Hervé Crès
email: cres@hec.fr

M Utku Ünver
email: uunver@ku.edu.tr

Ideology and existence of 50%-majority equilibria in multidimensional spatial voting models *

Keywords: Spatial voting, super majority, ideology, mean voter theorem, random point set

When aggregating individual preferences through the majority rule in an n-dimensional spatial voting model, the 'worst-case' scenario is a social choice conÞguration where no political equilibrium exists unless a super majority rate as high as 1 -1/n is adopted. In this paper we assume that a lower d-dimensional (d<n ) linear map spans the possible candidates' platforms. These d 'ideological' dimensions imply some linkages between the n political issues. We randomize over these linkages and show that there almost surely exists a 50%-majority equilibria in the above worst-case scenario, when n grows to inÞnity. Moreover the equilibrium is the mean voter. The speed of convergence (toward 50%) of the super majority rate guaranteeing existence of equilibrium is computed for d =1and 2.

Introduction

It is well known since [START_REF] Plott | A notion of equilibrium and its possibility under the majority rule[END_REF] that a 50%-majority stable political equilibrium typically does not exist in a multidimensional voting setup. A way to restore existence of a stable outcome is to require a super majority rule to overrun the status quo, thus giving rise to the concept of ρ-majority equilibrium, where ρ ∈ [1/2, 1] is the proportion of the voting population a challenger must rally to take over. It is widely admitted that the smaller the rate of super majority needed to secure existence of an equilibrium (i.e., the less conservative the voting rule), the better.

There is a wide literature on the level of super majority required for existence, both in deterministic or probabilistic setup (see, e.g., [START_REF] Ferejohn | On a class of rational decision procedures[END_REF], [START_REF] Caplin | Aggregation and social choice: a mean voter theorem[END_REF]Nalebuff (1988, 1991) and [START_REF] Balasko | Condorcet cycles and super-majority rules[END_REF]). In a standard social choice setup where agents, endowed with continuous and convex preferences, have to choose among political alternatives in a non-empty, compact and convex subset of R n , [START_REF] Greenberg | Consistent majority rules over compact sets of alternatives[END_REF] shows that a necessary and sufficient condition for the existence of a ρ-majority equilibrium is ρ ≥ n n+1 . To show that this bound is tight, [START_REF] Greenberg | Consistent majority rules over compact sets of alternatives[END_REF] constructs a voting conÞguration where no incumbent is stable with respect to a super majority rule with rate smaller than n n+1 . It follows: Take n +1 independent points in R n and interpret them as the ideal political choices of n +1 voters endowed with euclidean preferences. Denote S n the n-dimensional simplex generated by the voters' ideal points. Fix an incumbent x/ ∈ S n ; then s(x)= argmin {kx -sk,s∈ S n } is unanimously preferred to x, hence x is not stable under any ρ-majority rule with ρ<1.N o w ,Þx an incumbent x ∈ S n ;t h e ni t is always possible to Þnd a challenger preferred by n out of the n +1 voters: indeed, denote Sn the (n -1)-dimensional simplex generated by the ideal points of these n voters ( Sn is a face of S n ), then one can reconduct the previous argument, and show that s(x)= argmin {kx -sk, s ∈ Sn } is preferred to x by all of these n voters.

This example is thus a 'worst-case' scenario. One easily sees that if the voters' ideal points are taken in a lower dimensional subspace, then the upper bound decreases. But the gain remains small though. And one gets existence of political equilibria for not too conservative voting rules only when the number, n, of political issues is very low. This bound is ρ =1/2 when n =1(the so-called 'median voter theorem'); ρ =2/3 when n =2; and for n ≥ 3, then the required rate of super majority must be above 3/4 (and converges the the unanimity criterion when n goes to inÞnity), a level very rarely observed in practice. Indeed, constitutions or corporate charters build on super majority rates which are very rarely above 70% 1 , although the number of political issues at stake in electoral processes is obviously often very large: it is not rare, when reading political platforms proposed by candidates in large elections, to denumerate several dozens of issues 2 . Hence the question: why, if there are so many issues, do we observe so reasonable super majority rates in practice?

A Þrst answer might be that one should not believe in Greenberg's worst-case scenario. A second answer can be found in the Hinich-Ordeshook spatial voting model 3 . According to the latter, there are only a few political dimensions underlying the platforms proposed by the candidates. These few dimensions are claimed to be ideological. Ideologies imply linkages 4 between political issues and thus span a lower dimensional linear space (dubbed the 'campaign space' in the sequel) on which the original distribution of voters' ideal points is projected.

The assumption that political platforms are based on ideology stems from the belief that the cleavages between candidates separate along simpler, more predictable lines than the n-dimensional policy space would imply. As [START_REF] Popkin | The reasoning voter: Communication and persuasion in presidential campaigns[END_REF] states it (p. 51): "Ideology is not the mark of sophistication and education, but of uncertainty and lack of ability to connect policies with beneÞts... Parties use ideologies to highlight critical differences between themselves, and to remind voters of their past successes". This approach has some empirical relevance: [START_REF] Poole | Changing minds? Not Congress![END_REF]Rosenthal (1991, 1996) show that in the USA, with the exception of the 32nd Congress, two dimensions are always capable of explaining more than 80% and up to 95% of the variation in the votes of elected officials on most issues. The same, [START_REF] Mccarty | Income Redistribution and the Realignment of American Politics[END_REF] and [START_REF] Mccarty | Income Redistribution and the Realignment of American Politics[END_REF] test the Hinich-Ordeshook spatial voting model on post World War II Congressional roll call voting and show that only two dimensions are required to account for most of the votes: the liberal-conservative continuum 5 and the dimension of conßi c to v e rr a c ea n d civil rights.

But one cannot exclude that the number of underlying ideological dimensions be larger than 3. In the political debate in France on the referendum for ratiÞcation of the European Treaty.

2 Everybody in France recalls the '110 propositions' of the candidate François Mitterrand for the presidential election of 1981.

3 This model was Þrst proposed by [START_REF] Cahoon | A multidimensional statistical procedure for spatial analysis[END_REF], [START_REF] Ordeshook | The spatial theory of elections: a review and a critique[END_REF] and Hinich and Pollard (1981) and then developed by [START_REF] Enelow | The spatial theory of voting,N e wY o r k : C a mbridge[END_REF] and Hinich and Munger (1994). 4 These linkages formalize the fundamental insight of [START_REF] Converse | The nature of belief systems in mass publics, in Ideology and discontent[END_REF] according to which ideology (the Conversian 'belief system') interrelates and bundles the political issues: ideology is fundamentally the knowledge of what-goes-whith-what. As [START_REF] Converse | The nature of belief systems in mass publics, in Ideology and discontent[END_REF] states it: ideology is "...a conÞguration of ideas and attitudes in which the elements are bound together by some form of constraint" (p. 207). 5 A judgemental dimension that has been "highly serviceable for simplifying and organizing events in most Western politics for the past century", Converse (1964, p. 214).

constitutional treaty during the Spring of 2005, one cannot explain the cleavages between and within parties through the traditional left-right dimension. One also needs the now classical ideological dimension 'sovereignist-federalist' to explain the split of the gaullist party; furthermore the possible future entry of Turkey was an element of the debate, and religion (the ideological position in the 'laïcity' dimension) was clearly the only way to explain another (orthogonal) split of the gaullist party.

Another type of political debates often builds on more than three underlying dimensions: proxy Þghts in publicly traded corporations in a context of market failures 6 .T h e stakes are probably simpler to grasp than in ordinary political debates; moreover, shareholders usually have access to a more measurable and precise information which is easier to aggregate. Yet corporate charters rarely choose rates of super majority beyond 65%.

The answer to the question "why do we observe so reasonable super majority rate in practice?" seems to be: not only the political competition articulates along fewer, simpler and more predictable lines than the n-dimensional policy space would imply, but also one should not believe in Greenberg's worst-case scenario. The present paper goes one step further. Its main contribution is an aggregation theorem that links the two latter arguments: one should not believe in Greenberg's worst-case scenario because the political competition happens in a lower dimensional subspace spanned by the underlying ideologies. Indeed, if we randomize on the linkages between issues imputed by ideologies, our main result (Theorem 1) states that the Hinich-Ordeshook approach almost surely transforms Greenberg's worst-case scenario into the best-case scenario of a symmetric distribution of voting characteristics. And as a consequence we obtain (Theorem 2) a mean-voter theorem: the mean voter happens to almost always be the unique 50%majority equilibrium, when the number of political issues grows large.

The paper is organized as follows: Section 2 introduces the model, Þrst the classical Downsian spatial voting model (Section 2.1), then its Hinich-Ordeshook sophistication (Section 2.2). Then Section 3 states and proves the aggregation theorems. Section 4 computes a lower bound to the speed of convergence of the expected min-max rate toward 50%; the computations give upper bounds on the expected rate of super majority necessary to sustain the mean voter as a political equilibrium, for any number of voters, when the number of underlying ideologies is smaller than 2. Section 5 ends the paper with some concluding comments.

The model 2.1 Voters, platforms and the majority rule

The setup to model the electoral process and voting mechanism is the classical Downsian multidimensional spatial voting model (Downs (1957)). There are n measurable criteria of political activity, so that a political platform in the policy space can be represented as an n-dimensional vector: x ∈ R n .T h e r ea r em voters in a set I. Each voter is endowed with an euclidean preference relation on R n :a g e n ti, 1 ≤ i ≤ m,h a sapreferred choice in the policy space, x i ∈ R n , and his/her utility function over the space of political c h o i c e si sd e c r e a s i n gw i t ht h ee u c l i d e a nd istance from his/her preferred choice:

∀x ∈ R n u i (x)=-kx i -xk
A society is a m-tuple X =(x i ) m i=1 . We measure the stability of a political platform in a given society through the Simpson-Kramer approach. Given two political choices (a, b) ∈ R n × R n , ρ(b, a) measures the ratio of the electorate that strictly prefers b to a:

ρ(b, a)= /{i ∈ I|u i (b) >u i (a)} m .
The score of a political choice a ∈ R n is: ρ(a)=max b∈R n ρ(b, a). Clearly, the score of any political choice taken outside the closed convex hull, hXi,o fX will be 1: the challenger b that minimizes the distance between a and hXi is unanimously preferred to a. Hence lookingforthe'best'statusquo,i.e.,theoneswithlo w estscore,w ecanreduceoursearc h to hXi.T h emin-max rate of society X is: ρ * =m i n a∈R n ρ(a).T h emin-max set of society X is:

S * (X)={a ∈ R n |ρ(a)=ρ * }.
The majority rule with rate ρ ∈ [0, 1] states that candidate b is preferred by society X to (or defeats) candidate a if and only if ρ(b, a) >ρ .A c a n d i d a t e a is said to be ρ-majority stable in society X if and only if there is no alternative that defeats it, i.e., if and only if its score is not larger than ρ: ρ(a) ≤ ρ. Such a candidate is a political equilibrium for the majority rule with rate ρ.

One knows since the seminal work of [START_REF] Plott | A notion of equilibrium and its possibility under the majority rule[END_REF] that 50%-majority stable equilibria generally do not exist when n ≥ 2. To recover existence of political equilibria, one has to impose a super majority voting rule, i.e., a voting rule with rate ρ>1/2.T h i sp a p e r deals with existence of such political equilibrium based on super majority voting. Along that search, political platforms in the min-max set have this appealing property that they are equilibria for the lowest rate of super majority, hence the less conservative voting rule.

The super majority rate one has to impose in order to recover existence of equilibrium can be quite high, though: As extensively explained in the introduction, suppose that m = n+1 and the m-tuple X are the vertices of an n-dimensional simplex, then obviously any political choice in hXi has a score of n/(n +1) (it is enough to choose a challenger closer to any of the n +1 (n -1)-dimensional faces of the simplex). Therefore, since any political choice outside hXi has score 1, the min-max rate is n/(n +1) and the min-max set is hXi: one has to impose a super-majority rule of rate ρ ≥ n/(n +1) to get a stable political choice, and then all choices in hXi are ρ-majority stable. [START_REF] Greenberg | Consistent majority rules over compact sets of alternatives[END_REF] proves that the condition ρ ≥ n/(n +1) -to get existence of a ρ-majority stable political equilibrium-is in fact sufficient as soon as the voter's preferences satisfy very mild properties of continuity and convexity. Hence, the latter case is a worst-case scenario,a s far as getting not too conservative min-max rate is concerned. The present paper can be read as an attempt to downside the relevance of this worst-case scenario.

Some convincing arguments along the same line are available in the social choice literature. One of the most important one is given in [START_REF] Caplin | Aggregation and social choice: a mean voter theorem[END_REF]Nalebuff (1988, 1991). They give a dimension-free upper bound to the min-max rate under the conditions that preferred choices of agents are selected from a σ-concave distribution with compact and convex support. This upper bound (which, asymptotically, is lower than 64%)isgiv enb y the score of the mean voter, the voter whose preferred choice is the barycenter of all x i 's. This literature can roughly be regarded as looking for multi dimensional versions of the median voter theorem.

Ideology, candidates and political campaigns

A central assumption of our model is that, although the number (here: n) of criteria for political activity can indeed be quite large, the political competition takes place in a subspace of lower dimension: d<n .In accordance with the Hinich-Ordeshook spatial voting model, this lower dimensional space is considered to be the ideological space, assumed to be R d without loss of generality. According to this approach, the ideologies are linked to the platforms by a linear map, L, from the ideological space to the policy space: a candidate,

π A ∈ R d , imputes a platform x A ∈ R n such that x A = x 0 + Lπ A ,
where x 0 is the platform of status quo policies. Finally, the d-dimensional affine subspace which is the image of R d by L translated by x 0 is called the campaign space, C ⊂ R n , in the sequel. Before developing the strength of the model, let us illustrate through an example how the linear map L operates. An illustration: Issues of political activity are often precise and technical; consider two such classical issues like (1) how much of the State's budget, x 1 ,m u s tb ea l l o c a t e dt o buy helicopters, and (2) how much of the State's budget, x 2 ,m u s tbea l l oc a t e dt oc r e a t e more slots in kindergartens. For the sake of simplicity, we limit the issues to these two, hence n =2. The assumption is made that platforms proposed by candidates in this two-dimensional policy space can be explained through a (say) one-dimensional underlying linear subspace, e.g., the classical liberal-conservative (left-right) dimension; hence d =1. Given a vector of status quo policies x 0 , the sensitivity of x j , j =1 , 2,t ot h ep o s i t i o nπ of the candidate in the ideological space is a Þxed scalar l j ∈ R, therefore x j is an affine function of π:

x j = x 0 j + l j π, j=1, 2 . Figure 1 (resp. 2) plots the relation between ideology and helicopters (resp. kindergartens). E.g., the policy regarding kindergarten is almost not sensitive to ideology, and only slightly decreases (l 2 is small and negative) with π: a leftist candidate wants to create slots in kindergartens because these structures are more used by low-class workers than by wealthy families; a rightist candidate uses slots' creation in kindergartens as an incentive to increase fertility. The policy regarding helicopter is more sensitive to ideology, and increasing (l 1 is positive): rightist candidate are usually more hawkish, and spending on helicopters rises as ideology moves right, as shown by the plain line L 1 .

Figure 1 Figure 2

L 1 L 0 1 ππ x 1 x 2 x 0 1 x 0 2 - - 6 6
The sensitivity of policies to ideology as depicted on Figures 1 and 2 implies a linkage between the two issues in the policy space: the induced campaign space, C (plain line on Figure 3), is going through the status quo x 0 with slope l 2 /l 1 . The induced 'ideal candidate' xi of voter i, whose preferred platform is x i , obtains by orthogonal projection of x i on the campaign space. And consequently, voter i votes for the candidate whose imputed platform is closest to his 'ideal candidate' xi . In the general (n, d) case, the euclidean structure of the original voting conÞguration gives rise, through the orthogonal projection on C, to a social choice problem involving m voters with euclidean preferences in R d . Hence we are dealing with a d-dimensional spatial voting problem with m voters and thus we are left with a combinatorial problem about m-tuples of points in R d rather than in R n .

The assumption that political platforms are based on ideology stems from the belief Figure 3 x

1 x 2 x 0 1 x 0 2 x 0 • C C 0 xi x0 i • x i • • - 6
that the cleavages between candidates separate along simpler, more predictable lines than the n-dimensional policy space would imply. Simplicity and predictability makes the voter's duty of voting easier. Not only ideology transmits information to voters, but also it creates enthusiasm for political action. This is a virtue of this approach on the candidate's side. Of course, reducing the dimensionality of the campaign space also makes communication easier and less costly for candidates. But on top of that, the credibility of its commitment is stronger when his/her actions are perceived by the voters to be based on ideology 7 (see [START_REF] Enelow | The spatial theory of voting,N e wY o r k : C a mbridge[END_REF] or [START_REF] Hinich | Analytical politics[END_REF] for development of these arguments). Last but not least, as underlined in the introduction, the empirical relevance of this approach has been underlined by [START_REF] Poole | Patterns of congressional voting[END_REF][START_REF] Poole | Congress: a political-economic history of rollcal l voting[END_REF], 1997) and [START_REF] Mccarty | Income Redistribution and the Realignment of American Politics[END_REF].

If the Hinich-Ordeshook spatial voting model has the virtue of offering a more realistic view of electoral competition, we argue in the present paper that it moreover has extremely nice properties as far as aggregation of individual preferences is concerned. Indeed, we prove in the sequel that the worst-case conÞguration of the society (worst case as far as aggregation is concerned) -i.e., when the point-set X is an (n+1)-tuple of points forming a n-dimensional simplex with equal voting rights on the vertices-transforms almost surely 7 As Hinich and Munger puts it, "classical spatial theory assumes that each politician chooses the position that maximizes his or her vote share given the expected platform of the opponents. (...) Such an approach (...) may be of little use in describing real world politics. For such an approach to work, voters must believe that a candidate who takes a position is likely to deliver that position. (...) The candidate's promise must be 'credible'. into a best-case conÞguration -i.e., the (n +1 ) -tuples of projected points in R d X is symmetrically distributed.

The Þr s ts t e po fo u ra r g u m e n ti st oq u a l i f yw h a tw em e a nb y' a l m o s ts u r e l y ' . L e tu s go back to the above illustration. Suppose now that an exogenous historical shock occurs, e.g., a terrorist attack. Most probably this event is going to impact the sensitivity of the Þrst issue (helicopters) to ideology: all candidates become hawkish and want to invest more into such a modern defense tool as helicopters, independently of his/her ideology. Hence a new line L 0 1 , with a much smaller sensitivity rate: l 0 1 <l 1 (see the -almost ßat-dotted line on Figure 1). It is probably going to be the case that everybody in the society, candidates and voters are going to prefer an absolute increase ∆x 1 > 0 in the political platform; we assume that in such an event the perturbed status quo becomes:

x 0 0 1 = x 0 1 + ∆x 1 ,
a n dt h a tf o ra l li, the preferred platform's Þrst component becomes:

x 0 i1 = x i1 + ∆x 1 .
Hence this general absolute increase ∆x 1 results in a global (rightward) translation of the spatial point-set conÞguration, and this translation has no impact on the geometric properties of our problem. Therefore, without loss of generality, we can consider ∆x 1 to be zero, and the only impact of this exogenous historical event is a drop in the sensitivity rate l 1 . This results into a new campaign space C 0 (dotted on Figure 3) going through the status quo x 0 with slope l 2 /l 0 1 . The new induced 'ideal candidate' x0 i of voter i obtains by orthogonal projection of x i on the new campaign space C 0 .

T h ei d e ai st h a ts u c hrandom shocks always happen, although fortunately not all as dramatic as a terrorist attack, and that their 'media' treatment and destiny can change the sensitivities of various issues to ideology. Then the central question is: How is this d-dimensional campaign space chosen? In the present paper, we take a purely Laplacian perspective and assume that C is selected at random, according to a 'uniform' distribution on the natural underlying space. We deÞne C as an element in the Grassmanian G(n, d) of oriented d-subspaces in R n . Random historical and mediatic shocks generate a probability distribution over G(n, d). Among the latter ones, one arises 'naturally': the unique rotation-invariant probability measure, µ(n, d) (known as the Haar probability measure), on G(n, d), which intuitively selects all d-dimensional campaign spaces 'with equal probability'. Hence µ(n, d) will be dubbed impartial in the sequel. The idea behind impartiality is that the main themes at stake in a political campaign depend heavily on the exogenous shocks of recent history, and the exogenous treatment by the media of these shocks.

3M a i n r e s u l t

For any selected campaign space C, the original social choice problem characterized by the point set X in R n gives rise to a lower dimensional social choice problem characterized by the (orthogonally projected) point set X in R d . Suppose X is a m-1 dimensional simplex in R n , such that for each i, x i (a column vector in R n )i st h ei th vertex of simplex X.W e say that X is regular if kx i -x j k = kx i -x k k for any i, j, k. The simplex is O-centered if P i x i =0. Note that for an O-centered regular simplex, we have x T i x j = x T i x k < 0 for any i, j, k. In this section, we mute the translational part of the political shock as explained in the previous section and focus on the rotational shock to the campaign space. First, we will consider rotations pivoted at the center of gravity of the simplex X, that is, when the mean-voter is the status-quo (i.e. it is always on the campaign space). Center of gravity is normalized to O, center of the coordinate system. As we will show, this has no cost in our approach, since the center of gravity is the unique 50%-political equilibrium (Theorem 2).

Theorem 1 Let the campaign space C be impartially randomly selected. The point set X coincides in distribution with a negatively correlated sample from a symmetric probability distribution in R d which becomes asymptotically independent as n →∞with the rate 1 n when X is a regular O-centered simplex. P r o o fo fT h e o r e m1 :Let n ≥ m>d . Take a regular (m -1)-dimensional O-centered simplex in R n whose vertices are column vectors of an n × m matrix X =[x 1 x 2 ... x m ]. 8We have x T i x j < 0 for any i and j.L e t Π be d × n with π ii =1for i ∈ {1, 2,...,n} and all other entries of Π are 0. We can denote the random d-subspace by C =ΠR where R is a random rotation matrix distributed with the Haar probability measure among the n × n rotation matrix group denoted by R(n), that is, every n × n rotation matrix is chosen with equal probability as a draw of R.N o t et h a tC is distributed with probability measure µ (n, d) in Grassmanian G (n, d) . Note that every rotation matrix is an orthogonal matrix. Let O (n) denote the n × n orthogonal matrix group. First, we will consider orthogonal matrices instead of rotation matrices. Let C * = ΠA be such that A is a random orthogonal matrix distributed with the Haar probability measure in O (n) , that is, every n × n orthogonal matrix is chosen with the same probability as a draw of A. Note that orthogonal transformations include rotoinversions (where det (A)=-1) and rotations (det (A)=1), we will rule out rotoinversions later. First note that every column of A has a symmetric distribution and EA =0, since if A is orthogonal then -A is orthogonal and both A =A and A = -A are equally likely events. Since orthogonal transformation preserves the inner-product of two we have P i a ij a ik =0for j 6 = k and P i a 2 ij =1for any orthogonal matrix A. Since every row permutation and column permutation of A is equally likely to occur, E ¡ a 2 ij ¢ and E (a ij a k+ ) is constant for every i, j, k and ?.F i r s t ,

P i E ¡ a 2 ij ¢ =1implies E ¡ a 2 ij ¢ = 1 n .M o r e o v e r , P i E (a ij a ik )=0 implies E (a ij a ik )=0
, implying with the symmetry argument that E (a ij a k+ )=0for every i, j, k, ? such that i 6 = k or j 6 = ?. We are interested in the distribution of the columns of P = C * X = ΠAX, the projection of X to the random subspace C * . We will show that each column vector has identical symmetric distribution and each pair of column vectors are negatively correlated. The two events P =ΠAX and P =Π (-A) X are equally like to occur, therefore each column of P has a symmetric distribution. Since each x i has the same length, each column vector Ax i is an identical random vector. Let

Σ ij = cov (Ax i , Ax j ) for i 6 = j.T h e(g, h) th entry of Σ ij is σ ij gh = E (( P k a gk x ki )( P + a h+ x +j )).F o rg 6 = h, σ ij gh = X k X + E (a gk a h+ ) | {z } =0 x ki x +j =0 For g = h, σ ij hh = X k X + E (a hk a h+ ) x ki x +j = X k E ¡ a 2 hk ¢ | {z } = 1 n x ki x kj + X k X +6 =k E (a hk a h+ ) | {z } =0 x ki x +j = 1 n x T i x j < 0
Since p i is the Þrst d coordinates of Ax i for each i,d i fferent coordinates of each pair of p i and p j are independently sampled and the same coordinates of each pair p i and p j are negatively correlated, where correlation goes to zero as n →∞(or m →∞since m ≤ n). We conclude our proof by observing that by Theorem 1 of [START_REF] Baryshnikov | Regular simplices and Gaussian samples[END_REF], above matrix A can be swapped with a random rotation matrix R,a n d point set X consists of draws of columns of P = CX. ¥ O u rn e x tr e s u l ts t a t e st h a tt h e r ei sau n i q u ep o l i t i c a le q u i l i b r i u ma tt h em e a nv o t e r for 50%-majority rule as the number of voters goes to inÞnity.

Theorem 2 (Mean Voter Theorem) Fix d.T a k em →∞then almost surely O,t h e mean voter, is the unique political equilibrium for the 50%-majority rule.

Proof of Theorem 2: Let f : R d → R + be the underlying limiting marginal probability density function (p.d.f.) for the columns of lim m→∞ X. Let Eρ (w) denote the expected value of the score of any point w ∈ R n .

First, we show that O (∈ R n ), the mean voter, is a political equilibrium for the 50% majority rule. Negatively correlated sampling of political positions has a smaller min-max rate than independent sampling of political positions from the same symmetric marginal Let Y be a set of points independently sampled from p.d.f. f.U n d e r independent sampling, by Theorem 3 in Caplin and Nalebuff (1988) the min-max rate of O for Y converges almost surely to the min-max rate of f at O ∈ R d when m →∞ . The min-max rate of O for f is 0.5, since f is symmetric around O ¡ ∈ R d ¢ by Theorem 1. Since min-max rate cannot smaller than 0.5, min-max rate of O for X also almost surely converges to 0.5, that is, lim m→∞ Eρ (O)=0.5, implying O (∈ R n ),t h em e a nv o t e r ,i sa n equilibrium point for the 50%-majority rule.

Next, we prove that O (∈ R n ) is the unique equilibrium issue. Take any inÞnite dimensional issue vector w 6 = O.F o rÞnite n, let the Þrst n entries of w be relevant. We will show that lim m→∞ Eρ (w) >0.5. Let C = ΠR be the random campaign space spanned by the Haar measure, like in the proof of Theorem 1 where R is the n × n impartial random rotation matrix distributed with Haar measure on R(n) and Π is the d × n matrix with π ii =1for i ≤ d and all other entries of Π are zero. The random projection of the issue vector w on the campaign space is ΠRw. Note that ΠRw = O ∈ R d with probability 0 and ΠRw 6 = O ∈ R d with probability 1, since w 6 = O ∈ R n . Hence, the only relevant draws of R for the calculation of

Eρ (w) are all R ∈ R(n) such that ΠRw 6 = O ∈ R d .F i x a rotation matrix R ∈ R(n) such that ΠRw 6 = O. Since f is symmetric around O ∈ R d
by Theorem 1, the min-max rate of the point ΠRw for f is greater than 0.5. Hence expected min-max rate of the projection ΠRw is greater than 0.5 when m →∞, implying lim m→∞ Eρ (w) > 0.5 and concluding that w cannot be stable under 50% majority voting rule. ¥

This result is inspired from a modern approach (in the literature from discrete and computational geometry) to generate random points. When dealing with a d-dimensional spatial voting problem with m voters, one is left with a combinatorial problem about m-tuples of (random) points in R d . Many 'natural' distributions of these random points have been proposed in the mathematical literature (see [START_REF] Schneider | Discrete aspects of stochastic geometry, Handbook of Discrete and Computational Geometry[END_REF]). Among them, the one described above takes a central place: Every conÞguration of m>dnumbered points in general position in R d is affinely equivalent to the orthogonal projection of the set of numbered vertices of a Þxed regular (m -1)-dimensional simplex onto a unique ddimensional linear subspace in R n . This construction builds a one-to-one correspondence between the (orientation-preserving) affine equivalence classes of such point set conÞgurations and an open dense subset of the Grassmanian G(n, d) of oriented d-spaces in R n . The so-called Grassmann approach (sometimes referred as the Goodman-Pollack model) considers the probability distribution on the set of affine equivalence classes of m-tuples in general position in R d that stems from the unique rotation-invariant probability measure on G(n, d). [START_REF] Baryshnikov | Regular simplices and Gaussian samples[END_REF] (following an observation of [START_REF] Affentranger | Random projection of regular simplices[END_REF]) proved that under the Grassmann approach, the resulting point set coincides in distribution with a standard sample in that subspace. As a consequence, an affine-invariant functional of m-tuples with this distribution is stochastically equivalent to the same functional taken at an i.i.d. m-tuple of standard normal points in R d .

Our model can be seen as giving another interpretation to the Grassmann approach: We do not use it as a random generation of social choice conÞgurations (m-tuples of points in R d ) according to a 'natural' probability distribution, but as a random generation of a lower dimensional campaign spaces for any original (higher dimensional) social choice conÞguration. The latter in particular can be a worst-case scenario as deÞn e di nt h e introduction and Section 2.

Remark: A direct consequence of the Grassmann approach is that if we depart from a (regular) worst-case scenario in R n and take X to be the vertices of a regular (n -1)dimensional simplex (which can as well be translated to be spherico-regular), then, up to an affine transformation, the resulting point set X coincides in distribution with a standard Gaussian sample in R d . In our approach, we do not need linear images and affine translations, since economically they do not correspond to any interpretations, and mathematically we do not require stochastic independence and an underlying Gaussian distribution.

Low dimensional social choice conÞgurations

Of course, the asymptotic result in Theorem 2 does not give much idea of the rate of super majority, ρ (O), that one should impose in order to have existence of a political equilibrium when d and m are 'small', nor does it give an idea of its expectation, Eρ (O). In this section, we give an upper-bound to distribution-free value of this expected score for d =1and d =2.W ec o n s i d e rindependent sampling of m points drawn from a symmetric distribution about O (∈ R d for d =1and d =2 )which approximate the sampling for n →∞. In our approach in Theorem 1, we need negatively-correlated sampling for Þnite n. Therefore, the results derived in this section are upper bounds to actual Eρ (O) for Þnite n when d =1and d =2where X is regular and O-centered. Let n →∞in this section. For uni-and bidimensional campaigns, the upper-bounds that we derive are plotted at the end of the section in Figure 7.

Eρ(O)= 1 2 + 1 2 2[ m 2 ]+1 Ã 2 £ m 2 ¤ £ m 2 ¤ !
where [x] denotes the integer part of x.

Proof of Proposition 1: Consider the following process which is due to [START_REF] Wendel | A problem in geometric probability[END_REF]: choose m random points in an interval centered at O: q 1 , q 2 , ..., q m .F o re a c hi, 1 ≤ i ≤ m, set p i equal to q i or to -q i with equal probability 1/2. The points p 1 , ..., p m are again i.i.d. random points in the interval (we ignore the degenerate conÞgurations, occurring with probability zero, where two points q i and q j would be equal or opposed).

Consider Þrst the case of an even m: m =2p.T h es c o r eo fO will be the highest ratio of points on either side of O. Obviously if there are k points on one side (and mk on the other side), 0 ≤ k ≤ p, this number is (mk)/m, and this happens with probability

2 1 2 m à m k ! when 0 ≤ k ≤ p -1, and with probability 1 2 m à m k ! when k = p. Hence Eρ(O)= 1 2 m à 2p p ! p 2p + 1 2 m-1 p-1 X k=0 à m k ! m -k m = 1 2 m+1 à 2p p ! + 1 2 m-1 p-1 X k=0 à m -1 k !
and the second term on the right-hand side is 1/2. Now if m =2p +1:

Eρ(O)= 1 2 m-1 p X k=0 à m k ! m -k m = 1 2 m-1 p X k=0 à m -1 k ! = 1 2 m-1 p-1 X k=0 à m -1 k ! + 1 2 m à m -1 p ! | {z } = 1 2 + 1 2 m à m -1 p !
All we used in this proof was that the original distribution is symmetric about O and that some degeneracies occur with probability zero. Hence the result, which is distribution-free. ¥

Bidimensional campaigns

We consider the same as in Proposition 1 and choose m random points in a disk centered at O: Q 1 , Q 2 , ..., Q m .F o re a c h i, 1 ≤ i ≤ m,w es e tP i equal to Q i or to -Q i with equal probability 1/2 (without loss of generality, we can choose the Q i 's on the same side of a hyperplane through 0 as in Figure 4 below; Figure 5 corresponds to the conÞguration: P i = Q i for i =1 , 3, 4 and P i = -Q i for i =2 , 5). The points P 1 , ..., P m are again i.i.d. random points in the disk. The original question answered by [START_REF] Wendel | A problem in geometric probability[END_REF], see also [START_REF] Wagner | A continuous analogue of the upper bound theorem[END_REF], was: what is the probability that O is not in the convex hull of the P i 's? (In other words, what is the probability that the score of O be 1?) The answer is: m/2 m-1 . Indeed, independently of the choice of the Q i 's (again, we ignore the degenerate conÞgurations, occurring with probability zero, where two vectors Q i and Q j would be collinear), there are 2m possibilities to choose the signs of the P i 's such that O can be separated from these points by a line (every partition of the Q i 's by a line through O gives two such possibilities). Again, all we used in this line of reasoning was that the original distribution is symmetric about O and that some degeneracies occur with probability zero. Hence the result is, once more, distribution-free.

Figure 4 Figure 5 •

• •• • • • • •• • G Q 1 Q 2 Q 3 Q 4 Q 5 -Q 1 -Q 2 -Q 3 -Q 4 -Q 5 • • • • • • G P 1 P 3 P 4 P 2 P 5
We now proceed along this line of reasoning and compute the probability that the score of O be j/m. E.g., in the social choice conÞg u r a t i o ns h o w no nF i g u r e5 ,t h es c o r e of O is 4/5 as shown by the dotted separation line.

Each social choice conÞguration can be described as a (q, p)-sequence (or random of plus ones and minus ones, according to whether P i is equal to Q i (then +) or not (then -), 1 ≤ i ≤ m: M 1 ,...,M m ,w i t h ,s a y ,q plus ones and p minus ones, q + p = m. The conÞguration on Figure 5 corresponds to the (3,2)-sequence: + -++-. The partial sum s k = M 1 + ...+ M k represents the difference between the number of pluses and minuses occurring at the Þrst k places, 0 ≤ k ≤ m,w i t hs 0 =0and s m = qp.D e Þne: s =max k s k and s =min k s k .

Lemma 1 In the social choice conÞguration represented by a (q, p)-sequence (M 1 ,...,M m ), To compute the probability that the score of O be j/m, we need to compute the number of (q, p)-sequences such that max{q-s,p+s} = j. T odothat,wefollowaclassical geometric method in the standard orthonormal basis where the x-axis in horizontal and the y-axis is vertical. Following [START_REF] Feller | An introduction to probability theory and its applications[END_REF], the sequence (M 1 ,...,M m ) is identiÞed with a path from the origin to the point (m, qp): this path is a polygonal line whose vertices have abscissa 0, 1,...,m and ordinates s 0 ,s 1 ,...,s m = qp; s is the highest point of the path and s the lowest. Obviously there are9 Ã m p ! such paths from the origin to the point (m, qp): a sm a n ya st h e r ea r ew a y so fc h o o s i n gt h ep places for the minuses out of the m possibilities. Note that for a (q, p)-sequence, s ≥ max{0,q -p} and s ≤ min{0,q -p} entail that max{q -s,p+s} ≥ max{q, p}, therefore we restrict attention to j ≥ max{q, p}.A n d i nt h ec a s ew h e nm is even and p = q = m/2, obviously max{q -s,p+s} ≥ m/2+1 therefore we restrict attention in that case to j ≥ m/2+1. Hence we consider j such that [m/2] + 1 ≤ j ≤ m. A (q, p)-sequence is such that max{q -s,p +s} = j whenever the associated path remains in the corridor between the lines y = jp and y = qj, and hits at least one of them (see Figure 6 drawn for the conÞguration of Figure 5 and j =4). The number of (q, p)-paths such that max{qs,p+s} = j is a m,q,j = ( A m,q,j+1 -A m,q,j if mj ≤ q ≤ j 0 otherwise where for mj ≤ q ≤ j A m,q,j = X

the score of O is ρ(O)= max{q -s,
k "Ã m q + k(2j -m) ! - Ã m j + k(2j -m) !# (1)
(the series extending over all integers k from -∞ to +∞, but having only Þnitely many non-zero terms) is the number of (q, p)-paths such that max{q -s,p +s} <j :t h o s e which hit neither y = jp nor y = qj,a n dt h en u m b e ro f(q, p) paths such that max{q -s,p+s} <m+1 is given by

A m,q,m+1 = Ã m q ! . (2)
P r o o fo fL e m m a2 : The equation relating the a m,q,j 's to the A m,q,j 's is immediate. The computation of the A m,q,j 's relies on repeated applications of the 'reßection principle' due to Désiré André (see, e.g., [START_REF] Feller | An introduction to probability theory and its applications[END_REF]

Concluding Comments

The present paper proposes a theorem of aggregation of individual preferences through the 50% majority rule in a multidimensional spatial voting model. Of course, the result is obtained at a non-negligible cost in terms of assumptions: Þrst the regularity of the simplicial distribution of voters' ideal points; second the 'uniform distribution' on the set of linkages between issues imputed by ideologies (the Haar probability measure on G(n, d)). Of course, the robustness of the results when one relaxes these two assumptions should be studied. But on the other hand it is quite strong since it gives existence for

campaigns Proposition 1

 1 Set d =1 .C o n s i d e ra nm-sample independently drawn from a distribution which is symmetric about O, then the expected value of the score of O is

-Lemma 2

 2 Figure 6

Figure 7 :

 7 Figure 7: The min-max rate for n →∞when d =1and d =2.F o rÞnite n, these Þg u r e sa r eu p p e rb o u n d sf o rt h ea c t u a lm i n -m a xr a t e s .

  Consider a line which separates the +Q i 's from the -Q i 's and passing through O.I th a sq of the P i 's on one side and p on the other side. Now turn this line by pivoting at O so that it goes in-between Q 1 and Q 2 :i th a sn o wqs 1 of the P i 's on one side and p + s 1 on the other side. Now turn it by pivoting at O so that it goes in-between Q 2 and Q 3 :i th a sn o wqs 2 of the P i 's on one side and p + s 2 on the other side. And so on. The maximum number of P

	m	p+s}	.
	P r o o fo fL e m m a1 :

i 's on one side of a line through O is therefore max{...,q-s k ,...,p+ s k ,...} =max{q -s,p+s}. Hence the result. 2

For decisions on issues which are delegated to the European Union, the rate was 72% in the Maastrich Treaty, it was decreased to a mix of 65% -of the States-and 55% -of the population-in the Constitutional

The heterogeneity of the shareholders' opinions can come from imperfect competition, the incompleteness of Þnancial market structure or the presence of externalities. 'Ideological' dimensions in corporate politics can be: the 'philosophy' with respect to debt vs equity, horizontal vs vertical integration, international diversiÞcation, expansion vs concentration...

Let (i, j) th entry of a matrix H be denoted by h ij and j th column vector of H be denoted by h j .

By convention, the combination number will be set to zero in case p<0 or p>m.

* This research started while H. Crès was visiting Koç University. H. Crès would like to thank the CASE for its hospitality and the Fondation HEC for Þnancial support. M. U. Ünver gratefully acknowledges support from the NSF and TÜBA-GEB ú IP (Turkish Academy of Sciences -Distinguished Young Scholar Program).

t h e5 0 %m a o r i t yr u l e . A n o t h e ri m p o r t a n ta s p e c ti st h a ti tÞngers the mean voter as the candidate most likely to be stable in the voting process. As underlined in [START_REF] Crès | Portfolio diversiÞcation and internalization of production externalities through majority voting[END_REF], mean voter theorems are very welcome in public economics because in many contexts the mean voter is the one who has the right incentives as far as making an economically efficient choice is concerned.

We would like to stress one last point. An important property of the approach chosen here is that it is compatible with the idea that politicians "die in their ideological boots". [START_REF] Poole | Changing minds? Not Congress![END_REF] shows a variety of evidence that members of the US Congress are ideologically consistent: they adopt an ideological position and maintain it over time. An interpretation of Theorem 2 is that i nt h el o n gr u n , ignoring the historical, sociological and mediatic shocks which are going to shape the linkages between political issues, this might be a good strategy not to change one's mind! A strategic politician should choose an ideological position that he/she believes will place, as frequently as possible over the years, his/her imputed platform at the center of gravity of the voters' ideal points. Different ideological positions come from different tastes, but also from different priors on the distribution of historical shocks (the so-called 'sens de l'histoire') and therefore of linkages between issues. Maintaining that ideological position over time is essential for their credibility, and thus an important asset for future political successes. Now, what is a good strategy in the short-run? A strategy here is neither the choice of an ideological position (basically chosen once for all at the beginning of one's career -although there might be more than one beginning...) nor the choice of a political platform (automatically imputed by the ideological position), but an action that impacts the linkages between issues in a way that places the candidate at the center of gravity of the projected set of voters' ideal points. This is left for future work.

Appendix

P r o o fo fL e m m a2 :

The numbers A m,q,j of (q, p)-paths such that max{q -s,p+s} <j remain to be computed. These computations are based on the 'reßection principle' due to Désiré André (see, e.g., [START_REF] Feller | An introduction to probability theory and its applications[END_REF], Chapter III). Let A =( α, a) and B =( β, b) be points in the positive orthant: β>α≥ 0, a>0, b>0.B yr e ßection of A on the x-axis is meant the point A 0 =(α, -a) (see Figure 8).

Reßection principle:

The number of paths from A to B which touch or cross the x-axis equals the number of all paths from A 0 to B.

to (m, c).L e t a and b be positive, and -b<c<a .B y t h e r e ßection principle, the number of paths from (0, 0) to (m, c) which touch or cross y = a is equal to the number of paths from (0, 2a) (the reßection of O on the axis y = a)t o(m, c), i.e., N(m, 2a -c).

By the same argument, the number from (0, 0) to (m, c) which touch or cross y = -b is

Now, by a double application of the reßection principle, a path from (0, 0) to (m, c) which touch or cross y = a and then y = -b (calledan'(ab)' path in the sequel) can be Þrst associated to a path from (0, 2a) to (m, c), itself associated to a path from (0, -2a -2b) to (m, c); hence N(m, c +2(a + b)) of 'ab' paths. A triple application allows through the same line of argument to denumerate the paths which touch or cross y = a,t h e ny = -b, then y = a again ('(ab)a' paths); their number is N(m, 2a -c +2(a + b)). An extension of this method gives:

• N(m, c +2k(a + b)) for the number of paths which touch or cross y = a and then y = -bktimes in a row ('k(ab)'p a t h s ) ;

• N(m, 2a -c +2k(a + b)) for the number of paths which touch or cross y = a and then y = -bktimes in a row and then y = a again ('k(ab)a'p a t h s ) ;

• N(m, c -2k(a + b)) for the number of paths which touch or cross y = -b and then y = aktimes in a row ('k(ba)'p a t h s ) ;

• N(m, 2a -c -2(k +1)(a + b)) for the number of paths which touch or cross y = -b and then y = aktimes in a row and then y = -b again ('k(ba)b'p a t h s ) .

Our aim is to compute the number of paths from (0, 0) to (m, c) which touch or cross neither y = a nor y = -b.T h i sc o m e sÞrst by exclusion of paths which touch or cross y = a and paths which touch or cross y = -b.B u tt h u s' (ab)' and'(ba)' paths are excluded twice and must be reincluded once. But then '(ab)a' and'(ba)b' are excluded twice, then reincluded twice, and therefore must be re-excluded once... This standard application of the inclusion-exclusion principle (see Comtet (1974) for the concerned number, which can be rewritten: (over all integers k from -∞ to +∞, but only Þnitely many non-zero terms). The formula 1 obtains readily by substitution of the right parameters. ¥


Catherine Mathieu

Henri Sterdyniak

International relocation and deindustrialisation: some French perspectives

Introduction

General public debates on international relocation in the EU-15 generally focus on the process in its narrow sense: a company closing a plant in one of EU-15 countries while simultaneously opening a plant in a low-wage emerging country where it will produce goods at lower costs to be offered on the same market as previously. We will consider here relocation processes in a broader sense: production relocation from 'Northern' countries to low-wage economies, initiated by a 'Northern' company choosing to locate or to outsource part of or all its production in a Southern area; a 'Northern' company leaving its 'Northern' supplier of intermediate consumption purchases for a supplier in a southern country; a 'Northern' company of the trade sector leaving a supplier in a 'Northern country' for a supplier in a southern country; a 'Southern' company winning market shares over 'Northern' companies, first in its domestic market, then in third countries and finally in 'Northern' countries themselves. International relocation processes are difficult both to measure and to stop.

International relocation or in other words 'delocalisation' will be addressed in this paper from a French policy perspective. The issue was already very topical 12 years ago and focused on how to tackle relocation of traditional production industries, mainly textile (as well as shoe and leather) industries, from France to Asian newly emerging economies. At that time a number of politicians raised the issue of introducing fiscal measures to stop delocalisation (see [START_REF] Arthuis | Rapport d'information sur les délocalisations hors du territoire national des activités industrielles et de service, fait au nom de la Commission des finances, du contrôle budgétaire et des comptes économiques de la nation[END_REF][START_REF] Lauré | les délocalisations : enjeux et stratégies des pays développés[END_REF]. Among the ideas were: lowering taxation on low-skilled labour, introducing a social VAT, i.e. compensatory duties (droits compensateurs) on imports from developing countries (see [START_REF] Lauré | les délocalisations : enjeux et stratégies des pays développés[END_REF], for a critical view, Mathieu andSterdyniak, 1994).

International relocation and the domestic job losses involved became again highly topical in 2004, as elsewhere in Europe, with several reports being released: [START_REF] Fontagné | Désindustrialisation -Délocalisations[END_REF], [START_REF] Grignon | Rapport d'information sur la délocalisation des industries de main d'oeuvre[END_REF] and [START_REF] Roustan | Rapport d'information sur la désindustrialisation du territoire', fait au nom de la[END_REF]. The 2005 budget contains several measures aiming at promoting the attractiveness of specific French areas. It is worth noting that over the last 10 years, the policy debate has moved from supporting low-skilled employment in traditional industries to supporting high-tech industries and possibly employment in 'sheltered' services (see [START_REF] Letournel | L'émergence de l'Asie en développement menace-telle l'emploi en France ?[END_REF]. Some French economists, like [START_REF] Messerlin | La France profite mal de la mondialisation[END_REF], think French labour, goods and services markets are too regulated and should simply be deregulated, with no other measures needed. But a majority of French economists think measures should be introduced either to support endangered industries or to favour the emergence of well-positioned industries.

Section 2 discusses the potential impacts of relocation from Northern to Southern countries and policy answers. Section 3 addresses global strategies aiming at promoting growth and competitiveness in line with the Lisbon Strategy based on research and innovation. Section 4 addresses proposals aiming at the promotion of top-priority sectors. Section 5 considers measures in favour of unskilled workers. Section 6 describes and discusses specific measures taken by the French government to stop relocation. Section 7 concludes.

Potential impacts of international relocation and policy answers

The emergence of low-wage countries in international trade has seven main impacts on Northern countries: 1) Imports from the 'South' rapidly increase their market share in Northern domestic markets, especially for goods requiring a high proportion of unskilled labour.

2) Southern countries rapidly increase their market share in Northern foreign markets, especially for goods requiring a high proportion of unskilled labour.

3) Southern countries increase their imports of capital intensive or skilled-labour intensive products made in Northern countries.

4) The low prices of Southern produced goods reduce price levels in Northern countries and thereby increase consumers' purchasing power in these countries.

5) The weakening of labour demand for unskilled workers and the threats of delocalisation exert a downward pressure on unskilled workers' wages. Depending on the country, these wages will either drop (which will raise inequalities and dampen consumption) or be maintained (which will contribute to raise unemployment). Depending on the country, taxation or social measures will be introduced, or not, to reduce the rise in inequalities.

6) Northern companies will try to sell on rapidly growing Southern markets and will invest less in the North. This explains part of the observed stagnation of company investments in some European countries.

7) Capital flows towards Southern countries and their exports' income lead to an appreciation of their exchange rate, except if there is a significant rise in Southern imports of capital goods produced in the North. This exchange rate appreciation limits the size of delocalisation. However Southern countries can avoid this to occur through accumulating foreign currency reserves, like China does.

The impact of the emergence of Southern countries on employment in the North must not only be considered from a global point of view in a long run perspective, but also in a medium term perspective (what are the adjustment costs for the productive system?) and in terms of employment structure: the cuts in unskilled jobs add to those resulting from technical progress. Even if they are smaller in numbers, they are more sudden and more concentrated in some specific sectors. Trading-off unskilled for high-skilled jobs is not necessarily good news for employment in the countries where unskilled unemployment is already high.

Should European countries implement policy measures to fight deindustrialisation, i.e. the decline of production industries both as a share of total output and in terms of employment levels? Three views are opposed. The free-market view states that deindustrialisation is a natural phenomenon for European economies. Production has already moved from agriculture to industry in modern economies. These countries should now rely on the one hand on high skilled jobs in research and development, innovation and management and on the other hand on employment in services. This is a sine qua non condition for raising productivity. No public policy should be implemented to stop the international relocation process. On the contrary, relocation should be encouraged through opening frontiers and allowing for free competition. The world market will set the level of wages of high skilled workers. Services sectors will remain widely sheltered from international competition, and hence the wage level of 'sheltered' workers in services will be set at a domestic level and will rise in line with high skilled workers employment. Wages should be set both in services sectors and for high skilled workers through a market process. There is no guarantee that the market equilibrium will be socially acceptable and that income inequalities will not be excessive. But there is no alternative (TINA, as Margaret Thatcher said).

'Reformists' think that there is a significant risk for Europe to be badly positioned in world competition. First, research and development as well as education spending are too low.

Second, there are a number of rigidities in terms of labour legislation, wage bargaining institutions and laws restricting dismissal. Second, the welfare state weighs strongly on sectors facing fierce competition from abroad. The European social model needs to be reformed in order to become more competitive. Risk taking firms and workers must be supported. This implies supporting well-performing and innovating industries. The growth of these sectors will compensate for the employment losses in unskilled industries. How should this support take place? Two views can be found. In one view, the general environment of companies needs to be improved, taxes need to be reduced, labour legislation must be reformed, incentives need to be introduced (through research tax credits for instance). In another view, support should be given to specific companies -helping the emergence of and supporting European champions -and to specific innovative sectors, like biotechnologies. To a certain extent this view is similar with the free-market one: more jobs in technology industries will increase the number of jobs available for 'medium' skilled workers, which in turn will make more jobs available for unskilled workers. Here also, the final degree of income inequalities is unknown. Last, the 'archaic' view asks for specific support to be implemented in favour of unskilled labour as the functioning of economies per se raises social inequalities and makes unskilled workers unemployed, which cannot be accepted. Here also two views can be found. In one view, industrial sectors and national companies being in difficulty should be supported, a certain level of protectionism should be maintained, unfair competition should be fought, areas where employment is strongly affected should be subsidised. In another view, unskilled workers only should be subsidised, without interfering with production choices. These transfers may be found counterproductive because they benefit the losers and must be paid by the winners. Moreover, globalisation makes it difficult to raise taxes on high income people and healthy companies with a view to help people on lowest incomes and companies in difficulty. Individuals and companies subject to taxation may decide to leave for a country where taxation is lower.

Promoting growth and competitiveness

The Lisbon Strategy and the Sapir Report ( 2003) claim for a successful 'way out' for Europe: making the European economy the most innovative of the world. The aim is to compete with the US but not with China; to export to rapidly growing markets rather than in slowly growing (or not growing at all) European markets and to specialise in high value added sectors.

More European and national resources should be devoted to higher research and development spending and to higher education, by creating 'excellence poles' in Europe. Structural reforms should take place to increase mobility and risk-taking incentives.

Specific taxation rules should be introduced for companies operating at a European level, in order to allow them to take advantage of economies of scale; markets should be more rapidly deregulated and open.

This strategy raises six issues:

1) The State should create a business-friendly environment rather than help specific companies. DG Competition prevents Member States from implementing industrial policies.

But, in many cases, supporting specific companies or research areas could prove more efficient to raise growth in Europe.

2) Is it possible to promote scientific education and research in a context where financial profitability is a major objective?

3) Should European economies accept the death of traditional industries that are the more labour-intensive, to concentrate on highly profitable activities involving a very limited number of very high skilled jobs? 4) Should European countries encourage their companies to develop mainly on external markets, at the risk of having the companies relocating their production abroad?

5) This policy leaves aside traditional sectors in difficulty. It gives an incentive to firms for leaving these sectors. Will the development of high-technology sectors be sufficient to bring back a satisfactory level of employment?

6) What kind of jobs can we offers to the manual-oriented young men, if no workers job are available in Europe ? Some aspects of the Lisbon strategy can be found in French reports, suggesting for instance to support R&D and innovation though taxation measures [START_REF] Grignon | Rapport d'information sur la délocalisation des industries de main d'oeuvre[END_REF].

According to [START_REF] Fontagné | Désindustrialisation -Délocalisations[END_REF], the importance of scientific and technical issues should be restored in the European culture and education. The European enterprise should be promoted. Funding innovation should be made easier. The Report lists 10 priorities. 7 of them are part of this strategy: developing technology universities (Universités de technologie); adopting a small business act; unifying the European innovation equity stock exchanges; increasing the role of the European Science Foundation; introducing a surveillance of the relocation of headquarters; introducing rules applying to 'European companies'; promoting mergers between European Member States companies.

Other suggested measures aim at increasing European attractiveness [START_REF] Grignon | Rapport d'information sur la délocalisation des industries de main d'oeuvre[END_REF]:

-Improving the legislation of productive activities.

-Making easier new company launch and company ownership transfers (by inheritance or sale).

-Strengthening competitive advantages (quality of infrastructure, low energy prices)

-Promoting employability (higher education, professional training); deregulating labour markets; increasing flexibility and mobility -Harmonising company taxation, public aid and social legislation in Europe.

-Promoting norms, patents and European labels.

-Promoting the harmonisation of social and environmental standards at world level and rewarding the fulfilment of these norms at an international level with the help of the consumers.

Promoting high-priority sectors, implementing industrial policies again

Among recent reports, [START_REF] Fontagné | Désindustrialisation -Délocalisations[END_REF] suggest that competition policies and industrial policies in Europe should be rebalanced. Specific (i.e. restricted to a small group of European companies), industrial co-operation should be promoted, 'European champions' should be promoted albeit increasing the SMEs' capacity to adapt.

An aggressive strategy would consist in helping firms to produce higher-technology products for rapidly growing markets. Industry and innovation should become national priorities. Large programmes in top-priority sectors should be favoured.

Three priorities are listed: giving a 'brand name' to competitiveness poles; defining priority sectors at the European Council's level; setting a group of 3 European industrial managers to suggest initiatives supporting these sectors.

The Roustan Report ( 2004) suggests that State implements again a strategy to increase research funding and piloting innovations. The report recommends the issue to be tackled at the European level: less competition versus more industrial policies, research subsidies, tax harmonisation and rebuilding local area attractiveness through local public action. The Report also underlines that European enlargement may provide higher growth.

The Grignon report ( 2004) calls for a 'new-colbertism', in other words for a European industrial policy: implementing a sector approach by launching large innovative projects in rapidly growing sectors; defining the sectors to be promoted; improving parallel conducts of competition and industrial policies. It suggests using the EU trade policy to help the industry. It also recommends that the State supervises production processes (filières de production). [START_REF] Beffa | Pour une nouvelle politique industrielle[END_REF] suggests the implementation of industrial policies again. The strategy would consist in launching programmes promoting industrial innovation ('PMII': programmes mobilisateurs pour l'innovation industrielle). Companies, experts and public servants would define innovative industrial sectors (like hydrogen production chain, biofuels, high-definition secured networks); big companies would receive subsidies if they develop research in these fields (50% of research expenses being covered by public funding) and would have to develop their network with small or medium size companies. So France would have 'national champions' in high-tech sectors (France currently has a too strong specialisation in nuclear, space and military sectors). The report leaves a question open: at what level should this policy be settled? Should it be at the French level (with a notification to European institutions) at the European level or within restricted cooperation between a small number of countries (like France with Germany)?

Competitiveness Poles

The development of 'competitiveness poles' (pôles de compétitivité) designed to help activity in given geographical areas is also a key measure of Datar's strategies, as is the support for the building of firms networks (clusters). Competitiveness poles should be given a 'brand name'; they should be advertised and benefit from lower taxation. Dedicated risk-capital should be developed, transport and telecommunication infrastructure should be provided. The link between national and European policies remains difficult to tackle.

Both Grignon and Roustan Reports suggest a strengthening of the role of local administrations in supporting local firms and employment.

Supporting unskilled employment

Two strategies are often suggested to compensate for the negative effect of foreign relocation on employment. The first one would be to increase the level of employment in sheltered services, like construction, tourism, hotels and restaurants sectors (see [START_REF] Letournel | L'émergence de l'Asie en développement menace-telle l'emploi en France ?[END_REF][START_REF] Grignon | Rapport d'information sur la délocalisation des industries de main d'oeuvre[END_REF] or to increase the size of the services to households' sector (see [START_REF] Debonneuil | Productivité et emploi dans le tertiaire[END_REF]. This would be done through tax incentives (allowing individual employers to benefit from lower social contributions as is already the case for companies), and through restructuring measures (making the services sectors more industrial and professional). The French government has introduced such measures in February 2005. But this strategy may be difficult to implement due to social habits (getting used to hiring employees is something that needs to be learnt by households) and because of the reluctance of young people to apply for such jobs. This strategy implies a rise in income inequalities.

The second strategy consists in cutting unskilled labour costs. Since 1985, companies get a rebate on their contributions on low wages, up to a maximum of 26 percentage points of employers' social contributions (in 2005, on a total of 40 percentage points) for workers at minimum wage levels. This reduces by 18.6 % the cost of the minimum wage for firms.

Besides, low-wage workers benefit from a refundable tax credit of 4.6 % of their wage earnings in order to increase the gap between the minimum wage and the minimum income entitled to adults unemployed and not entitled to unemployment benefits (RMI: revenu minimum d'insertion). This is a 'making work pay' measure. The social contributions' rebate measure gives companies an incentive to hire unskilled workers (especially in services sectors) and can also help keep companies hiring a high level of low skilled labour stay in France. These measures cost 17.5 billions of euros by year in the French budget (1% of GDP).

The social contribution's rebate is estimated to have increased the number of jobs by 400,000. These rebates have the advantage of breaking the link between unskilled labour costs and the living standard of unskilled people. But they are costly for the Social Security budget on which they are funded and they lead to the creation of specific low-wage jobs with no career prospects.

The Grignon Report ( 2004) recommends a measure often suggested in France in past years, which would consist in abolishing (or significantly cutting) employers' social contributions and simultaneously increasing VAT. This reform aims at making the social security system being funded by consumption expenditures (and especially by imports) rather than by wages. The reform would entail competitiveness gains and cut wage costs in labour intensive sectors hence raising employment. In our opinion, the reform would be totally ineffective because the VAT weighs on wages (like employers' social contributions) and not on capital goods (see [START_REF] Sterdyniak | Pour une réforme du financement de la Sécurité sociale[END_REF]. Competitiveness gains would arise only if the rise in the prices of imported consumer goods resulting from higher VAT had no impact on wages, meaning that consumers would see their purchasing power reduced.

The French fiscal measures

The French government has introduced a significant number of measures, more or less successful, to prevent foreign relocation. These measures aim at providing an incentive for companies settling or to staying in France. They compensate partly for the high level of taxation in France. At the same time, they add to tax competition.

Since 2003, bonuses paid to foreign 'high skilled' foreign employees working in France have been exempted from income tax. This measure aims at attracting headquarters and research departments of big companies in France. The risk is that all countries introduce similar measures and raise tax competition at the expense of the distributional goals of income taxation. 'High skilled employees' will be less taxed if they work abroad than if they work in their country. These measures will be increased in 2005. In that field, European harmonisation would be useful.

A research tax credit has been introduced in 2004 allowing companies to benefit from a subsidy equal to 45% of the annual increase in their research and development expenses plus 5% of the amount spent. But, due to the European legislation, the tax credit has a relatively low ceiling. It would be useful to allow States to subsidize more significantly private research.

The 2005 budget has introduced several measures against the relocation process. First, companies relocating in France their existing production in non-EU countries benefit from a tax credit. Due to the European rules, this credit has a maximum of euros 100,000 in a threeyear period by company (50% of labour costs the first year, then 40%, 30%, etc.)

Second, the French budget contains a number of measures designed to improve the attractiveness of French areas ('competitiveness poles'). The idea is to give fiscal incentives to companies investing in high technology production in specific geographic areas. These poles must bring together Universities or training centres, public or private research centres, small and big firms, working together in innovating or specific production processes. These networks are expected to stop foreign relocation of these specific production activities.

Last, companies operating in areas said to be 'exposed to foreign relocation' (areas with a high level of unemployment rate and a high share of industrial employment) are entitled to a tax credit of euros 1000 per worker.

In all three cases, the measures have a limited amount. The costs expected in the 2005 Budget are respectively of 10, 50 and 330 million euros in 2005. Until now, these strategies are purely national and European rules are there to prevent their expansion. Is this a sustainable position for industrial policies in Europe?

France has suggested increased tax harmonisation in Europe, which would raise difficult issues, because Member States wish to keep domestic fiscal autonomy and also because low taxation rates are sometimes a convenient way to compensate for insufficient infrastructure, low levels of economic development or company subsidies not being allowed. Making sure that the profit tax is based on the source principle, and not on the origin principle, would avoid purely tax-motivated relocation. More, the fight against tax heavens should be strengthened at a European or world levels. Companies and financial institutions should not be allowed to have branches in or to transfer funds to these countries.

Company taxation amounted to 6.6% of GDP in France in 2002, ranking France 2 nd in the EU-15, behind Luxemburg, against 6.4% in Italy, 5.2% in the UK, 3.5% in the US, 3% en Spain, 0.9% in Germany (where the figure was exceptionally low due to the tax reform). The current profit tax rate is 34.3% in France, at the level of the European average. It may be necessary to cut it down to 33.3%, but the effort to be done is limited (1.7 billion euros). The problem is not the level of profit taxation itself, but the fact that the wage tax (taxe sur les salaries), the tax on real estate (taxe foncière) and the professional tax (taxe professionnelle) add to profit taxation.

The professional tax is currently based on tangible assets (real estate and equipment) and particularly affects production industries. Reforming this tax has been debated for a long time.

Recently it has been suggested for instance in the Roustan Report (2004) and the Grignon Report (2004). The government has promised to reform the tax in 2006, without affecting local administration funding and without increasing households' taxation. The new tax could possibly be based on value added (at the expense of trade sectors and financial institutions).

The French government refuses to undertake a global competitiveness strategy that would consist in cutting directly wages or reducing company taxation together with lower public and social expenditures. In general, the search for competitiveness gains may be worthless, since the rise in competitiveness will be cancelled between European countries while European demand will be lower. But the recent example of Germany cutting domestic labour costs to win competitiveness and hence depressing domestic demand illustrates that an isolated and non co-operative strategy can always be a temptation for a country.

Conclusion

The French government introduced several years ago cuts in employers' social contributions on unskilled work. It has recently opted for helping high-tech sectors and specific French areas, the 'competitiveness poles', rather than implementing additional measures in favour of unskilled labour in traditional industries (textile, etc). It remains to be seen how many jobs this new policy will generate. Besides, the French government is in favour of promoting the emergence of 'European champions'.

The interconnection between national and European policies remains an issue to be tackled. Three answers to deindustrialisation may be considered in Europe, mirroring different political and social issues. First, some governments could choose to implement domestic measures, but would be stopped by European authorities, in the name of free competition rules. Second, some governments could implement measures without the European authorities interfering, but this would not be optimal: is it reasonable or advisable that a Member State introduces alone domestic taxation measures strengthening tax competition in Europe? The other risk is that national measures would be limited by the fear of European reactions. The best strategy would be that European countries adopt a common strategy at the European level. European authorities could reduce the importance they give to competition policy, initiate a reflection on the future of productive activities and employment in Europe and promote a European industrial policy as part of the Lisbon strategy.