Hippolyte D'albis

Emmanuelle Augeraud-Veron

Alain Venditti

Business cycle fluctuations and learning-by-doing externalities in a one-sector model *

Keywords: One-sector infinite-horizon model, learning-by-doing externalities, inelastic labor, business cycle fluctuations, Hopf bifurcation. Journal of Economic Literature Classification Numbers: C62, E32, O41

We consider a one-sector Ramsey-type growth model with inelastic labor and learning-by-doing externalities based on cumulative gross investment (cumulative production of capital goods), which is assumed, in accordance with Arrow [5], to be a good index of experience. We prove that a slight memory effect characterizing the learning-by-doing process is enough to generate business cycle fluctuations through a Hopf bifurcation. This is obtained for reasonable parameter values, notably for both the elasticity of output with respect to the externality and the elasticity of intertemporal substitution. Hence, contrary to all the results available in the literature on aggregate models, we show that endogenous fluctuations are compatible with a low (in actual fact, zero) wage elasticity of the labor supply.

Introduction

Explaining the economic fluctuations associated with the business cycle is one of the main goals of modern macroeconomic theory. Two complementary explanations coexist in the literature: the endogenous cycle theory and the real business cycle theory based on exogenously-driven fluctuations. The common framework used in these two theories is given by the Ramsey [START_REF] Romer | Increasing returns and long-run growth[END_REF] optimal growth model. The standard aggregate formulation is characterized by the existence of a unique monotonically convergent equilibrium path and thus business cycle fluctuations can only be obtained if exogenous shocks on the fundamentals are introduced. Contrary to this, multisector optimal growth models easily exhibit endogenous fluctuations without any stochastic perturbation. However, depending on whether time is discrete or continuous, the number of goods matters. In a discrete-time model, the consideration of two sectors with both consumption goods and investment goods is sufficient to generate period-two cycles through a flip bifurcation as shown by Benhabib and Nishimura [START_REF] Benhabib | Competitive equilibrium cycles[END_REF]. 1 In a continuous-time model, Benhabib and Nishimura [START_REF] Benhabib | The Hopf bifurcation and the existence and stability of closed orbits in multisector models of optimal economic growth[END_REF] show that at least three sectors with one consumption good and two capital goods need to be considered to generate endogenous fluctuations through a Hopf bifurcation. 2More recently, endogenous fluctuations through the existence of local indeterminacy and sunspot equilibria have been shown to occur even in onesector models. Building on the work by Romer [START_REF] Romer | Increasing returns and long-run growth[END_REF], Benhabib and Farmer [START_REF] Benhabib | Indeterminacy and increasing returns[END_REF] consider a Ramsey-type continuous-time aggregate model augmented to include economy-wide externalities in the production function measured by the aggregate stock of capital and total labor, which are assumed to be a proxy for some learning-by-doing process. It is indeed assumed that by using capital over time, agents increase their experience and are thus able to increase their productivity. Within such a framework, Benhabib and Farmer [START_REF] Benhabib | Indeterminacy and increasing returns[END_REF] show that local indeterminacy and fluctuations derived from agents' self-fulfilling expectations can occur. However, besides external effects in production with large enough increasing returns at the social level, the basic model also has to be increased by the consideration of endogenous labor supply, 3 whose wage elasticity is sufficiently high, i.e., close enough to infinity. 4 Since the elasticity of the aggregate labor supply is usually shown to be low, 5 it follows that the occurrence of local indeterminacy relies on parameter values that do not match empirical evidence.

In this paper, we consider a continuous-time aggregate model with inelastic labor and learning-by-doing externalities in the production process. We depart significantly from most existing contributions in the literature and notably from Romer [START_REF] Romer | Increasing returns and long-run growth[END_REF] in which the average level of capital is used as a proxy of experience. We assume, in accordance with Arrow [START_REF] Arrow | The economic implications of learning by doing[END_REF], that cumulative gross investment (cumulative production of capital goods) is a better index of experience. More precisely, the learning-by-doing effects are measured by the whole gross investment process over some fixed period of time. 6 This last assumption, which will importantly shape the equilibrium dynamics, represents a memory effect suggesting that investments made a long time ago do not have the same impact on the index of experience as recent ones. This can be justified, for instance, by the finite longevity of the workers. 7 For computational convenience, we have chosen a memory process similar to a one-hoss shay depreciation: the weight of a given vintage in the index of experience is one during a given time interval, then zero. Our results extend to more general specifications.

Given this assumption, the equilibrium path is described by a system of functional differential equations. It is worth noting that our formulation closely resembles a time-to-build model, apart from the fact that the cu- 3 As shown by Boldrin and Rustichini [START_REF] Boldrin | Growth and indeterminacy in dynamic models with externalities[END_REF], endogenous fluctuations cannot occur within an aggregate model if labor is inelastic, even in the presence of strong externalities. 4 Nishimura et al. [START_REF] Nishimura | Indeterminacy in aggregate models with small externalities: an interplay between preferences and technology[END_REF] show that this is a generic condition for obtaining local indeterminacy in one-sector models. 5 Most econometric analyses available in the literature conclude that the wage elasticity of labor lies within the range (0, 0.3) for men and to (0.5, 1) for women (see Blundell and MaCurdy [START_REF] Blundell | Labour supply: a review of alternative approaches[END_REF]). 6 D'Autume and Michel [START_REF] Michel | Endogenous growth in Arrow's learning by doing model[END_REF] consider the original formulation by Arrow in which society's stock of knowledge, measured as the cumulative gross investment from -∞, acts as an externality in the production of all firms. They prove that endogenous growth can occur as a result of this. 7 Nevertheless, in the present paper, we do consider an infinitely-lived representative individual. For the dynamics of an economy with learning-by-doing externalities and a continuum of finitely-lived individuals, see d'Albis and Augeraud-Véron [START_REF] Albis | Competitive growth in a life-cycle model: Existence and dynamics[END_REF].

mulative process of gross investment is not internalized within the agents' decisions. Some papers have already studied functional differential equations with Ramsey-type aggregate models, notably with time-to-build investment.

As initially shown by Kalecki [START_REF] Kalecki | A macroeconomic theory of the business cycle[END_REF], some production lag is a possible source of aggregate fluctuations. Benhabib and Rustichini [START_REF] Benhabib | Vintage capital, investment and growth[END_REF] and Boucekkine et al. [START_REF] Boucekkine | Replacement echoes in the vintage capital growth model[END_REF] thus show that vintage capital leads to oscillatory dynamics governed by replacement echoes. 8 More recently, Bambi [START_REF] Bambi | Endogenous growth and time-to-build: The AK case[END_REF] considers an endogenous growth model based on some AK technology with time-to-build; his work shows that damped fluctuations occur, but that persistent endogenous fluctuations through a Hopf bifurcation are ruled out. A similar result has been obtained by Bambi and Licandro [START_REF] Bambi | In)determinacy and time-tobuild[END_REF] in the Benhabib and Farmer [START_REF] Benhabib | Indeterminacy and increasing returns[END_REF] model augmented to include time-to-build. 9The main difficulty of the use of vintage capital comes from the fact that since the production of the final good depends on the lagged capital stock, the optimality conditions are formulated as a system of mixed functional differential equations in which delayed (the capital stock) and advanced (the shadow price) terms are considered simultaneously. 10 Besides the additional complexity involved in studying this kind of differential equations, persistent fluctuations derived from a Hopf bifurcation do not appear to be an outcome. 11 In our paper, introducing a lagged capital stock through a Romer-type externality leads to optimality conditions formulated as a system of delay functional differential equations. 12 aggregate model with non-linear utility and vintage capital but do not provide any formal proof of its existence and do not discuss the stability of the bifurcating solutions. 10 Rustichini [START_REF] Rustichini | Hopf bifurcation for functional differential equations of mixed type[END_REF] considers a two-sector optimal growth model in which delays are introduced on both the control (investment and output) and state (capital stock) variables and derives a system of mixed functional differential equations. He shows that endogenous fluctuations can occur through a Hopf bifurcation. 11 Asea and Zak [START_REF] Asea | Time-to-build and cycles[END_REF] consider an exogenous growth model with time-to-build and claim that the steady state can exhibit Hopf cycles. However, their result is puzzling because a time-to-build assumption should lead to a system of mixed functional differential equations with both delay and advance, whereas they only consider delay in their model. 12 Almost all papers dealing with functional differential equations use control theory to derive the optimality conditions. When a lagged state variable is considered, the corresponding shadow price appears with an advance as a result of perfect foresight. In a recent paper, Fabbri and Gozzi [START_REF] Fabbri | Solving optimal growth models with vintage capital: the dynamic programming approach[END_REF] show that if the optimality conditions are derived using the dynamic programming approach, functional differential equations with delay only

From a mathematical point of view, the main purpose of our paper is to demonstrate the existence of a Hopf bifurcation, study the stability of the closed orbits and characterize the dynamics on the center manifold. The very first results on Hopf bifurcation theory date back to 1971 with a contribution by Chafee [START_REF] Chafee | A bifurcation problem for a functional differential equation of finitely retarded type[END_REF], who considered a situation where the origin remains uniformly asymptotically stable. According to Hale [START_REF] Hale | Introduction to Functional Differential Equations[END_REF], the first proof of the Hopf bifurcation theorem for functional differential equations was presented by Chow and Mallet Paret [START_REF] Chow | Integral averaging and bifurcation[END_REF]. Since then, much progress has been made, notably by Hassard et al. [START_REF] Hassard | Theory and Application of Hopf Bifurcation[END_REF] who provide an algorithm to compute coefficients that determine the stability of the periodic orbits.

We apply this methodology to an aggregate growth model with learningby-doing externalities in order to show the existence of a Hopf bifurcation and to establish the conditions under which the equilibrium paths converge towards the periodic solution. In particular, we show that persistent endogenous fluctuations can occur, first without considering endogenous labor and external effects coming from the labor supply, and second with a standard CES preferences and technology characterized by small values for the elasticity of intertemporal substitution in consumption and a capital-labor elasticity of substitution in line with recent empirical estimates. We hence prove that a simple aggregate model may generate business cycle fluctuations under plausible parameterization of the fundamentals and mild elasticity of the output with restpect to an externality based on a Arrow-type learning-by-doing process.

The economic intuition for such fluctuations is the following. Let us suppose, for instance, that the initial level of experience is low. Then, private returns to capital are high as well as the level of investment. This increases the experience and therefore reduces the returns to capital. Investments are consequently slowed down. However, due to the memory function, experience is reduced, which subsequently increases the return to capital. Permanent fluctuations are then possible, whereas they are ruled out with Romer [START_REF] Romer | Increasing returns and long-run growth[END_REF]'s assumption.

The paper is organized as follows: Section 2 presents the model and defines the intertemporal equilibrium. Section 3 contains the main results, i.e., proof of the existence of a Hopf bifurcation, analysis of the local stability properties of the periodic orbits, and presentation of a numerical example.

are obtained since the shadow price equation is no longer taken into account.

Section 4 contains concluding comments. Finally, the stability Theorem proposed by Hassard et al. [START_REF] Hassard | Theory and Application of Hopf Bifurcation[END_REF] is provided in the Appendix together, with all our proofs.

The model 2.1 The production structure

Let us consider a perfectly competitive economy in which the final output is produced using capital K and labor L. Although production takes place under constant return-to-scale, we assume that each of the many firms benefits from positive externalities due to learning-by-doing effects. We consider indeed that by using capital over time, agents increase their experience and are thus able to increase their productivity. Contrary to most contributions in the literature derived from Romer [START_REF] Romer | Increasing returns and long-run growth[END_REF], in which the average level of capital is used as a proxy of experience, we assume, in accordance with Arrow [START_REF] Arrow | The economic implications of learning by doing[END_REF], that cumulative gross investment (cumulative production of capital goods) is a better index of experience. "Each new machine produced and put into use is capable of changing the environment in which production takes place, so that learning is taking place with continually new stimuli " (Arrow [START_REF] Arrow | The economic implications of learning by doing[END_REF], page 157.). However, like Romer [START_REF] Romer | Increasing returns and long-run growth[END_REF], we consider that these learning-by-doing effects enter the production process as external effects.

The production function of a representative firm is thus F (K, L, e), where F (K, L, .) if homogeneous of degree one with respect to (K, L) and e ≥ 0 represents the externalities. Denoting the capital stock per unit of labor by k = K/L for any L = 0, we define the production function in intensive form as f (k, e).

Assumption 1. f (k, e) is C r over R 2 ++ for r ≥ 4 with f 1 (k, e) > 0, f 11 (k, e) < 0, f 2 (k, e) > 0 over R 2 ++ , lim x→0 f 1 (x, .) = +∞ and lim x→=∞ f 1 (x, .) = 0.
The interest rate r(t) and the wage rate w(t) then satisfy:

r(t) = f 1 (k(t), e(t)) -δ, w(t) = f (k(t), e(t)) -k(t)f 1 (k(t), e(t)) (1)
with δ ≥ 0 being the depreciation rate of capital. We also compute the share of capital as a proportion of total income:

s(k, e) = kf 1 (k,e) f (k,e) ∈ (0, 1) (2)
the elasticity of capital-labor substitution:

σ(k, e) = -(1-s(k,e))f 1 (k,e) kf 11 (k,e) > 0 (3)
and the following share and elasticity related to the externalities e: ε e (k, e) = ef 2 (k,e) f (k,e) , ε ke (k, e) = ef 12 (k,e)

f 1 (k,e) (4)
The share ε e (k, e) provides a measure of the size of the externalities and ε ke (k, e) is the elasticity of the rental rate of capital with respect to e. We will consider positive but small externalities, as shown in the next assumption:

Assumption 2. ε e (k, e) ∈ (0, 1 -s(k, e)) and ε ke (k, e) ≥ 0.

The first part of Assumption 2 implies that the externalities are small enough to be compatible with a demand for capital that is decreasing with respect to the rental rate, and the second part implies that the marginal productivity of capital is an increasing function of the externalities.

Considering the aggregate consumption C(t), the capital accumulation equation is then

K(t) = L(t)f (k(t), e(t)) -δK(t) -C(t) (5)
with K(0) = K 0 , which is given. For L(t) = e nt L(0) with n ≥ 0 and L(0) = L 0 , the capital accumulation equation in per capita terms becomes:

k(t) = f (k(t), e(t)) -(δ + n)k(t) -c(t) (6)
As explained previously, we assume that the externalities are generated by a learning-by-doing process in the sense described by Arrow [START_REF] Arrow | The economic implications of learning by doing[END_REF], and correspond to the per capita cumulative gross investment, namely

Assumption 3. e (t) = t t-τ k (s) + (δ + n)k (s) ds ≥ 0, with t ≥ τ ≥ 0.
The parameter τ is exogenous and represents a memory effect. We assume indeed that workers improve their experience by using capital over time but their memory is limited in the sense that after some time τ , experiences that are too old are forgotten. It is worth noting that a different formulation for the depreciation of memory could be considered. For instance Boucekkine et al. [START_REF] Boucekkine | Crecimiento económicos y generaciones de capital[END_REF] assume an exponential depreciation rate.

Remark 1. Assumption 3 encompasses the Ramsey [START_REF] Ramsey | A mathematical theory of saving[END_REF] model when δ = n = τ = 0 and the Romer [START_REF] Romer | Increasing returns and long-run growth[END_REF] model when δ = n = 0 and τ → +∞ as particular cases

Preferences and intertemporal equilibrium

The economy is populated by a large number of identical infinitely-lived agents. We assume without loss of generality that the total population is constant and normalized to one, i.e. n = 0 and N = 1. At each period, a representative agent supplies a fixed amount of labor l = 1 and derives utility from consumption c according to a function u(c) that satisfies:

Assumption 4. u(c) is C r over R + for r ≥ 2 with u (c) > 0, u (c) < 0, lim x→0 u (x) = +∞ and lim x→+∞ u (x) = 0.
We then define the elasticity of intertemporal substitution in consumption c ∈ (0, +∞) as follows:

c (c) = -u (c) u (c)c (7)
Since N (t) = 1 for all t ≥ 0, we obtain L(t) = 1 and C(t) = c(t). The intertemporal maximization program of the representative agent is given by: max

c(t),k(t) +∞ t=0 e -ρt u(c(t))dt s.t. k(t) = f (k(t), e(t)) -δk(t) -c(t) k(t) = k 0 (t) for t ∈ [-τ, 0] and {e(t)} t≥0 given (8)
where ρ > 0 denotes the discount factor. By substituting c(t) from the capital accumulation equation into the utility function we derive the following equivalent dynamic optimization program max

k(t) +∞ t=0 e -ρt u f (k(t), e(t)) -δk(t) -k(t) dt s.t. (k(t), k(t)) ∈ D({e(t)} t≥0 ) k(t) = k 0 (t) for t ∈ [-τ, 0] and {e(t)} t≥0 given (9)
with

D({e(t)} t≥0 ) = (k(t), k(t)) ∈ R + × R|f (k(t), e(t)) -δk(t) -k(t) ≥ 0, ∀e(t) ≥ 0
being the convex set of admissible paths. An interior solution to problem (9) satisfies the Euler equation

(f 1 (k(t), e(t)) -δ) k(t) + f 2 (k(t), e(t)) ė(t) -k(t) u (c(t)) + [f 1 (k(t), e(t)) -δ -ρ] u (c(t) = 0 (10)
and the transversality condition

lim t→+∞ u (c(t)) k (t) e -ρt = 0 (11)
for all given e(t) ≥ 0. At the individual level, a solution of the Euler equation ( 10) is thus a path of capital stock parameterized by a given path of externalities, namely k(t, {e(t)} t≥0 ). At the aggregate level, as the externalities are defined according to Assumption 3, an equilibrium path has to satisfy a fixed-point property such that

e (t) = t t-τ k (s, {e(t)} t≥0 ) + δk (s, {e(t)} t≥0 ) ds
for all t ≥ 0. Assuming that such a fixed-point problem has a solution, 13 the capital dynamics are characterized by the following non-linear functional differential equation with distributed delays:

k (t) = f 1 k (t) , k (t) -k (t -τ ) + δ t t-τ k (s) ds -δ k (t) + f 2 k (t) , k (t) -k (t -τ ) + δ t t-τ k (s) ds × k (t) -k (t -τ ) + δ [k (t) -k (t -τ )] + u (f(k(t),k(t)-k(t-τ)+δ R t t-τ k(s)ds)-δk(t)-k(t)) u (f(k(t),k(t)-k(t-τ)+δ R t t-τ k(s)ds)-δk(t)-k(t)) × f 1 k (t) , k (t) -k (t -τ ) + δ t t-τ k (s) ds -δ -ρ (12)
together with the transversality condition [START_REF] Benhabib | The Hopf bifurcation and the existence and stability of closed orbits in multisector models of optimal economic growth[END_REF]. Our strategy now consists of focusing on the existence of an interior steady state in the neighborhood of which an equilibrium path exists by continuity.

Steady state and characteristic polynomial

We consider the dynamics in the neighborhood of the steady state. Along a stationary path k(t) = k for any t ≥ 0, Assumption 3 implies that e(t) = ē = δτ k. An interior steady state is thus a k that solves: 13 In a continuous-time framework, the existence of a solution of this kind of fixed-point problem is a difficult issue. When e(t) is assumed to be given by the aggregate capital stock K(t), the existence of a solution is studied in Romer [START_REF] Romer | Increasing returns and long-run growth[END_REF] (see also d'Albis and Le Van [START_REF] Albis | Existence of a competitive equilibrium in the Lucas (1988) model without physical capital[END_REF] for a similar analysis in a simplified version of the Lucas [START_REF] Lucas | On the mechanics of economic development[END_REF] model without physical capital).

f 1 k, δτ k = δ + ρ (13)
and the corresponding stationary consumption level is

c = f k, δτ k -δ k > 0 (14)
In order to obtain the existence of an interior steady state, we introduce an additional assumption on technology:

Assumption 5. The aggregate production function F(x) = f (x, δτ x) satisfies the following properties: i) There exists x > 0 such that F(x) > x for 0 < x < x and F(x)

< x for x > x. ii) f 1 (x, δτ x) < 1 and lim x→0 f 1 (x, δτ x) > δ + ρ.
We immediately obtain:

Proposition 1. Let Assumptions 1-5 hold and δ +ρ < 1. In this case, there exists a steady state. If, moreover, f 1 (x, δτ x) is a non-increasing function of x, the steady state is unique.

In order to simplify the analysis and consider the elasticity of intertemporal substitution in consumption c (c) as the bifurcation parameter, we focus on a particular but standard class of utility functions: Assumption 6. The utility function is CRRA, i.e., c (c) = c for any c > 0.

Equation [START_REF] Benhabib | Competitive equilibrium cycles[END_REF] may then be rewritten as:

k (t) = f 1 k (t) , k (t) -k (t -τ ) + δ t t-τ k (s) ds -δ k (t) + f 2 k (t) , k (t) -k (t -τ ) + δ t t-τ k (s) ds × k (t) -k (t -τ ) + δ [k (t) -k (t -τ )] -c f k (t) , k (t) -k (t -τ ) + δ t t-τ k (s) ds -δk (t) -k (t) × f 1 k (t) , k (t) -k (t -τ ) + δ t t-τ k (s) ds -δ -ρ (15)
Let us use the following notations:

f = f k, δτ k , f 1 = f 1 k, δτ k , f 2 = f 2 k, δτ k , f 11 = f 11 k, δτ k , f 12 = f 12 k, δτ k Lemma 1. The characteristic equation is D (λ) = 0 with D (λ) = λ 2 -ρλ + c cf 11 -f 2 λ 2 + (f 2 δ -c cf 12 ) λ -c cf 12 δ 0 -τ e λs ds (16)
Using the shares and elasticities (2), ( 3), ( 4) and [START_REF] Bambi | Endogenous growth and time-to-build: The AK case[END_REF], all evaluated at the steady state, we obtain the following expression of ( 16):

D (λ) = λ 2 -ρλ -c(1-s)(δ+ρ)[δ(1-s)+ρ] sσ -εe(δ+ρ) sδτ λ 2 + εe(δ+ρ) sτ -εeε ke (δ+ρ)[δ(1-s)+ρ] sδτ λ -εeε ke (δ+ρ)[δ(1-s)+ρ] sτ 0
-τ e λs ds = 0 When τ > 0, the characteristic equation is transcendental and there exist an infinite number of roots, some of which are complex with negative real parts.

Remark 2. If τ = 0, we have f 2 = f 12 = 0. There is no externality and the characteristic equation is written as:

D (λ) = λ 2 -ρλ -c(1-s)(δ+ρ)[δ(1-s)+ρ]
sσ There are two real roots of opposite sign and the steady state is saddle-point stable. Moreover, if δ = 0 and τ → +∞, the characteristic equation becomes

D (λ) = λ 2 -(ρ + f 2 ) λ + c c (f 11 + f 12
) . With small externalities, i.e. f 11 + f 12 ≤ 0, the characteristic roots are always real with opposite signs. On the contrary, with strong externalities, i.e. f 11 + f 12 > 0, complex characteristic roots can occur but the real part is always non-zero and the Hopf bifurcation is ruled out.

We first derive a conclusion on some characteristic root with an additional condition on f 1 that ensures the uniqueness of the steady state:

Lemma 2. If f 1 (x, δτ x) is a non-increasing function of x, then D (λ) has at least one positive real root λ. Remark 3. If f 1 (k, e) is homogenous of degree θ < 0, Lemma 2 holds as θf 1 /k = f 11 + f 12 δτ < 0. With a general production function, this property is satisfied if ε ke < (1 -s)/σ.
According to Dieckman [START_REF] Diekmann | Delay Equations: Functional-, Complex-, and Nonlinear Analysis[END_REF], Lemma 2 implies that the limit cycle is unstable in the initial state of continuously differentiable functions on [-τ, 0]. As the transversality condition rules out divergent paths, the initial conditions should be chosen to belong to the direct sum of the stable space and center space. In the following, we will focus on the problem of stability on the center manifold.

Adding the extra root λ = 0 and letting ∆ (λ) = λD (λ), ∆ (λ) can then be rewritten as a third-order quasi-polynomial

∆ (λ) = P (λ) + Q (λ) e -λτ ( 17
) with P (λ) = λ 3 + p 2 λ 2 + p 1 λ + p 0 , Q (λ) = q 2 λ 2 + q 1 λ + q 0 and q 0 = u u f 12 δ = -cεke(δ+ρ)[δ(1-s)+ρ] sτ = -p 0 q 1 = f 2 δ + u u f 12 = εe(δ+ρ) sτ -cεke(δ+ρ)[δ(1-s)+ρ] sδτ q 2 = f 2 = εe(δ+ρ) sδτ = -(ρ + p 2 ) p 1 = -u u f 11 + q 1 = -c(1-s)(δ+ρ)[δ(1-s)+ρ] sσ + q 1 ( 18
)
This kind of quasi polynomial has been studied by Xiao and Cao [START_REF] Xiao | Stability and Hopf bifurcation in a delayed competitive web sites model[END_REF] and Crauste [START_REF] Crauste | Étude mathématique d'équations aux dérivées partielles hyperboliques modélisant les processus de régulation des cellules sanguines -Applications aux maladies hématologiques cycliques[END_REF]. Ours is a special case of their with p 0 = -q 0 . However, there is one major difference, in that the bifurcation parameter that they choose (which is the delay) appears in the coefficients (p i ) i=0..2 and (q i ) i=0..2 .

Endogenous business cycle fluctuations

The existence of business cycle fluctuations is obtained through the existence of a Hopf bifurcation giving rise to periodic cycles. The analysis is conducted in two steps: first, we study the existence of a Hopf bifurcation and second we provide conditions for the occurrence of locally-stable periodic cycles.

Hopf bifurcation: existence

This first part of the analysis concerns conditions that ensure the existence of a critical value H c > 0 for the elasticity of intertemporal substitution in consumption such that when c = H c , a pair of purely imaginary roots is the solution of the characteristic equation. Let us then consider the following expression:

ψ = c 2(1-s)(δ+ρ) 2 [δ(1-s)+ρ] 2 s 2 σδτ (1-s)δτ 2σ -ε ke ( c -c ) (19) with c = δsσ(ε ke ρ+(1-s)εe δ+ρ sσ ) (1-s)(δ+ρ)[δ(1-s)+ρ] " ε ke - (1-s)δτ 2σ " (20)
With Remark 3 in mind, we introduce the following restrictions:

Assumption 7. τ < 2/δ, ε ke ∈ ((1 -s)δτ /2σ, (1 -s)/σ) and c > c .
Assumption 7 implies that f 1 (x, δτ x) is a non-increasing function of x around the steady state and ψ < 0. Therefore Lemma 2 holds. We then provide the following result: Theorem 1. Let Assumptions 1-7 hold. In this case, there exists a critical value H c > c such that when c = H c a Hopf bifurcation occurs generically.

In actual fact, H c is obtained as the value of c such that ±iω 0 is an imaginary root of [START_REF] Boucekkine | Crecimiento económicos y generaciones de capital[END_REF] with ω 0 defined such that Q (iω 0 ) = 0. Note that Assumption 7 shows that endogenous business cycle fluctuations are compatible with small externalities as ε ke is bounded above by (1 -s)/σ, but require, as usual, a sufficiently high elasticity of intertemporal substitution in consumption. In Section 3.3, however, we will show that the lower bound c can be quite low and the critical value H c can remain compatible with plausible values. It is also worth noting that this result still applies to endogenous labor as long as the wage elasticity of the labor supply remains close enough to zero.

Hopf bifurcation: stability

We are now interested in the stability and the direction of the periodic orbit. In the previous section, we obtained conditions in which system [START_REF] Chafee | A bifurcation problem for a functional differential equation of finitely retarded type[END_REF] undergoes a Hopf bifurcation at H c . For c = H c , the characteristic equation has a pair of eigenvalues ±iω 0 . Using the normal form theory and the center manifold according to Hassard et al. [START_REF] Hassard | Theory and Application of Hopf Bifurcation[END_REF], we are able to determine the Hopf bifurcation direction and the properties of the bifurcating periodic solution. Our strategy can be described by the following steps:

-We write our system of delay functional differential equations as a system of ordinary differential equations but on a particular space (of functions C 1 ([-τ, 0], R 2 )), on which we define a bilinear form.

-We look for the tangent space of the central manifold.

-We project the solution of the delay functional differential equations system on this tangent space and look at the dynamics that are described by an ordinary differential equation.

-Some coefficients of the Taylor approximation of this ordinary differential equation give the conditions for stability.

Let y (t) = k (t) -k and let us write equation ( 15) by considering the variable y instead of k. The resulting dynamic system admits 0 as a steady state. Let ϕ = (ϕ 1 , ϕ 2 ) t with y (t) = ϕ 2 (t) and dy(t)/dt = ϕ 1 (t). System (15) may be rewritten as:

dϕ 1 (t) dt = f 1 ϕ 2 (t) + k, X (t) -δ ϕ 1 (t) + f 2 ϕ 2 (t) + k, X (t) × [ϕ 1 (t) -ϕ 1 (t -τ ) + δ [ϕ 2 (t) -ϕ 2 (t -τ )]] -c f ϕ 2 (t) + k, X (t) -δϕ 2 (t) -δk -ϕ 1 (t) × f 1 ϕ 2 (t) + k, X (t) -δ -ρ dϕ 2 (t) dt = ϕ 1 (t) (21)
where

X (t) = ϕ 2 (t) -ϕ 2 (t -τ ) + δτ k + δ t t-τ ϕ 2 (s) ds
Let c = H c +ε, and C be the space of continuous functions φ : [-τ, 0] → R 2 . System (21) can be rewritten as:

φ (t) = G (ε, ϕ t ) (22)
A Taylor expansion in the neighborhood of the steady state allows us to split this system into linear and non linear parts:

Lemma 3. System ( 22) can be written as the following functional differential equation in C:

φ (t) = Λ ε ϕ t + F (ε, ϕ t ) ( 23
)
where ϕ t (θ) = ϕ (t + θ) and Λ ε : C → R 2 is given by

Λ ε ϕ = L H c + ε ϕ (0) + R H c + ε ϕ (-τ ) + M H c + ε 0 -τ ϕ (u) du with L ( c ) = -p 2 -p 1 1 0 , R ( c ) = -q 2 -q 1 0 0 , M ( c ) = 0 -p 0 0 0
the coefficients p i , q i being defined in [START_REF] Boucekkine | Vintage capital and the dynamics of the AK model[END_REF], and

F (ε, ϕ t ) = G (ε, ϕ t ) -Λ ε ϕ t .
A Taylor expansion of F is given in the proof of Lemma 3. Moreover Λ ε ϕ can be rewritten as follows:

Λ ε ϕ = 0 -τ dη (u) ϕ (u) with dη ( c , u) = L ( c ) δ (u) + R ( c ) δ (u + τ ) + M ( c ) du
To determine the normal form, the projection method is used as in Hassard et al. [START_REF] Hassard | Theory and Application of Hopf Bifurcation[END_REF]. We first need to compute the eigenvector relative to the eigenvalue iω 0 . Instead of writing the delay dynamic system, we use the infinitesimal generator expression, as is usually done for delay functional differential equations. For ϕ ∈ C 1 [-τ, 0] , R 2 , let us define

A (ε) ϕ =    dϕ(θ) dθ , if θ ∈ [-τ, 0[ L H c + ε ϕ (0) + R H c + ε ϕ (-τ ) + M H c + ε 0 -τ ϕ (u) du, if θ = 0 and G (ε) ϕ = 0, if θ ∈ [-τ, 0[ F (ε, ϕ) , if θ = 0 It follows that (23) is equivalent to φt = A (ε) ϕ t + G (ε) ϕ t (24)
Remark 4. A (ε) is the operator that is used in general despite A (ε) C C. Nevertheless, Adimy et al. [START_REF] Adimy | Stability of limit cycles in a pluripotent stem cell dynamics model[END_REF] have shown in a different framework that using a new operator on an appropriate space gives the same result.

To compute the normal form on the center manifold, we use the projection method, which is based on the computation of the eigenvector relative to iω 0 and the corresponding adjoint eigenvector. The computation of the adjoint eigenvector requires the definition of the adjoint space and adjoint operator of A (ε) .

We define the adjoint space C * of continuously differentiable functions χ : [0, τ ] → R 2 with the adjoint operator A * (ε).

A * (ε) χ =    -dχ(σ) dσ , for σ ∈ ]0, τ ] 0 -τ dη t (ε, t) χ (-t) for σ = 0 Remark 5. As dη (ε, t) is real, we have dη t (ε, t) = dη t (ε, t).
We consider the bilinear form

(v, u) = v t (0) u (0) - 0 θ=-τ θ ξ=0 v t (ξ -θ) dη H c + ε, θ u (ξ) dξ = v t (0) u (0) + 0 ξ=-τ v t (ξ + τ ) R H c + ε u (ξ) dξ - 0 θ=-τ θ ξ=0 v t (ξ -θ) M H c + ε u (ξ) dξdθ
The following Lemma now provides a basis for the eigenspace and adjoint eigenspace.

Lemma 4. Let q (θ) be the eigenvector of A associated with eigenvalue iω 0 , and q * (σ) be the eigenvector associated with -iω 0 . Then

q (θ) = iω 0 1 e iω 0 θ , q * (θ) = u 1   1 - " p 1 +q 1 e -iω 0 τ -ip 0 1-e -iω 0 τ ω 0 " iω 0   e iω 0 θ with u 1 =   iω 0 + i (-p 1 -q 1 e iω 0 τ +ip 0 (-1+e iω 0 τ ) ω 0 ) ω 0 -τ (iq 2 ω 0 + q 1 )e -iω 0 τ -p 0 (-1+iω 0 τ e -iω 0 τ +e -iω 0 τ ) ω 2 0   -1
Remark 6. Computations lead to (q * , q) = 0.

Let ϕ t be a solution of equation ( 24) when ε = 0. We associate a pair (z, w) where z (t) = (q * , ϕ t )

Solutions ϕ t (θ) on the central manifold are given by

ϕ t = w (z, z, θ) + z (t) q (θ) + z (t) q (θ) (26)
Let us denote w (t, θ) = w (z, z, θ), where z and z are local coordinates for the center manifold in direction q * and q * , and F 0 (z, z) = F (0, w (z, z, 0) + 2Re (z (t) q (0))). Hassard et al. [START_REF] Hassard | Theory and Application of Hopf Bifurcation[END_REF] then show that the dynamics on the central manifold are the ones given in the following Lemma:

Lemma 5. The dynamics on the center manifold are given by

ż (t) = iω 0 + g (z, z) ẇ (t, θ) = A (0) w (t, θ) -2Re (g (z, z) q (θ)) if θ ∈ [-τ, 0) ẇ (t, θ) = A (0) w (t, θ) -2Re (g (z, z) q (θ)) + F 0 (z, z) if θ = 0
Let us consider a Taylor expansion of g (z, z) such that: g (z, z) = g 20

z 2 2 + g 11 zz + g 02 z 2 2 + g 21 z 2 z 2 + h.o.t.
According to Theorem 2 given in Section 5.1, we only need to compute (g 02 , g 11 , g 20 , g 21 ) to characterize the bifurcation, as

C 1 = i 2ω 0 g 20 g 11 -2|g 11 | 2 -1 3 |g 02 | 2 + g 21 2 , µ 2 = -Re(C 1 )
Re(λ (0))

τ 2 = - Im(C 1 )+µ 2 Im " λ (0) " ω 0 , β 2 = 2Re (C 1 ) (27)
Lemma 5 does indeed allow us to compute these coefficients explicitly.

A CES illustration

Let us consider an economy that is characterized by the CRRA utility function defined in Assumption 6 and by the following CES production function

f (k, e) = αk -ν + (1 -α)e -βν -1 ν
with α ∈ (0, 1) and ν > -1. The elasticity of capital-labor substitution is thus given by σ = 1/(1 + ν). We will assume in the following that ν > 0. This restriction ensures that over the business cycles, the labor share is countercyclical while the capital share is pro-cyclical. 14 Assumptions 3 and 5 are satisfied since:

x = (δτ ) β/(1-β) , f 1 (x, δτ x) = α, lim x→0 f 1 (x, δτ x) = +∞ It follows that there exists a unique steady state such that k = (δτ )

β 1-β " α δ+ρ " ν 1+ν -α 1-α 1 (1-β)ν (28)
We can easily derive from (2) and (4) that at the steady state:

s = α δ+ρ α ν 1+ν , ε e = β(1 -s), ε ke = β(1 + ν)(1 -s)(1 -α)
Considering a yearly calibration, we assume that the fundamental parameters are set to the following values: ν = 1, α = 0.5, δ = 10%, ρ = 0.0808, and τ = 0.1. It follows that the share of capital is, as usual, s = 30% and the elasticity of capital-labor substitution is σ = 0.5. Such a value for σ is in line with recent empirical estimates which show that σ is in the range of 0.4 -0.6. 15 The size of externalities in all the following simulations is contained between 15% and 20%, an interval which is in line with the estimations of Basu and Fernald [START_REF] Basu | Returns to scale in US production: estimates and implications[END_REF]. We also note that the learning-by-doing process is based on a rather small memory lag τ . But, in the following, we show that this small departure from the standard Ramsey model is enough to generate endogenous business cycle fluctuations. 16 While many standard RBC models such as that of Hansen [START_REF] Hartl | A simple proof of the monotonicity of the state trajectories in autonomous control problems[END_REF] or King, Plosser & Rebelo [START_REF] King | Production, growth and business cycles: II. New directions[END_REF] usually assume a unitary elasticity of intertemporal substitution in consumption, recent empirical estimates provide divergent views. On the one hand, Campbell [START_REF] Campbell | Asset prices, consumption and the business cycle[END_REF] and Kocherlakota [START_REF] Kalecki | A macroeconomic theory of the business cycle[END_REF] suggest the following plausible interval ∈ (0.2, 0.8). More recently, Vissing-Jorgensen [START_REF] Vissing-Jorgensen | Limited asset market participation and the elasticity of intertemporal substitution[END_REF] partially confirmed such findings by showing that the estimates of this elasticity are around 0.3 -0.4 for stockholders and around 0.8 -1 for bondholders, and are higher for households with larger asset holdings within these two groups. On the other hand, Mulligan [START_REF] Mulligan | Capital interest and aggregate intertemporal substitution[END_REF] repeatedly obtained estimates of one and above, i.e. around 1.1 -1.3, using different estimation methods. In the following simulations, we illustrate all of these different possible cases.

i) Let β = 0.286 and thus ε e ≈ 20%. We obtain H c ≈ 0.6 as the Hopf bifurcation value. The bifurcating periodic orbit solutions exist when 15 See Chirinko [START_REF] Chirinko | σ: the long and short of it[END_REF], Klum, McAdam and Willman [START_REF] Klump | Factor substitution and factor-augmenting technical progress in the United-States: A normalized supply-side approach[END_REF] and Chirinko, Fazzari and Meyer [START_REF] Chirinko | That elusive elasticity: a long-panel approach to estimating the capital-labor substitution elasticity[END_REF]. 16 Similar results can be obtained with larger values of τ .

These numerical illustrations prove that with standard values of the fundamental parameters, persistent endogenous fluctuations easily arise with an elasticity of intertemporal substitution in consumption that is sufficiently high but that still remains compatible with the recent empirical estimates of Campbell [START_REF] Campbell | Asset prices, consumption and the business cycle[END_REF] and Kocherlakota [START_REF] Kocherlakota | The equity premium: It's still a puzzle[END_REF]. It is also worth noting that similar results still apply to bifurcation values such that H c ∈ (0.5, 1.3) when different sizes of externalities are considered, or with endogenous labor as long as the wage elasticity of the labor supply remains close enough to zero, a property that is compatible with the empirical studies of the labor market. 17

Concluding comments

We have considered a one-sector Ramsey-type growth model with inelastic labor and learning-by-doing externalities based on cumulative gross investment, which is assumed, in accordance with Arrow [START_REF] Arrow | The economic implications of learning by doing[END_REF], to be a better index of experience. We have proven that a slight memory effect characterizing the learning-by-doing process and a small amount of externality are enough to generate business cycle fluctuations through a Hopf bifurcation if the elasticity of intertemporal substitution is high enough but remains within limits compatible with recent empirical estimates. Moreover, contrary to all the results available in the literature on aggregate models, we have shown that endogenous fluctuations are compatible with a zero wage elasticity of the labor supply.

One aspect that is missing from our analysis is a discussion of local indeterminacy and fluctuations based on self-fulfilling prophecies. It is indeed well-known that in Ramsey-type continuous-time aggregate models with standard externalities, i.e. leading to necessary conditions given by ordinary differential equations, if the two characteristic roots have negative real parts, there exists a continuum of equilibrium paths that converge towards the steady state. As a result, the existence of a Hopf bifurcation is intimately related to the existence of local indeterminacy. 18 In infinite dimensional problems, Hopf bifurcations are also related to indeterminacy but the analogy with the "sink case" has still not been proven when there are an infinite number of characteristic roots with negative real parts. This is an interesting field for future research.

Appendix

5.1 The theorem of Hassard et al. [START_REF] Hassard | Theory and Application of Hopf Bifurcation[END_REF] Let us consider the non-linear delay differential equation:

k (t) = g k (t) , k (t -τ ) , k (t) , k (t -τ ) , t t-τ k (s) ds, c , (29)
where k and k stand for the first and second derivative with respect to time respectively. Let us suppose that there exists a steady state k. A simple change of variables allow us to rewrite the dynamics with a zero steady state:

φ (t) = L ( c ) ϕ (t) + R ( c ) ϕ (t -τ ) + M ( c ) t t-τ ϕ (u) du + f (ϕ (t) , ϕ (t -τ ) , c ) , (30)
where ϕ ∈ R 2 , and L ( c ) , R ( c ) and M ( c ) are 2 × 2 real matrices and c ∈ R. Our aim is to study the local dynamics around the steady state.

The linear part of equation ( 30) is

φ (t) = L ( c ) ϕ (t) + R ( c ) ϕ (t -τ ) + M ( c ) t t-τ ϕ (u) du, (31)
In the following, we will write ω 0 = q( H c ). Conditions (32) are very similar to those needed for ordinary differential equations (ODE). The first one gives the existence of a bifurcating branch and the second one is a transversality condition that ensures the local uniqueness of cycles.

The stability analysis of the cycles is based on the projection on the central manifold and the computation of a normal form as for high-dimensional ODEs. The method that we will use to compute the dynamics on the normal form is based on Hassard et al. [START_REF] Hassard | Theory and Application of Hopf Bifurcation[END_REF]. Before computing the eigenvector q relative to the imaginary root and the adjoint eigenvector q * , the initial system has to be written as a functional differential equation in C where C is the space of a continuously differentiable function φ : [-τ, 0] → R 2 . We consider the bilinear form

v, u = v t (0) u (0) + 0 ξ=-τ v t (ξ + τ ) R H c + ε u (ξ) dξ - 0 θ=-τ θ ξ=0 v t (ξ -θ) M H c + ε u (ξ) dξdθ
For each solution φ t of the functional differential equation, we associate a pair (z, w) where z (t) = (q * , φ t ) and w (t, θ) = φ t (θ) -q (θ) z (t) -q (θ) z (t) = φ t (θ) -2Re (q (θ) z (t)) Hassard et al. [START_REF] Hassard | Theory and Application of Hopf Bifurcation[END_REF] prove that the dynamics on the central manifold solve an ordinary differential equation ż (t) = iω 0 z (t) + g (z, z). Let us write a Taylor approximation of the last term of this ODE:

g (z, z) = g 20 z 2 2 + g 11 zz + g 02 z 2 2 + g 21 z 2 z 2 + ... (33)
where

g ij belongs to C. Let us define C 1 = i 2ω 0 g 20 g 11 -2|g 11 | 2 -1 3 |g 02 | 2 + g 21 2 , µ 2 = -Re(C 1 )
Re(λ (0))

τ 2 = - Im(C 1 )+µ 2 Im " λ (0) " ω 0 , β 2 = 2Re (C 1 ) (34)
Hassard et al. [START_REF] Hassard | Theory and Application of Hopf Bifurcation[END_REF] then provide the following Theorem:

Theorem 2. When c = H c the system exhibits a Hopf bifurcation. The stability of the periodic solution is determined by formula [START_REF] King | Production, growth and business cycles: II. New directions[END_REF].

1. µ 2 determines the direction of the Hopf bifurcation. If µ 2 > 0 then the Hopf bifurcation is supercritical and the bifurcating periodic solutions exist for H c > c . 2. β 2 determines the stability of the bifurcating periodic solutions. If β 2 < 0 then the bifurcating periodic solutions are stable 3. Period T of the periodic solutions is given by

T = 2π ω 0 1 + τ 2 ( c -H c ) 2 + O ( c -H c ) 4
According to this theorem, we must first prove the existence of a Hopf bifurcation value H c that satisfies condition [START_REF] Hassard | Theory and Application of Hopf Bifurcation[END_REF], then provide a stability analysis of the cycle by computing g 20 , g 11 , g 02 , and g 21 .

Proof of Lemma 1

Linearizing system (15) around the steady state k and defining

k (t) = k + εx (t) leads to ẍ (t) = [f 1 -δ + f 2 ] ẋ (t) -f 2 ẋ (t -τ ) + [f 2 δ -c c (f 11 + f 12 )] x (t) -[f 2 δ -c cf 12 ] x (t -τ ) -c cf 12 δ t t-τ x (s) ds (35)
The characteristic equation D (λ) = 0 is obtained by replacing x (t) = x (0) e λt and rearranging using f 1 = δ + ρ.

Proof of Lemma 2

From the definition of D (λ) , we get lim λ→∞ D (λ) = ∞, and D (0) = ε c c (f 11 + τ δf 12 ) , which is negative when f 1 (x, xδτ ) is a non increasing function of x. The result follows.

Proof of Theorem 1

The proof of the theorem is given through the next three lemmas. Lemma 5.1. Under Assumption 7, there exists q > 0 such that |Q(iq)| = 0.

Proof : We study the occurrence of imaginary roots of the characteristic equation. Let λ = p + iω and then rewrite equation D (λ) = 0 such that:

-iω 3 -p 2 ω 2 + iωp 1 + p 0 + -q 2 ω 2 + iq 1 ω + q 0 (cos (ωτ ) -i sin (ωτ )) = 0
We are looking for ω 0 > 0 such that Q (iω 0 ) = 0. Separating real and imaginary parts, we have p 2 ω 0 -p 0 = q 0 -q 2 ω 2 0 cos (ω 0 τ ) + q 1 ω 0 sin (ω 0 τ ) -ω 3 0 + p 1 ω 0 = q 0 -q 2 ω 2 0 sin (ω 0 τ ) -q 1 ω cos (ω 0 τ ) Squaring both sides of the previous equation yields to

ω 4 0 + p 2 2 -2p 1 -q 2 2 ω 2 0 + p 2 1 + 2p 0 ρ -q 2 1 = 0 which rewrites: x 2 + 2ηx + ψ = 0 (36) where x = ω 2 0 and η = -c c (f 11 + f 12 ) + ρ 2 2 + (ρ + δ) f 2 ψ = [-c cf 11 ] 2 -2 c cf 11 [f 2 δ -c cf 12 ] + 2ρδ c cf 12 (37)
The discriminant of ( 36) is ∆ = η 2 -ψ and the roots are x 1,2 = -η ± η 2 -ψ. A first condition for the existence of a real root is ∆ ≥ 0. Then there are two cases depending on the sign of ψ:

-if ψ < 0 then ∆ ≥ 0 and there exists a unique positive real root for any sign of η.

-if ψ ≥ 0, then the existence of a positive real root requires η < 0 and η 2 ≥ ψ.

Note that using the shares and elasticities (2), ( 3), ( 4) and ( 7), η and ψ can be written after straightforward simplifications as

η = c (δ + ρ) δ(1-s)+ρ sδτ (1-s)δτ σ -ε ke + (δ+ρ) 2 sδτ ε e + ρ 2 2 ψ = 2 c(1-s)(δ+ρ) 2 [δ(1-s)+ρ] 2 " (1-s)δτ 2σ -ε ke " 0 @ c- δsσ ( ε ke ρ+(1-s)εe δ+ρ sσ ) (1-s)(δ+ρ)[δ(1-s)+ρ] " ε ke - (1-s)δτ 2σ « 1 A s 2 σδτ
We then consider Assumption 7 which implies that ψ < 0. It follows that the positive root of

x 2 + 2ηx + ψ = 0 is x 1 = -η + η 2 -ψ.
As η and ψ are functions of c , let us denote ω 0 = ω ( c ) = √ x 1 . We also have cos (ω 0 τ ) = ω 4 0 (q 1 -p 2 q 2 )+ω 2 0 (p 0 q 2 -p 1 q 1 +q 0 p 2 )-p 0 q 0 " (q 1 ω 0 ) 2 +(q 0 -q 2 ω 2 0 )

2 "

sin (ω 0 τ ) = ω 5 0 q 2 +ω 3 0 (p 2 q 1 -p 1 q 2 -q 0 )+ω 0 (p 1 q 0 -p 0 q 1 ) " (q 1 ω 0 ) 2 +(q 0 -q 2 ω 2 0 )

2 " [START_REF] Mulligan | Capital interest and aggregate intertemporal substitution[END_REF] It follows that the bifurcation value H c is obtained as the value of c that solves the following equation:

cos τ √ x 1 ≡ G 1 ( c ) =
x 2 1 (q 1 -p 2 q 2 )+x 1 (p 0 q 2 -p 1 q 1 +q 0 p 2 )-p 0 q 0 (q 2 1 x 1 +(q 0 -q 2 x 1 ) 2 )

≡ G 2 ( c ) (39)
Recall from Assumption 7 that c ∈ ( c , +∞). We can show easily that

lim c→ c G 1 ( c ) = lim c→ c G 2 ( c ) = 1
We can also compute a series expansion of G 2 ( c ) in order to compute the limit when c → +∞. We obtain:

G 2 ( c ) = -1 + δsτ c h 2q 2 " ε ke - (1-s)δτ 2σ " +δε ke +ρ " ε ke - (1-s)τ 2σ "i 2 ε 2 ke (δ+ρ)[δ(1-s)+ρ] " ε ke - (1-s)δτ 2σ " + o 1 2 c
It follows that under Assumption 7, lim c→+∞ G 2 ( c ) = -1 + . Moreover, as c increases from c to +∞, the function G 1 ( c ) oscillates continuously between 1 and -1. It follows that there necessarily exists at least an infinite number of solutions of equation ( 39) for c large enough.

Let us then consider the lowest solution of equation ( 39), denoted H c , which corresponds to the Hopf bifurcation value such that ±iω 0 is an imaginary root of [START_REF] Boucekkine | Crecimiento económicos y generaciones de capital[END_REF].

Lemma 5.2. ±iω 0 is generically a simple root.

Proof : If we suppose by contradiction that it is not a simple root, we have

P (iω 0 ) + Q (iω 0 ) -τ Q (iω 0 ) e -iω 0 τ = 0
separating imaginary and real part, and squaring each member, we have:

ω 4 0 τ 2 q 2 2 -9 + ω 2 0 6p 1 -4p 2
2 + 2τ q 2 (q 1 -τ q 0 ) + (2q 2 -τ q 1 ) 2 + (q 1 -τ q 0 ) 2 -p 2 1 = 0 As it is also a root of the characteristic equation, we also have

ω 4 0 + p 2 2 -2p 1 -q 2 2 ω 2 0 + p 2 1 + 2p 0 ρ -q 2 1 = 0 That implies -p 2 2 -2p 1 -q 2 2 + p 2 2 -2p 1 -q 2 2 2 -4 p 2 1 + 2p 0 ρ -q 2 1 =       -6p 1 -4p 2 2 + 2τ q 2 (q 1 -τ q 0 ) + (2q 2 -τ q 1 ) 2 - 6p 1 -4p 2 2 + 2τ q 2 (q 1 -τ q 0 ) + (2q 2 -τ q 1 ) 2 2 -4 (q 1 -τ q 0 ) 2 -p 2 1 τ 2 q 2 2 -9      
Such equality is non generic.

To complete the proof, we have to prove the transversality condition. Let H c be the value for c for which we have an imaginary root.

= -c c ∆ (λ, c ) = λ 3 + p 2 λ 2 + p 1 ( c ) λ + p 0 ( c ) + q 2 λ 2 + q 1 ( c ) λ + q 0 ( c ) e -λτ = 0
As iω 0 is a simple root, we can use the implicit function theorem

dλ d c | c= H c = - "" p 1 ( H c )λ+p 0 ( H c ) " + " q 1 ( H c )λ+q 0 ( H c ) " e -λτ " (P (λ)+(Q (λ)-τ Q(λ))e -λτ ) q H c = -ε ke (δ+ρ)(δ(1-s)+ρ) sδτ = q 1 H c -εe(δ+ρ) H c sτ p H c = -((1-s)(δ+ρ)(δ(1-s)+ρ)) sσ + q 1 H c -εe(δ+ρ) H c sτ p H c = p 1 H c + εe(δ+ρ) H c sτ q 0 = q 0 H c = -p 0 = -p 0 H c p 1 H c λ + p 0 H c + q 1 H c λ + q 0 H c e -λτ = p 1 H c + εe(δ+ρ) H c sτ λ + p 0 H c + q 1 H c -εe(δ+ρ) H c sτ λ + q 0 H c e -λτ = p 1 H c λ + p 0 H c + q 1 H c λ + q 0 H c e -λτ + 1 -e -λτ εe(δ+ρ) H c sτ λ = -λ 2 λ+p 2 +q 2 e -λτ -( 1-e -λτ ) λ δq 2 ! H c
Substituting λ = iω 0 , we get:

   p 1 H c λ + p 0 H c + q 1 H c λ + q 0 H c e -λτ    |λ=iω 0 = ω 2 H c   p 2 + q 2 cos (ω 0 τ ) -sin(ω 0 τ ) ω δq 2 +i ω 0 -q 2 sin (ω 0 τ ) + 1-cos(ω 0 τ ) ω 0 δq 2   Moreover P (λ) + Q (λ) -τ Q (λ) e -λτ = P (λ) + Q (λ) e -λτ + τ P (λ) = τ λ 3 + (3 + τ p 2 ) λ 2 + (2p 2 + τ p 1 ) λ + (p 1 + τ p 0 ) + e -λτ (2q 2 λ + q 1 ) P (λ) + Q (λ) -τ Q (λ) e -λτ |λ=iω 0 =         -(3 + τ p 2 ) ω 2 0 + (p 1 + τ p 0 ) + cos (ω 0 τ ) q 1 +2q 2 ω sin (ω 0 τ ) +i -τ ω 3 0 + (2p 2 + τ p 1 ) ω 0 -sin (ω 0 τ ) q 1 + 2q 2 ω 0 cos (ω 0 τ )         Let us then consider Re           -(3 + τ p 2 ) ω 2 0 + (p 1 + τ p 0 ) + cos (ω 0 τ ) q 1 +2q 2 ω 0 sin (ω 0 τ ) +i 0 @ -τ ω 3 0 + (2p 2 + τ p 1 ) ω 0 -sin (ω 0 τ ) q 1 + 2q 2 ω 0 cos (ω 0 τ ) 1 A p 2 +q 2 cos(ω 0 τ )- sin(ω 0 τ ) ω 0 δq 2 +i " ω 0 -q 2 sin(ω 0 τ )+ 1-cos(ω 0 τ ) ω 0 δq 2 "           =    -(3 + τ p 2 ) ω 2 0 + (p 1 + τ p 0 ) + cos (ω 0 τ ) q 1 +2q 2 ω 0 sin (ω 0 τ )    p 2 + q 2 cos (ω 0 τ ) -sin(ω 0 τ ) ω 0 δq 2 + -τ ω 3 0 + (2p 2 + τ p 1 ) ω 0 -sin (ω 0 τ ) q 1 + 2q 2 ω 0 cos (ω 0 τ ) ω 0 -q 2 sin (ω 0 τ ) + 1-cos(ω 0 τ ) ω 0 δq 2 = p 1 p 2 + q 1 q 2 + τ p 0 p 2 + 2δp 2 q 2 + τ δp 1 q 2 -τ ω 4 0 -ω 2 0 p 2 + τ ω 2 0 p 1 -τ δω 2 0 q 2 -τ ω 2 0 p 2 2 -2δq 2 2 + 3δω 0 q 2 -ω 0 q 1 -τ ω 0 p 1 q 2 + τ δω 0 p 2 q 2 + τ ω 3 0 q 2 -δ ω 0 p 1 q 2 -δ ω 0 q 1 q 2 -τ δ ω 0 p 0 q 2 sin τ ω 0 + p 1 q 2 + p 2 q 1 + τ p 0 q 2 -2δp 2 q 2 -τ δp 1 q 2 +2δq 2 2 -ω 2 0 q 2 + τ δω 2 0 q 2 -τ ω 2 0 p 2 q 2 cos τ ω 0 = H (ω 0 ) H (ω 0 ) rewrites H (ω 0 ) = H 0 + H 2 ω 2 0 + H 4 ω 4 0 + H 6 ω 6 0
where expressions of H i can be computed. As

ω 4 0 + p 2 2 -2p 1 -q 2 2 ω 2 0 + p 2 1 + 2p 0 ρ -q 2 1 = 0, H (ω 0 ) = H (ω 0 )
where H is defined by

H (ω) = H (ω) -H 6 ω 2 + H 4 - H 6 (p 2 2 -2p 1 -q 2
2 )

ω 4 + p 2 2 -2p 1 -q 2 2 ω 2 + p 2 1 + 2p 0 ρ -q 2 1 = A 2 ω 2 + A 0 So replacing ω 2 0 we can compute Re dλ d c | c= H c as a function of c Re dλ d c | c= j c = ϕ 0 + ϕ 1 j c + ϕ 2 j c 2
where ϕ 0 , ϕ 1 , ϕ 2 are independent of j c . Then there exists j c such that Re

Proof of lemma 3

Equation in the y variable writes

ÿ (t) = f 1 y (t) + k, y (t) -y (t -τ ) + δk + δ t t-τ y (s) ds -δ ẏ (t) + f 2 y (t) + k, y (t) -y (t -τ ) + δk + δ t t-τ y (s) ds × ẏ (t) -ẏ (t -τ ) + δ y (t) -y (t -τ ) -c f y (t) + k, y (t) -y (t -τ ) + δk + δ t t-τ y (s) ds -δk -δy (t) -ẏ (t) × f 1 y (t) + k, y (t) -y (t -τ ) + δk + δ t t-τ y (s) ds -δ -ρ
The linearization of the system at (0, 0, 0) is

φ1 (t) = [ρ + f 2 ] ϕ 1 (t) -f 2 ϕ 1 (t -τ ) + [f 2 δ -c c (f 11 + f 12 )] ϕ 2 (t) -[f 2 δ -c cf 12 ] ϕ 2 (t -τ ) -c cf 12 δ (ϕ 3 (t) -ϕ 3 (t -τ )) φ2 (t) = ϕ 1 (t) φ3 (t) = ϕ 2 (t) (40)
Let F : R×C → R and denote the partial derivatives of f as

f i = f i k, δτ k , f ij = f ij k, δτ k , f ijk = f ijk k, δτ k , f ijkl = f ijkl k, δτ k , i, j, k, l = 1, 2.
The following Lemma gives a Taylor expansion up to order three of

F ε, ϕ 1t (0) , ϕ 1t (-τ ) , ϕ 2t (0) , ϕ 2t (-τ ) , 0 -τ ϕ 2t (u) du . Lemma 5.4. Let ϕ 1t (0) , ϕ 1t (-τ ) , ϕ 2t (0) , ϕ 2t (-τ ) , 0 -τ ϕ 2t (u) du = (x 1 , x 2 , x 3 , x 4 , x 5 ). Then F ε, ϕ 1t (0) , ϕ 1t (-τ ) , ϕ 2t (0) , ϕ 2t (-τ ) , 0 -τ ϕ 2t (u) du = 5 i=1 5 j=i a ij x i x j + 5 i=1 5 j=i 5 m=j a ijm x i x j x m
with a ij and a ijm some coefficients that depend on the second, third and fourth order derivatives of the production function evaluated at the steady state. 20

Proof of lemma 4

As q (θ) is the eigenvector of A associated with eigenvalue iω 0 , q (θ) solves, for θ = 0 20 The expressions of these coefficients are available upon request. dq dθ = iω 0 q ⇒ q (θ) = q (0) e iω 0 θ For θ = 0, initial conditions write:

L H c q (0) + R H c q (-τ ) + M H c 0 -τ q (u) du = iω 0 q (0) Let q (0) = v = (v 1 , v 2 ) t .
Replacing the expression we first obtained in the second equation yields to

L H c q (0) + R H c ve -iω 0 τ + M H c v 1 -e -iω 0 τ iω 0 = iω 0 v that is -p 2 + q 2 e -iω 0 τ v 1 + p 1 + q 1 e -iω 0 τ + p 0 1-e -iω 0 τ iω 0 v 2 = iω 0 v 1 v 1 = iω 0 v 2
Substituting v 1 obtained in the second equation as a function of v 2 we have

-v 2 -ω 2 0 + p 2 + q 2 e -iω 0 τ iω 0 + p 1 + q 1 e -iω 0 τ + p 0 1-e -iω 0 τ iω 0 = 0 v 1 = iω 0 v 2 which rewrites v 2 [D (iω 0 )] = 0 v 1 = iω 0 v 2
As iω 0 is a root of the characteristic equation, we can choose v 2 as we want (for example v 2 = 1), so v is completely determined. Similarly we obtain: Let η (0) = u = (u 1 , u 2 ) t , the previous expression rewrites:

-u 1 p 2 + q 2 e -iω 0 τ + u 2 = iω 0 u 1 -u 1 p 1 + q 1 e -iω 0 τ + p 0 0 -τ e iω 0 u du = iω 0 u 2 Substituting u 2 in the first expression we have -u 1 +ω 2 0 + iω 0 p 2 + q 2 e -iω 0 τ + p 1 + q 1 e -iω 0 τ + p 0 0 -τ e iω 0 u du = 0 -u 1 p 1 + q 1 e -iω 0 τ + p 0 0 -τ e iω 0 u du = iω 0 u 2 u 1 D (iω 0 ) = 0 -u 1 p 1 + q 1 e -iω 0 τ + p 0 0 -τ e iωu du = iω 0 u 2

As iω 0 is a root of the characteristic equation, we can choose u 1 as we want, so u rewrites. u = u 1 1 i (p1+q1e -iω 0 τ +p 0 R 0 -τ e iωu du) ω 0

We now compute u 1 thanks to equation (q * , q) = 1, which leads to: As on the central manifold we have: ϕ t (θ) = w (t, θ) + 2Re (q (θ) z (t)) and q (θ) = (iω 0 , 1) T e iω 0 θ , coefficients of the solution can be expressed as: + z (t) e iω 0 θ + z (t) e -iω 0 θ + O (|z, z|) 21 The expressions of these functions are available upon request.

We use the preceding formula to compute (ϕ jt (.)) j=1..3 . Then replacing (ϕ jt (.)) j=1..3 in ( 41) we obtain coefficients φ 20 , φ 02 , φ 11 and φ 21 as in the lemma.

It is worth noting that φ 20 , φ 02 , φ 11 are obtained as constants, while φ 21 depends on w 20 (.) , w 11 (.), which will be computed later on.

Lemma 5.6. g 20 = u 1 φ 20 , g 11 = u 1 φ 11 , g 02 = u 1 φ 02. and g 21 = u 1 φ 21 .

Proof : As q * (0) = u 1   1, i p 1 + q 1 e -iω 0 τ + p 0 0 -τ e iωu du ω 0    and using g (z, z) = q * (0) F 0 (z, z) we obtained easily the result of this lemma.

To end the computation of coefficients (g 02 , g 11 , g 20 , g 21 ) , we need to compute w 11 (θ) and w 20 (θ). w 20 (θ) = E 1 e 2iω 0 θ + ig 20 q 0 q (0) e iω 0 θ + ig 02 3q 0 q (0) e -iω 0 θ w 11 (θ) = -ig 11 q 0 q (0) e iω 0 θ + ig 11 q 0 q (0) e -iω 0 θ + E 2

with

E 1 = 2iω 0 φ 20 [∆(2iω 0 )] φ 20 [∆(2iω 0 )]
and E 2 = -φ 11 (p2+q2e -2iτ ω 0 ) 0

Proof : We rewrite ẇ (t, θ) = Aw -2Re (g (z, z) q (θ)) if θ ∈ [-τ, 0) ẇ (t, 0) = Aw -2Re (g (z, z) q (0)) + F 0 (z, z) if θ = 0 as ẇ = Aw + H (z, z, θ)

And we consider a Taylor expansion of H (z, z, θ) = H 20 2 + h.o.t. As H (z, z, θ) = -g (z, z) q (θ) -g (z, z)q (θ) if θ ∈ [-τ, 0) that is H 20 (0) = -g 20 q (0) -g 02 q (0) + φ 20 0

H 11 (0) = -g 11 q (0) -g 11 q (0) + φ 11 0 Remembering the definition of A and (2iq 0 -A) w 20 (θ) = -g 20 q (θ) -g 02 q (θ) Aw 11 (θ) = g 11 q (θ) + g 11 q (θ) we have 2iω 0 w 20 (0) + g 20 q (0) + g 02 q (0) = L ( c ) w 20 (0) + R ( c ) w 20 (-τ ) + M ( c ) 0 -τ w 20 (u) du + φ 20 0 w 20 (0) = E 1 + ig 20 ω 0 q (0) + ig 02 3ω 0 q (0) w 20 (-τ ) = E 1 e -τ 2iω 0 + ig 20 ω 0 q (0) e -τ iω 0 + ig 02 3ω 0 q (0) e -τ iω 0 w 20 (u) = E 1 e 2iω 0 u + ig 20 ω 0 q (0) e iω 0 u + ig 02 3ω 0 q (0) e -iω 0 u

Using the fact that L H c q (0) -R H c q (-τ ) -M H c 0 -τ q (u) du = iω 0 q (θ) and that q (θ) is the eigenvector of A according to iω 0 , we derive Similarly, we have -p 2 + q 2 e -2iτ ω 0 E 1 2 + p 1 + q 1 e -2iτ ω 0 + p 0 0 -τ e 2iω 0 u du E (p2+q2e -2iτ ω 0 ) E 2 2 = 0 From all this we derive formula [START_REF] Fabbri | Solving optimal growth models with vintage capital: the dynamic programming approach[END_REF] and the result follows.

  c > H c and are orbitally stable. For any c in the right neighborhood of H c , the period of the bifurcating solutions is proportional to T ≈ 12.35. ii) Let β = 0.245 and thus ε e ≈ 17.2%. We obtain H c ≈ 1 as the Hopf bifurcation value. The bifurcating periodic orbit solutions exist when c > H c and are orbitally stable. For any c in the right neighborhood of H c , the period of the bifurcating solutions is proportional to T ≈ 4.03. iii) Let β = 0.244 and thus ε e ≈ 17.1%. We obtain H c ≈ 1.1 as the Hopf bifurcation value. The bifurcating periodic orbit solutions exist when c > H c and are orbitally stable. For any c in the right neighborhood of H c , the period of the bifurcating solutions is proportional to T ≈ 3.6. iv) Finally, let β = 0.243 and thus ε e ≈ 17%. We obtain H c ≈ 1.2 as the Hopf bifurcation value. The bifurcating periodic orbit solutions exist when c > H c and are orbitally stable. For any c in the right neighborhood of H c , the period of the bifurcating solutions is proportional to T ≈ 3.3.

  and its characteristic equation solves det D (λ, c ) = 0, where D (λ, c ) = λI -L ( c )+R ( c ) e -λτ +M ( c ) 0 -τ e -λu du. There exists a Hopf bifurcation if and only if there exists H c such that the characteristic equation D (λ, c ) = 0 has two simple imaginary roots λ ( c ) = p ( c ) ± iq ( c ) that cross the imaginary axis transversely at c = H c , 19 i.e.

Lemma 5 . 3 .

 53 Re dλ( c)Proof : Let us differentiate the following equation according to c , noting that c only appears in u u

η 1 1 (c η 1

 111 (σ) = e iω 0 σ with initial conditions:L t H c η 1 (0) + R t H c η (u) du = iω 0 η 1 (0)

 0 ) ω 0 - 1 5. 7 20 z 2 2 + φ 11 zz + φ 02 z 2 2 + φ 21 zz 2 2 5 j=i a ij x i x j + 5 i=1 5 j=i 5 m=ja

 0017202225555 iω 0 + i (-p 1 -q 1 e iω 0 τ +ip 0 (-1+e iω 0 τ ) ω τ (iq 2 ω 0 + q 1 )e -iω 0 τ -p 0 (-1+iω 0 τ e -iω 0 τ +e -iω 0 τ ) Proof of lemma 5Letw (z, z, θ) = w 20 (θ) z 2 2 + w 11 (θ) zz + w 02 (θ) z 2 2 + h.o.t.Lemma 5.5.F 0 (z, z) = φ + h.o.t0with φ 20 , φ 11 , φ 02 , φ 21 some complex functions of the coefficients a ij and a ijm derived in Lemma 5.5.21 Proof : We know from the proof of lemma 3 that F ε, ϕ 1t (0) , ϕ 1t (-τ ) , ϕ 2t (0) , ϕ 2t (-τ ) , ijm x i x j x m[START_REF] Romer | Increasing returns and long-run growth[END_REF]

ϕ 2 +

 2 1t (θ) = w 1 20 (θ) z 2 2 + w 1 11 (θ) zz + w 1 02 (θ) z 2 z (t) iω 0 e iω 0 θ -z (t) iω 0 e -iω 0 θ + O (|z, z|) ϕ 2t (θ) = w 2 20 (θ) z 2 2 + w 2 11 (θ) zz + w 2 02 (θ) z 2 2

z 2 2 +H 11 zz+H 02 z 2

 22

0 -τ e 2iω 0 u du E 2 1 = φ 20 2iω

 0120 2iω 0 -L H c -R H c e -τ 2iω 0p 2 + q 2 e -2iτ ω 0 E 1 1 + p 1 + q 1 e -2iτ ω 0 + p 0

The consumption goods sector needs to be more capital intensive than the investment goods sector.

The optimal path necessarily converges monotonously to the steady state in lowerdimensional continuous-time models. See Hartl[START_REF] Hartl | A simple proof of the monotonicity of the state trajectories in autonomous control problems[END_REF] for a proof of this result in general autonomous control problems with one state variable.

See also Boucekkine et al.[START_REF] Boucekkine | Vintage capital and the dynamics of the AK model[END_REF] and Boucekkine et al.[START_REF] Boucekkine | Endogenous vs exogenously driven fluctuations in vintage capital models[END_REF].

Benhabib and Rustichini [13] mention the possibility of a Hopf bifurcation in an

These properties are shown to match empirical evidence from the US economy over the period 1948-2004 (see Guo and Lansing[START_REF] Guo | Capital-labor substitution and equilibrium indeterminacy[END_REF]).

See, for instance, Blundell and MaCurdy[START_REF] Blundell | Labour supply: a review of alternative approaches[END_REF].

See Nishimura et al.[START_REF] Nishimura | Indeterminacy in aggregate models with small externalities: an interplay between preferences and technology[END_REF].

See in particular Diekmann and Van Gils[START_REF] Diekmann | Delay Equations: Functional-, Complex-, and Nonlinear Analysis[END_REF].

* This work was supported by the French National Research Agency grant No. ANR-08-BLAN-0245-01. We would like to thank M. Adimy, M. Bambi, J. Benhabib, R. Boucekkine, C. Ghiglino and C. Le Van for useful suggestions and comments.

we have H 20 (θ) = -g 20 q (θ) -g 02 q (θ) if θ ∈ [-τ, 0) H 11 (θ) = -g 11 q (θ) -g 11 q (θ) if θ ∈ [-τ, 0) [START_REF] Vissing-Jorgensen | Limited asset market participation and the elasticity of intertemporal substitution[END_REF] Moreover, we have on the central manifold

This rewrites, as ż = iω 0 z + g (z, z)

Coefficient identification in ( 42) and ( 44) leads to

Comparing [START_REF] Vissing-Jorgensen | Limited asset market participation and the elasticity of intertemporal substitution[END_REF] with the last expression, we have (2iω 0 -A) w 20 (θ) = -g 20 q (θ) -g 02 q (θ)

Aw 11 (θ) = g 11 q (θ) + g 11 q (θ) which rewrites, using definition of operator A,

Solving this equation, we obtain w 20 (θ) = E 1 e 2iω 0 θ + ig 20 q 0 q (0) e iω 0 θ + ig 02 3q 0 q (0) e -iω 0 θ In a similar way, we have ẇ11 (θ) = g 11 q (θ) + g 11 q (θ) if θ ∈ [-τ, 0) which implies w 11 (θ) = -ig 11 q 0 q (0) e iω 0 θ + ig 11 q 0 q (0) e -iω 0 θ + E 2 where E 1 and E 2 can be determined with initial conditions H (z, z, 0) = -2Re (g (z, z) q (0)) + f 0 (z, z)


Greqam

Carine Nourry
email: carine.nourry@univmed.fr

Alain Venditti
email: alain.venditti@univmed.fr

G Bloise

J P Drugeon

R Farmer

C Ghiglino

J M Grandmont

LOCAL INDETERMINACY UNDER DYNAMIC EFFICIENCY IN A TWO-SECTOR OVERLAPPING GENERATIONS ECONOMY

Keywords: Two-sector OLG model, dynamic efficiency, gross substitutability in consumption, local indeterminacy, stabilization policy Journal of Economic Literature Classification Numbers: C62, E32, O41

We consider a two-sector two-periods overlapping generations model with inelastic labor, consumption in both periods of life and homothetic CES preferences. Assuming gross substitutability and a capital intensive consumption good, we prove that when dynamic efficiency holds, local indeterminacy and sunspot fluctuations occur with low enough values for the sectoral elasticities of capital-labor substitution and we illustrate this finding within a standard example. This result shows that some fiscal policy rules can prevent the existence of business-cycle fluctuations in the economy by driving it to the optimal steady state as soon as it is announced, and thus shows that Reichlin's [9] influential conclusion is a robust property in a two-sector OLG economy.

Introduction

The existence of local indeterminacy and sunspot fluctuations under gross substitutability are well established facts whithin OLG economies. Considering an aggregate model with endogenous labor and consumption in the second period of life only, Reichlin [START_REF] Reichlin | Equilibrium Cycles in an Overlapping Generations Economy with Production[END_REF] has shown that Hopf cycles and thus local indeterminacy arise with a unique Pareto optimal steady state when the technology is Leontief. This conclusion has had a strong echo in the literature as it implies that the introduction of a public policy based on taxes and transfers could at the same time stabilize the economy and reach the Pareto optimal steady state along which all generations get an equal level of utility.

However, Cazzavillan and Pintus [START_REF] Cazzavillan | Dynamic Inefficiency in an Overlapping Generations Economy with Production[END_REF] have recently proved that the coexistence of local indeterminacy and dynamic efficiency is not robust to the consideration of any positive elasticity of capital-labor substitution. Indeed, apart from the very special case of Leontief technology, the steady state is always characterized by an over-accumulation of capital when local indeterminacy holds, 1 and policies targeting the steady state allocation generically leave room for welfare losses associated with productive inefficiency.

As initially shown in Galor [START_REF] Galor | A Two-Sector Overlapping-Generations Model: a Global Characterization of the Dynamical System[END_REF], local indeterminacy also arises in twosector OLG economies under gross substitutability. However, the Pareto optimality of the equilibrium has not been precisely discussed in such a framework. On the one hand, Reichlin [START_REF] Reichlin | Endogenous Fluctuations in a Two-Sector Overlapping Generations Economy[END_REF] proves local indeterminacy of a dynamically inefficient steady state through the existence of periodic, quasiperiodic and chaotic dynamics under the assumption of Leontief technologies. On the other hand, Drugeon et al. [START_REF] Drugeon | Does Dynamic Efficiency Rule Out Expectation-Driven Fluctuations ?[END_REF] show that any dynamically efficient equilibrium path is locally determinate if the sectoral elasticities of capital-labor substitution are large enough.

Our main objective in this paper is to show that Reichlin's [START_REF] Reichlin | Equilibrium Cycles in an Overlapping Generations Economy with Production[END_REF] result is a robust property in a two-sector OLG economy. Assuming gross substitutability and a capital intensive consumption good sector, we prove indeed that local indeterminacy can occur under dynamic efficiency when intermediary values for the elasticity of intertemporal substitution in consumption and low enough but positive sectoral elasticities of capital-labor substitution are considered. This conclusion shows that contrary to the aggregate framework, in two-sector OLG economies with capital-labor substitutability, some fiscal policy rules can eliminate business-cycle fluctuations in the economy

The economy consists in one consumption good y 0 and one capital good y produced from capital and labor through standard constant returns to scale technologies:

y 0 = f 0 (k 0 , l 0 ), y = f 1 (k 1 , l 1 ) with k 0 + k 1 ≤ k and l 0 + l 1 ≤ (1)
k being the total stock of capital and the total amount of labor.

Assumption 1. Each production function f i : R2 + → R + , i = 0, 1, is C 2 , increasing in each argument, concave, homogeneous of degree one and such that for any x > 0,

f i 1 (0, x) = f i 2 (x, 0) = +∞, f i 1 (+∞, x) = f i 2 (x, +∞) = 0. As y ≤ f 1 (k, ), Assumption 1 implies that there exists k( ) > 0 solution of k -f 1 (k, ) = 0 such that f 1 (k, ) > k when k < k( ), while f 1 (k, ) < k when k > k( ).
The set of admissible 3-uples (k, y, ) is thus defined as

K = (k, y, ) ∈ R 3 + |0 < , 0 ≤ k ≤ k( ), 0 ≤ y ≤ f 1 (k, ) (2)
For any (k, y, ), profit maximization in the representative firm of each sector is equivalent to solving the following problem of optimal allocation of productive factors between the two sectors:

T (k, y, ) = max k 0 ,k 1 ,l 0 ,l 1 f 0 (k 0 , l 0 ) s.t. y ≤ f 1 (k 1 , l 1 ) k 0 + k 1 ≤ k l 0 + l 1 ≤ k 0 , k 1 , l 0 , l 1 ≥ 0 (3)
The social production function T (k, y, ) gives the maximal output of the consumption good along interior temporary equilibria (k, y, ) ∈ K. Under Assumption 1, T (k, y, ) is homogeneous of degree one, concave and C 2 over K. 2 Denoting w the wage rate, r the gross rental rate of capital and p the price of investment good, all in terms of the price of the consumption good, we derive from the envelope theorem that

r = T 1 (k, y, ), p = -T 2 (k, y, ), w = T 3 (k, y, ) (4)
The share of capital in total income is then given by s(k, y, ) = rk T (k,y, )+py ∈ (0, 1)

(5)

Consumption and savings

In each period t, N t agents are born, and they live for two periods. In their first period of life (when young), the agents are endowed with one unit of labor that they supply inelastically to firms. Their income results from the real wage and is allocated between current consumption and savings which are invested in the firms. In their second period of life (when old ), they are retired and their income resulting from the return on the savings is entirely consumed. The utility function of a representative agent, defined over his consumption bundle (c t , when he is young, and d t+1 , when he is old), is

u(c t , d t+1 ) = c 1-1/γ t + δ(d t+1 /B) 1-1/γ γ/(γ-1)
with δ ∈ (0, 1) the discount factor, γ > 0 the elasticity of intertemporal substitution in consumption and B > 0 a scaling parameter. Each agent is assumed to have 1 + n > 0 children so that N t+1 = (1 + n)N t . Under perfect foresight, and considering the wage rate w t and the gross rate of return R t+1 as given, a young agent maximizes his utility function over his life-cycle as follows:

max ct,d t+1 ,φt u(c t , d t+1 ) s.t. w t = c t + φ t R t+1 φ t = d t+1 c t , d t+1 , φ t ≥ 0 (6)
Solving the first order conditions gives:

c t = wt 1+δ γ (R t+1 /B) γ-1 ≡ α(R t+1 /B)w t (7)
with α(R/B) ∈ (0, 1) the share of first period consumption over the wage income. Under gross substitutability, γ > 1 and the saving function

φ t = (1 -α(R t+1 /B))w t is increasing in R.

Perfect-foresight competitive equilibrium

Total labor is given by the number N t of young households, i.e., t = N t , and is increasing at rate n, i.e., t+1 = (1 + n) t . We also assume complete depreciation of capital within one period.

Definition 1. A sequence {k t , y t , t , c t , d t , r t , w t , p t } ∞ t=0 , with (k 0 , 0 ) = ( k0 , ˆ 0 ) given, is a perfect-foresight competitive equilibrium if: i) c t = α(R t+1 /B)w t ; ii) t (1 -α(R t+1 /B))w t = p t y t ; iii) y t = k t+1 ; iv) t+1 = (1 + n) t ; v) t [c t + d t /(1 + n)] = T (k t , y t , t ); vi) (r t , w t , p t ) is given by (4); vii) R t+1 = r t+1 /p t .
Let κ = k/ and κ be the solution of κ -f 1 (κ, 1) = 0. The set of admissible paths given by ( 2) can be redefined as follows

K = (κ t , κ t+1 ) ∈ R 2 + |0 ≤ κ t ≤ κ, 0 ≤ κ t+1 ≤ f 1 (κ t , 1)/(1 + n) (8)
A perfect-foresight competitive equilibrium then satisfies

(1 + n)κ t+1 + T 3 (κt,(1+n)κ t+1 ,1) T 2 (κt,(1+n)κ t+1 ,1) 1 -α -T 1 (κ t+1 ,(1+n)κ t+2 ,1) T 2 (κt,(1+n)κ t+1 ,1)B = 0 (9)
with α(R/B) given by ( 7), (κ t , κ t+1 ) ∈ K and κ 0 = κ0 = k0 / ˆ 0 given.

Steady state and dynamic efficiency 3.1 A normalized steady state

A steady state is defined as κ t = κ * for all t with κ * solution of

α -T 1 (κ,(1+n)κ,1) T 2 (κ,(1+n)κ,1)B = 1 + (1 + n) κT 2 (κ,(1+n)κ,1) T 3 (κ,(1+n)κ,1) ≡ Φ κ ∈ (0, 1) (10)
We consider a family of economies parameterized by γ = 1. We follow the same procedure as in Drugeon et al. [START_REF] Drugeon | Does Dynamic Efficiency Rule Out Expectation-Driven Fluctuations ?[END_REF]: we use the scaling parameter B to ensure the existence of a normalized steady state (NSS) κ * ∈ (0, κ) which remains invariant as γ is varied. From ( 7) and ( 10) we get:

Proposition 1. Under Assumption 1, let γ = 1 and κ * ∈ (0, κ). Then there exists a unique value B(κ * ) > 0 as given by

B(κ * ) = -T 1 (κ * ,(1+n)κ * ,1)
T 2 (κ * ,(1+n)κ * ,1)

-(1+n)κ * T 2 (κ * ,(1+n)κ * ,1) δ γ [T 3 (κ * ,(1+n)κ * ,1)+(1+n)κ * T 2 (κ * ,(1+n)κ * ,1)] 1 1-γ such that κ * is a steady state if and only if B = B(κ * ). Proof : See Appendix 5.1.
In the rest of the paper we assume that B = B(κ * ) so that s = s(κ * , κ * , 1) and α = α(R * /B(κ * )) = Φ κ * remain constant as γ is made to vary.

Dynamic efficiency

From Definition 1 and the homogeneity of T (k, y, ), considering that κ * T 2 /T 3 = (T 2 /T 1 )(κ * T 1 /T 3 ) = -s/R * (1 -s), we derive the stationary gross rate of return along the NSS:

R * = (1+n)s (1-α)(1-s) (11)
Under-accumulation of capital is obtained if and only if R * > 1 + n. As shown in Drugeon et al. [START_REF] Drugeon | Does Dynamic Efficiency Rule Out Expectation-Driven Fluctuations ?[END_REF] we have:

Proposition 2. Under Assumption 1, let γ = 1 and α = 1 -s/(1 -s). Then:
i) the NSS is characterized by an under-accumulation of capital if and only if α ≥ α;

ii) an intertemporal competitive equilibrium converging towards the NSS is dynamically efficient if α ∈ (α, 1) and dynamically inefficient if α ∈ (0, α).

If the labor income is relatively larger than the capital income (s < 1/2), α > 0 and young agents receive enough wage resources to provide a large amount of savings. But over-accumulation of capital can be avoided provided the share of first period consumption over the wage income is large enough, i.e. the agent does not save too much.

Local indeterminacy under dynamic efficiency

Let us introduce the relative capital intensity difference across sectors

b ≡ k 1 /l 1 -k 0 /l 0 l 1 /y (12)
and the elasticity of the rental rate of capital

ε rk = -T 11 (κ * , (1 + n)κ * , 1)κ * /T 1 (κ * , (1 + n)κ * , 1) (13)
evaluated at the NSS. From now on we consider a positive value for α = 1s/(1 -s), we assume gross substitutability, a capital intensive consumption good, 3 and we consider dynamically efficient paths:

Assumption 2. s ∈ (0, 1/2), γ > 1, b < 0 and α ≥ α.
We then prove that contrary to the aggregate OLG model, local indeterminacy arises under dynamic efficiency in a two-sector model with positive capital-labor substitution. Proof : See Appendix 5.2.

Theorem 1 is based on a large elasticity ε rk . This restriction can be interpreted through the aggregate elasticity of capital-labor substitution Σ which is linked to ε rk as follows (see [START_REF] Drugeon | On Consumptions, Inputs and Outputs Substitutabilities and the Evanescence of Optimal Cycles[END_REF]):

Σ = y 0 +py pyky 0 pyk 0 l 0 σ 0 + y 0 k 1 l 1 σ 1 , ε rk = l 0 /y 0 2 w(y 0 +py) Σ (14)
with σ 0 and σ 1 the sectoral elasticities of input substitution. Local indeterminacy under dynamic efficiency then requires a low enough aggregate elasticity Σ, i.e. low enough but still positive sectoral elasticities. Note also that the consumption good sector has to be sufficiently capital intensive. If b is too close to 0, i.e. if the model is close to the aggregate formulation, we get the same conclusion as Cazzavillan and Pintus [START_REF] Cazzavillan | Endogenous Business Cycles and Dynamic Inefficiency[END_REF][START_REF] Cazzavillan | Dynamic Inefficiency in an Overlapping Generations Economy with Production[END_REF]: local indeterminacy under dynamic efficiency can never arise with any positive elasticity of capital-labor substitution.

Remark 1: Note that γ is generically a flip bifurcation value giving rise to period-two cycles which are locally indeterminate (or unstable) in a right (or left) neighborhood of γ, while γ is generically a transcritical bifurcation value leading to the existence of a second steady state which is locally unstable (saddle-point stable) in a right (left) neighborhood of γ (see Figure 1 in Appendix 5.2).

Through the NSS, the conditions on α, s, b and ε rk in Theorem 1 are based on joint restrictions of the technologies. We need therefore to show that they can be satisfied with standard production functions. Consider the following CES formulations:

f 0 (k 0 , l 0 ) = χ(k 0 ) -ς + (1 -χ)(l 0 ) -ς -1 ς f 1 (k 1 , l 1 ) = 1 2 k 1 η -ρ + (l 1 ) 2 -ρ -1 ρ (15)
with χ ∈ (0, 1), ς, ρ > -1 and η > 0. The sectoral elasticities of capital-labor substitution are given by σ 0 = 1/(1 + ς) and σ 1 = 1/(1 + ρ). We assume for simplicity a constant population, i.e. n = 0. Note that when σ 1 = 0 the investment good technology becomes Leontief as lim ρ→+∞ f 1 (k 1 , l 1 ) = min{k 1 /η, l 1 }. Assume in a first step that σ 1 = 0. As y = k 1 /η = l 1 , we get

T 0 (k, y, ) = χ(k -ηy) -ς + (1 -χ)( -y) -ς -1/ς
Following Proposition 1, we consider a NSS κ = κ * ∈ (0, 1) and we compute

α 0 = (1-χ)(1-η) 1+ς κ * ς (1-κ * )-ηχ(1-κ * ) 1+ς (1-χ)(1-η) 1+ς κ * ς , s 0 = χ(1-κ * ) 1+ς (1-χ)(1-η) 1+ς κ * ς α 0 = (1-χ)(1-η) 1+ς κ -χ(1-κ * ) 1+ς (1-χ)(1-η) 1+ς κ * ς , b 0 = η-κ * 1-κ * and ε 0 rk = (1+ς)(1-χ)κ * ς (1-η) h χ " 1-κ * κ * (1-η) " ς +1-χ i together with the bounds b = -1, b 0 = -(1-χ)(1-η) 1+ς κ * 1+ς +ηχ(1-κ * ) 1+ς (1-χ)(1-η) 1+ς κ * ς (1-κ * )-ηχ(1-κ * ) 1+ς , ε 0 rk = (κ * -η)(1-κ * )(1-χ) (2κ * -1-η)(1-η)ηχ χ(1+η)+(1-χ) " κ * (1-η) 1-κ * " 1+ς χ " 1-κ * κ * (1-η) " ς +1-χ
In a first step we derive that if

κ * ∈ (1/2, 1), χ ∈ (χ 0 1 , χ0 ) with χ 0 1 = (1-η) ς κ * 1+ς (1-κ * ) 1+ς +(1-η) ς κ * 1+ς , χ0 = (1-η) 1+ς κ * ς (1-κ * ) 1+ς +(1-η) 1+ς κ * ς
and η < η0 = min{1 -κ * , 2κ * -1}, then s 0 ∈ (1/3, 1/2), α 0 ∈ (α 0 , 1/2) and b 0 ∈ (b 0 , -1). In a second step, we derive that if ς > ς 0 with

ς 0 = κ * [1-η(1-η)-κ * 2 ] (2κ * -1-η)η
then there exists χ 0 2 ∈ (0, χ) given by

χ 0 2 = (κ * -η)(1-κ * ) " κ * (1-η) 1-κ * " 1+ς ς(2κ * -1-η)η-κ * [1+η 2 -κ * (1+η)]+(κ * -η)(1-κ * ) " κ * (1-η) 1-κ * " 1+ς
such that ε rk > ε 0 rk when χ > χ 0 2 . Denoting χ 0 = max{χ 0 1 , χ 0 2 }, we have then proved: Lemma 1. Let the production functions be given by (15) with σ 1 = 0. If κ * ∈ (1/2, 1), η < η0 , ς > ς 0 and χ ∈ (χ 0 , χ0 ), there exist γ 0 > 1 and γ0 > γ 0 such that the NSS is locally indeterminate for any γ ∈ (γ 0 , γ0 ).

Assume now that σ 1 > 0. The social production function derived as the value function of program ( 3) is parameterized by σ 1 , namely T σ 1 (k, y, ). Considering the same NSS κ = κ * ∈ (0, 1), we can similarly compute α σ 1 ,

s σ 1 , α σ 1 , b σ 1 , ε σ 1
rk , b σ 1 and ε σ 1 rk . We have the following property: Lemma 2. The social production function T σ 1 (k, y, ) is continuous in (k, y, , σ 1 ), and as σ 1 → 0, converges to T 0 (k, y, ) uniformly in (k, y, ), i.e. for any > 0, there is ξ > 0 such that if 0 < σ 1 < ξ, |T σ 1 (k, y, ) -T 0 (k, y, )| < for any (k, y, ) ∈ K.

Proof : See Appendix 5.3.

It follows from these Lemmas that lim

σ 1 →0 (α σ 1 , s σ 1 , α σ 1 , b σ 1 , ε σ 1 rk , b σ 1 , ε σ 1 rk ) = (α 0 , s 0 , α 0 , b 0 , ε 0 rk , b 0 , ε 0 rk ).
We have then proved:

Proposition 3. Let the production functions be given by ( 15). If κ * ∈ (1/2, 1), there is σ1 > 0 such that for all 1 ∈ [0, σ1 ) there exist ησ 1 ∈ (0, 1), ς σ 1 > 0, χ σ 1 ∈ (0, 1), χσ 1 ∈ (0, 1), γ σ 1 > 1 and γσ 1 > γ σ 1 such that when η < ησ 1 , ς > ς σ 1 and χ ∈ (χ σ 1 , χσ 1 ), the NSS is locally indeterminate for any γ ∈ (γ σ 1 , γσ 1 ).

Theorem 1 and Proposition 3 show that in a two-sector OLG economy, the existence of local indeterminacy under dynamic efficiency is compatible with positive capital-labor substitution in each sector. Following Reichlin [START_REF] Reichlin | Equilibrium Cycles in an Overlapping Generations Economy with Production[END_REF], our conclusion suggests that the Pareto optimal steady state can be considered as a target of the policymaker. Indeed the introduction of a public policy based on taxes and transfers could at the same time rule out business-cycle fluctuations in the economy and drive the equilibrium to the optimal steady state which provides an equal level of utility to all generations.

Let us prove now along the line of Reichlin [START_REF] Reichlin | Equilibrium Cycles in an Overlapping Generations Economy with Production[END_REF] that such a policy exists under the assumption that agents and public authority do not make forecasting mistakes. Assume that the public authority buys goods, levies taxes and makes transfers. Let g t be the flow of consumption goods which is bought, τ y t < (>)0 the taxes (transfers) on the income of the young and τ o t < (>)0 the taxes (transfers) on the income of the old. We assume a balanced budget rule, i.e. g t + τ y t + τ o t = 0. The agent's first and second period budget constraints become w t +τ y t = c t +φ t and R t+1 φ t +τ o t+1 = d t+1 . From utility maximization, we get the optimal saving

φ t = B(δR t+1 /B) γ (wt+τ y t )-τ o t+1 R t+1 +B(δR t+1 /B) γ (16)
and a perfect-foresight competitive equilibrium satisfies

(1 + n)κ t+1 p t = B(δR t+1 /B) γ (wt+τ y t )-τ o t+1 R t+1 +B(δR t+1 /B) γ (17)
Now consider the expression of the normalization constant as given in Proposition 1 evaluated at some point κ ∈ (0, κ), namely B = B(κ) such that:

B(κ) = r(κ) p(κ) (1+n)κp(κ) δ γ [w(κ)-(1+n)κp(κ)] 1 1-γ with r(κ) = T 1 (κ, (1 + n)κ, 1), p(κ) = -T 2 (κ, (1 + n)κ, 1) and w(κ) = T 3 (κ, (1 + n)κ, 1). Let τ o t+1 = (1 + n) κp(κ) [R(κ) + B(κ)(δR(κ)/B(κ)) γ ] R t+1 R(κ) γ -κ t+1 p t [R t+1 + B(κ)(δR t+1 /B(κ)) γ ] τ y t = w(κ) -w t (18)
with R(κ) = r(κ)/p(κ). Plugging τ y t = τ y t , τ o t+1 = τ o t+1 and τ y t+1 = τ o t+2 = 0 into (17) gives

(1 + n)κp(κ) = (δR(κ)/B(κ)) γ B(κ)w(κ) R(κ)+(δR(κ)/B(κ)) γ (19)
As shown in Proposition 1, there exists a solution κ = κ * which corresponds to the normalized stationary Pareto optimal perfect-foresight competitive equilibrium. It follows that if agents believe in this announced policy rule, they will expect the optimal NSS allocation to hold in the future. This expectation in turn drives the sytem to the NSS and keep it there forever. Note however that contrary to Riechlin [START_REF] Reichlin | Equilibrium Cycles in an Overlapping Generations Economy with Production[END_REF], when local indeterminacy holds, the NSS is not the unique stationary solution of equation ( 19). Indeed, as mentioned in Remark 1, a second steady state generically exists through a transcritical bifuration occurring at γ and is locally unstable for γ > γ. It follows that the stabilization policy (18) has to be parameterized by the right NSS in order to jump on the optimal stationary allocation.

Appendix

Proof of Proposition 1

Consider the set K as defined by [START_REF] Nishimura | Interior Optimal Chaos with Arbitrarily Low Discount Rates[END_REF] and the expression of α(R/B) as given by [START_REF] Grandmont | Capital-Labor Substitution and Competitive Nonlinear Endogenous Business Cycles[END_REF]. κ * ∈ (0, κ) is a solution of (10) if

1 1+δ γ " - T 1 (κ * ,(1+n)κ * ,1) T 2 (κ * ,(1+n)κ * ,1)B " γ-1 = 1 + (1 + n) κ * T 2 (κ * ,(1+n)κ * ,1) T 3 (κ * ,(1+n)κ * ,1) ∈ (0, 1) (20)
Then there exists a unique positive value of B solution of (20) given by

B(κ * ) = -T 1 (κ * ,(1+n)κ * ,1)
T 2 (κ * ,(1+n)κ * ,1)

-(1+n)κ * T 2 (κ * ,(1+n)κ * ,1) [T 3 (κ * ,(1+n)κ * ,1)+(1+n)κ * T 2 (κ * ,(1+n)κ * ,1)]δ γ 1 1-γ
and κ * is a steady state if and only if B = B(κ * ).

Proof of Theorem 1

From (7), we get

α (R/B) = (1 -γ)α(R/B)(1 -α(R/B))B/R (21)
Under Assumptions 1 and 2, we get from the first order conditions of program ( 3)

T 12 = -T 11 b < 0, T 22 = T 11 b 2 < 0, T 31 = -T 11 a > 0, T 32 = T 11 ab > 0 (22)
with a ≡ k 0 /l 0 > 0, b as defined by (12) and T 11 < 0. Considering (13) together with (22),

T 1 κ * /T 3 = s/(1 -s), -T 1 /T 2 = R * = s/(1 -α)(1 -s)
and the fact that homogeneity of T (k, y, ) implies a = [1 -(1 + n)b]κ * , total differenciation of ( 9) using ( 4), ( 5) and ( 21) evaluated at the NSS gives the characteristic polynomial

P(λ) = λ 2 -λT (γ) + D(γ) with D(γ) = s[(1+n)bα(γ-1)+1-α+α(1+n)b] (1+n)b(1-α)(1-s)α(γ-1) , T (γ) = 1 (1+n)bε rk α(γ-1) + 1+D(1+n) 2 b 2 (1+n)b
When B = B(κ * ), the NSS, α, s and ε rk remain constant for any γ = 1. We then study the variations of T (γ) and D(γ) in the (T , D) plane as γ varies within (1, +∞). Solving T (γ) and D(γ) with respect to α(γ -1) yields to

D = ∆(T ) = ε rk s[1-α+α(1+n)b]T (1-α)(1-s)+ε rk s(1+n)b[1-α+α(1+n)b] - ε rk s[1-α+α(1+n)b]-s(1+n)b (1+n)b[(1-α)(1-s)+ε rk s(1+n)b(1-α+α(1+n)b)]
and allows to use the methodology introduced by Grandmont et al. [START_REF] Grandmont | Capital-Labor Substitution and Competitive Nonlinear Endogenous Business Cycles[END_REF]. As γ spans the interval (1, +∞), T (γ) and D(γ) vary linearly along the line ∆(T ). The fundamental properties of ∆(T ) depend on its extremities. The starting point of the pair (T (γ), D(γ)) is obtained when γ = +∞:

D(+∞) = D ∞ = s (1-α)(1-s) , T (+∞) = T ∞ = (1-α)(1-s)+(1+n) 2 b 2 s (1+n)b(1-α)(1-s) (23)
while the end point is obtained when γ converges to 1 from above. Let

b = -1/(1 + n), b = -(1 -α)/(1 + n)α and b = -(1 -α)(1 -s)/(1 + n)s.
As lim γ→1 + D(γ) = lim γ→1 + T (γ) = ∞, ∆(T ) is a half-line starting from (T ∞ , D ∞ ) and pointing downwards or upwards depending on the sign of The result follows.

D (γ) = s(b-b) b(1-α)(1-s)(γ-1) 2 (24) Under Assumptions 1-2, we get from (23) that D ∞ = R * /(1 + n) > 1. Local indeterminacy then requires that D (γ) > 0, i.

Proof of Lemma 2

We use the same kind of argument as Nishimura and Yano ([8], Lemma 1, p. 229). Define the set E(k, y, , σ 1 ) = {(k 0 , l 0 ) ≥ 0|y ≤ f 1 (k-k 0 , -l 0 )}. Then T σ 1 (k, y, ) = max f 0 (k 0 , l 0 ) s.t. (k 0 , l 0 ) ∈ E(k, y, , σ 1 ). Since E(k, y, , σ 1 ) is lower-semi-continuous in (k, y, , σ 1 ), T σ 1 (k, y, ) is continuous. It follows also that the functions (k 0 σ 1 (k, y, ), l 0 σ 1 (k, y, )) = argmaxf 0 (k 0 , l 0 ) s.t. (k 0 , l 0 ) ∈ E(k, y, , σ 1 ) are continuous. Moreover, as lim ρ→+∞ f 1 (k 1 , l 1 ) = min{k 1 /η, l 1 }, we can find a sub-sequence σ 1i such that when σ 1i → 0, (k 0 σ 1 (k, y, ), l 0 σ 1 (k, y, )) → (k -ηy, -y) for any (k, y, ) ∈ K. Therefore, the uniform convergence follows and lim σ 1 →0 T σ 1 (k, y, ) = T 0 (k, y, ).

Theorem 1 .

 1 Under Assumptions 1-2, let b = -1/(1 + n) and b = -(1α)/(1 + n)α. Then there exist ε rk > 0, γ > 1 and γ > γ such that the NSS is locally indeterminate if and only if s ∈ (1/3, 1/2), α ∈ (α, 1/2), b ∈ (b, b), ε rk > ε rk and γ ∈ (γ, γ).

1 ( 2 (Figure 1 :

 121 Figure 1: Local indeterminacy with dynamic efficiency.This requires that 1+T∞ +D ∞ = ( b-b)(b-b)/b b < 0 ⇔ b ∈ (-∞, b)∪( b, 0), and that D = -1 implies T > 0. Solving D = -1 gives α(1 -γ) = αs(b-b) b[1-α(1-s)]

See also Cazzavillan and Pintus[START_REF] Cazzavillan | Endogenous Business Cycles and Dynamic Inefficiency[END_REF] for an extension of this result to a model with positive first-period consumption.

See Benhabib and Nishimura[START_REF] Benhabib | Stability of Equilibrium in Dynamic Models of Capital Theory[END_REF].

When γ > 1, this is a necessary condition for local indeterminacy (See Galor[START_REF] Galor | A Two-Sector Overlapping-Generations Model: a Global Characterization of the Dynamical System[END_REF]).

* This work was supported by French National Research Agency Grant (ANR-08-BLAN-0245-01).