In this paper charity brings some joy of giving; it yields more contributions to public goods than standard "subscription", but its creation is costly. We compare the laissez-faire number of charities with both the second and the first-best level. In general, laissez-faire implies an underprovision of both charities and public goods.Charities, Public Good

Barla, Philippe

Pestieau, P.

English

Research Papers in Economics

The Optimal Number of Charities
P. Barlaa,P .P e s t i e a u b,∗
aDépartement d’économique and GREEN, Université Laval
bCREPP, Université de Liège, CORE, PSE and CEPR
June 24, 2005
Abstract
In this paper charity brings some joy of giving; it yields more
contributions to public goods than standard "subscription", but its
creation is costly. We compare the laissez-faire number of charities
with both the second and the ﬁrst-best level. In general, laissez-faire
implies an underprovision of both charities and public goods.
Keywords: Charities, Public Good
JEL classiﬁcation: L31, H41
1 Introduction
Charitable organizations represent in several countries a sizeable and growing
sector of the economy. In 2003, an estimated 800,000 US charities (double
the 1990 ﬁgure) received over 180 billions dollars in individual donations
representing about 2% of personal income (The Chronicle of Philanthropy,
January 6, 2005 and Andreoni, 2005). Despite its importance, the economic
functioning of charitable markets is still largely unexplored (Andreoni, 2005).
For example, while private ﬁrms’ entry determinants have been extensively
studied, there is relatively little research on entry decision by charities or on
the optimal number of these organizations. One exception is Rose-Ackerman
(1982) that develops a theoretical framework where free entry leads to too
much entry and excessive fund-raising as charities ignore the diverting impact
of their fund-raising eﬀorts on the donation received by others.
∗Corresponding author: Tel.: + 32 4 366 3109; Fax: + 32 4 366 3106; E-mail address:
p.pestieau@ulg.ac.be
1In this note, we underline other forces that may lead to a divergence
between the laissez-faire outcome and the optimal number of charities. We
develop a simple model of voluntary contributions where individuals can
contribute to the public good directly or through some charities, in which
case it brings some joy of giving. The creation of a new charity entails a
private ﬁxed cost of entry. After characterizing the laissez-faire outcome,
we derive the ﬁrst best optimum allowing for the social planner to include
or not the joy of giving into its objective function. We also explore the
second best solution where the social planner only controls the number of
charities. Our main ﬁndings are: i) while the optimal number of charities is
zero when the social planner ignore the joy of giving eﬀect, the second best
solution calls for the presence of charities; ii) in this case, the laissez-faire
number of charities is suboptimal unless the joy of giving is very high; iii)
when the social planner includes the joy of giving component, the laissez-
faire solution always involve too few charities compared to the second best
solution. The suboptimal number of charities results from the public good
aspect of creating a charity and from the fact that an individual ignores
that, by creating a new charity and thus increasing the number of donors, it
reduces the cost supported by existing contributors.
2M o d e l
The economy consists of N individuals who are ex ante alike. They have
an endowment equal to y and devote it to private consumption ci or to a
contribution si to some public good Z. The contribution to the public good
can be either direct or go through some charity, in which case it brings some
joy of giving. More formally, let us assume that we have p charities, involving
each a single individual. The utility of each of these p individuals is written
as:
U (cs,s,Z)=u(y − s − f)+h(s)+v(Z)
where the subscript s denotes individuals having a charity, u(·), h(·) and v(·)
are strictly concave utility functions. f is the ﬁxed cost associated with the
creation of a charity, h(·) represents the joy of giving and v(·) the utility for
the public good Z. The public good Z is the sum of individual contributions,
that are chosen in a non-cooperative way.
T u r n i n gt ot h eN − p individuals who have no charity and are given a
subscript 0, their utility is given by:
U (c0,Z)=u(y − s0)+v(Z).
2For the sake of simplicity we assume that for these individuals s0 =0as soon
as there is a charity. We thus distinguish two cases: p =0and 0 <p≤ N.
i) p =0 . Then each individual chooses so that
Maxs u(y − s0)+v(Ns0),
given the s0’s of the other N − 1 individuals. This yields the following
FOC:
−u
0 (c0)+v
0 (Ns0)=0 . (1)
ii) p ≤ N.T h e ns results from the following problem solved by each of the
p contributors:
Maxs u(y − s − f)+h(s)+v(ps).
In equilibrium, we obtain the following condition:
−u
0 (cs)+h
0 (s)+v
0 (ps)=0 . (2)
As to the other N−p individuals, as s0 =0by assumption, their utility
reduces to
u(y)+v(ps).
From (2) , one can easily show that s = s(p) with s0 (p) < 0.F o rf u r t h e r
u s ew ed e ﬁne the elasticity of s with respect to p:
0 ≤ η = −s
0 (p)p/s =
pv00(ps)
u00(cs)+h00(s)+pv00(ps)
≤ 1.
S of a rw eh a v ea s s u m e dt h a tp was given. We now turn to its determina-
tion in a laissez-faire (LF) economy. Starting with p =0 ,t h e nac h a r i t yi s
introduced if:
u(y − s − f)+h(s)+v(s) >u(y − s0)+v(Ns0) (3)
With p>0 an additional charity is introduced as long as
u(y)+v(ps(p)) <u(y − s(p +1 )− f)+h(s(p +1 ) )+v((p +1 )s(p +1 ) )
(4)
or
u(y − s(p +1 )− f) − u(y)+h(s(p +1 ) )+v
0 (ps) ∆(ps(p)) > 0.
3Expressed in continuous terms, free entry should stop when the following
condition is satisﬁed:
u(y − s(p) − f) − u(y)+h(s(p)) + v
0 (ps)
µ
s(p)+
∂s
∂p
¶
=0 . (5)
One clearly sees from (3) and (4) that the creation of a charity is made
desirable when the ﬁxed cost is not too high and the joy of giving is important.
In the following section we compare this laissez-faire solution with the ﬁrst-
best optimal solution.
3T h e ﬁrst-best optimum
To deﬁne the ﬁrst-best we take the sum of individual utilities. The key
issue is whether or not social welfare includes the joy of giving component of
individual utilities. In other words, are we going to launder out individual
utilities from the warm glow eﬀect of charities? To keep the problem open,
we will consider the two alternatives.
Social welfare is thus expressed as:
(N − p)u(c0)+p[u(cs)+εh(s)] + Nv(Z)
where ε ∈ {0,1} and Z i st h ea m o u n to fp u b l i cg o o d .W ea s s u m et h a tt h e
social planner can allocate the available resources Ny to consumption and
to contributions. Contributions are used to ﬁnance the public good: Z = ps.
The social planner can also determines the number of charities.
This problem can be represented by the following Lagrangean expression:
L =( N − p)u(c0)+p[u(xs − s)+εh(s)] + Nv(Z)
−µ[Z +( N − p)c0 + p(xs + f) − Ny]
where µ is the multiplier associated with the resource constraint and xs is
the resource allocated to each contributor. Maximizing L with respect to
c0,x s,s,Z and p, we obtain:
u
0 (c0)=u
0 (cs)=εh
0 (s)=Nv
0 (Z)=µ
and
∂L
∂p
= u(cs) − u(c0)+εh(s) − µ(xs + f − c0)
= εh(s) − u
0 (c0)(s + f). (6)
4If ε =0 , there is no reason to go through costly charities to ﬁnance the public
good: s =0and p =0 .
However if ε =1 ,a n yv a l u eo fp can occur depending mainly on h(·) and
f.
T og e tm o r ei n s i g h tl e tu sl o o ka tt h eS O Co f(6) for ε =1 .
∂2L
∂p2 = h
0 (s)
∂s
∂p
− u”(c0)
∂c0
∂p
(s + f) − u
0 (c0)
∂s
∂p
.
If we take the particular case where u(·)=h(·), s = c0.T h e n
∂2L
∂p2 = −u”(c0)(s + f)
∂s
∂p
< 0.
We now examine whether the FB solution can be decentralized.
When ε =0 , the FB implies p =0 . In that case there is no joy of giving.
Individuals contribute to the public good in the traditional way. Assume
that this contribution beneﬁts from a subsidy σ ﬁnanced by a lump-sum tax
T. Then, the problem of each contributor is:
Maxs u(y − s(1 − σ) − T)+v
0 (Ns)
where each agent considers that the contributions of the other p−1 ones are
given. At the equilibrium, there are all equal. The FOC are:
−u
0 (c0)(1− σ)+v(Ns)=0 .
One can easily see that if 1 − σ =
1
N
or σ =
N − 1
N
o n eg e t st h eo p t i m a l
solution.
When ε =1 ,i ft h eo p t i m a lp is positive, decentralization is more diﬃcult.
One needs a subsidy on the p contributions and interpersonal transfers to
guarantee equal consumption between contributors and non contributors.
One also needs to control p.
4 The second-best problem controlling only
the number of charities
We now consider the following question: assuming that the social planner can
only control the number of charities, can we say that the number it chooses is
higher or lower than the LF number? To see that, we diﬀerentiate the social
welfare function with respect to a continuous p. Social welfare is given by:
SW =( N − p)u(y)+p[u(y − s − f)+εh(s)] + Nv(ps).
5And thus
∂SW
∂p
= u(cs) − u(y)+εh(s)+p[εh
0 (s) − u
0 (cs)]
∂s
∂p
+Nv
0 (Z)
·
s + p
∂s
∂p
¸
.
Substituting (5) into this expression, we write
∂W
∂p
¯ ¯ ¯ ¯
LF
=( ε − 1)h(s)+p[εh
0 (s) − u
0 (cs)]
∂s
∂p
+(N − 1)
·
s + p
∂s
∂p
¸
=( ε − 1)h(s)+ηs[u
0 (cs) − εh
0 (s)] + (N − 1)v
0 (Z)s(1 − η).
From this expression we can compare the level of p in the LF and in the
second-best (SB). We have:
pSB >p LF if ηs[u
0 (cs) − εh
0 (s)]+(N − 1)
0 (Z)s(1 − η)−(1 − ε)h(s) > 0.
The ﬁrst term reﬂects the positive externality that an additional charity has
on the utility of the individuals having already a charity: their contribution
decreases, but this is partially oﬀset by less joy of giving. From (2) we know
that this term is positive.
The second term is the positive externality that an additional charity has
on the whole society by allowing for more public good. Finally, the last term
is the joy of giving that in the case where ε =0is not taken into account by
the social planner and then plays in favor of pLF.
To sum up, when ε =1 , pSB >p LF;w h e nε =0 ,t h i sw i l ln o tb et h ec a s e
if the joy of giving is very high.
5 Concluding remarks
To conclude, we discuss the impact of relaxing some of our hypothesis. First,
allowing each charity to provide a warm glow eﬀect to several individuals
should not signiﬁcantly aﬀect the contribution outcome. It may however
make the creation of a new charity even more diﬃcult as each individual has
an incentive to wait for others to bear the ﬁxed cost (see Bilodeau and Slivin-
ski, 1996). Second, allowing for the individual without charity to contribute
should not signiﬁcantly aﬀect the nature of our results. Third, if charities
6produce diﬀerentiated public goods, this could reinforce the private incentive
for creating additional charities. However, the ﬁrst and second best number
of charities is also likely to increase at least if the social planner takes into
account the individuals’ taste for diversity.
Acknowledgment
We thank Christos Constantatos for useful comments. Philippe Barla
acknowledges ﬁnancial support from the Social Science and Humanities Re-
search Council of Canada.
References
[1] Andreonis, J., 2005, Philantropy, in: Gerard-Varet L-A., S-C. Kolm and
J. Mercier Ythier, eds, Handbook of Giving, Reciprocity and Altruism,
Elsevier/North-Holland, forthcoming.
[2] Bilodeau, M. and A. Slivinski, 1996, Toilet cleaning and department
chairing: Volunteering a public service, Journal of Public Economics 59,
299-308.
[3] Rose-Ackerman, S., 1982, Charitable Giving and ’Excessive’ Fund-raising,
Quarterly Journal of Economics, 97, 193-212.
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