Frédéric Gruy
email: fgruy@emse.fr

Soong-Hyuck Suh

The chord length distribution of a dumbbell shaped aggregate: Analytical expression

Keywords: Chord Length Distribution (CLD), Dumbbell, Aggregation, Anomalous Diffraction

Dumbbell shaped aggregates are small particles synthesised in precipitation reactors. Their characterization by optical methods needs the chord length distribution (CLD) of such a shape. We present in this paper the analytical calculation of two CLD's corresponding to two different definitions of CLD. Comparison with Monte-Carlo simulations is presented. Good agreement is found between the exact calculation and simulations.

Introduction

Many manufacturers use solid micro-particles in suspension for various applications: ceramics, paintings, pharmaceutics, cosmetics, food and chemicals. Particle sizing can be performed by physical methods based on the scattering between the particles and an incident electromagnetic wave. The scattered wave is depending on the particle morphology and on the ratio between the refractive indices of particle material and suspending medium.

Depending on the particle material and the selected method the measured signal may be straightforwardly related to the chord length distribution (CLD) of the randomly orientated particles set. This is illustrated by three examples:

-Small-Angle Scattering (SAS) measurements [START_REF] Guinier | Small-Angle scattering of X-rays[END_REF] are suitable for nano-and microparticles interacting with X rays. This method can be extended to optically soft microparticles interacting with light [START_REF] Van De Hulst | Light scattering by small particles[END_REF].

-Focused Beam Reflectance Measurements (FBRM) are among the most widely techniques for particle sizing [START_REF] Li | [END_REF]. It uses a focused beam of laser light that scans across particles passing in front of the probe window to measure a chord length distribution.

The interpretation of the signal is only based on the reflected light. This method is suitable for particle size higher than 5µm.

-Spectral Turbidimetry, i.e extinction measurement, is an optical method to measure the light scattering or extinction cross section of particles. In the case of large microparticles (> 1µm) and very small optical contrast, extinction or forward scattering can be explained in the framework of anomalous diffraction approximation [START_REF] Van De Hulst | Light scattering by small particles[END_REF]: then the scattering cross section is expressed as an integral including the particle CLD.

Anomalous diffraction approximation was applied to a sphere, an infinitely long circular cylinder, a prism column, a hexagonal crystal of ice, ellipsoids and a finite cylinder [START_REF] Kokhanovsky | Optics of light scattering media[END_REF].

Depending on the physical principles of measurement the definition of the CLD for a given object changes: 3D isotropic uniform flow of infinite straight lines in the case of SAS and turbidity measurements, or 2D isotropic uniform flow of infinite straight lines for each projected area in the case of FBRM.

The CLD of convex and non-convex bodies has been studied from a mathematical point of view. Explicit expressions have been obtained for bounded 2D or 3D convex domains: disc, triangle, rectangle, regular polygon [START_REF] Ohanyan | Tomography of bounded convex domains[END_REF], sphere, hemisphere [6], cylinders of various cross sections [START_REF] Gille | Chord length distributions of infinitely long geometric figures[END_REF][START_REF] Gille | Chord length distribution density of a triangular rod[END_REF], spheroids, polyhedron [START_REF] Sukiasian | Three-dimensional Pleijel identity and its application[END_REF]. Moreover, Aharonyan [START_REF] Aharonyan | Chord length distribution functions for polygons[END_REF] obtained an explicit expression for the orientation-dependent CLD for any bounded convex body.

Non-convex bodies have paid less attention than convex ones. Mazzolo et al [START_REF] Mazzolo | On the properties of the chord length distribution, from integral geometry to reactor physics[END_REF] discussed the CLD in the context of reactor physics. They show that some relations between lower moments of CLD and simple geometric properties as volume, surface, … of the body remain valid for non-convex bodies whereas higher CLD moments do not obey the simple relations valid for convex bodies. Gille [START_REF] Gille | Chord length distribution density of an infinitely long circular hollow cylinder[END_REF] studied the CLD of an infinitely long circular hollow cylinder that is a special case of non convex body; the corresponding calculation is based on basic principles. Gille [START_REF] Gille | The small angle scattering correlation function of two infinitely long parallel circular cylinders[END_REF] also considered two parallel circular cylinders separated by a short distance and calculated the 3D-correlation function that is related to CLD. Vlasov [START_REF] Yu | Signed chord length distribution[END_REF] introduced the notion of signed chord distribution for convex and non-convex bodies. He started from the work of Dirac transforming the six-dimensional integral of pairwise interaction potential for a convex body into a simpler expression including the CLD; he extended it to a non convex body. He showed that the expression of the integral is much more complicated than the one for convex body: it can be decomposed into several terms (integrals), each one related to the various segments of the given chord inside the non convex body. He formally deduces the expression of the CLD for the non-convex case.

Among the particle shapes observed in industrial processes, small clusters of spherical particles are often present. By the past we calculated [START_REF] Gruy | the chord length distribution of a two-sphere aggregate[END_REF] the CLD for a two-sphere aggregate. In this paper we extend this calculation to a set of two spheres penetrating each other. For instance, these dumbbell-like particles appear in the precipitation of inorganic compounds performed at high supersaturation and weak aggregation conditions. Polymeric colloids with such a morphology are also synthesised [START_REF] Vutukuri | Synthesis of fluorescent monodisperse non-spherical dumbell-like model colloids[END_REF].

The section two of this paper develops a methodology in order to calculate the CLD of a dumbbell shaped aggregate. It is followed by a comparison with Monte Carlo Simulations in the section three. Section four is devoted to concluding remarks.

Calculation of the chord length distribution

A straight line may intersect more than one time across a non convex body. As a consequence, two CLD can be defined:

-The multiple chord distribution (MCD) where each segment interval on the same line is considered as one chord length separately. FBRM measurements are associated to MCD.

-The one chord distribution (OCD) where the sum of chord lengths for all intersected intervals is used as the definition of the chord length. SAS and turbidity measurements are associated to OCD.

Even if the latter ones only consider OCD, we will present both OCD and MCD calculations with 3D uniform flow of lines. The corresponding procedure is similar to the one used for a two-sphere aggregate [START_REF] Gruy | the chord length distribution of a two-sphere aggregate[END_REF]. Throughout the paper, the chord length distribution (density) is written ( ) D l . ( )

D l dl is the number of chords within the l-range [ ] , l l dl + . ( ) D l is normalized, i.e ( ) max 0 1 l D l dl = ∫ .

Definition of the different geometrical areas

In the following of the paper, points will be denoted by lower-case letter (except the origin O of the coordinates system), line by upper-case letter, area by upper-case letter within parentheses and a volume by upper-case letter within brackets.

The dumbbell projection on a plane is considered.

θ ( 0 / 2 θ π ≤ ≤
) is the angle between the line binding the centres of the two spheres (radius value equal to one) and the projection plane (coordinates x,y). The distance between the two sphere centres is denoted δ. The centre of one of the two spheres is chosen as the origin O of the coordinates system (Fig. 1). The projection of the dumbbell is represented in the figures 2a-b for two values of the θ angle and for a semiplane. The circles C1 and C2 represent the projection of the spheres (disks (C1) and (C2))

whereas the dashed curve (ellipse later called E1 in the paper) represents the projection of the circular junction J between the two spheres. Let us consider a chord (perpendicular to the projection plane). According to the location of its intersection (x,y) point with the plane this line may cross:

-one single sphere: the point belongs to (P1)

-the junction between the two spheres: the point belongs to (PJ)

-or successively the two spheres: the point belongs to (P12).

The (Pf) area that is bounded by C2 (and C1) and E1 is fictitious. Its definition will be detailed hereafter.

The previously defined areas correspond to a full plane.

The ellipse E1 has some interesting properties:

-the coordinates of E1 centre are ( ( )

2 cos δ θ ,0)
-the ellipse intersects the circles at the two b1 and b2 points (in fact four points if one considers the other semi-plane). The coordinates of the b2 point are

OCD P PJ C C C PJ C P C Pf C C C D l D l D l D l D l D l D l D l + ∩ = + + - - - - (2)
( ) X D l is the CLD for the chords perpendicularly crossing the (X) projected area.

( )

1, C X D l is
the contribution of the (X) area to the CLD of a single sphere.

1, 1, 12 1,
, ,

C PJ C P C Pf D D D are calculated for 1 E θ θ < whereas 1, 1 2 C C C D ∩ is calculated for 1 E θ θ > .
These partial distributions are equal to 0 outside the previously defined range of θ-value.

Calculation of the CLD (MCD)

The chord length distribution of the dumbbell consists of the contributions of the various areas:

( ) ( ) ( ) ( ) ( ) 1 2 1, 1, 2 2 MCD PJ C C C PJ C Pf D l D l D l D l D l + = + - - (3) 1, C Pf D is calculated for 1 E θ θ < whereas 1, C PJ D is calculated for any θ .
In the following of the paper, the various contributions are calculated.

As the number of chords perpendicularly crossing a projection plane and the particle with a chord length within the range [l,l+dl] is proportional to the corresponding projected area element, the relation between the projected area S X and the chord length distribution ( )

X D l is: ( ) X X dS D l N dl = ± (4)
N is a normalization factor which will be specified later. Depending on the calculation, the area S X is the part of the projected area connected with a chord length smaller or higher than a given l value. The equation 4 is integrated over the solid angle domain.

The calculation of ( )

X
D l depends at once on the δ and l values. For instance, certain l values are not available for a given δ value; moreover, the shapes of (P1), (PJ), (P12), (Pf) projected areas depend on the δ values. As a consequence, a careful analysis of the various projected areas has to be performed before calculating the CLD's.

Analysis of the various projected areas

Description of area (P12) and the corresponding chord length

The length of the chord (l) which crosses the two spheres obeys the equations:

( ) ( ) 1/ 2 1/ 2 2 2 2 1 2 1 ' l d d = - + - if 1, ' 1 d d < < with ( ) 1/ 2 2 2 d x y = + and ( ) ( ) ( ) 1/ 2 2 2 ' cos d x y δ θ = - +
The locus of the points (x,y) with constant l is a segment of the E2 ellipse:

( ) ( ) δ θ δ θ -   - - =   +   , ( )
1/ 2 2 2 1 cos / 4 b T δ θ   = - -   and 2 /16 T l =
The dotted curve in the figure 3a represents the ellipse E2. For small value of l ( 1 l

) the endpoint of E2 is close to the intersection point between C1 and C2. As the l value increases the endpoint of E2 is going away from this intersection point. E1 and E2 are matching for a particular value of l ( ( )

2 sin δ θ = ).
The ellipse E2 exists if ( 2 0 b > ):

-

( ) ( ) ( ) 1/ 2 0 cos cos 2 1 / T θ θ δ < = - and 2 1 / 4 T δ > - or - 2 1 / 4 T δ < -
The intersection of the ellipse E2 with the circle C1 exists if A<1 with ( ) ( )

2 cos cos 2 T A δ θ δ θ = +
The coordinates of the intersecting point are: 2

x A = and ( ) 3b).

1/ 2 2 2 2 1 y x = - . If 1 A > , there is no intersection (figure
One defines the angles θ + and θ -as:

( ) ( ) ( ) 1/ 2 cos 1 1 4 / X T θ δ ± ± = = ± - which are the roots of 1 A = ( if 2 l < ).
The two angles θ ± exist if 0 1 X ± ≤ ≤ . In fact the conditions are more strict:

( )

1 2 0 cos E E X θ ± ≤ ≤ .
As only the part of E2 into the region bounded by C1, C2 and E1 is relevant, then the condition on θ is ( )

2 sin l δ θ ≤ i.e.

( ) (

) ( )

1/ 2 2 1 2 sin sin 4 / E E T θ θ δ ≥ = . It can be proved that 1 2 0 E E θ θ > . As ( ) 1 2 sin 1 E E θ ≤ , then 2 l δ ≤
. We emphasize that a chord goes across the two spheres if

1 2 / 2 E E θ θ π < <
. This leads to an additional condition on T and δ.

So, the figure 4 defines the different areas in the ( 2 / 4,T δ

)-space: Each one corresponds to a specific configuration concerning C1, C2, E1 and E2. Each area also corresponds to a specific expression of the CLD.

Area 1: intersection between C1 and E2; ,

θ θ + - ∃ ∃ ; 1 2 / 2 E E θ θ π - < <
Area 2: no intersection between C1 and E2: θ ± ∃ Area 3: intersection between C1 and E2; θ ± ∃ ;

1 2 0 / 2 E E θ θ θ π + - < < < < .

Description of area (PJ) and the corresponding chord length

(PJ) area behaves as the (P12) area.

The length of the chord (l) which crosses the two spheres obeys the equations:

( ) ( ) ( ) 1/ 2 1/ 2 2 2 1 1 ' sin l d d δ θ = - + - + if 1, ' 1 d d < < with ( ) 1/ 2 2 2 d x y = + and ( ) ( ) ( ) 1/ 2 2 2 ' cos d x y δ θ = - +
The locus of the points (x,y) with constant l is the E3 ellipse:

( ) ( ) 2 2 2 2 cos / 2 1 ' ' x y a b δ θ - + = (6) with ( ) ( )
( ) ( ) 1/ 2 1/ 2 1/ 2 2 1/ 2 2 1/ 2 ' 4 sin 1 / 4 4 2 sin / / 4 4 2 sin / 2 a T T T T T θ δ δ θ δ δ θ     = - - -+ + -     ( ) 1/ 2 2 1/ 2 ' 1 / 4 4 2 sin b T T δ δ θ   = - -+  
The figure 5 represents the E3 ellipse (dotted line). The properties of E3 are the following:

-E3 corresponds to the inequality ( )

2 sin l δ θ ≥ -E3 exists for any θ if 2 4 1 / 4 T δ < - -E3 exists for 2 θ θ ≥ with ( ) ( ) ( ) 2 2 2 sin 4 / 2 l l θ δ δ = + - if 2 4 1 / 4 T δ > - -E3=E1 for ( ) 2 sin l δ θ = -E3 may intersect E1. The x-coordinate of the intersection point is: ( ) ( ) ( ) ( ) 13 cos / 2 tan sin / 2 x l δ θ θ δ θ = ± - -E3 intersects E1 if ( ) ( ) ( ) ( )(
) ( ) ( )

1/ 2 2 tan sin / 2 sin 1 / 4 l θ δ θ θ δ - ≤ - , i.e. ( )
( ) 2
1/ 2 1/ 2 2 2 sin 1 4 / 4 X T T T θ δ δ δ ± ± = = ± + -- which are the roots of 0 B = ( if 2 l < ).
The two angles θ ± exist if 0 1 X ± ≤ ≤ . In fact the inequalities are more strict:

( ) ( ) 2 1 2 sin sin E E X θ θ ± ≤ ≤
. This leads to an additional condition on T and δ. As only the part of E3 inside E1 is relevant, then the condition on θ is ( )

2 sin l δ θ ≥ i.e.

( ) (

) ( )

1/ 2 2 1 2 sin sin 4 / E E T θ θ δ ≤ = . As ( ) 1 2
sin 1

E E θ ≤ , then 2 l δ ≤ .

Description of area (P1) and the corresponding chord length

Area (P1) corresponds to the path of a line (chord) across a single sphere. The simplest way to calculate the corresponding CLD is to consider the various contributions of each projection area knowing that ( ) ( ) ( ) ( ) ( ) ( )

1 12 1 2 P PJ P Pf C C + + + = =
. However two cases appear:

-E1 is tangential to the circles C1 and C2: we have to calculate the contribution of (PJ), (P12) and (Pf) zones.

-E1 is not tangential to the circles C1 and C2: ( ) ( ) ( ) ( ) ( )

1 1 2 1 2 P C C C C + ∩ = = . Note that ( ) ( ) ( ) ( ) 12 1 2 PJ P C C + ≠ ∩ .
The transition between these two cases happens for

( ) cos / 2 θ δ = ( 1 E θ θ = ).

Contribution of (C1)∩(C2)

(

1 E θ θ > )
The figure 7 describes the different areas in the ( 2 / 4,T δ )-space: each one corresponds to a specific configuration concerning C1 and C2. Each area also corresponds to a specific expression of the CLD. Each area corresponds to a set of inequalities:

1 δ > or 1 δ < ( ) ( ) 2 2 / 2 1 1 l δ + - > or ( ) ( ) 2 2 / 2 1 1 l δ + -< ( ) ( ) ( ) 1/ 2 1 cos 1 1 4 / cos / 2 E T θ δ θ δ ± = ± - ≤ = θ ± is such as ( ) ( ) ( ) 2 2 / 2 cos 1 1 l δ θ + - = ,i.e. A=1
Each area is characterized by several angular intervals 1 ;

j j k θ θ +     (j=1, 2) defined as: Area 1: [ ] [ ] 1 1 2 ; ; / 2 E θ θ θ π - - Area 2: [ ] 1 2
; / 2

E θ π Area 3: [ ] [ ] 1 2
; ; / 2

θ θ θ π + - -
The index k corresponds to the angular distance k ψ ∆ :

1: 1 ψ ϕ ∆ = 2: 2 ψ π ∆ = Where ( ) ( ) ( ) ( ) 1/ 2 2 cos / 2 2 / cos acos 1 / 4 T l δ θ δ θ ϕ - = -

Contribution of (PJ) (for any θ)

We are looking for the intersection between E1 and a circle denoted C3 with radius ( )

1/ 2 2 1 / 4 l - (l is constant). Depending on θ-value, 0,
+ + + +             Area 2: 0 1 3 0; ; ; / 2 l l r r θ θ θ θ π - - - -             Area 3: 0 1 0 0; ; ; / 2 l l r r θ θ θ θ π - - + +             Area 4: 2 1 3 0; ; ; / 2 l l r r θ θ θ θ π + + - -            
The index k corresponds to the angular distance k ψ ∆ :

0: 0 0 ψ ∆ = 1: 1 ψ ψ - ∆ = 2: 2 ψ ψ ψ - + ∆ = - 3: 3 ψ π ∆ = Where ( ) ( ) 1/ 2 2 cos / 1 / 4 a u l ψ ± ± = -

Contribution of (PJ)

( 1 E θ θ < )
The various areas in the ( 2 / 4,T δ

) sub-space are already described in the figure 6.

Each area has several angular intervals 1 ;

j j k θ θ +     defined as: Area 1a: 1 2 1 0; ; l l E θ θ θ + +         Area 1b: 1 0 1 0; ; l l E θ θ θ - -         Area 2a: 1 2 1 3 0; ; ; l l r r E θ θ θ θ θ + + - -             Area 2b: 1 0 1 3 0; ; ; l l r r E θ θ θ θ θ - - - -             Area 3: [ ] 1 0 0; E θ Contribution of (P12) ( 1 E θ θ < )
The allowed area for the calculation of the CLD is the area denoted 1 (figure 4).

Contribution of (Pf)

(

1 E θ θ < )
The allowed areas for the calculation of the CLD are the areas denoted 1a and 2a (figure 6).

Contributions of the different areas to the CLD

Contribution of (P12) area to the CLD

The fraction of the projected area, such as the chord length is smaller than l, obeys the equations:

-if C1 and C2 intersect E2

( ) ( ) 2 1/ 2 2 cos / 2 1/ 2 2 2 1 0 4 1 cos 1 / 2 x S x b x a dx δ θ δ θ -         = -+ - -               ∫ (7a) -if C1 and C2 do not intersect E2 ( ) ( ) ( ) ( ) 1/ 2 1/ 2 2 2 1/ 2 2 2 2 2 sin 2 cos cos 4 4 S T T T δ δ α α π θ θ -       = - - + -+               (7b) With ( ) ( ) 1/ 2 2 2 sin 1 cos 4 δ α θ   = -    
The average CLD over all the orientations will be such as:

-area 1:

( ) ( ) ( ) θ θ θ θ θ θ - + + - = + + ∫ ∫ ∫ (8-3)

Contribution of (PJ) area to the CLD

The fraction of the projected area, such as the chord length is smaller than l, obeys the equations:

-if E3 intersects E1 ( ) ( ) ( ) ( ) ( ) 13 cos / 2 1/ 2 1/ 2 1/ 2 2 2 2 2 2 1 0 4 1 / 4 1 / sin ' 1 / ' x S x b x a dx δ θ δ θ -   = - - - -     ∫ (9a) -if E3 does not intersect E1 ( ) ( ) 2 2 1 / 4 sin ' ' S a b π δ θ π = - - (9b)
The average CLD over all the orientations will be such as:

-area 1a :

( ) ( )

1 2 1 0 / cos E E PJ D l N dS dl d θ θ θ = ∫ (10-1a)
-area 1b:

( ) ( ) ( )

1 2 1 2 2 / cos / cos E E PJ D l N dS dl d N dS dl d θ θ θ θ θ θ θ θ - - = + ∫ ∫ (10-1b)
-area 2a:

( ) ( ) ( )

1 2 2 1 0 / cos / cos E E PJ D l N dS dl d N dS dl d θ θ θ θ θ θ θ + + = + ∫ ∫ (10-2a) area 2b : ( ) ( ) ( ) ( ) 1 2 1 2 2 2 / cos / cos / cos E E PJ D l N dS dl d N dS dl d θ θ θ θ θ θ + - - + = + + ∫ ∫ ∫ (10-2b)
-areas 3 and 4:

( ) ( ) 1 2 2 2
/ cos

E E PJ D l N dS dl d θ θ θ θ = ∫ (10-3) If 2 l δ > then 1 2 / 2 E E θ π = 2.

Contribution of (P1) area to the CLD

Contribution of (C1)∩(C2)

(

1 E θ θ > )
Let us consider the sphere of centre O and its projection. The area element dS belonging to the intersection of the two disks (projected spheres) and having the chord length in the range [ ]

, l l dl + , is: -if ( ) cos 1/ θ δ > 1 1 dS r dr ϕ ϕ = with ( ) ( ) 1 cos 1 ,1 r δ θ   ∈ -   -if ( ) cos 1/ θ δ < 1 1 dS r dr π π = with ( ) ( ) 1 0, 1 cos r δ θ   ∈ -   1 1 dS r dr ϕ ϕ = with ( ) ( ) 1 1 cos ,1 r δ θ   ∈ -   With ( ) ( ) 1/ 2 1/ 2 2 1 1 / 4 1 4 r l T = - = - .
The average (for the 2 spheres) over all the orientations is written:

-area 1:

( ) ( ) ( ) / 2 1, 1 2 1 2 / cos 2 / cos C C C E D l N dS dl d N dS dl d θ π ϕ π θ θ θ θ θ θ - ∩ - = - - ∫ ∫ (11-1)
-area 2:

( ) ( )

/ 2 1, 1 2 1 2 / cos C C C E D l N dS dl d π π θ θ θ ∩ = -∫ (11-2) -area 3: ( ) ( ) ( ) / 2 1, 1 2 2 / cos 2 / cos C C C D l N dS dl d N dS dl d θ π ϕ π θ θ θ θ θ θ - + - ∩ = - - ∫ ∫ (11-3)
Contribution of (PJ) (

/ cos

j j C PJ k j j D l N r dr dl d θ θ ψ θ θ + = = - ∆ ∑ ∫ (12)
With ( )

1/ 2 2 1 1 / 4 r l = - This relation is valid at once for ( 1 E θ θ < ) and ( 1 E θ θ < and 1 E θ θ > ) Contribution of (P12) ( 1 E θ θ < )
One considers the 1 2 d d arc (or the corresponding angle) defined by the two d1 and d2 points: d1 is the intersection point of C3 and E1, d2 is the intersection of C3 and C2.

( )

1 2 1 cos / d d a u r ϕ - = -
The swept area by increasing the radius of the C3 circle is: Contribution of (Pf) (

1 1 2 1 dS r d d dr = ( ) ( ) ( ) ( ) 1
1 E θ θ < )
The swept area by increasing the radius of the C3 circle is:

1 1 dS r dr ψ + = ( ) ( ) 1, 0 2 / cos l C Pf D l N dS dl d θ θ θ + = -∫ (14)

Contribution of (C1)+(C2)

The chord length distribution of the 2 spheres considered alone is obtained while taking ϕ π = :

( ) ( )

1 2 2 / 2 C C D l N l π + = (15)

Normalization factor

The normalization factor N is the inverse of the dumbbell projected area averaged over all the orientations:

( )

/ 2 1 0 cos p p N S S d π θ θ θ -= = ∫ (16) With ( ) ( ) ( ) ( ) ( ) 1

Comparison with Monte Carlo Simulations

Two procedures for Monte Carlo Simulations (MCS) have been used:

2 i
Q is the length of the chord in the sphere i. d is the distance between the two end points of the line crossing the two spheres.

The MC sampling distribution may be visually represented as the discrete probability histogram. The chord length between zero and the maximal possible length is divided on mbins with the equal size of ∆l. All the simulation runs have been carried out by generating 10 7

unbiased random lines. The sampling error is smaller than 10 -4 .

ii) An alternative algorithm is suitable for application in optics (AD approximation, see [START_REF] Gruy | the chord length distribution of a two-sphere aggregate[END_REF]).

It completes the second step of the preceding algorithm by scanning the discretized projected area and by calculating the corresponding chord length for each point belonging to the grid.

The other steps of the algorithm are not modified. If this sequence is repeated 10 5 times, the sampling error is smaller than 10 -3 . The two algorithms are in very good agreement. However this variant will not be used in this work.

The analytical CLD 1 (section 2) is compared to the data coming from MCS. Figure 11 shows such a comparison for δ=0.4 (MCD). The agreement between the two methods is very good.

Similar results are obtained for the two types of CLD (MCD and OCD) and for all δ-values. ( ) ( ) ( )

1/ 2 2 1 1 / 1 N MCS i i i D l D l N =   Σ = -     ∑
The summation is performed with the chords in the ranges [0; 1.98] and [2.02; 4].

The Σ-value has been calculated for each δ-value and for the two types of CLD. The Σ-value is always within the [0.002-0.004] range.

Conclusion

The CLD for the two types of chord length (OCD and MCD) has been analytically calculated.

A good agreement was found between the analytical expression of the CLD and Monte-Carlo simulations. The analytical CLD for dumbbell-like aggregates is an illustration of the ideas of Vlasov [START_REF] Yu | Signed chord length distribution[END_REF]: non convexity leads to complicated expressions for CLD that are built by considering the contributions coming from several parts of the body. This work will be used for studying systematically the effect of the δ-parameter on the CLD and its moments. It also permits to compare OCD and MCD. The corresponding results will be soon published.

So, the figure 6

 6 defines the different areas in the ( 2 / 4,T δ )-space: Each one corresponds to a specific configuration concerning E1 and E3. Each area also corresponds to a specific expression of the CLD.Area 1a: no intersection between E1 and E3;

-

  1 or 2 intersection points are present. In case of 2 intersection points, their x-coordinates are a diagram in the ( , x θ ) space showing the intersection of C3 with E1.Each line segment of the dash-dot curve corresponds to an angle interval with 0, 1 or 2 intersection points. We called , , , the boundary angles between segments (r: right branch; l: left branch; +: solid line; -: dashed line).Depending on (δ, l) values, one identifies several areas in the (2

Figure 12

 12 Figure12represents the relative difference σ between the two methods: §

-

  It can be observed that the relative deviation is very small except about l=2. This last feature is due to the sharp discontinuity for D(l) at l=2 (Fig.11). The range of l-value corresponding to this large deviation is [1.98; 2.02]. MCS and our analytical calculations (including numerical integration) fail together within this l-range. For comparing the two methods by a quantitative way we have calculated the standard relative deviation as:

Figure 1 :Figure 2a :Figure 2b :

 12a2b Figure 1: 3D drawing of the dumbbell shaped aggregate

Figure 3a :Figure 3b :Figure 4 :Figure 5 :Figure 6 :Figure 8 :

 3a3b4568 Figure 3a: projection (x,y) of the dumbbell shaped aggregate. δ=1.5 and θ=0.5rad;

Figure 10 :Figure 11 :

 1011 Figure 10: the dashed area (projection plane) is C1∩C2.

Figure 12 :

 12 Figure 12: relative deviation between the CLD (MCD) calculated from Monte Carlo simulations and the analytical method (section 2). δ=0.4.

The calculation of the intercept I depends on the value of Q i given by ( ) ( )

with i=1,2 and where ( ,

x y ) are the coordinates of the centres of the projected spheres and (x, y) are the coordinates of intersection of the line and the plane.

Then,


Marie-Estelle Binet

Fabrizio Carlevaro

Michel Paul

ESTIMATION OF RESIDENTIAL WATER DEMAND WITH IMPRECISE PRICE PERCEPTION

Keywords: D12, C26, Q25 billing information, price perception, residential demand for water, time unbalanced panel data

. The empirical results support the hypothesis that households respond to the average price perceived from the latest water bill. Households facing an increasing block rate schedule perceive a price of water that is generally lower than its actual marginal price. This conclusion emphasizes the relevance of a marginal price information policy to promote water saving.

Introduction

As shown by the number of surveys recently published in the literature, the estimation of residential demand for water is a major issue in environmental economics (see [START_REF] Arbuès | Estimation of residential water demand: a state-of-the-art review[END_REF], [START_REF] Dalhuisen | Price and income elasticities of residential water demand: a meta analysis[END_REF], [START_REF] Worthington | An empirical survey of residential water demand modelling[END_REF], [START_REF] Nauges | Estimation of water demand in developing countries: An overview[END_REF]). A large part of this empirical literature aims to obtain consistent estimates of the price elasticity of demand for water as a prerequisite to analyze the relevance of pricing policies for a demand-side management of water consumption.

As water tariff schedules are often complex, with increasing or decreasing block rates and fixed charges, an important issue discussed in the literature is the provision of a relevant specification of the price variable for residential water consumption analysis. The discussion generally focuses on whether to use the average price of water or the marginal price measured as the block rate charged on the last unit consumed. A perfectly informed consumer should react to the marginal price, but there is a strong presumption that imperfect information may confuse consumer perception of block rates. Therefore, in the case of incomplete information, the consumer may respond to other price indicators, in particular the average price which can be obtained easily by dividing the water bill by water consumption.

To date, the determination of the price variable to which consumers respond has been tackled as an empirical issue. [START_REF] Howe | The impact of price on residential water demand and its relation to system design and price structure[END_REF] were the first to discuss and compare average and marginal prices for water demand analysis. In a first approximation, the price providing the best fit is presumed to be the price perceived by consumers (see [START_REF] Foster | On the specification of price in studies of consumer demand under block price scheduling[END_REF]). [START_REF] Ruijs | Demand and distributional effects of water pricing policies[END_REF] provide a more recent study comparing the use of average versus marginal prices in water demand modelling.

At least two formal statistical tests of average versus marginal prices have been developed in the literature to gain a better insight into the issue of consumer price perception. The first was suggested by [START_REF] Opaluch | Urban residential demand for water in the United States: further discussion[END_REF]. Unfortunately, this test can lead to the rejection of both water price assumptions by preventing one from discriminating between these two price specification, as occured in the study of [START_REF] Ruijs | Demand and distributional effects of water pricing policies[END_REF]. For this reason, we did not use

Opaluch"s test in our study.

The second test was suggested by [START_REF] Shin | Perception of price when price information is costly: Evidence from residential electricity demand[END_REF] to analyze residential electricity demand of households faced with a declining block rate schedule. The test is based on a specification of the electricity price perception, which is a weighted geometric average of marginal and average prices, where the weight plays the role of a price perception parameter leading to one of these two prices depending on whether its value is 0 or 1. Thus, the relevant price perception specification can be identified by estimating and testing the value of the price perception parameter. [START_REF] Nieswiadomy | A note of price perception in water demand models[END_REF] first used this methodology to identify, through a time series approach, the price perception of residential water consumers faced with a multistep block rate schedule. To the best of our knowledge, with the exception of the working paper by [START_REF] Kavezeri-Karuaihe S T | Perceived water prices and estimated water demand in the residential sector of Windhoek, Namibia: an analysis of different water market segments[END_REF], no recent studies have used Shin"s methodology.

This paper intends to contribute to the literature on empirical residential water demand by applying Shin"s methodology to a unique micro data set collected on an island where the use of water resources has become a source of increasing controversy. We use a sample of 449 useful water bills (between 1 and 3 water bills per household) collected from a household survey in the French overseas territory of Réunion. We have time unbalanced panel data as residential water consumption is observed for durations and periods of time that change across the surveyed households.

The conclusions that we draw from this empirical study also have innovative implications for policy analysis. In the case of imperfect information and increasing block rates, consumers may be led to think that each additional unit of water consumed is paid at an average price which turns out to be smaller than the actual price paid, namely the marginal price. Therefore, an effective water saving policy can consist in improving consumer"s price information, i.e. by providing households information about the marginal price they pay on their water bill.

Using our econometric results, we assess by simulation the magnitude of the impact of such an information pricing policy on water conservation. Our simulation shows that if households set their water demand according to marginal price instead of perceived price, then average water consumption would decline substantially. This paper proceeds as follows. In section 2, the model specification is presented. Section 3 describes the data used in our empirical application. Section 4 examines and discusses the empirical results. Section 5 concludes by outlining recommendations intended to improve the information provided to households on the cost of water, thereby allowing them to rationally respond to pricing policies aiming to promote household water saving behaviour.

Modelling residential water demand

In this section, a description of the pricing model is followed by a presentation of the water demand specification.

Pricing model

On Réunion island, as in numerous other locations, the pricing of residential water consists of several increasing consumption blocks (between 1 and 4 depending on the municipality considered).

To simplify the presentation, consider a block rate schedule consisting of two consumption blocks with increasing prices 1  and 2  and a fixed charge F . The consumer"s budget constraint can be written as follows:

, ) 1 ( 1 Y X p q F x    
if the consumer"s actual water consumption, denoted by q , is located in consumption block 1 and as:

, ) ( ) 2 ( 1 2 1 1 Y X p b q b F x        if q is located in consumption block 2, with 1
b the highest consumption level in block 1.

x p and X are respectively a price index and the corresponding consumption of other private goods, while Y is the household income.

According to [START_REF] Nordin | A proposed modification on Taylor"s demand analysis: Comment[END_REF], in both cases the budget constraint can be rewritten as a standard budget constraint of the form:

D F Y X p q x     ) 3 (  with         1 1 1 2 1 if , ) ( 0 if , 0 ) 4 ( b q b b q D  
and  is the "marginal price" of water, namely: either

1  if 1 0 b q   or 2  if 1 b q  . This
formalization can be easily generalized to the case of a multi block rate schedule. Formula (3) claims that a perfectly informed water consumer should react not only to marginal price, but also to changes in intra-marginal prices through an income increase measured by Nordin"s difference variable D, expressing the refunding to which the consumer is entitled whether he had paid his entire water consumption at the marginal price.

But, if consumers are not perfectly informed about the method used by the residential water provider to price water consumption, their price perception may lie between marginal and average price  , so defined:

, ) 5 ( q F D     
which is lower than marginal price  when there is no fixed charges or when Nordin"s

F D  .
According to [START_REF] Shin | Perception of price when price information is costly: Evidence from residential electricity demand[END_REF], the perceived price to which an imperfectly informed consumer responds may be expressed by the following formula:

. ) ( ) 6 ( k p    
According to this formula the consumer responds to marginal price  when the perception parameter k is equal to zero and to average price  when k is equal to one. Assuming    , under an increasing block rate schedule, the perceived price lies between average and

marginal price if 1 0   k , whereas     p when 1  k and     p when 0  k .
Parameter k was estimated for the first time by [START_REF] Shin | Perception of price when price information is costly: Evidence from residential electricity demand[END_REF] for electricity demand, then by [START_REF] Nieswiadomy | A note of price perception in water demand models[END_REF], [START_REF] Nieswiadomy | Estimating urban residential water demand: effects of price structure, conservation, and education[END_REF] and [START_REF] Nieswiadomy | Impact of pricing structure selectivity on urban water demand[END_REF] for water demand. These studies had mixed results. Under a declining rate schedule, [START_REF] Shin | Perception of price when price information is costly: Evidence from residential electricity demand[END_REF] found a price perception parameter equal to 1.007, with the null hypothesis 1  k not rejected. He concluded that electricity consumers respond to the average price. [START_REF] Nieswiadomy | Estimating urban residential water demand: effects of price structure, conservation, and education[END_REF] and [START_REF] Nieswiadomy | Impact of pricing structure selectivity on urban water demand[END_REF] reported similar results. [START_REF] Nieswiadomy | A note of price perception in water demand models[END_REF] obtained a non significant parameter estimate under an increasing water consumption block rate schedule and concluded that the consumers respond to marginal prices. However, for a decreasing block rate schedule, they found a price perception parameter estimate not significantly different from that in [START_REF] Shin | Perception of price when price information is costly: Evidence from residential electricity demand[END_REF].

Residential water demand specification

The residential water demand function proposed by [START_REF] Shin | Perception of price when price information is costly: Evidence from residential electricity demand[END_REF] assumes a constant elasticity with respect to household income and perceived price. He used a partial adjustment model with time series data. We estimate a static form of Shin"s specification as we use time unbalanced panel data. So, consumption data were converted to a daily consumption per household. It is written as a regression equation of the form:

          x p Y q ln ln ln ) 7 ( 2 1 0
where 1

 and 2

 denote income and perceived price elasticities, x a vector of additional explanatory variables and  a random disturbance.

Replacing in this equation the unobservable perceived price p by functional form (6) leads to the derived regression equation:

               x k Y q ) ln( ln ln ln ) 8 ( 2 2 1 0
which is linear with respect to parameters 0  , 1  and  , but nonlinear with respect to the perceived price elastricity 2  and the perception parameter k.

In addition to price and income, one important explanatory variable is household size (N)

including the number of children and the number of adults in the family. If we consider the entire family"s daily needs, we expect that water consumption will increase with household size. However, we expect a greater impact of the percentage of non-working adults (PNWA), as they spend more time at home than other family members.

Generally, outdoor water use is a major determinant of residential water consumption. The presence of a garden is an important factor on Réunion island as in 2004, 77% of households lived in a house with a garden, compared with 30% in mainland France. The presence of a garden is defined here as a dichotomous variable that takes the value of one if the family has a garden, zero otherwise. We would have liked to use data giving the size of the garden, but our data set only noted the presence or absence of a garden without specifying garden size.

Outdoor uses are supposed to be influenced by weather conditions. We therefore decide to analyze the impact of the presence of a garden conditional to climate effects on water consumption (WEATHER). To this purpose, an interaction variable (GARD.WEATHER) defined by multiplying the dichotomous garden variable by the WEATHER variable has also been introduced into regression equation ( 8), leading to the residential water demand specification:

. . ln ) ln( ln ln ln ) 9 ( 5 4 3 2 2 1 0                    WEATHER GARD PNWA N k Y q
WEATHER is measured by the percentage of non rainy days over the billing period.

Household survey on Réunion island

The analysis is based on a unique survey dataset covering the entire territory of Réunion island. We first offer a description of the survey and then introduce the data selected for empirical analysis.

Household survey

Réunion, a French overseas territory lying in the Indian Ocean, is 70 km long and 50 km wide, with the population in 2004 estimated to be approximately 700,000 inhabitants. In 2004, a great part of the population was quite young (40% are under 40 years old). Furthermore, the population growth rate and the unemployment rate (about 30%) are both high. The climate is rather humid and tropical. The rainy season (from December to April) follows the dry season (from May to November). Rainfall differs considerably according to the geographical location: the northeast of the island receives about 70% of the total rainfall. Urban development mainly occurs in the northwest of the island, where the weather is dry. Lastly, household use of water in 2004 appears quite high, as the daily water consumption level on Réunion, computed with aggregate data, is 269 litres per inhabitant compared to an average of 150 litres on mainland France, [START_REF] Coutelier | La facture d"eau domestique en 2004, 177 Euros par personne et par an. Le 4 pages ifen[END_REF].

Water therefore has become the source of increasing controversy on Réunion because supply is failing to meet demand in many areas, especially in the western part of the island. In this context, the Regional Directorate for the Environment (DIREN) was given the important job of setting up an overall water management system based on a law passed in 1992 intended to secure the future provision of water on Réunion. The long term objective of the water management plan is to reduce water consumption by 30% over 20 years (or about 1.32% per year).  Consumption habits (washing frequency, business activity at home to verify that only domestic users are targeted).

Carried out by telephone, this first step survey was followed by a mailing to 1000 volunteer households intended to collect information on the volume of water consumption displayed on the last three bills that they had received (covering one year). Unfortunately, this second step survey provided 173 reliable responses supplying us with 449 useful water bills. Since the billing period varies across municipalities, consumption data were converted to a daily consumption per household (in litres).

Corresponding daily weather indicators as precipitation, temperature were taken from Meteo France Agency.

Description of the variables

Price of water

Data on rates and the length of consumption blocks were available from DIREN. The block rate schedule varies across municipalities (between 1 and 4 consumption blocks). As discussed by Taylor et al. ( 2004), the fixed charges are subtracted from the total bill to compute the average price to prevent bias in the estimated price elasticity. When the fixed portion of the bill is removed, the average variable price includes only the portion which varies with consumption.

Household income

The DIREN survey recorded household income level 1 as an ordered qualitative variable, namely as belonging to one of the following five income intervals (in Euros per month):

[0;750], [750;1500], [1500;3000], [3000;4500] and [4500;. +  ].

Unfortunately, this income information is not relevant to estimate the income elasticity of the water demand specification (9). Therefore, we used a quantitative estimate of the household income levels developed by [START_REF] Carlevaro | Econometric modeling and analysis of residential water demand based on unbalanced data[END_REF]. These estimates are based on an econometric model describing the observed qualitative information on household income according to an ordered polychotomous econometric model, where the unobserved household income level is specified as a latent variable. This unobserved variable is assumed to be distributed, within the household population, according to a log-normal random variable.

Furthermore, the household income distribution is influenced by some household income indicators, recorded by the DIREN survey, to characterize the household standard of living.

Finally, an individual income level estimate for each household of the DIREN sample is obtained by computing an estimate of the mean square error predictor of this latent variable, namely its expected value given all the available information at hand, including the income interval the household declared to belong to.

Note, that the household income level Y of model ( 9) has been defined as the former household income level estimate adjusted by substracting the fixed charges of the water tariff schedule and adding the Nordin"s difference variable (4), as explained in section 2.1.

Climate variables

The impact of climate on residential water use can be measured in different ways. In the literature, precipitation and temperature often are assumed to influence residential water demand. [START_REF] Schleich | Determinants of residential water demand in Germany[END_REF] provided some evidence that households respond to whether it rains or not rather than to the total amount of rainfall. More precisely, we chose to use the percentage of non rainy days over the billing period, with the expectation that demand for garden water will be higher when the percentage of non rainy days is high.

We used daily observations recorded by Meteo France Agency (about one hundred stations set up on Réunion). These geographical distributed observations allowed us to compute the number of days without rainfall for each bill collected according to the observations recorded at the closest weather station.

Water-using capital stock and other variables collected

We tested other explanatory variables, particular those measuring billing frequency, housing characteristics, consumption habits and the presence of a swimming-pool, dishwasher or washing machine. However, the parameters associated with all these variables turned out to be highly not significant (p-value greater than 30%), probably due to the absence of information other than presence or absence. As variables such as appliance stocks are particularly useful in analyses of long term reactions to a price shock, which was not the focus of our study, we did not address this issue.

The set of available variables measured in 2004 and summary statistics for all variables are displayed in Table 1: In our sample, average residential water consumption is 675 litres per household per day (which correspond to 253 litres per capita on average). [START_REF] Coutelier | La facture d"eau domestique en 2004, 177 Euros par personne et par an. Le 4 pages ifen[END_REF] confirm the pertinence of our estimates as they measured water consumption to be 269 litres per inhabitant on Réunion, which is similar to the highest consumption levels in the OECD.

For example, the average daily per capita water use in the UE-15 countries ranges from 115 litres in Belgium to 265 litres in Spain, EWA (2002). On Réunion, per capita residential water consumption also is about 60% greater than in mainland France, on average. Water bills furthermore account for 2.8% of household income, on average, and up to 30% for the poorest consumers.

Distinctive features such as lifestyle (a high proportion of households live in a house with a garden) and climate may help explain these high water consumption levels. Water prices also are very low on Réunion (around one Euro per m 3 in average). To conclude, the fixed charges represent a small part of household income, equal to 0.22% on average.

Empirical estimation

The estimation strategy is described in section 4.1, and empirical results are reported and discussed in section 4.2.

Estimation method

One problem which must be addressed with multipart tariff pricing is simultaneity because consumers select the quantity of water and the price simultaneously. As an application of the the Breush-Pagan test also revealed the presence of heteroscedasticity, we chose to implement a GMM (Generalized Method of Moments) estimator, using an appropriate set of instruments 2 , in order to deal with both econometric issues. Heteroskedasticity-robust standard errors estimates of parameter estimates were computed.

In the spirit of [START_REF] Hausman | The evaluation of results from truncated samples: The New Jersey income maintenance experiment[END_REF], prices associated with fixed levels of consumption (the three first quartiles of the water consumption distribution) were used as instruments for marginal and average prices. Instrumental variables must satisfy two requirements. They must be correlated with the endogenous variable and be uncorrelated with the error term. We use the [START_REF] Bound | Problems with instrumental variables estimation when the correlation between the instruments and the endogeneous explanatory variable is weak[END_REF] test to select relevant instruments and we perform the Hansen test of overidentifying restrictions to choose valid instruments.

Results of model estimation

The results of the GMM estimation of water demand model ( 9) are presented in Table 2:



indicates that the population moment conditions are not rejected, implying the validity of the choosen instruments. The adjusted R 2 is quite low but this is typical with microeconomic survey data.

The parameter estimates deserve the following comments.

 The income elasticity has the expected positive sign but is not statistically significant, probably due to the errors inherent in the indirect method of measurement used for quantifying this explanatory variable. Its low numerical value, equal to 0.29, reflects the feature of basic need of water consumption for the majority of households.

 The estimate of the perceived price elasticity obtained is equal to -0.31 and is statistically different from zero. This value is in the range of the estimates found in the applied econometric literature and expresses the difficulty of substituting water with other goods, except through an household saving behavior.

 We observe a positive and highly significant impact of household size on its water consumption. According to our estimate, an increase of 10% in the family size will result in an increase of 4.7 % in its daily consumption of water. This result can be understood by the existence of household economies of scale in the use of water [START_REF] Garcia-Valiñas | Efficiency and equity in natural resources pricing: A proposal for urban water distribution service[END_REF].

 As expected, a working adult consume less water at home than a non-working adult.

Hower, the impact of an increase of 10 percentage points in the share of non-working adults with respect to the total number of adults in the household increases the household consumption of water by only 0.044% and this impact is not statistically significant.

 As expected, the presence of a garden, combined with a low occurrence of rainfall, has a positive impact upon water consumption. More precisely, a 10% increase of the days without rainfall increases the household water consumption of 3.5%, but this impact is weakly significant.

 The perception price parameter estimate is equal to 1.53, reflecting a perceived price not only less than the marginal price but also less than the average price. This indicates that Réunion households highly underestimate the price of water. From a statistical point of view, the price perception parameter is significantly different from zero (the value for which perceived price is equal to marginal price), but not significantly different from 1 (the value for which perceived price is equal to average price). This result contrast with [START_REF] Nieswiadomy | A note of price perception in water demand models[END_REF], as they concluded that under increasing block rates, water consumers react to marginal price. But their analysis is based on a monthly time series between 1976 and 1985 of aggregate water consumption of the city of Denton, Texas, U.S.A. Therefore, their conclusion could be the result of an aggregation bias. Our explanation is that households of Réunion receive billing information, on the amount of water bill and of water consumption, enabling them to assess only a rough average price perception.

As a consequence, increasing the billing information of Réunion households on the actual marginal price of water they pay may noticeably reduce their consumption of water and therefore contribute to a sustainable use of this scare resource of the island. To our knowledge, [START_REF] Gaudin | Effect of price information on residential water demand[END_REF] has been the first to analyze the effects of billing price information on residential water demand. Her study shows that billing price information increases the value of water price elasticity.

To gain a better insight on the effectiveness of such an information policy on residential water saving, we simulate the impact of the presence of marginal price information on the bill on water consumption. We compare simulated water consumption levels obtained through estimated equation ( 9), by assuming that consumers respond to a perceived price defined by formula (6) with a perception price parameter k=1.53, to consumption levels obtained by assuming that consumers respond to marginal price (k=0). Using an average price as defined in section 3.2.1 (by excluding fixed charges), our simulations show that the household sample average water consumption level would decrease from 432 to 362 litres per household per day. These simulated results correspond to a drop in residential water consumption of 16%.

We conclude that the use of clear marginal price information as a water conservation policy could be advocated, in order to significantly reduce residential water consumption on Réunion.

Conclusions

This article examines the household"s perception of the price of water under an increasing block rate schedule by assuming that residential water consumers are not well-informed about the marginal price at which a rational consumer should respond. To estimate an unbiased value for the price elasticity of water, we use a methodology developed by [START_REF] Shin | Perception of price when price information is costly: Evidence from residential electricity demand[END_REF]. This methodology is based on a specification of the water price perception as a weighted geometric average of marginal and average prices, where the weight plays the role of a price perception parameter leading to one of these two prices depending on whether its value is 0 or 1. Thus, the relevant price perception specification can be identified by estimating and testing the value of the price perception parameter within an econometric specification of residential water demand.

Using a unique sample of water bills collected from a household survey on the French overseas territory of Réunion, our main finding suggests that the perceived price to which consumers respond is not only less than the marginal price but also less than the average price. This indicates that Réunion households highly underestimate the price of water.

Therefore, the addition of marginal price information on the bill alongside the quantity consumed could be an effective simple device to help reduce water consumption.

Using our model estimate, we assess by simulation the magnitude of the impact of such an information pricing policy on water conservation. Our simulation shows that if the survey households set their water demand according to marginal price instead of perceived price, then their average water consumption would decline substantially (from 432 to 362 litres per household per day, on average). As the block rates schedules remain the same, the distributional effects of such a clearer price policy are negligible compared to those resulting from a traditional pricing policy. The welfare and distribution effects of changes in block price systems are evaluated by [START_REF] Ruijs | Welfare and distribution effects of water pricing policies[END_REF] for example. We conclude that the use of clearer information on marginal prices should be considered in conservation policy as it should lead to lower water bills for all the consumers who reduce water consumption. Furthermore, it does not compromise the financial situation of the water companies as block rates are unchanged.

From a policy perspective, this price information policy could be extended, as recommended by [START_REF] Thaler | Nudge: improving decisions about health, wealth, and happiness[END_REF], to give consumers incentives to adopt ("to nudge") by themselves water conservation behaviours. The general idea is that water can be saved simply by suggesting the right options to households without imposing constraints. For example, the Southern California Edison company succeeded in reducing household electricity consumption by 40% by providing an "ambient orb" which turns red when consumption is excessive and green when moderate. Another application of such a policy also involves electricity consumption. In California, electricity bills include information about small water users" average consumption to encourage electricity saving by mimicking these consumption targets. To conclude, future analyses based on behavioral economics could provide more insights into the sensibility of water users to billing information and effective changes in water consumption behavior.

Notes

1. Households were asked to include all their income sources, including wages, welfare benefits, property revenues,…

2. Once the endogeneity in the price variable had been purged via IV or GMM estimators, the price coefficient estimate acquired the expected sign, whereas it was positive with the simple OLS technique.

  This paper analyzes the data of a survey sample performed by the authors on Réunion on behalf of the Regional Directorate for the Environment. The objective of the survey was to identify the reasons for the comparative over-consumption of water by island inhabitants compared to those on mainland France. The stratified random survey was financed by DIREN and conducted in 2004 on a sample of households living on Réunion. The survey was designed as a proportionate stratified random survey according to municipalities, on a sample of 2,000 households representing 1% of the total household population of Réunion. Carlevaro et al. (2007) provide further description of the survey. The questionnaire included 25 questions concerning:  Household socio-economic characteristics (sex, family head age and occupation, family income and size, number of working adults and children, and if they are property owners or tenants). Housing characteristics (detached house or flat, age, number of rooms, altitude). Water consumption equipment (swimming pool, washing machine, dishwasher, garden ownership).

Table 1 .

 1 Data set description, 449 water bills in 2004

	Description of variables	Mean	Min	Max
	Daily water consumption per household	675	102	5204
	(litres)
	Monthly household income (Euros)	2096	426	7374
	Average price of water without fixed	0.00089	0.000010	0.00218
	charges (Euros per litre)
	Marginal price of water (Euros per litre)	0.00118	0.00013	0.00346
	Household number of children	0.91	0	4
	Household number of working adults	0.98	0	3
	Household number of non-working adults	1.29	0	5
	Share of days without rainfall (%)	58	0	100
	Share of fixed charges in total income (%)	0.22	0.01	1.86
	Share of water bill in total income (%)	2.8	0.19	30.6

Table 2 .

 2 GMM estimation of water demand model (9)

	Model parameters and tests	Parameter estimates a and test statistics
			0	2.08
	Model intercept	(1.11)
		1	0.29
	Income elasticity	(0.82)
			2	-0.31 ***
	Perceived price elasticity	(2.72)
		3 '	0.47 ***
	Household size elasticity	(7.87)
		'	4	0.0044
	Rate of impact on household daily water	(1.33)