Saïd Hanchane
email: said.hanchane@univmed.fr

Tarek Mostafa
email: tarek.mostafa@univmed.fr

School Choice: Income, Peer effect and the formation of Inequalities

Keywords: JEL Classification: I20, I21, H52 Education market, majority voting equilibrium, peer group effect, pricing discrimination, educational opportunity

In this paper, we analyze the equilibrium on the market for schooling where both public and private schools coexist and where individuals are differentiated by income and ability. We introduce a non linear in means model of peer effect by shedding the light on the fact that school quality is not solely dependent on mean ability but also on the dispersion of abilities. We study the distribution of students across sectors while examining the conditions for the existence of a majority voting equilibrium in the context of non single peaked preferences. Finally, we examine the presence of a hierarchy of school qualities. In the paper we shed the light on equity problems related to the access to educational quality while analyzing the functioning of the educational system.

Reducing inequalities both in the access to education and in the outcomes of education has become a major driving force of educational reforms in most developed countries.

Unequal access to education, unequal outcomes, disparities of earnings, disparities of educational qualities between communities, and the quality of public education constituted an important subject of debate in the recent French presidential elections.

Recent and ongoing international surveys by the OECD -PISA and IALS -showed that the outcomes of education in major western countries tend to have large levels of inequalities. For instance in PISA2000, Belgium, Germany, the USA, The UK and France turned out to be the most unequal states in term of the dispersion of students' achievements over reading, numeracy and basic science tests1 . The ongoing debate over these issues provides the motivation for our article as well as to the growing literature on school choice, college choice and urban economics.

The aim of our paper is to study the formation of inequalities in the access to education and in educational outcomes through an analysis of stratification on the market for schooling when both private and public education alternatives coexist. We analyze stratification by contrasting individual behavior in term of voting over tax rates and private schools profit maximizing behavior. We shed the light on the fact that school quality and students' achievements depend not only on mean ability in the school but also on the distribution of students thus on the dispersion of ability2 . One reads for instance in a mimeo of Caroline Hoxby and Gretchen Weingarth (Hoxby, Weingarth, 2005 pp. 2 and 30).

"The linear-in-means model assumes that each student has the same effect on each other student (a homogeneous treatment effect). It also assumes that a single student whose achievement raises a class's mean achievement by two points has precisely the same effect as several students whose combined achievement raises the class's mean by two points (that is, all effects operate through one moment: the mean of peers)… Moreover, most applications of peer effects-school desegregation, school choice, college choice, urban economics-need to have non-linear peer effects to generate results that are interesting and that mimic the facts." pp.30 "Our finding support for the Boutique and Focus models suggests that schools, colleges, and workplaces should be wary of creating peer groups in which some people are isolated. However, they should also avoid creating critical mass around a certain type of person. Some focus is good".

The philosophy behind the introduction of peer ability dispersion in the quality function can be understood in two ways3 . The first is related to the impact of the dispersion on students' achievements. Schools with low ability dispersions tend to be more appreciated by students. When teaching, a teacher normally targets the mean ability students; students with ability higher than the mean understand more easily than those with mean ability and conversely for students with ability lower than the mean. In a highly dispersed class we have two groups, very high ability and very low ability students; when aiming at the mean ability holders, the high dispersion will negatively affect the transmission of knowledge since the teacher will no longer be able to cater to the needs of each group4 .

The second is understood through the signaling effect of education when information about an individual's ability is incomplete5 ; belonging to a school with very high dispersion of ability reduces the quality of the signal that this individual generates, since outsiders with limited information on ability will no longer be able to identify his type.

Thus, we can consider that school quality is decreasing in the variance of abilities.

Our model constitutes an integrated analysis of school choice, stratification, and the provision of education by public and private sectors. It coincides and supports the empirical findings of Hoxby and Weingarth; in fact, since mean ability and the variance of abilities will be working in opposite directions, schools tend to have relatively homogenous student composition, in other terms students will not be isolated and a critical mass will not be built around a particular type. This was found by Hoxby and Weingarth to be an efficient form of mixing.

The analysis of public versus private school choice started with Yoram Barzel (1973); in a critical study of an article of Robin Barlow (1970) he introduced what is known as the "Ends against the middle phenomenon" for which we have a coalition of the rich and the poor against the middle class. The study of Barzel constituted an introduction to a more formalized theoretical approach of school choice, starting with Joseph Stiglitz in (1974), [START_REF] Clotfelter | School desegregation, Tipping and private school enrollment[END_REF] and [START_REF] Sonstelie | Public school quality and private school enrollments[END_REF].

Stiglitz analyzed the demand for schooling under different institutional arrangements. In his study education is a complex good; it is consumption and an investment good as well as a screening instrument. He notes that preferences are not single peaked when public and private alternatives coexist. All these characteristics may prevent the existence of a majority voting equilibrium. At this level, two categories of articles are relevant to our analysis.

The first concerns the article written by [START_REF] Fernandez | Income distribution, communities, and the quality of public education[END_REF] The second concerns the series of articles written by Epple et al (1998, and2006), two major characteristics are the linearity in mean ability and the single crossing assumption.

In 1998, Epple and Romano considered an economy where individuals are differentiated by income and ability; the private sector is active and private schools maximize their profit, this strategic behavior combined to a single crossing assumption leads to the existence of a hierarchy of school qualities. In their article quality is only dependent on peer group effects. In 2006 the authors added an income diversity measure in order to estimate a model for higher education. The objective of these articles was the analysis of stratification between schools. Some related literature may include [START_REF] Calabrese | Local public good provision: Voting, peer effects, and mobility[END_REF], [START_REF] Epple | Equilibrium and local redistribution in an urban economy when households differ in both preferences and income[END_REF], Epple, Filimon andRomer (1984-1993).

In this paper, we carry on the work made by [START_REF] Glomm | Opting out of publicly provided services: A majority voting Equilibrium[END_REF], [START_REF] Epple | Competition between public and private schools, Vouchers, and peer group effects[END_REF], by combining the study of stratification and the provision of public services when private alternatives exist. Our model differs in a number of ways, in comparison to the study of Glomm and Ravikumar; we consider an economy where individuals are differentiated by both income and ability. We present the necessary conditions for the existence of a majority voting equilibrium in the context of non single peaked preferences. We analyze the distribution of students across sectors and schools, while presenting two equity related problems. First, private schools will operate to the detriment of public schools by siphoning high ability high income students. Second, inequalities of educational opportunity exist when a fraction of the population does not have an equal access to educational achievement, given its allocation of income and ability.

In our model, similarly to Epple and Romano, the private sector is active. Private schools maximize profits under a quality constraint. Unlike them, we consider a school quality that is dependent on peer effect and the dispersion of peer ability in a school and we do not impose a single crossing assumption. Individual utility will be maximized through a utility taking assumption. Peer group effect and the dispersion of peer ability are built into our model; these factors are the corner stone of the discriminating pricing strategy.

The inclusion of peer group effect and the dispersion effect is consistent with the literature on this subject; [START_REF] Summers | Do schools make a difference?[END_REF], [START_REF] Henderson | Peer group effects in educational production functions[END_REF], [START_REF] Sorensen | Effects of ability grouping on growth in academic achievement[END_REF], [START_REF] Kulik | Effects of ability groupings on secondary school students: A Meta analysis of evaluation findings[END_REF], [START_REF] Benabou | Heterogeneity, Stratification, and Growth: Macroeconomic implications of community structure and school finance[END_REF] and [START_REF] Hoxby | Taking race out of the equation: School reassignment and the structure of peer effect[END_REF] have provided evidence that such an effect exists.

Our equilibrium will be characterized by the existence of an indifference point between the two sectors determined by a couple of income and ability. The distribution of students across sectors and schools, given the discriminating pricing strategy will happen to the detriment of public schools. Individuals with high ability and income levels will be in the private sector; individuals with high ability and low income levels will be in the private sector and will receive financial aides; individuals with high income and low ability levels will be in the private sector and will pay tuition premia; individuals with low ability and income levels will be in the public sector. Individual behavior and preferences will determine the existence conditions of a majority voting equilibrium over tax rates.

Section I presents the model, equilibrium conditions follow in section II, section III examines the distribution of students across sectors, the conditions for a majority voting equilibrium and the existence of a hierarchy of school qualities, section IV concludes. An appendix contains mathematical details.

I-The model

In our model, we consider an economy populated by a continuum of agents; the number of types is largely superior to that of schools, we do not make a difference between households and students, each household has one student and decisions are made by the household. Students have identical preferences over consumption; they are differentiated by their income and ability levels, an individual i (with i = 1, 2, 3, 4, …, P) has an income i y and an ability i b , i is also used to denote a type of individuals. Income and ability are distributed in the population according to

= ∞ → q c U c
, in other terms individuals always choose the highest school quality they can afford given their disposable income.

School quality is determined by school expenditure per pupil E, by peer group effect j θ , and by the dispersion of peer ability in a school 2 j σ . Quality is increasing in the first two arguments and decreasing in the last. j is an index denoting a particular school, (with j = 1, 2, 3, …, j). Educational achievement is given by ) , ( q b a a =

; a is continuous and increasing function of both arguments. Thus, the access to a higher quality school is translated through higher achievements. In term of inequalities, unequal access to education generates unequal outcomes.

Education production cost depends only on the number of students enrolled in a particular school, it is denoted

C (k) = V (k) + F with 0 > ′ V and 0 < ′ ′ V or 0 > ′ ′ V 6 .
k is the number of students in this school; F is a constant reflecting a fixed cost when no student are enrolled. The existence of economies of scale in the production of educational services is realistic; it will prevent the existence of an infinite number of private schools.

In our model we suppose that the number of student types is largely superior to the number of schools.

Each private school will retain a fraction of the student population that has chosen the private sector, by applying tuition and admission policy permitting the maximization of each individual's utility through a utility taking assumption. The pricing strategies will be characterized by a meritocratic system.

In the private sector, students with ability higher than the mean receive a tuition reduction, conversely, students with ability lower than the mean pay a tuition premia.

Similarly, students with ability close to the mean or (an ability pushing the variance downward receives tuition reductions, conversely, students with ability far from the mean (an ability pushing the mean upward) pay tuition premia. This discriminating pricing strategy is justified in the context of internalizing the externality generated by the position of a student's ability in the distribution of abilities in the school.

It is important to note that there will be no supplementary educational investments in students with low ability levels, as a consequence school expenditure per student will be the same for all students in a particular school.

Individuals in the private sector can always choose public sector schools, conversely individuals obliged to enroll in the public sector given their allocation of income and ability do not have the opportunity to enroll in private schools.

The public Sector: .

Public
P: The number of students in the economy.

u is an index denoting a particular public school.

The quality of education in the public sector for a school j is written as follows: Individual utility in the public sector is of the following form: ] ), 1 ( [

      = 2 , ,
ju i iu q t y U U - =
Indirect utility can be written as:

( ) i ju ju i iu iu b N Y y t W W U , , , , , , 2 σ θ = =
In the public sector, utility maximization is trivial. A student will allocate his disposable income (after the deduction of taxes dedicated to public school finance) to private consumption.

The private sector:

In the private sector, schools are tuition financed. Parents pay a positive tuition equal to ijr p . Private school resources can be determined by ∑ ijr p . These resources are not entirely distributed over students; per student expenditure is determined by dividing the production cost of education by the number of students in the school. The difference between resources and expenditure represent private school's profit. This positive profit will encourage new schools to enter the market until it vanishes (in a competitive market).

Expenditure per student is given by:

jr jr jr jr k F k V k k C + = ) ( ) (
r is an index denoting a particular private school. The quality of education in the private sector for a school j is written as follows:

        = 2 , , ) ( jr jr jr jr jr k k C q q σ θ
It should be noted that private school quality is positively correlated with individual income; when income increases, the private quality that can be purchased is much higher.

When an individual compares quality in both sectors; his ability level has no influence because it is the same in all schools.

Private sector quality is heterogeneous between schools in all arguments. Private school expenditure is homogenous for students attending the same school, and heterogeneous between schools. Peer group effect and dispersion effect are heterogeneous between schools.

Individual utility in the private sector is of the following form:

] , ) 1 ( [ jr ijr i ir q p t y U U - - =
Individuals are going to choose an optimal combination between consumption goods and school quality in order to maximize their utility.

Max ] , ) 1 ( [ jr ijr i ir q p t y U U - - = Subject to ( ) i i ijr c t y p - - = 1
with c as consumption.

Indirect utility can be written as:

( ) i jr jr i ir ir b y t W W U , , , , 2 * σ θ = =
, it represents the only solution to this optimization.

Private schools behave strategically on the market; they maximize profits as utility takers by conditioning admission and tuition according to ability levels. The utility obtained in a school should be at least equal to that obtained elsewhere. This behavior is similar to that of private clubs with non anonymous crowding, see [START_REF] Scotchmer | Competitive equilibrium and the core in club economics with non anonymous crowding[END_REF]. It should be noted that individual ability is perfectly observable by private schools.

Private schools have to maximize the following function: Constraint (1c) determines the umber of students enrolled in a particular private school.

[ ] F k V dbdy y b f y b y b p Max j s jr ijr jr - - = ∫∫ ) ( ) , ( ) , ( ) , ( α π (1) Subject to: ( ) [ ] N P y b jr - ∈ , 0 , α ( ) y b, ∀ (1a) ( ) [ ] ( ) [ ] jr ijr i jr ijr i r q y b p t y MaxU q y b p t y U U ), , ( 1 ), , ( 1 - - ≥ - - = ( ) y b, ∀ (1b)
Constraint (1d) determines peer group effect given by mean ability. Constraint (1e)

determines dispersion effect given by the variance of peer ability. Constraint (1b)

represents the utility taking assumption.

II-Equilibrium conditions

Three types of conditions are necessary for the existence of equilibrium:

A-The conditions for private sector equilibrium are:

• Individual utility maximization:

[ ] • New entries are expected so long as private schools expect to make a profit; the private sector equilibrium will be defined by.

jr ijr i q p t y MaxU y b U , ) 1
∑ ∑ = - + = + P N P N k k jr ju ) ( ( ) y b, ∀
This condition represents market clearance; public schools are preferred to no schooling.

III-Theoretical results:

A-Private schools' profit maximization results.

The optimal function combining tuition and individual utility is given by:

) , , , ( ) , ( * ] , ) 1 ( [ * i jr i ir jr ijr i ir b y t W y b U q p t y U θ = = - - ( ) y b, ∀ (2)
The optimal level of tuition is:

[ ] N y b ju , 0 ) , ( ∈ α (3) [ ] N P y b jr - ∈ , 0 ) , ( α ] ) [( ) ( ) ( 2 2 * jr jr jr jr jr jr ijr b b k V p σ θ µ θ µ - - ′ + - + ′ = ( ) y b, ∀
With jr µ and jr µ′ are the Lagrangian multipliers 8 .

8 Result (3) are obtained by forming a Lagrangian function to take account of constraints (1c) and ( 1d) and (1e), then by optimizing over jr α . Mathematical details are to be found in the appendix.

This price is what an individual of type (b,y) should be able to pay in order to be admitted in school j. This price is formed of two parts; fraction ) ( jr k V ′ represents a homogenous price for all students resulting from the production cost of education, fraction An interesting result is related to the difference between peer group effect and the dispersion effect; peer effect has these two properties

σ θ µ σ θ µ - - ′ = - - ′ → and 2 2 2 ] ) [( lim jr jr jr jr jr b b σ µ σ θ µ θ ′ - = - - ′ → .
In other words, for a student the benefit of having ability largely higher than the mean is high in term of peer effect and the cost is also high in term of dispersion effect; similarly, the cost of peer effect attains its In our pricing function, schools admit all students who are able to pay a price * ijr p p ≥ , however no student would accept to pay a price superior to the Pareto optimal one * ijr p unless there is no school that cater to his needs. The existence of economies of scale in the production of education ( ) 0

< ′ ′ j k V
limits the number of schools and thus some students will be obliged to find a second best solution. Otherwise, ( ) 0

> ′ ′ j k V
the market produces a large number of schools catering to each type. For the rest of the paper we consider ( ) 0

> ′ ′ j k V
to be true 10 .

It is of a great importance to note that the price of private education can be negative. For some students we may have

0 ) ( < -b jr jr θ µ , 0 ] ) [( 2 2 < - - ′ jr jr jr b σ θ µ and ] ) [( ) ( ) ( 2 2 * jr jr jr jr jr jr ijr b b k V p σ θ µ θ µ - - ′ + - + ′ =
<0. This negative price represents a financial aid to students permitting an important amelioration of school quality. This pricing strategy represents the corner stone of our meritocratic system.

Our results converge towards the empirical findings of [START_REF] Hoxby | Taking race out of the equation: School reassignment and the structure of peer effect[END_REF] schools looking for a quality peer effect tend to have a high mean ability and a low variance. However since mean ability and the variance of abilities operate in opposite directions, schools have to make a compromise between the two effects, this leads to a relative homogenization of the students' population without creating a critical mass around one type. These results are consistent with the Boutique and the focus models and diverge considerably from the linear in means models 11 .

10 Lawrence Kenny 1982 provided some evidence that economies of scale do exist in the production of education. A counterargument was provided by [START_REF] Ferris | Economies of scale, school violence and the optimal size of schools[END_REF]; it relies on the existence of external costs that increase with size and can be related to social problems present in large schools. 11 Note that in the work of Epple and Romano 1982, subsidization between low ability high income students and high ability low income students is pushed to the limits. In our model schools can no longer admit students with abilities so far from the mean since the variance has a negative effect on pricing and quality.

B-Majority voting results.

Proofs of propositions and lemmas are to be found in the appendix.

Notation remark: If ( ) Proposition 1: we suppose that 0 ) , ( lim

y b ′ ′, is better than ( ) y b , ˆ. It is denoted ( ) y b ′ ′, > (
= ∞ → q c U c 0 > ∀q .
Given [ ]

1 , 0 ∈ t , ( ) P N , 0 ∈ and + ℜ ∈ Y
. There exists a unique couple ( ) In this proposition, we ignore extreme cases. For t=0, we have N=0, nobody will choose the public sector. Conversely for t=1, we have N=P, everybody will choose the pubic sector. An important remark to take into account is that the positioning of ( ) y b , ˆ will depend on income and ability distribution and the quality of education in the public sector determined by t Y and N.

y b , ˆ with 0 ˆ> b and 0 ˆ> y such that ( ) i ju ju i iu b N Y y t W W , , , , , , 2 σ θ =
Lemma 1: as in proposition 1 we suppose that 0 ) , ( lim In this lemma we show that given our discriminating private pricing function, compensation between ability and income is possible. Sufficiently high abilities allow a student to compensate his lack of income by having a tuition reduction or a scholarship.

= ∞ → q c U c 0 > ∀q . Given [ ] 1 , 0 ∈ t , (
Conversely, sufficiently high levels of income allow a student to compensate his lack of ability by giving him a higher purchasing power in order to buy private school quality.

We can see that compensation will allow private schools to operate to the detriment of public schools by siphoning high ability high income students. Two uncommon cases shall be mentioned: individuals with abilities weakly higher than b ˆand very low income levels, and individuals with incomes weakly higher than y ˆand very low ability levels will not be admitted in the private sector since b and y are not sufficient to compensate the lack on income and ability respectively.

Lemma 2: in this lemma we establish the properties of the couple ( )

y b , ˆ
relatively to N, Y and t. (i) For ( )

P N , 0 ∈ , the couple ( ) y b , ˆ is decreasing in N. (ii) For + ℜ ∈ Y , the couple ( ) y b , ˆ is increasing in Y. (iii) For [ ] 1 , 0 ∈ t , the couple ( ) y b , ˆ
is increasing in t.

Proofs for (i) and (ii) are based on the fact that private school quality is not related to N and Y. These factors only influence school choice through public school quality.

Proof for (iii) is based on the fact that t has the same influence on consumption in both sectors. t also has an effect on public school quality without affecting that of private schools.

When we say ( ) .

Joseph [START_REF] Stiglitz | The demand for education in public and private school systems[END_REF] has presented the fact that, preferences over tax rates are not single peaked. For low tax rates, public school quality is low and individuals prefer private schools. A marginal increase of the tax rate will not increase public school quality that much, but will reduce consumption; private schools will still be preferred and utility will be reduced. An important increase of the tax rate will induce an important increase of public school quality; at ) , ( ˆi i i y b t individuals will be indifferent between the two sectors; from this point on, an increase in the tax rate will reverse tendencies and the public sector becomes preferable, utility is increasing in t. When the tax rate reaches ) , (

i i iu y b t
utility is maximal in the public sector; from this point on utility is decreasing in t. Thus we find that utilities may have two peaks, one in each sector. This is represented in figure 1.

For extreme cases where individuals have low ability and income levels; they can not choose private schools and thus they are obliged to enroll in public ones; they have single peaked preferences. Conversely, individuals with very high income and ability levels will choose private schools (See Lemma 1); they have single peaked preferences. Households with old or no children can be regarded as individuals with high ability and income levels; they have single peaked preferences and always vote for a zero tax rate, this can be seen on the preference curve A.

Assumption 1: students with higher abilities can get higher educational qualities.

This assumption is intuitive, even if in the same school a higher ability may have contradicting effects, because mean ability is higher and dispersion is higher too.

Since ( ) 0

> ′ ′ j k V
, the market produces a large number of schools catering to each type.

High ability students will always be able to find a higher quality school willing to enroll them given their endowments in ability and income and the discriminating pricing function.

Assumption 2: students with higher income levels can get higher educational qualities.

This assumption is simple to understand, higher income students have higher purchasing power. They can afford higher levels of education in the private sector.

For the same level of income; when b increases, the quality that can be obtained increases. In the public sector the quality that can be obtained increases with b until reaching its maximum in the school with the highest public quality12 . From this point on any increase in ability provokes a tipping point, students will opt out of public schools and will join the private sector. If for a sufficiently high ability, a student stays in the public sector; he will have his utility reduced. We can write: For the same level of ability; when y increases, the quality that can be purchased increases. In the public sector, since education is free, an income increase permits only a higher consumption and thus a higher utility. Since

0 ) , ( lim = ∞ → q c U c
, when income becomes sufficiently large, students opt out of the public sector and join the private one.

If for sufficiently high income, a student stays in the public sector; he will have a high consumption and a low school quality, and thus he will have his utility If the first equilibrium condition is not satisfied, then a majority voting equilibrium may not exist. If the second condition is not satisfied then ) 0 , 0 ( is a majority voting equilibrium.

] , , , , 0 [ 1 2 1 1 b y W jr jr r σ θ > ] , , , , , ), , ( [ 1

C-The existence of a hierarchy of school quality.

In their analyses, [START_REF] Epple | Competition between public and private schools, Vouchers, and peer group effects[END_REF] have demonstrated the existence of a hierarchy of school quality in the private sector. They have argued that if two schools have the same level of quality, they can increase their profit by modifying the student composition of the school by exchanging the same number of different ability and income students. Quality will increase for one and decrease for the other. This strategy will allow a profit increase following a differentiation of quality. The single crossing condition is central to the demonstration; it implies that higher ability higher income students value increased quality by more than lower ability lower income types.

Two major differences exist in our model relatively to Epple and Romano.

The first difference is related to the fact that school quality in our model is dependent on expenditure per pupil, mean ability and the dispersion of abilities. Expenditure per pupil is not an additional variable relatively to Epple and Romano since it is determined by

jr jr jr jr k F k V k k C + = ) ( ) (
, exchanging the same number of students would not have an effect on expenditure. However the inclusion of the dispersion of abilities has an important effect, it is no longer sufficient to exchange ) , ( . Two private schools with the same quality will charge the same price 13 .

School hierarchy in the public sector:

Following the equilibrium between the two sectors, the number of students enrolled in the public sector is given by N. Since public schools are free entry free of tuition schools, the income level of individuals will not have an effect on their distribution between 13 See Epple and Romano 1998, pp. 56. schools14 . We have previously seen that expenditure per student is the same for all public schools and depend on income tax levels, thus public school quality can only be differentiated through peer effect or dispersion effect. In the public system, the absence of restrictions on entry encourages individuals to choose the school (or the number of schools) with maximum quality levels, denoted ju q max . If all N individuals have the same behavior, they will all end up in the school with ju q max , and then

u ju q q = max
with u q the level of quality when the N individuals are enrolled in the single existing public school u15 .

Following this kind of equilibrium, a suboptimal solution is reached. Individuals providing a non-internalized positive externality are disadvantaged and individuals providing non internalized negative externalities are advantaged. In this situation, a hierarchy of qualities will not exist in the public sector, and public school quality is given by u q .

In what follows, we identify two different solutions for the problem. ( )

] ) [( ) ( ) , , ( 2 2 ju ju ju ju ju ju ju i i b b k V y b mc σ θ µ θ µ θ - - ′ + - + ′ =
. Since public schools are free of tuition, the resources collected through income taxes should cover the total cost of students in the public sector. In this case the funding condition is

[ ] ∑ ∑ - - ′ + - + ′ = ] ) [( ) ( ) ( 2 2 ju ju ju ju ju ju b b k V E σ θ µ θ µ
; expenditure per pupil should be able to compensate positive externality providers through financial aids, and should counterbalance the negative impact of negative externality providers, since they would not pay (free public schools) in order to internalize their negative externality. This solution does not lead to the creation of a quality hierarchy; instead, we still have the same large public school with u q . It is not fully optimal, since negative externality providers do not pay to compensate their negative impact on quality. The optimal solution would be like the one in private schools, negative externality providers compensate directly positive externality providers, tax revenues should only cover the part related to the functioning of public schools

) ( ju k V ′
. This may seem unacceptable since optimality contradicts the free entrance assumption to public schools; public schools will become private to a certain degree. It should be noted that a combination of screening by examination and compensation schemes can lead to the existence of a non-strict hierarchy of school quality with each school regrouping at least two different types of individuals. It should be noted that the equilibrium point between public and private sectors will be modified since public schools behave strategically through selection or resources maximization; this entails a study of competition on the market of schooling which is not the object of this paper.

IV-Conclusion

In this paper, we have constructed an equilibrium on the market for education where individuals are differentiated by income and ability and where educational quality is not linear in mean ability. We have studied the distribution of students across sectors while mentioning the fact that private schools will operate to the detriment of public sector schools by siphoning high income high ability students. We have presented the necessary conditions for the existence of a majority voting equilibrium determined by the individual with the median couple of income and ability. Finally, we have analyzed the presence of a hierarchy of quality in both sectors.

According to our results, stratification by income and ability will prevent a fraction of the student population from having access to a more performing education, given its allocation of income and ability. In future studies, we may want to revisit models of urban economics contrasting school choice with residential choice in order to get a more generalized view of the provision of educational services, stratification, and the formation of inequalities in the access to education.

Appendix

Optimization in a private school:

Private schools profits maximization:

[ ] , results are the following:

] ( )       - - ′ -       - - - - = Φ ∫∫ ∫∫ ∫∫ ∫∫ ∫∫
( ) [ ] 0 ) ( ) ( ) , ( 2 2 * = - - ′ - - - ′ - = ∂ Φ ∂ jr jr jr jr jr jr ijr b b k V p y b σ θ µ θ µ α ] ) [( ) ( ) ( 2 2 * jr jr jr jr jr jr ijr b b k V p σ θ µ θ µ - - ′ + - + ′ = Proof of proposition 1:
The public sector is chosen when:

( ) i ju ju i iu b N Y y t W W , , , , , , 2 σ θ

=

It should be noted that private school quality is always superior to that of public schools. This is consistent with the reality; if public schools have better qualities than private schools, rational individuals will choose free public schools and the private sector will not exist. A private school with a quality lower than that of a public school will be eliminated from the market by the simple functioning of competition. Thus we can write: jr ju q q < In order to demonstrate the existence of ( )

y b , ˆ
we analyze two extreme cases.

First case:

For ). As a consequence we have

* * * ) 1 ( ) 1 ( ijr i ir iu i p y t c c y t - - = > = - but the difference is small. Or ( * * * ) 1 ( ) 1 ( ijr i ir iu i p y t c c y t - - = ≤ = - ) 0 ) 1 ( * * > - - = ijr i ir p y t c
since income is high. . Individuals choose the private sector.

We have seen that for low levels of income and ability, the public sector is preferred; conversely for high levels of income and ability the private sector is preferred. The continuity of f(b,y) implies that when ability and income levels increase (going from one extreme to the other) it is clear that there exists a point for which preferences will be reversed. This equilibrium is determined by a couple ( ) For the first case:

* * * ) 1 ( ) 1 ( ijr i ir iu i p y t c c y t - - = > = - .
A necessary condition for admission in the private sector will be:

0 ) 1 ( * * > - - = ijr i ir p y t c in other terms ) 1 ( * t p y ijr i - ≥ . 0 ) , ( lim = ∞ →
q c c implies that this student prefers higher school quality over higher consumption, thus he chooses the private sector. For the second case:

* * * ) 1 ( ) 1 ( ijr i ir iu i p y t c c y t - - = ≤ = -
The student is admitted in the private sector without any condition.

b N Y y y b t W ju ju m m m mu u σ θ > ] , , , , , , [ 1 2 1 1 b N Y y t W ju ju u σ θ 2-For an individual with ) , ( 1 1 y b we suppose that ) , ( ˆ1 1 1 y b t > ) , ( 1 1 1 y b t u .
From figure 1, this individual will be in the private sector.

As private sector's utility is decreasing in t.

  on the provision of public goods in a multicommunity model, the one written by Epple and Romano1996, and the one written by Glomm and Ravikumar. The latter constructed a majority voting equilibrium determined by the individual with median income in the context of non single peaked preferences; they provided the necessary conditions for the existence of this equilibrium. In their article individuals are differentiated only by income, and quality is only dependent on public expenditure per pupil. The objective of these articles was the study of the provision of public services.

  for all schools and students. However peer group effect and dispersion effect may be different or homogenous depending on whether public schools

••

  The number of students in a public school is given by: Public school peer ability variance is given by: students in the age for schooling are enrolled in one of the two sectors:

.

  resulting from the positioning of an individual's ability around the mean. Students with ability levels lower than school mean ability Students with ability levels higher than school mean ability that the distance of one's ability from the mean determines the importance and as we go away from the mean this price reduction decreases until it becomes a positive price 9 . An example of the distribution of externalities in a school is given in figure2.is always positive and has the effect of increasing price.

  b′ is weakly inferior to b ˆ while y′ is largely superior to y ˆ, or finally b′ is largely superior to b ˆ while y′ is weakly inferior to y ˆ. With ( ) ability and income. For the last two cases the superiority of one of the couples depends on the structural form of the utility function.

.

  enrolled in the private sector and will receive tuition discounts. (iiii) Individuals with y y i > and b b i < will be enrolled in the private sector and will pay tuition premia. (iii) and (iii) depend on certain conditions.(i) And (ii) represent the outcome of proposition 1. The proof of (iii) and (iiii) depends on the fact that the disposable income should cover the price of private education when it is positive; secondly on the fact that 0 When consumption becomes largely superior to school quality (when most of individual income is dedicated to consumption), individual utility tends to decrease. In other words, Individuals accept a decrease in consumption if school quality increases; they will choose the highest level of quality given their disposable income.

  Admitting higher ability students has contradicting effects on quality, it increases quality through peer effect and it reduces quality through the dispersion of abilities. The final effect of this exchange depends on the structural form of educational quality. The second difference is that in our model we did not impose a single crossing assumption. These two differences may result in the absence of a strict hierarchy of private school qualities. Different schools may have the same quality through different

a-

  Screening by examination: this is a trivial solution, if public schools are allowed to screen individuals according to their ability; we will have a number of schools that is at least equal to the number of types i b in the population. If a school defines an access score by ju s that is measured by ability units, it will accept all students with ju i s b ≥ . An individual looking for a high level of educational quality will choose a school with i ju b s ≥ in order to be enrolled with students with at least the same ability level. The combination of Thus a hierarchy of qualities will exist and it follows the distribution of ability types in the population. It should be noted that a number of different schools can share a group of students of the same type; in this case, these schools will have the same quality and the hierarchy will not be strict. b-Compensation schemes: another optimality improving solution to the problem can be obtained through a compensation scheme; in this context public schools have to maximize a resources function. Similarly to private schools' profit maximization, resources maximization yields the marginal cost function of admitting a student in a public school. It is of the following form:

  The Lagrangian function is of the following form:

[

  have no choice between the two sectors. They are obliged to enroll in public schools.Proof of proposition 3:In this proposition, we have to demonstrate that over the interval )

  means that both b ˆand y ˆincrease, or one of them increases while he other remains the same, or one of them largely increases while the other weakly decreases. The reverse is true when we say ( ) So far we have considered an exogenous tax rate; in the following sections we endogenize the tax rate through majority voting. For an individual i, the most preferred

											y b , ˆ				increases, this y b , ˆ	decreases.
	Proposition 2: for [ ] 1 , 0 ∈ t	and	Y	∈	ℜ	+	. There exists a unique N* which determines the
	number of students in the public sector as a result of majority voting. The number of
	individuals with	y y i < and	b b i < must be exactly equal to N*.
	tax rate is denoted as:
	t	*	( b i	,	y	i	)	=	Arg	max{max[ W iu	( t	,	y	i	,	Y	,	N	,	θ	ju	,	σ	2 ju	,	i b	),	ir W	( t	,	y	i	,	θ	jr	,	σ	2 jr	,	i b	)]}
	This tax rate will maximize individual utility in the sector offering the highest indirect
	utility.
	We denote	t	iu	( b i	,	y	i	)	the tax rate that maximizes utility in the public sector for an
	individual i with a couple	, i y b (	i		)	.		t	iu	( b i	,	y	i	)	=	arg	max	iu W	( t	,	y	i	,	Y	,	N	,	θ	ju	,	i b	)	.
	We denote	, ˆi i ( i y b t	)	the tax rate for which an individual is indifferent between the two
	sectors. In this case	iu W	( t	,	y	i	,	Y	,	N	,	θ	ju	,	i b	)	=	ir W	( t	,	y	i	,	θ	jr	,	i b	)	. It is clear that for each
	couple of ability and income, there exists a critical tax rate which determines the state of
	indifference. For low levels of t, the private sector is preferred; for high level of t, the
	public sector is preferred. While passing from an extreme to the other, there exists a
	tipping point of preferences determined by the tax rate	, ˆi i ( i y b t	)

  very low income levels, individuals can not pay tuition for the private school with the lowest quality. For very low abilities, individuals can not benefit from a financial aide or

	a tuition discount; we have	p	* > ijr	0	and	1 (	-	t	)	y	i	-	c	* ir	<	p	* ijr	for this school; which means
	that the price for the lowest quality private school exceeds the disposable income.
																			ir W	= 0	<	W	iu
	Second Case:
	For very high incomes, individuals can pay private school tuition. For very high abilities,
	individuals can obtain financial aide (scholarship) in a certain private school. We
	have	p	* > ijr	0	and small (or		0

These individuals can not pay the price in the private sector, and no private school accepts to admit them. Thus they are obliged to choose the public sector * . * < ijr p

  As public sector's utility is increasing in t over this interval. The individual will choose the highest tax rate over this interval, thus he chooses ) , (

	Proof of Lemma 5: For ∈ t			[ t	m	( b m	,	y	m	),	t	mu	( b	m	,	y	m	)]	, we consider an individual, he has
	, 1 y b (	1	)	<	, m y b (	m	)	.
	t	∈	[ t	m	( b m	,	y	m	),	t	mu	( b	m	,		y	m	)]	=>		t <	t	mu	( b m	,	y	m	)
																																											t	mu	m b	y	m	.
	In this case, we have:	1 W u	[ t	mu	( b m	,	y	m	),	1 y	,	Y	,	N	m	,	θ	ju	,	σ	2 ju	,	1 b	]	>	1 W u	[ t	,	1 y	,	Y	,	N	,	θ	ju	,	σ	2 ju	,	1 b	]
	Over the interval		[ t	m	( b m	,		y	m	),	t	mu	( b	m	,	y	m	)]	, there does not exist a tax rate different from
			(			,				)
																																											The individual will prefer the lowest tax rate
	over the interval, thus he chooses	t	mu	( b m	,	y	m	)	.
	In this case, we have:	1 W r	[ t	mu	( b m	,	y	m	),	1 y	,	θ	jr	,	σ	2 jr	,	1 b	]	<	1 W r	[ t	,	1 y	,	θ	jr	,	σ	2 jr	,	1 b	]
	Over the interval	[ t	mu	( b m	,	y	m		] 1 ),	, there does not exist a tax rate different from
	t	mu	( b m	,	y	m	)	which represents a majority voting equilibrium.

Chapter 5 in "Education, Equality and Social Cohesion" by Green, Preston and Janmatt provides interesting results on outcomes' inequalities in five groups of countries: East Asia, the Nordic countries, the Germanic region, the Mediterranean countries and the predominantly English speaking countries.

In our model, we account for nonlinearity in mean ability by taking the variance of abilities into consideration.

[START_REF] Dertouzos | Made in America: regaining the productive edge[END_REF] pp. 84-85) underlined the need to take the dispersion in test scores into account. R.[START_REF] Benabou | Heterogeneity, Stratification, and Growth: Macroeconomic implications of community structure and school finance[END_REF] explored the effect of human capital dispersion on economic growth.

This is consistent with the Boutique and the Focus models. SeeHoxby and Weingarth 2005, pp 6.

A school known to be highly selective can be imagined to have a high mean ability and a low ability variance, since selection leads to the homogenization of the student population.

The existence or nonexistence of economies of scale in the production of education is discussed in the optimization section.

The analysis of public schools' strategical behavior will be seen in the last section.

The fact that public schools may have different quality levels will be discussed in the last section.

In this paper, residential choice is not built into the model. In a multicommunity model, income will have an effect on the distribution of students between public schools through residential choice.

This public school can operate on multiple geographical locations (multiple campuses), but it can be regarded as a large single public school since expenditure per pupil, peer effect and dispersion effects would be the same.

Proof of Lemma 2:

(i) For ( )

is decreasing in N. An increase of N represents a congestion effect of public schools; thus public expenditure per pupil N tY decreases. As a consequence public school quality decreases. Students on the margin between sectors will opt out of the public sector to join the private one. In this case ( ) y b , ˆ decreases.

(ii) For

is increasing in Y. An increase of Y implies a growth of public school quality following a growth of public expenditure per pupil N tY . Students on the margin between sectors will opt out of the private sector to join the public one. In this case ( )

is increasing in t. On one hand, an increase of t implies a decrease of consumption in both sectors, thus the effect of t on consumption as a factor of individual utility is neutral. On the other hand an increase of t induces a growth of public expenditure per pupil. As a consequence public school quality increases. Students on the margin between sectors will opt out of the private sector to join the public one. In this case ( )

increases.

Proof of proposition 2:

In Lemma 2 we have presented the fact that public school quality is decreasing in N.

Conversely, private school quality is not dependent on N 16 . Indifference is given by :

The left hand side of the equation is decreasing in N; while the right hand side of the equation is independent of N. For N very low, public school quality is very high. For N very high, public school quality is very low. 16 Each school chooses its size according to its profit maximization problem.

32

While passing from one extreme to the other. The intersection between the decreasing and the steady function determines the unique equilibrium number of students N*.

Proof of Lemma 3:

We consider two individuals, the first has a couple ) , ( . We consider that ) , (

For the individual with ) , ( 22 y b , we have:

From assumption 1 and 2 we have for ) , (

. This is a contradiction with (d).

In


Hakim Idjis
email: hakim.idjis@ecp.fr

Danielle Attias
email: danielle.attias@ecp.fr

Jean Claude Bocquet
email: jean-claude.bocquet@ecp.fr

Sophie Richet
email: sophie.richet@mpsa.com

Designing a Sustainable Recycling Network for Batteries from Electric Vehicles. Development and Optimization of Scenarios

Keywords: Recycling, Value chain, System analysis and design, Lithium batteries

Since the 2008 crisis, the automotive industry is shifting towards a new paradigm and repositioning around the green mobility, using mainly lithium battery technology. Given this new development, the issue of recycling batteries arises for ecological, economic and geostrategic reasons. Our work consists in formalizing a methodology to help design a sustainable recycling network applied to batteries, under uncertainty. The proposed approach involves two steps. The first step is about modeling the recycling network and the characterization of the problem's elements, using systemic analysis. The second step consists in developing scenarios about the configurations of this value chain, and then optimizing their functioning. We'll choose different positioning for the actors and make assumptions about logistic data. In this paper, we will try to summarize the issues of recycling lithium battery, explain in detail this approach and present the first results of application.

Introduction

Everyone agrees that we are at the beginning of a second automotive revolution, induced by the decline of the life cycle of the internal combustion engine (ICE) vehicle, the limits of the economic and financial model of the automotive industry and the effect of the current mobility system on the environment. These factors raise a reflection about a new mobility system and its impact on territories, resources allocation and regulations. These new constraints have led the automotive industry to reposition itself around the green mobility. Several technologies exist, including fuel cell technology. However, they are not yet marketable. Thus, the industry turned to hybrid or fully electric vehicles (EVs), using mainly lithium batteries.

Given this new developments, the issue of recycling batteries arises for ecological, economic and geostrategic reasons. This recycling network will be implemented by the original equipment manufacturers (OEMs). This is because, firstly, they have a legal responsibility on batteries at end-of-life [1] and secondly, they are the actors who have the most power and resources in the automotive value chain. Currently, the industrial-scale recycling of these batteries is non-existent. The prospective nature of the study encompasses uncertainties, caused by the development of electric vehicle market, the evolution of battery technology and the maturity of recycling processes [START_REF] Hoyer | A Strategic Framework for the Design of Recycling Networks for Lithium-Ion Batteries from Electric Vehicles[END_REF]. The interest of this subject lies in the identification of a configuration of the value chain associated to batteries at the end-of-life. This value chain should be able to comply with regulations while controlling the overall cost of batteries. This paper will be focused on lithium ion batteries (Li-ion), because it's the most promising technology for vehicles electrification [START_REF] Grosjean | Assessment of world lithium resources and consequences of their geographic distribution on the expected development of the electric vehicle industry[END_REF] - [START_REF] Kushnir | The time dimension and lithium resource constraints for electric vehicles[END_REF]. In this article, "recycling" will be referred to as the treatment and recovery of batteries after the automotive use, including: reparation, second life applications and materials recycling. The term EV includes: hybrid-electric vehicles (HEV), plug-in hybrid vehicles (PHEV) and battery-electric vehicles (BEV)

Next section provides an overview of the existing literature about end-of-life products recovery, as well as Li-ion battery recycling. The third section illustrates the specificity of our topic. We will identify the drivers for recycling batteries from EVs. We will also describe the related value chain, in order to highlight its complexity, caused by uncertainties and the multicity of stakeholders. The last section justifies and explains in detail the proposed approach. It also presents the first results of application.

Literature Review

The issue of recycling end-of-life products has been treated by several strands of literature: Environmental value chain management (EVCM), green supply chain management (GrSCM), and reverse logistics (RL), may also be called: Closed-loop supply chains (CLSC). Reverse Logistics is emerging as an area suitable for the study of recycling networks. Rogers and Tibben-Lembke [START_REF] Rogers | Going Backwards: Reverse Logistics Trends and Practices[END_REF] define RL as the process of planning, implementing and controlling the efficient, cost-effective flow of raw materials, in-process inventory, finished goods and related information from the point of consumption to the point of origin for the purpose of recapturing or creating value, or for proper disposal. Fleischmann, one of the biggest contributors on the logistics of recovery systems, described in [START_REF] Fleischmann | A characterisation of logistics networks for product recovery[END_REF] a typical structure for products recovery, as a network converging from disposer markets, going through recovery facilities and diverging to re-use market.

Research carried out in products recovery concerns mainly; either the assessment whether or not the recovery of used products is economically more attractive than disposal, or the organization of the recovery network if the disposal option is prohibited by legislation. Many products specific studies were conducted regarding recycling, such as sand, carpet, bottles and waste electrical and electronic equipment (WEEE). We found it; the most pertinent ones for our work are either end-of-life vehicles (ELV) or products disassembled from ELVs (plastics, lead batteries, etc.). Kumar and Sutherland [START_REF] Kumar | Sustainability of the automotive recycling infrastructure: review of current research and identification of future challenges[END_REF] explored the effect of innovative vehicle designs (material composition, new power-train technologies) on the sustainability of the automotive recovery infrastructure. Maudet-Charbuillet [START_REF] Maudet-Charbuillet | Proposition d'outils et démarches pour l'intégration de filières de recyclage de matières plastiques dans la "supply chain" automobile[END_REF] investigated the issue of integrating recycling plastics networks in the automotive supply chain, by proposing a modeling tool used in order to stabilize these networks. Mathieux and Brissaud [START_REF] Mathieux | End-of-life product-specific material flow analysis. Application to aluminum coming from end-of-life commercial vehicles in Europe[END_REF] used material flow analysis to get insights about aluminum material flows and stocks, identification and quantification of relevant processes. Farel et al. ([10] - [START_REF] Farel | A cost and benefit analysis of future end-of-life vehicle glazing recycling in France: A systematic approach[END_REF]) provided a five steps method to design a value chain for ELV glazing recycling: modeling material and information flow; establishing value network; structural analysis; scenario generation and simulation and evaluation. Farel et al. [START_REF] Farel | Finding best practices for automotive glazing recycling: a network optimization model[END_REF] for the same issue in [START_REF] Farel | A method to design a value chain from scratch[END_REF] used a linear programming model to simulate the conditions of costs and benefits situations for the upcoming glazing recycling chain in 2015 in France. Kannan et al. [START_REF] Kannan | A genetic algorithm approach for solving a closed loop supply chain model: A case of battery recycling[END_REF] proposed a heuristics based genetic algorithm (GA) to solve a closed loop supply chain model for spent lead-acid batteries in India.

Regarding Li-ion battery recovery, the number of articles attests to the recent emergence of this issue. Several industrial, academic or joint initiatives have emerged to explore the end-of-life of EV batteries, mainly concentrated in the USA, Europe and Japan. In the USA, the National Renewable Energy Laboratory's Second Use Project Objective is modeling the actual battery degradation behavior; identification, assessment, and verification of PHEV/EV Li-ion second use profitability. One of the results of this project, published in [START_REF] Neubauer | The ability of battery second use strategies to impact plugin electric vehicle prices and serve utility energy storage applications[END_REF] is focused on estimating the possible value of battery second use strategies to reduce the cost and accelerate adoption of PHEV/EVs. LithoRec and LithoRec II, a consortium of 12 academic and industrial partners, financed by the German federal government aims to assess the development of an industrial scale structure and process for used Li-ion batteries in Germany, in an economic and environmentally-friendly way. The strategic framework for designing a recycling network developed in LithoRec is highlighted in [START_REF] Hoyer | A Strategic Framework for the Design of Recycling Networks for Lithium-Ion Batteries from Electric Vehicles[END_REF]. It involves three steps: analyzing problem characteristics; description of actors and requirements that need to be considered and an integrated planning approach of the network.

The reading we make about this literature reveals the following common features:

• Focus on material recycling, actually few articles deal with multiple recovery options at the same time • Deals mainly on logistics aspects; location of recovery facilities, transport planning and inventory control; • Considers only quantitative uncertainty; volume and quality of returns, costs and prices; • Does not consider conflicting objectives from multiple stakeholders • Focus on the economic objective We will see in the next section, that the existing literature is not suitable for our topic. The recycling drivers imply the consideration of all the recovery options (reuse and recycling). The value chain complexity implies to go beyond the logistics aspects, and consider the actors positioning and expectations in this value chain.

Focus on Electric Vehicle Batteries

The Geostrategic Issue

Demand and production of lithium have increased substantially since 2000, with nearly 6% average annual growth [START_REF] Roskill | The Economics of Lithium[END_REF]. Lithium is traditionally used in glass, ceramics, metallurgy, pharmacy... etc. According to the US Geological Survey, these batteries held 21% of the market share in 2009 [START_REF] Goonan | Lithium use in batteries[END_REF]. This proportion will be higher in the future, with the widespread introduction of electric vehicles.

Market development of lithium, which is expected to double in 10 years, raises the question of the availability of resources and the security of supply. In fact, these resources are more than 70% located in the "ABC" zone: Argentina, Bolivia and Chile. Today, the availability of lithium is not an issue. In EV Traction batteries, lithium carbonate (Li2CO3), accounts for less than 10% of a battery weight and lithium price was nearly 6 $/kg in 2010 [START_REF] Grosjean | Assessment of world lithium resources and consequences of their geographic distribution on the expected development of the electric vehicle industry[END_REF]. The recycled lithium is more expensive than extracted lithium. So, we understand that today it is not interesting to retrieve lithium from used batteries. It will not be the case in the future, when recycled lithium will represent the largest cumulative source [START_REF] Kushnir | The time dimension and lithium resource constraints for electric vehicles[END_REF]. Although experts do not consider a scenario of lithium depletion [START_REF] Kesler | Global lithium resources: Relative importance of pegmatite, brine and other deposits[END_REF], the question of emancipation from suppliers remains crucial. Miedema and Moll [START_REF] Miedema | Lithium availability in the EU27 for battery-driven vehicles: The impact of recycling and substitution on the confrontation between supply and demand until2050[END_REF] confronted the supply capacity of lithium to the needed demand from the automotive industry. They concluded that undersupply can be expected in the near future, before a large scale recycling of batteries. The challenge in the future is to meet the growing demand without causing price volatility and to implement early enough recycling networks to provide a capital of lithium in countries using it.

The Regulatory Issue

Taking into account the effects of human activities on the environment has become one of the criteria for assessing decision-making process in the industrial world. In this context, the automotive industry is fully involved, as it was among the first industries subject to environmental regulations. Directive 2000/53/EC [19] lays down requirements for the treatment of end-of-life vehicles (ELV) in Europe. In addition, Directive 2006/66/EC [1] sets the legal framework for the treatment of batteries and accumulators in Europe. This directive requires particularly for electric vehicle batteries the implementation of:

• A dedicated collection system at no cost to the end user;

• A system of treatment, recycling and disposal of batteries waste.

The Economic Issue

The economic stakes are induced by the possibility of a second life use of batteries. In fact, these batteries; when out of use for optimum automotive propulsion could be used as energy storage for stationary or other embedded applications based on their technical, economic and environmental feasibility. Among the possible outcomes, the use in cars exploited over shorter distances is considered, as well as stationary energy storage. The topic of the second life of batteries is attracting interest worldwide, but it is still the object of research projects at an early stage [START_REF]Étude de la seconde vie des batteries de véhicules électriques et hybrides rechargeables[END_REF].

The Related Value Chain

Since the EV technologies are recent, the recycling of these batteries is not yet industrialized. Industry players do not necessarily exist today, which raises the question of their emergence and the disruption of the actual value chain. In addition, legislation (recycling targets and the Extended Producer Responsibility "EPR" concept) is pushing towards more cooperation between the OEM, the ELV recovery infrastructure, the consumer, and other stakeholders. OEMs are shifting from traditional paradigms to innovative business models [START_REF] Kumar | Sustainability of the automotive recycling infrastructure: review of current research and identification of future challenges[END_REF].

Value chains for end-of-life products recovery are not considered as "natural value chains", in the sense that their implementation does not meet a market demand. This description of value chains and their origins, explains the difficulties of different actors to project economically viable systems and organize themselves [START_REF] Maudet-Charbuillet | Proposition d'outils et démarches pour l'intégration de filières de recyclage de matières plastiques dans la "supply chain" automobile[END_REF].

The key factor to the sustainability of this recovery chain is the ability to prove its economic profitability as a whole. This assumption seems idealistic, considering the differences in the business logics of the stakeholders and the issue of redistributing profit among them.

Our Approach

Bringing together the remarks formulated about the literature and the specificity of our topic, we formalize a methodology to help design a sustainable recycling network applied to batteries, under uncertainty. This methodology encompasses the following features:

• Uncertainty and business models for value capture will be modeled by scenarios • The scenarios will reflect: (i) the governance and the several positioning of the actors in the considered value chain, and (ii) assumptions about quantitative data (volume of returns, batteries quality, etc.) • Considering the multicity of stakeholders, by optimizing objectives deriving from expressed requirements

• Considering the hole recovery options (reparation, second life applications, recycling) • Designing a sustainable network, considering economic, social and ecologic performances. Our approach involves two steps:

• Network modeling & problem characterization • Scenarios development and optimization

Network Modeling and Problem Characterization

If we consider the recycling network as a complex system with multiple stakeholders, the tool that seems most suitable for the design and deployment of such a system is the systemic analysis. Applying systemic concepts to the design of complex logistic networks under uncertainty, Patay et al. [START_REF] Patay | Systemics for the Design of Complex Distribution Networks with uncertainties[END_REF] developed a method called SCOS'M; Systemics for Complex Organizational Systems' Modeling. It comprises five steps: • Isolate the system and its subsystems to define the scope • Describe the phases of the system's life cycle • Describe for each phase of its life cycle, the expectations on the system in terms of satisfaction and performance • Develop the functions to be performed by the system to meet these expectations • Determine the parameters and variables of the system allowing the satisfaction of expectations. This method seems ideal, since it meets some of the requirements of our approach, namely, the multicity of stakeholders, the multicity of performances and the identification of parameters that allow scenarios simulation. At the end of Step 5, we have identified the variables and parameters influencing the economic dynamics of the value chain.

This step is crucial, since it will lead to characterizing a network type and the elements of the problem, by defining the objectives, constraints, variables and parameters. As well as values (economic, environmental, flexibility, security) expected by stakeholders. These elements will help address the second step of this approach.

Scenarios Development and Optimization

The second step in our approach consists in a projection to the future to formalize possible value chains for recycling batteries, and then optimize their functioning. It is an approach coupling simulation and optimization. Scenarios development is about identifying several configurations for the value chain, by making assumptions about uncertainties, which we could categorize into two kinds: (i) quantitative (volume of returns, batteries quality, costs, etc.) and (ii) qualitative (monopoly of a recycler, internalization of the recycling activity by OEMs, recycling opportunities, etc.).

Regarding quantitative uncertainties, Hoyer et al. [START_REF] Hoyer | A Strategic Framework for the Design of Recycling Networks for Lithium-Ion Batteries from Electric Vehicles[END_REF] identify the following ones:

• Sources (quantity, spatial distribution, composition of battery returns);

• Outputs (reuse opportunity: products, components, materials);

• Process (combination and configuration of processes). In order to tackle these uncertainties, firstly, we'll link the parameters (results of SCOS'M) to these uncertainties, and then use system dynamics (SD) approach to predict the system's future behavior. The choice of SD has been made given the complexity of interactions existing within the system. Donnadieu & Karsky [START_REF] Donnadieu | La systémique, penser et agir dans la complexité[END_REF] define SD as a science of change and evolution. It is concerned with understanding the phenomena and their causes, identifying the factors that create change, or otherwise oppose it. System dynamics simulation is a renowned approach to simulate the effect of future changes, several system dynamics-based studies were conducted in the area of recycling ( [START_REF] Farel | A cost and benefit analysis of future end-of-life vehicle glazing recycling in France: A systematic approach[END_REF], [START_REF] Zhao | A system dynamics model for evaluating the alternative of type in construction and demolition waste recycling center -The case of Chongqing, China[END_REF], [START_REF] Georgiadis | Sustainability in electrical and electronic equipment closed-loop supply chains: A System Dynamics approach[END_REF]).

Regarding qualitative uncertainties, based on the work of Williamson and the work on global value chains [START_REF] Gereffi | The governance of global value chains[END_REF], we will imagine different positions of OEMs in this value chain, as well as those of the other actors. Examples:

• OEM will internalize transportation from collection points to recovery facilities; • There will be a unique recycler in Europe (monopoly) Assessing the sustainability of a supply chain goes by identifying the appropriate performance indicators, which proves to be a difficult task. The lack of social metrics is a serious challenge [START_REF] Tang | Research advances in environmentally and socially sustainable operations[END_REF], while there is a multicity of environmental indicators (energy efficiency, greenhouse gas emissions, waste generated, waste recycling rate, etc.). In our work, we identified the metrics that seem to be most relevant to our topic, namely the economic and environmental balance. "Fig. 1" gives an overview of the causal loop diagram modeling the recycling network. This latter includes the interaction between virgin and recycled lithium, the macroscopic view of costs, benefits and environmental performances, which are also developed in other diagrams.

These scenarios will be used to develop linear programming models for the network design problem. These models will be addressed by an approach coupling simulation and optimization. The simulation enables us to test the scenarios with economic and industrial data, by the mean of SD. The optimization will allow, for each scenario, identifying the optimum solution in terms of: location and sizing of treatment facilities, planning of their implementation, as well as transport planning.

In summary, step 1 of the approach will be used to characterize the elements of the problem modeling the dimensioning of the value chain and optimizing its flows. Step 2 will adopt assumptions about the role of actors, as well as data to enable the optimization of the developed models. The results will be used to estimate sustainability of this value chain. At last, we'll synthesize the results of scenarios analysis in order to bring out the conclusions and recommendations for the recycling network deployment. This approach features some characteristics; developed in this section, that enable meeting the requirements to establish a recycling network for batteries from EVs, such as identifying the actors, the choice of governance and implementation planning.

Conclusion

By regulatory constraints or economic opportunism, the recycling network of batteries from EVs will be implemented. The prospective dimension of the study is a significant challenge, which requires the use of an efficient approach, able to overcome the various uncertainties and provide useful results. The emergence of this topic and the limits of the existing literature have driven us to propose a novel approach, more suitable to the specificity of our topic. This approach is multidisciplinary, involving knowledge from:

• Complex systems engineering • Business and economic sciences • Operational research and logistics optimization In this paper, only modeling results are exhibited. More findings will be included in our future contributions. The perspectives after this first contribution are:

• Data collection and simulation using stock & flow diagrams. The exposed SD model being validated by the car manufacturer, the next step is data collection by the mean of interviews. Meetings are scheduled with experts within the car manufacturing company and its partners; • The scenarios building methodology and the scenarios elaboration;

• Integration of the multi-objective optimization in the whole approach. Some of the variables identified in the causal loop diagrams are decision variables such as "investment in collection centers" and "Upgrade recycling process". The idea is to optimize these variables, while evaluating the economic and environmental balances of the recycling network; • Recommendations about the collaborative governance of the recovery chain, including role-play definition and profit redistribution. We will strive to ensure the robustness of the results and the inclusion of possible new data. Since we are in a context of value chain emergence, several factors can disrupt established patterns in our work, namely the composition of EV batteries, the arrival of new actors or the evolution of legislation. This dynamic aspect of the subject will be treated with a permanent monitoring of the real value chain, allowing us to make adjustments in our scenarios.

Fig. 1 .

 1 Fig. 1. Overview of the recycling network

Acknowledgments. This work was benefited from the support of the chair "PSA Peugeot Citroen Automobile: Hybrid technologies and Economy of Electromobility. So-called Armand PEUGEOT Chair", led by Ecole Centrale Paris, ESSEC and SUPELEC, and sponsored by PEUGEOT CITROEN Automobile.