Journal of Public Economics 72 (1999) 213–242
Modeling charitable contributions of time and money
Brian Duncan*
Department of Economics, University of Colorado at Denver, Suite 460, 1380 Lawrence Street,
Denver, CO 80204, USA
Received 1 April 1997; received in revised form 1 May 1998; accepted 2 June 1998
Abstract
Public goods theory predicts that government spending on charity can perfectly crowdout charitable contributions. Empirical research has found little support for the perfect
crowd-out hypothesis. However, the empirical work only measures part of a contributor’s
total contribution: gifts of money. Contributors also volunteer labor. This article extends the
public goods model to the case in which individuals contribute both time and money to a
charity that in turn produces a public good. The model suggests that charitable contributions
of time and money are perfectly substitutable in equilibrium. This result offers new
interpretations for existing empirical observations, and suggests a new test of the crowd-out
hypothesis. The implications of the model are empirically tested using a national survey of
charitable activity.  1999 Elsevier Science S.A. All rights reserved.
Keywords: Public goods; Crowd-out; Volunteer labor; Charitable contributions; Charity;
Nonprofit organizations
JEL classification: H41; L3
1. Introduction
When researching charitable contributions of time and money, the interpretation
of empirical evidence is significantly affected by the model through which the
researcher views it. The two benchmark models researchers most often use are the
public goods model and the private consumption model. The difference between
*Tel.: 11-303-556-6763; Fax: 11-303-556-3547.
E-mail address: [email protected] (B. Duncan)
0047-2727 / 99 / $ – see front matter  1999 Elsevier Science S.A. All rights reserved.
PII: S0047-2727( 98 )00097-8
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B. Duncan / Journal of Public Economics 72 (1999) 213 – 242
these models lies in what motivates contributors to give. In the public goods
model, a desire to increase the public good motivates contributors to give, and
thus, a charitable gift is only meaningful if it increase the supply of the public
good. In the private consumption model, the act of giving itself motivates
contributors to give, and thus, a charitable gift is always meaningful. Researchers
often mix the two motivations together creating what is perhaps a more realistic
view of the world, one in which contributors are motivated by what their gifts
produce as well as by how giving makes them feel.1 This article extends the public
goods model by allowing individuals to contribute both time and money to a
charity that uses contributions to produce a public good. In equilibrium, contributors view gifts of time and money as perfectly substitutable. The model offers
new interpretations for empirical observations found in both the crowding-out
literature and the volunteer labor supply literature.
When a public good is financed through voluntary contributions, government
spending on the public good will crowd-out voluntary contributions. Warr (1982);
Roberts (1984); Bergstrom et al. (1986) all present public goods models in which
government spending on a public good is met with equal reductions in voluntary
contributions. Inspired by the theory, empirical research puts the suggested
ineffectiveness of government spending to the test. While several researchers have
found partial evidence of crowding-out, none, with the notable exception of
Roberts (1984), have found evidence of perfect crowding-out. However, much of
this empirical work, such as Abrams and Schitz (1978); Roberts (1984); Khanna et
al. (1995); Kingma (1989); Steinberg (1985); Reece (1979); Schiff (1985), only
examine charitable contributions of money. The theoretical models do not
distinguish between gifts of time and gifts of money. The model developed in this
article suggests that this distinction is significant and that previous measures of
crowding-out are incomplete because they fail to account for volunteer labor. Data
from a national survey of charitable activity suggests that omitting volunteer labor
reduces the estimate of crowding-out by 27%.
Researchers of the supply of volunteer labor motivate their empirical specification using the private consumption model. Menchik and Weisbrod (1987);
Brown and Lankford (1992); Schiff (1990) each estimate price and income
elasticities for volunteer labor. Using an individual’s net wage as the price of
volunteering and one minus an individual’s marginal tax rate as the price of money
donations, empirical researchers estimate negative price and cross-price elasticities
for volunteer labor. Thus, the voluntary labor supply literature has concluded that
charitable gifts of time and money are gross complements. The model developed
in this article suggests just the opposite, that gifts of time and money are perfectly
substitutable. While estimates of the price and cross-price elasticity found in the
volunteer supply literature are correctly specified using the private consumption
model, using the public goods model they are not. When interpreted through the
1
The impure altruist model was developed by Andreoni (1990).
B. Duncan / Journal of Public Economics 72 (1999) 213 – 242
215
public goods model developed in this paper, the existing empirical evidence is
consistent with contributors who view charitable gifts of time and money as
perfectly substitutable in equilibrium.
The model developed in this article suggests new interpretations for previous
empirical observations and provides a framework for additional empirical work.
Though the model I find that: (1) In equilibrium, individuals view gifts of time and
money as perfectly substitutable; (2) even if perfect crowd-out holds, a one-dollar
increase in government spending will not result in a one-dollar decrease in money
contributions; (3) in the public goods model, an individual’s net wage is an
inappropriate measure of the price of volunteering; and (4) in the public goods
model, one minus an individual’s marginal tax rate is an inappropriate measure of
the price of money contributions. The implications of the model are empirically
tested using a national survey of volunteer activity. Although the crowd-out
estimate that includes charitable contributions of both time and money is 27%
larger than the estimate that includes only money donations, the data does not
support the perfect crowd-out hypothesis. However, the data is consistent with the
public goods prediction that charitable contributions of time and money are
perfectly substitutable in equilibrium.
2. The production of charity
The current economic literature that focuses on charitable contributions makes
little reference to how charity is actually produced. This is justified considering
that production can often be thought of as simply measuring inputs. For many
public choice applications, the results do not change if the public good is measured
as the output of the production process or as the inputs to the production process.
For example, in Bergstrom et al. (1986), the public good is measured as the sum
of all individuals’ contributions. Their results do not change if the public good is
instead measured as a function of all individuals’ contributions.
This article, however, presents a model that explores the relationship between
different types of charitable inputs. It is therefore necessary to examine how these
inputs are related in the production process. To this end, consider a charity firm
that uses capital (K) and labor (H ) to produce charity through the technology
function C(H, K). The technology function exhibits positive marginal products of
capital and labor. Charity is non-rival and non-excludable, and is thus a public
good. The charity firm’s goal is to maximize the production of charity given its
financial resources. To finance the production of charity, the firm relies on
charitable contributions of time and money from n individuals living in the
community.
Let vi and d i represent individual i’s charitable contribution of time and money
given to the charity firm. The charity firm is free to use money donations to buy
capital at a fixed price r or labor at a fixed price w. However, the charity firm
B. Duncan / Journal of Public Economics 72 (1999) 213 – 242
216
cannot sell volunteer labor to buy more capital. Let h i represent the total amount of
time individual i spends working at the charity firm. Some of this time may be
volunteered (vi ), and some may be paid for (h i 2 vi ). Furthermore, let D 5 o ni 51 d i
represent the total amount of money donated to the charity firm, V 5 o in51 vi
represent the total amount of time volunteered to the charity firm, and H 5 o ni 51 h i
represent the total amount of labor (volunteer and paid) used by the charity firm.
Finally, let G 5 D 1 wV represent the total value of charitable gifts given to the
charity firm. The charity firm chooses combinations of capital and labor that solve
the production maximization problem:
MAX
C(H, K)
s.t.
G 5 wH 1 rK
rK # D
hH,K j
(1)
The second constraint in Eq. (1) reflects the fact that the charity firm can be
capital constrained if contributors volunteer a disproportionate amount of time.
However, in the public good model, it will be shown that contributors will never
leave the charity firm capital constrained in equilibrium. Moreover, while omitting
the second constraint in Eq. (1) can affect the equilibrium allocation of time and
money contributed, this simplification will never affect the equilibrium supply of
charity.
To pin down exactly when the charity firm is capital constrained, consider
omitting the second constraint from Eq. (1). This omission effectively creates an
unrestricted world in which the charity firm can sell volunteer labor. Gifts of time
and money only enter the charity firm’s unrestricted problem in G. The supply of
charity in the unrestricted world is therefore a function of the total value of
charitable gifts, not contributions of time and money separately as it is in the
restricted world. The charity firm’s optimal choice of capital and labor are h u (G)
and k u (G) in the unrestricted world, and h r (D, V ) and k r (D, V ) in the restricted
world. The total supply of charity in the unrestricted and restricted worlds is,
respectively:
Z u (G)5C(h u (G), k u (G)),
Z r (D, V )5C(h r (D, V ), k r (D, V )).
(2)
(3)
Capital constrained means that the charity firm would like to sell some of its
volunteer labor and buy more capital. If the total amount of labor volunteered to
the charity firm is less than or equal to the firm’s demand for labor in the
unrestricted world, then the charity firm is not capital constrained in the restricted
world. Thus,
If V # h u (G) then the charity firm is not capital constrained.
(4)
B. Duncan / Journal of Public Economics 72 (1999) 213 – 242
217
3. The consumer in a public goods model
The charity firm views charitable gifts of time and money as exogenous
parameters. For the consumer, however, these are choice variables. The assumption that defines the public goods model is that a contributor’s gift does not
directly enter his or her utility function. Instead, contributors use their gifts to
affect the supply of charity. Each person is endowed with t units of time, a fixed
wage w, and non-wage income y i . Preferences are represented by the utility
functions Ui [x i , l i , Z r (D, V )], where x i represents private consumption, l i represents
leisure, and Z r (D, V ) represents the supply of charity. Private consumption, leisure,
and charity all exhibit positive marginal utility. In addition, the marginal
propensity to consume the public good is positive and less than one.2 Consumers
choose combinations of private consumption, leisure, volunteer labor, and money
donations that solve the utility maximization problem:
MAX
hx i ,l i ,v i ,d i j
s.t.
Ui [x i , l i , Z r (D, V )]
w(t 2 l i 2 vi ) 1 y i 5 x i 1 d i ,
x i $ 0, 0 # l i # t, 0 # vi # t, d i $ 0.
(5)
3.1. Nash equilibrium
People independently decide their own gift to the public good. The charity each
person consumes, however, is determined by the value of all gifts. An individual’s
optimal choice of private consumption, leisure, volunteer labor, and money
donations are therefore determined, in part, by the gifts of others. A Nash
equilibrium is an allocation of private consumption, leisure, volunteer labor and
money donations such that, given the charitable gifts of others, every individual is
contributing an optimal amount.
In the Nash game, the charity firm is a passive player, picking only H and K.
The firm does not take an active role in determining a contributor’s allocation of
volunteer labor and money donations. In actuality, a charity can turn away a
volunteer in an attempt to force him or her to give money instead. Would the
charity firm ever need to do this? Specifically, does the possibility of capital
constraints necessitate the charity firm’s active role in determining an individual’s
gift allocation? In a public goods model, the answer is no. Contributors will never
leave the charity firm capital constrained. The proof of this is based on the
marginal product of a volunteer’s time. If the charity firm is left capital
constrained, then the marginal product of a volunteer’s time is greater when
working than when volunteering. The volunteer could therefore increase his or her
consumption of all goods by working more and volunteering less. Working more
2
Bergstrom et al. (1986) show that the Nash equilibrium supply of a public good is unique if the
marginal propensity to consume the public good is positive and less than one.
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B. Duncan / Journal of Public Economics 72 (1999) 213 – 242
increases the volunteer’s income, but if it comes at the expense of volunteer labor,
then it also decreases the supply of charity. However, because the volunteer’s
marginal product of time is greater when working than when volunteering, the
decrease in charity is more than offset by the increases in income. A volunteer can
consume the same amount of charity by volunteering less labor and donating more
money, and still have additional income left over to increase his or her
consumption of the private good. Any allocation of time and money in which the
charity firm is capital constrained cannot be utility maximizing for volunteers and
is thus not a Nash equilibrium.
Proposition 1. If h(v i* , d *i , x *i , l i* )j is a Nash equilibrium, then V *#h u (G*) and
the charity firm is not capital constrained.
Proof. See Appendix A.
Proposition (1) holds because, in a public goods model, contributors value the
supply of charity, not their personal gifts of time and money. Thus, the goals of
contributors are perfectly harmonious with the goals of the charity firm. This
harmony implies that contributors can, within a feasible range, trade volunteer
labor for donations of money at a constant rate w, without affecting the supply of
the public good or their total utility. For example, if a contributor reduces his or
her volunteer labor and increases his or her money donation, the charity firm will
simply offset these changes by increasing its purchases of labor, leaving the supply
of charity unaffected. Consumers are indifferent to any feasible gift allocation that
holds the total value of their gift constant. Each individual’s specific allocation of
time and money, and thus the total allocation of time and money given to charity
firm, is arbitrary.
Proposition 2. If h(v i* , d i* , x i* , l i* )j is a Nash equilibrium and ( i) wv i* 1d i* 5
wv i9 1d 9i , ( ii) v 9i , d 9i $0, and ( iii) V 9#h u (G*), ;i then h(v 9i , d 9i , x *i , l *i )j is also a
Nash equilibrium.
Proof. See Appendix B.
Expressed as conditions ( ii) and ( iii) in Proposition (2), a feasible range means
that contributors never leave the charity firm capital constrained and, of course, an
individual’s gift cannot be negative. Condition ( i) in Proposition (2) implies that
charitable gifts of time and money are perfectly substitutable in equilibrium. Any
gift allocation h(v i9 , d i9 , x i* , l i* )j that satisfies conditions ( i) – ( iii) of Proposition (2)
is a Nash equilibrium, and thus, there are an infinite number of Nash equilibrium
gift allocations.
While the equilibrium allocation of time and money given to the charity firm is
not unique, the equilibrium supply of charity is. A unique Nash equilibrium supply
B. Duncan / Journal of Public Economics 72 (1999) 213 – 242
219
of charity exists even though the specific allocation of time and money is
arbitrary; the total value of all charitable gifts is not arbitrary. To prove this,
consider a specific Nash equilibrium in which contributors only give money. Gifts
of time and money are perfectly substitutable, and thus, for any Nash equilibrium
there exists an equivalent Nash equilibrium in which every individual contributes
only money. The money only Nash equilibrium is, by definition, within every
individual’s feasible range. Furthermore, extending the uniqueness proof in
Bergstrom et al. (1986), there exists a unique money only Nash equilibrium. Thus,
the equilibrium supply of charity is unique, and all Nash equilibria are essentially
equivalent.
Propositions (1) and (2) imply that simplifying the firm’s production maximization problem by assuming the firm operates in the unrestricted world does not
affect the Nash equilibrium supply of charity. This simplification can affect the
specific allocation of time and money contributed to the charity firm. In fact,
letting the charity firm operate in the unrestricted world adds additional Nash
equilibrium gift allocations to the infinite number which already exist. However,
there exists only one Nash equilibrium supply of charity regardless of whether the
charity firm operates in the restricted or unrestricted world.
3.2. The demand for volunteers
The charity firm uses two types of workers to produce charitable output: paid
employees and volunteers. Steinberg (1990) discusses the potential productive
relationship between these two types of workers. He points out that, their name
notwithstanding, volunteers are not free. The charity firm must recruit, train, and
supervise volunteers. Using a survey of nonprofit organizations, Emanuele (1996)
estimates a downward sloping demand curve for volunteer labor and essentially
concludes that Steinberg is correct, volunteers are not free goods. The presence of
employment costs associated with volunteer labor does not violate the assumptions
of the public goods model. However, Eq. (1) assumes that paid employees and
volunteers with the same skills are perfect substitutes. If the difference in
employment costs between paid and volunteer labor is not equal to w, then the
public good model predicts that contributors will only give time or only give
money, depending on which is relatively cheaper for the charity firm. Duncombe
and Brundney (1995) use data on volunteer and paid firefighters to explore the
productive relationship between paid and unpaid employees. They conclude that
volunteer and paid fire fighters are imperfect substitutes. The Duncombe and
Brundney (1995) results do not support the public goods model.3 However, what is
true for volunteer firefighters may not be true for volunteers in general. Volunteer
3
Actually, this result is only inconsistent with the public goods model if we observe volunteer
firefighters also donating money to their fire departments.
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B. Duncan / Journal of Public Economics 72 (1999) 213 – 242
firefighters may represent a type of volunteering that is not addressed by the public
goods or private consumption model. Menchik and Weisbrod (1987) develop a
model in which workers use volunteer labor as an investment mechanism. In the
investment model, volunteering does not support a public good or directly enter a
contributor’s utility function. Instead, individuals volunteer to gain labor market
experience, or to signal their ability to prospective employers.
3.3. The charity wage and the opportunity cost of time
Steinberg (1990) argues that the opportunity cost of a volunteer’s time does not
necessarily equal the value of the volunteer’s production to the charity firm. This
inequality can exist for two reasons. First, a person might volunteer to perform a
service for a charity that is less valuable than the service he or she provides to the
general labor market. An example of this is a doctor who volunteers to work in a
soup kitchen. Second, the charity firm may be able to pay less compensation to
their workers than other firms pay for similar services. An example of this is a
doctor who is willing to accept less compensation from a non-profit hospital than
from a for-profit hospital. However, both of these explanations are motivated, not
by the public goods model, but by the private consumption model. In the private
consumption model, contributors value something other than the production of
charity. Namely, contributors value their gifts themselves. Consequently, the wage
the charity firm would have to pay someone else to perform the services provided
by a volunteer is not necessarily equal to the volunteer’s opportunity cost of time.
In the public goods model, a contributor will not volunteer to a charity that
values the service he or she provide less than his or her opportunity cost of time.
This is not an assumption of the public goods model, but rather a result. The
doctor in a soup kitchen example, in a public goods model, will not happen. The
reason is that the doctor gets more ‘‘bang for the buck’’ by giving money. If a
contributor’s opportunity cost of time is greater than the value of his or her
marginal product to the charity firm, then the contributor can consume more
charity, and thus more utility, if he or she gives money instead of time.4
The second example of the doctor accepting less compensation from a nonprofit hospital than from a for-profit hospital also will not happen in the public
goods model. It has sometimes been observed, however, that non-profit workers
earn lower wages than similar for-profit workers do (see Goddeeris (1988);
4
This implicitly assumes that a person’s opportunity cost of time is equal to his or her opportunity
wage. As Brown and Lankford (1992) point out, the observed market wage is not always a good
measure of the opportunity wage, particularly if hours of work are constrained. For this reason, the
variable w in Eq. (3.1) should be thought of as the individual’s opportunity wage, which may or may
not equal the individual’s observed market wage.
B. Duncan / Journal of Public Economics 72 (1999) 213 – 242
221
Preston (1989)).5 While this evidence, on its face, supports the private goods
model, it is unpersuasive. As Preston (1989) points out, a wage differential does
not imply that that non-profit workers are less compensated than for-profit
workers. A compelling case for a compensating differential, even beyond the usual
claim of non-pecuniary compensation, can be made for non-profit employees.
An individual’s decision of where to work, for a charity firm or for a for-profit
firm, does not affect the supply of charity. Thus, the public goods model implies
that the charity firm does not have any advantage or disadvantage over for-profit
firms in the labor market. If an individual’s market wage represents his or her
opportunity cost of time, then it also represents the wage the charity firm must pay
its employees. However, individuals value the output produced by charity firm and
so, to increase this output, they donate their time and money. Being employed by
the charity firm does not increase the supply of charity; however, charity firm
employees have an additional way to contribute: They can agree to work for lower
wages. Consider an example of a charity firm employee who works 40 hours a
week and earns his or her market wage of $10 an hour. Suppose that the employee
wishes to give a $10 per week gift to charity. Proposition (2) implies that the
employee is indifferent between any specific allocation of time or money so long
as the total value of his or her gift is $10. Thus, the employee can contribute many
ways: (a) He or she could work 40 hours a week for the charity, earn $400, and
give back $10; (b) he or she could work 39 hours a week for the charity, earn
$390, and volunteer an hour; or (c) he or she could work 40 hours a week for the
charity at $9.75 an hour, earn $390, and give no additional time or money. Both
the charity firm and the employee are indifferent between scenarios (a), (b), and
(c). In each scenario, the time the employee spends at the charity, the total supply
of charity, and the employee’s net income are the same.6 In scenario (c), however,
an observer may incorrectly conclude that non-profit firms compensate workers
less than for-profit firms do. In fact, non-profit workers are simply using lower
wages as part of their gift to charity. In addition, there is a compelling reason to
believe that we would observe scenario (c) over scenarios (a) and (b). The U.S.
Fair Labor Standards Act prohibits non-profit workers, as well as for-profit
workers, from donating a substantial amount of time or money to their employers
(see Smith and Steinberg (1990)). These regulations are aimed at employers who
would attempt to avoid minimum wage regulation by pressuring workers into
contributing time and money to the firm.
5
Evidence of a negative non-profit wage differential is far from conclusive. Goddeeris (1988);
Preston (1989) find evidence that non-profit workers earn less than for-profit workers do. However,
Preston (1988); Leete (1998); Borjas et al. (1983) find evidence that non-profit workers earn more than
for-profit workers do. Results in this literature are highly dependent on the occupation and industry of
the workers.
6
In this case, money donations are tax deductible and the individual itemizes. If the individual does
not itemize, then it is better to volunteer labor or work for a lower wage.
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B. Duncan / Journal of Public Economics 72 (1999) 213 – 242
3.4. Whether to volunteer or donate
How do contributors decide between volunteering time and donating money?
Proposition (2) suggests that the decision is arbitrary up to a point. A contributor
will not volunteer if by doing so he or she causes the charity firm to become
capital constrained. If the charity firm is not capital constrained, then contributors
are indifferent between equal-valued contributions of time and money. How does a
contributor know if the charity firm is capital constrained? One way a charity firm
can reveal its capital constraint is by turning away volunteers and urging them to
give money instead. This sends a very clear signal to the potential volunteer to
give money instead of time. The charity firm can also send a signal to contributors
by deciding whether to ask potential contributors to volunteer labor or to donate
money. Freeman (1997) explores the importance of being asked. He shows that
many contributors only volunteer when requested to do so. Freeman argues that
this evidence suggests that volunteering is a ‘‘conscience good or activity,’’ which
is correct when volunteering is viewed through the private consumption model.
When viewed through the public goods model the evidence has another interpretation. Being asked to volunteer signals that the charity firm is not capital
constrained. If asked, a contributor is assured that the charity firm is not capital
constrained and can always offset volunteering more by donating less.
3.5. The role of fundraising
The public goods model does not provide an explicit role for fundraising
activities. Fundraising, however, can play a role as an information mechanism.
Fundraisers might tell contributors if the charity firm is capital constrained. A
fundraiser might also reveal the charity firm’s production function to potential
contributors in an attempt to increase charitable contributions. However, the public
goods model suggests that the effectiveness of fundraisers can be deceiving. For a
fundraiser, the easiest way to increase donations of money is to ask those who
already volunteer labor. Conversely, the easiest way to increase volunteer labor is
to ask those who already donate money. This fundraiser might appear very
successful, but if we assume that the current contributors are already familiar with
the charity firm’s production function, then this success is deceptive. According to
Proposition (2), volunteers are always willing to increase their donations of
money, but if they do, they will offset this increase by volunteering less. Similarly,
money contributors are always willing to increase their volunteer labor, but if they
do, they will offset this increase by donating less money. Thus, a fundraiser that
appears very successful might not be.
4. Mixing in private consumption
The defining assumption of the public goods model is that an individual’s gift
B. Duncan / Journal of Public Economics 72 (1999) 213 – 242
223
provides them with utility only if that gift supports the public good. Propositions
(1) and (2), as well as the results that follow, were derived under this public goods
assumption. However, relaxing this assumption by allowing contributors to receive
additional private utility from the act of giving itself will not necessarily invalidate
Propositions (1) and (2).
The private consumption model assumes that a contributor’s gift provides them
with utility directly, regardless of what the gift supports. In the private consumption model, the gifts themselves are private goods consumed by the giver. A
defining question of the private consumption model is therefore, ‘‘what are the
private goods?’’ How researchers answer this question determines both their
empirical specification and the interpretation of their results. One definition views
charitable gifts of time and money as two separate goods. This is the approach
currently used by the volunteer labor supply literature. Menchik and Weisbrod
(1987) develop a private consumption model in which individuals care only about
their personal gifts of time and money. The model implies that contributors are just
as happy if their gifts produce nothing, because they are motivated only by the act
of giving itself. Defining charitable gifts of time and money as two separate goods
offers no testable implications of the relationship between these gifts. If this type
of private consumption, or ‘warm-glow’ utility, is added to the public goods
model, then Propositions (1) and (2) no longer hold. Contributors may not view
gifts of time and money as perfectly substitutable and the charity firm can be left
capital constrained.
Another definition of the private consumption model views the private good as
the value of a person’s charitable contribution. This is the implicit assumption
made in the impure altruist model in which contributors directly give units of the
public good. A contributor derives personal satisfaction from the value of his or
her gift, which is equal to: gi 5d i 1wvi . This definition of the private consumption
model implies that gifts of time and money are perfect substitutes. Combining this
type of ‘warm-glow’ utility with Eq. (3.1), the consumer’s utility maximization
problem becomes:
MAX
Ui [x i , l i , Z u (G), G 2 G2i ]
s.t.
wt 1 y i 1 G2i 5 x i 1 wl i 1 G
G $ G2i ,
hx i ,l i ,G j
(6)
where G2i 5G2gi .
The Nash assumption in Eq. (6) implies that consumers view the total value of
charitable gifts given by everyone else as fixed. Thus, individual i effectively
decides the total supply of charity. The left-hand side of the first constraint in Eq.
(6) represents individual i’s social income. Because gifts of time and money are
perfect substitutes in the private consumption aspect of the consumer’s utility
function, adding this type of ‘warm-glow’ utility to the public goods model does
not affect Propositions (1) or (2). In equilibrium, gifts of time and money remain
perfectly substitutable and the charity firm is never left capital constrained.
B. Duncan / Journal of Public Economics 72 (1999) 213 – 242
224
5. Government policy
In a large part of the theoretical literature, government spending on public goods
is financed with lump-sum taxes. In empirical studies, government spending is
financed with income taxes. This difference, as it turns out, does not present a
problem because both tax mechanisms produce similar results (see Bernheim
(1986); Bagwell and Bernheim (1988)). Let ti [[0, 1], represent individual i’s
income tax rate. Assuming charitable contributions are tax deductible, the
consumer’s utility maximization problem described by Eq. (6) becomes:
MAX
Ui [x i , l i , Z u (G), G 2 G2i ]
s.t.
(1 2 ti )[wt 1 y i ] 1 G2i 5 x i 1 w(1 2 ti )l i 1 (1 2 ti )G,
G $ G2i
hx i ,l i ,G j
(7)
The notation in Eq. (7) was simplified by including government spending in G
as the n11 individual. The left-hand side of Eq. (7) represents the individual’s
social income. The right-hand side of the constraint in Eq. (7) implies that the
price of charitable contributions is one minus the individual’s income tax rate.7
The current economic literature contains no estimates of the price elasticity of
total charitable contributions. Previous research estimates price elasticities for
charitable contributions of time and money separately. However, knowing how
government policy affects contributions of time and money separately is not
equivalent to knowing how that policy affects the supply of the public good. To
identify policy effects on the public good, a researcher must first estimate the value
of the volunteer labor to the charity firm.
6. Implications for empirical work
Eq. (7) motivates a researcher to estimate the total value of charitable
contributions, not gifts of time and money separately. An individual’s demand for
charity is a function of his or her social income and the price of charitable
contributions. If ‘warm-glow’ utility derived from the value of an individual’s gift
is added to the public goods model, then an individual’s full income and the
contributions of others can affect demand separately. Thus, an individual’s demand
for charity is estimated as:
7
The tax price is not necessarily the price of charity, but is rather the price of charitable
contributions. As Callen (1994); Posnett and Sandler (1989) point out, the price of charity is how much
it costs a contributor to increase charity one unit. This price depends both on the contributor’s marginal
tax rate and on the charity firm’s technology function. However, the price of charity is simply a
transformation of the price of charitable contributions, and is the same for charitable contributions of
time or money.
B. Duncan / Journal of Public Economics 72 (1999) 213 – 242
G* 5 BX i 1 d1 (1 2 ti )(w i t 1 y i ) 1 d2 G2i 1 d3 (1 2 ti ) 1 ´i ,
225
(8)
where X i represents a vector of personal characteristics and a constant, d1
represents an income effect, d2 represents a reaction effect, d3 represents a price
effect, and ´i represents a random error term. Subtracting G2i from both sides for
Eq. (8) yields an individual’s reaction function. Incorporating the fact that gifts
cannot be negative, an individual’s reaction function is estimated as:
g˜ *i 5 BX i 1 d1 (1 2 ti )(wt 1 y i ) 1 a G2i 1 d3 (1 2 ti ) 1 ´i ,
g i* 5
H
0
g˜ *i
if g˜ i* # 0
if g˜ *i . 0
(9)
where a 5d2 21, and g˜ *i represents an individual’s index reaction function. The
empirical section uses a Tobit model to consistently estimate the parameters of the
reaction function described by Eq. (9).8
6.1. Crowding-out
Public goods theory predicts that government spending crowds-out voluntary
contributions. Two caveats to this prediction are that the taxes collected from each
individual must not exceed his or her original charitable contribution, and the tax
revenue must finance the public good. This prediction has stimulated a flurry of
empirical work most of which finds little support for the crowd-out hypothesis. For
example, using tax return data, Abrams and Schitz (1978) estimate a 28%
crowd-out effect. Using the National Survey of Philanthropy, Schiff (1985)
estimate a 60% crowd-out to 30% crowd-in effect, depending on the type of
government expenditures. Using public radio data, Kingma (1989) estimate a 14%
crowd-out effect. Finally, using data from the United Kingdom, Steinberg (1985)
estimate 0.5% crowd-out while Khanna et al. (1995) find no evidence of crowdout. For a comprehensive survey of empirical work on crowding-out, see Steinberg
(1991).
Interpreted through the private consumption, the empirical findings suggest that
government spending and gifts to charity are imperfect substitutes. The private
consumption model assumes that gifts of time and money are privately consumed
goods. The model makes no assumptions about government spending; it can be a
private good or a public good. In either case, government spending is a distinct
good from charitable contributions. Increasing government spending can affect
charitable contributions through income and substitution effects. The income effect
comes from the fact that increasing government spending changes the individual’s
endowment, both because it increases the supply of charity and because it
increases taxes. The substitution effect comes from the fact that government
8
See Maddala (1983).
226
B. Duncan / Journal of Public Economics 72 (1999) 213 – 242
spending and charitable gifts can be substitutes or complements. The fact that
empirical research finds a small positive crowd-out effect is not surprising when
viewed through the private consumption model, because the model make no
predictions at all.
The public goods model developed in this article offers another interpretation
for the empirical findings. The model suggests that government spending can
perfectly crowd-out total contributions, not money contributions. Steinberg (1991)
suggests that estimates of crowding-out should include contributions of both time
and money. The public goods model developed in this article shows that dollarfor-dollar crowd-out holds only if money is the only way to contribute to the
public good. Contributors also volunteer labor. By omitting volunteer labor
previous empirical research only estimates one part of a person’s total contribution. This omission has an ambiguous affect on crowd-out estimates. Proposition
(2) implies that the specific allocation of time and money is arbitrary but, as an
example, consider a few possible allocation rules. Suppose that contributors tend
to give half time and half money. Then, under perfect crowd-out, a one-dollar
increase in government spending would reduce money contributions by fifty cents.
On the other hand, suppose contributors tend to give a fixed amount of labor,
satisfying the remainder of their demand for charity with contributions of money.
Then, under perfect crowd-out, a one-dollar increase in government spending
would reduce money contributions by nearly one dollar. Thus, the theoretical
affect that omitting volunteer labor has on crowd-out estimates is ambiguous.
An important observation of Bergstrom et al. (1986) is that perfect crowd-out
only occurs if every individual is taxed less than or equal to his or her original
charitable contribution. A test for perfect crowd-out is therefore not equivalent to a
test of the crowd-out hypothesis. An appropriate test of the crowd-out hypothesis
is derived from Eq. (8). In a pure public goods model, government spending,
contributions of others and an individual’s full income all have the same effect on
an individual’s demand for the public good. If ‘warm-glow’ utility is added, then
government spending and full income can affect demand differently. Thus, from
Eq. (8), Ho : d1 5d2 is a test of the crowd-out hypothesis. Empirically, researchers
estimate an individual’s reaction function expressed by Eq. (9), not an individual’s
demand function expressed as Eq. (8). A test of the crowd-out hypothesis based on
Eq. (9) is:
Ho : d1 5 a 1 1.
(10)
Failure to reject the null-hypothesis suggests that government spending and full
income have the same affect on an individual’s demand for the public good.
Rejecting suggests that consumers receive ‘warm-glow’ utility from giving.
However, rejecting the null-hypothesis does not invalidate Propositions (1) and
(2). If contributors derive ‘warm-glow’ utility from the value of their charitable
gifts, then the crowd-out hypothesis will fail, but contributors still view contribu-
B. Duncan / Journal of Public Economics 72 (1999) 213 – 242
227
tions of time and money as perfectly substitutable in equilibrium. Propositions (1)
and (2) are only invalidated if contributors receive ‘warm-glow’ utility separately
from their gifts of time and money.
6.2. The price elasticity of volunteering
Motivated by the private consumption model, the volunteer labor supply
literature uses an individual’s net wage to measure of the price of volunteering.
Menchik and Weisbrod (1987) estimate a price elasticity of demand of 20.40.
Similarly, Schiff (1990) estimates a price elasticity of demand of 20.11, but not
statistically significant.
Eq. (7) offers the public goods interpretation of an individual’s net wage. The
net wage represents an individual’s opportunity cost of time; but time is not what
contributors are buying. Contributors buy charity. Exactly how much it costs to
increase charity one unit depends on the charity firm’s technology function.
However, if a one-unit increase in charity costs one dollar, then it also costs one
dollar’s worth of time. An individual’s net wage plays no role in his or her
allocation of time and money contributions. Eq. (7) does not imply that the
individual’s wage has no effect at all. Changing an individual’s wage changes his
or her social income, which will affect his or her demand for charity. However, a
consumer who views gifts of time and money as perfectly substitutable can
accommodate a change in his or her demand for charity with a change in volunteer
labor or with a change in money donations.
6.3. The cross-price elasticity
In the private consumption model, the price of donating one dollar to charity is
one minus the individual’s marginal tax rate. Empirical research finds a negative
cross-price elasticity between the price of donating money and volunteering time.9
Menchik and Weisbrod (1987) estimate a cross-price elasticity of 21.25. Brown
and Lankford (1992) estimate a cross-price elasticity of 21.27 for men and 21.62
for women. Finally, Schiff (1990) estimates a cross-price elasticity of 20.36. The
literature has concluded that gifts of time and money are gross complements.
The public goods model developed in this article offers another interpretation:
the demand for charity is downward sloping. Eq. (7) implies that one minus the
marginal tax rate is the price of total charitable contributions. If the price
increases, then the quantity demanded will decrease. Consumers accommodate this
decrease by reducing volunteer labor, money donations, or both. Therefore, the
9
A notable exception to this trend is Wolff et al. (1993) who estimate a positive cross-price elasticity
for volunteer hospital workers.
228
B. Duncan / Journal of Public Economics 72 (1999) 213 – 242
fact that charitable contributions of time and money respond to changes in the tax
price by moving in the same direction does not mean that they are complements.
In fact, the public goods model suggests just the opposite. A negative cross-price
elasticity in the public goods framework is not surprising because it is not a
cross-price elasticity at all, it is one piece of an own-price elasticity.
7. Empirical results
The data used in this section comes from the National Study of Philanthropy
(1974). This study uses the National Study of Philanthropy with the understanding
that it is not the ideal data set with which to estimate the implications of the public
goods model. Wages, for example, are not directly observed and therefore must be
estimated. In addition, charitable gifts are measured only in aggregate. Kingma
(1989) points out that when estimating crowding-out, it is best to use data on an
individual’s gift to specific charities rather than data on an individual’s aggregate
giving. However, the National Study of Philanthropy is one of the few sources of
data that includes detailed information on charitable contributions of both time and
money. In addition, the data is national, which makes it possible to measure the
effect of government policy across different States and localities. Both Schiff
(1990); Menchik and Weisbrod (1987) use the National Study of Philanthropy to
estimate price and income elasticities for volunteer labor. Schiff (1985) also uses
the study to estimate crowd-out effects for donations of money. The sample
includes 2917 responses, of which 1892 were interviewed by the Survey Research
Center (SRC) and 1025 were interviewed by the U.S. Census Bureau. The
questionnaires used by the two agencies are virtually identical. However, due to
Census Bureau policy, the household’s location is not identified in the Census
sample. For this reason, the Census sample cannot be used to identify the effect of
government policy.
Important data, such as income and money donations, are only reported at the
household level. This makes it impossible to identify charitable contributions for
married individuals. It also makes it impossible to identify wages for individuals
living in dual income households. A researcher can work around these limitations
by restricting the sample to single earner households, and attributing all money
donations to the head of the household.10 This approach, however, can potentially
create a selection bias, as the observations dropped are probably correlated with
giving. Another approach is to measure charitable gifts at the household level.
Regardless of the data’s limitations, measuring charitable contributions at the
household level is arguably more appropriate. For married couples, giving is likely
to be a joint decision, particularly in the public goods model. For this reason, in
addition to the realities of the data at hand, this study measures all variables at the
household level.
10
This approach is taken by Schiff (1985); Menchik and Weisbrod (1987).
B. Duncan / Journal of Public Economics 72 (1999) 213 – 242
229
7.1. Household contributions to charity
The dependent variable used in the regressions is the total value of a
household’s charitable contribution (value5gi ). Consistent with the above public
goods model, value is calculated as the household’s wage multiplied by the
number of hours volunteered plus contributions of money. Although they are not
considered correctly specified in the above public goods model, the household’s
contribution of money (money5d i )11 and the value of the household’s volunteer
labor (time5w i vi ) are also used as dependent variables. In the sample, 88% of
households gave money, 45% gave time, while 90% gave either time or money.
Table 1 reports household averages for value, money and time. The averages are
calculated for the entire sample in column (1) and for givers only in column (2).
The summary statistics and following regressions use sample weights that reflect
the fact that the survey over-sampled older and upper-income households.12
Table 1
Summary statistics (standard deviation in parentheses)
Dependent variables:
Value
Money
Time
Independent variables:
Price
Full income
Other gifts
Local govt. expenditures
Number of observations:
a
Full sample
(1)
Givers only a
(2)
848.61
(1570.41)
327.69
(515.45)
520.92
(1276.59)
945.64
(1629.88)
365.16
(531.40)
580.49
(1334.74)
0.83
(0.16)
14 751.62
(5756.73)
820.82
(270.27)
680.17
(261.44)
0.82
(0.16)
15 178.70
(5669.65)
821.12
(279.02)
676.24
(250.43)
1428
1309
Observations with value greater than zero.
11
The dependent variable money includes money and property gifts given to charity.
Including sample weights does not significantly affect the regression estimates, as the regressions
already control for age and income.
12
230
B. Duncan / Journal of Public Economics 72 (1999) 213 – 242
7.2. Determinants of household gifts
Eq. (9) implies that charitable contributions are a function of price and social
income. All of the regressions also control for demographic characteristics. Table 2
briefly describes all of the variables used in estimation. The tax price of gifts
( price) is measured as one minus the household’s marginal tax rate for those who
itemize, and as one for those who do not. This assumes that the decision to itemize
Table 2
Variable description
Dependent variables:
Value
The total value of the household’s charitable contributions ( gi )
Money
The household’s charitable contribution of money (d i )
Time
The value of household’s charitable contributions of time (w i vi )
Independent variables:
Price
One minus the household’s marginal tax rate.
fullinc
The household’s full income.
othgifts
Per household value of gifts of others, by state.
locexp
Per household local government direct general expenditures.
kid0 2
Youngest child is between 0 and 2 years old.
]
kid3 5
Youngest child is between 3 and 5 years old.
]
kid6 10
Youngest child is between 6 and 10 years old.
]
kid11 17
Youngest child is between 11 and 17 years old.
]
support
Equal to one if the household supports a child not living at home.
married
Married household.
white
Household head is white.
age
Age of household head.
age 2
Age of household head squared.
wifeage
Age of wife (0 if single).
wifeage 2
Age of wife squared (0 if single).
edu 0 6
Household head has between 0 and 6 years schooling.
] ]
edu 7 10
Household head has between 7 and 10 years schooling.
] ]
edu aa
Household head earned an AA degree.
]
edu babs
Household head earned a BA or BS degree.
]
edu maph
Household head earned an MA or Ph.D. degree.
]
edu miss
Household head education is unknown.
]
wedu0 6
Wife has between 0 and 6 years schooling.
]
wedu7 10
Wife has between 7 and 10 years schooling.
]
weduaa
Wife earned an AA degree.
wedubabs
Wife earned a BA or BS degree.
wedumaph
Wife earned an MA or Ph.D. degree.
wedumiss
Wife education is unknown.
bigcity
Household is located in a large city.
suburbs
Household is located in the suburbs.
smalcity
Household is located in a small city.
farm
Household is a farm.
areamiss
Household location is unknown.
B. Duncan / Journal of Public Economics 72 (1999) 213 – 242
231
is exogenous to charitable contributions. A household’s marginal tax rate (ti ) is
measured as the sum of its State and Federal marginal tax rates, corrected for the
fact that State income tax is deductible from Federal income tax. Social income is
broken down into its components: full income, gifts of others, and government
spending.
Full income ( fullinc) is measured as (12ti )(wˆ i t j 1y i ), where wˆ i represents the
household’s estimated wage, t j represents the household’s available time (2000
hours for single households, 4000 hours for married households), and y i represents
the household’s non-wage income. The data does not report the household’s wage,
so wages are estimated as:
Taxable Income 2 Asset Income
w 5 ]]]]]]]]].
Hours Worked
(11)
Households report three types of assets: stocks and mutual funds, fixed investment
instruments, and investments in real estate. The calculation of Asset income uses
the 1974 average rate of return for each type of asset.13
The household’s wage is an exogenous parameter, however, the estimate of the
household’s wage includes hours worked, an endogenous choice variable. As
pointed out in Section 3.3, if hours of work are constrained, then an individual’s
observed market wage may not equal his or her opportunity wage. In addition,
18% of households report zero hours worked. Simply dropping these observations
would create a selection bias, as the households dropped are likely correlated with
giving. To address both of these problems, a Heckman selection model is used to
estimate wages. The wage estimates do not rely on data omitted from the Census
sample and so, to increase efficiency, the Heckman model is estimated over the
entire sample (i.e., the Census sample and the SRC sample). The Heckman wage
estimates are assigned to the entire SRC sample that is later used to estimate the
giving equations. Table 3 reports the results of the Heckman selection model. Due
to the small number of single female workers in the sample, it is impossible to
estimate female wages and their respective gifts to charity reliably. For this reason,
it is necessary to limit the sample to single men and married couples.
Gifts of others (othgifts) is measured as the average, per household charitable
contribution, by state. Othgifts is constructed from the data by calculating the State
average household contribution, excluding the current household. Government
spending (locexp) is measured as the per household local government direct
general expenditures, excluding capital outlays, in 1967.14 To convert government
expenditures from per capita to per household, an average household size of three
13
An annual rate of return of 3.06% is used for stocks and mutual funds, 6.92% for fixed investment
instruments, and 3.06% for real estate. See Schiff (1985).
14
Local government expenditure data comes from item 109 of the U.S. Bureau of the Census, County
and City Data Book, 1972 (A Statistical Abstract Supplement), 1973, U.S. Government Printing Office,
Washington, DC.
232
B. Duncan / Journal of Public Economics 72 (1999) 213 – 242
Table 3
Heckman wage regression results (standard errors in parentheses), N52291, x 2 (52)5824.75
Probit
(1)
kid0 2
]
kid3 5
]
kid6 10
]
kid11 17
]
support
married
white
age
age 2
wifeage
wifeage 2
edu 0 6
] ]
edu 7 10
] ]
edu aa
]
edu babs
]
a
20.045
(0.160)
0.236
(0.194)
20.199
(0.144)
0.248 a
(0.135)
0.249 a
(0.103)
1.687 a
(0.566)
0.367 a
(0.111)
0.174 a
(0.023)
20.002 a
(0.000)
20.033
(0.025)
0.000
(0.000)
20.409 a
(0.161)
20.012
(0.100)
20.483 a
(0.208)
20.221
(0.141)
Wage
(2)
edu maph
]
edu miss
]
wedu0 6
]
wedu7 10
]
weduaa
22.734
(2.584)
0.902 a
(0.489)
0.019
(0.120)
0.000
(0.001)
0.216 a
(0.123)
20.002 a
(0.001)
23.220 a
(0.808)
21.109 a
(0.393)
0.643
(0.868)
1.067 a
(0.522)
wedubabs
wedumaph
wedumiss
bigcity
suburbs
smalcity
farm
areamiss
constant
Probit
(1) Cont.
Wage
(2) Cont.
20.298 a
(0.169)
1.228
(2.166)
20.161
(0.213)
20.161
(0.106)
0.369
(0.257)
0.017
(0.159)
0.707
(0.459)
0.864
(3.993)
20.119
(0.135)
0.136
(0.168)
20.208 a
(0.123)
20.273 a
(0.142)
20.107
(0.254)
22.591 a
(0.515)
3.372 a
(0.668)
1.072
(5.156)
0.070
(1.061)
20.540
(0.426)
21.042
(0.792)
20.178
(0.601)
2.443 a
(1.170)
1.638
(4.115)
0.522
(0.532)
0.942 a
(0.565)
20.336
(0.467)
20.274
(0.542)
21.596
(1.093)
2.146
(2.684)
Statistically significant at the 90% confidence level.
is assumed (the national average for 1974). Table 1 reports summary statistics for
price, fullinc, othgifts and locexp.
Using local per household government expenditures to represent government
spending on the public good is not entirely consistent with Eq. (7). First, there is
no guarantee that all of this type of government spending is actually spent on
public goods, and, even if it were, it must be public goods to which households
can contribute. Second, government spending is measured per household, where
the model measures government spending in total dollars. The distinction is not an
artifact of the data, but rather reflects the nature of the public good itself. Charity
is a composite good, the consumption of which is rather vague. For example,
community size, an essential component of social income, is arbitrary. Measuring
B. Duncan / Journal of Public Economics 72 (1999) 213 – 242
233
government spending in total dollars implies that, simply calling North Dakota and
South Dakota one state, doubles every household’s social income and consumption
of charity. To avoid this problem, it is assumed that households value the density
of charity in their community. Although government spending clearly includes
money spent on private goods, it is used as the best available proxy for
government spending on public goods.
7.3. Testing the public goods model
The public goods Eq. (7) motivates a researcher to estimate the value of a
household’s charitable contributions as:
g *i 5 B 1 X i 1 a11 (1 2 ti )(w i t 1 y i ) 1 a12 (1 2 ti ) 1 a13 G2i .
(12)
Government spending and the contributions of others, both included in G2i , enter
Eq. (12) separately from full income. This accounts for ‘warm-glow’ utility that is
derived from the total value of the household’s charitable contribution. If
households derive ‘warm-glow’ utility from gifts of time and money separately,
then the private consumption model correctly specifies charitable contributions. In
the private consumption model, contributions of time and money are separate
goods. Thus, the private consumption model motivates a researcher to estimate a
household’s charitable contribution of time and money separately as:
d *i 5 B 2 X i 1 a21 (1 2 ti )(w i t 1 y i ) 1 a22 (1 2 ti ) 1 a23 G2i 1 a24 w i (1 2 ti ),
(13)
v *i 5 B3 Xi 1 a31 (1 2 ti )(w i t 1 y i ) 1 a32 (1 2 ti ) 1 a33 G2i 1 a34 w i (1 2 ti ).
(14)
The net-wage terms are added to represent the price of volunteer labor, which is
appropriate in the private consumption model. The value of a household’s total
charitable contribution is estimated by combining Eqs. (13) and (14):
d *i 1 w i v *i 5 B 2 X i 1 B 3 w i X i 1 a21 (1 2 ti )(w i t 1 y i )
1 a31 w i (1 2 ti )(w i t 1 y i ) 1 a22 (1 2 ti ) 1 (a24 1 a32 )w i (1 2 ti )
1 a34 w i2 (1 2 t i ) 1 a23 G2i 1 a33 w i G2i .
(15)
The public goods giving equation, represented by Eq. (12), is nested in Eq. (15).
In the public goods model, the household’s wage is not a determinant of its
charitable contribution, and thus, wages only enter Eq. (12) as part of full income.
In the private consumption model, the household’s wage is a determinant of its
charitable contribution, and thus, wages enter Eq. (15) separate from full income.
A test of private consumption model based on Eq. (15) is therefore:
Ho : B 3 5 a31 5 (a24 1 a32 ) 5 a33 5 a34 5 0.
(16)
234
B. Duncan / Journal of Public Economics 72 (1999) 213 – 242
Rejecting the null-hypothesis suggests that households receive ‘warm-glow’ utility
derived from gifts of time and money separately. The null-hypothesis is empirically tested by estimating Eq. (15) twice: once as is, and a second time imposing the
restrictions from Eq. (16). This likelihood-ratio test produces a chi-squared
statistic of 41.31 with 31 degrees of freedom, which means that the test fails to
reject the null-hypothesis at the 90% confidence level.
The likelihood-ratio test fails to find evidence that households receive ‘warmglow’ utility derived from charitable contributions of time and money separately.
However, this result should be viewed with some caution. First, the test assumes
that Eq. (15) is correctly specified. More importantly, Eq. (16) poses the public
goods hypothesis as the null-hypothesis, rather than as the alternative hypothesis.
Thus, while the test fails to find evidence of ‘warm-glow’ utility, it does not reject
the presence of ‘warm-glow’ utility. In fact, it will never be possible to reject the
private consumption model because the model makes no predictions to be rejected.
Even if a researcher uncovers conclusive proof that contributions of time and
money are perfect substitutes, this would not provide evidence against the private
consumption model. Thus, the strongest statement that can be made in light of the
likelihood-ratio test is that the data is consistent with the public goods prediction
that households view charitable contributions of time and money as perfectly
substitutable in equilibrium.
7.4. Results of the public goods model
Table 4 reports the results of the Tobit household giving models. The
dependent variables in columns (1), (2) and (3) of Table 4 are value, money and
time. The money and time equations are the same as private consumption Eqs. (13)
and (14) except that the net wage effects (a24 and a34 ) are set to zero. These
restrictions reflect the fact that the money and time equations are public goods
giving equations motivated by Eq. (9). They are reported in Table 4 so they can be
compared to the value equation. However, the money and time equations are
deliberately mis-specified, because they are public goods giving equations but
estimate contributions of time and money separately. The public goods model
suggests that contributions of time and money should be estimated together.
Therefore, the money and time equations have little intrinsic value. No a priori
predictions can be made between the results of the correctly specified value
equation and the incorrectly specified money and time equations. The purpose of
estimating these two equations is to determine whether or not omitting contributions of money or time significantly effects the estimation results. Specifically, is a
researcher significantly misled if he or she draws conclusions based solely on
contributions of time or money?
Schiff (1985) also uses the National Study of Philanthropy to estimate
charitable contributions of money. His money equation, however, is different from
the money equation estimated here in two ways. First, Schiff limits the sample to
Table 4
Tobit regression results (standard errors in parentheses), N51428
price
othgifts
locexp
married
age
age 2
wifeage
wifeage
2
edu 0 6
] ]
edu 7 10
] ]
edu aa
]
edu babs
]
edu maph
]
wedu0 6
]
Money
(2)
Time
(3)
22414.454 a
(300.950)
0.002
(0.017)
20.098
(0.155)
20.381 a
(0.174)
21521.937 a
(656.753)
218.190
(26.254)
0.329
(0.266)
99.086 a
(32.048)
21.042 a
(0.343)
2254.125 a
(257.664)
2210.240 a
(120.505)
383.001 a
(229.290)
419.822 a
(157.050)
1087.930
(236.270)
276.010 a
(248.798)
21211.517 a
(98.572)
20.006
(0.006)
0.043
(0.051)
20.234 a
(0.057)
2399.146 a
(215.895)
20.446
(8.597)
0.061
(0.087)
27.547 a
(10.493)
20.271 a
(0.112)
2128.573
(84.226)
257.421
(39.412)
177.561 a
(75.371)
99.387 a
(51.478)
182.915 a
(77.450)
21.449
(81.062)
22792.589 a
(458.761)
0.014
(0.025)
20.118
(0.234)
20.677 a
(0.273)
22587.744 a
(1082.101)
244.890
(43.551)
0.546
(0.440)
160.775 a
(51.928)
21.665 a
(0.555)
2627.781
(402.840)
2413.054 a
(184.728)
446.818
(328.423)
587.221 a
(222.637)
1202.789 a
(335.767)
59.290
(402.308)
wedu7 10
]
weduaa
wedubabs
wedumaph
wedumiss
kid0 2
]
kid3 5
]
kid6 10
]
kid11 17
]
bigcity
suburbs
smalcity
farm
areamiss
constant
x 2 (29)
a
Money
(2) Cont.
Time
(3) Cont.
2199.960
(117.239)
356.719
(224.711)
519.352 a
(177.698)
678.718 a
(375.711)
2866.306
(1132.247)
173.486
(156.309)
180.505
(157.876)
306.679 a
(151.698)
158.250
(134.832)
48.309
(147.777)
37.311
(164.793)
2137.633
(127.102)
2220.124
(148.966)
2426.418
(292.347)
2589.755 a
(726.693)
358.56
258.288
(38.391)
49.431
(73.447)
167.130 a
(58.341)
246.423 a
(123.040)
2242.235
(369.652)
45.146
(51.354)
7.293
(51.816)
57.741
(49.784)
46.634
(44.040)
23.494
(48.307)
29.731
(53.838)
269.901 a
(41.656)
286.404 a
(48.673)
2149.240
(95.415)
1213.275 a
(238.583)
419.10
2164.987
(180.013)
452.614
(314.847)
631.955 a
(244.875)
626.058
(511.838)
24430.828
(5093.067)
343.129
(239.917)
482.505 a
(234.275)
541.365 a
(222.402)
101.220
(202.568)
244.020
(228.445)
216.585
(245.463)
21.937
(191.735)
2157.968
(225.202)
91.033
(420.766)
1984.409 a
(1167.466)
296.13
235
Statistically significant at the 90% confidence level.
Value
(1) Cont.
B. Duncan / Journal of Public Economics 72 (1999) 213 – 242
fullinc
Value
(1)
236
B. Duncan / Journal of Public Economics 72 (1999) 213 – 242
single earner households. Second, Schiff, citing Menchik and Weisbrod’s (Menchik and Weisbrod, 1981) model, includes an individual’s net wage as an
independent variable to measure the price of volunteering. However, the Menchik
and Weisbrod’s (Menchik and Weisbrod, 1981) model is a private consumption
model. Crowding-out is a public goods hypothesis. In the public goods model, the
household’s wage is not a determinant of charitable contributions and does not
belong on the right-hand side of a giving equation.
7.5. Price and income effects
Table 5 reports price and income effects for the value, money and time
equations. The price elasticity of household gifts to charity is 21.6, and the
income elasticity is 0.02. Both of these elasticities are smaller in magnitude when
compared with the money equation. The price elasticity reported in column (1)
estimates the relationship between tax policy and a household’s total charitable
contribution. Previous research has estimated this elasticity, but only for charitable
contributions of time or money separately. The price elasticity that is reported in
column (1) of Table 5 implies that a 10 percent increase in the marginal tax rate
will increase the value of the average household’s contribution by 5 percent.15
Table 5
Partial price and income effects (calculated at the mean of the data)
Value
(1)
Money
(2)
Time
(3)
Marginal effects:
Price
21655.641
(2259.321)
Income
0.001
(0.015)
2868.714
(288.472)
20.004
(20.005)
21056.776
(2297.252)
0.005
(0.019)
Partial elasticity:
Price
21.629
(20.255)
Income
0.023
(0.268)
22.213
(20.225)
20.182
(20.220)
21.693
(20.476)
0.150
(0.534)
a
Standard errors were calculated using the delta method (see Greene, 1990).
15
The tax elasticity is a partial effect. It is not an estimate of crowding-out because it holds
≠gi ≠gi ≠pi
≠gi
government spending constant. It is derived by noting that ] 5 ] ] 5 2 ], where pi 512ti .
≠ti ≠pi ≠ti
≠pi
≠gi ti
This implies that the tax elasticity of gifts is equal to: 2 ] ].
≠pi gi
B. Duncan / Journal of Public Economics 72 (1999) 213 – 242
237
Table 6
Crowd-out effect (calculated at the mean of the data)
Marginal effect:
Gifts of others
Local government
Partial elasticity:
Gifts of others
Local government
Crowd-out
a
Value
(1)
Money
(2)
Time
(3)
20.067
(20.138)
20.261
(20.167)
0.031
(0.054)
20.168
(20.053)
20.045
(20.165)
20.256
(20.215)
20.065
(20.133)
20.209
(20.134)
0.077
(0.135)
20.348
(20.109)
20.071
(20.259)
20.334
(20.280)
20.244
20.177
20.240
Standard errors were calculated using the delta method (see Greene, 1990).
7.6. Crowd-out
Table 6 reports marginal effects and partial elasticities for household charitable
contributions with respect to gifts of others and local government spending.
Column (1) reports that a one-dollar increase in the average gifts of others will
decrease the value of charitable contributions by 7 cents, which translates into an
elasticity of 20.07. Likewise, a one-dollar increase in per household government
expenditures will decrease the value of charitable contributions by 26 cents, which
translates into an elasticity of 20.21. As expected in the public goods model, the
marginal effect of government spending and the marginal effect of gifts of others
are not statistically different from each other.
The marginal effects and elasticities reported in Tables 5 and 6 are all partial
effects. Table 6 also reports the total crowd-out parameter. The total crowd-out
parameter estimates the impact a one-dollar increase in local government expenditures will have on the average household’s charitable contribution. In the
calculation of the crowd-out parameter, lump-sum taxes fund the government
spending, and thus, government expenditures, full income, and the gifts of others
are all allowed to vary.16 The last row of Table 6 reports that a one-dollar increase
in local government spending crowds-out 24 cents of charitable contributions. The
crowd-out parameter for the money equation is 27% smaller in magnitude than the
crowd-out parameter for the value equation. On the other hand, the crowd-out
parameter for the time equation is nearly identical to the crowd-out parameter for
the value equation. This suggests that volunteer labor is more reactive to
government policy than donations of money.
16
See Appendix C for the calculation of the total crowd-out parameter.
238
B. Duncan / Journal of Public Economics 72 (1999) 213 – 242
The fact that the crowd-out parameter is not equal to one is not evidence against
the crowd-out hypothesis. Section 6.1 outlines a test of the crowd-out hypothesis.
The null-hypothesis from Eq. (10) is: H + : d1 5 a 11, where d1 represents the
coefficient on fullinc and a represents the coefficient on locexp. With an F-statistic
of 12.73, the value equation rejects the null-hypothesis at the 95% confidence
level. This test suggests that households derive ‘warm-glow’ utility from the total
value of their charitable contributions.
8. Conclusion
This article presents a public goods model that incorporates the fact that
individuals can contribute both time and money to the production of a public good.
The model suggests that, in equilibrium, gifts of time and money are perfectly
substitutable. Crowd-out estimates are therefore incomplete if they fail to account
for volunteer labor. While the estimates generated by omitting volunteer labor and
including only money donations are the same sign as those generated by including
both time and money, they are considerably different in magnitude. Failing to
account for volunteer labor reduces the estimated crowding-out by 27%. The
omission of volunteer labor also increased the estimated price and income
elasticities. The same is not true when omitting contributions of money. The
estimates generated by omitting money donations are very similar to those
generated by including contributions of both time and money.
The data is consistent with the public goods prediction that contributors view
charitable gifts of time and money as perfectly substitutable in equilibrium.
However, the data fails to support the crowd-out hypothesis. This failure suggests
that households do derive ‘warm-glow’ utility from their charitable contributions.
The failure of the crowd-out hypothesis, however, does not distinguish whether the
households derive the ‘warm-glow’ utility from the total value of their charitable
contributions, or from their gifts of time and money separately. However, the fact
that the data is consistent with the public goods prediction of perfectly substitutable gifts suggests that households derive ‘warm-glow’ utility from the total value
of their charitable gifts, and not from gifts of time and money separately.
Acknowledgements
I would like to thank Jon Sonstelie for his thoughtful advice on previous
versions of this paper. I would also like to thank an anonymous referee for helpful
comments.
B. Duncan / Journal of Public Economics 72 (1999) 213 – 242
239
Appendix A
Proof of Proposition (1)
Proposition 1. If h(v i* , d *i , x *i , l i* )j is a Nash equilibrium, then V *#h u (G*) and
the charity firm is not capital constrained.
The Lagrangian for the firm’s production problem in the restricted world is:
L 5 C(H, K) 1 l(G 2 wH 2 rK) 1 m (H 2V ).
(A1)
Using the Envelope Theorem, Eq. (A1) yields:
≠Z r
]
≠v
m
]]ir 5 w 2 ]
l
≠Z
]
≠d i
(A2)
u
On the consumer’s side, if V .h (G) then at least one individual must be
volunteering time to the charity. That individual’s first order conditions yield:
≠Z r
]
≠v
]]ir 5 w
≠Z
]
≠d i
(A3)
u
If V .h (G), m .0 and thus, Eqs. (A2) and (A3) cannot hold simultaneously.
Therefore, V must be less than or equal to h u (G) in equilibrium and, by Eq. (4) the
charity firm is not capital constrained. Q.E.D.
Appendix B
Proof of Proposition (2)
Proposition 2. If h(v i* , d *i , x *i , l i* )j is a Nash equilibrium and ( i) wv i* 1d i* 5
wv i9 1d 9i , ( ii) v 9i , d 9i $0, and ( iii) V 9#h u (G*), ;i then h(v 9i , d 9i , x *i , l *i )j is also a
Nash equilibrium.
Let h(v i* , d i* , x i* , l i* )j represent a Nash equilibrium. Let V * represent the total time
volunteered, D* represent the total money donated and G* represent the total
value of all charitable gifts. In addition, g i* 5d i* 1wv i* represents the value of
each individual’s charitable contribution. In the restricted world, the charity firm
B. Duncan / Journal of Public Economics 72 (1999) 213 – 242
240
cannot sell volunteer labor. However, if V #h u (G) then the firm is not effectively
restricted. Under these circumstances, the total supply of charity is equal in the
restricted and unrestricted worlds.
If V # h u (G), then Z r (D,V ) 5 Z u (G).
(B1)
Proposition (1) implies that V *#h u (G*), and thus, from Eq. (B1), Z r (D*,
V *)5Z u (G*). Suppose that h(v 9i , d 9i , x *i , l *i )j is another allocation in which some
individuals change their gifts of time and money. Similarly, g 9i 5d 9i 1wv 9i
represents the value of each individual’s charitable contribution, and G9 represents
the total value of all charitable gifts in the new allocation. Furthermore, let the
value of each individual’s gift in the two allocations be the same (i.e., g *i 5g i9 ;i).
Finally, let V 9#h u (G9), which, by Eq. (B1), implies that Z r (D9, V 9)5Z u (G9).
Individual i’s budget constraint in Eq. (5), ignoring the second line, is re-written
as: wt1y i 5x i 1wl i 1gi . Because g i9 5g i* ;i, the new allocation is affordable.
After including the second line of the budget constraint in Eq. (5), the new
allocation is only affordable if v 9i , d 9i $0, ;i. In addition, because the value of each
individual’s gift is unchanged (i.e., g *i 5g 9i ;i), the total value of all gifts is
unchanged (i.e., G*5G9), the supply of charity is unchanged (i.e., Z u (G9)5
Z u (G*)), and each individual’s utility is unchanged (i.e., U(x *i , l *i , Z u (G*))5U(x *i ,
l *i , Z u (G9)) ;i). Finally, because the original allocation, U(x *i , l *i , Z u (G*)), and
the new allocation, U(x *i , l *i , Z u (G9)), yield the same utility, the new allocation
must also be utility maximizing. This is true for every individual, and therefore,
the new allocation is a Nash equilibrium. Q.E.D.
Appendix C
Calculation of total crowd-out parameter
The full crowd-out effect is calculated as:
U
≠gi
≠gi
≠gi
≠G2i ±n11 ≠gi ≠y i
]]
5 ]] 1 ]]] ]]] 1 ] ]]
≠gn11 ≠gn11
≠G2i ±n11 ≠gn11
≠y i ≠gn 11
U
(C1)
≠gi
where, ]] represents the partial effect of government expenditures on
≠gn11
household gifts, G2i ±n11 represents the gifts of other households, gn11 represents
government expenditures, and y i represents non-wage income. Assuming identical
≠G2i ±n11
≠y i
≠g
individuals, ]] 5 ]]], and lump sum taxes, ]] 5 1. Thus, Eq. (C1)
≠gn11
≠gn11
≠gn11
is written as:
B. Duncan / Journal of Public Economics 72 (1999) 213 – 242
U
≠gi
≠gi
]]
1]
≠gi
≠gn11
≠y i
]]
5 ]]]]
.
≠gi
≠gn11
1 2 ]]]
≠G2i ±n11
241
(C2)
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