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:[email protected] (A. Venditti). 25 (2001) 765}787
Intersectoral external e!ects,
multiplicities & indeterminacies
qJean-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
1The case of negative externalities leaves room for indeterminacies, a computed example of which was provided by Kehoe (1991).
2It has just been extended by Benhabib and Nishimura (1998).
Keywords: Intersectoral external e!ects; Multiple steady states; Endogenous # uctu-ations; 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 asindeterminateas 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 intertem-poral 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.
3Benhabib 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. 4Their 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.Videthe argument of Lemma A.1 in Appendix A.3.
5These conclusions have been con"rmed in a non-stationary context by Drugeon and Venditti (1996).
Contemplating a broader perception of the indeterminacy issue, the present contribution will focus on a continuous time version of the canonical multisec-tor growth model with one capital good.3The 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 themyopyormultiple consumption goodsargument 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 multiplic-ities or endogenous#uctuations relate to or even underlie local indeterminacies are assessed.
6Precise 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).
7Assumptions concerning preferences will be denoted as P, restrictions involving technology will be referred to as T and joint restrictions will be designated as PT.
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
Cassigning a consumption#owc(t) att3R
`:
U(C)"
P
`=t/0
u[c(t)] exp (!ot) dt, o3R
` (1)
foru()) an instantaneous utility function such that:7 Assumption P.1.u3C2(RH
`,R) and satis"esu@'0,uA(0 for anyc'0,u(0)"0, u(R)"R, lim
c?0u@"R, limc?=u@"0.
Any consumeri3[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, orj"1. Any individual holds a"rm in any of the sectors and is endowed with an initial capital stock x
i(0)"x0. Both capital and labour are freely
shiftable from one sector to the other.
The production technology of a given"rmFjin any of the sectors will depend on privately held capital and labour, orxj(t) andlj(t), as well as on the aggregate
value of the capital stockX(t) at the same date:
Assumption T.1.Fj3C2(RH
`]RH`]R`,R`),j"0, 1, satis"es, for a givenX3R`:
8E.g., available in Madden (1986).
9Negative-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).
Lety(t) denote the production of the capital good sector andd3R
`feature
the depreciation rate of the capital stock. The investment resource constraint is given byx5(t)"y(t)!dx(t) at datet3R
`. The maximum sustainable value of the
capital stock states as
Assumption PT.1.For a givenX3R
`, there existsx6 such thatF1(x, 1,X)'dx
for anyx(x6, andF1(x, 1,X)(dxfor anyx'x6. In addition,F1(0,l1,X)"0 for anyl13[0, 1].
The de"nition of theproduction possibility frontierthen results from solving the following problem:
c"¹(x,y,X) :"Max
Mxj,l
jN
F0(x0,l0,X)
s.t. y4F1(x1,l1,X),
x0#x14x,
l0#l141,
l0,l1,x0,x150.
S(X)
¸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 stocksX:
X:"M(x(t),x5(t))3R
`]RD04x(t)4x6 andx5(t)4f[x(t),X(t)]!dx(t)
for any givenX(t)3R
`N, (2)
for X a convex set withXsO0as long as f(),)) is not identical to zero over [0,x6]]R`. It is convenient at this stage to introduce anindirect 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 overXfor any givenX3R
10See Kehoe et al. (1991) and Mitra (1995) for a detailed assessment in the discrete time case. 11For recent and enlightening discussions on this topic, see Blot and Cartigny (1995). 12In order to lighten the presentation, the dependency with respect to the discount rate will be omitted when it is not explicitly considered.
Competitive market capital-pathssassuming a time-tvalue ofx(t) solve, for a given external-paths6, the following problem:10
Max
The set of admissible externality paths is restrained to those which preserve a"nite value for the integral of welfare:
Assumption PT.3.:=
t/0exp(!ot);[x(t),x5(t),X(t)](#R.
Also, assuming that an interior solution to P
=exists, the latter is
character-ised by satisfying11of the Euler}Lagrange equation
;
1(x,x5,x)!(d/dt)[;2(x,x5,x)]#o;2(x,x5,x)"0, (4)
plus a transversality condition on the boundary, i.e., lim
t?=M!;2[x(t),
The existence of such a position is ensured by extra minor restrictions on the production technology of the capital good:
Assumption PT.4. Letx1(xH,dxH,xH) andl1(xH,dxH,xH) denote the steady input
demand functions. The productivity of the investment good
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
13It is understood thatZ
o(xH)"0 is parameterised by a given valueXH"xHfor the externality.
14Following 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.
The subsequent de"nition will provide a useful benchmark for these multipli-city phenomena:
Dexnition 2.Consider an equilibrium trajectorysconverging to a steady-state (xH, 0,xH) for a given x(0)"x
03[0,x6]. Globally indeterminate equilibrium
dynamics emerge if there exists another paths@which assumes ((xH)@, 0, (xH)@) as its asymptotic position and satis"esx@(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 foro"0,
there exists a unique stationary state (xH, 0,xH). Then if [(;
11#o;21)#
(;
13#o;23)](0is satisxed over[0,o6[,there exists a unique stationary state for anyo3[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 existsvalues ofo3R
`such that (¹11#¹13)!o(¹21#¹23)50,multiple solutions can no 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 detailedexpose&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
15It is understood that;
ij"L2;/LiLj,i,j"1, 2, 3, is evaluated at (xH, 0,xH).
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 ;
ij" ;ij(xH, 0,xH),i,j"1, 2, 3 and R the spectrum of the characteristic
polynomial:
Lemma 2.Under Assumptions P.1, T.1, PT.1}4, letk3R, there then exists ak@3R
such thatR(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#kto any rootk3R, remains valid in the present setting for external e!ects such that
;
3O0,;13O0 but;23"0. At this stage, a rough sketch of the indeterminacy
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 wheno!(;
23/;22)(0.
A benchmark structure with a dimension for the stable manifold equal to the number of state variables is stated as follows:
Dexnition 3.Letsdenote an equilibrium trajectory for a givenx(0)"x
0and
consider a steady state (xH, 0,xH), for xH"XH and with R"Mk,k@N. It is a regular saddlepoint ifkk@(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(;
11#o;21)#(;13#o;23)(0.
Proof. See Appendix A.3. h
Hence, the saddlepoint condition of an optimal growth environment, i.e.,
;
11#o;21(0, appears as a mere part of the condition that pertains to
a competitive equilibrium with externalities. Let us now turn to the way this relates to the uniqueness concerns mentioned earlier in this paper:
16The requesite (iii) in Proposition 4 is"lled whenDo(0.
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 steady-state position:
Dexnition 4. Let s denote an equilibrium trajectory for a given
x(0)"x
0,x03[0,x6] and with a steady-state position of (xH, 0,xH),xH"XH. It
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) ;
23(0holds,
(ii) ;
23/;22'oholds,
(iii) o;
21#;13'D;11#o;23D.
Proof.See Appendix A.4. h
The scope for indeterminacies is related to the occurrence of;
23(0. On the
contrary, any sign for ;
13 may a priori leave room for indeterminacies.16
A more detailed examination of the properties on preferences and sectoral production technologies which underlie the signs of ;
13 and;23 is given in
Section 5.
Remark 2. Benhabib and Farmer (1996) and Benhabib and Nishimura (1998) considersector-specixc externalities. On a technical basis, internal factors of the production technologies are duplicated by external ones as Fj(xj,li,x6j,lMj)"
(xj)aj(lj)bj(x6j)aj(lMj)bj, witha
j#bj#aj#bj"1. The constant returns property
17See Guckenheimer and Holmes (1986, p. 151) for an exposition.
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 C5parameterised byo, there exists a uniqueoHsuch that o![;
23(o)/;22(o)]~0 foro~oHand the eigenvalues
remain complex in an open neighbourhood de"ned aroundoH.
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 bifurca-tion Theorem for#ows17inR2, a new type of attractor, i.e., closed curves around the steady state, appears:
Proposition 5. Under AssumptionsP.1, T.1, PT.1}5,a Poincare&}Hopf bifurcation occurs ato"oHand 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)]/doDo/oH
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. Theirendogenous cyclesresult is based upon a two-capital goods envi-ronment and an asymmetric structure for the matrix ;
12, which is out of
purpose when;
12is a scalar. In opposition to this, the current argument hinges
on an environment augmented by external e!ects but with a unique capital good.
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 functionu()) and a production possibility frontier¹(),),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 ;
13 and;23are decisive in the area for indeterminacies. Immediate
computations deliver, ford"0:
;
13"uA¹3¹1#u@¹13, (5a)
;
23"uA¹3¹2#u@¹23. (5b)
A thorough understanding of the indeterminacy issue is obtained by directly building on the properties of theexternalities-augmentedproduction 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
13"F011(Lx0/LX)#F012(Ll0/LX)#F013, (6a) ¹
23" !¹
13F11#[F111(Lx1/LX)#F112(Ll1/LX)#F113]F01
(F11)2 , (6b)
for
(Lx0/LX)"!(Lx1/LX), (7a)
(Ll0/LX)"!(Ll1/LX), (7b)
(Lx0/LX)"[F13!F12(Ll0/LX)/F11]. (7c) A noticeable clari"cation emerges:
Lemma 3. Foro3RH
`andd"0,the indeterminacy condition(iii) of Proposition 4
no longer depends on¹
13.
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.,;
23. Such an explanation is directly related to intersectoral arbitrages:
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) Fj3(xj,lj,X)50, (ii) Fj13(xj,lj,X)50 and (iii) Fj23(xj,lj,X)50 in sectors j"0, 1. Then and from the role of ;
23(0 in the indeterminacy
outcome of Proposition 4, the expression of ;
23 and the concavity ofu()), it
becomes clear that the occurrence of;
23(0 requires the parallel holding of ¹
23(0.
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. LetFj(),),)),j"0, 1,be homogeneous of degreeaj'1.The deriv a-tive ¹
23 is strictly negative at the steady-state if the following conditions are satisxed:
(i) Ll0/LX(0.
(ii) F013!F113(F01/F11)'0,
(iii) F013!F113(F10/F11)#(F13/F11)[F011#F111(F01/F11)]50.
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. Condi-tion (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
positiveexternal e!ects in the consumption good sector clearly favour indeter-minacy whereas similar e!ects in the investment good sector have opposite implications.
6. Global indeterminacies:An illustration
18See Liviatan and Samuelson (1969) for a related argument in an environment with a unique homogeneous good and wealth e!ects in the utility function.
19See Drugeon and Venditti (1996).
20The de"nition of the stationary values of the captial stock remain unaltered for anyg. 21From a formal point of view, the conclusion obtained signi"es that it is necessary to establish the nullity of all the coe$cients associated with terms of order greater than or equal to 2 in the
normal formof the dynamical system. See Guckenheimer and Holmes (1986) and Chenciner (1981).
(1970).18In order to clarify the exposition, the more technical details have been relegated to the working paper version available as Drugeon and Venditti (1996).
Consider the following speci"cation for the indirect utility function of the Sutherland (1970) type:
;(x,x5,X)"!7x2!12xx5!38x52#6xX!x5X
#(18/3)x3!(9/4)x4#(g/4)x54, (8) withga parameter whose sign is discussed later on. It satis"es, at least locally, the whole range of properties under which the previous analysis has been performed. It is proved19that;
1'0,;2(0,;11(0,;22(0,;12(0 hold
in a neighbourhood of a steady position for whichx5"0.
Consider now the implications of the externality. First, note that
;
3"6x!x5. The focus is on positive externalities around stationary states.
Also, note that;
13"6'0 and;23"!1(0: from Proposition 4(i),
indeter-minacies cannot a priori be ruled out. Lettingy"x5, the dynamical system is derived as
x5"y; (9a)
y5" 1
3gy2!76[(!8!13o#18x!9x2)x!(76o!goy2!1)y]. (9b) Hence a"rst stationary position xHlocated at the origin. The set of interior steady-states is given by the values ofx'0 which solve a quadratic form, or
x2!2x#((8#13o)/9)"0. This leads to a pair of candidate solutions for the capital stock de"ned fromxHH,HHH"1$[1!(8#13o)/9][email protected] 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
HH"1/76. However two con"gurations are to be distinguished
22Note that, had lower values ofgbeen considered, the real part of the coe$cient would have been signi"cantly weaker and the orbit would have only been weakly attracting. In order to reach a clear graphical illustration of the preceding theoretical assessement, an admittedly high value of ghas been retained. For more reasonable values ofg, similar conclusion hold, but the convergence speed towards the periodic orbit is limited while its basin of attraction becomes smaller.
23The stability properties of the steady state are entirely derived from the non-linear component of the system,xHHbeing avague attractorifg'0.
or hamiltonian, a detailed examination of such a system is available in Mat-suyama (1991). The subsequent argument assumes the opposite and focuses on the stability properties of closed orbits in the generic case.
Consider the non-linear dynamical system (9) foro
HH"1/76. As assessed in
Proposition 5, the essential di$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 steady-state 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), anormal formof the dynamical system emerges as
z5"0)22375iz!z2z6[0.0000032499g#(0.198468!0.000018422g)i]
#O(DzD4) (10)
withzas thecomplexixcationof the expression of the variablesxandyin the coordinates of the eigenspace. The stability properties of the periodic orbit result from the sign of the real part of the coe$cient associated with the cubic term. Here, the cubic component is summarised byz2z6 and this coe$cient is such that its real part isRe"!0.0000032499g. From Guckenheimer and Holmes (1986, pp. 151}152), the PoincareH }Hopf bifurcation is supercritical * the periodic orbit is a limit cycle*forg'0 and subcritical forg(0. In a particular case, forg"0, one recovers the degenerated example*a critical bifurcation* pre-viously mentioned.
All these results are con"rmed by numerical simulations completed from the fully non-linear dynamical system and forg"1000.22 Firstly, considering the stability properties of the steady statexHHo , from Fig. 2a, it reveals that the steady state is strongly attractive foro"1/100. In opposition to this, for the critical bifurcation value of oHH"1/76, attraction towards the steady state become much weaker in Fig. 2b.23 In this latter con"guration, a supercritical Poin-careH }Hopf bifurcation emerges.
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 andy(0)"0.001); (c) Stable periodic orbit (&from the outside') (Discount factor: o"1/73; Initial conditions: x(0)"1.295 andy(0)"0.001); (d) Stable periodic orbit (&from the inside') (Discount factor: o"1/73; Initial conditions: x(0)"0.696 andy(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 andy(0)"0.001).
latter. Secondly, in Fig. 2d, the initial conditions lie within the closed curve. As expected,xHHis locally repulsive where the periodic orbit is attractive.
A further interesting phenomenon occurs for (o"1/68) and a steady-state of
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
;
or, recalling the de"nition of¹
1and¹2in terms of the fundamentals:
F01[x0(x,dx,x),l0(x,dx,x),x]
!(o#d)F01[x0(x,dx,x),l0(x,dx,x),x]
F11[x1(x,dx,x),l1(x,dx,x),x]"0. Finally simplifying in the case of an interior solution:
F11[x1(x,dx,x),l1(x,dx,x),x]!(o#d)"0. h
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
A.3. Proofs of Lemma 2 and Proposition 3
23)]. It is "rst of some use to provide a clari"cation about the actual
possibility of complex eigenvalues:
Lemma A.1. Under Assumptions P.1, T.1, PT.1}4, the eigenvalues of the character-istic polynomial are real in a Model M1,X,X,;,oN if one of the following
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
the statement follows. For (iii), remark that the expression of Do may be rearranged as
As detailed in the preceding derivation, had externalities been omitted from the analysis, i.e., for;
24This line of argument is borrowed from Venditti (1998).
have backed. Under the strict concavity restriction of Assumption PT.2,
D(o)'0.24The statement follows from noticing that (;
23)2'0. h
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!(;
23/;22). Hence, ifk3R, P[!k#o!(;23/;22)]"0 and the statement
follows. The proof of Proposition 3, as for itself, is immediate from the expres-sion 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 (;
11#o;21)#(;13#o;23)50, having integrated (i). h 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 givenX. 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],
withy"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
1(x,y,X)"F01[x0(x,y,X),l0(x,y,X),X]
"q(x,y,X)F11[x1(x,y,X),l1(x,y,X),X]
"u(x,y,X) ,
¹
2(x,y,X)"!
F0
j [x0(x,y,X),l0(x,y,X),X]
F1
j[x1(x,y,X),l1(x,y,X),X]
foru(),),X) andq(),),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
3(x,y,X)"F03[x0(x,y,X),l0(x,y,X),X]
#q(x,y,X)F13[x1(x,y,X),l1(x,y,X),X]. The remaining issue has then to do with the determinants of¹
13and¹23. From
the above, they, respectively, back to ¹
13"Lu/LX and ¹23"!Lq/LX and
their full expression appears in the main text. Restate Proposition 1(iii):
23expresses as a function of¹13along ¹
23" !¹
13F11#[F111(Lx1/LX)#F112(Ll1/LX)#F113]F01
(F11)2 ,
and that the expression of the discount rate reduces to
o"!;
at the steady-state, condition (iii) in Proposition 4 reformulates as
A.6. Proof of Proposition 6
The derivation of the second equation from the"rst hinges on the use of the equilibrium constraint, or Lx0/LX"[F13!F12(Ll0/LX)/F11] as well as on the de"nition of the relative price of the investment good, or F12/F11"F02/F01and the homogeneity of degree one ofF0(),),X). The simpli"ed expression in the third equation results from the nullity of the determinant of the Hessian matrix ofF0(),),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
23. Along the same approach,
incorporating the third equation in the expression of¹
13and letting nowa1'1
Under the positive signs for Fj12,j"0, 1, that result from the homogeneity assumption T.1 andpositiveexternal e!ects along De"nition 6, the details of the statement follow. h
References
Benhabib, J., Farmer, R., 1994. Indeterminacy and growth. Journal of Economic theory 63, 19}41. Benhabib, J., Farmer, R., 1996. Indeterminacy and sector-speci"c externalities. Journal of Monetary
economics 37, 421}443.
Benhabib, J., Nishimura, K., 1979a. The Hopf bifurcation and the existence and stability of closed orbits in multisector models of optimal economic growth. Journal of Economic Theory 21, 421}444.
Benhabib, J., Nishimura, K., 1979b. On the uniqueness of steady states in an economy with heterogeneous capital goods. International Economic Review 20, 59}82.
Benhabib, J., Nishimura, K., 1998. Indeterminacy and sunspots with constant returns. Journal of Economic Theory 81, 58}96.
Blot, J., Cartigny, P., 1995. Bounded solutions and oscillations of concave Lagrangian systems in presence of discount rate. Journal of Mathematical Analysis and its Applications 14, 731}750. Boldrin, M., 1989. Paths of optimal accumulation in two-sector models. In: Barnett, W., Geweke, J., Shell, K. (Eds.), Economic complexity: Chaos, Sunspots, Bubbles and Nonlinearity. Cambridge University Press, Cambridge.
Boldrin, M., Rustichini, A., 1994. Growth and indeterminacy in dynamic models with externalities. Econometrica 62, 323}342.
Brock, W., 1973. Some results on the uniqueness of steady states in multisector models of optimum economic growth when future utilities are discounted. International Economic Review 14, 535}559.
Cazavillan, G., Lloyd-Braga, T., Pintus, P., 1998. Multiple steady states and endogenous# uctu-ations with increasing returns to scale in production. Journal of Economic Theory 80, 60}107. Chenciner, A., 1981. Bifurcations de di!eHomorphismes deR2au voisinage d'un point"xe elliptique. In: Helleman, R., Iooss, G., Stora, R. (Eds.), Comportements chaotiques de syste`mes deH ter-ministes, Les Houches, session XXXVI. North-Holland, Amsterdam, pp. 273}348.
Drugeon, J.-P., Venditti, A., 1996. On externalities, indeterminacies & homothetic growth paths in a Canonical model of capital accumulation. G.r.e.q.a.m. Working paper 96A, 40.
Guckenheimer, J., Holmes, P., 1986. Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Second Printing, Revised and Corrected. Springer, New York.
Hirota, M., Kuga, K., 1971. On an intrinsic joint production. International Economic Review 12, 87}98.
Kehoe, T., 1991. Computation and multiplicity of equlibria. In: Werner H., Sonnenschein, H. (Eds.), Handbook of Mathematical Economics, Vol. IV, North-Holland, Amsterdam pp. 2049}2145. Kehoe, T., Levine, D., Romer, P., 1991. On characterizing equlibria of economies with externalities
and taxes as solutions to optimization problems. Economic Theory 2, 43}68. Kuga, K., 1972. The factor-price equalization theorem. Econometrica 40, 723}736.
Levhari, D., Liviatan, N., 1972. On stability in the saddle point sense. Journal of Economic Theory 4, 88}93.
Liviatan, N., Samuelson, P., 1969. Notes on Turnpikes : stable and unstable. Journal of Economic Theory 1, 454}475.
Madden, P., 1986. Concavity and Optimization in Microeconomics, Basil Blackwell, Oxford. Matsuyama, K., 1991. Increasing returns, industrialization and indeterminacy of equilibrium.
Quarterly Journal of Economics 106, 617}650.
Milnor, J., 1965. Topology from the Di!erentiable Viewpoint. The University Press of Virginia, Charlottesville.
Mitra, T., 1995. On Equilibrium dynamics under externalities in a model of economic development. Manuscript, Cornell University.
Sutherland, W.R., 1970. On optimal development in a multi-sectoral economy: the discounted case. Review of Economic Studies 46, 585}589.
Venditti, A., 1997. Strong concavity properties of indirect utility functions in multisector optimal growth models. Journal of Economic Theory 74, 349}367.
