Toward a Theory of Charitable Fund-Raising

James Andreoni
University of Wisconsin—Madison

Private providers of public goods, such as charities, invariably enlist
fund-raisers to organize and collect contributions. Common incharitable fund-raising is seed money, either from a government
grant or from a group of ‘‘leadership givers,’’ that launches thefund drive and generates additional gifts. This paper provides atheoretical basis for fund-raisers and seeds to charity. The primary
assumption is that there is a range of increasing returns at low
levels of provision of the public good. It is shown that fund-raisers
have a natural and important role, and that sometimes only a small
amount of seed money can grow into a substantial charity.

I. Introduction
Economic research on altruism, public goods, and charitable giving
has flourished over the past decade. The analysis to date has focused
almost exclusively on donors—the supply side of charity—and has
left unexplored the role played by fund-raisers—the demand side. Yet
fund-raising is a vibrant, innovative, and highly professional industry.
According to one estimate, about 115,000 organizations hire fundraising staff and consultants, spending $2 billion per year on fundraising. In 1995 the 25 largest charities spent an average of over $25
million each on fund-raising, or about 14 percent of charitable gifts.1
I am grateful to the National Science Foundation and the University of Wisconsin
Vilas Associates Fund for financial support; to Marc Bilodeau, Peter Cramton, Preston McAfee, Larry Samuelson, Al Slivinski, Rich Steinberg, Lise Vesterlund, the editor, and an anonymous referee for helpful comments; and to Deena Ackerman forvaluable research assistance.
1
The figures on the overall industry are taken from Kelly (1997). The 25 largest
charities are those identified by Money Magazine Online: http://money/features/charity 1196/top25.html.
[Journal of Political Economy, 1998, vol. 106, no. 6]
 1998 by The University of Chicago. All rights reserved. 0022-3808/98/0606-0003$02.50

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Understanding fund-raising may be important for improving policy predictions. In the 1980s several policy changes resulted in reduced government grants to charities and a reduced tax preference
for donations. While economists predicted dire consequences forcharities, contributions continued to rise in the 1980s. Several authors have conjectured that, had the models accounted for the vigorous response by fund-raisers, the predictions might have been
better.2
This paper begins the task of including fund-raisers by modelingthem alongside donors as active participants in the market for privately provided public goods. In doing so, I distinguish between two
different types of fund drives, each with different incentives, constraints, and strategies. The first type is capital campaigns. They characterize new charities, or major new initiatives of existing charities,
and, as the name implies, involve projects with large fixed costs ofcapitalization. Examples include buying expensive equipment or
constructing new buildings. The second type is continuing campaigns. They raise the operating funds for items such as salaries,
supplies, and maintenance for charities that are already capitalized.
This paper, the first in a research program, will focus exclusively
on capital campaigns. I begin here because capital campaigns are
generally the origins of charitable organizations. In addition, they
are characterized by several features that appear to conflict with theintuition of free riding. Foremost among them is that capital campaigns rely heavily on ‘‘seed grants’’ and large ‘‘leadership gifts’’that are publicly announced before the general fund drive begins.
In fact, a well-known rule of thumb for capital campaigns is that onethird of the goal must be raised in a ‘‘quiet phase’’ before the publicfund drive is launched. One might guess that such leadership gifts
would only encourage free riding among later givers, whereas fundraisers surely believe that they encourage gifts. By contrast, continuing campaigns turn directly to general fund-raising, without relying
on the leadership phase.
While the discussion will be carried out in terms of capital campaigns, the application of the model is much broader. The key tothe model is the assumption that there are initially some economies
of scale in producing the public good. This is obviously true of capital investments, such as buildings, which must meet some criticalthreshold before they become productive. However, any public good2
See Clotfelter (1990) and Auten, Cilke, and Randolph (1992) for discussions of
the effects of the tax reforms of the 1980s on charitable giving. Economists did
accurately predict the effect of tax changes on the timing of gifts, however, as seenin Randolph (1995).

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that has increasing returns at low levels of provision will fit into themodel presented here and can form the basis for fund-raising andleadership giving.
Note that one obvious way to explain leadership gifts is to consider
social pressures and warm glows to giving. While they are clearly
important, this paper will begin with the focus on purely economic
variables. This is done for simplicity and to motivate more general
results. Adding social effects, as is shown later in the paper, only
strengthens the findings.
In Section II, I discuss some central features of capital campaigns
that the model must explain. Section III provides important theoretical background, and Section IV presents the basic model. Sections
V and VI look at the role of seed grants and leadership gifts. Section
VII expands the model to include social effects, and Section VIII
considers the possibility of seed subsidies.
II. Capital Campaigns
In the spring of 1995, Wisconsin Governor Tommy Thompson offered $27 million in state bonds to finance a new $72 million basketball arena for the University of Wisconsin, on the condition that therest of the money be raised by private donations. A few days later,
on April 1, 1995, Wisconsin’s U.S. Senator Herb Kohl, who is alsoa wealthy entrepreneur, pledged $25 million to the project, which
would now be called the Kohl Center. On June 27, 1995, Ab Nichols,a former University of Wisconsin basketball star, pledged $10 million. In November of 1995 the Kellner family pledged $2.5 million.
By the time the university formally announced its public fund-raising
campaign in February of 1996, it needed only $7 million to reachits goal. Building for the arena began in May of 1996, a little morethan a year after the governor’s offer.3
The Lawrenceville School, a private high school in New Jersey,
just completed a 5-year, $125 million capital campaign. The drive
was described in the October 1992 issue of Fund Raising Management
magazine as ‘‘a technical and diplomatic juggling act requiring atimetable divided into three phases: Preparation, Nucleus Fund andthe Public Campaign’’ (p. 21). During the preparation phase, thefund-raisers gathered information about potential donors, especiallythose who would be capable of contributing $25,000 or more. In the
3This information was drawn from various issues of Wisconsin newspapers: theWisconsin State Journal ( June 27, 1995, p. 1A; September 12, 1995, p. 1B; October
28, 1995, p. 2D), the Capital Times ( January 17, 1996, p. 1B), and the Milwaukee
Sentinel (November 13, 1995, sports page 4).

charitable fund-raising1189

nucleus fund stage, the school set out to raise one-third of the goalfrom these large donors. After quietly securing $42 million in the
nucleus fund stage, the fund-raisers were ready for the public campaign phase. With a lavish kickoff banquet and public relations blitz,
the school announced its campaign. It then relied on small contributions raised from mailings, phone solicitations, and class reunions
to complete the drive (see Fund Raising Management, October 1992,
pp. 19–25; August 1996, pp. 21–24).
These two examples are typical of many capital campaigns. Skimming recent issues of Fund Raising Management yields many similarcases in which large donations from governments or from a small
group of ‘‘leadership givers’’ act as a seed to grow a successful major
capital fund drive. The phenomenon is so prevalent that experts infund-raising have developed rules of thumb to describe it. For instance, Robert F. Hartsook in Fund Raising Management (August
1994, p. 32) advised that ‘‘the leadership commitment . . . should
represent no less than 20 percent of the capital campaign goal.’’Jerold Panas, a professional fund-raiser and author of ‘‘how-to’’books for fund-raisers, suggests that ‘‘if . . . there is not the potential
for receiving 40 percent of your objective from your top twenty gifts,
your campaign is likely to be moribund’’ (Fund Raising Management,
August 1994, p. 27). The Lawrenceville School was following a similar and often-quoted rule: raise one-third of the goal in a ‘‘quiet
phase’’ before announcing the general public fund drive.
In the economic models, charities are generally treated as inert
organizations without goals, strategies, or influence.4 However, thefund-raising apparatus appears to play a very important role in providing public goods.5 Complicating the role of fund-raisers in capital
campaigns is the fact that the charitable goods require significant
fixed costs. For instance, a basketball arena or a school building
needs to be of a certain minimum size and quality before it can be
useful. Similarly, a public television or radio station needs expensive
equipment before a minute of broadcasting can be produced. Unless there is a single large benefactor who is willing to guarantee thisminimum quality, an equilibrium will spring up at zero contributions, even though another equilibrium exists at an interior point.
4
Several authors have looked at nonprofit entrepreneurship. Recent contributions include Bilodeau and Slivinski (1996a). Earlier theoretical models of fundraising have focused on maintaining the proper incentives of the fund-raisers andmanagers from the point of view of donors (see Rose-Ackerman 1982, 1987).
5
Studies of charitable organizations have found fund-raising to be positively related to donations. The most recent study, by Khanna, Posnett, and Sandler (1995),
shows that government grants and large ‘‘legacies’’ also appear to increase the donations of others. The type of effect will be illustrated in Sec. V.

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Until the charity is sure to reach the threshold, no one has an incentive to give, meaning, of course, that without certain efforts thethreshold may never be exceeded.
This paper will discuss the origins of privately provided public
goods and, in particular, the role of seed money. In one version of
the model, a government grant provides seed money. In a second
version, fund-raisers solicit the seed money from donors. To makethe model interesting, I consider two additional assumptions. First,
the public good must meet a minimum quality threshold before it
yields any services. Second, fund-raising is costly. The result is thatthe seed money is often necessary, but need be only a small fractionof the ultimate equilibrium level of contributions, and in fact need
be only a small fraction of the minimum threshold, in order to push
the economy off the zero provision level of the public good. Just
as in the Kohl Center or the Lawrenceville School, help from thegovernment or from large private donors—each well below theamount required to build the good—can seed the millions of dollarsof small private contributions needed to complete the project.
It is important to note that this is not a paper about mechanism
design. Rather, this is a positive look at the role of seeds and fundraisers in starting charitable organizations. Why don’t charities simply adopt one of the many mechanisms suggested in the economics
literature? Aside from the fact that these mechanisms can be complicated and abstract, charities rely on many donors, perhaps thousands, and the organizational activity of the charity is costly. Organizing such a mechanism with so many people involved could beprohibitively expensive. The objective here is to understand why
charities invest in organizing only a relatively small band of major
contributors and then count on them to jump-start a general funddrive.
III.

Theoretical Foundation

This section reviews the model of privately provided public goods
that forms the basis for the later analysis. This model was introducedby Warr (1982, 1983) and developed further by others.6 Individuals
are each endowed with money m i , which they can allocate between
consumption of a private good, x i , and gifts to the public good, g i .
Assume that units are normalized so that x and g can be expressedin dollars; hence each person faces a budget x i ⫹ g i ⫽ m i . Let G ⫽∑ in⫽1 g i be the total contributions to the public good. Individuals are
6
Examples include Roberts (1984, 1987), Bergstrom, Blume, and Varian (1986),
and Andreoni (1988). See Sandler (1992) for a detailed discussion of this literature.

charitable fund-raising1191

taken to have preferences u i ⫽ u i (x i , G ), which are continuous andstrictly quasi-concave. Note that so far I have not assumed that there
is a minimum threshold for the production of the public good;hence there is no range of increasing returns. As we shall see, theassumption of a threshold will create an important nonconvexity.
For that reason, I shall refer to this standard model as the convex
case throughout the paper. It will serve as an important reference
point.
The provision of the public good is modeled as a simultaneous
play Nash equilibrium game. Define G⫺i ⫽ ∑ j ≠i g j as the contributions of everyone except person i. Alternatively, we can write G ⫽G⫺i ⫹ g i . Under Nash equilibrium, each person’s strategy is a contribution g i , 0 ⱕ g i ⱕ m i , with G⫺i taken as given. With this framework,
an individual’s maximization problem can be written as
max u i (x i , g i ⫹ G⫺i )x i , gi

subject to x i ⫹ g i ⫽ m i ,g i ⱖ 0.
Since gifts of others are treated as exogenous in the individual’s maximization problem, it is equivalent to add the constant G⫺i to bothsides of the constraints and then to think of individuals as choosing
G rather than g i . Hence, the maximization above is equivalent tomax u i (x i , G )x i ,G

subject to x i ⫹ G ⫽ m i ⫹ G⫺i ,G ⱖ G⫺i .
Ignoring the inequality constraint, one can easily see that eachindividual will have a solution to this problem of the form G ⫽f i (m i ⫹ G⫺i ) or, equivalently, g i ⫽ f i (m i ⫹ G⫺i ) ⫺ G⫺i . However, keeping the inequality constraint in mind, we can write the individual
supply function as(1)
g i ⫽ max{ f i (m i ⫹ G⫺i ) ⫺ G⫺i , 0}.
We can add the further assumption that both the public good andthe private good are normal; hence 0 ⬍ f i′ ⬍ 1.
Given these supply functions, we can define a Nash equilibrium
as a vector of gifts (g 1* , g 2* , . . . , g n* ) such that the supply functionsin (1) map this vector into itself. Under the assumptions made, it
can be shown that a Nash equilibrium will exist and that it will be
unique. Both of these propositions have been shown by Bergstrom

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et al. (1986) and follow directly from the assumption of normal
goods. For brevity, interested readers are referred to the original
source.
Before continuing with the new model, I need to add two moreassumptions. First, assume that if we were to remove any threshold,the equilibrium G* ⫽ ∑ ni⫽1 g i* would always be positive. Furthermore,
let C* be the equilibrium set of contributors to the public good in
the convex case. Then assume that the number of people in this set
is at least two.
IV. A ModelIn this section and in the remainder of the paper, I shall often referto the model above as an important reference point. Hence, any
mention of G * or g *i will be referring to the Nash equilibrium inthe convex case.
For the new model, keep all the same assumptions used above,
but now define the technology for producing the public good G as
including an important nonconvexity:

G⫽

冦

n

n

冱 g if 冱 g ⱖ G
i

i

i⫽1

0

i⫽1
n

if

冱 g ⬍ G.
ii⫽1

There is now a minimum threshold G that contributions must meetor surpass before any benefits of the public good can be consumed.7
Note that this implies that there are some fixed costs involved in
producing the good or, alternatively, there are increasing returns
around the point at which contributions approach G.8
The interesting feature of having fixed costs is that the economy
could get ‘‘stuck’’ at an equilibrium of zero contributions. First notice that when G ⫽ 0, we are back in the convex case; hence g i ⫽g i* for all i is the unique equilibrium. As G increases from zero, thethreshold may have absolutely no impact on g i . In fact, it is easy to
7Note that a more general model would assume that G ⫽ α ∑ ni⫽1 g i if ∑ ni⫽1 g i ⬍G , where 0 ⬍ α ⬍ 1. For instance, if the project fails to reach the threshold, thenan inferior alternative public good will be built. Similar results would follow as longas α is small enough to create an additional equilibrium at G ⫽ 0.
8
Clearly, there are many other less dramatic ways in which increasing returns
could have been introduced. However, it is easy to show that equilibria would neveroccur on an area of increasing returns; hence the assumption of this technology isa simplifying, if extreme, form of increasing returns.

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Fig. 1.—Defining g oi and τ isee that if G is below the g i* of any individual, then G will have noeffect on the existence or uniqueness of any equilibrium.
How big does G need to be? Define g oi as the solution to the following for all i: u i (m i ⫺ g oi , g oi ) ⬅ u i (m i , 0). Then g oi is the thresholdsuch that if G ⫽ g oi , person i would be just willing to provide all the
public good alone, if need be; whereas if the threshold is above
g oi, this person would not. The solution for g oi can be shown graphically in figure 1. Clearly, an equilibrium will spring up at G ⫽ 0 if
no one is willing to make a solo gift. Hence, define g omax ⫽ max{g o1 ,g o2 , . . . , g on } as the highest of these values across all individuals. Then
we can make the following proposition.
Proposition 1. The allocation G ⫽ 0 is a Nash equilibrium iff
g omax ⱕ G .
Proof. If g oi ⱕ G, then the best response to G⫺i ⫽ 0 will be g i ⫽ 0
for all i. Hence, g i ⫽ 0 for all i is a Nash equilibrium. Likewise, suppose that G ⫽ 0 is a Nash equilibrium but g omax ⬎ G . Then for atleast one person k there exists a g k ⬎ G that yields higher utility
than g k ⫽ 0, which contradicts the assumption that G ⫽ 0 is a Nashequilibrium. Q.E.D.
If G is too large, then there may be only one equilibrium at G ⫽0. The interesting case is one in which G is large enough to possibly
prevent a charity from springing up, but not so large as to keep itfrom ever being provided. It turns out that if G ⬍ G*, then we are
guaranteed to have two and only two Nash equilibria. This is statedformally in the following proposition.
Proposition 2. If g omax ⱕ G ⱕ G *, then there will be exactly two
Nash equilibria: one at G ⫽ 0 and another at G ⫽ G *.

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Proof. If follows obviously from proposition 1 that both G ⫽ 0 and
G ⫽ G * are Nash equilibria. What we need to establish is that there
are no other Nash equilibria. Since any G ⬎ G would have been
available in the convex case (where G ⫽ 0), we have to rule out
possibilities for G as an equilibrium only when G ⬍ G *.
Consider a G ⬍ G * and a vector (g 1 , g 2, . . . , g n) such that g i ⱖ
0 for all i and ∑ ni⫽1 g i ⫽ G . Suppose that this allocation is an equilibrium. Let ĝ i ⫽ f i (m i ⫹ G⫺i ) ⫺ G⫺i . If g i is an equilibrium, then it
must be that the best response to G⫺i is not a number greater thang i ; that is, ĝ i ⱕ g i for all i. Since f i (m i ⫹ G⫺i ) ⱕ G ⬍ G * ⫽ f i (m i ⫹G *⫺i ) for all i ∈C * and since f is monotonically increasing, we knowthat G⫺i ⬍ G *⫺i for all i ∈C *. Finally, by the assumption that all goods
are normal, ⫺1 ⬍ f ′ ⫺ 1 ⬍ 0. It follows that ĝ i ⬎ g i* for all i ∈ C *.
Since g i ⱖ ĝ i , it follows that g i ⬎ g i* for i ∈C *. But this would imply
that G ⬎ G *, a contradiction. Q.E.D.
The next two results follow easily from the discussion above.
Corollary 1. If G ⫽ G *, then (g 1* , g 2* , . . . , g n* ) is the uniqueinterior equilibrium.
Proof. Consider any allocation (g 1, g 2, . . . , g n) not equal to (g 1* ,g 2* , . . . , g n* ) such that ∑ in⫽1 g i ⫽ ∑ ni⫽1 g i*. Suppose that (g 1, g 2, . . . ,g n) is a Nash equilibrium. Then there must be at least one j ∈ C *such that g j ⬍ g *j , and hence G⫺j ⬎ G *⫺ j. By monotonicity, f j (m j ⫹G⫺j ) ⬎ f j (m j ⫹ G *⫺j ) ⫽ G* for j, which contradicts the assumptionthat (g 1 , g 2, . . . , g n ) is a Nash equilibrium. Q.E.D.
Corollary 2. If g omax ⬎ G , then (g 1* , g 2* , . . . , g n* ) is the unique
Nash equilibrium.
Proof. When g omax is above the threshold G , then for some k there
is a g k ⬎ 0 that is a best response to G⫺k ⫽ 0. Hence, G ⫽ 0 is nolonger a Nash equilibrium. The proof that (g 1* , g 2* , . . . , g n* ) is theunique Nash equilibrium is exactly the same as that in corollary 1,
which showed it to be the unique interior equilibrium.
Proposition 2 shows that if the threshold G is big enough, it will
create another Nash equilibrium at G ⫽ 0; but as long as the threshold is not above G*, it will not create any more Nash equilibria. Thisis true even if by chance the threshold actually equals G *. Of course,
it is still possible for two Nash equilibria to exist if G ⬎ G *. It is
obvious that a threshold G ⬎ G * means that G * is unavailable as anequilibrium. It is also clear that no equilibrium G will develop above
G . However, it is possible that G itself may become an equilibrium.
In particular, as long as there exists a Pareto-improving way to allocate gifts such that ∑ g i ⫽ G , there will exist a Nash equilibrium at
G . This is shown next.
Proposition 3. Assume G ⬎ G *. If there exists a (g 1 , g 2 , . . . , g n )charitable fund-raising1195

such that g i* ⬍ g i ⱕ g oi for each i and ∑ in⫽1 g i ⫽ G, then (g 1 , g 2 , . . . ,g n ) is a Nash equilibrium.
Proof. First note that since ∑ in⫽1 g i ⫽ G , everyone is pivotal; that is,
if anyone provides less than g i , then no public good will be provided.
But since g i ⱕ g oi for all i, no one has an incentive to provide lessthan g i ; that is, G is Pareto improving. Furthermore, since g i* ⬍ g i
for all i, G *⫺ i ⬍ G⫺i for all i. By normal goods, no one can increase
utility by providing more than g i . Hence, g i is a best response for alli. Q.E.D.
An interesting contrast between propositions 2 and 3 is that whenG ⱕ G*, there is only one nonzero equilibrium at G *. However,when G ⬎ G *, any set of contributions that are between g oi and g i*for all i and sum to G may be a Nash equilibrium. In general, there
will be a continuum of such equilibria. This case is very similar tothe ‘‘provision point’’ models considered by Bagnoli and Lipman(1989). Typically in these models the public good is provided onlyat the threshold, not above or below, and any contributions above
the threshold are either wasted or refunded in some manner. If aninterior equilibrium exists, however, it is always at contributions thatsum to exactly G . In our model, the case in which G ⬎ G * is much
like the Bagnoli and Lipman framework since no one has an incentive to give more than the assigned g i ; hence models of provision
points can be applied to the fund-raising problem.
This naturally raises the point of what to assume in the event that
contributions fail to meet the threshold, ∑ g i ⬍ G . There are several
possibilities. Funds could be wasted, refunded, or applied to someother (inferior) project. The provision point models often also allowthat contributions below the threshold are refunded to the donor.
Applying the funds to an alternative project would require a moregeneral model than that presented here but would have consequences similar to those of a refund as long as utility is continuous
in the contributions to the inferior project. As far as equilibria areconcerned, they are the same regardless of whether a refund isgiven, the funds are wasted, or an alternative project is built. Because
it is simpler, and perhaps more realistic, assume that if gifts do notmeet the threshold, the funds are used for some other project thatthe contributors do not care about and in this sense are wasted.9
The next two sections examine how a small grant by government
9
It is interesting to note, however, that experiments run by Bagnoli and McKee(1991) reveal that a refund greatly increases the chance of providing the good. In
practice, therefore, charities may find it in their interest to get pledges that will be
refunded if the goal is not met. I return to this point in Sec. VI.

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or individuals can result in big growth for charity, that is, can removeG ⫽ 0 as an equilibrium. It is assumed throughout that G is large
enough that a Nash equilibrium exists at G ⫽ 0.
V. Government Grants
Imagine that the government levies lump-sum taxes t i on each individual and contributes a grant T ⫽ ∑ ni⫽1 t i to the public good. It is
well known that if t i ⱕ g i* for all i, then the lump-sum tax will becompletely crowded out. That is, total giving G ⫹ T will remain unchanged (Warr 1982). Since the individual’s voluntary gift, g i*, isgreater than the involuntary gift, t i , the individual can simply reducethe voluntary component of the gift so that the total gift, voluntaryplus involuntary, equals the value of the pretax gift g i*. By the definition of the equilibrium, this is optimal for everyone, and hence, private contributions are completely crowded out by the public grant.
This crowding-out hypothesis will greatly simplify the analysis below.
Note that the crowding-out proposition has been the subject ofmany papers, which show that it is unrealistic and easily rejected
when confronted with data. It is used here for its analytical ease.
Later I show that qualitatively similar, although technically more
complicated, results follow from more general and realistic models
of giving, such as warm glow and prestige giving. Likewise, using
income or commodity taxes would be more descriptive than simple
lump-sum taxes. Similar results follow from more complex taxes but
are much more easily shown using lump-sum taxation.10
For each i let τ i be the amount of exogenous giving at which person i is just willing to bring the public good up to the threshold valueby acting alone. That is, τ i is determined implicitly as the solution tou i (m i ⫹ τ i ⫺ G, G ) ⬅ u i (m i , 0),(2)where g i ⫽ G ⫺ τ i is the individual’s gift. An illustration of τ i is shownin figure 1.
The quantity τ i will be an important variable in determining thenecessary size of seeds to charitable giving. This is illustrated in thenext proposition.
Proposition 4. Suppose G ⬍ G * and g omax ⬍ G. For any lump-sum
tax scheme (t 1 , t 2 , . . . , t n ) such that t i ⱕ g i* for all i ∈ C *, t i ⫽ 0
for all i ∉C *, and ∑ j≠k t j ⬎ τ k for at least one k ∈C *, there will exist
a unique interior Nash equilibrium G ′, where g i′ ⫽ g i* ⫺ t i for all iand G * ⫽ G ′ ⫹ T.
10
Andreoni (1988, 1993) examines the validity of the crowding-out hypothesis,
and Andreoni and Bergstrom (1996) discuss distortionary taxes in models of privately provided public goods.

charitable fund-raising

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Proof. As in proposition 1, for some k there exists a g k ⬎ 0 that is
a best reply to G⫺k ⫽ 0; hence G ⫽ 0 is no longer a Nash equilibrium.
It is also clear, by the crowding-out hypothesis, that g i′ ⫽ g i* ⫺ t i for
all i is also a Nash equilibrium. We need to show that this equilibriumis unique.
As before, we need to verify that the threshold is not an equilibrium. Consider person k for whom ∑ j ≠k t j ⬎ τ k . For this person,G⫺k ⫹ ∑ j ≠k t j ⬎ τ k . By the assumption of normal goods and by thedefinition of τ, person k will have a best response g k ⬎ G ⫺ τ k . Hence,
there is not an equilibrium at the threshold.
To complete the proof, we need to show that it is indeed possible
to find a tax vector that satisfies the conditions of the proposition,that is, t i ⱕ g i* for all i and ∑ j ≠k t j ⬎ τ k for some k. Suppose τ k ⫽min{τ 1 , τ 2 , . . . , τ n }. It is easy to show (see Andreoni and McGuire
1993) that the assumption of normal goods implies that G ⫺*k ⬎ τ k .
Hence, set t i ⫽ g i* ⋅ [(τ k ⫹ ⑀)/G *⫺k] for all i ≠ k and for some positive
⑀ arbitrarily close to zero, and let t k be any nonnegative number lessthan g *k . Then t i ⬍ g i* for all i, and ∑ j≠k t j ⬎ τ k . Q.E.D.
This proposition states that if the government raises taxes on people
other than i by an amount at least as high as τ i , then there will be aunique Nash equilibrium in which the total provision, private gifts
plus taxes, will be identical to the Nash equilibrium in the original
convex version of this economy. Notice that, by definition, τ i will beless than g oi for each i ; hence, τ i will be less than G . The implicationof this is that the seed provided by the government can be below thethreshold G. In fact, the seed can be chosen with the most generous
person, that is, with the lowest τ, in mind.
Several other interesting things can be noted. First, if G ⬎ G *but there still exists an interior Nash equilibrium at G and if thegovernment contributes a grant T that is large enough to induce
one person to give to the public good, then in equilibrium that person need not be the sole contributor to the public good. As in proposition 3, as long as there is a Pareto-improving way to allocate contributions such that the threshold is met, such an allocation can be an
equilibrium. In general there may be a continuum of such equilibria. All that the government seed does is to guarantee that G ⫽ 0 is
no longer an equilibrium.11
A second interesting point is that giving can also be seeded insituations less restrictive than those spelled out in this proposition.
In fact, as shown by Bergstrom et al. (1986), taxing nongivers will11
At this point one could consider other assumptions on fund-raising, such as
sequential giving among givers. As Admati and Perry (1991) show, this will also leadto an equilibrium that meets the threshold. This will be discussed more later.

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increase the equilibrium level of public goods. The implication forthis model is that if nongivers are taxed, then the seed will generate
even greater gifts. All that is needed is that the tax on individuals
other than i be greater than τ i for at least one i in order to rule out
a Nash equilibrium at the origin. If this is accomplished, the resulting Nash equilibrium will be whatever the unique interior Nashequilibrium would have been for that tax scheme in the ordinary
convex case. As we know from Bergstrom et al., if nongivers are taxed
or if individuals are taxed in excess of their original gift, then the
Nash equilibrium will be higher than if only givers had been taxed
and at an amount lower than the original gift. This is shown next.
Corollary 3. Assume G ⬍ G *. Let G ** be the interior Nash equilibrium in the convex case that would result from a lump-sum tax
scheme (t 1 , t 2 , . . . , t n ). If t i ⱖ 0 for all i and if ∑ j≠k t j ⬎ τ k for at least
one k, then G ** will be the unique Nash equilibrium andG ** ⫹ T ⱖ G *.
Proof. First find a vector of taxes that satisfies the conditions inproposition 4. Hence, only givers are taxed. Now adjust that vectorof taxes to the desired new vector. Since all changes either will be
neutral (by completely crowding out givers) or will be increases intaxes on nongivers and hence cause a nonneutral increase in total
giving, G ** ⫹ T ⱖ G*. Q.E.D.
This corollary has the natural interpretation that if free riders aretaxed, we can get an even bigger bang for each tax dollar collected.
What is important, and perhaps surprising, is that τ need not belarge compared with G . If G is close to g oi , then the τ may bequite small. As G moves farther above g oi , the τ will grow as well,
but the return from government tax dollars in terms of seeding andpromoting private gifts may still be substantial. Only when G ismany times g oi will the seed need to be a significant fraction of thetotal public good.
These propositions can be illustrated with a simple example. Consider an economy of 100 individuals with identical utility functionsu i ⫽ (1 ⫺ α)ln x ⫹ α ln(G ⫹ γ). In this example, set α ⫽ 0.75, γ ⫽200, and income m ⫽ 100 for all 100 individuals. These parameters
imply an interior Nash equilibrium, in the case with no threshold,at G* ⫽ 97.09; hence g i* ⫽ 0.9709 for all i, which is about 1 percent
of income. The critical value for the threshold is g oi ⫽ 46.75. Suppose
for simplicity that the government levies an identical tax on eachindividual such that (n ⫺ 1)t ⫽ τ, and hence T ⫽ nτ/(n ⫺ 1). Table
1 shows a number of different outcomes based on different assumptions about the threshold G .
Look first at case 1, where G is 50, only 3.25 above g oi . Here a total
tax of only T ⫽ 1.21 will guarantee that G * ⫽ 97.08 is the uniquecharitable fund-raising

1199
TABLE 1
Exampleτt i ⫽ τ/(n ⫺ 1)
Equilibrium G ⫹ T
Bang for the buck: G/T

Case 1:G ⫽ 50Case 2:G ⫽ 75

Case 3:G ⫽ 100

Case 4:G ⫽ 150

Case 5:G ⫽ 200

1.20
.012
97.08
79.10

13.47
.136
97.08
6.14

29.62
.299
100
2.34

68.65
.693
150
1.16

112.50
1.136
200
.76

.75
Note.—u i ⫽ x .25
for all i, m ⫽ 100, n ⫽ 100, G * ⫽ 97.08, and g oi ⫽ 46.75. Identical individualsi (G ⫹ 200)
and identical taxation t i ⫽ τ/(n ⫺ 1) are assumed.

Nash equilibrium. Every dollar of taxes collected generates $79.10
of private charity, a substantial bang for the buck. In case 2, a G of
75 is 28.5 above g oi , and here taxes T ⫽ 13.61 are enough to guarantee the unique equilibrium at G *, with each dollar of taxes generating over $6.00 of charity. As the threshold G rises above G *, as in
the last three cases, the threshold itself becomes the equilibrium.
But as we can see in case 5, even when the threshold is more than
twice G *, each dollar of tax revenue still raises an additional 76 centsof private charity.
I can also illustrate corollary 3. Suppose in the example above thatan amount of income equal to g i* is transferred from half the population to the other half; hence the richer half will have income of$100.97 and the poorer half $99.03. Now, because of the crowdingout hypothesis, only the rich half will give, but the equilibrium G *will stay the same. If we retain the assumption of taxing everyone
equally, then in case 1 the τ i for the givers falls to 0.73; hence thetax t i falls as well and the bang for the buck rises to $131.35. Providing any level of G now requires less total taxation. For instance, thebang for the buck in the case G ⫽ 75 is 6.46, for G ⫽ 100 is 2.42,
and for G ⫽ 200 is 0.77.
VI. Fund-Raising
Imagine again the convex case. Here there is no role for fundraising. To understand it in this context we might turn to modelsof advertising and information, or perhaps appeal to ad hoc models
of social pressure. With the existence of a nonconvexity, however,fund-raising plays a natural role in coordinating givers to overcome
the equilibrium at zero contributions.
Fund-raising is modeled in three stages. The first stage is the selection stage. Here the charity selects a subset of the population to act
as leaders. This stage is done ‘‘quietly’’ by the charity. One can think1200

journal of political economyof this as the time at which the fund-raiser assesses the feasibility ofthe project and does ‘‘market research’’ into the potential donors.
In the second stage, called the leadership phase, the charity organizes
the leaders and allows them to make pledges of contributions. Just
as with real fund-raising, these pledges are binding.12 The leadershipphase is thus a game among the leaders. An interesting aspect of
the leadership phase is that being a leader is voluntary; hence thecharity cannot make people worse off by being leaders. Models of
the sequential provision of public goods can be exploited to analyze
this stage of the game. It should also be added that organizing leaders is costly, and the greater the number of leaders the greater thecost of the fund-raising. This gives charities (and leaders) an incentive to keep the group small. The final stage is called the contribution
stage. In this stage the charity turns to the general public for massive
fund-raising. This stage is modeled as a simultaneous contribution
public goods game. The charity announces the leadership pledges
and then collects contributions to the public good. The crux of themodel is that the charity will choose leaders, and select a game torun among the leaders, so that when they turn to the general contributions stage they have omitted zero as a Nash equilibrium.
A.

Fund-Raising in a Pure-Altruism Model

Define the set of leaders, ᏸ, as those who are approached by thecharity. Let l be the number of people in the set ᏸ. Then define Ᏺ
to be the set of followers, where Ᏺ is the complement of ᏸ. Since
the people who are chosen to be leaders must agree to do it, theymust all be at least as well off being a leader as not. Of course, there
are many ways a charity can make a leader better off than nonleaders. It can, for instance, offer a quid pro quo such as choice seatsin return for a major gift to the opera. Or it can provide a warm
glow or prestige in the form of an award or mention in the symphonyprogram. At this point I shall not consider any of these effects, although I shall return to this possibility later. Instead, assume thatthe charity can affect utility only through consumption of x and G .
Selecting the leaders amounts to letting this group collude to be
first movers. What is the advantage in this? Varian (1994) analyzed
a very simple model of sequential provision of public goods andfound that the first mover is always better off. The reason is that a
12
It is not uncommon for fund-raisers who gather large pledges to have a legal
contract drawn up between the donor and the organization. The reason for this isthat fund-raisers make many plans contingent on the anticipation of the promised
cash; hence they cannot risk a change of heart. As we shall see, the leaders would
actually prefer to have their pledge binding as well.

charitable fund-raising1201

first mover can commit to more free riding than a second mover.
Through a similar means, the charity can give its first movers justsuch an advantage.
Before I provide a more general model, let us look at some examples to illustrate the issues. For ease of exposition, assume that everyone has identical preferences and identical wealth and, hence, everyone has identical τ. In each of the three examples to follow, the funddrives will be allowed to have a leadership stage in which a gameamong the leaders results in binding commitments. These pledges
are then announced to the community, at which time a Nash equilibrium contribution game is played among the rest of society. For
these examples the cost of fund-raising will be ignored.
Example 1: ∑ i∈ᏸ g i* ⫽ τ. In this case the charity would ask all themembers of the group of leaders to pledge g i* . Looking ahead, theleaders know that if they all do this, then the result will be an equilibrium with total contributions of G *. Since this is better than the
Nash equilibrium of G ⫽ 0 and since none of the leaders would
rather be followers, all members of the leaders will pledge to giveg *i . Note that there is also an equilibrium at g i ⫽ 0 for all the leadersin the leadership stage. However, if the fund-raiser collects pledgesof the form ‘‘I will pledge y1 only if all other leaders pledge (y2 , . . . ,y l ),’’ then, as discussed later, a subgame perfect equilibrium in theleadership stage will have every leader pledging g i* , conditional on
the pledge of g *j from other leaders (Admati and Perry 1991).
Example 2: ∑ i∈ᏸ g i* ⬍ τ. In this case there is no way for the charity
to specify any contributions by the leaders that would both meet thethreshold of τ and make each of the leaders better off than if they
had been followers. Hence, the group of leaders is too small.
Example 3: ∑ i∈ᏸ g i* ⬎ τ. Here the leaders will each be better off
committing to some g i such that g i* ⱖ g i and ∑ i∈ᏸ g i ⱖ τ. Since thosemembers of Ᏺ, the followers, each have donation functions g i ⫽ f i (m i
⫹ G⫺i ) ⫺ G⫺i and ⫺1 ⬍ f i′ ⫺ 1 ⬍ 0, if the members of ᏸ give a little
less than g i* , then the members of Ᏺ will each give a little more.
Hence, G will not fall in equilibrium by the amount by which theleaders reduce their contributions. Since we know from the definition of g *i that each individual’s marginal rate of substitution equals
one when everyone gives g i* , the members of ᏸ can be made betteroff by committing to giving an amount below g i* . As long as theirgifts are large enough to surpass τ, they will all be better off foragreeing to be leaders, and the public good will be provided at aninterior point.
Notice that the outcome of example 3 is that, while the public
good is provided, it is somewhat below G *. This result matches thepredictions in Varian (1994). Hence, while the provision of public1202

journal of political economy

goods will be greater than zero in equilibrium with fund-raisers, it
may also be less than G *. Next this model will be solved in moredetail.
B. The Game among the Leaders
Generalize the examples above in two ways. First, imagine that individuals are heterogeneous; hence each g i* and τ i is unique. Second,
assume that the charity faces costs of fund-raising that increase withthe size of the leadership group,13 so c ⫽ c(l) and c ′ ⬎ 0. Furthermore, assume that the cost of organizing everyone, c(n), is prohibitively high.
Assume that the costs of fund-raising must be paid with the funds
collected. So if l is the number of leaders and total contributions
are G , then consumption of charity by everyone will be G ⫺ c(l ).
Also assume that the objective of the charity is to provide the greatest
level of these net contributions, G ⫺ c(l ), as possible. This now
changes the calculus of the problem slightly. For a given cost, theequilibrium in the convex case will now be below G *. This adds one
more reason for the charity to want to minimize on costs. For complete generality we should now think of G * as a function of the costs,
that is, G * ⫽ G*(c) and G *′ ⬍ 0. Likewise, we should also think of
τ as depending on costs: τ i ⫽ τ i (c) and τ′i ⬎ 0.
In constructing the game among the leaders, we need to keep inmind the subgame played by the followers. They will take the pledgesof the leaders as given and play a simultaneous contributions game.
For instance, for a given c(l ), if the leaders by coincidence were eachto give g i* (c(l )) and if ∑ i ∈ᏸ g i* ⫺ c(l ) ⱖ τ k, for some k in Ᏺ, thenthe followers would each give g *i (c(l )) as well. This follows from thedefinition of Nash equilibrium. However, if the leaders were eachto give an amount below g i* (c(l )), but with a total still above the τnecessary to get an interior equilibrium, then, as shown earlier, thefollowers would respond with gifts greater than g i* (c(l )) and hence
make up some of the difference. As long as ∑ i ∈ᏸ g i* ⫺ c ⬎ τ k , there
will be room for the leaders to make smaller gifts and increase theirown utility.
The objective of the fund-raiser will be to find the set of leaders,set a goal for leadership contributions, and choose a game among
the leaders that will guarantee that the goal is met. The ultimate
13
For instance, the more people there are to organize as leaders, the higher thecost in terms of meetings, negotiations, and legal advice; whereas certain economies
of scale, such as billboards, advertisements, and mailings, will prevent much costsavings among the followers as l grows.

charitable fund-raising1203

objective is to reach the highest net contributions. Once again, let
us build up the logic of the model by starting with a special case.
Suppose that the charity has identified the optimal set ᏸ and that∑ i∈ᏸ g i* (c(l )) ⫺ c(l ) ⫽ τ min (c(l )), where τ min is the lowest τ among
the set of followers. Now the charity needs to implement a gameamong the leaders that will have each member giving g i* .
There are at least two games that the charity could choose from.
First, notice that this problem is identical to a game in which a groupof individuals ᏸ must provide a discrete public good at the levelτ k ⫹ c(l ). A mechanism suggested by Bagnoli and Lipman (1989)
shows that in a simultaneous play game, g i ⫽ g i* is an equilibrium.
However, g i ⫽ 0 is also an equilibrium. An alternative is presented
by Admati and Perry (1991). They derive a model with two agents
who alternate making gifts. On an agent’s turn he can increase, but
not decrease, his contribution. Admati and Perry show that underreasonable assumptions this game will have an equilibrium at thedesired level, although it also cannot be guaranteed. However, if
agents are allowed to make contributions with conditional statements of the form ‘‘I will give X if the other person gives Y,’’ then thedesired equilibrium will always be met. Moreover, the equilibrium is
identical to that of an alternating offers bargaining problem.14
There are many other mechanisms that the charity could use inthis case to get the members of ᏸ to give g i* (c(l )). Which mechanism the charities use is beside the point of this analysis. Hence, themechanism used among the leaders will be treated as a black box.
How do things change when the optimal ᏸ is such that ∑ i∈ᏸg i* (c(l )) ⫺ c(l ) ⬎ τ min (c(l ))? The leaders can look ahead to the contribution stage and know that their contributions will determine theultimate equilibrium. They could collude to set their contributions
at the level that will maximize their own utility, taking the followers
as Nash responders to their leadership gift. Now define τ* ⫹ c(l )as the collusive level of giving by the leadership givers that maximizes
their collective welfare as first movers. That is, if the leadership givers
commit to giving an aggregate level τ*, with an implied allocation ofgifts so that ∑ i ∈ᏸ g i ⫺ c(l ) ⫽ τ*, then when the followers move to thecontribution stage, the result will maximize the utility of the leaders.
Hence, in the leadership phase, the charity should set the goal of14
A related mechanism designed by Cornelli (1996) deals with providing a private
good with large fixed costs in which individuals have unitary demands, such as aconcert series. Cornelli finds individuals paying different amounts in equilibrium.
She considers a generalization of her model to privately provided public goods.
Unfortunately, she shows that her mechanism is impractical for large populations.
If we think of τ min as a public good among leaders, however, Cornelli’s mechanism
could also be considered as another means to raise the critical threshold.

1204

journal of political economy

the leaders at τ L ⫽ max{τ*, τ min }. With the goal for the leaders at τ L,
the charity then runs a game among these leaders to meet this goal.
Finally, how does the charity choose the members of ᏸ? The charity now faces a difficult nonlinear problem. One complication is thetension between τ and g *. To keep the contributions of the leaders
as high as possible, the charity should choose those with the highest
g *’s. But they should leave out people with low τ’s since they needto surpass the τ of only one individual in Ᏺ. But people with highg *’s are also likely to have low τ’s, and they are likely to be thewealthy. Hence, depending on the distribution of preferences andincomes, the charity can adopt any number of strategies. Unless wemake stricter assumptions about preferences and income distributions, it is impossible to present a solution to the charity’s problem.15
The model now generally characterizes the behavior of charities
and fund-raisers in the economy. Fund-raisers organize their mostgenerous givers and coordinate their contributions before announcing a general fund drive. The fund-raiser’s rule of thumb—variously
stated as raising one-third of the goal or as getting commitmentsfrom the 20 top contributors before announcing the general funddrive—is also captured by this model. The only assumptions madehere to distinguish the model from prior models of privately provided public goods are, first, that there is a range of increasing returns in the production of the public good and, second, that fundraising is costly.
C. A Note about Commitment
A key assumption in the model just described is that the first movers
are able to commit to their actions. There has been an active discussion in the game theory literature on commitment in sequential
games. Bagwell (1995), for instance, shows that in a two-person
game, if the second mover has any degree of uncertainty (over an
infinite support) about the first mover’s choice, then the simultaneous play Nash equilibrium will result, even though the game is played
sequentially. Intuitively, this holds because the second mover must
be willing to choose the sequential-game equilibrium move regardless of the information he gets about the choice of the first mover.
This in turn gives the first mover an incentive to cheat on the sequen15
It is, however, possible to characterize that solution. Consider some arbitrary
partition of individuals into leaders and followers. Define τ min ⫽ min{τ j: j ∈ Ᏺ} as
the lowest τ among the followers. Then if ∑ i∈ᏸ g *i (c(l )) ⫺ c(l ) ⱖ τ min (c(l )), wherel is the number of members of ᏸ, this partition is ‘‘viable’’; i.e., it is capable ofovercoming the zero equilibrium. The charity must then search among all the viable
partitions such that it chooses the partition that will result in the highest G ⫺ c(l ).

charitable fund-raising1205

tial-game equilibrium outcome. As a result, the second mover willassume, regardless of information, that the first mover has indeed
cheated. Hence, subgame perfection leads to the simultaneous play
equilibrium. Bagwell’s results are naturally extended to games withmany first and second movers. The only way for first movers to maintain an advantage is to make their moves perfectly observable.
In the models presented above, the first mover is either the government or the leadership givers. The government can make its contribution observable by issuing the cash ex ante. Indeed, this is often
how the government works. Even in so-called challenge grants thegovernment will typically give money contingent only on a plan toraise privately three or four times the challenge grant funds.16
For leaders to commit is a bit more complicated. The fact thatgeneral fund drives are often initiated with large press conferences
and major donors make legally enforceable pledges can be seen as
an effort to make the leadership gifts credible and known. However,
it is difficult to credibly prevent first movers from giving again. Whileleadership givers may not be able to commit to not giving more thanthe leadership gift, they can clearly commit to giving no less.17 If weallow that leaders, in the contribution stage, can increase their gifts
as part of the simultaneous provision game, then the results in thispaper remain the same. The only difference would be that leaders
will have no particular first-mover advantage, except the possible social effects discussed below. If, on the other hand, leaders and followers are allowed to alternate moves ad infinitum, then the game becomes far more complex.18 There may be a benefit to future researchthat explores these alternatives.
VII. Warm Glow, Prestige, and Social Effects
Several authors have argued that a model of warm-glow giving, inwhich individuals behave as though there is some private goods benefit from the act of giving, captures the data much better than the‘‘pure altruism’’ model just described.19 The warm glow may enter16
See Challenge (1996) regarding challenge grants. The only penalty for failure
to meet goals for private fund-raising is a lower likelihood of future federal grants.
17
Large gifts are often secured with contracts and legally binding letters of intent.
Failure to meet these obligations has resulted in lawsuits.
18
Work by Bilodeau and Slivinski (1996a) indicates that such a situation may giveleaders an incentive to actually increase their gift in the first stage. Also, games withalternating moves in two-person provision point models, such as Admati and Perry(1991), show a first-mover advantage, although the mechanism is not generally Pareto efficient.
19
See Cornes and Sandler (1984), Steinberg (1987), and Andreoni (1989, 1990).
This view has been supported by various experimental and econometric studies (see,e.g., Kingma 1989; Andreoni 1995).

1206

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the model through any number of mechanisms, but the importantcomponent is that utility depends, at a minimum, on one’s own contribution, so u i ⫽ u i (x i , G , g i ).
How would warm-glow giving affect the results just stated? The
answer is that the basic result would remain unchanged: if the government or fund-raisers can reach some critical mass of provision,
they can eliminate the zero equilibrium. The difference is that such
efforts can have an even bigger bang for the buck with a warm glow.
The reason is that, as shown in Andreoni (1989), lump-sum taxes
on warm-glow givers will generally be incompletely crowded out.
Hence, if the government raises an amount of tax T to get the economy off the zero equilibrium, the resultant equilibrium will be above
the G * defined as the unique interior equilibrium in the convex
case.20
Related work by Harbaugh (1998) introduces the concept of prestige as the benefit individuals might get from having the size of theirgifts known to others. A distinguishing feature of leadership gifts isthat the givers are often named publicly and, as Harbaugh notes,
the sizes of their gifts are often revealed. Prestige may matter because, for instance, there are benefits to business owners in advertising their companies or to lawyers or accountants in signaling theirhonesty and integrity; or, as Glazer and Konrad (1996) hypothesize,
people may simply get consumption value from sending signals of
their wealth. All of these are various ways of saying that if there is awarm glow, then there may be an additional glow to the membersof ᏸ; hence they may be willing to pay more to the public good as
leaders than as followers. As a result, an even smaller group of leaders may be needed in order to reach the threshold.21 Moreover, theamount collected from these people may easily exceed the convex
case; hence, total contributions could be larger as well. Thus warm
glow and prestige can generate an even bigger impact for fundraising than in the case of perfect altruism and pure public goods.
A third possible effect is ‘‘social comparison.’’ Sociologists and
20
Technically, all that we need to get results similar to those discussed above is
to assume that all goods are normal. Preferences with a warm glow, u i (x i , G, g i ),generate a demand function g i ⫽ f i (m i ⫹ G⫺i , G⫺i ) ⫺ G⫺i . As long as ⫺1 ⬍dg i /dG⫺ i ⫽ [∂ f/∂(m i ⫹ G⫺i )] ⫹ (∂ f/∂G⫺i ) ⫺ 1 ⬍ 0, the principal results hold, withthose that rely on complete crowding out excluded. The propositions using complete crowding out are even stronger than those using incomplete crowding. Hence,
the general result regarding government taxation (under the assumption that there
is no warm glow from paying taxes) has the flavor of corollary 3. It is interesting
to note that this holds whether one assumes that the warm glow is felt only whenG ⱖ G or for all levels of G.
21
In fact, if the value of being a leader is large enough, then adding leaders may
have other benefits above simply meeting a threshold. Such strategies are, unfortunately, beyond the scope of this paper.

charitable fund-raising1207

other social scientists stress how one person’s gift may act as a model
for another’s. People who want to do their ‘‘fair share’’ may look toothers around them to get an idea of what gift is appropriate. One
might imagine that leadership gifts serve to set some standard forfair share. Let G ᏸ be the contributions of the leadership group, andlet d i ⫽ d(g i , G ᏸ ) be some distance function between one’s own gift
and a standard set by the leadership gifts. Then utility could looksomething like u i ⫽ u i (x i , G , 1/d i ). Analysis of this model would
proceed just as in the warm-glow case, except now G * and τ i will
also depend on the G ᏸ determined in the leadership stage, with G *increasing and τ i decreasing in the leadership gifts.
One final word about social effects is that they may be creating anonconvexity in themselves. If certain charities become fads or if
people want to get involved only if enough others show interest, thenthere may be a cascading effect of participation in charity. If this isso, then it shows one more role for seed money, that is, to createsocial momentum behind a cause.
VIII.

Subsidizing Gifts

In addition to direct grants, another instrument available to bothgovernment and leaders is subsidies or, as they are sometimes called,
matching grants. Subsidies are, of course, already a part of government policy. By giving to tax-exempt organizations, givers who itemize their taxes gain a subsidy. Much less common, however, arematching grants by the government to specific charities.
In private fund-raising campaigns, matching gifts are occasionallyobserved. Some corporations offer to match their employees’ contributions. A common tool in fund-raising for public broadcasting isthe ‘‘challenge grant,’’ where an individual promises to give, for instance, $1,000 if a certain number or amount of pledges is received
by a specified deadline. Still, general offers of matching grants,
where the more an individual gives the more another will give in
return, remain relatively rare.
In this section I discuss generalizations of the model above to include the possibility for subsidies in addition to direct grants. I shalllook at both government subsidies and leadership subsidies. The result is that indeed subsidies should be rare.
A.

Government Subsidies

There are several ways in which the government could offer a subsidy. It could, for instance, earmark a certain group of philanthropists and offer a subsidy to just those people. In a way this is like

1208

journal of political economy

making the government part of the group of leaders described in
the last section. Rather, what I have in mind here is a general offerby the government to give s dollars to the charity for every dollar
the charity raises privately, with the subsidy paid from general tax
revenues. This subsidy could be combined with direct grants or usedalone. It is important to note that the question being asked here is
not whether there is an optimal subsidy. This has been addressed
elsewhere. Instead, the question is whether the government could
get a bigger bang for the buck by using some of its money on subsidies rather than on direct grants.22
First, could government remove the zero equilibrium with a subsidy alone? The answer here is clearly no. From the logic of Section
V, to guarantee that G will be met, the government must offer a
subsidy of at least s, where s solves
u i (m i ⫺ (1 ⫺ s)G , G ) ⬅ u i (m i , 0)for at least one i. Since G may be many times m i for most i, thesubsidy may need to be quite high. But to get one person to preferto give, the government will have to subsidize all individuals by s .
This is an unlikely scenario.
Next turn the question to one of incremental changes. Imagine
that the government has chosen lump-sum taxes t i as described in
Section V so that they just reach the minimum τ necessary to get aninterior equilibrium. Now let us ask whether they can shave somerevenue from the t i’s to devote the money to subsidies and increase
welfare.
An individual’s budget constraint is now x i ⫹ (1 ⫺ s)g i ⫽ m i ⫺ t i .
Next, determine the taxes on others, τ i , necessary to make an individual willing to provide G for this given s. To provide G he must
make a gift of g i ⫽ G ⫺ τ i ⫺ t i . Substituting this into the budget
constraint, we see that τ i ⫽ τ i (s) is defined implicitly as the solution tou i (m i ⫺ st i ⫺ (1 ⫺ s)(G ⫺ τ i ), G ) ⬅ u i (m i , 0).
This subsidy will imply a new Nash equilibrium, GN (s). Hence, the
new subsidy will cost the government revenue in the amount R(s) ⫽sGN (s).
Does τ(s) fall fast enough to free up tax revenues to pay for thesubsidy R(s)? It is easy to verify that dτ(s)/ds ⫽ (T ⫺ G)/(1 ⫺ s)
22
There has been discussion in the literature about whether subsidies can crowdout other giving. Andreoni and Bergstrom (1996) show that, while complete crowding out is possible, it is also possible to choose subsidy schemes that are effective.
The details of any subsidy scheme are not essential for this discussion, although it
is clearly assumed that an effective subsidy would be chosen.

charitable fund-raising1209

and dR(s)/ds ⫽ G N (s) ⫹ [sdG N (s)/ds]. Evaluating both of these ats ⫽ 0, we see that

冷

dτ(s)
⫽ ⫺(G ⫺ T ),ds s ⫽0

冷

(3)

dR(s)
⫽ G N.
ds s⫽0However, by definition, G N ⱖ G , so clearly G N ⬎ G ⫺ T. This meansthat there is no room for the government to convert tax dollars fromseed grants to subsidies.23 If the total budget that the government
can devote to the charity is T, it is best to spend it all as a direct
grant. Only when the government’s budget is large should it consider adding subsidies.
The result is that indeed government matching grants to specificcharities are likely to be rare and will not replace seed grants. One
should emphasize again that this is not a normative exercise, but
a positive one. If a government is constrained and cannot afford
Pigouvian subsidies to all deserving public goods, it may be best to
provide seed grants to several charities to nudge them into existence.
This indeed appears to capture the behavior of the government.
B. Leadership Subsidies
Now consider whether leadership givers would ever choose to subsidize the followers by offering a matching grant. Note first that wemust look only at subsidies that operate at the margin. For instance,
an offer to match dollar for dollar the first million dollars of gifts
to a $10 million fund drive is inframarginal and hence should operate as an ordinary direct grant. To be true subsidies, they must matchthe last dollars given.
As in the case with government subsidies, it is unlikely that subsidies alone will be sufficient for the leaders to eliminate the zero equilibrium. So again, we can start at a solution found in Section V tosee whether it can be adjusted to include a subsidy from the leaders
to the followers.
Begin by redefining τ i (s) among the followers as the solution tou i (m i ⫺ (1 ⫺ s)(G ⫺ τ i ), G ) ⬅ u i (m i , 0).
23This does not rule out more complicated subsidy schemes, such as subsidizing
only gifts above a certain minimum. Such a step subsidy is implicit in the test foritemized deductions. However, designing such a step subsidy for specific charities
would be unwieldy and, again, is virtually never done.

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Assume for simplicity that the leaders provide the optimal τ given
an announced s. Then the cost of the subsidy to the leaders will beR(s) ⫽ s[G N (s) ⫺ τ(s)]. Analogously to (3), differentiating τ(s) andR(s) and evaluating at s ⫽ 0, we find

冷

dτ(s)
⫽ ⫺(G ⫺ τ),ds s⫽0冷dR(s)⫽ G N ⫺ τ.
ds s⫽0Hence, the net cost to the leaders of introducing a subsidy is simply
G N ⫺ G . Again by definition, G N ⱖ G , so adding a subsidy will costthe leaders more money.24 This clearly stacks the deck against a leadership-sponsored subsidy. However, the shape of the G N (s) function
may restore some chance. As s increases, presumably G N will rise as
well. There may be some s ⬎ 0 such that the leaders are willing to
pay the extra cost in order to get the extra public goods. I cannot
tell for sure whether this will happen, however, without more assumptions on preferences and distributions of income. Hence, except for extremely small s, one cannot rule out subsidies by leaders,
although one can say that subsidies, if they are used at all, will complement and not replace ordinary seed gifts. Again, this is in linewith the observed facts.
IX.

Conclusion

This paper shows that models of privately provided public goods can
include a role for fund-raising. Moreover, the results of the modelaccord quite nicely with the actual role that fund-raising and seedmoney seem to play. Also, many of the troubling features of priormodels of charitable giving, such as complete crowding out and neutrality of income redistribution, no longer hold under some circumstances in this model, and the addition of warm-glow givers has anatural application. Hence, further work that looks more carefullyat the role of fund-raising may ultimately help us to gain better understanding of charitable institutions, private initiatives, and government policies to encourage them.
The model presented here relied entirely on the fact of a noncon24
Notice that this mathematics implicitly assumes that leaders subsidize their own
direct gifts as well as those of the followers. There is no loss of generality in doing
so if the gifts of the leaders are redefined to account for this ‘‘double giving’’ ofsubsidizing themselves.

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vexity in the production of the public good for the result. While it
was assumed that the nonconvexity came from the technology of
producing the public good, it might also be possible for the nonconvexity to be the result of preferences as well. For instance, supposethat individuals want to give to a fund-raising campaign but would
value their gift more if the campaign were ‘‘successful,’’ that is, met
its goals. Hence, the marginal rate of substitution could be increasing with donations over a range until it becomes clear that indeed
the campaign goals will be met. In the extreme case, the marginalutility could be zero until the goal, G , is reached and positive thereafter. Thus this nonconvexity in preferences could be represented ina model formally identical to the model in which the nonconvexity
comes from technology. In this sense, continuing campaigns that areable to create ‘‘bandwagon effects’’ of this kind could also exhibit
qualities of leadership givers even in the absence of any increasing
returns in the production of the public good.
In addition to exploring nonconvexities in preferences, there aremany other fruitful ways for research on fund-raising to continue.
One important avenue is to study continuing campaigns. Harbaugh(1998) has noticed that law schools and operas, for instance, often
announce contributions of donors and that the donations are reported in ‘‘bracket’’ amounts rather than the actual amounts. He
finds that, under certain assumptions on preferences, such a strategy
is optimal for fund-raisers. Other attempts to capture the stylized
facts of fund-raising in a theoretical model of continuing campaigns
would be useful. Ultimately, the models can address important policy questions, such as how fund-raisers’ strategies change as levels ofgovernment grants and tax subsidies change.
Another possible area of research is to model more carefully thesequential nature of donations. Perhaps early donations or large donations convey some information about the quality of the charity.25
How a charity chooses the sequence of donor solicitations, or theorder in which individuals volunteer to contribute, may be a criticalfactor in the fund-raising strategy.
Finally, it will also be important to devote more research, boththeoretical and empirical, to understanding the motives and objectives of charitable entrepreneurs, and whether they are compatible
with the views of donors.26 There are also more subtle issues, such as25
For instance, Vesterlund (1998) examines a model in which asymmetric information about the quality of a charity leads to gifts as signals of that quality.
26
Some authors have begun to ask these questions. Bilodeau and Slivinski (1996a),for instance, ask whether it is ‘‘rational’’ to impose a nondistribution constraint
voluntarily, i.e., to choose to charter the firm as nonprofit rather than for profit.
Bilodeau and Slivinski (1996b) look at selection of heterogeneous agents into becoming charity managers. Finally, Bilodeau and Slivinski (1997) study charities that

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whether a charitable organization is incentive compatible, whether
charities are able to act as private implementation mechanisms, whysome charity markets are ‘‘contested’’ by competing charities versus‘‘collusive’’ with a single charity, and whether united fund drives are
better than competing fund drives. Addressing these issues fully will
require a fundamental understanding of charities and their interactions with donors. Understanding this interaction will help form better policy regarding the charitable sector of the economy.
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