Pascal Bégout
email: begout@ann.jussieu.fr

Maximum Decay Rate for the Nonlinear Schr ödinger Equation

In this paper, we consider global solutions for the following nonlinear Schrödinger equation

). We show that no nontrivial solution can decay faster than the solutions of the free Schrödinger equation, provided that u(0) lies in the weighted Sobolev space H 1 (R N ) ∩ L 2 (|x| 2 ; dx), in the energy space, namely H 1 (R N ), or in L 2 (R N ), according to the different cases.

Introduction and notations

We consider global solutions of the following nonlinear Schrödinger equation,

     i ∂u ∂t + ∆u + λ|u| α u = 0, (t, x) ∈ [0, ∞) × R N , u(0) = ϕ, in R N , (1.1)
where λ ∈ R, 0 α < 4 N -2 (0 α < ∞ if N = 1) and ϕ a given initial data.

It is well-known that if we denote by T (t) the Schrödinger's free operator, then for every r ∈ [2, ∞] and for every ϕ ∈ L r ′ (R N ),

∀t ∈ R \ {0}, T (t)ϕ L r (4π|t|) -N ( 1 2 -1 r ) ϕ L r ′ , (1.2)
where r ′ = r r-1 . Note that for every r ∈ 2, 2N N -2 (r ∈ [2, ∞) if N = 1), if ϕ ∈ L 2 (R N ) ∩ L 2 (|x| 2 ; dx) then ϕ ∈ L r ′ (R N ) and ϕ L r ′ C( ϕ L 2 , xϕ L 2 ). Furthermore, the estimate (1.2) is optimal in the following sense. For every r ∈ [1, ∞] and for every ϕ ∈ S ′ (R N ), if ϕ ≡ 0 then lim inf

t→±∞ |t| N ( 1 2 -1 r ) T (t)ϕ L r > 0.
(1.3)

2000 Mathematics Subject Classification: 35Q55,(35B40)

For the proof, see Strauss [START_REF] Strauss | Nonlinear scattering theory[END_REF] (r 2) and Kato [START_REF] Kato | An L q,r -theory for nonlinear Schrödinger equations[END_REF] (general case). In the same way, there exist some solutions of the nonlinear Schrödinger equation (1.1) which have a linear decay (in the sense of (1.2)). See for example Cazenave [START_REF] Cazenave | An introduction to nonlinear Schrödinger equations, volume 26 of Textos de Métodos Matemáticos[END_REF], Theorem 7.2.1; Hayashi and Naumkin [START_REF] Hayashi | Asymptotics for large time of solutions to the nonlinear Schrödinger and Hartree equations[END_REF]. In particular, these solutions lie in H 1 (R N ) ∩ L 2 (|x| 2 ; dx). On the other hand, we know that there exist solutions of some heat, Ginzburg-Landau and Schrödinger type equations which have a decay rate faster than the corresponding linear problem (Hayashi, Kaikina and Naumkin [START_REF] Hayashi | Large time behavior of solutions to the Landau-Ginzburg type equations[END_REF][START_REF] Hayashi | Large time behaviour of solutions to the dissipative nonlinear Schrödinger equation[END_REF]). Take an example. Let u be a classical solution of the heat equation u t -∆u + |u| α u = 0, (t, x) ∈ [0, ∞) × R N , with initial datum u 0 ∈ L ∞ (R N ), u 0 ≡ 0 and u 0 0 a.e. Then we have by the maximum principle, u(t) L ∞ (αt) -1 α , for every t > 0, whereas for every t 1, e t∆ u 0 L ∞ Ct -N 2 , for some constant C > 0. Thus, if 0 < α < 2 N then u(t) decays faster than e t∆ u 0 . So, it is natural to wonder if some solutions of the nonlinear Schrödinger equation (1.1) may have faster decay than the solutions of the linear equation.

We will see that such solutions do not exist (except the trivial solution). There exist partial results in this direction. This is the case for α = 2

N and N = 1 (Hayashi and Naumkin [START_REF] Hayashi | Asymptotics for large time of solutions to the nonlinear Schrödinger and Hartree equations[END_REF]), for α = 4 N (Cazenave and Weissler [START_REF] Cazenave | The structure of solutions to the pseudo-conformally invariant nonlinear Schrödinger equation[END_REF], Theorem 2.1 (a)) or for some self-similar solutions for α > α 0 , where α 0 is given by α

0 = -(N -2)+ √ N 2 +12N +4 2N
(Cazenave and Weissler [START_REF] Cazenave | Scattering theory and self-similar solutions for the nonlinear Schrödinger equation[END_REF], Corollary 3.9). This paper is organized as follows. In Section 2, we give the main results concerning the solutions lying in H 1 (R N ) ∩ L 2 (|x| 2 ; dx) and in H 1 (R N ). In Section 3, we give the main results concerning the solutions lying in L 2 (R N ). In Section 4, we give several estimates for large times and establish Lemma 4.5, which asserts that the existence of a scattering state in L 2 (R N ) implies a maximum rate decay which is linear (in the sense that the solution satisfies (1.3)). Lemma 4.5 is at the heart of the results of this paper. Finally, we will prove the results for solutions in H 1 (R N ) ∩ L 2 (|x| 2 ; dx) and H 1 (R N ) in Section 5, and those for solutions in L 2 (R N ) in Section 6.

We finish this section by giving some notations and we recall an embedding property of the weighted Sobolev space L 2 (R N ) ∩ L 2 (|x| 2 ; dx), which will be used to prove the results for solutions lying in this space, and some results of the solutions of the nonlinear Schrödinger equation (1.1).

We design by z the conjugate of the complex number z and ∆ =

N j=1 ∂ 2 ∂x 2 j . For p ∈ [1, ∞], we denote by p ′ the conjugate of p defined by 1 p + 1 p ′ = 1 and by L p (R N ) = L p (R N ; C), with norm . L p , the Lebesgue spaces. H 1 (R N ) = H 1 (R N ; C) with norm . H 1 ,
is the well-known Sobolev space and we use the convention

W 0,p (R N ) = L p (R N ) and H 0 (R N ) = W 0,2 (R N ) = L 2 (R N ). We define the Hilbert spaces Y = ψ ∈ L 2 (R N ; C); ψ Y < ∞ with norm ψ 2 Y = ψ 2 L 2 (R N ) + R N |x| 2 |ψ(x)| 2 dx and X = ψ ∈ H 1 (R N ; C); ψ X < ∞ with norm ψ 2 X = ψ 2 H 1 (R N ) + R N |x| 2 |ψ(x)| 2 dx. For a functional space E ⊂ S ′ (R N ) with norm . E , we write f E = ∞ if f ∈ S ′ (R N ) and if f ∈ E.
We design by (T (t)) t∈R the group of isometries (e it∆ ) t∈R generated by i∆ on L 2 (R N ; C) and by C the auxiliary positive constants. Finally C(a 1 , a 2 , . . . , a n ) indicates that the constant C depends only on parameters a 1 , a 2 , . . . , a n and that the dependence is continuous.

It is clear that Y is a separable Hilbert space and that Y ֒→ L r ′ (R N ) with dense embedding, if

r ∈ 2, 2N N -2 (r ∈ [2, ∞] if N = 1
). We recall that for every ϕ ∈ H 1 (R N ), (1.1) has a unique solution u ∈ C((-T * , T * );

H 1 (R N ))
which satisfies the conservation of charge and energy, that is, for all t ∈ (-T * , T * ), u(t

) L 2 = ϕ L 2 and E(u(t)) = E(ϕ), where E(ϕ) def = 1 2 ∇ϕ 2 L 2 -λ α+2 ϕ α+2 L α+2 .
Moreover, for every admissible pair (q, r) (see Definition 1.1 below), u ∈ L q loc ((-T * , T * );

W 1,r (R N ))). In addition, if λ 0, if α < 4 N or if ϕ H 1 is small enough then T * = T * = ∞ and u L ∞ (R;H 1 ) < ∞. Finally, if ϕ ∈ X then u ∈ C((-T * , T * ); X)
. See Ginibre and Velo [START_REF] Ginibre | On a class of nonlinear Schrödinger equations. III. Special theories in dimensions 1, 2 and 3[END_REF][START_REF] Ginibre | On a class of nonlinear Schrödinger equations. I. The Cauchy problem, general case[END_REF][START_REF] Ginibre | On a class of nonlinear Schrödinger equations. II. Scattering theory, general case[END_REF][START_REF] Ginibre | On the global Cauchy problem for some nonlinear Schrödinger equations[END_REF][START_REF] Ginibre | The global Cauchy problem for the nonlinear Schrödinger equation revisited[END_REF], Kato [START_REF] Kato | On nonlinear Schrödinger equations[END_REF][START_REF] Kato | Nonlinear Schrödinger equations[END_REF]. See also Cazenave and Weissler [START_REF] Cazenave | The Cauchy problem for the nonlinear Schrödinger equation in H 1[END_REF][START_REF] Cazenave | Some remarks on the nonlinear Schrödinger equation in the subcritical case[END_REF]. We are also interested by solutions in L 2 (R N ). We recall that if 0 < α

4 N then for every ϕ ∈ L 2 (R N ), (1.1) has a unique solution u ∈ C((-T * , T * ); L 2 (R N ))∩L q loc ((-T * , T * ); L α+2 (R N ))
, where q = 4(α+2)

N α , which satisfies the above conservation of charge. In addition, for every admissible pair (q, r), u ∈ L q loc ((-T * , T * ); L r (R N )). Finally, if α < 4 N then T * = T * = ∞. See Tsutsumi [START_REF] Tsutsumi | L 2 -solutions for nonlinear Schrödinger equations and nonlinear groups[END_REF]. See also Cazenave and Weissler [START_REF] Cazenave | Some remarks on the nonlinear Schrödinger equation in the critical case[END_REF][START_REF] Cazenave | The Cauchy problem for the critical nonlinear Schrödinger equation in H s[END_REF]. Definition 1.1. We say that (q, r) is an admissible pair if the following holds.

(i) 2 r 2N N -2 (2 r < ∞ if N = 2, 2 r ∞ if N = 1), (ii) 2 q = N 1 2 -1 r . Note that in this case 2 q ∞ and q = 4r N (r -2)
.

Finally, we recall the Strichartz' estimates. Let I ⊆ R, be an interval, let t 0 ∈ I, let (q, r) and (γ, ρ) be two admissible pairs, let ϕ ∈ L 2 (R N ) and let f ∈ L γ ′ (I; L ρ ′ (R N )). Then the following integral equation defined for all t ∈ I, u(t) = T (t)ϕ + i t t0

T (t -s)f (s)ds, satisfies the following inequality

u L q (I,L r ) C 0 ϕ L 2 +C 1 f L γ ′ (I;L ρ ′ )
, where C 0 = C 0 (N, r) and C 1 = C 1 (N, r, ρ). For more details, see Keel and Tao [START_REF] Keel | Endpoint Strichartz estimates[END_REF].

2 Sharp lower bound

Theorem 2.1. Let λ 0, 0 α < 4 N -2 (0 α < ∞ if N = 1), ϕ ∈ H 1 (R N ) and u be the corresponding solution of (1.1). If α 4 N then assume further that ϕ ∈ X. If ϕ ≡ 0 then for every r ∈ [2, ∞], lim inf t→∞ |t| N ( 1 2 -1 r ) u(t) L r > 0. Theorem 2.2. Let λ > 0, 0 α < 4 N -2 (0 α < ∞ if N = 1) and ϕ ∈ H 1 (R N ) be such that the corresponding solution u of (1.1) is positively global in time. If α 4 N then assume further that ϕ ∈ X. If ϕ ≡ 0 then for every r ∈ [α + 2, ∞],        lim inf t→∞ |t| N ( 1 2 -1 r ) u(t) L r > 0, if α 4 N , lim sup t→∞ |t| N ( 1 2 -1 r ) u(t) L r > 0, if α > 4 N .

And if there exists

ρ ∈ [α + 2, ∞] such that lim sup t→∞ |t| N ( 1 2 -1 ρ ) u(t) L ρ < ∞ then for every r ∈ [2, ∞], lim inf t→∞ |t| N ( 1 2 -1 r ) u(t) L r > 0.
Remark 2.3. Theorems 2.1 and 2.2 assert that if u is a solution of (1.1) with λ ∈ R, α = 0 and initial data ϕ ∈ X, then for every r ∈ [2, ∞], lim inf

t→∞ |t| N ( 1 2 -1 r ) u(t) L r > 0.
In the attractive case and when α > 4 N +2 (α > 2 if N = 1), we may obtain an optimal lower bound. It is sufficient to choose ϕ X small enough (see corollary below).

Corollary 2.4. Let λ > 0, 4 N + 2 < α < 4 N -2 (2 < α < ∞ if N = 1)
, ϕ ∈ X and u be the corresponding solution u of (1.1). If ϕ ≡ 0 and if ϕ X is small enough then u is global in time and

for every r ∈ [2, ∞], lim inf t→∞ |t| N ( 1 2 -1 r ) u(t) L r > 0.
When α = 4 N , we may suppose that ϕ ∈ H 1 (R N ) instead of ϕ ∈ X, as shows the following proposition, provided that ϕ H 1 is small enough.

Proposition 2.5. Let λ ∈ R \ {0}, α = 4 N , ϕ ∈ H 1 (R N )
and u be the associated solution of (1.1). If ϕ ≡ 0 and if ϕ H 1 is small enough then u is global in time and for every r ∈ [2, ∞],

lim inf t→∞ |t| N ( 1 2 -1 r ) u(t) L r > 0.
Remark 2.6. The lower bounds obtained in Theorems 2.1 and 2.2 are optimal with respect to the decay. In particular, if u is any nontrivial solution of (1.1), then the following estimate never occurs.

∀t > 0, u(t) L r C|t| -N ( 1 2 -1 r ) (ln |t|) -δ , (2.1)
for some r > 2 and δ > 0. This is very surprising since some above results are established for solutions of some heat or Ginzburg-Landau equations (see the beginning of Section 1). For example, such estimates are obtained for the solutions u of the Schrödinger equation

u t -u xx + i|u| 2 u = 0, (t, x) ∈ [0, ∞) × R,
if u(0) X is sufficiently small (Theorem 1.1 of Hayashi, Kaikina and Naumkin [START_REF] Hayashi | Large time behaviour of solutions to the dissipative nonlinear Schrödinger equation[END_REF]). Furthermore, if α > 4 N then Theorems 2.1 and 2.2 are optimal with respect to the assumption on the initial data ϕ, that is ϕ ∈ H 1 (R N ), in the sense that H 1 (R N ) is the smallest functional space in which we must take ϕ to have a solution. On the other hand, when α 4 N , we have to make the additional assumption on initial data ϕ, that is ϕ ∈ X. This request is very reasonable since this is in this functional space that we obtain solutions of (1.1) which have a linear decay (see the references cited in Section 1).

Remark 2.7. Note that all the results of this section and Section 3 hold for t < 0 as soon as the solution u is negatively global in time. Indeed, it is sufficient to apply the case t > 0 to the solution positively global in time ũ of (1.1) with initial data ϕ. Since ũ(t) = u(-t), the result for t < 0 follows.

Main results in the Lebesgue space

As show the results of Section 2, if we suppose a suitable asymptotic behavior of the initial value

(u(0) ∈ X if α 4 N , u(0) ∈ H 1 (R N ) if α > 4
N ), then we have a sharp lower bound. In particular, under the hypotheses of Section 2, such results do not allow estimates of type (2.1), for any nontrivial solution of (1.1), for some r > 2 and δ > 0 (see Remark 2.6). In this section, we establish some lower bounds which eventually allow estimates on the above type, only if α is small enough (see Theorem 3.5 below). The loss of sharp estimate is compensated by a weaker assumption on u(0), that

is u(0) ∈ L 2 (R N ) if α 4 N .
As we can see, this hypothesis is optimal with respect to the integrability of the initial data, in the sense that we make the minimal assumption on u(0) to have existence of a solution. But when α > 4 N , Theorems 2.1 and 2.2 are optimal with respect to the lower bound and to the assumption on u(0). So we only have to consider the case α (α > 2 if N = 1), then the sharp estimate still holds (see Theorems 3.1 and 3.2 below). However, we have to make an additional decay assumption on the solution u (u must satisfy (3.1)).

Theorem 3.1. Let λ ∈ R \ {0}, 4 N + 2 < α 4 N (2 < α 4 if N = 1), ϕ ∈ L 2 (R N )
and u be the corresponding solution of (1.1). If α = 4 N then assume further that u is positively global in time.

Suppose that for every r ∈ 2, 2N

N -2 (r ∈ [2, ∞) if N = 1), a.e. t > 0, u(t) L r C|t| -N ( 1 2 -1 r ) . (3.1)
Then we have for every r ∈ [2, ∞], estimates that u ∈ L q loc ([0, ∞); L r (R N )), for every admissible pair (q, r). This yields, u(t) ∈ L r (R N ), for almost every t > 0 and for all r

lim inf t→∞ |t| N ( 1 2 -1 r ) u(t) L r > 0. Theorem 3.2. Let λ ∈ R \ {0}, let α 0 = -(N -2)+ √ N 2 +12N +4 2N , α 0 < α 4 N , ϕ ∈ L 2 (R N )
∈ 2, 2N N -2 (r ∈ [2, ∞) if N = 2, r ∈ [2, ∞] if N = 1).
Remark 3.4. As shows Lemma 4.3, Theorem 3.2 has less restrictive assumptions than Theorem 3.1 when α > α 0 . Indeed, we do not have to suppose that u satisfies (3.1) for all r. We may only assume that it is satisfied for r = α + 2. Furthermore, estimates of type (2.1) do not occur. Finally, Theorem 3.2 can be extended for α = α 0 in the following sense. If there exists r ∈ [α 0 + 2, ∞] and ε > 0 such that for almost every t > 0,

u(t) L r C|t| -N ( 1 2 -1 r )-ε , (3.2)
then for all t ∈ R, u(t) ≡ 0. See the proof of Theorem 3.2 for the justification.

Remark 3.5. In the case where α 4 N +2 (α 1 if N = 1), we have the following result. Let

λ ∈ R \ {0}, 0 < α 4 N + 2 (0 < α 1 if N = 1), ϕ ∈ L 2 (R N ) and u be the corresponding solution of (1.1). Let r ∈ [2, ∞]. If there exists ε > 0 such that          u(t) L 2N N -2 C|t| -(1+ 4-α(N +2) 4α +ε) , if N 3, u(t) L r C|t| -2( 1 2 -1 r )-r(1-α)+α rα -ε , if N = 2, u(t) L ∞ C|t| -( 1 2 + 2-α 2α +ε) , if N = 1, (3.3)
for almost every t > 0, then for all t ∈ R, u(t) ≡ 0. See the proof of Theorem 3.2 for the justification.

When N 3 and α = 4 N + 2

, the result is the same if we have (3.3) for some r ∈ 2N N -2 , ∞ . Indeed, by Lemma 4.3 this hypothesis leads to (3.3)

with r = 2N N -2 . When α < 4 N +2 (α 1 if N = 1), estimate (3.
3) is a very strong assumption since it implies that u decays faster than the solution of the linear equation. Furthermore, there is a gap between the admissible and the non-admissible powers of decay (compare (3.1) with (3.3)).

Estimates at infinity

Proposition 4.1. Let λ ∈ R \ {0}, m ∈ {0; 1}, 0 < α 4 N -2m (0 < α < ∞ if N = m = 1 and 0 < α < 4 N -2 if N 2 and m = 1), ϕ ∈ H m (R N ) and u ∈ C((-T * , T * ); H m (R N
)) be the unique corresponding solution of (1.1). Assume that T * = ∞. If there exist t 0 0 and (γ, ρ) an admissible pair with γα γ-2 < ∞ and

2 ρα ρ-2 2N N -2 (2 ρα ρ-2 < ∞ if N = 2, 2 ρα ρ-2 ∞ if N = 1) such that u ∈ L γα γ-2 ((t 0 , ∞); L ρα ρ-2 (R N ))
, then the following properties hold.

1. For every admissible pair (q, r), u ∈ L q ((0, ∞); W m,r (R N )),

There exists u

+ ∈ H m (R N ) such that lim t→∞ T (-t)u(t) -u + H m = 0.
A similar result holds for t < 0.

Proof. By remark 2.7, we only have to show the case t > 0. We proceed in 2 steps. Set f (u) = λ|u| α u.

Step

1. f (u) ∈ L γ ′ ((t 0 , ∞); W m,ρ ′ (R N )).
We first show that u ∈ L γ ((0, ∞); W m,ρ (R N )). We already know that u ∈ L q loc ([0, ∞); W m,r (R N )), for every admissible pair (q, r). We have the following integral equation.

∀S 0, ∀t 0, u(t) = T (t -S)u(S) + i t S T (t -s)f (u(s))ds.
So we have by the Hölder's inequality (applied in space-time) and Strichartz' estimates,

f (u) L γ ′ ((t0,t);W m,ρ ′ ) C u α L γα γ-2 ((t0,∞);L ρα ρ-2 ) u L γ ((0,t);W m,ρ ) , (4.1)
u L γ ((S,t);W m,ρ ) C + C 0 u α L γα γ-2 ((S,∞);L ρα ρ-2 ) u L γ ((S,t);W m,ρ ) , (4.2)
for every

t 0 S < t < ∞. Since u ∈ L γα γ-2 ((t 0 , ∞); L ρα ρ-2 (R N )), there exists S 0 > t 0 large enough such that C 0 u α L γα γ-2 ((S0,∞);L ρα ρ-2 )
1/2, where C 0 is the constant in (4.2). So with (4.2), we obtain

u L γ ((S0,t);W m,ρ ) 2C, for every t > S 0 . It follows that u L γ ((S0,∞);W m,ρ )
2C and so we have

u ∈ L γ ((0, ∞); W m,ρ (R N )).
Hence the result by letting t ր ∞ in (4.1).

Step 2. Conclusion.

By

Step 1 and Strichartz' estimates, u ∈ L q ((0, ∞); W m,r (R N )), for every admissible pair (q, r). Then 1 follows. From the Strichartz' estimates and by the fact that T (t) is an isometry on H m (R N ), we obtain for every τ > t > t 0 ,

T (-t)u(t) -T (-τ )u(τ ) H m C f (u) L γ ′ ((t,τ );W m,ρ ′ ) t,τ →∞ ----→ 0,
by Step 1. Hence 2. This concludes the proof.

Remark 4.2. Note that by assumption, one always has γα γ-2 > 0. However, it may happen that γα γ-2 < 1. This is clearly not a problem since the above proof still holds and that we do not use the triangular inequality.

Lemma 4.3. Let λ ∈ R, m ∈ {0; 1}, 0 α 4 N -2m (0 α < ∞ if N = m = 1 and 0 α < 4 N -2 if N 2 and m = 1), ϕ ∈ H m (R N ) and u ∈ C((-T * , T * ); H m (R N )) be the corresponding solution of (1.1). Assume that T * = ∞. If there exist r ∈ (2, ∞], ε 0 and a constant C = C(t) > 0 such that u(t) satisfies (3.
2) for some t > 0, then for every ρ ∈ (2, r], there exist ε(ρ) 0 and C 0 (t) > 0 such that

u(t) L ρ C 0 (t)t -N ( 1 2 -1 ρ )-ε(ρ) , (4.3)
where the function ρ -→ ε(ρ) is continuous from (2, r] to [0, ∞) and satisfies ε(ρ) > 0 ⇐⇒ ε > 0. If C is independent on t then C 0 is also independent on t. Finally, if (3.2) is satisfied for every t > 0 then (4.3) is satisfied for every t > 0, and if lim inf t→∞

C(t) = 0 then lim inf t→∞ C 0 (t) = 0. Proof. Let ρ ∈ (2, r]. Set θ = r ρ ρ -2 r -2 , ε(ρ) = εθ and C 0 (t) = C(t) θ . Then θ ∈ (0, 1] and θ satisfies 1 ρ = 1 -θ 2 + θ r
. By Hölder's inequality and conservation of charge, we obtain

u(t) L ρ u(t) 1-θ L 2 u(t) θ L r C(t) θ |t| -N ( 1 2 -1 2 )(1-θ)-N ( 1 2 -1 r )θ-εθ C 0 (t)|t| -N ( 1 2 -1 ρ )-ε(ρ) .
Hence the result.

Lemma 4.4. Let λ ∈ R \ {0}, m ∈ {0; 1}, 0 < α 4 N -2m (0 < α < ∞ if N = m = 1 and 0 < α < 4 N -2 if N 2 and m = 1), ϕ ∈ H m (R N ) and u ∈ C((-T * , T * ); H m (R N )) be the corresponding solution of (1.1). Assume that T * = ∞. If u satisfies (3.1) for every r ∈ 2, 2N N -2 (r ∈ [2, ∞) if N = 1) and if α > 4 N +2 (α > 2 if N = 1)
, then there exists an admissible pair (γ, ρ) with

1 < γα γ-2 < ∞ and 2 < ρα ρ-2 < 2N N -2 (2 < ρα ρ-2 < ∞ if N = 1) such that u ∈ L γα γ-2 ((1, ∞); L ρα ρ-2 (R N )).
Proof. We distinguish 3 cases : N 3, N = 2 and N = 1.

Case N 3. Set ρ * = 4N 2N -α(N -2) . Since 0 < α < 4 N -2 then 2 < ρ * < 2N N -2 . Let γ * > 2 be such that (γ * , ρ * ) is an admissible pair. For this choice of ρ * , we have ρ * α ρ * -2 = 2N N -2 and γ * γ * -2 = 4 4-α(N -2) . When α < N +2 N , we have ρ * < 2N (N +2)-N α ⇐⇒ α > 4 N +2 . Let ρ > ρ * , ρ sufficiently close to ρ * to have ρα ρ-2 > 2. If α < N +2
N , then we also choose ρ < 2N (N +2)-N α . Since ρ > ρ * then ρα ρ-2 < 2N N -2 and so there exists γ > 2 such that (γ, ρ) is an admissible pair. Then γ γ-2 = 2ρ 2N -ρ(N -2) . By (3.1) we have,

u γα γ-2 L γα γ-2 ((1,∞);L ρα ρ-2 ) = ∞ 1 u(t) γα γ-2 L ρα ρ-2 dt C ∞ 1 t -N ρα-2(ρ-2) 2N -ρ(N -2) dt < ∞. Indeed, if α < N +2 N then N ρα-2(ρ-2) 2N -ρ(N -2) > 1 ⇐⇒ ρ < 2N (N +2)-N α and if α N +2 N then we always have N ρα-2(ρ-2) 2N -ρ(N -2) > 1. So, for this choice of (γ, ρ), u ∈ L γα γ-2 ((1, ∞); L ρα ρ-2 (R N )).
Case N=2. Since α > 1 is fixed, we can choose ρ > 2 sufficiently close to 2 to have α > 2(ρ-1) ρ .

In particular, this implies that ρα ρ-2 > 2. Moreover, γ γ-2 = ρ 2 where γ > 2 is such that (γ, ρ) is an admissible pair. By (3.1) we have,

u γα γ-2 L γα γ-2 ((1,∞);L ρα ρ-2 ) = ∞ 1 u(t) ρα 2 L ρα ρ-2 dt C ∞ 1 t -ρα-2(ρ-2) 2 dt < ∞, since ρα-2(ρ-2) 2 > 1 ⇐⇒ α > 2(ρ-1) ρ . So u ∈ L γα γ-2 ((1, ∞); L ρα ρ-2 (R 2 )
) for this choice of (γ, ρ).

Case N=1. Since α > 2 is fixed, we can choose ρ > 2 sufficiently close to 2 to have α > 3ρ-2 ρ . In particular, this implies that ρα ρ-2 > 2. Moreover, γ γ-2 = 2ρ ρ+2 where γ > 2 is such that (γ, ρ) is an admissible pair. By (3.1) we have,

u γα γ-2 L γα γ-2 ((1,∞);L ρα ρ-2 ) = ∞ 1 u(t) 2ρα ρ+2 L ρα ρ-2 dt C ∞ 1 t -ρα-2(ρ-2) ρ+2 dt < ∞, since ρα-2(ρ-2) ρ+2 > 1 ⇐⇒ α > 3ρ-2 ρ . So for this choice of (γ, ρ), u ∈ L γα γ-2 ((0, ∞); L ρα ρ-2 (R)).
As seen in Section 1, the crux of the proof of results of this paper is based on the following lemma.

Lemma 4.5. Let λ ∈ R \ {0}, m ∈ {0; 1}, 0 α 4 N -2m (0 α < ∞ if N = m = 1 and 0 α < 4 N -2 if N 2 and m = 1), ϕ ∈ H m (R N ) and u ∈ C((-T * , T * ); H m (R N
)) be the corresponding solution of (1.1). Assume that T * = ∞. If ϕ ≡ 0 and if there exists

u + ∈ L 2 (R N ) such that lim t→∞ T (-t)u(t) -u + L 2 = 0, then for every r ∈ [2, ∞], lim inf t→∞ |t| N ( 1 2 -1 r ) u(t) L r > 0.
The proof of Lemma 4.5 is based on the pseudo-conformal transformation.

For every positively global solution u of (1.1) with initial data ϕ ∈ L 2 (R N ), we define the function

v ∈ C([0, 1); L 2 (R N )) by ∀t ∈ [0, 1), a.e. x ∈ R N , v(t, x) = (1 -t) -N 2 u t 1 -t , x 1 -t e -i |x| 2 4(1-t) . (4.4)
A straightforward calculation gives for every p ∈ [1, ∞] and for all t ∈ [0, 1),

v(t) L p = (1 -t) -N ( 1 2 -1 p ) u t 1 -t L p , (4.5)
v(t) L 2 = ϕ L 2 , (4.6)
where the last identity comes from (4.5) and from conservation of charge for u. Note that (4.5) makes sense as soon as u t 1-t ∈ L p (R N ). When ϕ ∈ X, we obviously have v ∈ C([0, 1); X) and so we may define for all t ∈ [0, 1),

E 1 (t) = 1 2 (1 -t) 4-N α 2 ∇v(t) 2 L 2 - λ α + 2 v(t) α+2 L α+2 , E 2 (t) = 1 8 (x + 2i(1 -t)∇)v(t) 2 L 2 - λ α + 2 (1 -t) N α 2 v(t) α+2 L α+2 .
Then for all t ∈ [0, 1), Proof. We argue by contraposition. Let v ∈ C([0, 1); X) be the function defined by (4.4). Assume there exists r 2 if λ 0, and r α + 2 if λ > 0, such that lim inf

d dt E 1 (t) = N α -4 4 (1 -t) 2-N α 2 ∇v(t) 2 L 2 , (4.7)
t→∞ t N ( 1 2 -1 r ) u(t) L r = 0.
Then, we have to show that ϕ ≡ 0.

By conservation of charge, if r = 2 then ϕ ≡ 0. So we may assume that r > 2. Furthermore, by Lemma 4.3, we also may assume that r <

2N N -2 (r < ∞ if N = 1) if λ 0 or if α = 0, and r = α + 2, if λ > 0 and if α > 0. Since lim inf t→∞ t N ( 1 2 -1 r ) u(t) L r = 0, it follows from (4.5) that lim inf tր1 v(t) L r = 0.
Thus, there exists a sequence (t n ) n∈N ⊂ (0, 1) satisfying

t n n→∞ ----→ 1 such that lim n→∞ v(t n ) L r = 0. ( 5
n ∈ N, ∇v(t n ) L 2 C(1 -t n ) N α-4 4
. It follows that,

(1 -t n ) ∇v(t n ) L 2 C(1 -t n ) N α 4 n→∞ ----→ 0,
v(t n ) Y < ∞. (5.2)
It follows that for λ ∈ R and for α ∈ 0, 4 N , we have (5.2). From (4.6), Hölder's inequality, from the embedding Y ֒→ L r ′ (R N ), from (5.2) and (5.1), we obtain

ϕ L 2 = v(t n ) L 2 v(t n ) 1 2 L r ′ v(t n ) 1 2 L r C v(t n ) 1 2 Y v(t n ) 1 2 L r C v(t n ) 1 2 L r n→∞ ----→ 0. So ϕ L 2 = 0 which is ϕ ≡ 0. Hence the result. Proof of Theorem 2.1. If α 4
N then the result comes from Lemma 5.1. So we may assume that α > 4 N . Since λ < 0 and α > 4 N , there exists

u + ∈ H 1 (R N ) such that lim t→∞ T (-t)u(t) -u + H 1 = 0
(Ginibre and Velo [START_REF] Ginibre | Scattering theory in the energy space for a class of nonlinear Schrödinger equations[END_REF], Nakanishi [START_REF] Nakanishi | Energy scattering for nonlinear Klein-Gordon and Schrödinger equations in spatial dimensions 1 and 2[END_REF][START_REF] Nakanishi | Remarks on the energy scattering for nonlinear Klein-Gordon and Schrödinger equations[END_REF]). The result comes from Lemma 4.5.

Proof of Theorem 2.2. We proceed in 4 steps. Let v ∈ C([0, 1); L 2 (R N )) be the function defined by (4.4).

Step 1. If α > 4 N and if lim sup

t→∞ t N ( 1 2 -1 α+2 ) u(t) L α+2 C then there exists u + ∈ H 1 (R N ) such that lim t→∞ T (-t)u(t) -u + H 1 = 0. ( 5.3)
embedding Y ֒→ L r ′ (R N ), we have for all t ∈ [0, 1),

ϕ L 2 = v(t) L 2 v(t) 1 2 L r ′ v(t) 1 2 L r C v(t) 1 2 Y v(t) 1 2 L r C v(t) 1 2 L r . Thus ϕ L 2 C lim inf tր1 v(t) 1 2
L r = 0 by (5.5) and so ϕ L 2 = 0, which is absurd. Case 2 : α > 4 N . By Step 1, there exists u + ∈ H 1 (R N ) satisfying (5.3), which gives ϕ ≡ 0 by Lemma 4.5. This result being absurd, Step 4 is true. This concludes the proof.

Proof of Corollary 2.4. By Cazenave and Weissler [START_REF] Cazenave | Rapidly decaying solutions of the nonlinear Schrödinger equation[END_REF], we know that if ϕ X is sufficiently small, then u is global in time and there exists u + ∈ X such that T (-t)u(t) Proof of Proposition 2.5. It is well-known that if ϕ H 1 is sufficiently small then u is global in time and u ∈ L α+2 (R; L α+2 (R N )) (Remark 7.7.6 of Cazenave [START_REF] Cazenave | An introduction to nonlinear Schrödinger equations, volume 26 of Textos de Métodos Matemáticos[END_REF]). Then T (-t)u(t)

H 1 (R N )
-----→ t→∞ u + , for some u + ∈ H 1 (R N ) (Proposition 4.1), and the result comes from Lemma 4.5.

Proof of the results of Section 3

Our strategy is the same as for Section 5. However, we could give an other proof as follows, without requiring the pseudo-conformal transformation. We would show that if a solution u of (1.1) had a decay rate too fast, then u would have a scattering state u ∞ whose corresponding solution of the linear problem (that is (1.1) with λ = 0) would have a decay rate of the same order of u. In particular, α > 2 N otherwise u ∞ ≡ 0 (Barab [START_REF] Barab | Nonexistence of asymptotically free solutions for a nonlinear Schrödinger equation[END_REF], Strauss [START_REF] Strauss | Nonlinear scattering theory[END_REF][START_REF] Strauss | Nonlinear scattering theory at low energy[END_REF]). This rate being too fast, we would have u ∞ ≡ 0 (by (1.3)). And from conservation of charge, we would deduce that u(t) ≡ 0, for all t ∈ R. Furthermore, in the case N = 1, we would have to make the additional assumption ϕ ∈ X when 1 < α 2 (in order to apply the result of Barab [START_REF] Barab | Nonexistence of asymptotically free solutions for a nonlinear Schrödinger equation[END_REF]). But this case falls into the scope of Theorems 2.1 and 2.2

where there is a better result. It follows that in this case, the result would not be interesting.

Proof of Theorems 3.1 and 3.2 and Remarks 3.4 and 3.5. We proceed in 2 steps.

Step 1. There exists u + ∈ L 2 (R N ) such that lim Case of Theorems 3.2 and Remark 3.4. Set q = 4(α+2) N α . Thus (q, α + 2) is an admissible pair. By

4 N.

 4 On the other hand, if α > 4 N +2

1 rRemark 3 . 3 .

 133 and u be the corresponding solution of (1.1). If α = 4 N then assume further that u is positively global in time. If there exists r ∈ [α + 2, ∞] such that u satisfies (3.1) then for every r ∈ [2, ∞], ) u(t) L r > 0. When ϕ ∈ L 2 (R N ), the condition (3.1) makes sense. Indeed, we have by the Strichartz'

d dt E 2

 2 (t) = 0. (4.8)

X

  ---→ t→∞ u + . Then, Lemma 4.5 gives the result.

t→∞T 2 (r ∈ [ 2 ,

 22 (-t)u(t) -u + L 2 = 0. Case of Theorems 3.1. Since u satisfies (3.1) for every r ∈ 2, 2N N -∞) if N = 1), it follows from Lemma 4.4 that there exists an admissible pair (γ, ρ) such that u ∈ L γα γ-2 ((1, ∞); L ρα ρ-2 (R N )). The result follows from Proposition 4.1.

Acknowledgments

The author wishes to thank his thesis adviser, Professor Thierry Cazenave, for having suggested this work and for helpful advices.

Note added in proof. Recently, a generalization of Theorem 2.1 has been established for a large class of nonlinearities, as soon as the solution is bounded in time in H 1 0 (Ω). Unfortunately, these results do not apply in the case of L 2 -solutions (which is the case in Section 3 of this paper). For more details, see [2].

For the proof, see Proposition 3.8 and formulas (3.20) and (3.21) of Cazenave and Weissler [START_REF] Cazenave | Rapidly decaying solutions of the nonlinear Schrödinger equation[END_REF].

Proof of Lemma 4.5. We argue by contradiction. Let v ∈ C([0, 1); L 2 (R N )) be the function defined by (4.4). Assume that there exists r 2 such that lim inf

Then, we shall show that ϕ ≡ 0.

By conservation of charge and Lemma 4.3, we may assume that 2 < r < 2N N -2 (2 < r < ∞ if N = 1). Since u(t) ∈ L r (R N ) for almost every t > 0, it follows that v(t) ∈ L r (R N ), for almost every t ∈ (0, 1). By (4.5), we have

Cazenave and Weissler [START_REF] Cazenave | Rapidly decaying solutions of the nonlinear Schrödinger equation[END_REF], this implies that there exists

(Although Proposition 3.14 is given with α > 0, the result still holds for α = 0 since the proof applies without any modification.) From (4.9) and (4.10) we deduce that lim tր1 v(t) L 2 = 0, from which it follows with the conservation of charge (4.6), ϕ L 2 = 0. This is absurd since ϕ ≡ 0.

Proof of the results of Section 2

Our strategy is the following. We show that if a solution u of (1.1) has a decay rate too fast, then the corresponding function v given by the pseudo-conformal transformation must converge to 0 in a Lebesgue space L p (R N ), for some 2 < p < ∞. But these functions also satisfy the conservation of charge. And by using the embedding Y ֒→ L p ′ (R N ) or the existence of a strong limit for v(t) in

In order to show Theorems 2.1 and 2.2, we split the proof in 2 cases, which are α

for every r ∈ [2, ∞] if λ 0, and for every r

4-α(N -2) > 1 and it follows that

Therefore, u ∈ L qα q-2 ((0, ∞); L α+2 (R N )) and the result comes from Proposition 4.1.

Step 2. If ϕ ≡ 0 and if α 4

N then for all r α + 2, lim inf

The result comes from Lemma 5.1.

Step 3. If ϕ ≡ 0 and if α > 4 N then for all r α + 2, lim sup

We argue by contraposition. Assume that there exists r α + 2 such that lim sup

Then, we have to show that ϕ ≡ 0.

By Lemma 4.3, we may assume that r = α+2.

Step 1 implies that there exists u + ∈ H 1 (R N ) satisfying (5.3). Then ϕ ≡ 0 by Lemma 4.5, which is the desired result.

Step 4. If ϕ ≡ 0 and if there exists ρ α + 2 such that lim sup

If α = 0 then Step 2 gives the result and so we consider the case α > 0. By Lemma 4.3, we may assume that ρ = α + 2. We argue by contradiction. Suppose that there exists r 2 such that lim inf

. Indeed, this comes from conservation of charge and Lemma 4.3. We obtain with (4.5),

Case 1 : 0 < α We set ε 0 = qα q-2 ε. And since

1 ⇐⇒ α α 0 , it follows from (3.1) or (3.2) that,

) and the result comes from Proposition 4.1.

Case of Remark 3.5. Let r 2 and ε > 0 be as in (3.3). By conservation of charge, r > 2. Furthermore when N = 2, we may assume that r < ∞ (Lemma 4. Step 2. Conclusion.

The result comes from Step 1 and Lemma 4.5. This achieves the proof.


Hippolyte D'albis

Emmanuelle Augeraud-Veron

Alain Venditti

R Benhabib

C Boucekkine

C Ghiglino

Le Van

Albis Hippolyte D'

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, local determinacy 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 [4], to be a better index of experience than the average capital stock. We prove that a slight memory effect characterizing the learning-by-doing process is enough to generate business cycle fluctuations through a Hopf bifurcation leading to stable periodic orbits. This is obtained for reasonable parameter values, notably for both the amount of externalities 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

Learning-by-doing externalities in the production process are well-known as a source of economic growth. In this paper, we show that they could also explain economic fluctuations. To do so, 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] that use the existing capital stock as a proxy for 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, we measure the learning-by-doing effects with the whole gross investment process over a given period of time. 1 This assumption, which will importantly shape the equilibrium dynamics, represents a memory effect suggesting that investments made some time ago do not have the same impact on the index of experience as recent ones. This can be justified by the finite life expectancy of the workers,2 by the job turn-over of employees, and also by the fact that the diffusion's speed of innovations in the economy is increasing. However, considering that the structural parameters are evaluated on a yearly basis, and that the time interval characterizing the memory process will be limited, our paper clearly focuses on business cycle fluctuations.

We study a continuous-time 3 aggregate model with inelastic labor and learning-by-doing externalities in the production process. For computational convenience, we represent the memory process as 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. The equilibrium path is described by a system of functional differential equations that is similar to those that can be found in the vintage capital literature. The main difficulty that emerges in the course of the analytical resolution of those models is that the optimality conditions are formulated as a system of mixed functional differential equa-tions in which delayed (the capital stock) and advanced (the shadow price) terms are considered simultaneously. In our paper, introducing a lagged capital stock through a Arrow-type externality leads to optimality conditions formulated as a system of delay functional differential equations. This simpler framework notably allows us to provide results on local stability and determinacy.

The main purpose of our paper is to show the existence of a Hopf bifurcation and to establish the conditions under which the equilibrium path converges towards the periodic solution in an aggregate growth model with learning-by-doing externalities. In particular, we prove that persistent endogenous fluctuations can occur, first without considering endogenous labor and external effects coming from the labor supply, and second with 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 demonstrate that a simple aggregate model may generate business cycle fluctuations under plausible parameterization of the fundamentals and a low amount of externalities based on a Arrow-type learning-by-doing process. Moreover, we show that the equilibrium, be it convergent toward the steady state or the periodic cycle, is locally determinate, i.e. unique.

The economic intuition for such fluctuations is the following. Assume, 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 reduces the returns to capital but increases the wage rate. Provided the labor supply is weakly elastic, labor adjustments cannot compensate the fall of the interest rate and investments are then slowed down. 4However, 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.

Our paper also proposes an analysis of the local determinacy around the steady-state. Even though this exercise is intuitively similar to the ones that are performed with systems of ordinary differential equations, one has here to deal with a continuum of initial values and with a stable manifold whose dimension is infinite. We show that the steady-state cannot be indeterminate and give a condition for local determinacy. However, non existence of equilibrium paths cannot be excluded in our framework.

Our paper contributes to the theory of endogenously-driven fluctuations that studies extensions of the Ramsey [START_REF] Romer | Increasing returns and long-run growth[END_REF] optimal growth model. In this framework, the aggregate dynamics are characterized by a unique monotonically convergent equilibrium path and 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 one consumption good and one 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]. 5In 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. 6 It is worth noting here that Hopf bifurcations explain endogenous fluctuations better than flip bifurcations. They are more likely to imply persistent, positively auto-correlated endogenous fluctuations (see Dufourt et al. [START_REF] Dufourt | Sunspot equilibria and the expectations-driven Phillips curve[END_REF]), while the latter generate several counterfactual characteristics in the simulated time series.

More 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, but not through a Hopf bifurcation. 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, whose wage elasticity is sufficiently high, i.e., close enough to infinity. 7 Since the elasticity of the aggregate labor supply is usually shown to be low, 8 it follows that the occurrence of local indeterminacy relies on parameter values that do not match empirical evidence.

To overcome this problem, some papers have studied Ramsey-type aggregate models notably with vintage capital, whose dynamics are represented by functional differential equations that are similar to the one we study. 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] show that vintage capital leads to oscillatory dynamics governed by replacement echoes. 9 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, and 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. However, once utility and production functions are non linear, Hopf bifurcations can occur (see Benhabib and Rustichini [START_REF] Benhabib | Vintage capital, investment and growth[END_REF] 10 , Rustichini [START_REF] Rustichini | Hopf bifurcation for functional differential equations of mixed type[END_REF] 11 , Asea and Zak [3] 12 ). 7 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 local indeterminacy in one-sector models. 8 Most econometric analyses available in the literature conclude that the wage elasticity of labor at the microeconomic level lies in (0, 0.3) for men and (0.5, 1) for women (see Blundell and MaCurdy [START_REF] Blundell | Labour supply: a review of alternative approaches[END_REF]). Moreover, Rogerson and Wallenius [START_REF] Rogerson | Micro and macro elasticities in a life cycle model with taxes[END_REF] show that the macroeconomic elasticity of the labor supply with respect to the wage is in the range of (2.25, 3.0). 9 See also Boucekkine et al. [START_REF] Boucekkine | Vintage capital and the dynamics of the AK model[END_REF], Boucekkine et al. [START_REF] Boucekkine | Endogenous vs exogenously driven fluctuations in vintage capital models[END_REF] and Fabbri and Gozzi [START_REF] Fabbri | Solving optimal growth models with vintage capital: the dynamic programming approach[END_REF]. 10 Benhabib and Rustichini [START_REF] Benhabib | Vintage capital, investment and growth[END_REF] mention the possibility of a Hopf bifurcation in an 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. 11 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. 12 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

The paper is organized as follows: Section 2 presents the model and defines the intertemporal equilibrium. Section 3 contains the main results on the existence of a Hopf bifurcation, the local stability properties of the periodic orbits, the local determinacy of the steady state, and presents a numerical example. Section 4 contains concluding comments and proofs are provided in a final Appendix.

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 at the firm level 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 following 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. In what follow, we consider a model with no endogenous growth but our framework could be easily extended to constant return to scale in the accumulation factor.

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

f (k, e) = A αk -ν + (1 -α)e -βν -1 ν ( 1
)
with both delay and advance, whereas they only consider delay in their model.

with α ∈ (0, 1), β ∈ (0, 1), ν > -1 and A > 0. At the optimum of the firm, 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)) (2)
with δ ≥ 0 the depreciation rate of capital. We also compute the share of capital in total income:

s(k, e) = kf 1 (k,e) f (k,e) = α α+(1-α)e -βν k ν ∈ (0, 1) (3)
the elasticity of capital-labor substitution σ = 1/(1 + ν), and the following share and elasticity related to the externalities e:

ε e (k, e) = ef 2 (k,e) f (k,e) = β(1 -s(k, e)), ε ke (k, e) = ef 12 (k,e) f 1 (k,e) = β(1-s(k,e)) σ (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. Note that the restriction β ∈ (0, 1) implies that the externalities are small enough to be compatible with a demand for capital which is decreasing with respect to the rental rate and that the marginal productivity of capital is an increasing function of the externalities. We will also assume in the following that ν > 0, i.e. σ ∈ (0, 1). This restriction ensures that over the business cycles, the labor share is countercyclical while the capital share is pro-cyclical. 13Assuming a stationary population, the dynamics of capital per capita writes:

k(t) = f (k(t), e(t)) -δk(t) -c(t) (5)
where c(t) denotes the consumption per capita. 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:

Assumption 1. e (t) = t t-τ k (s) + δk (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 bounded 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. The formula in Assumption 1 encompasses the Ramsey [START_REF] Ramsey | A mathematical theory of saving[END_REF] model when δ = τ = 0 and the Romer [START_REF] Romer | Increasing returns and long-run growth[END_REF] model when δ = 0, τ → +∞ and lim t→-∞ k(t) = 0 as particular cases.

Preferences and intertemporal equilibrium

The representative infinitely-lived individual supplies a fixed amount of labor l = 1 and derives utility from consumption c according to a CRRA function such that

u(c) = c 1-1 c 1-1 c (6)
with c ∈ (0, +∞) that denotes the elasticity of intertemporal substitution in consumption.

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 (7)
where ρ > 0 denotes the discount factor. By substituting c(t) from the capital accumulation equation into the utility function we obtain a problem of calculus of variations 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 (8)
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 (8) 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 (9)
and the transversality condition

lim t→+∞ u (c(t)) k (t) e -ρt = 0 ( 10
)
for all given e(t). At the individual level, a solution of the Euler equation ( 9) is thus a path of capital stock parameterized by a given path of externalities, namely k(t, {e(s)} s≥0 ). At the aggregate level, as the externalities are defined according to Assumption 1, an equilibrium path is the solution of a fixed-point problem defined such that

e (t) = t t-τ k (s, {e(z)} z≥0 ) + δk (s, {e(z)} z≥0 ) ds (11)
for all t ≥ 0. Assuming that such a fixed-point problem has a solution, 14the capital dynamics are characterized by the following non-linear functional differential equation with distributed delays. By deriving equations ( 5) and ( 9) as well as e (t) with respect to time, and replacing the elasticity of intertemporal substitution by the parameter defined above, we obtain:

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 (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].

We now study the existence of an interior steady state in the neighborhood of which an equilibrium path exists by continuity.

Steady state and characteristic equation

We consider the dynamics in the neighborhood of the steady state. Along a stationary path k(t) = k for any t ≥ 0, Assumption 1 implies e(t) = ē = δτ k provided that τ +∞. An interior steady state is thus a k that solves:

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

c = f k, δτ k -δ k > 0 (14)
We immediately obtain:

Proposition 1. Under Assumption 1, there exists a unique steady state k = (δτ )

β 1-β " αA δ+ρ " ν 1+ν -α 1-α 1 (1-β)ν (15)
We easily derive from ( 3) and ( 4) that at the steady state:

s = α δ+ρ αA ν 1+ν , ε e = β(1 -s), ε ke = β(1-s) σ (16)
The elasticity of intertemporal substitution in consumption c will be used as the bifurcation parameter. Let us now establish the characteristic equation.

Lemma 1. The characteristic equation is D (λ) = 0 with D (λ) = λ 2 -ρλ -c(1-s)(δ+ρ)[δ(1-s)+ρ] sσ -β(1-s)(δ+ρ) sτ λ 2 δ + 1 -c[δ(1-s)+ρ] σδ λ -c[δ(1-s)+ρ] σ 0 -τ e λs ds = 0 (17)
When τ ∈ (0, +∞), the characteristic equation is transcendental and there exist an infinite number of roots, some of them being complex with negative real part. When τ / ∈ (0, +∞), one may in some cases derive characteristic polynomials that are easy to study. For instance, when τ = 0, there is no externality and the characteristic equation becomes:

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. Similarly, if δ = 0, τ → +∞ and lim t→-∞ k(t) = 0, the characteristic equation becomes

D (λ) = λ 2 -ρλ -c(1-s)ρ 2 sσ
and again there are two real roots of opposite sign. When there is no externality (τ = 0), the formulation corresponds to a standard optimal growth model with one state variable, the capital stock, and one forward variable, the associated implicit price, or equivalently the time derivative of the capital stock k(t). One negative, i.e. stable, characteristic root is then associated to the state variable, while one positive, i.e. unstable, characteristic roots is associated with the forward variable. Starting from a given initial value for the capital stock k(0), the unique optimal path is obtained from the existence of a unique value for the implicit price which allows to select the unique converging path.

When externalities are present (τ > 0), while there is still one state variable, we have now to define a continuum of initial values for the capital stock, i.e. k(t) = k 0 (t) for t ∈ [-τ, 0]. Moreover, as we will show in the next section, the initial condition for the forward variable is no longer a real number but is now a continuously differentiable function of time t that has also to be defined on t ∈ [-τ, 0].

Because of the delays, the spectral decomposition features an infinite number of characteristic roots with negative real parts. Let us now establish an important result: Lemma 2. D (λ) = 0 has at least a positive real root.

The existence of characteristic roots with positive real parts implies that any path is unstable in the initial state of continuously differentiable functions on [-τ, 0]. As the transversality condition rules out divergent paths, 15the initial conditions for the forward variable should be chosen such that the equilibrium path belongs to the space associated with the stable characteristic roots. 16 As it will be shown below, the restrictions imposed on this choice will then depend on the (finite) number of unstable characteristic roots.

Endogenous business cycle fluctuations

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

Second, we analyze the local determinacy property of the steady state and the bifurcating cycles.

Hopf bifurcation: existence and stability

In this first part of the analysis, we propose 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 lower bound:

c = δσβ (1-s)(δ+ρ)(β-δτ 2 ) (18)
that is positive if τ < 2β/δ. We introduce the following restrictions:

Assumption 2. c > c > 0.
We provide the following result: Theorem 1. Under Assumptions 1-2, there exists a critical value H c such that when c = H c a Hopf bifurcation occurs generically.

We prove that endogenous business cycle fluctuations are compatible with small externalities as ε ke is bounded above by (1 -s)/σ, but cannot be obtained for arbitrarily low elasticities of intertemporal substitution in consumption. In Section 3.3, we nevertheless show that 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 low enough.

We are now interested in the stability and the direction of the periodic orbit. Indeed, two different cases may occur: the periodic orbit may arise in the right neighborhood of H c and be stable, or may arise in the left neighborhood of H c and be unstable. We use the methodology of Hassard et al. [START_REF] Hassard | Theory and Application of Hopf Bifurcation[END_REF] to compute coefficients that determine the Hopf bifurcation direction and the stability properties of the bifurcating periodic solution. Our strategy can be described as follows: 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 ( 12) 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 (12) becomes:

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) (19)
where 19) can be written as:

X (t) = ϕ 2 (t) -ϕ 2 (t -τ ) + δτ k + δ t t-τ ϕ 2 (s) ds Let c = H c + ε. System (
φ (t) = G (ε, ϕ t ) (20)
A Taylor expansion in the neighborhood of the steady state allows us to split this system into linear and non linear parts. Let z(t) = ϕ t be a solution of equation ( 20) when ε = 0. Following Hassard et al. [START_REF] Hassard | Theory and Application of Hopf Bifurcation[END_REF], the projection of this dynamical system on the center manifold gives the following equation

ż (t) = iω 0 + g (z, z) (21)
which allows to discuss the stability of the limit cycle. Let us indeed 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.
From the terms (g 02 , g 11 , g 20 , g 21 ), applying the main results of Hassard et al. [START_REF] Hassard | Theory and Application of Hopf Bifurcation[END_REF], we can compute three coefficients µ 1 , µ 2 and µ 3 which will allow to characterize the bifurcation.

Theorem 2. There exist three coefficients µ 1 , µ 2 and µ 3 such that the stability of the periodic solution is determined as follows:

1. µ 1 determines the direction of the Hopf bifurcation. If µ 1 > 0 then the Hopf bifurcation is supercritical and the bifurcating periodic solutions exist for c > H c .

2. µ 2 determines the stability of the bifurcating periodic solutions. If µ 2 < 0 then the bifurcating periodic solutions are stable.

3. Period T of the periodic solutions is given by

T = 2π ω 0 1 + µ 3 ( c -H c ) 2 + O ( c -H c ) 4
In the Appendix 5.4, we show how to compute these coefficients explicitly and provide, in the section 3.3, some numerical illustrations.

Local determinacy

Externalities often generate local indeterminacy, i.e. the existence of a continuum of equilibrium paths from a given initial stock of capital. On a more formal basis, this is obtained when the number of stable characteristic roots is larger than the number of pre-determined variables, or equivalently, when the number of unstable characteristic roots is lower than the number of forward variables.

The main difference of our framework with the standard optimal growth model hinges on the initial condition associated with the forward variable, i.e. the time derivative of the capital stock k(t). Contrary to the framework with ordinary differential equations, the initial value in presence of delays is a continuously differentiable function k0 (t) that is defined over the interval [-τ, 0]. Therefore, k0 (t) has to be compatible with the fact that the initial value for the capital stock is given by k(t) = k 0 (t) for t ∈ [-τ, 0]. As a result the only degree of freedom to compute the initial condition is to define a value for k0 (0). Now, using the result obtained in Lemma 2 such that there exists at least one positive real root, we define the equilibrium path on a particular sub-space. Denoting by n ≥ 1 the (finite) number of unstable characteristic roots, it can be shown that the function k0 (t) also has to satisfy n constraints. We then get the following result: Proposition 2. If n = 1, the steady state is locally determinate, i.e. there exists a unique equilibrium path, while there does not exist any equilibrium path when n > 1.

Under Assumptions 1-2, the existence of the critical value H c allows now to discuss more precisely the local determinacy property of the steady state. Theorem 3 is based on the fact that, when τ ∈ (0, τ ) and c ∈ [ c , H c ), the characteristic equation admits exactly one root with positive real part, i.e. n = 1. As a result, there does not exist any fluctuations based of selffulfilling expectations although external effects occur into the production process.

The local determinacy of periodic cycles can be also characterized. When c is in a right neighborhood of H c , we get n = 3 and the steady state is now unstable. But using similar arguments, we conclude that if µ 1 > 0 and µ 2 < 0, the periodic cycle occurs when c > H c and is locally determinate, in the sense that there exist a unique initial value for the forward variable such that the equilibrium converges toward the limit cycle.

A numerical illustration

Considering a yearly calibration, we assume that the fundamental parameters are set to the following values: ν = 1, α = 0.5, δ = 0.1, ρ = 0.0808, and τ = 0.1. 17 It follows that the share of capital is, as usual, s = 0.3 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. 18 The size of externalities in all the following simulations is contained between 0.15 and 0.20, 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], although the latter consider a different production function. 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.

Recent empirical estimates of the elasticity of intertemporal substitution in consumption provide divergent views. Some authors like Campbell [START_REF] Campbell | Asset prices, consumption and the business cycle[END_REF] and Vissing-Jorgensen [START_REF] Vissing-Jorgensen | Limited asset market participation and the elasticity of intertemporal substitution[END_REF] argue that it is definitely below one, while others, like Mulligan [START_REF] Mulligan | Capital interest and aggregate intertemporal substitution[END_REF] and Gruber [START_REF] Gruber | A Tax-Based Estimates of the Elasticity of Intertemporal Substitution[END_REF] repeatedly obtained estimates above the unity. However, all acknowledge that the elasticity should belong to the range 0. In each configuration, the bifurcating periodic orbit solutions exist when c > H c (i.e. µ 1 > 0) and are orbitally stable (i.e. µ 2 < 0). Moreover, for any c in the right neighborhood of H c , the period of the bifurcating solutions is proportional to T . Note that the lower the amount of externalities ε e , the higher the Hopf bifurcation value H c and the lower the period of the cycle T .

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]. It is also worth noting that similar results still apply to bifurcation values such that H c ∈ (0.5, 2.1) when different sizes of externalities are considered, or with endogenous labor as long as the wage elasticity of the labor supply remains low enough, a property that is compatible with the empirical studies of the labor market. 19Note finally that using a low value for the parameter τ , namely τ = 0.1, has allowed us to prove that only a small departure from the standard Ramsey model is enough to generate endogenous fluctuations. All our results remain valid for longer memory processes. In particular, using the same values for the structural parameters, the previous numerical results are basically not affected as long as τ < 0.7.

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 than the Romer-type formulation based on the instantaneous aggregate capital stock. 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 (or at least low enough) wage elasticity of the labor supply. Finally, we have shown that the equilibrium, be it convergent toward the steady state or the limit cycle, is always locally unique.

Appendix

Proof of Lemma 1

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
Linearizing system [START_REF] Benhabib | Vintage capital, investment and growth[END_REF] 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 (22)
The characteristic equation D (λ) = 0 is obtained by replacing x (t) = x (0) e λt and rearranging using f 1 = δ + ρ together with the shares and elasticities (3), (4).

Proof of Lemma 2

From the definition of D (λ), we get lim λ→∞ D (λ) = +∞, and D (0) < 0. The result follows.

Proof of Theorem 1

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

∆ (λ) = P (λ) + Q (λ) e -λτ
(23) 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 = -cβ(1-s)(δ+ρ)[δ(1-s)+ρ] sτ σ = -p 0 q 1 = β(1-s)(δ+ρ) sτ 1 -c[δ(1-s)+ρ] δσ q 2 = β(1-s)(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 (24)
The proof of Theorem 1 is given through the next three lemmas.

Lemma 5.1. Under Assumption 2, 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 ∆ (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 2 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 0 cos (ω 0 τ ) Squaring both sides of the previous equations and summing them 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 (
η = c (1-s)(δ+ρ)[δ(1-s)+ρ] σsδτ (δτ -β) + β(1-s)(δ+ρ) 2 sδτ + ρ 2 2 ψ = c(1-s) 2 (δ+ρ) 2 [δ(1-s)+ρ] 2 (δτ -2β) c- δβσ (1-s)(δ+ρ) ( β-δτ 2 ) ! s 2 σ 2 δτ
The discriminant of ( 25) 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. Assumption 2 implies 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 " (26)
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 ) (27)
Recall from Assumption 2 that c ∈ ( c , +∞). We can easily show 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 2, 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 exist an infinite number of solutions to equation ( 27) for c large enough.

Let us then consider the smallest solution of equation ( 27), denoted H c , which corresponds to the Hopf bifurcation value such that ±iω 0 is an imaginary root of (23).

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.

Let H c be the value for c for which we have an imaginary root.

Lemma 5.3. Re dλ( c)

d c | c= H c = 0
Proof : This proof is computational and consists in three steps. In step 1, we compute dλ ,c) ; in step 3, we use the specific form of the pure characteristic root iω 0 to conclude.

d c = D 2 ∆(λ, c) D 1 ∆(λ, c) ; in step 2, we evaluate Re dλ d c = Re D 2 ∆(iω, c) D 1 ∆(iω
Step 1. Let us differentiate the following equation according to c , noting that c only appears in

u u = -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 the root we consider is a simple root, we can use the implicit function theorem

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
To simplify the computation, we can rewrite the numerator of dλ

d c | c= H c as N λ, H c : N λ, H c = -λ 2 λ + p 2 + q 2 e -λτ - (1-e -λτ ) λ δq 2 H c
Indeed:

N λ, H c = 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
Step 2. We substitute λ = iω 0 respectively in the numerator and in the denominator of dλ

d c | c= H c
. Substitution in the numerator yields to:

N iω, H c = a + ib where a = ω 2 H c p 2 + q 2 cos (ω 0 τ ) - sin (ω 0 τ ) ω δq 2 b = ω 2 H c ω 0 -q 2 sin (ω 0 τ ) + 1 -cos (ω 0 τ ) ω 0 δq 2 Substitution of λ = iω 0 in the denominator D λ, H c of dλ d c | c=
= -(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 ) d = -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 ) We then compute Re D 2 ∆ iω 0 , H c D 1 ∆ (iω 0 , H c ) = ac + bd c 2 + d 2
As iω 0 is a simple root, c 2 + d 2 = 0. Let us then compute ac + bd and express it as a function of ω 0 that we will denote H (ω 0 )

H (ω 0 ) = 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.

Step 3. We use that fact that iω 0 is a root of the characteristic equation and thus solves ω 4 0 + p 2 2 -2p 1 -q 2 2 ω 2 0 + p 2 1 + 2p 0 ρ -q 2 1 = 0, to simplify the expression of H (ω 0 ) . We obtain H (ω 0 ) as a polynomial of order two in

j c , H (ω 0 ) = ϕ 0 + ϕ 1 j c + ϕ 2 j c 2
where ϕ 0 , ϕ 1 , ϕ 2 are independent of j c . This equation has only two real roots, but we have seen in proof of lemma 5.1 that there were a large number of j c for which the hopf the bifurcation occurs. Thus, the transversality crossing condition is satisfied.

Proof of Theorem 2

We follow the procedure described by Hassard et al. [START_REF] Hassard | Theory and Application of Hopf Bifurcation[END_REF] and we provide various Lemmas to derive the dynamics of the system on the center manifold.

Lemma 5.4. System (20) can be written as the following functional differential equation in C:

φ (t) = Λ ε ϕ t + F (ε, ϕ t ) (28)
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] Guo | Capital-labor substitution and equilibrium indeterminacy[END_REF], and

F (ε, ϕ t ) = G (ε, ϕ t ) -Λ ε ϕ t .
Proof : 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) (29)
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 ) , 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 We can also formulate Λ ε ϕ as follows:

0 -τ ϕ 2t (u) du . Lemma 5.5. 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
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 ( 28) is equivalent to

φt = A (ε) ϕ t + G (ε) ϕ t (30)
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 2. As dη (ε, t) is real, we have dη t (ε, t) = dη t (ε, t). 20 The expressions of these coefficients are available upon request.

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 5.6. 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
Proof : As q (θ) is the eigenvector of A associated with eigenvalue iω 0 , q (θ) solves, for θ = 0 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:

η 1 (σ) = e iω 0 σ
with initial conditions:

L t H c η 1 (0) + R t H c η 1 (-τ ) + 0 -τ M t H c η 1 (u) du = iω 0 η 1 (0)
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:

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 3. Computations lead to (q * , q) = 0.

Let ϕ t be a solution of equation ( 30) 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) (32)
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 center manifold is given by

ż (t) = iω 0 + g (z, z)
and in the neighborhood of H c the complete dynamic of the system is given by Proof : We know from the proof of lemma 5.4 that

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 w (z, z, θ) = w 20 (θ) z 2 2 + w 11 (θ) zz + w 02 (θ) z 2 2 + h.o.t. Lemma 5.7.
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 (33)
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:

ϕ 1t (θ) = w 1 20 (θ) z 2 2 + w 1 11 (θ) zz + w 1 02 (θ) z 2 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 (t) e iω 0 θ + z (t) e -iω 0 θ + O (|z, z|)
We use the preceding formula to compute (ϕ jt (.)) 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). Lemma 5.9.

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
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) 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 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 2 Re(λ ( H c ))

2iω 0 -L H c -R H c e -τ 2iω 0 -M H c 0 -τ e 2iω 0 u du E 1 = φ 20 0 which implies 2iω 0 + p 2 + q 2 e -2iτ ω 0 E 1 1 + p 1 + q 1 e -2iτ ω 0 + p 0 0 -τ e 2iω 0 u du E 2 1 = φ 20 2iω 0 E 2 1 -E 1 1 = 0 and thus E 2 1 = φ 20 [∆(2iω 0 )] E 1 1 = 2iω 0
µ 3 = - Im(C 1 )+µ 2 Im " λ ( H c ) " ω 0 , µ 2 = 2Re (C 1 ) (37)
and the result follows from Hassard et al. [START_REF] Hassard | Theory and Application of Hopf Bifurcation[END_REF].

Proof of Proposition 2

Let us linearise the equation ( 12) round the steady state and rewrite the dynamics as k (t) = y (t) ẏ (t) = α Initial conditions for this dynamics are such that k (t) = k 0 (t) and y (t) = y 0 (t), both functions in C ([-τ, 0]). Moreover for t ∈ [-τ, 0], we need to have y 0 (t) = k0 (t). It follows that k 0 (t) has to be given in C 1 ([-τ, 0[) ∩ C ([-τ, 0]), and that it contraints y 0 (t) for t ∈ [-τ, 0[ . The characteristic matrix is

J (λ) = λ -1 -α 0 -τ e λx φ (x) dx -β λ -a
Let n ≥ 1 be the number of eigenvalues with Re (λ) ≥ 0. Assume first that we have a simple eigenvalue λ 0 > 0 and that this is the only eigenvalue with Re (λ) ≥ 0, i.e. n = 1. The criteria for an initial condition (k 0 (t), y 0 (t)) with t ∈ [-τ, 0] to be continued to a bounded solution for t ≥ 0 is then given by Q λ 0 (k 0 (.), y 0 (.)) = 0 where Q λ is the spectral projection (The reasonning is the same if we have a set Σ with a finite number n > 1 of roots with positive real part, exept that we have to consider the spectral projection on the set vect e θ , θ ∈ ).

According to the spectral projection formula given by equation (IV.3.3) in [START_REF] Diekmann | Delay Equations: Functional-, Complex-, and Nonlinear Analysis[END_REF], then (Q λ (k 0 (.), y 0 (.))) (θ) = e λθ H (k 0 (0), y 0 (0)) + 0, α 0 -τ φ (x) e λ 0 x 0 x e -λ 0 u k 0 (u) dudx

where H has to be now determined. We are looking for some H of the form H = pq t , with some vectors p ∈ R 2 and q ∈ R 2 , such that J (λ 0 ) p = 0 J (λ 0 ) t q = 0 where J (λ 0 ) t is the transpose matrix of J (λ 0 ). We thus find that q = (λ 0 -a, 1) and p = (1, λ 0 ). Expliciting H, Q λ 0 (k 0 (.), y 0 (.)) = 0 implies that y 0 (0) + (λ 0 -a)k 0 (0) + α Considering now that k(t) = k 0 (t) is given for t ∈ [-τ, 0] and that y 0 (t) must satisfy y 0 (t) = k0 (t) for t ∈ [-τ, 0], when n = 1 the only degree of freedom to solve (locally) our problem is to choose y (t) ∈ C ([-τ, 0]) with the condition that y 0 (0) is specified as y 0 (0) = -(λ 0 -a)k 0 (0)α 0 -τ φ (x) e λ 0 x 0 x e -λ 0 u k 0 (u) dudx. On the contrary, if n > 1 we derive, applying the same argument as above, n constraints similar to (38). But as y 0 (0) is the only unknown, there does not exist any solution.

Proof of Theorem 3

For τ = 0 the characteristic equation admits one positive real root and one negative real root. Assuming that τ > 0 implies that an infinite number of roots with negative real part will now exist. Our aim here is to prove that for a non empty set of values of τ , the characteristic equation admits only one positive real root. To prove this, we show that there exists τ > 0 such that for 0 < τ < τ , there are no pure imaginary roots. According to the proof of Lemma 5.1, under Assumption 2, if there exists a pure imaginary roots iω 0 , with ω 0 > 0, it has to solve 2η ( c ) + x = 0 ω 2 0 = x Let us consider the critical value c as defined by [START_REF] Campbell | Asset prices, consumption and the business cycle[END_REF]. We get It follows that there exists τ 1 , such that for τ ∈ (0, τ 1 ), we have

η ( c ) = c
(1 -s)(δ + ρ) 2 -δβ[δ(1-s)+ρ] β-δτ 2 > 0 if δ(1 -s) + ρ < (1-s)(δ+ρ) 2 δ ⇔ 1 2+ ρ δ < 1 -s
If s ∈ (0, 1/2), the previous condition is always satisfied. We then conclude that when τ ∈ (0, τ 1 ), η ( c ) > 0. Let us now consider τ ≡ min{τ 1 , 2β/δ}. According to Theorem 1, for c ∈ [ c , H c ), the characteristic polynomial admits no pure imaginary roots. Thus for τ ∈ (0, τ ) and c ∈ [ c , H c ), there exists only one root with positive real part which corresponds to the unique positive real root.

13 2 ) 3 .

 23 Documents de Travail du Centre d'Economie de la Sorbonne -2012.26 Indeed, we can provide simple conditions on the share of capital s and the delay τ to get n = 1 when c ∈ [ c , H c ). Assumption 3. s ∈ (0, 1/Theorem Under Assumptions 1-3, there exists τ ∈ (0, 2β/δ] such that when τ ∈ (0, τ ) and c ∈ [ c , H c ), the steady state is locally determinate.

  25) where x = ω 2 0 and, using the shares and elasticities (3)-(4): 17 Documents de Travail du Centre d'Economie de la Sorbonne -2012.26

19

  Documents de Travail du Centre d'Economie de la Sorbonne -2012.26

  dλ

  du Centre d'Economie de la Sorbonne -2012.26

  c

= 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 22 Documents

 5555522 de Travail du Centre d'Economie de la Sorbonne -2012.26

F 0 (z, z) = φ 20 z 2 2 + φ 11 zz + φ 02 z 2 2 + φ 21 zz 2 2

 222 + h.o.t 0 with φ 20 , φ 11 , φ 02 , φ 21 some complex functions of the coefficients a ij and a ijm derived in Lemma 5.4.21

  du Centre d'Economie de la Sorbonne -2012.26

  Lemma 5.3 and let us denote λH c = (dλ/d c ) | c= H c .From all these computations we derive the formulaC 1 = i 2ω 0 g 20 g 11 -2|g 11 | 2 -1 3 |g 02 | 2 + g 21 2 , µ 1 = -Re(C 1 )

0

  -τ k (t + x) φ (x) dx + bk (t) + ay(t) witha = ρ + β(1-s)(δ+ρ) s 1 -c[δ(1-s)+ρ] σδ (1 -β(1-s)(δ+ρ) sδ )

e λ 0 x 0 xe

 0 -λ 0 u k 0 (u) dudx = 0 (38)

( 1 - 2 sδτ + ρ 2 2 = 1 τ

 1221 s)(δ+ρ)[δ(1-s)+ρ] σsδτ (δτ -β) + β(1-s)(δ+ρ) β sδ (1 -s)(δ + ρ) 2 -δβ[δ(1-s)+ρ] β-δτ 2 + O (1)

  2-2.2. In the table that follows, we present the Hopf bifurcation value H c and the period of the cycle T computed for various combinations of parameters β and ε e .

	β	ε e	H c	T
	0.286 20% 0.6 12.35
	0.245 17.2% 1	4.03
	0.243 17% 1.2	3.3
	0.24 16.8% 2	2.17

  j=1..[START_REF] Asea | Time-to-build and cycles[END_REF] . Then replacing (ϕ jt (.)) j=1..3 in[START_REF] Rustichini | Hopf bifurcation for functional differential equations of mixed type[END_REF] 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 5.8. 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 .

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 over an infinite past, acts as an externality in the production of all firms. They prove that endogenous growth can occur as a result of this.

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].

It is possible to consider a discrete time framework, but this would yield to a large scale dynamical system for which the analytical characterization is difficult. 1 Documents de Travail du Centre d'Economie de la Sorbonne -2012.26

We assume in this paper that the labor supply is inelastic, but all our results are robust to the consideration of a positive but low enough wage elasticity of labor. 2 Documents de Travail du Centre d'Economie de la Sorbonne -2012.26

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

The optimal path necessarily converges monotonically to the steady state in twodimensional 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. 3 Documents de Travail du Centre d'Economie de la Sorbonne -2012.26

Documents de Travail du Centre d'Economie de la Sorbonne -2012.26

These properties are shown to match empirical evidences from the US economy over the period 1948-2004 (see Guo and Lansing[START_REF] Guo | Capital-labor substitution and equilibrium indeterminacy[END_REF]). 6 Documents de Travail du Centre d'Economie de la Sorbonne -2012.26

In a continuous-time framework, the existence of a solution of this kind of fixed-point problem is a difficult issue. Simple cases in endogenous growth settings have nevertheless been studied by Romer[START_REF] Romer | Increasing returns and long-run growth[END_REF] and d'Albis and Le Van[START_REF] Albis | Existence of a competitive equilibrium in the Lucas (1988) model without physical capital[END_REF]. 8 Documents de Travail du Centre d'Economie de la Sorbonne -2012.26

This property can be shown using standard methods.

From a mathematical point of view, the initial conditions have to be chosen to belong to the direct sum of the stable space and the center space.

We have also checked that in a neighborhood of those parameters, the fixed point as defined by[START_REF] Benhabib | Competitive equilibrium cycles[END_REF] exists and satisfies k (t) > 0 and e (t) > 0.

See Chirinko[START_REF] Chirinko | σ: the long and short of it[END_REF] for a review of the many studies that have attempted to estimate the elasticity of capital-labor substitution using various econometric methods.

See Blundell and MaCurdy[START_REF] Blundell | Labour supply: a review of alternative approaches[END_REF] for micro elasticities and Rogerson and Wallenius[START_REF] Rogerson | Micro and macro elasticities in a life cycle model with taxes[END_REF] for macro elasticities.

The expressions of these functions are available upon request.

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.

And we consider a Taylor expansion of H (z, z, θ) = H 20 z 2

2 +H 11 zz+H 02

Moreover, we have on the central manifold

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

Coefficient identification in [START_REF] Vissing-Jorgensen | Limited asset market participation and the elasticity of intertemporal substitution[END_REF] and (36) leads to

Comparing (35) with the last expression, we have

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)