Aurélie Chapron

A local Blaschke-Petkantschin formula in a Riemannian manifold

In this paper, we show a local Blaschke-Petkantschin formula for a Riemannian manifold. Namely, we compute the Jacobian determinant of the parametrization of (n + 1)-tuples of the manifold by the center and the radius of their common circumscribed sphere as well as the (n + 1) directions characterizing the positions of the n + 1 points on it. We deduce from it a more explicit two-term expansion when the radius tends to 0. This formula contains a local correction with respect to the flat case which involves the Ricci curvatures in the (n + 1) directions.

Introduction and result

A Blaschke-Petkantschin formula is a rewriting of the (m + 1)-fold product of the volume measure for 1 ≤ m ≤ n which is based on a suitable geometric decomposition of a m-tuple of points, see the historical papers [START_REF] Blaschke | Integralgeometrie 1. Ermittlung der Dichten für linear Unterräume im En[END_REF] and [START_REF] Petkantschin | Integralgeometrie 6. Zusammenhänge zwischen den Dichten der linearen Unterräume im n-dimensionalen Raum[END_REF]. In particular, it provides the calculation of the Jacobian of the associated change of variables in an integral. Classically, the geometric decomposition consists in fixing the linear or affine subspace which contains all the points, integrate over all m-tuples in that subspace and then integrate over the Grassmannian of all subspaces, see [SW08, Chapter 7]. There exists another kind of formulas of Blaschke-Petkantschin type with a spherical decomposition: the sphere containing all the points is fixed, we integrate over the positions of the points on the sphere and then integrate over the Grassmannian of all subspaces spanned by the sphere and over the radius and center of the sphere.

This can be described as follows in the Euclidean space. For almost every set of points x 0 , . . . , x n ∈ R n , there exists a unique circumscribed ball. This makes it possible to define the change of variables (x 0 , . . . , x n ) → (r, z, u 0 , . . . , u n ), given by the relations

x i = z + ru i
where z ∈ R n and r > 0 are the circumcenter and radius respectively of the circumscribed ball of x 0 , . . . , x n and where u 0 , . . . , u n ∈ S n-1 indicate the respective positions of the points x 0 , . . . , x n on the boundary of that ball, S n-1 being the unit-sphere of R n . The calculation of the Jacobian of this change of variables dates back to Miles [START_REF] Miles | A synopsis of "Poisson flats in Euclidean spaces[END_REF]Formula (70)]. It provides the following equality of measures, see also [START_REF] Møller | Lectures on random Voronoȋ tessellations[END_REF] Proposition 2.2.3] for a more general statement with (m + 1) points, 1 ≤ m ≤ n: dx 0 . . . dx n = n!∆ n (u 0 , u 1 , . . . , u n )r n2 -1 drdz dvol (S n-1 ) (u 0 ) . . . dvol (S n-1 ) (u n )

(1)

where ∆ n (u 0 , u 1 , . . . , u n ) is the n-dimensional Lebesgue measure of the simplex spanned by the unit vectors u 0 , u 1 , . . . , u n and where vol (S n-1 ) is the spherical Lebesgue measure of S n-1 . In general, for any Riemannian manifold M , we denote by vol (M) the measure associated with the Riemannian volume form of M . The formula (1) was extended by Miles to the case of the unit-sphere in dimension two [Mil71b, Theorem 3.2] then in any dimension [START_REF] Miles | Isotropic random simplices[END_REF]Theorem 4]. Let S n k be the n-dimensional sphere centered at the origin and of radius 1/k. Again, for almost all (n + 1)-tuple of points in S n k , we denote by z and r ∈ (0, π 2 ) the center and radius respectively of the associated geodesic circumscribed ball and by u 0 , . . . , u n the respective positions of the points on the boundary of that ball. The equality of measures below is then a slight modification of Miles's formula when replacing S n with S n k :

dvol (S n k ) (x 0 ) . . . dvol (S n k ) (x n )

= n!∆ n (u 0 , u 1 , . . . , u n ) sin(kr) k n 2 -1 dr dvol (S n k ) (z) dvol (S n-1 ) (u 0 ) . . . dvol (S n-1 ) (u n ).

(2)

The aim of this paper is to extend the spherical Blaschke-Petkantschin formulas (1) and (2) to a general Riemannian manifold M , i.e. decompose the product measure dvol (M) (x 0 ) . . . dvol (M) (x n ) through a geometric decomposition based on the smallest geodesic circumscribed ball containing the (n + 1) points x 0 , . . . , x n . One of our goals is to use this formula for calculating mean values of Poisson-Voronoi tessellations in a manifold [START_REF] Calka | Mean asymptotics for a Poisson-Voronoi cell in a Riemannian manifold[END_REF]. Indeed, Blaschke-Petkantschin formulas are already a key tool to study the characteristics of such tessellations in the Euclidean case [START_REF] Møller | Lectures on random Voronoȋ tessellations[END_REF] and in the spherical case [START_REF] Miles | Random points, sets and tessellations on the surface of a sphere[END_REF]. In the hyperbolic case, Isokawa used for n = 2 [START_REF] Isokawa | Some mean characteristics of Poisson-Voronoi and Poisson-Delaunay tessellations in hyperbolic planes[END_REF] and n = 3 [START_REF] Isokawa | Poisson-Voronoi tessellations in three-dimensional hyperbolic spaces[END_REF] formulas of the flavour of Blaschke-Petkantschin even though they were not clearly stated.

We start by completing the work for manifolds with constant sectional curvature. For k > 0, let H n k be the n-dimensional hyperbolic space of curvature -k 2 . The next proposition provides the spherical Blaschke-Petkantschin formula when M = H n k .

Proposition 1.1. For every k > 0 and n ≥ 1, the following equality of measures is satisfied.

dvol (H n k ) (x 0 ) . . . dvol (H n k ) (x n ) = n!∆ n (u 0 , u 1 , . . . , u n ) sinh(kr) k n 2 -1 dr dvol (H n k ) (z) dvol (S n-1 ) (u 0 ) . . . dvol (S n-1 ) (u n ). (3)
Now let M be a Riemannian manifold of dimension n and vol (M) be its associated Riemannian measure. For any z ∈ M , we denote by T z M the tangent space of M at z, by •, • z the associated scalar product and by exp z the exponential map at z. Since the geometric transformation and the Jacobian calculation will be done in a vicinity of a fixed point z, we can assume that M is a compact set even if it means replacing M by a compact neighborhood of z. We recall that thanks to the Hopf-Rinow theorem, the compacity of M guarantees that M is geodesically complete, which implies that the exponential map exp z is defined on the whole tangent space T z M . Let us now define the set

T n M = {(z, u 0 , . . . , u n ) such that z ∈ M, u i ∈ T z M with norm 1, i = 0, . . . , n}
and let us introduce the function

Φ n : (0, ∞) × T n M → M n+1 (r, z, u 0 , . . . , u n ) → (x 0 , . . . , x n ) where x i = exp z (ru i ). (4)
For fixed (z, u 0 , . . . , u n ) ∈ T n M and fixed 1 ≤ i ≤ n, we denote by γ i the geodesic γ i : t → exp z (tu i ) so that x i = γ i (r). We consider an orthonormal basis of

T z M , V (i) = {v (i) 1 , . . . , v (i) n } where v (i) 1 = u i . The orthonormal basis of T xi M denoted by V (i) (r) = {v (i) 0 (r), . . . , v (i) n-1 (r)} is obtained by parallel transport of V (i) along γ i .
We denote by J Φn the Jacobian of Φ n , i.e. the function which satisfies the equality of measures

dvol (M) (x 0 ) . . . dvol (M) (x n ) = |J Φn (r, z, u 0 , . . . , u n )|dr dvol (M) (z) dvol (S n-1 ) (u 0 ) . . . dvol (S n-1 ) (u n ).
We provide a formula for J Φn in terms of several explicit Jacobi fields. To do so, we use a precise connection between Jacobi fields and derivatives of curves defined through the exponential map, see Lemma 2.1 below.

Theorem 1.2. The Jacobian determinant J Φn of Φ n satisfies

|J Φn (z, r, u 0 , u 1 , . . . , u n )| = n!∆ n (u 0 , . . . , u n ) n i=0 | det B (i) | (5)
where for any

0 ≤ i ≤ n, B (i) = (B (i) l,m ) 1≤l,m≤(n-1)
, is a (n -1) × (n -1) real matrix which satisfies for l, m = 1, . . . , n -1,

B (i) l,m = v (i) l+1 (r), J(i) m+1 (r) γi(r) , J(i) m being the Jacobi field along γ i with J(i) m (0) = 0 and J(i)′ m (0) = v (i) m
Moreover, when r tends to 0,

|J Φn (z, r, u 0 , u 1 , . . . , u n )| = n!∆ n (u 0 , u 1 , . . . , u n ) r n 2 -1 - n i=0 Ric z (u i ) 6 r n 2 +1 + o(r n 2 +1 ) (6)
where Ric z (u i ) denotes the Ricci curvature at z of u i and f (r) = o(g(r)) means that lim r→0 f (r) g(r) = 0. As expected, since the manifold can be approximated at first order by the tangent space at z, the first term of the expansion (6) corresponds to the Euclidean case. Moreover, when M has constant sectional curvature, (6) is consistent with both (2) and (3) for small r. Let us note that for r small enough, the Jacobian given by ( 5) is different from zero almost everywhere which implies thanks to the inverse function theorem that Φ n is a local C 1 -diffeomorphism.

As emphasized earlier, the Euclidean spherical Blaschke-Petkantschin formula was extended to provide the geometric decomposition of the (m + 1)-fold product of the Lebesgue measure on R n , for any m = 1, . . . , n. We recall below this general formula, see e.g. [Møl94, Proposition 2.2.3]. Let G(n, m) denote the Grassmanian of m-dimensional space of R n endowed with its Haar measure vol (G(n,m)) and let us consider the set

G(n, m) = {(L m , u 0 , . . . , u m ), L r ∈ G(n, m), u i ∈ L m with norm 1}.
Then the change of variables defined by

R + × R n × G(n, m) → (R n ) m+1
(r, z, L m , u 0 , . . . , u m ) → (x 0 , . . . , x m ) with x i = z + ru i , satisfies the equality of measures

dx 0 . . . dx m = c (n) m (m!∆ m (u 0 , . . . , u m )) n-m+1 r nm-1 drdz dvol (G(n,m)) (L m ) dvol (S n-1 ) (u 0 ) . . . dvol (S n-1 ) (u m )
where c

(n)

m is an explicit constant depending only on n and m. Going back to our manifold setting, it seems delicate to extend Theorem 1.2 to the decomposition of an (m + 1)-fold measure for any m. However, we are able to derive formulas of this type for the manifold M in the special cases m = 1 and m = (n -1) in Propositions 1.3 and 1.4 respectively. Indeed, in these particular cases, we can identify the Grassmanian with the unit sphere S n-1 and this makes it possible to write expansions of the Jacobian determinant.

Proposition 1.3 (Case m = 1). The Jacobian J Φ1 of the function

Φ 1 : R + × T 1 M → M 2 (r, z, u) → (x 0 = exp z (ru), x 1 = exp z (-ru)) satisfies |J Φ1 (r, z, u)| = 2 n (r n-1 - 2 3 Ric z (u)r n+1 + o(r n+1 )) (7)
In particular, we notice that the first term of the expansion in (7) is exactly the Jacobian in the Euclidean case since ∆ 1 (u, -u) is equal to 2.

Let us define the set T ⊥ n M = {(z, v, u 0 , . . . , u n-1 ), x ∈ M, v, u 0 , . . . , u n-1 ∈ T z M with norm 1 and v⊥{u 0 , . . . , u n-1 }}.

Proposition 1.4 (Case m = (n -1)). The Jacobian J Φn-1 of the function

Φ n-1 : R + × T ⊥ n M → M n (r, z, v, u 0 , . . . , u n-1 ) → (x i = exp z (ru i )) i=0,...,n-1 satisfies |J Φn-1 (z, r, v, u 0 , . . . , u n-1 )| = ((n -1)!∆ n-1 (u 0 , . . . , u n-1 )) 2 r n(n-1)-1 -r n(n-1)+1 ( n-1 i=0 Ric v z (u i ) + Tr(∆ -1 K)) + o(r n(n-1)+1 ) (8)
where

∆ =    1 -u T 0 . . . . . . 1 -u T n-1    and K =    K(u0,v) 2 u T 0 K(u0,v) 6 . . . . . . K(un-1,v) 2 u T n-1 K(un-1,v) 6   
The paper is structured as follows: we start with some geometrical preliminaries in Section 2. Section 3 is devoted to the case m = n with the proofs of Theorem 1.2 and Proposition 1.1 which is a corollary of Theorem 1.2. The particular cases m = 1 and m = (n -1) are derived in Sections 4 and 5 respectively. The paper ends with some concluding remarks.

Geometrical preliminaries

In this section, we introduce some useful notation and we survey several fundamental definitions and results from the theory of Riemannian geometry. For more details, we refer the reader to the reference books such as [START_REF] Do | Riemannian geometry. Mathematics: Theory & Applications[END_REF], [START_REF] Lee | Riemannian manifolds[END_REF] and [START_REF] Berger | A panoramic view of Riemannian geometry[END_REF].

Exponential map. Let x ∈ M and v ∈ T x M . There exists a unique geodesic γ v such that γ v (0) = x and γ ′ v (0) = v. The exponential map of v at x, denoted by exp x (v) is defined by the identity

exp x (v) = γ v (1) (9)
For sake of simplicity, we omit in the notation of the exponential map the dependency on the manifold M which should be implicit anyway.

Riemann curvature tensor and curvatures. Let us denote by R (M) the Riemann curvature tensor of M that is for x ∈ M and for u, v, w ∈ T x M ,

R (M) x (u, v)w = ∇ u ∇ v w -∇ v ∇ u w -∇ [u,v] w,
where ∇ denote the Levi-Civita connection. Let u,v be two unit vectors of the tangent space T x M with u, v x = 0. The sectional curvature of the plane spanned by u and v is defined through the identity

K (M) x (u, v) = v, R (M) x (u, v)u x .
Now let u be a unit vector of T x M and let us extend it to an orthonormal basis {u 1 , . . . , u n-1 , u} of T x M . The Ricci curvature of M at x in direction u, denoted by Ric (M) x (u) is defined by the identity

Ric (M) x (u) = n-1 i=1 K (M) x (u, u i ). (10)
Note that Ric (M)

x (u) does not depend on the choice of the basis.

Jacobi fields. A Jacobi field along a geodesic γ is a vector field J verifying the Jacobi equation

J ′′ (t) = R (M) γ(t) (γ ′ (t), J(t))γ ′ (11)
where the derivative of J is understood in the sense of the covariant derivative with respect to the Levi-Civita connection. In particular, along γ, there exists a unique Jacobi field with given J(0) and J ′ (0). We recall without proof the following general result which connects the derivative of the exponential map to the Jacobi fields [START_REF] Do | Riemannian geometry. Mathematics: Theory & Applications[END_REF]p. 119]. This is a key tool of the proofs of Theorem 1.2, Proposition 1.3 and Proposition 1.4.

Lemma 2.1. Let γ be a geodesic, c a curve on M such that c(0) = γ(0) and V a vector field along c such that

V (0) = γ ′ (0). Then the function f (t, s) = exp c(s) (tV (s)) satisfies ∂f ∂s (t, 0) = J(t) ( 12
)
where J is the unique Jacobi field along γ with J(0) = c ′ (0) and J ′ (0) = V ′ (0).

In the case of manifolds with constant sectional curvature, Jacobi fields have an exact expression [START_REF] Do | Riemannian geometry. Mathematics: Theory & Applications[END_REF]p113]. The particular case of the hyperbolic space H n k , stated in the following lemma, is the main argument of the proof of Proposition 3.

Lemma 2.2. Let γ be a geodesic of H n k and V be a parallel vector field along γ with norm 1 such that γ ′ (t), V (t) γ(t) = 0 and V (t) = 1. Then the vector field defined by

J(t) = sinh(kt) k V (t) (13)
is the unique Jacobi field of H n k along γ which satisfies J(0) = 0 and J ′ (0) = V (0).

Parallel transport. Let V be a vector field along a curve γ. V is called a parallel vector field if V ′ (t) = 0 for all t, in the sense of the covariant derivative. Now, for any u ∈ T γ(0) , there exists a unique parallel vector field V along γ such that V (0) = u, called the parallel transport of u along γ. In this paper, V (t) will be denoted by u(t). Note that the parallel transport is a linear isomorphism from T γ(0) M to T γ(t) M which preserves the scalar product.

3 Proof of Theorem 1.2 and Proposition 1.1

3.1 Proof of (5)

Let us fix (z, u 0 , . . . , u n ) ∈ T n M . We endow the tangent space T z M with an orthonormal basis {e 1 , . . . , e n } and write

u i = n j=1 u j i e j .
In order to write the Jacobian matrix of Φ n , we need to introduce well-adapted bases of the tangent spaces T xi M for i = 0, . . . , n. To this end, for each i we consider an orthonormal basis of

T z M , V (i) = {v (i) 1 , . . . , v (i)
n } where v

(i) 1 = u i . We consider the basis of T xi M , V (i) (r) = {v (i) 1 (r), . . . , v (i) n (r)}, obtained by parallel transport of V (i) along γ i (t) = exp z (tu i ).
Step 1: derivatives with respect to r.

Since ∂xi ∂r = P z→xi (u i ) = v (i)
1 (r), we notice that the column vector of the coordinates of ∂xi ∂r in the basis

V (i) (r) is ∂x i ∂r =      1 0 . . . 0      =     1 0 n-1 T     (14) z xn x 0 x i r r r u 0 un u i v (0) l (r) v (0) 0 (r) v (n) l (r) v (n) 0 (r) v (i) l (r) v (i) 0 (r)
V (i) (r)
where 0 n-1 = (0, . . . , 0). In the rest of the paper, we identify for sake of simplicity a vector (resp. a linear transformation) with its column vector in a natural basis (resp its matrix with well-chosen natural bases).

Step 2: derivatives with respect to z. Let us consider the submatrix with size n × n ∂xi ∂z . First, we notice that the entry ∂xi ∂z l,m is the projection onto v (i) l (r) of the derivative of x i with respect to z in the direction e m . Secondly, applying Lemma 2.1 to γ(t) = γ i (t) = exp z (tu i ), c(s) = γ(s) = exp z (se m ) and V as the parallel transport of u i along c, we obtain that the derivative of x i with respect to z in the direction e m is J

(i) m (r) where J (i) m is the unique Jacobi field along γ i such that J (i) m (0) = c ′ (0) = e m and J (i)′ m (0) = V ′ (0) = 0.
We deduce from these two observations that

∂x i ∂z l,m = ∂x i ∂z l,m = v (i) l (r), J (i) m (r) xi .
Now thanks to [DC92, Chapter 5, Proposition 3.6], we can do the following calculation:

v (i) 1 (r), J (i) m (r) xi = u i , J (i)′ m (0) z r + u i , J (i) m (0) z = u i , e m z = u m i . (15)
In particular, (15) shows that the first line of the submatrix

A (i) is equal to u T i .
In other words, we can rewrite A (i) as

∂x i ∂z =     u T i n/a     (16)
where 'n/a' only means that the submatrix of A (i) under its first line does not need to be explicit in the rest of the proof.

Step 3: derivatives with respect to u i . The submatrix ∂xi ∂ui has size n × (n -1). Again the entry ∂xi ∂ui l,(m-1)

for 1 ≤ l ≤ n and 2 ≤ m ≤ n, is the projection onto v (i) l (r) of the derivative of x i with re- spect to u i in the direction v (i) m . Moreover, applying Lemma 2.1 to γ = γ i , c(s) = z and V (s) = v (i) 1 +sv (i)
m , we get that the derivative of x i with respect to u i in the direction v

(i) m is J(i) m (r) where J(i) m is the unique Jacobi field along γ i such that J(i) m (0) = c ′ (0) = 0 and J(i)′ m (0) = V ′ (0) = v (i) m . Consequently, this implies that ∂x i ∂u i l,(m-1) = ∂x i ∂u i l,m = v (i) l (r), J(i) m (r) xi .
An application of [DC92, Chapter 5, Proposition 3.6] shows that v (i) 1 (r), J(i) m (r) = 0 which means that J(i) m is a normal Jacobi field. In particular, this implies that the first line of B (i) is identically equal to zero. In the sequel, we denote by B (i) the (n -1) × (n -1)-matrix under the first line of ∂xi ∂ui , i.e. such that

∂x i ∂u i =     0 n-1 B (i)     . (17)
In particular, we get for every 2 ≤ l, m ≤ n,

B (i) l-1,m-1 = v (i) l (r), J(i) m (r) xi (18)
Step 4: rewriting of the Jacobian determinant. The Jacobian determinant J Φn of Φ n can be written as

J Φn (z, r, u 0 , . . . , u n ) = det           
          
Combining this equality with (14), ( 16) and (17), we obtain that

J Φn (z, r, u 0 , . . . , u n ) = det                    1 u T 0 0 0 0 0 n/a B (0) 0 0 . . . . . . 0 . . . 0 1 u T n 0 0 0 0 n/a 0 0 B (n)                    (19)
where for sake of simplicity, we have written 0 for any zero matrix independently of its size. We can then apply a permutation of the lines so that the lines (1|u T i |0) appear in the first (n + 1) lines of a new matrix which is a block lower triangular matrix and which has the same determinant as the Jacobian determinant up to a possible minus sign:

|J Φn (z, r, u 0 , . . . , u n )| = | det           1 u T 0 . . . . . . 0 1 u T n 0 B (0) . . . . . . 0 B (n)           | (20) Using the fact that n!∆ n (u 0 , • • • , u n ) = det    1 u T 0 . . . . . . 1 u T n   
, we obtain (5).

Proof of (6)

We now derive from (5) the expansion (6) for small values of r. This only requires to expand det B (i) .

To do so, we expand the coefficients of B (i) given by (18) in Lemma 3.1 below.

Lemma 3.1. The coefficients of B (i) satisfy, for l, m = 2, . . . , n,

B (i) (m-1),(m-1) = v (i) m (r), J(i) m (r) xi = r - K(u i , v (i) m ) 6 r 3 + o(r 3 ) (21)
B (i) (l-1),(m-1) = v (i) l (r), J(i) m (r) xi = o(r 2 ), l = m. ( 22
)
where

K(u i , v (i)
m ) is the sectional curvature at z of the vectors u i and v

(i) m .
Proof of Lemma 3.1. Let us define the function f

(i) l,m (r) = v (i) l (r), J(i) m (
r) xi for fixed i, l and m. We only need to write the third order Taylor expansion of f

(i) l,m in the neighborhood of 0. Since J(i) m (0) = 0, we have f (i) l,m (0) = v (i) l , J(i) m (0) = 0. ( 23
) Because v (i) l (r) is a parallel transport, v (i)
′ l (r) = 0 and the derivative f

(i)′ l,m satisfies f (i)′ l,m (r) = v (i) l (r), J(i)′ m (r) xi .
Consequently, using

J(i)′ m (0) = v (i)
m , we get

f (i)′ l,m (0) = v (i) l , v (i) m z = δ l,m . (24)
The Jacobi field J(i) m (r) along the curve γ i satisfies the Jacobi equation, see e.g. [DC92, Chapter 5, §2]:

J(i)′′ m (r) + R( J(i) m (r), γ ′ i (r))γ ′ i (r) = 0
where R is the Riemann curvature tensor. Combining this with the facts that v

(i) l (r) is a parallel transport and that γ ′ i (r) = v (i) 1 (r), we obtain f (i)′′ l,m (r) = v (i) l (r), J(i)′′ m (r) xi = -v (i) l (r), R (M) xi ( J(i) m (r), v (i) 1 (r))v (i) 1 (r) xi and in particular f (i)′′ l,m (0) = 0. (25)
Finally, using again the fact that v

(i) l
is a parallel transport, we show that the third derivative of f

(i) l,m is f (i)′′′ l,m (r) = v (i) l (r), J(i)′′′ m (r) xi .
Thanks to [DC92, p.115] and the equality

γ ′ i (r) = v (i) 1 (r), we notice that J(i)′′′ m (r) = -R (M) xi ( J(i)′ m (r), v (i) 1 (r))v (i) 1 (r). (26)
Consequently, using that

J(i)′ m (0) = v (i)
m , we deduce from the two previous equalities that when l = m,

f (i)′′′ m,m (0) = -v (i) m , R (M) z (v (i) m , v (i) 1 )v (i) 1 xi = -K (M) z (u i , v (i) m ). ( 27
)
where the last equality comes from the definition of the sectional curvature, see e. Going back to the proof of (6), let us show that the determinant of B (i) has the required expansion

det B (i) = r n-1 - Ric (M) z (u i ) 6 r n+1 + o(r n+1 ). (28)
Indeed, we can rewrite the determinant as

det B (i) = σ∈Sn-1 sgn(σ) n-1 m=1 B (i) σ(m),m = n m=2 v (i) m (r), J(i) m (r) xi + σ∈Sn-1\{Id} sgn(σ) n m=2 v (i) σ(m) (r), J(i) m (r) xi ,
where S n-1 denotes the set of permutations of {1, • • • , (n -1)} and sgn(σ) is the signature of the permutation σ. For sake of simplicity, we use the same notation σ for a permutation of either the set

{1, • • • , (n -1)} or the set {2, • • • , n}.
It then follows from ( 21) and (10) that

n m=2 v (i) m (r), J(i) m (r) xi = r n-1 - n m=2 K (M) z (u i , v (i) m ) 6 r n+1 + o(r n+1 ) = r n-1 - Ric (M) z (u i ) 6 r n+1 + o(r n+1 ).
(29) It remains to show that for all σ = Id,

lim r→0 1 r n+1 n m=2 v (i) σ(m) (r), J(i) m (r) xi = 0. (30)
Since σ = Id, at least two indices, say 2 and 3 satisfy 2 = σ(2) and 3 = σ(3). Let us rewrite the product as

1 r n+1 n m=2 v (i) σ(m) (r), J(i) j (r) xi = 1 r n+1 v (i) σ(2) (r), J(i) 2 (r) xi v (i) σ(3) (r), J(i) 3 (r) xi n j=4 v (i) σ(m) (r), J(i) m (r) xi . (31)
Now, thanks to (22), the first two terms satisfy

lim r→0 1 r 2 v (i) σ(2) (r), J(i) 2 (r) xi = lim r→0 1 r 2 v (i) σ(3) (r), J(i) 3 (r) xi = 0. (32)
Looking at both (21) and ( 22), we observe that the remaining terms behave like r n-3 at most, i.e. there is a positive constant C such that for r small enough,

n j=4 v (i) σ(m) (r), J(i) m (r) xi ≤ Cr n-3 . ( 33
)
Thus inserting (32) and ( 33) into (31), we obtain (30) which, combined to (29), implies in turn (28). The expansion ( 6) is now a direct consequence of both ( 5) and (28).

Proof of (3)

Recall that J Φn (z, r, u 0 , u 1 , . . . , u n ) denote the Jacobian determinant of the function Φ n . To prove (3) it suffices to prove that, when

M = H n k , |J Φn (z, r, u 0 , u 1 , . . . , u n )| = n!∆ n (u 0 , . . . , u n ) sinh(kr) k n 2 -1 .
We proved in Section 3.1 that in a general manifold,

|J Φn (z, r, u 0 , u 1 , . . . , u n )| = n!∆ n (u 0 , . . . , u n ) n i=0 | det B (i) | (34)
where B (i) are (n -1) × (n -1) matrices with coefficients

B (i) l-1,m-1 = v (i) l (r), J(i) m (r) xi (35)
and J(i) m (r) is the unique Jacobi field with J(i) m (0) = 0 and

J(i)′ m (0) = v (i)
m . Now Lemma 2.2 provide exact expression of these Jacobi fields. Indeed, applying Lemma 2.2 to γ = γ i and V (t) = v (i) m (t), we obtain

J(i) m (r) = sinh(kr) k v (i) m (r). (36)
Inserting ( 36) into (35), it follows that

B (i) l-1,m-1 = v (i) l (r), sinh(kr) k v (i) m (r) xi = sinh(kr) k δ l,m (37)
so that

B (i) = sinh(kr) k Id (n-1) (38)
where Id (n-1) denotes the identity matrix of size (n -1). Combining ( 38) and (34), we obtain

|J H n k Φn (z, r, u 0 , u 1 , . . . , u n )| = n!∆ n (u 0 , . . . , u n ) sinh(kr) k n 2 -1
which establishes the equality of measures (3).

Proof of Proposition 1.3

Let us calculate the Jacobian determinant of the function Φ 1 given in Proposition 1.3. To do so, let us consider an orthonormal basis of T z M , V = {v 1 , . . . , v n } where v 1 = u. We consider the bases

V (i) (r) = {v (i) 1 (r), . . . , v (i)
n (r)} obtained by parallel transport of the basis V along γ i (t) = exp z ((-1) i tu).

Step 1: derivatives with respect to r. As in the proof of (5), the column vector of the coordinates of ∂xi ∂r in the basis

V (i) (r) is ∂x i ∂r =     (-1) i 0 n-1 T     . ( 39
)
Step 2: derivatives with respect to z. Again, we show that the submatrix with size n × n, ∂xi ∂z satisfies

∂x i ∂z l,m = v (i) l (r), J (i) m (r) xi .
where

J (i)
m is the unique Jacobi field along γ i such that J

v (i) 0 (r), J (i) m (r) xi = u, J (i) m ′ (0) xi r + u, J (i) m (0) xi = u, v m xi = δ 0,m ,
which shows that the shape of ∂xi ∂z is

∂x i ∂z =     1 0 n-1 n/a A (i)     . ( 40
)
where A (i) is a (n -1) × (n -1)-matrix.

Step 3: derivatives with respect to u i . ∂xi ∂u with size n × (n -1) satisfies

∂x i ∂u l,m = v (i) l (r), J(i) m (r) xi , l = 0, . . . , n -1, m = 1, . . . , n -1
where J(i) m is the unique Jacobi field along γ i such that

J(i) m (0) = 0 and J(i)′ m (0) = (-1) i v m .
Again, the first line of B (i) is identically equal to zero, which means that we can rewrite B (i) as

∂x i ∂u =     0 n-1 B (i)     . ( 41
)
Step 4: rewriting of the Jacobian determinant. We recall that the Jacobian determinant is 40), ( 41) and the exact same permutation of lines as for (20), we can rewrite the matrix up to a possible minus sign in front of the determininant as the following block lower triangular matrix:

J Φ1 (z, r, u) = det
|J Φ1 (z, r, u)| = det     1 1 -1 1 0 0 n/a A (0) B (0) A (1) B (1)     = 2 × det A (0) B (0) A (1) B (1) = 2 det D where D = A (0) B (0) A (1) B (1) .
Step 5: expansion of the coefficients of the determinant. As in the proof of (6), the required expansion (7) of the determinant follows from the expansion of each of its coefficients. This is done in the following lemma.

Lemma 4.1. The coefficients of A (i) and B (i) satisfy for l, m = 2, . . . , n

A (i) (m-1),(m-1) = v (i) m (r), J (i) m (r) xi = 1 - K (M) z (u, v m ) 2 r 2 + o(r 2 ), A (i) (l-1),(m-1) = v (i) l (r), J (i) m (r) xi = o(r), l = m, B (0)
(m-1),(m-1) = v (0) m (r), J(0) m (r) xi = r - K (M) z (u, v m ) 6 r 3 + o(r 3 ), B (1)
(m-1),(m-1) = v (1) m (r), J (1) m (r) xi = -(r - K (M) z (u, v m ) 6 r 3 ) + o(r 3 ), B (i) (l-1),(m-1) = v (i) l (r), J (i) m (r) xi = o(r 2 ), l = m.
Proof of Lemma 4.1. We omit the proof for the coefficients of B (i) as it is very similar if not identical to the proof of Lemma 3.1.

Let us show the first two expansions. To do so, we consider the function f

(i) l,m (r) = v (i) l (r), J (i)
m (r) xi and apply Taylor's theorem to it at the second order in the neighborhood of 0.

Since

J (i) m (0) = v m , we have f (i) l,m (0) = δ l,m . (42)
Because v (i) l is a parallel transport, its derivative is equal to zero and so

f (i)′ l,m (0) = v (i) l (0), J (i)′ m (0) z = 0. (43)
The required estimate of A

(l-1),(m-1) for l = m follows from (42) and (43). We calculate now the second derivative of f

(i)
m,m at 0: since J (i) m satisfies the Jacobi equation, we get

f (i)′′ l,m (0) = -v (i) m (0), R (M) z (J (i) m (0), u)u xi = -v m , R (M) z (v m , u)u z = -K (M) z (u, v m ). (44)
It remains to combine (42), ( 43) for l = m and (44) with Taylor's theorem to deduce the required expansion of A

(m-1),(m-1) .

Step 6: conclusion. Let us write the determinant of D as

det D = σ∈S 2(n-1) sgn(σ) 2(n-1) j=1 D σ(j),j = σ∈S sgn(σ) 2(n-1) j=1 D σ(j),j + σ∈S 2(n-1) \S sgn(σ) 2(n-1) j=1 D σ(j),j (45)
where

S = {σ ∈ S 2(n-1) , ∀j, D σ(j),j = A (i) m,m or D σ(j),j = B (i)
m,m , for some i, m}. We expect the contribution of the permutations in S to be dominant in the sum above. Indeed, using the expansions contained in Lemma 4.1, the fact that the cardinality of S is 2 n-1 and the identity (10), we get σ∈S sgn(σ)

2(n-1) j=1 D σ(j),j = 2 n-1 n m=2 (1 - K(u, v m ) 2 r 2 + o(r 2 ))(r - K (M) z (u, v m ) 6 r 3 + o(r 3 )) = 2 n-1 (r n-1 - 2 3 Ric (M) z (u)r n+1 + o(r n+1 )) (46)
Moreover, for any σ ∈ S 2(n-1) \S, there exist at least two j such that D σ(j),j is equal to

A (i) m,l or B (i) m,l
for different m and l. Because of Lemma 4.1, this means that the contribution of the product is at least of order r n-2 × r 2 × r 2 . Consequently, this yields that

lim r→0 1 r n+1 σ∈S 2(n-1) \S sgn(σ) 2(n-1) j=1 D σ(j)j = 0. (47)
Inserting ( 46) and ( 47) into (45), we get the required expansion (7).

Proof of Proposition 1.4

Let us calculate the Jacobian determinant of the function Φ n-1 given in Proposition 1.4. To do so, we consider an orthonormal basis of T z M , V = {v 1 , . . . , v n } where v n = v. In a very similar way to the proof of Theorem 1.2, we write each vector u i , 0 ≤ i ≤ (n -1) in this basis, i.e.

u i = n j=1 u j i v j .
In particular, we notice that u n i = 0 for every 0 ≤ i ≤ (n -1). For sake of simplicity, we will use in the rest of the paper the notation u i for the (n -1)-dimensional line vector (u 1 1 , . . . , u n-1 1 ). Then for each i, we introduce an orthonormal basis of

T z M V (i) = {v (i) 1 , . . . , v (i) n } where v (i) 1 = u i and v (i) n = v,
and we do the parallel transportation of this basis V (i) (r) = {v (i) 1 (r), . . . , v (i) n (r)} along the curve γ i (t) = exp z (tu i ).

Step 1: derivatives with respect to r. Again, the column vector of the coordinates of ∂xi ∂r in the basis

V (i) (r) is ∂x i ∂r =     1 0 n-1 T     . (48)
Step 2: derivatives with respect to z. We consider the submatrix ∂xi ∂z . We recall that for 1 ≤ l, m ≤ n

∂x i ∂z l,m = v (i) l (r), J (i) m (r) xi ,
where

J (i)
m is the unique Jacobi field along γ i such that

J (i) m (0) = v m and J (i)′ m (0) = 0.

Arguments similar to

Step 2 of the proof of (5) show that ∂xi ∂z is equal to

∂x i ∂z =   u T i 0 n/a A (i)   (49)
where A (i) is a (n -1) × 1-column vector with transpose

A (i) T = ( v (i) 2 (r), J (i) n (r) xi , . . . , v (i) n (r), J (i) n (r) xi ).
Step 3: derivatives with respect to u i . We first recall that u i is a vector from the (n -1)-dimensional subspace {v} ⊥ of T z M . The submatrix ∂xi ∂ui is then equal to

∂x i ∂u i =     0 n-1 B (i)     . ( 50
)
where B (i) is the (n -1) × (n -2)-matrix such that for every 1 ≤ l ≤ (n -1) and 1 ≤ m ≤ (n -2),

B (i) l,m = v (i) l+1 (r), J(i) m+1 (r) xi , J(i) m being unique Jacobi field along γ i such that J(i) m (0) = 0 and J(i)′ m (0) = v (i) m .
Step 4: derivatives with respect to v. When comparing to the proof of (5), we observe that this step is new. The submatrix ∂xi ∂v has size n × (n -1) and ∂xi ∂v l,m is the projection onto v (i) l (r) of the derivative of x i with respect to v in the direction v m . Let m, i be fixed, s > 0 and let us define the new basis of

T z M , V(s) = {v 1 (s), . . . , v n (s) = v(s)} by v(s) = v n (s) := v + sv m v + sv m , v m (s) := v m -sv v m -sv , v j (s) := v j , j = m, j = n.
In this new basis, we define the vector u s i by

ũi (s) = n-1 j=1 u j i v j (s) = u i + u m i (v m (s) -v m ) = u i + u m i v m -sv v m -sv -v m .
Now, the derivative of x i with respect to v in the direction v m , is obtained by applying Lemma 2.1 with γ = γ i , c(s) = z and V (s) = ũi (s) that is

∂x i ∂v m = J(i) m (r) (51)
where J(i) m is the unique Jacobi field along γ i such that J(i) m (0) = 0 and J(i)′ m (0) = V ′ (0). We have

V ′ (s) = u m i -v v m -sv - (v m -sv) -v, v m -sv z v m -sv 3 so V ′ (0) = -u m i v.
This, with (51), implies that , for every 1 ≤ l, m ≤ (n -1),

∂x i ∂v l,m = v (i) l (r), J(i) m (r) xi ,
where J(i) m is the unique Jacobi field along γ i such that J(i) m (0) = 0 and

J(i)′ m (0) = -u m i v.
As for ∂xi ∂ui , the first line of ∂xi ∂v is identically equal to zero. Consequently, we get

∂x i ∂v =     0 n-1 C (i)     (52)
where

C (i) is a (n -1) × (n -1)-matrix satisfying C (i) l,m = v (i) l+1 (r), J(i) m (r) xi .
Step 5: rewriting of the Jacobian determinant. Thus, the Jacobian determinant of Φ n-1 , J Φn-1 is

J Φn-1 (z, r, v, u 0 , . . . , u n-1 ) = det             ∂x0
            . (53)
Considering ( 48), ( 49), ( 50) and (52), we can then apply the same permutation of the lines as in the proof of (5) so that the determinant is up to a possible minus sign equal to the determinant of a new matrix which is a block lower triangular matrix:

|J Φn-1 (z, r, v, u 0 , . . . , u n-1 )| = | det               1 u T 0 . . . . . . 0 1 u T n-1 A (0) C (0) B (0) 0 n/a . . . . . . 0 . . . A (n-1) C (n-1) B (n-1)               | = (n -1)!∆ n-1 (u 0 , . . . , u n-1 )| det D| (54)
the final expansion. To this end, let us define the set of permutations

S = {σ ∈ S n(n-1) , D σ(1),1 = A (i)
(n-1),1 for some i = 0, . . . , (n -1), D σ(j),j = C (τ (j))

(n-1),m for j = 2, . . . , n, and τ : {2, . . . , n} → {0, . . . , (n -1)}\{i} is some bijection, D σ(j),j = B (k) m,m for j = n + k(n -2) + m where k = 0, . . . , (n -1), m = 1, . . . , (n -2)}.

Since only the integer i and the bijection τ have to be chosen, the cardinality of S is n × (n -1)! = n! . We can then split det D into two terms

det D = σ∈S n(n-1) sgn(σ) n(n-1) j=1 D σ(j),j = σ∈S sgn(σ) n(n-1) j=1 D σ(j),j + σ / ∈S sgn(σ) n(n-1) j=1 D σ(j),j (65)
Observe that considering σ∈S sgn(σ)

n(n-1) j=1

D σ(j),j instead of det D is equivalent to considering the determinant of a new matrix D with entries Dl,m = D l,m as soon as there is a permutation σ ∈ S , such that σ(m) = l and Dl,m = 0 otherwise, that is σ∈S sgn(σ)

n(n-1) j=1 D σ(j),j = det D = det                     B (0) 1,1 0 0 . . . 0 0 B (0) (n-2),(n-2) A (0) (n-1),1 C (0) (n-1),1 • • • C (0) (n-1),(n-1) 0 • • • 0 . . . . . . B (n-1) 1,1 0 0 0 . . . 0 B (n-1) (n-2),(n-2) A (n-1) (n-1),1 C (n-1) (n-1),1 • • • C (n-1) (n-1),(n-1) 0 • • • 0                     (66)
Now, rearranging the lines so that the lines (A

(n-1),1 , C

(n-1),1 , . . . , C

(n-1),(n-1) |0) appear in the first n lines of a block upper triangular matrix. Then, up to a possible minus sign

| σ∈S sgn(σ) n(n-1) j=1 D σ(j),j | = | det(E)| × | det(F )|
where E is the n × n-matrix defined by

E =     A (0) (n-1),1 C (0) (n-1),1 • • • C (0) (n-1),(n-1) . . . . . . . . . A (n-1) (n-1),1 C (n-1) (n-1),1 • • • C (n-1) (n-1),(n-1)     (67)
and F is a n(n -2) × n(n -2)-diagonal matrix with diagonal entries

(B (0) 1,1 , . . . , B (0)
(n-2),(n-2) , . . . , B

(n-1) 1,1 , . . . , B

(n-1)

(n-2),(n-2) ).

where

Ric v z (u i ) = n-1 m=2 K(u i , v (i) m ) = Ric z (u i ) -K (M) z (u i , v).
To conclude for the expansion of det D, it suffices now to prove that the second sum in (65) has a negligeable contribution, that is lim r→0 1 r n(n-1)+1 σ / ∈S sgn(σ) n(n-1) j=1 D σ(j),j = 0.

(73) Indeed, let σ / ∈ S . Then, there exists at least two indices j such that D σ(j),j = A l,m for some k = n -1, l = m. As in the proof of (6), because of Lemma 5.1, the product n(n-1) j=1 D σ(j),j is at least of order r n(n-1)+2 thus (73) holds. Combining (72) and (73), we obtain | det D| = (n -1)!∆(u 0 , . . . , u n-1 ) r n(n-1)-1 -r n(n-1)+1 ( n-1 i=0 Ric v z (u i ) + Tr(∆ -1 K)) + o(r n(n-1)+1 ) .

which inserted in (54) concludes the proof of Proposition 1.4.

Concluding remarks

1. General Blaschke-Petkantschin formula Theorem 1.2 and Propositions 1.3 and 1.4 provide asymptotic Blaschke-Petkantschin formulas for a m-tuple of points where m = n + 1, 2 and n respectively. Determining similar estimates for general m is still open. The general method developed in this paper, goes along the following steps:

-the choice of suitable bases for the respective tangent spaces of the m points and for the Grassmannian, -the writing of the Jacobian matrix in function of several Jacobi fields, -the expansion of each entry of the matrix with the Jacobi equation, -the expansion of the Jacobian determinant by elementary row and column operations.

This could be applied in principle to any m. However, the practical difficulty comes from the calculations of the partial derivatives with respect to the vector space which belongs to the Grassmanniann. We would require a simple analytic way to describe the elements of the Grassmannian and rewrite the associated Haar measure, only it seems out of reach. For that matter, the classical Blachke-Petkantschin formula in the Euclidean space was not derived via a direct analytic calculation of the Jacobian but through the use of a classical Fubini-type exchange formula for the Haar measures on Grassmannians, see e.g. [SW08, Theorem 7.1.1], combined with an induction argument, see [SW08, Theorem 7.2.1]. Moreover, when it becomes possible to make explicit the Haar measure as in the case m = n, we observe that it adds extra Jacobi fields as min(m, n+ 1 -m) increases and that the final expansion of the determinant involves a more and more intricate second term which does not seem to have a simple geometrical meaning.

Uniformity of the expansions

When M is a compact Riemannian manifold, it is possible to show that the expansion of the Jacobian determinant provided by ( 6) is uniform with respect to z, u 0 , u 1 , . . . , u n . To do so, we need the two following steps:

-check carefully that each remainder in the expansions of the coefficients in Lemma 3.1 is uniform, -show that this uniformity of the remainder is preserved during the calculation of the determinant.

The first step requires in particular to rewrite a Taylor expansion with a Lagrange-type remainder for each scalar product involving a Jacobi field.

The uniformity of the Jacobian expansion plays a key role in the on-going work [CCE] on mean asymptotics for a Poisson-Voronoi tessellation in M , when applying a Blaschke-Petkantschin change of variables and integrating over u 0 , . . . , u n and/or over z.

Figure 1 :

 1 Figure 1: The function Φ n and the bases V (i) (r)

  g. [DC92, Proposition 3.1]. The estimates (21) and (22) follow now by (23), (24), (25) and (27) combined with Taylor's theorem applied to f (i) l.m in the neighborhood of 0.

Laboratoire MODAL'X, EA

3454, Université Paris Nanterre, 200 avenue de la République, 92001 Nanterre, France. E-mail: achapron@parisnanterre.fr

Acknowledgements. The author thanks Pierre Calka and Nathanaël Enriquez, her PhD advisors, for helpful discussions, comments and suggestions. This work was partially supported by the French ANR grant PRESAGE (ANR-11-BS02-003), the French research group GeoSto (CNRS-GDR3477) and the Institut Universitaire de France.

where D is the n(n -1) × n(n -1)-matrix given by

C (0) B (0) 0 . . . . . . 0 . . .

Step 6: expansion of the coefficients of the determinant. In Lemma 5.1 below, we provide estimates for the coefficients of the matrix D when r → 0.

Lemma 5.1. The coefficients of D satisfy for l = 2, . . . , n, m = 2, . . . , (n -1)

Proof. We only provide a proof for the last two expansions since the proof for the coefficients of A (i) and B (i) is very similar to the proofs of Lemmas 3.1 and 4.1. Let us define the function f

xi and write its third order Taylor expansion. To this end, let us compute the successive derivatives of f

The Jacobi field J(i) m satisfies the Jacobi equation thus

Now, (61), ( 62) and (63) with Taylor's theorem imply the expansion of C (i) (l-1),m , for l = n. For the case l = n, let us compute the third derivative of f (i) l,m . As in the proof of Lemma 3.1, the identity (26) is satisfied. This with

Combining (61), (62) and (63) for l = n with (64) and Taylor's theorem, we obtain the expansion of

Step 7: expansion of the determinant. To find the expansion of det D, we identify the dominant terms involved in the determinant, that is, we isolate the permutations providing the minimal power of r in Using Lemma 5.1, we get expansions of det E and det F . From (57), we have

Then from (55) and (59), we have

where

In order to obtain the expansion of | σ∈S sgn(σ)

D σ(j),j |, it remains to expand det G.

Lemma 5.2. The determinant of G has the following expansion

where

Proof of Lemma 5.2. The matrix G can be written as

. . . . . .

Since, ∆ is an invertible matrix for almost every (u 0 , . . . , u n-1 ), the expansion of the determinant function yields

where Tr denotes the trace operator. Now, observe that det ∆ is (n -1)! times the volume of the (n -1)dimensional simplex spanned by the vectors -u 0 , . . . , -u n-1 which has same volume as the simplex spanned by u 0 , . . . , u n-1 that is det ∆ = (n -1)!∆(u 0 , . . . , u n-1 ). = (n -1)!∆(u 0 , . . . , u n-1 ) r n(n-1)-1 -r n(n-1)+1 ( n-1 i=0 Ric v z (u i ) + Tr(∆ -1 K)) + o(r n(n-1)+1 ) (72)


Fabien Candau

Florent Deisting

Julie Schlick

How Income and Crowding Effects influence the World Market for French Wines

whether they are published or not. The documents may come   L'archive ouverte pluridisciplinaire

Introduction

What is the impact of per capita GDP growth on prices? On one hand, richer agents are more willing to pay high prices, but on the other hand richer markets attract more companies and this new competition can reduce prices. What is the net impact? Do rms generally set higher prices on richer markets? 1 These questions are both signicant and controversial in terms of international economics. In standard models of monopolistic competition with homothetic preferences (e.g. [START_REF] Krugman | Scale Economies, Product Dierenciation, and the Patterns of Trade[END_REF], the law of one price is veried ; markups, prices and the elasticity of demand are invariant to competition and income. The price of one type of goods diers between countries because of trade costs or preferences. In models with strategic interactions along a nite product space (e.g. [START_REF] Lancaster | Variety, Equity, and Eciency[END_REF], more rms entering the market causes a crowding eect, varieties become more substitutable and the price elasticity of demand, although invariant to income, increases with market size (i.e. competition). The empirical interpretation of this result is that, once we control for competition, the elasticities of the demand would be identical in rich and poor countries. The same result is obtained in models with heterogeneous rms and quasi-linear preferences ( e.g. [START_REF] Melitz | Market Size, Trade and Productivity[END_REF]. In contrast, by extending the Lancasterian approach, Hummels and Lugovskyy (2009) propose a model where per capita GDP growth attracts more sellers, leads to more competition and forces rms to set lower prices. This result has been challenged by [START_REF] Simonovska | Income Dierences and Prices of Tradables: Insights from an Online Retailer[END_REF], both by proposing a new model 1 and also by controlling for transport costs in her empirical analysis. The fact of disregarding logistics companies, that apply lower transport costs to both richer and larger markets (according to economies of scale, competition) can bias the result.

By working with 245 identical products sold exclusively on the Internet and by taking into account shipping prices, [START_REF] Simonovska | Income Dierences and Prices of Tradables: Insights from an Online Retailer[END_REF] nds that rms set higher prices for identical goods in richer destinations but are not inuenced by the market size.

Our paper pursues these studies with two goals, the rst is to use insight from this existing literature to analyse the wine market, the second is to use the wine market to test competing models of international economics. We use a unique dataset for 20012011 that reports bilateral export prices for French producers selling to all importers worldwide. The panel data structure makes it possible to identify price levels that are specic to an importer-product (taste, culture, etc) and price changes that are specic to an exporter-product (quality, etc), 2 relating changes in prices over-time for 1 [START_REF] Simonovska | Income Dierences and Prices of Tradables: Insights from an Online Retailer[END_REF] proposes a model with non-homothetic preferences from a hierarchic-choice of consumption [START_REF] Jackson | Hierarchic Demand and The Engel Curve for Variety[END_REF]. In this model where the marginal utility is bounded (consumers may have zero demand for some varieties, see also [START_REF] Sauré | Bounded Love of Variety and Patterns of Trade[END_REF], the relative price of a variety is higher in relatively richer markets which contradicts Hummels and Lugovskyy's (2009) results.

2 The current paper is not interested in vertical dierentiation i.e. we wish to control for the high price of wine on richer market due to the fact that rich importers demand goods of high quality. See [START_REF] Verhoogen | Trade, Quality Upgrading and Wage Inequality in the Mexican Manufacturing Sector[END_REF] and [START_REF] Fajgelbaum | Income Distribution, Product Quality, and International Trade[END_REF] for models of international trade focusing on product quality and more particularly Crozet, Head and Mayer (2012) on Champagne wine. See also Di Comite, Thisse and Vandenbussche (2014) for a model of vertical and horizontal dierentiation applied to the an importer/exporter-product to changes in importer characteristics. While the Lancasterian approach has been applied in many hedonist studies on wine characteristics 3 , its extension by Hummels and Lugovskyy (2009) to analyse international wine trade has, to our knowledge, never been carried out. In comparison to the wine economics literature that distinguishes the relevance of various characteristics mainly in terms of supply, 4 here we focus on determinants of demand. We conrm Hummels' and Lugov- skyy's (2009) conclusion 5 by nding that the price elasticity of demand is inuenced by GDP per capita as well as importer GDP, contrasting with standard results based on monopolistic competition with homothetic preferences. [START_REF] Barclays | Prot or Pleasure? Exploring the motivations behind treasure trends[END_REF] The market size, approx- imated by importer GDP, has a negative eect on the price dierential as well as on export share, revealing a competition eect on external markets for French producers.

While our data does not enable us to control for shipping costs as in Simonovska (2015), we do know the means of transport; trade ows can therefore be separated and analyzed in detail in order to reconcile standard models with the data. We show that a standard gravity equation performs poorly for the wine sector. However, we nd that a more precise specication of trade costs helps to improve this performance. Imitating pro-competitive eects, it is possible that shipping prices react to change in market size. This change is due to economies (or dis-economies) of scale in transportation. With endogenous trade costs, an increase in income per-capita has a stronger impact on the trade of high quality wines exchanged under modes of transport with high economies of scale. We quantify to which degree wines exported by water and land benet from a smaller income eect than wine exported by air. To our knowledge, such an analysis beer sector.

3 For instance to quote just a few contributions (see Cardebat, Figuet and Paroissien (2014) for a review), [START_REF] Nerlove | Hedonic Price Functions and the Measurement of Preferences: The Case of Swedish Wine Consumers[END_REF] discusses the standard hedonic price equation to study preferences of Swedish wine consumers, [START_REF] Combris | Estimation of a Hedonic Price Equation for Bordeaux Wine : Does Quality Matter ?[END_REF] apply this method to Bordeaux wines to analyze the impact of sensory characteristics (provided by a jury under blind tasting conditions) on prices, and Roma, Di Martino and Perrone (2013) use this method to explain the price of Sicilian wines.

4 [START_REF] Ashenfelter | Predicting the Quality and Prices of Bordeaux Wine[END_REF] updates its "Bordeaux wines equation" where prices are explained by weather conditions, wine age and expert judgments. See Storchmann (2011) for a survey on Ashenfelter's works and on recent developments in wine economics. Furthermore, by using data on endowments (e.g. soil qualities, weather conditions) as well as data on technology (such as manual operations e.g. picking and selecting grapes, the process of bottling wines, etc), [START_REF] Gergaud | Natural Endowments, Production Technologies and the Quality of Wines in Bordeaux. Does Terroir Matter?[END_REF] succeed in discriminating between determinants of quality (by using instrumental variables) in favor of technology.

5 Hummels and Lugovskyy (2009, Table 2, p.16) test their predictions for thirty-eight 2-digit indus- tries, including HS-code 22, beverages, spirits and vinegar, which include wine. [START_REF] Barclays | Prot or Pleasure? Exploring the motivations behind treasure trends[END_REF] The most recent work in wine economics with CES preferences and monopolistic competition between heterogeneous rms is Crozet, [START_REF] Crozet | Quality Sorting and Trade: Firm-level Evidence for French Wine[END_REF]. By working on Champagne wine, they justify the CES assumption by emphasizing that (in contrast to other wines studied here), producers blend several years of grapes to reproduce a constant quality over time.

of export linked to transport has never before been studied in wine economics.

Our paper is in four parts. Firstly we give a brief introduction and presentation of our database. Secondly, by applying the Hummels and Lugovsky (2009) methodology, we show the empirical results for the wine sector. Thirdly, we analyse alternative models and show our investigations into the role of (dis)economies of scale in wine transport.

Finally we draw our conclusions.

Data and preliminary results

In this section, we briey present our data set and our rst results based on the Generalized Model of Ideal Variety (GMIV) proposed by Hummels and Lugovskyy (2009). This database contains the SIREN number that allows each exporter to be identied (address and economic features of each unit). We match this database with the SIREN register and we only retain rms found under the label "culture" that includes wine producers. The value and volume of each product are reported monthly, we compute the sum annually, by product, exporter and destination market. This database also contains information concerning the mode of transport. More precisely we know at the individual level and for each destination market if the wine has been exported by plane, boat, road, rail, river, postal services or by private mode. Export by plane has sharply increased, particularly in terms of value. For instance the value of red Bordeaux wines exported by plane has soared and is 10 times higher than in 2001. In contrast the volume of Bordeaux exported by plane has remained constant, revealing a striking increase of the unit value of this wine that cannot simply be explained by an increase in quality. Moreover this increase in the value of wine masks contrasting price-setting in destination markets. Figure [START_REF] Alvaredo | The World Top Incomes Database[END_REF] plots the value of red Bordeaux exported by air with respect to GDP per capita of destination countries.

For each year, a positive correlation can be observed. The cross-country variation in price seems directly related to GDP per capita. However if we now compare pricechanges over a particular time-period for a particular importer with changes in the countries' GDP per capita, very dierent results can be observed. The price of wine sold in China strongly increased between 2001 and 2011, while in Brazil, a country also with a strong growth in terms of GDP per capita, the value of Bordeaux wine exported stagnated. In Argentina, a country that developed over that period to become the fth world wine producer, the nominal exportation of Bordeaux even declined. According to the GMIV model, two opposing eects related to income and income per-capita are crucial to the analysis of trade elasticities. On one hand, a rise of income generates more demand and the entry of new varieties. This leads to more competition between rms which proceed to reduce their mark-ups in order to stay in the market.

Ceteris paribus, there are lower prices in larger markets.

On the other hand, consumers are increasingly nicky regarding the gap between current consumption and the ideal variety when consumption of a typical variety increases. Consequently, the richer the consumer, the more they appreciate value of a variety that is close to the ideal. This reaction encourages rms to set higher prices.

All else being equal (in particular market size), rms set higher prices when consumers are richer. While the market size generates a pro-competitive eect, increasing price elasticity, rising income per worker reduces this elasticity.

To capture these opposed eects in the wine sector we consider that changes in the price set by a French producer j on a product k in the country i at time t depends on destination GDP and GDP per capita:

ln p k ij,t p k ij,t-1 = a 0 + a 1 ln Y i,t Y i,t-1 + a 2 ln Y i,t /L i,t Y i,t-1 /L i,t-1 + f j + f k + k ij,t . (1)
where Y i measures the market size of the partner country i, approximated by GDP, and L i,t the population (GDP and population statistics are from the WDI database).

A negative coecient a 1 is expected to conrm that the market size reduces price due to a competition eect that increases price elasticity. In contrast, a positive coecient a 2 conrms that, depending on market size, a rise in GDP per capita increases prices due to the income eect (that reduces price elasticity). Fixed eects at the product and rm level are introduced to control for cost and quality variations. We do not introduce destination xed eects which would control for variations that are specic to importers. In fact these eects would capture the eect of economic growth of the partners we intend to measure. In other words, we exploit cross-importer variation to assess the income and competition eect. Whatever the period considered, Table [START_REF] Alvaredo | The World Top Incomes Database[END_REF] supports the conclusion of the GMIV.

Income per capita fosters prices, while market size impacts negatively on the price differential. Using exporters and product xed eects and exploiting time series variation in importer characteristics, we nd that prices fell with importer GDP growth but rose with importer GDP per capita. A 1% increase in GDP per-capita generated an increase in price of 1.13% between 2001 and 2011, while a 1% increase in market size reduced prices by 1.10% over the same period. In Appendix A we show that these results hold at a more disaggregated level (i.e. at the French regional level distinguishing wines by reputation). Additional robustness checks were performed here, including changes in variables capturing the income and market size eect.

9

The model seems more pertinent for a long term analysis than over a short period of time, explaining almost 60% of the average price variations between 2001 and 2011, while only 10% per year on average.

Competing theories

The above result illustrates that the main predictions of the GMIV models are conrmed for the wine sector. Here we go beyond this rst step to discriminate between this model and more standard models of international trade. To illustrate this we consider the gravity equation proposed by Markusen (2010) (see also Frankel, Stein and Wei, 1998):

x k ij = (Y j Y i ) α Y j L j Y i L i β τ 1-σ ij P i P j . (2)
Where Y i and Y j are incomes (GDPs), L i and L j are the populations in i and j, P i and P j are price indices, τ ij represents bilateral trade costs and σ is the elasticity of substitution between two varieties. With α = 1 and β = 0 the gravity equation is similar to that obtained in [START_REF] Anderson | Gravity with Gravitas : A Solution to the Border Puzzle[END_REF] and [START_REF] Krugman | Scale Economies, Product Dierenciation, and the Patterns of Trade[END_REF], there is no income eect, in contrast with β = 0 by which luxury and goods of basic necessity can be analysed.

To discriminate between this general equation and the GMIV prediction, we simply have to use this formula to compute imports of the rest of the world from j, i.e. x k rj :

x k rj = (Y j Y r ) α Y j L j Y r L r β τ 1-σ rj P r P j
and we use this expression to compute the trade-share s k ij , which is the i's import from j on imports of the rest of the world, r from country j. Taking the logarithm of this share nally provides the following equation:

ln s k ij ≡ ln x k ij x k rj = (1 -σ) ln τ ij + ln Y α+β i P i L β i + ln P r L β r Y α+β r τ σ-1 rj (3)
The interesting feature of this computation is that all the variables specic to French producers have been eliminated. Moreover, by using partner-product xed eects, denoted by f k i , we can capture specic characteristics of importers (GDP per capita, price index, etc). In short, taking the logarithm of (3) using (2) enables the following equation to be estimated:

ln(s k ij ) = (1 -σ) ln(τ ij ) + f k i + k ij (4)
The only variables that explain s k ij are trade costs τ ij approximated by bilateral distance d ij . This result contrasts with the GMIV model, where the distance to the market depends on the number of competitors, which itself varies according to GDP and GDP per-capita. Thus, by introducing distance in interaction with other variables, Hummels and Lugovskyy ( 2009) obtain what they call "a test of the CES null hypothesis":

ln s k ij,t = a 0 + a 1 ln d ij + a 2 ln d ij ln (Y i,t ) + a 3 ln d ij ln (Y i,t /L i,t ) + f k i + k ij,t . (5)
To validate the CES model (and also other models where GDP per-capita enters in a multiplicative form as we have shown), only the coecient of distance should be signicant (with a 2 = a 3 = 0 Equation ( 3) is reduced to Equation (4) obtained from the general gravity equation).

They nd that a 2 < 0 and a 3 > 0 are statistically signicant, which validates their model. We have applied this methodology to our database. Geographical distance between countries is taken from the CEPII database. Lastly Column 4 and 5 use the wine production in the destination market to test the robustness of our result regarding the competition eect. Once again, the interaction with distance is signicant.

Non-homothetic preferences and trade costs

Surprisingly the GMIV has a better predictive power than alternative models based on the widely-used standard/powerful gravity equation ( 2). Consequently we return to this equation and after rearrangement and by taking the log of ( 2) we obtain the following expression:

ln x k ij = (α + β) ln (Y j Y i ) -β ln (L j L i ) + (1 -σ) ln (τ ij ) + ln (P i P j ) . ( 6
)
From this we estimate the following equation by separating wines exported by air, water and land:

ln x k ij,t = a 1 ln (Y j,t Y i,t ) + a 2 ln (L j,t L i,t ) + a 3 ln (τ ijt ) + a 4 ln (P it P jt ) + k ijt (7)
We expect to obtain support for non-homothetic preferences with a positive impact of GDP per capita for wine exported by air. These wines may be of better quality than wine exported via other transport systems and consequently may be more sensitive to GDP per capita. The crucial coecient is that of population; indeed with a negative sign, a 2 < 0 we verify that β > 0 and thus the gravity equation (2).

To control for price index, we refer to a wide range of literature using consumer price indices (e.g. [START_REF] Bergstrand | The Generalized Gravity Equation, Monopolistic Competition, and the Factor-Proportions Theory in International Trade[END_REF][START_REF] Baldwin | Gravity Chains: Estimating Bilateral Trade Flows When Parts And Components Trade Is Important[END_REF]. We carried out many robustness checks concerning this last variable. 14 [START_REF] Anderson | Gravity with Gravitas : A Solution to the Border Puzzle[END_REF] and [START_REF] Baldwin | Gravity for Dummies and Dummies for Gravity Equations[END_REF] recommended the setting of partner-xed eects to control for price index and to obtain an unbiased coecient of distance (and border eects). These xed eects have been used here and provide similar results to the consumer price index. However, because our main interest lies in GDP and population (and not in dyadic variables such as distance) it seems logical to remove these xed eects. This is also the empirical strategy adopted by Baldwin and Taglioni (2011) who write: 'If the econometrician is only interested in estimating Furthermore, estimating the gravity equation by type of transport allows us to partially treat heterogeneity in terms of both products and destination markets. A selection eect linked to distance and to product quality certainly leads to the choice of one mode of transport rather than another. We have considered a standard form for trade costs:

τ ijt = dist ij e b ij
where b ij includes dummies representing common language, lang ij , and past colonial links, col ij , such as b ij = dist ij col ij . Common language and colonial history appear crucial in explaining bilateral trade but direct measures are riddled with intrinsic errors.

By using a constructed 0-1 index we limit the extent of these errors. In wine economics the importance of past colonial links has been studied for instance by Melonni and Swinnen (2014) who detail the rise and fall of Algeria as the largest wine-exporter, worldwide during the French colonization.

We use the OLS estimator, as well as the PPML and Gamma estimators. the impact of a pair-specic variable such as distance or taris the standard solution is to put in time-varying country-specic xed eects. [...] Plainly we cannot use this approach to investigate the impact of using GDPs as the economic mass proxies'. We have also conducted estimations with dierent functional forms such as: 1) all variables concerning France (including price index) reported on the right hand side of the gravity equation (a trade adjusted measure) 2) xed eects and the population variable are introduced on the left hand side to analyse whether the sign of population changes when multicollinearities between variables are reduced to the minimum 3) introduction of unit Estimations realized with year, rm and product xed eects a: signicant at 1%, b: signicant at 5%, c: signicant at 10% By referring to this literature, we assume that transport costs take the following form:

τ ij = x k ij,t η d ij e b ij
where x k ij,t represents the wine exports in value and η density (dis)economies. There are economies of scale with η negative, and diseconomies in the opposite case. Inserting this function in the gravity equation and resolving for x k ij,t to eliminate the endogeneity bias gives:

ln x k ij,t = α + β 1 -(1 -σ)η ln (Y j,t Y i,t ) + β 1 -(1 -σ)η ln (L j,t L i,t ) + (1 -σ)η 1 -(1 -σ)η ln d ij e b ij + 1 1 -(1 -σ)η ln (P i,t P j,t ) .
This last gravity equation is helpful in revisiting the previous results. In particular in comparison with Equation ( 7) the coecient of distance now consists of the following parameters:

a 3 = (1 -σ)η 1 -(1 -σ)η (8)
For air transport according to the estimation using the Gamma PML estimator we have a 3 = -0.143, thus assuming an elasticity of substitution equal to 5 (which is a realistic value of σ according to estimations of [START_REF] Broda | Globalization and the Gains from Variety[END_REF], we obtain η = -1.988. From such a result we can now deduce β, we have:

a 2 = β 1 -(1 -σ)η (9)
17 There also are interesting theoretical papers, for instance [START_REF] Duranton | Rising trade costs? Agglomeration and trade with endogenous transaction costs[END_REF] propose a model whereby the decrease in transport costs can generate an increase in trade costs. In their model with vertical dierentiation and where the quality of input is not contractible, they show that a decrease in transport costs leads to exchange of higher quality goods for which trade costs increase.

Lastly the assumption of endogenous transport costs with respect to trade is not innocuous in terms of specialization and location choice (see [START_REF] Matsuyama | Beyond Icebergs: Towards a Theory of Biased Globalization[END_REF][START_REF] Behrens | Industry location and welfare when transport costs are endogenous[END_REF][START_REF] Behrens | Transportation, Freight Rates, and Economic Geography[END_REF] and from Table [START_REF] Ashenfelter | Predicting the Quality and Prices of Bordeaux Wine[END_REF]Gamma PML) the coecient of population gives a 2 = 0.075 thus using η and σ we arrive at β = -0.524. This result conrms the income eect that was previously rejected. Lastly by using the coecient of GDPs from:

a 1 = α + β 1 -(1 -σ)η (10)
with a 1 = -0.034 we get α = 0.762 which is relatively close to the unit elasticity of GDP obtained in many trade gravity equations. We note that our calculation of α and β does not depend on existing estimations of σ. In other words these parameters only depend on our estimations in table [START_REF] Ashenfelter | Predicting the Quality and Prices of Bordeaux Wine[END_REF]. Resolving the system [START_REF] Behrens | Transportation, Freight Rates, and Economic Geography[END_REF][START_REF] Bekkers | Import Prices, Income, and Inequality[END_REF][START_REF] Bergstrand | The Generalized Gravity Equation, Monopolistic Competition, and the Factor-Proportions Theory in International Trade[END_REF] gives expressions of (η,β,α) where η depends on σ but where α and β only depend on estimates: Expected results are obtained. Wine exported by ship and road benets from a smaller income eect than wine exported by air. The coecient of β is not strictly equal to zero but is however, closer to this value, in particular for wine transported by ship. Interestingly economies of scale are observed for transport by plane and ship (η < 0) but not for road. A 10% increase in the value of wine exported by road leads to a rise in transport costs of 0.5%.

η = a 3 (1 + a 3 )(1 -σ) , β = a 2 1 + a 3 , α = a 1 -a 2 1 + a 3

Conclusion

Hummels and Lugovskyy (2009) proposed a model generalizing the ideal variety approach of [START_REF] Lancaster | Variety, Equity, and Eciency[END_REF][START_REF] Lancaster | Protection and Product Dierentiation[END_REF]. In applying this model to the wine sector, we have conrmed the majority of its conclusions. A 1% increase in GDP per capita generates, on average, an increase in price dierential of between 0.55% and 1.13%. Trade-share is strongly inuenced by all variables approximating wealth concentration such as income earned at the top of the distribution. Lastly a gravity-trade equation supports the view that income eects play an important role in explaining wine exports but also raise questions about transport costs. Depending on economies of scale and market structure in this sector, changes in price and volume exported may partly be explained by transport costs interacting with rising global demand.

2. 1

 1 Data description Our data set on wine exports comes from the Single Administration Declarations (SAD) concerning the period 2001-2011, collected by the French customs and assembled by INSEE. The monthly database reports wine exports, mode of transport, exporters on each market at the 8-digit Harmonized System. Such a large database of French wine has never been used up to now.

7

 7

Figure 1 :

 1 Figure 1: Bordeaux wine (exported by air) and GDP per capita

14 13

 14 One can observe that measures of income inequalities are related to the GMIV model, for instance Bekkers, Francois and Manchin (2012) generalize the Lancaster model more by considering that the compensation function is not only dependent on the consumption of a variety (as Hummels and Lugovskyy, 2009) but on total consumption. They then demonstrate that the elasticity of trade decreases with an Atkinson index of income inequality.

15 UsingMonte

 15 Carlo simulations, Santos[START_REF] Silva | The Log of Gravity[END_REF] show that the Poisson pseudo-MLE performs better than the traditional OLS linear-in-logs.Head and Mayer (2013) recommend the use of OLS, Poisson and Gamma PML, and state that if the sample is large enough then Poisson and Gamma PML should give approximately the same result.

Table ( 1

 ( ) reports results. Column 1 explains price growth over the whole period by only using extrema of the period, i.e. years 2001 and 2011. Column 2 uses price dierential yearly (2011-2010, 2010-2009, and so on). Year xed eects are introduced for estimations reported in the last Column.

		Price dierential	Yearly price dierential
		(using years 2001 and 2011)	(2001-2011)
	GDP per capita	1.13	0.55
		(0.498) b	(0.160) a
	GDP	-1.10	-0.49
		(0.481) b	(0.158) a
	Year xed eect	No	Yes
	Product xed eect	Yes	Yes
	Firm xed eect	Yes	Yes
	R-square	0.587	0.091
	Obs	3281	106508

OLS Estimations (RSE in brackets) a: signicant at 1%, b: at 5%. All variables are in Log.

Table 1 :

 1 Price regressions of French wines, 2001-2011

Table (

 (

	dep var:		Share of trade
	Distance	-1.004	-1.115	-0.938	-1.580	-0.054
		(0.011) a (0.026) a (0.019) a (0.135) a (0.015) a
	GDP*Distance	0.032	0.037	0.031
		(0.000) a (0.001) a (0.001) a
	GDP per capita*Distance	0.010				0.020
		(0.001) a				(0.001) a
	Income share of top 10%*Distance		0.797		0.474
			(0.069) a	(0.037) a
	Income share of top 1%*Distance			0.710
				(0.019) a
	Production*Distance				-0.018	-0.002
				(0.002) a (0.000) a
	R-square	0.368	0.386	0.388	0.572	0.328
	Obs	203 375	111 383	113 383	8 751	67 489
	OLS Estimations (RSE in brackets) with years, rms and product-xed eects
	a: signicant at 1%, b: signicant at 5%, c: signicant at 10%
				2) illustrates results.
	Column 1 is the benchmark with similar independent variables to those used in the
	previous section (GDP and GDP per capita) in interaction with distance.

Table 2 :

 2

	11 , to

Share of trade and wealth

To validate the CES model, distance must be the only signicant coecient, which is clearly not the case here. These results disqualify the CES assumption used with a simple model of monopolistic competition.

10 Interactions between distance, market size and GDP per capita are essential in order to explain the share of wine exported as predicted by the GMIV model.

As a robustness check we have also used alternative measures of GDP per capita taking into account the income eect (Column 2 and 3). We have chosen specicially the top income shares (top 10% and top 1%) from the World Top Incomes Database 12

[START_REF] Dimson | The Price of Wine[END_REF]

for instance wrote that among wealthy individuals, ne wine is a mainstream investment and

[START_REF] Barclays | Prot or Pleasure? Exploring the motivations behind treasure trends[END_REF]

reports that wine represents 2% of the wealth of about one quarter of high-net-worth individuals around the world that own a wine collection.

  value of wines instead of price index. Whatever the specication, results reported in the text table are still valid.[START_REF] Combris | Estimation of a Hedonic Price Equation for Bordeaux Wine : Does Quality Matter ?[END_REF] As recommended for instance by Head and Mayer (2013) who write "if all three estimates are similar, then we can relax because the model appears to be well specied [...] Therefore the OLS results are the maximum likelihood estimates".

	Dep var:	French Wine Exportation (adjusted by french GDP per-capita)
	Mode:		Air-plane		Water-ship	Land-road & rail
	Estimator:	OLS	PPML	Gamma	OLS	OLS
	GDPs	-0.007	-0.012	-0.034	0.290	0.385
		(0.036) b (0.028) (0.029)	(0.007) a	(0.018) a
	POPs	0.083	0.116	0.075	-0.063	-0.115
		(0.039) b (0.027) a (0.030) b	(0.009) a	(0.017) a
	Distance	-0.105	-0.001	-0.143	0.108	-0.180
		(0.075) (0.064) (0.063) b	(0.009) a	(0.013) a
	CPIs	-1.792	-3.764	-2.573	-1.941	1.570
		(1.779) (1.207) a (1.474) a	(0.385) a	(0.724) b
	Colony	-0.411	-0.799	-0.620	0.348	-0.047
		(0.102) a (0.083) a (0.081) a	(0.015) a	(0.044)
	Common Language	-0.661	-0.192	-0.448	0.155	0.332
		(0.134) a (0.118) (0.110) a	(0.019) a	(0.025) a
	R²/Pseudo R²	0.762	0.672	0.086	0.401	0.389
	Obs	3971	3971	3971	69079	39619

Table 3 :

 3 Gravity equation Transport costs of course vary with investment in 16We carried out estimations with Poisson and Gamma PML with similar results more ecient technology. By using price data from UPS, they nd that a 10% increase in exports decreases transport prices by 0.8%. Hummels, Lugovskyy and[START_REF] Hummels | The trade reducing eects of market power in international shipping[END_REF] show that shipping costs decrease according to the number of competitors, with low taris and product prices and with high demand elasticities.

	Columns 1, 2 and 3 give results for wine exported by plane using the three dierent
	estimators. Distance and price index have the expected sign, but GDP and population
	contradict the theory. A positive sign is obtained for population, rejecting the idea
	that wines exported using this mode of transport are luxury goods. On the contrary,
	the theory is validated for wines exported by ship and road reported in Table 3 in
	column 4 and 5 using OLS.	16 This validation of non-homothetic preferences is surprising
	since wines exported by road can be considered as basic goods not characterized by
	a strong income eect. After various robustness checks providing similar results, we
	conclude that these initial results are correct and that the theoretical model needed to be
	revisited. To reconcile our data with theory we decided to introduce economies of scale
	in transport. Depending on export-value, obviously rms pay dierent transport costs.
	There is a wide range of literature on this topic. For instance Skiba (2007) considering
	economies of scale in transport, nds that a 10% increase in the volume of trade brings
	about a 2.5% reduction in trade costs. Clark et al. (2004) nd that transport costs are
	lower when trade volumes are high. In Kleinert and Spies (2011) total export gures
	determine the mode of transport.

Table ( 4

 ( ) summarizes the numerical expressions derived from estimations and also reports results for wine exported by road and ship since we expect a zero value for β that was not obvious until now.

	Mode:	Air	Water	Land-road & rail
	Transp econ scale (η)	-1.988	-0.024	0.054
	Income per cap (β)	-0.524	-0.056	-0.140
	GDPs (α)	0.762	0.318	0.609
	Calculation for η using σ = 5

Table 4 :

 4 Economies of scale in transportation

For instance, following[START_REF] Simonovska | Income Dierences and Prices of Tradables: Insights from an Online Retailer[END_REF] we have also used population instead of GDP. While this variable appears insignicant in her analysis we nd a coecient of -0.743 signicant at 1%.

Appendix A: Price elasticity and geographical reputation To investigate whether the previous results hold at a more disaggregated level, we pursue the econometric exercise by focusing on French regional wines. This decision was taken in order to treat specic heterogeneous behaviour not fully captured by xed eect and/or to consider aggregation bias. By estimating the previous equation with rm and time xed eects and by separating wine by region we can compare the eects of both competition and income on wines produced in dierent locations.

We consider three regions: Bordeaux, Alsace and Languedoc Roussillon.

Bordeaux is recognized worldwide for its wine and the wines produced there have the most prestigious reputation in our sample.

We chose Alsace as a region producing white wines with a clear dierentiation including the reputable dry Riesling and Gewürztraminer wines. This region also has the advantage of being located near the urbanisation corridor of Western Europe, with a population of around 111 million, oering producers an advantage in terms of market access.

Lastly, we consider the Languedoc Roussillon region which is one of the main wine producers in France, with annual production volumes that have been known to surpass the production of nations like the United States. However the production is heterogeneous in terms of quality and the region suers from a poorer reputation than wines produced elsewhere (e.g. in Bordeaux).

Table [START_REF] Baldwin | Gravity Chains: Estimating Bilateral Trade Flows When Parts And Components Trade Is Important[END_REF] presents the results. It is worth noting that while Hummels and Lugovskyy (2009) pool over multiple exporters and provide results at an industrial level (HS2), we have enough data variation to lead the estimation at the HS8 level maintaining rmxed eects. This detailed analysis conrms the previous results, a rise in GDP per capita favours wine exportation for many varieties, while GDP growth, theoretically associated with more competition, is detrimental.