Inequality and club formation

Journal of Public Economics 87 (2003) 931–955 www.elsevier.com / locate / econbase

Inequality and club formation Fernando Jaramillo, Hubert Kempf*, Fabien Moizeau ´ , 106 -112 Boulevard de l’ Hopital, EUREQua, Bureau 303, Universite´ Paris-1 Pantheon-Sorbonne 75013 Paris, France Received 9 April 2000; received in revised form 11 July 2001; accepted 28 July 2001

Abstract We study the relationship between social segmentation and income inequality by means of the economic theory of clubs with private provision of the club good. After having characterized the core partition of society in clubs and investigating its characteristics, we show how the clubs’ sizes depend on income distribution and compare segmentation profiles arising in societies characterized by different inequality patterns.  2001 Elsevier Science B.V. All rights reserved. Keywords: Club goods; Inequality; Core; Social segmentation JEL classification: D3; D71; H40; H41

1. Introduction The issue of social segmentation is central to the understanding of human societies and attracts interest both from philosophers (cf. Rousseau, 1985 (1755)), sociologists (cf. Bourdieu, 1987) as well as from economists. For economists, the issue at stake is not just an understanding of the social contract, but also an investigation in the relationships between social segmentation and the pattern of income distribution of a given society. In particular, recent developments in political economy currently aim at studying the formation of communities.1 *Corresponding author. Tel.: 133-1-4407-8200; fax: 133-1-4407-8202. E-mail address: [email protected] (H. Kempf). 1 Cf. e.g. Alesina and Spolaore (1997), Bolton and Roland (1997), Alesina et al. (2000b). 0047-2727 / 01 / $ – see front matter  2001 Elsevier Science B.V. All rights reserved. PII: S0047-2727( 01 )00162-1

F. Jaramillo et al. / Journal of Public Economics 87 (2003) 931–955

932

The aim of the present paper is to understand how the participation mechanisms to ‘local’ or ‘partial’ communities hinge on the inequality characteristics of a given society. As noted by Max Weber almost one century ago, the creation of what he called the ‘social closure’ represents an attempt by richer people to exclude the less fortunate individuals from some common enterprises or social communities, whereas the latter ones try to create their own social structures in order to collectively compensate this eviction. Interestingly, Max Weber stressed the fact that social closure depends on well understood individual interests more than on tradition, class ethos or shared common values. Max Weber wrote: that ‘in the following type of situation, a social relationship may provide the parties to it with opportunities for the satisfaction of spiritual or material interests. If the participants expect that the admission of others will lead to an improvement of their situation, an improvement in degree, in kind, in security or the value of satisfaction, their interest will be in keeping the relationship open. If, on the other hand, their expectations are on improving their position by monopolistic tactics, their interest is in a closed relationship’ (Weber, 1978 (1922), p. 43). Recent empirical studies have confirmed this statement and stress the close relationship between inequality, social segmentation and the workings of communities. Yitzhaki and Lerman (1991), developing a new index of stratification, show that groups depend on income inequality and tend to be homogeneous, in the sense that income inequality tends to be relatively similar within groups and different from those outside the group. Epple and Sieg (1999) focusing on the spatial distribution of population within the Boston Metropolitan Area observe substantial stratification according to income. Alesina and La Ferrara (2000) investigate the relationship between group formation and the degree of participation when the population is heterogeneous. They conclude that participation in social activities is significantly lower in more unequal and (ethnically) divided societies. Alesina et al. (2000a), in a study over American local jurisdictions, show that the number of school districts tend to increase with the Gini coefficient in a given county. In the same spirit, we address the issue of the link between income inequality characteristics and social segmentation by means of a formal model of community formation. In other words, we want to develop a theory of endogenous community formation based on inequality characteristics. This problem can be decomposed into two important issues. The first one is to understand how the pattern of community formation depends on the characteristics of inequality. The second issue is to know whether more inequality fosters a more divided society. This amounts to comparing the social segmented patterns arising in two societies such that one is more unequal than the other. We are able to characterize the partition of society as depending on the endowment distribution among agents and then to address both issues. Firstly, for

F. Jaramillo et al. / Journal of Public Economics 87 (2003) 931–955

933

a given society, we characterize the pattern of communities as the number, the composition and the sizes of communities and are able to relate both the number and the sizes of communities to the variations of the income / endowment gaps between individuals. Secondly, comparing two societies differing in inequality, we are able to show that a more unequal society is not necessarily more segmented than a more egalitarian one, but that again depends on the properties of endowment gaps. If in both societies the income gap between two successive agents is constant or growing with income, then a more unequal society is unambiguously more segmented than a more egalitarian one. The impulse toward community formation comes from the fact that there is a common interest in forming a community, formalized as a public good. Hence we study the relationship between social segmentation and income inequality, by means of the economic theory of clubs with private provision of the club good, following Barham et al. (1997). A club according to Cornes and Sandler (1996) is defined as ‘a voluntary group of individuals who derive mutual benefit from sharing one or more of the following: production costs, the members’ characteristics, or a good characterized by excludable benefits’ (p. 347). Hence it nicely formalizes the idea of closed communities, or communities with explicit membership. We shall consider clubs providing public goods to their members, with the assumption that in each club, these goods are voluntarily funded. This assumption is standard in public economics 2 and corresponds to many actual situations. The importance of voluntary contributions (both in time and money) of individuals to communities cannot be neglected. Overall, the relative size of voluntary pecuniary contributions to public goods is evaluated at some 2% of GDP in the US in 1989, a rising number.3 In some sectors or activities, voluntary contributions are crucial for the financing of public goods. According to the Council for Aid to Education, private contributions to America’s colleges and universities reached $23.2 billion in fiscal year 2000, a 13.7% increase over the 1999 figure. Cornell University received some $340 million in gifts during academic year 1998-1999, with 79.7% of the total coming from individual donors.4 We shall focus on income inequality by assuming that income is the only factor differentiating agents.5 We shall adopt a fairly standard representation of individual 2

The seminal contribution to the literature on the voluntary provision of public goods is Bergstrom et al. (1986). 3 See Glomm and Lagunoff (1999). 4 See Ehrenberg and Smith (2001). The Council for Aid to Education publishes a yearly report on Voluntary Support to Education. See www.cae.org / vse. 5 About the influence of individual income on contribution behavior, Independent Sector, a coalition of non-profit institutions, reports that ‘when measured as a percentage of total household income, households at either end of the income scale were the most generous. Households earning under $10,000 a year gave 2.5% of total household income and households with income over $100,000 gave 1.9%. However, many of those with incomes under $10,000 were retired with little regular income and gave from their accumulated wealth’ (Independent Sector, 2001). Moreover, we cannot forget that many voluntary contributions are in time. Over all, it is clear that voluntary contributions are related to wealth and income.

934

F. Jaramillo et al. / Journal of Public Economics 87 (2003) 931–955

preferences with a public good and we shall not a priori assume anything on the division of society. Formally, not only the size but also the number of communities will be endogenously determined. Our paper is related to a strand of literature stressing the effects of income distribution on coalition formation. Focusing on firms, Bennett and Wooders (1979) investigate the formation of firms by agents, with unequal individual endowments: they are then able to relate the stability of the partition in firms to the characteristics of the endowment distribution. Farrell and Scotchmer (1988) investigate ‘partnerships’ formation. A partnership is a productive grouping of individuals the output of which is to be equally divided. Farrell and Scotchmer study the characteristics of the core partition of a society in partnerships, in particular under uniform distribution of individual abilities. Here considering the case of clubs and using a simple model of voluntary provision of local public goods, we study the core partition of individuals into clubs and their contributing behaviors. Furthermore, the analytical tractability of our setting allows us to investigate the effects of relative inequality between individual endowments on social segmentation. In that respect, our research is clearly different from these previous studies. The plan of the paper is as follows. Section 2 presents the model. In Section 3, we characterize the core partition of society in clubs and we investigate its characteristics. The relationship between inequality and social segmentation is thoroughly investigated in Section 4. Concluding comments are offered in Section 5.

2. The model Consider a set I 5h1, . . . , Nj of individuals forming a society. Each individual i [ I has an initial strictly positive endowment vi . Agents are ordered so that v1 . v2 . . . . . vN . The aggregate level of endowment will be given by V 5 o i [I vi . Each agent is able to freely allocate its endowment between the consumption of a private good and a contribution to the funding of a public good. The latter good is a club good, that is, in order to benefit from it, an agent must belong to a club. The number and boundaries of clubs are not given ex ante but will be endogenously determined.

2.1. Clubs and local public goods A club is characterized by its boundaries and the production of its specific public good. In the present paper, we assume that each member voluntarily defines his contribution to the financing of the public good characterizing the club to which he belongs. Hence society is divided into clubs which form a partition of I. We denote Sj the jth club in a partition in J clubs (or a club-partition) of I such that:

F. Jaramillo et al. / Journal of Public Economics 87 (2003) 931–955

935

J

< Sj 5 I

j51

Sj > Sj 9 5 5 for j ± j9. The characteristics of the partition of society, and especially the number of clubs is not assumed a priori but will be determined through the behavior of individuals. It is supposed that the production of the public good in club Sj is given by the following equation: Gj 5

Og

i

(1)

i [S j

where Gj is the quantity of the public good produced in Sj and gi is the voluntary contribution of agent i to the production of the club good in Sj . The club good is homogeneous, that is it is identically produced in each club. A crucial difference with Bergstrom et al. (1986) is that they consider a unique public good, benefiting all individuals in the society, whereas we consider local public goods, the amount of which can vary according to the characteristics of the club in which they are produced. The functioning of the economy is defined in two stages: (i) in the first stage, a partition of society is formed: stable clubs are created on the basis of individual characteristics, in such a way that no individual and no group of individuals wants to defect and join any other possible coalition of agents; (ii) in the second stage, individuals voluntarily contribute to the club to which they belong, club goods are produced and consumed within the boundaries of clubs. This game is similar to the one studied by Barham et al. The first stage is analyzed as a cooperative game without side payments, leading to a partition of players; the second stage is modelled as a non-cooperative game on voluntary contributions. Barham et al. justify this modelling strategy by noticing that it is more appropriate than modelling the first-stage as a non-cooperative game. It would have to rely on an ad hoc rule of club formation or would lead to implausible multiple equilibria as this game would take the form of a coordination game.6 The resolution of this dilemma would imply the use of an implicit cooperative argument. This ‘hybrid’ approach amounts to assuming that it is possible to behave cooperatively in the first stage as the endowment of an agent is known, but afterwards, it is impossible to cooperate in forcing an agent to contribute to the

6

On coordination games, cf. Cooper (1999).

F. Jaramillo et al. / Journal of Public Economics 87 (2003) 931–955

936

public good because of the lack of commitment and the impossibility to punish her in any way, for example in excluding him from the club. In other words, nobody is forced to commit to the amount of resources she will devote to the club good once she is admitted in a given club before the formation of clubs. Of course, this does not mean that the definition of the club partition is made without reference to the funding of the club goods: on the contrary, it will be based on the expectation of funding capacities of individuals.

2.2. Agents Individuals have the same preferences and differ only according to their initial endowment. Consider an agent i belonging to the club Sj . His preferences are given by the following utility function: U(c i , Gj ) 5 ln c i 1 b ln Gj 2 a n j

i [ Sj , b . 0, a . 0

(2)

where c i represents the private consumption of agent i and n j is the number of agents in Sj . His budget constraint is: vi 5 c i 1 gi . The important point in (2) is that the utility obtained by i in club Sj only depends on the internal decisions taken by the set of members of club Sj , independently of what happens in the rest of society, that is, independently of what is taking place in the other clubs composing the partition of society. Hence, only the composition and individual behaviors in a club matter for its members. The last term in (2) represents congestion costs in the use of the club good. These costs are proportional to the number of agents in a given club: the larger is the club, the less enjoyable are the conditions in which each individual benefits from the club good.7 An alternative specification would be to assume that congestion costs enter the production function of the club good: the more members, the less efficient is the production of the club good.8 Nothing is changed in the analysis except on the influence of b on the partition. Hence, we use the simplest specification.9 Assuming the club-partition being defined in the first stage, agent i when he belongs to Sj chooses his level of voluntary provision so as to maximise (2) subject to the budget constraint, the production function for the club good (1) and the condition of positivity of gi . The first-order condition is given by: 7

When a equals 0, we get the Bergstrom et al. (1986) model. Then the production function (1) is written as Gj 5 o i [Sj gi /e a n j . It follows that the size of the club enters with a coefficient ab in the individual utility function. 9 This specification is such that the second-order conditions of the optimal size programme are met (see Barham et al.). In fact, most results happen to be quite robust to the choice of the specification of congestion costs in the production function (1). In a previous draft, we proceeded to our analysis using two alternative specifications: Gj 5 o i [Sj gi /n aj or Gj 5 o i [Sj gi 2 a n j . As long as the coefficient a is not too large in the first case, and for any value of a in the second specification, we obtained almost the same results, especially on the properties of the stable partition. 8

F. Jaramillo et al. / Journal of Public Economics 87 (2003) 931–955

gi 5

1 vi 2 ] b

5

SO D gz

if positive

z [S j

0

i [ Sj

937

(3)

otherwise

Moreover, the second-order condition is satisfied as: 21 b ]]]2 2 ]]]2 , 0 v 2 g s i gz id

SO D

i [ Sj .

z [S j

Following Bergstrom et al. (1986) it is known that a Nash equilibrium of this game exists. Denoting S˜j the set of contributing members of Sj and n˜ j its cardinal, we get:

O g 5 O v 2 ]nb˜ O g j

z

z [S j

z

z

z [S j

z [S˜ j

and hence:

O

b Gj 5 ]] v b 1 n˜ j z [S˜ z

(4)

j

We observe that the amount of club good available in the jth club is an increasing function of the coefficient b which denotes the relative preference for the club good of any individual, and a decreasing function of the size of the club, other things being equal. The contribution of each individual is determined in accordance with the ‘perimeter’ of the club to which he belongs. Since there is no precommitment mechanism on the level of contributions, an agent does not take into account when computing his contribution in a given club, the potential contributions he would make in other (potential) clubs. Finally, in conformity to the standard theory of private provision to a public good, the indirect utility is identical for each member of a club Sj . Indeed, according to the budget constraint and (3), and anticipating the result to be proved later that any member of a club positively contributes to the funding of the club good (i.e. n˜ j 5 n j ), it is clear that the consumption of each agent is identical for all members and equal to (1 /b ) Gj . Hence, replacing this value in the expression of individual utility given by (2), we get:

Ov

1 2 z

z [S j

VisSjd 5s1 1 bd ln ]] 1 b ln b 2 a n j b 1 nj 5VhsSjd 5VsSjd

;i, h [ Sj .

(5) (6)

This equilibrium property is identical to the one associated by Farrell and Scotchmer (1988) to a partnership. But the crucial difference is that it is not

F. Jaramillo et al. / Journal of Public Economics 87 (2003) 931–955

938

imposed ex ante as the rule guiding the formation of communities, or coalitions, as in the case of partnership. Within a club, externalities exist as (5) makes it clear. On the one hand, any member generates gains for the whole club membership because of her individual contribution to the club good. On the other hand, any member in a club contributes to the congestion effect and to the well-known free-riding phenomenon associated with voluntary provision. Hence, the adding of a new agent to a club should be such that the marginal benefit enjoyed by existing members should be at least as large as the marginal inconvenience they suffer from this adding. Finally we observe that a higher endowed agent in a club generates more net utility than a less endowed one as she equally contributes to the congestion costs and the net effect of her contribution on the club good is higher. Hence everybody is better off the more wealthy people there are in her club. This will help us to understand the condition on the marginal individual accepted in a club.

3. Characterizing the core partition Having defined in Section 2 the behavior of optimizing agents when they belong to a club, we now turn to the issue of characterizing the segmentation of society into clubs. We aim at studying the stability problem linked to segmentation. We are interested in studying core partitions of society, that is allocations of individuals to clubs such that there is no mutual advantage for anybody, individually or collectively, to modify them by forming new clubs. Capturing the notion of the right to exclude, the core is a useful way to formalize social segmentation. Moreover, while admittedly a reduced form for explicit coalition formation, the core will prove a powerful tool for generating comparative statics results in the sequel of the paper. For that purpose, we can provide the following formal definition. Definition 1. A core partition S E 5hS 1E , . . . , S jE , . . . j is such that: '⁄ £ , I such that ;i [ £, Vis£d .VisS Ed where VisS

(7)

E

d denotes the utility for agent i associated with partition S E .

According to this definition, a core partition is such that there exists no blocking coalition such that each member of this coalition prefers it to the club to which he belongs in S E . Denote S Esid the club in a core partition to which belongs i. Because of the existence of congestion costs linked to the size of any club, no non-contributing individual can be accepted in a club of S E . Consider a club Sj such that at least one of its members does not contribute. Then, the club Sj 9 defined as Sj without this

F. Jaramillo et al. / Journal of Public Economics 87 (2003) 931–955

939

member will benefit from the same sum of contributions and less congestion costs. It is obvious that Sj 9 will give a higher welfare to all its members or equivalently that Sj does not belong to the core. Hence we know that for any i and any j, his endowment satisfies the following inequality:

Ov

z

z [S˜ j

vi . ]], ;i [ S jE b 1 n˜ j

(8)

and n˜ j 5 n j . Let us now characterize the core partition. We can then offer the following Proposition 1. Let S E 5hS E1 , . . . , S Ej , . . . j be a core partition. S E satisfies the following properties: (i) Consecutive structure: a club S Ej is consecutive, that is if i and i˜ belong to ˜ then ;i*, i . i* . i, ˜ i* [ S E . S Ej , i . i, j (ii) Uniqueness: the core partition is unique. (iii) Welfare ordering: consider two clubs S Ej and S Ej 9 such that ;i [ S jE , ;i9 [ S Ej 9 , i , i9, then VisS Ed .Vi 9sS Ed. Proof. See Appendix A. h Given these properties, a core partition amounts to the definition of ‘pivotal agents’. A pivotal agent is the poorest member of a club whose income is high enough to allow him to positively contribute to the club good and just balance the negative effects of its inclusion. This allows us to offer a simple characterization of the core partition. Proposition 2. S E 5hS 1E , . . . , S jE , . . . j can be defined as a sequence of pivotal agents h p1 , . . . , pj , . . . , pJj where S Ej 5h pj 21 1 1, . . . , pjj, and pj are such that: p j 21

O

S

E

D

b 1 nj vp j $ vz ]]]] e a / 11 b 2 1 E b 1 n 2 1 z 5p j 21 11 j

(9)

and pj

vp j 11 ,

S

D

b 1 n Ej 1 1 a / 11 b vz ]]]] e 21 b 1 n jE z 5p j 21 11

O

(10)

where n Ej denotes the size of S Ej , p0 1 1 5 1 and pJ 5 N. We remark that there exists a club such that its poorer agent is agent N, satisfying inequality (9) but of course there cannot be any higher indexed individual,

940

F. Jaramillo et al. / Journal of Public Economics 87 (2003) 931–955

satisfying inequality (10). Hence the club is ‘residual’ and its ranking determines the endogenous number of clubs. Admitting it is the Jth club, it implies that there exist J clubs in the core partition. Later, we shall refer to the J 2 1-th first clubs as the ‘non-residual’ clubs. The notion of a ‘consecutive’ structure has been used and developed by Greenberg and Weber (1986). Fernandez and Rogerson (1996) call this property of a partition ‘stratification’. Any club in the core partition is a consecutive coalition. The definition of this coalition only depends on the differentiated endowments. In other words, individuals coalesce in a club by means of ‘neighborhood effects’ in the income distribution. The richest individuals belong to a club, then the less rich ones to another club . . . until the existence of a ‘residual club’, which includes the poorest individuals of society. The social segmentation in this model is only linked to the characteristics of the unequal distribution of income, in conformity with the intuition of many sociologists who insist on the economic origin of ‘class’ antagonisms.10 In this economy, there is a unique partition in the core. In Appendix A, we show that the pivotal agent of a consecutive club is uniquely defined. In fact, the type of our core partition is exactly the same as the one obtained by Farrell and Scotchmer. Once the first club, S E1 , is formed containing all the richest individuals until the first pivotal agent, it gives the highest possible level of welfare. There is no blocking coalition that could contain any member of this best club because it is impossible to give them more than what they are currently getting. Then we can repeat the process for the second club and so on. We know that any individual richer than the pivotal agent in a club S Ej of the core partition generates more utility to the other members than the pivotal agent herself (by inspection of the value of the indirect utility level). The pivotal agent is therefore the marginal member: her endowment is such that her contribution to the club good will just be above the increased costs she inflicts on the existing members. Denoting pj 1 1 the following agent in I, (that is vp j 11 , vp j ), this agent, if included in the club would generate a smaller benefit than pj as he would contribute less than pj and a larger marginal cost: the balance turns out to be negative for him. We know as well that a club containing richer agents gives a higher welfare than a poorer group because the level of public good is higher in this club thanks to the highest financing capacity of its members. This directly comes from the fact that everybody enjoys more utility the more wealthy people there are in his club.11 10

In their empirical study of the spatial distribution within the Metropolitan Boston Area, Epple and Sieg observe that the single-crossing condition, implying consecutiveness, is satisfied. However, they conclude that, in addition to income inequality, heterogeneity in preferences is crucial for explaining decisions over location. 11 This welfare ordering property is also in accordance with Epple and Sieg’s empirical results.

F. Jaramillo et al. / Journal of Public Economics 87 (2003) 931–955

941

Notice that the equilibrium segmentation of society relies on a partial redistribution effect, through the individual financing of the club goods. Within a club, because an individual contribution is proportional to income, the richest members contribute more than poorer agents, and ‘subsidize’ them (without any altruism) by means of the provision of the club good. A member in a club gets larger benefits from her membership to this club, the poorer she is relative to the richest member of the club: the sacrifice she makes through her own contribution is small relative to the amount of public good she enjoys. But this redistribution between members is of course limited to them, and does not extend to the whole society. Two corollaries can now be proved. Corollary 1. The contribution of an agent increases with income within a club. But comparing two individuals belonging to two different clubs, the contribution made by the richer individual in his club may be smaller than the contribution made by the other poorer individual in her own club. Proof. Consider two individuals i and i9, i , i9. i belongs to club Sj and i9 belongs to club Sj 9 . From (3), it is deduced: 1 gi . gi 9 ⇔svi 2 vi 9d . ]sGj 2 Gj 9d. b

(11)

Within the same club of the core partition, i.e. Sj 5 Sj 9 , it is immediate that: gi . gi 9 with vi . vi 9 . For two different clubs, if condition (11) is not satisfied, we get: gi , gi 9 whereas vi . vi 9 . h This is quite paradoxical at first hand. Indeed two effects are at work in determining individual voluntary contributions: the endowment distribution and the size of clubs. In our framework, these two effects are closely linked as changing the size of a club corresponds to allowing poorer individuals to join. It is precisely the subject of the subsequent section to explore this link and relate the sizes of clubs in the core partition to the characteristics of the endowment distribution. Given the pattern of income distribution, the size effect clearly matters in the funding of public goods when individual contributions are voluntary: it is well known that the free-riding phenomenon increases with the number of individuals in the economy as proved by Andreoni (1988). It exacerbates when the contributions of other individuals increase, i.e. when they are richer and more able to pay. This free-riding phenomenon is at work in the present

942

F. Jaramillo et al. / Journal of Public Economics 87 (2003) 931–955

setting with heterogenous agents as will be explained in the subsequent section.12 Within a club, members contribute according to their endowment, and the pivotal agent does contribute a minimum. But she may contribute less than what the agent just after her in the endowment sequence contributes in his club (of which he happens to be the richest member). Put differently, there is no monotone relation between the level of endowment and the level of voluntary contribution to the club good. This turns out to come from the no-commitment assumption we made. Consider the pivotal agent pj of club Sj . Corollary 1 states that agent pj may contribute less in Sj than agent pj 1 1 in club Sj 11 , even though pj 1 1 was not accepted in Sj ! If pj 1 1 had committed to contribute in club Sj the amount he gives in club Sj 11 , ceteris paribus, he would have been included in Sj . Therefore he would have forced pj to modify her contribution and most likely to contribute more than pj 1 1. But this is time-inconsistent and hence pj is immune to any claim made by pj 1 1 for positive contribution if accepted in Sj as this claim is not credible. Corollary 2. There is a unique club encompassing the whole society (i) if a 5 0; (ii) when a ± 0, if and only if the income of the poorest agent in society satisfies the following inequality:

b 1N O v S]]] e b 1N21

N 21

vN $

i

i 51

a / 11 b

D

21

(12)

Proof. (i) is due to the superadditivity of the game; (ii) is immediate from the definition of a pivotal agent. h If there is no congestion cost, the larger the utility of any member is, the more people belong and contribute to a club. Therefore, the optimal club for anyone is clearly society as a whole. In the presence of congestion costs, this is no more valid but there may exist a unique club if the endowment sequence is sufficiently egalitarian and a small enough so that condition (12) is fulfilled. 12 As suggested by an anonymous referee, one simple way to emphasize the size effect through free-riding is to consider two homogeneous groups, S1 and S2 . n 1sn 2d agents with endowment v1 sv2 , v2 , v1d belong to S1sS2d. Assume that n 1 . tn 2 where t is a positive integer. Then we obtain the following optimal individual contributions g1 and g2 within each club g1 5 bv1 /( b 1 n 1 ) and g2 5 bv2 /( b 1 n 2 ). Hence, if t is large enough, the free riding problem is worse in S1 than in S2 . On the other hand, experimental studies on the voluntary provision of public goods (Isaac and Walker, 1988; Hackett et al., 1994) show that heterogeneity in income reduces voluntary contributions to public goods under the assumption that agents can communicate prior to making contribution decisions.

F. Jaramillo et al. / Journal of Public Economics 87 (2003) 931–955

943

4. Inequality and the endogenous formation of clubs In our framework, the key factor of social segmentation is the sequence of endowment. Denote L 5hv1 , v2 , . . . , vNj a given sequence of endowment. This puts to the forefront the issue of the incidence of resource inequality on segmentation. What can be said about the relationship between resource inequality and club formation, that is the core partition? We already know that whatever the degree of resource inequality, clubs are consecutive, i.e. include ‘neighbors’ in the resource distribution. Now we would like to understand its impact on the size and number of endogenously formed clubs. We shall proceed in two steps. (i) Firstly, for a given sequence of endowment we shall investigate the relationship between its characteristics and segmentation. (ii) Secondly, we shall compare two societies, differing by their sequence of endowment hence by their inequality, and study how the increase of endowment inequality modifies segmentation.

4.1. Inequality and the characteristics of segmentation We denote li,k the ratio of agent i’s endowment to agent k’s endowment in a given sequence L:

vi li,k ; ]. vk We shall restrict the analysis to endowment sequences in which the ratio li,i 11 varies monotonously with respect to i. The pivotal agent for a club Sj in which agent i is the richest member, is defined by (9) and (10). Denoting n *j the optimal size of Sj , we can deduce that pj 5 i 1 n *j 2 1. We can express (10) in terms of endowment ratios. Dividing both sides of (10) by vp j 11 , we obtain:

S

n* j

DO l

b 1 n *j 1 1 1 , ]]]] e a / 11 b 2 1 b 1 n j*

z 51

i 1n j *2z,i 1n j *

.

(13)

We can then establish the following proposition. Proposition 3. The size of clubs in the core partition depends on endowment ratios as follows. 1. Whatever j [h1, . . . , J 2 1j, n Ej 5 n Ej 21 if and only if li,i 11 is constant on the rank, i.e. li,i 11 5 l, ;i [ I. 2. Whatever j [h1, . . . , J 2 1j, n jE $ n jE21 if and only if li,i 11 decreases with the rank, i.e. li,i 11 $ li 11,i 12 , ;i [ I.

944

F. Jaramillo et al. / Journal of Public Economics 87 (2003) 931–955

3. Whatever j [h1, . . . , J 2 1j, n Ej # n Ej 21 if and only if li,i 11 increases with the rank, i.e. li,i 11 # li 11,i 12 , ;i [ I. Proof. See Appendix B. h This proposition relates social segmentation to inequality. It can be explained as follows. The equilibrium level of the local public good is the main determinant of the level of indirect utility and depends on the individual contribution of the members of the community. From Corollary 1, the key dimension for the endogenous formation of clubs is heterogeneity measured by the endowment ratios. That is, what matters for an agent when determining his own (noncooperative) contribution is the membership of the club to which he belongs, hence its endowment compared to the sum of endowments of members: a rich individual i can contribute less in his club Sj than a poorer individual i9 (i9 . i) belonging to another club Sj 9 if his endowment relative to the sum of endowments in Sj is much smaller than the endowment of i9 relative to the sum of endowments in Sj 9 . This explains why marginal benefits associated with the poorest individual admitted in a club negatively depend on the endowment ratio between the richest individual and this individual li,i 1n j * , as can be seen from Eq. (13). This amounts to say that what is critical for the admission of any member of a club depends on the endowment ratios li,z (z 5 i 1 1, . . . , i 1 n j* ) between members of the club and not on her absolute level of endowment: what matters is how many rungs down the ladder she is. On the other hand, congestion costs associated with this individual do not depend on his endowment (his characteristics) but merely on the crude fact that there is one more member in the club. First, consider case (1). Then the endowment ratio between the richest individual in any consecutive club and a poorer member indexed by z can be expressed as: li,z 5 l z 2i . It is then immediate to observe that the arbitrage condition is similar for any individual, generating the same decision over the size of any club. Consider now case (2). Suppose two clubs Sj and Sj 9 , of optimal size n *j and n *j 9 respectively; i and i9 (i , i9) are the richest members of Sj and Sj 9 , respectively. Under the assumption li,i 11 $ li 11,i 12 , the endowment ratio li 9,i 91n j 9 * is smaller than the endowment ratio li,i 1n j 9 * ; hence the marginal benefits associated with i 1 n *j 9 in Sj are at most equal to the marginal benefits associated with i9 1 n j*9 in club Sj 9 . As the marginal cost to the inclusion of i 1 n *j 9 in Sj is equal to the marginal cost associated with the inclusion of i9 1 n *j 9 in Sj 9 , it results that the optimal size of Sj is at most equal to the optimal size of Sj 9 . This reasoning applies to the core partition. Case (3) can be understood by reversing this reasoning.13 13

Alesina et al. (2000a) in their study of school districts in the US, stress the importance of tradeoffs between heterogeneity and economies of scale in understanding the borders of school districts. Indeed, in our setting, such a tradeoff is at the root of the endogenous sizes of clubs.

F. Jaramillo et al. / Journal of Public Economics 87 (2003) 931–955

945

4.2. Difference in inequality and segmentation comparison We now turn to the comparison in terms of segmentation of two societies I1 and I2 , encompassing an equal number of individuals but characterized by different values of the parameters or endowments. In other terms we want to study the impact of a variation of income inequality on social segmentation, that is to relate the inequality characteristics of two different endowment sequences with the endogenous segmentations they support. We will compare the endogenous formation of clubs between two societies characterized by two different endowment sequences L1 5hv 11 , v 21 , . . . , v N1 j and L2 5hv 12 , v 22 , . . . , v N2 j. With respect to segmentation, we shall use the following: Definition 2. A society I2 is more segmented than a society I1 if the number of non-residual clubs in the core partition S E 2 associated with I2 , J2 2 1, is at least E equal to the number of non-residual clubs in the core partition S 1 associated with E2 I1 , J1 2 1, and the jth club in S is never larger than the jth club in S E 1 , for j , J1 . We consider that a society I2 is more segmented than a society I1 if in its core partition, at least one individual with rank i who does not belong to the residual club, is a member of a smaller club than the individual with the same ranking in society I1 and no individual in society I2 belongs to a larger club than the individual with the same rank in society I1 . Any voluntarily formed neighborhood or club in I2 is then necessarily at most as large as the voluntarily formed club with the same rank in the ranking of clubs in I1 . But this decrease in the sum of the sizes of the J1 2 1 voluntary clubs does not necessarily lead to an increase in the number of clubs as it may just imply an increase in the size of the residual club in society I2 . This explains the formal definition of a more segmented society. This definition of an increase in segmentation is obviously restrictive and other definitions could be conceived. Nevertheless we think that it is both simple and logically consistent with the idea of an increase in segmentation. With respect to inequality, we shall consider two sequences L1 and L2 such that L1 is Lorenz-dominating L2 .14 Because of this Lorenz-dominance, the endowment sequence L2 corresponds to a more unequal society than L1 . Even though this is clearly a restriction, it allows us to prove an important point: social segmentation depends on income distribution and an increase in inequality, as exemplified by the dominance criterion, modifies this segmentation. We can then prove the following. Proposition 4. Assuming two sequences L1 and L2 , where L1 Lorenz-dominates L2 , a society I2 associated with L2 is not necessarily more segmented than a society I1 associated with L1 . 14

For an illustration of the Lorenz-dominance, cf. Foster and Ok (1999).

946

F. Jaramillo et al. / Journal of Public Economics 87 (2003) 931–955

Proof. See Appendix C. h This proposition states that more inequality does not necessarily lead to a more segmented society. We remark that the example relies on a property of increasing endowment ratios with the rank of individuals. It can then be easily understood. On the one hand, the first club is smaller under L2 than under L1 because the endowment ratio l1,11n for any n is higher under L2 . On the other hand, as the ranking of the richest individual in the second club is lower under L2 than under L1 , this implies from Proposition 3, that the second club under L2 can be larger than the second club under L1 . Hence the two effects going in opposite directions, we cannot say which is the largest second club. This generalizes to any successive club. It is then interesting to consider the two other cases where the endowment ratios li,i 11 are either constant or decreasing with the rank of individuals. We now consider two endowment sequences L1 and L2 satisfying assumption (A1) or (A2) defined as follows: A1: both sequences are characterized by ratios l ki,i 11 which are equal for any i [ I and such that l 2 . l 1 . A2: both sequences are characterized by ratios l ki,i 11 which are strictly higher than l ki 11,i 12 , for any i [ I and such that l 2i,i 11 . l 1i,i 11 , ;i [ I. Then we prove the following. Proposition 5. Assuming two sequences L1 and L2 , satisfying either (A1) or (A2), a society I1 associated with L1 is less segmented than a society I2 associated with L2 . Proof. See Appendix D. h This proposition means that under some conditions with regard to relative endowment inequality, an increase in inequality (as defined above by means of Lorenz-dominance) induces an increase in social segmentation, as the number of clubs increases, and the clubs’ sizes are smaller. This proposition can be explained as follows. Consider first the case where each of the two different sequences satisfies the assumption of equal constant relative inequality (Assumption A1). It is clear that the whole core partition does not depend on the absolute levels of income but on the relative levels of income (see (13)). We know from Proposition 3 that the endogenous segmentations in both cases will be such that in each society, clubs will be of same size. In this case, as the endowment ratio l1,11n for any n is higher under L2 than under L1 , the first club is smaller under L2 . But because clubs are of equal size, the number of clubs increases in more unequal

F. Jaramillo et al. / Journal of Public Economics 87 (2003) 931–955

947

societies. A similar reasoning applies when we consider two endowment sequences characterized by (A2).15

4.3. Core partitions, congestion and preferences for the club good Lastly, there are two crucial features in our framework: the congestion effect and the relative preference for the public good. Therefore it is of interest to understand how the parameters a and b corresponding to these two effects, respectively, affect social segmentation. Assuming the endowment sequence to be constant, we can prove the following. Proposition 6. (i) An increase in a cannot lead to a less segmented society. (ii) On the other hand, an increase in b cannot lead to a more segmented society. Proof. Immediate from (13). h An increase in congestion costs makes any potential member in a club less profitable for the club as a whole. Consider the richest club: if a increases, without modifying individual contributions, the benefit drawn from the marginal poorest member of the club decreases. It may then become optimal to let him not join the richest club. In any case, the richest club’s size cannot be increased. If the richest club is reduced, the society cannot be less segmented according to Definition 2. If its size is not reduced, repeating the argument on the following clubs eventually leads to the first part of the proposition. In the opposite, an increase in the relative desirability of the local public good increases the benefits drawn for any member of a club from the voluntary contribution of any member in the club. Consider the richest club: if b increases, without modifying individual contributions, the benefit from an additional unit of the club good increases. Hence each existing member is willing to let in an additional contributor who is necessarily poorer. It also increases the marginal utility of any contribution and induces each member to increase it. In any case, given that the congestion effect is not modified if the number of members is constant, there is no incentive to decrease the size of the richest club. In other words, the richest club either increases or remains identical. If the richest club is enlarged, the society cannot be more segmented according to Definition 2. If its size is not increased, repeating the argument on the following clubs eventually leads to the second part of the proposition. 15

It is interesting to note that Alesina et al. (2000a) obtain that ‘a county with near maximum income inequality has 2.2 more districts than a county with near minimum income inequality’ (p. 20). In their study on segregation of high- and low-skill workers into separate firms which can be assimilated to coalitions of workers, Kremer and Maskin (1996) emphasize an empirical finding by Kremer and Troske (1995) that US states with higher skill-dispersion tend to be more highly segregated.

948

F. Jaramillo et al. / Journal of Public Economics 87 (2003) 931–955

5. Conclusion The purpose of the present paper was to investigate the link between inequality and the endogenous formation of clubs when members voluntarily contribute to the funding of the club good, once they belong to a club. The formation of clubs can be seen as a formal feature corresponding to social segmentation. This allowed us to obtain several results both on the core partition and the impact of inequality on this partition. With regard to the core partition, it appears that it is uniquely defined. Clubs are consecutive, i.e. they are formed according to a ‘neighborhood’ rule and members in a given club are close in income terms. The utility level is the same for all members of a given club and the utility level characterizing a club differs according to income inequality: it is higher in richer clubs than in a poorer club. With respect to the influence of inequality, restricting the analysis to simple, regular patterns of relative endowment inequality among society, there is a clear connection between the property of this pattern and the characteristics of the endogenous core partition we are able to obtain. For a given distribution, satisfying some monotonicity property of the relative inequality pattern, the relative size of clubs depends on the degree of relative inequality. Higher (smaller) endowment ratios among richer agents than among poorer ones lead to smaller (larger) rich clubs and larger (smaller) poor clubs. Clubs are of equal size when relative inequality is constant at any level of income. Inequality happens to be a major factor of segmentation. We also analyze the difference in segmentation generated by differing patterns of inequality. For two endowment sequences, being characterized either by a decreasing or a constant relative inequality, and such that one Lorenz-dominates the other, the more ‘inegalitarian’ one generates a higher number of clubs, with smaller ‘non-residual’ clubs. In other words, an increase in inequality creates more social segmentation. However, it may happen that a more unequal society generates a less segmented society: again this hinges on the property of the endowment sequences. Our analytical framework relies on simplifying assumptions. In particular, agents differ only in their initial endowment. Apart from differing incomes, they are all alike. We also assume the absence of precommitment mechanisms on contributions, and we do not consider other decision rules than individually voluntary decisions on the level of contributions to the club good. Moreover, we only consider anonymous crowding effects.16 Finally, several recent studies have emphasized the role of neighborhood formation on human capital accumulation ´ and income distribution dynamics (see, for instance, Benabou, 1993, 1996a,b; Durlauf, 1996). A possible extension of this simple and tractable static setting in a 16 Recent advances in local public economics take into consideration that members’ identity matters for the formation of clubs (see Conley and Wooders, 1996).

F. Jaramillo et al. / Journal of Public Economics 87 (2003) 931–955

949

dynamic perspective could be developed so as to relate the dynamics of income inequality to the formation of communities and their properties. These various extensions are left to further research.

Acknowledgements ´ We are very indebted to Claude d’Aspremont, Roland Benabou, Francis Bloch, Russell Cooper, Michel Le Breton, Herve´ Moulin, Zvika Neeman, Antoine Soubeyran, Jean-Marc Tallon, Jacques Thisse, Shlomo Weber and Myrna Wooders, to participants to the 4th workshop on ‘Coalition Theory’ (Universite´ AixMarseille, January 1999), as well as to participants to seminars in Boston University and Universite´ Paris-1, for their comments on a first draft of this paper and to an anonymous referee for insightful suggestions.

Appendix A Proof of Proposition 1 Consecutive structure Proof. Consider a partition S and a club S1 [ S. We define:

vs 1 5 Min fvz , vz [ S1g ]

v]s 1 5 Max fvz , vz [ S1g.

Let us assume that S1 is not a consecutive set, i.e.: 'vi , vs 1 , vi , vs] 1 , vi [ S2 [ S, S2 ± S1 ]

Then we show that partition S is not a core partition. A. Consider first the case where VsS1d .VsS2d. We define:

vs 2 5 Min fvz , vz [ S2g ]

v]s 2 5 Max fvz , vz [ S2g.

We form the club S3 5sS1 \s] 1d < i. Then, since vi . vs 1 according to (5), we can ] deduce: V(S3 ) .V(S1 ) .V(S2 )⇔;vh [ S3 ,Vh (S3 ) .Vh (S) where Vh (S) denotes individual h’s utility for the club-partition S. Hence, partition S does not satisfy condition (7) and is not a core partition. B. Consider the case where V(S1 ) ,V(S2 ). Since vi , vs] 1 , it follows that vs 2 , v]s 1 . Then consider the club S4 5 (S2 \s] 2 ) < j, j [ S1 such that vj . vs 2 . ] ] Hence, from (5), it follows that:

F. Jaramillo et al. / Journal of Public Economics 87 (2003) 931–955

950

V(S4 ) .V(S2 ) .V(S1 )⇔;vh [ S4 ,Vh (S4 ) .VhsSd. Hence the partition S does not satisfy condition (7) and is not a core partition. h Uniqueness Proof. We shall focus on consecutive clubs only. Consider a given consecutive club S1 from which the richest type is v1 . A pivotal agent associated to S1 is such that the welfare for this club is maximal. Formally, this pivotal agent indexed p1 is such that: VsS1d .VsS9d where S9 5h1, . . . , i9j with i9 ± p1 . This pivotal agent is such that adding it to the club formed by agents from v1 to vp 1 increases the level of utility associated with the club, but adding the agent immediately after him decreases the club utility, that is:

Ov 1v

1

i

Ov

h 21

h21

h

i 51

2

1

2

i

i51

2 a n 1 $s1 1 bd ln ]]] 2 asn 1 2 1d s1 1 bd ln ]]] b 1 n1 b 1 n1 2 1 which implies: p 1 21

vp 1 $

b 1n O v S]]] e b 1n 21 1

a / 11 b

z

21

1

z 51

D

and

Ov 1 v

Ov

h

1

h

h 11

i

i 51

2

1 2 i

i 51

2 asn 1 1 1d ,s1 1 bd ln ]] 2 a n 1 s1 1 bd ln ]]]] b 1 n1 1 1 b 1 n1 which implies: p1

vp 1 11 ,

b 1n 11 O v S]]] e b 1n 1

a / 11 b

z

z 51

D

21 .

1

Is this maximum unique? Suppose on the contrary that there exists another pivotal agent p 91 . So, we have two clubs S1 5h1, . . . , p1j and S 91 5h1, . . . , p 91j with vp 1 9 , vp 1 and p 1 21

vp 1 $

b 1n O v S]]] e b 1n 21 1

1

z 51

p1

b 1n 11 O v S]]] e b 1n 1

z

z 51

a / 11 b

z

1

a / 11 b

D

21

D

2 1 . vp 1 11 . . . . . vp 1 9 .

F. Jaramillo et al. / Journal of Public Economics 87 (2003) 931–955

951

We have the following equivalence:

O v ssb 1 n 1 1de i

a / 11 b

2s b 1 n 1dd 1 vp 1 11 . vp 1 11s b 1 n 1 1 1d.

1

i [S 1

We can deduce easily:

O v ssb 1 n 1 1d e 2sb 1 n dd 1 v ,S O v 1 v Dssb 1 n 1 1d e 2sb 1 n dd ,S O v 1 v Dssb 1 n 1 2d e 2sb 1 n 1 1dd. a / 11 b

i

1

p 1 11

1

i [S 1

a / 11 b

i

p 1 11

1

i

p 1 11

1

i [S 1

1

a / 11 b

i [S 1

1

This implies that:

sb 1 n 1 2d e S O v 1 v DS]]]] sb 1 n 1 1d 1

i

i [S 1

p 1 11

a / 11 b

1

D

2 1 . vp 1 11 .

Reiterating the reasoning for vp 1 12 until vp 1 9 leads to the following inequality:

s d e S O v DS]]]] sb 1 n 9d b 1 n 19 1 1

i [S 91

i

1

a / 11 b

D

2 1 . vp 1 9

which contradicts the assumption that vp 1 9 too is a pivotal agent. h Welfare ranking Because agent p1 gives the maximal welfare for the club S1 , we have: Vsh1, . . . , p1jd .VsS9d, ;i [ S1 . Moreover we know that this club S9 5h1, . . . , ij gives more utility than a club S0 5h2, . . . , i0j which has the same number of agents but does not contain the richest agent. For example, p 1 11

1O 2

vz z 52 Vsh1, . . . , p1jd .s1 1 bd ln ]] 1 b ln b 2 a n 1 . b 1 n1 Hence, we deduce that the welfare of h1, . . . , p1j is the highest possible welfare achievable in this economy. So, it is straightforward that h1, . . . , p1j belongs to the core partition and will be denoted S 1E in it. If we consider now the subset I\S 1E and iterate the steps above, it is very easy to show that S 5h p1 1 1, . . . , p2j, where p2 is the pivotal agent of this club, is unique and belongs to the core. We will repeat this reasoning until we achieve agent N. At this moment, the whole core partition will be obtained. h

F. Jaramillo et al. / Journal of Public Economics 87 (2003) 931–955

952

Appendix B Proof of Proposition 3 1. We denote n*sid and n*si9d the optimal sizes of clubs when i (resp. i9, i9 , i) is the richest member. Hence n*sid and n*si9d are the smallest integers such that the following inequalities are satisfied:

D OS P D OS P

S S

b 1 n*(i) 1 1 1 , ]]]] e a / 11 b 2 1 b 1 n*(i)

b 1 n*si9d 1 1 1 , ]]]] e a / 11 b 2 1 b 1 n*si9d

n*(i )21

i 1n*sid21

v 50

x 5v 1i

D

lx,x 11

n*si 9d21

i 1n*si 9d21

v 50

x 5v 1i 9

(B.1)

D

lx,x 11 .

(B.2)

If li,i 11 $ li 11,i 12 , ;i, then we have:

D OS P l D D OS P l D

Sb b

1 n*si9d 1 1 ]]]] e a / 11 b 2 1 1 n*si9d

S

n*si 9d21

i 91n*si 9d21

v 50

x 5v1i 9

b 1 n*si9d 1 1 $ ]]]] e a / 11 b 2 1 b 1 n*si9d

x,x 11

n*si 9d21

i 1n*si 9d21

v 50

x 5v1i

x,x 11

(B.3)

(B.4)

which implies that n*si9d # n*sid, ;i9 , i, i9 and i [ I. Now suppose that n*si9d # n*sid, ;i9 , i, i9 and i [ I. Given that n*sid and n*si9d fulfil inequalities (B.1) and (B.2), it implies:

DOS P l D DOS P l D

Sb b

1 n*si9d 1 1 ]]]] e a / 11 b 2 1 1 n*si9d

S

n*si 9d21

i 91n*si 9d21

v 50

x 5v1i 9

b 1 n*si9d 1 1 $ ]]]] e a / 11 b 2 1 b 1 n*si9d

x,x 11

n*si 9d21

i 1n*si 9d21

v 50

x 5v1i

x,x 11

.

(B.5)

(B.6)

It is then straightforward to deduce that:

OS P

n*si 9d21 v50

i 91n*si 9d21 x 5v 1i 9

D OS P n*si 9d21

lx,x 11 $

v 50

D

(B.7)

;i, i9 [ I, i9 , i.

(B.8)

i 1n*si 9d21 x 5v 1i

lx,x 11 .

Equivalently, we get: n*si9d # n*sid ⇒ li,i 11 $ li 11,i 12

2. The proof is similar for li,i 11 5 li 11,i 12 5 l and li,i 11 # li 11,i 12 , ;i [ I. h

F. Jaramillo et al. / Journal of Public Economics 87 (2003) 931–955

953

Appendix C Proof of Proposition 4 Consider two societies which differ only according to individual endowments. We assume N 5 10, b 5 0.5 and a 5 0.01. The following table gives the individual endowments in both societies and the corresponding values of the endowment ratios: i v 1i

8 3.5

7 4.5

6 5.5

5 6.5

4 7.5

3 8.5

2 9.5

1 10.5

v 2i 1.5 3.5 5.5 1 l i,i 11 1 1.66 1.4

7.5

9.5

11.5

13.6

15.5

17.5

19.5

1.28

1.22

1.18

1.15

1.133

1.11

1.10

1.36

1.26

1.21

1.18

1.139

1.129

1.11

l

10 1.5

2 i,i 11

1

9 2.5

2.33

1.57

2

1

First we show that if L1 and L2 are such that l i,i 11 . l i,i 11 , ;i [ I, then L1 Lorenz-dominates L2 . The definition of a Lorenz curve associated with a sequence L is:

O SP D O SP D N

N

vN li,i 11 m i 5v v5m F ] 5 ]]]]]] N N N vN li,i 11

S D

v51

i5v

with lN,N 11 5 1. It is equivalent to:

S D

m 1 F ] 5 ]]]]]]]]] m21 m21 N N li,i 11 li,i 11 i 5m v51 i 5v 1 1 ]]]]]]]] N N

SP D O S P D O SP D v5m

i 5v

li,i 11

Defining A as follows:

SP D O S P D O S P D O SP D O SP D N

m 21

m 21

m 21

m 21

li,i 11 li,i 11 li,i 11 i 5m v 51 i 5v v 51 i 5v ]]]]]] A 5 ]]]]]]]] 5 N N 21 N v 1 ]] li,i 11 11 l v5m i 5v v 5m i 5m i,i 11 it is obvious that when li,i 11 increases, A increases. Hence, the Lorenz curve associated with L2 is below the one associated with L1 . In our example, the second society is Lorenz-dominated by the first one. Using the formulas developed in the main part of the paper, the two core partitions are as follows:

F. Jaramillo et al. / Journal of Public Economics 87 (2003) 931–955

954

S E 1 5hh1, 2, 3j, h4, 5j, h6, 7j, h8, 9j, h10jj S E 2 5hh1j, h2, 3, 4j, h5, 6j, h7, 8j, h9j, h10jj Hence the second society is not more segmented than the first one. h

Appendix D Proof of Proposition 5 Denoting n ksid the size of the consecutive club whose richest member is i and maximizes i’s welfare under Lk , k 5 1, 2, it is deduced from (13) that n 2sid # n 1sid, ;i [ I. This implies that the first pivotal agent under L2 denoted by p1sL2d has a lower index than under L1 s p1sL2d , p1sL1dd. Hence, n 2s p1sL2dd # n 1s p1sL2dd. Under A1 (A2), using Proposition 3, we know that n 1s p1sL2dd 5 1 s ,dn s p1sL1dd. Combining these two inequalities, we get that the second club is smaller under L2 than under L1 . Repeating this reasoning completes the proof of Proposition 5. h

References Alesina, A., Spolaore, E., 1997. On the number and sizes of nations. Quarterly Journal of Economics 112, 1027–1055. Alesina, A., La Ferrara, E., 2000. Participation in heterogeneous communities. Quarterly Journal of Economics 115, 847–904. Alesina, A., Baqir, R., Hoxby, C., 2000a. Political jurisdictions in heterogeneous communities, Mimeo, http: / / post.economics.harvard.edu / faculty / alesina / papers.html. Alesina, A., Spolaore, E., Wacziarg, R., 2000b. Economic integration and political disintegration. American Economic Review 90, 1276–1296. Andreoni, J., 1988. Privately provided public goods in a large economy: the limits of altruism. Journal of Public Economics 35, 57–73. Barham, V., Boadway, B., Marchand, M., Pestieau, P., 1997. Volunteer work and club size: Nash equilibrium and optimality. Journal of Public Economics 65, 9–22. ´ Benabou, R., 1993. Workings of a city: location, education, and production. Quarterly Journal of Economics 108, 619–652. ´ Benabou, R., 1996a. Heterogeneity, stratification and growth: macroeconomic implications of community structure and school finance. American Economic Review 86, 584–609. ´ Benabou, R., 1996b. Equity and efficiency in human capital investment: the local connection. Review of Economic Studies 63, 237–264. Bennett, E., Wooders, M., 1979. Income distribution and firm formation. Journal of Comparative Economics 3, 304–317. Bergstrom, T., Blume, L., Varian, H., 1986. On the private provision of public goods. Journal of Public Economics 29, 25–50. Bolton, P., Roland, G., 1997. The break up of nations: a political economy analysis. Quarterly Journal of Economics 112, 1057–1090.

F. Jaramillo et al. / Journal of Public Economics 87 (2003) 931–955

955

Bourdieu, P., 1987. Distinction: a Social Critique of the Judgment of Taste. Harvard University Press, Cambridge. Conley, J., Wooders, M., 1996. Taste-homogeneity of optimal jurisdictions in a Tiebout economy with crowding types and endogenous educational investment choices. Ricerche in Economiche 50, 367–387. Cooper, R.W., 1999. Coordination Games: Complementarities and Macroeconomics. Cambridge University Press. Cornes, R., Sandler, T., 1996. The Theory of Externalities, Public Goods and Club Goods. Cambridge University Press. Durlauf, S., 1996. A theory of persistent income inequality. Journal of Economic Growth 1, 75–93. Epple, D., Sieg, H., 1999. Estimating equilibrium models of local jurisdictions. Journal of Political Economy 107, 645–681. Ehrenberg, R.G., Smith, C.L., 2001. The sources and uses of annual giving at private research universities. NBER, Working paper no. 8307. Farrell, J., Scotchmer, S., 1988. Partnerships. Quarterly Journal of Economics 103, 279–297. Fernandez, R., Rogerson, R., 1996. Income distribution, communities and the quality of public education: a policy analysis. Quarterly Journal of Economics 111, 135–165. Foster, J.E., Ok, E.A., 1999. Lorenz-dominance and the variance of logarithms. Econometrica 67, 901–907. Glomm, G., Lagunoff, R., 1999. A dynamic Tiebout theory of voluntary vs. involuntary provision of public goods. Review of Economic Studies 66, 659–678. Greenberg, J., Weber, S., 1986. Strong Tiebout equilibrium under restricted preferences domain. Journal of Economic Theory 38, 101–117. Hackett, S.C., Schlager, E., Walker, J., 1994. The role of communication in resolving commons dilemmas: experimental evidence with heterogeneous appropriators. Journal of Environmental Economics and Management 27, 99–126. Independent Sector, http: / / www.independentsector.org. Isaac, R.M., Walker, J., 1988. Communication and free-riding behavior: the voluntary contribution mechanism. Economic Inquiry 24, 585–608. Kremer, M., Troske, K., 1995. Notes on segregation by skill, unpublished. Kremer, M., Maskin, E., 1996. Wage inequality and segregation by skill, NBER working paper no. 5718. Quarterly Journal of Economics, in press. Rousseau, J.J., 1985. In: A Discourse on Inequality. Penguin Classic, First published in 1755. Weber, M., 1978. Economics and Society. University of California Press, First published in 1922. Yitzhaki, S., Lerman, R.I., 1991. Income stratification and income inequality. Review of Income and Wealth 37 (3), 313–329.