Philippe Boyer
email: philippe.boyer@univ-fcomte.fr

Jones matrices of perfectly conducting metallic polarizers

We deduce from Monomode Modal Method the analytical expressions of transmission and reflexion Jones matrices of an infinitely conducting metallic screen periodically pierced by subwavelength holes. The study is restricted to normal incidence and to the case of neglected evanescent fields (far-field) which covers many common cases. When only one non-degenerate mode propagates in cavities, they take identical forms to those of a polarizer, with Fabry-Perot-like spectral resonant factors depending on bigrating parameters. The isotropic or birefringent properties are then obtained when holes support two orthogonal polarization modes. This basic formalism is finally applied to design compact and efficient metallic half-wave plates.

Introduction

Metallic metamaterials made of subwavelength holes are now designed to exhibit new polarization properties [START_REF] Iwanaga | Photonic metamaterials : a new class of materials for manipulating light waves[END_REF]. Single periodically pierced metallic screen provides a compact linear polarizer [START_REF] Ren | Polarization properties of subwavelength hole arrays consisting of rectangular holes[END_REF], doublelayer fishnet metamaterials reveal optical activities [START_REF] Zhang | Asymmetric fishnet metamaterials with strong optical activity[END_REF], and multi-layer structures allow polarization conversion [START_REF] Xu | Manipulating optical polarization by stereo plasmonic structure[END_REF]. One topical issue consists in developping efficient theoretical tools to describe with accuracy polarization properties of stacked subwavelength metallic bigrating (SMBG).

In this paper, we extract from the well-known Momonode Modal Method (MMM) [START_REF] Lalanne | One-mode model and Airy-like formulae for one-dimensional metallic gratings[END_REF][START_REF] Garcia-Vidal | Light passing through subwavelength apertures[END_REF][START_REF] Boyer | Analytical study of resonance conditions in planar resonators[END_REF][START_REF] Mcphedran | On the theory and solar application of inductive grids[END_REF][START_REF] Petit | Electromagnetic Theory of Gratings[END_REF] the analytical expressions of reflexion and transmission Jones matrices in far-field approximation of one SMBG pierced by subwavelength holes with arbitrary cross section (see fig. 1). The working wavelengths are chosen higher than the first Rayleigh-Wood wavelength in order to consider only one propagative diffracted wave (0 th -order) in incidence and transmission regions. In accordance with the Jones formalism, we assume that an incident planewave falls on the SMBG in normal incidence. We show that these Jones matrices can be basically expressed in this way :

J T,R = α T,R J (pol,ex) θ -ξ T,R I d (1)
where the superscripts (T, R) refer to transmission and reflection respectively, I d is the identity matrix, ξ T = 0 and ξ R = 1. The terms α T,R are Fabry-Perot-like spectral resonant scalar factors and

J (pol,ex) θ =    cos 2 θ cos θ sin θ cos θ sin θ sin 2 θ    (2)
is the Jones matrix of a x-axis linear polarizer rotated to θ (e x being one transverse unit vector of the SMBG and θ depending on SMBG and pattern rotations, see fig. 1). The exact expressions and meanings of α T,R and θ will be given in Section 2 (see eq. ( 14)) which deals with the theoretical background of this work. The eq. [START_REF] Iwanaga | Photonic metamaterials : a new class of materials for manipulating light waves[END_REF] clearly reveals that such a SMBG is equivalent to a spectral resonant linear polarizer in transmission mounting [START_REF] Ulrich | Tunable Submillimeter Interferometers of the Fabry-Perot Type[END_REF][START_REF] Ulrich | Far-infrared properties of metallic mesh and its complementary structure[END_REF]. Note that this efficient formalism may be used to accurately analyse the important role of the reflected waves via the reflection Jones matrix in multilayer polarizing systems as polarization converters [START_REF] Li | Cavity-involved plasmonic metamaterial for optical polarization conversion[END_REF][START_REF] Zhang | Asymmetric fishnet metamaterials with strong optical activity[END_REF][START_REF] Xu | Manipulating optical polarization by stereo plasmonic structure[END_REF]. The case of rectangular apertures and particularly the role of the pattern rotation in resonance properties are then discussed in Section 3. In Section 4, we extend these equations to study polarization properties of SMBG supporting two modes in holes. We show that the transmission Jones matrix reduces to a sum of two monomode metallic polarizer's Jones matrices when two modes propagates in apertures without mode coupling via evanescent diffracted waves. Consequently, the SMBG behaves as spectral resonant isotropic layer for degenerate modes in one cavity, or reveals birefringence otherwise (two monomode cavities in patterns). Finally, we take advantage of this formalism to design thin half-wave plates with optimized transmission when the SMBG is patterned with two orthogonal rectangular holes [START_REF] Baida | Enhanced-transmission metamaterials as anisotropic plates[END_REF][START_REF] Boutria | Tunable artificial chirality with extraordinary transmission metamaterials[END_REF].

2 Analytical expressions of Jones matrices

Presentation of the problem

We consider a metallic screen with thickness h periodically pierced by subwavelength holes described in Cartesian coordinate system Oxyz with (e x , e y ) as unit vectors in transverse plane (determined by bigrating interfaces) and e z as longitudinal unit vector (see fig. 1). We restrict our analyze to biperiodic structures as depicted in fig. 1 with Ox and Oy as periodic axes, then with d x and d y as periods respectively. The metal is assumed perfectly conducting and the refractive index of hole medium is denoted n 2 .

The planar object is surrounded by two semi-infinite homogenenous regions (j) with refractive indices n j , j ∈ {1, 3}. An incident plane wave falls on the SMBG from region (1) or (3) in normal incidence and with ϕ inc as polarization incident angle. We introduce the rotation angle ϕ G with the x-axis of the SMBG in Oxy plane. The far-field approximation consists to neglect evanescent waves in electromagnetic field description sufficiently distant from the SMBG (half-wavelength about). This hypothesis allows the equivalence between ϕ inc and ϕ G = -ϕ inc since the light polarization far from the SMBG is given by polarizations of the specular diffracted waves. That is why only the angle ϕ G is used in the following theory (and not ϕ inc ) in order to respect the independence of Jones matrices to the incident wave. It is worth noticing that the present theory may easily be used for monoperiodic objects as subwavelength metallic gratings [START_REF] Boyer | Analytical study of resonance conditions in planar resonators[END_REF] and objects under in oblique incidence.

The present theory is derived from the MMM's basic equations [START_REF] Boyer | Analytical study of resonance conditions in planar resonators[END_REF] extended to biperiodic structures.

The electromagnetic fields are described as Fourier-Rayleigh (FR) expansions in homogenenous regions (j), i.e. as sums of Floquet modes. To simplify notations of FR-orders, p-orders stands for (n, m)-orders

with n ∈ [-N, N ], m ∈ [-N, N ]
and N is the truncation order of FR expansions. p = 0 refers to (0, 0)-order. Then, we assume that only the non-degenerate fundamental mode (q = 1) can propagate in cavities (monomode approximation). This hypothesis restricts the spectral validity domain to [λ c,2 , λ c,1 ] with λ c,q the cut-off wavelength of the q th mode. Futhermore, this conditions deals with apertures with C i (i ∈ N), C 1v or C 2v cross section symmetry [START_REF] Mcisaac | Symmetry-induced modal characteristics of uniform waveguides I : summary of results[END_REF]. Figure 2 depicts common examples gathered in E1-set as rectangular [START_REF] Mary | Theory of light transmission through an array of rectangular holes[END_REF][START_REF] Ren | Polarization properties of subwavelength hole arrays consisting of rectangular holes[END_REF], ellipsoidal [START_REF] Zhang | Asymmetric fishnet metamaterials with strong optical activity[END_REF], split-ring [START_REF] Wei | Broadband polarization transformation via enhanced asymmetric transmission through arrays of twisted complementary split-ring resonators[END_REF] or chiral [START_REF] Kanda | Terahertz wave polarization rotation with double layered metal grating of complimentary chiral patterns[END_REF] hole profiles.

The Jones matrices are now denoted J T j and J R j when the incident wave is placed in region (j). In the present case of biperiodic objects, the well-known polarizations (te, tm) or (p, s) are used to describe electric field of diffracted FR-waves. Thus, the analytical terms of J T,R j directly identify to the zeroorder (far-field approximation) transmitted and reflected amplitudes given by eqs. ( 41) and (42) in ref.

[7] extended to biperiodic metallic gratings :

J T,R j = f T,R j    g 0,tm g * 0,tm g 0,tm g * 0,te g 0,te g * 0,tm g 0,te g * 0,te    -ξ T,R    1 0 0 1    , (3)
where g p,σ are the overlap integrals between FR-waves and the cavity modes (expressed below), and σ = {te, tm} denotes the polarization of the transverse basis-vectors of Floquet mode wavevectors. The factors f T,R j are spectral resonant Airy-like functions :

f T j = 4uη (j) 0 η C (1) + η C (3) + η -u 2 C (1) -η C (3) -η , (4)
and

f R j = 2η (j) 0 C (j ′ ) + η + u 2 η -C (j ′ ) C (1) + η C (3) + η -u 2 C (1) -η C (3) -η , (5)
where

C (j) = p h (j) p • g p , (6)
and

j ′ = 1 if j = 3 and j ′ = 3 if j = 1. η (j)
p,σ and η are the relative admittances of the p th FR-order in region (j) and of the cavity mode respectively. We use the following notations : η

(j) 0 = η (j) 0,te = η (j)
0,tm = n j and u = exp(i γh) where γ are the propagation constant of the cavity mode. The terms g p,σ and

h (j) p,σ = η (j)
p,σ g * p,σ for σ ∈ {te, tm} are components of vectors g p and h

(j) p , respectively. It is worth noticing that coefficients C (j) are computed for n ∈ [-N, N ] and m ∈ [-N, N ], and so it takes into account coupling between cavity mode and evanescent diffracted waves.

Overlap integrals between FR-waves and the cavity modes

A detailed analysis of g p,σ expressions provides the basic and analytic formulation of J T,R j given in eq. ( 1) from the eq. ( 3). In this aim, we have first to pose transverse (in (e x , e y )-plane) field expressions of Floquet modes and the ones of cavity modes.

The transverse field profiles of the well-known Floquet modes in homogenenous regions are given by

     E p,σ (x, y) = e ikp .ρ √ S e p,σ , H p,σ (x, y) = η 0 η (j) p,σ e z ∧ E p,σ (x, y), (7)
where ρ = xe x + ye y , η 0 = ǫ 0 /µ 0 is vaccum admittance, S the transverse surface area of the periodic cell, k p = n2π/d x e x + m2π/d y e y the transverse component of p th FR-wavevector. The polarization vectors e p,σ is

e p,tm =      kp kp if |k p | = 0, cos ϕ G e x + sin ϕ G e y if |k p | = 0, (8)
and

e p,te =      e z ∧ kp kp if |k p | = 0, -sin ϕ G e x + cos ϕ G e y if |k p | = 0. (9) Note that |k p | = 0 is equivalent to p = 0.
Concerning field expressions inside apertures, we note E(x, y) the transverse electric field profiles of the fundamental mode. The corresponding transverse magnetic field H(x, y) is expressed as in eq. ( 7) substituting η (j) p,σ by η and E p,σ (x, y) by E(x, y).

Thus, the overlap integrals between FR-orders and the cavity mode are defined by

g p,σ = S E * p,σ (x, y) • E(x, y)ds = e p,σ • g p . ( 10
)
The integration is only computed on the surface S of cavity cross section since the fields in bigrating are different to zero only on S. The g p -vector is

g p = S E(x, y) e -ikp.ρ √ S ds = g p v p , (11)
v p is the unit polarization vector of the overlap integrals g p . These overlap vectors cause the linear polarization filtering of the metallic screen which is described in detail below. We introduce the polarization angle ψ p such as v p = cos(ψ p )e x + sin(ψ p )e y . We easily obtain that

e p,te • v p =      mdx cos(ψp)-ndy sin(ψp) √ n 2 d 2 y +m 2 d 2 x if |k p | = 0, -sin(ϕ G -ψ p ) if |k p | = 0, (12)
and

e p,tm • v p =      ndy cos(ψp)+mdx sin(ψp) √ n 2 d 2 y +m 2 d 2 x if |k p | = 0, cos(ϕ G -ψ p ) if |k p | = 0, (13)
which is required in eq. ( 10) to compute g p,σ .

Final expressions of Jones matrices

Making explicit the 0 th orders overlap integrals finally leads to eq. ( 1) where

α T,R (λ, ψ) = f T,R j (λ, ψ) | g 0 | 2 . ( 14
)
The 0 th orders overlap integrals are obtained from eq. [START_REF] Ulrich | Tunable Submillimeter Interferometers of the Fabry-Perot Type[END_REF] in which e p,σ are expressed in eqs. ( 8) and and α T,R on λ and ψ. Note that coefficients α T,R (λ, ψ) don't depend on ϕ G since the coupling coefficients C (j) don't (but still vary with ψ, see eqs. ( 12) and ( 13)). In fact, coefficients C (j) also take the form

C (j) = n j | g 0 | 2 + p =0 h (j) p • g p , (15)
knowing that | g 0 | 2 and the summation for p = 0 don't depend on ϕ G .

It is interesting to remark that the determinant of J T j given by eq. ( 3) is equal to zero meaning that the metallic array behaves in transmission as a linear polarizer. This result is confirmed by the final expressions of J T j given by eq. ( 1). We can also remark that the resonant factor α T (λ, ψ) depends on bigrating parameters (via k p -vectors) but polarization properties given by a polarizer's Jones matrix J (pol,ex) ψ0-ϕG don't. Indeed, the expressions of J (pol,ex) ψ0-ϕG terms are obtained from eqs. ( 12) and ( 13) for p = 0 (|k 0 | = 0). They are thus only depending on ψ 0 and ϕ G .

Case of rectangular apertures : Role of pattern rotation

As shown in fig. 2 (E1-set), many hole shapes can be considered. The shape of the apertures first changes the cut-off of each mode and thus causes wavelength shifts of resonance peaks related to cut-off.

Secondly, the effective indices of modes (or their admittances particularly required in calculation of overlap integrals) are also affected which equally provokes shifts of Fabry-Perot-like resonance wavelengths.

Besides, the radiative losses are changed due to different reflection and transmission conditions at each interface of the bigrating. Consequently, the quality factors are different for each hole shape. We analyse here the influence of the pattern rotation on transmission behaviour of one basic sample. We consider that the SMBG pattern is made of one rectangular hole (first profile of E1-set, see fig. 2). In this case, analytical expressions of overlap integrals can be easily obtained. For the most other cases (except for the ones for which a pertubtive process may be used), overlap integral calculations directly depending on the transverse field profile require a numerical treatment.

We let d = d x = d y . The width and length are a x /d = 0.2 and a y /d = 0.7 respectively. ψ identifies here to the angle between the length side and the e y -axis. Other parameters are h/d = 0.8 and N = 5.

The geometrical parameters are chosen such that resonant peaks appears for λ/d ∈ [1.0, 2.0]. In this λ-range, the cavities can be effectively assumed monomode and the polarization angles of g p is given by the linearly polarized electric field direction of the T E 01 -mode (∀p, ψ p = ψ). The fig. 3 depict the spectra of α T,R (λ, ψ) for SMBG patterns rotated by ψ = 0 o , 15 o , 30 o and 45 o . The spectra of the terms of the reflexion Jones matrices are directly deduced from those of α R (λ, ψ) in eq. ( 1). Their analysis reveal interesting properties due to the identity matrix. The fig. 3 shows two peaks for α R (λ, ψ) at resonances. This induces common deep peaks (reflexion-like) for diagonal terms of J R j , whereas the extra-diagonal terms behaves as transmission ones (peaks at resonances). This property may cause some special polarization and transmission effects when several SMBG are piled up.

The peak close to λ/d = 1.41 is related to the resonance at T E 01 -mode cut-off whereas the peak close to λ/d = 1.18 refers to the first Fabry-Perot-like resonance [START_REF] Boutria | Tunable artificial chirality with extraordinary transmission metamaterials[END_REF]. The resonant functions α T,R (λ, ψ) little depend on SMBG pattern rotation angle ψ. The variation of ψ does not affect maxima values but causes wavelength shifts of peaks. The variations of λ max /d at each peak maxima of α T,R (λ, ψ) according to ψ are plotted in fig. 4. We remark that small blueshifts (< 2.10 -2 ) occurs when the angle ψ increases from 0 o to 45 o and redshifts from 45 o to 90 o . The dephasing between diffracted waves and the incident one given by arg α T,R (λ max , ψ) (see fig. 3) is close to -π for first resonant peak (λ/d ≈ 1.18) and -2π for the second one (λ/d ≈ 1.41), whereas it remains equal to zero for reflected waves. In view of α T,R (λ, ψ) spectra, we equally show that transmitted and reflected waves don't resonate exactly at the same wavelengths. This may causes special resonance properties of stacks of metallic polarizers.

Extension to bimodal systems

For some cavity cross sections and/or frequency-ranges, two modes have to be considered in cavities of the SMBG pattern. The first case deals with one cavity allowing two modes (any cross section a priori). We highlight the particular case of one degenerate mode (two modes with the same effective index). Knowing that the mode field symmetries are independent of the bigrating lattice's ones only for the studied case of perfectly conducting metals, the degeneracy can be obtained for holes with C iv (i > 2) cross section symmetry [START_REF] Mcisaac | Symmetry-induced modal characteristics of uniform waveguides I : summary of results[END_REF] as circular [START_REF] Ortuno | Role of surface plasmon polaritons on optical transmission through double layer metallic hole arrays[END_REF][START_REF] Nguyen | Engineering the properties of terahertz filters using multilayer aperture arrays[END_REF], square [START_REF] Mary | Holey metal films : From extraordinary transmission to negative-index behavior[END_REF][START_REF] Oates | Optical activity in sub-wavelength metallic grids and fishnet metamaterials in the conical mount[END_REF] and annular [START_REF] Baida | Light transmission by subwavelength annular aperture arrays in metallic films[END_REF][START_REF] Chen | Quarter-wave plate with subwavelength rectangular annular arrays[END_REF] hole cross sections (E2-set in fig. 2), or can accidentally occur. The second case deals with two monomode cavities as for any combinaison of geometries in E1-set as example. The working spectral range determines the monomode or bimode regime of apertures. However, we focus on the particular case of non-coupled modes via evanescent waves which implies basic expressions of Jones matrices.

Similarly to eq. ( 3) for monomode SMBG exposed in Section 2, the eqs.( 30), (31), ( 33) and (34) in ref.

[7] applied to bimode holes (q ∈ {1, 2}) lead to semi-analytical Jones matrices after tedious calculations :

                 J T 1 = 2 [(u g) t M 11 + g t M 21 ] g * η (1) , J R 1 = 2 [ g t M 11 + (u g) t M 21 ] g * η (1) -I d , J T 3 = 2 [(u g) t M 12 + g t M 22 ] g * η (3) , J R 3 = 2 [ g t M 21 + (u g) t M 22 ] g * η (3) -I d , (16)
where the superscript t stands for transpose, u is a 2 × 2-matrix such as (u) q,q ′ = u q δ q,q ′ with u q = exp(i γ q h) (q and q ′ ∈ {1, 2}), then (η

(j) ) σ,σ ′ = η (j)
σ δ σ,σ ′ and ( g) q,σ = g q,σ with g q,σ the overlap integral between FR (0, σ)-order and the q th mode. To simplify notations, we consider here that σ ∈ {tm, te} (and σ ′ ) stands for the subscript (0, σ), i.e. for p = 0 (and (0, σ ′ ) respectively). The 2×2-matrices M χ,χ ′ , with χ and χ ′ ∈ {1, 2}, are 2 × 2-blocks of the 4 × 4-matrix M = M -1 with

M =          C (1) 1,1 + η 1 C (1) 1,2 C (1) 1,1 -η 1 u 1 C (1) 1,2 u 2 C (1) 2,1 C (1) 2,2 + η 2 C (1) 2,1 u 1 C (1) 2,2 -η 2 u 2 C (3) 1,1 -η 1 u 1 C (3) 1,2 u 2 C (3) 1,1 + η 1 C (3) 1,2 C (3) 2,1 u 1 C (3) 2,2 -η 2 u 2 C (3) 2,1 C (3) 2,2 + η 2          . ( 17
)
This matrix linking field amplitudes of both cavity modes depends on the cross-coupling coefficients C (j) q,q ′ between q th -mode and q ′th -mode via FR-orders in (j) region. These coefficients are similarly defined as in eq. ( 6) :

C (j) q,q ′ = p h (j) p,q • g p,q ′ , (18)
with p = (n, m), n ∈ [-N, N ] and m ∈ [-N, N ], then q and q ′ ∈ {1, 2}. The numerical inversion of M makes the theory semi-analytical for most cases.

Nevertheless, we are specially interested in the basic form of the Jones matrices in the case of noncoupled modes. Indeed, both modes are not coupled via evanescent waves when their cross-coupling coefficients C

(j)

1,2 and C (j)

2,1 nullify. Consequently, the matrices M and so M become block-diagonalizable, and M can be analytically inverted. In this case ( C

(j) 1,2 = C (j)
2,1 = 0), the eqs. ( 16) and ( 17) lead to the following analytical expressions of Jones matrices :

J T,R j = 2 q=1 α T,R (λ, ψ (q) )J (pol,ex) ψ0,q-ϕG -ξ T,R I d , (19)
with

ψ (q) = (• • • , ψ p,q , • • • ).
To resume, the transmission Jones matrices of bimode system with noncoupled modes is simply written as the sum of those related to each mode (see eq. ( 1)).

We have now to clarify non-coupling conditions of modes. A tedious analysis of C (j) q,q ′ terms from eqs. ( 12) and [START_REF] Baida | Enhanced-transmission metamaterials as anisotropic plates[END_REF] shows that two modes are not coupled via evanescent diffracted waves when v p,1 • v p,2 = 0 ∀p, and when v p,1 and v p,2 vectors coincide with e x and e y respectively. Moreover, the SMBG pattern's cross section must respect C 1v symmetry (E3-set in fig. 2). These assertions reduce to ψ 0,1 = 0 and ψ 0,2 = π/2 when the mode fields are linearly polarized for which ψ (p) ≡ ψ ∀p (rectangular or square apertures as example). Other geometries inducing C (j) q,q ′ = 0 may exist but remain difficult to obtain. We thus deduce from the eq. ( 19) that J T,R j is a diagonal matrix for ϕ G = 0 :

J T,R j =    α R,T (λ, ψ (1) ) -ξ T,R 0 0 α R,T (λ, ψ (2) ) -ξ T,R    . (20)
Thus, such metallic plates are divided into two sets :

i. α R,T (λ, ψ (1) ) = α R,T (λ, ψ (2)
) for the case of one cavity in SMBG pattern with one degenerate mode (E2-set). Consequently, the SMBG behaves as an Fabry-Perot-like isotropic resonator in transmission.

ii. α R,T (λ, ψ (1) ) = α R,T (λ, ψ (2) ) for other cases, i.e. for patterns made of one cavity allowing two non-degenerate modes (E1-set) or two monomode cavities (E3-set). Consequently, the SMBG behaves as an Fabry-Perot-like birenfringent resonator in transmission.

Application to metallic half-wave plate

According to the results obtained in the last section, we know that one SMBG with periodic cells made of two orthogonal monomode cavities behaves as an metallic birenfringent plates in transmission which allows design of compact waveplates [START_REF] Baida | Enhanced-transmission metamaterials as anisotropic plates[END_REF][START_REF] Boutria | Tunable artificial chirality with extraordinary transmission metamaterials[END_REF]. In order to valid our formalism, we consider quasiidentical metallic plates studied in [START_REF] Baida | Enhanced-transmission metamaterials as anisotropic plates[END_REF] but with C 1v -pattern made of two orthogonal rectangular holes as depicted in fig. 2 (first pattern of E3-set). Actually, the geometry proposed in [START_REF] Baida | Enhanced-transmission metamaterials as anisotropic plates[END_REF][START_REF] Boutria | Tunable artificial chirality with extraordinary transmission metamaterials[END_REF] including both rectangular profiles does not respect C 1v -symmetry (but respects L-shape) which induces non-nil coupling between modes. We thus introduce the following ratios

τ (j) q,q ′ |q ′′ = C (j) q,q ′ / C (j) q ′′ ,q ′′ , (21)
with q ′ = q to evaluate the significance of these couplings. As shown in Fig. 5 (calculus made with the complete bimode theory, see eq. ( 17)), they are τ

(1)

1,2|1 = τ (1)
2,1|1 = 13% and τ

(1)

1,2|2 = τ (1)
2,1|2 = 25.8% for the case studied in [START_REF] Baida | Enhanced-transmission metamaterials as anisotropic plates[END_REF] at the transmission peak maxima with λ max /d close to 1.186. The values of geometrical parameters are h/d = 0.83, a x /d = 0.73, c x /d = 0.067, a y /d = 0.58, b y /d = 0.2 and c y /d = 0.45. All media are filled with air. As in [START_REF] Baida | Enhanced-transmission metamaterials as anisotropic plates[END_REF], the transmission (T ) is computed for an incident light with ϕ G = 45 o (and n 1 = n 3 ) :

T = 1 2 |t xx + t xy | 2 + |t yx + t yy | 2 , (22)
using the same notations introduced in [START_REF] Baida | Enhanced-transmission metamaterials as anisotropic plates[END_REF]. In our theory, t ρ,ρ ′ , ρ and ρ ′ ∈ {x, y}, are J T j -terms (given in eq. ( 16)). The structure behaves as a quasi half-wave plate such as the transmission maxima (T max ) reaches 92% and the phase difference (PD) between t xx and t yy is approximatively equal to 3.15 radians at λ max . The transmission Jones matrix is indeed not exactly the same as the one of a perfect half-wave plate knowing that t xy = 0 and t yx = 0 since C (j) 1,2 = 0 and C (j) 2,1 = 0 (see Fig. 3 in [START_REF] Boutria | Tunable artificial chirality with extraordinary transmission metamaterials[END_REF]). But for our proposed structure, t xy and t yx exactly nullifies when pattern respect C 1v -symmetry (c y = (a x -b y )/2).

For this case, the Fig. 6 depicts the argument of α T (λ, 0 o )/ α T (λ, 90 o ) versus λ/d and a y /d where α T (λ, 0 o ) and α T (λ, 90 o ) identify to t xx and t yy respectively (identical parameters as the previous ones).

We remark that the figure is similar to Fig. 3 We propose now to take the advantage of our very efficient analytical model to improve performances of such metallic waveplates. Our goal is to achieve a more compact system (lower thickness) with better transmission. In this aim, the designed object must satisfy to the three following conditions simultaneously :

           L 1 = arg(t xx /t yy )/π -1 = 0 : PD condition, L 2 = |t xx | -|t yy | = 0 : identical transmission moduli condition, L 3 = |t xx | -1 = 0 : total transmission condition, (23)
which are gathered in the global following condition :

L = 3 l=1 |L l | . ( 24
)
The value of h/d is changed from 0.5 to 0.85. For each value of h, we determine the point A (and so the values of λ max /d and a y /d) as in fig. 6. The variations of L and L 1 to L 3 according to h/d are shown in Fig. 7. We see that conditions are satisfied for many values of h/d (hollow peaks). Then, the discontinuities close to h/d = 0.55 correspond to a y /d = 0.5, i.e. when the cut-off wavelength of one cavity mode (position of the first |t yy | peak) is equal to the Rayleigh wavelengths. Thus, half-wave plate cannot be designed for h/d < 0.55 about. We also remark that L 1 = 0 and L 2 = 0 cannot occur for the same value of h/d (see subfigure), and L 3 = 0 never occurs. We so deduce that perfect half-wave plates cannot be obtained in general with metallic plates made of subwavelength rectangular holes. The variations of a y /d, T max , λ max /d and P D at each minimum of L are plotted in Fig. 8. In order to achieve our goal, we have chosen the most compact system : h/d = 0.5484 for one minimum of L such as L 1 ≈ 0 and L 3 reaches one of minima. Finally, the transmission of the retained metallic plate is T max = 96.16% at λ max /d = 1.073 and with P D = 3.1324 rad and a y /d = 0.5008. To complete the analysis, the corresponding transmission and P D spectra are plotted in Fig. 9.

Conclusion

We provide an efficient theoretical tool to analyse polarization features of subwavelength metallic bigratings in monomode and different bimode regimes. The considered geometries cover a wide part of applications studied in litterature. This model has especially been used to optimize thin metallic halfwave plates with high transmission (patterns with two orthogonal rectangular apertures). The analytical Jones matrices for one metallic plate and the scattering-matrix propagation algorithm can be combined in an analytical reccurence way. This basic process allows the computation of the global Jones matrices of stacked structures and forms an extended Jones-like formalism for metallic plates. Futher works are in progress to show with the help of this new formalism that an efficient polarization conversion with total transmission occurs for stacked twisted metallic polarizers.

q,q ′ |q ′′ and C q,q ′ versus λ/d. The transmission spectrum is plotted in grey color (scale not mentioned).

( 9 )

 9 for |k p | = 0 (p = 0). The square matrix in eq. (1) is the Jones matrix after | g 0 | 2 being factored out and with θ = ψ 0 -ϕ G . To simplify writing, we introduce the row matrix ψ = (• • • , ψ p , • • • ) containing all ψ p -values. The notations highlight here the dependencies of f T,R j

  in [13]. Point A defines values of λ and a y to achieve one half-wave plate : arg (t xx /t yy ) ≈ π and |t xx | ≈ |t yy |. The other geometrical parameters have been chosen to maximize transmission maxima (red lines related to |t xx | maxima).

Fig. 1 -Fig. 2 -|ig. 3 -

 123 Fig. 1 -Metallic screen periodically pierced by subwavelength holes.

2 αT 2 Fig. 4 -

 224 Fig. 4 -(Color online) Variation of λ max at α T,R (λ, ψ) maxima according to ψ. See fig. 3 for parameter values.

Fig. 5 -

 5 Fig. 5 -Coupling analysis between modes of each cavities for L-shape pattern made of two orthogonal rectangular apertures : τ

Fig. 6 -Fig. 7 -

 67 Fig. 6 -Phase difference PD between t xx and t yy versus λ/d and a y /d for SMBG with C 1v -pattern made of two orthogonal rectangular apertures. The white contour plots give the couples (λ,a y ) that correspond to |t xx | = |t yy |. The black line corresponds to P D = π, the blue line to P D = π/2 and the red lines to |t xx | maxima. The point A answers the case of a half-wave plate. The parameter are h/d = 0.83, a x /d = 0.73, c x /d = 0.067, a y /d = 0.58, b y /d = 0.2 and c y /d = 0.45.

Fig. 8 -Fig. 9 -

 89 Fig. 8 -Variations of a y /d, T max , λ max /d and P D at each minimum of L depicted in Fig. 7.

Acknowledgements : I would like to thank Lifeng Li from Dept. of Precision Instruments (Tsinghua University, China) and Daniel Van Labeke from the FEMTO-ST institute (Besançon, France) for their helpful advices.


Jean-Philippe Garnier
email: jean-philippe.garnier@univ-lille3.fr

Sunspots, cycles and adjustment costs in the two-sectors model

Keywords: Two-sector growth model, externality, adjustment costs, endogenous fluctuations.} JEL classification: C62, E32, 041

Do adjustment costs able to modify the dynamic of the two sectors model?

We examine the impact of adjustment costs in capital on the properties of long-run equilibrium. We propose to analyse how the positive and negative degrees of adjustment costs could interplay with the local indeterminacy mechanism coming from the presence of sector specific externalities. When the adjustments costs function is convex there exists a Höpf bifurcation and the trajectory describes a cycle around the steady state. We give an heuristic economic explanation of the role of the adjustment costs leading to economic cycles.}

Introduction

Many interesting findings have emerged from the study of indeterminacy and endogenous fluctuations, i.e.the existence of a continuum of equilibria that arises in dynamic economies with some market imperfections. In a major contribution, Benhabib and Nishimura [START_REF] Benhabib | Indeterminacy and sunspots with constant returns[END_REF] examined the two-sector model with different Cobb-Douglas technologies at the private level with sector-specific externalities and constant social returns to scale. They prove, with a separable utility function which is linear in consumption and strictly concave with respect to labor, that local indeterminacy arises if and only if technological externalities allow factor intensities between the private and social levels to be reversed (i.e. the consumption good is capital intensive at the private level and labor intensive at the social level). Consequently, there is a technological mechanism arising from externalities which breaks the duality between the Rybczynsky and Stolper-Samuelson effects and leads to 1 indeterminacy. Garnier, Venditti and Nishimura [START_REF] Garnier | Intertemporal substitution in consumption, labor supply elasticity and sunspot fluctuations in continuous-time models[END_REF] subsequently introduce non-linear utility in consumption. Under factor intensity reversal between private and social levels, they prove that sunspot fluctuations exist if and only if the elasticity of intertemporal substitution in consumption is high enough and the elasticity of labor supply is low enough (even equal to zero).

Given these findings, it appears relevant to ask whether introducing an adjustment cost in capital might affect dynamic of the model. The idea that the installation of a new capital could generate additional costs (positive or negative) is widely viewed as an important feature of the investment decision analysis. Neoclassical studies of investment behavior often ignore variations in capacity utilization.

This paper shows how investment adjustment costs interact with positive sector specific externalities in the two-sector model à la Benhabib and Nishimura [START_REF] Benhabib | Indeterminacy and sunspots with constant returns[END_REF]. In the one sector model, the introduction of adjustment costs make it difficult for indeterminacy to occur: the required degree of increasing returns is higher in the presence of such costs as in the paper of Jinill Kim [START_REF] Jinill | Indeterminacy and adjustment costs: An analytic result[END_REF]. Nevertheless, in the two sector model with constant returns to scale at the social level and decreasing at the private level, the indeterminacy mechanism is different and the presence of adjustment costs interacts positively with the indeterminacy mechanism. In this way, we show that the presence of such costs make it possible the Höpf bifurcation in the standard two-sector model (with only positive sector specific externalities, exogenous labor and linear utility function in consumption) whereas it's not possible otherwise. The unique contribution of this paper is to present this relationship in a simple analytic way and to prove the existence of the Höpf bifurcation.

The rest of the paper is organized as follows. Section 2 describes the economy. Section 3 characterizes the competitive equilibrium. Section 4 analyzes the mechanism that leads to equilibrium indeterminacy. Section 5 gives an example of utility function that allows the existence of indeterminacy and illustrates our main result through a standard parametrization of the model. Section 6 concludes. All the proofs are collected in the appendix.

The economy

We consider an infinite horizon, continuous time, two-sector model with Cobb-Douglas technologies, inelastic labor supply and non linear utility function in consumption. The economy consists of competitive firms and a representative household.

Firms

We assume that consumption good y 0 and capital good y 1 are produced by capital x 1j and labor x 0j , j =0 , 1, through a Cobb-Douglas technology which contains sector specific externalities e j . The representative firm in each industry faces the following technology called private production function:

y j = F j (x 0j ,x 1j )=x β 0j 0j x β 1j 1j e j (X 0j ,X 1j ) for j =0, 1 (1)
with β ij 2 [0, 1] and X ij the average use of input i in the sector j.

The positive sector externalities are such that:

e j (X 0j ,X 1j )=X b 0j 0j X b 1j 1j (2)
We assume that this economy wide average are taken as given by each individual firms. At the equilibrium, since all firms of sector j are identical, we have X ij = x ij and we may define the social production function as follows:

y j = x β0j 0j x β1j 1j for j =0, 1 (3)
with βij = β ij + b ij we assume that the returns to scale are constant at the social level and decreasing at the private level: in each sector j =0 , 1, β0j + β1j =1.

The labor is exogenous, therefore the total labor, normalized to one, is given by:

x 00 + x 01 =1 (4)
and the total stock of capital is given by:

x 10 + x 11 = x 1 (5)
Choosing the consumption good as the numeraire, i.e. p 0 =1 ,afi r mi n each industry maximizes its profit given the output price of the investment p 1 , the rental rate of capital w 1 and the wage rate w 0 . The first order conditions subject to the private technologies (1) give

x ij /y j = p j β ij /w i ⌘ a ij (w i ,p j ),i , j =0, 1 (6)
We call a ij the input coefficients from the private viewpoint. If the agents take account of externalities as endogenous variables in profit maximization, the first order conditions subject to the social technologies (3) give on the contrary

x ij /y j = p j βij /w i ⌘ âij (w i ,p j ),i , j =0, 1 (7)
We call âij the input coefficients from the social viewpoint. As we will show below, the factor-price frontier, which gives a relationship between input prices and output prices, is expressed with the input coefficients from the social viewpoint.

Based on these input coefficients it may be shown that the factor-price frontier is determined by the input coefficients from the social viewpoint while the factor market clearing equation depends on the input coefficients from the private perspective: ⇤ Lemma 1 :D e n o t ep =( 1 ,p 1 ) 0 , w =( w 0 ,w 1 ) 0 and Â(w, p)=[â ij (w i ,p j )]. Then p = Â0 (w, p)w.

Lemma 2 :D e n o t ex

=( 1 ,x 1 ) 0 , y =( y 0 ,y 1 ) 0 and A(w, p)=[ a ij (w i ,p j )]. Then A(w, p)y = x.
Note that at the equilibrium, the rental rate is function of the output price only, i.e. w 1 = w 1 (p 1 ), while outputs are functions of the capital stock, total labor and the output price, y j =ỹ j (x 1 ,p 1 ), j =0, 1.

Household

We assume that the population is constant and normalized to one. At the date t, the representative agent derives his utility U (.) from consumption ⇤ See Garnier, Nishimura and Venditti [START_REF] Garnier | Capital-labor substitution and indeterminacy in continuous-time two-sector models[END_REF] for the proofs of these results. c(t). Considering the external effects as given, profit maximization in both sector gives demand functions as function of capital stock x 1 (t), production level of the investment good y 1 (t) and external effects (e 0 ,e 1 ), namely xij = x ij (x 1 ,y 1 ,e 0 ,e 1 ) for i, j =0, 1. The production frontier is then defined as:

y 0 = T (x 1 ,y 1 ,e 0 ,e 1 )=Max xij xβ 00 00 xβ 10 10 e 0 s.t. (1) (3) (4)
From the envelop we get: ∂T ∂x 1 = w 1 and ∂T ∂y 1 = -p 1 . In this model, the representative agent consumes the whole consumption good therefore we have c = y 0 so he solves the following intertemporal maximization problem † :

max y 1 (t),x 1 (t) ´1 0 e -ρt c(t)dt s.c. s.c. ẋ(t)=x 1 (t)Φ ⇣ y 1 (t) x 1 (t) ⌘ x 1 (0) = x 1 and {e 0 (t),e 1 (t)} t≥0 given (8)
Where ρ > 0 is the subjective discount rate and Φ the function of investment adjustment costs. We incorporate the investment adjustment costs in the capital accumulation equation in a way similar to Lucas and Prescott [START_REF] Lucas | Investment under uncertainty. Econometrica,1 9 7 1 . Documents de travail récents François Langot, Lise Patureau and Thepthida Sopraseuth[END_REF]. In this specification, the adjustment costs occur when the level of capital stock changes. We note that the classical expression of capital accumulation corresponds to the particular adjustment costs function Φ

⇣ y 1 x 1 ⌘ = y 1
x 1 -g with g the constant depreciation rate of capital. The adjustment costs could be thought as a mesure of the efficiency of the investment i.e. efficiency index of the investment. For example, in the extrem cases, if the investment is so efficient, an investment per capital unit y 1

x 1 < 1 gives an efficiency index Φ

⇣ y 1 x 1 ⌘
> 1 and the capital stock increases (the adjustments costs are negative) and if it is so inefficient an investment per capital unit y 1

x 1 > 1 gives an efficiency index Φ

⇣ y 1 x 1 ⌘
< 1 and the capital stock decreases (the adjustments costs are positive). † We suppose that the utility function is linear in consumption: Garnier, Venditti and Nishimura [START_REF] Garnier | Intertemporal substitution in consumption, labor supply elasticity and sunspot fluctuations in continuous-time models[END_REF] have shown that the parameter preference have to be small (close to 0) to allow indeterminacy in the two sector model with sector specific externalities.

For the later analysis of the local dynamics, we make two assumptions on the specific form of the adjustment costs function.

Assumption 1 : The adjustment costs function satisfies:

1. Φ (g)=0 2. Φ 0 (g)=1
The first assumption defines the depreciation rate, g, as the ratio between investment and capital at the steady state. The second assumption makes the steady state of our model with adjustment costs the same that the one with linear capital accumulation equation. We don't impose convexity or concavity of the function of adjustment costs, consequently Φ 0 (g) > 0 or Φ 0 (g) < 0.

The Hamiltonian in current value of ( 8) is:

H = T (x 1 ,y 1 )+q 1 (t) ✓ x 1 (t)Φ ✓ y 1 (t) x 1 (t) ◆◆ (9)
The first order conditions are

p 1 (t)=q 1 (t)Φ 0 ✓ y 1 (t) x 1 (t) ◆ ( 10
)
q1 (t)=q 1 (t)  ρ -Φ ✓ y 1 (t) x 1 (t) ◆ + y 1 (t) x 1 (t) Φ 0 ✓ y 1 (t) x 1 (t) ◆ -w 1 (11) ẋ1 = x 1 Φ ✓ y 1 (t) x 1 (t) ◆ (12)
with the transversality condition:

lim t!+1 x 1 (t)p 1 (t)e -ρ.t =0 (13)
Where q 1 is the co-state variable which corresponds to the utility price of capital in current value.

The competitive equilibrium

We use the parameter ϕ = -

y 1 x 1
Φ 00 Φ 0 to express the degree of the investment adjustment costs i.e. the elasticity of the investment adjustment costs. Therefore, ϕ could be seen as a mesure of the efficiency index of the investment per capita.

To obtain the dynamic equations characterizing the equilibrium path, we combine [START_REF] Garnier | Capital-labor substitution and indeterminacy in continuous-time two-sector models[END_REF] and ( 11) (after a total differentiation of ( 10)) and we obtain two equations of motion which describe the dynamic of equilibrium paths

‡ ẋ1 = x 1 Φ (14) ṗ1 = 1 E  p 1 ✓ ρ + y 1 p 1 Φ 0 -Φ ◆ -Φ 0 w 1 + ϕ p 1 y 1 ✓ ∂y 1 ∂x 1 - y 1 x 1 ◆ ẋ1 (15)
with E =1+ϕ p 1 y 1 ∂y 1 ∂p 1 . Any solution {x 1 (t),p 1 (t)} t>0 of the system ( 14) satisfying the transversality condition (13) will be called equilibrium path.

Steady state

We want to study the dynamical system [START_REF] Hercowitz | Endogenous fluctuations[END_REF] in the neighborhood of the steady state.

Proposition 1 Under assumption 1 there exists a unique steady state

(x ⇤ 1 ,p ⇤ 1 ) > 0 solution of : ẋ1 =0 () y 1 (x 1 ,p 1 )=gx 1 ṗ1 =0 () w 1 (p 1 )=p 1 (δ + g)
We note that the steady state is the same that the one of the model with linear capital accumulation. ‡ We note that all function depends on x1 and p1 :w1 = w1(p1), y1 = y1(x1,p1) and

Φ = Φ ⇣ y 1 (x 1 ,p 1 ) x 1 ⌘

The linearized system

In order to study the indeterminacy properties of equilibrium, we linearize the system ( 14) around (x ⇤ 1 ,p ⇤ 1 ) which gives the following Jacobian matrix § :

J = 0 @ ∂y 1 ∂x 1 -g ∂y 1 ∂p 1 -g ϕ⇤ p 1 y 1 ⇣ ∂y 1 ∂x 1 -g ⌘⇣ g+ρ- ∂y 1 ∂x 1 ⌘ E⇤ n δ+g- ∂w 1 ∂p 1 -ϕ⇤ p 1 y 1 h⇣ ∂y 1 ∂x 1 -g ⌘⇣ ∂y 1 ∂p 1 -g ⌘ -ρ ∂y 1 ∂p 1 io E⇤ 1 A (16
) Given initial capital stock x 1 (0) if there is more than one initial price p 1 (0) in the stable manifold of (x ⇤ 1 ,p ⇤ 1 ), the equilibrium path coming from x 1 (0) will not be unique. In particular, if the Jacobian matrix J (16) has two eigenvalues with negative real part (the locally stable manifold of the steady state (x ⇤ 1 ,p ⇤ 1 ) is two dimensional), there will be a continuum of converging paths and thus a continuum of equilibria: (x ⇤ 1 ,p ⇤ 1 ) is said to be locally indeterminate.

The dynamics of the model around the steady state can be fully derived from the eigenvalues of Jacobian matrix [START_REF] Lucas | Investment under uncertainty. Econometrica,1 9 7 1 . Documents de travail récents François Langot, Lise Patureau and Thepthida Sopraseuth[END_REF]. If we denote T and D the trace and the determinant of the Jacobian matrix [START_REF] Lucas | Investment under uncertainty. Econometrica,1 9 7 1 . Documents de travail récents François Langot, Lise Patureau and Thepthida Sopraseuth[END_REF], we know that the steady state is locally indeterminate if and only if T<0 et D>0. Therefore, we need to study the sign of T and D given by:

T = 1 E⇤ ⇢ ∂y 1 ∂x 1 -∂w 1 ∂p 1 -ρ + ϕ ⇤ p 1 y 1 h g ⇣ ∂y 1 ∂x 1 -g ⌘ + ρ ∂y 1 ∂p 1 i (17) D = 1 E⇤ ⇣ ∂y 1 ∂x 1 -g ⌘⇣ ρ + g -∂w 1 ∂p 1 + ϕ ⇤ gρ p 1 y 1 ⌘ (18)

Existence of local indeterminacy

Our main objective is to study the impact of adjustment costs measured by the elasticity ϕ⇤ on the indeterminacy mechanism coming from sector § At the steady state, under the assumption 1 the elasticity ϕ becomes: ϕ⇤ = -gΦ 00 (g) ≥ 0

. specific externalities.

Solving the system (17-18) with respect to ϕ⇤ gives a linear relationship between T (ϕ⇤) and D (ϕ⇤): when ϕ⇤ varies on ]-1, +1[, T (ϕ⇤) and D (ϕ⇤) move along the line called in what follows ∆ ϕ , which is defned by ¶ :

D = S ϕ T + M ϕ with S ϕ = ⇣ ∂y 1 ∂x 1 -g ⌘h ∂y 1 ∂p1 ⇣ ρ + g -∂w 1 ∂p 1 ⌘ + ρg i g ⇣ ∂y 1 ∂x 1 -g ⌘ + ρ ∂y 1 ∂p 1 + ∂y 1 ∂p 1 ⇣ ρ + g -∂w 1 ∂p 1 ⌘ (19) M ϕ = h g ⇣ ∂y 1 ∂x 1 -g ⌘ + ρ ∂y 1 ∂p 1 i⇣ ∂y 1 ∂x 1 -g ⌘⇣ ρ + g -∂w 1 ∂p 1 ⌘ -ρg ⇣ ∂y 1 ∂x 1 -g ⌘⇣ ρ + g -∂w 1 ∂p 1 ⌘ g ⇣ ∂y 1 ∂x 1 -g ⌘ + ρ ∂y 1 ∂p 1 + ∂y 1 ∂p 1 ⇣ ρ + g -∂w 1 ∂p 1 ⌘ (20)
Note that S ϕ and M ϕ depend only upon technological parameters. We use the geometrical method of Grandmont, Pintus and De Vilder [START_REF] Grandmont | Capital-labor substitution competitive nonlinear endogenous business cycles[END_REF] in order to study the variations of T (ϕ) and D (ϕ) in the (T,D) plane, when ϕ⇤ varies continuously on ]-1, +1[.

Condition for local indeterminacy without adjustment costs i.e. ϕ⇤ =0

In the case of ϕ⇤ =0there is no adjustment costs what it's correspond to the linear accumulation of capital, we get E⇤ =1and:

T (0) = ∂y 1 ∂x 1 - ∂w 1 ∂p 1 -ρ and D(0) = ✓ ∂y 1 ∂x 1 -g ◆✓ ρ + g - ∂w 1 ∂p 1 ◆
with:

∂y 1 ∂x 1 =
a 00 a 00 a 11 -a 01 a 10 and ∂w 1 ∂p 1 = â00 â00 â11 -â01 â10

We note that ∂y 1 ∂x 1 represents the Rybczynsky effect (i.e. quantity effect) and ∂w 1 ∂p 1 the Stolper Samuelson effect (i.e. price effect).

We can caracterize both partial derivatives in terms of capital intensity differences across private and social levels as in Benhabib and Nishimura [START_REF] Benhabib | Indeterminacy and sunspots with constant returns[END_REF]. Using the input coefficients given 6 and7, we give the following definition:

Definition 1 :T h ec o n s u m p t i o ng o o di ss a i dt ob e : i) capital (labor) intensive at the private level if and only if:

a 00 a 11 -a 01 a 10 < (>)0

ii) capital (labor) intensive at the social level if and only if:

â00 â11 -â10 â01 < (>)0
At the steady state, it's possible to give these condition i) and ii) only with the technological parameters β ij and βij . ii) â00 â11 -â10 â01 < (>)0 , b<(>)0

It follows that ∂y 1 /∂x 1 corresponds to the factor intensity difference from the private viewpoint (Rybczynski effects), while ∂w 1 /∂p 1 corresponds to the factor intensity difference from the social viewpoint (Stolper-Samuelson effects). Therefore, we can give the indeterminacy condition given by Benhabib and Nishimura [START_REF] Benhabib | Indeterminacy and sunspots with constant returns[END_REF] in the two-sectors model with exogenous labor, linear utility function, sector specific externalities and linear capital accumulation:

Proposition 3 :T h es t e a d ys t a t ei sl o c a l l yi n d e t e r m i n a t ei fa n do n l yi ft h e consumption good is capital intensive at the private level (b<0) and labor intensive at the social level ( b>0) i.e. there is a factor intensities reversal between the private and the social perspective.

This factor intensities reverseal corresponds to a break of the duality between Rybczynszy and Stolper Samuelson effects.

With the proposition 3 we know that the ∆ ϕ line reachs the local indeterminacy area of (T,D) plane when ϕ⇤ =0,i fa n do n l yi fb<0 and b>0. Therefore, the presence of adjustment costs only is not sufficient to lead to local indeterminacy without sector specific externalities but we will see they can interplay with externalities to provide Höpf bifurcation. Now, to ensure we have the steady state locally indeterminate when ϕ⇤ =0, we have to make the following assumption: Assumption 2 : b<0 and b>0.

Infinite degree of adjustment costs i.e. ϕ⇤!±1

In the case of infinite degree of adjustment costs i.e. ϕ⇤!±1 the Trace and the Determinant (17,18) become:

T (1)=g ⇣ ∂y 1 ∂x 1 -g ⌘ ∂y 1 ∂p 1 + ρ D(1)=gρ ⇣ ∂y 1 ∂x 1 -g ⌘ ∂y 1 ∂p 1
with:

∂y 1 ∂p 1 = ∂y 1 ∂x 1 1 p 1 2 4 x 1 1 - β01 β00 ! -1 + a 10 a 00 1 - β11 β10 ! -1 3 5 - y 1 p 1
Consequently, it's possible to express D(1) as linear function of T (1) that we note 4 1 :

D(1)=ρT (1) -ρ 2
This line 4 1 represents both the start points and the end points set of the segment 4 ϕ . Therefore, we can see immediatly that this line 4 1 can not get throught the indeterminacy area of the (T,D)plane for any values of b and b since both D(+1) and T(+1)h a v ea l w a y st h es a m es i g n( po s i t i v e or negative) for small values of the parameter ρ. This sign is rely on the sign of the both derivatives ∂y 1 ∂x 1 (which depends on the sign of the parameter b) and ∂y 1 ∂p 1 . Now, we focus on the case charactherized by the assumption 2. Indeed, on this assumption, we have the pair (T (0),D(0)) in the indeterminacy area of the (T,D) plane. Consequently, if we can verify that T (1) > 0 and D(1) > 0 then we know that the line 4ϕ cuts the D-axes that is the trace of the Jacobian matrix is nul and we have a Höpf bifurcation (i.e. cycles exist).

Proposition 4 :O nt h ea s s u m p t i o n2 ,T (1) > 0 and D(1) > 0 if and only if β01 < β00 .

We make the following assumption:

Assumption 3 : β01 < β00 .
This assumption ensures that the derivative ∂y 1 ∂p 1 is negative when assumption 2 is verified.

∂D ∂ϕ = P 1 y 1 ⇣ ∂y 1 ∂x 1 -g ⌘h gρ -∂y 1 ∂p 1 ⇣ ρ + g -∂w 1 ∂p 1 ⌘i E 2
On assumptions 2 and 3, it's easy to check that ∂D ∂ϕ > 0. Conséquently, when ϕ⇤ increases from -1 to +1,t h ep a i r(T (ϕ),D(ϕ)) increases along the line 4 ϕ from the starting point such that T (1) > 0 and D(1) > 0, cuts the D-axes, gets through the point (T (0),D(0)), gets out of the indeterminacy area and returns on the starting point from below. we can give the following proposition:

k Indeed, we have the starting point, the middle point (T (0),D(0)) and the end point therefore if we know only the sign of the derivative ∂D ∂ϕ we know the way of move of the pair (T (ϕ),D(ϕ)) on the line 4ϕ.

Proposition 5 :O nt h ea s s u m p t i o n1 ,2a n d39 ϕ c < 0 and ϕ > 0 such that:

i) T (ϕ c )=0and D (ϕ c ) > 0 ii) 8 ϕ⇤2]ϕ c , ϕ[ : T (ϕ⇤) < 0 and D (ϕ⇤) > 0.
The case i) ensures the existense of the value ϕ c < 0 such that we have a Höpf bifurcation: the steady state is a center and there exists periodic stable trajectory. This case is possible for a negative degree of adjustment costs only i.e. Φ is convex adjustment costs function. The presence of convex adjustment costs does not allow the firm to make instantaneous changes in the stock of capital when the price of investment good is modified since this adjustment of capital will have an infinite cost. Consequently, the firm have to adjust progressively the stock of capital.

The case ii) explains the local indeterminacy of the steady state is possible for an interval of values of adjustment costs degree which includes 0. If the negative degree of adjustment costs is too large (i.e. 0 < ϕ c < ϕ⇤) the steady state becomes instable and if it the positive degree of adjustment costs is too large (i.e. ϕ⇤ > ϕ > 0) the steady state is a saddle point stable. Now, we try to explain the economic intuition of impact of the adjustment costs on the indeterminacy mechanism coming from factor intensities reverseal between the private and social level that is what is the role of adjustment costs in the trajectories switching which are the source of sunspots.

Starting from an arbitrary equilibrium, consider that the agent expects another equilibrium with a larger rate of investment due to an instantaneous increase in the relative price of investment good p 1 . The only way for this other equilibrium path to become a new equilibrium path is to find a mechanism which reverses the price toward the equilibrium and offsets this initial increase. But the rise in the stock of capital, due to a higher rate of investment depends on the adjustment costs function. If the adjustment costs function is convex (resp. concave) i.e. ϕ⇤ < 0 (resp. ϕ⇤ > 0), the stock of capital will increase more than proportional (resp. it will increase less than proportional) for a low increase of investment. Consequently, from the Rybczynsky theorem ( ∂y 1 ∂x 1 < 0), since the investment good is labor intensive at the private level (b<0) i.e. the consumption good is capital intensive at the private level, there is a more than proportional decrease in its output ⇤⇤ . This more than proportional decrease is amplified (resp. diminished) by the adjustments costs through the variation of the capital stock. Morevover, from the Stolper Samuelson effect ( ∂w 1 ∂p 1 > 0), since the investment good is capotal intensive at the social level ( b>0), a rise in initial price of the investment good leads to an increase in the rate of return of capital. This increase have to be offset by a new decrease in the investment price to maintain the overall return of capital such that † † :

ṗ1 p 1 + w 1 p 1 = ρ + g + ϕ ⇤ ✓ ∂y 1 ∂x 1 -g ◆ ẋ1
The amplitude of this new decrease depends on the degree of adjustment costs: it can be offset the initial rise i.e ϕ⇤ > 0 but it can be also greater than the initial rise and leads to another fluctuations i.e. ϕ⇤ < 0. There is a limit case where this amplitude leads to reverse mechanism where the variables take exacly the opposite values that is we have a trajectory which describes a cycle i.e ϕ⇤ = ϕ c .

Concluding comments

In this paper we have prove that convex adjustment costs function interplays with sector specific externalities to lead to Höpf bifurcation and then periodic cycles.

Proposition 2 :

 2 L e tb ⌘ β 00 β 11 -β 01 β 10 and b ⌘ β00 β11 -β10 β01 .A tt h e steady state we have: i) a 00 a 11 -a 01 a 10 < (>)0 , b<(>)0

4. 3

 3 General case: finite degree of adjustment costs i.e. ϕ⇤ > 0 Finally, We have to study the way of move of the pair (T (ϕ),D(ϕ)) on the line 4 ϕ .I ts u fficients to compute the derivative ∂D ∂ϕ k :

¶ Note that (x ⇤ 1 ,p ⇤ 1 )does not depend on ϕ⇤ and remains the same along line ∆α.


Jean-Philippe Garnier

Keeping-up with the Joneses, a new source of fluctuations in the two-sector continuous-time models

Keywords: Two-sector models, continous-time models, consumption externality, keeping up with the Joneses, local indeterminacy, endogenous fluctuations

Our main objective is to study the impact of consumption externality like keeping of with the Joneses on the properties of long-run equilibrium in the two-sector optimal growth model. Does this consumption externality lead to a new mechanism of local indeterminacy and endogenous fluctuations ? We will see that, in two-sector growth models with exogenous labor and without technological externalities, if the representative agent is able to give more value to his social status than his own consumption, this is the keys of a new mechanism for endogenous fluctuations. Moreover, by opposition with the other endogenous fluctuation mechanisms, we will see that this one doesn't need to have restriction on the factor intensity configuration of the consumption sector.

Introduction

The study of endogenous fluctuations, i.e. the existence of a continuum of equilibria that arises in dynamic economies with some market imperfections, have given a lot of results. In general equilibrium model, the existence of a continuum of equilibria which converges to the same steady state is relied to the local indeterminacy property of the steady state. Generally, indeterminacy emerges with small market distortions as a type of coordination problem. Following Benhabib and Nishimura [START_REF] Benhabib | Indeterminacy and Sunspots with Constant Returns[END_REF], these distortions lead to a mechanism such that, starting from an arbitrary equilibrium, if all agents were simultaneously increse their investment in an asset, the rate of return on the asset would tend to increase, and in turn set off relative price changes that would drive the economy back towards steady-state. In the one-sector models, increasing returns, via external effects on production or monopolistic competition, are able to generate this mechanism. In the two-sector models, the rate of return and marginal products depend not only on stocks of assets, but also on the composition of output across sectors. Increasing the production and the stock on a capital, say due to an increase in its price, may well increase its rate of return. It is possible therefore to have constant aggregate returns in all sectors at the social level (but decreasing at the private level) and to still obtain indeterminacy. For that, it has to have external effect, through sector specific externalities, at least in one of the sectors. Consequently, the major contributions of Benhabib and Nishimura [START_REF] Benhabib | Indeterminacy and Sunspots with Constant Returns[END_REF] and Benhabib, Nishimura and Venditti [START_REF] Benhabib | Indeterminacy and Cycles in Two-Sector Discret-Time Model[END_REF], in the two-sector model with different Cobb-Douglas technologies at the private level with sector-specific externalities and constant social returns to scale, Benhabib and Nishimura [START_REF] Benhabib | Indeterminacy and Sunspots with Constant Returns[END_REF] prove, with a separable utility function which is linear in consumption and strictly concave with respect to labor, that local indeterminacy arises if and only if technological externalities allow to have a reversal of factor intensities between the private and social levels (i.e. the consumption good is capital intensive at the private level and labor intensive at the social level). Therefore, there exists technological mechanism coming from externalities which brokes the duality between Rybczynsky and Stolper-Samuelson effects and 1 leads to indeterminacy. In all these contributions, an implication of decreasing private returns is positive profits. The presence of such profits (even quite small) would invite entry, and unless the number of firms cannot be constant along the equilibrium and a fixed cost of entry has to be assumed to solve this problem, but it is not really satisfying.

More recently, Nishimura, Takahashi and Venditti [START_REF] Nishimura | Endogenous Fluctuations in Two-Sector Models : The Role of Preferences[END_REF] have proved that indeterminacy is possible without any market distortions, so in the twosectors optimal growth discret time models with CES technologies, non linear preferences are able to generate a mechanism such that for high enough values of elasticity of intertemporal substitution in consumption, indeterminacy is led, but only if the consumption good is capital intensive. In this case, this is the intertemporal arbitrage behavior of the representative agent and his capacity to substitute future consumption to the present one that is the source of the endogenous fluctuations. Indeed, oscillations in the consumption and investment goods output, led by fluctuations, require some oscillations in the consumption levels, that is the case when the elasticity of intertemporal substitution in consumption is high enough (i.e. the degree of concavity of the utility function is low enough). In this line, Garnier, Venditti and Nishimura [START_REF] Garnier | Intertemporal Substitution in Consumption, Labor Supply Elasticity and Sunspot Fluctuations in Continuous-Time Models[END_REF] have introduced non linear utility in consumption in the two-sector continous-time model with sector specific externalities such that there are constant returns to scale at the social level but decreasing at the private level. With the breaking of Rybczynsky and Stolper-Samuelson effects (like in Benhabib and Nishimura [START_REF] Benhabib | Indeterminacy and Sunspots with Constant Returns[END_REF] and Benhabib, Nishimura and Venditti [START_REF] Benhabib | Indeterminacy and Cycles in Two-Sector Discret-Time Model[END_REF]), they prove the existence of sunspot fluctuations is obtained if and only if the elasticity of intertemporal substitution in consumption is large enough and the elasticity of labor supply is low enough (even equal to zero). Therefore, the role played by non linear preferences seems the same than the one of Nishimura, Takahashi and Venditti [START_REF] Nishimura | Endogenous Fluctuations in Two-Sector Models : The Role of Preferences[END_REF], that is, it allows the representative agent to smooth his consumption over time.

In all these contributions, a necessary condition to obtain indeterminacy is the presence of capital intensive consumption good since the coordination problem needs that all agents increase their investment in the capital asset what leads to a first increase (before the future decreases coming from the fluctuations) of the more capital intensive good : the consumption. Whereas the output of the consumption good increases, the consumption of agents too. Then, even if this rise of the consumption will be followed by future decreases, the representative agent has a present time preference and is able to smooth his consumption over time. For these reasons, this indeterminacy mechanism is accepted by the agents. The major problem of the existence of capital intensive consumption good is that the capital share in the production function has to be greater than the labor share, but we know that this fact is not consistent empirically. Moreover, whereas the consumption good is capital intensive, then the investment good has to be labor intensive.

In order to show that it is possible to have a new mechanism of endogenous fluctuations led only by the preferences and that, even if the consumption good is not capital intensive, we introduce a new type of externality called keeping up with the Joneses. This consumption externality operates through the utility function by adding an exogenous variable that represents the consumption standard of the economy. The idea that the happiness of an individual depends upon the consumption of others is widely viewed as an important feature of our shared social existence. The introduction of consumption standard in the utility function generates an externality which increases the felicity that each individual obtains from his own consumption (i.e. the utility function is increasing according to the consumption standard). Preferences exhibit "keeping up with the Joneses" since we assume that the marginal utility of consumption is increasing with respect to the externalities.

Liu and Turnovsky [START_REF] Liu | Consumption Externalities, Production Externalities and Long-Run Macroeconomic Efficiency[END_REF] have showed that the consumption externalities do not generate indeterminacy, in one-sector growth model with exogenous labor supply. Moreover, we know from Alonso-Carrera, Caballé and Raurich [START_REF] Alonso-Carrera | Can consommation spillovers be a source of equilibrium indeterminacy ?[END_REF] that consumption externalities are a source of indeterminacy in one-sector growth model with endogenous labor supply. So, in this paper, we will show, in the two-sector growth model that endogenous labor supply is not necessary to have equilibrium indeterminacy with consumption externalities. Moreover, we will see the indeterminacy results are only linked to consumption externality in preferences (i.e. keeping up with the joneses). The major difference between the results of Alonso-Carrera, Caballé and Raurich [START_REF] Alonso-Carrera | Can consommation spillovers be a source of equilibrium indeterminacy ?[END_REF] and ours results is that we do not make any assumption on the weight of consumption externality (i.e. on the sensibility of the representative agent according to the consumption standard) whereas they suppose the consumption externality plays a neglicted role compared to the consumption. Consequently, we choose to interpret the relative consumption of the representative agent (i.e. the ratio between his own consumption and the consumption standard) as his social status. Hence, we do not make any assumption on the weight of consumption externality so we can suppose that the social status may have the same kind of importance than the consumption for the representative agent. We also let the representative agent have a behaviour which maintains his social status over time. We will see, such situations create a new mechanism for sunspot fluctuations. First, to obtain general results we do not specify the utility function. Then, we take a Cobb-Douglas utility function to illustrate the relationship between social statuss and own consumption.

Consequently, keeping up with the Joneses in the optimal growth continous-time model with two-sector leads to a new mechanism of sunspot fluctuations even if the consumption good is labor intensive. The labor intensive configuration for consumption good in the continous-time two setor model is quite new in the litterature on endogenous cycles but it seems more satisfying according the empiral results.

The rest of the paper is organized as follows. Section 2 describes the economy. Section 3 characterizes the competitive equilibrium. Section 4 gives formal conditions to obtains local indeterminacy. Section 5 analyzes the mechanism that leads to equilibrium indeterminacy. Section 6 gives an example of utility function that allows the existence of indeterminacy and illustrates our main result through a standard parametrization of the model. Finally, Section 7 concludes. All the proofs are collected in the appendix (section 8).

The economy

We consider an infinite horizon, continous-time, two-sector model with Cobb-Douglas technologies, inelastic labor supply and consumption externalities through "keeping up with the Joneses" preferences. The economy consists of competitive firms and a representative household.

Firms

We assume that consumption good y 0 and capital good y 1 are produced by capital x 1j and labor x 0j , j =0 , 1, through a Cobb-Douglas technology. Hence, the "private production function" used by the representative firm in each industry is given by :

y j = F j (x 0j ,x 1j )=x β j 0j x 1-β j 1j for j =0, 1 (1)
with

β j ∈ [0, 1].
Therefore the returns to scale are constant, in each sector j =0, 1.

We normalize the price of consumption good to one i.e. p 0 =1 . The representative firm in each sector j =0 , 1 maximizes its profit ⇡ j given the output price p j of the good produced in sector j,t h ew a g er a t ew 0 and the rental rate of capital w 1 , subject to production function 1. The first order conditions give :

a 0j (w 1 ,p j )= β j p j w i = x 0j y j a 1j (w 1 ,p j )= (1-β j )p j w i = x 1j y j (2)
for the factor i in the sector j. We call a ij the input coefficients. The factor-price frontier, which gives a relationship between input prices and output prices is expressed with these input coeffcients.

Lemma 1 :D e n o t ep =( 1 ,p 1 ) 0 , x =( 1 ,x 1 ), y =( y 0 ,y 1 ), w =( w 0 ,w 1 ) 0 and A(w, p)=[a ij (w i ,p i )].T h e n: p = A 0 (w, p)w x = A(w, p)y
Note that at the equilibrium the wage rate and the rental rate are functions of the output price only i.e. w 0 = w 0 (p 1 ) and w 1 = w 1 (p 1 ) whereas outputs are functions of the capital stock and the output price i.e. y j = y j (x 1 ,p 1 ) for j =0, 1.

Household

We assume that the population is constant and normalized to one. The representative agent derives his utility from consumption according to function

U (c(t),z(t))
,where c(t) is the individual consumption, z(t) the consumption externality given by the consumption average of the economy which can be interpreted as the consumption standard. The function U satisfies the standard hypothesies on the behavior of the consumer with respect to the consumption : the marginal utility in consumption is positive and decreasing. Moreover, the introduction of consumption standard implies that consumption spillovers affect the household's utility. Indeed, we assume that preferences correspond to the "keeping up with the Joneses" formulation such that the marginal utility of consumption rises with the consumption standard. Hence the following assumption holds :

Assumption 1 : U (., z(t)) is increasing and concave ∀z(t) ∈ R + , the first partial derivatives satisfy U z (c(t),z(t)) > 0 and U c (c(t),z(t)) > 0 and the second partial derivatives satisfy U cz (c(t),z(t)) > 0 and U cc (c(t),z(t)) < 0
We introduce the following elasticities :

✏ cc = - U c (c(t),z(t)) c(t)U cc (c(t),z(t)) > 0 (3)
✏ cz = U c (c(t),z(t)) c(t)U cz (c(t),z(t)) > 0 (4)
where ✏ cc is the elasticity of intertemporal susbstitution in consumption and ✏ cz is the absolute value of the elasticity of substitution between consumption and consumption standard.

The enhanced value of the consumption standard by the agent is positively correlate to ✏ cz . We introduce the ratio c(t) z(t) as the relative consumption of the agent (i.e. the ratio between his own consumption and the consumption standard of the economy). Note that this ratio can be interpreted as his social status.

The elasticity ✏ cz can be interpreted as a measure of the importance given by the agent to his social status. In most of the models using "keeping up with the Joneses" formulations, it is common to assume that ✏ cz <✏ cc i.e. U cc + U cz < 0 in order to have a consumption externality that has smaller effects on the preference of the representative agent compared to his own consumption. Consequently we do not make any assumption on the weight of consumption externality since we suppose that the social status may have the same kind of importance than the consumption for the representative agent 1 . Therefore, we can have ✏ cc < ✏ cz i.e. U cc + U cz > 0 that is, we consider that the agents may give more weight or more importance to their social status rather than to their own present consumption.

The objective of the representative agent is to solve the following intertemporal optimization problem by taking z(t) as given :

max y 1 (t),x 1 (t) ´1 0 e -δt U (c(t),z(t)) dt s.c. ẋ(t)=y 1 (t) -gx 1 (t) x 1 (0) = x 1 given (5)
Where δ > 0 is the subjective discount rate and g ∈ (0, 1) the depreciation rate of capital. The production frontier is defined as :

c = T (x 1 ,y 1 )=x β 0 00 x 1-β 0 01
1. We note that there is no problem about the concavity of the utility function when we suppose that ✏cc < ✏cz as z(t) is take as given by the representative agent.

From the enveloppe theorem we easily get w 1 = T 1 (x 1 ,y 1 ) and

p 1 = -T 2 (x 1 ,y 1 ) 2 .
The Hamiltonian in current value of ( 5) is :

H = U (T (x 1 ,y 1 ) ,z(t)) + q 1 (t)(y 1 (t) -gx 1 (t)) (6)
The first order conditions are :

p 1 (t)U c (c(t)) = q 1 (t) (7)
q 1 (t)(δ + g) -w 1 U c (c(t)) = q1 (t) (8)
y 1 -gx 1 =ẋ 1 ( 9
)
lim t!+1 x 1 (t)U c (c(t)) p 1 (t)e -ρ.t =0 (10)
Where q 1 is the co-state variable which corresponds to the utility price of capital in current value.

The competitive equilibrium

Let us denote :

↵ = 1 ✏ cc - 1 ✏ cz (11)
If ✏ cc < (>) ✏ cz then we have ↵ > (<)0.

To obtain the dynamic equations characterizing the symetric equilibrium i.e z = c, we combine [START_REF] Benhabib | Indeterminacy and Sunspots with Constant Returns[END_REF] and [START_REF] Benhabib | Indeterminacy and Cycles in Two-Sector Discret-Time Model[END_REF] and since c = c(x 1 ,p 1 ), after a total differentiation of (7), we have the two equations of motion which describe the dynamic of equilibrium paths :

ẋ1 = y 1 (x 1 ,p 1 ) -gx 1 (12) ṗ1 = 1 E(x 1 ,p 1 )  p 1 (δ + g) -w 1 (p 1 )+↵ p 1 c @c @x 1 (y 1 (x 1 ,p 1 ) -gx 1 ) with E(x 1 ,p 1 )=1-↵ p 1 c @c @p 1 (13)

Steady state

Any solution {x 1 (t),p 1 (t)} t>0 of the system (12) satisfying the transversality condition (10) will be called equilibrium path.

Proposition 1 There exists a unique steady state (x ⇤ 1 ,p ⇤ 1 ) > 0 solution of :

ẋ1 =0 ⇐⇒ y 1 (x 1 ,p 1 )=gx 1 ṗ1 =0 ⇐⇒ w 1 (p 1 )=p 1 (δ + g)
See appendix 8.1 for details.

Caracteristic polynomial

In order to study the indeterminacy properties of equilibrium, we linearize the system (12) around (x ⇤ 1 ,p ⇤ 1 ) which gives the following Jacobian :

J = ∂y 1 ∂x 1 -g ∂y 1 ∂p 1 α E p 1 c ∂c ∂x 1 ⇣ ∂y 1 ∂x 1 -g ⌘ 1 E h δ + g -∂w 1 ∂p 1 + ↵ p 1 c ∂c ∂x 1 ∂y 1 ∂p 1 i ! (14)
Any solution from (12) that converges to the steady state (x ⇤ 1 ,p ⇤ 1 ) satisfies the transversality condition and is an equilibrium. Therefore, given initial capital stock x 1 (0) if there is more than one initial price p 1 (0) in the stable manifold of (x ⇤ 1 ,p ⇤ 1 ), the equilibrium path coming from x 1 (0) will not be unique. In particular, if the Jacobian matrix J (14) has two eigenvalues with negative real part (the locally stable manifold of the steady state (x ⇤ 1 ,p ⇤ 1 ) is two dimensional), there will be a continuum of converging paths and thus a continuum of equilibria :

(x ⇤ 1 ,p ⇤ 1
) is said to be locally indeterminate. The dynamics of the model around the steady state can be fully derived from the eigenvalues of Jacobian matrix [START_REF] Guesnerie | Endogenous fluctuations[END_REF] or from the roots λ 1 and λ 2 of the caracteristic polynomial :

P (λ)=λ 2 -T λ + D ( 15
)
where T and D are respectively the trace and the determinant of the Jacobian matrix [START_REF] Guesnerie | Endogenous fluctuations[END_REF].

The trace and the determinant T and D are given by :

T = λ 1 + λ 2 = 1 E ⇢ ∂y 1 ∂x 1 + δ -∂w 1 ∂p 1 + ↵ p 1 c ⇣ ∂c ∂x 1 ∂y 1 ∂p 1 -∂c ∂p 1 ⇣ ∂y 1 ∂x 1 -g ⌘⌘ (16
)

D = λ 1 λ 2 = 1 E ⇣ ∂y 1 ∂x 1 -g ⌘⇣ δ + g -∂w 1 ∂p 1 ⌘ (17)
We know that the steady state is locally indeterminate if and only if

T<0 et D>0.

Existence of local indeterminacy

Our main objective is to study the impact of consumption externality measured by the elasticity ✏ cz on the local determinacy properties of the long-run equilibrium.

Solving the system [START_REF] Lloyd-Braga | Indeterminacy with Small Externalities : the Role of Non-Separable Preferences[END_REF][START_REF] Liu | Consumption Externalities, Production Externalities and Long-Run Macroeconomic Efficiency[END_REF] with respect to ↵ gives a linear relationship between T (↵) and D (↵) : when ↵ varies, T (↵) and D (↵) move along the line called in what follows ∆ α (see appendix 8.3 for the proof), which is defined by 3 :

D = S α T + M α with S α = ∂c ∂p 1 ⇣ ∂y 1 ∂x 1 -g ⌘⇣ δ + g -∂w 1 ∂p 1 ⌘ ∂c ∂x 1 ∂y 1 ∂p 1 + ∂c ∂p 1 ⇣ δ + g -∂w 1 ∂p 1 ⌘ (18)
M α = ⇣ ∂y 1 ∂x 1 -g ⌘⇣ δ + g -∂w 1 ∂p 1 ⌘⇣ ∂c ∂x 1 ∂y 1 ∂p 1 -∂c ∂p 1 ⇣ ∂y 1 ∂x 1 -g ⌘⌘ ∂c ∂x 1 ∂y 1 ∂p 1 + ∂c ∂p 1 ⇣ δ + g -∂w 1 ∂p 1 ⌘ (19)
Note that S α and M α depend only upon technological parameters.

In their previous paper, Garnier, Venditti and Nishimura [START_REF] Garnier | Capital-Labor Substitution and Indeterminacy in Continuous-Time Two-Sector Models[END_REF] have already studied the case where the utility function is non linear, under the presence of sector-specific externalities such there is capital intensity reversal between the private and the social level, they have proved the existence of sunspot fluctuations is obtained if and only if the elasticity of intertemporal substitution in consumption is large enough and the elasticity of labor supply is low enough (even equal to zero). Therefore, we will see that without externalities (i.e. constant returns to scale), it is possible to have a room of local indeterminacy led by keeping up with the Joneses. The case where ↵ > 0 cannot provide local indeterminacy, indeed, in the two-sector model without consumption externalities, Garnier, Venditti and Nishimura have shown that it must have sector specific externalities to have local indetermincy even if the utility function is non linear (which corresponds to the case where ↵ =1 /✏ cc ). Therefore, we can deduct that whereas the level of consumption externality is low (i.e.✏ cc < ✏ cz and ↵ > 0), the presence of consumption externality cannot change the results obtained by Garnier, Venditti and Nishimura [START_REF] Garnier | Capital-Labor Substitution and Indeterminacy in Continuous-Time Two-Sector Models[END_REF] . Consequently, we focus only on the case where the level of consumption externality is sufficiently large to have ✏ cc > ✏ cz and thus ↵ < 0.

We use the geometrical method of Grandmont, Pintus and De Vilder [START_REF] Grandmont | Capital-Labor Substitution and Competitive Nonlinear Endogenous Business Cycles[END_REF] in order to study the variations of T (↵) and D (↵) in the (T,D) plane, when ↵ varies continuously on ]-∞, 0[. We make the following assumption :

Definition 1 :T h ec o n s u m p t i o ng o o di ss a i dt ob ec a p i t a l( l a b o r )i n t e n s i v e if and only if :

a 00 a 11 -a 01 a 10 < (>)0

At the steady state, it's possible to give this last condition only with the technological parameters β ij : Proposition 2 :L e tb ≡ β 0 -β 1 .A tt h es t e a d ys t a t ew eh a v e:a 00 a 11a 01 a 10 < (>)0 ⇔ b<(>)0 So, when ↵ gets from -∞ to 0,t h ep a i r(T (↵) ,D(↵)) moves along the half line ∆ α 4 characterized by a starting point (T (∞) ,D(∞)) :

T (∞)=- ∂c ∂x 1 ∂y 1 ∂p 1 ∂c ∂p 1 + @y 1 @x 1 -g (20)
D (∞)=0 (21)
and a end point (T (0) ,D(0)) such that :

D (0) = ✓ @y 1 @x 1 -g ◆✓ δ + g - @w 1 @p 1 ◆ (22)
T (0) = δ (23)
Consequently, as the ending point is characterized by D (0) < 0 (since

∂y 1 ∂x 1 = ∂w 1 ∂p 1 )
and T (0) = δ then indeterminacy is ruled out for ↵ =0whereas the starting point is located on the abscissa area since D (∞)=0.

If we can verify both T (∞) < 0 and D is an increasing function of ↵ then the half line ∆ α gets by the indeterminacy area (T<0 and D>0). We have to give conditions to have T (∞) < 0 and dD dα =

p * 1 c * E 2 ∂c ∂p 1 D (0) > 0.
We can see immediately that dD dα depends only on the sign of the derivative ∂c ∂p 1 since D (0) < 0 and then we must verify ∂c ∂p 1 < 0. In this case, the pair (T (↵) ,D(↵)) enters in the indeterminacy area when ↵ gets from -∞ to a critical value ↵ then it gets out of the indeterminacy area and finishes on the point Otherwise, T (∞) > 0 implies dD dα < 0 and the pair (T (↵) ,D(↵)) moves in the wrong way : local indeterminacy is ruled out (see appendix 8.5 and 8.4).

Proposition 3 :S u p p o s et h a tβ

1 > β 1 ≡ δ 2δ+g ,

t h e nt h e r ee x i s t sβ

⇤ β 1 ≡ - g δ+g β 1 (1-β 1 ) (1-2β 1 -g (1-β 1 ) δ+g ) > 0 such that ∀β 0 > β ⇤ β 1 we have dD dα > 0 and T (∞) < 0.

Where β *

β 1 is a critical value that depends on the value that we have choosen for the labor share in the investment sector β1.

Assumption 2 : The labor share of the investment sector verifies :

β 1 > β 1 ≡ δ 2δ+g .
Finally, when 3 is held and since ∆ α is a half-line and D (↵) is an increasing function of ↵ and T (∞) < 0,t h ep a i r(T (↵) ,D(↵)) gets by the indeterminacy area when ↵ increases from -∞ to 0 and finishs on the point (T (0) ,D(0)) such that T (∞) < 0 and D (∞)=0 + (as the pair (T (↵) ,D(↵)) comes from the half-plan where the values of D(↵) are positive). More precisely, there exists ↵ ∈ ]-∞, 0[ such that the steady state is locally indeterminate ∀↵ ∈ ]-∞, ↵[. Hence, we give the following lemma :

Lemma 2 :U n d e ra s s u m p t i o n s1a n d2:s u p p o s et h a tβ

1 > δ 2δ+g , ∀β 0 > β ⇤ β 1 ∃↵ ≡ 1 p 1 c dc dp 1 ∈ ]-∞, 0[ such that ∀↵ < ↵ the steady state (x ⇤ 1 ,p ⇤ 1 ) is locally indeterminate and ∀↵ > ↵ the steady state (x ⇤ 1 ,p ⇤ 1 ) is saddle point.
This lemma gives a condition set which gives endogenous fluctuations without any restriction on the factor intensity of the consumption good. Indeed, there is just a critical value on the labor share in the consumption sector β ⇤ β 1 (linked on the labor share of the investment sector β 1 ) and a critical value of the labor share in the investment sector through assumption 2 (linked on the discount time and depreciation rate) ; it is sufficient to choose β 0 > β ⇤ β 1 to obtain local indeterminacy for any quite small value of ↵ (i.e. ↵ must be smaller than ↵). Therefore, endogenous fluctuations don't depend on the factor intensity of the consumption sector. This result is in sharp constrast with the ones of Benhabib, Nishimura and Venditti [START_REF] Benhabib | Indeterminacy and Cycles in Two-Sector Discret-Time Model[END_REF] or Nishimura, Takahashi and Venditti [START_REF] Nishimura | Endogenous Fluctuations in Two-Sector Models : The Role of Preferences[END_REF].

Endogenous fluctuations mechanism

This result means that consumption externality like "keeping up with the Joneses" gives rise to new mechanism of indeterminacy which is not linked to the one coming from technological externalities.

We can provide an heuristic explanation to understand how the "keeping up with the Joneses" leads to local indeterminacy. We know that ↵ has to be negative to lead to the equilibrium indeterminacy. When ↵ < 0, the agent favours his social status c(t) z(t) more than his own consumption c(t) over time. Now, starting from an arbitrary equilibrium, consider that the agent expects another one with a higher rate of investment and higher level of capital stock coming from an instantaneous increase in relative price of investment good p 1 . The only way that this other equilibrium path becomes a new equilibrium path is to find a mechanism which reverses the price toward the equilibrium and offsets this initial increase. Since the labor supply is exogenous, if one of the both goods (consumption or investment) increases (decreases) then the other one decreases (increases). Suppose, for example, that the investment good is capital intensive (b>0), then a higher capital stock implies an increase (more than proportional) of output of the investment good and a decrease (more than proportional) of output of the consumption good and thus in the own consumption of the representative agent and in his social status. But the representative agent, who gives more value to his social status than his consumption, wants to keep his social status over time and thus has to increase his future consumption. The level of this rise in the consumption depends on the level of the externality measured by ✏ cz .I ndeed, there are two opposite effects playing through the decreasing marginal utility in consumption (i.e. U cc < 0)andthroughthepositiv eeffect of externality on this marginal utility (i.e. U cz > 0 ). Since, we have assumed that U cc + U cz > 0, the overall effect of this decrease in the present consumption and in the social status leads to a decrease in the marginal utility. Therefore, the representative agent has to increase more than proportional his future consumption to conserve his level of utility. Consequently, the present decrease in the consumption of the representative leads to a large increase of his future level of consumption. To support this future level of consumption the output in this sector will have to increase and as the labor is exogenous, that will lead to a large decrease of the output in the investment sector what will reverse its price p 1 toward the equilibrium and offsets its initial rise 6 . But, to ensure that this increase of level of consumption has a sufficient decreasing impact on the output of the investment sector, the labor share in each sector has to be greater than a critical value which depends only on the depreciation rate g and discount time δ for the investment sector (i.e assumption 2) but depends also on the labor share β 1 of the investment sector for the consumption sector (i.e. β 0 > β ⇤ β 1 ).

Examples

Let us consider Cobb-Douglas formulation between the social status c(t)

z(t)
and the consumption c(t) such as :

U (c(t),z(t)) = 1 1 -σ " c(t) 1+γ ✓ c(t) z(t) ◆ -γ # 1-σ
where β > 0 represents the weight of the consumption externality or the sensibility of the representative agent according to the standard of consumption z(t), and σ > 0 represents the individual risk aversion. This function satisfies assumption 1. We can derive the following elasticities :

✏ ⇤ cc = 1 σ ✏ ⇤ cz = - 1 γ(1 -σ) Therefore : ↵ = σ(1 + γ) -γ
Note that for any γ > 0,i fσ =0(i.e. the utility is linear with respect to the consumption) then ↵ < 0 and if σ =1then ↵ =1Therefore ∀γ > 0, ∃σ ∈ [0, 1[ such that ∀σ ∈ [0, σ[ we have ↵ < 0.

We want to illustrate the lemma 2) for both capital and labor intensive consumption good

Labor or capital intensive consumption good

We use a parametrization according empirical results on the labor share in european countries :

β 1 =0.62 g =0.05 δ = 0.01
This configuration givesβ 1 =0 .142 and β ⇤ β 1 =0 .35, that means that we can take any value greater than 0.142 for the labor share of the investment sector and greater than 0.35 for the consumption sector to obtain endogenous fluctuations. Moreover we can choose this value in order to have labor intensive or capital intensive consumption good. For example, if we set β 0 =0 .6 then we have b = -0.02 and the consumption good is capital intensive, moreover, we obtain the following critical value : ↵ = -0.05, that is we can set, for example, σ =0 .2 and γ =0 .32 (i.e. all value of σ and γ such that + γ) -γ < -0.05. Otherwise, if we set β 0 =0.65 then we have b =0.03 and from now, the consumption good is labor intensive, moreover, we obtain a new critical value for the inverse of the elasticity of intertemporal substitution in consumption : ↵ = -0.01, for example σ =0.2 and γ =0.27.

Concluding comments

In this paper we have shown that a new mechanism can bring endogenous fluctuations. It comes only from preferences and the behavior of the representative agent toward his social status. Since Benhabib and Nishimura [START_REF] Benhabib | Indeterminacy and Sunspots with Constant Returns[END_REF], endogenous fluctuations were the result of market imperfections coming from technological externalities leading to a capital intensity reversal between the private and social level. Our new mechanism is totaly independent of that of Benhabib and Nishimura. Moreover, the consumption good may be capital or labor intensive by opposite to the model with market imperfection where the endogenous fluctuation need a capital intensive consumption good. This mechanism depends on the behavior of the representative agent : he has to give more value to his social status than his own present consumption.

The most important results is that we have a mechanism of endogenous fluctuations that not implies capital intensities reversal or capital intensive consumption goo, but only the assumption that the representative agent gives some value to his social status. The direct consequence is that, for the first time in this type of model, we have endogenous fluctuations under constant returns to scale, that is without any type of technological externalityor market imperfections, moreover with labor intensive sectors.

8 Appendix 8.1 Proof of existence of (x ⇤ 1 ,p ⇤ 1 )
The maximization of profit gives the following first order conditions :

w i = p j β ij y j x ij for i, j =0, 1 (24)
The steady state is characterized by : y 1 = gx 1 and w 1 =( δ + g) p 1 ,s o that :

x 11 = 1 -β 1 δ + g gx 1 (25)
Moreover we have :

x 01 = gx 1 ✓ 1 -β 1 δ + g ◆ -1-β 1 β 1 (26)
The stock equations : x 1 = x 10 + x 11 et 1=x 01 + x 00 allow to give :

x 10 = x 1 ✓ 1 -g 1 -β 1 δ + g ◆ (27) x 00 =1-gx 1 ✓ β 11 δ + g ◆ -1-β 1 β 1 (28)
From (24) we have :

x 00 x 11 x 01 x 10 = (1 -β 1 )β 0 (1 -β 0 )β 1 (29)
From ( 25), ( 26), ( 27), ( 28), (29), we have :

x ⇤ 1 = ⇣ 1-β 1 δ+g ⌘ 1 β 1 1-β 1 δ+g g ⇣ 1 -(1-β 1 )β 0 (1-β 0 )β 10 ⌘ + (1-β 1 )β 0 (1-β 0 )β 1 > 0 (30) For i =1and j =0,( 2 4
) ,( 2 7 )a n d( 3 0 )g i v e:

w ⇤ 1 =(1-β 0 ) ✓ (1 -β 1 )β 0 (1 -β 0 )β 1 ◆ β 00 ✓ 1 -β 1 δ + g ◆ -β 0 β 1 (31)
We derive :

p ⇤ 1 = 1 -β 0 δ + g ✓ (1 -β 1 )β 0 (1 -β 0 )β 1 ◆ β 00 ✓ 1 -β 1 δ + g ◆ -β 0 β 1 > 0 (32)
8.2 Computation of derivatives used in T (α) and D(α)

In order to compute ( 16) and ( 17) we need the following partial derivatives : ∂y 1 ∂x 1 , ∂c ∂x 1 , ∂w 1 ∂p 1 , ∂y 1 ∂p 1 , ∂c ∂p 1 . To compute ∂y 1 ∂p 1 and ∂c ∂p 1 we begin by the total differentiation of the quantity equations given by : a 00 y 0 + a 01 y 1 =1 a 10 y 0 + a 11 y 1 = x 1

The total differentiation gives : a 00 dy 0 + a 01 dy 1 + @a 00 @w 0 y 0 dw 0 + y 1 ✓ @a 01 @w 0 dw 0 + @a 01 @p 1 dp 1 ◆ =0 (33)

a 10 dy 0 + a 11 dy 1 + @a 10 @w 1 y 0 dw 1 + y 1 ✓ @a 11 @w 1 dw 1 + @a 11 @p 1 dp 1 ◆ = dx 1 (34)
After, we need ∂c ∂x 1 and ∂y 1 ∂x 1 with c = y 0 and dw 0 = dw 1 = dp 1 =0. Then, we have : @y 1 @x 1 = a 00 a 11 a 00 -a 10 a 01 (35) @c @x 1 = -a 01 a 11 a 00 -a 10 a 01 (36)

These derivatives correspond to the Rybczynsky effect. Now, compute ∂y 1 ∂p 1 and ∂c ∂p 1 . With the price equations given by :

a 00 w 0 + a 10 w 1 =1 a 01 w 0 + a 11 w 1 = p 1 So : @w 1 @p 1 = @y 1 @x 1 (37)
@w 0 @p 1 = -a 10 a 11 a 00 -a 10 a 01

On the other hand we derive from

a 0j (w 1 ,p j )= β j p j w i = x 0j y j a 1j (w 1 ,p j )= (1-β j )p j w i = x 1j y j that : @a ij @w i = - a ij w i (39) @a ij @p j = a ij p j (40)
We substitute (37), (38), (39) et (40) in (33) and (34). Hence, the resolution of the system (33) and (34), with dx 1 =0give :

a 00 dy 0 + a 01 dy 1 + dp 1 a 01 p 1 y 1 - ✓ p 1 - â11 â10
◆ -1 ! =0 (41)
a 10 dy 0 + a 11 dy 1 + dp 1 a 11 p 1 y 1 -x 1 ✓ p 1 - â01 â00
◆ -1 ! =0 (42)
Using ( 41) and (42) we have thus :

@y 1 @p 1 = a 00 a 11 a 00 -a 10 a 10 With ( 16) and ( 17) we can extract ↵ : where S α is given by ( 18) and M α is given by [START_REF] Nishimura | Indeterminacy and the Role of Factor Substitutability[END_REF] The computation of the derivative dT dα and dD dα give :

1 p 1 ✓ β 0 β 0 -β 1 x 1 - a 10 a 00 1 -β 0 β 0 -β 1 ◆ - gx 1 p 1 (43) @c @p 1 = - a 01 a 11 a 00 -a 10 a 10 1 p 1 ✓ β 0 β 0 -β 1 x 1 - a 11 a 01 1 -β 0 β 0 -β 1 ◆ (
↵ = D (↵) - ⇣ ∂y 1 ∂x 1 -g ⌘⇣ δ + g -∂w 1 ∂p 1 ⌘ p 1 c ∂c ∂p 1 D (↵) = T (↵) - ⇣ ∂y 1 ∂x 1 + δ -∂w 1 ∂p 1 ⌘ p 1 c ∂c ∂p 1 T (↵)+ p 1 c ⇣ δ + g -∂w 1 ∂p 1 ⌘ (
dT d↵ = p ⇤ 1 c ⇤ E 2  @c @x 1 @y 1 @p 1 + @c @p 1 ✓ δ + g - @w 1 @p 1 ◆ (47)
dD d↵ = p ⇤ 1 c ⇤ E 2 @c @p 1 ✓ @y 1 @x 1 -g ◆✓ δ + g - @w 1 @p 1 ◆ (48)
When technological parameters are fixed, only E depends on ↵ in the expression of these derivatives. Hence, it is easy to show that the sign of ∂T ∂α and ∂D ∂α does not depend on ↵ and remains constant ∀ ↵.

8.4 Proof of Proposition 3 dD dα < 0

We have to prove that dD dα > 0. Since we can rewrite 48 as dD dα =

p * 1 c * E 2 ∂c ∂p 1 D(0) and since D(0) is always negative we have to give conditions to have dc dp 1 < 0. From 40, we can rewrite ∂c ∂p 1 with the parameters of the model :

@c @p 1 = x 1 (δ + g) ⇢ β 1 (1 -2β 0 ) b 2 + ✓ 1 -g 1 -β 1 δ + g ◆ 1 b (49)
When b<(>)0 we have β 0 < (>)β 1 and since 1 -g 1-β 1 δ+g > 0 (since g δ+g < 1 and 1 -β 1 < 1), consequently, we have several cases where ∂c ∂p 1 < 0.

Let β ⇤ β 1 ≡- g δ+g β 1 (1-β 1 ) (1-2β 1 -g (1-β 1 ) δ+g )
a critical value which depends on the labor share in the investment sector such that we have ∂c ∂p 1 > 0, ∀β 0 > β ⇤ β 1 , where the critical value β ⇤ β 1 depends on the labor share in the investment sector β 1 , that we have setted but not on the factor intensity configuration of sectors. Moreover, we have to ensure that this critical value is positive, that is garanted by β 1 > δ 2δ+g . Consequently : when β 1 > δ 2δ+g is checked, then ∀β 0 > β ⇤ β 1

we have ∂c ∂p 1 < 0 and thus dD dα > 0.

8.5 Proof of Proposition 2 T (∞) < 0 T (∞) (eq.15) is linked on the sign of the derivative ∂c ∂p 1 , ∂y 1 ∂p 1 and b (as we know that b<0 gives ∂y 1 ∂x1 < 0 and ∂c ∂x 1 > 0). As we know conditions which give ∂c ∂p 1 positive, we have to study the impact of these ones on the sign of T (∞).

From 43, we can rewrite ∂y 1 ∂p 1 with the parameters of the model :

@y 1 @p 1 = gx 1 p 1  1 b 2 δ + g g β 0 (2β 0 -1) -1+ 1 -β 0 b (50)
Now, from 20, 49 and 50 we can write T (∞) with the parameters of the model as :

T (∞)=
Since ∂c ∂p 1 has to be negative we have to check that the sign of the numerator of T (∞) is positive. Moreover, to verify ∂c ∂p 1 < 0 we have to check that β 0 > β ⇤ β 1 , which implies the numerator of T (∞) is positive and if ∂c ∂p 1 < 0 then T (∞) is always negative. Consequently, it explains why T (∞) must be negative and why T (∞) > 0 rules out local indeterminacy.

  Therefore, we have to study the sign of T and D in function of b and ↵.

5 (

 5 Proof : see appendix 8.5 and 8.4) 4. When ↵ > 0,t h ep a i r(T (↵) ,D(↵)) moves along the another part of the half line ∆α :s e eG a r n i e r ,V e n d i t t ia n dN i s h i m u r a[ 1 2 ] .

44) 8 . 3

 83 Computation of ∆ αConsider the expressions T (↵)[START_REF] Lloyd-Braga | Indeterminacy with Small Externalities : the Role of Non-Separable Preferences[END_REF], D(↵)[START_REF] Liu | Consumption Externalities, Production Externalities and Long-Run Macroeconomic Efficiency[END_REF] and E =1-↵. p 1 c . ∂c ∂p 1

x 1 b

 1 2 w 0 [gβ 1 (1 -β 0 -b)+β 0 (1 -β 0 )(δ + gβ 1 )] ∂c ∂p 1 -g

Note that y0 = c(x1,p1)=T (x1,y1(x1,p1)).

Note that (x * 1 ,p * 1 ) does not depend on ↵ and remains the same along the line ∆α.

We note that there is a symetric mechanism, but in the opposite way, when the investment good is labor intensive.

Bibliography

Références