Multiple Steady States and Endogenous Fluctuations with Increasing Returns to Scale in Production

Journal of Economic Theory  ET2384 journal of economic theory 80, 60107 (1998) article no. ET972384

Multiple Steady States and Endogenous Fluctuations with Increasing Returns to Scale in Production* Guido Cazzavillan Department of Economics, University of Venice, Ca'Foscari, Italy

Teresa Lloyd-Braga  Department of Economics, Universidade Catolica Portuguesa, Lisbon, Portugal

and Patrick A. Pintus 9 THEMA, Department of Economics, University of Cergy-Pontoise, 95011 Cergy-Pontoise Cedex, France Received September 9, 1997; revised November 17, 1997

The purpose of this paper is to study the influence of the elasticity of capitallabor substitution and of the labor supply elasticity on the existence of multiple Paretoranked stationary equilibria, local indeterminacy, bifurcations, and expectations-driven fluctuations in economies with external and internal increasing returns to scale in production. This is done through general geometrical methods that do not depend upon particular specifications for preferences and technology. For the sake of concreteness, the analysis focuses on the model with heterogeneous agents and financial constraints presented in Woodford (J. Econ. Theory 40 (1986), 128137) and extended by Grandmont, Pintus, and de Vilder (``CapitalLabor Substitution and Competitive Nonlinear Endogenous Business Cycles,'' CREST Working Paper 9728, 1997) to account for capitallabor substitution. Increasing returns to scale are incorporated in two different market structures: * The authors thank Jean-Michel Grandmont very much for valuable comments, enlightening remarks, and constant encouragement. Special grateful thanks to Robin G. de Vilder and Duncan J. Sands for comments and for helping with the computer work. E-mail: guidounive.it. Part of the work of the author was done at CEPREMAP. Financial support from the CNRS is gratefully acknowledged.  E-mail: tlbeuropa.fcee.ucp.pt. Part of the work of the author was done at CEPREMAP. Financial support from the CNRS is gratefully acknowledged. 9 E-mail: pintusu-cergy.fr. Part of the work of the author was done at CEPREMAP and at the University of Venice. Financial support from the CNRS and the University of Venice is gratefully acknowledged.

60 0022-053198 25.00 Copyright  1998 by Academic Press All rights of reproduction in any form reserved.

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 an economy with aggregate production externalities and perfectly competitive markets,  an economy with increasing returns internal to the firm and imperfect competition in the product(s) market(s) only. Journal of Economic Literature Classification Numbers: C61, E32.  1998 Academic Press

1. INTRODUCTION In this paper, we study the influence of the elasticity of capitallabor substitution and of labor supply elasticity with respect to real wages on the existence of multiple Pareto-ranked stationary equilibria, local indeterminacy, bifurcations, and expectations-driven cycles in economies with internal and external increasing returns to scale in production. Many recent contributions which show the possibility of indeterminacy and endogenous fluctuations through the introduction of either external increasing returns (see, among others, Benhabib and Farmer [5], Boldrin [9], Cazzavillan [12], Farmer and Guo [18], Matsuyama [32]) or internal increasing returns and imperfect competition (see, among others, Benhabib and Farmer [5], Farmer and Guo [18], Kiyotaki [30] for models with capitallabor substitution; d'Aspremont et al. [2], Rivard [37] for models without productive capital) rely on particular specifications for preferences and technology. In particular, the extensive and almost exclusive use of the unit factor substitution elasticity case does not provide enough information to evaluate the sensitivity of those findings when the class of technologies is more general (see Rotemberg and Woodford [38]). This may appear a limit once it is recognized that the reported estimates of the elasticity of factor substitution are not conclusive as they actually fall within a quite large range of values. 1 In contrast to the existing literature, we use simple qualitative geometric methods, adapted from Grandmont et al. [22], which allow full characterization of the behavior of the orbits around a steady state as a function of the elasticities of capitallabor substitution and of labor supply, without particular specifications for preferences and technology. When increasing returns are small, we find that local indeterminacy, periodic or quasi-periodic endogenous fluctuations, and stochastic equilibria occur when factors are highly complementary, whereas local determinacy is bound to prevail when the elasticity of capitallabor substitution is slightly larger, as in models based on constant returns (see, for example, Benhabib and Laroque [7], Grandmont [19, 21], Reichlin [36], de Vilder [44, 45], Grandmont et al. [22], Woodford [48], and also Farmer [17]). More surprising, however, 1 See, for instance, Hamermesh [26, Chap. 3] for a survey of direct and indirect estimates, Epstein and Denny [16] for estimates arising from a labor and capital dynamic adjustment model.

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is the fact that when the elasticity of input substitution is increased further and made arbitrarily large, local indeterminacy and endogenous fluctuations reappear (in agreement with the unit elasticity examples reviewed above). Moreover, the higher the elasticity of factor substitution, the lower the labor supply elasticity required to get endogenous cycles (the same sort of results were obtained through computational methods in CES specifications and in the context of nonmonetary overlapping generations models with internal increasing returns, productive capital, and imperfect competition by Lloyd Braga [31, part of which was circulated in 1994]). We also provide a thorough analysis of global uniqueness versus multiplicity of stationary equilibria that does not depend on specific production and utility functions (in contrast with d'Aspremont et al. [2], Benhabib and Farmer [5], Benhabib and Rustichini [8], Boldrin [9], Cazzavillan [12], Kiyotaki [30], Rivard [37], Rotemberg and Woodford [39, 40]), and also Weil [46] and Silvestre [42, 43] for an overview). It is shown that the steady state is bound to be unique when increasing returns are sufficiently high, while multiple Pareto-ranked stationary equilibria typically occur otherwise. The main framework we investigate is the model with heterogeneous agents and financial constraints studied by Woodford [48] and, more recently, by Grandmont et al. [22] to account for capitallabor substitution, where the length of the period can be interpreted as short. We introduce increasing returns in two different market structures:  an economy with aggregate production externalities and perfectly competitive markets, 2  an economy with increasing returns internal to the firm and imperfect competition in the product market only. The way we introduce productive externalities is close to that proposed in Baxter and King [4], Benhabib and Farmer [5], Farmer and Guo [18], and King et al. [29]. According to this view, the private total factor productivity is positively affected by an increase in the aggregate capital and labor stocks. At least two informal arguments exist to legitimate such a formulation. The first accounts for the contribution of aggregate capital to externalities and is now standard: positive externalities come from learning spillovers created through the production activities (see Arrow [1]). The accumulation of productive capital increases public knowledge and hence factor productivities. The same sort of argument may be used to justify part of the positive externality coming from the aggregate labor stock through 2 In Cazzavillan and Pintus [13], the dynamics arising in OLG economies with current consumption, endogenous labor supply, and increasing returns through aggregate externalities are analyzed by applying the very same geometric approach.

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learning by doing, associated to the level of aggregate production rather than the aggregate capital stock. A second justification for labor externalities is related to the exchange process in a market economy, and in particular to the thick market externalities modeled in Diamond [14]. As in Murphy et al. [33] or Howitt and McAfee [27], we can interpret the externalities as the result of the following mechanism, taking place in the labor market. 3 The higher the level of employment, the higher the probability of a fast matching between sellers and buyers in the labor market, in which case the search costs decrease or equivalently the total factor productivity increases. When dealing with short run fluctuations, thick market externalities may appear more relevant than those originated through learning by doing, so we should expect externalities coming from labor to be more important than capital externalities. This is precisely the kind of condition we shall use to generate indeterminacy and endogenous fluctuations for values of the elasticity of capitallabor substitution which belong to the reported range of estimated values, when labor supply is sufficiently elastic. The rest of the paper is organized as follows. The general framework and the characterization of perfect foresight competitive equilibria with aggregate productive externalities are presented in the next section, whereas Section 3 focuses on the steady state analysis. The local dynamics of the first model are analyzed in Section 4, while Section 5 reports the results obtained from the simulations performed on the class of the CES economies with externalities. The economy with an imperfectly competitive product market is introduced and studied in Section 6. The main conclusions are summarized in Section 7.

2. THE INFINITE HORIZON AGENTS MODEL WITH PRODUCTIVE EXTERNALITIES In this section we present the benchmark model, following the lines set out in Woodford [48] and Grandmont et al. [22]. The economy consists of two classes of agents who trade competitively, consume, and have perfect foresight during their infinite lifetime. On one side, identical agents called workers consume and work during each period. They supply a variable quantity of labor hours and may save a fraction of their income by holding two assets: productive capital and nominal outside money. Rental returns on capital and wage are paid by the firms at the end of the period while workers buy the unique consumption good at the outset of it. In addition, a labor market imperfection, due to 3 The argument is easily transposed to the output market and it actually constitutes the foundation of the search approach in which multiple equilibria and endogenous fluctuations have been shown to occur by those authors.

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incomplete or imperfect information, prevents workers from borrowing against their human capital, while productive capital is accepted as collateral to secure a loan. Therefore, a financial constraint is imposed on workers. Indeed, they cannot consume out of their money balances held at outset of the period or out of the returns earned on productive capital. On the other side, identical capitalists run the production process, consume, and save an income composed of money balances and returns on capital. Here an important assumption is made: capitalists discount future utility less than workers. Therefore, at the nonautarkic steady states and nearby, capitalists end up holding the whole capital stock. The resulting nonautarkic steady states are characterized by the modified golden rule (with constant population): the stationary real rental rate on productive capital is equal to the capitalists' discount rate plus the constant depreciation rate for capital. Therefore, at these steady states and nearby, the real return on capital is positive and larger than that of money balances which is close to zero, as long as there is positive discounting andor depreciation: capitalists choose not to hold outside money. To summarize, the steady state (and nearby) savings structure is the following: capitalists own the whole capital stock and workers hold the entire nominal money stock, assumed to be constant over time. Finally, the financial constraint faced by workers becomes a liquidity constraint which is obviously binding at the steady states and nearby. In that framework, Woodford [48] showed that although workers have an infinite lifetime, they behave like two-period-living agents: they choose optimally their labor supply for today and consequently their next period consumption demand. Equivalently, workers know the current nominal wage and the next period price for the consumption good (along an intertemporal equilibrium with perfect foresight) and choose the (unique under usual assumptions) optimal bundle of labor today and consumption tomorrow on their offer curve. Therefore, the liquidity constraint allows one to interpret the length of the period as a quarter or a year, and eventual endogenous fluctuations occur at business cycle frequency. The reminder of this section gives the equations which describe the economy. 2.1. The Technology with Externalities In each period t # N, a consumption good is produced combining labor l t 0 and the capital stock k t&1 0 resulting from the previous period. The gross production function F exhibits constant returns to scale and allows for input substitution. We assume that each entrepreneur-capitalist benefits from positive productive externalities: while he rents l and k, the input services of labor and capital are respectively A(K, L) l and A(K, L) k, where K and L are respectively the average capital stock and the average number of

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worked hours of the economy for the current period. 4 Moreover, when the average capital or labor stock goes up, the productivity of each input moves up: A(K, L) is increasing in K and increasing in L, while the case where A is constant is obviously equivalent to the absence of positive externalities. Because we assume an external effect compatible with perfect competition in all markets, each (small enough) identical entrepreneur takes K and L as given when optimizing. Of course, K=k and L=l in equilibrium. Therefore, the quantity of good produced is F(A(k, l ) k, A(k, l ) l )=A(k, l ) F(k, l )=A(k, l ) lf (a).

(1)

The second and third equalities are deduced from the constant returns to scale assumption, and the latter equality defines the standard production function in intensive form defined upon the capital labor ratio a=kl. On technology, we shall assume the following. Assumption 2.1. The intensive production function f (a) is continuous for a0, C r for a>0 and r large enough, with f $(a)>0 and f "(a)<0. def def Therefore, \(a) = f $(a) is decreasing in a, while |(a) = f (a)&af $(a) is increasing in a. The total factor productivity function A(k, l ) is continuous on R 2+ , C r on R 2++ for r large enough, homogeneous of degree & and can be written, therefore, as A(k, l )=Al &(a), where the scaling factor A>0. Moreover, A(k, l ) is increasing in k, i.e. the contribution of capital to the externalities is = (a)>0, and increasing in l, i.e. the contribution of labor to the def externalities is &&= (a)>0, where = (a) = a$(a)(a). We can interpret the contribution of aggregate capital to externalities, measured by = (a), as coming from a learning by doing process, while the contribution of aggregate labor, i.e. &&= (a), may be resulting from learning by doing andor thick market externalities. Resulting from these two sources of economies of scale, social returns to scale are equal to 1+& and are increasing, while private returns to scale are constant: the assumption of perfectly competitive markets can be maintained. Firms take real rental prices of capital and labor as given and determine their input demands by equating the private marginal productivity of each input to its real price. Accordingly, the real competitive equilibrium wage is def

0=A(k, l) |(a)=A(ka) & (a) |(a) = 0(a, k),

(2)

4 For notational convenience, we shall assume that the number of capitalists is equal to that of workers.

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while the real competitive gross return on capital is def

R=A(k, l ) \(a)+1&$=A(ka) & (a) \(a)+1&$ = R(a, k),

(3)

where 0$1 is the constant depreciation rate. Since there are increasing returns to scale, i.e., &>0, the wage and the return on capital depend not only on the capitallabor ratio a but also on the scale of production, as expressed by the term k & in Eqs. (2) and (3). 2.2. The Agents and the Intertemporal Equilibria To complete the description of the model, we must characterize the behavior of both classes of agents.5 A representative worker solves the utility optimization problem maximize[V 2(c wt+1 B)&V 1(l t )]

such that p t+1 c wt+1 =w t l t , c wt+1 0, l t 0,

(4)

where B>0 is a scaling factor, c wt+1 is the next period consumption, l t is the labor supply, p t+1 >0 is the next period price of consumption, assumed to be perfectly foreseen, and w t >0 is the nominal wage rate, i.e., w t = p t 0 t . We only consider the case where leisure and consumption are gross substitutes and assume therefore the following. Assumption 2.2. The utility functions V 1(l) and V 2(c) are continuous for 0ll* and c0, where l *>0 is the (possibly infinite) workers' endowment of labor. They are C r for, respectively, 00, and r large enough, with V$1(l )>0, V"1(l )>0, lim l  l * V$1(l )=+, and V$2(c)>0, V"2(c)<0. Moreover, consumption and leisure are gross substitutes, i.e., &cV"2(c)0 and consequently the next period consumption c wt+1 >0, from v 1(l t )=v 2(c wt+1 ) def

and

p t+1 c wt+1 =w t l t ,

def

(5)

where v 1(l ) = lV$1(l ) and v 2(c w ) = cV$2(cB)B. Assumption 2.2 implies that v 1 and v 2 are increasing while v 1 is onto R + . Therefore, Assumption 2.2 def allows one to define from Eqs. (5) the function # = v &1 b v 1 , whose graph 2 is the offer curve, i.e., c wt+1 =#(l t ). Moreover, # is defined on a nonempty open interval, and leisure and consumption are gross substitutes, i.e., = #(l)>1. 5

See Woodford [48] and Grandmont, Pintus, and de Vilder [22] for more details.

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Equivalently, the labor supply is an increasing function of the real wage, def whose elasticity is = l (0 t p t p t+1 ) = 1(= #(l t )&1)>0. 6 Capitalists maximize the discounted sum of utilities derived from each period of consumption. They consume c ct 0 and save k t 0 from their income, i.e., the real gross return on capital. To fix ideas, we assume, following Woodford [48], that the capitalists' instantaneous utility function is logarithmic. Their optimal choices are then c ct =(1&;) R t k t&1 ,

k t =;R t k t&1 ,

(6)

where 0<;<1 is the capitalists' discount factor and R t >0 is the real gross rate of return on capital. As usual, equilibrium in capital and labor markets is ensured through Eqs. (2) and (3). Since workers save wage income in the form of money, the equilibrium money market condition is 0(a t , k t&1 ) l t =M p t ,

(7)

where M>0 is the constant money supply and p t >0 is the current nominal price of consumption. Finally, Walras' law accounts for the equilibrium in the good market, i.e., A(k t&1 , l t ) l t f (a t )=c wt +c ct +k t &(1&$) k t&1 . From the equilibrium conditions in Eqs. (2), (3), (5)(7), one easily deduces that the variables c wt+1 , c wt , l t , p t+1 , p t , c ct , and k t are known once (a t , k t&1 ) are given. This implies that intertemporal equilibria are actually summarized by the dynamic behavior of the two variables a and k. Definition 2.1. An intertemporal perfectly competitive equilibrium with perfect foresight its a sequence (a t , k t&1 ) of R 2++ , t=1, 2, ..., such that

{

0(a t+1 , k t )a t+1 =#(k t&1 a t )k t , k t =;R(a t , k t&1 ) k t&1 .

(8)

3. STEADY STATES ANALYSIS: UNIQUENESS AND MULTIPLICITY The first step in analyzing the dynamical system (implicitly) given by Eqs. (8) consists in finding its stationary points. Existence is easily established by scaling the two parameters A and B so as to ensure the proper boundary conditions and to normalize the steady state values of capital and labor. To study globally uniqueness and multiplicity, we shall first reduce the dimension of the problem to one. The general result states that a unique 6

See Grandmont, Pintus, and de Vilder [22, Sect. 2] for more details.

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steady state is bound to prevail whenever externalities are uniformly small or uniformly large, while in the intermediate range, multiple Pareto-ranked stationary equilibria may occur and may be arbitrarily close. The analysis is then applied to the standard parametric family of the CES economies. It turns out, in agreement with the general result, that the steady state is bound to be unique when externalities are sufficiently large. If externalities are very low, however, two Pareto-ranked steady states are shown to coexist generically, maybe in a very small neighborhood. In particular, the result of uniqueness obtained in the case of the Cobb Douglas technology, for which the elasticity of capitallabor substitution is equal to one, is actually a very particular configuration and indeed not a stable property: if externalities are low enough, uniqueness of the steady state does not persist after any arbitrary small perturbation, i.e., if the elasticity is as close as we want to one. On the contrary, two stationary equilibria generically co-exist when the elasticity of factor substitution is different from, but infinitely close to one. In view of Eqs. (8) and of the definition a=kl, the nonautarkic steady states are the solutions (a, l ) in R 2++ of 0(a, al )=#(l )l and ;R(a, al )=1. Equivalently, in view of Eqs. (2) and (3), the steady states are given by

{

Al 1+&(a ) |(a )=#(l ), Al &(a ) \(a )+1&$=1;.

(9)

We shall solve the existence issue by choosing appropriately the scaling parameters A and B, so as to ensure that one stationary solution coincides with, for instance, (a, l )=(1, 1). This will occur if and only if A(1) \(1)=1;&1+$ and A(1) |(1)=#(1), or equivalently v 2(A(1) |(1))=v 1(1). The first equality is achieved by scaling the parameter A, while the second is achieved by scaling the parameter B. Indeed, from Assumption 2.2, v 2 is decreasing in B, and the latter equation will be satisfied for some (then unique) B>0 if and only if limc  0 v 2(c)
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can write the two-dimensional system (9) in the form l=l 1(a), l=l 2(a), where the two functions l 1(a), l 2(a) are defined and positive on the two open intervals I 1 and I 2 , respectively. Finding a steady state is then equivalent to solving the much simpler one-dimensional equation l 1(a )l 2(a )=1, where l 1(a)l 2(a) is defined on the intersection I of I 1 and I 2 . At this stage, the interval I may be empty, but we know that the assumptions of Proposition 3.1 ensure that it contains a =1 (i.e. l 1(1)=l 2(1)). We shall focus on two configurations, essentially because they are those which arise in the specification analyzed in Subsection 3.2, where the production and the utility functions exhibit constant elasticities. First of all, it is clear that if l 1(a)l 2(a) is monotone (either increasing or decreasing everywhere), there exists at most one steady state (hence unique under the assumptions of Proposition 3.1). On the other hand, if l 1(a)l 2(a) is either single-peaked (see Fig. 1) or single-caved (see Fig. 2), there are at most two steady states. Under the assumptions of Proposition 3.1, there exist then exactly two steady states under appropriate boundary conditions (namely l 1(a)l 2(a)&1 has the same sign when a is close to the lower and the upper bounds of the interval I), at least generically, i.e., provided that the derivative of l 1(a)l 2(a) does not vanish at a =1. The use of this method requires that we put Eqs. (9) in the form l=l 1(a), l=l 2(a). The function l 1(a) is easily obtained by solving for l the second equation of (9) and it is defined for all a>0, i.e., I 1 =(0, +). The same procedure, however, does not work for the first equation since #(l )l 1+& may not be monotone. Dividing the first equation of (9) through by the second one yields (1;&1+$) |(a)\(a)=#(l)l. Since from Assumptions 2.1 and 2.2 both functions |(a)\(a) and #(l )l are increasing, solving that equation for l yields the second function l 2(a) we sought. It is defined, positive, differentiable, and increasing, i.e., l $2(a)>0, on an open interval I 2 (which contains a =1, under the assumptions of Proposition 3.1). It follows that the multiple stationary equilibria are ordered: if (a 1 , k 1 ) and (a 2 , k 2 ) are solutions of Eqs. (9), then a 1 >a 2 if and only if k 1 >k 2 , as

Fig. 1.

A single-peaked map.

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Fig. 2.

A single-caved map.

k=al. We shall soon see (Proposition 3.5) that this property implies, rather trivially, that multiple stationary equilibria are Pareto-ranked. To study the variation of l 1(a)l 2(a) on the intersection I of I 1 and I 2 , it is convenient to look at its elasticity, i.e., =l1(a)&= l2(a), where =l1(a)=al$1(a)l1(a) and = l2(a)=al $2(a)l 2(a) are the elasticities of l 1(a) and l 2(a). Indeed, the derivative of l 1(a)l 2(a) is positive if and only if = l1(a)&= l2(a)>0 or equivalently = l1(a)= l2(a)>1. Straightforward computations from the two equations defining l 1(a) and l 2(a), i.e., Al &(a) \(a)=1;&1+$ and (1;&1+$) |(a) \(a)=#(l )l, lead to = l1(a)=(|= \(a)| &= (a))&,

= l2(a)=(= |(a)+ |= \(a)| )(= #(l 2(a))&1). (10)

It is perhaps a little more illuminating to restate these expressions in terms of the elasticity of factor substitution _(a) and the capital share in def total income 0
= l2(a)=1(_(a)(= #(l 2(a))&1). (11)

With this notation, the condition = l1(a)&= l2(a)>0 (or = l1(a)= l2(a)>1) is equivalent to (1&s(a)&_(a) = (a))(= #(l 2(a))&1)>&.

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(12)

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One therefore concludes that l 1(a)l 2(a) is increasing and that there exists at most one steady state if we assume (1&s(a)&_(a) = (a))(= #(l )&1)>&

for all a, l>0.

(13)

The above condition is satisfied (as expected in view of the results in Grandmont, Pintus, and de Vilder [22, Sect. 2]) in the limit case where there are no externalities, i.e, when &=0 and = (a)=0 for all a. 7 Accordingly, it will tend to be fulfilled when the externalities are uniformly small, i.e., when & and = (a) are small for all a>0. In the constant elasticity case studied in Subsection 3.2, we shall see that, apart from the CobbDouglas specification, Eq. (13) cannot be met for all a>0 even when =(a), which is now independent of a, is small: this is due to the fact that the capital share s(a) then varies monotonically between zero and one, implying that the left-hand side of Eq. (13) becomes negative for some small or large a's. We also get that l 1(a)l 2(a) is decreasing and that there is at most one steady state in the opposite configuration (1&s(a)&_(a) = (a))(= #(l )&1)<&

for all a, l>0.

(14)

This condition will tend to be satisfied for uniformly large externalities, i.e., either when & only, or both & and = (a), are large for all a>0. This will be in particular the case if &>= #(l )&1 for all l>0, or if _(a) = (a)>1&s(a) for all a>0. Finally we note that when = # , =  , _ and s are constant (which will occur when both the production and the externalities functions are CobbDouglas while the utility functions have constant first derivatives elasticities), one is bound to be in the case described either in Eq. (13) or in Eq. (14), with the exception of the special case where (1&s&_= )(= # &1)=& (in the latter configuration l 1(a)l 2(a) is constant and there is, under the assumptions of Proposition 3.1, a continuum of steady states). The above arguments can be summarized in the following proposition. Proposition 3.2 (Uniqueness of the Steady State). Under Assumptions 2.1 and 2.2, there is at most one steady state, i.e., at most one stationary solution (a, k ) in R 2++ of the dynamical system in Eqs. (8), if one of the following two conditions is satisfied: (i)

(uniformly small externalities) (1&s(a)&_(a) = (a))(= #(l )&1)>&

for all a, l>0,

7 We conclude from Eqs. (11) that the function l 1(a) is then infinitely elastic and, under Assumption 2.1, that the steady state is unique.

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(ii)

(uniformly large externalities) (1&s(a)&_(a) = (a))(= #(l )&1)<&

for all a, l>0.

The steady state (a, k )=(1, 1) is then the unique interior steady state under the assumptions of Proposition 3.1. Condition (ii) is in particular satisfied if _(a) = (a)>1&s(a) for all a>0 or if &>= #(l )&1 for all l>0. In the special case where = # , =  , _ and s are constant, either case (i) or (ii) occurs, except when (1&s&_=  )(= # &1)=&. In the latter configuration, there is, under the assumptions of Proposition 3.1, a continuum of steady states. The above result states that one gets a unique interior steady state when externalities are uniformly small or uniformly large. The question to be addressed, next, is how many stationary equilibria one may get when these conditions are violated. We shall see that there are simple assumptions leading to the configurations described in Figs. 1 and 2, and ensuring that there are at most two steady states. These assumptions arise naturally in the constant elasticity specifications considered in Subsection 3.2. The picture of Fig. 2 emerges when l 1(a)l 2(a) is single-caved, i.e., when = l1(a)&= l2(a) goes through zero (or = l1(a)= l2(a) goes through one) at most once, and from below when it does. In view of Eq. (13) and since l 2(a) is increasing, we get this configuration when 1&s(a)&_(a) = (a) is increasing in a, and = #(l ) is nondecreasing in l. We shall see in Subsection 3.2 that this will indeed arise for small externalities when the constant elasticity of input substitution _ is less than one. The symmetrical picture of Fig. 1, where l 1(a)l 2(a) is single-peaked, is obtained when = l1(a)&= l2(a) goes through zero at most once, and from above when it does. This case will occur if 1&s(a)&_(a) = (a) is decreasing in a and = #(l) is nonincreasing in l. This configuration will arise for small externalities when the constant elasticity of input substitution _ exceeds one. Proposition 3.3 (Uniqueness versus Multiplicity of Steady States). Let Assumptions 2.1 and 2.2 hold. There are at most two steady states, i.e., at most two stationary solutions (a, k ) in R 2++ of the dynamical system in Eqs. (8), if one of the following conditions is satisfied: (i)

1&s(a)&_(a) = (a) is increasing in a and = #(l ) is nondecreasing in l,

(ii)

1&s(a)&_(a) = (a) is decreasing in a and = #(l ) is nonincreasing in l.

The above proposition implies the existence of exactly two steady states, provided that appropriate boundary conditions for l 1(a)l 2(a) are satisfied (as in Figs. 1 and 2), whenever the derivative of l 1(a)l 2(a) does not vanish

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at a =1. In the alternative case where a =1 is a critical point of l 1(a)l 2(a), (1, 1) is the unique steady state (see Fig. 3 for an illustration when l 1(a)l 2(a) is single-caved). This is, however, a nongeneric configuration: uniqueness of the steady state does not persist if we perturb the derivative of l 1(a)l 2(a) slightly at a =1. In particular, suppose that starting from the configuration represented in Fig. 3 (which arises when = l1(1)== l2(1) or equivalently, in view of Eq. (13), def when = #(1)=1+&(1&s(1)&_(1) = (1)) = = #S (1)) we fix the technology, i.e., _(1), s(1), and = (1), and increase = #(1) slightly so that the derivative of l 1(a)l 2(a) at a =1 is slightly positive. There exists then a second stationary equilibrium close and to the left of a =1 (see Fig. 4). In the symmetrical case where we decrease slightly = #(1), starting from the configuration in Fig. 3, another steady state appears close and to the right of a =1. In Section 4 where the local dynamics around a steady state is studied, the phenomenon we just described geometrically is alternatively interpreted as a transcritical bifurcation involving an exchange of stability between two stationary equilibria, when = #(1) is increased and goes through = #S (1). We therefore conclude that if = #(1) is close to but different from def = #S (1) = 1+&(1&s(1)&_(1) = (1))>1, the two steady states which occur under the assumptions of Proposition 3.3 (and appropriate boundary conditions) are infinitely close to each other. To appreciate the plausibility of such a configuration, we may evaluate the elasticity of the offer curve = #(1) or the corresponding labor supply elasticity = l (A(1) |(1)) = 1(= #(1)&1) = (1&s(1)&_(1) = (1))&. For instance, we may set & at 0.2 (& should not be too large in the aggregate according to the estimates of Baxter and King [4], Caballero and Lyons [11], and Hall [25]; see also Basu and Fernald [3], Burnside [10]), s(1) at 0.4, and _(1) at 0.5, 1, or 2 (in agreement with the estimates reported in Hamermesh [26]; see also Rotemberg and Woodford [38]).

Fig. 3.

A non-transversal intersection leading to the unique steady state a =1.

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Fig. 4.

A transversal intersection leading to two steady states 0
Then if externalities are due exclusively to labor, i.e., if = (1)=0, the two steady states will be close to each other if the labor supply elasticity = l (A(1) |(1)) is close to 3 no matter the elasticity of factor substitution _(1). On the other hand, if externalities come exclusively from capital, i.e., if = (1)=0.2, the two steady states will be near each other when = l (A(1) |(1)) is close to 2.5 when _(1)=0.5, to 2 when _(1)=1, and to 1 when _(1)=2. 8 On the other hand, if we admit larger externalities by setting & at 0.6, and if externalities are due exclusively to labor (= (1)=0), the two steady states will be close to each other if the labor supply = l (A(1) |(1)) is close to 1, independent of the value of _(1). If externalities are due to capital only (= (1)=0.6), the two steady states will be close to each other when = l (A(1) |(1)) is close to 0.5 when _(1)=0.5 and to 0 when _(1)=1. 3.2. A Standard Example: the CES Economies We now specialize the above analysis to the benchmark case where all functions involved display constant elasticities. This specification will be the basis for the simulations we shall perform, in particular in Section 5. We shall see that the steady state is unique when externalities are large. For small externalities, the rule is the existence of two steady states, with the exception of the very special case of a CobbDouglas production function (_(a)=1 for all a) where uniqueness happens to prevail. Let f be a CES production function, let A be a CobbDouglas externalities function homogeneous of degree &, and let V 1 and V 2 be two utility functions with constant elasticity of marginal utility. As the elasticity of capitallabor substitution, the intertemporal elasticities of consumption and leisure substitution, and therefore the labor supply elasticity are all constant, we shall call 8

These values of the labor supply elasticity are admittedly high in view of the reported estimates (see, for instance, Killingsworth and Heckman [28] and Pencavel [34]), but not much higher than the usual values employed in business cycles models (see King, Plosser, and Rebelo [29] and Schmitt-Grohe [41]).

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these economies the CES economies. In this case, = l2(a) is constant while = l1(a) is monotone. For the standard CES production function, one has f (a)=

{

(sa &' +1&s) &1' as

if '> &1, '{0, if '=0,

(15)

where 0
(16)

where A>0, &>>0. One verifies by direct inspection that the capital share in total income s(a)=a\(a)f (a) is given here by s(a)=s(s+(1&s) a ' ).

(17)

The real wage 0(a, k) and the real gross return R(a, k) which appear in Eqs. (8) are, in view of Eqs. (2) and (3), accordingly 0(a, k)=A(k, ka)(1&s(a)) f (a), R(a, k)=A(k, ka)(s(a) f (a)a)+1&$.

(18)

Moreover, in the case at hand, one obtains |= \(a)| =(1&s(a))_ and = |(a) =s(a)_. As for the workers, we shall assume V 1(l )=

l 1+:1 , 1+: 1

V 2(cB)=

(cB) 1&:2 , 1&: 2

(19)

where B>0, : 1 >0, 0<: 2 <1. As easily shown, one has #(l )=v &1 b v 1(l ) 2 =Bl #, with #=(1+: 1 )(1&: 2 )>1. The functions V 1 and V 2 satisfy the assumptions of Proposition 3.1, and one can verify that the choices of scaling parameters A=(1;&1+$)s,

B=(1;&1+$)(1&s)s,

(20)

guarantee that (a, k )=(1, 1) is a steady state, whatever the values of # and '. The share of capital in total income s(a) coincides with s if and only if a =1, i.e., in the persistent steady state, and is constant and equal to s when '=0, i.e., in the CobbDouglas case _=1. From Eqs. (11), we derive = l1(a)=(1&s(a)&_)(&_),

= l2(a)=1((#&1) _).

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(21)

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CAZZAVILLAN, LLOYD-BRAGA, AND PINTUS

Therefore, three cases have to be covered, according to s(a) decreasing, constant, or increasing. '>0 (_<1): In this case, s(a) decreases from one to zero (see Eq. (17)). If (1&_)(#&1)<&, the case (ii) of uniformly large externalities in Proposition 3.2 applies, and the steady state (a, k )=(1, 1) is unique. For smaller externalities, i.e., when (1&_)(#&1)>&, case (i) of Proposition 3.3 is obtained and there are at most two steady states. One can verify by direct inspection that the single-caved function l 1(a)l 2(a) goes then to + when a tends to zero or to +. So, there must be exactly two steady states, provided that the derivative of that function does not vanish at a =1, i.e, provided that (1&s&_)(#&1){&. The two steady states coincide with (a, k )=(1, 1) when (1&s&_)(#&1)=&, as shown in Fig. 3. '=0 (_=1): In this case, s(a) is constant and equal to s, and therefore we are in either case (i) or (ii) of Proposition 3.2, and the steady state (a, k )=(1, 1) is unique. In the case where (1&s&)(#&1)=&, there is a continuum of steady states. '<0 (_>1): In this case, s(a) increases from zero to one (see Eq. (17)). If externalities are large, i.e., (1&_)(#&1)<&, the steady state (a, k )=(1, 1) is unique. For smaller externalities, i.e., (1&_)(#&1)>&, the function l1(a)l 2(a) is single-peaked and goes then to zero as a tends to zero or to +. One has therefore exactly two steady states, if (1&s&_)(#&1){&. Here again, the two steady states coincide with (a, k )=(1, 1) when (1&s&_) (#&1)=&, as shown in Fig. 3. The next proposition summarizes the previous discussion about the number of steady states in the CES economies. Proposition 3.4 (Uniqueness and Multiplicity in the CES Economies). Assume (15), (16), (19), and (20). Then the following cases hold. 1. '=0, i.e., _=1: If &{(#&1)(1&s&), then (a, k )=(1, 1) is the unique steady state in R 2++ . 2. '{0, i.e., _{1: If externalities are large, i.e., &>(#&1)(1&_), then (a, k )=(1, 1) is the unique steady state in R 2++ . If externalities are small, i.e., &<(#&1)(1&_), then, generically, i.e., when &{(#&1)(1&s&_), there are two distinct steady states in R 2++ . The two steady states coincide with (a, k )=(1, 1) when &=(1&s&_)(#&1). If we consider the results stated in Proposition 3.4, it appears that the CobbDouglas case _=1 is very peculiar and indeed structurally unstable, not only in this model but, it should be expected, also in most dynamical models with increasing returns: we can find parameters constellations for which uniqueness of the steady state does not persist after any arbitrary

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small C  perturbation. In particular, Proposition 3.4, part 2, tells us that multiplicity is generically bound to occur if _ is (arbitrarily) close to (but different from) one, when &(#&1)+_<1. This inequality will be met if externalities are small (& and  small), and if the labor supply elasticity 1(#&1) is small. Moreover, as shown at the end of Subsection 3.1, the two steady states can be made arbitrarily close to each other if the labor supply elasticity = l (|(1)) is close to (1&s&_)&. As shown there, this configuration is obtained for _ close to 1 when aggregate externalities are small (&=0.2), if = l (|(1)) is close to 3 when externalities are due solely to labor (=  =0) and to 2 when externalities come exclusively from capital (&==  =0.2). 9 3.3. Welfare Analysis of the Multiple Steady States Due to the presence of productive externalities, we expect the First Welfare Theorem to fail and some intertemporal equilibria to Pareto-dominate others. This is indeed proved for the multiple steady states exhibited in the previous subsections. Proposition 3.5 (Pareto-Ranked Steady States). Under the assumptions of Proposition 3.3, let (a 1 , k 1 ) and (a 2 , k 2 ) in R 2++ be two steady states. It follows that a 1 >a 2 if and only if k 1 >k 2 , and therefore that (a 1 , k 1 ) strongly Pareto-dominates (a 2 , k 2 ). Proof. The ordering of the two vectors (a 1 , k 1 ), (a 2 , k 2 ) follows from the fact that l 2(a) is increasing, in view of Assumptions 2.1 and 2.2. From Eqs. (6) and (9), the capitalists' steady state consumption is c c =(1&;) k;. Therefore, capitalists strictly prefer (a 1 , k 1 ) to (a 2 , k 2 ). The fact that workers strictly prefer the highest steady state follows from the observation that the contemporaneous workers' utility function is strictly quasi-concave and from the fact that the slope of the offer curve is positive. As a result, workers strictly prefer (a 1 , k 1 ) to (a 2 , k 2 ), since l 1 >l 2 . It follows that both capitalists and workers strictly prefer the highest steady state, i.e., that (a 1 , k 1 ) strongly Pareto-dominates (a 2 , k 2 ). K Even without any information about the stability of each steady state, Proposition 3.5 shows that although agents have perfect foresight, they face a serious coordination problem, at least if cooperative behavior is ruled out. Even though the steady states are strongly Pareto-ranked, we cannot predict, at this stage, which stationary equilibrium the economy will end up with. However, the next section will bring out the information about the (local) stability property at each fixed point of the map defined by Eqs. (8), 9 Again, these values of the labor supply elasticity are relatively high, but not too high compared to the usual calibrations of business cycles model (see King, Plosser, and Rebelo [29] and Schmitt-Grohe [41]).

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under the assumption of perfect foresight. In particular, it will be shown that agents may select a Pareto-dominated steady state equilibrium.

4. LOCAL DYNAMICS AND BIFURCATION ANALYSIS We wish to study the dynamics of Eqs. (8) around one of its interior stationary points (a, k ). These equations define locally a dynamical system of the form (a t+1 , k t )=G(a t , k t&1 ) if the derivative of 0(a, k)a with respect to a does not vanish at the steady state, or equivalently if = (a )+= |(a ) &1&&{0. Thus, the usual procedure to study the local stability of the steady states is to use the linear map associated to the Jacobian matrix of G, evaluated at the fixed point under study. In view of the HartmanGrobman Theorem, 10 this procedure is indeed valid if the Jacobian at the steady state is invertible and has no eigenvalue on the unit circle. From now on, the def following notation is used: For example, = R, a = (aR(a, k )a)R(a, k ) denotes the elasticity of R(a, k) with respect to a evaluated at the steady state (a, k ) under study. Similarly, = R, k is the elasticity of R(a, k) with respect to k at the steady state. Analogous interpretations hold for = 0, a and = 0, k . Proposition 4.1 (Linearized Dynamics around a Steady State). Let = R, k , = R, a , = 0, k , = 0, a , = # and =  be the elasticities of the functions R(a, k), 0(a, k), #(l ), and (a) evaluated at a steady state (a, k ) of the dynamical system in Eqs. (8), and assume = 0, a {1, i.e., = | +=  {1+&. The linearized dynamics for the deviations da=a&a, dk=k&k are determined by the linear map

{

= += R, a (1+= 0, k ) a = &(1+= 0, k )(1+= R, k ) da t+1 =& # da t + # dk t&1 , = 0, a &1 k = 0, a &1 (22) k dk t = = R, a da t +(1+= R, k ) dk t&1 . a

The associated Jacobian matrix evaluated at the steady state under study has trace T and determinant D, where T=T 1 &

= # &1 |= | &1&= R, k += R, k = 0, a += 0, k |= R, a | , with T 1 =1+ R, a , = 0, a &1 = 0, a &1

D== # D 1 , 10

with D 1 =

|= R, a | &1&= R, k . = 0, a &1

Exposed in Grandmont [20, Theorem B.4.1], Guckenheimer and Holmes [23, Theorem 1.4.1].

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Moreover, T 1 =1+D 1 +4, with 4=

= R, k = 0, a += 0, k |= R, a | . = 0, a &1

4.1. A Geometrical Method of Local Stability and Bifurcation Analysis We shall assume throughout that a steady state exists in the whole range of parameter values that will be considered. To fix ideas, we may assume without loss of generality that the steady state has been normalized at (a, k )=(1, 1) through the scaling procedure of Proposition 3.1. Confronted with the complicated expression of the Jacobian in Proposition 4.1, one may think that tedious computations are necessary to study the local dynamics. Indeed, we shall show that this way of analysis is not unavoidable. As in Grandmont et al. [22], we shall use a geometrical method to get a very simple picture of how local (in)determinacy, local bifurcations, and endogenous fluctuations emerge as a function of the fundamental parameters of the system. This procedure aims at studying how the trace and the determinant of the Jacobian, i.e., respectively the sum and the product def of the roots of the associated characteristic polynomial Q(*) = * 2 &*T+D, vary in the (T, D) plane when one changes continuously the values of some parameters of the model. In particular, if the trace and the determinant are such that the point (T, D) lies in the triangle ABC represented in Fig. 5, we conclude that the eigenvalues have modulus less than one and therefore that the steady state is asymptotically stable, i.e., locally indeterminate in the present context where the capital stock is the single predetermined variable. In contrast, in the complementary region of the triangle ABC in the (T, D) plane, the stationary state is locally determinate, i.e., either a saddle when |T | > |1+D|, or a source (see Fig. 5). A first valuable feature of this method appears if, at the steady state, we fix the technology (i.e., = R, k , = R, a , = 0, k , = 0, a ) and vary the parameter representing workers' preferences = # >1. 11 In other words, consider the parametrized curve (T(= # ), D(= # )) when = # describes (1, +). The direct inspection of the expressions of T and D in Proposition 4.1 shows that this locus is a half-line 2 that starts from (T 1 , D 1 ) when = # is close to 1, and whose slope is 1+= R, k & |= R, a |, as shown in Fig. 5. The value of 4= T 1 &1&D 1 , on the other hand, represents the deviation of the generic point (T 1 , D 1 ) from the line (AC) of the equation D=T&1, in the (T, D) plane. The examples presented in Fig. 5 illustrate the types of local bifurcations one should (generically) expect when = # varies, the technology being fixed, 11 It is shown below that the technological parameter values are independent, at a steady state, of workers' preferences.

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Fig. 5.

The half-line 2 in the (T, D) plane.

so that (T(= # ), D(= # )) describes 2. Suppose that for some value of = # larger than 1, 2 intersects the line (AC). At this point of intersection, one root of the characteristic polynomial Q(*)=* 2 &*T+D is equal to 1; i.e., Q(1) =1&T+D=0 (see Fig. 5). If the analysis is applied to the steady state (a, k )=(1, 1), which is assumed to persist, one generally expects an exchange of stability between that steady state and another one through a transcritical bifurcation. Moreover, if the half-line 2 intersects the interior of the segment [BC] in Fig. 5, the two eigenvalues of the Jacobian in Proposition 4.1 are complex conjugate and have modulus 1 at the intersection point. In this case, therefore, the family of maps G =# , parametrized by = # , generically undergoes a Hopf bifurcation. When 2 crosses the line (AB), one of the eigenvalues is equal to &1, i.e. Q(&1)=1+T+D=0, at the crossing point and one generically expects a flip bifurcation when = # varies (see Fig. 5). For a thorough treatment of local bifurcations, see, e.g, Grandmont [20, Sect. C], Guckenheimer and Holmes [23, Sect. 3.5], and Wiggins [47, Sect. 3.2]. 12 The core of the method we shall use consists of locating the half-line 2 in the plane (T, D), i.e., its origin (T 1 , D 1 ) and its slope 1+= R, k & |= R, a |, as a function of the parameters of the system. The parameters we shall 12 The present approach enables one to verify directly whether the modulus of the bifurcating eigenvalue, i.e. the real eigenvalue in absolute value or the square root of the complex eigenvalues product, (D(= # )) 12 here, goes through one at a non-zero speed when the bifurcation parameter is increased. It is clearly the case for the model we are focusing on when = # is taken as the bifurcation parameter.

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focus on are the depreciation rate for the capital stock 0$1, the capitalist's discount factor 0<;<1, the share of capital in total income 00 and =  == (a )>0, and the elasticity of input substitution _=_(a )>0, all evaluated at the steady state (a, k ) under study. We have therefore to relate the elasticities of the functions 0 and R, with respect to a and k, to these parameters. By definition, 1_(a) is the elasticity with respect to a of the ratio of the rental prices of labor and capital. In view of Eqs. (2) and (3), 1_(a)= = |(a)&= \(a). Moreover, the derivative of the Euler identity, i.e., of the zero profit condition f (a)=|(a)+a\(a), with respect to a yields |$(a)=&a\$(a). All this leads to = 0, a (a, k)=s(a)_(a)&&+= (a), |= R, a | (a, k)=%(a, k)((1&s(a))_(a)+&&= (a)), = 0, k(a, k)=&,

(23)

= R, k(a, k)=%(a, k) &, def

where %(a, k) = (R(a, k)&1+$)R(a, k). Therefore, we derive from Eqs. (23) and Proposition 4.1 the expressions D 1 =(%(1&s)&_(1+%=  ))(s&_(1+&&=  )), 4=%&(s&_(1+&&=  )), T 1 =1+D 1 +4,

(24)

slope 2 =1+%(=  &(1&s)_), def

where % = 1&;(1&$)>0 and all these expressions are evaluated at the steady state under study. Our aim now is to locate the half-line 2, i.e., its origin (T 1 , D 1 ) and its slope in the (T, D) plane when the capitalists' discount rate ;, as well as the technological parameters $, s, &, and =  in the steady state are fixed, whereas the elasticity of factor substitution _ is made to vary. In the benchmark case with no externalities, &==  =0, studied by Grandmont et al. [22], the origin (T 1 , D 1 ) of 2 is located on the line (AC); i.e., 4#0 (see Fig. 6). In particular, under appropriate assumptions on the parameters, D 1 is a decreasing function of _, from some value between zero and one to & as _ is increased from 0 to s, and from + to 1 as _ goes from s to +. Moreover, the slope of 2 increases from & to 1 as _ moves from 0 to + (see Fig. 6). The immediate implication of this geometrical representation is that indeterminacy and endogenous fluctuations can emerge only for low values of _, i.e., for _<_ I , and indeed for _ less than 0.2 when the other

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parameters take reasonable values, while, in contrast, local determinacy is bound to prevail for larger values of _ (see Fig. 6). We are now going to show that in the case with (small) positive externalities, the origin (T 1 , D 1 ) of 2 still describes part of a line 2 1 which is steeper than (AC) and intersects (AC) at a point I whose ordinate is between zero and one, as shown in Fig. 7. In particular, D 1 decreases from a positive value less than one while the slope of 2 increases from & when _ goes up. Indeterminacy, therefore, is still expected for _ close to zero and indeed disappears even more quickly in comparison with the no externality case. In addition, 2 crosses (AC) when _ is sufficiently low, leading generically to an exchange of stability between two steady states when = # varies. More important, however, is the fact that for large _'s, the origin of 2 is such that D 1 is between zero and one (see Fig. 7). Therefore, 2 intersects the triangle ABC where the steady state is asymptotically stable: there is then local indeterminacy and therefore stochastic equilibria can be constructed, for _>_ H 2 and =# close to one, in any small neighborhood of the stationary state. Moreover, 2 also crosses the interior of [BC] when _ is large enough and, accordingly, a Hopf bifurcation is generally expected, generating periodic and quasiperiodic dynamics around the steady state, when = # is increased.

Fig. 6.

The local dynamics regimes in the (T, D) plane when there are no externalities.

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ENDOGENOUS CYCLES AND INCREASING RETURNS

Fig. 7.

83

The local dynamics regimes in the (T, D) plane when there are positive externalities.

Therefore, both indeterminacy and endogenous fluctuations are possible when the elasticity of factor substitution is sufficiently high. To prove these claims, formally, our first task is to show that the point (T 1(_), D 1(_)), as a function of _, indeed describes part of a line 2 1 . From the fact that T 1(_)=1+D 1(_)+4(_) and D 1(_) are fractions of first degree polynomials in _ with the same denominator (see Eq. (24)), we conclude that the ratio of their derivatives D$1(_)T$1(_) or D$1(_)(D$1(_)+4$(_)) is independent of _. 13 Straightforward computations show that the slope of 2 1 is slope 21 =

%(1&s)(1+&&=  )&s(1+%=  ) D$1(_) = . T $1(_) %(1&s+&)(1+&&=  )&s(1+%=  )

(25)

13 The numerator of the derivative of a homographic function f (x)=(a+bx)(c+dx) is in fact independent of x, since f $(x)=(bc&ad )(c+dx) 2.

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From Eq. (24), we conclude that 4(_) vanishes when _ goes to infinity. It follows that 2 1 intersects the line (AC) at a point I of coordinates (T1(+), D1(+)), where D1(+)=(1+%= )(1+&&= )>0 (see Fig. 7). We shall focus throughout on the configuration presented in Fig. 7 where D 1(+)<1 and the slope of 2 1 is greater than 1. We shall ensure the latter condition by imposing, as in the no externality case, that T 1(_) is a decreasing function, i.e. T $1(_)=D$1(_)+4$(_)<0. Indeed, since 4(_) is increasing (see Eq. (24)), it follows that D 1(_) is then decreasing and, accordingly, that the slope of 21 is greater than one. From the denominator in Eq. (25), T$1(_)<0 is equivalent to %(1&s+&)s<(1+%=  )(1+&&=  ). The conditions that D 1(+)<1 and T $1(_)<0 (implying that the slope of 2 1 is greater than one) are therefore equivalent to %(1&s+&) 1+%= 
(26)

The right inequality is equivalent to = (1+%)<&. On the other hand, the expression D 1(+) is an increasing function of =  . Therefore, to ensure that D 1(+) satisfies the left inequality in Eq. (26) for all 0<=  <&(1+%), it is necessary and sufficient that it satisfy it with a weak inequality sign for def =  =0, or equivalently that P(&) = %& 2 +%(2&s) &+%(1&s)&s0. Under the assumption %(1&s)
The geometrical method can be applied as well to every case out of Lemma 4.1.

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slope of 2, when _ moves from 0 to +. In particular, T 1(0) and D 1(0)= %(1&s)s are positive and the corresponding point is below I on the line 2 1 . As _ increases from 0, T 1(_) and D 1(_) are decreasing and tend to & when _ tends to _ 0 =s(1+&&=  ) from below. 15 When _ increases from _=_ 0 to +, T 1(_) and D 1(_) are still both decreasing, from + to (T 1(+), D 1(+)), which is represented by the point I in Fig. 7. On the other hand, the intersection of 2 1 with [BC] is characterized by D 1(_)=1 def which leads, in view of Eq. (24), to _ H 2 = (s&%(1&s))(&&(1+%) =  ). In addition, the slope of 2 as a function of _ increases monotonically from & to 1+%=  >1 as _ moves from 0 to +, and vanishes when D 1(_)=0. Moreover, the half-line 2 is above 21 when _<_ 0 , and below it when _>_ 0 . The only remaining feature to be determined in order to carry out our geometrical analysis is when the slope of 2 is equal to one. From Eq. (24), the value of _ for which slope 2(_)=1 is _ S =(1&s)=  . That feature is important because for _<_ S the half-line 2 crosses the line (AC) in the (T, D) plane and therefore a transcritical bifurcation generally occurs when = # is made vary, whereas 2 does not cross (AC) and no transcritical occurs when _>_ S . In order to fix ideas, we shall focus on the case where _ S >_ H 2 , which can be restated as follows. def

Assumption 4.1. The slope of the half-line 2 for _=_ H 2 = (s&%(1&s)) (&&(1+%) =  ) is less than one, i.e. =  <&(1&s). Given that the value of s is about 0.4 in industrialized economies, this requires that the relative contribution of capital in the overall externalities, i.e., =  &, not exceed 0.6, or equivalently that the relative contribution of labor (&&=  )& is larger than 0.4. Accordingly, this assumption does not appear implausible and, at any rate, its influence on the geometric analysis is only peripheral. We give a few hints in Remark 4.1 on how to modify the analysis when Assumption 4.1 is violated. All these geometrical features are summarized in Table I. Direct inspection of Table I and of Fig. 7 shows that indeterminacy occurs first for low values of _: as _ increases from zero, D 1(_) moves down from 0
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CAZZAVILLAN, LLOYD-BRAGA, AND PINTUS TABLE I Variations and limits of the critical expressions involved in the geometrical analysis, when _ varies.

_

0

Slope of 2 Position of 2

&

D1 4

%(1&s) 1+%=  Z

%(1&s) z s %& s

_0 =

Z

_H 2 _ S =

Z &

0 above 2 1 0

s 1+&&= 

z

&& +

z

1&s =

+

Z

1 below 2 1

1

z

1+%=  1+&&= 

Z

0

+& &

1+%= 

values of _ defining these regimes are respectively defined as the value of _ such that the slope of 2 is equal to &1 (for _ F 1 ), and the value for which 2 goes through the point B (for _ H1 ), in Fig. 7. These configurations arising for low values of _ are described in cases 1, 2, and 3 of Proposition 4.2 below and are similar to those analyzed in the no externality case. The expressions of the critical values _ F1 , _ H 1 , and _ I are given in the proof of that proposition, in Appendix. In the case of positive externalities, a new regime occurs when _ I <_<_ F 2 : the steady is generically either a saddle or a source and undergoes a (reverse) flip bifurcation since 2 intersects (AB) (see Fig. 7 and case 4 of Proposition 4.2). The critical value _ F 2 is such that the origin of 2, i.e., (T 1(_), D 1(_)), belongs to the line (AB) (see the Appendix). For higher values of _, in fact for _ F 2 <_<_ 0 and for _ 0 <_<_ H2 , the steady state is generically either a saddle or a source and there are no endogenous fluctuations in its neighborhood (see Fig. 7). This configuration is reviewed in case 5 of Proposition 4.2. The only qualitative difference occurring when positive externalities are considered is that the half-line 2 crosses the line (AC) for some value = # == #S . This implies that there generically exist two steady states when = # is close enough to = #S , and that a transcritical bifurcation involving an exchange of stability between the two steady states is generally expected to occur. Interesting regimes, however, arise when _ is increased further, i.e., when _>_ H 2 , since D 1(_) is then positive but less than one: for = # close enough to one, the point (T, D) lies in the triangle ABC (see Figs. 7, 8, and 9). Accordingly, the steady state is locally indeterminate and there exist stochastic equilibria in any of its (small) neighborhoods. As far as the Hopf bifurcation is concerned, since the slope of 2 is positive when _ is slightly larger than _ H 2 , 2 is bound to cross the interior of [BC] (see Fig. 8). From Proposition 4.1, this

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def

occurs when D==# D 1 =1, i.e., for = #H = 1D 1 . Therefore, a Hopf bifurcation is generically expected for a fixed _ slightly larger than _ H 2 , when = # goes through = # H . Since the slope of 2 tends to 1+%=  >1 as _ goes to infinity, the same phenomenon should be observed for large _'s (see Fig. 9). One can show analytically (see the proof of Proposition 4.2 in Appendix) that, in fact, the half-line 2 crosses the interior of [BC ], and therefore that a Hopf bifurcation should emerge, for every fixed _>_ H 2 : deterministic as well as stochastic expectations driven fluctuations should generically occur when the elasticity of factor substitution is large. Two subcases arise depending upon whether the slope of 2 is less than one (_<_ S ), in which case 2 crosses (AC) and a transcritical bifurcation occurs when = # goes through = #S (see Fig. 8), or whether the slope of 2 exceeds one (_>_ S ), in which case there is no transcritical bifurcation whatever = # >1 (see Fig. 9). To these regimes correspond cases 6 and 7 of Proposition 4.2. In view of the preceding discussion, one can therefore order seven possible different dynamic regimes as _ is made to vary from 0 to +. Among those, the last two are the most interesting since they show the compatibility of indeterminacy and endogenous fluctuations with a high enough elasticity of factor substitution. We summarize, in Proposition 4.2 below, the characteristics of each regime. Proposition 4.2 (Local Stability and Bifurcations of the Steady States). Consider a steady state that is assumed to be set at (a, k )=(1, 1) through

Fig. 8.

Case 6 (_ H2 <_<_ S ) of the analysis.

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Fig. 9.

Case 7 (_ S <_) of the analysis.

the procedure in Proposition 3.1. Under the assumptions of Lemma 4.1, and under Assumptions 2.1, 2.2, and 4.1, the following generically holds. 16 1. 0<_<_ F 1 : The steady state is a saddle when 1<= # <= #S , where = #S is the value of = # for which the half-line 2 crosses the line (AC). The steady state undergoes a transcritical bifurcation (one characteristic root goes through 1) at = # == #S , and the steady state is a sink for = #S <= # <= #H , where = #H is the value of = # for which 2 crosses [BC]. The steady state undergoes a Hopf bifurcation (the complex characteristic roots cross the unit circle) at = # == #H , and is a source when = # >= #H . 2. _ F 1 <_<_ H 1 : The steady state is a saddle when 1<= # <= #S , undergoes a transcritical bifurcation at = # == #S , and becomes a sink when = #S <= # <= #H . Then the steady state undergoes a Hopf bifurcation at = # == #H and is a source when = #H <= # <= #F , where = #F is the value of = # for which 2 crosses the line (AB). A flip bifurcation occurs (one characteristic root goes through &1) at = # == #F and the steady state is a saddle when = # >= #F . 3. _ H 1 <_<_ I : The steady state is a saddle when 1<= # <= #S , undergoes a transcritical bifurcation at = # == #S , and is a sink when = #S <= # <= #F . A flip bifurcation occurs at = # == #F and the steady state is a saddle if = # >= #F . 16

The expressions _ F 1 , _ H 1 , _ I , _ F 2 , = #S , and = #F (or how to compute them) are given in the proof of the proposition in Appendix. The expressions = #H =(s&_(1+&&=  ))(%(1&s)& _(1+%=  )), _ 0 =s(1+&&=  ), _ H 2 =(s&%(1&s))(&&(1+%) =  ), and _ S =(1&s)=  were given in the text.

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4. _ I <_<_ F 2 : The steady state is a saddle when 1<= # <= #F , undergoes a flip bifurcation at = # == #F , and is a source when = #F <= # <=#S . A transcritical bifurcation occurs at = # == #S , and the steady state is a saddle when = # >= #S . 5. _ F 2 <_<_ H 2 and _{_ 0 : The steady state is a source when 1<= # <= #S , undergoes a transcritical bifurcation at = # == #S , and is a saddle when = # >= #S . 6. _ H 2 <_<_ S : The steady state is a sink when 1<= # <= #H , undergoes a Hopf bifurcation at = # == #H , and is a source when = #H <= # <= #S . At = # == #S , a transcritical bifurcation occurs, and the steady state is a saddle when = # >= #S . 7. _ S <_: The steady state is a sink when 1<= # <= #H , undergoes a Hopf bifurcation at = # == #H , and is a source when = # >= #H . Remark 4.1. In Assumption 4.1, we imposed that the slope of 2 is less than one for _=_ H 2 , i.e, that _ H 2 <_ S , in order to simplify the exposition by focusing on a single configuration. One easily adapts cases 5, 6, and 7 of Proposition 4.2 when alternative assumptions are made as, for instance, _ 0 <_ S <_ H 2 . Using the expressions of these critical values, one gets that the latter condition is equivalent to the inequalities &(1&s)<= <(1+&)(1&s), which are less restrictive than the condition in Assumption 4.1. Since, as in the analysis of the text, 2 cannot cross (AC) and a transcritical bifurcation cannot occur when _>_ S , the steady state being generically either a sink or a source, cases 1, 2, 3, and 4 of Proposition 4.2 then remain unchanged, while the configuration of case 5 occurs for _ F 2 <_<_ S . When _ S <_<_ H 2 , the steady state is a source for all = # >1, and finally, case 7 occurs for _ H2 <_. The investigation of the different regimes when _ S <_ 0 , which can be performed by adopting the very same tools, is left to the reader. 4.2. Local Indeterminacy for Large Elasticities of Input Substitution A significant outcome of the above analysis is that, under the Assumptions of Lemma 4.1 (Assumption 4.1 is not really needed as indicated in Remark 4.1), local indeterminacy (and thus stochastic sunspot fluctuations) as well as the Hopf bifurcation (and therefore deterministic endogenous fluctuations) occur for large _'s. More precisely, when _>_ H 2 =(s&%(1&s))(&&(1+%) = ), local indeterminacy prevails if 1<= # <= #H =(s&_(1+&&=  ))(%(1&s)& _(1+%=  )) whereas the Hopf bifurcation occurs at = # == #H . This configuration is economically interesting, however, only if _ H 2 is not too large and if = #H is not too close to one, i.e., if the critical value of the labor supply elasticity = lH =1(= #H &1) is not too high. Our approach, being based upon general specifications (as compared to studies focusing on the CobbDouglas case or on numerical simulations), permits a systematic discussion of this point.

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The first general observation is that both _ H 2 and = lH are decreasing in & and increasing in =  : a higher overall degree of increasing returns (1+&) and a lower contribution of capital (=  ) to short run externalities make local indeterminacy for large elasticities of factor substitution more likely (geometrically, the region for which there is local indeterminacy in Figs. 8 and 9 expands as & goes up and =  moves down: the ordinate D 1(+)=(1+%=  ) (1+&&=  ) on the intersection of 2 1 with (AC) then decreases while the slope of 2 1 given in Eq. (25) increases). One is nevertheless limited here by the fact that & cannot be empirically very large, and presumably is lower than 0.6 in view of the estimates of Baxter and King [4], Caballero and Lyons [11], and Hall [25] (see also Basu and Fernald [3] and Burnside [10]). On the other hand, the contribution of capital to short run externalities appears a little bit controversial, according for instance to Benhabib and Jovanovic [6]. The next reward is that if the period is short (so that %=1&;(1&$) is quite small and can be neglected), an approximate evaluation of _ H 2 is s(&&=  ), and for __ H 2 , = #H is approximately given by 1+&&=  &s_, the corresponding labor supply elasticity being = lH =1(= #H &1)r_((&&=  ) _&s).

(27)

The above expression decreases when the contribution of labor to short run externalities (&&=  ) is higher. If we are willing to admit relatively large externalities by setting &&=  at 0.5, while s=0.4, then = lH is approximately equal to 10 when _=1, to 4.7 when _=1.5, and to 3.3 when _=2. 17

5. SIMULATIONS OF CES ECONOMIES WITH EXTERNALITIES We now present the simulations based on the class of CES economies defined by Eqs. (15), (16), (19), and (20), in Subsection 3.2, building on the parameter values, on quarterly basis, usually used in the literature, i.e. s=0.4, ;=0.988 and $=0.025, and finally choose A, B as in Eqs. (20) so as to fix the steady state under consideration to (a, k )=(1, 1). In the general setting of the previous section, we cannot easily infer whether the local bifurcations are actually supercritical or subcritical. Considering the CES case will allow us to derive more precise conclusions. One could apply general methods to study these nonlinear maps: the center manifold theory to reduce the dimension of the system so as to determine more easily the stability of a periodic orbit when some eigenvalues have modulus one, 17 These labor supply elasticities are relatively high but not much higher than those assumed in usual business cycles models (see, for instance, King, Plosser, and Rebelo [29], and Schmitt-Grohe [41]).

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and the method of normal forms to derive the nature of the local bifurcations encountered (see, e.g., Grandmont [20], Guckenheimer and Holmes [23], and Wiggins [47]). We made instead a computer assisted analysis by using the program Dunro [15], elaborated by D. J. Sands and R. G. de Vilder, to check the direction and stability of each bifurcation of the fixed point (a, k )=(1, 1). It turns out that the case of a CobbDouglas production function not only leads to a very specific (structurally unstable) configuration where the steady state is generically unique, as shown in Proposition 3.4, but it also implies very peculiar dynamics around a steady state. More precisely, the Hopf bifurcation that is generally expected at = # == #H when _>_ H 2 (see cases 6 and 7 of Proposition 4.2) is found to be supercritical when _<1 while, on the contrary, it is subcritical when _1, _ being close to 1. This means that when = # is increased, an invariant closed curve, on which the dynamics is either periodic or quasiperiodic, appears slightly after the supercritical Hopf bifurcation, surrounds the unstable steady state, and is attracting when _<1. In contrast, an invariant closed curve occurs slightly before the subcritical Hopf bifurcation, around the stable steady state, and repels all points nearby, when _1. 18 In the first case (_<1), endogenous perfect foresight fluctuations are therefore locally indeterminate, while they are locally determinate when _1. In view of the general results presented in Grandmont et al. [22, Sect. 3], we know that this heavily affects the possibility of stochastic (sunspot) equilibria around a steady state. In particular, after the supercritical Hopf bifurcation (_<1), there are no truly stochastic sunspot equilibria staying in a compact neighborhood of the unstable steady state, if that neighborhood lies in the interior of the (locally indeterminate) invariant closed curve. But there are infinitely many nondegenerate stochastic equilibria staying in a compact neighborhood containing in its interior the attracting closed curve. On the other hand, there exist infinitely many nondegenerate stochastic equilibria, before a subcritical Hopf bifurcation (_1), but their support has to be contained in the interior of the repelling closed curve. These two typical cases are illustrated below by several examples where _ is close to one. 5.1. Multiple Steady States and the Transcritical Bifurcation In Proposition 4.2, cases 1 to 6, where _<_ S , show that one eigenvalue of the Jacobian matrix evaluated at (a, k )=(1, 1) crosses 1, when = # goes through = #S . Using Dunro [15], we found that a transcritical bifurcation 18 The theory of local bifurcations is presented in, for instance, Grandmont [20, Sect. C], Guckenheimer and Holmes [23, Sect. 3.5], and Wiggins [47, Sect. 3.2].

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of the fixed point (a, k )=(1, 1) actually occurs. 19 To illustrate this bifurcation, 20 we now fix &=0.6, =0.1 and consider two examples. Example 1. '=0.05 (_r0.95). In that configuration, we observe that for = # slightly smaller than = #S , (1, 1) is a source (see Fig. 8, case 6 of Proposition 4.2). In the (k, a) plane, a second positive fixed point is located north-east of (1, 1) and is a saddle (see Fig. 10a). As the bifurcation parameter = # is increased towards = #S , the saddle approaches the source (1, 1) from north-east. The two steady states actually merge at the bifurcation point = # == #S , where one eigenvalue is equal to one. Finally, for = # larger than = #S , the two positive fixed points have exchanged stability: (1, 1) is henceforth a saddle while the second steady state located south-west of (1, 1) is a source (see Fig. 10b). Numerical simulations revealed that this configuration occurs for _<1 close enough to one. Example 2. '=&0.05 (_r1.05). In view of Fig. 8, and case 6 of Proposition 4.2, (1,1) is a source when = # is slightly smaller than = #S and becomes a saddle after the transcritical bifurcation. The second positive fixed point is located south-west of (1, 1) and is a saddle before the transcritical bifurcation, becomes a source through the bifurcation and moves towards north-east of (1, 1) when = # increases from = #S , (see Figs. 11a and 11b). Again, numerical simulations revealed that this configuration occurs for _>1 close enough to one. These two examples illustrate how special the case _=1 (the CobbDouglas production function) is, as far as the number of stationary equilibria is concerned: a slight departure from this specification leads to multiple steady states. From a more technical point of view, these two examples also allow the comparison between two possible ways of proving multiplicity. Proposition 3.4 explores a global approach, while Proposition 4.2 uses the local dynamics information to provide simple sufficient conditions for arbitrarily close steady states (generically speaking). In Subsections 3.1 and 3.2, we argued that the possibility of multiple steady states cannot be rejected on empirical grounds. Multiplicity of steady states has therefore to be taken seriously, as it is responsible for coordination failures. For instance, two locally determinate positive steady states may coexist in a small neighborhood, as the two previous examples illustrated. 19

Because we assume that Eq. (20) holds, the steady state (a, k )=(1, 1) persists whatever the values of # and _ are. Moreover, Proposition 3.4 showed that there exist at most two steady states. Therefore, we should expect that neither the generic saddle-node bifurcation nor the pitchfork bifurcation can occur. 20 See Grandmont [20, Subsect. C.2] and Wiggins [47, Subsect. 3.2A] for a theoretical treatment and bifurcation diagrams.

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Fig. 10. The example 1 ('=0.05) in the (k, a) plane. The multiple steady states are shown (a) slightly before the transcritical bifurcation, i.e., for = # <= #S and |= # &= #S | small, (b) slightly after the transcritical bifurcation, i.e., for = # >= #S and |= # &= #S | small.

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Fig. 11. The example 2 ('=&0.05) in the (k, a) plane. The multiple steady states are shown (a) slightly before the transcritical bifurcation, i.e., for = # <= #S and |= # &= #S | small, (b) slightly after the transcritical bifurcation, i.e., for = # >= #S and |= # &= #S | small.

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More importantly, the two stationary points are strongly Pareto-ranked: the highest is strictly preferred by both classes of agents, as shown above in Proposition 3.5. The two previous examples suggest that a small departure from the Cobb Douglas configuration (_=1) leads to very different pictures: agents select a Pareto-optimal steady state in Figs. 10 (for _<1 close to one), whereas a Pareto-dominated stationary state is chosen in Figs. 11 (for _>1 close to one). 5.2. The Hopf Bifurcation and Expectations Driven Fluctuations We used the program Dunro [15] to determine the direction and stability of the Hopf bifurcation, 21 at = # == #H (see cases 6 and 7 in Figs. 8 and 9, and in Proposition 4.2). We keep the values &=0.6, =0.1 and give two examples. Our numerical findings will show again that the CobbDouglas case (_=1) is very special and has to be handled with care. Indeed, for _<1 close to one, the Hopf bifurcation turns out to be supercritical, with a stable (locally indeterminate) invariant closed curve appearing near the unstable steady state after the bifurcation, whereas it is on the contrary subcritical for _1 close to one, with an unstable closed curve near the stable steady state before the bifurcation. A slight departure from the specific CobbDouglas case (_=1) has accordingly dramatically different consequences on endogenous cycles and stochastic sunspot equilibria when the system displays close to unit roots, whether _ is slightly above or below one. Example 3. '=0.05 (_r0.95). Proposition 4.2 shows that slightly before the Hopf bifurcation, (1, 1) is a sink, i.e. it is (locally) asymptotically stable, while slightly after the bifurcation, (1, 1) is a source (see Fig. 8). Figure 12 reveals that for = # larger than and close to = #H , (1, 1) is surrounded by an attracting and invariant closed curve, homeomorphic to a circle, created through the Hopf bifurcation. The Hopf bifurcation is then supercritical. Numerical simulations confirmed that this configuration occurs for _<1 close to one.

Example 4. '=0 (_=1). In that case, using Dunro [15], we found that the Hopf bifurcation is now subcritical: a repelling and invariant closed curve surrounds the asymptotically stable fixed point (1, 1) before the Hopf bifurcation, i.e. for = # <= #H and |= # &= #H | small. Slightly after the bifurcation, (1, 1) is a source. 21 See Grandmont [20, Subsect. C.4] and Wiggins [47, Subsect. 3.2C] for a theoretical treatment and bifurcation diagrams.

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Fig. 12. The example 3 ('=0.05) in the (k, a) plane. Quasiperiodic dynamics on the attracting invariant closed curve created through a supercritical Hopf bifurcation, i.e., for =# >= #H and |= # &= #H | small.

Example 5. '=&0.05 (_r1.05). The Hopf bifurcation is here again subcritical and numerical simulations confirmed that this configuration occurs for _>1. As already mentioned above, the existence of deterministic and stochastic expectations driven fluctuations depends critically on the direction and stability of the Hopf bifurcation. We refer the reader to Grandmont et al. [22, Sect. 3] for a geometrical method of proving the existence of stochastic equilibria around a locally indeterminate steady state as well as along local bifurcations. Appealing to the results of these latter authors, we next characterize those regions in the (k, a) plane where endogenous fluctuations occur. In Example 3, the steady state (a, k )=(1, 1) is locally indeterminate before the supercritical Hopf bifurcation: infinitely many nondegenerate stochastic equilibria with a compact support can be constructed in any arbitrarily small neighborhood of the sink. From Grandmont et al. [22, Sect. 3], we know that, after the supercritical Hopf bifurcation, there are no truly stochastic sunspot equilibria staying in a compact neighborhood of the unstable steady state, if that neighborhood lies in the interior of the (locally indeterminate) invariant closed curve. But there are infinitely many nondegenerate stochastic equilibria staying in a compact neighborhood containing in its interior the attracting closed curve.

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In Examples 4 and 5, the Hopf bifurcation is subcritical: before the bifurcation, there exist infinitely many nondegenerate stochastic equilibria near the steady state, but their support has to be contained in the interior of the repelling closed curve. Bounded (locally determinate) deterministic endogenous fluctuations only occur on this latter repelling set, therefore solely before the subcritical Hopf bifurcation. After this bifurcation, the steady state is locally determinate: there exists a neighborhood where no endogenous fluctuations occur. 22

6. INTERNAL INCREASING RETURNS TO SCALE AND IMPERFECT COMPETITION In this section, we modify the model introduced in Section 2 in only one of its dimensions: the unique good is now produced under internal increasing returns to scale. In order for the market structure to be compatible with internal increasing returns, we introduce imperfect competition in the product market, while money and production factors are still traded in a perfectly competitive way. Increasing returns to scale are now represented by a single parameter, i.e., the degree of internal increasing returns. The critical assumption we adopt is that costless entry and exit eliminate the possibility of profits, so that the dynamical system in Eqs. (8) still summarizes the intertemporal equilibria. Ignoring ``Ford effects,'' it follows that the markup is constant and that the elasticities of factor rental prices depend, as above, only on technology. 23 It turns out that a change in the functions representing increasing returns, related also to the labor share, leads one back to the previous elasticities of factor prices: it is now as if each factor originated an external effect proportional to its share in total income, given the constant level of social, or equivalently internal, increasing returns. Consequently, as far as the steady states and local dynamics analysis are concerned, the above methods apply equally well to the model with an imperfectly competitive product market and yield qualitatively similar results, i.e., multiple Pareto-ranked steady states and endogenous fluctuations for large elasticities of factor substitution. 24 22

With the local dynamics analysis presented in Section 4 as a starting point, one may wish to get information about global dynamics, i.e. about orbits away from the steady state. We refer the reader to the techniques presented in de Vilder [44, 45] and Pintus, Sands, and de Vilder [35]. 23 This formulation follows the specification of LloydBraga [31] which considered CES production functions. A similar formulation has been used by Benhabib and Farmer [5] with a CobbDouglas technology. 24 The results obtained under internal increasing returns and imperfect competition are equivalent to the ones obtained under aggregate externalities and perfect competition as long as the endogenous number of firms (driving profits to zero) and the endogenous markup are constant. In the absence of a representative agent in this model, the existence of positive profits might change the results.

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Especially, we consider Cournot oligopolistic competition between the entrepreneur-capitalists: each producer knows the aggregate inverse demand and chooses the optimal amount to produce, taking as given the quantities chosen by all the other competitors. We introduce imperfect competition through the Cournotian approach for simplicity of the exposition, but we could consider, instead, Cournotian monopolistic competition by adapting the formulation used in d'Aspremont et al. [2]. 25 6.1. The Technology with Increasing Returns and Symmetric Cournot Equilibria Each identical entrepreneur-capitalist is endowed with the same technology producing from a capital stock k t&1 0 and labor l t 0, possibly in variable proportion. The main difference from the technology in Subsection 2.1 is that the gross production F now exhibits internal increasing returns to scale and, moreover, is homogeneous of degree e>1. Therefore, the quantity of good produced is given by AF(k, l )=Al ef (a).

(28)

The second equality defines the production in intensive form defined upon the capital labor ratio a=kl. On technology, we shall assume that f (a) is smooth and increasing. These assumptions, however, do not ensure that the elasticity of capital def def def labor substitution 1_(a) = = |(a)&= \(a), where \(a) = f $(a) and |(a) = ef(a)&a\(a), is positive. One can verify that _(a)>0 is equivalent to (e&1)e > f (a) f "(a)( f $(a)) 2. In particular, the second derivative f "(a) is not necessarily negative. Assumption 6.1. f is a continuous function of a0, C r for a>0 and r large enough, with f $(a)>0. In addition, the elasticity of capital def def def labor substitution 1_(a) = = |(a)&= \(a), where \(a) = f $(a) and |(a) = ef (a)&a\(a), is positive, i.e. (e&1)e> f (a) f "(a)( f $(a)) 2, for all a>0. The optimal amount of each factor is determined by a Cournot competitor in the following manner. Each firm knows the inverse demand function p( y i + y &i ), where y i 0 and y &i 0 are respectively the residual demand addressed to the firm i and the aggregate production resulting from the decisions of all the other competitors, whereas the map p gives the associated price of the good. Taking y &i as given, the firm i determines its factor demands 25

We could introduce differentiated goods, each produced by several firms, as long as the same CES aggregator is considered for consumption of workers and for capital and consumption of capitalists. Then, if identical firms take as given the quantities of the competitors in its sector (Cournot oligopolistic competition within a sector) as well as the prices of the other goods (monopolistic competition between sectors) and know the inverse demand function, we would get the same qualitative results.

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li 0 and k i 0 by maximizing the profit p(AF(k i , l i )+ y &i ) AF(k i , l i )& wli &rki , where w>0 and r>0 are respectively the nominal rental prices of labor and capital, taken as given by firm i under the assumption of perfectly competitive factor markets. As usual, the critical information needed for a Cournot competitor is the elasticity of the inverse demand function: this elasticity determines the level of the price over marginal cost markup. Indeed, we shall consider symmetric CournotNash equilibria 26 and ignore the so-called Ford effects reflecting the dependence of the demand to the income paid by the firms: the priceelasticity of the aggregate demand for the good equals &1 at the steady state and nearby, expressing the fact that the total nominal revenue, i.e., the stock of money and the returns on capital, is taken as given by the firms. Accordingly, at a symmetric equilibrium involving n firms, the familiar first order conditions of that problem are therefore P=(1&1n) AF k( } ),

0=(1&1n) AF l ( } ),

(29)

where P and 0 denote respectively the real competitive rental prices of capital and labor. 27 Together with the necessary condition (29), the nonnegative profit condition for each active firm is required for a positive production. 28 Indeed, we shall assume that costless entry and exit push the level of profit towards zero, i.e. AF(k, l )=Pk+0l. Under this assumption, one easily deduces from the homogeneity of the production function, hence the Euler relation eF( } )=kF k( } )+lF l ( } ), and the conditions (29) that the level of increasing returns determines then the number of active firms to be e(e&1). Without loss of generality and only for convenience, we shall suppose that the number of entrepreneur-capitalists actually equals e(e&1), and that it is also the number of workers. 29 We then derive the respective real rental prices of capital and labor arising from the symmetric Cournot 26

Using the same kind of arguments as those in d'Aspremont, Dos Santos Ferreira, and Gerard-Varet [2, Lemma 3], it can be proved that any intertemporal equilibria is symmetric in quantities relatively to the active firms. 27 Of course, one can alternatively consider the dual minimizing cost problem, since the homogeneity of the production function allows the total cost function C( y i , w, r)= y 1e i c(w, r), to be derived easily for given w and r. The well-known outcome of this procedure is the equality between marginal revenue and marginal cost. 28 It is easily shown that when the first order and the nonnegative profit conditions are satisfied at an interior optimum, the Hessian matrix of the profit function of each firm, as a function of k i and l i , is negative definite, i.e., the profit function of each firm p( y i + y &i ) y i & y 1e i c(w, r) is strictly concave in y i : the first-order conditions (29) are then sufficient and characterize the optimum quantity chosen by each firm. 29 The limit case of constant returns to scale e=1 implies an infinite number of firms and therefore the absence of market power in the product market.

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equilibrium on the product market and the perfectly competitive equilibria on the factor markets, def

where

\(a)= f $(a),

def

where

|(a)=ef (a)&a\(a).

P(a, k) = A(ka) e&1 \(a)e, 0(a, k) = A(ka) e&1 |(a)e,

(30)

As in the case of the previous technology with externalities, a remarkable consequence of increasing returns is that both marginal productivities depend on the scale of production, i.e., on k in Eqs. (30). We shall show that this is of some importance for the number of steady states and the dynamics of the model. Although the current model is different from the previous one with externalities, it will turn out that, under Assumption 6.1, the structures of both models are sufficiently similar so that the methods and the results of our previous analysis, concerning multiplicity of steady states and local stability, can be applied after a change of functions. This similarity between the two models relies essentially on a proper correspondence between the two technologies, which we now explicit. The implication of these findings seems to be that the deep reason for the phenomena studied in this paper is the presence of internal or external increasing returns, rather than the product market structure. 6.2. Steady States, Local Dynamics, and Bifurcation Analysis Apart from the technology and the product market structure, the remaining part of the model is identical. More precisely, the behaviors of agents, as expressed in Eqs. (5) and (6), are unchanged. On the other hand, Eq. (7) still describes the money market clearing condition, when there are no profits. Accordingly, these equations define, in view of the definitions given in Eqs. (30), the imperfectly competitive intertemporal equilibria as follows. Definition 6.1. An intertemporal imperfectly competitive equilibrium with perfect foresight is a sequence (a t , k t&1 ) of R 2++ , t=1, 2, ..., such that

{

0(a t+1 , k t )a t+1 =#(k t&1 a t )k t , k t =;R(a t , k t&1 ) k t&1 ,

(31)

def

where R(a, k) = P(a, k)+1&$ denotes the real gross return on capital. The results concerning the number of steady states and the local dynamics presented in Sections 3 and 4 for the perfectly competitive economy with externalities rely heavily on the elasticities of the functions P and 0 with respect to a and k. Although these functions now depend on a new parameter e and on the (differently defined) functions \ and |, in Eqs. (30), it turns out

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that the methods in Sections 3 and 4 apply equally well here, after a slight change of functions, and that the results for the perfectly competitive economy with externalities are directly transposable to the present economy with an imperfectly competitive product market. Preliminary computations are required to get the correspondence announced just above. The objective is again to express the elasticities of the functions P and 0, with respect to a and k, in terms of our parameters. From the definition 1_(a)== |(a)&= \(a) (see Assumption 6.1) and the derivative of the Euler identity with respect to a yielding |$(a)+a\$(a)=(e&1) \(a), we derive = |(a)=(e&1) s(a)+s(a)_(a) and = \(a)=(e&1) s(a)&(1&s(a))_(a). Together with Eqs. (30) and the definition R(a, k)=P(a, k)+1&$, this leads to the expressions = 0, a (a, k)=s(a)_(a)&(1&s(a))(e&1), |= R, a | (a, k)=%(a, k)(1&s(a))(1_(a)+e&1) = 0, k(a, k)=e&1,

(32)

= R, k(a, k)=%(a, k)(e&1), def

where %(a, k) = P(a, k)R(a, k). By comparing Eqs. (23) and Eqs. (32), one easily deduces that the expressions are actually identical if the change of functions e&1=&,

(1&s(a))(e&1)=&&= (a),

hence

s(a)(e&1)== (a), (33)

is made in Eqs. (32). This change is natural since it amounts to identify the level of internal increasing returns in the present economy to the level of social increasing returns in the economy with externalities, i.e., e to 1+&. Moreover, the external effect due, respectively, to the aggregate capital and labor stocks is identified here to the contribution of each factor to the internal increasing returns proportional to its share in total production, i.e., to s(a)(e&1) and (1&s(a))(e&1). We have seen in Sections 3 and 4 that each contribution separately matters, at the aggregate level, to account for the existence of multiple steady states and expectations driven aggregate fluctuations. Therefore, using an homogeneous technology with increasing returns gives one degree of freedom less than introducing increasing returns through aggregate productive externalities: in the latter case, each factor contributes to the social increasing returns independently of the other. In conclusion, one expects that the change of functions in Eqs. (33) makes the analysis of the preceding sections directly transposable, as a special case, to the present economy with an imperfectly competitive product market.

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In view of Eqs. (31) and (30), the stationary states are given by

{

Al e|(a )e=#(l ), Al e&1\(a )e+1&$=1;.

(34)

One can then ensure the existence of a steady state by setting appropriately the scaling parameters A and B, as shown in Proposition 3.1. For instance, the steady state is fixed at (1, 1) if and only if A=e(1;+$&1)\(1), while B>0 is, under Assumptions 2.2 and 6.1, the unique solution of A|(1) V$2(A|(1)(eB))(eB)=V$1(1). In view of the change of functions expressed in Eqs. (33), one can carry out the very same analysis and derive the same results as those stated in both Propositions 3.2 and 3.3. In particular, the CES example is, in the present economy, easily adapted from Subsection 3.2 and yields the analog of Proposition 3.4. Finally, Proposition 3.5 still holds and is proved by the same arguments since, in view of Assumption 6.1, the function |(a)\(a) is still increasing in a: the multiple steady states of the imperfectly competitive economy are still ordered and, accordingly, Pareto-ranked. In summary, the main result that there may exist multiple Parero-ranked steady states, possibly in a very small neighborhood, in the economy with externalities remains true in the economy with internal increasing returns and imperfect competition. Finally, in view of Eqs. (31) defining the dynamical system, Proposition 4.1 still expresses the dynamics around a steady state. Therefore, one deduces, here again, the existence of the half-line 2, in the (T, D) plane, when = # is made to increase. Moreover, the change of functions in Eqs. (33), when evaluated at the steady state under study and applied to Eqs. (24), allows one to derive the existence of the line 2 1 generated when _ increases: the two key ingredients of the geometrical method of analyzing the dynamics around a steady state are equally useful to study locally the orbits of this model. Therefore, one adapts Lemma 4.1, Table I, and the Appendix by using Eqs. (33) and gets the counterpart of Proposition 4.2. Accordingly, Figs. 7, 8, and 9 still summarize the different local dynamics regimes: through local indeterminacy and the Hopf bifurcation, deterministic and stochastic expectations driven fluctuations are shown to exist (generically) for high elasticities of capitallabor substitution, in the imperfectly competitive economy.

7. CONCLUSION Multiple steady states, local indeterminacy, and expectations-driven cycles have been shown to exist under external or internal increasing returns to scale in production, within particular specifications for preferences and

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technology. The contribution of this paper is to study, under quite general assumptions, the sensitivity of these phenomena to variations of key parameters whose estimates are not conclusive. In particular, we showed that endogenous fluctuations occur, at business cycle frequency, for high elasticities of capitallabor substitution. Moreover, we showed that the use of Cobb Douglas specifications, which is pervasive in the literature, presumably because it allows one to get explicit closed-form solutions, can be very misleading. The qualitative methods used in this paper, by contrast, make it possible to deal easily with more general specifications. Although our results are primarily obtained in a relatively simple model because perfect competition is maintained, it is shown that the same tools are useful to study models with internal increasing returns and imperfect competition in the product market, and that they yield, in some specific configurations, qualitatively similar results. It remains to be seen how the methods employed in this paper can be put to use to analyze alternative, theoretically and empirically relevant, models of business cycles.

APPENDIX Proof of Proposition 4.2 The Hopf Bifurcation in Case 6: _ H 2 <_<_ S In this case, the appraisal of the values of _ for which a Hopf bifurcation occurs at = # == #H is not immediate. Table I and Fig. 8 show that, by continuity, 2 intersects [BC] in its interior for _ slightly larger than _ H 2 and for _ slightly smaller than _ S (i.e., when the slope of 2 is slightly lower than one). Between these small neighborhoods, however, geometrical arguments cannot account for the occurrence of the Hopf bifurcation whatever _ H 2 < _<_ S . The critical inequality to check, which is in fact a necessary condition for a Hopf bifurcation to occur at = # == #H , is that T(= #H )<2 when D=1, i.e. when = # == #H =1D 1 . In view of the expressions given in Proposition 4.2 and Eqs. (24), the last inequality is restated as Q(_)=a_ 2 +b_+c>0, where the coefficients of Q(_) are the following: a== (&&(1+%) =  )>0, b=s(&&=  )+= (1&s+%(2(1&s)+&))>0, c=(1&s)(s&%(1&s+&))>0. The signs are derived from the assumptions of Lemma 4.1 and allow one to conclude that the roots of Q(_) are either real negative or complex.

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Therefore, Q(_)>0 whatever _>0 and, accordingly, the half-line 2 intersects the interior of the segment [BC], whenever _ H 2 <_<_ S . The Local Bifurcation Values In this section, we derive all bifurcation values as functions of the structural def def def parameters. We define % = 1&;(1&$), : = %(1&s)s, and s 2(_) = 1+ %(=  &(1&s)_) as the slope of the half-line 2. An eigenvalue of 1: The condition s 2(_)=1 gives the bifurcation value _ S =(1&s)=  , so that s 2(_)1 when __ S . Second, the equation of the line (AC) in Fig. 7, i.e. 1&T(= # )+D(= # )=0, yields that 2 intersects (AC) for = #S =1+&(1&s&=  _), which is larger than one when _<_ S . An eigenvalue of &1: The flip bifurcation. The equality s 2(_)=&1 allows one to derive _ F 1 =%(1&s)(2+%=  ), so that s 2(_) &1 when __ F 1 . The line (AB) in Fig. 7, of the equation 1+T(= # )+D(= # )=0, yields that 2 intersects (AB) for = #F =(s(2+:)+%&&_(2+2&+(%&2) =  )) (_(2+%=  )&%(1&s)). The condition that (T 1 , D 1 ) belongs to the line (AB), i.e., that 1+T 1(_)+D 1(_)=0 or, equivalently, = #F =1, gives the last flip bifurcation value _ F 2 =(s(2+:&%)+%(1+&))(4+2&+(2%&2) =  ), so that = #F >1 when _<_ F 2 . A pair of eigenvalues of modulus 1: the Hopf bifurcation. The condition that T(= #H )=&2 when D=1, i.e., when = # == #H =1D 1 , is rewritten as def QH (_) = a_ 2 +b_+c, the roots of which contain the bifurcation value _ H 1 . In the configuration of Lemma 4.1, it is easily shown in Fig. 7 that there must exist two distinct real roots, and that _ H 1 is the lower. The coefficients of Q H (_) are a=4+4(&&=  )+%= (4+3&+(%&3) =  ), b= &s(4+%= (3+:)+:(4+3&+(%&3) =  ))&%&(1+%=  ), c=:s(s(3+:)+%&). The condition D 1(_)=1 yields, in view of Eqs. (24), _ H 2 =(s&%(1&s)) (&&(1+%) = ). Finally, the bifurcation value = #H =(s&_(1+&&= ))(%(1&s) &_(1+%=  )) follows from D== # D 1 =1, i.e., = #H =1D 1 . Indeterminacy of the steady states. The condition that the point A belongs to 2, i.e., that = #S == #F , as shown in Fig. 7, is given by def Q I (_) = a_ 2 +b_+c, where

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a=%= (2+&+(%&1) =  ), b= &:s(2+&+(%&1) =  )&%&&%= (s(1+:)+%&), c=:s(s(1+:)+%&). Accordingly, 1<= #S = #F if _ F 1 <__ I , while 1<= #F = #S if _ I _<_ F 2 . Again, a geometrical argument shows that the roots of Q I (_) must be real and distinct, and that the lowest is _ I .

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