurrence.

Appendix A
A.1. Proof of Lemma 1
From De"nition 2, a steady-state position is de"ned from
for

; (xH, 0, xH)#o; (xH, 0, xH)"0,
1
2

; "u@(¹ #d¹ ),
1
1
2
; "u@¹ ,
2
2
or, recalling the de"nition of ¹ and ¹ in terms of the fundamentals:
1
2
F0[x0(x, dx, x), l0(x, dx, x), x]
1
F0[x0(x, dx, x), l0(x, dx, x), x]
!(o#d) 1
"0.
F1[x1(x, dx, x), l1(x, dx, x), x]
1
Finally simplifying in the case of an interior solution:
F1[x1(x, dx, x), l1(x, dx, x), x]!(o#d)"0. h
1
Under Assumption PT.4, the statement follows.
A.2. Proof of Proposition 2
Consider the reexpression of the de"nition of a steady state along
;
! 1"o
;
2
or, and from the de"nition of ;( ) , ) , X),!¹ /¹ "o. The slope of the L.H.S. of
1 2
this equation states as
!(¹ #¹ )¹ #(¹ #¹ )¹
(¹ #¹ )!o(¹ #¹ )
11
13 2
21
23 1"!¹
11
13
21
23 .
2
(¹ )2
(¹ )2
2
2
When, as in the statement, this expression encovers a positive sign, the locus of
steady-states results from the intercrossing of two increasing curves, so that
multiplicity can no longer be discarded. h

J.-P. Drugeon, A. Venditti / Journal of Economic Dynamics & Control 25 (2001) 765}787 783

A.3. Proofs of Lemma 2 and Proposition 3
The characteristic polynomial states as
P(k)"k2; #k(; !o; )![(; #o; )#(; #o; )]"0,
22
23
22
11
21
13
23
with a discriminant D "(; !o; )2#4; [(; #o; )#; #
o
23
22
22
11
21
13
o; )]. It is "rst of some use to provide a clari"cation about the actual
23
possibility of complex eigenvalues:
Lemma A.1. Under Assumptions P.1, T.1, PT.1}4, the eigenvalues of the characteristic polynomial are real in a Model M1, X, X, ;, oN if one of the following
conditions holds:
(i) [(; #o; )#(; #o; )]40,
11
21
13
23
(ii) ; #o; #; 40,
11
21
23
(iii) ; [2o#4(; /; )]40.
23
13 23
Proof. (i) Considering the expression of the characteristic polynomial, this
follows from the expression of the discriminant. (ii) Note that the discriminant
may be reexpressed along
D "(o; )2!2o; ; #(; )2#4; (; #o; #; #o; ),
o
22
22 23
23
22 11
21
13
23
"(o; )2#2o; ; #(; )2#4; (; #o; #; ),
22
22 23
23
22 11
21
13
"(o; #; )2#4; (; #o; #; ),
22
23
22 11
21
13
the statement follows. For (iii), remark that the expression of D may be
o
rearranged as

C
C

A BD
A BD

;
;
23#2o#4 13
D "[o2(; )2#4(; #o; )]#; ;
o
22
11
21
22 23 ;
;
23
22

C

DC D

;
;
12
"; [2 o] 11
22
;
;
21
22

C

2

;
;
23#2o#4 13
#; ;
22 23 ;
;
o
23
22

A BD

;
;
23#2o#4 13
"D(o)#; ;
22 23 ;
;
23
22

,

,

.

As detailed in the preceding derivation, had externalities been omitted from the
analysis, i.e., for ; "; "; "0, D(o) is the expression to which D would
3
13
23
o

784 J.-P. Drugeon, A. Venditti / Journal of Economic Dynamics & Control 25 (2001) 765}787

have backed. Under the strict concavity restriction of Assumption PT.2,
D(o)'0.24 The statement follows from noticing that (; )2'0. h
23
From the expression of the characteristic polynomial, the proof of Lemma A.2
builds upon the fact that the sum of the eigenvalues derives as: k#k@"
o!(; /; ). Hence, if k3R, P[!k#o!(; /; )]"0 and the statement
23 22
23 22
follows. The proof of Proposition 3, as for itself, is immediate from the expression of P(k).
A.4. Proof of Proposition 4
(i) and (ii) This follows from Assumption PT.2 and the expression of the
trace in the Proof of Lemma 2. (iii) This follows from the requirement
(; #o; )#(; #o; )50, having integrated (i). h
11
21
13
23
A.5. Proof of Lemma 3
Let x0(x, y, X), l0(x, y, X), x1(x, y, X), l1(x, y, X) denote the allocations of
capital and labour between the two sectors that derive from S(X) for a given X.
Along Boldrin (1989) and from arguments detailed in Hirota and Kuga (1971)
and Kuga (1972), the solution of S(X) may be expressed as
¹(x, y, X)"F0[x0(x, y, X), l0(x, y, X), X],
with y"F1[x1(x, y, X), l1(x, y, X), X]. Hinging on these expressions and on the
reformulation of the binding resource constraints as
x0(x, y, X)#x1(x, y, X)"x,
l0(x, y, X)#l1(x, y, X)"1,
static optimisation conditions lead to
¹ (x, y, X)"F0 [x0(x, y, X), l0(x, y, X), X]
1
1
"q(x, y, X)F1 [x1(x, y, X), l1(x, y, X), X]
1
"u(x, y, X) ,
F0 [x0(x, y, X), l0(x, y, X), X]
¹ (x, y, X)"! j
2
F1[x1(x, y, X), l1(x, y, X), X]
j
"!q(x, y, X)(0, j"1, 2

24 This line of argument is borrowed from Venditti (1998).

J.-P. Drugeon, A. Venditti / Journal of Economic Dynamics & Control 25 (2001) 765}787 785

for u( ) , ) , X) and q( ) , ) , X) the rental and the price of the capital good in terms of
the price of the consumption good. Resting on standard envelope arguments
and from the above set of equations, the direct e!ect of the externality on the
production possibility frontier can be characterised as
¹ (x, y, X)"F0[x0(x, y, X), l0(x, y, X), X]
3
3
#q(x, y, X) F1 [x1(x, y, X), l1(x, y, X), X].
3
The remaining issue has then to do with the determinants of ¹ and ¹ . From
13
23
the above, they, respectively, back to ¹ "Lu/LX and ¹ "!Lq/LX and
13
23
their full expression appears in the main text.
Restate Proposition 1(iii):
o; #; 'D; #o; D.
21
13
11
23
Building on the de"nition of ;( ) , ) , X), immediate computations give, for d"0:
; "uA(¹ )2#u@¹ ,
11
1
11
; "uA¹ ¹ #u@¹ ,
12
1 2
12
; "uA¹ ¹ #u@¹ ,
13
3 1
13
; "uA¹ ¹ #u@¹ .
23
3 2
23
Noticing that ¹ expresses as a function of ¹ along
23
13
!¹ F1#[F1 (Lx1/LX)#F1 (Ll1/LX)#F1 ]F0
13 1 ,
12
11
13 1
¹ "
23
(F1)2
1
and that the expression of the discount rate reduces to
o"!; /;
1 2
"!u@¹ /u@¹
1
2
"!F0 ) [!(F1/F0)]
1 1
1
"F1
1
at the steady-state, condition (iii) in Proposition 4 reformulates as
F1(uA¹ ¹ #u@¹ )#uA¹ ¹ #u@¹
1
1 2
12
1 3
13
'!uA(¹ )2!u@¹ !F1uA¹ ¹
1
3 2
1
11
#u@M¹ ![F1 (Lx1/LX)#F1 (Ll1/LX)
12
11
13
#F1 ](F0/F1)N.
13 1 1
The statement follows.

786 J.-P. Drugeon, A. Venditti / Journal of Economic Dynamics & Control 25 (2001) 765}787

A.6. Proof of Proposition 6
The formal expression of ¹

reduces to
13
¹ "F0 (Lx0/LX)#F0 (Ll0/LX)#F0
13
12
11
13
F1 Ll0 X(F0 F0 !F0 F0 )
12 13 #F0 .
11 23
"F0 3#
13
11 F1 LX
(1!a )F1
0 1
1
The derivation of the second equation from the "rst hinges on the use of the
equilibrium constraint, or Lx0/LX"[F1!F1(Ll0/LX)/F1] as well as on the
1
2
3
de"nition of the relative price of the investment good, or F1/F1"F0/F0 and
2 1
2 1
the homogeneity of degree one of F0( ) , ) , X). The simpli"ed expression in the
third equation results from the nullity of the determinant of the Hessian matrix
of F0( ) , ) , M) for any X3R .
`
Consider now and similarly the way the relative price is in#uenced by the
externality, i.e., the sign of the derivative ¹ . Along the same approach,
23
incorporating the third equation in the expression of ¹ and letting now a '1
13
1
denote the degree of homogeneity of F1( ) , ) , X), it expresses as
¹ "F0 (Lx0/LX)#F0 (Ll0/LX)#F0 ,
13
12
11
13
!¹ F1#[F1 (Lx1/LX)#F1 (Ll1/LX)#F1 ]F0
13 1
12
11
13 1
¹ "
23
(F1)2
1
1 F1
F0
F0
3 F0 #F1 1 #F0 !F1 1
"!
11
13
11
13
F1 F1
F1
F1
1 1
1
1
Ll0 X F0 F0 !F0 F0
F0 F1 F1 !F1 F1
12 13# 1 11 23
11 23
12 13 .
! )
(1!a0)F0
(1!a )F1
F1
LX F1
1
1 1
1
1
Under the positive signs for Fj , j"0, 1, that result from the homogeneity
12
assumption T.1 and positive external e!ects along De"nition 6, the details of the
statement follow. h

G A
C

B

H

D

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