Inequality, Growth and the Dynamics of Social Segmentation. Hubert Kempf

Fabien Moizeau

Banque de France and Paris School of Economics.y

Université de Rennes 1 (CREM).z

Revised version, March 2009.

Abstract In this paper we investigate the link between the dynamics of society segmentation into communities and the growth process, based on a simple human capital growth model. Using coalition theory, we study the socioeconomic dynamics of an economy over time, characterize it and prove that the economy converges to a steady state partition which may be segmented. Eventually the whole economy tends to a balanced growth path, exhibiting persistent inequality in the case of segmentation. We then provide su¢ cient conditions on initial inequality and the technology parameters generating local and global externalities for obtaining a segmented society in the long run. On the whole, the relationship between inequality and growth cannot be assessed without taking into consideration the strati…cation phenomena at work in society over time. Keywords: Coalition Theory, Inequality, Growth. JEL Classi…cation: C71, D71, D3, O40.

We are very grateful to Matt Jackson, Fernando Jaramillo, Frank Page, as well as to participants to seminars in universities of Austin, Barcelona (UAB), Caen, Marseille, Taïpeh, the European University Institute and various conferences (EEA Amsterdam 2005, Theories and Methods in Macroeconomics Lyon 2005, Association Française de Science Economique Paris 2005, Coalition Theory Network Paris 2005, SED Budapest 2005, APET Marseille 2005), for their comments on earlier drafts of this paper. y Address: Centre d’Economie de la Sorbonne, 106-112 Boulevard de l’Hopital, 75013 Paris, France. email address: [email protected] z Address: Université Rennes 1, CREM (UMR CNRS 6211), 7 Place Hoche, 35000 Rennes, France. [email protected]

1

email address:

1

Introduction.

Accumulation of knowledge is both central to the growth process and a social activity, taking place in communities. In his seminal paper on endogenous growth theory, Lucas stressed the role of human clusters on growth. In Lucas’s words, “we know from ordinary experience that there are group interactions that are central to individual productivity and that involve groups larger than the immediate family and smaller than the human race as a whole. Most of what we know we learn from other people. ... All of intellectual history is the history of such e¤ects” (Lucas, 1988, p.38). More recently, Bénabou (1993, 1996a,b) provided a theoretical framework to explore the role of existing communities on human capital accumulation and growth. The insights of Lucas and Bénabou about the critical role of local interactions on growth are the starting point of our research.1 In this paper, we show how the dynamics of social segmentation is at the root of the relationship between inequality and growth. The key feature of our analysis is that individual growth both depends on productive public goods provided by communities, or equivalently clubs, and on an index of total capital formation.2 In a given period, individuals sort themselves into clubs according to their endowments, giving rise to a segmented society. Since the club good in‡uences individual productivities of club members and thus their human capital accumulation, it also a¤ects the subsequent endowment distribution. In turn, this new distribution leads to a new partitioning of society. The recurring process of club formation thus both determines the growth process and continuously reshapes the endowment distribution over time. We then prove that endogenously formed segmentation may be sustained in the long run, despite the presence of a global externality e¤ect which plays an homogenizing role in society over time. This is consistent with a balanced growth path, where all clubs grow at the same rate, but at di¤erent human capital levels. In other words, permanent segmentation is related with persistent inequality. This result stresses the crucial in‡uence of endogenous formation of groups on the dynamics of inequality. To investigate the nexus between social segmentation, inequality and growth, we set up an endogenous growth model with the following features. Agents live for one period and care about the bequest left to their unique o¤spring, because of a “joy-of-giving” motive. This motive generates the dynamic linkage between periods. As the production of human capital depends on the provision of a “club good”, agents in a given period form clubs, leading to a partition of society. As each generation forms its own clubs, the partitioning of society may be altered over time. 1 It

is worth stressing that Easterly and Levine (2001) documented that the concentration of economic activity is one of the

stylized facts of growth. Interestingly, they stress that economic concentration has a fractal-like quality. Factors of production tend to congregate whatever the level of analysis: the globe, countries, regions, cities, ethnic groups, etc. 2 Clubs that we have in mind are any group of individuals that provides internally human capital interactions to its members and that excludes outsiders from the consumption of these interactions. In various arenas of social life such as workplaces, schools, neighborhoods, individuals choose their peers and sort themselves along the socioeconomic basis (see for instance Fernandez, 2002, and references therein). The reason is that group interactions are a key determinant for individuals’productivity (see for instance the recent works of Mas and Moretti, 2007, Goux and Maurin, 2007, and among others, the survey of Mo¢ tt, 2001).

2

The decision over membership re‡ects a trade-o¤ between economies of scale and an heterogeneity e¤ect.3 On the one hand, any additional member increases the size of a club and this creates economies of scale in the production of the club good. On the other hand, her relative capacity to contribute to the provision of the club good is taken into consideration: a poorer agent depresses the average human capital available in the club, and hence the tax base of the club. This acts against the inclusion of poor agents in a club. In other words, endowment heterogeneity limits the size of a club. This trade-o¤ explains why there are boundaries to clubs. On the whole, inequality leads to a partition into clubs. With respect to the formation of clubs, we consider the core: to belong to a club, individuals must be willing to enter it and unanimously accepted by its other members. In each period, an equilibrium exists with a partition belonging to the core of the coalition formation game. This partition is unique, consecutive with respect to human capital endowments, and welfare-ordered. In each period, within each club, there is endowment convergence: The di¤erence between incomes of two members of a club at the end of a period (after the provision of the club good) is reduced with respect to the di¤erence in their initial incomes (upon which the membership is decided). As members of the same club get more homogeneous and the distribution of endowments varies over time, subsequent decisions (by o¤springs) with respect to the formation of clubs may therefore alternate catching-up and income divergence episodes. Despite the complexity of club formation and human capital accumulation dynamics, we prove that any economy, given its initial human capital distribution, eventually establishes a unique permanent partition: after a certain …nite date, clubs’borders do not move, even though agents have the possibility to change these borders period after period. It follows that endowment convergence happens within a club. Importantly, there is no systematic tendency to convergence between communities over time. Depending on initial inequality and the size of the global externality with respect to the club externality, the long run distribution of human capital may exhibit poverty traps preventing poorer groups from catching-up richer ones. But club growth rates converge to the same steady-state value, due to the presence of the global externality. In brief, there is intra-club but not inter-club convergence. Our paper builds on static models of endogenous formation of coalitions (see among others Wooders, 1978, Guesnerie and Oddou, 1981, Farrell and Scotchmer, 1988, Barham, Boadway, Marchand and Pestieau, 1997, Banerjee, Konishi and Sönmez, 2001, Jaramillo, Kempf and Moizeau, 2003, 2005). In particular, Jaramillo, Kempf and Moizeau4 (2005) addresses the link between inequality and growth, o¤ering the notion of “growth clubs”. A growth club is a cluster of individuals bonding together as they share a common resource, which makes them grow together. In JKM (2005), individuals are in…nitely-lived and clubs are formed in the …rst period and last forever. Within each club, a club good is provided once-and-for-all. As a result, the initial distribution of capital a¤ects the entire growth process over time, through the partitioning of society. Moreover convergence cannot be taken for granted. Thus, inequalities do not necessarily vanish. In contrast to this paper, where segmentation happens once and forever, we enrich the picture as we are able to exhibit a much more complex and two-dimensional dynamics. There are the social dynamics of the evolving partitions, and the economic dynamics of capital accumulation. These two dynamics interact and 3 This

is in line with the evidence on political jurisdictions presented in Alesina, Baqir and Hoxby (2004). JKM.

4 Hereafter,

3

generate a di¤erentiated growth pattern. Recently, Konishi and Ray (2003) explore the dynamics of coalition formation: they study coalition formation as an on-going dynamic process where individuals are long-lived and have expectations about the future value of coalitional moves. Their perspective is di¤erent from ours as we assume that agents live one period and coalitions form at each period. Finally, our paper relates to a strand of literature that stresses the role of neighborhood e¤ects on the evolution of income inequality and productivity growth (see de Bartolome, 1990, Bénabou 1993, 1996a,b and Durlauf, 1996). Bénabou assumes that communities are initially formed and endure forever. Durlauf explores the dynamics of income inequality by studying the evolution of neighborhood choice of individuals, without taking into account the presence of a global externality. Here we introduce both the possibility of an endogenous reshu- ing of temporary clubs and the presence of a global externality in addition to local externalities, and prove that these features do not contradict the idea that inequality may persist over time, even in a growing economy. The plan of the paper is as follows. In the following section we set up the model. In section 3, we characterize the equilibrium attained in period t and study its properties. Then in section 4, we study the dynamic sequence of these equilibria, proving that a steady state exists, with both a permanent core partition of the economy and a steady-state growth pattern. We also discuss the in‡uence of integration on the dynamics of inequality and growth performance. Section 5 concludes.

2

The economy.

We consider a model of successive generations of individuals. Each individual lives one period and has a unique o¤spring at the end of his life. Population is of constant size N: A dynasty is an in…nite sequence of succeeding individuals. In each period, society is comprised of the N individuals S = f1; :::; N g. At

date t, each individual i is endowed with a level of human capital hit . At t = 0, agents are ordered so that N 1 h10 > h20 > :::: > hN 0 . We refer to St = ht ; :::; ht

as the endowment schedule.5

Individuals only di¤er according to their human capital endowment. Agents’preferences are the same. For any individual born in t, preferences depend on private consumption cit and the bequest left to her o¤spring hit+1 : U (cit ; hit+1 ) = ln cit + ln hit+1 Each individual is endowed with 1 unit of time. (1

(1)

) is the fraction of time the individual devotes to

the transmission of human capital to his child.

is the fraction of time devoted to work. For simplicity, is N X assumed to be constant. We assume an aggregate production function such that Yt = hit : This implies i=1

that hit equals the hourly wage wti and thus the income of individual i is yti = hit : The consumption level is equal to the after-tax net income. Agents are willing to form or join a club because it provides a productive club good. More precisely, for 5 We

adopt the following convention: in the case of a state variable, t refers to the beginning of period t:

4

each individual i belonging to the j

club Ctj , the human capital technology is de…ned by:

hit+1 = [ (1

1

) hit

Gjt

]1

(H t )

(2)

with Gjt the level of public good in club Ctj ; and H t the average level of human capital in society: H t = N P 1 hit . We assume > 0; 2 (0; 1); 2 (0; 1) so that all factors exhibit diminishing returns. N i=1

The amount of human capital left to any o¤spring depends on parental contribution, i.e. (1

) hit ; the

level of the club good and the average global human capital. The quality of education itself depends on the available individual human capital. The club good (e.g. think of education provided by a “school”) helps a member of the club to accumulate human capital. This corresponds to the presence of a “local externality”. Finally the impact of average human capital formed in the whole society captures the in‡uence of a “global externality”. The higher is the elasticity of individual human capital with respect to H t , measured by ; the more important is the “global externality”. The club good is …nanced through taxes levied on members’incomes with a proportional tax rate denoted by

j 6 t.

The tax rate

later. We denote by

njt

j t

is speci…c to each club and is endogenously chosen in a manner to be developed card Ctj

the number of individuals belonging to Ctj : The amount of club good

Gjt is generated by the following production function: P

j t

hit

i2Ctj

Gjt =

A(njt )

:

(3)

The club good technology is hampered by congestion e¤ects, captured by A(njt ). We assume that A0 (njt ) > 0 and that A(njt ) is a log-convex function.7 Importantly, congestion e¤ects are anonymous. The harm in‡icted by other members of a club to any individual member is related to their number, not to their precise identity or characteristics. It is illuminating to rewrite (3) as follows:

Gjt =

j t

P

i2Ctj njt

hit

njt

(4)

A(njt )

The …rst ratio corresponds to the average human capital in the club, which we denote by assume that the second ratio,

njt , A(njt )

increases with respect to

njt ,

whatever

njt

j Ht

N , i.e. 1 >

=

P

j i2Ct

hit

njt njt A0 (njt ) . A(njt )

: We The

larger is a club, the more e¢ cient it is in providing the club good. Accepting a poorer agent as a member in a club is bene…cial to the club because his entry generates “economies of scale”.8 However, as she is poorer, this depresses the average club’s human capital which has a negative e¤ect on the club good provision since 6 An

alternative assumption would be that each member voluntarily contributes a fraction of his income to the provision of

the club good (see Barham, Boadway, Marchand and Pestieau, 1997, for a model of club formation with private provision of the club good). In fact, the taxing aspect of the model is not crucial for our results on coalition formation. All we need is that the technology of the club good provides incentives for individuals to segregate into clubs. 7 This property will be useful for the proof of uniqueness of the equilibrium. 8 The

log-convexity of A(n) implies that the ratio

nA0 (n) A(n)

increases. The assumption that A(:) is such that

allows us to have economies of scale whatever the group size.

5

N A0 (N ) A(N )

< 1

it amounts to reduce the club’s tax base. As accepting a poorer agent increases the social diversity within a club, we call this e¤ect the “static heterogeneity” e¤ect. Notice that if individuals were endowed with the same level of human capital, it would be e¢ cient to form the group encompassing the whole society as it would maximize economies of scale. The “static heterogeneity” e¤ect limits the size of the group. In brief, the formation of clubs relies on a trade-o¤ between “economies of scale”and the “static heterogeneity”e¤ect. As agents may willingly form clubs, at any date t the society S may be partitioned: o n De…nition 1 A nonempty subset Ctj of S is called a club and Ct = Ct1 ; :::; Ctj ; :::; CtJ for j = 1; :::; J

is called a partition of S if: J S (i) Ctj = S; j=1 (ii) Ctj

T

0

Ctj = ; for j 6= j:0

Given the one-period duration of individual live, clubs form for one period only. Hence a partition Ct is de…ned for a given period. Finally, there is no capital market: Agents cannot borrow and lend freely so as to alter their decision to accumulate human capital.9 As it is, the present model is reminiscent of the models used by Glomm and Ravikumar (1992), Saint-Paul and Verdier (1993) and Bénabou (1996 a,b). We adapt the basic speci…cation used by Glomm and Ravikumar to an economy with clubs, introducing local human capital externalities as in Bénabou.

3

Segmentation, growth and inequality.

3.1

Inequality and the core partition at time t.

In each period, given the endowment schedule

; the functioning of the economy is as h1t ; :::; hit ; ::::; hN t

follows: 1. In the …rst stage, clubs form. This implies some agreement over the membership, the amount of club good to be provided by its members and the tax rate to be chosen by the constituency formed by the sole members of the club. The tax rate chosen within a club is decided through a simple majority rule. The partition into clubs must belong to the core. 2. Then, in the second stage, individuals pay taxes, produce and leave bequests in the next period to their o¤spring. We shall prove the existence of an equilibrium when agents form clubs and will use the following de…nitions: 9 From

this perspective, the ability to lend and borrow is a way for individuals to shape their bonding and thus, a¤ects the

segmentation of society. This issue is beyond the scope of this paper.

6

De…nition 2 A partition Ct = if: @$

o n Ct1 ; :::; Ctj ; :::; CtJ belongs to the core of the coalition-formation game S such that 8i 2 $; V i ($) > V i (Ct )

where V i (Ct ) denotes the utility for agent i associated with partition Ct : According to this de…nition, a partition belongs to the core when it is immune against any defection, i.e. no member of the deviating group obtains more than what he is currently getting in the partition.10 b 1 ; :::; C btj ; :::; C btJt g; bjt De…nition 3 At date t, an equilibrium Cbt = fC t (i)

bjt

is chosen in club

btj C

j2f1;:::;Jt g

satis…es:

according to the majority rule and such that:

b jt = bjt G

P

bj z2C t n bjt

hzt

n bjt

A(b njt )

and

(5)

(ii) Cbt belongs to the core of the coalition-formation game.

According to this de…nition, the equilibrium we are looking for is such that in each club, the provision of the club good is fully …nanced through taxes, and no agent has any interest to propose or accept a defection from any club, as the partition belongs to the core. In the sequel, we refer to Cbt as a core partition. It is

indexed by t since the inequality schedule changes from period to period, modifying incentives the individuals face while forming groups.

Solving backwards allows us to characterize the equilibrium as follows: Proposition 1 At date t, the equilibrium exists and is characterized by the following: (i) The tax rate in club j is equal to: bjt = b =

1+

(1 ) : (1 )

(ii) The core partition Cbt is unique and satis…es :

btj , then 8i ; i > i > i0 ; i 2 C btj . (a) Consecutivity: if i and i0 both belong to C 0 btj and i 2 C btj ; then V i (C btj ) > (b) Welfare ordering: for two individuals such that i0 > i, i0 2 C 0 0 0 btj ) and if j 6= (=)j 0 ; G b jt > (=)G b jt . V i (C Proof See Appendix.

In any club at any date, due to our assumption that the utility function is log-linear, any individual votes for the same tax rate. It implies that for any agent, her private consumption does not depend on the 1 0 To

enter a club, an individual needs the unanimous consent of its members. In many situations, groups are able to exclude

outsiders. If we consider the case of local jurisdictions, we could view this admission rule as a reduced form of zoning regulations. In particular, a zoning regulation requires individuals to purchase a minimum level of housing as a condition for residence in the community (see Fernandez and Rogerson, 1997, for a study on the impact of zoning rules, determined by a voting process, on communities formation). If we assimilate a club to a nation, this admission rule could correspond to a constitutional rule that constrains migration among nations (see Jéhiel and Scotchmer, 2001, for a study of jurisdiction formation under di¤erent migration rules).

7

membership of her club. Moreover the indirect utility is a sum of two components, one linked to private consumption and the other linked to the amount of club good. Since the tax rate is independent from the partition, the comparison by an individual between two clubs only depends on the amount of club good btj ), the indirect utility for individual i belonging to C btj , we have: provided in a given club. Denoting by V i (C 0 P z 1 ht j j b n bt C B z2Ct btj ) = Fti + (1 (6) V i (C ) ln @ A j n bt A(b njt ) with Fti = ln

1+

1 (1

)

hit + ln

(1

) hit

(1

)(1

)

+

(1

) ln ( 1+

(1 ) (1 ) )

+

ln H t :

Since only the club good component in the indirect utility matters for comparing coalitions, all agents display the same preference ordering of coalitions. Our framework features the “top coalition property”which immediately ensures the existence of a core partition (see Banerjee, Konishi and Sönmez, 2001). As stressed before, the formation of a club relies on the trade-o¤ between “economies of scale” and the “static heterogeneity”e¤ect.11 The richer is a member in a club, the more she contributes to the …nancing of the club good whereas any additional agent to a club generates the same “economies of scale”(which depends on the number of members and not on their characteristics). Hence the net marginal bene…t generated by the entry of an individual in a club is higher for richer agents. It turns out that any individual is in favor of accepting in her club the richest possible agent rather than poorer ones. This tends to support a drive toward closed and homogeneous clubs: the richest individuals in society will accept into their club additional therefore poorer agents until the net marginal bene…t of an entry is nil. Then the richest individuals among the non-accepted will form their own closed club, etc.12 Hence, this endogenous formation of clubs will lead to a core partition characterized by the properties o¤ered in Proposition 1. Uniqueness of the core partition at date t directly derives from the fact that the net marginal bene…t linked to an agent i to a club monotonously decreases with her endowment, that is the rank of i. The consecutivity property is a direct consequence of the fact that the net marginal bene…t of a member is increasing in her endowment. Then if i0 is accepted by i, any i who is richer than i0 is also accepted by i. We adopt the convention that clubs are indexed according to the wealth of their members, or equivalently the 0 0 btj ) > V i0 (C btj ); btj such that V i (C btj and C ranking of utilities associated with them: Considering two clubs C 0

btj and any i0 2 C btj ; then j < j 0 . Given the consecutivity property of the core partition, it for any i 2 C bt1 ; the next n bt2 ; amounts to say that the n b1t richest agents form the club C b2t richest agents form the club C

etc. The richer the members of a club, the more tax incomes they generate. As, the richest individual, 1; bt1 generates more utility for 1 is welcome in any club, she will choose the most pro…table club. Therefore C

btj ; j > 1: Leaving aside agent 1; we can adapt this reasoning to agent 2, and so on. Hence, the than any C welfare ordering property is true for any couple of individuals.

Equation (6) allows us to o¤er a simple characterization of the core partition valid at t:

1 1 This

trade-o¤ is standard in endogenous club formation setups. It could also be obtained in a framework where individual

most preferred tax rates di¤er. In such a case, the costs of heterogeneity while forming a club come from the distance between the individual bliss point and the tax rate chosen by the median voter (see for instance Alesina and Spolaore, 1997). 1 2 The reasoning we apply to construct the core partition employs the same logic as in Farrell and Scotchmer (1988).

8

bt1 ; :::; C btj ; :::; C btJt g can be de…ned as a set of pivotal agents fp1t ; :::; pjt ; :::; pJt t g Proposition 2. 8t; Cbt = fC j btj = fpjt 1 + 1; :::; pjt g and hpt t is such that: where C pjt

X1

pj ht t

z=pjt

1

A(b njt )

hzt

A(b njt

+1

1)

!

1

(7)

and j

pj +1 ht t

pt X

<

z=pjt

1

A(b njt + 1)

hzt

1

A(b njt )

+1

!

(8)

where p0t + 1 = 1 and pJt t = N: btj . The marginal “static heterogeneity” e¤ect she in‡icts A pivotal agent is the poorest agent of a club C

on all members is covered by economies of scale generated by her entry. The next agent, following the pivotal agent in the sequence of endowments, does not have a su¢ cient human capital endowment to compensate btj . Depending on the human capital dynamics, pivotal these costs, and thus is not accepted by members of C agents may change over time. Checking whether inequalities (7) and (8) are still satis…ed at date t + 1 allows

us to know whether individual pjt is still the pivotal agent of group j at date t + 1. Hence, the number of btJt is called the “residual”club. Its size is not “optimal”as its last clubs Jt depends on time. The last club C agent is the economy’s poorest agent N , so inequality (8) is meaningless for this club.

The endowment heterogeneity in society may be characterized by the sequence of human capital ratios between two succeeding individuals i and i + 1; at date t; de…ned as as i is richer than i + 1, any ratio

i t

i t

=

hit : hi+1 t

Given our ranking convention,

is larger than 1. Hence the conditions de…ning pivotal agents given in i t

Proposition 2 can be expressed in terms of the ratios pjt

1

j

X1

pt Y

j

pt Y

z=pjt

1

z t

+1 z

: A(b njt ) A(b njt

1)

!

(9)

!

(10)

1

and

1<

pt X

z=pjt

1

j

z t

+1 z

A(b njt + 1) A(b njt )

1

where p0t + 1 = 1 and pJt t = N . Remark that the static heterogeneity e¤ect depends on the relative human capital of an additional member, with respect to the human capital of the other members. The further down an agent i is in the income ladder, the higher is the static heterogeneity e¤ect that her entry in a richer club would generate. In other words, the lower (higher) is the human capital ratio between two individuals the more (the less) they are willing to interact in the same club.

9

To summarize, size is a factor of bonding, whereas human capital heterogeneity is a factor of separation. Since the tax rate is independent of the distribution of human capital at any date, these two opposing forces are the only e¤ects to take into consideration as far as club formation is concerned.

3.2

Growth and inequality dynamics: the convergence issue.

Given the segmentation put in place in a given period, we are interested in its economic consequences in terms of growth and inequality. This is the …rst step of the analysis of the socioeconomic dynamics of this economy. In this setting, the course of human capital accumulation impacts on the dynamics of segmentation. With respect to income distribution dynamics, we will distinguish income dynamics between clubs and income dynamics within clubs. “Inter-club convergence” refers to catching-up episodes of richer clubs by poorer ones. “Intra-club convergence” corresponds to homogenization of individuals within a club at date t. Given what we said about the impact of human capital heterogeneity on incentives to form a group, it is likely that inter-club convergence favors the reduction of social segmentation. Given the value of Gjt in (5) and using (2), the human capital hit+1 is equal to:13 hit+1 with

= (1

)(1

)(1

= )

(hit )(1

)(1

(1

)

1+

) (1

)

j Ht

njt A(njt )

!

(1

)

(H t )

(11)

:

We give some properties of the core partition of a given period t and its consequences on growth in the following: Proposition 3. (i) Intra-club human capital convergence: At any date t, within clubs, there is human capital convergence: hit+1 hi < it0 ; 8i; i0 > i 2 Ctj ; 8t: 0 i ht+1 ht (ii) Dynasty ordering: Whatever t; the initial individual ordering remains unchanged: 0

hit+1 > hit+1 ; 8i; i0 > i 2 S; 8t: Proof See appendix. Point (i) states that within any club formed in a given period, there is convergence in human capital/endowment between members of a given club: The di¤erences between members’ human capitals are reduced. As the human capital technology exhibits diminishing returns, a poorer agent in a club bene…ts more from the club good than a richer member. For the equilibrium at t, from (11), the individual-humancapital growth rate

i t

for an agent i who belongs to Ctj can be written as follows: i t

1 3 We

hit+1 = hit

njt

j

Ht j hi A(nt ) t

!

(1

)

Ht hit

:

drop the hat symbol for ease of reading. Now on, we only refer to equilibrium values.

10

(12)

j

This ratio is increasing in the average wealth of the club H t for any individual, the richer is the club she belongs to, the higher is the level of the club good she bene…ts from, and the higher is the human capital she bequests to her child. But remark also that it is increasing in the ratio between the aggregate club wealth and the individual current human capital: the poorer is a member, the more she bene…ts from the club good. Finally, any individual bene…ts from the economy-wide knowledge spillovers, and the larger , the more so: This e¤ect is the more e¤ective, the poorer the agent is. Hence the global externality generates a catching-up e¤ect. Point (ii) states that, even though there is a catching-up mechanism at work inside clubs, it can never lead to an inversion of the ordering of dynasties. Agent i will always be richer than i0 , if i0 > i: This directly comes from the fact that i at date t inherited from a larger human capital; given the consecutivity property of the core partition at t, she bene…ts from a club good provision at least equal to the provision enjoyed by i0 at date t: Therefore her o¤spring is endowed with a larger human capital at t + 1 than i0 ’s. In other words, there is no “dynasty overtaking”. The ratio hit+1 =hit di¤ers between individuals, even if they belong to the same club. We denote by Ctj ’s

j t

the

j t

growth rate of the total level of human capitals of club members. is such that: X 0 (1 )(1 ) 1 (hzt ) ! ! (1 ) j C H B 1 z2C j Ht+1 njt B C t t j = : B j C t @ nt (H j )(1 )(1 ) A H j Htj A(njt ) t t We will refer to

j t

as the club Ctj ’s growth rate.14 We use

j t

(13)

as a convenient index of the whole process

of growth characterizing a club in the core partition. This expression makes clear that three e¤ects play a role in the growth process. Leaving

aside, the …rst term corresponds to economies of scale: more agents

in the club ceteris paribus produce a higher level of club good. The second term relates to a “dynamic heterogeneity”e¤ect: we will see below that the more heterogenous the club is and the lower its growth rate. Finally, there is a global externality e¤ect which may favor “inter-club convergence”: a poorer club bene…ts relatively more from the global externality e¤ect than a richer one. The higher is the parameter

governing

the global externality e¤ect, the more e¤ective is this catching-up phenomenon. Rewriting

j t

as follows allows us to highlight the role played by the heterogeneity e¤ect on the club

growth rate: pjt

j t

=

A(njt )

!

(1

)

(njt )

Ht j

Ht

!

X1

z=pjt

00 @@

1

pjt

1

X

z=pjt

1 4 We

+1

1

+1

pjtQ 1 x=z

pjtQ 1 x=z

x t

!(1 1

xA t

)(1

)

+1 1(1

)(1

)

:

(14)

+ 1A

could use di¤erent indicators of the aggregate growth properties of a club. In particular, a possible indicator could be

the average of individual growth rates for individuals belonging to Ctj . Due to the non-linearities at work in this economy, these indicators are likely to di¤er. However, in the long run,

j t

tend to the same value.

11

and the average of individual growth rates for members of Ctj

Heterogeneity of endowments in club Ctj ; measured by the ratios Appendix that

j t

is decreasing in any

x t ):

x t;

lowers the growth rate (we prove in

This comes from the concavity of the human capital technology 0

which magni…es the discrepancy between hit and hit . Remark that, for a given size, from Jensen’s inequality, the maximal value for

j t

x t

obtains when all the

are equal to 1 as then the dynamic heterogeneity e¤ect

fully vanishes. In brief, the more homogenous a club is, the more rapidly does it grow.15 Hence, the size of a club has a priori an ambiguous e¤ect on the club growth rate as it a¤ects economies of scale but also interferes with the membership of this club, that is the individual human capitals of its members. Inter-club convergence is not granted. There does not exist a systematic catching-up process taking place between individuals belonging to di¤erent clubs. Consider two individuals i and i0 with i0 > i , i 2 Ctj and 0

i0 2 Ctj . Thus from (2) we may have:

hit+1 hit > 0 0 : hit+1 hit

This is true when hit+1 =hit 0 0 = hit+1 =hit

Gjt =hit 0 0 Gjt =hit

(15) (1 (1

) )

>1

which cannot be ruled out. Moreover we do not know a priori whether

j t+1 j0 t+1

>

j t j0 t

; 8j < j 0 . Therefore we

cannot identify which clubs are more likely to catch-up along the transition path. As a consequence, even if the gap between two agents who belong to the same club at t reduces, it may increase in the sequel as their o¤spring may belong to di¤erent clubs. Phases of catching up and rising inequalities can alternate. If there is a long enough period of inter-club catching-up, sooner or later, the partition will change. But in turn this will a¤ect the dynamics of accumulation and may end the catching-up process, at least temporarily.

4

Segmentation in the long run.

Up to now, we have only investigated some of the short-run properties of the dynamics of social segmentation and its economic consequences. Now we would like to address the long-run properties of the dynamics of social segmentation and the growth process. Here the dynamics are more complex than in standard growth models since they involve a possible change in social segmentation, that is ‡uctuating boundaries of clubs. Hence, we are interested in three issues: First, is there a steady state for this economy? Second, does it entail permanent social segmentation, that is the coexistence forever of di¤erent clubs as engines to steady-state growth? Third, are there bene…ts in terms of growth performance of integration? 1 5 This

negative e¤ect of inequality on accumulation is reminiscent to Bénabou’s (1996a) work. It has been emphasized by

Persson and Tabellini (1994). See also the surveys of Aghion et al. (1999) and Bénabou (1996c).

12

4.1

The permanent core partition.

A critical issue is to know whether the dynamics tend to a steady-state. Our analysis has both a social and an economic dimension. Hence the analysis of the steady state is twofold: 1. Do the social segmentation dynamics reach a steady state? In our setting, even though clubs are allowed to last one period, their borders may be permanent and sustained forever by successive generations. Then we want to know whether the society ends-up polarized in di¤erent groups or integrated in a single group. 2. Does there exist a balanced growth path characterizing the rate of human capital accumulation? Here there is an interplay between these two dimensions. If the grand coalition does not emerge in the long run and the society remains segmented forever, steady-state clubs will be characterized by their own growth rate. Then the balanced growth path is de…ned by the set of steady state growth rates of the various steady state clubs. We notice that there is a logical precedence of permanent social segmentation: as long as the borders of clubs vary, there cannot be a balanced growth path since growth rates refer to temporary clubs. A steady state partition may be explained as follows. There may exist a given date t such that from t onwards, the core partition does not change anymore: all pivotal agents remain the same over time. We refer to this partition as the t

permanent partition and we o¤er the following:

De…nition 4. A core partition C is said to be t

permanent when Ct = C; 8t

t .

We can prove the following: Proposition 4. For any society, there always exists a unique date t such that a permanent core partition forms. Proof See appendix. The intuition underlying this proposition is as follows. Let us reason on the …rst club. The …rst club can be modi…ed over time only because the rank of its pivotal agent increases. The reason is twofold. First, given the intra-club convergence property, members of the …rst club become more homogenous; thus increasing their willingness to interact in the same club. Second, given the welfare ordering property, the highest level of club good is provided in the …rst club, so every individual wishes to belong to it. Hence, the size of the richest club can only expand. This enlargement process reaches a limit at a given date, either because the …rst club becomes the grand coalition or its members do not want to ever accept any additional member. Starting at this date, and leaving aside the …rst club’s members, we can reason on the remaining individuals, and focus on the second club. Iterating this reasoning leads to the existence of a t Beyond this, we cannot say much about the transitional path toward the t

permanent partition.

permanent partition: clubs

may increase or decrease in size over time, except the …rst club which can only weakly expand. However, we know that on the transition path, the number of clubs decreases or remains constant. The reason is that 13

when some richer members quit their club, it is because they are allowed to enter a richer group. Due to the intra-cub convergence property, the case where those richer members would quit their current club in order to form their own group and expel the poorer members is impossible. Hence, we deduce that Jt

J0 ;

whatever t > 0. The proposition makes clear that this process, unless special circumstances, does not go toward the eventual disappearance of any social segmentation, that is the creation of the grand coalition. At some period, the society necessarily reaches a permanent partition, which in general, will imply several clubs. This will become apparent in the sequel. Importantly, once the t

permanent partition has formed, this does not mean that the economy as

such has reached a steady-state. At t , once memberships have stabilized, individual endowments still di¤er, raising the issue of the balancedness of the growth path. Actually we can prove that the economy will eventually converge to a unique steady state. This steady state exhibits growth rates for clubs which are constant and can di¤er. In the following, variables without a subscript t refer to the t

permanent partition. It is characterized

as follows:

Proposition 5. (i) Whithin clubs belonging to the t

permanent partition, individual human capitals

tend to be equal: lim

t!1

(ii) If the t

hit j

Ht

= 1; 8i 2 C j :

(16)

permanent partition is composed by more than one club (i.e. the grand coalition does not

form), then human capitals do not converge. For two clubs j < j 0 j

Ht

lim

t!1

j0

0

=

Ht

nj A(nj ) A(nj ) nj 0

!

(1

)

> 1:

(17)

(iii) For each club C j there is a balanced growth path:

j 1

lim

t!1

j t

=

(

)

(1

2 J X nz )4 N z=1

z

n A(nz )

(1

)

3

5 :

(18)

Proof See appendix. This proposition assumes that there are several clubs in the permanent partition. This possibility is explored in the next subsection. Point (i) is the direct consequence of the convergence e¤ect that takes place inside a club. Once a club’s membership is stabilized, there is full convergence between individual endowments. Point (ii) states that when there is more than one club in the permanent core partition convergence in human capital levels does not occur. In other words, permanent social segmentation implies persistent human capital inequality. A poorer club persists as its members are turned down by the members of a richer club. Hence they do not bene…t from the intra-club convergence occurring over time in this richer club: in other words, they are caught in a permanent poverty trap, never able to catch-up with members of 14

a richer club. Conversely, equality at the steady state requires that the t

permanent partition be formed

of one club only, the grand coalition. Remark that the di¤erence in steady state human capital between two clubs depends on the sizes of these clubs. Hence, for j < j 0 , limt!1

j

Ht

j0 Ht

0

> 1 is guaranteed by nj > nj given

there are economies of scale. In other words, richer clubs exploit larger economies of scale in the long run. Point (ii) also stresses the crucial role played by local interactions necessary for inequality to persist in the long run. If there are no local interactions, i.e.

= 0, no club forms but the global externality e¤ect ensures

that all agents will eventually bene…t from the same steady-state human capital. Hence, income inequality vanishes when only global externality is at work (see for instance Tamura, 1991). As a consequence of Point (ii), the economy converges toward a balanced growth path, with a permanent growth rate common to all clubs, which depends on the sizes of clubs (Point (iii)). This comes from the fact that once the memberships are stabilized, the intra-club convergence e¤ect is such that eventually the club good contribution to growth is stabilized. The fact that there is eventually a common club growth rate comes from the presence of a global externality.16 Indeed, the global externality plays an homogenizing role: eventually, every individual human capital grows at the same rate because all agents bene…t from global interactions. But remark that the various club sizes enter the expression for the steady state club growth rate: the global accumulation of human capital depends on the characteristics of clubs, since human capital are formed within clubs. In other words, the permanent segmentation of society impinges on the balanced-growth rate.

4.2

Initial conditions and coexistence of clubs in the long run.

In the previous subsection, we took for granted that there could be distinct clubs in the long run, permanently partitioning society but we did not tackle this issue directly. In this subsection, we want to prove that such a feature may characterize the steady state. Is the permanent partition consistent with the coexistence of distinct clubs or is the “grand coalition”, encompassing all agents into a single club, the unique outcome of the dynamic complex process which spins together economic accumulation and the repeated formation of clubs? What are the eventual consequences of the segmentation of society into more than one club? These are the issues we shall address in this section. In our framework, di¤erent inequality distributions may generate di¤erent history-dependent steady states. Along the transitional path, the interplay between human capital accumulation and social segmentation can lead to complex inequality dynamics. Potentially, as long as human capital distribution evolves, the core partition can change. The t

permanent partition that emerges depends on the initial pattern of

human capital inequality. Hence to answer these various questions, we have to consider the initial human capital distribution. Here we shall prove that despite the presence of a global externality e¤ect linking all clubs together, the t

permanent partition may be segmented into more than one club. We o¤er the following 1 6 With

equal to 0, corresponding to the absence of any economy-wide knowledge spillovers, it can be shown that the steady

state club growth rates may di¤er. Precisely, when

= 0; it is required at the t

grow at least as fast as poorer ones.

15

permanent partition that the richer clubs

for j = Proposition 6 For an initial endowment distribution hi0 2 h10 ; :::; hJ0 ; 8i = 1; :::; N , hj0 > hj+1 0 1; :::; J

1; nj0 being the number of individuals with initial human capital hj0 such that (i) nj0 > nj+1 whatever 0

j = 1; :::; J k

hj+1 0 hj0

1; (ii)

(nk 0 ) A0 (nk )) 0 A(nk 0) 0 1+ k+1 n0 A(nk 0)A ln@ k+1 nk A(n0 ) 0 ln(

nj0 A(nj0 )

<

A(nj0 + 1)

A(nj0 ) ; and (iii)

<

with

= mink=1;:::J

1

f kg ;

2 (0; 1); then there exists a 0-permanent partition. This partition is formed of

J clubs; the j-th club includes all agents endowed with an initial human capital hj0 ; for all j: Proof See appendix. This proposition studies a particular endowment distribution such that in the …rst period, only homogenous clubs form. Thus, the dynamic heterogeneity e¤ect is eliminated. However, this also singles out the size e¤ect. Per se, the size of a club is favorable to growth as it allows agents to pool more resources in the club good which bene…ts human capital accumulation. Given the initial distribution, richer clubs have bigger size and thus grow faster. This counters the global externality e¤ect. This latter e¤ect depends positively on the size of the global externality, . In Proposition 6, we prove that if the magnitude of the global externality is not too high, that is

<

, the size e¤ect overcomes the global externality. For low enough values of ;

the global externality e¤ect is too weak to overcome this di¤erence in the provision of club goods; thus it is impossible for the poorer club to catch-up the richer club in the …rst period. The fact that a richer club C0j grows faster than a poorer club C0j+1 between date 0 and date 1 can be formally written as follows: nj+1 0

A(nj0 )

A(nj+1 0 )

nj0

!

(1

)

<

hj+1 0 hj0

This inequality means that the heterogeneity between individuals in club C0j and in club C0j+1 must not be too high in order to lessen the catching-up process generated by the global externality. Hence, the pivotal agent of club C0j does not change whenever both following inequalities are satis…ed nj+1 0

A(nj0 )

A(nj+1 0 )

nj0

!

(1

)

<

hj+1 0 hj0

<

nj0 A(nj0 )

A(nj0 + 1)

A(nj0 ) :

(19)

On the one hand, the …rst inequality means that individual human capital in club C0j+1 must not be too low from individual human capital in club C0j in order to slow down the catching-up process. On the other hand, individual human capital in C0j+1 must not be too close to individual human capital in C0j so that an individual belonging to C0j+1 is turned down from club C0j . The left hand side being an increasing function of , choosing satis…ed nj+1 0

A(nj0 )

A(nj+1 0 )

nj0

we know that for every club <

C0j

j

!

(1

j

such that the following inequality is

j) j

=

nj0 A(nj0 )

A(nj0 + 1)

A(nj0 ) '

nj0 A(nj0 )

A0 (nj0 + 1);

inequalities in (19) are satis…ed. At date 1, the identity of the pivotal agent of

remains the same. In order for all the pivotal agents to remain identical at date 1, one must have that minj=1;:::;J

1 f j g.

In every subsequent period, the same phenomenon is repeated and therefore

16

the segmentation remains unchanged. This explains why the permanent core partition is established in the …rst period and entails multiple clubs. On this account, it can be noticed that in Bénabou (1996a), the levels of individual human capital may converge to the same steady-state value. Hence, if individuals were allowed to change along the transitional path their group membership, the grand coalition will eventually form and segmentation will disappear17 . In contrast, here we do not need to assume exogenous communities to obtain the coexistence of segmentation and persistent inequality. Despite the possibility given to individuals to modify their belongings over time, as opposed to Bénabou’s frameworks (1996a,b) and the presence of a global externality, a case not considered in Durlauf (1996), the dynamics may tend to a steady state where some groups are caught in poverty traps and inequality persists forever. In order to get more insight on the dynamic properties of the model, we provide in the Appendix a numerical example that shows how a di¤erence in the initial distribution has a strong impact on the dynamics pattern.

4.3

Growth and the Bene…ts of Integration.

Finally, in order to assess costs and bene…ts of social segmentation, we compare growth performance between a segmented society and an integrated society where the whole population interacts in the same group. More speci…cally, we address the following issue: is complete integration bene…cial for growth in the long run, when compared to a society whose agents are free to shape their communities? To this aim, we consider a society with an initial endowment schedule such that the partition in the core is never the grand coalition. We then compare the growth process of the sequence of core partitions and the one obtained when the society is “forced” by some coercive law18 to form the grand coalition at date 0 and forever. In other words, the grand coalition does not correspond to the individually optimal decisions on club membership. In this case, each agent bene…ts from the society’s total human capital, but the club good provision su¤ers from a maximal heterogeneity e¤ect, at least in the …rst periods. Comparing the balanced growth rates obtained for these two scenarios, we are able to prove the following Proposition 7 Consider a given initial inequality schedule such that the t

permanent partition is not

the grand coalition. Then, if society is “constrained” to form the grand coalition at all dates, the long run economy growth rate is bigger than the one obtained with the t

permanent partition.

Proof See appendix. Proposition 7 sheds some light on the consequence of integration on economic growth. In the long run, the grand coalition unambiguously maximizes the growth rate as it generates the largest economies of scale and makes inequality vanish. Conversely, the core partition with several groups may lower the growth rate as it can lead to persistent inequality, preventing the society from bene…ting from the largest economies of scale. In the short run, by pooling total human capital resources in the same group, the grand coalition 1 7 See 1 8 A¢

both Figure 2 and Proposition 2, p. 594, Bénabou (1996a). rmative-action policies, desegregation programs in education are examples of laws that intend to favor social mixity in

groups. Here, we do not address the origin of such laws.

17

induces heterogeneity costs that overcome economies of scale. Hence, the grand coalition initially slows down growth. However, the ine¢ ciency generated by heterogeneity is eroded overtime due to the intra club convergence property. Integration is a force toward homogenization of the society as a whole. Thus, the inequality dynamics of the grand coalition are quite di¤erent from the trajectories followed by the sequence of core partitions and lead to a balanced growth path where the economy’s growth rate is maximal.19 This result stresses the impact of our assumption on limited altruism. As individuals do not have a fully in…nite optimizing horizon, they do not internalize the long run bene…t of interacting in the same group. A proper intergenerational altruism, in place of the joy-of-giving motive, or a more farsighted behavior would strengthen incentives for individuals to form bigger groups and reap a higher portion of these bene…ts.

5

Conclusion.

In this paper, we o¤er an approach to study the socioeconomic dynamics of growing economies, based on coalition theory. We develop a view on the growth process, taking into account the fact that agents form communities, and that neither borders nor the shape of a growing economy are a priori given. Growth is based on human capital accumulation process. This process depends both on local, communities-based, externalities and global externalities. With explicit micro-foundations, we are able to show that the dynamics of social segmentation generated by inequality play a critical role on the growth process. Social dynamics and growth constantly interact and cannot be separated: growth alters the way society is segmented; segmentation itself a¤ects the way the durable factor accumulates and therefore impacts on the growth pattern. We prove the existence of a socioeconomic equilibrium arising in each period. The sequence of temporary equilibria leads to a steady state, characterized by a permanent partition of society, which is not necessarily the grand coalition, and a balanced growth path, with di¤erences in individual human capital levels in the long run. There is no full income convergence unless society eventually forms a unique community. Given the coexistence of both local and global external e¤ects, and their opposite e¤ects on club growth rates, the question is whether eventually, more than one club forms, that is whether a society can remain segmented in several communities for ever. We answer positively to this question. At a given period, the growth rate of a community (or club) depends on its size, its membership and the average human capital in this club with respect to the economy-wide average human capital. In particular, the more heterogeneous is the membership of a community, the less is its growth rate; on the other hand, there is a “global externality” e¤ect: the poorer is a club with respect to the rest of society, the higher it is. It is therefore possible that the catching-up process generated by the global externality, is too small to counter the size e¤ect and the heterogeneity e¤ect which may hinder the growth of a poorer agent. In other words, the mere presence of a global external e¤ect is insu¢ cient to guarantee the eventual social homogenization of society. Lastly, even though communities are voluntarily formed at each period and thus, the partition at each period belongs to the core, all external e¤ects are not internalized over the course of time. This is seen when we compare the steady-state outcome related to the dynamics of segmentation and what would be achieved 1 9 This

result is reminiscent to Bénabou (1996a) who emphasizes that integration involves an intertemporal trade-o¤ such

that it may slow down growth in the short run yet raise it in the long run.

18

if society were “forced” to form a unique coalition. In this case, the economy-wide growth rate would be eventually larger than what is obtained in a (voluntarily) segmented society. Several extensions of our model are worth investigating. A limitation of our model is that the ranking of individuals/families is not modi…ed over time. In the real world, we witness much less stability in the social fabrics. Family trajectories may cross over time. A possible approach to this phenomenon would be to introduce shocks on human capital and investigate at which conditions these shocks lead to the disappearance of the constant dynasty ordering property. We relied on a simple endogenous growth mechanism, based on human capital à la Lucas (1988). Other mechanisms are able to generate endogenous growth, linked to various externalities, and can be linked to the gathering of individuals, working together or sharing some resources. It would be interesting to apply the endogenous formation of communities to these alternative frameworks. In particular, R&D clusters play a role in Schumpeterian growth theory and appear to be empirically e¤ective: the consensus is that part of the growth gap between the US and European countries is due to di¤erences in the …nancing and the use of innovations. The relationship between technological communities and growth could be investigated using the type of analysis developed here.20 Here …nancial markets play no role. This assumption is crucial as it ensures that an individual can only rely on his club’s mates for capital accumulation. Actually …nance can be seen as a way to overcome physical barriers and extend the reach of an agent outside her neighborhood. At the same time, we know how segmented is the …nancial sphere and that …nancial clusters (through …rms, markets and locations) exist. Hence …nancial matters could have opposite e¤ects, both enlarging communities and creating new forms of segmentation. Finally, congestion is linked to the mere number of agents in a club, not to their characteristics.21 It would be worth to explore the dynamic consequences of non anonymous crowding out. These various extensions are left for future research.

References [1] Aghion, Ph., E. Caroli and C. Garcia-Penalosa, 1999, “Inequality and Economic Growth: the Perspective of New Growth Theories”, Journal of Economic Literature, vol. 37, 1615-1660. [2] Alesina, A. and E. Spolaore, 1997, “On the Number and Sizes of Nations”, Quarterly Journal of Economics, vol. 112, 1027-1055. [3] Alesina, A., R. Baqir and C. Hoxby, 2004, “Political Jurisdictions in Heterogeneous Communities”, Journal of Political Economy, vol. 112, 348-396. 2 0 Howitt

and Mayer-Foulkes (2005) investigate the emergence of convergence clubs based on Schumpeterian growth theory.

However, they do not adress the issue of the dynamic formation of R&D clusters. 2 1 On non-anonymous crowding out, see Conley and Wooders (2001).

19

[4] Banerjee, S., H. Konishi and T. Sönmez, 2001, “Core in a Simple Coalition Formation Game”, Social Choice and Welfare, vol. 18, 135-153. [5] Barham,V., B. Boadway, M. Marchand and P. Pestieau, 1997, “Volunteer Work and Club Size: Nash Equilibrium and Optimality”, Journal of Public Economics, vol. 65, 9-22. [6] Bénabou, R., 1993, “Workings of a City: Location, Education, and Production”, Quarterly Journal of Economics, vol. 108, 619-652. [7] Bénabou, R., 1996a, “Heterogeneity, Strati…cation and Growth: Macroeconomic Implications of Community Structure and School Finance”, American Economic Review, vol. 86, 584-609. [8] Bénabou, R., 1996b, “Equity and E¢ ciency in Human Capital Investment: the Local Connection”, Review of Economic Studies, vol. 63, 237-264. [9] Bénabou, R., 1996c, “Inequality and Growth”, NBER Macroeconomics Annual, B. Bernanke and J. Rotenberg eds., 11-74. [10] Bénabou, R., 2002, “Tax and Education Policy in a Heterogenous Agent Economy: What Levels of Redistribution Maximize Growth and E¢ ciency?”, Econometrica, vol. 70, 481-517. [11] Conley, J. and Wooders M., 2001, “Tiebout Economies with Di¤erential Genetic Types and Endogenously Chosen Crowding Characteristics”, Journal of Economic Theory, vol. 98, 261-294. [12] De Bartolome, Ch., 1990, “Equilibrium and Ine¢ ciency in a Community Model with Peer Group Effects”, Journal of Political Economy, vol. 98, 110-133. [13] Durlauf, S.N., 1996, “A Theory of Persistent Income Inequality”, Journal of Economic Growth, vol. 1, 75-93. [14] Easterly, W. and R. Levine, 2001, “It’s not Factor Accumulation: Stylized Facts and Growth Models”, World Bank Economic Review, vol. 15, 177-219. [15] Farrell, J. and S. Scotchmer, 1988, “Partnerships”, Quarterly Journal of Economics, vol. 103, 279-297. [16] Fernandez, R., 2002, “Education, Segregation and Marital Sorting: Theory and an Application to the UK”, European Economic Review, vol. 46, 993-1022. [17] Fernandez, R. and Rogerson R., 1997, “Keeping People Out: Income Distribution, Zoning and the Quality of Public Education”, International Economic Review, vol. 38, 23-42. [18] Glomm, G. and B. Ravikumar, 1992, “Public versus Private Investment in Human Capital: Endogenous Growth and Income Inequality”, Journal of Political Economy, vol. 100, 818-834. [19] Goux, D. and E. Maurin, 2007, “Close Neighbours Matter: Neighborhood E¤ects on Early Performance at School”, Economic Journal, vol. 117, 1193-1215.

20

[20] Guesnerie, R. and C. Oddou, 1981, “Second Best Taxation as a Game”, Journal of Economic Theory, vol. 25, 67-81. [21] Jéhiel, Ph. and S. Scotchmer, 2001, “Constitutional Rules of Exclusion in Jurisdiction Formation”, Review of Economic Studies, vol. 68, 393-413. [22] Jones-Lee, M.W., 1992, “Paternalistic Altruism and the Value of Statistical Life ”, Economic Journal, 102, p. 80-90. [23] Howitt, P. and D. Mayer-Foulkes, 2005, “R&D, Implementation and Stagnation: A Schumpeterian Theory of Convergence Clubs”, Journal of Money, Credit and Banking, vol. 37, 147-177. [24] Jaramillo, F., H. Kempf and F. Moizeau, 2003, “Inequality and Club Formation”, Journal of Public Economics, vol. 87, 931-955. [25] Jaramillo, F., H. Kempf and F. Moizeau, 2005, “Inequality and Growth Clubs”, in G. Demange and M. Wooders (eds), Group formation in economics, Cambridge (UK): Cambridge University Press. [26] Konishi, H. and D. Ray, 2003, “Coalition Formation as a Dynamic Process”, Journal of Economic Theory, vol. 110, 1-41. [27] Lucas, R. E., 1988, “On the Mechanics of Economic Development”, Journal of Monetary Economics, vol. 22, 3-42. [28] Mas A., and E. Moretti, 2007, “Peers at Work”, mimeo. [29] Mankiw, N.G., D. Romer and D. Weil, 1992, “A Contribution to the Empirics of Economic Growth”, Quarterly Journal of Economics, vol. 107, 407-437. [30] Mo¢ tt, R., 2001, “Policy Interventions, Low-level Equilibria and Social Interactions”, in S.N. Durlauf and H. Peyton Young (eds), Social Dynamics, Cambridge (Mass.): MIT Press. [31] Persson, T. and G. Tabellini, 1994, “Is Inequality Harmful for Growth? Theory and Evidence”, American Economic Review, vol. 84, 600-621. [32] Saint-Paul, G. and T. Verdier, 1993, “Education, Democracy and Growth”, Journal of Development Economics, vol. 42, 399-407. [33] Tamura, R., 1991, “Income Convergence in an Endogenous Growth Model”, Journal of Political Economy, vol. 99, 522-540. [34] Wooders, M., 1978, “Equilibria, the Core and Jurisdiction Structures in Economies with a Local Public Good”, Journal of Economic Theory, vol. 18, 328-348. [35] Wooders, M., 1980, “The Tiebout Hypothesis: Near Optimality in Local Public Good Economies”, Econometrica, vol. 48, 1467-1485.

21

APPENDIX A

Proof of Proposition 1.

1. Proof of (i). Let us compute the preferred tax rate by individual i in club Cj : The corresponding …rst-order condition is: 1

(1

+

i t

1

) i t

=0

which leads to the following: bit =

1+

(1 ) : (1 )

(20)

There is unanimity across time, clubs and agents. Hence this is the solution chosen by the majority rule. 2. Existence of the core partition. btj can Replacing (20) in (1) and using (2) and (3), the indirect utility function for agent i in coalition C

be expressed as follows:

btj ) V i (C

=

ln

1+ 0 P

1 (1

B z2Cbtj + ln B @

hit

)

( 1+

+ ln

1 (1 (1 ) z (1 ) )ht

A(b njt )

C C A

(1

) hit

(1

)(1

)

)

+ ln H t :

(21)

Given this indirect utility function, our coalition formation game satis…es the top coalition property and hence the core partition exists, following Banerjee, Konishi and Sönmez (2001). 3. Proof of uniqueness of the core partition. The proof of uniqueness is as follows. Consider a consecutive club whose richest member i is endowed with ht and the poorest i with ht : An agent i is admitted in a consecutive club Ctj ; formed with njt members, as long as the bene…t from the tax rate covers the congestion e¤ect, i.e. given (6): 2 0 1 0 13 h i X X 4ln @ hzt A5 ln A(njt + 1) ln A(njt ) > 0: hzt + hit A ln @ z2Ctj

(22)

z2Ctj

Since the …rst term in brackets is monotonously decreasing with the membership in a club and the second term in brackets is monotonously increasing with njt , given the log-convexity assumption made on congestion costs, there is a unique individual i such that (22) is true for any i; i uniqueness of the core partition follows. 4. Proof of (ii)a.

22

i

i and untrue for i < i: The

Since (6) is increasing in the individual human capital and decreasing in the size of the clubs, the proof of the consecutivity property is identical to Proposition 1’s in JKM (2003). 5. Proof of (ii)b. b1 , C b 2 and the consecutive club C 1 de…ned as: f1; ::::; i g with i = n2 . Since they belong to Consider C t t t t P z P z the core partition, and ht > ht ; we deduce that: z2Ct1

b2 z2C t

bt1 ) > V i (Ct1 ) > V i (C bt2 ): V i (C

bti and C bti+1 : This completes the proof of (ii)b. The same argument may be repeated for any C

B

Proof of Proposition 3.

In the following, as we always study the core partition for period t, the superscripts ^ are omitted for ease of reading. 1. Proof of (i). At any date t, consider two individuals i and i0 with i0 > i , i 2 Ctj and i0 2 Ctj : Thus from (2) we have hit+1 = 0 hit+1

(1

hit 0 hit

)(1

)

<

hit 0 : hit

0

Still, hit > hit ; 8t: 0

2. Proof of (ii). Given the welfare ordering property of the core partition, Gjt

Gjt for j

j 0 : This implies

0

that, for i0 > i, i 2 Ctj and i0 2 Ctj ; we have hit+1 = 0 hit+1

hit 0 hit

(1

)(1

)

Gjt Gjt

0

!

(1

)

(1

hit 0 hit

)(1

)

> 1:

This equation means that the ordering of individuals according to wealth remains unaltered through time.

C

The impact of heterogeneity on the club growth rate

Here we prove that rewrite

j t

j t

decreases with any

i

; i 2 Ctj : For convenience, the time index is omitted. Let us

as follows:

j

=

1 A(nj )

(1

)

nj

j pX 1

z=pj

00 @@

1 A(nj )

(1

)

nj

pj nj

pjQ 1 x=z

1 +1

pj

X1

z=pj

=

j t:

1

+1

1 +1

H H

23

j

pjQ 1 x=z

:

x

!(1 1

xA

)(1

)

+1 1(1

+ 1A

H )(1

)

H

j

pj nj

We de…ne the ratio

1

+1

:

j

p nj

1

+1

j pX 1

z=pj

= 00 @@

pjQ 1 x=z

1 +1

j 1 pX

z=pj

x

1

pjQ 1

)(1

)

+1 1(1

xA

x=z

1 +1

!(1

)(1

)

:

(23)

+ 1A

W.l.o.g., let us consider the …rst club (growth rate), of size n1 :

1 n1

1 nX 1

1 nQ 1

x=z

z=1

=

x

1 1 nX 1 1 nQ

x

z=1 x=z

!(1 !

)(1

)

+1 !(1

)(1

)

+1

which is equivalent to: 1 1 n1

where

(1

2

:::

n1 1

=

)(1

We will denote )+(

Nn11 2

2 1

(

n1 1

::: 2

n1 1

:::

2

+

n1 1

:::

)+(

2

+

2

n1 1

:::

n1 1

+ ::: + ) + ::: +

n1 1

+1 :

+1

): 1

=

:::

2

n1 1

n1 1

:::

) + ::: +

n1 1

d

n1 1

:::

1 n1

1 nX 1

=

Let us …rst show that for any i = 1; :::; n1

@ @

1;

@ @

1 n1 i

1 n1 z

z

d

< 0 whatever n1 . 1 i+z

for clubs starting with agent 1 of

i: We will proceed recursively.

Let us consider the …rst term of this sequence: 1 i+1

=

i v

Ni1

Di1 i

+1

+1

:

We thus obtain the following derivative: @ @ which is negative as with

i

1 i+1 i

1

< 0:

By taking a particular i; we will consider the sequence of values size i + z, with z = 1; :::; n

+ 1 and Dn1 1 = (

+ 1. We want to show that

z=1

1

n1 1

+ ::: +

= v Di1

> 1; we have Ni1

i

+1

v 1

(Ni1 (

i v 1

=

Di1 i

< Ni1 < Di1 :

Let us now consider the z th term of the sequence: 1 i+z

i v 1

1 Ni+z 1 Di+z

24

Di1 ) v

+ 1 )2

:

i

and let us assume that its derivative with respect to @ @

1 i+z i

=

1 @Ni+z @ i

1

1 Di+z

is negative, that is: 1 @Di+z 1 Ni+z @ i 2

1 Di+z 1 Di+z

< 0: i

We thus have to show that the (z + 1)th term of the sequence also decreases with respect to

: It can be

expressed as follows: 1 i+z+1

=

i+z

1 Ni+z

i+z 1 Di+z

+1 +1

:

So we compute:

@

1 i+z+1 i

@

=

1 Di+z

i+z

1

+1

:

1 @Ni+z 1 Di+z @ i

i+z 1+v

1 @Di+z @ i

1 Ni+z +

1 Di+z

i+z

1 @Ni+z @ i 2

i+z v

1 @Di+z @ i

i+z

v :

+1

By assumption, the …rst term of the numerator is negative. It thus su¢ ces to show that the second term is negative. As 1 @Ni+z

@ = v

1 @Di+z

i+z v

i 1

i

"

i X

i+z Y

k=1

i+z

i

@ x

x=k

!v

v

i X

k=1

It is straightforward to deduce that it is negative as by de…nition of can conclude that for any i = 1; :::; n

D

1

1;

@ @

1 n1 i

i+z Y

x=k x

x

!#

and ;

x

>(

x v

1

) > 1; 8x: Thus we

< 0 whatever n .

Proof of Proposition 4.

1. Step 1. We focus on the …rst clubs of the successive (i.e. over time) core partitions. In this step we prove that, for any t, the …rst club of the core partition at t + 1 is at least as large as the …rst club in the core partition at t: n1t

n1t 1 ; 8t:

Consider the “initial” pivotal agent p10 of the “initial” …rst club C01 of the core partition. We can prove that he will always belong to the sequence of …rst clubs Ct1 ; for any t > 0, i.e. the size of the …rst club can only (weakly) increase. This is true if the following property is satis…ed: p1

ht 0

p1

ht t ;

8t > 0:

(24)

Let us start from the de…nition of the initial pivotal agent p10 for C01 given in Proposition 2: p1 h0 0

p10 1

X z=1

hz0

A(n10 ) A(n10 1)

p10 1

1

,1

25

X hz 0 z=1

p10

h0

A(n10 ) A(n10 1)

1 :

Obviously: p10 1

1

X hz 0

p10 z=1 h0

p10 1

A(n10 ) A(n10 1)

1

>

X

hz0 p1 h0 0

z=1

!(1

)(1

)

A(n10 ) A(n10 1)

1 :

Hence, from the individual human capital growth (2), p10 1

X

p1 h1 0

A(n10 ) A(n10 1)

hz1

z=1

1

which ensures that the above condition is satis…ed at t = 1. Hence p10 is accepted in C11 : By recurrence, this is true for Ct1 : This implies that once agent p10 lives in the club formed of the richest agents in the initial period 0; he will always be accepted by them at any date t. 2. Step 2. We now prove that there exists a e t1 such that n10 know that

n1t

n1t 1 ;

n1t = n1t+1

8t > 0: Hence, either there exists t1 such that

not, there exist dates t1 such that n1t1 < n1t1 +1

n1t

N; 8t > e t1 : From Step 1, we

= n1t+1 < N; 8t > t1 ; or not. If

N . But since max(n1t ) = N; this implies that there

exists a t1 such that n1t = N; 8t t1 : This completes the proof of Step 2 and proves that at some …nite date e t1 ; a club C 1 = Ct1 ; 8t > e t1 ; forms.

3. Step 3. We now prove that as long as the …rst club has not reached his permanent con…guration, some clubs cannot have reached theirs. From Step 1, if Ct1 6= C 1 ; then 9t; t t < e t1 ; such that p1t > p1t 1 : The consecutivity property implies, that p1t

date t to Ct1 : Hence, since Ct2

1

1

1 belonged to Ct2 1 ; belongs at 6 C 2 : Starting at t e = t1 ; consider Ct2 : This club is

+ 1; which at time t

6= Ct2 ; then Ct2

1

1

the …rst club of the subset Ct nC of the successive core partitions of Ct nC 1 : Hence we can apply the t2 and n2t = n2t+1 N; 8t > e previous Steps 1-2 and deduce that there exists a date e t2 e t1 that ne2t 2 2 2 e therefore Ct = C ; 8t > t2 :

Iterating Step 3 until club CtJ yields to the existence of J permanent clubs. Thus there exists a date t , where all clubs have reached their permanent con…guration. Let us show that t is unique. First, let us focus on the …rst club. Suppose that there are two sequences of club formation. According to the …rst sequence, C 1 forms at date e t1 , and according to the other, it forms at ee t1 . Wlog we assume that e t1 < ee t1 . We know that the …rst club can only grow in size. Hence in the second sequence, the …rst club formed at date t = e t1 is characterized

by a size less than its permanent size. However this permanent size is reached by the …rst club at this date according to the …rst sequence. This contradicts the uniqueness property of the core partition at any date.

Hence, there is an inconsistency. The same argument can be adapted to any higher indexed club. Hence there exists a unique t after which no club changes.

E

Proof of Proposition 5. 1. Proof of (i). At the t from (2), we have at t

permanent partition, for two individuals i and i0 with i0 > i and i; i0 2 C j , t hit+1 = 0 hit+1

hit 0 hit 26

(1

)(1

)

<

hit 0 hit

and thus, limt!1

hit 0 hit

2. Proof of (ii): At the t

= 1: We deduce that limt!1

hit

j

Ht

= 1; 8i 2 C j ; 8C j 2 C:

permanent partition and due to intra-club convergence, we have for any club

j: j

j

(H t )(1

H t+1 =

(1

)

1+

(1

nj ) A(nj )

) (1

)

(H t )

The dynamics of the human capital ratio between two clubs j and j 0 can be written: j

j

H t+1

Ht

=

j0

j0

H t+1 As 0 < 1

Ht

!(1

)

0

nj A(nj ) A(nj ) nj 0

!

(1

)

< 1; the human capital ratio reaches a steady state: j

Ht

lim

j0

t!1

0

=

Ht

nj A(nj ) A(nj ) nj 0

!

(1

)

:

3. Proof of (iii). The growth rate of each club C j derives from (13), intra club convergence and the above result: j 1

nj A(nj )

=

which leads to: j 1

F

=

(

)

(1

2 J X nz )4 N z=1

(1

)

"

Ht

lim

t!1

j

Ht

#

(1

z

n A(nz )

)

3

5 :

Proof of Proposition 6.

A- At t = 0; let us consider the following initial human capital distribution hi0 2 h10 ; :::; hJ0 ; 8i = 1; :::; N , hj0 > hj+1 for j = 1; :::; J 0

1; nj0 being the number of individuals with initial human capital hj0 such that

nj0 > nj+1 0 . Human capital endowments are such that: hj+1 0

<

nj0 hj0

!

A(b nj0 + 1)

1 :

A(b nj0 )

A su¢ cient condition for this inequality to hold at date 1 is: hj+1 1

hj1

hj+1 0

hj0

which is equivalent to (1 (hj+1 0 )

)(1

Gj+1 0

)

(1

)

Gj0

(1

hj+1 0

)

(hj0 )(1 hj0

)(1

)

.

De…ning the following ratio: Rj ( ; t + 1)

Gjt Gj+1 t

!

(1

)

(hjt )(1 (hj+1 )(1 t

27

)(1 )(1

) 1 ) 1

;

2 (0; 1);

(25)

the inequality

hj+1 1 hj+1 0

hj1 hj0

is equivalent to Rj ( ; 1)

1:

First, for two clubs j and j + 1; let us study Rj ( ; 1)

Gj0

=

Gj+1 0

=

!

(1

)

(hj0 )(1

)(1

(1 (hj+1 0 )

nj0

A(nj+1 0 )

A(nj0 )

nj+1 0

!

) 1

)(1

(1

) 1

)

hj+1 0 hj0

So Rj ( ; 1) S 1 , Due to economies of scale and as

nj+1 0

hj+1 0 hj0

<

S

nj+1 0

A(nj0 )

A(nj+1 0 )

nj0

nj+1 0 A(nj+1 ) 0

nj0 ;

(1

A(nj0 ) nj0

!

!

(1

)

)

is an increasing function of

from

[0; 1] to [0; 1]. If we want that Rj ( ; 1)

1 and that the pivotal condition being satis…ed, the following inequalities must

hold: nj+1 0

A(nj0 )

A(nj+1 0 )

nj0

!

(1

)

hj+1 0 hj0

(nj0 )

<

A(nj0 + 1)

A(nj0 )

A(nj0 ) '

(1

Thus there exists a j (n0 ) j 0 j A (n0 )) A(n0 ) 0 1+ j+1 j n0 A(n0 ) A ln@ j j+1 n0 A(n0 )

j;

j

nj+1 0 A(nj+1 ) 0

2 (0; 1) such that

A(nj0 ) nj0

j) j

(nj0 ) A(nj0 )

A0 (nj0 )

(nj0 ) A0 (nj0 ): A(nj0 )

=

For

(26)

<

j

; both inequalities are consistent.

ln(

B - At any t: (i) Assume that at t, the partition is the same as at t = 0; and that Rj ( ; t) Rj ( ; t + 1)

1: Does this imply that

1?

Let us now consider Rj ( ; t + 1):

Rj ( ; t + 1) = As hj+1 t hjt

=

hj+1 t 1 hjt

1

nj0

A(nj+1 0 )

A(nj0 )

nj+1 0

!1

!

(1

)

nj0

A(nj+1 0 )

A(nj0 )

nj+1 0

hjt

!

(1

)

hj+1 t

!

we can see that Rj ( ; t + 1) and lim Rj ( ; t + 1) t!1

Hence, if Rj ( ; t)

1 then Rj ( ; t + 1)

= =

1

(1

(Rj ( ; t)) 1 , lim

1; 8t

= (Rj ( ; 1))

hj+1 t

t!1

1.

28

hjt

=

)t

nj+1 0

A(nj0 )

A(nj+1 0 )

nj0

!

(1

)

:

(ii) Let us check now whether 1 is equivalent to

hjt+1 hjt

hj+1 t+1 hj+1 t

hj+1 t hjt

(nj0 ) A0 (nj0 ) A(nj0 )

<

: So when Rj ( ; t + 1) nj+1 0 A(nj+1 ) 0

Following items (B)-(i) and (B)-(ii), hj+1 t+1 hjt+1

<

(nj0 ) A0 (nj0 ); A(nj0 )

Hence, when that inequalities

G

8t

1; 8j = 1; :::; J

< mink=1;:::J nj+1 0 A(nj+1 ) 0

implies

A(nj0 ) nj0

hjt+1

1 and

A(nj0 ) nj0

(1

<

hj+1 t hjt )

(nj0 ) A0 (nj0 ): A(nj0 ) (nj0 ) A(nj0 )

hj+1 t hjt

We know that Rj ( ; t+1)

A0 (nj0 ) then

(nj0 ) A0 (nj0 ) A(nj0 )

<

hj+1 t+1 hjt+1

(nj0 ) A0 (nj0 ): A(nj0 )

imply that

nj+1 0 A(nj+1 ) 0

(nj0 ) A0 (nj0 ); A(nj0 )

we know

1.

1 f k g and that (1

hj+1 t+1

)

hj+1 t hjt

<

nj+1 0 A(nj+1 ) 0

A(nj0 ) nj0

(nj0 ) A0 (nj0 ) A(nj0 )

(1

)

hj+1 0 hj0

<

are satis…ed at any date t

1:

A numerical example.

We compare inequality dynamics of two societies di¤ering with respect to their initial endowment distribution. We choose values of some key parameters according to some estimates provided in the literature. In particular, regarding the human capital technology, we follow Bénabou (2002) and choose the elasticity of children’s income to education spending (1

); equal to 0:25. We arbitrarily set the scale parameter

= 3:2 such that individual human capital is growing through time. We choose the working time

= 0:2

which lies in the range of working time per day observed for OECD countries in 2007, the highest level being equal to 0.26 for Korea and the lowest being equal to 0.15 for Germany. Following Jones-Lee (1992) estimation of the value of life, the altruism parameter

is set to 0.1. We assume the following log-convex

j

congestion function A(nj ) = exp(0:05 n ) which satis…es the condition

nj A0 (nj ) A(nj )

< 1 for any nj < 20:

We consider the two following initial endowment distributions. 1. First, society S is characterized by n1 = 10 individuals with human capital h10 = 15000; n2 = 5 individuals with h2 = 7500 and n3 = 4 individuals with h3 = 1922. Given the parameter values, we

obtain the following core partition at date 0 C0 = fC01 ; C02 ; C03 g with C0i comprised of the ni individuals with human capital hi0 .

1 02 03 2. Second, society S 0 is characterized by an endowment distribution with h01 0 = h0 , h0 = 9675, h0 = 1922

and ni = n0i ;for i = 1; 2; 3. This endowment distribution is similar to the …rst one except that we have

a 29% increase of h20 .22 It turns out that C00 = fC001 ; C002 g where now individuals with human capital 01 02 02 h01 0 and h0 interact together in C0 relegating the poor individuals in C0 :

Finally, we choose

such that, for both societies, the core partition at date 0 is a 0-permanent partition.

Precisely, for society S, given our parameter values, it turns out that minf j g

2 2 We

j n0 j 0 j A (n0 )) A(n0 ) + j+1 j n0 A(n0 ) ln( j ) j+1 n0 A(n0 )

= 0:0318.

ln(

need a 29% increase in order for the 5 individuals with human capital h2 to be accepted by the 10 richer individuals.

This allows us to avoid the existence problem of the core partition that would arise when only a fraction of individuals with human capital h2 were accepted by the 10 richer individuals (see Wooders, 1980, for the issue of existence of the core partition). If we assumed only one individual per human capital level, we would only need to consider a slight di¤erence of the human capital distribution to obtain a di¤erent social segmentation.

29

A(nj0 ) nj0

(1

)

We …x

= 0:03
) = 0:25; we get

= 0:25773. Hence, conditions for Proposition

6 to hold are satis…ed leading C0 to be a 0-permanent partition23 . Considering society S 0 , we can check

along the transitional dynamics that the h03 individuals are never allowed to enter club C001 . Hence, C00 is a 0-permanent partition. The following …gure displays inequality dynamics for both societies over 16 periods.

Inequality Dynamics 40 35 30 25 20 15 10 5 0

h1/h2 h2/h4 h'1/h'2 h'3/h'4

1

2

3

4

5

6

7

8

9 10 11 12 13 14 15 16

Periods

First, considering society S, we see that human capital ratios date 0, we have

h10 h20

= 2; respectively

h20 h30

= 3:09, and at date 15,

hit hi+1 t

h115 h215

increase monotonously over time. At

= 6:009; respectively

h215 h315

= 4:02: Hence,

pivotal agents remain identical over time con…rming Proposition 6. According to Proposition 5, we know that limt!1

h1t h2t

=

exp(0:05 5) 10 exp(0:5) 5

0:25 0:03

' 40:16 and limt!1

h2t h3t

0:25

=

Second, considering society S 0 , we observe intra-club convergence in and inter-club divergence

h02 ( h003 0

= 5:03 and

h02 15 h03 15

exp(0:05 4) 0:03 5 exp(0:05 5) 4 01 h01 01 h0 C0 ( h02 = 1:55 while h15 02 0 15

' 4:2328. = 1:003)

= 35:59). Hence, we observe the deep impact of social

segmentation on inequality dynamics. While at date 0, the ratio

h20 h30

is approximately 61% of

h02 0 ; h03 0

this

value falls to 11% after …fteen periods. The gap between both clubs gets larger over time and converges to 1

limt!1

Ht h3t

=

exp(0:05 4) 15 exp(0:05 15) 4

0:25 0:03

= 620:98. Thus, poor agents are stuck in a poverty trap.

We can also compute Gini indexes and examine the evolution of the ratio of the Gini index of endowment distribution of society S over the Gini index of endowment distribution of society S 0 . This ratio increases 2 3 Proposition

6 is satis…ed for a value of

which seems too small with regard to the estimated impact of human capital

(see for instance Mankiw, Romer and Weil 1992, who estimated in a three-factor production function that the share of human capital is 0.3.) However, in our framework, a low value of

does not imply that the human capital externalities are absent as

they also play a role via club interactions.

30

monotonously from 1.12 to 1.93. Gini index (society S)/Gini index (society S') 2,5 2 1,5 1 0,5 0 1

2

3

4

5

6

7

8

9

10 11 12 13 14 15 16

Periods

H

Proof of Proposition 7.

Let us denote by

C=fSg 1

the long run growth rate of the grand coalition. Then let us show that

According to the previous de…nitions of growth rates, we have: C=fSg 1

=

(

and C0 1

=

(

)

(1

)

(1

)

N A(N )

2 J X nj )4 N j=1

j

(1

(1

)

)

n A(nj )

Due to economies of scale, it is easy to deduce that 0

C 1

<

(

)

(1

)

(1

N A(N )

This completes the proof.

31

)

3 2 J j X n 5 = 4 N j=1

3 5

C=fSg : 1

C=fSg 1

C0 1: