Hung Truong
email: dinh-hung.truong@univ-amu.fr

Thomas Engels
email: thomas.engels@ens.fr

Dmitry Kolomenskiy
email: dkolomenskiy@jamstec.go.jp

Kai Schneider
email: kai.schneider@univ-amu.fr

Fluid-structure interaction using volume penalization and mass-spring models with application to flapping bumblebee flight

Wing flexibility plays an essential role in the aerodynamic performance of insects due to the considerable deformation of their wings during flight under the impact of inertial and aerodynamic forces. These forces come from the complex wing kinematics of insects. In this study, both wing structural dynamics and flapping wing motion are taken into account to investigate the effect of wing deformation on the aerodynamic efficiency of a bumblebee in tethered flight. A fluid-structure interaction solver, coupling a mass-spring model for the flexible wing with a pseudospectral code solving the incompressible Navier-Stokes equations, is implemented for this purpose. We first consider a tethered bumblebee flying in laminar flow with flexible wings. Compared to the rigid model, flexible wings generate smaller aerodynamic forces but require much less power. Finally, the bumblebee model is put into a turbulent flow to investigate its influence on the force production of flexible wings.

Introduction

In recent years, the effect of wing flexibility on aerodynamic performance of flapping wings has drawn attention of researchers, scientists and engineers. Compared to conventional airplanes with fixed wings, flapping wings have several aerodynamic advantages with the ability to create lift even at high angles of attack due to the delayed stall of the leading edge vortex (LEV) [START_REF] Ellington | Leading-edge vortices in insect flight[END_REF]. These extraordinary capabilities of hovering and manoeuvrability have made bio-inspired flapping wings a strong candidate for developing human-engineered micro-air-vehicles (MAVs) with possible applications in environmental monitoring, surveillance and security. However, in previous studies, the wings were usually considered as rigid to simplify the problem. The sophisticated interaction between the anisotropic wing structures and the surrounding unsteady flow makes the analysis of flapping flexible wings challenging, but at the same time intriguing. In the last two decades, with the dramatic improvement of measuring equipment as well as computing power, many experimental and numerical studies have investigated the effect of wing flexibility and drawn contradictory conclusions. Mountcastle and Combes [START_REF] Mountcastle | Wing flexibility enhances load-lifting capacity in bumblebees[END_REF] showed that passive deformations enhance lift production in bumblebees by artificially stiffening their wings using a micro-splint. Campos et al. [START_REF] Campos | Flow around flapping flexible flat plate wings[END_REF] and Fu et al. [START_REF] Fu | Effects of flexibility and aspect ratio on the aerodynamic performance of flapping wings[END_REF] used experimental methods and found that highly flexible wings show significant tip-root lag which weakened vortices and reduce the force production. Du and Sun [START_REF] Du | Sun: Effects of wing deformation on aerodynamic forces in hovering hoverflies[END_REF] solved the Navier-Stokes equations coupled with measured wing deformation data and compared with the rigid counterparts. They obtained a 10% increase in lift caused by the camber deformation and a 5% reduction in required power.

In our previous work [START_REF] Truong | A mass-spring fluid-structure interaction solver: application to flexible revolving wings[END_REF], we considered a flexible bumblebee wing rotating around a hinge point at the angle of attack equal to 45 • . The stiffness of the wing was varied to get two cases: flexible and highly flexible. We found that the flexible wing produces less lift than the rigid wing but it has a better lift-to-drag ratio. On the other hand, the highly flexible wing experienced a strong tip-root lag caused by twisting and behaves poorly in term of aerodynamic performance with a much smaller lift and lift-to-drag ratio. Although the study provided us some ideas about the influence of wing flexibility on the force generation, the revolving motion remains too simple to fully represent the complicated dynamics of a flapping wing. After a short transition period, the revolving wing attains its steady state and its dynamic deformation hardly plays a role in the force production. The wing kinematics of insects is in reality more intricate with many characteristic features such as flapping amplitude, wingbeat frequency, angle of attack, etc. These features have strong impact on the ability of generating force of the wings. Kang and Shyy [START_REF] Kang | Scaling law and enhancement of lift generation of an insect-size hovering flexible wing[END_REF] showed that the ratio between the flapping frequency and the first natural frequency of a flexible wing can yield advanced, symmetric or delayed rotation modes which in turn alter the resulting lift. Zhao et al. [START_REF] Zhao | Aerodynamic effects of flexibility in flapping wings[END_REF] conducted experiments of simple isotropic flapping wings with varied stiffness values at different angles of attack. They found that at low angles of attack (20 • to 60 • ), flexible wings have relatively the same aerodynamic performance as rigid wings but they outperformed their rigid counterparts at high angles of attack (up to 90 • ).

Consequently, in this work, we will investigate the aerodynamic efficiency of the flexible bumblebee wing model [START_REF] Truong | A mass-spring fluid-structure interaction solver: application to flexible revolving wings[END_REF] within a tethered flight context where the wing motion is real bumblebee wing kinematics measured by the work of Dudley and Ellington [START_REF] Dudley | Mechanics of forward flight in bumblebees I. kinematics and morphology[END_REF]. The resulting force and power are compared with those of a rigid flat wing computed by Engels et al. [START_REF] Engels | Bumblebee flight in heavy turbulence[END_REF]. The comparison shows the effects of the wing deformation on the aerodynamic forces of a flapping flight insect.

The remainder of the manuscript presents in section 2 the numerical methods used for solving the governing equations of the flexible wing, the fluid flow and its coupling. The numerical set-up and the bumblebee model are given in section 3 and the results of the numerical simulations are discussed in section 4. Finally some conclusions are drawn in section 5.

Numerical methods and governing equations

Modeling insect flight is a delicate topic due to the fact that one needs to model both the mechanical behavior of the wings by a solid solver and the surrounding flow by a fluid solver. The two solvers must then be coupled to study their interaction. This section presents how these three aspects can be handled numerically.

Solid solver using mass-spring system

Insect wings are sophisticated structures consisting of membranes and veins. The wings get their nonlinear anisotropic properties from a truss framework composed of horizontal and vertical veins connected by membranes [START_REF] Shyy | An introduction to flapping wing aerodynamics[END_REF]. This along with their small wing lengths (from to ) makes it extremely challenging to model the mechanical behavior of insect wings. In our work, a mass-spring system is employed to mimic the dynamics of the complicated membrane-vein network by taking the different mechanical properties between veins and membranes into account [START_REF] Truong | A mass-spring fluid-structure interaction solver: application to flexible revolving wings[END_REF]. While veins can be considered as rods which resist mainly the torsion and bending deformation, a membrane is fabric-like and behaves like a piece of cloth which resists against the extension deformation.

The mass-spring system has been around since the end of the 20 ℎ century and it is well-known for its computational efficiency and ability of handling large deformation [START_REF] Nealen | Physically based deformable models in computer graphics[END_REF]. The wing is discretized using mass points ( = 1, ..., ) connected by springs. Among the different types of springs, our model is built only on extension and bending springs. The dynamic behavior of the mass-spring system, at a given time , is defined by the position x and the velocity v of the mass point and they are governed by the eqns. (1) below: [START_REF] Ellington | Leading-edge vortices in insect flight[END_REF] where is the number of mass points, F int is the internal force and F ext is the external force acting on the ℎ mass point, and a are mass and acceleration of the ℎ mass point, respectively.

F int + F ext = a for i = 1 . . . n v ( = 0) = v 0, x ( = 0) = x 0,
The system (1) is then advanced numerically in time by applying a second order backward differentiation scheme with variable time steps [START_REF] Berger | A second order backward difference method with variable steps for a parabolic problem[END_REF]:

q +1 - (1 + ) 2 1 + 2 q + 2 1 + 2 q -1 = 1 + 1 + 2 Δ f (q +1 ) (2)
where q = x , v is the phase vector containing positions and velocities of all mass points and f (q) = v , -1 (F int + F ext ) is the right hand side function, = Δ /Δ -1 is the ratio between the current time step Δ and the previous one Δ -1 . The phase vector of the system at the current time step q +1 is found by solving eqn. (2) using the Newton-Raphson method. All details of this solver are explained in [START_REF] Truong | A mass-spring fluid-structure interaction solver: application to flexible revolving wings[END_REF].

Fluid solver and volume penalization method

Due to their small sizes and elevated flapping frequencies, insect flight is normally categorized in the Reynolds number regime between O (10 1 ) and O (10 4 ). For example, for hawkmoth we have = 6000, bumblebee = 2000, fruit fly = 100 or thrips = 10 [START_REF] Shyy | Aerodynamics, sensing and control of insect-scale flapping-wing flight[END_REF][START_REF] Engels | Impact of turbulence on flying insects in tethered and free flight: High-resolution numerical experiments[END_REF]. The flow can be considered as incompressible and governed by:

u + × u = -∇Π + ∇ 2 u - (u -u ) penalization term - 1 sp ∇ × ( ) ∇ 2 sponge term (3) ∇ • u = 0 (4) u(x, = 0) = u 0 (x) x ∈ Ω, > 0 (5)
The above equations [START_REF] Campos | Flow around flapping flexible flat plate wings[END_REF][START_REF] Fu | Effects of flexibility and aspect ratio on the aerodynamic performance of flapping wings[END_REF][START_REF] Du | Sun: Effects of wing deformation on aerodynamic forces in hovering hoverflies[END_REF] are called the penalized Navier-Stokes equations [START_REF] Angot | A penalization method to take into account obstacles in incompressible viscous flows[END_REF] where u is the fluid velocity, = ∇ × u is the vorticity, Π = + 1 2 u • u is the total pressure and is the kinematic viscosity. Except for all the terms found in the classical incompressible Navier-Stokes equations, it appears two more terms which are called the sponge and the penalization terms. The former is added to remove the periodicity of the Fourier discretization which affects the upstream inflow. The penalization term is used to impose the no-slip boundary conditions on the fluid-solid interface in [START_REF] Engels | Flusi: A novel parallel simulation tool for flapping insect flight using a Fourier method with volume penalization[END_REF]. All geometrical information of the solid is encoded in the mask function given by:

( ) =          1 ≤ -ℎ 1 2 1 + cos ( -+ℎ) 2ℎ -ℎ < < + ℎ 0 ≥ + ℎ (6)
where is the signed distance field of the bumblebee skeleton and represents here the distance from the skeleton to the outer surface, i.e. the fluid-solid interface.

The skeleton of the bumblebee is a curvilinear centerline along which we sweep an elliptical section of variable size to draw the insect's body, legs and antennae [START_REF] Engels | Numerical modeling of fluid-structure interaction in bioinspired propulsion[END_REF]. However, to avoid the force oscillation when dealing with moving solid body, a smoothing layer with a thickness 2ℎ is added right at the fluid-solid interface to prevent the discontinuity of the mask function [START_REF] Kolomenskiy | A Fourier spectral method for the Navier-Stokes equations with volume penalization for moving solid obstacles[END_REF].

For solving the fluid equations (3-5), a Fourier pseudospectral discretization with semi-implicit time stepping is employed, implemented in the FLUSI code [START_REF] Engels | Flusi: A novel parallel simulation tool for flapping insect flight using a Fourier method with volume penalization[END_REF]. The general idea consists of representing quantities (velocity, pressure, vorticity) as truncated Fourier series,

(x, ) = -1 =0 -1 =0 -1 =0 ˆ (k, ) exp( k • x) (7)
where k = [ , , ] T is the wavevector, = √ -1 and are the discrete complex Fourier coefficients of . The Fourier coefficients can be computed with the fast Fourier transform (FFT) using the P3DFFT library. The main motivation of using a Fourier discretization is the simplicity of inverting a diagonal Laplace operator and the high numerical precision reflected in the absence of numerical diffusion and dissipation in the discretization. The gradient of a scalar can, for instance, be obtained by multiplying with the wavevector and the complex unit, ∇ = k ˆ . The Laplace operator becomes a simple multiplication by -|k| 2 , it is thus diagonal in Fourier space. For further details, we refer the reader to the reference article on the FLUSI solver [START_REF] Engels | Flusi: A novel parallel simulation tool for flapping insect flight using a Fourier method with volume penalization[END_REF].

Fluid-structure interaction

For time-stepping, the coupled fluid-solid system is advanced by employing a semiimplicit staggered scheme, as proposed in [START_REF] Engels | Numerical modeling of fluid-structure interaction in bioinspired propulsion[END_REF]. On the one hand, we advance the fluid by using the Adam-Bashforth second order (AB2) scheme with exact integration of the viscous term. On the other hand, the Backward Differentiation Formula of second order (BDF2) is used for the time discretization of the solid solver. The two modules are then coupled by the algorithm presented in the flowchart shown in figure 1. For the range of Reynolds numbers (75-4000), Dickinson et al. [START_REF] Dickinson | Wing Rotation and the Aerodynamic Basis of Insect Flight[END_REF][START_REF] Dickinson | Unsteady Aerodynamic Performance of Model Wings at Low Reynolds Numbers[END_REF][START_REF] Roccia | Modified unsteady vortex-lattice method to study flapping wings in hover flight[END_REF] showed that pressure forces dominate the shear viscous forces. Hence, for calculating the solid deformation, the viscous fluid tension is considered negligible compared to the static pressure. Moreover, the scheme is called a weak coupling method since the static pressure is computed from the previous state of the solid model. This makes the system conditionally stable only if the structure is heavy enough with respect to the fluid density. However, the scheme is efficient because the fluid and the solid need to be advanced only one time at the current time level. Full details of the fluid-structure interaction (FSI) framework as well as detailed validation of the results can be found in our previous work [START_REF] Truong | A mass-spring fluid-structure interaction solver: application to flexible revolving wings[END_REF].

Numerical set-up and bumblebee model

To study the influence of wing flexibility on the aerodynamic forces, we compare the flexible wings with rigid ones using the same numerical set up in previous work in [START_REF] Engels | Bumblebee flight in heavy turbulence[END_REF].

Flow configuration

The computational domain, shown in figure 2, is 6 × 4 × 4 large, where is the bumblebee wing length, discretized by 1152 × 768 × 768 grid points. The bumblebee is tethered (both translational and rotational motion of the body are inhibited) at x cntr = (2 , 2 , 2 ) and exposed to a head wind with a mean flow accounting for the insect's forward velocity u ∞ = (1.246 , 0, 0) , where is the wingbeat frequency. Due to the periodicity inherent to the spectral method, a thin vorticity sponge outlet, covering the last 4 grid points in -direction, is used to minimize the upstream influence of the computational domain. The sponge penalization parameter is usually set to a value larger than the permeability , normally = 10 -1 . By construction, the sponge term is divergence-free to avoid the influence on the pressure field, which in turn would be modified even in regions far away from the sponge due to its nonlocality. A detailed discussion on the influence of the vorticity sponge can be found in [START_REF] Engels | Flusi: A novel parallel simulation tool for flapping insect flight using a Fourier method with volume penalization[END_REF].

In nature, insects do not always fly in a calm, quiescent environment. Instead, they face, most of the time, many kinds of aerial perturbations such as gusty wind, vortices or turbulent flow generated by surrounding obstacles. Taking this into account, both laminar and turbulent flows are investigated here to study the role of wing flexibility under these two circumstances. For the laminar case, in the entire computational domain a mean flow u ∞ is imposed by simply setting the zeroth Fourier mode of the velocity u [START_REF] Engels | Numerical modeling of fluid-structure interaction in bioinspired propulsion[END_REF]. On the other hand, information on turbulent flow conditions, which are experienced by flying insects in nature, remains an open question with limited data [START_REF] Crall | Foraging in an unsteady world: bumblebee flight performance in field-realistic turbulence[END_REF]. However, for indoor wind tunnel experiments, isotropic or near-isotropic turbulence generated by a grid has been used as inflow condition to study the impact of turbulence on insect flight performance. Consequently, a homogeneous isotropic turbulence (HIT) is chosen as turbulent inflow in our present work in order to compare with the results obtained for rigid wings in [START_REF] Engels | Impact of turbulence on flying insects in tethered and free flight: High-resolution numerical experiments[END_REF]. For this purpose, in the inlet region containing the first 48 grid points along the axial direction, a precomputed HIT velocity field u is added into the mean flow as velocity fluctuations u = u ∞ +u . The HIT field is then transported downstream by the mean flow and evolves dynamically like grid turbulence. In order to compare with the results from [START_REF] Engels | Bumblebee flight in heavy turbulence[END_REF], we use here a HIT field characterized by the same parameters which are the turbulent intensity = / ∞ = 0.33, the integral length scale Λ = 0.77 and the turbulent Reynolds number = / = 129, based on the Taylor-micro scale = 0.18. More technical details on this approach can be found in [START_REF] Engels | Bumblebee flight in heavy turbulence[END_REF][START_REF] Engels | Impact of turbulence on flying insects in tethered and free flight: High-resolution numerical experiments[END_REF][START_REF] Engels | Numerical modeling of fluid-structure interaction in bioinspired propulsion[END_REF]. , 0, 0) . The blue inlet region is used to impose a precomputed HIT velocity field which is constantly advected downstream by the mean flow. Another vorticity sponge region is placed at the outlet to damp out vortices. Periodic boundary conditions are set for the four other sides of the domain.

Bumblebee model

The bumblebee model here is the same as the one used in [START_REF] Engels | Impact of turbulence on flying insects in tethered and free flight: High-resolution numerical experiments[END_REF] and derived from case BB01 in [START_REF] Dudley | Mechanics of forward flight in bumblebees I. kinematics and morphology[END_REF], except for the wings which will be introduced later in section 3.3.

The animal's body mass, , is 175 , the gravitational acceleration = 9.81 / 2 and wing length , amounts to [START_REF] Engels | Impact of turbulence on flying insects in tethered and free flight: High-resolution numerical experiments[END_REF] . The bumblebee is composed of linked rigid bodies including the head, the thorax, the abdomen, all legs, the proboscis and the antennae. These parts are circular elliptical or cylindrical sections joined by spheres, and the bilateral symmetry of the insect is assumed. The Reynolds number is = / = 2685, where = 2Φ = 9.15 / is the mean wingtip velocity, = 4.6 the mean chord length, = 1.568 • 10 -5 2 / is the kinematic viscosity of air, = 152 ( = 1/ = 6.6

) is the wingbeat frequency ( is duration) and = 115 • is the wingbeat amplitude. The wingbeat kinematics are prescribed based on the work of Dudley and Ellington [START_REF] Dudley | Mechanics of forward flight in bumblebees I. kinematics and morphology[END_REF].

Flexible wing model

The two flexible wings of the insect are modeled using the mass-spring system as detailed in [START_REF] Truong | A mass-spring fluid-structure interaction solver: application to flexible revolving wings[END_REF]. In the following we describe the venation pattern, the mass distribution and the flexural rigidity of the veins.

Venation pattern

The venation architecture is claimed to be responsible for the anisotropy of the wing and it plays a crucial role on the wing dynamics during flight. Consequently, the functional approach is used to take into account the venation pattern in our model. The wing contour and the vein network are adapted from [START_REF] Kolomenskiy | The dynamics of passive feathering rotation in hovering flight of bumblebees[END_REF] and encoded into the mass-spring system. The wing is then discretized by a triangular mesh with 1065 mass points, as shown in figure 3, using SALOME , an open-source integration platform for mesh generation. A mesh convergence study comparing between two wings, discretized by 465 and 1065 mass points, was performed in [START_REF] Truong | A mass-spring fluid-structure interaction solver: application to flexible revolving wings[END_REF] for the revolving motion. Looking at the aerodynamic forces generated, the coarse-mesh wing showed no major difference with respect to the fine-mesh wing. However, for the flapping motion, the pressure field is expected to be more unstable and a fine-mesh wing is needed for the pressure interpolation in this case.

Mass distribution

The mass distribution represents the inertia of the system and the position of the mass center has a connection with the wing dynamics during flight. The mass distribution is calculated based on the measured wing mass data from [START_REF] Kolomenskiy | The dynamics of passive feathering rotation in hovering flight of bumblebees[END_REF] and the vein pattern. For our numerical simulations, the total wing mass is chosen as the same used by Kolomenskiy et al. [START_REF] Kolomenskiy | The dynamics of passive feathering rotation in hovering flight of bumblebees[END_REF],

= 0.791 . The mass is then distributed into vein and membrane parts based on their geometry and material.

For the vein structure, each vein is considered as a rod composed of cuticle, = 1300 / 3 [START_REF] Kolomenskiy | The dynamics of passive feathering rotation in hovering flight of bumblebees[END_REF], with a circular cross section of constant diameter [START_REF] Kolomenskiy | The dynamics of passive feathering rotation in hovering flight of bumblebees[END_REF] and length , calculated directly from the model. The mass of each vein is then Fig. 3 Illustration of the mass-spring model which is meshed based on measured data of real bumblebee wings [START_REF] Kolomenskiy | The dynamics of passive feathering rotation in hovering flight of bumblebees[END_REF]. The black and white marker represents the mass center. Color codes (red, green and blue) are used for identifying the veins and the membranes are represented by cyan circles. calculated and shown in table 1. Both diameter and mass are dimensionless quantities, normalized by wing length and air density 3 , respectively. For the mass distribution of the membrane, the same optimization method as in [START_REF] Truong | A mass-spring fluid-structure interaction solver: application to flexible revolving wings[END_REF] is applied where the objective function is the difference between the mass center of the wing measured in the experiment [START_REF] Kolomenskiy | The dynamics of passive feathering rotation in hovering flight of bumblebees[END_REF] and the one calculated from the massspring model.

Flexural rigidity of veins

Because the bending stiffness of the membrane is neglected, the flexural rigidity of the wing comes solely from the flexural rigidity of veins which is calculated based on their material and geometry. While the estimation of their second moments of inertia is straight forward using the diameter data from table 1, determining the Young's modulus is not trivial. In our present work, the veins are considered to be made of cuticle which is reported to have a Young's modulus in the range of 1 to 50 [START_REF] Vincent | Design and mechanical properties of insect cuticle[END_REF]. The wing needs to be flexible enough to reveal the influence of wing flexibility to the aerodynamic performance of insects but it cannot be too flexible to show unrealistic mechanical behaviors. For the purpose of our study, the value = 700 is chosen.

Forewing Hindwing

# Nominal diameter

Nominal mass # Nominal diameter Nominal mass 1 0.0070 0.0209 1 0.0065 0.0180 2 0.0074 0.0237 2 0.0043 0.0071 3 0.0055 0.0076 3 0.0046 0.0024 4 0.0070 0.0063 4 0.0011 0.0001 5 0.0040 0.0031 5 0.0038 0.0043 6 0.0048 0.0094 6 0.0037 0.0005 7 0.0040 0.0019 7 0.0020 0.0012 8 0.0038 0.0009 9 0.0041 0.0023 10 0.0048 0.0064 11 0.0045 0.0017 12 0.0038 0.0018 13 0.0042 0.0010 14 0.0038 0.0020 15 0.0034 0.0008 16 0.0032 0.0005 17 0.0032 0.0004 18 0.0044 0.0009 19 0.0015 0.0001 20 0.0018 0.0001 21 0.0020 0.0009 Table 1 Dimensionless vein diameter (adapted from [START_REF] Kolomenskiy | The dynamics of passive feathering rotation in hovering flight of bumblebees[END_REF]) and their corresponding dimensionless mass .

Results and discussion

The forces generated by the bumblebee model with flexible wings as well as the required aerodynamic power will be presented in this section. Furthermore, they will be compared with the results obtained in [START_REF] Engels | Bumblebee flight in heavy turbulence[END_REF] where the same bumblebee with rigid wings was considered. This allows us to have some insight into the wing flexibility influence on the insect aerodynamic performance.

Tethered flight in laminar flow

The vertical and horizontal forces produced by the flapping motion of the flexible wings are shown by red curves in figure 4 (a,b) while blue curves are those generated by rigid wings. The forces are normalised by = 4 2 . Here, the sideways force is small and not presented, since the animal is modeled with the assumption of symmetry. The simulation is computed for 4 strokes with 28776 time steps using 32 processors on Intel Xeon Gold 6142 (Sky Lake) 2.6 and consumed 8128 CPU hours. For each cycle, the cycle-average values are calculated and presented in table 2. While the wing flexibility has minor effect on the average thrust with a decline of 11%, it accounts for a 28% drop of the average lift. These losses can be explained as a result from the decrease of the effective angle of attack caused by wing deformation. The shape adaptation of the wing during the flapping motion alters the instantaneous angle of attack which is claimed to play a significant role in the force generation [START_REF] Fu | Effects of flexibility and aspect ratio on the aerodynamic performance of flapping wings[END_REF]. However, these negative impacts do not necessarily mean that the rigid wings outperform aerodynamically their flexible counterparts. Although the flexible wings generate smaller forces, they consume much less energy, with almost 36% required aerodynamic power is reduced. The cycle-averaged lift-to-power ratio of flexible wing is 0.026, 35% larger than the one of rigid wing which is 0.019.

Nevertheless, regarding the time evolution of the forces during one wingbeat, the instant surges of the forces at the ends of upstroke and downstroke, observed in the rigid case, are significantly weakened. The sudden rotation of the rigid wings at the midstrokes and the end of strokes are the reason for these large force peaks [START_REF] Sane | The aerodynamics of insect flight[END_REF]. This effect has now little impact due to the fact that the wing inertia are now taken into account. The inertial force makes the wing deform and streamline its shape to the airflow. This shape adaptation helps to mitigate the large pressure jump between upper and lower surfaces, especially at the trailing edge [START_REF] Kang | Scaling law and enhancement of lift generation of an insect-size hovering flexible wing[END_REF] and provides a smoother flight [START_REF] Ifju | Flexible-wing-based micro air vehicles[END_REF][START_REF] Ifju | Analysis of a flexible wing micro air vehicle[END_REF]. This finding has more advantages in term of stabilizing generated forces, rather than lift-enhancement effect.

Tethered flight in turbulent flow

We then study the influence of an isotropic turbulence on the aerodynamic performance of a bumblebee by putting it in a turbulent flow. The simulation is computed for 4 strokes with 29000 time steps using 32 processors on Intel Xeon Gold 6142 (Sky Lake) 2.6 and consumed 9000 CPU hours. Figure 5 presents the flow structure of the bumblebee flying in a turbulent flow visualized by the normalized vorticity isosurfaces at two levels = 15 and = 100. The aerodynamic forces and the corresponding power in this turbulent condition are shown in figure 4. The results demonstrate insignificant differences between turbulent and laminar flow conditions. The aerodynamic forces generated by the bumblebee are almost identical to those derived during unperturbed, laminar inflow, with the same required energetic cost. For > 100, the aerodynamic forces are mainly produced by the differential dynamics pressure across the wing [START_REF] Sane | The aerodynamics of insect flight[END_REF]. Figure 6 shows the normalized pressure distribution on top and bottom wing surfaces of the two cases just before the stroke Normalised aerodynamic power Fig. 4 Normalized vertical force, horizontal force and aerodynamic power generated by a bumblebee with rigid wings (black) [START_REF] Engels | Impact of turbulence on flying insects in tethered and free flight: High-resolution numerical experiments[END_REF] and flexible wings in laminar flow (red) and turbulent flow (blue). Circles represent the cycle-averaged value of forces and power. reversal = 0.45 . The effect of turbulence can hardly be seen here which explains the negligible change of aerodynamic forces. The outcome here is consistent with the one observed in the rigid case in [START_REF] Engels | Bumblebee flight in heavy turbulence[END_REF].

Conclusions and perspectives

Following our previous work on revolving flexible wings [START_REF] Truong | A mass-spring fluid-structure interaction solver: application to flexible revolving wings[END_REF], the impact of wing flexibility was now studied in the context of tethered flight using flapping wing kinematics measured in experiments by Dudley and Ellington [START_REF] Dudley | Mechanics of forward flight in bumblebees I. kinematics and morphology[END_REF]. High-resolution numerical simulations on massively parallel machines were carried out to solve the fluid-structure interaction problem between the fluid solver FLUSI and the solid solver based on a mass-spring system. Both laminar and turbulent inflows were considered to investigate diverse flight conditions of insects. The preliminary results obtained in this work allow us to have some understanding about the role of wing flexibility in flapping flight.

In laminar flow, the aerodynamic forces and the required power have been calculated and compared with the ones obtained for rigid wings. We found that wing flexibility hardly contributed to lift or thrust enhancement. However, the significant reduction of the required power suggested that wing flexibility plays an important role in saving flight energetic cost. Moreover, the wing inertia also helped to damp out the fluctuation of the aerodynamic force and helped thus the insect to stabilize during flight.

In turbulent flow, although the ability of shape adaptation of flexible wings makes them more sensitive to fluctuation of the flow structure than their rigid counterparts, the impact of turbulence is still negligible under the considered flight conditions. Nevertheless, due to costly computational time, the statistical property of the turbulent flow is not considered because only one simulation is done to obtain the results for the turbulent case.nMoreover, due to the expensive computational cost, especially in the turbulent case, the mesh convergence study was not performed in this paper and we refer readers to [START_REF] Truong | A mass-spring fluid-structure interaction solver: application to flexible revolving wings[END_REF].

Despite of these findings, we have to keep in mind that the wing kinematics has an essential effect on the aerodynamic performance of wings and we have considered only one set of wing motion in this study. In perspective, these limitations can be overcome by examining other species with different wing kinematics or including flight control in our model. This is planned for our work in the future where we will study Calliphora with its wing kinematics measured from experiments.

Finally, although the wing flexibility was calculated based on the geometrical property of the veins, the estimation of veins' Young's modulus remains somewhat limited due to the vast range of known cuticle's property. This can be improved by using mathematical optimization for determining the right elastic properties of the wing model. To this end the equilibrium state of the wing model under external static force as a function of wing stiffness will be calculated and compared with data measured from experiments done by our team.

Intensif d'Aix-Marseille is acknowledged for granting access to its high performance computing resources financed by the project Equip@Meso (No. ANR-10-EQPX-29-01). The authors thankfully acknowledge financial support granted by the ministères des Affaires étrangères et du développement international (MAEDI) et de l'Education nationale et l'enseignement supérieur, de la recherche et de l'innovation (MENESRI), and the Deutscher Akademischer Austauschdienst (DAAD) within the French-German Procope project FIFIT. D.K. gratefully acknowledges financial support from the JSPS KAKENHI Grant No. JP18K13693.

  //github.com/pseudospectators/FLUSI).

Fig. 1

 1 Fig. 1 Flowchart: Semi-implicit staggered scheme of the time advancement for the fluid-structure interaction problem.

Fig. 2

 2 Fig. 2 Illustration of the computational domain of size 6 × 4 × 4 used in all simulations. A bumblebee model with flexible wings is tethered at x = (2 , 2 , 2 ) in a flow with the mean flow velocity u ∞ = (1.246, 0, 0) . The blue inlet region is used to impose a precomputed HIT velocity field which is constantly advected downstream by the mean flow. Another vorticity sponge region is placed at the outlet to damp out vortices. Periodic boundary conditions are set for the four other sides of the domain.

  For a mass point belonging to the membrane at position [ , ], we get: = 9.14 • 10 -5 -3.48 • 10 -5 + 2.48 • 10 -4 (8) Differences, between two mass centers, of 3.85 • 10 -3 [ ] in the -direction and 0.93 • 10 -3 [ ] in the -direction are obtained. These are negligible compared to the reference wing length .

Fig. 5

 5 Fig. 5 Visualization of flow generated by a tethered flapping bumblebee with flexible wings in turbulence = 0.33 showing normalized absolute vorticity isosurfaces at two levels | | | | = 15 (light blue) and | | | | = 100 (dark blue). The flow fields are plotted at time = 0.45/ and the weaker vortices are only shown in the region 3.7 ≤ ≤ 4 .

Fig. 6

 6 Fig. 6 Normalized pressure distribution on the wing for top and bottom surfaces, plotted at time = 0.45/ just before the stroke reversal, for the laminar (top) and turbulent cases (bottom).

Table 2

 2 Cycle-averaged forces and power in the laminar case.

	Flow	Thrust	Lift	Aerodynamic power
		Rigid Flexible Rigid Flexible Rigid Flexible
	Laminar 0.17 0.15 2.09 1.51 8.84	5.67

https://www.salome-platform.org/

Acknowledgements

Financial support from the Agence Nationale de la Recherche (ANR Grant No. 15-CE40-0019) and Deutsche Forschungsgemeinschaft (DFG Grant No. SE 824/26-1), project AIFIT, is gratefully acknowledged. The authors were granted access to the HPC resources of IDRIS under the Allocation No. 2018-91664 attributed by GENCI (Grand Équipement National de Calcul Intensif). For this work, Centre de Calcul


Dominique Ami

Frédéric Aprahamian

Olivier Chanel

Robert-Vincent Joulé

Stéphane Luchini
email: stephane.luchini@univmed.fr.

Willingness To Pay of Committed Citizens: A Field Experiment ‡

Keywords: Field Experiment, Citizen, Consumer, Contingent Valuation, Willingness to pay, Commitment, Air Pollution

In this paper, we propose a behavioural approach to determine the extent to which the consumer/citizen distinction aects interpretations of monetary values. We perform a eld experiment dealing with air pollution, where some (randomly selected) subjects are given the opportunity to behave politically by signing a petition for environmental protection prior to stating their private preferences in a standard contingent valuation exercise. We show that the petition has the potential to inuence respondents' willingness to pay and that whether the eect is negative or positive depends on the degree of (dis)similarity between the petition and the scenario in which willingness to pay are elicited. We interpret the results using the theory of commitment borrowed from social psychology.

Introduction

From the outset, the interpretation of respondents' behaviour in CV questionnaires has provoked debate. In particular, some authors have questioned the interpretation of respondents as consumers, proposing instead a more political interpretation of respondents as citizens [START_REF] Nyborg | Homo economicus and homo politicus: interpretation and aggregation of environmental values[END_REF][START_REF] Lewinson-Zamir | Consumer preferences, citizen preferences, and the provision of public good[END_REF][START_REF] Blamey | Respondents to contingent valuation surveys: consumers or citizens[END_REF]. As citizens, the assumption is that respondents express political attitudes rather than expressing monetary values in line with consumer surplus [START_REF] Orr | Values, preferences, and the citizen-consumer distinction in cost-benet analysis[END_REF][START_REF] Blamey | Decisiveness, attitude expression and symbolic responses in contingent valuation surveys[END_REF]. This is primarily because attitudes do not correspond to people's consumer surplus [START_REF] Kahneman | Experienced utility as a standard of policy evaluation[END_REF]. It has also been argued, for example, that respondents may adopt dierent 'roles' and that if they are in their citizen role, they may consider overall social welfare instead of their own needs or wants (Sago 1988). If this is the case, it would of course cast doubt on the use of respondents' answers in CV surveys in standard cost-benet analysis [START_REF] Nyborg | Homo economicus and homo politicus: interpretation and aggregation of environmental values[END_REF].

The question is how to operationalize the distinction between respondents in CV surveys as consumers and as citizens. One way of doing this is to adopt an ex-post approach that looks at determinants of observed behaviour, by correlating willingness to pay (WTP) values with appropriate attitudinal questions [START_REF] Nunes | Identifying the warm glow eect in contingent valuation[END_REF]. In this paper, however, we adopt an ex-ante and behavioural approach in the spirit of [START_REF] Boyce | An experimental examination of intrinsic values as a source of the WTA-WTP disparity[END_REF] that relies on commitment theory borrowed from social psychology [START_REF] Kiesler | The psychology of commitment. Experiments linking behavior to belief[END_REF]. Experiments on the social psychology of commitment have shown that when people agree to perform an initial action, even when it appears to be innocuous, this can have strong attitudinal and behavioural consequences [START_REF] Kiesler | A test of a model for commitment[END_REF][START_REF] Joule | La soumission librement consentie[END_REF]. Our experiment follows this route in that some (randomly selected) subjects are given the choice to behave politically prior to stating their private preferences in a CV exercise. 1 We call subjects who agree to perform the political action committed citizens and we compare their behaviour to that of a control group of respondents who merely answer the CV questionnaire directly.

Our eld experiment focuses on a specic environmental issue: air pollution.

First, some subjects are given the opportunity to sign a petition calling on (the then) future candidates in the French 2007 presidential election, to be held six months later, to take an ocial stand (regarding their program and respective commitment) on environmental preservation. Second, all subjects are asked to respond to a CV questionnaire aimed at eliciting willingness to pay for a reduction in air pollution. Our sample is therefore composed of two groups of subjects: one asked to sign a petition before performing the CV exercise and one performing this exercise directly.

We investigate the articulation between political behaviour and stated willingness to pay by examining values under three typical scenarios of the CV literature on air pollution: a new drug that prevents adverse health eects of air pollution exposure for the respondent alone [START_REF] Alberini | Does the value of a statistical life vary with age and health status? Evidence from the US and Canada[END_REF][START_REF] Krupnick | Age, health and the willingness to pay for mortality risk reductions: a contingent valuation study of Ontario residents[END_REF], moving the whole household to an already less polluted city (Viscusi et al. 1988;[START_REF] Aprahamian | Modeling starting point bias and unobserved heterogeneity in contingent valuation surveys: an application to air pollution[END_REF]) and new regulations on air pollution that will potentially increase market prices [START_REF] Desaigues | Final report on the monetary valuation of mortality and morbidity risks from air pollution[END_REF]. One feature of these scenarios is they dier in scope. This means that benets range from the individual (drug scenario) to the family level (move scenario) and up to the society level (regulation scenario). A second feature is that the drug scenario does not imply any particular measure to protect the environment whereas the two other scenarios do. The move scenario implies that local action has already been taken with the result that one of the locations has less air pollution.

The regulation scenario presupposes national, or even international, measures to protect the environment.

The experiment took place over three days in October 2006 in the council chamber of Regional Council in the city of Marseilles which is equipped for electronic voting. It involved more than four hundred subjects under six sets of experimental conditions. Our results are as follows. First, more than eighty percent of subjects agreed to sign the petition when it was presented to them.

Second, the petition had a signicant, although varying, impact on subjects' willingness to pay. Econometric analysis shows that subjects who were oered the petition were more likely to pay more in the move and regulation scenarios. In the drug scenario, subjects were less willing to pay when they had been oered the petition prior to the valuation exercise.

The paper is organized as follows. In the rst section, we dene our experimental hypotheses based on the social psychology of commitment. In the second section, we give a detailed description of the experimental design. The third section provides information on the sample characteristics and presents the empirical results. In the nal section, we discuss the results and draw conclusions.

Three experimental hypotheses

Our rst hypothesis is that subjects who have previously performed a political act, that is, who have signed the petition, will be willing to pay more for a good or a policy relevant to the protection of the environment than subjects who were not asked to sign the petition in the rst place. Our second hypothesis is that subjects who have signed the petition will state a lower WTP in the drug scenario than subjects who have not been oered the petition. These two hypotheses can be understood in the light of the theory of commitment [START_REF] Kiesler | The psychology of commitment. Experiments linking behavior to belief[END_REF][START_REF] Joule | La soumission librement consentie[END_REF], whereby commitment means the pledging or binding of the individual to behavioural acts (Kiesler and Sakumura 1966, p. 349). Foot-in-the-door experiments are situations that typically produce this kind of commitment [START_REF] Joule | How can people be induced to willingly change their behavior? The path from persuasive communication to binding communication[END_REF], by asking subjects to comply with an initial request and later on making a second request which is thematically in line with the initial request. In commitment theory in general, a rst request acts as a commitment device that puts people in a certain mindframe regarding the action that is going to follow [START_REF] Joule | La soumission librement consentie[END_REF]. That is, when they are asked to comply with the second request, subjects may consider themselves as being the kind of person [...] who does this sort of things (Freedman and Fraser 1966, p.101) in our case, a citizen who is concerned about environmental protection because they have already complied with the initial request. 2 Be- ing this kind of person means that they are committed to a certain cause and hence are more likely to comply with the second request. Foot-in-the-door experiments have shown that subjects agree to a second request much more easily if they have already agreed to an initial request of a similar kind (see [START_REF] Freedman | Compliance without pressure: the footin-the-door technique[END_REF] for the original foot-in-the-door experiments, 3 [START_REF] Burger | The foot-in-the-door compliance procedure: A multipleprocess analysis and review[END_REF][START_REF] Joule | La soumission librement consentie[END_REF]for reviews, and Katzev and Wang 1994and Wang and Katzev 1990 for applications to the environment).

In our experiment, we ask some subjects to sign a petition in favour of environmental protection and later on we ask subjects for their WTP on the one hand for living in a less polluted city (move ) or for the implementation of new rules and laws against air pollution (regulation ) and on the other hand for a drug (drug) that does not imply any protection of the environment but only reduces the eects of air pollution exposure on individual health. Because,

[c]ommitment in the guise of signing a petition [has] a powerful eect on the expression of attitude, leading to a more extreme attitude (Kiesler, 1971, p. 79), we expect higher WTP values in the move and regulation scenarios (reecting a more extreme attitude to environmental protection) but lower WTP values in the drug scenario, because the latter does not imply better protection of the environment. Indeed, taking the drug could even be taken to mean not protecting the environment, which is incompatible with being a citizen committed to environmental protection.

Our third hypothesis is that the impact of signing the petition on WTP should be dierent in the move and regulation scenarios. More precisely, the impact of the petition should be stronger in the regulation scenario and weaker in the move scenario. The foot-in-the-door literature shows that compliance with the second request increases with similarity to the initial request [START_REF] Burger | The foot-in-the-door compliance procedure: A multipleprocess analysis and review[END_REF][START_REF] Joule | La soumission librement consentie[END_REF][START_REF] Freedman | Compliance without pressure: the footin-the-door technique[END_REF]. Similarity here means the same level of identication for the rst and second request [START_REF] Joule | La soumission librement consentie[END_REF]. For example, in their second experiment, [START_REF] Freedman | Compliance without pressure: the footin-the-door technique[END_REF] consider initial requests which involve either putting a small sign for safe driving or keeping California beautiful on their window. In the second request, they ask subjects to install in front of their door a very large sign which says Drive carefully. The initial request related to safe driving is said to have a higher degree of similarity with the second request. In our experiment, subjects are given the opportunity to sign a petition concerning nationwide environmental protection. This rst act is a political commitment inducing a nationwide identication level of environmental protection. The move and regulation scenarios, however, induce dierent levels of identication regarding environmental protection. The move scenario implies a lower eect since only some local protective measures have been implemented. The regulation scenario implies a higher level because it presupposes national, or even international, measures to protect the environment and hence presents the highest degree of similarity with the petition. We therefore expect the highest eect in this particular scenario.

Experimental design

The experiment is a 2 × 3 design aimed at investigating the interplay between a specic political act and stated willingness to pay (WTP) for air quality improvements. Under the experimental conditions, subjects are either oered a petition before the valuation exercise or not, and are presented with three dierent scenarios.

Let us rst describe the three scenarios. The empirical aim of the eld experiment was to elicit willingness-to-pay for a decrease in air pollution. To this end, we considered three typical scenarios from the CV literature devoted to the valuation of air pollution eects. In the rst scenario, called Drug scenario, we adopted the methodology proposed by [START_REF] Alberini | Does the value of a statistical life vary with age and health status? Evidence from the US and Canada[END_REF] and [START_REF] Krupnick | Age, health and the willingness to pay for mortality risk reductions: a contingent valuation study of Ontario residents[END_REF]. The scenario presented subjects with a new drug that has to be taken on a monthly basis. This drug would reduce, by half, the long-term health eects of air pollution exposure.

4 The direct benets of this new drug are therefore essentially at the individual level 5 and air pollution itself is not reduced (see Appendix A for the scenario). The second scenario, called Move scenario (Viscusi et al. 1988;[START_REF] Aprahamian | Modeling starting point bias and unobserved heterogeneity in contingent valuation surveys: an application to air pollution[END_REF], is a choice of moving with the subject's household to one of two cities which have exactly the same characteristics (city size, housing, weather, public services etc.) apart from the cost of living and the level of air pollution. 6 By moving to a less polluted place, the subject was oered the opportunity to improve air quality for him/herself and other members of his/her household (see Appendix B for the scenario). We borrowed the third scenario, called Regulation scenario, from the New Energy Externalities Developments for Sustainability project (NEEDS) supported by the European Commission [START_REF] Desaigues | Final report on the monetary valuation of mortality and morbidity risks from air pollution[END_REF]). This scenario involves new rules and laws to be applied to polluting rms and activities (industries, transport, etc.). Since the introduction of these new laws and rules would increase the cost of living, through market prices, the subject is asked to state the maximum s/he would be willing to pay each month to implement this policy.

7 In this scenario, not only the subject and his/her family but also the whole community would benet from the implementation of this new policy (see Appendix C for the scenario). It is worth noting that the changes in air pollution proposed in each scenario were identical: air pollution health eects reduced by half with the same situation of reference.

Respondents were recruited through advertisements in local newspapers and on local TV news -the survey was described as being about quality of life.

Each participant was paid A C 20 (about US $26) in gift vouchers. Respondents were unaware of the exact topic of the survey prior to the experiment. The experiment was conducted in the Regional Council council chamber. The voting room is equipped with an electronic voting system that allows some of the information to be collected in real time from up to 120 participants and ensure that experiments are carried out in the same manner in each session. On the rst day, three sessions were implemented, each devoted to one particular scenario (scenarios were drawn randomly). On the second day, three sessions were also implemented in the same way but, prior to the CV experiment itself, subjects were asked to sign a petition.

8 The petition asked future candidates in the 2007 presidential election to take an ocial stand concerning environmental preservation. 9 Three students from the Department of Psychology, who presented themselves as apolitical students, asked subjects to sign the petition before entering the voting room. The students were given a script to learn by heart. The script was therefore the same each time they presented the petition to subjects (see appendix D for a full translation). To correlate the information on subjects' willingness to sign the petition with their respective answers in the CV experiment, we assigned to each subject a number corresponding to their voting seat when they registered. Two other experimenters noted the number worn on the subject's chest, and whether or not s/he signed the petition (CV answers were also identied using this number). 10 Our design thus allowed us to keep track of subjects throughout the experimental procedure.

The CV experiment itself was divided into three parts. The rst part of the experiment contained a self-administered survey with questions on the socioeconomic background of the respondent and knowledge of air pollution. Thereafter the scenario was presented to respondents. Their willingness to pay regarding air pollution was elicited using a sequential procedure. In this procedure, scientic information on the eects of air pollution 11 and information on other subjects' willingness to pay (gathered through the electronic voting system) was progressively introduced. Finally, subjects were asked about their willingness to pay for a reduction in the eects of air pollution in an open-ended question at the end of the sequential procedure (see [START_REF] Chanel | Does public opinion inuence WTP? Evidence from the eld[END_REF] for a complete description of the elicitation procedure). In the present article, we focus on the answer to the nal open-ended elicitation question. Follow-up questions allowing protest responses to be identied were also provided at the end of the experiment.

The CV experiment provides us with two dierent tracks to investigate the impact of the political action taken through the petition. The rst track is whether exposure to the petition aects the likelihood of protest responses, and if so, whether this varies with the scenario. The second track is whether taking a political action before the valuation exercise aects willingness to pay.

Results

The experiment was carried out in October 2006 with a total number of 441 participants for the six sessions. As registration via the Public Economics Institute website or phoneline was not mandatory, the total number of participants in each session varies (see Table 1 for the number of subjects in each session and in Appendix E for the sample means and standard deviations of key socioeconomic variables).

Unconditional petition eect

Table 1 presents descriptive statistics of the experiment. A large proportion of respondents agreed to sign the petition asking the future candidates to take an ocial stand in their program regarding environmental preservation: 82.3%. This is not surprising given that, when the experiment took place, this issue was receiving a lot of attention in the French media. Moreover, numerous studies in social psychology show that when people are given the opportunity to perform a small inexpensive act (in our case, signing a petition related to a current issue and involving no monetary cost), many people freely choose to do so [START_REF] Katzev | Can commitment change behavior ? A case study of environmental actions[END_REF][START_REF] Pallack | Commitment and energy conservation[END_REF] 12 The number of protests diers between the drug scenario and the move and regulation scenarios (p-values lower than 0.0001 for both comparisons). It is markedly higher for the former, regardless of whether the petition has previously been oered (16.9% vs 2.0% for move and 2.4% for regulation on average). There is also a small but non-signicant decrease in protest responses when the petition has been oered, for the move and regulation scenarios.

Let us now consider the WTP statistics.

13 Table 1 shows rst that mean WTP diers depending on which scenario has been implemented. The mean WTP obtained in the move scenario, taking both with and without petition responses together(A C83.78), is signicantly and substantially higher than in the drug (A C23.01 with p < 0.0001) and the regulation (A C27.13 with p < 0.0001)

scenarios. There are no signicant dierences between mean WTP of the drug or regulation scenarios (p = 0.206 14 ). Second, the impact of the petition on mean WTP diers according to the scenario evaluated in the session. The impact of the petition on mean WTP is negative in the drug (p = 0.043) and move (p = 0.092) scenarios, whereas it is positive in the regulation scenario (p = 0.034). However, such signicant dierences in mean WTP could be due to the heterogeneity of respondents between each session. One primary source of heterogeneity in respondents' WTP is income. Table 1 also provides mean WTP divided by income to account for potential heterogeneity. Results show that the dierences in WTP induced by the petition also apply to the drug (3.77% vs. 1.51% -p = .054) and regulation (2.73% vs. 6.22% -p = .038) scenarios whereas the impact of the petition on the ratio of WTP over income in the move scenario is not signicant (8.48% vs. 8.98% p = .204).

Conditional petition eect

We have so far only considered dierences in mean WTP or taken into account respondents' income when analysing dierences in mean. Conrmation that the results still hold when overall potential heterogeneity of respondents is taken into account is provided by regressing explanatory variables as well as experimental condition variables on the logarithm of WTP both on the subsample of

Note.

Percentage changes are derived from the econometric model and have to be read as marginal changes. That is, the WTP elicited in the Drug scenario is the referent for scenario eects and the WTP elicited without the petition in each scenario is the referent for petition eects.

Regressing logarithm of WTP on the various explanatory variables gave the following results. In particular, the logarithm of income is highly signicant, indicating that the higher the income, the higher the WTP. This is reassuring (see [START_REF] Hausman | Contingent Valuation : A Critical Assessment, Contribution to Economic Analysis[END_REF] for a discussion on this issue) and provides evidence of the validity of the stated preference experiment (which Bishop and Woodward 1995 dened as theoretical construct validity). Having Senior High School Education level (Bac), Having a complementary private health insurance (P rivHealthIns), Going to the countryside regularly to breath pure air (F reshAir) and Declaring good or very good health (GoodH ealth) have a positive impact on WTP. Living in Marseilles (LiveM ars), Declaring that air quality in Marseilles is good (GoodQualiM ars) and Never sorting waste for recycling (N evSortW aste) have a negative impact on WTP values. Interestingly, the latter is also connected to social awareness.

We now describe the eect of the scenarios on WTP values (referent is the drug scenario). Econometric results conrm the unconditional mean tests: WTP values elicited in the move scenario are higher than those elicited in the drug scenario (p < 0.001). WTP values elicited in the regulation scenario are higher, but not signicantly, than those elicited in the drug scenario (p = .203).

The percentage changes in WTP values induced by the move and regulation scenarios with respect to WTP elicited in the drug scenario (referent) are presented in Figure 1. 16 In particular, it shows that WTP elicited in the move scenario are in average 210.6% higher than those elicited in the drug scenario.

Three dierent dummies are introduced in the WTP equation to investigate the eect of signing the petition on WTP values one for each scenario. The results are as follows. The petition has a negative eect on WTP values elicited in the drug scenario: a one-sided test rejects non-negativity with p = .084. The petition has no signicant eect in the move scenario (p = .316), either positive or negative. The impact of the petition in the regulation scenario is positive and signicant: a one-sided test strongly rejects non-positivity of the eect with p=.017. The percentage changes in WTP values induced by the petition in the drug, move and regulation scenarios are presented in Figure 1 and are -26.72%, 17.55% and 38.69% respectively.

Discussion

In this eld experiment, we shed new light on the consumer/citizen distinction often used in the literature to explain respondents' answers in CV surveys.

We do so by giving respondents the opportunity to behave politically by signing a petition about protection of the environment prior to a standard CV questionnaire dealing with air pollution using three dierent scenarios. Inspired by social psychology research, we argue that signing the petition is likely to induce an attitude more oriented toward environmental protection by turning subjects into citizens concerned with environmental protection. Accordingly, we suggest that signing the petition should have a positive impact on willingness to pay values in the regulation and move scenarios (rst hypothesis) but a negative one in the drug scenario. This is because the drug scenario does not protect the environment in any strict sense and is therefore incompatible with being a citizen committed to environmental protection (second hypothesis). Moreover, the impact of signing the petition should be greater in the regulation scenario than in the move scenario because the regulation scenario is much more closely related to the petition (third hypothesis).

Having presented in the previous section a number of results, via unconditional tests and econometrics, we will now comment on them further, highlighting certain results. First, there is a low number of protest responses in our sample (7.8% for the whole sample). This result is in line with what [START_REF] Chanel | Does public opinion inuence WTP? Evidence from the eld[END_REF] found using a similar valuation procedure also dealing with air pollution and based on the same referent population. The petition has no signicant impact on the protest responses in our sample, whatever the scenario considered. The petition is therefore not a determinant of respondents' agreeing to participate in the valuation exercise. Since up to ninety percent of subjects agreed to pay that control groups without petition, it would have been surprising to nd that the petition had an impact on protest responses. Second, committed citizens, that is, subjects who signed the petition, are more likely to pay more in the move and regulation scenarios (although the eect is not significant in the move scenario). Our rst hypothesis is therefore partly supported by our sample. The eect of the petition on willingness to pay values elicited in the drug scenario is negative, therefore in accordance with our second hypothesis. The strongest positive eect of the petition is found in the regulation scenario and this supports our third hypothesis.

How do our results relate to previous ndings? Experimental studies show that invoking moral values or responsibility indeed increases individuals' willingness to pay. For instance, [START_REF] Boyce | An experimental examination of intrinsic values as a source of the WTA-WTP disparity[END_REF] induce a moral dimension in a valuation experiment by destroying unsold goods (small Norfolk Island pine trees). If the subject submits a bid to purchase the tree lower than the selling price (unknown by the subject when s/he submits his/er bid), the pine tree is killed by the experimenter. They show that subjects endowed with a moral responsability, that is in the 'kill' experimental condition, pay more than in the 'no-kill' condition (where unsold goods are not destroyed). In a paired comparison study, [START_REF] Clarke | The inuence of attitude priming and social responsability on the valuation of environmental public goods using paired comparisons[END_REF] show that when the decision is public goods are higher as compared to a situation where the decision is shared between subjects. This is what we, too, obtain in the move and regulation scenarios. Our experimental ndings, as well as previous ndings, are compatible with the view that the respondent adopts a social welfare perspective when s/he acts as a citizen (rather than considering his/er own private interest only when s/he acts as a consumer). Such a respondent will have a higher willingness to pay such as in [START_REF] Nyborg | Homo economicus and homo politicus: interpretation and aggregation of environmental values[END_REF]'s model because s/he cares about the welfare of others when s/he acts as a citizen.

Results however dier for the drug scenario, where subjects who have signed the petition tend to express lower willingness to pay values. This behavior cannot be explained by considering that subjects who act as citizen adopt a social welfare perspective because welfare of other members of the society is not considered in this scenario (remember that the drug scenario involves only personal health benets). We rather argue, based on the theory of commitment borrowed from social psychology, that the petition in our experiment has induced a commitment to environmental preservation. Taking the drug would mean decreasing the health eects of air pollution but not protecting the environment, and this is incompatible with being a citizen who is concerned about environmental protection. This incompatibility between the drug scenario and the attitudes of respondents as citizens would explain the decrease in stated willingness to pay.

Conclusion

Does the consumer/citizen distinction matter in stated preferences surveys?

First, our results suggest that respondents would give a dierent answer depending on whether they reason from a consumer or citizen perpective. Second, the sign and the size of the eect on stated willingness to pay of being a citizen rather than a consumer depend on the (in)compatibility and the degree of similarity between attitudes as a citizen and the hypothetical scenario implemented in the survey.

What are the consequences of these ndings for the use of stated preferences methods in public decision making? If cost-benet analysis is interpreted as a market simulation, practionners of CV would favour the consumer perspective (Sugden 2005). However, our results indicate that respondents may adopt a citizen perspective even in a typical consumer perspective scenario, such as in the drug scenario in our study. The hypothetical scenario alone may therefore not guarantee that respondents answer as consumers to the willingness to pay question. Hence, one may want to identify respondents who act as citizens. To this end, the real meaning of monetary values can be analysed by using attitude questions that can be correlated to willingness to pay values (see for instance [START_REF] Blamey | Respondents to contingent valuation surveys: consumers or citizens[END_REF][START_REF] Blamey | Decisiveness, attitude expression and symbolic responses in contingent valuation surveys[END_REF][START_REF] Nunes | Identifying the warm glow eect in contingent valuation[END_REF]. We argue that questions which investigate past behaviour of respondents as citizens (such as belonging to an environmental association or to a union, or having signed a petition,...) should also systematically be added to the analysis.

How do we then account for willingness to pay of respondents acting as citizens? One the one hand, one may remove these answers from the analysis because adding personal and social values is like adding apples and oranges (Nyborg 2000, p.319). Another alternative would be to argue that what matters is the willingness to pay at the moment of decision (Sugden 2007). To follow this route, consider a person who enters a furniture shop to have a look around but with no clear intention to buy anything right now. When s/he enters, the salesman oers him/er a coee and suggests s/he may sit on a sofa s/he was looking at. The person accepts the coee and sits down on the sofa. Research in social psychology evoked sooner in this paper explains that the invitation of the salesman is a typical foot-in-the-door technique (of which the salesman may be aware or not). This technique dramatically increases the probability that the person eectively buys the sofa. Proponents of willingness to pay at the moment of decision approach would certainly consider that this purchase is a valid economic decision and it is this purchase that should be taken into account if the government were to consider subsidizing national sofa production.

18

However, if this is accepted, then so should the idea that the willingness to pay of our committed citizens can be taken into account for public policy.

mortality and morbidity risks induced by air pollution [START_REF] Künzli | Public-health impact of outdoor and trac-related air pollution: a European assessment[END_REF]. In the move and regulation scenarios, we also reminded subjects of the non-health eects of air pollution.

12 For instance: I do not agree with the principle of paying, I would not pay since I will only move to live in the country, I do not agree to pay to move to a less polluted place when I can die tomorrow crossing the street or I do not want to pay because the factories are the major polluters).

13 WTP statistics are computed using non-protest responses only.

14 In the following, unconditional mean tests are two-sided. 15 The model is estimated in Stata 9.0 and p-values are computed using the robust variancecovariance matrix. Some observations are dropped from the analysis due to missing values, mainly on income. We also estimate a Box-Cox linear model where the WTP is transformed according to a Box-Cox transformation (see [START_REF] Davidson | Estimation and Inference in Econometrics[END_REF]. Tests on the estimated transformation parameter (θ = .0208) reject equality to one (p < .0001) or minus one (p < .0001) but fail to reject nullity (p = .547), that is using the logarithm of the WTP.

In doing so, we however lose some observations, that is, true zero WTP, 11 observations hence 2.72% of the sample. We also exclude protest responses (34 subjects 7.8% of the total sample). As we only consider subjects who agreed to sign the petition (when given the opportunity) and subjects who gave a strictly positive WTP, we also test for potential selection biases that would undermine our previous results. We do so by estimating Heckman selection models [START_REF] Heckman | Sample selection bias as a specication error[END_REF], rst considering the selection of subjects and then, in a second equation, estimating the WTP model. We introduce into the selection equation the same covariates as in the WTP equation (covariates in the WTP equation are kept the same).

Wald tests of independent equations show that there is no selection bias induced by protest responses (p = .2972), nor for subjects who did not sign the petition (p = .923) the joint test is p = .491.

16 Because the WTP equation is semilogarithmic, the percentage changes in WTP values induced by the petition are computed as per [START_REF] Kennedy | Estimation with correctly interpreted dummy variables in semilogarithmic equations[END_REF].

17 The subject is told that s/he has been randomly selected to make the decision for all other participants.

18 So called assymetric paternalists (see, for instance, [START_REF] Camerer | Regulation for conservatives: behavioral economics and the case for 'asymmetric paternalism[END_REF] may not be happy with this behavioural approach that induces people to make purchases without their prior intention and hence would contest the validity of the consumer surplus' evaluation based on those purchases. Arguing pro or contra assymetric paternalism however is beyond the scope of this paper.

The problem is that the cost of living is higher in LESSPOL (the less polluted city): housing, local taxes, public transport, etc. are more expensive. This means that if you choose to move to LESSPOL, you will have to pay more to have the same standard of living as in POL.

We would like to know how much you would be willing to pay per month for you and your household to move to LESSPOL (the less polluted city) rather than to POL (the town as polluted as Marseilles). Do not forget that this money will be drawn from your household's budget! You will therefore have less money at the end of the month for consumption and savings.

C Scenario 3: Regulation scenario You are going to be the main actor in our scenario. You will have to take the best decision for yourself and your household.

Let's imagine that new laws and rules are to be adopted to limit air pollution. Therefore industries, manufacturers of consumer products, public or private transport, will have to adopt less polluting technologies. Studies have shown that these new laws and rules will make it possible to reduce by half the number of highly polluted days in the PACA region, and particularly in Marseilles.

The implementation of these new technologies will induce higher costs in every-day life: energy, food and other goods, transport. This means that you will have to pay more to enjoy the same standard of living as before the implementation of these new laws and rules.

We would like to know how much you would be willing to pay per month for these new laws and rules to be implemented. Do not forget that this money will be drawn from your household's budget! You will therefore have less money at the end of the month for consumption and savings.

D Petition

Soon, all the candidates for the next presidential election will be known. You may know that for several days, since a few days, a petition has been circulating in France. The aim of this petition is to obtain a commitment from each candidate, whatever his/her political background, to include the questions of environmental conservation in his/er program, because it is important that each of us is able to make informed choices.

We need a lot of signatures. Therefore, if you think, like us, that issues of environmental conservation are essential issues on which candidates for the presidential election should take a clear stance, we invite you to sign this petition. Thank you.

Table 1 :

 1 Summary statistics by treatment condition and scenario (N = 441)

	Scenario		Standard	Petition	Both
	Drug	of respondents	49	117 (82.9% signatures)	166
		of protests (rate)	8 (16%)	20 (17%)	28 (16.9%)
		Mean WTP (sd)	A C30.22 (8.18)	A C16.62 (1.67)	A C23.01
		Mean WTP/Income	3.77 %	1.51 %	2.26 %
	Move	of respondents	86	63 (84.1% signatures)	149
		of protests (rate)	3 (3.5%)	0 (0%)	3 (2.0%)
		Mean WTP	A C94.76 (12.64)	A C72.62 (9.25)	A C83.78
		Mean WTP/Income	8.48 %	8.98 %	8.87 %
	Regulation	of respondents	63	63 (79.3% signatures)	126
		of protests (rate)	2 (3.2%)	1 (1.6%)	3 (2.4%)
		Mean WTP	A C24.66 (3.58)	A C36.85 (5.03)	A C27.13
		Mean WTP/Income	2.73 %	6.22 %	4.63 %
	All scenarios	of respondents	198	243 (82.3% signatures)	441
		of protests (rate)	13 (6.5%)	21 (8.6%)	34 (7.8%)
		Mean WTP (sd)	A C57.34 (6.54)	A C38.16 (3.42)	A C46.88
		Mean WTP/Income	4.82 %	5.56 %	5.30 %

Table 2 :

 2 Semi-logarithmic WTP Model (N =341) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

		Parameter
	Variable		(p-value)
		Estimate
	Subject's characteristics
	Constant term	.4632	(0.521)
		M ale	-.0585	(0.577)
	log(Age)	.0129	(0.948)
		Bac	.2545	(0.041)
	log(N P ers)	.1287	(0.200)
	log(Income)	.2055	(0.004)
	LiveM ars	-.2042	(0.091)
		Owner	.0944	(0.423)
	P rivHealthIns	.3419	(0.016)
	GoodQualiM ars	-.3067	(0.081)
	KnowAir	-.0418	(0.745)
	KnowAtmo	.0328	(0.818)
	KnowAirmaraix	-.1966	(0.115)
	N evSortW aste	-.2969	(0.013)
		Hab	.0803	(0.428)
	F reshAir	.2785	(0.048)
		Smoker	.0768	(0.637)
	Goodhealth	.2392	(0.036)
	Scenario eects
		Move	1.1563	(0.000)
	Regulation	.2639	(0.203)
	Petition eects
	Petition × Drug	-.2890	(0.168)
	Petition × Move	.1773	(0.316)
	Petition × Regulation	.3399	(0.035)
	Wald joint nullity test: p < 0.0001
	if p-value<0.01,	if p-value<0.05 and	if p-value<0.1
	respondents who signed the petition and on those who only answered the CV
	questionnaire. 15 Table 2 contains the estimation results (the rst column con-
	tains the parameter estimates and the second column contains the associated
	p-values). The set of covariates is left relatively large to avoid misspecication
	biases.

% . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

The presidential election was to be held 6 months later. At the time of the experiment, a petition, promoted in the French media, was actually circulating in France. Our petition was eectively sent to each candidate.

When we were unsure whether or not certain subjects had signed the petition, they were identied later using the video system of the voting room.

Information was presented simultaneously on individual screens and aloud to ensure that all respondents had the same information. It was based on epidemiological data on long-term

Financial support of French Research Ministry (n 04 5 666 ACI Santé et Environnement) and technical support from the Provence-Alpes-Côte-d'Azur

sity Press. [START_REF] Kahneman | Experienced utility as a standard of policy evaluation[END_REF]. Coping with preference anomalies in cost-benet analysis: a market simulation approach. Environmental and Resource Economics, 32, 129160. Sugden, R (2007). Cost-benet analysis as market simulation: A new approach to the problem of anomalies in environmental evaluation. Resources for the future discussion paper. RFF DP 07-28-REV.

Viscusi, W., W. A. Magat, and A. Forrest (1988). Altruistic and private valuations of risk reduction. Journal of Policy Analysis and Management, 7,

227245.

Wang, T.H. and R.D. Katzev (1990). Group commitment and resource conservation : two eld experiments on promoting recycling. Journal of Applied Social Psychology, 20, 265275. Notes 1 Here, we mean by political behaviour a behaviour relying on social preferences.

2 This argument stems from self-attribution theory [START_REF] Bem | Self-perception theory[END_REF] and is central to self- signaling models in economics. In self-signaling models, agents derive utility from the outcome of actions, outcome utility, but also derive diagnostic utility from the information that the action provides on some underlying trait or disposition of themselves (see for instance [START_REF] Bodner | Self-signaling in a neo-calvinist model of everyday decision making[END_REF].

3 In their real experiment, [START_REF] Freedman | Compliance without pressure: the footin-the-door technique[END_REF] telephoned housewives in Palo Alto, California. They asked them if they would be willing to answer a few questions about the kind of soaps they used. Two or three days later, the subjects were asked if they would agree to have ve or six men come to their house for two hours or so to classify the household products they used. In the control group, only the second request was made. In the control 22.2% accepted, while with foot-in-the-door, 52.8% accepted.

4 Long-term health eects of air pollution data are derived from current air pollution in Marseilles. Note also that our approach diers from [START_REF] Alberini | Does the value of a statistical life vary with age and health status? Evidence from the US and Canada[END_REF] and [START_REF] Krupnick | Age, health and the willingness to pay for mortality risk reductions: a contingent valuation study of Ontario residents[END_REF]. When they present the new drug to respondents, they do not refer explicitly to air pollution the drug is only associated with a reduction in mortality risks. In that sense, their approach is non-contextual.

5 The scenario may also involve indirect benets to the extent to which relatives of the subject can also benet from his or her better health.

6 Air quality in Marseilles, the largest city in the area, was used as a referent for all respondents.

7 Here again, air quality in Marseilles was used as a referent.

8 We chose to introduce the petition only on the second day in order not to contaminate the sessions without the petition. Three other sessions related to other topics were implemented in between.

A Scenario 1: Drug scenario You are going to be the main actor in our scenario. You will have to take the best decision for yourself and your household.

Let's imagine that the Ministry of Health has authorized a new drug, dierent from those currently available. It allows the long-term eects of air pollution exposure to be reduced by half. It simply involves taking a pill every month, which, by the combined action of several vitamins, stimulates immunological responses. It has no side eects and no contra-indications. It can be taken by any person who is more than 5 years old. This new drug is reimbursed neither by the social security system nor by the CMU (state means-tested health cover) nor by the complementary health insurances policies. This implies that, if you choose to buy it, you will bear the full cost. We would like to know how much you would be willing to pay to use this drug, which would reduce by half the long-term diseases and mortality risks associated with air pollution. Do not forget that this money will be drawn from your household's budget! You will therefore have less money at the end of the month for consumption or savings.

B Scenario 2: Move scenario

You are going to be the main actor in our scenario. You will have to take the best decision for yourself and your household.

Let's imagine that you and your household have to move. You can choose between two cities which are exactly equivalent in terms of inhabitants, working conditions, schools, climate, public services, cultural life, transport, housing, surroundings, etc. There is only one dierence between them: the level of atmospheric pollution. The rst city -let's call it POL -is as polluted as Marseilles. And the second city -let's call it LESSPOL -is half as polluted as Marseilles.

E Sample characteristics