Frédéric Dufourt

Kazuo Nishimura

Alain Venditti

H D'albis

S Bosi

R Boucekkine

R Dos

Santos Ferreira

J P Drugeon

J M Grandmont

C Le Van

T Lloyd-Braga

L Modesto

C Nourry

T Seegmuller

Y Vailakis

Sunspot Fluctuations in Two-Sector Models: New Results with Additively-Separable Preferences

Keywords: Indeterminacy, sunspots, two-sector model, sector-specific externalities, real business cycles Journal of Economic Literature Classification Numbers: C62, E32, O41

We analyze local indeterminacy and sunspot-driven fluctuations in the standard two-sector model with additively separable preferences. We provide a detailed theoretical analysis enabling us to derive relevant bifurcation loci and to characterize the steady-state local stability properties as a function of various structural parameters influencing the degree of increasing returns to scale, the amount of intertemporal substitution in consumption, and the elasticity of the aggregate labor supply curve. On the theoretical side, we prove the existence of both a flip and a Hopf bifurcation locus in the corresponding parameter space. We also show that local indeterminacy can be obtained under any labor supply elasticity or under an arbitrarily low elasticity of intertemporal substitution in consumption. On the empirical side, we find that indeterminacy and sunspot fluctuations are robust features of two-sector models, prevailing for most empirically plausible calibrations for these parameters.

Introduction

The literature on one-sector Real Business Cycle models with productive externalities and increasing returns to scale (IRS) offers today a relatively exhaustive picture of the conditions required for the existence of local indeterminacy and sunspot fluctuations. 1 Local indeterminacy typically requires a large enough elasticity of intertemporal substitution in consumption (EIS), a large enough degree of increasing returns to scale (IRS), and a large enough elasticity of aggregate labor supply. These conditions obviously interact together: for a given labor supply elasticity, a lower degree of IRS must be combined with a larger EIS in consumption in order to obtain indeterminacy, and vice-versa. Likewise, for any given degree of IRS, a lower EIS in consumption must be combined with a larger elasticity of aggregate labor supply for indeterminacy to prevail. Despite these tradeoffs in the relative intensities of these economic mechanisms, a standard conclusion from one sector models is that indeterminacy hardly occurs for empirically plausible calibrations of the parameters unless other features such as a variable capital utilization rate are introduced (Wen [25], Benhabib and Wen [START_REF] Benhabib | Indeterminacy, Aggregate Demand, and the Real Business Cycle[END_REF]).

One noticeable feature of two-sector Real Business Cycle models is that local indeterminacy typically requires much lower degrees of IRS than their one-sector equivalents. This is well known from the canonical two-sector model of Benhabib and Farmer [START_REF] Benhabib | Indeterminacy and Sector Specific Externalities[END_REF] -featuring a separable utility function with a unitary EIS -in which only 7% of IRS are required for indeterminacy compared to about 50% in the corresponding one-sector model of Benhabib and Farmer [START_REF] Benhabib | Indeterminacy and Increasing Returns[END_REF]. However, the literature on two-sector models is far from being as exhaustive on the required combinations in terms of labor supply elasticity, intertemporal substitution effects and externalities consistent with indeterminacy as the literature on one-sector models is. Actually, many of the results obtained in two-sector models have been derived under relatively narrow specifications for technology and/or preference, without much systematic analysis of the interplays between the relevant underlying economic mechanisms, and often through numerical simulations. 2 Our aim in this paper is to contribute to fill this gap by providing an exten- 1 See among others Benhabib and Farmer [START_REF] Benhabib | Indeterminacy and Increasing Returns[END_REF], Lloyd-Braga et al. [START_REF] Lloyd-Braga | Indeterminacy with Small Externalities: the Role of Non-Separable Preferences[END_REF], Nishimura et al. [START_REF] Nishimura | Indeterminacy in Aggregate Models with Small Externalities: an Interplay Between Preferences and Technology[END_REF],

Pintus [START_REF] Pintus | Indeterminacy with Almost Constant Returns to Scale: Capital-Labor Substitution Matters[END_REF], Wen [START_REF] Wen | Capacity Utilization Under Increasing Returns to Scale[END_REF].

2 For example, in Benhabib and Farmer [START_REF] Benhabib | Indeterminacy and Sector Specific Externalities[END_REF], the utility function is restricted to be logarithmic in consumption (unitary EIS). In Harrison [START_REF] Harrison | Indeterminacy in a Model with Sector Specific Externalities[END_REF], a more general utility function for consumption is considered, but the analysis is restricted to the case of an infinitely elastic labor supply. These analyses thus do not cover the set of empirically credible calibrations for these parameters.

sive analysis of the local stability properties of two-sector optimal growth model, considering a fairly general class of additively separable preferences. 3 Starting from the Benhabib and Farmer [START_REF] Benhabib | Indeterminacy and Sector Specific Externalities[END_REF]'s formulation with increasing social returns to scale but considering a more general additively separable utility function, we analyze the interplays between the degree of IRS, the EIS in consumption and the labor supply elasticity in the emergence of local indeterminacy. Assuming in a first step a sufficiently elastic labor supply, which includes the range of empirically credible values for this elasticity, we prove that local indeterminacy occurs quite generally, in particular for arbitrarily low EIS in consumption, provided that the degree of increasing social returns is larger than some (empirically plausible) lower bound. This conclusion is drastically different from what is known from the previous literature, in which a large enough EIS was always assumed in order to get indeterminacy with empirically plausible amounts of externalities (Garnier et al. [START_REF] Garnier | Intertemporal Substitution, Labor Supply Elasticity and Sunspot Fluctuations in Continuous-Time Models[END_REF][START_REF] Garnier | Local Indeterminacy in Continuous-Time Models: The Role of Returns to Scale[END_REF], Harrison [START_REF] Harrison | Indeterminacy in a Model with Sector Specific Externalities[END_REF]). We show that changes in the local stability properties of the model occur through both flip and Hopf bifurcations, and we provide the analytical expressions for these bifurcation values. We also prove that local indeterminacy occurs no matter how elastic or inelastic the labor supply is, provided the EIS in consumption and the amount of externalities are in an intermediary range still compatible with empirically relevant values. As a result, we show that indeterminacy and sunspot driven fluctuations can occur under a wide range of empirically credible calibrations for all the structural parameters.

The rest of this paper is organized as follows. We present the model and we characterize the intertemporal equilibrium and the steady state in the next Section. In Section 3, the complete set of conditions for indeterminacy are derived and some numerical illustrations are provided. Some concluding remarks are stated in Section 4, whereas all the technical details are given in an Appendix.

The model

We consider a standard infinite-horizon two-sector real business-cycle model à la Benhabib and Farmer [START_REF] Benhabib | Indeterminacy and Sector Specific Externalities[END_REF], with productive externalities in the investment sector. 4

Production

The economy produces a consumption good, c, and an investment good, I, with constant returns to scale Cobb-Douglas technologies at the private level in both sectors and output externalities in the investment sector only. We denote by Y c and Y I the outputs of sectors c and I, and by A the external effects. The production functions at the private level are thus:

Y ct = K α ct L 1-α ct , Y It = A t K α It L 1-α It ( 1
)
where K ct and L ct are capital and labor units allocated to the consumption sector, and K It and L It are capital and labor units allocated to the investment sector. The externality parameter A t depends on KI,t and LI,t , the average levels of capital and labor in sector I, such that

A t = KαΘ It L(1-α)Θ

It

(2) with Θ ≥ 0. These economy-wide averages are taken as given by individual firms. Assuming that factor markets are perfectly competitive and that capital and labor inputs are perfectly mobile across the two sectors, the first order conditions for profit maximization of the representative firm in each sector are

r t = αYct Kct = p t αY It K It , ω t = (1-α)Yct Lct = p t (1-α)Y It L It (3)
where r t , p t and ω t are respectively the rental rate of capital, the price the investment good and the real wage rate at time t, all in terms of the price of the consumption good.

Preferences

We consider an economy populated by a continuum of unit mass of identical infinitely-lived agents. At each period, a representative agent supplies elastically an amount l t of labor, consumes c t and invests I t so as to accumulate capital. He derives current period utility from consumption and labor according to a standard additively separable utility function given by

U (c, l) = c 1-σ 1-σ -l 1+χ 1+χ (4)
with σ ≥ 0 and χ ≥ 0 which are respectively the inverse of the EIS in consumption and the inverse of the Frisch wage elasticity of labor supply.

Denoting by k t the household's capital stock and by Y t the GDP, the budget constraint faced by the representative household is

c t + p t I t = Y t = r t k t + ω t l t (5)
Assuming that capital depreciates at rate δ ∈ (0, 1) in each period, the law of motion of the capital stock is:

k t+1 = (1 -δ)k t + I t (6)
with k 0 given. Combining ( 5) and ( 6), the representative household then maximizes its present discounted lifetime utility

max {ct,lt,k t+1 } t=0...∞ +∞ t=0 β t c 1-σ t 1-σ - l 1+χ t 1+χ s.t. k t+1 = (1 -δ + r t )k t + ω t l t -c t , t = 0...∞, k 0 given (7)
with β ∈ (0, 1) the discount factor. The first-order conditions are, for t = 0...∞,

c -σ t = βc -σ t+1 r t+1 +(1-δ)p t+1 pt (8)
ω t c -σ t = l χ t (9)
Equation ( 8) is the standard Euler equation, and (9) corresponds to the trade-off between consumption and leisure.

Intertemporal equilibrium and steady state

We consider symmetric perfect-foresight equilibria which consist of prices {r t , p t , ω t } t≥0 and quantities {c t , l t , I t , k t , Y ct , Y It , K ct , K It , L ct , L It } t≥0 that satisfy the household's and the firms' first-order conditions as given by ( 3) and ( 8)-( 9), the technological and budget constraints (1)-( 2) and ( 5)- [START_REF] Campbell | Asset Prices, Consumption and the Business Cycle[END_REF], and the market equilibrium conditions. All firms of sector I being identical, we have KIt = K It and LIt = L It for any t. The production function at the social (aggregate) level in the investment good sector is defined as

Y It = K α(1+Θ) It L (1-α)(1+Θ) It (10)
We thus have increasing returns at the social level with size given by Θ.

The market clearing conditions for the consumption and investment goods are c t = Y ct and I t = Y It , while the market clearing conditions for capital and labor yield

K ct + K It = k t and L ct + L It = l t .
Any solution that also satisfies the transversality condition lim

t→+∞ β t c -σ t k t+1 = 0 is called an equilibrium path.
A steady state is defined by constant equilibrium quantities and prices. We provide the following Proposition: Proposition 1. Assume that Θ = (1 -α)/α and χ = χ ≡ σ(1-α)+αΘ 1-α(1+Θ) . Then there exists a unique steady state.

Proof : See Appendix 5.1.

We can now turn to the analysis of the local stability properties of the model, considering a family of economies parameterized by the EIS in consumption 1/σ, the degree of IRS Θ and the wage elasticity of labor supply 1/χ.

Local stability analysis

After some manipulations, the two-sector model described above can be reduced to a two-dimensional dynamic system in (k t , c t ) . Linearizing this system in a neighborhood of the steady state yields a Jacobian matrix for which the characteristic polynomial is given in Appendix 5.2. In this Appendix, we also show that for a given value of χ, the Trace and Determinant of the Jacobian matrix are linear functions of σ. This means that when σ is varied over (0, +∞), the Trace and Determinant move along a line, denoted by ∆ χ , whose location depends on the parameters, in particular on the size of externalities Θ and the (inverse of) the elasticity of the labor supply χ. Thus, we can analyze the local stability properties of the model by using the geometrical methodology described in Grandmont et al. [START_REF] Grandmont | Capital-Labor Substitution and Competitive Nonlinear Endogenous Business Cycles[END_REF].

Let us introduce at this stage the following parameter restrictions, which enable us to simplify the analysis by restricting the number of possible configurations:

Assumption 1. α ∈ (1/4, 1/2), β ∈ ( β, 1), δ < δ and Θ ∈ (Θ, Θ) with β ≡ max{(1 -2α)/[(1 -δ)(1 -α) 2 ], (1 -α -δ)/(1 -δ)(1 -α)}, δ ≡ [β(1 -α) -2(1 - β)]/β(2 -α), Θ = δ/(1 -δ) and Θ = α/(1 -α). 5
These restrictions are sufficient to consider the whole range of empirically credible values for these parameters. Estimates for the labor share (equal to 1 -α in the model) in industrialized economies are typically in the range 60-70%. Estimates for the quarterly depreciation rate are typically close to 2.5%, and estimates for the subjective discount factor are typically around 0.99. Using a standard calibration of RBC models compatible with quarterly data, namely (α, δ, β) = (0.3, 0.025, 0.99), Assumption 1 holds and is compatible with mild external effects since β ≈ 0.837, δ ≈ 0.4, Θ ≈ 0.0256 and Θ ≈ 0.4286. The interval for Θ largely covers the range of available empirical estimates for the amount of IRS to scale in the US economy. 6Note also that Assumption 1 implies that all our results are compatible with standard negative slopes for the capital and labor equilibrium demand functions.

Denoting θ = β(1 -δ), let us also introduce the following bounds on χ and Θ, important for the local stability properties of the model (see below):

χ = αΘ 1-α(1+Θ) , Θχ = α(1+β)[1-θ(1-α)+χ] 1-θ + (1-α)(1-θ) 2 (1-βαδ 1-θ ) α( χ-χ) with χ = (1-δ)(1+β)(1-α)-δα δα
Since Θχ is increasing in χ, there are two main cases to consider: (i) χ is not too large, i.e. the elasticity of the labor supply curve is not too small, so that Θχ ∈ (Θ, Θ) ; (ii) χ is large (the elasticity of the aggregate labor supply curve is small), so that Θχ > Θ. The next subsection is devoted to case (i) which, we argue, covers all the empirically relevant configurations. Subsection 3.2 will consider instead the case of an arbitrary value for the elasticity of aggregate labor supply, restricting then the range of values considered for the degree of IRS Θ. This second case is mostly important for theoretical purposes as it covers the case of a fixed labor supply.

Local indeterminacy for small elasticities of intertemporal substitution in consumption

Let us first consider the case of a not too small labor supply elasticity, so that Θχ ∈ (Θ, Θ). For practical purposes, we assume that χ ∈ [0, 2/3), an interval for which the condition Θχ ∈ (Θ, Θ) is always satisfied under Assumption 1. 7 This interval covers values for the elasticity of the aggregate labor supply curve ranging from 3/2 to +∞, which includes the range of empirically credible values for this elasticity according to Prescott and Wallenius [START_REF] Prescott | Federal Reserve Bank of Minneapolis[END_REF] and Rogerson and Wallenius [START_REF] Rogerson | Micro and Macro Elasticities in a Life Cycle Model with Taxes[END_REF], who concluded for values typically larger than 2 and probably around 3. 8 This range also covers Hansen's [START_REF] Hansen | Indivisible Labor and the Business Cycle[END_REF] and Rogerson's [START_REF] Rogerson | Indivisible Labor, Lotteries and Equilibrium[END_REF] models of indivisible labor (with employment lotteries), corresponding to χ = 0. We can now apply the geometrical methodology of Grandmont et al. [START_REF] Grandmont | Capital-Labor Substitution and Competitive Nonlinear Endogenous Business Cycles[END_REF]. In appendix 5.2, we show that when σ increases from 0 to +∞, the value of the pair (T , D) varies along a line ∆ χ , whose starting and ending points depend on the values 7 Using again as an example the standard calibration (α, δ, β) = (0.3, 0.025, 0.99), we get Θχ ≈ 0.1033 when χ = 0, and Θχ ≈ 0.316 when χ = 2/3. Note also that the threshold χ lies in [0, 2/3) in all cases.

of structural parameters, in particular regarding the amount of externalities Θ and the labor supply elasticity, 1/χ. 9In Appendix 5.3, we show that when externalities are weak, Θ ∈ (Θ, Θχ ), we have the following geometrical configurations (see Figure 1,a,b) Figure 1(a) corresponds to the case of a large (possibly infinite) labor supply elasticity: χ ∈ [0, χ). The line ∆ χ crosses the triangle ABC in which both characteristic roots have a modulus less than 1 and local indeterminacy arises. Indeed, when σ increases from 0, the value of the pair (T , D) varies along ∆ χ . The steady state is first saddle-point stable for σ ∈ [0, σT ), becomes unstable for σ ∈ (σ T , σH ), then locally indeterminate when σ ∈ (σ H , σF ) and is finally saddle-point stable for σ > σF . When σ crosses σT , one positive characteristic root crosses the value 1 and a transcritical bifurcation occurs. 10 When σ crosses σH , one pair of complex characteristic roots crosses the unit circle and a Hopf bifurcation occurs generating quasi-periodic endogenous fluctuations. When σ crosses σF , one negative characteristic root crosses the value -1 and a flip bifurcation occurs generating period-two cycles.

Figure 1(b) covers the case of a smaller labor supply elasticity: χ ∈ (χ, 2/3). As can be observed, the local stability properties of the steady-state are the same, except that the starting point of the line ∆ χ (associated to σ = 0) is now located above the triangle ABC, where the steady-state is unstable, so that the transcritical bifurcation no longer exists. The steady-state is thus unstable for σ ∈ (0, σH ), locally indeterminate for σ ∈ (σ H , σF ) and saddle-point stable for σ > σF .

The Hopf, flip and transcritical bifurcation values are respectively given by (see Appendix 5.3):

σH = (1-βδα 1-θ ) Θ(χ+α)(1-δ)(1-β) δ - Θ[(1-α)(1-θ)-δ-χδα] δ + α(1-β)[1-θ(1-α)+χ] 1-θ (11) σF = (1-βδα 1-θ ) Θ[α(1+θ)(2-δ)+χ[2(1-δ)(1+β)+δα(1-θ)]]-δχ(1-α)(1-θ) δ(1-α) 2 - Θ[(1-δ)(1-α)(1+β)-δα(1+χ)] δ(1-α) + α(1+β)[1-θ(1-α)+χ] (1-α)(1-θ) +(1-θ)(1-βδα 1-θ ) (12)
and σT =

[1-α(1+Θ)](χ-χ) (1-α) (13)
When the amount of externalities is larger, Θ ∈ ( Θχ , Θ), we get the following geometrical configurations. Comparing Figure 2(a) to Figure 1(a), we observe that the stability properties of the model are the same, except that the line ∆ χ now has a terminal point (for σ = +∞) which is located in the interior of the triangle ABC (where the model is locally indeterminate), and no longer crosses the line generated by AB, so that the flip bifurcation no longer occurs. The steady-state is thus locally indeterminate for any σ > σH , i.e. for arbitrarily low values for the EIS in consumption. Likewise, in Figure 2(b) where χ ∈ (χ, 2/3), the situation is the same as in Figure 1(b) except that the flip bifurcation no longer exists. Indeterminacy thus also occurs for arbitrarily low values for the EIS.

We summarize these results in the following Proposition: Proposition 2. Under Assumption 1, let χ ∈ [0, 2/3). Then, there exist δ, δ, with 0 < δ < δ ≤ δ, and β ∈ [ β, 1) such that if δ ∈ (δ, δ) and β ∈ (β, 1), the following results hold: i) For Θ ∈ (Θ, Θχ ), when χ ∈ [0, χ), the steady state is saddle-point stable when σ ∈ [0, σT ), undergoes a transcritical bifurcation at σ = σT , becomes locally unstable when σ ∈ (σ T , σH ), undergoes a Hopf bifurcation at σ = σH , becomes locally indeterminate when σ ∈ (σ H , σF ), undergoes a flip bifurcation at σ = σF , and becomes again saddle-point stable when σ ∈ (σ F , +∞). When χ ∈ (χ, 2/3), the transcritical bifurcation disappears so that the model is locally unstable for σ ∈ [0, σH ).

ii) For Θ ∈ ( Θχ , Θ), when χ ∈ [0, χ), the steady state is saddle-point stable when σ ∈ [0, σT ), undergoes a transcritical bifurcation at σ = σT , becomes locally unstable when σ ∈ (σ T , σH ), undergoes a Hopf bifurcation at σ = σH , and becomes locally indeterminate when σ ∈ (σ H , +∞). When χ ∈ (χ, 2/3), the transcritical bifurcation disappears so that the model is locally unstable for σ ∈ [0, σH ).

Proof : See Appendix 5.3.
In Proposition 2, the bounds δ, δ and β are complicated expressions of the structural parameters, obtained from second-order polynomials derived in Appendix 5.3. Of course, the conditions β ∈ (β, 1) and δ ∈ (δ, δ) are always satisfied under Assumption 1 by our benchmark calibration (α, δ, β) = (0.3, 0.025, 0.99), and are also always satisfied in a significantly large neighborhood of this calibration.

Comments. Proposition 2 provides new clear-cut conditions for the occurrence of local indeterminacy in two-sector RBC models with additively separable preferences, generalizing the results obtained by Benhabib and Farmer [START_REF] Benhabib | Indeterminacy and Sector Specific Externalities[END_REF] in the particular case of utility function that is logarithmic in consumption (unitary EIS). First, we prove the existence of a Hopf and, in some cases, of flip and transcritical bifurcations in the parameter space. 11 Second, we prove that local indeterminacy can arise in two-sector models for an arbitrarily small EIS in consumption 1/σ, provided that the amount of IRS in the investment sector is in an intermediary range, namely Θ ∈ ( Θχ , Θ). Note that this range includes the empirical estimates of Basu and Fernald [START_REF] Basu | Returns to Scale in US Production: Estimates and Implications[END_REF] for the degree of IRS in the US durable manufacturing industry. Third, we easily derive from equations ( 11) that ∂σ H /∂Θ > 0 and ∂σ F /∂Θ > 0. We conclude from case ii) of Proposition 2 (where indeterminacy requires σ ∈ (σ H , +∞)) that for any labor supply elasticity χ ∈ (0, 2/3), decreasing the amount of externalities actually favors the emergence of indeterminacy by increasing the range of values for σ for which the steady-state is locally indeterminate. The conclusion is different in case i) of Proposition 2, where indeterminacy requires σ ∈ (σ H , σF ), since both bifurcation parameters are decreasing in Θ.

As an application from Proposition 3 , Figures 3(a) and (b) display the determinacy/indeterminacy areas in the (σ, Θ) plane for two different values for the elasticity of the aggregate labor supply curve: χ = 0, which corresponds to Hansen's [START_REF] Hansen | Indivisible Labor and the Business Cycle[END_REF] assumption of indivisible labor, and χ = 1/3, which corresponds to a labor supply elasticity of 3, the value recommended by Prescott and Wallenius [START_REF] Prescott | Federal Reserve Bank of Minneapolis[END_REF] and Rogerson and Wallenius [START_REF] Rogerson | Micro and Macro Elasticities in a Life Cycle Model with Taxes[END_REF] to calibrate business cycle models. We observe that indeterminacy prevails for a wide range of empirically plausible values for σ and Θ. For example, when χ = 0 and the amount of IRS is calibrated to Θ = 0.33 (the point estimate obtained by Basu and Fernald [START_REF] Basu | Returns to Scale in US Production: Estimates and Implications[END_REF]) indeterminacy requires σ > σH ≈ i.e. an EIS in consumption smaller than 1.2. When χ = 1/3, indeterminacy requires σ > σH ≈ 0.65, i.e. an EIS in consumption smaller than 1.54. This range of values is consistent with most, although not all, empirical estimates for this coefficient. 12 There is no agreement in the empirical literature about the precise value of the EIS in consumption, since most estimates typically vary between 0 and 2. See in particular Campbell [START_REF] Campbell | Asset Prices, Consumption and the Business Cycle[END_REF] , Kocherlakota [START_REF] Kocherlakota | The Equity Premium: It's Still a Puzzle[END_REF] and Vissing-Jorgensen [START_REF] Vissing-Jorgensen | Limited Asset Market Participation and the Elasticity of Intertemporal Substitution[END_REF] for estimates smaller than 1, and Mulligan [START_REF] Mulligan | Capital Interest and Aggregate Intertemporal Substitution[END_REF] and Gruber [START_REF] Gruber | A Tax-Based Estimate of the Elasticity of Intertemporal Substitution[END_REF] for estimates ranging between 1 and 2.

Local indeterminacy for any elasticity of the labor supply

In the previous Section we have shown that by focusing on an empirically realistic subset of values for the wage-elasticity of labor supply, we can get local indeterminacy for arbitrarily low values of the EIS and mild externalities. We now consider the possibility of obtaining local indeterminacy for any labor supply elasticity, enabling us in particular to cover the case of a fixed labor supply (obtained as the limiting case where χ tends to +∞). Following such a route requires to restrict the size of externalities, and we consequently introduce the following bound on Θ:

Θ = δα[1-θ(1-α)] (1-α)(1-θ)(1-δ) (14)
In appendix 5.4, we prove that when Θ ∈ (Θ, Θ), depending on whether χ is larger or lower than χ, we get the same geometrical configurations as those described by Figure 1,a,b. We can then establish the following Proposition: Proposition 3. Under Assumption 1, let Θ ∈ (Θ, Θ). Then there exist δ, δ, with 0 < δ < δ ≤ δ, and β ∈ [ β, 1) such that when β ∈ (β, 1) and δ ∈ (δ, δ), the following results hold: i) If χ ∈ [0, χ), the steady state is saddle-point stable when σ ∈ [0, σT ), undergoes a transcritical bifurcation at σ = σT , becomes locally unstable when σ ∈ (σ T , σH ), undergoes a Hopf bifurcation at σ = σH , becomes locally indeterminate when σ ∈ (σ H , σF ), undergoes a flip bifurcation at σ = σF , and becomes again saddle-point stable when σ ∈ (σ F , +∞).

ii) If χ ∈ (χ, +∞), the steady state is locally unstable when σ ∈ [0, σH ), undergoes a Hopf bifurcation at σ = σH , becomes locally indeterminate when σ ∈ (σ H , σF ), undergoes a flip bifurcation at σ = σF , and becomes saddle-point stable when σ ∈ (σ F , +∞).

Proof : See Appendix 5.4. Proposition 3 also generalizes previous results obtained in the literature on twosector models with additively separable preferences. 13 Considering the case of a utility function with a unitary EIS in consumption, Benhabib and Farmer [START_REF] Benhabib | Indeterminacy and Sector Specific Externalities[END_REF] show how varying the wage-elasticity of labor supply affects the range of values for which indeterminacy occurs. They show in particular that considering an infinitely elastic labor supply allows to minimize the amount of externalities required for indeterminacy. We prove here that adjusting the EIS in consumption is another way to favor the occurrence of local indeterminacy since, provided the amount of externalities is mild enough (i.e. Θ ∈ (Θ, Θ)), indeterminacy can occur for any elasticity of labor supply. 14 This conclusion thus covers the case of a fixed labor supply.

Concluding comments

Although two-sector infinite-horizon models require smaller degrees of increasing returns to scale for indeterminacy than aggregate models, they are usually criticized on the fact that they rely on too large values for the EIS in consumption, and too large values for the elasticity of the labor supply with respect to empirically plausible estimates. However, most of the contributions are based on relatively narrow specifications for technology and preferences and/or often rely on numerical simulations, preventing from getting a full picture of the configurations giving rise to local indeterminacy and sunspot fluctuations. We have proved that local indeterminacy occurs through flip and Hopf bifurcations for any value of the elasticity of the labor supply, and can even be compatible with an arbitrarily low EIS in consumption. Moreover, the existence of expectation-driven fluctuations is consistent with a mild amount of increasing returns.

Appendix

Proof of Proposition 1

Equation ( 6) evaluated at the steady state gives I = Y I = δk. Moreover, we derive from (3) that r/p = αY I /K I = αδk/K I . It follows from (8) that K I = βδαk/(1 -θ) with θ = β(1 -δ). Merging equations (3) gives L I /K I = L c /K c = l/k and thus we get from [START_REF] Grandmont | Capital-Labor Substitution and Competitive Nonlinear Endogenous Business Cycles[END_REF]

k = l (1-α)(1+Θ) 1-α(1+Θ) δ Θ 1-α(1+Θ) βα 1-θ 1+Θ 1-α(1+Θ) ≡ l (1-α)(1+Θ) 1-α(1+Θ) κ ( 15
)
assuming of course that Θ = (1 -α)/α. Consider now c = Y c = K α c L 1-α c = (k -K I ) l k 1-α . Substituting (15) into this expression gives c = 1 -βδα 1-θ κα l 1-α 1-α(1+Θ) ≡ ψl 1-α 1-α(1+Θ) (16)
From the expression of the prices in (3) we derive 14 For an analysis of the role of the labor supply elasticity in two sector models where technologies have constant social returns to scale, see Benhabib and Nishimura [START_REF] Benhabib | Indeterminacy and Sunspots with Constant Returns[END_REF], Garnier et al. [START_REF] Garnier | Intertemporal Substitution, Labor Supply Elasticity and Sunspot Fluctuations in Continuous-Time Models[END_REF][START_REF] Garnier | Local Indeterminacy in Continuous-Time Models: The Role of Returns to Scale[END_REF] and Nishimura and Venditti [START_REF] Nishimura | Indeterminacy in Discrete-Time Infinite-Horizon Models with Non-Linear Utility and Endogenous Labor[END_REF].

ω = (1 -α)κ α l αΘ 1-α(1+Θ) ≡ φl αΘ 1-α(1+Θ) , r = αl -(1-α)Θ 1-α(1+Θ) κα-1 , p = β 1-θ r (17)
Equation ( 9) now gives using ( 15)-( 17):

(1 -α) 1 -βδα 1-θ σ δ Θα(1+σ) 1-α(1+Θ) βα 1-θ (1+Θ)α(1+σ) 1-α(1+Θ) = l χ-χ (18)
with χ ≡ σ(1-α)+αΘ

1-α(1+Θ)
Assuming now that χ = χ, we derive

l = (1 -α) 1 χ-χ 1 -βδα 1-θ σ χ-χ δ Θα(1+σ) [1-α(1+Θ)](χ-χ) βα 1-θ (1+Θ)α(1+σ) [1-α(1+Θ)](χ-χ) (19)

Computation of the linearized dynamical system

Let us introduce the following elasticities:

cc = -U 1 (c,l) U 11 (c,l)c = 1 σ , ll = -U 2 (c,l) U 22 (c,l)l = 1 χ (20)
Consider the first-order condition (9) together with the expression of the wage rate as given by (3). We easily get

l t = (1 -α) 1 α+χ k α α+χ t c -σ α+χ t ≡ l(k t , c t )
Substituting this function into the expressions of the prices (3) allows to get r t = r(k t , c t ), ω t = ω(k t , c t ) and p t = p(k t , c t ). Consider now the first-order condition [START_REF] Garnier | Intertemporal Substitution, Labor Supply Elasticity and Sunspot Fluctuations in Continuous-Time Models[END_REF] with the capital accumulation equation. We have the following two equations

c -σ t+1 [r(k t+1 , c t+1 ) + (1 -δ)p(k t+1 , c t+1 )] = βp(k t , c t )c -σ t k t+1 = (1 -δ)k t + r(kt,ct)kt+ω(kt,ct)l(kt,ct)-ct p(kt,ct)
Using ( 17), total differentiation of these equations in a neighborhood of the steady state gives after simplifications the following linear system

A   c t+1 k t+1   = B   c t k t   with A =    σ -σ(1-α)(1-θ) α+χ + Θθ 1 - σ(1-α)(1-θ)(1-βδα 1-θ ) (α+χ)βδα χ(1-α)(1-θ) α+χ + Θ(1-δ)(1-θ) 1- χ(1-α)(1-θ) α+χ δα 0 1    and B =     σ + Θ 1 - σ(1-α)(1-θ)(1-βδα 1-θ ) (α+χ)βδα Θ(1-θ) 1- χ(1-α)(1-θ) α+χ βδα -1-θ β(α+χ) σ(1-α) α + 1 -βδα 1-θ 1 β 1 + (1-α)(1-θ) α+χ     It follows that the linearized system is   c t+1 k t+1   = J   c t k t   with J = A -1 B.
Denoting by T and D the Trace and Determinant of J, tedious but straightforward computations allow therefore to compute the following characteristic polynomial P σ (λ) = λ 2 -T λ + D = 0 with

D = D χ (σ) = σ -1-δ δ Θ+(1+Θ)α+ (χ+α)α (1-α) 1+Θ(1-θ) 1-θ -(1-βαδ 1-θ ) (1-δ)Θ(χ+α) δ(1-α) σ -θΘ δ +βα+ (χ+α)βα (1-θ)(1-α) -(1-βαδ 1-θ ) θΘ(χ+α) δ(1-α) T = T χ (σ) = 1 + D χ (σ) + (1-θ)(1-βαδ 1-θ ) σ+ χ(1-α(1+Θ))-Θα 1-α σ -θΘ δ +βα+ (χ+α)βα (1-θ)(1-α) -(1-βαδ 1-θ ) θΘ(χ+α) δ(1-α)
We derive from this that when σ is varied over the interval [0, +∞), D and T are linked through a linear relationship D = ∆ χ (T ) = T S χ + C with a slope S χ = (∂D χ (σ)/∂σ)/(∂T χ (σ)/∂σ). Let us introduce the following notation:

N = (1-βαδ 1-θ ) σ -θΘ δ +βα+ (χ+α)βα (1-θ)(1-α) -(1-βαδ 1-θ ) θΘ(χ+α) δ(1-α) 2
We easily derive that

∂Dχ(σ) ∂σ = -N θΘ 2 (χ+α)α(1+χ) δ(1-α) 2 < 0 ∂Tχ(σ) ∂σ = -N βα Θ(χ+α)(1+χ) (1-α)(1-δ)(1-θ)(1+Θ) χ+α +(1-δ)(1+Θ)-1 +χδ(1-α)(1+Θ)(1+χ-θ(1-α)) δ(1-α) 2
and thus

S χ = (1-δ)Θ 2 (χ+α)(1+χ) Θ(1+χ) (1-α)(1-δ)(1-θ)(1+Θ)+(χ+α)[(1-δ)(1+Θ)-1] +χδ(1-α)(1+Θ)(1+χ-θ(1-α))
Under Assumption 1, we conclude that if Θ > Θ, then ∂T χ (σ)/∂σ < 0 and thus S χ > 0. We also derive that S χ < 1 if and only if

χ 2 δ (Θα -(1 -α)) -χ Θ 2 (1 -α)(1 -δ)(1 -θ) + Θ (1 -α)(1 -θ) -δ[1 + α(1 -θ(1 -α))] + δ(1 -α)(1 -θ(1 -α)) -Θ (1 -α)(1 + Θ)(1 -θ)(1 -δ) -δα ≡ g(χ) < 0 (21)
Under Assumption 1, there exists δ ≤ δ such that g(χ) < 0 for any χ ≥ 0 if δ < δ.

In other words, ∆ χ (T ) corresponds to a half-line in the (T , D) plane with the starting point (T χ (+∞), D χ (+∞)) obtained when σ = +∞ such that

D χ (+∞) = - (1-δ)Θ δ +(1+Θ)α+ (χ+α)α (1-α) 1+Θ(1-θ) 1-θ -θΘ δ +βα+ βα(χ+α) (1-θ)(1-α) T χ (+∞) = 1 + D χ (+∞) + (1-θ)(1-βαδ 1-θ ) -θΘ δ +βα+ βα(χ+α) (1-θ)(1-α)
and the end-point (T χ (0), D χ (0)) obtained when σ = 0 such that:

D χ (0) = 1 β , T χ (0) = 1 + D χ (0) -δ(1 -θ) χ(1-α(1+Θ))-αΘ θΘ(χ+α)
For the starting point, when σ = +∞, i.e. cc = 0, we get:

P +∞ (1) = - (1-θ)(1-βαδ 1-θ ) -θΘ δ +βα+ βα(χ+α) (1-θ)(1-α) P +∞ (-1) = 2 - (1-δ)(1+β)Θ δ +(1+β+Θ)α+ (χ+α)α (1-α) 1+β+Θ(1-θ) 1-θ +(1-θ)(1-βαδ 1-θ ) -θΘ δ +βα+ βα(χ+α) (1-θ)(1-α)
For the end point, when σ = 0, i.e. cc = +∞, we get:

P 0 (1) = (1-θ)δ[χ(1-α(1+Θ))-αΘ] (χ+α)θΘ P 0 (-1) = χ{Θ[2(1-θ(1-δ))+δ(1-θ)α]-δ(1-θ)(1-α)}+α(1+θ)(2-δ)Θ (χ+α)θΘ
It follows immediately that P 0 (1) > 0 if and only if χ > αΘ/[1 -α(1 + Θ)] ≡ χ, while it can be easily shown that P 0 (-1) > 0 if Θ > Θ.

Proof of Proposition 2

We assume first that χ ∈ [0, χ). We know that the end point satisfies D χ (0) = 1/β > 1, P 0 (1) < 0 and P 0 (-1) > 0. The starting point can be written:

D χ (+∞) = δα(χ 1 -χ)(Θ-Θ 1 ) β(1-α)(1-δ)(Θ-Θχ) P +∞ (1) = δ(1-θ)(1-βαδ 1-θ ) β(1-δ)(Θ-Θχ) P +∞ (-1) = 2δα( χ-χ)(Θ-Θχ) β(1-α)(1-δ)(Θ-Θχ) with Θχ = δα[1-θ(1-α)+χ] (1-α)(1-θ)(1-δ) , Θχ = α(1+β)[1-θ(1-α)+χ] 1-θ + (1-α)(1-θ) 2 (1-βαδ 1-θ ) α( χ-χ) , Θ 1 = 1-θ(1-α)+χ (1-θ)(χ 1 -χ) and χ 1 = 1-α-δ δα , χ = (1-δ)(1+β)(1-α)-δα δα
Under Assumption 1 we get χ < 2/3 < χ 1 < χ and Θ < Θχ < Θχ < Θ 1 < Θ for any χ ∈ [0, 2/3). Let us denote by σH the value of σ that solves D χ (σ) = 1. a) Assume first that Θ < Θ < Θχ . We get D χ (+∞) > 1/β = D χ (0), P +∞ (1) < 0 and P +∞ (-1) > 0. Therefore, assuming β ∈ ( β, 1) and δ ∈ (δ 0 , δ), provided T χ (σ H ) ∈ (-2, 2), we conclude that the ∆ χ line is located as in Figure 1(a). b) Assume then that Θχ < Θ < Θ 1 . We get D χ (+∞) < 0, P +∞ (1) > 0 and P +∞ (-1) < 0 if Θ ∈ ( Θχ , Θχ ) while P +∞ (-1) > 0 if Θ ∈ ( Θχ , Θ 1 ). Provided T χ (σ H ) ∈ (-2, 2), we conclude that when Θ ∈ ( Θχ , Θχ ), the ∆ χ line is located as in case (a) of the following Figure: and when Θ ∈ ( Θχ , Θ 1 ), the ∆ χ line is located as in Figure 2(a). It is worth noting that the local stability properties of the steady state in Figure 4(a) are exactly the same as those implied by Figure 1(a). c) Assume finally that Θ ∈ (Θ 1 , Θ). We get under Assumption 1 D χ (+∞) ∈ (0, 1) and P +∞ (-1) > 0. Therefore, provided T χ (σ H ) ∈ (-2, 2), the ∆ χ line is again located as in Figure 2(a).

We need now to prove that T χ (σ

H ) ∈ (-2, 2). Solving D χ (σ) = 1 gives σH = (1-βδα 1-θ )Θ(χ+α)(1-δ)(1-β) -Θ[(1-α)(1-θ)-δ-χδα]+ δα(1-β)[1-θ(1-α)+χ] 1-θ (22)
Under Assumption 1, there exists δ 0 ∈ (0, 1) as given by

δ 0 = (1 -β) -{Θ[1+χα-2β(1-α)]+α[1+χ-β(1-α)]}+ √ Λ 2β{Θ[1+χα-β(1-α)]+α(1-α)(1-β)} (23) with Λ = {Θ [1 + χα -2β(1 -α)] + α [1 + χ -β(1 -α)]} 2 + 4Θβ(1 -α) {Θ [1 + χα -β(1 -α)] + α(1 -α)(1 -β)} such that σH > 0 if and only if δ ∈ (δ 0 , 1
). Moreover, there exists β ∈ [ β, 1) such that δ 0 < δ ≤ δ when β ∈ ( β, 1). Assuming β ∈ ( β, 1) and δ ∈ (δ 0 , δ), we then derive

T χ (σ H ) = 2 - (1-θ)(1-βαδ 1-θ ) σH + [1-α(1+Θ)](χ-χ) 1-α σH θ(Θ-Θ) δ - χβα (1-θ)(1-α) +(1-βαδ 1-θ ) θΘ(χ+α) δ(1-α)
Under Assumption 1, the denominator of the ratio in T χ (σ H ) is positive for all χ ≥ 0.

Let β = β and β ∈ (β, 1). As χ ∈ [0, χ), we need to study the numerator of the ratio in T χ (σ H ). Note that σH +

[1-α(1+Θ)](χ-χ) 1-α > σH -αΘ 1-α = (1-βδα 1-θ )Θ(χ+α)(1-δ)(1-β)(1-α)-αΘdenσ H (1-α)denσ H
with denσ H the denominator of σ H . We have shown previously that denσ H > 0 if and only if δ ∈ (δ 0 , 1), with lim δ→δ 0 denσ H = 0. Moreover, we derive from the expression of σH that lim δ→1 σH = 0. Therefore, we conclude that there exists δ 1 ∈ (δ 0 , δ] such that the numerator of the ratio in T χ (σ H ) is positive for all χ ≥ 0 if δ ∈ (δ 0 , δ 1 ). It follows that T χ (σ H ) < 2 for all χ ≥ 0 if δ ∈ (δ 0 , δ 1 ). Let us then denote δ = δ 1 . As in case i), since lim δ→0 Θ = 0, we get lim δ→0 S = 0 and thus lim δ→0 T χ (σ H ) = -∞. Therefore, there exists δ ∈ [δ 0 , δ) such that when δ ∈ (δ, δ), T χ (σ H ) ∈ (-2, 2).

Finally, we may compute the bifurcation values of σ. The Hopf bifurcation value σH is given by [START_REF] Gruber | A Tax-Based Estimate of the Elasticity of Intertemporal Substitution[END_REF], while the flip bifurcation value σF is such that P σ (-1) = 1 + T χ (σ) + D χ (σ) = 0, which leads to the expression given by [START_REF] Hansen | Indivisible Labor and the Business Cycle[END_REF]. The transcritical bifurcation value σT is such that P σ (-1) = 1 -T χ (σ) + D χ (σ) = 0, leading to the expression given by [START_REF] Harrison | Indeterminacy in a Model with Sector Specific Externalities[END_REF].

We assume now that χ ∈ (χ, 2/3). The same proof as in the case χ ∈ [0, χ) applies except that now P 0 (1) > 0. Provided T χ (σ H ) ∈ (-2, 2), it follows that depending on the value of Θ, the ∆ χ line is again located as in The last step consists finally in showing that T χ (σ H ) ∈ (-2, 2). Let us note first that T χ (σ H ) can be written as follows: As we have shown previously, σH > 0 if and only if δ ∈ (δ 0 , 1) with δ 0 as given by [START_REF] Rogerson | Micro and Macro Elasticities in a Life Cycle Model with Taxes[END_REF]. Moreover, there exists β ∈ [ β, 1) such that δ 0 < δ ≤ δ when β ∈ ( β, 1). From now on, let β ∈ ( β, 1) and δ ∈ (δ 0 , δ). Under these restrictions, when Θ ∈ (Θ, Θχ ), Assumption 1 ensures that the denominator of the ratio in T χ (σ H ) is positive. Moreover, when χ ∈ (χ, 2/3), the numerator of the ratio in T χ (σ H ) is positive. It follows that T χ (σ H ) < 2. Let us then denote δ = δ and β = β. Note that, as lim δ→0 Θχ = 0, we get lim δ→0 S = 0 and thus lim δ→0 T χ (σ H ) = -∞. Therefore, there exists δ ∈ [δ 0 , δ) such that when β ∈ (β, 1) and δ ∈ (δ, δ), T χ (σ H ) ∈ (-2, 2).

T χ (σ H ) = 2 -
Assume now that Θ > Θχ . The denominator of the ratio in T χ (σ H ) is positive and thus T χ (σ H ) < 2 for all χ ∈ (χ, 2/3). We have shown previously that denσ H > 0 if and only if δ ∈ (δ 0 , 1), with lim δ→δ 0 denσ H = 0. Moreover, we derive from the expression of σH that lim δ→1 σH = 0. Therefore, we conclude that there exists δ 1 ∈ (δ 0 , δ] such that the numerator of the ratio in T 0 (σ H ) is positive and T 0 (σ H ) < 2 if δ ∈ (δ 0 , δ 1 ). Let us then denote δ = δ 1 and consider the case δ = δ 0 . As denσ H = 0 we derive Θ = δ 0 α(1-β)[1-θ(1-α)+χ] [(1-α)(1-θ)-δ 0 (1+χα)](1-θ) ≡ Θ δ 0 and σH = ∞ with Θ δ 0 > Θχ for any χ ∈ (χ, 2/3). It follows therefore that

T χ (σ H ) δ=δ 0 = 2 - (1-θ) 1- βαδ 0 1-θ Θ δ 0 -Θχ Straightforward computations show that T χ (σ H )| δ=δ 0 > -2 is equivalent to 4(δ 0 α) 2 [1 -θ(1 -α) + χ] (1 + χ) > [1 -θ -βδ 0 α](1 -θ)(1 -α)(1 -δ 0 ) × [(1 -α)(1 -θ) -δ 0 (1 + χα)]
As this equality is satisfied when β = 1, there exists β ∈ [ β, 1) such that T χ (σ H )| δ=δ 0 > -2 for any β ∈ (β, 1). Denoting δ = δ 0 we conclude that when δ ∈ (δ, δ) and β ∈ (β, 1), T χ (σ H ) ∈ (-2, 2).

Proof of Proposition 3

From the proof of Proposition 2, let Θ = Θ0 and assume that Θ ∈ (Θ, Θ). It follows easily that D χ (+∞) > 1/β = D χ (0), P +∞ (1) < 0 and P +∞ (-1) > 0 for any χ ≥ 0. Therefore, assuming β ∈ ( β, 1) and δ ∈ (δ 0 , δ), provided T χ (σ H ) ∈ (-2, 2), we conclude that i) when χ > χ, the ∆ χ line is located as in Figure 4(a), and ii) when χ < χ, the ∆ χ line is located as in Figure 4(b). Depending on whether χ is larger or lower than χ, the arguments to prove that T χ (σ H ) ∈ (-2, 2) are the same as those presented in the proof of Proposition 2.

Figure 1 :

 1 Figure 1: Local indeterminacy with Θ ∈ (Θ, Θχ ).

Figure 2 :

 2 Figure 2: Local indeterminacy with Θ ∈ ( Θχ , Θ).

12

 12

Figure 3 :

 3 Figure 3: Indeterminacy areas in the (σ, Θ) plane

Figure 4 :

 4 Figure 4: Local indeterminacy with Θ ∈ (Θ, Θχ ).

  Figures 1(b), 4(b) or 2(b).

( 1 -

 1 θ)(1-βαδ 1-θ ) σH + [1-α(1+Θ)](χ-χ)

For a thourough analysis of two-sector model with GHH preferences -with no-income effects on labor supply -see Dufourt et al.[START_REF] Dufourt | Indeterminacy and sunspots in two-sector RBC models with generalized no-income effect preferences[END_REF].

As is well-known, externalities in the consumption sector tend to increase the aggregate degree of IRS required for indeterminacy Moreover, a constant returns to scale technology in the consumption sector and an increasing returns to scale technology in the investment sector are consistent with the empirical findings of Basu and Fernald[START_REF] Basu | Returns to Scale in US Production: Estimates and Implications[END_REF] and Harrison[START_REF] Harrison | Returns to scale and Externalities in the Consumption and Investment Sectors[END_REF].

Note that under Assumption 1, Θ < (1 -α)/α.

For example, Basu and Fernald[START_REF] Basu | Returns to Scale in US Production: Estimates and Implications[END_REF] obtain a point estimates for the degree of IRS in the durable manufacturing industry in the US economy of 0.33, with standard deviation 0.11.

See Prescott and Wallenius[START_REF] Prescott | Federal Reserve Bank of Minneapolis[END_REF] for a discussion of the factors that make the wage elasticity of aggregate labor supply significantly different from the corresponding elasticity at the micro level.

The slope of the line is also affected by a change in the values of these parameters. Yet, these changes are sufficiently contained under our parameter restrictions that they do not lead to additional conceptual configurations.

Note that a transcritical bifurcation is usually associated with two-steady-states. However, as proved by Proposition 1, the steady state is unique. It follows that this transcritical bifurcation is degenerate and only associated with a loss of stability of the unique steady state.

In Benhabib and Farmer [3] the existence of the Hopf bifurcation is mentioned but not proved, while Harrison[START_REF] Harrison | Indeterminacy in a Model with Sector Specific Externalities[END_REF] focuses exclusively on the flip bifurcation through a numerical analysis.

See in particular Benhabib and Farmer[START_REF] Benhabib | Indeterminacy and Sector Specific Externalities[END_REF] and Harrison[START_REF] Harrison | Indeterminacy in a Model with Sector Specific Externalities[END_REF].


High Frequency Surface Wave Radar in the French Mediterranean Sea: an element of the Mediterranean Ocean Observing System for the Environment

Keywords: High Frequency Surface Wave Radar, Coastal Observing System, Northern Current, North Western Mediterranean Sea

A Mediterranean Ocean Observing System for the Environment (MOOSE) has been set up as an interactive, distributed, and integrated observatory system of the North West Mediterranean Sea in order to detect and identify long-term environmental anomalies. In this framework, the Mediterranean Institute of Oceanography (MIO) operates two couple of High Frequency Surface Wave Radars (HFSWR). HFSWRs provide synoptic observations of sea surface currents with high temporal (0.25-1 h) and spatial (1-3 km) resolution and long range (30-100 km). They can therefore be used to study the variability of the current as well as eddy dynamics. One site, based on the WERA (Helzel Messtechnik) technology and installed near Toulon, is operational and monitored in real-time since more than 3 years. It is composed of two WERA systems featuring respectively a non-linear receiving array and a full bistatic configuration, both working in Direction Finding mode. The second site, intended to extend further East the coverage of the Observatory, was set up in 2014 near Nice with two SeaSonde (Codar) systems. We present here an overview of the HFSWR network, the surface current mapping facility offered by the system, and recent observational results and applications.

INTRODUCTION

The Mediterranean Ocean Observing System for the Environment (MOOSE) focuses on documenting the long-term evolution of the North West (NW) Mediterranean Sea in the context of climate change and anthropogenic pressure. Furthermore, it targets building efficient indicators of the health of the basin. Finally, MOOSE will provide a large flux of realtime data to facilitate validation of operational oceanographic models.

The MOOSE concept is based on a multisite system of continental shelf and deep-sea fixed stations as well as on an autonomous mobile platform network devoted to the observation of the space-time variability of processes interacting between the coastal/open ocean and the ocean/atmosphere. It supplies and maintains long-term time series, thus allowing to evidence climatic trends. These data sets are included in the MISTRALS project and shall use the MyOcean data distribution infrastructure through Coriolis.

The predominant and most energetic circulation feature of the NW Mediterranean Sea is the Northern Current, a border current flowing cyclonically above the slope roughly over the 1000-2000 m isobaths. Its activity is highly variable. It is narrower, deeper and more intense during wintertime [START_REF] Albérola | On the seasonal and mesoscale variabilities of the northern current during the PRIMO-0 experiment in the western mediterranean sea[END_REF] with an intense mesoscale activity generating meanders, filaments, and eddies. Recent oceanographic studies in the Gulf of Lions (NW Mediterranean) have indeed shown the existence of vortex structures and instabilities in a wide range of scales. Namely, they have been well evidenced in the context of the ECCOLO campaign [START_REF] Schaeffer | Generation mechanism of mesoscale eddy in the Gulf of Lions: radar observation and modelling[END_REF], where a certain number of in situ and remote sensors have been conjointly used, such as SST and water color satellite imagery, satellite-tracked drifting buoys and HF radar.

Today, High Frequency Surface Wave Radar (HFSWR) is indeed routinely used for the remote sensing of the ocean surface. Current measurements are obtained from the Doppler shift of the first-order Bragg-resonant echoes [START_REF] Crombie | Doppler spectrum of sea echo at 13.56 Mc./s[END_REF]. The Mediterranean Institute of Oceanography (MIO) has proven its expertise in radar measurement of surface currents [START_REF] Broche | VHF radar for ocean surface current and sea state remote sensing[END_REF][START_REF] Molcard | Comparison between VHF radar observations and data from drifter clusters in the Gulf of La Spezia (Mediterranean Sea)[END_REF] and has been assigned the task to deploy the HFSWR network for the MOOSE program.

DESCRIPTION OF THE SITES

The sites chosen for the HF radar network have been identified in order to cover the position of two instrumented lines (Fig. 1) with long time series of ocean parameters: ANTARES (Astronomy with a Neutrino Telescope and Abyss environmental RESearch), and DYFAMED (DYnamics of Atmospheric Fluxes in the MEDiterranean sea). The main difficulties for deploying HFSWR lie in hardware and maintenance costs, frequency licenses, and site finding, especially when the coast is irregularly shaped, steep, or close to touristic and crowded regions. [START_REF] Millot | Circulation in the Mediterranean Sea[END_REF] (cartographer GRID-Arendal) with the two site locations of the HFSWR measurements around the areas of interest: 1-Antares (near Toulon), 2-Dyfamed (near Nice).

First area of interest: ANTARES (Toulon)

The first target area off the coast of Toulon, related to the ANTARES instrumented line, is a key zone conditioning the behaviour of the North Current just upstream of the Gulf of Lions. It displays significant cross-shelf exchanges correlated to the strong northwesterlies present in the region (Mistral, Tramontane). This fully operational site is composed of two Wellen Radar (WERA [START_REF] Gurgel | Wellen Radar (WERA), a new ground-wave based HF radar for ocean remote sensing[END_REF]) systems manufactured by Helzel Messtechnik GmbH providing real-time data. The first radar system has been installed at the Cap Sicié in 2010 and works in quasi-monostatic configuration with a non-linear, W-shaped, 8-antenna receiving array and a single emitting antenna. Such irregular configuration of the array was the only solution to cope with the insufficient space available. This site has been complemented in May 2012 with a fully bistatic second system, a pioneering configuration for WERA at the time of the setup. The receiver, a regular linear 8-antenna array, is located at Cap Bénat while the transmitter, GPS-synchronized, is installed in the Porquerolles Island, 17 km away from the receiver, in order to circumvent the presence of several large islands. The bistatic configuration has required some dedicated and original hardware and software processing. It also allowed us to experimentally study the effects of bistatism on the HF Doppler spectra, namely evidencing good potential for the purpose of wave spectrum inversion [START_REF] Grosdidier | Simulation of HF bistatic ocean radar system in experimental conditions[END_REF].

Our systems are continuously working in the frequency band of 16.1 to 16.2 MHz allocated by the ITU (International Telecommunication Union) to the oceanographic radar operators. They sweep over a 50 kHz bandwidth, i.e. half of the allocated bandwidth, resulting in a 3 km range resolution. A 2 deg azimuthal resolution is achieved through a Direction Finding method based on Music [START_REF] Lipa | SeaSonde Radial Velocities: Derivation and Internal Consistency[END_REF][START_REF] Molcard | Comparison between VHF radar observations and data from drifter clusters in the Gulf of La Spezia (Mediterranean Sea)[END_REF]. The integration time can vary from 20 minutes to 1 hour. The radial velocities maps are transmitted every 20 minutes. Cartesian velocities are then reconstructed on a regular 2 x 2 km 2 grid (Fig. 2) and displayed in near real-time on a dedicated website: http://hfradar.univ-tln.fr.

Second area of interest: DYFAMED (Nice)

The installation of the second HFSWR site related to the DYFAMED buoy (Fig. 3) in the Ligurian Sea area,, extended the observation zone to the full coastal area between Toulon and Nice, allowing a much larger coverage of the Northern Current.

The selected equipment is a pair of more compact HFSWR system, namely 2 SeaSondes from CODAR Ocean Sensors [START_REF] Barrick | CODAR: A coastal HF radar for real-time current mapping[END_REF] Our pair of SeaSondes works in the 13.5 MHz frequency band allocated by the ITU with a 50 kHz bandwidth. The parameters are similar to those of the WERA systems, except for the azimuthal resolution set at 5 deg.

The ever-increasing deployment of HFSWRs has led the international community to build a frequency sharing plan. Although our radars follow the ITU's recommendations and operate over the frequency bands specified for oceanographic purposes, strong Radio Frequency Interference (RFI) is often present, due to official and non-official radio services.

The result is a radical reduction in coverage and the presence of outliers in the velocity maps.

DATA FLOW AND QUALITY CONTROLS

The first product of the measurement made by a HFSWR is the radial velocity. Total vector is a combination of less two HFSWR measurements.

The quality insurance of the total vector map is depending of all the steps of the processing chain, so the quality control must be hierarchized as following with in first the control of the instrument and in second the control of the measurement on each component related to each site.

Level 0 -Availability of the instrument

Our first indicator is related to the first level of the data acquisition chain and determines if the data is available or not. A first diagnostic is done on the HFSWR system: controlling if emitter and receiver are properly functioning (and well synchronized if necessary using GPS timing service), and if the contamination by radio interference is not prejudicial to the measurement. Theses diagnostics alert the operator of troubleshooting on the hardware system.

The coverage of the radial velocity measurement is then analysed. If the coverage is not sufficient the radial file is rejected. The number of radial files per ten days is a good indicator (Fig. 4) on the availability of the data.

Level 1 -Control of the instrument

Best practices for the operation and maintenance of HFSWRs prescribe regular measurement of directional antenna pattern of the receiver as a necessary component of the quality assurance and quality control [START_REF] Cook | Deployment and maintenance of a high-frequency radar for ocean surface current mapping: Best practices[END_REF]. An idealised antenna pattern can be used for a first guess, but it doesn't take into account the environment of the site. We perform at least every year an Antenna Pattern Measurement (APM) on each HFSWR site. The comparison between different measurements ensured the stability of the antenna system and characterised the performance of the system.

Level 2 -Measurement performance

On ANTARES area, different campaigns (TOSCA, SUBCORAD and BOMBYX) allow us to validate the HFSWR radial velocities measurements by comparison with in situ measurements as lagrangian drifters and ADCP (tracked and moored).

The comparisons give good agreement in the limits of the intrinsic radar accuracy, in the order of 5 cm.s -1 [START_REF] Fraunié | Experimental investigation of the relationship between HF radar measurements of currents and the dynamical properties of the upper ocean[END_REF].

APPLICATIONS

The applications of our HFSWR measurements have spanned the fields of ocean circulation to operational oceanography to methodological development.

In the first step of the ANTARES implementation (2010-2011), when only one radar site was operational (Cap Sicié), [START_REF] Marmain | Derivation of ocean surface current properties from a single-site HF/VHF radar[END_REF] developed and successfully tested the Vortex Identification Method (VIM) to reveal the presence of eddy-like structures through radial current velocity maps. During the PHYOCE experiment (March, 31 -April, 3 2011) the variability of the current flow was deduced from the radial velocity field and was used in complement to in-situ measurements to confirm the existence of a mesoscale meander. This was also visible in the numerical model of circulation GLAZUR-64 developed at MIO [START_REF] Guihou | A case study of the meso-scale dynamics in the North-Western Mediterranean Sea: a combined data-model approach, Ocean Dynamics[END_REF]. [START_REF] Berta | Estimating Lagrangian transport blending drifters with HF radar data and models: results from the TOSCA experiment in the Ligurian Current (North Western Mediterranean Sea)[END_REF].

Full
Finally, hydrological data have complemented HFSWR current velocity maps in order to study the Northern Current variability under wind forcing [START_REF] Bellomo | Observational evidence of mesoscale variability of the Northern Current (North-Western Mediterranean Sea): a combined study via gliders, HF RADAR, surface drifters, and vessel data[END_REF].

The potential of this real-time observation lies also in the possibility of assimilating the surface current velocities in numerical oceanic models. As an example, [START_REF] Marmain | Assimilation of HF radar surface currents to optimize forcing in the North Western Mediterranean Sea: a combined data-model approach[END_REF] successfully used the HFSWR data to correct hydrographic boundary conditions as well as the wind forcing.

Finally, two projects are currently focusing on the physical understanding of the HFSWR current measurement. Namely, SUBCORAD is investigating the impact of sub-grid variability, whereas within STRING surface drifters designed to sense the vertical shear of the current as the HFSWR does have been built and experimentally tested.

CONCLUSIONS

An HFSWR network has being deployed by the MIO along the Mediterranean coast of France, including both WERA and SeaSonde systems. For the former ones, some innovative technical solutions have allowed to solve limitations due to the geographical configuration of the sites, including the successful use of a non-linear array of antennas and the exploitation of a full bistatic configuration. Furthermore, a Direction Finding algorithm based on MUSIC has for the first time been used with a WERA system, and adapted to the bistatic case.

Antenna Pattern Measurements are routinely performed to take into account the specificity of the environment where the antennas are placed as well as their ageing. Validation experiments have been performed, demonstrating the accuracy of the HFSWR-derived surface current velocity maps as well as their importance for operational oceanography as well as circulation studies.

With a dataset of more than four years of radial velocity maps and two years of total ones for the first site (ANTARES), this work is the start of a long-term observational effort with HFSWR aimed at providing continuous surface current data in the context of the MOOSE program.

Fig. 1 :

 1 Fig.1: Surface circulation scheme in the Western Basin of the Mediterranean Sea[START_REF] Millot | Circulation in the Mediterranean Sea[END_REF] (cartographer GRID-Arendal) with the two site locations of the HFSWR measurements around the areas of interest: 1-Antares (near Toulon), 2-Dyfamed (near Nice).

Fig. 2 :

 2 Fig. 2: ANTARES site. An example of Cartesian velocity map reconstructed on a grid 2x2 km 2 from the two radial velocity maps measured at the Cap Sicié and Cap Bénat sites using a Direction Finding method based on MUSIC.

  . The two locations are the lighthouse of Cap Ferrat in Saint-Jean Cap Ferrat (set up on October 2013) and the semaphore of Cap Dramont in Saint-Raphaël (set up on May 2014), resulting in a 50 km baseline. The latter was struck by a lightning in June 2014 and is currently off service.

Fig. 3 :

 3 Fig. 3: DYFAMED site. Simulation of the coverage of each HFSWR station (Cap Dramont in red points, and Cap Ferrat in blue points) and coverage of the Cartesian velocity map (pink area).

Fig. 4 :

 4 Fig. 4: ANTARES: availability of the data given by the number of radial files per ten days on each couple of receiver/transmitter according to the different mono and bi-static configurations. The main ones are in blue PEY-collocated receiver and transmitter, and POB-Porquerolles transmitter/Cap Bénat receiver.

Acknowledgements

The National Institute for Earth Science and Astronomy (INSU) of CNRS supports MOOSE with the Alliance ALLENVI on behalf of the French research organizations on the environment. The HF radar deployment near Toulon was initially funded with national programs such as LEFE IDAO/ECCOP and by the EU MED Program TOSCA. We are grateful to P. Guterman and K. Bernadet from the DT-INSU (CNRS) who supervised the deployment of the Seasonde in Saint-Jean Cap Ferrat and helped to develop the data transfer by GSM. We thank the cartographer GRID-Arendal for his illustration scheme on the circulation in Mediterranean sea (http://www.grida.no/graphicslib/ detail/surfacecirculation-in-the-mediterranean-sea_052b).


Raphaël Brière

Adélaïde Feraille

Yannick Tardivel

Olivier Baverel

ENVIRONMENTAL APPROACH OF THE END-OF-LIFE OF BUILDINGS

Keywords: LCA, end-of-life of buildings, demolition

Due to a worldwide increasing population, the construction sector is key in the international economy. This activeness implies consequences from an environmental point of view. Numerous life cycle assessments (LCA) studies have been made on buildings and pointed out the importance of the impacts caused by the use phase. The development of low energy buildings has been a step forward to decrease these impacts. The environmental burdens associated with the end of life of buildings are low so the interest showed to this particular phase is because it allows closing the loop in the material and good life cycles. Indeed, reclaimed and reused components and recycled materials can bring environmental benefits. To assess these potential benefits, it is essential to know the current practices in terms of demolition and the parameters which have an influence on the final LCA result.

Based on data of a recent demolition site, a partial LCA has been realised that underlines two points. Nowadays, mechanical demolition is commonly used implying fuel consumption then waste management induces waste to be transported to recycling facilities, wastetransfer stations or landfills. The remaining shadow stays on the future of the different materials sorted once they left the station. On this case study chosen, fuel burnt in the machines has the highest contribution to the final impacts. However, this result is not taken as a conclusion but as step to a more complete LCA. The final objective is to assess the consequences in terms of environmental impacts of a change of practices in the demolition process.

INTRODUCTION

In 2010, the French construction industry has produced 360 millions of tons of waste representing more than 40% of the total waste generated on the territory. Building construction sites which account for 50 millions tons of waste can be divided in three categories: renovation, construction and demolition sites. The study focuses on the latter one which generates about 65% of the waste [1]. Those waste are mostly composed of inert (mainly concrete and bricks) and non dangerous ones like untreated wood, plastics and metals [START_REF]Déconstuire les bâtiments[END_REF]. The environmental impacts of this end of life stage are relatively small even negative sometimes compared to the material production stage or the use phase [START_REF] Andrea | Life cycle of buildings, demolition and recycling potential: a case study in Turin, Italy[END_REF]. However, demolition is a key stage as it permits to reduce the three main impacts that can be attributed to the building sector: depletion of non-renewable natural resources (minerals or fuels), air pollution from manufacturing processes and road transport, and decrease of the pressure on landfill sites [START_REF] Addis | Reclaimed components and materials[END_REF]. To complete these goals, it seems crucial to adopt the best possible strategy in order to reuse, recycle and increase the valorisation of the different goods and components of the buildings.

Life cycle assessment (LCA) is an environmental tool to assess the impact potentials of a product or a system from the material production to the end of life. Two main approaches can be distinguished: attributionnal and consequential. Attributionnal LCA (ALCA) takes into account the impacts of all the flows inside the studied system whereas consequential LCA (CLCA) focuses on the environmental consequences due to a change in the system. Thus, only processes that are modified by this change are modelled in CLCA.

This LCA tool will be applied to the end of life stage of the buildings. The objective of this work is to realize an ALCA to better understand this system and lay the foundations of a further CLCA which goal will be to evaluate the impact potentials due to a change of practices in the demolition of buildings.

METHODOLOGY

ALCA

The ALCA tool is standardised by norm ISO 14 040 [START_REF]Environmental management -LCA framework and principles[END_REF] and is divided in 4 phases:

 Goal and scope: in this key step, the objectives and the system studied have to be clearly defined. This is done by detailing the functional unit (reference which explicits and quantifies the service delivered by a product system), the boundaries, the assumptions made and their justification…  Life cycle inventory (LCI): that second step corresponds to the data collect that is to say creating an inventory of flows (inputs and outputs) including raw materials, energy, emissions to air and water...  Life cycle impact assessment: flows collected in the previous phase are transformed into potential environmental impacts and represented through a set of indicators.  Interpretation: based on the different indicators, the results are analyzed to draw conclusions and recommendations in accordance with the objectives defined in the first step.

Demolition steps

A demolition process can be separated in 6 main steps:  Preparation of the site: organization of the site, temporary installations (fence), pollution risk assessment…At the same time, a site analysis can be done to discover hidden and potentially dangerous materials and to quantify the materials and components to deal with.  Decontaminationasbestos removal: as asbestos is considered as a dangerous waste, its removal must be dealt with great care and special equipment  Removal of the light-work: interior and external works, mechanical and electrical equipment are removed  Cutting of the structure: depending on the location, the structure, or the knowledge of the company a demolition technique is chosen.  Waste management: waste from the removal of the light-work or of the cutting has to be taken care of. They can be sent to a landfill site, to a recycling facility, potentially reused in another place or incinerated.  Reconditioning of the site: the foundations can be embanked, the temporary installations are uninstalled.

Figure 1: Main steps of the end of life building management [START_REF]Déconstuire les bâtiments[END_REF] Figure 1 shows two aspects that are needed to correctly model the end of life stage of a building. The first one is energy. Indeed, diesel is burnt in machines to complete the demolition so it is essential to have an idea of how much fuel has been used during the all process. The second aspect deals with transport: it is important to know where and how far waste goes.

RESULTS AND DISCUSSIONS

Thanks to the EPFIF (Etablissement Public Foncier d'Ile de France) which takes care of demolition in Paris and its surroundings, we had access to data about some of their construction site. One of those took place in 2012 in the city of Brou-sur-Chantereine (77) and information about the consumption of the machines and the destination of the different waste flows were available and are summed up in table 1: Figure 2: environmental impacts of the Brou demolition site

In this particular example, the energy used during the process has the highest contribution to environmental impacts in every indicator of the chosen method. This simple model underlines two recommendations to decrease the impacts of this phase: optimizing the transport distance of the reclaimed components and waste flows and to reduce the quantity of fuel used during the demolition.

One parameter has been offset of this modelling because of a lack of information: the possible reuse, recycling or incineration of some components or materials. For instance, in a LCA study, the incineration of timber with energy recovery should be accounted for. One possible way of doing it, is to consider that the energy recovered avoids the production of more natural gas and consequently avoids the impacts linked to this production. Likewise, the recycling or the reuse of a metal beam should induce the avoiding of virgin metal production. That's the reason why it is important to precisely know how each material and component is dealt with after the demolition.

The demolition of each building is different depending on its characteristics (structure, materials, height…), its localisation, the time and economic constraints. However, in a short term perspective, we would like to increase our knowledge about the waste management and the possibilities of reuse in order to achieve a more complete ALCA. Based on these ALCA, the next step will be to switch to the CLCA to assess the environmental consequences of a more selective demolition especially during the light work removal.

En raison d'une population mondiale toujours grandissante, le secteur de construction a une contribution non négligeable dans les impacts environnementaux. Les différentes études d'Analyse de Cycle de Vie (ACV) menées sur les bâtiments ont montré l'importance de la production des matériaux et de la phase d'usage sur ces impacts. Notre intérêt s'est pourtant porté sur la fin de vie des bâtiments malgré la faible contribution de cette étape en raison de son importance dans une démarche éco-conceptive. En effet, la valorisation, le recyclage et la réutilisation d'un maximum de composants en fin de vie peuvent entraîner de potentiels bénéfices environnementaux.

L'objectif de cette étude est de réaliser une première ACV d'un chantier de démolition afin d'augmenter les connaissances sur cette phase et déterminer les paramètres à connaître. Ce premier bilan permettra de réaliser une ACV conséquentielle pour estimer les bénéfices possibles d'un changement de pratiques dans la démolition des bâtiments et dans les traitements des déchets.

La démolition d'un édifice peut être décomposée en 6 étapes comme le montre la figure 1 suivie d'une phase de gestion des déchets.

Figure 1: Les différentes étapes de la gestion de la fin de vie des bâtiments [START_REF]Déconstuire les bâtiments[END_REF] Grâce à l'EPFIF (Etablissement Public Foncier d'Ile de France) qui est responsable de démolitions dans Paris et sa banlieue nous avons pu mener une première étude. Celle-ci prenait en compte la consommation des engins de chantiers et les distances de transport des différents flux de déchets qui étaient les seules informations disponibles.

Ce simple modèle souligne le besoin de limiter la quantité de diesel utilisé ainsi que d'optimiser les distances de transport. Cependant, un paramètre important n'a pour l'instant pu être pris en compte : le devenir final des déchets après être passés par la plateforme de tri et de recyclage. Ainsi, la réutilisation d'éléments et de matériaux ou bien le recyclage peuvent entraîner des bénéfices environnementaux qu'il faudra prendre en compte dans les études à venir. Cette ACV attributionnelle, complétée par plus d'informations sur la gestion des déchets permettra de débuter une ACV conséquentielle pour estimer les bénéfices potentielles d'un changement de pratiques dans la gestion de la fin de vie des bâtiments.

Table 1 :

 1 Waste flows of the Brou demolition siteBeing the only data available, these material and transport flow were inputted in a LCA software called SimaPro to assess the environmental impacts as a first approximation. The results are shown in figure2:

	Waste flows	Quantity (tons)	Destination	Distance (km)
	Masonry	100	Recycling facility	169
	Reinforced concrete	50	Recycling facility	169
	Timber	6,54	Recycling facility	14
	Plaster	11,76	Landfill	217
	Metals	1	Recycling facility	169
	Mixed inert waste	17,6	Landfill	217
	Asbestos	2,5	Landfill	227


Aya Mabrouki Eseo

Mohamed Latrach
email: mohamed.latrach@eseo.fr

Vincent Lorrain

High Efficiency Low Power Rectifier Design using Zero Bias Schottky Diodes

Keywords: Schottky, rectenna, energy harvesting I

In this paper we present the design of high efficiency low power rectifier for microwave energy harvesting. The proposed circuit is based on a voltage booster formed by a voltage doubler type Latour structure. The circuit topology including parasitic elements and microstrip lines has been studied and optimized for high efficiency energy conversion dedicated to low input power operations (below -10dBm). Measurement results show 21% and 38% RF-DC conversion efficiencies for, respectively, -20dBm and -10dBm input power for 10KΩ resistor load at 850MHz. Experimental performances of the rectifier are in good agreement with the simulated ones.

INTRODUCTION

The widespread diffusion of remotely powered devices has led to a growing interest in wireless energy harvesting techniques. This concept can be used to supply low power electronic devices like sensors or RFID tags over distances of several meters [START_REF] Mikeka | Design issues in Radio Frequency Energy harvesting System[END_REF]. One approach of wireless power transmission uses the electromagnetic (EM) waves and energy transfer process can be described by three main stages. First, the power signal is generated and sent by an emitting antenna. Then, it is transported under the form of a free propogating EM wave towards the receiver where it is collected by the rectenna (rectifying antenna) to be transformed into a DC power. Rectenna efficiency optimization is still a hard task for RF designers when the incident power levels are very low [START_REF] Mikeka | Design issues in Radio Frequency Energy harvesting System[END_REF] [START_REF] Dolgov | Power management system for online low power rf energy harvesting optimization[END_REF].

Several rectification structures have been proposed to improve the RF-DC power conversion efficiency. In [START_REF] Imai | Efficiency and harmonics generation in microwave to dc conversion circuits of half-wave and full-wave rectifier types[END_REF], [START_REF] Roberg | High-Efficiency Harmonically Terminated Diode and Transistor Rectifiers[END_REF], and [START_REF] Falkenstein | Low-power wireless power delivery[END_REF] harmonic terminations, employed in RF power amplifier (PA) designs, have been used to improve the efficiency of a single stage shunt diode rectifier. However, optimized results are obtained only for input power beyond 0dBm. Other research works, [6][7][8], deal with the improvement of the DC output filter of the rectenna to boost the DC voltage and the conversion efficiency but they still not very efficient for low power detection. The conventional rectifying structures that have been widely studied, measured, and discussed are mainly single shunt and series mounted diode, voltage doubler and bridge rectifiers. In this work, we propose a voltage doubler structure type Latour, which to the author knowledge, has not been previously used in rectenna design. The paper is organised as follows: First, we review the different rectifying topologies. Then, we present the proposed circuit and its characteristics. In section III, we give the design guidelines and the simulated results. Finally, the measurement results are shown and discussed in section IV.

II. RECTENNA CONFIGURATIONS

A. Conventional rectifier structures

Rectifier circuits are built around diodes or diodes mounted transistors. In Fig. 1 we present three conventional rectifying structures the most used in litterature [START_REF] Zbitou | Hybrid Rectenna and monolithic integrated zero bias microwave rectifier[END_REF] [START_REF] Douyère | high efficiency microwave rectenna circuit: Modeling and design[END_REF]. The circuit is generally composed of 4 blocks: The input filter to preserve the antenna re-irradiating the high order harmonics generated by the rectifier, the input matching network (not presented in Fig. 1), the rectifying diode, and the output DC filter to filter harmonics in order to reduce voltage and current overlap. Diodes threshold voltage is a key factor when designing the rectenna. In fact, under high power levels, diode threshold is not important since it is very low compared to incident high frequency voltage amplitude. However, it becomes critical for low incident power levels due to losses. In this work, we use a zero bias shottky diode HSMS2852 with low threshold voltage of 150mV and a low junction capacitance C j0 of 0.18pF. The main characteristics that should be optimized when designing rectennas are the DC output voltage V out and the RF-DC power conversion efficiency defined as [START_REF] Marian | Efficient design of rectifying antennas for low power detection[END_REF]:

2 2 2 . . . 4 . λ π η G E Z R V P P air L out RFin DCout = = (1)
Where R L is the load resistance, Z air is air characteristic impedance (120π ohms), E is electric field RMS value at receiver position, G is the receiving antenna gain, and λ is the wavelength.

RF designers should find a trade-off between a high output voltage and good power conversion efficiency when choosing the rectifier structure. A comparison between the performance of conventional rectifier topologies shown in Fig. 1, as a function of input power, has been widely discussed in several works. [START_REF] Marian | Efficient design of rectifying antennas for low power detection[END_REF] used a Rectenna Figure of Merit (RFoM), described in (2), to compare the different rectifying topologies performances.

d optimalLoa uit DCopencirc in V P RFoM η . ) ( = (2)
It reported that series mounted diode rectifier seems to have the highest RFoM for low input power (below -5dBm) and offers a best compromise between DC output level and power conversion efficiency.

B. The Proposed rectifier topology

The rectifier topology we propose in this work is depicted in Fig. 2 For open circuit case, the output voltage is twice the input peak voltage. The input and output waveforms of the circuit are illustrated in Fig. 2(b). This structure will be studied and optimized to design a high efficiency low power rectifier at 915MHz.

III. DESIGN GUIDELINES AND SIMULATION RESULTS

The global circuit optimization technique should take into account several factors such as passive components dimension including Q factor and tolerance, mircostrip lines added for matching and the non-linear behavior of the diode. These opimizations were made using the software ADS (Momentum EM simulations) from Agilent Technologies.

A. Circuit Design

The circuit architecture including passive components and microstrip lines is described in Fig. 3. An input inductance and a shunt capacitor of 3 pF have been added to improve the input matching. Special attention should be given to the choice of passive component in order to reduce the effect of dispersion on the sensitivity of the circuit. The input inductance has a value of 39nH and a Q factor of 35 at 800MHz. The output capacitors C 1 and C 2 are equals to 100pF. The proposed structure is designed on 1.6mm thickness FR4 substrate (εr=4.5 and tanδ=0.025).

B. Simulation results

The circuit has been simulated under Harmonic balance (HB) routine from Agilent ADS. S-paramter simulations have been caried out to optimize the matching network of the rectifier to a 50Ω input impedance at 915MHz (for input power below -10dBm). The simulated input return loss is illustrated in Fig. 4 and -19dB S11 is obtained at 915MHz. As can be seen, 0.163 V and 27% RF-DC conversion efficiency are obtained at -20dBm input power. At -10dBm almost 50% efficiency can be achieved.

IV. RECTIFIER MEASUREMENT RESULTS

The rectifier has been designed and printed on a FR4 substrate. A photograph of the circuit is given in Fig. 6. The circuit occupies a total area of 5cm² which still very small in comparison with the area of rectifier structures using microstrip stub lines either in output or input filters [6] [9]. The measured input return loss S11 is shown in Fig. 7. The rectifier is slightly shifted but it is well matched with -13dB S11 at 850MHz for 10KΩ resistor load. The measurement setup, used to characterize the circuit, is described in Fig. 8. The directional coupler monitors the second harmonic power level. The power meter is used to measure the reflected power. The DC output voltage V out is obtained by subtracting the voltage V -from voltage V + . The power conversion efficiency has been computed according to (3) and plotted as a function of frequency in Fig. 10. A 21% maximum power conversion efficiency is measured at 850MHz for -20dBm input power and 10KΩ resistor load. The circuit performances are in good agreement with the predicted simulted results for input power below -15dBm. At -10dBm input power, measured power conversion efficiency is reduced of 13% in comparison with the simulated one due to ohmic losses of the diodes. Where P inc is the incident RF power, R L is the DC load and V out is the output DC voltage. Measurement of the rectifier performances as a function of input power have been also carried out and plotted in Fig. 11 and Fig. 12. 610mV and 1.17V DC output voltages are measured respectively at -10dBm and -5dBm input powers and 46% maximum RF-DC conversion efficiency is achieved at 0dBm input power. The proposed rectifier presents good performances for low power detection in comparison with the state of the art [START_REF] Marian | Efficient design of rectifying antennas for low power detection[END_REF]. Most of the reported researches in rectifier designs operate at 2.4GHz. In [START_REF] Marian | Efficient design of rectifying antennas for low power detection[END_REF], 300mV DC output voltage is obtained at 2.45GHz for -15dBm input power and 6KΩ resistor load.

V. CONCLUSION

A voltage doubler rectifier, type Latour, has been proposed and measured. The circuit is dedicated to low power detection applications at 850MHz. The experimental results validated the circuit high efficiency operation at low input power (below -10dBm). 21% RF-DC conversion efficiency is achieved at 850 MHz for -20dBm input power and 10KΩ resistor load. Future works involve the increase of the voltage doubler stages to boost the output DC voltage and improve the power conversion efficiency.

Fig. 1 .

 1 Fig. 1. Conventional rectenna topologies: (a) series; (b) shunt; (c) single stage voltage doubler

Fig. 2 .

 2 Fig. 2. (a) The proposed rectifier topology: "Latour" doubler; (b) the corresponding input and output waveforms

Fig. 3 .

 3 Fig. 3. Global circuit topology

Fig. 4 .Fig. 5 .

 45 Fig. 4. Simulated S11

Fig. 6 .

 6 Fig. 6. Photograph of the rectifier

  Fig. 7. Measured S11

Fig. 8 .

 8 Fig. 8. Measurement setup of the rectifierWe measured the output DC voltage at three different input power levels (-20dBm, -15dBm, -10dBm) for 10 KΩ resistor load and we plotted it as a function of frequency in Fig.9. Maximum DC voltages of 144mV and 610mV are obtained at 850MHz for -20dBm and -10dBm respectively. Good correlation with simulation results is observed at -20dBm and -15dBm input powers.

Fig. 9 .

 9 Fig. 9. Measured DC output voltage as a function of frequency

Fig. 10 .

 10 Fig. 10. Measured efficiency as a function of frequency

Fig. 11 .Fig. 12 .

 1112 Fig. 11. Measured DC output voltage as a function of input power