Denisse Sciamarella
email: denisse@limsi.fr

Fabrice Silva

LFD G Artana
email: gartana@fi.uba.ar

Similarity analysis of a glottal-like jet

Keywords: glottal flow, self-similarity, high aspect ratio jets, axis switching

published or not. The documents may come

Introduction

Glottal flow is a pulsating jet with lateral confinement. Pulsation is produced by the flow-induced motion of the vocal folds which open and close repetitively during voice production, defining a time-varying jet exit section with a lenslike shape. Confinement is introduced by the presence of the vocal tract, which extends from the glottis to the mouth, inhibiting the phenomenon of flow entrainment from the atmosphere as observed in free jets.

The glottal jet presents a number of characteristics that may be of interest for biomechanical modeling and other applications, such as the development of bio-inspired devices for flow control purposes. Recent studies [START_REF] Triep | Three-dimensional nature of the glottal jet[END_REF][START_REF] Krebs | A three-dimensional study of the glottal jet[END_REF] have stressed its three-dimensional axis switching properties: the jet has a crosssection evolution in which the minor and major axes switch identity as the downstream distance from the jet exit increases. This study deals with selfsimilarity properties that are found to hold locally, including the zone where axis switching occurs.

Why and how self-similarity properties can be exploited is thoroughly discussed in [START_REF] Cantwell | Introduction to Symmetry Analysis[END_REF].When the equations governing a phenomenon present a certain invariance, the invariance is also present in the solution, which can be expressed in terms of a reduced set of combination of the basic variables. The variables in this set are called similarity variables, and the analysis leading to them is called a similarity analysis. Once the similarity variables are known, alternative techniques such as numerical analysis can be applied more effectively and with a better basic understanding of the problem. Because the invariance requirements inevitably force the suppresion of certain parameters of the real problem, any claims for the generality of a similarity property must be tempered by comparison with experiment.

In the field of flow control, self-similarity has been reported both for continuous and synthetic jets [START_REF] Agrawal | Similarity analysis of planar and axisymmetric turbulent synthetic jets[END_REF]. Similarity analyses have been provided for continuous turbulent round jets, for instance in [START_REF] Burattini | Synthetic jets[END_REF], and for elongated synthetic jets [START_REF] Krishnan | An experimental and analytical investigation of rectangular synthetic jets[END_REF]. The elongated jet in [START_REF] Krishnan | An experimental and analytical investigation of rectangular synthetic jets[END_REF] is modeled as a continuous planar jet without special consideration of the non planar properties associated to the axis switching phenomenon. This work presents a similarity analysis for a glottal-like jet taking into account its three-dimensional nature, and using experimental observations as guide.

Time-averaged profiles documented in [START_REF] Krebs | A three-dimensional study of the glottal jet[END_REF] are such that a simple rescaling of the streamwise velocity yields a collapse onto a single curve for a wide range of streamwise distances. Since self-similarity is mostly deduced by the good collapse of mean velocities and Reynolds stresses, results in [START_REF] Krebs | A three-dimensional study of the glottal jet[END_REF] suggest that a similarity analysis may lead to interesting results for the glottal jet.

Self-similarity can be defined in more than one way, including different types and degrees [START_REF] George | The self-preservation of turbulent ows and its relation to initial conditions and coherent structures[END_REF]. A strong kind of self-similarity, called self-preservation, is said to occur when a single length and velocity scale is sought to bring into congruence all the velocity profiles (or other quantities). A weaker type of self-similarity is said to occur if every dependent variable is allowed to have its own scale, with each scale evolving downstream in a manner that is determined by the equations of motion. In such cases, self-similarity results from the constraint that all terms in the governing equations should evolve in the same manner [START_REF] George | The decay of homogeneous isotropic turbulence[END_REF]. As in [START_REF] Burattini | Synthetic jets[END_REF], this work will adopt the less restrictive definition.

For the high-aspect-ratio geometry of our case study, it is convenient to restrict the similarity analysis to certain privileged (symmetry) planes that coincide, physiologically, with the mid-sagittal and mid-coronal planes. We introduce the term in-plane similarity to refer to cases in which self-similarity occurs at least within certain flow planes. This analysis discloses normalization scalings leading to the collapse of all the mean velocity components and of the mean Reynolds stresses profiles of the glottal jet data. In addition, it provides relationships indicating the different self-similar ranges and predicting characteristics of the axis switching phenomenon.

The article is organized as follows. Section 2 presents the source experiment and the particular features of the generated glottal jet, which pose the constraints of the similarity analysis that is undertaken. Section 3 derives the governing equations for a high-aspect-ratio jet in the two planes of symmetry of the flow in the assumption of self-similarity. Section 4 combines the experimental data with the scaling relations derived from the flow equations. Section 5 discusses how the similarity properties condition the axis switching phenomenon. Conclusions of the analysis are drawn in Section 6.

Glottal Flow

During voice production, the glottal source has the shape of an elongated orifice with sharp corners at the commissures, that presents a time-varying crosssection as the vocal folds oscillate. Vocal-fold vibration can include complete or incomplete glottal closure. If there is a phase of complete glottal closure, the airflow during this phase is interrupted. In the open phase, a high-aspect-ratio jet is formed that is laterally confined due to the presence of the vocal tract. For experimental characterization with velocimetry techniques, the glottal jet can be generated using a physical model of the vocal folds.

The experimental data used as guide for this work were obtained in a Stereoscopic Particle Image Velocimetry (SPIV) experiment presented in [START_REF] Krebs | A three-dimensional study of the glottal jet[END_REF]. The jet in this study was generated with a life-size vocal fold model based on a design developed in [START_REF] Ruty | An in vitro setup to test the relevance and the accuracy of low-order vocal folds models[END_REF][START_REF] Ruty | Influence of acoustic waveguides lengths on selfsustained oscillations: Theoretical prediction and experimental validation[END_REF]. A sketch of the experimental setup is given in The SPIV measurements were conducted with the vocal fold valve attached to a uniform square-sectioned duct (20 mm× 20 mm× 17 cm). The inspected flow region was 20 mm long and was chosen to lie as close as possible (x = 6 mm) to the valve exit (x = 0 mm). Time averaged flow fields are the mean of the 1000 instantaneous velocity fields captured for each of the 9 coronal (xy) slices throughout the SPIV experiment -data acquisition lasted 200 seconds per slice.

Previous studies regarding high-aspect-ratio jets, whether pulsating or nonpulsating [START_REF] Ho | Vortex induction and mass entrainment in a small-aspect-ratio elliptic jet[END_REF][START_REF] Hussain | Elliptic jets. part 1. characteristics of unexcited and excited jets[END_REF][START_REF] Chua | Measurements of a confined jet[END_REF][START_REF] Smith | The formation and evolution of synthetic jets[END_REF][START_REF] Yoon | Investigation of the near-field structure of an elliptic jet using stereoscopic particle image velocimetry[END_REF][START_REF] Quinn | Experimental study of the near field and transition region of a free jet issuing from a sharp-edged elliptic orifice plate[END_REF][START_REF] Murugappan | Flow-structure interaction effects on a jet emanating from a flexible nozzle[END_REF], generally include a check for the collapse of the time-averaged streamwise velocity profile in the minor plane. These studies, however, use a single length and velocity scale in the normalization of the profiles, as mentioned in our reference to self preservation. Two remarks are necessary regarding these studies. On one hand, mean velocities are poor indicators of self-similarity, while other quantities, such as Reynolds stresses, provide a far more sensitive indication of the self-similar nature of a wide range of flows cf. [START_REF] Wygnanski | Some measurements in the self-preserving jet[END_REF][START_REF] Browne | The interaction region of a turbulent plane jet[END_REF][START_REF] Chua | Measurements of a confined jet[END_REF]. On the other hand, in the absence of an analysis leading to plane-dependent length scales, no property of self-similarity comes out at all. This work will show that further insight on the similarity properties of this type of jet can be gained combining the in-plane flow equations with the experimental profiles of the three components of the mean velocity and Reynolds stresses.

An analytical derivation of the self-similarity from the equations of motion and conservation of momentum is seldom possible for complex flows [START_REF] Chua | Measurements of a confined jet[END_REF]. Restricting the study to certain privileged planes will allow us to overcome this limitation for the investigated jet. Complexity in the glottal jet is associated to three aspects. The first one is geometrical in nature, the second one re-gards the temporal evolution of the jet and the third one is associated to the boundary conditions. Let us make a brief review of how self-similarity may be affected by these flow features. During the open phase, the outlet presents a time-varying cross section, with an aspect ratio that is also time-varying. Jet aspect ratio may play an important role in determining, for instance, the switching location of an axisswitching jet [START_REF] Gutmark | Flow control with noncircular jets[END_REF]. In line with this trend, the three-dimensional reconstruction in [START_REF] Krebs | A three-dimensional study of the glottal jet[END_REF] shows that the switching location of the glottal jet exhibits an evolution which varies in time in a manner that seems to follow the time-dependence of the jet aspect ratio. This suggests that the glottal jet keeps memory of the time-varying jet exit conditions, at least in the axis-switching region. A similarity analysis of the glottal jet in this region, even if conducted on the time-averaged fields, should comply with a non loss of memory of the averaged jet exit conditions.

(c) Boundary conditions:

In the jet developing region examined here, and further downstream, the glottal jet is laterally confined by the walls of the vocal tract. Confinement is known to affect the centerline position, the expansion rate and the stability of a jet [START_REF] Abramovich | The theory of turbulent jets[END_REF][START_REF] Rees | The effect of confinement on the stability of viscous planar jets and wakes[END_REF]. In unconfined jets, the net rate of mass flow increases with downstream distance due to flow entrainment, thus favoring self-similarity. In laterally confined jets, the absence of this effect is expected to reinforce the jet memory of the source conditions.

The next section shows that an analytical treatment is possible conducting the analysis in the two planes where the similarity equations can be simplified invoking symmetry arguments. Taking up the term used in the introduction, the similarity analysis restricted to these symmetry planes is hereafter termed in-plane similarity.

In-plane Similarity Equations

In this section we derive similarity equations of a high-aspect-ratio jet in the minor and major planes of the jet exit. To do this, we follow firstly the approach in [START_REF] Tennekes | A First Cours in Turbulence[END_REF] and secondly the approach in [START_REF] George | The self-preservation of turbulent ows and its relation to initial conditions and coherent structures[END_REF]. The starting point is to write out the standard Reynolds Decomposition for the time-averaged velocity in the two symmetry planes sketched in Fig 2 . The standard Reynolds Decomposition is typically considered for turbulent flows, but has been previously used for cyclic jet flows [START_REF] Agrawal | Similarity analysis of planar and axisymmetric turbulent synthetic jets[END_REF][START_REF] Krishnan | An experimental and analytical investigation of rectangular synthetic jets[END_REF].

If (U, V, W ) denote the time-averaged flow velocities in (x, y, z), we may assume that W = 0 for plane (z = 0) and that V = 0 for plane (y = 0). It is also assumed that the viscous terms can be dropped. If P is the pressure and ρ the density, the time-averaged streamwise momentum equation in these planes can be written as follows:

(z = 0) U ∂U ∂x + V ∂U ∂y + 1 ρ ∂P ∂x + (1) + ∂ < u 2 > ∂x + ∂ < uv > ∂y + ∂ < uw > ∂z = 0 (y = 0) U ∂U ∂x + W ∂U ∂z + 1 ρ ∂P ∂x + (2)
+ ∂ < u 2 > ∂x + ∂ < uv > ∂y + ∂ < uw > ∂z = 0
with the continuity equation:

∂U ∂x + ∂V ∂y + ∂W ∂z = 0 (3)
Estimating the orders of magnitude of the various terms, it is possible to obtain equations relating the velocity scales (U s , V s , W s ) with the length scales (L x , L y , L z ) and the fluctuating components of the velocities (u s , v s , w s ).

(z = 0) U 2 s L x + U s V s L y + 1 ρ P s L x + u 2 s L x + u s v s L y + u s w s L z ∼ 0 (4) (y = 0) U 2 s L x + U s W s L z + 1 ρ P s L x + u 2 s L x + u s v s L y + u s w s L z ∼ 0 (5)
Notice that the fluctuating velocities are condensing both, phase-averaged and turbulent fluctuations. From (3):

U s L x + V s L y + W s L z ∼ 0 (6)
Similar scaling identities can be written using the cross-stream momentum equations. Taking into account that W is constant and zero in plane (z = 0), and that V is constant and zero in plane (y = 0), we obtain:

(z = 0) 1 ρ ∂P ∂z + ∂ < uw > ∂x + ∂ < vw > ∂y + ∂ < w 2 > ∂z = 0 (7) (y = 0) 1 ρ ∂P ∂y + ∂ < uv > ∂x + ∂ < vw > ∂z + ∂ < v 2 > ∂y = 0 (8) so that, (z = 0) 1 ρ P s L z + u s w s L x + v s w s L y + w 2 s L z ∼ 0 (9) (y = 0) 1 ρ P s L y + u s v s L x + v 2 s L y + v s w s L z ∼ 0 (10)
Replacing ( 7) and ( 8) into ( 4) and ( 5) respectively:

(z = 0) P s ρ ∼ -w 2 s - L z L x u s w s - L z L y v s w s (11) (y = 0) P s ρ ∼ -v 2 s - L y L x u s v s - L y L z v s w s (12)
Now, [START_REF] Hussain | Elliptic jets. part 1. characteristics of unexcited and excited jets[END_REF][START_REF] Yoon | Investigation of the near-field structure of an elliptic jet using stereoscopic particle image velocimetry[END_REF] can be used to eliminate the pressure scale from (4-5), so that:

(z = 0) U 2 s L x + V s U s L y + u 2 s L x + u s v s L y + (13) + (1 - L 2 z L 2 x ) u s w s L z -( L z L x ) v s w s L y - w 2 s L x ∼ 0 (y = 0) U 2 s L x + W s U s L y + u 2 s L x + u s w s L z + (14) + (1 - L 2 y L 2 x ) u s v s L y + ( L y L x ) v s w s L z - v 2 s L x ∼ 0
The analysis is undertaken at some distance from the exit orifice for regions in which L y ∼ L z ≪ L x . This leads to a simplified equation for the streamwise time-averaged momentum equations in the mid-coronal (z = 0) and the midsagittal (y = 0) planes:

(z = 0) U ∂U ∂x + V ∂U ∂y + (15)
+ ∂ < u 2 > ∂x + ∂ < uv > ∂y + ∂ < uw > ∂z - ∂ < w 2 > ∂x = 0 (y = 0) U ∂U ∂x + W ∂U ∂z + (16) + ∂ < u 2 > ∂x + ∂ < uv > ∂y + ∂ < uw > ∂z - ∂ < v 2 > ∂x = 0
Notice that if we had considered the planar jet assumption for planes (z = 0) and (y = 0), the simplified equations would differ from the above equations in the last terms.

To seek for a similarity solution, one assumes that:

U = U s (x)f (ξ, η) (17) V = V s (x)h(ξ, η) (18)
W = W s (x)b(ξ, η) (19)
< u 2 > = R uu (x)g uu (ξ, η) (20)
< uv > = R uv (x)g uv (ξ, η) (21) < uw > = R uw (x)g uw (ξ, η) (22)
where f, h, b, g uu , g uv , g uw are some functions (to be determined empirically), ξ, η are the normalized cross-stream coordinates (ξ = y/L y , η = z/L z ) and U s , V s , W s , R uu , R uv , R uw are functions of x only.

For the fluctuations, different length scales l y and l z could be used so that g uu (ξ ′ , η ′ ), g uv (ξ ′ , η ′ ), g uw (ξ ′ , η ′ ), with ξ ′ = y/l y , η ′ = z/l z . Introducing this distinction does not affect the subsequent derivation of the similarity equations as far as

[l y,x /l y ] ∼ [L y,x /L y ] and [l z,x /l z ] ∼ [L z,x /L z ].
On substituting expressions [START_REF] Rees | The effect of confinement on the stability of viscous planar jets and wakes[END_REF][START_REF] Ruty | Influence of acoustic waveguides lengths on selfsustained oscillations: Theoretical prediction and experimental validation[END_REF][START_REF] Ruty | An in vitro setup to test the relevance and the accuracy of low-order vocal folds models[END_REF][START_REF] Smith | The formation and evolution of synthetic jets[END_REF][START_REF] Tennekes | A First Cours in Turbulence[END_REF][START_REF] Triep | Three-dimensional nature of the glottal jet[END_REF] into [START_REF] Murugappan | Flow-structure interaction effects on a jet emanating from a flexible nozzle[END_REF][START_REF] Quinn | Experimental study of the near field and transition region of a free jet issuing from a sharp-edged elliptic orifice plate[END_REF], one obtains:

(z = 0) [U s U s,x ]f 2 + [ U 2 s L y L y,x ](-ξf f ,ξ )+ (23) [ U s V s L y ](hf ,ξ ) + [ R uv L y ]g uv,ξ + [ R uw L z ]g uw,ξ + [ R uu L y L y,x ](-ξg uu,ξ ) + [R uu,x ]g uu -[R ww,x ]g ww + [ R ww L y,x L y ](ξg ww,ξ ) = 0 (y = 0) [U s U s,x ]f 2 + [ U 2 s L z L z,x ](-ηf f ,η )+ (24) [ U s W s L z ](bf ,η ) + [ R uv L y ]g uv,η + [ R uw L z ]g uw,η + [ R uu L z L z,x ](-ηg uu,η ) + [R uu,x ]g uu -[R vv,x ]g vv + [ R vv L z,x L z ](ηg vv,η ) = 0
where commas before x, ξ or η denote differentiation with respect to these variables. The terms in brackets group the terms that are functions of x only.

For self-similarity to hold, we do not require as in standard similarity analysis applied to round or planar jets (see for instance [START_REF] Agrawal | Similarity analysis of planar and axisymmetric turbulent synthetic jets[END_REF]) that the terms in brackets should be constant. The approach in [START_REF] Tennekes | A First Cours in Turbulence[END_REF] is here abandoned by the proposal in [START_REF] George | The self-preservation of turbulent ows and its relation to initial conditions and coherent structures[END_REF] where a less restrictive condition is required, namely, that all the terms in brackets should have the same x-dependence. Equating the terms in brackets and dividing all the terms by U 2 s , we obtain:

(z = 0) [ U s,x U s ] ∝ [ L y,x L y ] ∝ [ R uv L y U 2 s ] ∝ (25) ∝ [ V s L y U s ] ∝ [ R uw L z U 2 s ] ∝ [ R uu L y,x U 2 s L y ] ∝ ∝ [ R uu,x U 2 s ] ∝ [ R ww,x U 2 s ] ∝ [ R ww L y,x L y U 2 s ] (y = 0) [ U s,x U s ] ∝ [ L z,x L z ] ∝ [ R uw L z U 2 s ] ∝ (26) ∝ [ W s L z U s ] ∝ [ R uv L y U 2 s ] ∝ [ R uu L z,x U 2 s L z ] ∝ ∝ [ R uu,x U 2 s ] ∝ [ R vv,x U 2 s ] ∝ [ R vv L z,x L z U 2 s ]
It is important to notice that the requirement that the terms in brackets should be constant would lead to a linear growth rate in both cross-stream directions (y and z). This does not hold for the glottal jet data: as shown in the following section, such a condition is fulfilled in the y direction but not in z direction. The looser requirement we adopt allows, instead, for many more classes of functions satisfying the self-similarity constraint.

Combining (25) and (26) one obtains the following requirement:

[L y,x /L y ] ∝ [L z,x /L z ] ∝ [U s,x /U s ] (27)
Note however that relations in (27) need not hold simultaneously in the whole range of x values. Relations (25-26) yield the scalings that will be used to rescale the profiles in the minor and major planes. For example, V s is such that V s ∝ U s L y,x in z = 0, and hence, V (x i , y, 0) at different streamwise distances x i with i = 1, 2, ... can be normalized dividing by U s L y,x . This is not the only normalization scale compatible with (25). Another possible choice for V in z = 0 would be V s ∝ U s,x L y . If there is in-plane self-similarity, these scalings should make the profiles for different x i with i = 1, 2, ... collapse onto a single curve within a certain range of the x axis. Because the dependent variable V of this example scales with U s L y,x , requirement (27

) reduces to [L y,x /L y ] ∝ [U s,x /U s ].
Similar scalings can be derived for other quantities (velocities or Reynolds stresses) as shown in Table 1. Notice that the similarity scalings are profileand plane-dependent, differing from the conventional use of a single length and velocity scale.

Profiles

In-Plane Similarity Scaling

z = 0 y = 0 U Us Us V UsLy,x - W - UsLz,x < uu > U 2 s U 2 s < vv > U 2 s U 2 s < ww > U 2 s U 2 s < uv > U 2 s Ly,x Us,xUsLy < uw > Us,xUsLz U 2 s Lz,x
Table 1: Normalization similarity scalings for velocities and Reynolds stresses.

In-plane Similarity Analysis of the Glottal Jet Data

In this section we apply the in-plane similarity relations derived in the previous section to the glottal jet data.

For high aspect ratio nozzles, data are often rendered dimensionless using the jet exit velocity U 0 and the equivalent diameter D e . The latter is defined as the diameter of a circular jet with momentum flux equal to that of the elongated jet [START_REF] Hussain | Elliptic jets. part 1. characteristics of unexcited and excited jets[END_REF]. For the nozzle in motion of the present study, D e can be defined following [START_REF] Murugappan | Flow-structure interaction effects on a jet emanating from a flexible nozzle[END_REF]. With this definition, the glottal jet experiment has an equivalent diameter D e ∼ 4 mm [START_REF] Krebs | A three-dimensional study of the glottal jet[END_REF].

In the context of the similarity analysis we wish to undertake, a natural choice for the velocity scale is the centerline velocity U c , since U c (x) is the maximum velocity of the profile at any x position. This choice already assures that all U/U s profiles will collapse at least on one point, located at the profile's maximum. The in-plane similarity analysis calls for one length scale per symmetry plane. These two length scales, labeled L y and L z in the previous section, need not coincide. For the glottal jet data, L y and L z can be conveniently chosen following two alternatives. The first one is to fix a second point of collapse for the U/U s profiles, given for instance by the jet half width. The second alternative is to infer L y and L z from the proportionality relations in the previous section, namely:

L y ∝ V s U s,x (28)
L z ∝ W s U s,x (29)
with V s (x) and W s (x) defined as the maxima of the V (x) and W (x) profiles. The first choice leads to more accurate results in combination with our data (the error in the determination of the maxima of W (x) is large in our experiment due to the separation between the different measurement planes).

In this section L y and L z are defined following the first choice, but further considerations regarding the second choice will be provided in Section 5.

As in [START_REF] Krebs | A three-dimensional study of the glottal jet[END_REF], the half widths are computed from the 2D gaussian profiles that best fit the experimental values of U for each x position. The empirical x-dependence of L y and L z follows respectively a linear and a quadratic law, as shown in Fig 3 . The same holds if one computes l y and l z as the half widths of the 2D gaussian profiles that best fit the experimental values of < u 2 > for each x position. 26). For the glottal jet data, the x-dependence of the similarity ratios is not the same throughout the whole inspected range, but the scalings involved in the similarity ratios are not required to hold simultaneously. Table 1 shows A remark is necessary before these tasks are undertaken. The in-plane analysis has the zero velocity condition for the velocity component that is normal to the correspondent symmetry plane, but the time-averaged jet centerline has a slight tilt (of 5 degrees at most) with respect to the x axis. Because of this, the orientation and origin of the local coordinate system satisfying the in-plane condition at each x position will not exactly coincide with the fixed laboratory coordinate system defining the mid-sagittal (y = 0) and mid-coronal (z = 0) planes. Coordinates ȳ and z are defined to include the shifts y c (x) and z c (x) correcting the centerline position of the jet. These coordinates compensate for the error in the origin of the local coordinate system, but not for the error in the orientation of the symmetry planes. The latter correction would have required a larger densification in the measurement planes of the inspected volume, but in view of the reduced tilt, this should not significantly alter the presented profiles.

Minor plane results

For the analysis of the profiles, let us first take the minor plane, that coincides with the physiological mid-coronal plane (z = 0). As mentioned above, the shift introduced by the centerline position of the jet, which is slightly tilted as the jet spreads, is corrected in the rescaled profiles using ȳ = yy c (x).

Using the similarity scalings, collapsing profiles are obtained for both nonzero velocity profiles U and V , as shown in Fig 5 and The asymmetry in the profiles with respect to the vertical axis can be attributed to more than one cause. There is, of course, the above-mentioned usage of a fixed coordinate system and the time-averaged jet tilt, but there is also the existence of moderate asymmetries in the aperture defined by the model folds during regular oscillations. This can be appreciated in Fig 1b: notice that a vertical line joining the two commissures does not separate the slit into two exactly equal black sections. The same holds for a centered horizontal line dividing the image into an upper and a lower part.

Collapsing profiles are also obtained for the mean Reynolds stresses rescaled in the y and z axes using length scales l y and l z . These length scales are applied because they are available from glottal jet data, but the collapse is not significantly modified if L y and L z are used instead. For < u 2 > two subregions can be distinguished, as shown in Fig 7 : subregion (I) 10 mm< x < 14 mm and subregion (II) 16 mm< x < 21 mm. The profile for < u 2 > changes shape approximately where L z (x) reaches its minimum value. Profiles < uv > are normalized using R uv , that scales as U 2 s L y,x . The range of validity is once again x > 10 mm, as shown in Fig 8 . The profile for < uw > is not shown because the measured values are too small (within the error bars of the experiment) to allow for a similarity analysis.

In spite of the moderate lack of symmetry of the minor plane profiles for the glottal jet, their shape compares well with the shape of the cross-stream distributions reported, for instance, for the high-aspect-ratio rectangular synthetic jet studied in [START_REF] Smith | The formation and evolution of synthetic jets[END_REF]. In the < u 2 > profile, where two profile shapes are distinguished, it is the shape in subregion (I) that compares well with Fig. 12c in [START_REF] Smith | The formation and evolution of synthetic jets[END_REF].

Major plane results

The major plane (y = 0) is not a measurement plane in the source experiment [START_REF] Krebs | A three-dimensional study of the glottal jet[END_REF]. Nevertheless, this near-symmetry plane can be reconstructed extracting the lines of data collected for the nine inspected coronal planes that approximately correspond to the mid-sagittal position. The fact that the major plane has to be reconstructed poses two difficulties that were absent for the minor plane: the lines of data will not necessarilly coincide exactly with the major axis position, and the major plane profiles will be made up of only nine points per x position. This section shows that the collapsing tendency is nonetheless verified when velocities and Reynolds stresses are rescaled in the similarity variables. As before, the shift introduced by the mean centerline position of where such shape is found (10 mm< x < 14 mm) is the same in both, major and minor planes.

The ranges in x where similarity applies in the major plane need not coindice with those of the minor plane. are too small (within the error bars of the experiment) to allow for a similarity analysis.

Axis-switching prediction

Axis switching is the most salient feature of the three-dimensional study presented in [START_REF] Krebs | A three-dimensional study of the glottal jet[END_REF]. The effect is typical of non circular jets [START_REF] Gutmark | Flow control with noncircular jets[END_REF]. For jets in which the exit cross section has two symmetry axes, axis switching means that the major and minor axes invert their roles as the downstream coordinate x increases. Information can be extracted from the derived similarity relations in connection with the axis switching phenomenon in more than one way. Let us consider two of them.

-In section 4, it was mentioned that functions L y (x) and L z (x) could be estimated defining V s (x) and W s (x) as the maxima of the V (x) and W (x) profiles and using the proportionality relations (28) and (29). can be used to rewrite the continuity equation along the centerline (divided by U s ) as follows:

U s,x U s + c 1 L y,x L y + c 2 L z,x L z = 0 (31)
The first term of (31) is a function of x that can be fitted from the jet centerline velocity data. If the jet spread in the mid-coronal plane L y is available, one can use equation ( 31) to obtain L z by integration. The result of the numerical integration of L z is shown in Fig 12. Note that the similarity analysis has the property of yielding the jet contraction/spread rate in one of the symmetry planes (the mid-sagittal plane) from data collected in the other symmetry plane (the mid-coronal plane). This procedure does not only allow for a prediction of the approximate axis switching location, but also of the position where the jet contracts most in the z direction, i.e. of the minimum of L z (x).

Conclusions

Glottal-like flow is subjected to a similarity analysis that has two specificities.

In the first place, the analysis is restricted to the two privileged planes of the flow: the minor plane and the major plane of the jet exit. In physiological terms these planes correspond to the mid-sagittal and mid-coronal planes. Such in-plane similarity analysis allows for a simplification of the derivation of the similarity variables. In the second place, the self-similarity condition that is used to derive the similarity variables is looser than the standard requirement: the dependent variables involved in the problem are allowed to have their own scale, with each scale evolving downstream in a manner that is determined by the in-plane equations of motion. These two specific features associate conveniently in the analysis of a high aspect ratio jet, rendering selfsimilarity compatible with the quadratic contracting-spreading rates observed in the minor plane of jets presenting axis switching. Imposing a similarity solution to the simplified flow equations, one obtains a series of proportionality relations holding within the minor and major planes. These relationships provide the similarity variables that can be used, in combination with glottal-like jet data, to infer the ranges where self-similarity may be expected to occur. Using the similarity scalings to normalize the timeaveraged profiles, and restricting the plots to the ranges in which self-similarity is expected, empirically non-collapsing mean profiles are shown to rescale as collapsing profiles. This is verified for the time-averaged streamwise velocity, the cross-stream velocity components and the Reynolds stresses, in both the minor and the major planes.

In combination with data, the similarity constraints are shown to predict characteristics of the axis switching phenomenon observed in the investigated jet. These characteristics include the mean crossover position at which the switch occurs and the location at which the jet contracts most in the midsagittal plane. Once self-similarity is assured as a property of a certain jet region, the degrees of freedom of the problem are reduced and therefore, information obtained at different measurement points becomes redundant. Experimental efforts can thus be economized and numerical validation can be simplified.

The in-plane similarity variables obtained in this work are not exclusive to glottal-like jets. They can be applied, more generally, to the time averaged velocities and Reynolds stresses of high aspect ratio jets. The rationale applied to this case study could be of use for flows presenting, for instance, several symmetry planes. On the other hand, the characterization of an axis-switching jet in terms of similarity properties can be exploited in complement with other techniques towards a better understanding of this kind of flows.

Fig 1 .

 1 During the experiment, the vocal-fold valve oscillated with a fundamental frequency of f 0 ∼ 55 Hz, a mean subglottal overpressure of 15 cm H 2 O, a mean flow rate of 5.5 × 10 -4 m 3 /s, an open quotient of about 0.75 and a reference jet velocity at the flow exit of U 0 ∼ 2∆p/ρ ∼ 52 m/s.

Fig. 1 :

 1 Fig. 1: Experimental setup used to generate the glottal jet (a), and close-up of artificial vocal folds at plane x = 0 defining the open/closed glottal outlet.

  (a) Local Reynolds number: Self-similarity is expected to occur differently if the local Reynolds number of the flow varies or not with downstream distance. The planar and the axisymmetric jet are examples of these two categories respectively. The glottal jet is neither axisymmetric nor planar: it is an example of local-Reynoldsvarying flow, but with different trends in the y and z directions. Being an axis-switching flow, the time-averaged local length scales are different in the major/minor axis. Local Reynolds increases with x in the y direction and decreases with x in the z direction. (b) Time-varying nozzle: The glottal outlet vibrates with an open and a close phase. The duration of the open phase with respect to the cycle duration is termed open quotient.

Fig. 2 :

 2 Fig. 2: Symmetry planes considered in the derivation of the similarity equations of the jet: plane (z = 0), also called mid-coronal plane, and plane (y = 0) or mid-sagittal plane.

Fig. 3 :

 3 Fig. 3: Length and velocity scales for the glottal jet data used in the similarity analysis. (a) Length scales L y , L z , l y and l z as a function of the streamwise distance x. (b) Streamwise component of the centerline velocity U c of the glottal jet as a function of the streamwise distance x.

Fig. 4 :

 4 Fig.4: Similarity ratios appearing in (25) and (26) (U s,x /U s , L y,x /L y , L z,x /L z , l y,x /l y and l z,x /l z ) as a function of x for the glottal-jet data. Ratios L z,x /L z and l z,x /l z are shown multiplied by (-1) (and shifted with a constant in the second case) to achieve a better visibility in the plot.

  Fig 6 respectively. The collapse sweeps the range x > 10 mm where U s,x /U s and L y,x /L y show a similar x-dependence.

Fig. 5 :Fig. 6 :

 56 Fig. 5: Dimensionless time-averaged profiles for U (x i , y, z = 0) using different scalings. (a) Scaling based on the jet exit velocity U 0 and the equivalent diameter D e [15, 13]. (b) In-plane similarity scaling: the jet centerline velocity U s = U c and L y for lengths.

Fig. 7 :

 7 Fig. 7: Dimensionless time-averaged profiles for < u 2 > (x i , y, z = 0) using different scalings. (a) Scaling based on the squared jet exit velocity U 2 0 and the equivalent diameter D e [15, 13]. (b-c) In-plane similarity scaling: U s L y,x for velocities and l y for lengths. Profiles are shape-preserving if two subregions in x are distinguished as shown in (b) and (c).

Fig. 8 :

 8 Fig. 8: Dimensionless time-averaged profiles for < uv > (x i , y, z = 0) using different scalings. (a) Scaling based on the squared jet exit velocity U 2 0 and the equivalent diameter D e [15, 13]. (b) In-plane similarity scaling: U 2 s L y,x for Reynolds stresses and l y for lengths.

Fig. 9 :Fig. 10 :

 910 Fig. 9: Dimensionless time-averaged profiles for U (x i , y ∼ 0, z) using different scalings. (a) Scaling based on the jet exit velocity U 0 and the equivalent diameter D e [15, 13]. (b) In-plane similarity analysis: the jet centerline velocity U s = U c and L z for lengths.

2 Fig. 11 :Fig. 12 :

 21112 Fig. 11: Dimensionless time-averaged profiles for < u 2 > (x i , y ∼ 0, z) using different scalings. (a) Scaling based on the squared jet exit velocity U 2 0 and the equivalent diameter D e [15, 13]. (b) In-plane similarity scaling: U 2 s for Reynolds stresses and l z for lengths.

Acknowledgements This research has been performed with the support of the LIA PMF-FMF (Franco-Argentinian International Associated Laboratory in the Physics and Mechanics of Fluids).


Gerard Mondello

Strict Liability, Capped Strict Liability And Care Effort Under Asymmetric Information By

Keywords: K0, K32, Q01, Q58) 1

This paper compares the effectiveness ofstrict liability and capped strict liability regimesin an agency relationshipamong a regulatory agency and operators of risky activities.Under a double asymmetric information assumption(wealth and efficiency in care effort), it showsthat capping liability is more efficient than keeping with strict liability,this at the price of an informational rent. Efficiency means that the efficient agent supplies the level of safety effort equivalent to the first best solution one. At the optimum, this rent is minimized by the efficient contract supplied by the principal.

Introduction

In April 2010, the BP's offshore drilling rig explosion spilled crude oil in the Mexico Gulf and polluted it on a large scale. This event reminded us that our contemporaneous industrial societies are highly sensitive to technological hazards. Productive activities generate potential huge harm with large ripple and irreversible effects on public health or natural resources. Hence, nowadays, one major task ofgovernments and risky activities corporate managers is to find effective tradeoffs between natural resources preservation and economic growth. Environmental and economic policy should achieve such a balance by combining optimally ex ante regulation instruments (taxes, permits, standards, etc.) and ex post liability regimes(KOLSTAD, UHLEN AND JOHNSON [1990]), [START_REF] Schmitz | On the joint use of liability and safety regulation[END_REF]).The range of exante and ex post environmental regulatory instruments is wide. Then, harmonizing these tools constitutes a major stake for environmental policies.

However, the present paper will not take the road of optimally combining these instruments. Its scope is limited to a comparative analysis within the field of strict liability regimes applied to environmental protection. Its object is to determinethe best strict liability regime that inducesthe operators of risky activities to supply the optimal care effort under asymmetric information. It studies the agency relationship between a principal (a government through for instance a regulatory agency) and an operator (agent) who exploits a risky facility. The principal has to induce the agent to provide the highest safety level against financial transfers. Confiningthe analysis range to the strict liability question allowsrestrictingthe field to the ex post liability question only. Indeed,after aharm occurrence, Courts do not seek the misconduct or the negligence of the polluter as under a negligence rule regime. Under strict liability, Courts have only tocheckthe causal link between the harm and the risky activity. Indeed, thiscausationis sufficient to assess whether the operator's actionshas involved the accident or not, this, regardless of the level of care exercised beforehand by the manager. Conversely, under negligence rule, Courts have to assess the operator's compliance with law, rules and the optimal level of care to determine whether afault has been committed or not. For instance, typically, KOLSTAD, UHLEN AND JOHNSON [1990] or, still,BOYER AND PORRINI [2006] and [2008]lead their analysis under a negligence rule.

Nowadays, strict liability regimes are implemented to protect the environment as under CERCLA in the United States 1 or, still, the directive on Environmental liability in European Union 2 . The enforcement of such regimes strives towards twofold objectives: first compensating and repairing damage and, second, inducing the potential polluters to take preventive measurestill reaching theoptimal level. Without a doubt, this regime advantages victims that can access rapidly to compensation without bearing the burden of the proof of the fault. However, its weaknesses are several. First, redresses can exceed the polluter's financial capacities and lead him to become judgment proof [START_REF] Summers | The Case of the Disappearing Defendant: An Economic Analysis[END_REF]), [START_REF] Shavell | The Judgment Proof Problem[END_REF]). This induces a de facto limited liability. Second, Society as a whole will endure the cost of the incomplete internalization. Third, as a consequence, the level of care could be undersized [START_REF] Shavell | The Judgment Proof Problem[END_REF]).However, this last point is controversial as shown below. Fourth, the potential polluter can strategically organize his judgment-proofness (VAN'T VELD, RAUSSER AND SIMON [1997], [START_REF] Van' | Hazardous-industry restructuring to avoid liability for accidents[END_REF]). Fifth, the level of investment can be discouraged when strict liability is extended to lenders (BOYER AND LAFFONT [1995] and [1997]).

Some sensitive sectors as the maritime transportation and the electronuclear industry have reacted to the negative effects of strict liability by putting caps on the level of repairs. The operator is exposed to a level of redress substantially lower than the amount of the harm. This lightened responsibility 1 Comprehensive Environmental Response, Compensation and Liability Act (CERCLA) (1980) and Superfund for cleaning-up dangerous waste sites, (see [START_REF] Roman | Comprehensive Environmental Response, Compensation, and Liability Act (CERCLA), United States[END_REF]). 2 Directive 2004/35/CE of the European parliament and of the Council of 21 April 2004 on environmental liability with regard to the prevention and remedying of environmental damage, L143/56, 30/4/04). See also OECD [2009].

should act as an investment incentive3 .For instance, the International Convention on Liability and Compensation for Bunker Oil Pollution Damage of 2001 stated the strict liability of ship-owners for all types of pollution damage caused by bunker oil and binds the repairs to an amount calculated in accordance with the amended 1976 Convention on Limitation of Liability for Maritime Claims (SIOPCF [2009]).. Concerning the maritime transport, compensation for oil pollution is regulated by the International Convention on Civil Liability for Oil Pollution (CLC) and the International Convention setting up. The Oil Pollution Compensation Fund (Fund Convention) establishes a two-tier liability system built upon the (limited) strict liability for the ship owner and a collectively financed fund which provides supplementary compensation to victims of oil pollution damage who have not obtained full compensation. This last notion applies only to people privately concerned by personal losses.After the Exxon Valdez disaster, the USA adopted the 1990 Oil Pollution Liability and Compensation Act. It states the ability to collect from companies for naturalresource damage and gives victims the right to make claims directly to the company. All claims for damages made under the 1990 act are capped at $75 million. The law also set up a trust fund to pay claims companies involved in oil spills decline to pay. However, after the Deepwater Horizon rig explosion, this fund proved to be too low.

Nuclear civil liability is also based on caps on the redresses4 . Developing nuclear industry involves relieving nuclear operators of the burden of potentially ruinous liability claims5 . They establish a strict liability regime channeled exclusively to the operators of the nuclear installations. If this liability is absolute, it is limited in time and amount which is set to €1.500M (seeWORLD NUCLEAR ASSOCIATION [2009]).At the political level, this analysis is echoed by opponents to the introduction of such liability regimes 6 .

From an economic viewpoint, industrial accidents are negative externalities that disturb the classical agency relationship put into evidence in the eighties by [START_REF] Maskin | Monopoly with Incomplete Information[END_REF], [START_REF] Baron | Regulating a Monopolist with Unknown Costs[END_REF]), [START_REF] Mussa | Monopoly and Product Quality[END_REF], [START_REF] Myerson | Optimal Auction Design[END_REF]). These disruptions involve that the most efficient agent will not get the first-best outcome.This paper shows that under asymmetric information, under a strict liability regime, the principal cannot expect from the operator the first rank level of prevention of symmetric information. Paradoxically this lastone is reached when is relaxed the strict liable regime for a less rigorous one: the capping of redress. Basically, our model assumes the existence of twofold information asymmetry: first, the operator's safety efficiency is unknown from the government and, second, theagent's level of wealth is private information. This twice uncertainty is detrimental in the supplying of the first-best level of safety. Then,it is shown that capping the amount of repairs lead the efficient agents to supply the first best solution against, as a price, a minimized informational rent. Indeed, capping the level of repairs removes one level of uncertainty.

This contribution brings some answers to the controversy about the effective impact of bound on repairs. Following SHAVELL [1986], some authors consider that caps induce operators to lower their safety effort because they proportionate it to the level of redress (FAURE AND HU[2006]), [START_REF] Faure | Financial caps for oil pollution damage: a historical mistake?[END_REF]). Furthermore, under the ceiling of repairs, the internalization process remains structurally incomplete because the victims' rights to full compensation are seriously impaired. It is a kind of limitation for a liabilityalready limited by wealth.

However, the debate is open because other authors consider that limiting institutionally the amount of the polluters' repair may induce them to increase the safety level beyond the optimal level(JOST [1996], MICELI AND SEGERSON [2003]), (DARI-MATTIACCI[2006]). These authors extend the analysis of ( [START_REF] Beard | Bankruptcy and Care Choice[END_REF]).These contributions insist on the tradeoff between the cost of precaution and the amount of wealth dedicated to redress. The liability caps are independent from the injurer's safety expenditures that can contribute to limit excessive precaution and reduce the insolvency risk. Hence, a potential insolvent agent may be induced to take too much precaution compared to the social optimum. This increases the total social costs of accident: the more is spent on prevention, the less for repairs. Bounding the liability allows the injurers to spare more for compensation (DARI-MATTIACCI[2006]), (DARI-MATTIACCI AND DE [START_REF] De Geest | When Will Judgment Proof Injurers Take Too Much Precaution?[END_REF]).Here, we join the conclusion of this literature: an appropriate ceiling of repairs gives better results than a strict liability regime. The main difference is that,under asymmetric information, this regime leads to the first best level of prevention ofa complete information situation under a standard strict liability regime (i.e. without cap).

This approach comes from the judgment-proof literature initiated by SUMMERS [1983] and SHAVELL [1986], followed byPITCHFORD [1995], BOYER AND LAFFONT [1997] or still HEYES [1996]. The object is to internalize the costs of damages byshifting the liability burden to vicarious or creditworthy third parties. Furthermore, operators have to choose an adequate level of preventive measures. This means the necessity of finding a trade-off between the amount dedicated to the repairs and the level of safety effort (BOYER AND [START_REF] Boyer | Environmental Risks and Bank Liability[END_REF]). Under complete information this goal is reached, however, under asymmetric information, only the second best level of prevention can be achieved7 .

In a first section, the basic features of the model are given. The first rank level of safety effort under symmetric information for standard strict liability regime is defined. A second section shows that information asymmetry breaks this scheme because efficient agents are deterred to exert the risky activity. In a third section, capped liability is introduced and, there,is studied howthe most efficient agent is led to supply the first rank level of effort. A fourth section concludes.

Economic environment: technology, preferences, information

This paper applies the methodology of asymmetric information theory, but, basically, it rests on the foundations of liability theory developed by [START_REF] Shavell | The Judgment Proof Problem[END_REF] because determiningthe optimal level of care is central. The principal corresponds to a government that expects from agents (operators of risky facilities) the highest level of safety. In our representation, agent and principal focus on safety mainly. That means that reference to the supplied quantity is only implicit. Indeed, the agents differ by the marginal costs of their safety effort.Regarding quantities, the principal considers that the marginal production costs of gross production are almost identical. The competitive difference is made by the level of safety brought by the operators. Putting it otherwise, if basic production technologies are roughly similar, they differ mainly by the levelof safety that the agent embeds in it.

The range of application and relationship is wide. This may concern as well the relationships between government and utilities that supply environmental services as water, waste treatment, etc, but also the regulation of electro-nuclear plants, or energetic facilities. The transfers remunerate the supplementary efforts that improve safety beyond what is expected which is reflected in differentiated marginal costs8 . This state of matter is a common feature considering most modern products and productions because knowledge about basic technology is available everywhere. Consequently, supplying a basic service as, for instance, fresh water, sewage, power, etc, can be achieved by any firm. However, efficiency in safety differs from an agent to another one. For instance electro-nuclear power may be supplied by highly secure plants or by less efficient ones as the Tchernobyl catastrophe in 1986 revealed it. Nowadays, for most products and production, competition bears more on quality or safety (here safety) than on quantity.

Consequently, in order to assess a relevant level of transfers to pay for safety (as an embedded part of the total price) the principal has to induce the conditions under which extending liability to the principal improves social welfare. If the principal has all the bargaining power, then extended liability favors the internalization of environmental damage and so improves welfare [START_REF] Balkenborg | How Liable Should the Lender Be? The Case of judgment-Proof Firms and Environmental Risks: Comment[END_REF]).

operators to supply the highest level of care. Let us explain the point. Most of risky activities are under the supervision of authorities or State Agencies. They give permits and administrative authorizations to operators that deal with dangerous activities (chemical plants, electro-nuclear production, waste treatment, etc.). This belongs to the set of ex ante regulation. That involves the regular checking up of the quality andthe safety of the products. For instance, steady inspections may verify the safety of facilities, the existence of installations etc. This may be assimilated to the checking of the level of the safety effort 𝑒. However, the Principalcannot appreciate the effective efficiency of the agent when managing this effort. For instance, the Deepwater Horizon rig explosion causing the Gulf of Mexico oil spill, had suffered a leak in the weeks leading up to the blast. This leak has been insufficiently appreciated by the BP's management. PATZEK [2010]considers that the problem is deeper than a simple monitoring question and involves the necessary revision of the whole production structure. Difference in marginal costs in safety indicates the degree of skill of the agent and this is private information. This is the root of asymmetric information in our model. Hence, if the principal can accede to the level of agents' safety effort, he cannot appreciate their effective skill. The more efficient an agent is, the lowest his safety marginal costs.

General feature of the model

A government through a regulatory agency (noted RA) wants to induce the operators of risky facilities to accompany the provision of their production or services with the highest level of safety. Compared to models that deals with liability and competition, as [START_REF] Pitchford R | How Liable Should the Lender be? The Case of Judgement-Proof Firms and Environmental Risks[END_REF]), (BOYER ANDLAFFONT [1997]), (BOYD AND [START_REF] Boyd | The search for deep pockets: is "extended liability" expensive liability?[END_REF]), (HIRIART AND MARTIMORT [2006b]), the level of safety effort is dissociated from the efficiency about the skill of the firm in its management of safety. This could correspond for instance to the relationships between a facility (energy as a nuclear plant), and the government. The government through a set of transfers corresponding either to subsidies allocated for R&D for safety for instance, or for other motives associated with a care effort or through credit policy, or other government intervention as fixation of tariffs [START_REF] Koplow | Subsidies to Energy Industries[END_REF]) and (KOPLOW [2010 p.17]) will induce the operator to supply the highest safety effort. These transfers remunerate or pay the supplement of safety that is above the standard product.

Rather, here, the tradition initiated by SHAVELL [1986] will be followed and the present analysis will focus on the care effort level 𝑒, (𝑒 ≥ 0). The Principal derives a benefit 𝑆 𝑒 from the acquisition of the good so long as it ensures an adequate level of safety𝑒.𝑆 𝑒 is defined on ℝ with 𝑆 𝑒 ′ > 0 and 𝑆 𝑒𝑒 ′′ < 0. Furthermore, this function satisfies the Inada condition: 𝑆 ′ (0) → +∞and lim 𝑒→0 𝑆 ′ (𝑒) 𝑒 = 0.

Both regimes will be analyzedrespectively, i.e. the strict liability one and the capped one.This last regime puts ceilings of the level of repairs. Let 𝑦be the level of the agent's wealth and,𝐷, the level of a majordamagewhich exceeds the agent's financial capacities (𝑦 < 𝐷). Under a standard liability regime, if a benevolent Court considers the agent liable, he will have to pay from his own assets. If these ones are insufficient, he becomes judgment proof [START_REF] Summers | The Case of the Disappearing Defendant: An Economic Analysis[END_REF]), [START_REF] Shavell | The Judgment Proof Problem[END_REF]). Under a capped liability scheme, the amount of damage is fixed to 𝐶, where 𝐶 < 𝐷 and the liable agents can escape the judgmentproof situation.The potential damage 𝐷 of the activity is common knowledge as the probability distribution of the environmental harm 𝑝(𝑒) where 𝑝 𝑒 ′ < 0and 𝑝 𝑒𝑒 ′′ > 0.

Informational asymmetries are twofold:i) the level of safety effort and ii) the agent's wealth that cannot be considered as public information.This last assumption seems quite natural because the operator's effective wealth is private information.

2.2The utility functions

When the regulatory agency acquires the public good, he requires also a given level of safety and the RA's utility function is:

(1) 𝑉 = 𝑆 𝑒 -𝑡, where, 𝑡 is the payment made to the agentby the RA,𝑡 has to behigh enough to cover the costs induced by the production activity and the safety effort. If 𝑈 is the profit function of the agent:

(2) 𝑈 = 𝑡 -𝐶 𝑒, 𝐾 . This writing means that the agent assesses both cost of safety and he has to internalize the possible damage to the environment 𝐾𝑝(𝑒) where𝐾 = {𝐶, 𝑦}is the amount of the redress under a capped strict liability regime for a fixed amount 𝐶and a 𝑦(the amount of his wealth) for a "standard" strict liability. The value 𝜃 stands for themarginal cost ofsafety efforts made by the agent where 𝜃 ∈ 𝜃, 𝜃 with𝜃 (respectively𝜃) the marginal safety effort cost of the efficient (resp. inefficient) agent). Asagent's efficiency is private information, the regulatory agency assesses the following probability distribution on the distribution between efficient agent(ϑ) and inefficient ones(1 -ϑ), ( 1 ≥ ϑ ≥ 0).

Then, is defined the firm'scost function considering the possibility of the occurrence of a severe accident (probability 𝑝(𝑒)). The expected cost of safety writes now:

(3) 𝐶 𝜃, 𝑒, 𝐾 = 𝜃𝑒 1 -𝑝 𝑒 + 𝜃𝑒 + 𝐾 𝑝 𝑒 or, after developing: (4) 𝐶 𝜃, 𝑒, 𝐾 = 𝜃𝑒 + 𝐾𝑝 𝑒 , for𝜃 ∈ 𝜃, 𝜃 and 𝐾 = {𝐶, 𝑦}. Then, the profit function becomes:

(5) 𝑈 = 𝑡 -𝐶 𝜃, 𝑒, 𝐾 = 𝑡 -𝜃𝑒 -𝐾𝑝(𝑒). Then, the society's welfare function is deduced (see section A.1 in the Appendix): (6) 𝑊 = U + V = 𝑆 𝑒 -𝜃𝑒 -𝑝 𝑒 𝐷for 𝜃 ∈ 𝜃, 𝜃 . This function is such that𝑊 ′ 𝑒 > 0, 𝑎𝑛𝑑 𝑊"(𝑒) > 0, because𝑆 ′ 𝑒 > 0, 𝐷 > 0 and𝑝 ′ 𝑒 > 0.As for standard asymmetric information theory, the contracting outcomecan be defined. Here, the relevant variables are the level of effort 𝑒which is necessary to achieve an acceptable level of safety and the transfer 𝑡 received by the agent. Let Ξbe the set of feasible allocations: 𝛯 = 𝑒, 𝑡 : 𝑒 ∈ ℝ +, 𝑡 ∈ ℝ , These variables are both observable and verifiable by a third party such as a benevolent court of law.Hence, the effectiveinformational asymmetries are the agent's level of wealth and his efficiency level of safety. Thisextendsthe models of SHAVELL [1984], [1986], and [1987]or LANDES AND POSNER [1989] toinformational asymmetries.

2.3The Complete Information Optimal Contract

2.3.1The first-best safety level

Let us assume first that there is no informationasymmetry between the principal and the agent(either in efficiency or in wealth). Then, the government can perform an appropriate transfer.The efficient care levels are obtained by equating the principal's marginal value and the agent's marginal cost and are deduced the following first-order conditions from ( 6): ( 7)

𝑆 ′ 𝑒 * = 𝜃 + 𝑝′ 𝑒 * 𝐷 and, (8) 𝑆 ′ 𝑒 * = 𝜃 + 𝑝′ 𝑒 * 𝐷.

The complete information efficient safety level 𝑒 * and 𝑒 * should be carried

2.3.2Implementing the first-best

For a successful delegation of the task, the principal has to offer the agent a utility level that is at least as high as the level the agent obtains outside the relationship. These arethe agent's participation constraints. Then, the quo-utility level or participation constraintswrite as: (10) 𝑡 -𝜃𝑒 -𝑝 𝑒 𝑦 ≥ 0 (11) 𝑡 -𝜃𝑒 -𝑝 𝑒 𝑦 ≥ 0.

To implement the first-best production levels, the principal makesa contract of a take-it or leave-it type to the agent and supply a 𝑡 * , 𝑒 * -contract for the efficient agent (𝜃) or a 𝑡 * , 𝑒 * -contractfor the inefficient one(𝜃). Hence,under symmetric information assumption, the principal needs to know perfectly the agent's wealth level to performan appropriate payment. Indeed, the transfer 𝑡includes both the safety price and the risk cover.

Asymmetric information and information rents: the case of standard strict liability

Now, we analyze the situation characterized by information asymmetries (efficiency and wealth) between the RA and the agent in a standard strict liability framework. Hence,if the agent isliable for the harm, he will have to repair by engaging the whole of his assets.

The agent's program under standard strict liability

The agent knows privately how efficienthe is and his wealth level.These values are ignored by the principal who has to design an incentive mechanism that will reveal this double information. Conform to standard asymmetric information theory (LAFFONT AND MARTIMORT[2002, chap.2]),a menu of contracts 𝒞 = 𝑡, 𝑒 , (𝑡, 𝑒) is incentive compatible when 𝑡, 𝑒 is weakly preferred to (𝑡, 𝑒) by the agent 𝜃 and (𝑡, 𝑒) is weakly preferred to 𝑡, 𝑒 by the agent 𝜃. This involves that the following constraints (incentive compatibility constraints) have to be respected: (12)

𝑡 -𝑐(𝜃 , 𝑒, 𝑦) ≧ 𝑡 -𝑐(𝜃, 𝑒, 𝑦)

(13) 𝑡 -𝑐(𝜃, 𝑒, 𝑦) ≧ 𝑡 -𝑐(𝜃, 𝑒, 𝑦). A supplementary condition is that participation constraints have to be respected too: (14) 𝑡 -𝑐(𝜃 , 𝑒, 𝑦) ≧ 0 (15) 𝑡 -𝑐(𝜃, 𝑒, 𝑦) ≧ 0. The menu of contracts is incentive feasible if the constraints ( 12)to (15)are satisfied. Contracts in 𝒞 are truthful, i.e. the firm is induced to report its true technological parameters. We define the information rents of the agent of each type as: ( 16) 𝑈 = 𝑡 -𝑐(𝜃 , 𝑒, 𝑦)

(17) 𝑈 = 𝑡 -𝑐(𝜃, 𝑒, 𝑦). Then we can define the amount that an efficient agent can capture by mimicking an inefficient agent. However, the risk question makesthis point more delicate. Hence, if the efficient agent can mimic the 𝜃 agent by adapting its supply of security service, a priori, he cannot imitate the 𝜃agent's level of wealth𝑦which is unknown to him... Furthermore, in the case of an accident, his effective wealth will be engaged. Hence, the informational rent depends only on the level of supplied safety which expresses as: 𝑈 = 𝑡 -𝑐 𝜃 , 𝑒, 𝑦 ≧ 𝑡 -𝑐 𝜃 , 𝑒, 𝑦 or, still: 𝑈 = 𝑡 -𝑐(𝜃, 𝑒, 𝑦) -𝑐 𝜃, 𝑒 + 𝑐(𝜃, 𝑒, 𝑦)

= 𝑈 -𝜃𝑒 + 𝑦𝑝 𝑒 + 𝜃𝑒 -𝑦𝑝 𝑒 (18) 𝑈 = 𝑡 -𝑐 𝜃 , 𝑒 ≧ 𝑈 + ∆𝜃𝑒 -∆𝑦𝑝(𝑒)

(Where∆𝜃 = 𝜃 -𝜃 > 0and ∆𝑦 = 𝑦 -𝑦). Knowinga prioriwhether the wealth difference∆𝑦 is positive or negative is impossible. Indeed, we cannot postulate that the efficient agent has to be richer than the inefficient one or the reverse. The consequences of bothdesigns have to be discussed.

The program of the principal under standard strict liability

To overcome the uncertainty induced by informational asymmetries, the principal offers a menu of contracts. Before defining his complete program, we have to define the regulator'sexpected gainwhich expresses as:

𝑆 𝑒 -𝑡 1 -𝑝 𝑒 + 𝑆 𝑒 -𝑡 -𝐷 -𝑦 𝑝 𝑒 = 𝑆 𝑒 -𝐷 -𝑦 𝑝 𝑒 -𝑡.

Then, taking into account the nature of the agent,the principal's program writes as: The principal aims, first, atmaximizing the net safety surplus and, second, minimizing the information rents. In general, following standard presentation (LAFFONT AND MARTIMORT [2002]), finding solution to this program involves choosingthe relevant constraints, i.e. the binding ones at the optimum.Hence, the relevant constraints are reduced from four to two: the incentive constraint of the efficient agent and the participation constraint of the 𝜃 agent. Now, taking into account the severe accident occurrence, this simplification has to be made cautiously because the agent's wealth is privately known and this adds a supplementary uncertainty.

PROPOSITION 2:Considering standard strict liability regime, when the probability of severe accident with social impact (health or environment) is introduced, the revelation mechanism depends on the wealth of each category's of agent.Considering the program (20) to(15'), the necessary condition for solving it is that𝑦 > 𝑦. PROOF: In appendix A3.

This proposition means that when the inefficient agent is richer than the efficient one, the usual mechanism that involves that efficient agent will supply the first best level of effortdoes not workanymore. Indeed,(15a) (𝑈 ≧ 0) cannot be respected(this value can be negative).The efficient agent ignores if his assets are higher than the ones of the inefficient agentand, logically, is deterred to participate.

If 𝑦 > 𝑦, (proposition 2 fulfilled), the remaining relevant constraints are (12a),and (15a),and both of them have to be binding. Consequently: (12b) 𝑈 = ∆𝜃𝑒 -∆𝑦𝑝(𝑒) and, (13b) 𝑈 = 0. Implementing them into the principal's program, we get:

(20a) 𝑀𝑎𝑥 (𝑒 ,𝑒 𝜗 𝑆 𝑒 -𝜃𝑒 -𝐷𝑝 𝑒 + (1 -𝜗)(𝑆 𝑒 -𝜃𝑒 -𝐷𝑝 𝑒 )- 𝜗 ∆𝜃𝑒 -∆𝑦𝑝 𝑒 .
From the analysis of the first order conditions,are deduced the informational rents that the efficient agent can capture. Indeed, if the inefficient agent gets no rent by mimicking the 𝜃 agent,the efficient agentmay acquire information rent. We note by "SB" the second best optimal values. The first order conditions are given by: (21) 𝑆 ′ 𝑒 𝑆𝐵 = 𝜃 + 𝐷𝑝′ 𝑒 𝑆𝐵 . This corresponds to the first best value of 𝑒 and 𝑒 𝑆𝐵 =𝑒 * . The informational rent of the principal is then equal to 𝑈 = ∆𝜃𝑒 𝑆𝐵 -∆𝑦𝑝(𝑒 𝑆𝐵 ). Concerning the inefficient agent: Now we can determine the level of the second best transfers taking into account the information rent. For that, we recall that from its definition:

(22) 1 -𝜗 (𝑆 ′ 𝑒 𝑆𝐵 -𝜃 -𝐷𝑝 ′ 𝑒 𝑆𝐵 ) = 𝜗(∆𝜃 -∆𝑦𝑝 ′ 𝑒 𝑆𝐵 ) (
𝑈 𝑆𝐵 = 𝑡 𝑆𝐵 -𝑐 𝜃 , 𝑒 𝑆𝐵 =∆𝜃𝑒 𝑆𝐵 -∆𝑦𝑝(𝑒 𝑆𝐵 ),

then, (24)

𝑡 𝑆𝐵 -𝜃𝑒 -𝑦𝑝 𝑒 = ∆𝜃𝑒 𝑆𝐵 -∆𝑦𝑝 𝑒 𝑆𝐵 .

As a consequence: ( 25)

𝑡 𝑆𝐵 = 𝜃𝑒 * + 𝑦𝑝 𝑒 * + ∆𝜃𝑒 𝑆𝐵 -∆𝑦𝑝(𝑒 𝑆𝐵 )
and, ( 26)

𝑡 𝑆𝐵 = 𝜃𝑒 𝑆𝐵 + 𝑦𝑝 𝑒 𝑆𝐵 .
These results differ slightlyfrom standard asymmetric information theory. They call for some remarks.

REMARK 1:It is legitimate to consider that ∆𝜃 = 𝜃 -𝜃 > 0, that expresses the difference in efficiency of agent 𝜃 compared to agent 𝜃 considering marginal costs. However,there is no economic legitimacy putting ∆𝑦 = 𝑦 -𝑦 > 0 (or the reverse) as an assumption. Proposition 2 results from a strong assumption. However, in general,there is no economic reason to consider that the efficient agent should be richer than the inefficient one or the reverse. If this condition is not met, then the value of the informational rent can be weak. Concretely, this condition means that the difference between the efforts brought by the efficient agent compared to the inefficient one, has to be higher than 𝑦 𝑦 𝑝 𝑒 𝑆𝐵 . The demonstration of this remark is made in Appendix A4.

REMARK 3: Under a standard strict liability regimeand asymmetric information, the efficient agents may be deterred to enter in the game. Indeed, two conditions have to be met to induce him to compete. The first one is necessary but insufficient (proposition 1) i.e. his level of wealth has to be higher than the one of the inefficient agent. The other condition, (sufficient) is that the level of safety effort has to be high enough such that the difference in the probability of accident will exceed

𝑦 𝑦 𝑝 𝑒 𝑆𝐵 .
This condition is particularly stringent because the efficient agent must know too much information before accepting the contract. Indeed, the efficient agent cannot know the nature of his opponent's wealth.

REMARK 4: The constraint [13a] (𝑈 ≧ 𝑈 -∆𝜃𝑒 + ∆𝑦𝑝(𝑒))means that the inefficient agent claims that he is efficient but he will fail to supply the promised level of safety. This is typically an adverse selection problem. However, it cannot be solved here because instruments that could induce the efficient agent to overcome his reluctance to produce when conditions are not favorable are lacking.

As a conclusion, standard strict liability is not a powerful instrument to protect public health and the environment. This result has long been known [START_REF] Shavell | The Judgment Proof Problem[END_REF]) and asymmetric information reinforces the point. We show furthermore that uncertainty about wealth level under this regime tends to favor the adverse selection effect. This state of matter introduces biases in the calculus of the efficient agent.

The Capped strict liability scheme and asymmetric information

In this section two points will be discussed: first, the way to get an acceptable solution for the strict liability scheme and second, the consequences for a better involvement of associated financing institution. Hence capped liability allow to secure investment and makes easier insuring investment.

A solution for the ceiling of liability

Now we make the assumption that Law limits the amount of repairs. The ceiling of damages should preserve the wealth of the agent:𝐶 < 𝑦 < 𝐷,𝐶 > 0. This induces to modify generically the cost function as:

(27)
𝐶 𝑒, 𝜃, 𝑦 = 𝜃𝑒 + 𝐶𝑝 𝑒 . As previously, the informational rent expresses as: (28)

𝑈 𝐶 = 𝑡 -𝑐(𝜃 , 𝑒, 𝑦) ≧ 𝑡 -𝑐(𝜃 , 𝑒, 𝑦)

and,processing as before when we got equation ( 18 As previously, we have to definewhich are the relevant constraints among the incentive compatibility and participation constraints. Relevancy means the binding ones at the optimum level. We consider contracts without collapse, i.e. 𝑒 > 0. This is verified when the Inada condition 𝑆 ′ (0) → +∞is satisfied andlim 𝑒→0 𝑆 ′ (𝑒) 𝑒 = 0. The participation constraint of the efficient agent in [34] is always satisfied because [32] and [35] involves [34]. In this context, the inefficient agent has no interest to mimic efficiency, then [33] is irrelevant. After this simplification, two constraints are remaining the 𝜃-agent's incentive compatible constraint [32] and the participation constraint of the 𝜃-agent [35]. Getting the optimum of the PC program involves that both constraint must be binding: (36)

𝑈 𝐶 ≧ ∆𝜃𝑒 and, (37) 𝑈 𝐶 = 0.

This reduces the objective function of the program (PC) becomes:

(38) 𝑀𝑎𝑥 𝑒,𝑒 𝜗 𝑆 𝑒 -𝜃𝑒 -𝐷𝑝 𝑒 + 1 -𝜗 𝑆 𝑒 -𝜃𝑒 -𝐷𝑝 𝑒 -𝜗∆𝜃𝑒.

As in standard representations, asymmetric information modifies the principal's optimization by the subtraction of the expected rent that has to be given up to the 𝜃 agent. This rent depends on the level of effort requested from the inefficient type. From the first order conditions is drawn the equilibrium values which are identical to the full information setting for the efficient agent. (39)

𝑆 ′ 𝑒 𝑆𝐵 = 𝜃 + 𝐷𝑝′ 𝑒 𝑆𝐵 , and for the inefficient one: It can be verified that with a similar argument made for the standard liability schemecan be defined the following relationship that follows from the monotony of the second-best schedule of safety level:

(41) 𝑒 * = 𝑒 𝑆𝐵 > 𝑒 * > 𝑒 𝑆𝐵′ (Where (". " 𝑆𝐵′ ) stands for the second best under the capped regime).In summary, the following propositioncan deduced:

PROPOSITION 3:Under asymmetric information, under a cap strict liability regime, the optimal contracts entail:

-No safety effort distortion for the 𝜃 agent in respect to the first best 𝑒 * = 𝑒 𝑆𝐵 and a downward distortion for the 𝜃 type, gives:

𝑆′ 𝑒 𝑆𝐵 -𝜃 = 𝜗 1-𝜗 ∆𝜃, with 𝑒 * > 𝑒 𝑆𝐵′ .
-Only the efficient type gets a positive information rent given by: (42)

𝑈 𝐶 = ∆𝜃𝑒 -The second best transfers are respectively: The proof is deduced from the previous argument.

The ceiling of liability allowsdroppingthe unknown level of wealth. Indeed, by [29] and [30] the value of the ceiling replaces the agent's wealth. Hence, the problem reduces to only one private information variable: the safety effortefficiency. The result that follows isquite standard. Under the ceiling of redress, the level of precautionary effort of the most efficient agent corresponds to the first rank of the symmetric information scheme. The counterpart is that this agent benefits of an informational rent that, however, is minimized by the optimal contract between the RA and the efficient agent.

Capped liability and insurance: an introduction

Conversely to a well shared opinion, the above results show that under asymmetric information, putting caps on redress issues on the same level of effort than the standard strict liability regime under symmetric information. After this initial result, many avenues must be explored. For instance, the issue of insurance has not beenaddressedalthough itis an important matterfor capped liability [START_REF] Shavell | Minimum asset requirements and compulsory liability insurance as solution to the judgment-proof problem[END_REF], (BOYD AND [START_REF] Boyd | The search for deep pockets: is "extended liability" expensive liability?[END_REF]). Subscribing policy insurance is compulsory for oil operators in the maritime sector and the nuclear industry. In this paper, the concern has been limited tothe scope and power of an ex post regulatory control based on the ceiling of redress under informational asymmetries.

By ceiling the redress, the principal reduces the uncertainty involved by the unknown polluters' wealth.Furthermore, itcan control the agents' activity by requiring that theyhave to own at least the amount of the cap as financial guarantee. This inducesthe withdrawal of the insufficient endowed agents. This can be achieved by resorting to insurance. For instance, if Q is that share which is insured, where:

𝐶 -𝑄 = 𝑤(𝑤theshare of the agent's wealth used as commitment). As a consequence, the agent has to cope with two principals: the RA and the insurance company. Indeed, the insurance premium is equivalent to 𝑄𝑝(𝑒) = 𝑚, that is to say the probability of an accident by the claim of the company. As a result, to reduce his premium the agent has to increase his level of effort. Indeed, the insurance company has to check that the level of safety corresponds to the level of the insurance premium. A further research will have to develop these relationships.

5.

Conclusion

Under asymmetric information, standard strict liability rules fail to provide the first best level of effortin safety. Thisfavors adverse selection emergenceand can induce inefficient operatorsto undertake risky activitiesbydeterring the efficient ones. Then, capping the level of redress can be an alternative to a standard strict liability regime. However, this switching does not guarantee automatically restoring efficiency. Indeed, some necessary conditions have to be fulfilled.

In real life, bounding the level of repairs raises strong oppositionwhen people consider that the level of ceiling is too low. Consequently, under asymmetric information, the Principal has to formulate relevant contracts that make a tradeoff between the level of repairs and the level of safety effort. These contracts are second bestcontracts compared to the certain case under strict liability, but they adjust the level of safety to the level of the cap. At equilibrium, the level of care has to be chosen such that the marginalcosts of care are offset by marginal reductions in expected damages.To be fully efficient, a capped strict liability scheme needs to associate the utility level of the principal to a relevant level of security. This involves establishing a tradeoff between a relevant safety effort and its associated costs and the level of redress designed by the level of the cap. Indeed, this tradeoff balances the risk level that the principal can accept and the amount of the fund dedicated to repairs.

Capping the repairs level does not mitigate the sharpness of the judgmentproof question even if an efficient contract is formulated. However, it locks up the debate by explicitly involving all the parties. Hence, at the equilibrium,implicitly, the principal accepts incomplete repairs but the potential loss is balanced by anincrease in safety. These one consists in two points. First, the equilibrium level of effort is calculated on the whole cost of damage that the society can endure. The effort level is identical to the one of the certainty case reached under strict liability. Second, the contract attracts theefficient agent in safety and avoids the adverse selection effect. This eviction effect of inefficient agent can be reinforced by the requirement of insurance policy that introduces a new principal in the scheme.

This paper brings a contribution to the debate about the reciprocal efficiency of capped or standard strict liability. It shows notably that under asymmetric information and a strict liability regime all opportunistic behavior can be adopted by inefficient operators and this leads to an inappropriate level of safety. Capping the level of redress induces to find again the care level of the symmetric information case under strict liability regime. The price to pay by the government is an information rent that can be minimized with an adequate level of transfer.This advocates for a generalization of strict liability with the capping of redresses as environmental policy. In the public's opinion, this choice could appear as counterintuitive because the level of redress is less than the damage involved by the harm. However, this solution means that the principal balances between the costs of improvement in safety and the costs of repairs and compensation. Furthermore, as in the nuclear industry, the level of the cap can be progressively raised. This involves to pool insurance companies and the deeppocket lenders. This opens the debate on the other stream of the literature dedicated to the understanding of the judgment-proof question. This will constitute a further step in this research program.

Hence, having 𝑈 ≧ 0[14'] cannot be considered as granted. Preliminary conditions have to be formulated. 𝑈 ≧ 0means that if 𝑈 ≧ 0 is binding ( 𝑈 = 0) then this involves that: 𝑈 = ∆𝜃𝑒 -∆𝑦𝑝(𝑒) ≧ 0 or, still,∆𝜃 ∆𝑦 ≧ 𝑝(𝑒) 𝑒 , by definition 𝑝(𝑒) 𝑒 > 0(with 𝑝(𝑒) 𝑒 → 0 ), furthermore, by definition, ∆𝜃 > 0 then, the condition for having ∆𝜃 ∆𝑦 > 0 is that ∆𝑦 > 0 i.e. 𝑦 > 𝑦 because ∆𝜃 > 0. Hence, the condition for having 𝑈 ≧ 0 is that 𝑦 > 𝑦. That means that if the efficient agent is less rich than the inefficient one, then the participation constraint cannot be fulfilled.

Q.E.D.

A.4.Proof of remark 2

Starting from 𝑡 𝑆𝐵 = 𝜃𝑒 * + 𝑦𝑝 𝑒 * + ∆𝜃𝑒 𝑆𝐵 -∆𝑦𝑝(𝑒 𝑆𝐵 )

We study the conditions for which: 𝑦𝑝 𝑒 * -∆𝑦𝑝(𝑒 𝑆𝐵 ) ≥ 0or still 𝑦𝑝 𝑒 * -(𝑦 -𝑦)𝑝(𝑒 𝑆𝐵 ) ≥ 0, under the respect of proposition 2, the results ensues:

𝑝 𝑒 * > 1 - 𝑦 𝑦 𝑝 𝑒 𝑆𝐵 .
Getting the program 𝑀𝑎𝑥 𝑡,𝑒 ,(𝑡,𝑒) 𝜗 𝑆 𝑒 -𝐷 -𝐶 𝑝 𝑒 -𝑡 + (1 -𝜗)(𝑆 𝑒 -𝐷 -𝐶 𝑝 𝑒 -𝑡 ) subjectto the constraints of incentive compatibility:

𝑡 -𝜃𝑒 -𝐶𝑝 𝑒 ≧ 𝑡 -𝜃𝑒 -𝐶𝑝 𝑒 𝑡 -𝑐(𝜃, 𝑒) ≧ 𝑡 -𝑐(𝜃, 𝑒)

and the supplementary condition of the participation constraints that have to be respected too:

𝑡 -𝜃𝑒 -𝐶𝑝 𝑒 ≧ 0 𝑡 -𝜃𝑒 -𝐶𝑝 𝑒 ≧ 0. As previously, we can cancel the transfers 𝑡, 𝑡 and replacing them by the informational rents, we get the PC program.

Q.E.D.

( 19 )

 19 𝑀𝑎𝑥 𝑡,𝑒 ,(𝑡,𝑒) 𝜗 𝑆 𝑒 -𝐷 -𝑦 𝑝 𝑒 -𝑡 + (1 -𝜗)(𝑆 𝑒 -𝐷 -𝑦 𝑝 𝑒 -𝑡 ),subject to the constraints(12) to (15). Considering the information rents 𝑈 = 𝑡 -𝑐(𝜃 , 𝑒, 𝑦)and 𝑈 = 𝑡 -𝑐(𝜃, 𝑒, 𝑦), we can replace the value of the transfers by the information rents, and,then,the program becomes: (20) 𝑀𝑎𝑥 𝑈,𝑒 , 𝑈,𝑒 𝜗 𝑆 𝑒 -𝜃𝑒 -𝐷𝑝 𝑒 + (1 -𝜗)(𝑆 𝑒 -𝜃𝑒 -𝐷𝑝 𝑒 )-(𝜗𝑈 + (1 -𝜗)𝑈), subject to the incentive constraints: (12a) 𝑈 ≧ 𝑈 + ∆𝜃𝑒 -∆𝑦𝑝(𝑒)

REMARK 2 :

 2 We can check that the informational rent of the efficient agent is positive only if: 𝑝 𝑒 * > 1

(

  Where the index 𝐶to 𝑈 𝐶 and 𝑈 𝐶 indicates that the new liability regime is capped strict liabilityand where ∆𝜃 = 𝜃 -𝜃 > 0).Using the same argumentfor -𝑐 𝜃, 𝑒, 𝑦 ≧ 𝑈 𝐶 -∆𝜃𝑒. The principal's program becomes now (simplification in the appendix) (Program PC): (31) 𝑀𝑎𝑥 𝑈 𝐶 ,𝑒 , 𝑈 𝐶 ,𝑒 𝜗 𝑆 𝑒 -𝜃𝑒 -𝐷𝑝 𝑒 + 1 -𝜗 𝑆 𝑒 -𝜃𝑒 -

  22)expresses the tradeoff between efficiency and rent extraction.

	Here 𝜗 ∆𝜃 -∆𝑦𝑝 ′ 𝑒 question is to know if this condition is compatible with the monotony condition 𝑆𝐵 > 0because 𝑝 ′ 𝑒 𝑆𝐵 < 0, ∆𝑦 > 0 and ∆𝜃>0. The that can be deduced from [12'] and [13']. It appears from them that: 0 ≧ ∆𝜃 𝑒 𝑆𝐵 -𝑒 𝑆𝐵 -∆𝑦 𝑝 𝑒 𝑆𝐵 -𝑝 𝑒 𝑆𝐵 . By assumption ∆𝑦 > 0, 𝑝 𝑒 𝑆𝐵 -𝑒 𝑆𝐵 < 0, then the proposition is verified and we getthe following relationship: 𝑆𝐵 -𝑝 𝑒 𝑆𝐵 >0 because 𝑒 𝑆𝐵 =𝑒 ∆𝜃 > 0and 𝑒 (23) 𝑒 * = 𝑒 𝑆𝐵 > 𝑒

* hence ∆𝑦 𝑝 𝑒 𝑆𝐵 -𝑝 𝑒 𝑆𝐵 > 0 and -(∆𝑦 𝑝 𝑒 𝑆𝐵 -𝑝 𝑒 𝑆𝐵 < 0). Furthermore, * > 𝑒 𝑆𝐵 .

See for instanceBOYD [2001, p.47]:"[…]that environmental costs above the cap will be uncompensated by responsible parties(emphasis added)."

See theIAEA's Vienna Convention of 1963, OECD's Paris Convention of 1960, the Convention on Supplementary Compensation for Nuclear Damage (CSC) of 1997 and 2003, the OECD Paris (and Brussels) amended in 2004. For the USA, the Price-Anderson Act limits insurance to $300 million and caps the operators' liability to $10.5 billion.

See OECD, Nuclear Energy Agency (NEA)[1982]"Exposé des Motifs, Motif 45" or still(SCHWARTZ [2006, p.39]).

In India, see the opposition in 2010 to the Proposed Civil Nuclear Liability [Cap] Bill http://www.sacw.net/article1288.html. See also[START_REF] Anderson | Assessing Environmental Damage under Indian Law[END_REF]).

HIRIART AND MARTIMORT [2006a] consider the optimal regulation of a risky project under a moral hazard assumption. By fines and rewards, the authors derive

Brands differ in their specific embedded equipment about safety or quality (see for instance, (GABSZEWICZ AND[START_REF] Gabszewicz | Price Competition, Quality, and Income Disparities[END_REF],(TIROLE [1988, chap.7],(TAY [2003]),(TOSHIMITSU AND JINJI [2007]).

Updated: August, 2011, (My last access: February 27, 2012).

-𝑝 𝑒 * 𝐷 and 𝑊 * = 𝑆 𝑒 * -𝜃𝑒 * -𝑝 𝑒 * 𝐷 are non-negative,then: (9) 𝑆 𝑒 * -𝜃𝑒 * -𝑝 𝑒 * 𝐷 ≥ 𝑆 𝑒 * -𝜃𝑒 * -𝑝 𝑒 * 𝐷 ≥ 𝑆 𝑒 * -𝜃𝑒 * -𝑝 𝑒 * 𝐷. when the agent is efficient than when it is not. * 𝐷 < 0, and 𝑆 𝑒 * -𝜃𝑒 * > 0, 𝑆 𝑒 * -𝜃𝑒 * > 0 and 𝑆 𝑒 * -𝜃𝑒 * ≥ 𝑆 𝑒 * -𝜃𝑒 * ≥ 𝑆 𝑒 * -𝜃𝑒 * , is verified then, 𝑆 𝑒 * -𝜃𝑒 * -𝑝 𝑒 * 𝐷 ≥ 𝑆 𝑒 * -𝜃𝑒 * -𝑝 𝑒 * 𝐷 ≥ 𝑆 𝑒 * -𝜃𝑒 * -𝑝 𝑒 * 𝐷is verified too. Q.E.D.

A.2.Proof of proposition 1

To see this point we note that because, 𝑝 𝑒 < 𝑝 𝑒 , then 𝑝 𝑒 * 𝐷 -𝑝 𝑒