Public Choice 111: 49-71, 2002.
© 2002 Kluwer Academic Publishers. Printed in the Netherlands.
49
Pumpkin pies and public goods: The raffle fundraising strategy
BRIAN DUNCAN
Departmentof Economics, Universityof Colorado at Denver,CO 80217-3364, U.S.A.;
e-mail: Brian.Duncan@cudenveredu
Accepted 9 October 2000
Abstract. Charitable organizations, such as schools and churches, often use raffles to raise
money. This article explores the economic incentives inherentin raffle fundraisers.Raffling off
a prize is comparedto simply asking for voluntarycontributions(i.e., a rafflewithout a prize).
Even if every contributoris risk-averse, offering a prize can increase contributionsto a public
good by more than the value of the prize. Thus, tying contributionsto a raffle can increase the
equilibriumsupply of a public good. Moreover,there exists a raffle prize that maximizes the
supply of public good over other prizes.
1. Introduction
Charitableorganizations commonly use raffles to raise money. Although the
prizes vary widely in type and value, most raffles follow the same format
of selling tickets, one of which the fundraiserrandomly draws to determine
the winner. For example, a church might raffle off a vacation cruise to the
Caribbean,or a booster club might raffle off a pumpkinpie at a college basketball game. Regardless of the prize, a ticket-holder'sprobabilityof winning
is equal to the number of tickets he or she owns divided by the total number
of tickets sold. A quick assessment of the probabilityof winning compared
with the estimated value of the prize reveals the truthabout raffles;they are
generally very unfair gambles. Yet, many people buy tickets. In the booster
club example, the fans at the college basketballgame may talk aboutthe raffle
as if it is all about winning the pie. However,it is likely thatboth ticket buyers
and fundraisersunderstandthat the raffle is really about supportingthe home
team. But if buying a raffle ticket is just another way to support the home
team, then why have a raffle at all? Why not simply ask the fans for voluntary
donations?
One reason a fundraiser might choose to use a raffle to raise money is
psychological. A raffle provides a painless way for fundraisersto ask both
friends and strangersfor charitabledonations. However, this article does not
addressthe psychological motivationsof fundraisers.Instead,this article analyzes the economic incentives inherentin a raffle fundraiser.In that context,
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50
the success of the school's basketballteam is a public good, and the role of
the booster club is to collect contributionsto finance that good.
To raise money, a booster club has two possible fundraising strategies:
it can ask for voluntary contributionsor it can raffle off a prize. The first
strategy is equivalent to a raffle without a prize. Without a prize, spectators
buy tickets motivatedby increasingthe supply of public good. With a prize,
spectatorsbuy tickets motivatedby increasing the supply of public good, and
by winning the prize. For a raffle to outperformvoluntary contributionsthe
increase in ticket sales must be greaterthan the value of the prize.
Will a raffle outperformvoluntarycontributions?Initially, consider a case
in which the standsare filled with risk-neutralspectators.In thatcase, it might
seem that if the booster club raffles off a six-dollar prize, then spectators
would simply spend six dollars more on tickets than they would have in the
absence of the raffle.Because fundraisersfinance both the prize andthe public
good, tying contributionsto a rafflewill not change the supply of public good.
In this article, I show that the above scenario will not happen. In fact, a
raffle can increase the supply of public good, even among risk-averseticketbuyers. The raffle increases the supply of public good because it lowers the
marginal price of contributing.For example, consider the story outlined at
the beginning of this article. Suppose that there are one hundredspectators
attendinga basketballgame, each a potential contributorto the booster club.
When fundraisersask for voluntarycontributions each spectatorgives onedollar, for a total of one hundreddollars. In that case, it costs each spectator
one-dollar to give one-dollar.What will happen if instead fundraisersraffle
off a one hundred-dollarprize? Suppose each spectatorbuys two tickets. The
raffle's gross proceeds would increase to two hundreddollars, but its net-ofprize proceeds would remain at one hundred dollars. In that situation, the
spectatorshave exactly offset the raffle prize. However, if at this breakeven
point one spectatorbuys three tickets, then his or her expected prize will increase from one-dollarto one-dollarand fortyninecents. Thus, in expectation,
it only cost the ticket-buyerfifty-one cents to give one-dollar. The raffle has
effectively cut the price of giving in half.
Of course, buying a raffle ticket introduces uncertainty as ticket-buyers
trade certain consumption for expected consumption. How a ticket-buyer
feels about this trade-off depends on his or her risk-preference. Because
the raffle lowers the marginalprice of the public good, offering a prize can
increase the supply of public good even among risk-adverse ticket-buyers.
However,the size of the prize matters.Depending on how ticket-buyersview
uncertainty,the supply of public good may not increase for all prize values.
Offer a prize too large, and the supply of public good can decrease. This
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51
suggests that there is a raffle prize that maximizes the supply of public good
over other prizes.
While the use of raffles is common, Morgan(1996) is the only study that
examines raffles as a supply mechanismfor public goods.' Using quasi-linear
utility functions, Morgan shows thatthe supply of a public good can increase
when fundraiserstie contributionsto a raffle.This paperanalyzes rafflesusing
general utility functions and shows that a raffle can increase or decrease the
equilibrium supply of a public good, depending on the value of the prize.
With general utility functions, the raffle affects the supply of public good in
two ways, each working in opposite directions. First, the raffle lowers the
marginalprice of giving, which leads to an increase in the supply of public
good. Second, the raffle introduces uncertaintyin private consumption, or
more importantly,uncertainty in the marginal utility of private consumption. With general utility functions, this uncertaintycan lead to a decrease
in the supply of public good. Quasi-linearutility functions have constant
marginalutility. Therefore, in the quasi-linearcase, the raffle has no effect
on an individual's marginalutility of privateconsumption,and increasing the
raffle prize will always increase the supply of public good until at least one
ticket-buyer spends his or her entire wealth on raffle tickets. With general
utility functions, a raffle will increase the supply of public good only if the
fundraiserchooses an appropriateraffleprize.
2. The private supply and efficient supply of a public good
As a starting point, consider a set of individuals, N = {1, 2,...,
i,...
n},
where each member has preferencesover combinationsof two items: private
consumption (xi E R+) and a public good (G E R+). Each individualhas an
endowed wealth of wi, which he or she may spendon private consumptionor
contributeto the public good (gi E R+). Preferencesare representedby the
utility functions U'[xi, G], where U'i > 0, UG > 0, Uiixi < 0, UG < 0, and
U'xiG > 0.
2.1. Private supply without a raffle
Warr(1983), Roberts (1984) and Bergstrom,Blume, and Varian (1986) develop models of voluntary provision of public goods. When a fundraiser
financesa public good throughvoluntarycontributionsthe equilibriumsupply
of public good will be less than socially efficient (see Samuelson, 1954; and
Stiglitz, 1987). Kaplow (1995) demonstratesthat subsidizing donations can
achieve a Pareto efficient outcome, however, that remedy is not available to
most fundraisers.
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52
In the context of raffles,the standardvoluntaryprovision model is equivalent to a raffle without a prize. A ticket-buyer purchases tickets motivated
by his or her desire to increase the supply of public good. Following the
Bergstrom et al. techniqueof substitutingthe individual's budget constraint
into his or her utility function, the consumer's utility maximizationproblem
becomes:
max U'[wi + G-i - G, G]
(1)
{G}
s.t. G - G_i > 0,
= G - gi representsthe level of public good without i's contriWhere
G-i
bution. The Nash assumptionis that each contributorviews the contributions
of others as fixed when deciding how much to give. Thus, each contributor
effectively chooses between combinations of private consumption and the
total supply of public good. Applying the Nash assumption, the first order
condition to choosing G thatmaximizes (1) is:
Ui - U'
0, with equality if gi > 0.
(2)
A Nash equilibriumallocationof privateconsumption and public good solves
(2) for every individual.Let G*(0) represent the equilibriumlevel of public
good derived from (2). Further,let N*(0) c N, where i E N*(0) if and only if
g (0) > 0, representthe set of individuals contributingto the public good in
the Nash equilibrium.Throughoutthis article, I assume that N*(0) is not an
empty set. Rearrangingterms in (2) yields:
mrsl(G*(0), (0)) = 1, if i E N*(0).
xi
(3)
2.2. Evaluating the supplyof public good between two allocations
Before exploring the economic effects of a raffle, it is useful to develop
a fact that can be used to compare the supply of public good between
two allocations. Consider two allocations (g, i) and (g, x) where g =
{gl, g2 ...
g,
x = Ixl,1,,
Xn,
g
=
-
glg2
...gn},
and
==
{xl,
X2,
2
... ,}. Let G and G representthe supply of public good in each allocation
and N C N and N C N represents the sets of individuals for whom
gi > 0 and gi > 0, respectively. Neither allocation is necessarily a
solution to any maximizationproblem. However, each person has the same
endowed wealth in both allocations (i.e., xi + gi = xi + gi, Vi). Fact 1 offers
a rule that can evaluatethe supplyof public good between the two allocations.
Fact 1. If mrsi(G, xi) < mrs'(G, x), Vi e N, then G > G.
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53
Proof: See Appendix A.
Fact 1 allows us to compare the supply of public good between two
allocations. The procedure is as follows. Take one allocation and call it
(k, x). Look only at the individuals contributingto the public good in that
allocation, N. If, in any other allocation, the marginalrate of substitution
between the public and private good goes down for everyone in N, then the
supply of public good has gone up (i.e., G > G).
3. A raffle with quasi-linear utility
The most familiar type of raffle has a fixed prize and a single winner. In a
fixed prize raffle, the prize is not a function of the number of tickets sold.
A ticket-holder'sprobability of winning the prize is equal to the number of
tickets he or she owns divided by the total numberof tickets sold. While it
is reasonable to predict that ticket-buyers will buy more tickets in a raffle
with a larger prize, a fundraisershould increase the value of the raffle prize
only if he or she believes it will result in an even greater increase in ticket
sales. In a fixed prize raffle, the value of the raffleprize affects the supply of
public good in two ways. First, it lowers the marginal price of the public
good, because every ticket purchased increases both the supply of public
good and the probability that the ticket-holderwill win the prize. Second,
it introduces uncertaintybecause a ticket-holdermight or might not win the
prize. However, only a ticket-buyer's private consumption is uncertain, not
the supply of public good. Furthermore,if the uncertaintyof a raffle does
not affect a ticket-buyer'sexpected marginalutility of private consumption,
then the uncertaintywill not affect the supply of public good. One example is
quasi-linearutility.
With quasi-linear utility, tying contributionsto a raffle will increase the
supply of public good. This result holds for any raffle prize, so long as no
ticket-buyer spends his or her entire wealth on raffle tickets. For example,
consider the following quasi-linearutility function:
(4)
U'[xi, G] = aoixi+f'(G),
the
of
where ai > 0 is a parametercorrespondingto
marginalutility private
consumption. The quasi-linear utility function (4) is linear in private consumption,and therefore,a ticket-buyeris always certainof his or her marginal
utility.With quasi-linearutility, thefixed prize rafflelowers the marginalprice
of the public good which, ignoring wealth constraints,will always increase
the supply of public good. To see this explicitly, write the consumer's utility
maximizationproblem for a fixed prize raffle with quasi-linearutility as:
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54
max aiE[xi] + f(G)
{IG
s.t. 0 <G - G-i < wi,
(5)
G represents i's expected private
+ r(G-Gi)
G+r
=
r
G
the
total supply of public good and
consumption,
represents
-n gi
=
G-gi representsthe total supply of public good minus i's contribution.
G-i
Assuming thatthe wealth constraintis not binding, the first orderequationfor
picking G that maximizes (5) is:
where E[xi] = wi + Gi
(
+
r))
r(G-i
(G + r)2
0
-f
(6) holds with equality if G - Gi > 0. Let G*(r) representthe equilibrium
supply of public good derivedfrom (6). Rearranging(6) gives:
f<1
r(G*_i(r) + r)
(7)
ai
(G*(r) + r)2
Described in more detail in Section 4.2, the right-handside of (7) represents
i's marginal price of contributingto the public good. The last term in (7)
is positive for any prize greater than zero. Therefore, the marginal price of
contributingto the public good is less than one and,
mrs'(G*(r),x*(r)) < 1 Vi.
(8)
Equations (3) and (8) imply that mrsi(G*(r),x*(r)) < mrs'(G*(0), x*(0)), Vi
e N*(0). Therefore,by Fact 1, G*(r) > G*(0) whenever r > 0.
One advantage of the quasi-linear utility function is that its separable
naturelends itself to a graphicalanalysis. Figure 1 diagramsthe quasi-linear
utility function described by (4). Panel (A) graphs the utility derived from
n
wealth spent on the public good (i.e., D =
gi), and panel (B) graphs
the utility derived from wealth spent on the private good (i.e., wi - gi). In
equilibrium,the marginalutility of public good is equal to marginalutility of
privateconsumption.When r = 0, the marginalutility of privateconsumption
is always equal to ai. Therefore,the Nash equilibriumsupply of public good,
labeled D*(0) in panel (A), is where the slope off' (D) is ai.
When r > 0 the utility received from spending on the public good shifts
to the right by the value of the prize (r). That shift is shown in panel (A). In
addition, the expected utility received from private consumption is equal to
the wealth a person spends on private consumption plus his or her expected
prize: ai(wi - gi + rg). Panel (B) graphs the expected utility function. The
vertical distance between the two utility functions in panel (B) is equal to
i's expected prize. When r > 0 the utility received from spending on private
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55
B. Private consumption
A. Public good
ai(wi - gi + rgi )
G+r
f-
ai(wi -gi)
rgi
G+r
001
f'(D)
:a.
-.Oo
rw.
G+r
f'(D - r)
r
D'(o)j
D'(r)
D
w-g-
Wi
Wi--gi
--gi
Figure 1. A fixed prize raffle with quasi-linearutility.
Wi
Wi- gi
consumption no longer passes through the origin because a person cannot
consume zero expected private consumption. That is, even if a person does
not directly buy privateconsumption,he or she still receives expected private
consumption from his or her raffle proceeds. With a positive raffle prize,
the marginal utility of spending on private consumption is ati(l which is less than ai whenever the raffle prize is greaterthan zero. _++),
Thus, in
equilibrium, the marginal utility of spending on the public good must also
be less than ai. Because the raffle prize simply shifts the utility function for
the public good to the right, the marginal utility of spending on the public
good is equal to ai at D*(0) + r. Therefore,the equilibriumcontributionsto
the public good must be greaterthan D*(0) + r. That logic implies that total
contributionsto the public good increase by more than the value of the prize,
and thus, the total supply of public good increases.
Provided that no ticket-buyer spends his or her entire wealth on raffle
tickets, a raffle will always increase the supply of public good regardless of
the size of the raffle prize. With general utility, this result will not hold.
4. A raffle with general utility
With general utility functions, thefixed prize raffleincorporatestwo effects; it
lowers the marginalprice of contributingto the public good and it introduces
uncertaintyin the marginal utility of private consumption. Let xw represent
i's consumption of privategood contingent on winning the raffle and xi represent i's consumption of private good contingent on loosing. Under a fixed
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56
prize raffle, the consumer's von Neumann-Morgensternutility maximization
problem is:
maxU
{x",x
,G)
(G+r)
i[x
G]
+
(1-
(G+r)
U'[x,
I, s.t. + G = G-i + wi
xi
w
1
G
'
(9)
xi -xi = r
0<xi-
(9) conceptualizes the individual as choosing between combinations of
privateconsumptionin the winning state, privateconsumption in the loosing
state, and public good (which is, of course, the same in both states). The first
constraintin (9) restrictsthe choice set to combinations that exhauststhe individual's wealth, the second constraintrestrictsthe choice set to combinations
of private consumption in the winning and loosing states that differ by the
value of the raffle prize, and the third constraint restricts the choice set to
non-negativevalues.
4.1. Twoheuristic raffles
In orderto deconstructthe differentways a raffle affects the supply of public
good, I analyze two heuristic raffles called the expected prize raffle and the
door prize raffle. Each of the heuristicraffles captures one of the two effects
incorporatedin thefixed prize raffle, while ignoring the other. For example,
the expected prize raffle removes uncertainty,so it captures the raffle's effect of lowering the marginalprice of the public good. Conversely,the door
prize raffle fixes the price of contributing,so it captures the raffle's effect of
introducinguncertaintyin the marginalutility of private consumption.
A fundraiseroffers a predeterminedprize in all three types of raffles:fixed
prize, expected prize and door prize. What distinguishes each raffle is how
the fundraiserdistributesthe prize. In the fixed prize raffle, the fundraiser
randomly selects a single winner. Conversely, in an expected prize raffle,
every participantwins a share of the total prize. For example, if the raffle
prize is a pumpkinpie, then underan expectedprize raffle every ticket-holder
receives a slice of pie equal to the percentage of tickets he or she owns. The
expectedprize raffle gets its name from the fact that ticket-holders have the
same expected consumptionas they do in thefixed prize raffle.The difference
is thatthereis no risk in the expectedprize raffle;each ticket-holderconsumes
his or her expected consumption with complete certainty. Finally, in a door
prize raffle, only one participantwins. In that respect, the door prize raffle
is similar to the fixed prize raffle. The difference between the fixed prize and
door prize raffles is that all of the participantsin a door prize raffle have an
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57
equal probabilityof winning the prize, regardlessof any additionalcontributions to the public good. The only link between the door prize raffle and the
supply of public good comes from the fact that the fundraisermust finance
the prize.
4.2. The expectedprize raffle
While the expected prize raffle is introducedfor illustrativepurposes only, it
is temptingto consider possible real-worldexamples. However, an important
feature of an expected prize raffle, that the total value of the prize is fixed,
rules out most examples. When a personbuys an expectedprize raffle ticket,
he or she literally takes consumption away from other ticket-buyers.2The
redistributionof private consumptionlowers the marginalprice of the public
good.
In additionto isolating a raffle's effect on the marginalprice of the public
good, the expected prize raffle offers an additional analytic feature: with
quasi-linearutility, it is identical to a fixed price raffle. With general utility,
the fixed prize raffle combines the effects of the expected prize and door
prize raffles. However, with quasi-linear utility, the door prize raffle has
no effect on the supply of public good, and thus, the fixed prize raffle is
equivalentto the expectedprize raffle.This is statedformally as Proposition 1.
Proposition 1. With quasi-linear utility functions, the fixed prize raffle
is equivalentto the expected prize raffle.
Proof:Using the quasi-linearutility functiondescribedby (4), the consumer's
utility maximization problem with an expectedprize raffle is:
max ai (wi - (G- G-i) +
s.t. 0 < G - G-i < wi.
+f(G)
(0)
Q.E.D.
Equations(4) and (10) are equivalent.
for
the
exderived
heuristic
result
that
1
Proposition implies
any general
with
raffle
also
raffle
holds
for
the
quasi-linearutility.
pected prize
fixed prize
The opposite is not always true. However,the quasi-linearresult that a raffle
will always increase the supply of public good generalizes to the expected
prize raffleregardless of the utility function.
The expectedprize raffle and thefixed prize rafflewith quasi-linearutility
both increase the supply of public good for the same reason: they lower the
marginalprice of contributing to the public good. Given the opportunityto
participatein an expected prize raffle,an individual'sbudget constraintis:
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58
wi + G-i +
G - G-i
r = xi + G.
(11)
G+r
The last term on the left-handside of (11) represents i's portion of the raffle
prize, which is equal to the percentage of tickets he or she owns multiplied
by the value of the prize. This term is responsible for the raffle's effect on
the marginal price of the public good. With a raffle, the marginal price of
contributingis less than one, because when individual i buys a raffle ticket
he or she redistributeswealth from other ticket-buyers. Individuali's budget
constraint defines the average price of the public good as 1 -
~r
However,
the marginalprice of the public good is:
r(G-i + r)
(G + r)2
(12)
Examining (12), if r = 0, then the raffle reduces to the voluntary contribution model described in Section 2.1 in which the price of the public
good is one. If r > 0, then the second term in (12) is positive and the price
of the public good is less than one. Thus, the public good is cheaper when
fundraisersraffle off a prize.
The decrease in the marginalprice of the public good leads to an increase in supply. To see how this works explicitly, write the consumer's
maximization problemwith an expectedprize raffle as:
max U'[xi, G]
{xi,G)
GG
s.t. wi + G-i +
=
0 < G - G-i < wi.
i + G,
(13)
Assuming that the wealth constraintis not binding, the first-orderequation
for picking combinationsof xi and G the maximize (13) is:
G)
h'(G, r) = U (G(xi,
Xi G)(1 1r)
U(,
r(Gi
(G
+
r)(14)
r)
2
U'(xi, G) > 0.
(14)
(14) holds with equality if G - G-i > 0. Let G*(r) representthe equilibrium
level of public good derivedfrom (14). Rearranging(14) gives:
+ r)
1 - r(G*_i(r)
r(15)
(G*(r)
(G*(r) + r)2
The last term in (15) is positive for any prize greaterthan zero. Therefore,
mrsI(G*(r),x (r))
mrs'(G*(r),
x*(r)) < 1 Vi.
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(16)
59
Equations (3) and (16) imply than mrs'(G*(r),x*(r)) < mrs'(G*(0), x*(0)),
Vi e N*(0). Therefore,by Fact 1, G*(r) > G*(0) wheneverr > 0.
Provided that no individual is pushed up against his or her wealth
constraint,the supply of public good is greaterwith an expected prize raffle
than without one. In fact, increasing the raffle prize will always increase the
supply of public good until at least one ticket-buyerspends his or her entire
wealth on raffle tickets.
Proposition 2. With an expected prize raffle (or, equivalently, with a
fixed prize raffle with quasi-linearutility), a largerprize will result in a higher
level of public good until one individual spends his or her entire wealth on
raffle tickets.
Proof: See Appendix B.
Provided that every ticket-buyer can afford to buy more tickets, the
supply of public good will increase whenever a fundraiser increases the
raffle prize. When the raffle prize becomes large enough relative to the
participants'wealth, the ticket-buyers' demandfor public good will become
more than they can afford. Once one ticket-buyeris pushed into a comer
solution, Proposition 2 no longer holds.
According to Proposition 2, the prize thatmaximizes the supply of public
good compels at least one ticket-buyerinto spendingall of his or her wealth
on raffle tickets. That person consumes privateconsumption solely from his
or her raffle proceeds. If all individuals have identical preferences and are
endowed with equal wealth, then the raffle prize that maximizes the supply
of public good has every individual spending his or her entire wealth on
raffle tickets.
Corollary2.1. If every individual in the communityhas identical preferences
and is endowed with equal wealth, then the raffle prize that maximizes the
supply of public good under an expectedprize raffle (or, equivalently, under
a fixed prize raffle with quasi-linear utility) is given by h(nw - r*, r*) = 0.
Every individualspends his or her entire wealth on raffle tickets.
Corollary2.1 follows directly from Proposition2.
Figure 2 illustrateswhat happens to the equilibriumsupply of public good
as a function of the raffle prize. The figurerepresentsa two-person economy
with identical Cobb-Douglas utility functions.The only constrainton the total
supply of public good is the wealth constraint.The fundraiser achieves the
maximum supply of public good when he or she sets the raffle prize at 70, at
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60
60
50
40
30
20
10
0
0
10
20
30
40
50
60
Prize
70
80
90
100
110
120
Figure 2. Supply of public good with an expected prize raffle; Cobb-Douglas utility:
U[ui, G] = xi5G'5; n = 2, wi = 60, i = 1, 2).
which point every individualspends all of his or her wealth on raffle tickets.
When the fundraiserincreasesthe value of the prize beyond 70, ticket-buyers
cannot afford to buy more tickets, and so the supply of public good decreases
one dollar for every one-dollar increase in the prize. The supply of public
good at the optimalprize is 49.67, whereas the supply of public good at r = 0
is 40. Therefore,tying contributionsto a raffle increased the supply of public
good by 24%. However,the Paretoefficient supply of public good is 60 (see
Appendix D). While the rafflecan increase the supply of public good, it does
not come arbitrarilyclose to the efficient supply.
4.3. The door prize raffle
A door prize raffle has a fixed prize and a single winner. However, unlike
the fixed prize raffle, a participant'sprobability of winning the prize is also
fixed. An example of a door prize raffle is an event in which every attendeeis
given one ticket as they enter.Each attendee's probabilityof winning is 1/n.3
Attendees can make additional contributions to the public good. However,
doing so does not alter his or her probability of winning the prize. The prize
is linked to the public good only by the fact that the fundraisersmust finance
the prize.
Both the door prize and fixed prize raffles introduce uncertainty in the
consumption of privategood; a participantmay win or loose the raffle. Under a door prize raffle, an individual's von Neumann-Morgensternutility
maximization is
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61
max
Ui[x
'G]
{xW,xi,G)(n)
+ (1 -
n)
U[x', G]
s.t. x1+ G = Gi + wi
w
1
1
x
(17)
wi.
Like in a fixed prize raffle, participantsin a door prize raffle choose between
combinations of privateconsumptionin the winning state, private consumption in the loosing state, and public good. Assuming thatthe wealth constraint
is not binding, the first-ordercondition for picking combinations of x', xi,
and G that maximize (17) yield:
EUxi
- EU' = 0.
(18)
Where EU'i represents i's expected marginalutility of private consumption
and EUh representsi's expected marginalutility of the public good. In a door
prize raffle, participantsequate their expected marginalutilities of public and
privateconsumption.
With quasi-linear utility, a door prize raffle will not influence the supply
of public good.4 With general preferences,however,the door prize raffle may
not be as forgiving. In fact, with general preferences,a door prize raffle can
increase or decrease the supply of public good, dependingon how participants
view tradingcertain consumptionfor expected consumptionat the margin. If
the participants' marginal utility of private consumption is decreasing and
concave in xi, then the door prize raffle will increase the supply of public good. On the other hand, if the participants'marginal utility of private
consumption is decreasing and convex in xi, then the door prize raffle will
decrease the supply of public good. For the purposesof this paper,I focus on
the latter case.
It is always possible for participantsto exactly offset a door prize raffle,
as they do in the quasi-linear case. That is, if a fundraiser increases the
door prize by $10, participants can always give $10 more and leave the
supply of public good unchanged. Will participantsdo this? The answer,
of course, depends on preferences, but for now suppose that they do. Let
g*(0) = {g*(0),gg(0),...,g*(0),
represent the equilibriumvector of gifts
to the public good when r = 0. Consider a hypothetical allocation of gifts
g(r) = {g,(r), g2(r), ..., gn(r)}, where gi(r) = g*(0) + I/n Vi. The vector of gifts g(r) is called the offset allocation because G(r) = G*(0) and
= xt Vi. The offset allocation is not necessarily an equilibriumallocaE[i]-xi]
tion. If, at the offset allocation, every participantdesires to contributemore to
the public good, then the raffle prize will increase the supply of public good.
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62
If, on the other hand,every contributordesires to contributeless to the public
good, then the raffleprize will decrease the supply of public good.
In practice, the first-orderequation determines whether every participant
desires more or less public good at the offset allocation. Using that logic it
becomes clear why the door prize raffle does not affect the supply of public
good with quasi-linearutility. The first-orderequation for the quasi-linear
utility function is:
ai f=
Vi.
(19)
(19) holds at the offset allocation. In fact, with quasi-linear utility, a participant's first-orderequation with a door prize raffle is identical to his or her
first-orderequationwithout a raffle.
With generalutility,(18) describes the first-orderequation.Although there
is no uncertaintyin the supply of public good, the right-hand side of (18)
representsthe expectedmarginalutility of public good, ratherthan simply the
marginalutility of public good, because of the assumptionthat Ui > 0. For
that U'i
=
clarity, assume, without loss of generality,condition
as is the case with
0,
Separable
functions.
The
first-order
becomes:
(18)
utility
functions.
becomes:
The
first-order
condition
(18)
separableutility
(20)
G'
Xi = U'.
EUix
The right-handside of (20) is the same at the offset allocation as it is
at the original equilibrium allocation. However, if the marginal utility of
private consumption is decreasing and convex in xi, then the left-hand side
of (20) is greater at the offset allocation than at the original equilibrium
allocation. Therefore,the offset allocation cannot be an equilibrium.In fact,
at the offset allocation, every participantdesires more private consumption
and less public good. Figure 3 diagrams how this works. At the original
equilibrium allocation, g*(0), the marginal utility of private consumption is
U'i. To reach the offset allocation, participantsgive up certain consumption,
x=.
(x* - xl), but receive expected consumption such that E[xi]
At the offset allocation, g(r), the marginal utility of private consumption
which is greaterthanU . Repeating this logic leads to Proposition3.
is
EUi,
Proposition 3. If every person's marginal utility of private consumption is
decreasing and convex, then a largerdoor prize raffle will result in less public
good.
Proof: See Appendix C.
Figure 4 illustrates what happens to the equilibrium supply of public
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63
EU'i
xi
Xi
-1
-W
xi
r
Figure 3. Marginalutility of privateconsumption.
42
40
38
36
3432
30
0
5
10
15
20
25
30
35
40
45
50
55
60
Prize
Figure 4. Supply of public good with a door prize raffle; Cobb-Douglas utility:
U[xi, G] = xi5G5; (n = 2, wi = 60, i = 1, 2).
good as a fundraiserincreases the raffleprize. The figure representsthe same
two-personeconomy diagrammedfor the expectedprize raffle (see Figure 2).
The fundraisermaximizes the supply of public good when he or she sets the
raffleprize at zero. Any increase in the raffle prize leads to a reductionin the
supply of public good.
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64
4.4. Thefixed prize rafflewith general utility
With general utility, afixed prize raffleincorporatesthe propertiesof both the
expectedprize and door prize raffles.That is, buying a raffle ticket lowers the
marginalprice of giving, but it also introduces uncertainty.The combination
of the two effects is seen explicitly in the first-orderequations. Assuming
that the wealth constraintis not binding, the first-ordercondition for picking
combinationsof xw, x!, and G that maximize (9) yield:
EU - EU
I + r)
(GEU
(G + r)(U'[x,
(G + r)2
G] - U'[xl, G]).
(21)
The left-hand side of (21) is identical to the first-orderconditions for participants in a door prize raffle.That side of the first-orderequationis derived
from the fact that the raffle introducesuncertaintyin the marginalutility of
private consumption. If the marginalutility of private consumption is convex in xi, then ticket-buyersmust buy fewer tickets in order to equate their
expected marginalutility of public and private consumption as a fundraiser
increases the raffleprize.
However, ticket-buyers do not equate their expected marginal utility of
public and privateconsumptionin afixed prize raffle as they do in a door prize
raffle. The right-handside of (21) representsthe marginal price effect. That
side of the first-orderequationis derivedfrom the fact that the probabilityof
winning the prize is a function of the numberof tickets sold. For any positive
raffle prize, the right-handside of (21) is positive. Therefore, any allocation
that equates the expected marginalutility of public and privateconsumption
will leave the right-hand side of (21) greater than the left. At that allocation, ticket-buyers desire more public good and less private consumption.
Therefore, a fixed prize raffle incorporatesboth the marginal price effect of
the expected prize raffle, which leads to an increase in public good, and the
uncertaintyeffect of the door prize raffle, which leads to a decrease in public
good.
Because the marginal price effect and the uncertainty effect can potentially work in opposite directions,it is possible that an offset allocation is an
equilibriumallocation.5The right-handside of (21), evaluatedat the offset allocation, increases as a fundraiserraffles off largerprizes. The marginalprice
effect leads participantsto buy more raffle tickets. However, the left-hand
side of (21), evaluatedat the offset allocation, can also increase as fundraisers
raffle off larger prizes. The uncertaintyeffect leads participantsto buy fewer
raffle tickets. If, by moving to the offset allocation, both sides of the firstorderequation increase by the same amount,then the offset allocation is also
an equilibriumallocation. Therefore,whether a fixed prize raffle increases or
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65
44
42
40
38
36
34
32
30
0
5
10
15
20
25
30
35
40
Prize
45
50
55
60
65
70
Figure 5. Supply of public good with a fixed prize raffle; Cobb-Douglas utility:
U[xi, G] = xi5G.5; (n = 2, wi = 60, i = 1, 2).
decreases the supply of public good depends both on preferences and on the
size of the raffle prize.
For example, consider the same two-personeconomy diagrammedfor the
expected prize and door prize raffles (see Figures 2 and 4). Figure 5 illustrates what happens to the equilibriumsupply of public good as a fundraiser
increases the prize of a fixed prize raffle. When the raffle prize is small, the
supply of public good increases. As the fundraiserincreases the raffle prize,
the uncertaintyeffect increases relativeto the marginalprice effect, eventually
resultingin a decrease in the supply of public good. The prize that maximizes
the supply of public good is 25, at which point the supply of public good
is 43.75. When r = 0, the supply of public good is 40. Therefore, tying
contributionsto a raffle increased the supply of public good by 9%
The raffle's effect on the supply of public good does not disappearas the
size of the economy grows. Holding the total wealth of the previous economy
constant, the maximum attainable supply of public good decreases as the
community size (n) increases. Thus, the raffle does worse as the size of the
economy grows. However, voluntarycontributionsalso do worse as the size
of the economy grows. In fact, as the size of the community grows, thefixed
prize raffle does worse in absolute terms,but betterrelative to privatesupply.
Figure 6 graphs the percentage change in the supply of a public good as a
fundraiserincreases the value of the raffleprize. Figure 7 graphsthe deviation
from the efficient supply of public good as a functionof the raffle prize. Both
figures show a two, three, and nine-person economy with the same CobbDouglas utility functions used earlier. Table 1 lists the efficient supply of
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66
50%
n=2
n=3
n=9
40%
30%
20%
10%
0%
-10% i
-20%
5
15
10
20
25
30
35
40
45
50
55
t,
_70
-30%
Prize
Figure 6. Percentage change in public good with a fixed prize raffle; Cobb-Douglas utility:
Vi)
--n-I
U[xi, G] = xi5G.5;
(wi =120
90%
80%
70%
60%
50%
40%
30%
20%
10%
0%
n=2
n=3
n=9
0
5
10
15 20
25
30
35 40
Prize
45
50
55
60
65
70
Figure 7. Percentagedecreasefrom the efficient supply of public good; Cobb-Douglasutility:
U[xi, G] = xi5G5; (wi = 120 Vi)
public good, the supply at r = 0, the prize that maximizes the supply of
public good, and the maximumattainablesupply of public good.6
Table1. Supply of public good with and without afixed prize raffle, by community size
n
Ge
G(0)
r*
G(r*)
G(r*)-G(0)
G(r*)Ge
2
3
9
60
60
60
40
30
12
25
30
12
43.75
35.00
16.69
9%
17%
39%
-27%
-42%
-73%
G(0)
Ge
Notes. Ge is the efficient supply of public good; G(0) is the supply of public good without a
raffle; r* is the raffle prize that maximizes the supply of public good over other prizes; and
G(r*) is the maximum public good afixed prize raffle can produce.
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67
Figure 6 shows that the percentage change in public good is greater for
larger economics. Thus, the raffle does bettercomparedto private supply as
the number of people in the economy grows. On the other hand, Figure 7
shows that the percentagedecrease in the supply of public good compared to
the efficient level gets largeras n increases.Therefore,the raffle does betteras
n increases relativeto private supply,but it does worse relative to the efficient
supply.
5. Conclusion
In general, a rafflecan increase or decreasethe supplyof public good, depending on the preferences of ticket-buyers and on the value of the raffle prize.
When a person buys a raffle ticket he or she redistributesexpected private
consumption away from other ticket-buyers.The redistributionof expected
private consumption lowers the marginalprice of contributingto the public
good and, therefore, leads to an increase in its supply. However, the redistributiontrades certain consumption for expected consumption. Depending
on how ticket-buyers view uncertainty,that tradeoffcan lead to a decrease
the supply of public good. It is, therefore,the fundraiser'stask to choose the
appropriateraffle prize that maximizes the supplyof public good.
Notes
1. Klein (1987) considers a more general case of tying a public good to the purchase of a
privateone. Under Tiebout (1956) assumptions,public good tie-ins can produce efficient
outcomes when the public good can be excluded from individualswho do not purchase a
particularprivategood.
2. Harbaugh(1998) develops a prestige model of philanthropy.If the total prestige available
is fixed, then his model would have features similar to an expected prize raffle. In effect,
making a contributionwould redistributeprestige.
3. In practice, a fundraisermight use a door prize to increaseattendance(i.e., n is a function
of r). A fundraisermight also use a door prize to impose an all-or-nothing level of contribution, similar to the provision point mechanism (see Isaac, Schmidtz, and Walker, 1999;
and Rondeau, Schulze, and Poe (1999). However, this paper introduces the door prize
raffle to explain features of the fixed prize raffle, and so, it assumes that n is independent
of r.
4. With quasi-linearpreferences, a participantin a door prize raffle has the utility function:
U'[xi, G] = aixi + fi(G) +
r/n.
The raffle prize constitutes a simple monotonic transformationof the utility function.
Therefore, assuming the wealth constraintis not binding, the raffle prize will not affect
the supply of public good.
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68
5. In afixed prize raffle,the offset allocation is defined such that
g o(0)
gi(r) = g* (0) + G* (0) Vi.
)r
6. See Appendix D for the derivationof the efficient supply of public good.
References
Bergstrom, T., Blume, L. and Varian,H. (1986). On the private provision of public goods.
Journal of Public Economics 29: 25-49.
Harbaugh,W. (1998). Whatdo donationsbuy? A model of philanthropybased on prestige and
warm glow. Journalof Public Economics 67: 269-284.
Isaac, R.M., Schmidtz, D. and Walker, J. (1989). The assurance problem in a laboratory
market.Public Choice 62: 217-236.
Kaplow, L. (1995). A note on subsidizing gifts. Journal of Public Economics 58: 469-477.
Klein, D. (1987). Tie-ins and the marketprovision of collective goods. Harvard Journal of
Law & Public Policy 10: 451-474.
Morgan, J. (1996). Financingpublic goods by means of lotteries. PrincetonWoodrow Wilson
School Discussion Paperin Economics 183 (September).
Roberts, R. (1984). A positive model of private charity and wealth transfers. Journal of
Political Economy 92: 136-148.
Rondeau, D., Schulze, W.D. and Poe, G.L. (1999). Voluntaryrevelation of the demand for
public goods using a provision point mechanism. Journal of Public Economics 72: 455470.
Samuelson, P.A. (1954). The pure theory of public expenditure. Review of Economics and
Statistics 36: 378-389.
Stiglitz, J. (1987). Paretoefficient and optimal taxation and the new welfare economics. In A.
Auerbac and M. Feldstein (Eds.), Handbook of public economics, II. Amsterdam:North
Holland.
Tiebout, C. (1956). A puretheoryof local expenditures. TheJournal of Political Economy 64:
416-424.
Warr,P.G. (1983). The privateprovision of public goods is independentof the distributionof
income. Economics Letters 13: 207-211.
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All use subject to JSTOR Terms and Conditions
69
Appendix A
Proof of Fact 1
Fact 1. If mrsi(G,
b i) < mrsi(G, xi), Vi e N, then G > G.
Fact 1 is proved by contradiction. Each individual has the same endowed
wealth in the two allocations. Therefore,the total expenditureson privateand public
consumption must be the same in both allocations:
n
n
i + G=
i=l
i+ G.
(Al)
i=l
Rearranging(Al) yields:
n
n
=G i=l
i--
G.
(A2)
i=l
Suppose that G < G. The right-handside of (A2) is non-negative, which implies
that ii > i for some i E N. That is, if the total contributionsto the public good go
down, then someone who was contributingto the public good (i.e., some i e N) must
now be contributingless. As that person contributesless to the public good, he or she
must consume more privateconsumption. Furthermore,if, for that someone, G < G
and Ri> Xi, then the marginalconditions on Ui imply:
UiG U-G
>
for some ie N,
Uixi - U!
xi
(A3)
which contradicts the assumption that mrsi(G, ii) < mrsi(G, xi)Vi E N. Therefore,
G > G.
Q.E.D.
Appendix B
Proof of Proposition 2
Proposition 2. With an expected prize raffle (or, equivalently, with a fixed
prize raffle with quasi-linear utility), a larger prize will result in a higher level of
public good until one individual spends his or her entire wealth on raffle tickets.
> 0 for all
Proof: To prove Proposition 2, it is sufficient to prove that dG*(r)
dr
ticket-buyers when ignoring the wealth constraint. For ticket-buyers, substituting
G*(r) into (14) yields the identity:
h'(G*(r), r) - O, if G - G_1 > 0.
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(B1)
70
(B1) withrespectto r, andcollectingtermsyields,
Differentiating
dG*dG*
i
-
-U xi a
-
xixi
xi--SP + PG
GU.xixi
O
p
i
GUxixi
xi r + UiGxi ar
xi
-
G
PIGUxiG
UGxi
> 0,
--
(B2)
UGG
where
i PG= 1
axi
G(G-Gi)
ar
ai
0
(G+r)2
-r
apG
a
> 0,
r(G-i+r)
(G+r)2
r
GG
r(G_-2G)(G+r)3 _i
i
G
0,
apG 2r(Gi + r)
aG
(G + r)3
>
- 0-.
Q.E.D.
Appendix C
Proof of Proposition3
Proposition 3. If every person's marginal utility of private consumption is
decreasing and convex, then a largerdoor prize raffle will result in less public good.
Proof: Ignoring the wealth constraints,the first-orderequations for participantsin a
door prize raffle are:
=
(C1)
EUxi EUG'.
Let g*(r) = {g*(r), g*(r),..., g*(r)}, represent the equilibrium vector of gifts
that solve Equation (20) for a given r. Proposition 3 suggests that G*(r) <
G*(0) for any r > 0. Consider a hypothetical allocation of gifts g(r) =
= g*(0) + rVi. The vector of gifts g(r) is
{gl (r), g2(r) . . .gn(r)), where gi(r)
called the offset allocation because G(r) = G*(0) and E[xi] =
Vi. By concavity
xi
>
0
and
0.
Therefore,
assumptionEUxixixi
EUGxixi<
EU i
Ei
U> and EU < G*
Uix,U,.
(C2)
(Cl) and (C2) imply that
EU'i > U'
Vi.
(C3)
Thus, the offset allocation is not an equilibrium allocation. In fact, at the offset
allocation, every participantwants more privateconsumption.
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71
An equivalent statement of Proposition3 is that G*(r) < G(r). Suppose alternatively thatG*(r) > G(r). If the total supply of public good stays the same or goes up,
then x* < xi, for at least one participant.Moving from the offset allocation to the
equilibriumallocation, at least one participant'sconsumptionof public good goes up
(or stays the same) while his or her consumptionof privategood goes down (or stays
the same). Therefore, by the marginalconditions placed on U':
and EU6- > EUiG
< EU'.
xi
EUijxi
(C3) and (C4) imply that for at least one participant,
-
EU'
> Ua*.
(C4)
(C5)
(C5) contradicts (C1). Therefore, G*(r) < G(r) and G*(r) < G*(0) for any r. This
Q.E.D.
logic holds for any initial prize value.
Appendix D
Efficient supply with Cobb-Douglas utility
Substituting the budget constraint into the utility function, the consumer's
utility maximization problem is:
max Ui[wi + G-i - G, G] = (wi + G-i - G)cGP
{G}
(D1)
s.t. G - G-i > 0
An efficient allocation of privateconsumptionand the public good is such that:
n
i=l
mrsi(Ge, xe) = 1.
(D2)
Calculatingthe marginalrate of substationfrom Equation(Dl) yields:
xi
(D3)
IaG
xe)
Let wi = wjVi,j. Then a particularefficient allocation, the one in which every
individualconsumes the same amountof the privategood, satisfies the equation:
mrsi(Ge,
P(W - G)
= 1,
aG
where W =
il
(D4)
wi. Some algebraicre-arrangementgives:
Ge=w
(a +0)
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(D5)