The number of organizations in heterogeneous societies∗
Jo Thori Lind†
November 2008
Abstract
I consider an economy with heterogeneous individuals who can form organizations
for the production of a differentiated good. A pattern of organizations is said to be
split up stable if there is no majority to split up within any of the organizations. If
all agents in society belong to an organization, the largest number of organizations
that is split up stable corresponds to the socially optimal number of organizations.
This no longer holds if membership is voluntary. Then organization membership is
higher among agents who derive a high utility of membership, and there are too few
organizations.
JEL codes: D71, D73, H49, L31
Keywords: Organizations, public goods, median voter, stability, endogneous membership
I am grateful for comments from Bård Harstad, Anthony McGann, Kalle Moene, and Fredrik Willumsen
as well as seminar participants at EPCS 2007 and EEA 2007.
†
Department of Economics, University of Oslo, PB 1095 Blindern, 0317 Oslo, Norway. Email:
[email protected]. Tel. (+47) 22 84 40 27.
∗
1
1
Introduction
Organizations are everywhere in society and different kinds of organizations provide a multitude of services, ranging from the local bridge club to nation-wide unions and international
organizations. Within each sphere of services, there are usually several organizations providing differentiated but comparable tasks. This is to accommodate that users have heterogeneous tastes and needs. A larger number of organizations assures that each user has access
to one close to her preferred variety. The flip side is higher total costs as there are fixed
costs associated with running organizations. It is interesting in itself to study the optimal
number of organizations for a given level of heterogeneity and for a given cost structure
of running organizations. Of greater importance, though, is the number of organizations
that arises if organizations form freely. The question is under which conditions one should
expect this decentralized solution to achieve an optimal number of organizations. If the
equilibrium number of organizations is sub-optimal, this may justify subsidizing the running
of organizations.
The analysis applies to several sorts of organizations. The most obvious application
is probably the organization of private provision of excludable public goods with different
varieties, such as churches (with different approaches to the same religion), newspapers (with
differentiated coverage), self-help groups, and sports and recreation facilities (with different
equipment and location in space). In all of these cases, there are good reasons to entrust
provision to private actors. Still, the usual welfare theorems do not apply, so we are not
guaranteed an optimal variation in service provision. Another application is to the number
of political parties. Particularly in ethnically or religiously heterogeneous societies, an large
number of political parties may be required to get a properly functioning democracy where
all interests are represented. A third application of this analysis is the formation of unions.
All employees have some common interests (high wages, good work conditions, and so on),
but a single union may not be efficient to promote the interests of all employees as there
will be differences in interests between industrial workers and office clerks, between private
sector and public sector employees, between high skilled and low skilled workers and so on.
Hence most countries have several unions.
In this paper I consider a heterogeneous population, which can join organizations that
provide certain services. The utility an agent derives from membership depends on how
well suited the services provided by the organization are to his needs, as well as the cost
of joining the organization. In highly heterogeneous societies, we need a large number of
organizations to provide suitable services for everybody, whereas less heterogeneous societies
can do with fewer. I first study the situation where all individuals are required to join one
organization, but has a choice of which organization to join. In any organization, a member
can propose to split the organization in two parts. If this proposal is only favored by a
2
minority of the members of the organization, I label it split up stable. I consider equilibria
where the number of organizations is found as the smallest number of organization that is
split up stable. In the case with exogeneous membership, the decentralized solution realizes
the socially optimal number of organizations.
In most cases of interest, however, membership in an organization is voluntary, so it
is unsatisfactory to simply assume that everybody belongs to an organization. To model
this, I let all agents have an outside opportunity with heterogeneous value. Agents with a
good outside opportunity remains outside the organizations, and agents with a bad outside
opportunity joins. This introduces a number of new features. First, the fraction of agents
joining an organization depends on the agent types; agents who can find an organization close
to what they prefer are more attracted to the organization, and hence join even when they
have relatively good outside opportunities. Hence a higher fraction of these join compared
to agents who are less satisfied with the organization. Also, as it is now optimal that some
agents, those with very good outside options, stay outside the organizations, the optimal
number of organizations is lower than with full membership. Finally, with decentralized
organization creation, we do not usually get a socially optimal number of organizations.
There are three factors behind this. First, when agents have outside opportunities,
agents whose preferred type of services are close to those provided by the organization tend
to be over-represented within the organization. These are less inclined to favor a split of
the organization, reducing the pressure for splitting it. Second, as the cost of running the
organization is split between the actual members, this also tend to limit the incentives to
form new organizations. These two factors tend to give to few organizations. However,
organizations are composed of agents with relatively bad outside opportunities, and when
deciding on whether to spilt the organization or not, they do not take the preferences of
agents with good outside opportunities into account. This third factor tends to give too
many organizations. Under general assumptions, one can show that the two first factors
dominate, though, so in most cases there is under provision of organizations.
There are two major novelties in this paper. First, the split up stability criterion has
not been used before. For the type of organizations I study, where the number of groups is
not determined in a coordinated way, and where it is reasonable to allow factions to leave
a group, this is the most satisfactory approach to use. Second, earlier studies have not
considered the issue of endogeneous membership, which is highly relevant for most groups
and in contrast to e.g. jurisdictions and countries.
The paper is related to several strands of literature. It is probably most closely related to
Cremer et al.’s (1985) model of the location of facilities in space. However, their approach is
based on a society-wise decision mechanism so no group can choose to form a new facility, and
membership is compulsory. There is also a large literature on local public goods provision
starting with Tiebout (1956); Parts of this literature, such as Westhoff (1977) and Jehiel
3
and Scotchmer (1997), ask similar questions to the present paper. There, however, the focus
is largely on the effect of mobility on equilibria in different jurisdictions whereas I use a
type of heterogeneoity where sorting is not possible. In political science, there is also a
small literature studying multi party systems in a similar fashion (McGann 2002), but this
literature pays little attention to the determination of the number of parties. The paper is
also related to the literature on the size and number of nations (Bolton and Roland 1997;
Alesina and Spolaore 1997, 2003; Le Breton and Weber 2003), but as secessions in countries
are different from secessions in organizations, the natural criteria for stability differ. Also,
membership in countries is usually not voluntary, so there is no question of endogeneous
membership. There is also a literature on the location and size of cities (Krugman 1993;
Tabuchi et al. 2005), but this literature, although conceptually close, is quite different in
the way the economy and the set of possible locations is modelled. Finally, it is related to
the literature on club goods (Buchanan 1965; Ellickson et al. 1999; and a lot of others; see
Scotchmer 2002 for an overview) as well as the literature on group formation (e.g. Milchtaich
and Winter 2002). This literature, however, is more centered on the problem of crowding,
which I disregard. Also, stability is not considered, as they do not consider the threat of
secession.
2
Basic setup
Consider a continuum of agents with type x uniformly1 distributed on [0, 1]. An agent’s
location on the unit interval describes her preferences for the services provided by the organizations. There are N organizations that provide services to its members. Initially, I
assume that each individual is a member of one and only one organization. This assumption is relaxed in Section 3. Organizations choose the variety of services they supply, also
described by a point qi ∈ [0, 1] on the unit interval. This is referred to as the organization’s
location. The services supplied by the organization are best suited for agents located in its
proximity. The difference |x − qi | describes an agent’s dissatisfaction with the organization.
How large the loss from dissatisfaction is depends on how heterogeneous society is, measured
by a parameter a ∈ [0, 1]. This parameter can also be interpreted as a transport cost.
A member located at x ∈ [0, 1], belonging to an organization i located at qi , derives
utility
U (x, qi ) = (1 − a |x − qi |)
There is a fixed cost C of running the organization2 , financed by imposing a fee c on each
A common finding in this class of models is that with general distributions, hardly any results exist; see
e.g. McGann (2002). An extended discussion is available upon request [see Appendix B.1].
2
More general cost structures do not fundamentally alter the conclusions; a detailed exposition is available
upon request [see Appendix B.2].
1
4
member. Utility is taken to be in money terms, so an agent’s objective is to maximize
U (x, qi ) − c.
2.1
The social optimum
The social optimum is found as the optimal trade off between higher costs of more organizations and on average higher dissatisfaction. A social planner’s objective is
max
Z 1
N,{qi } 0
U [x, q (x)] dx − N C
where q (x) = arg min |x − q̃| is the optimal organization to join for an agent located at x.
q̃∈{qi }
Notice first that with N organizations, it is optimal to position the organizations to
R
minimize 01 |x − qi | dx, which implies
q1 =
1
1
1
1
, q2 =
+ , . . . , qN = 1 −
.
2N
2N
N
2N
This means that dissatisfaction varies between 0 and
objective for the number of organizations becomes
1−a
1
2N
with a mean of
(1)
1
.
4N
Hence the
1
− N C,
4N
and the optimal number of organizations is given by3
N ∗∗ =
2.2
r
a
.
4C
(2)
Decentralized solution
The next step is to see how organizations are created and located when they are controlled
by their members. The location of an organization is determined by popular vote among the
organization’s members, but agents can choose themselves which organization to join. As
the organization’s location is a unidimensional decision and preferences for the location are
single peaked, the median voter theorem applies.
Proposition 1. For a given number of organizations N , the locations of organizations in
the decentralized solution are socially optimal, i.e. given by (1).
The number of organizations should be an integer, so strictly speaking it’s either the integer number
above or below N ∗∗ .
Notice, however, that if we were to double the population, the optimal number of organizations would be
the integer closest to 2N ∗∗ . Hence, a difference between two optimal sizes is bounded between the same
integers may result in different outcomes in larger populations.
3
5
To see this, notice first that if all organizations have the same size and all agents belong
to their best choice of organization, the median voter theorem assures the locations given
by (1). Next, assume that organization 1 is larger than an adjacent organization 2. Then
a member at the border between the two will strictly prefer to go to organization 2, so this
cannot be an equilibrium. Hence the location are optimal.
There are different ways to model how the number of organizations is determined. Here,
I use an equilibrium concept I call split up stability:4
Definition 2. A number of organizations is said to be split up stable if no organization has
a majority for splitting into two organizations.
It is easily seen that large organization split up more easily than smaller organizations.
i
h
Consider without loss of generality organization 1 which covers the interval 0, N1 . The
1
. If the organization splits in
membership fee is CN , and the organization is located at 2N
1
two equal parts,the new membership fee becomes 2CN , and the locations become 4N
and
3
. The distribution of preferences between splitting and not is shown in Figure 1.
4N
1
3
We see that if the members located at 4N
and 4N
favour a split, then so will the members
1
3
at locations x < 4N
and x > 4N
. These constitute a majority. If on the contrary the members
3
1
3
1
, 4N
. These
located at 4N and 4N oppose a split, then so will the members located in 4N
also constitute a majority. Hence the smallest number of organizations which is split up
1
3
stable is when the members at 4N
and 4N
are indifferent between a split and no split. This
is the case when
1
1−a
− CN = 1 − 2CN ⇔ N =
4N
r
a
4C
(3)
which corresponds exactly to (2), the number chosen by the social planner. Hence with
fixed costs and no subsidies, the decentralized solution realizes the social planner’s optimal
plan.
One could object that the requirement of a majority vote to split is too strong. But one
could easily verify that no single group would unilaterally want to quit the organization if
1
is the most reluctant to a split, so
(3) holds. The reason is that the member located at 2N
1
he will not support any split up. Hence the new organization would have a size below 2N
,
which would give it larger costs than the majority vote of splitting in two. Hence no such
group could be formed.
One can show that under split up stability, no other secessionist groups can succeed either, nor can a new
group be formed by secessionists from two adjacent groups. Details available upon request [see Appendix
B.3].
4
6
Utility
Figure 1: Utility from splitting up (dashed line) and not splitting up (solid line)
0
3
1/4N
1/2N
Type
3/4N
1/N
Endogeneous membership
3.1
Social optimum
The model where everybody belongs to an organization is unrealistic unless the service
provided is extremely valuable or there are legal obligations to belong to an organization.
To analyse endogeneous membership, we need to introduce an outside option to capture the
value of not joining any organization, for instance by abstaining from consuming the good
provided by the organization or providing a substitute privately. We can then extend the
types of agents to a two-dimensional space (x, ε) where as before x is the optimal location
with a uniform distribution on [0, 1] and ε is a variable with cumulative distribution function
F representing the utility of not belonging to an organization.5 An individual will join an
organization if, for the optimal choice of organization i, U (x, q(x)) − ci > ε where ci is the
membership fee in organization i. Hence the fraction of agents of type x who belong to an
organization is F [U (x, q(x)) − ci ].
As before, the social planner’s problem is to choose the number of organizations and their
5
It seems reasonable to assume that x and ε are independent.
7
Utility
Figure 2: Expected utility with outside option g (solid line) and without outside option u
(dotted line)
0
1/2N
1/N
Type
3/2N
2/N
locations to maximize
max
Z 1 Z +∞
N,{qi } 0
−∞
max (1 − a |x − q (x)| , ε) − N C dF (ε) dx.
(4)
Define the expected utility of an agent located at a distance y ≥ 0 from the centre of the
organization when the optimal choice of membership is taken into account as
g (y) =
Z +∞
−∞
max (1 − ay, ε) dF (ε) .
(5)
Define also the parallel function
u (y) = 1 − ay
as the utility when the choice of membership is not allowed. The two utility functions are
illustrated in Figure 2.
Proposition 3. In the social planner’s problem with endogeneous membership (4), the optimal location of organizations for given N is equally spaced as in (1).
Proof. The proof is quite straightforward but fairly tedious, and is given in Appendix A.1.
By maximizing the socially optimal number of organizations (4), and using
8
R 1/N
0
1/2N g x −
1
2N
=
R 1/2N 0 1
g
0
2N
− x , it is easily seen that the optimal number of organization satisfy
Z 1/N ∗
0
g 1
1
− x − g
∗
2N
2N ∗
dx = C.
(6)
This expression states that the average advantage relative to being at the worst location
of an organization should be equal to the marginal cost of an additional organization. For
comparison, notice that the optimal solution with compulsory membership may be expressed
as
Z 1/N ∗∗
0
u 1
1
− x − u
∗∗
2N
2N ∗∗
dx = C.
(7)
Comparing expressions (6) with (7) we can show the following result:
Proposition 4. The optimal number of organizations with endogeneous membership is lower
than the optimal number of organizations with full coverage.
Proof. It suffices to show that
R 1/N
0
1
To see that this is the case, notice that at x = 0, both g 2N
− x − g
1
u 2N
1
− x − u 2N
=
R u(y)
u0 (y) < 0 and 0 ≤
so we have
1
u 2N
−∞
− x
R 1/N 1
1
1
1
dx < 0 u 2N
dx.
− x −g 2N
− x −u 2N
g 2N
1
2N
= 0 and
R u(y) 0
R u(y)
0. Also, g 0 (y) = −∞
u (y) dF (ε) = u0 (y) −∞
dF (ε) ≥ u0 (y) as
1
1
1
dF (ε) ≤ 1. Hence u0 2N
− x < g 0 2N
− x for all x ∈ 0, 2N
,
1
1
> g 2N
− x for all x ∈ 0, 2N
.
As there are usually some agents with better outside than inside opportunities, and hence
choose not to join any organization, the social gain from an additional organization is lower
when we consider the possibility of outside opportunities. Hence the optimal number of organizations is also lower. Particularly, agents with ε > 1 never joins an organization however
high N is, and agents with ε ∈ (1 − a/2N, 1) only join if a sufficiently good organization
exists, i.e. for a high N .
3.2
Decentralized solution
Before we can discuss split up stability in the decentralized solution, we need to study the
pattern of membership for a given number N of organizations. Without loss of generality,
i
h
1
. As
consider the organization located at 0, N1 , where the good produced is of type 2N
membership is now endogeneous and members located around the center of the organization
derive more utility from joining than members closer to the boundaries, a larger fraction of
1
agents with types close to 2N
are members. Particularly, for an organization covering an
interval of length 1/N , the fraction of agents of type x who are members is implicitly defined
by


C
.
ψ N (x) = F U (x, q(x)) − R 1
N
0 ψ N (x) dx
9
Figure 3: Determination of the membership of organizations ΨN
µ
1
F 1−
N
Λ(ΨN )
µ
1
F 1−
N
C
ΨN
¶
a
C
−
2N
ΨN
¶
N
Λ (Ψ )
1/N
1/N
ΨN
1
1
As ψ N is symmetric around 2N
, the median member of the organization is located at 2N
.
The median voter theorem still applies, so the variety chosen by the location is the variety in
1
the center, 2N
. To study which ψ N arise in equilibrium, we first consider total membership
in an organization ΨN , given by
ΨN =
Z 1/N
0
ψ N (x) dx = 2
Z 1/2N
0
C
F 1 − ax −
dx.
ΨN
h
If ΨN exists, the existence of ψ N follows trivially as ψ N (x) = F U (x, q(x)) −
equilibrium, ΨN must be a fixed point of
Λ (ΨN ) = 2
Z 1/2N
0
F 1 − ax −
C
ΨN
i
. In
C
dx.
ΨN
Notice that Λ it is bounded above and below by
1
a
C
F 1−
−
N
2N
ΨN
1
C
< Λ (ΨN ) < F 1 −
.
N
ΨN
(8)
Also, Λ is increasing and Λ0 (ΨN ) → 0 as ΨN → 0. The typical shape of the function Λ is
illustrated in Figure 3.
There is always a fixed point at Ψ = 0, which is stable as Λ0 (ΨN ) → 0 when ΨN → 0.
10
There are parameter configurations where this is the only fixed point of Λ, but as this is a
fairly uninteresting case, I focus on cases where there are stable interior equilibria.6 There
could also be multiple stable interior equilibria. If the distribution of ε is single peaked,
however, Λ will inherit the shape of F and assure at most one stable equilibrium. In what
follows, I will disregard the possibility of multiple interior equilibria.
As the number of organizations N increases, the possible membership basis for each
organization shrinks, and so does the number of members:
h
i
C
a
1 F 1 − 2N − ΨN
dΨN
=− 2
<0
dN
N
1 − Λ0 (ΨN )
as stability of ΨN = Λ (ΨN ) implies Λ0 (ΨN ) < 1.7 Also, we have the following result on
total membership:
C
Lemma 5. If a < 2 N
, total membership is decreasing in the number of organizations, i.e.
ΨN
dN ΨN
8
< 0.
dN
Proof. See Appendix A.2.
Hence if heterogeneity is low relative to the cost born by each organization member, total
memberships is decreasing in the number of organizations. As ΨN is decreasing in N , the
inequality must hold for all N above a certain treshold.
We are now ready to study whether a number N of organizations is split up stable.
An organization will have a majority to split if the members with median preferences for
splitting prefer to split. As the fraction of the population belonging to the organization is
1
, the positions of the
higher closer to the centre of the organization, i.e. as ψ N (x) < ψ N 2N
median voters is in general not 1/4N and 3/4N as in the case with exogeneous membership.
Let us abstract from this at the moment and condition on the pivotal voters being located
g0 . These
at 1/4N and 3/4N . Call the equilibrium number of organizations in this case N
1
− ΨCN = ΨC2N , i.e. when the the equilibrium number of organizations
voters are indifferent if 4N
g0 is determined by
N
a
g0
4N
=C
ΨN − Ψ2N
,
ΨN Ψ2N
(9)
From (8) it follows that a sufficient condition
for the existence of at least one stable interior equilibrium
a
C
is that there is some Q ∈ (0, 1] such that F 1 − 2N − Q > N Q.
7
A special case arises when Λ0 (ΨN ) = 1 so Λ is tangent to the 45◦ -line. Then a marginal increase in
N makes the equilibrium cease to exist, moving the solution to a new equilibrium with a lower ΨN (which
could be ΨN = 0).
8
If the condition does not hold, notice that it can be weakened to
C
a
C
C
C
−F 1−
−
<F 1−
−F 1−2
F 1−2
Ψ
2N
Ψ
Ψ
Ψ
6
11
We now have
f0
g0 < N ∗ .
Lemma 6. In equilibria with ΨNf0 < 2C Na , we have N
Proof. See Appendix A.3
There are two forces working in opposite directions here. On the one hand, the agents in
the decentralized solution realize that a splitting of the organization first leads to increased
costs per person, which again will reduce membership, hence the effect of the per member fee
is exacerbated relative to the case of exogeneous membership. The social planner considers
only total costs, so she ignores this effect. This effect pulls in the direction of too few
organizations in the decentralized solution.
On the other hand, as seen from Proposition 4, the effect of increasing the number of organizations as judged by the social planner is lower than in the case of exogeneous membership.
The reason is that agents who are among the less satisfied with the provided organizations
(i.e. those located far from the centre) now has an outside opportunity, reducing the gain
from increasing the number of organizations. In the decentralized solution, however, the
decision is only made by insiders. As they ignore the welfare of outsiders, they will not take
this effect into consideration. This tends to make the number of organizations too large. In
f0
equilibria with ΨNf0 < 2C Na , the first effect dominates.
The next step is to see what happens when the pivotal members are not forced upon the
m
organization but rather the outcome of a median voter type process. Denote by xm
l and xh
the location of the two median members, defined by
Z xm
l
ψ (x) dx =
0
Z
1
N
xm
h
1
ψ (x) dx = ΨN .
4
Call the equilibrium number of organizations in this case N 0 . The organization will have
1
a majority from splitting if these two members prefer to split. As ψ N is maximized at 2N
1
1
3
m
and non-increasing when we move away from 2N
, it follows that xm
l ≥ 4N and xh ≤ 4N ,
m
with strict inequalities unless ψ N is constant on the interval [xm
l , xh ]. It is straightforward
m
to show that agents located at xm
l and xh will agree whether to split or not. These agents
are indifferent between splitting and not splitting if
U
xm
l ,
1
2N
C
1
−
= U xm
l ,
ΨN
4N
−
C
,
Ψ2N
leading to the equilibrium condition
ΨN − Ψ2N
1 a
=C
+ 2a Xlm −
.
0
4N
ΨN Ψ2N
4N g0 . This leads to the following Proposition:
It is easily seen that N 0 < N
12
(10)
Proposition 7. We have N 0 < N ∗ , i.e. the equilibrium number of organizations in the
decentralized solution is below the socially optimal number of organizations in equilibria with
0
ΨN 0 < 2C Na
As for Lemma 6, there are the effect of reduced group size that inflates costs per member,
hence reducing the push for more organizations and the effect of insiders versus outsiders,
increasing the push for more organizations that oppose each other. In addition, there is
the new effect of the pivotal members being more conservative to splits than the members
located at 1/4N and 3/4N , which also tends to make the number of organizations too small.
4
Conclusions
We have seen that in a heterogeneous society where new organizations are created if and
only if there is a majority within an organization to split, we obtain a socially optimal
number of organizations if either organization membership is compulsory or not dependent
upon individual location. If location, and hence utility of membership, affects the decision
to join or not, the decentralized solution will generally lead to too few organizations. As the
number of organizations arising in equilibrium depends on the cost of running organizations,
one remedy would be to subsidize organizations if one believes that the world is best seen
as one where membership is endogeneous. In the case of full membership, however, public
intervention through subsidies are not necessary for efficiency purposes and distorts the
creation of organizations unless it is given as a linear subsidy per member. Another option
to assure an optimal number of organizations could be to allow organizations to charge a
higher membership fee for members closer to the centre of the organization. It is unclear,
however, whether this would attract a majority within the organization, so I leave this
extension to future research.
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A
A.1
Proofs
Proof of Proposition 3
For a vector of locations q = (q1 , . . . , qN ), define
q i (q) =


0
 qi−1 +qi
2
if i = 1
if 1 < i ≤ N
and q̄i (q) =

 qi +qi+1
if 1 ≤ i < N
if i = N
2

1
Then organization i located at qi captures members in the interval q i (q) , q̄i (q) . It is easy
to verify that g is non-increasing on R+ . Then we can rewrite (4) as
W (qi ) =
N Z q̄i (q)
X
i=1 q(q)
14
g (|x − qi |) dx .
Hence for 2 ≤ i ≤ N − 1, we have
Z qi
Z q̄i
∂W (q)
=
− g 0 (|x − qi |) dx +
g 0 (|x − qi |) dx
∂qi
q
qi
i
∂q ∂ q̄i−1
+
g (|q̄i−1 − qi−1 |) − i g q i − qi ∂qi
∂qi
∂q
∂ q̄i
+ g (|q̄i − qi |) − i+1 g q i+1 − qi+1 .
∂qi
∂qi
(11)
For the boundaries,
Z q1
Z q̄1
∂W (q)
g 0 (|x − q1 |) dx
=
− g 0 (|x − q1 |) dx +
∂q1
q
q1
1
∂q
∂ q̄1
+
g (|q̄1 − q1 |) − 2 g q 2 − q2 ∂q1
∂q1
and
(12)
Z qN
Z q̄N
∂W (q)
0
g 0 (|x − qN |) dx
=
− g (|x − qN |) dx +
∂qN
q
qN
N
∂q ∂ q̄N −1
+
g (|q̄N −1 − qN −1 |) − N g q N − qN .
∂qN
∂qN
(13)
For q defined by (1), q̄i−1 (q) − qi−1 =qi (q) − qi so the two last lines of (11) and the last
¯∂q
q̄i−1
= ∂q¯ii = 12 . For q defined by (1), it also follows that
line of (12) and (13) are zero as ∂∂q
i
q (q)+q̄ (q)
qi = ¯i 2 i , i.e. qi is the centre of its membership interval. Then
R q̄N 0
∂W
qN g (|x − qN |) dx, so ∂qi = 0.
A.2
R qN
qN
¯
g 0 (|x − qN |) dx =
Proof of Lemma 5
We have
dN ΨN
dΨN
= ΨN + N
=
dN
dN
ΨN − ΨN Λ0 (ΨN ) −

 Z 1/2N
2
=
1 − Λ0 (ΨN ) 
0
2C
C
−
F 1−
aΨN
ΨN
1
F
N
h
1 − Λ0 (ΨN )
0
C
F 1 − ax −
ΨN
−
C
ΨN
i
C
a
C
F 1 − ax −
−F 1−
−
ΨN
2N
ΨN
a
C
−F 1−
−
2N
ΨN



As
Z 1/2N
a
2N
1−
a
C
−F 1−
−
2N
ΨN
15
dx =
dx
.
Z 1/2N Z 1−ax− C
Ψ
N
0
1
1− 2N
− ΨC
N
dF (ε) dx,
we get by interchanging the integration signs and integrating that the expression equals
Z 1− C
Ψ
N
1
1− 2N
− ΨC
1−
=
N
C
ΨN
a
1−
C
ΨN
a
C
F 1−
ΨN
−ε
dF (ε)
a
C
−F 1−
−
2N
ΨN
1 Z 1− ΨN
−
ε dF (ε) .
1
a 1− 2N
− ΨC
N
C
Hence
dN ΨN
dN


2
2C
=
1−
0

a [1 − Λ (ΨN )]
ΨN
−
=
If
a
2N
A.3
−
C
ΨN
2
a
Z 1− C
Ψ
N
1
1− 2N
− ΨC
N
R 1− ΨCN
a
1− 2N
− ΨC
N


C
F 1−
ΨN
ε dF (ε) 
1 − 2 ΨCN − ε dF (ε) −
1 − Λ0 (ΨN )
C
a
−
−F 1−
2N
ΨN
a
− ΨC
R 1− 2N
N
1
1− 2N
− ΨC
ε dF (ε)
N
< 0, the integrand of the first integral is always negative, and hence
dN ΨN
dN
< 0.
Proof of Lemma 6
We have
dx = C
1
1
1
= u (0) − u
=u
−u
4N
4N
2N
Z 1/N 1
1
= N
u − x − u
dx.
2N
2N
0
1
a
4N
so N 0 could also be characterized as
Z 1/N
0
1
1
u − x − u
2N
2N
1 ΨN − Ψ2N
.
N ΨN
Ψ2N
Recall that N ∗ is defined by
0
g Z 1/N
As
Z 1/N
0
g 1
1
− x − g
2N
2N
1
1
− x − g
2N
2N
dx = C.
1
a Z 1/N
1
dx > F 1 −
u dx,
− x − u
2N 0
2N
2N
16
we have N 0 < N ∗ if
C
F 1−
a
2N
<C
1 ΨN − Ψ2N
.
N ΨN
Ψ2N
2N
This expression holds if ΨNΨ−Ψ
> 0 and N ΨN < F 1 −
2N
a
2N
Proof that ΨNΨ−Ψ
> 0: As ΨN < 2C Na , so 2N
<
2N
1
Ψ2N < 2 ΨN so
ΨN − Ψ2N
1≤
.
Ψ2N
Proof that N ΨN < F 1 −
α
2N ∗
α
2N ∗
C
,
ΨN
. I prove these in turn:
we have dNdNΨN < 0. Hence
: As we have N ΨN < F 1 −
C
F 1−
ΨN
α
< F 1−
2N ∗
C
α
⇔
>
ΨN
2N ∗
C
ΨN
, this is satisfied if
which holds when
ΨN
2C
2CN ∗
=
<
a
a
=
s
C
a
s
C
<
C̃
s
s
a
4C̃
C
a
where the last inequality holds as C < C̃.
2N
As 1 ≤ ΨNΨ−Ψ
, (9) entails that N < N ∗∗ , where N ∗∗ is the social optimum with full
2N
participation. Hence we have 1 <
so condition (14) holds.
N ∗∗
N0
=
q
a 1
4C N 0
17
=
q
C
a
.
a 2CN 0
Therefore
q
C
a
>
2CN 0
a
> ΨN ,