Laurent Simula
email: laurent.simula@ehess.fr

PSE, GREQAM Laurent Simula Ehess

Optimality conditions and comparative static properties of non-linear income taxes revisited

Keywords: Optimal tax, income tax, comparative statics Optimal Tax, Income Tax, Comparative Statics. JEL classi cation: H21; H31

Optimality conditions and comparative static properties of the optimal Mirrleesian nonlinear income tax are obtained for a nite population and quasilinear-in-consumption preferences. Contrary to Weymark (1987) who considers quasilinear-in-leisure preferences, the linearity with respect to gross income, which is observed by the government and used as a tax base, is lost. A reduced-form optimal income tax problem is derived, in which consumption levels are obtained as functions of gross incomes. The contribution of this new reduced form is twofold. First, the optimal allocation can be characterized geometrically in a simple way. Second, comparative static results with respect to individual productivities are easy to obtain.

INTRODUCTION

This paper uses a reduced form of the optimal non-linear income tax problem to derive a geometric characterization of the social optimum in the gross-income/consumption space as well as comparative static properties. Following [START_REF] Guesnerie | Nonlinear Pricing in a Finite Economy[END_REF] and Weymark (1987), it considers a nite population version of [START_REF] Mirrlees | An Exploration in the Theory of Optimum Income Taxation[END_REF]'s model. All individuals have the same preferences over consumption and leisure, but differ in skill levels. The government wants to redistribute income from the more to the less productive individuals. However, if the distribution of this parameter within the population is common knowledge, each agent's productivity is private information. Accordingly, the government is restricted to setting taxes as a function of earnings and faces an adverse selection problem when designing the optimal income tax schedule.

The optimal tax structure is the product of different sorts of interacting in uences. It basically depends on the skill distribution [START_REF] Diamond | Optimal Income Taxation: An Example with U-Shaped Pattern of Optimal Marginal Tax Rates[END_REF][START_REF] Saez | Using Elasticities to Derive Optimal Income Tax Rates[END_REF], on the government's aversion to income inequality, re ected by the welfare weights in the social objective function, but also on the responsiveness of labour supply. In addition, the way in which all these in uences interact is affected both by the incentive-compatibility constraints and the tax revenue constraint, which restrict the possibilities for income redistribution. Because of the complexity of the relationship between the optimal tax schedule and the set of underlying parameters, investigations must usually resort to numerical simulations [START_REF] Tuomala | Optimal Income Tax and Redistribution[END_REF]. This is an unfortunate state of affairs because some features of the model are necessarily left somewhat obscure by such an approach, which is very useful since it allows the optimal tax rates to be quanti ed, but is not ideally suited for shedding light on the economic intuition behind the results.

In a pioneering paper, Weymark (1987) has derived a number of comparative static results of optimal non-linear income taxes for the case in which individual preferences are quasilinear in leisure and the population is discrete. Their derivation uses a reduced form of the optimal tax problem, which does only involve the choice of the consumption good (Weymark, 1986b) and concentrates on the fully separating social allocation distinguished from those involving bunching in Weymark (1986a). This methodology has been adapted to obtain comparative static properties for a model in which the government both designs an optimal income tax and provides a public good optimally [START_REF] Brett | Public Good Provision and the Comparative Statics of Optimal Nonlinear Income Taxation[END_REF]. The assumption that individual preferences are quasilinear in leisure is maintained in this paper. The disutility of effort is therefore constant.

In other words, when a price is varied, the change in individual consumption does only depend on the substitution effect while all income effects are absorbed by the labour supply. This preference speci cation has also been employed to derive an explicit solution to the optimal income tax problem with a continuous population [START_REF] Lollivier | Bunching and Second-Order Conditions: A Note on Optimal Tax Theory[END_REF], the properties of which have been investigated by [START_REF] Boadway | Optimal Income Taxation With Quasi-Linear Preferences Revisited[END_REF] and [START_REF] Boone | The Simple Economics of Bunching: Optimal Taxation with Quasi-Linear Preferences[END_REF]. [START_REF] Hamilton | Optimal Income Taxation and the Ability Distribution : Implications for Migration Equilibria[END_REF] have used it to derive some comparative static results with respect to individual productivity in an economy with two classes of agents where the government adopts a maximin or maximax objective function. Its tractability has been exploited by [START_REF] Ebert | A Reexamination of the Optimal Nonlinear Income Tax[END_REF] to provide a complete example in which different types of individuals are bunched together, establishing that the rst-order approach to [START_REF] Mirrlees | An Exploration in the Theory of Optimum Income Taxation[END_REF]'s model can be misleading.

Quasilinear-in-leisure preferences offer technical advantages. They are indeed linear with respect to gross income, i.e. to the variable observed by the government and used as the tax base. This allows the reduced-form optimal income tax problem to have an explicit solution (Weymark, 1986b). When quasilinear-in-consumption preferences are considered, the linearity with respect to the observable variable is lost, the social objective of the reduced-form depends on productivity through the disutility of labour and its maximization does not yield an explicit solution.

Although working with them is less tractable, quasilinear-in-consumption preferences are worth examining for at least three reasons. First, from the theoretical viewpoint, assuming that all income effects are absorbed by consumption is a more satisfying assumption. Otherwise, as is made clear in the continuous population framework, the optimal tax schedule does only depend on the skill distribution and on the social weights [START_REF] Boadway | Optimal Income Taxation With Quasi-Linear Preferences Revisited[END_REF], but not on the labour response. The ef ciency/rent-extraction trade-off re ected by the income tax schedule is thus very speci c. On the contrary, when the income effects on the labour supply are omitted, the optimal tax scheme basically depends on the elasticity of the labour supply as well. Second, most of the empirical studies, though not all, gives credence to small income effects relative to substitution effects as regards labour supply [START_REF] Blundell | Labour Supply and Taxation: A Survey[END_REF][START_REF] Blundell | Labor Supply: A Review of Alternative Approaches[END_REF]. Accordingly, the case with no income effects on labour supply provides a useful benchmark, which has been theoretically studied by [START_REF] Atkinson | Public Economics and the Economic Public[END_REF], Boadway andPestieau (2007), d'Autume (2000), [START_REF] Diamond | Optimal Income Taxation: An Example with U-Shaped Pattern of Optimal Marginal Tax Rates[END_REF], [START_REF] Piketty | La redistribution scale face au chômage[END_REF], [START_REF] Salanié | Un Exemple de Taxation Optimale[END_REF] or Saez (2002) and used in the numerical part of other papers [START_REF] Saez | Using Elasticities to Derive Optimal Income Tax Rates[END_REF]. Third, the comparative static properties of the optimal non-linear income tax problem could differ signi cantly from those obtained under quasilinear-in-leisure preferences.

This paper derives a reduced-form optimal non-linear income tax problem involving only the allocation of gross incomes within the population. Consumption levels are thus obtained as a function of gross incomes. This reduced form can be seen as a special case of Chambers (1989) "concentrated" objective function derived for separable preferences. However, thanks to our quasilinearity assumption, it clearly re ects the trade-off between equity and ef ciency. On the one hand, it incorporates the fact that the only incentive-compatibility constraints which matter are the downward adjacent ones. Accordingly, in the optimum, each individual is indifferent between his own bundle and that of his nearest less productive neighbour. This is in accordance with the optimality features derived in the principal/agent model and re ects ef ciency considerations. On the other hand, at any given gross-income/consumption bundle, the angle between the indifference curve of the individual for whom the bundle is designed and the indifference curve of the nearest more productive individual is entirely determined by the social weights. Altogether, these features allow a very simple geometric characterization of the optimal allocation and of the bunching pattern. In particular, it is suf cient to know the indifference curves and the individual social weights to construct the complete optimal allocation geometrically. The reduced form is then used to provide the comparative statics of the optimal tax schedule with respect to the marginal utility of money and the weights in the welfare function, as in Weymark (1987), but also to individual productivity. It appears that varying the skill level of an individual only alters the optimal allocation locally, through three channels: it involves a local substitution effect, an incentive effect and an informational externality which modi es the behaviour of the nearest less productive individual. This analysis can be regarded as a special case but also as a generalization of [START_REF] Stiglitz | Self-Selection and Pareto Ef cient Taxation[END_REF]'s model. First, it investigates quasilinear preferences instead of well-behaved generic preferences and focuses on the maximization of social weighted functions rather than characterizing Pareto ef cient tax schedules. This latter restriction amounts to investigating the case [START_REF] Stiglitz | Self-Selection and Pareto Ef cient Taxation[END_REF] refers to as "normal" as soon as the individual social weights are declining with individual productivity. Second, it casts light on a population consisting of more than two classes of individuals, which is of interest for the applicability of the model to real situations.

The paper is organized as follows. Section 2 sets up the model. Section 3 derives the reduced form of the optimal non-linear income tax problem and provides a geometric characterization of the optimal allocation. Section 4 examines the comparative statics of the solution to the optimal income tax problem. Section 5 concludes.

THE MODEL

The population consists of I 2 individuals, indexed by i 2 I := f1; :::; Ig : There are two goods, consumption and leisure. Person i's consumption and labour supply are denoted x i and `i, respectively. The economy is competitive, with constant-returns-to-scale technology; so person i's wage rate is xed and equal to his productivity θ i : For convenience, only one person has a given productivity level. Individuals are thus indexed in terms of productivity. This simpli cation is not particularly restrictive as the distance between two productivity levels is free to vary. Without loss of generality, the vector of productivities θ := (θ 1 ; :::; θ I ) is taken to be monotonically increasing, 0 < θ 1 < ::: < θ I :

(1) An individual with productivity θ i working `i units of time has gross income

z i := θ i `i; i 2 I : (2)
All individuals have the same preferences over consumption and leisure, represented by the utility function U : R 2 + ! R,

U (x i ; `i) := γx i v (`i) ; i 2 I ; (3)
where γ 2 R ++ is the marginal utility of money. It is assumed that the disutility of labour v (`i)

is a C 3 -function which satis es v 0 > 0; v 00 > 0; v 000 > 0; v (0) = 0; v 0 (0) = 0; and v 0 ` ! ∞
where `is the time endowment of each individual.

By (2), the utility function (3) can be rewritten as U (x i ; `i) = U (x i ; z i =θ i ) : Individuals have therefore personalized utility functions u : R 2 + ! R in the gross-income/consumption space,

u (x i ; z i ; θ i ) := γx i v z i θ i ; i 2 I : (4)
The marginal rate of substitution s (z i ; θ i ) of the θ i -individual at the (x i ; z i )-bundle only depends on gross income, with

s (z i ; θ i ) := u 0 z i (x i ; z i ; θ i ) u 0 x i (x i ; z i ; θ i ) = v 0 (z i =θ i ) γθ i ; i 2 I : (5)
In particular, the higher is γ; the atter are the indifference curves and thus the lower is the increase in consumption required to compensate for an increase in gross income while keeping utility constant.

A social allocation speci es the consumption and gross income levels for each individual. It is represented by a vector a = (x; z) 2 R I + R I + , with x = (x 1 ; :::; x I ) and z = (z 1 ; :::; z I ). The tax policymaker knows the functional form of the utility function and the distribution of wages in the population. He is however unable to observe each individual's productivity. As a result, he is restricted to setting taxes as a function of gross income z i . By the taxation principle, a non-linear income tax schedule is therefore a mapping

( θ ! a R + R + θ i (x i ; z i ) ; (6)
which satis es the incentive compatibility constraints u (x i ; z i ; θ i ) u (x j ; z j ; θ i ) ; 8 (i; j) 2 I 2 ;

(7) and the tax revenue constraint

I ∑ i=1 z i I ∑ i=1 x i : (8)
An allocation a is production ef cient if the budget-balanced constraint (8) is binding.

The social welfare function W : R

I + R I + ! R is a weighted sum of individual utilities, W (a) := I ∑ i=1 λ i u (x i ; z i ; θ i ) ; (9)
in which λ := (λ 1 ; :::; λ I ) are individual social weights. The tax policymaker's taste for redistribution from the high to the low productive individuals is captured through the requirement that the higher the individual productivity the less the weight in the social objective, i.e. 0 < λ I < ::: < λ 1 :

If I = 2; this assumption amounts to considering the "normal" case studied by [START_REF] Stiglitz | Self-Selection and Pareto Ef cient Taxation[END_REF] in which only the incentive compatibility constraint of the high type is binding. When I > 2, one can therefore expect that the only binding incentive compatibility constraints will be the downward adjacent ones stating that the θ i+1 -individual must be indifferent between his own bundle and that of the θ i -individual, for i = 1; :::; I 1:

As W (a) is homogeneous of degree one in λ , the sum of the social weights can be normalized without loss of generality. It is convenient to de ne Λ (θ i ) as the cumulative social weight of the i less productive individuals, and to set

Λ (θ I ) = I: (11)
Consequently, admissible parameters (θ ; γ; λ ) belong to the set 10) and (11) are satis edg :

P := R I ++ R ++ fλ j(
The optimal non-linear income tax problem can thus be formulated as follows:

Problem 1 (Optimal Non-linear Income Tax Problem). For (θ ; γ; λ ) 2 P; choose an allocation a 2 R I + R I + to maximize W (a) under the self-selection constraints (7) and the tax revenue constraint (8).

For xed values of the parameters (θ ; γ; λ ) 2 P, there is a unique solution to Problem 1. I denote by g x : P ! R I + and g z : P ! R I the functions which relate (θ ; γ; λ ) to the optimal consumption and gross income vectors for Problem 1 respectively, with g x (θ ; γ; λ ) := (g x 1 (θ ; γ; λ ) ; ::: ; g x I (θ ; γ; λ )) ;

(13) g z (θ ; γ; λ ) := (g z 1 (θ ; γ; λ ) ; ::: ;

g z I (θ ; γ; λ )) : (14)
The indirect utilities V i : P ! R are thus obtained as

V i (θ ; γ; λ ) := u (g x i (θ ; γ; λ ) ; g z i (θ ; γ; λ ) ; θ i ) ; i 2 I : (15)

THE REDUCED-FORM PROBLEM

The optimal non-linear income tax problem involves two sets of control variables, gross income z and net income x: It can however be transformed into a reduced-form problem in which the policymaker chooses only one of these variables. The reduced-form problem makes it easier to interpret the social value function, to interpret the optimality conditions and to derive comparative static results. For this purpose, Problem 1 is separated into two subproblems. In the rst one, gross income is arbitrarily chosen within the set of incentive-feasible gross income levels Z .

Subproblem 1. Given a gross income vector z 2 Z and the parameters (θ ; γ; λ ) 2 P; choose the consumption vector x 2 R I + to maximize the social welfare function W (a) subject to the self-selection constraints (7) and the tax revenue constraint (8).

Let X (z; θ ; γ; λ ) be the set of maximizers. Then, if there is a unique consumption vector

x (z; θ ; γ; λ ) in X (z; θ ; γ; λ ), the solution in z to Problem 1 is obtained as arg max z2Z W (x (z; θ ; γ; λ ) ; z) : (16)
So, the reduced-form problem can be stated as follows.

Subproblem 2. Given the parameters (θ ; γ; λ ) 2 P; choose z 2 Z to maximize the social welfare function W (x (z; θ ; γ; λ ) ; z).

For this two-stage reasoning to hold, it remains to clarify why all implications of the selfselection constraints, except z 2 Z , are taken into account in Subproblem 1 and to establish that the function x (z; θ ; γ; λ ) is unique and differentiable.

Implications of the Self-Selection Constraints

The self-selection constraints (7) place structure on the solution to Problem 1. Indeed, incentive compatibility of the income tax schedule requires the indirect utility to increase at a speci c rate and the gross income to be non-decreasing in productivity. These restrictions can be used to derive suf cient conditions under which an allocation a satis es the incentive-compatibility constraints (7). We proceed in two steps.

First, if an allocation a satis es (7), then gross income and net income must be non-decreasing in productivity, i.e. (x 1 ; z 1 ) :::

(x I ; z I ) ; (17)
with

(x i 1 ; z i 1 ) < (x i ; z i ) if (x i 1 ; z i 1 ) 6 = (x i ; z i
) ; i = 2; :::; I: Therefore, the set Z in which the solution in z to Problem 1 must lie is de ned as

Z := z 2 R I j0 z 1 ::: z I : (18)
The condition that z belongs to Z corresponds to the second-order condition for incentive compatibility derived in the continuum model.

Second, given z 2 Z , a suf cient condition for an allocation to satisfy the incentive compatibility constraints (7) employs the concept of simple monotonic chain to the left. Following [START_REF] Guesnerie | Nonlinear Pricing in a Finite Economy[END_REF], a simple monotonic chain to the left is an allocation a such that u (x i+1 ; z i+1 ; θ i+1 ) = u (x i ; z i ; θ i+1 ) ; i = 1; :::; I 1:

Given quasilinear-in-consumption preferences, ( 19) is equivalent to

u (x i+1 ; z i+1 ; θ i+1 ) u (x i ; z i ; θ i ) = v z i θ i v z i θ i+1 : (20)
In words, the adjacent downward incentive compatibility constraints are active for all i = 2; :::; I.

Proposition 1. Let an allocation a 2 R I + R I + be a simple monotonic chain to the left and z 2 Z : Then a satis es the incentive compatibility constraints (7).

Proof. [START_REF] Guesnerie | Nonlinear Pricing in a Finite Economy[END_REF].

This pattern expresses a speci c ef ciency/rent-extraction trade-off. Indeed, (20) expressed at which rate utility must be increased for the tax schedule to induce individual truthtelling. For each pair of adjacent productivity levels (θ i ; θ i+1 ) ; this rate basically depends on

R i+1 := v z i θ i v z i θ i+1 ; i = 1; :::; I 1; (21)
which may be regarded as the marginal rent the policymaker has to leave to the more productive individuals because of the informational externality. Consequently, (20) constitutes the discrete analogue of the rst-order condition for incentive compatibility obtained in the models with a continuum of individuals.

Optimal Consumption Given Fixed Levels of Income

The properties of the solution in x to Subproblem 1 are now investigated. The next lemma establishes that there exist solutions to Subproblem 1 for all gross income vector z 2 Z and all (θ ; γ; λ ) 2 P. In addition, each of them is a simple monotonic chain to the left for which the tax revenue constraint ( 8) is binding.

Lemma 1. Given z 2 Z and (θ ; γ; λ ) 2 P; there is at least one solution to Subproblem 1 and any allocation a = (x ; z) where x 2 X (z; θ ; γ; λ ) ; is a simple monotonic chain to the left which is production ef cient.

Proof. See the Appendix.

The implications are twofold. First, combined with Proposition 1, Lemma 1 ensures that all implications of the incentive-compatibility constraints (7) are embedded in any solution to Subproblem 1, provided z 2 Z . Second, the fact that a is a simple monotonic chain to the left gives rise to a speci c consumption pattern. Indeed, by ( 19),

x i = x i 1 + 1 γ v z i θ i v z i 1 θ i ; i = 2; :::; I; (22)
and so

x i = x 1 + 1 γ i ∑ j=2 v z j θ j v z j 1 θ j ; i = 2; :::; I: (23)
As any solution to Subproblem 1 is production ef cient, by Lemma 1, the binding tax revenue constraint (8) can be substituted in ∑ I i=1 x i ; obtained from (23), to get

I ∑ i=1 z i = Ix 1 + 1 γ I ∑ i=2 i ∑ j=2 v z j θ j v z j 1 θ j = Ix 1 + 1 γ I ∑ i=2 (I + 1 i) v z i θ i v z i 1 θ i : (24)
This equation admits a unique solution in x 1 : Substituting the latter in ( 23) and proceeding sequentially show that there is a unique consumption vector in X (z; θ ; γ; λ ), which is independent of the social weights λ and inherits the differentiability properties of v:

Proposition 2. Given z 2 Z and (θ ; γ; λ ) 2 P; the unique function solution to Subproblem 1 is twice continuously differentiable, de ned by x :

Z R I ++ R ++ ! R I + with x 1 (z; θ ; γ) = 1 I ( I ∑ j=1 z j 1 γ I ∑ j=2 (I + 1 j) v z j θ j v z j 1 θ j ) ; ( 25
)
x i (z; θ ; γ) = x 1 (z; θ ; γ) + 1 γ i ∑ j=2 v z j θ j v z j 1
θ j ; i = 2; :::; I:

(26)

The Reduced Form

We can now take stock of the previous results to give a more compact formulation of Subproblem 2. For this purpose, it is convenient to introduce the new vector of social parameters β = (β 1 ; :::; β I ) with

β i := Λ (θ i ) i; i 2 I : (27)
Because of ( 10) and ( 11), the graph of i ! Λ (θ i ) is hump-shaped and above the 45 -line. Hence, β i > 0 for all i = 1; :::; I 1: The parameters β i summarize in a transparent way the redistributive taste of the government.

First, all λ i would be equal if the government adopted pure utilitarianism as a social objective.

In this case, β i = 0 for every i: Consequently, the social parameters β i express the policymaker's strict aversion to income inequality. Second, to get further insight into β i ; it is instructive to consider the effects of the government's decision to give each of the i less productive individuals one extra euro of consumption. Since γ is the marginal utility of money, the utility of each of them is increased by γ: Accordingly, the gross social bene t amounts to γΛ i (θ i ) : However, the tax revenue is decreased by i euros, which corresponds to a social cost γi: Summing both effects, it appears that γβ i is the net social bene t of marginally increasing the consumption of the i less skilled individuals. So, β i is this net social bene t expressed in monetary units. Third, the parameters β i can alternatively be de ned as

β i := I i I ∑ j=i+1 λ j ; i = 1; :::; I 1; ( 28
)
and β I = 0; because Λ (θ I ) = I. They thus also corresponds to the net social cost, expressed in euros, of marginally increasing the consumption of the I i most productive individuals. That is why they are henceforth referred to as net cumulative social weights. They allow us to rewrite the reduced-form optimal non-linear income tax problem (i.e. Subproblem 2) as follows.

Problem 2 (Reduced Form). For (θ ; γ; λ ) 2 P, choose z in Z so as to maximize the social objective function W (z; θ ; γ; λ ) :

Z P ! R, with 1 W (z; θ ; γ; λ ) := I ∑ i=1 γz i v z i θ i I ∑ i=1 β i R i+1 : (29)
This problem is called a reduced form of Problem 1 because the optimal solution in gross income of the former, and the consumption pattern it generates through Proposition 2, are the optimal solutions of the latter.

Proposition 3. For (θ ; γ; λ ) 2 P, the optimal solution to Problem 2 is g z (θ ; γ; λ ), the gross income vector solution to Problem 1, and the optimal consumption vector for Problem 1 is

g x (θ ; γ; λ ) = x (g z (θ ; γ; λ ) ; θ ; γ) : (30)
Proof. See the Appendix.

An important implication of Proposition 3 is that the social allocation solution to the optimal non-linear income tax problem is a monotonic chain to the left. In consequence, the optimal tax schedule is not differentiable at each observed gross income level z i : It is nevertheless possible to use the differentiability of the indifference curves in order to de ne implicit marginal tax rates. Since at the optimum only the adjacent downward self-selection constraints are binding, two implicit marginal tax rates are of particular interest at each observed gross income level z i :

the implicit marginal tax rate T 0 (z i ; θ i ) faced by the θ i -individual for whom the (x i (z i ; θ ; γ) ; z i )bundle is designed, on the one hand, and the implicit marginal tax rate T 0 (z i ; θ i+1 ) the nearest more productive θ i+1 -individual would face if he were mimicking the θ i -individual. They are formally de ned as

T 0 (z i ; θ j ) := 1 s (z i ; θ j ) = 1 v 0 (z i =θ j ) γθ j ; (i; j) 2 I 2 : (31)
The implicit marginal tax rates allows us to get further understanding of the social objective function of the reduced-form optimal income tax problem W (z; θ ; γ; λ ) : Indeed, let z be a xed gross income vector and consider that the gross income z i of the θ i -individual is increased at the 1 Since β I = 0; R I+1 is an arbitrary number.

margin. As

∂ W (z; θ ; γ; λ ) ∂ z i = γT 0 (z i ; θ i ) β i ∂ R i+1 ∂ z i ; i 2 I ; (32)
by ( 29) and ( 31), the impact on social welfare may be thought of as proceeding in two steps.

In the rst step, the θ i -individual pays T 0 (z i ; θ i ) additional euros in taxes, which relaxes the tax revenue constraint (8). As γ is the marginal utility of money, the positive effect on social welfare amounts to γT 0 (z i ; θ i ) : In the second step, the effect on incentives is taken into account. The ∂ z i euros. Because γβ i is the net social cost of marginally increasing the consumption of the I i most productive individuals, social welfare is reduced by

β i ∂ R i+1
∂ z i : If the social optimum is interior, it is therefore obtained when the positive effect on social welfare due to the relaxation of the tax revenue constraint offsets the negative one stemming from private information, i.e.

∂ W (z; θ ; γ; λ ) ∂ z i = 0 , T 0 (z i ; θ i ) = β i γ ∂ R i+1 ∂ z i ; i 2 I ; (33)
whose sign is given by β i because γ > 0 and ∂ R i+1 =∂ z i > 0: Consequently, as β i > 0 for i < I and β I = 0; the optimal marginal tax rates T 0 (z i ; θ i ) faced by all but the most productive individuals is strictly positive. The labour supply is thus distorted except at the top.

Characterization of the Social Optimum

Characterizing the social optimum requires to know whether it involves bunching or is fully separating. If the solution z = g z (θ ; γ; λ ) to Problem 2 is such that 0 < z 1 < ::: < z I ; then it must satisfy (33). Using (31), ( 32) and ( 21), this is equivalent to

z i = α 1 i (1=β i ; θ ; γ) ; i = 1; :::; I 1; (34)
where

α i (z i ; θ ; γ) = T 0 (z i ; θ i+1 ) T 0 (z i ; θ i ) T 0 (z i ; θ i ) ; i = 1; :::; I 1; ( 35
)
FIGURE 1: Geometric interpretation of the rst-order conditions: at the optimum and at the (x i ; z i )-bundle, the angle α i between the indifference curves i and i + 1 of the θ i and θ i+1individual respectively is determined by the cumulative social weights β i .

and z I = θ I (v 0 ) 1 (γθ I ) : As α i (z i ; θ ; γ) is a strictly increasing function of z i 2 , z i is a strictly decreasing function of β i ; with z i = α 1 i (1=β i ; θ ; γ). Hence, 0 < z 1 < ::: < z I is possible if and only if 0 < β I < ::: < β 1 : Proposition 4. The optimal allocation is fully separating if and only (θ ; γ; λ ) 2 P 0 := f(θ ; γ; λ ) 2 Pj0 < β I < ::: < β 1 g :

(36)

This necessary and suf cient condition has a very clear interpretation. Bunching does not occur when giving one euro to each of the i less productive individuals is socially more effective than giving one euro to each of the i + 1 less productive ones, for i = 1; :::; I 1: This corresponds to an additional restriction on the individual social weights λ i : From now on, attention is restricted to the case where this restriction is met. The optimality condition can then be written in a strikingly simple form.

Proposition 5. For (θ ; γ; λ ) 2 P 0 ; z is socially optimal if and only if

α i (z i ; θ ; γ) = 1
β i ; i = 1; :::; I 1; (37) 2 As d [T 0 (z i ; θ i+1 ) T 0 (z i ; θ i )] =dz i = 1 γ h v 00 (z i =θ i ) θ 2 i v 00 (z i =θ i+1 ) θ 2 i+1 i > 0 while dT 0 (z i ; θ i ) =dz i < 0; (35) implies
dα i (z i ; θ ; γ) =dz i > 0 for i = 1; :::; I 1:

and T 0 (z I ; θ I ) = 0: For a gross income z i ; α i (z i ; θ ; γ) tells us to which extent the indifference curves of the θ i+1individual must be atter than those of the θ i -individual. Geometrically, it thus corresponds to the angle between the tangents to the indifference curves of the θ i and θ i+1 -individuals depicted in Figure 1. This angle is closely related to the single-crossing condition and thus henceforth referred to as the Spence-Mirrlees angle. The single-crossing condition is a restriction on the sign of α i (z i ; θ ; γ) ; which must be strictly positive. Here, this condition is automatically satis ed because individual preferences are quasilinear in consumption. The conditions for social optimality (37) introduce an additional restriction on α i (z i ; θ ; γ) : an allocation is socially optimal only if, at each observed gross income level z i , the Spence-Mirrlees angle is entirely determined by the exogenously given cumulative social weight β i : In addition, the labour supply of the more productive individuals is not distorted since the marginal tax rate at the top is equal to zero.

Thanks to Proposition 5, the optimal allocation can be constructed geometrically in two steps illustrated in Figure 2. In the rst step, the tax revenue constraint is ignored. Starting from zero, gross income is gradually increased until α 1 = 1=β 1 and the bundle (x 1 ; z 1 ) is determined.

Then, gross income is increased along the indifference curve of the θ 2 -individual through the (x 1 ; z 1 )-bundle until the angle with the indifference curve of the θ 3 -individual is equal to 1=β 2 :

Proceeding recursively, a monotonic chain to the left (x; z) is obtained. This allocation is incentive compatible, but not necessarily budget-balanced. That is why, in the second step, each x i is varied by a same amount ε so as to get a binding tax revenue constraint. The resulting allocation (x + ε; z) ; which is both incentive compatible and production ef cient, is socially optimal.

Before going further and derive comparative static properties, it is instructive to examine one main source of differences between our results and those derived in Weymark (1986aWeymark ( ,b, 1987)).

In the latter papers, the quasilinear-in-leisure utility function u (x i ; z i ; θ i ) := h (x i ) γz i =θ i is replaced by its monotone transform e u (x i ; z i ; θ i ) = θ i h (x i ) γz i in order to sum e u (x i ; z i ; θ i ) over all i in I , get

I ∑ i=1 e u (x i ; z i ; θ i ) = I ∑ i=1 θ i h (x i ) γ I ∑ i=1 z i ; (38)
and replace ∑ I i=1 z i by ∑ I i=1 x i : This step is required to obtain a reduced-form optimal income tax problem. Consequently, skill-normalized social weights e λ i := λ i =θ i are used in the social objective ∑ I i=1 e λ i e u (x i ; z i ; θ i ) : The rst-order conditions of the reduced-form problem involve therefore skilled-normalized cumulative social weights ∑ i j=1 e λ j instead of Λ i . So, the impact of the policymaker's taste for redistribution is less transparent because social weights and productivity levels are mixed together.

FIGURE 2: Two-step geometric construction of the optimal allocation.

Step 1: the simple monotonic chain to the left (x; z) is obtained by setting α i = 1=β i for i = 1; 2 and choosing z 3 such that the θ 3 -individual is not taxed at the margin.

Step 2 : (x; z) is translated by (0; ε) so as to obtain a production ef cient allocation.

COMPARATIVE STATIC PROPERTIES

Besides providing a geometric interpretation of the optimality conditions, the reduced form makes it possible to derive comparative static results of the optimal income tax allocation. For this purpose, it is rst necessary to examine the differentiability properties of the main variables.

Since the disutility of labour v is C 2 , the implicit function theorem implies that g z i (θ ; γ; λ ) is C 1 : It thus follows from Proposition 2 that g x i (θ ; γ; λ ) = x (g z i (θ ; γ; λ ) ; θ ; γ; λ ) is also C 1 : These results can be summarized as follows.

Proposition 6. The functions g x ; g z ; T 0 and V are C 1 at every (θ ; γ; λ ) in P 0 :

The effects of changing an underlying parameter at the margin can now be investigated.

Comparative Statics for the Marginal Utility of Money

The marginal utility of money corresponds to the unit of count in welfare of our economy.

Namely, one additional euro of consumption for a given individual increases his well-being by γ utils. Varying γ at the margin is thus likely to modify the individual labour supply.

Proposition 7. For (θ ; γ; λ ) 2 P 0 and i 2 I ;

∂ g z i (θ ; γ; λ ) ∂ γ > 0: (39)
Proof. See the Appendix.

A small increase in the marginal utility of money γ raises the gross income of every individual. Indeed, when γ goes up, an extra unit of consumption contributes more to individual well-being as previously. So, the indifference curves become atter in the (z; x)-space. Every θ iindividual is thus willing to work more in order to increase his consumption by a given amount.

Unfortunately, the comparative statics of the optimal implicit marginal tax rates cannot be obtained in the general case. In fact, as T 0 (z i ; θ j ) = 1 v 0 (z i =θ j ) = (γθ j ) ; the reduction in 1=γ goes in the opposite direction to the associated increase in gross income.

Comparative Statics for Individual Productivities

Varying the skill levels has more subtle effects on the optimal allocation. This is of particular interest since productivities are probably the most basic ingredients of the Mirrleesian optimal income tax model. They are indeed the sole source of heterogeneity within the population and give rise to the adverse selection problem which is the key of Mirrleesian income taxation. The fact that the productivity vector θ is strictly monotonically increasing ensures that (1) remains satis ed once a given individual productivity is changed at the margin. The effects of a variation in θ i+1 can be summarized as follows.

Proposition 8. For (θ ; γ; λ ) 2 P 0 and (i; j) 2 f1; :::; I 1g I ;

∂ T 0 (z i ; θ i ) ∂ θ i+1 > 0; ∂ T 0 (z i ; θ i+1 ) ∂ θ i+1 > 0; (40) ∂ T 0 (z i+1 ; θ i+1 ) ∂ θ i+1 < 0; ∂ T 0 (z i+1 ; θ i+2 ) ∂ θ i+1 < 0; (41) ∂ T 0 (z j ; θ j ) ∂ θ i+1 = ∂ T 0 (z j ; θ j+1 ) ∂ θ i+1 = 0 for j = 2 fi; i + 1g ; ( 42
)
and

∂ z i ∂ θ i+1 < 0; (43) ∂ z i+1 ∂ θ i+1 > 0; (44) ∂ z j ∂ θ i+1 = 0 for j = 2 fi; i + 1g ; (45)
where z g z (θ ; γ; λ ) :

Proof. See the Appendix.

Increasing the productivity of the θ i+1 -individual does only alter his gross income and that of his nearest less productive neighbour. Indeed, by Proposition 5, only α i and α i+1 depend on θ i+1 : So, the optimality condition α j = 1=β j , which implicitly de nes z j as a function of θ i+1 ; is unaffected except for the θ i and θ i+1 -individuals. Accordingly, the gross income levels of all other individuals remain unaltered. As regards the θ i and θ i+1 -individuals, the adjustment process combines three effects.

First, the variation in θ i+1 gives rise to a local substitution effect. The increase in the productivity of the θ i+1 -individual results in a rise in his net-of-tax wage rate, which leads him to increase his labour supply in ef ciency units, z i+1 :

Second, changing θ i+1 has an incentive effect. As he becomes more ef cient, the θ i+1individual has to provide less effort if he wants to imitate the θ i -individual. Consequently, his indifference curve through the gross-income/consumption bundle of the θ i -individual attens.

This corresponds to an increase in the implicit marginal tax rate T 0 (x i ; z i ; θ i+1 ) he would face if he were cheating.

Third, the θ i -individual incurs an informational externality induced by the incentive effect.

Since the cumulative social weight β i is unaltered, the angle α i between the indifference curves of the θ i and θ i+1 -individuals through the (x i ; z i )-bundle must stay constant (Proposition 5).

Consequently, the increase in T 0 (x i ; z i ; θ i+1 ) must be associated with an increase in the implicit marginal tax rate T 0 (x i ; z i ; θ i ) and thus with a reduction in the net-of-tax wage rate of the θ iindividual. Finally, the substitution effect leads the θ i -individual to work less.

The changes in gross income ensure that a new monotonic chain to the left is obtained.

However, this incentive-compatible allocation is not necessarily budget-balanced. Therefore, in a second step, the consumption levels are adjusted in order to obtain a production-ef cient allocation. However, the comparative static results as regards consumption cannot be derived in the general case.

Comparative Statics for the Social Weights

The geometric characterization of the solution to the optimal income tax problem found in Proposition 5 basically involves the cumulative social weights β i , and thus the individual social weights λ i : As emphasized previously, these social weights express the government's strict aversion to income inequality. Changing them marginally is thus likely to alter the progressivity of the tax schedule.

The impact of a change in the cumulative social weight β , with i < I3 ; is examined rst because it will be useful when deriving the comparative statics with respect to the individual social weights. As z i = α 1 i (1=β i ; γ; λ ) where α 0 i > 0; an increase in β i is associated with a reduction in z i : Indeed, given T 0 (z i ; θ i+1 ) ; an increase in β i requires the implicit marginal tax rate T 0 (z i ; θ i ) faced by the θ i -individual to be raised. The induced substitution effect leads the θ i -individual to work less. Proposition 9. For (θ ; γ; λ ) 2 P 0 and (i; j) 2 f1; :::; I 1g I with i 6 = j;

∂ g z i (θ ; γ; λ ) ∂ β i < 0 and ∂ g z j (θ ; γ; λ ) ∂ β i = 0 (46)
Proof. See the Appendix.

This result can now be used to consider the impact of an increase in the individual social weight of the θ i -individual to the detriment of a more productive θ j -individual. By de nition of Λ (θ k ) ; every β k is increased for k 2 fi; :::; j 1g while all other β k remain unaltered. By Proposition 9, it is thus optimal to decrease the gross income z k of each θ k -individual, with k 2 fi; :::; j 1g ; and to hold that of the others constant. The impact on the consumption levels and indirect utilities can also be signed for all k = 2 fi; :::; j 1g : Proposition 10. Let θ ; γ; λ 2 P 0 ; i 2 f1; :::; I 1g and j 2 fi + 1; :::; Ig : Let λ : S ! R I ;

where S = ( 1; 1) ; be C 1 with 8 > < > :

λ k (0) = λ k ; k = i; j; λ k (s) λ k ; 8s 2 S; 8k 6 = i; j; dλ i (s) =ds = dλ j (s) =ds; 8s 2 S: (47)
Then, if λ i is increased to the detriment of λ j4 , ( dz k =ds < 0; 8k 2 fi; :::; j 1g ; dz k =ds = 0; 8k = 2 fi; :::; j 1g ;

( dx k =ds > 0; 8k < i; dx k =ds < 0; 8k j; (48)
( dV k =ds > 0; 8k < i; dV k =ds < 0; 8k j; (49)
where z g z θ ; γ; λ and x x z; θ ; γ :

Proof. See the Appendix.

Before interpreting these results, it is worth examining the impact of this change in the social weights on the implicit optimal marginal tax rates. By (31), they only depend on λ through gross income z: It thus follows from (48) that T 0 x k ; z k ; θ k and T 0 x k ; z k ; θ k+1 are unaltered for k = 2 fi; :::; j 1g whilst T 0 x k ; z k ; θ k and T 0 x k ; z k ; θ k+1 are increased for k 2 fi; :::; j 1g :

For concreteness, let us consider that the population consists of three individuals and that the social weight of the θ 2 -individual is increased at the expense of the θ 3 -individual. Let (x; z) be the initial allocation and denote by (x; z) the new one. The changes in gross income have been explained previously. The adjustments in consumption can be thought of as proceeding in two steps. In the rst step, the budget constraint ( 8) is left aside. By (48), the gross income levels of the θ 1 and θ 3 -individuals are held xed, i.e. z 1 = z 1 and z 3 = z 3 ; while z 2 is reduced (by dz 2 ): In consequence, the requirement that the θ 3 -individual is indifferent between his own bundle and that of the θ 2 -individual induces a decrease in the consumption levels x 2 and x 3 of both more productive individuals (by dx 2 and dx 3 respectively) as well as in the indirect utility of the θ 3 -individual. A new monotonic chain to the left is obtained. As the θ 2 -individual faces a strictly positive marginal tax rate, he reduces his gross income by a smaller amount than his consumption, i.e. dx 2 > dz 2 : So, dx 3 can be suf ciently small for 3

∑ i=1 x i dx 2 dx 3 < 3 ∑ i=1 z dz 2 ; (51)
which means that the new monotonic chain to the left is not production ef cient. As ∑ 3 i=1 x i = ∑ 3 i=1 z, the second step consists therefore in giving

ε = 1 3 (dx 2 + dx 3 dz 2 ) > 0 (52)
euros of consumption to each individual. The new consumption levels are thus the following:

x 1 = x 1 + 1 3 (dx 2 + dx 3 dz 2 ) > x 1 ; (53)
x 2 = x 2 2 3 dx 2 + 1 3 (dx 3 dz 2 ) ; (54)
x 3 = x 3 2 3 dx 3 + 1 3 (dx 2 dz 2 ) < x 3 : (55

CONCLUSION

Thanks to the absence of income effects on labour supply, the trade-off between equity and efciency is very pure when individual preferences are quasilinear in consumption. This case has been investigated in depth in the continuous population version of Mirrlees model [START_REF] Atkinson | Public Economics and the Economic Public[END_REF], [START_REF] Diamond | Optimal Income Taxation: An Example with U-Shaped Pattern of Optimal Marginal Tax Rates[END_REF], [START_REF] Piketty | La redistribution scale face au chômage[END_REF], [START_REF] Salanié | Un Exemple de Taxation Optimale[END_REF][START_REF]L'imposition optimale du revenu: une application au cas français[END_REF]), but the analysis carried out for a nite population has concentrated on the situation where preferences are quasilinear in leisure. In this extent, the present paper contributes to lling this gap.

When preferences are quasilinear in consumption, it is not necessary to work with skillednormalized social weights. Therefore, the respective in uences of individual productivities and social weights are easier to separate in the social objective function of the reduced-form optimal income tax problem. This offers two advantages. First, the link between the social weights and the conditions for social optimality is very transparent. Second, clear-cut comparative statics properties can easily be derived as regards changes in the productivity levels which are the key parameters of the optimal income tax model.

APPENDIX

Proof of Lemma 1. The proof proceeds in three steps.

(i) a is a simple monotonic chain to the left.

The proof is ad absurdum. Assume a is not a simple monotonic chain to the left, i.e. u x j ; z j ; θ j 6 = u x j 1 ; z j 1 ; θ j : As u x j ; z j ; θ j < u x j 1 ; z j 1 ; θ j implies u x j ; z j ; θ j < u x j 1 ; z j 1 ; θ j 1 which violates (7), our assumption is equivalent to considering that there exists j 2 for which

γx j v z j θ j > γx j 1 v z j 1 θ j : (56)
By ( 17), it must be x j > x j 1 and z j > z j 1 for (56) to be satis ed. If z does not satisfy the second inequality, then (i) is established.

Let x i = x i + ε 1 for i = 1; :::; j 1 and x i = x i ε 2 for i = j; :::; I; where ε 1 > 0 and ε 2 > 0:

The incentive compatibility constraints remain satis ed for suf ciently small ε 1 and ε 2 : ε 1 and ε 2 are chosen such that x = x ; i.e.

( j 1) ε 1 = (I j + 1) ε 2 : (57)

Therefore, using (57), one gets:

W (x; z) W (x ; z) = γ I ∑ i=1 λ i (x i x i ) = γ " j 1 ∑ i=1 λ i ε 1 I ∑ i= j λ i ε 2 # ; (58)
which can be minored thanks to (10):

W (x; z) W (x ; z) γ [λ j 1 ( j 1) ε 1 λ j (I j + 1) ε 2 ] = γ (I j + 1) ε 2 [λ j 1 λ j ] > 0: (59)
This contradicts the fact that x is in X (z; θ ; γ; λ ) :

(ii) a is production ef cient.

Fix z in Z . The self-selection constraints (7) are satis ed, with z i = z i and x i = x i for every i 2 I : The proof proceeds by contradiction. Assume (8) is not binding. Hence, (8) remains satis ed if all x i are increased by a suf ciently small ε > 0: But the the self-selection constraints (7) are equivalent to

γx i + γε v z i θ i γx j + γε v z j θ i ; 8 (i; j) 2 I 2 : (60)
Increasing every x i by ε is thus incentive compatible. Since it is also Pareto improving, x cannot be in X (z; θ ; γ; λ ) : A contradiction.

(iii) Existence of an optimal allocation.

Fix z in Z and pick an arbitrary value x 1 for x 1 : Then, proceed sequentially to select x which solves

x i+1 = x i + 1 γ v z i+1 θ i+1 v z i θ i+1 ; i = 1; :::; I 1: (61)
By construction, a is a monotonic chain to the left. Since z 2 Z , it follows from Proposition 1 that a satis es (7). If a does not satisfy (8), it is suf cient to change each x i by a suf cient large amount. If ( 8) is not binding, the argument used in (i) applies: it is suf cient to increase each x i by a well-chosen ε > 0: Consequently, the constraint set is not empty. Now, write (7) for j = 1 to obtain

γx i v z i θ i γx 1 v z 1 θ i , γ [x i x 1 ] v z i θ i v z 1 θ i 0; 8i 2 I : (62)
Since x 1 0; all x i are bounded from below. In addition, all x i must be bounded from above for (8) to be binding. Consequently, if the set X (z; θ ; γ; λ ) is non-empty, it is a bounded subset of the feasible set.

Finally, W is continuous while the constraint set is compact (because the inequalities are weak)

and non-empty. Hence, by Weierstrass theorem, X (z; θ ; γ; λ ) 6 = / 0.

Proof of Proposition 3. It is suf cient to establish that substitution of x (z; θ ; γ) into W yields W for all z 2 Z : By (20),

u (x i ; z i ; θ i ) = u x i 1 ; z i 1 ; θ i 1 + v z i 1 θ i 1 v z i 1 θ i ; i = 2; :::; I; (63) from which u (x i ; z i ; θ i ) = u (x 1 ; z 1 ; θ 1 ) + i 1 ∑ j=1 v z j θ j v z j
θ j+1 ; i = 2; :::; I:

Consequently,

I ∑ i=1 u (x i ; z i ; θ i ) = Iu (x 1 ; z 1 ; θ 1 ) + I 1 ∑ i=1 (I i) v z i θ i v z i θ i+1 : (65)
In addition, summing (4) over i on I and employing (8),

I ∑ i=1 u (x i ; z i ; θ i ) = γ I ∑ i=1 z i I ∑ i=1 v z i θ i : (66)
Plugging ( 66) in ( 65) and solving for u (x 1 ; z 1 ; θ 1 ) ;

u (x 1 ; z 1 ; θ 1 ) = 1 I ( γ I ∑ i=1 z i I ∑ i=1 v z i θ i I 1 ∑ i=1 (I i) v z i θ i v z i θ i+1 ) : (67)
By ( 64) and ( 67), the social value function reads

W = u (x 1 ; z 1 ; θ 1 ) I ∑ i=1 λ i + I ∑ i=2 i 1 ∑ j=1 λ i v z j θ j v z j θ j+1 ; = u (x 1 ; z 1 ; θ 1 ) I + I 1 ∑ i=1 I ∑ j=i+1 λ j ! v z i θ i v z i θ i+1 ; = I ∑ i=1 ( γz i v z i θ i + i i ∑ j=1 λ j ! v z i θ i v z i θ i+1 ) ; (68)
in which v (z I =θ I+1 ) is an arbitrary number.

Proof of Proposition 7. Let z i g z i (θ ; γ; λ ) and write the rst-order conditions given in Proposition 5 as Proof of Proposition 8. Let z i g i z (θ ; γ; λ ) and use (69) to de ne φ θ j (z j ; θ i+1 ) = 0 for j = 1; :::; I 1: Hence,

γ 1 + β i θ i v 0 z i θ i + β i θ i+1 v 0 z i θ i+1 = 0; i =
∂ φ θ i (z i ; θ i+1 ) ∂ θ i+1 = β i θ 2 i+1 v 0 z i θ i+1 + z i θ i+1 v 00 z i θ i+1 < 0; (72) ∂ φ θ i+1 (z i+1 ; θ i+1 ) ∂ θ i+1 = 1 + β i+1 θ 2 i+1 ! v 0 z i+1 θ i+1 + z i+1 θ i+1 v 00 z i+1 θ i+1 > 0; (73)
∂ φ θ j (z j ; θ i+1 ) ∂ θ i+1 = 0 for j = 2 fi; i + 1g :

By the implicit function theorem, for every θ i+1 , i = 1; :::; I 1; φ θ j (z j ; θ i+1 ) = 0 has a unique solution which de nes z j as a C 1 -function z j = ϕ θ j (θ i+1 ) ; with derivative

dϕ θ j (θ i+1 ) dθ i+1 = ∂ φ θ j (z j ; θ i+1 ) =∂ θ i+1 ∂ φ θ j (z j ; θ i+1 ) =∂ z j : (75)
As ∂ φ θ j (z j ; θ i+1 ) =∂ z j ∂ φ γ j (z j ; γ) =∂ z j < 0 by (70), it follows from ( 72)-( 75) that Proof of Proposition 9. Let z i g i z (θ ; γ; λ ) and use (69) to de ne 5 to de ne φ β j (z j ; β i ) = 0 for j = 1; :::; I 1: Since θ i < θ i+1 and v 00 > 0;

∂ g z j (θ ; γ; λ ) ∂ θ i+1 dϕ θ j (θ i+1 ) dθ i+1 = 8 > < > : < 0 if j = i; > 0 if j = i + 1; = 0 if j = 2 fi; i + 1g : (76)
∂ φ β j (z j ; β i ) ∂ β i = ( 1 θ i v 0 z i θ i + 1 θ i+1 v 0 z i θ i+1 < 0 if j = i; 0 otherwise. ( 82
)
By the implicit function theorem, for every θ i+1 , i = 1; :::; I 1; φ β j (z j ; β i ) = 0 has a unique solution which de nes z j as a C 1 -function z j = ϕ Proof of Proposition 10. Since β k is increased for all k 2 fi; :::; j 1g and unaltered otherwise, Proposition 9 implies (48). We then prove (49). By Proposition 2,

x k (z; θ ; γ) = 1 I " I ∑ h=1 z h 1 γ I ∑ h=2 (I + 1 h) v z h θ h v z h 1 θ h # + 1 γ k ∑ h=2 v z h θ h v z h 1 θ h : (85)
For k < i; differentiating (85) and using (48),

dx k (z; θ ; γ) ds = 1 I j 1 ∑ h=i 1 1 γθ h v 0 z h θ h + I h γ 1 θ h+1 v 0 z h θ h+1 1 θ h v 0 z h θ h dz h ds = 1 I j 1 ∑ h=i T 0 (z h ; θ h ) β h [β h I + h] dz h ds : (86)
As, for h = 1; :::; j 1; (i) β h > 0, (ii) β h I + h < 0 by ( 28) and ( 1), (iii) T 0 (z h ; θ h ) > 0 [cf.

(33)] and (iv) dz h =ds < 0; one gets dx k (z; θ ; γ) =ds > 0 for k < i.

For k j; d ds

( 1

γ k ∑ h=2 v z h θ h v z h 1 θ h ) = 1 γ j 1 ∑ h=i 1 θ h v 0 z h θ h 1 θ h+1 v 0 z
As, for h = i; :::; j 1; (i) β h + h = Λ (θ h ) > 0; (ii) T 0 (z h ; θ h ) > 0 [cf. (33)] and (iii) dz h =ds < 0, one obtains dx k (z; θ ; γ) =ds < 0 for k j:

By ( 15), ( 50) is a direct implication of ( 49) and ( 48).

  θ iindividual receives 1 T 0 (z i ; θ i ) extra euro of consumption. As a result, the I i more productive individuals have to sacri ce less consumption when they decide to mimic the θ i -individual. So, cheating becomes more attractive to them. In order to restore individual truthtelling and obtain a new monotonic chain to the left, the policymaker has to increase the marginal information rent left to each of them by 1 γ ∂ R i+1

  ) Hence, the θ 1 -individual enjoys greater consumption, contrary to the θ 3 -individual. The change in the consumption of the θ 2 -individual is ambiguous. It is positive if and only if dz 2 < dx 3 2dx 2 : The variations (50) in the indirect utilities directly follow from those in gross income and consumption.

  i ; γ) = 0: Since v 000 > 0 and 0 < θ i < θ i+1 ; function theorem, for every γ 2 R ++ ; φ γ i (z i ; γ) = 0 has a unique solution which de nes z i as a C 1 -function z i = ϕ γ i (γ) ; with derivative ∂ g z i (θ ; γ; λ )

∂

  T 0 (z j ; θ j+1 ) ∂ θ i+1 = 0 for j = 2 fi; i + 1g : z i ; θ i ) ; i = 1; :::; I 1; (80)where β i > 0: Therefore, (77) and (78) imply∂ T 0 (z i ; θ i+1 ) ∂ θ i+1 > 0 and ∂ T 0 (z i+1 ; θ i+1 )

:

  j ; β i ) =∂ β i ∂ φ β j (z j ; β i ) =∂ z j j ; β i ) =∂ z j ∂ φ γ j (z j ; γ) =∂ z j < 0 by (70), it follows from (82)-(83) that ∂ g z j (θ ; γ; λ )

A change in β I is impossible since, by de nition, β I 0:

The inequalities in (48)-(50) are reversed if λ i is decreased to the bene t of λ j :


Antoine Billot

Social consistency and individual rationality

Keywords: A13, D71, J15 preferences, social interaction, communities, group loyalty JEL Classi…cation: A13-D71-J15 Preferences, social interaction, communities, group loyalty

This paper aims at proving that social interactions can easily be rationalized by individual preferences as de…ned in standard microeconomic theory. For that purpose, we show individual choice rationality to be logically equivalent to social consistency, when individual rationality means that individual preferences are completely ordered and social consistency that there is a one-to-one mapping between a given family of social communities and the existence of a particular (unique, re ‡exive and symmetric) interaction relation between individuals. Moreover, continuity and monotonicity of individual preferences are shown to …t the modeling of group loyalty when group loyalty is de…ned as the ability to freely accept a personal loss for the global gain of a particular population.

Cet article vise à montrer que les interactions sociales peuvent être déduites des relations individuelles de préférence (et vice versa). Nous montrons ainsi que le postulat de rationalité individuelle est logiquement équivalent à celui de cohérence sociale dès lors que le postulat de rationalité individuelle signi…e que les préférences des agents sont des préordres complets et celui de cohérence sociale, qu'il existe une bijection entre chaque système de coalitions et la relation d'interaction (unique, ré ‡exive et symétrique) qui le soutend. De plus, on montre que la continuité et la monotonie des préférences individuelles permettent de modéliser la notion de loyauté communautaire dès lors que la loyauté communautaire est dé…nie en tant que la capacité des agents à accepter une perte personnelle d'utilité au pro…t d'un gain collectif pour une coalition particulière.

1 Introduction 1.1. Social Interaction within Economic Theory. Standard economic theory is generally considered unable to incorporate social phenomena because I thank Friedrich Schneider, Tzachi Gilboa and Jacques-François Thisse for helpful comments and suggestions. For correspondence : Université Panthéon-Assas, Paris 2, 92 rue d'Assas, 75006, Paris, France, e-mail address : billot@u-paris2.fr 1 it rests on the methodological assumption of individualism. Hence, to build on maximizing behavior, most economic models require that individuals' utility must depend on a quite limited set of arguments: while workers are supposed to be concerned with wages, relationships with colleagues or supervisors are assumed to be indi¤erent; while consumers are supposed to be concerned with prices, neighbors'consumptions are assumed to be irrelevant; while voters are supposed to be concerned with candidates, friends'political preferences are assumed to be without in ‡uence. In other words, people are supposed to choose jobs, goods or representatives independently of their social environment. Conformism or mimetism, fashion or tradition are then hardly seen as economic phenomena. As noticed by [START_REF] Postlewaite | The Social Basis of Interdependent Preferences[END_REF]: 'At the same time that singleminded materialistic focus has made economics successful as a discipline, it has led to a belief that economics methodology is inadequate to understand important aspects of human behavior, particularly in settings in which individuals are concerned about the opinion of others.' Even though most economists are sympathetic to the idea that social relations are important, they are generally reluctant to explicitly incorporate them within economic theory. Two reasons could explain this reluctance: …rst, increasing the number of variables which a¤ect individuals'utility necessarily decreases the predictive power of the models, especially if coordination problems arise; second, when interactions produce externalities which can be directly captured within groups (or social networks), interactions can be considered negligible (see Becker, 1971).

1.2. An Emerging Theory of Social Interaction. In the last decade, several authors have nevertheless proposed new themes and new analyses which try to explicitly link sociological considerations and economic ones: for instance, [START_REF] Akerlof | Social Distance and Social Decisions[END_REF] with the concept of social distance, Becker (1991) with that of social in ‡uence, [START_REF] Benabou | Workings of a City: Location, Education, and Production[END_REF] for the notion of community structure, [START_REF] Manski | Identi…cation of Endogenous Social E¤ects: The Re ‡ection Problem[END_REF] for the so-called 're ‡ection problem'... and so forth. Empirical researchers have also investigated the importance of peer e¤ects and neighborhood e¤ects in crime [START_REF] Sacerdote | Crime and Social Interactions[END_REF], unemployment [START_REF] Topa | Social Interaction, Local Spillovers, and Unemployment[END_REF] or group loyalty [START_REF] Luttmer | Group Loyalty and the Taste for Redistribution[END_REF]. The key point of all these contributions, say the emerging theory of social interaction [START_REF] Akerlof | Social Distance and Social Decisions[END_REF](Akerlof , p. 1005)), is the fact that individuals are assumed to be deeply in ‡uenced in choices and economic behaviors by the social environment, that is they are no longer considered as Robinsons but rather as the members of particular groups, class, city... However, the link between individuals and social groups is typically symmetric: individuals'choices are determined by the social environment as well as the environment is determined by individuals'choices (see e.g. Stau¤er and Sahini, 2006). While basic economic theory is just concerned with individual preferences and their consequences on choices, the economic theory of social interaction is also concerned with identities and the way these identities in ‡uence choices (see [START_REF] Kasher | On the Question "Who is a J?": A Social Choice Approach[END_REF]Samet and Schmeidler, 2002): while the former theory only wonders 'what do individuals prefer?', the latter also asks 'who are they?', this last question meaning …rst 'with whom do they interact?' and second 'how does social interaction in ‡uence their choices?'.

Methodologically, the great majority of the theoretical papers in this new …eld have chosen to use a common modeling framework stemmed from the analysis of interacting particle systems in physics: they assumed that any individual's optimal choice varies positively with the average choice of his group. However, several recent contributions (see e.g. [START_REF] Morris | Interaction Games: A Uni…ed Analysis of Incomplete Information, Local Interaction and Random Matching[END_REF]2000) try to emphasize social interactions outside of this current methodology in using another formal analogy: the one existing between populations -or communities -within a social structure and events -or beliefs -within an information structure. Actually, we can observe …rst that a social group is a subset of individuals linked by a social relation while an event is a subset of possible worlds linked by an accessibility relation [START_REF] Kripke | Semantical Analysis of Modal Logic[END_REF][START_REF] Aumann | Agreeing to Disagree[END_REF][START_REF] Billot | A Mixed Knowledge Hierarchy[END_REF]. Second, we can also observe that beliefs and communities are conceptually twins since they are both based on elements -worlds on one side, individuals on the other -which are su¢ ciently joined by a relation -of accessibility on one side, of interaction on the other -to be …nally 'isolated'from the others elements. Then, the basic intuition of this paper is the following: while [START_REF] Kripke | Semantical Analysis of Modal Logic[END_REF] describes these isolated elements as 'indistinguishable'when de…ned as abstract worlds, we describe the individuals as 'interdependent' when preferences are mutually sensitive. This way, we suggest that social interactions between individuals are depending on their preferences as well as the individual preferences are depending on the social interactions. In other words, individual preferences and social interactions are two possible representations of the same evidence. In order to prove it, we establish that the standard assumption in microeconomics of complete ordered individual preferences is logically equivalent to the consistency of the social system, that is the existence of a unique re ‡exive and symmetric binary relation between individuals which perfectly describes all the social interactions.

1.3. Community and Loyalty. In this paper, we use three main tools: interaction, interdependence and group loyalty. The two …rsts are not original. They already exist in the literature [START_REF] Akerlof | Social Distance and Social Decisions[END_REF] and the third one is used as an illustration of the intrinsic relevance of microeconomics to model some social evidences.

An interaction is a social situation where two individuals are su¢ ciently connected to consider themselves as social neighbors and we de…ne a social community as a subset of individuals whose social neighborhood is fully included in the same particular population. When assuming the social in ‡uence to be encapsulated within social neighbors, a social community is therefore an exclusive concept in the sense where it forbids to be in ‡uenced by anyone belonging to another population [START_REF] Lickel | Intuitive Theories of Group Types and Relational Principles[END_REF]. In fact, belonging to what we call a 'native population'-to be gay or hispanic, protestant or catholic -is just a given individual data which does not necessarily imply to belong to the associated social community. 1 Belonging to a social community is much more demanding than belonging to a particular population: it requires to never signi…cantly interact with people of other native populations. An individual may be French and catholic and may have jewish friends, work with muslim people and be married with an Italian woman, i.e. may deeply interact outside his native population. Then, because of his various social interactions, he may be sensitive to various populations' lot, jewish, muslim, European... In this case, it is reasonable to consider that he does not belong to the French catholic community even if he belongs to the French catholic native population. On contrary, a Chinese immigrant who only interacts within his native population (because, for instance, of the language), does not regularly see and meet individuals outside the Chinese population and then he is a natural candidate to be a member of the Chinese community.

Interdependence is used to describe an individual preference focused on a particular population in such a way that this individual feels himself only concerned with what economically happens to this particular population. We then de…ne an economic community as a subset of interdependent individuals, that is a community of economic interests. A Chinese immigrant whose preferences translate that he only cares about the Chinese population issues (and then neglects the other populations'interest) obviously belongs to the Chinese economic community. On the contrary, even if he belongs to the French catholic population, an individual may neglect the very interests of this population and may be particularly sensitive to what happens to the jewish or muslim populations; this way, it is reasonable to consider that he has no real true common interest with French catholics and consequently that he does not belong to the economic community they form.

A priori, the two kinds of communities are conceptually independent since they do not involve the same kind of relationships. Nevertheless, our purpose is precisely to show that this conceptual independence is no longer valid when individual preferences are constrained to satisfy standard properties. More precisely, when individuals are assumed to be rational in the standard sense, that is when their preferences are completely ordered as in an Arrow-Debreu framework, we prove that the economic communities form a social consistent system. This result also proves that the language of preferences can be used to describe some social basic evidences. In order to illustrate this last thesis, we later show how to translate the concept of group loyalty as de…ned by [START_REF] Luttmer | Group Loyalty and the Taste for Redistribution[END_REF] in terms of preferences properties, that is continuity and monotonicity.

1.4. The Results. The organization of the paper is the following: in Section 2, the formal analogy between information structure and social structure is used to de…ne social communities in terms of neighborhoods based on interactions between individuals, that is a re ‡exive and symmetric binary relation. Then, we prove (Theorem 1) that a social system, i.e. the set of all social communities within a great set of individuals called society, is consistent w.r.t the individual interactions if and only if (i) the society is a social community itself and (ii) the social system is monotone. In Section 3, while de…ning an economic community as a subset of individuals who are interdependent in terms of preferences (over alternatives), we prove (Theorem 2) that an economic system, i.e. the set of all economic communities, is consistent w.r.t the individual interactions if and only if individual preferences are completely ordered. This way, we show the conditions under which social interactions can be rationalized by individual preferences. In Section 4, we introduce the notion of social loyalty and prove (i) (Theorem 3) that any neighborhood is made of loyal individuals if preferences are continuous and monotone and (ii) (Theorem 4) that any social community is made of loyal individuals if preferences are continuous and monotone and if individuals are empathic.

Interactions and Social Communities

Consider a …xed …nite set S of n individuals. Call society the set S and population any nonempty element of 2 S . Take as a simple representation of social interaction a binary relation between individuals denoted N : write j 2 N (i) if individual j socially interacts with (i.e. is a social neighbor of) individual i. This way, the subset N (i) de…nes the (social) neighborhood of i. We assume that the interaction N is (1) re ‡exive: 8 i 2 S, i 2 N (i) and ( 2) symmetric: 8i; j 2 S, i 2 N (j) ) j 2 N (i).

One natural way to interpret the social relation N is to consider that two individuals i and j are said to be neighbors if the intensity of their relationship is 'signi…cant': for instance if they spend together a signi…cative part of their time (greater than a given amount under which they consider that there is no genuine relation) but also if they share a signi…cative number of cultural values meaning that they both belong to the same circumscribed population of individuals within S even if they seldom meet. The …rst kind of interaction is rather spatial while the second is rather cultural but in both cases, the concept of interaction translates a certain sort of 'proximity'. Then, what we called 'interaction' throughout these pages is conceptually and even formally more general than Akerlof's since not necessarily related to a distance (even social). Actually, this de…nition leads us to distinguish people who meet at random in the bus or in the street for a short while (who then do not really interact) from people who are truly in interaction such like those who work in the same o¢ ce, those who share regular cultural activities and so forth... We simply propose to de…ne a social community (based on a native population A) as the larger 1-cohesive 2 nonempty subset of A.

Formally, we denote by soc (A) the social community based on native population A: soc (A) = fi 2 S : N (i) Ag .

(1) Indeed, each social community soc (A) forms a more cohesive population than its complement in S. A social community is then a population of individuals locally closed by the social interaction. Each social community is a population of S but each population S does not necessarily form a social community. Empirically, except the set of all human beings, there is nowadays very few 'native populations'which can be said 1-cohesive (except maybe some fully isolated social organizations such like primitive tribes who do not even know that they are not alone on Earth ...); generally, in each potential native population, there exist individuals who socially interact with 'outside'people, that is individuals who then do not belong to the community associated with their native population. In our words, this means that communities are w.l.o.g. smaller than 2 According to Ellison (1993) and [START_REF] Morris | Contagion[END_REF], a population A of individuals is said to be p-cohesive if every member i 2 A is such that there exists at least a proportion p of his social neighbors who belong to A. Hence, a population A is 1-cohesive whenever the subset of all individuals i 2 S such that N (i) A is equal to A. However, for every nonempty population A S, it is possible (at least for a modeler) to point out individuals i 2 S such that N (i) A and to isolate them from individuals j 2 S for which there exist individuals k = 2 A such that k 2 N (j). Note that no one can be sure that the latter individuals are su¢ ciently cohesive to form a true community (since p 2 [0; 1]) while the former ones can always revendicate a strong speci…c link (p = 1).

populations since more demanding (exclusive) in terms of social interactions. 3This criterion -allowing to distinguish a native population from its communityis not purely formal. In most big western cities, Paris, London or New York City..., one can easily observe the di¤erent local communities (jewish, Chinese, Irish, gay...) living in some speci…c areas ('Le Marais'for Parisian gays or 'East London'for Indian people) where the density of the corresponding native populations is so high that the possibility to socially interact with someone belonging to another population is low. Nevertheless, in the same time, one can …nd gays in each Parisian borough even far from Le Marais as well as Indian people in the West London, which illustrates the empirical di¤erence between a population and a community. Note also that nothing prevents to understand the de…nition of a social community in a broader sense where the native population is not racially or sexually but spatially (e.g., a village) or culturally (e.g., a linguistic minority) speci…ed.

Formally, a social system4 is a mapping soc from 2 S towards 2 S . In short, we say that the social system soc represents the social interaction N if (1) holds for any A S.

De…nition 1 : A social system soc is consistent if there exists an interaction N such that soc represents the relation N .

Theorem 1 : A social system soc is consistent if and only if ( A1) the society S forms a social community: soc (S) = S, ( A2) for two populations A; B S such that A B: soc (A) soc (B).

(All proofs are in Appendix.) A1 is called inclusivity and means that the society S is assumed to be 1-cohesive whatever happens to smaller communities. A1 requires that any individual in S knows that S exists and is the greatest social community he belongs to. A1 has also a ‡avor of a 'civil peace' requirement as explicitly shown by Lemma 1 below.

A2 corresponds to monotonicity of the social communities with respect to the size of the associated sequence of nested native populations. This means that a community based on a big native population is bigger than a community based on a smaller nested one. For instance, the American population is obviously bigger than the Afro-American one and the two associated communities, i.e. the individuals who revendicate and essentially de…ne themselves as Americans or Afro-Americans are ranked the same way. This axiom then introduces some minimal regularity in the sociological phenomena we study which seems to be empirically relevant (see e.g. Wellman and Berkowitz, 1988, ch. 5).

Theorem 1 states that if a social system consistently represents an interaction, it satis…es the two conditions.5 Conversely, if a given social system satis…es the two conditions, then there exists an interaction which generates the social system. Actually, the proof illustrates the uniqueness of the interaction for a given system.

Remark 1 : When characterized by N (i) = fj 2 S : i = 2 N (S j )g (for all individuals i of S), the interaction N means that an individual j is a social neighbor of individual i i¤ the complement population S j does not form a social community.

Preferences and Economic Communities

Consider a …xed …nite set X of options a; b; c::: Call an issue any vector x of X n such that the ith coordinate x i corresponds to a particular option of X that is allocated to (or chosen by) individual i. Basically, any option can be allocated to (or chosen by) any individual. The set X n is the n-size set of all possible individual issues. 6 For any population A S, x A denotes the tuple fx i g i2A . Furthermore, if z = (x A ; y A ), it means that z i = x i 2 X for all individuals i 2 A and z j = y j 2 X for all the other individuals j = 2 A. Each individual i 2 S is assumed to rank the issues according to a preference relation < i , that is a complete binary relation de…ned over X n X n : 8 (x; y) 2 (X n ) 2 : x < i y or y < i x: This way, x < i y means that for i, issue x is at least as good as issue y. Two issues x; y are considered indi¤erent, that is denoted x i y when, simultaneously, x < i y and y < i x. A complete relation < is ordered if it satis…es transitivity: for all issues x; y; z 2 X n , x < y and y < z implies x < z. Note that any complete binary relation which is ordered is also re ‡exive: for all issues x 2 X n , x < x.

Let an individual i 2 S be such that

(x A ; y A ) i (x A ; z A ) for all x; y; z 2 X n . (2)
Such an individual i is said to be interdependent with the population A and the subset of all individuals interdependent with A is called the economic community (based on A) and denoted eco(A).

An economic community is a set of individuals who share a common interest for a particular population (without necessarily being members of it). More precisely, the individuals who belong to the same economic community reveal that they are always sensitive to what happens to a particular population while never sensitive to the rest of the society; they expand (or export) the sel…sh device beyond themselves but untill the border of a given population. For instance, parents are generally sensitive to the economic condition in life of their children and little children. If they restrict their interest to this small subset of individuals, they form an economic community who can behave in a way which makes the whole family happy (in sharing by advance the capital with equity, in helping the youngest...). If a family (even spatially and then socially scattered) is a natural candidate for being an economic community, note that an individual can form his own economic community when he is sel…sh (this situation corresponds to the Senian liberal one, see [START_REF] Billot | How Liberalism Kills Democracy: Sen's Theorem Revisited[END_REF]. By the way, (2) also can be seen as an equation for the …rst Marxists-socialists who considered the industrial proletariat, that is here the native A, as the only population whose economic situation, that is x A , fully determines the historical and political evolution of a society. In short, there are numerous potential economic communities: friends circle, families, …rms, classes, villages, minorities...

An economic system is a mapping eco from 2 S towards 2 S .

In what follows, we investigate the relationship between the properties of the social system and the properties of the economic system, that is the theoretical link between the interaction N via the social communities with the individual preferences via the economic communities. The intuition is that standard microeconomics requirements about individual rationality implicitly correspond to a particular organization of the social system: Theorem 2 : The economic system eco forms a consistent social system if and only if preference relations are completely ordered.

It is also possible to prove a converse result: for any consistent social system, there exist complete ordered preference relations such that each social community is also an economic community.

From Theorem 1, we know that a social system is consistent i¤ A1 and A2 are satis…ed. Hence, one way to prove Theorem 2 is to prove that, when the agents are interdependent, complete ordered relations lead to set up a system of economic communities which satisfy A1 and A2. For that purpose, we establish the two following lemmas:

Lemma 1 : Preference relations are re ‡exive if and only if A1 holds.

By [START_REF] N (j), Then N (i) N (j) And ; Symmetry | Proof of Theorem 4. The result is immediate by (1) and Lemma 3 since eco (N (i)) = N (i). This way, smallest communities exactly correspond to neighborhoods as implied by the social interaction and then[END_REF], preference re ‡exivity must be understood as an interdependence constraint for all individuals in S. This way, re ‡exivity ensures that the society S is formed with people who de…ne a community from both sides, social and economic.

Lemma 2 : Preference relations are transitive if and only if A2 holds.

Preference transitivity implies a kind of consistency between a …rst pair of issues, a second, a third, etc... and a last one which collapses all the intermediate steps. The property of monotonicity corresponds somehow to the same kind of consistency: inclusion of a sequence of native populations can be collapsed in saying that the biggest native population (which is actually the society itself by A1) includes all other communities with respect to the same nesting rule. The intuition is simple. Consider the French last presidential elections:7 the global set of individuals who 'choose among several candidates and vote for Jospin'is included in the set of individuals who 'choose among several parties and vote for the Socialist one'which in turn is included in the set of individuals who 'choose between Right and Left and vote for the Left'. In terms of native populations, being in the …rst population naturally implies to be in the second one which in turn implies to be in the third one. Then, by [START_REF] N (j), Then N (i) N (j) And ; Symmetry | Proof of Theorem 4. The result is immediate by (1) and Lemma 3 since eco (N (i)) = N (i). This way, smallest communities exactly correspond to neighborhoods as implied by the social interaction and then[END_REF], the community based on the …rst population, say the Jospinist trend within the Socialist Party, cannot be greater than that based on the second population, say the voters of the Socialist Party itself, nor that based on the third. Hence, preferences transitivity and communities monotonicity translate the same kind of consistency in their respective context. Theorem 2 exactly states that the interaction N is fairly represented by the economic system eco when individual preferences are completely ordered. Furthermore, if preference relations are completely ordered, each economic community exactly de…nes a social one, i.e. the two mappings, soc and eco, are identical:

[(x A ; y A ) i (x A ; z A ) , 8x; y; z 2 X n ] , [N (i) A] . (3)
In this case, when soc (A) = eco (A), we simply call it a 'community'. Hence, the right interpretation of Theorem 2 seems to be the following: standard basic assumptions about the individual behavior (that is, complete ordered preferences as a signal of rationality) are logically equivalent to an assumption about the social organization of the society (that is, consistency of the social system).

In other words, individual rationality and social consistency can be considered as the two faces of the same coin.

re ‡exivity of preferences transitivity of preferences () () inclusivity of communities monotonicity of communities i.e.:

individual rationality () social consistency.

Beyond this formal equivalence, Theorem 2 means that the two following statements are equivalent:

When rational, individuals are interdependent with individuals with whom they interact;

When rational, individuals choose to interact with individuals with whom they are interdependent.

In short, to be sensitive for instance in France to the so-called second generation's situation in life requires to be in social contact with the immigrant population. Hence, the existence of suburban ghettos (where immigrants often represent the 2/3 of the inhabitants) prevent any real global sensitivity (of the rest of the society) to the problems of this population. In the same time, being a priori interdependent with such a population prevents to interact with people outside and then can induce some 'communitarist' behaviors such as religious fundamentalism, political extremism... Individual rationality, that is social consistency, seems then to lead to a segregated (at least partitioned) society where the di¤erent communities are economically and socially disjoint, except at the level of the society itself, that is for some public events such as elections or wars for which votes and lifes are totally aggregated within the national community (A1).

Since individual rationality (in terms of preferences) is logically equivalent to social consistency of the interaction system (in terms of communities), we propose to go on the analysis in pointing out other kinds of equivalence between well-known properties of the preferences (e.g. continuity and monotonicity) and speci…c properties of the social system, e.g. group loyalty.

Remark 2 : Note that, thanks to Theorem 2, Theorem 3 can be rewritten as follows: if preference relations are continuous and monotone, then any neighborhood is a subset of mutually loyal individuals. This result means that, in order to be loyal with someone, an individual needs to socially interact with him: loyalty is not altruism, it is rather the economic counterpart of social sensitiveness.

Community Loyalty

Are continuity and monotonicity of preference relations su¢ cient to ensure that the result established in Theorem 3 can be naturally extended from neighborhoods to communities? Intuitively, since a community requires more structure than a neighborhood, the answer should be negative. Hence, we propose the following characterization of preference relations in order to extend the result. Given preference relations and a particular neighborhood N (i), write x < N (i) y when x < j y for all individuals j 2 N (i) and x N (i) y when x < N (i) y and x j y for some j 2 N (i).

De…nition 3 : Let an individual i 2 S such that if x < i y, then x < N (i) y, for all x; y 2 X n . Such an individual i is said to be empathic with his neighborhood N (i).

Empathy requires that the preferences of a given individual are sensitive (in the most minimal way) to what other individuals' preferences are when these individuals belong to the social neighborhood of the …rst individual. It can be also written as: if x N (i) y, then x i y, for all x; y 2 X n .

Theorem 4 : If preference relations are continuous and monotone and if individuals are empathic with their neighborhoods, then each smallest community is a subset of mutually loyal individuals.

The following lemma is central for proving the theorem.

Lemma 3 : If all individuals are socially empathic with their neighborhood, then i 2 N (j) implies N (i) = N (j), for all i; j 2 S.

We use the term 'native'as a synonymous for 'exogeneous'. A native population gathers people who have not previously decided to be gathered but whose idiosyncratic identity, according to a particular criterion, is very similar.

This is consistent with Lévi-Strauss's analysis (e.g., 1961) and overall the recent Guérin's works (2001):

This concept is close to that of a local interaction system as de…ned by[START_REF] Morris | Contagion[END_REF].

Note that inclusivity and monotonicity can be seen as the set-theoretic translation of the two conditions that de…ne, in decision theory, a set function as a Choquet capacity.

Hence, if S = f1; 2g and X = fa; bg, then the set X 2 of issues is:f(a; 1; a;2) ; (a; 1; b; 2) ; (b; 1; b; 2) ; (b; 1; a; 2)g where x = (a; 1; b; 2) means that x 1 = a while x 2 = b, that is 1 chooses a while 2 chooses b.

In 2002.

Group Loyalty

When assuming preference relations to be completely ordered, we know by [START_REF] Now | x N (i) 0 n for all i 2 S and 0 n 3 x is a contradiction[END_REF] that if an individual i socially interacts with an individual j and belongs to a community eco (A), then j belongs to the same community eco (A). This relationship is clearly strong. Similarly, economic interdependence is modeled by means of the individual's indi¤erence between all situations where the native population, say A, gets the same issue, i.e. x A , whatever happens to the rest of the society, i.e. y A or z A ... To weaken the possible relationship between two individuals, we propose to introduce the concept of group loyalty. Actually, in a recent paper, [START_REF] Luttmer | Group Loyalty and the Taste for Redistribution[END_REF] shows that individual preferences for redistribution are not only determined by self-interest but also a¤ected by the characteristics of the social environment (district, family...). One of the main explanations for this empirical feature is the racial group loyalty: individuals increase their support for welfare spending as the share of local recipients from their own racial group rises. In our (more abstract terms), this phenomenon means that when individuals are loyal with a given native population, they can accept to loose something within a particular issue if it is bene…cial to the rest of this native population (eventually a subset of it). Let us consider an individual who is the born-in-France son of a Maghrebian immigrant, that is he belongs to the 'second generation'. Assume he becomes a famous and rich actor or a soccer player (like Zinedine Zidane for instance). If he accepts (in supporting or voting for a leftist party) to pay more taxes (i.e., to decrease his own utility) in order to improve the standards of life of the French Muslims by means of the redistribution system (i.e., to increase their utilities), he can be said loyal with his native population. In short, group loyalty may translate a link which is weaker than interdependence but nevertheless signi…cant.

Hence, in what follows, we seek for the conditions under which the previous social entities (neighborhood and community) can constitute a population of individuals who are 'loyal'. Write x y if x i y i for all i 2 S, x > y if x y and x i > y i for some i 2 S and x y if x i > y i for all i 2 S.

De…nition 2 : Let an individual i 2 S. If for all issues x y, there exist issues z y such that the individual j 6 = i is better o¤ with (x j ; z i ; z ij ) rather than with y ((x j ; z i ; z ij ) i y with x j y j , z i y i and z ij y ij ), then such an individual i is said to be loyal with the individual j 2 S.

Neighborhood Loyalty

In the same spirit than previously, we search for proving that the concept of group loyalty can be encapsulated within preferences properties.

A preference relation < is continuous if the set of issues fx 2 X n : x < yg is closed for every y 2 X n and monotone if (x i ; z i ) < (y i ; z i ) for some issues x; y; z 2 X n implies that (x i ; z i ) (y i ; z i ) for all x; y; z 2 X n such that x i > y i .

Theorem 3 : Let a consistent social system soc. If preference relations are continuous and monotone, then any neighborhood is a subset of mutually loyal individuals.

Appendix

Proof of Theorem 1. Suppose that a social system soc represents an interaction N . Then, soc (A \ B) = fi 2 S : N (i) A \ Bg which is obviously equal to fi 2 S : N (i) Ag \ fi 2 S : N (i) Bg, that is soc (A) \ soc (B). Now, suppose that sc satis…es A1-A2. By A2, A B ) soc (A) soc (B) for all populations A; B S. Write S j for Sn fjg. De…ne the interaction N as follows: N (i) = fj 2 S : i = 2 N (S j )g. Show that the system soc represents the interaction as de…ned above. (i) Suppose i 2 soc (A). Then, A S j implies by A2 i 2 soc (S j ) which implies j = 2 N (i). Thus, j 2 N (i) implies A * S j which implies j 2 A. Hence, N (i)

A. (ii) Conversely, suppose N (i) A. If A = S, i 2 soc (S) by A1. Suppose then that we consider a population A 6 = S. Then, A = \ j = 2A S j . Since i = 2 soc (S j ) implies j 2 A, j = 2 A implies i 2 soc (S j ). Hence, i 2 soc (S j ) for all j = 2 A implies, by the above de…nition of A, i 2 soc (A).

Proof of Lemma 1. Write y ; for y S . We have by de…nition of an economic community: eco (S) = fi 2 S : (x S ; y S ) < i (x S ; z S ) for all x; y; z 2 X n g = fi 2 S : (x S ; y ; ) < i (x S ; z ; ) for all x; y; z 2 X n g = fi 2 S : x < i x for all x 2 X n g since x S = x. Then, it is obvious that eco (S) = S i¤ each preference relation is re ‡exive.

. Hence, take any individual i 2 S and two populations A and B such that i 2 eco (A) \ eco (B). By de…nition of an economic community, for all pro…les x; y; z 2

. Thus by transitivity: x A\B ; y (A\B) < i x A\B ; z (A\B) for all x; y; z 2 X n . So, i 2 eco (A \ B). (ii) Suppose now an agent i 2 S such that i 2 eco (A \ B). Then, for all x; y; z 2 X n : x A\B ; y (A\B) < i x A\B ; z (A\B) .Thus, (x A ; y A ) < i (x A ; z A ) and (x B ; y B ) < i (x B ; z B ) which implies: i 2 eco (A) \ eco (B).

Proof of Theorem 2: By Lemmas 1 and 2, we know that preference relations are ordered if and only if A1 and A2 hold. By Theorem 1, we know a social system is consistent if and only if A1 and A2 hold. Then, the economic system eco is a consistent social system.

Proof of Theorem 3. (i) Let an individual j 2 N (i). By monotonicity of i's preference relation < i , we have: (x j ; y j ) i y for all issues x; y 2 X n where x j > y j . Now, by continuity, there always exists an issue z y such that (x j ; z j ) i y. Hence, i is loyal with j. (ii) Suppose now that i is loyal with j. Then, by de…nition, (x j ; z j ) i y for some x y z. Thus, by monotonicity of i's preference relation < i , we have: (x j ; z j ) < i (y j ; z j ) i y. Hence, i 2 eco (j) by ( 2), that is j 2 N (i) since the interaction N is re ‡exive and symmetric. Hence, any neighborhood is a subset of mutually loyal individuals.

Proof of Lemma 3. Empathy can also be written: if x N (i) y, then x i y, for all x; y 2 X n . With symmetry of the relation N , if i 2 N (j) then j 2 N (i). Hence, N (i) = N (j), for all i; j 2 S, requires that if i 2 N (j), then N (i) N (j), namely the transitivity of the social interaction N . Prove the contrapositive of the required property. Then, suppose that there exists three individuals 1; 2; 3 2 S such that 1 2 N (2), 2 2 N (3) but 1 = 2 N (3). Now, let x 1 = 1 while x i = 0 for all individual i 6 = 1. By continuity of preference relations, we have x 2 0 n where 0 n is the null vector of X n . Then, x < i 0 n for all i 2 S. Thus, it is true that x N (i) 0 n . But, in the same time, since 1 = 2 N (3), then 13