DEPARTMENT OF ECONOMICS
Philanthropy, multiple equilibria and
optimal public policy
Sanjit Dhami, University of Leicester, UK
Ali al-Nowaihi, University of Leicester, UK
Working Paper No. 12/08
April 2012
Philanthropy, multiple equilibria and optimal public
policy
Ali al-Nowaihiy
Sanjit Dhami
08 April 2012
Abstract
Let a large-number of small-individuals contribute to charity. We show that
‘strong aggregate complementarity’is necessary for multiple equilibria in a competitive equilibrium. Consider two equilibria with low (L) and high (H) levels of giving.
Suppose that society is stuck at L and wishes to move to H using welfare-maximizingpublic-policy. Subsidies are ine¤ective when comparative statics at L are ‘perverse’
(subsidies reduce giving). Public policy in the form of temporary direct government
grants to charity can engineer a move from L to H. We contribute to the broader
question of using public policy to engineer moves between multiple equilibria.
Keywords: Charitable giving; multiple equilibria; strong aggregate substitutes; optimal
mix of public and private giving; subsidies and direct grants; redistribution; privately
supplied public goods.
JEL Classi…cation: D6, H2, H4.
Corresponding author. Department of Economics, University of Leicester, University Road, Leicester,
LE1 7RH. Tel: +44-116-2522086. E-mail: [email protected].
y
Department of Economics, University of Leicester, University Road, Leicester. LE1 7RH, UK. Phone:
+44-116-2522898. Fax: +44-116-2522908. E-mail: [email protected].
1
1
Introduction
Charitable donation is a signi…cant economic activity. For instance in 2003 for the USA,
89 percent of all households gave to charity with the average annual gift being $1620,
which gives an aggregate total of about $100 billion; see, Mayr et al. (2009). A recent
World Giving Index published by the Charities Aid foundation used Gallup surveys of
195,000 people in 153 nations. It found that more than 70% of the population gave money
to charity in Australia, Ireland, United Kingdom, Switzerland, Netherlands, Malta, Hong
Kong, Thailand.1
We …rst highlight some relevant stylized facts, S1-S5, associated with charitable giving; see Andreoni (2006) for more details. S1: There is substantial heterogeneity in giving
between countries.2 S2: Individual private donors are the largest contributors.3 S3: Government direct grants are signi…cant in terms of magnitude.4 S4: Contributions to charity
are typically tax deductible. For instance, the rate of charitable deductions is 50% in the
US and 17-29% for Canada. S5: Direct grants in the form of seed money or leadership contributions (which precede private giving) made by governments, foundations, the national
lottery (as in the UK) or exceptionally rich individuals are e¢ cacious.5
1.1
Behavioral assumptions about small donors
In this paper we are interested only in the behavior of small donors to charity (stylized
fact S2). How do these individuals make choices on their private giving, conditional on the
choices made by the government and the charity? There are the following two possibilities.
1. Strategic behavior. Each giver chooses own giving so as to maximize own utility conditional on the vector of donations of all other givers. For tractability, many models
of charity typically restrict attention to a symmetric Nash equilibrium.6 The strategic approach assumes critically that when a small donor decides to contribute, say
1
See https://www.cafonline.org/research/publications/2010-publications/world-giving-index.aspx for
the index.
2
As a percentage of GDP, for 1995-2000, non-religious philanthropic activity was in excess of 4% for
the Netherlands and Sweden; 3-4% for Norway and Tanzania; 2-3% for France, UK, USA; and less than
0.5% for India, Brazil, and Poland; see Salamon et al. (2004).
3
For US data, for 2002, individuals accounted for 76.3 percent of the total charitable contributions.
Other givers are: foundations (11.2%), bequests (7.5%), corporations (5.1%); see Andreoni (2006).
4
For non-US data, governments are typically the single most important contributors to charities. On
average, in the developed countries, charities receive close to half their total budget directly as grants from
the government, while the average for developing countries is about 21.6 percent; See the Johns Hopkins
Comparative Nonpro…t Sector Project (http://www.jhu.edu/~cnp/).
5
See, for instance, Karlan and List (2007), Potters et al. (2007), and Rondeau and List (2008).
6
But there are also several exceptions. For instance, Bergstrom et al. (1986) consider the case of
heterogenous individuals. However, in order to obtain sharper results, even they consider the symmetric
case in their section 4.
1
£ 10, to the Red Cross, they are engaged in a strategic game in charitable contributions with respect to all other givers. While large donors may behave strategically7 ,
no evidence has been provided to the best of our knowledge that small donors who
provide the bulk of contributions (stylized fact S2), behave strategically.
2. Competitive equilibrium: In this view, which we subscribe to in this paper, there
is a large number of contributors who are individually small. It would then seem
that an analysis based on competitive equilibrium in giving is more compelling. In a
competitive markets view, the giving of an individual is su¢ ciently small compared
to total giving so that each giver takes total giving as exogenous. One way to …rm
up this idea is to assume a continuum of consumers, each of measure zero. This is
the approach we take.
1.2
Equilibrium: Unique or Multiple?
The literature has typically focussed on a unique strategic equilibrium in giving, thereby
ruling out multiple equilibria by assumption. In contrast, multiple equilibria are endemic
in many important economic phenomena. This would seem particularly to be the case
for economics of charity. It is quite conceivable that the uncoordinated giving of a large
number of dispersed small donors gives rise to multiple equilibria, depending on the beliefs
held by the donors.
Suppose that the marginal utility of contributing an extra unit is increasing in the level
of aggregate giving. Now, if contributors believe that aggregate giving will be high (low),
then they are also induced to contribute a large (small) amount. Ex post, aggregate giving
would be high (low), con…rming the givers’expectations. Hence, there could be several,
self-ful…lling, equilibria. In some equilibria, giving is high (H), while in other equilibria,
giving is low (L).
We allow for multiple competitive equilibria in this paper. In Cooper and John (1988)
where all goods are private, strategic complementarity is a necessary condition for multiple
equilibria. On the other hand, when some goods have the nature of a public good, we show
that strong aggregate complementarity, a new concept that is a modi…cation of strategic
complementarity, is a necessary condition for multiple equilibria.
1.3
Some implications of multiple equilibria
If multiple equilibria exist then one could rank them by the amount of aggregate giving or
according to some social criterion. There are two immediate implications.
7
Large donors, on the other hand, may behave in a strategic manner (say, to establish green credentials)
but their share in the total contributions is small (5.1% for the US).
2
1. Multiple equilibria can potentially explain heterogeneity in giving. Among similar
societies, some can achieve the high equilibrium, H, while others can be stuck at the
low equilibrium, L.
2. At some equilibria one might obtain perverse comparative static results (i.e., an
increase in subsidy reduces contributions). There should be no presumption that
this equilibrium is unstable in any sense (see the discussion in section 7 below). At
other equilibria, the comparative static results could be normal, in the sense that
contributions respond positively to incentives. Policy makers could, understandably,
be interested in engineering a move from a low equilibrium with perverse comparative
statics (LP) to a high equilibrium with normal comparative statics (HN), if such
equilibria exist.
Suppose that we have two, and only two, non-zero equilibria, LP and HN, but that the
economy is stuck at LP. Consider a policy maker who desires to engineer a move from LP
to HN. How should the policy maker proceed? This question is of fundamental importance
to economics and has lacked satisfactory answers. We provide a concrete answer in the
context of the economics of charity. Clearly incentives for charitable giving, in the form of
higher subsidies, will not work at the equilibrium LP because of the perverse comparative
static e¤ects. Suppose, instead, that the government gives a temporary direct grant to the
charity …nanced by an income tax.8 Let this direct grant exceed the level of aggregate
contributions at LP. This makes the equilibrium at LP unfeasible (as we shall show in
greater detail below), leaving HN as the only feasible equilibrium.
Once the economy arrives at the equilibrium HN (where the comparative statics are
normal), the government can withdraw the temporary direct grant and successfully stimulate private giving through greater subsidies. This is welfare improving because, unlike
direct government grants, increased private giving confers warm glow on the contributors. The applicability of these ideas is illustrated by examples where voluntary giving
contributes towards public redistribution or towards public goods (Section 6).9
8
Empirical evidence shows that crowding-out of private contributions by direct government grants, if
any, is quite small. See Andreoni (2006) for a discussion of the empirical evidence. Cornes and Sandler
(1994, p.419) show that crowding out is always likely to be less than full whenever there is a private
bene…t (e.g., warm glow) that arises from the act of donating. It is also likely that some of the observed
crowding-out is due to moral hazard issues on account of the fund-raising activities by the charity. These
issues lie beyond the scope of our paper.
9
These results also provide another explanation for the e¤ectiveness of seed money or leadership donations. Seed money, leadership contributions and national lottery money in the UK play a role similar to
the direct grant, D, in our framework.
3
1.4
A brief comparison with some other models of multiple equilibria
The rationale and implications of multiple equilibria in our paper are quite di¤erent from
Andreoni’s (1998) strategic-Nash framework, which requires a special type of non-convexity
in production. Andreoni (1998) distinguishes between two di¤erent kinds of charities. (1)
Start-up charities that require some initial amount of contributions G (the set-up or …xed
costs) before the charity can even begin operations. (2) Continuing charities that have
already invested G in the past and are now running concerns.
Andreoni’s (1998) focus is only on start-up charities. There are two kinds of equilibria
for start-up charities. In one equilibrium, beliefs are that the charity will fail to raise
the amount G, hence, the consistency of actions with beliefs requires that g = 0. In the
second equilibrium, beliefs are that the charity will successfully raise the amount G, and so
positive individual contributions become optimal, ful…lling the original beliefs. Hence, if
leadership contributions are equal to or greater than G, then positive individual donations
are induced.
The reader will immediately note that our analysis di¤ers very signi…cantly from Andreoni’s (1998) important contribution. First, our analysis applies both, to start-up charities and to continuing charities, but particularly the latter where in Andreoni’s framework
there cannot be multiple equilibria. Second, Andreoni (1998) uses a non-cooperative symmetric Nash equilibrium analysis while we use a competitive analysis that is not restricted
to symmetric equilibria. Third, the critical feature that generates multiple equilibria in
our paper is not non-convexities in production but strong aggregate complementarity, a
new concept that we introduce below. Fourth, the mechanisms to engineer moves between
various equilibria are di¤erent in the two models. Fifth, the issue of perverse comparative
statics is critical in our case but not in Andreoni’s (1998). Sixth, unlike Andreoni (1998),
we perform a welfare analysis in a general equilibrium model with a government budget
constraint where the …scal parameters (such as s and t) are optimally determined.
The idea of engineering a move from a low equilibrium characterized by poverty traps
to an equilibrium with greater prosperity is important in development economics but it
relies on a di¤erent mechanism.10
1.5
Results
We make seven main contributions. (1) Our proposed condition strong aggregate complementarity is a necessary condition for multiple equilibria in a competitive equilibrium
10
See Rosenstein-Rodan (1943) for an early idea. Murphy et al. (1989) focus on the mechanism of aggregate demand spillover arising from a coordinated increase in outputs in a range of imperfectly competitive
industries. However, our mechanism for engineering moves between equilibria is very di¤erent.
4
in charitable contributions. (2) Multiple equilibria provide a possible explanation of heterogeneity in charitable giving. (3) Using temporary direct grants, a policy maker can
engineer a move from the low to the high equilibrium. This is particularly desirable when
comparative statics at the low equilibrium are perverse and those at the high equilibrium
are normal. (4) When comparative statics at the low equilibrium are normal and those at
the high equilibrium are perverse, the government can do better than simply encouraging
subsidy-induced giving at the low equilibrium. Indeed, once the government successfully
engineers a move to the high equilibrium using temporary direct grants, the perverse comparative statics at the high equilibrium ensure that a reduction in subsides will induce
even greater private giving. (5) By carrying out a welfare analysis, we give conditions that
specify the optimal mix of public contributions and private contributions to charity. (6)
We show that our results are equally applicable to redistributive and public goods contexts.
(7) A key technical device is our proposed aggregate desire to give function; this may prove
useful in other contexts.
Section 2 formulates the theoretical model. Section 3 derives the equilibria of the
model and their comparative static results. Section 4 examines multiple equilibria in
aggregate giving in more detail. Section 5 performs a welfare analysis and characterizes the
normatively optimal public policy. Section 6 gives two illustrative examples of voluntary
private contributions to redistribution and public good provision, respectively. Section 7
discusses dynamic issues. Section 8 concludes. Most proofs are in the appendix.
2
Formal model
There are three types of players in the economy, (1) consumers, (2) a …scal authority or
government, and (3) a single charity (or a group of identical charities in which case we
may speak of a representative charity).
There is a continuum of consumers located on the unit interval, [0; 1], each consumer
is identi…ed by her location, x 2 [0; 1] and, hence, we shall simply refer to her as consumer
x. The utility function of consumer x is
u (c; g; G; x) ,
(1)
where c is private consumption expenditure of individual x, g
0 is her contribution
to charity, it re‡ects the warm glow or prestige (also known as impure altruism) from
own contribution11 and G
0 is the aggregate (which is also the average) level of giv11
The introduction of a warm glow motive was suggested by Cornes and Sandler (1984) and Andreoni
(1989, 1990). The presence of a warm glow term re‡ects the fact that individuals no longer consider their
contributions to be perfect substitutes for the contributions of others. Hence, there is extra utility from
one’s own contribution, which mitigates the free rider problem arising from purely altruistic considerations,
5
ing to charity. There is overwhelming experimental and …eld evidence and also growing
neuroeconomic evidence that justi…es such a formulation.12
Let m (x) 0 be the income of consumer x. We assume m is exogenous. The aggregate
(which is also the average) income is
Z 1
mdx < 1.
(2)
M=
x=0
Since aggregate giving cannot exceed total income, we must have
M.
G
(3)
The government levies a tax on income at the rate t 2 [0; 1], hence, income tax revenue
R1
equal t x=0 mdx = tM . The tax revenues are used to subsidize private donations to
charity at the rate s 2 [0; 1] and to …nance direct grants to charities, D 0. Therefore,
the (balanced) government budget constraint is
Z 1
gdx; s; t 2 [0; 1] , D 0:
(4)
tM = D + s
x=0
Note from (4), that ‘inaction’, s = 0, t = 0, is always feasible. The government chooses
its instruments s and t to maximize a social welfare function, which we shall introduce in
Section 5, below.
R1
The charity collects all donations from private consumers, x=0 gdx, and the direct
grant, D, from the government. Hence, the aggregate level of giving to the charity is
Z 1
gdx.
(5)
G=D+
x=0
The charity uses G to perform two traditional functions of charities: …nance transfers to
individuals and …nance provision of public goods. Consumer xh receives an amount
(x) G
i
R1
R1
from the charity. The balance (if any), G
Gdx = 1
dx G is used to
x=0
x=0
…nance provision of public goods. We assume that
Z 1
(x) 0 and
dx 1.
(6)
x=0
i.e., from a utility function of the form u (c; G). It also obviously implies that government grants to charities
do not crowd out private donations completely because the two are imperfect substitutes from the point
of view of givers.
12
For the evidence on altruism, see Andreoni (2006). For experimental evidence on warm glow preferences, see Andreoni (1993, 2006), Palfrey and Prisbrey (1997) and Andreoni and Miller (2002). For the
neuroeconomic evidence see, for instance, Harbaugh et al. (2007), Moll et al. (2006), and for a survey of
the neuroeconomic evidence see Mayr et al. (2009).
6
R1
Example 6.1, below, considers the special case x=0 dx = 1. Thus, the charity uses it’s
entire budget for income redistribution. Example 6.2, below, considers the case (x) = 0
for all x so that the entire budget, G, …nances the provision of public goods.
In order to focus on the determinants of private giving and the in‡uence of public
policy, we assume that the charity is a passive player in the game. Thus we take to be
exogenous (see Remark 3, below).
The budget constraint of consumer x is given by
c + (1
s) g
(1
(7)
t) m (x) + (x) G.
The LHS of (7) is total expenditure which is made up of private consumption, c, whose
price is normalized to unity, plus the (net of subsidy) private charitable giving. The
RHS of (7) is total income which consists of the after-tax income, (1 t) m (x), plus the
individual-speci…c transfer, (x) G, received from the charity.
2.1
Restrictions on consumer preferences
2
@u
@ u
, u2 = @u
, u3 = @G
, u12 = @c@g
... etc.).
Denote partial derivatives by subscripts (u1 = @u
@c
@g
We use measurability and integrability in the sense of Lebesgue (see, for example, Cohn
(1980) and Kharazishvili (2006)). Our assumptions are quite standard:
2.1.1
Measurability and Integrability assumptions
Income, m (x) and (x) are measurable functions of x 2 [0; 1].13 The consumption, c,
of consumer x is bounded below by a measurable and integrable function, c (x)
0.
u (c; g; G; x) is a measurable and integrable function of x 2 [0; 1].
2.1.2
Technical assumptions for existence and for an interior solution
u (c; g; G; x) is a C 2 function of c, g, G for g > 0 and c > c (x). The condition
u1 > 0; u11
0; u2
0; u22
0; u3
0
holds for all consumers and u2 > 0 for a set of consumers of positive measure.
For all s 2 [0; 1),
(1 s)2 u11 2 (1 s) u12 + u22 < 0.
(8)
The marginal utility of consumption tends to in…nity as consumption tends to its lower
bound from above:
(9)
u1 " 1 as c # c (x) .
13
The Lebesgue integral of a measurable function always exists but, however, it may be in…nite. If the
integral is …nite, the function is said to be integrable.
7
Either
u is extended to the boundary, g = 0, as a C 1 function, or
(10)
u2 " 1 as g # 0.
(11)
Aggregate income is su¢ cient to cover subsistence consumption:
Z 1
c (x) dx < M .
(12)
x=0
For each consumer, x 2 [0; 1],
Z
m (x) 1
c (y) dy + (x) M
c (x) <
M
y=0
Z
1
c (y) dy .
(13)
y=0
The above assumptions are satis…ed by Examples 6.1 and 6.2 in Section 6. In particular,
(10) holds for Example 6.1 and (11) holds for Example 6.2.
By (2), aggregate income, M , is …nite but, by (12), more than adequate to cover aggregate subsistence consumption, (13) then guarantees that a tax rate exists that redistributes
income so that each consumer can a¤ord a consumption level greater than her subsistence
level, c (x), see Lemma (1) below.
2.2
Sequence of moves.
The charity moves …rst to announce the parameters (x). The government moves next
to announce the parameters s, t and D.14 Finally, the consumers move simultaneously.
Consumer x chooses c and g so as to maximize her utility, u (c; g; G; x), subject to her
budget constraint, c + (1 s) g
(1 t) m (x) + (x) G. Since each consumer is of
measure zero, she takes G, s, t, m (x) and (x) as given. An equilibrium, G , is a value
of G that equates supply and demand for charitable giving.
We solve the model backwards. Each consumer x 2 [0; 1] chooses c (x) and g (x) so
as to maximize her utility, u (c; g; G; x), subject to her budget constraint, c + (1 s) g
(1 t) m (x) + (x) G, and given G, s, t, m (x) and (x). The government chooses s and
t given (x), x 2 [0; 1], correctly anticipating the consumers’choices of g (x), x 2 [0; 1].
2.3
A discussion of some assumptions
We here o¤er some clarifying remarks that may be useful.
Remark 1 : If s = 1, then the budget constraint (7) of consumer x reduces to c
(1 t) m (x) + (x) G. If u2 > 0 (and we have assumed this to be the case for a set of
consumers of positive measure) then it would follow that G = 1. This is, clearly, not
feasible. Hence, s < 1. Thus warm glow is never a free good.
14
We shall see below that announcing D can help coordinate private expectations of the level of G.
8
Remark 2 (Warm glow, taxes and private giving): In our model direct private giving to
charities yield a warm glow while paying taxes does not. One reason is that giving to
charity is voluntary while paying taxes is not. Another reason is empirical. People do
enjoy warm glow from giving but, generally, resent paying taxes. Also, in actual practice,
when an individual makes a tax payment he does not know what fraction will be used by
the government towards support of his chosen charity. Hence, we have chosen to model
warm glow arising only through direct giving although our results survive if there is some
warm glow arising through indirect giving, so long as the warm glow through direct giving
is relatively greater.
Remark 3 (Active and passive charities): In our paper charities are passive players.
However, in actual practice, charities could act strategically to attract extra donations or
respond to direct government grants by raising less money; see, for instance, Andreoni
(2006). Treating charities as active or passive players is not critical to our paper. Suppose
charities were active players, and suppose that the government and the charities simultaneously or sequentially choose their respective policies (taxes, redistribution and even
fund-raising) followed by private giving of individuals. So long as charities cannot completely negate the e¤ect of direct government grants (an assumption that is supported by the
evidence) then all our results go through. However, our approach is pedagogically simpler
and clearer.
Remark 4 (The nature of redistribution): In our model, redistribution is carried out by
charities only. One could extend the model to allow for direct public redistribution in addition to charitable redistribution. In this richer model, a part of tax revenues …nance public
redistribution, while the remaining part …nances direct grants to charities and subsidizes
private charitable giving. However, that adds nothing substantial to our framework or to
the main insights that we o¤er. Hence, we have chosen the current level of abstraction to
make our points as clearly as possible.
Remark 5 (A special case): Suppose that individuals make voluntary contributions towards the provision of a public good and that (x) = 0 for all x. Then our model reduces
to the case of privately provided public goods, …nanced by voluntary contributions. Indeed,
over human history many important public goods have been provided in this manner at
some point in time or the other.15
15
Examples of such public goods include, defence, lighthouses, information on marine navigation, shipping intelligence, mail services, education, public works, support for the poor and care for the sick.
9
2.4
Some preliminary and intermediate results.
R1
Lemma 1 : Consider s = 0, t = 1 M1 y=0 c (y) dy, D = tM then 0 < t
each consumer, x 2 [0; 1], c (x) < (1 t) m (x) + (x) G.
1 and for
Lemma 1 shows that by a feasible choice of s and t, the government can ensure that
each consumer can consume more than the subsistence level of consumption, c (x).
Since u1 > 0, the budget constraint (7) holds with equality. Hence, we can use it to
eliminate c from (1). Letting U (g; G; s; t; x) be the result, we have
U (g; G; s; t; x) = u ((1
t) m (x) + (x) G
(1
s) g; g; G; x) .
(14)
Lemma 2 : (a) m (x) and (x) are integrable.
(b) U (g; G; s; t; x) is a C 2 function of g; G; s; t and an integrable function of x.
(c) The …rst and second partial derivatives of u (c; g; G; x) with respect to c, g, G are
measurable functions; and the …rst and second partial derivatives of U (g; G; s; t; x) with
respect to g; G; s; t are measurable functions.
From (8) and (14) it follows that
U11 < 0.
(15)
Choose g so as to maximize U (g; G; s; t; x) ,
1
[(1 t) m (x) + (x) G c (x)] ,
subject to 0 g
1 s
given s; t; G; m (x) ; (x) ; c (x) .
(16)
Consumer x’s objective is16
The constraint follows from (7) and from the assumption that g, c are bounded below by
zero and c (x), respectively.
If c (x) = (1 t) m (x) + (x) G then the maximization problem (16) has the unique
solution, g = 0. But if c (x) > (1 t) m (x) + (x) G then it has no solution. In this
case we set g = 0 and U = 1. The more interesting case, c (x) < (1 t) m (x) +
(x) G, is considered in Proposition 3, which also considers the existence of a solution, its
measurability and its integrability.
Proposition 3 : Suppose c (x) < (1 t) m (x) + (x) G.
(a) The consumer x’s maximization problem (16) has a unique solution, g (G; s; t; x).
(b) g is a measurable function of (G; s; t; x).
16
Recall from Remark 1 that s < 1.
10
(c) 0
(d) 0
(e) g
(f) If,
g (G; s; t; x) < 1 1 s [(1 t) m (x) + (x) G c (x)].
R1
tM + (1 s) x=0 g dx < 2M .
is an integrable function of x.
in addition, (11) holds, then g (G; s; t; x) > 0.
In Lemma 4 we now consider the comparative static results.
Lemma 4 : Suppose that g (G0 ; s0 ; t0 ; x0 ) > 0. Let c (G; s; t; x) = (1 t) m (x) +
(x) G (1 s) g (G; s; t; x). Then, at G0 , s0 , t0 , x0 , g (G0 ; s0 ; t0 ; x0 ), c (G0 ; s0 ; t0 ; x0 ):
s)u13 + [u12 (1 s)u11 ]
= UU1211 = u232(1(1 s)u
,
(a) U1 = 0, (b) (1 s) u1 = u2 , (c) @g
@G
(1 s)2 u
u
12
(d)
(f)
+[u12 (1 s)u11 ]g
@g
, (e) @g
= UU1311 = 2(1u1s)u
= UU1411
2
@s
@t
12 (1 s) u11 u22
@g
, @g
, @g
are measurable functions of s; t; G; x.
@G
@s
@t
2.5
=
11
22
m[(1 s)u11 u12 ]
,
2(1 s)u12 u22 (1 s)2 u11
Complements and Substitutes
De…nition 1 (Bulow, Geanakoplos and Klemperer, 1985): g and G are strategic comple@2U
ments (substitutes) at s0 , t0 if @g@G
> 0 (< 0) at s0 , t0 for all g; G; x.
2
@ U
Suppose that g and G are strategic complements at s0 , t0 . From De…nition 1, @g@G
>0
@g
at s0 , t0 for all g; G; x. From Lemma 4c we then get that for all x, @G > 0. Aggregating
R1
over all consumers we get x=0 @g
dx > 0.
@G
De…nition 2, below, introduces a new concept that will be fundamental to our paper.
De…nition 2 (Strong aggregate complementarity): g and G are strong aggregate compleR1
ments at s0 , t0 , G0 if x=0 @g
dx 1 1s0 .
@G
Hence, strategic complementarity (De…nition 1) is not su¢ cient for strong aggregate
complementarity (De…nition 2). Furthermore, strategic complementarity ( @g
> 0 for
@G
all x) is not necessary for strong aggregate complementarity either. This is because, in
1
holds on average, not at each x.17
De…nition 2, we require that @g
@G
1 s0
3
3.1
Equilibrium giving and public policy
The aggregate desire to give to charity18
When consumer x makes her charity decision, she takes as given aggregate donations to
charity, G, and determines her optimal charitable contributions, g (G; s; t; x). Just as the
17
Strategic complementarity, strictly increasing di¤ erences, single crossing and supermodularity are increasingly more general concepts, but they all coincide for our model.
18
Our solution method has some similarities with the techniques developed in Cornes and Hartley (2007).
11
aggregate demand in competitive markets need not equal actual aggregate supply for all
R1
price vectors, the aggregate of all desired donations, D + x=0 g (G; s; t; x) dx, need not
equal actual aggregate contributions, G. Therefore, we introduce a new function, F , which
represents the aggregate of all desires (public and private) to give to charity:
Z 1
F =D+
g (G; s; t; x) dx.
(17)
x=0
The government budget constraint, (4), can be written as
Z 1
g (G; s; t; x) dx; s; t 2 [0; 1] , D
D (s; t; G) = tM s
0:
(18)
x=0
From (17) and (18) we can substitute out D to get
Z 1
F (s; t; G) = tM + (1 s)
g (G; s; t; x) dx:
(19)
x=0
Remark 6 : In (19) we have eliminated D and we will focus on the remaining two instruments of government policy, s; t. The government budget constraint always holds and
s; t 2 [0; 1] and D 0 in (18).
From (19) and Proposition 3c a simple calculation shows that
0
F (s; t; G) < 2M .
(20)
For given s and t, G 7! F (s; t; G) is a mapping from [0; 2M ] to [0; 2M ].
The above discussion suggests the following de…nition.
De…nition 3 : The aggregate desire to give is de…ned by the mapping, F (s; t; G),
Z 1
F (s; t; G) = tM + (1 s)
g (G; s; t; x) dx.
x=0
From De…nition 3, we get that
@F (s; t; G)
FG (s; t; G) =
= (1
@G
s)
Z
1
x=0
@g (G; s; t; x)
dx.
@G
(21)
Lemma 5 gives a useful alternative characterization of “strong aggregate complements”
(De…nition 2).
Lemma 5 : g and G are strong aggregate complements at s0 , t0 , G0 if, and only if,
FG (s0 ; t0 ; G0 )
12
1.
(22)
3.2
Equilibria
De…nition 4 (Equilibrium in giving): The economy is in an equilibrium if the aggregate
of all desires to donate to charity, F , equals the aggregate of all donations, G, i.e., G 2
[0; 2M ] is an equilibrium if
G = F (s; t; G ) ;
(23)
where the government budget constraint, (18) holds.
De…nition 5 (Isolated equilibrium): An equilibrium, G , is isolated if there is a neighborhood of G in which it is the only equilibrium.
Proposition 6 : (a) An equilibrium, G 2 [0; 2M ], exists and satis…es 0 G < 2M .
(b) If FG < 1 for all G 2 [0; 2M ] or if FG > 1 for all G 2 [0; 2M ], then an equilibrium,
G , is unique.
(c) If [FG ]G 6= 1, then G is an isolated equilibrium.
Proof of Proposition 6: Let H (s; t; G) = G F (s; t; G).
(a) Recall that at an equilibrium, G , G = F (s; t; G ).
(i) F (s; t; 0) < 0 is not feasible.
(ii) If F (s; t; 0) = 0 then, clearly, 0 is an equilibrium.
(iii) Now, suppose that F (s; t; 0) > 0. Since F (s; t; 0) > 0, it follows that H (s; t; 0) =
F (s; t; 0) < 0. From (20), F (s; t; G) < 2M , so, in particular, F (s; t; 2M ) < 2M . Hence,
H (s; t; 2M ) = 2M
F (s; t; 2M ) > 0. Since H (s; t; G) is continuous, it follows that
H (s; t; G ) = 0, for some G 2 [0; 2M ). Hence G is an equilibrium and 0 G < 2M .
(b) Since H is continuous, the set of equilibria fG : H (s; t; G) = 0g is a closed subset of
[0; 2M ] and, hence, is compact. By (a), it is not empty. Hence, it must contain minimum
Gmax , H (s; t; Gmin ) = 0,
and maximum elements, Gmin and Gmax , respectively, Gmin
H (s; t; Gmax ) = 0. If Gmin < Gmax then, since HG is continuous, it follows that HG = 0 for
some G 2 (Gmin ; Gmax ). Thus, FG = 1 for some G 2 (Gmin ; Gmax ), which cannot be the
case because of the restriction FG < 1 or FG > 1 in Proposition 6(b). Hence, Gmin = Gmax ,
which implies that the equilibrium is unique.
(c) Suppose [FG ]G 6= 1. Then [HG ]G 6= 0. Hence, either [HG ]G < 0 or [HG ]G > 0.
Since HG is continuous, it follows that HG < 0 (or HG > 0) in some neighborhood of
G . Thus, using an argument similar to that deployed in (b), G can be shown to be an
isolated equilibrium. .
Figure 1 illustrates the results in Proposition 6. Three possible shapes of the function
F (s; t; G) are shown. Along the curve AED, the su¢ cient condition for uniqueness, FG < 1
for all G 2 [0; 2M ] holds, and we have a unique equilibrium at E. Along the two paths,
ABCD and AHD, this su¢ cient condition is violated. In particular, along these curves
13
Figure 1: Unique and multiple equilibria.
we observe values of G for which FG < 1 and other values for G for which FG > 1. In
this case, since the condition in Proposition 6b is a su¢ cient condition, we could have a
unique equilibrium (as in the case of curve AHD) or multiple equilibria (as in the case of
curve ABCD); see also our examples 6.1 and 6.2. All equilibria shown in Figure 1 satisfy
[FG ]G 6= 1, hence, they are all isolated equilibria.
3.3
Equilibrium analysis: Normal, neutral and perverse comparative statics
We now investigate how aggregate equilibrium giving to charity, G, responds to the policy
instruments, s; t. We, therefore, consider an equilibrium, G , at which FG 6= 1. By Proposition 6c, such an equilibrium is isolated. We can then regard G as a C 1 function, G (s; t),
of s and t in that neighborhood (this is a special case of the implicit function theorem).
Letting Gs , Gt be the partial derivatives of G with respect to s and t respectively. The
following lemma gives a useful identity.
Lemma 7 :
R1
x=0
@g
@s
+
@g
G
@G s
dx =
1
1 s
Gs +
R1
x=0
g dx .
Proposition 8 gives some comparative static results for an isolated equilibrium.
Proposition 8 : Let G be an equilibrium at which FG (s; t; G ) 6= 1. Then G is isolated
and
(F +F G )(1 F )+F (F +F G )
(a) Gs (s; t) = 1 FFs G , (b) Gt (s; t) = 1 FFt G , (c) Gtt (s; t) = tt tG t (1 GF )2t tG GG t .
G
We now de…ne the critical concepts of normal, neutral and perverse comparative statics,
which have to do with the response of G to changes in the subsidy, s.
14
De…nition 6 (Normal, neutral and perverse incentives): Comparative statics are normal
if Gs > 0, neutral if Gs = 0 and perverse if Gs < 0.
From De…nition 6, comparative statics are perverse when an increase in subsidy reduces
total giving in equilibrium. Proposition 8 and De…nition 6 immediately imply Corollary
9, below.
Corollary 9 : (a) Comparative statics are normal if (i) Fs > 0 and FG < 1 or if (ii)
Fs < 0 and FG > 1.
(b) Comparative statics are neutral if Fs = 0.
(c) Comparative statics are perverse if (i) Fs > 0 and FG > 1 or if (ii) Fs < 0 and FG < 1.
For ease of reference and to facilitate the discussion in subsequent sections, we have
found it helpful to organize the results in this subsection in Propositions 10 and 11 below.
Proposition 10 : Let G be an equilibrium at which FG (s; t; G ) 6= 1.
(a) Let Fs > 0 at G .
(i) If FG < 1 at G then Gs (s; t) > 0 (normal comparative statics). (ii) If FG > 1 at G
then Gs (s; t) < 0 (perverse comparative statics).
(b) Let Ft > 0 at G .
(i) If FG < 1 at G then Gt (s; t) > 0. (ii) If FG > 1 at G then Gt (s; t) < 0.
Proposition 11 : Let G be an equilibrium at which FG 6= 1. Then
(a) Let Fs < 0 at G .
(i) If FG < 1 at G then Gs (s; t) < 0 (perverse comparative statics). (ii) If FG > 1 at G
then Gs (s; t) > 0 (normal comparative statics).
(b) Let Ft < 0 at G .
(i) If FG < 1 at G then Gt (s; t) < 0. (ii) If FG > 1 at G then Gt (s; t) > 0.
Figure 2 illustrates the two cases of normal and perverse comparative statics in Proposition 10a for a particular case. We assume that Fs > 0, for all values of G and s1 < s2 .
From Remark 6 recall that D has been substituted out but the condition s; t 2 [0; 1] and
D 0 ensures that the government budget constraint holds. Thus, an increase in subsidies could be …nanced by extra taxation, say, with the tax rate increasing from t1 to t2
or simply by reducing direct government grants, D, or a combination of both. In Figure
2 we have assumed that taxes increase from t1 to t2 but the net impact of an increase in
s; t on F is positive. In Figure 2, the 45o line, F = G, is shown as the dark line. The case
0 < FG < 1 is illustrated by the two, thin, continuous, parallel straight lines, while the
other case, FG > 1 is shown by the two dashed lines. Thus, in the latter case, and from
Lemma 5, g and G are strong aggregate complements.
15
Figure 2: Fs > 0; s1 < s2 . 0 < FG < 1 (continuous, light lines). FG > 1 (dashed lines)
A. Normal comparative statics (Gs > 0): Figure 2 illustrates Proposition 10ai for the
case Fs > 0, 0 < FG < 1 (although only FG < 1 is required) by an upward shift of
the continuous, light, curve F (s1 ; t1 ; G) to F (s2 ; t2 ; G). The equilibrium moves from
A to B. Hence, the optimal level of aggregate giving, G, is increasing in the subsidy
to individual giving, i.e., Gs > 0.
B. Perverse comparative statics (Gs < 0). This case is shown in Figure 2, by an upward
shift of the dashed curve F (s1 ; t1 ; G) to F (s2 ; t2 ; G), which assumes Fs > 0, FG > 1,
as required in Proposition 10aii. The equilibrium moves from B to A. Equilibrium
aggregate contributions, G, decrease as the price of giving reduces (larger s). Because
FG > 1, consumers over-react to an increase in G. Thus, paradoxically, G needs to
fall in order to ensure equilibrium in the market for charity. The tax reforms of
the 1980’s in the US increased the price of charitable giving by reducing subsidies,
yet charitable contributions continued to rise in the following years; see, Clotfelter
(1990).
The following lemma will be useful in welfare analysis, below.
Lemma 12 :
4
@s
@D
=
R1 1 s
.
sGs + x=0
g dx
Equilibrium analysis with multiple equilibria
Suppose we have two equilibria in aggregate giving, G < G+ . Suppose that the economy
is at G and we wish to move it to G+ . Clearly, if it is possible for the government to give
a direct grant, D, G < D < G+ , then, from (5), we see that G+ becomes the only feasible
candidate. But we wish to go further. Does such a D exist? Once G+ is established, can
we phase out D? Would this cause G+ to decline or increase further? Could the economy
revert back to G ? It is questions like these that we address below.
16
There is no reason to suppose that the equilibrium with perverse comparative statics
is ‘unstable’and that with normal comparative statics is ‘stable’; see section 7 below.
We investigate one case in detail: Fs > 0; Ft > 0; FG > 0; FGG < 0 (subsection 4.2)
The case Fs = Gs = 0 (neutral comparative statics) is considered in detail via Example 2
in subsection 6.2, below. All other cases can be dealt with in a similar fashion. We begin
with a reasonable assumption which we call “stability of beliefs”.
4.1
Stability of beliefs
Suppose one has two equilibria, G < G+ . In some sense, suppose that one of the two
equilibria is ‘good’or more desirable and the other is ‘bad’. Ideally, one would like to adopt
an explicitly dynamic model in which beliefs endogenously evolve over time as individuals
engage in learning.19 In the absence of a satisfactory resolution of this problem, one needs
to fall back on some ad-hoc (though hopefully plausible) assumption about the stability
of beliefs. This is an issue in all static economic models.20
Consider a problem in coordination where people have to decide whether to drive on
the left or the right of the road. In the UK, people drive on the left, safe in their beliefs
that all others will also drive on the left. A similar observation applies to people in the
US, except that expectations are to drive on the right. One could compare each of these
countries at a point of time in history when expectations were not coordinated on the
good equilibrium (all drive on one side of the road). This example illustrates powerfully
the idea that when a good equilibrium gets established, people come to expect that it will
prevail. In other words once established, beliefs may exhibit inertia to change, i.e., they
may be stable. In a charity context, once the Red Cross is established as a “large”charity
then beliefs are that it will be large next period.
Weber (2006) …nds that coordination in groups can be grown gradually (as group size
increases) from a small enough size of groups where cooperation is already existent but
not otherwise. The norm of cooperation is then learnt by successive entrants, who also
cooperate. These …ndings again illustrate the stability of beliefs on a good equilibrium
when a good equilibrium is established.
In the context of our speci…c model, we illustrate the stability of beliefs in Figure 3.
Suppose that we have two equilibria, G (s; t) and G+ (s; t) such that G (s; t) < G+ (s; t) .
Suppose that the comparative statics along G are perverse, i.e., Gs < 0, while those
21
along G+ are normal, i.e., G+
s > 0.
19
Most traditional learning models do not perform too well when taken to the evidence although some
behavioral models of learning do better; see Camerer (2003).
20
See Section 7, below for further discussion of these issues.
21
Since we have assumed Fs > 0; Ft > 0; FG > 0; FGG < 0; from Proposition 10a, this occurs when
FG > 1 on G and FG < 1 on G+ .
17
Figure 3: Persistence in beliefs
Suppose that we begin at point ‘a’ on the G locus, where Gs < 0. As we vary
s we will simply trace out various points on the G locus. Now suppose that we can
somehow engineer a jump to the G+ locus, say, to point ‘b’where G+
s > 0 (point b need
+
not be directly above point a). Once the economy is on the G locus (at point, b, say)
for a su¢ cient length of time then the stability of beliefs requires that as we adjust s the
economy moves along the G+ locus.
4.2
Engineering moves between equilibria: The case Fs > 0, Ft >
0, FG > 0, FGG < 0
Suppose that the objective of the government is to move the economy from a low equilibrium with perverse comparative statics, G , to a high equilibrium with normal comparative
statics, G+ . We now discuss this central question in our paper. We shall consider in detail
the case, Fs > 0, Ft > 0, FG > 0, FGG < 0. The case Fs = Gs = 0 (neutral comparative
statics) is considered in detail via Example 2 in subsection 6.2, below. All other cases can
be dealt with in a similar fashion, so we omit a discussion of these cases.
Figure 4, below, sketches the case F (s; t; G) = 0 for G 2 [0; G (s; t)], F (s; t; G) > 0,
FG > 0, FGG < 0 for G > G (s; t). Thus F is strictly increasing and strictly concave in
G, for G > G0 (s; t). We assume that s1 < s2 . Since Fs > 0, it follows that the graph
of F (s2 ; t1 ; G) is strictly above that of F (s1 ; t1 ; G). There are four equilibria: a and d
corresponding to the parameter values (s1 ; t1 ); and b and c corresponding to the parameter
values (s2 ; t1 ). Furthermore, we see that FG > 1 at points a and b but FG < 1 at points c
and d.
Suppose now that the economy is at point a, with G = G (s1 ; t1 ), in 4. Also suppose
that (for whatever reason) the government wants to shift the economy from a to d, where
the latter point corresponds to G = G+ (s1 ; t1 ). Since Fs > 0 and FG > 1 at point a it
18
Figure 4: The case s1 < s2 with t1 …xed
follows, from Corollary 9ci, that comparative statics at point a are perverse (Gs < 0).
Suppose that the equilibrium G = G (s1 ; t1 ) is established, so using the argument
about stability of beliefs and Figure 3, any change in the parameter s would move the
economy along the G locus. Hence, an increase is s would make things worse by reducing
aggregate giving, G. In particular, an increase in s, from s1 to s2 , would reduce aggregate
giving, from G (s1 ; t1 ) to G (s2 ; t1 ). A decrease in s would not enable the economy to
move to the G+ locus either. Thus the instrument, s, is ine¤ective in moving the economy
from a to d. We will now show that, by contrast, the other instrument, t, can be e¤ective.
Starting from the policy parameters, s1 ; t1 , at point a, we alter the policy parameters to
0; t2 and at the same time give a public grant equal to D > G (0; t2 ). Using (5), this leaves
G+ (0; t2 ) as the only equilibrium. We know that this equilibrium will exist because of the
existence result in Proposition 6. Once the economy is allowed to get established at this
new equilibrium, the stability of beliefs argument (see subsection 4.1) can be used to argue
that the economy is now on the G+ locus. One can then choose the policy parameters
optimally. In particular, one can choose the parameters s1 ; t1 in which case the economy
reaches the equilibrium G+ (s1 ; t1 ) corresponding to point d. We now elaborate on this.
Figure 5 plots the locus of F (s1 ; t1 ; G), which is exactly as in Figure 4. We also sketch
the graph of F (0; t1 ; G) in Figure 5, which lies strictly below the graph of F (s1 ; t1 ; G), if
s1 > 0, because Fs > 0. Figure 5 shows four equilibria a, d, e and f . Equilibria a and d,
in Figure 5, are exactly the same as in Figure 4 and correspond to the parameter values
(s1 ; t1 ). However, e and f correspond to the parameter values (0; t1 ).
Figure 6 plots the locus of F (0; t1 ; G), which is exactly as in Figure 5. We also sketch the
graph of F (0; t2 ; G), t1 < t2 , in Figure 6, which lies strictly above the graph of F (0; t1 ; G),
because Ft > 0. Figure 6 shows four equilibria e, f , g and h. Equilibria e and f , in Figure
6, are exactly the same as in Figure 5 and correspond to the parameter values (0; t1 ).
However, g and h correspond to the parameter values (0; t2 ). We now consider a policy
19
Figure 5: The case of decreasing subsidies from s1 to 0 with t1 …xed.
Figure 6: The case of increasing taxes from t1 to t2 with s = 0 …xed.
that moves the economy from a to d.
4.2.1
Moving the economy from a to d.
Suppose that the economy is at point a of Figures 4, 5 and its policy instruments are given
by (s1 ; t1 ). The equilibrium level of contributions to charity are given by G (s1 ; t1 ). Once,
G is established, changes in s,t will move the economy along the G locus (see Figure 3).
In the …rst step, the government changes the values of the instruments from (s1 ; t1 )
to (0; t1 ). At the new policy (0; t1 ), the government budget constraint (4) is given by
t1 M = D. This will give rise to a new equilibrium G (0; t1 ) and the corresponding level
direct grants by the government are D (0; t1 ; G (0; t1 )). Thus, the government budget
D(0;t1 ;G (0;t1 ))
.
constraint is given by t1 =
M
In the second step, the government changes the values of the instruments from (0; t1 )
to (0; t2 ). The government budget constraint (4) is given by t2 M = D. Let us determine
(0;t1 )
t2 such that the direct grant to the charity is D = G (0; t1 ). Thus, t2 = G M
. If s1 > 0
then G (0; t1 ) > G (s1 ; t1 ) (see Figure 5) because of the perverse comparative statics,
20
Gs < 0, on the locus G (s; t1 ).
By de…nition (see (5)), when the policy is (0; t1 ) we get
Z 1
g (0; t1 ; G (0; t1 ) ; x)dx;
G (0; t1 ) = D(0; t1 ; G (0; t1 )) +
x=0
D(0;t1 ;G (0;t1 ))
so, if some g > 0 then G (0; t1 ) > D(0; t1 ; G (0; t1 )). Thus, t1 =
<
M
G (0;t1 )
Ft
= t2 . Since Gt < 0 (because Ft > 0, FG > 1 and from Proposition 8, Gt = 1 FG ),
M
it follows that G (0; t2 ) < G (0; t1 ) = D as in Figure 6. Hence, the sole equilibrium,
consistent with D, is now the high equilibrium, G+ (0; t2 ), point h of Figure 6. Once
G+ (0; t2 ) is established, each individual will make her private giving decision conditional
on G+ (0; t2 ), thereby, raising her own giving to a higher level such that equilibrium beliefs
about G+ (0; t2 ) become self-ful…lling.
We can now invoke the stability of beliefs assumption, outlined in subsection 4.1 above
as beliefs about G+ (0; t2 ) become self-ful…lling. As one varies the policy parameters, the
economy moves along the G+ locus in Figure 3. If the government desires, it can now
reduce the tax rate from t2 back to its original value, t1 , and increase s back to its original
value, s1 (we consider these issues in section 5 below). The economy then moves to point
d of Figure 4, hence, completing the desired move from point a to point d.
Garrett and Rhine (2007) report that in the US, the growth in private charitable
giving over time has been paralleled by similar growth in expenditure by various levels
of government. In particular, between 1965 and 2005, the most rapid growth in private
giving (g in our set up) has been in charities associated with health, education and social
services. These are precisely the areas in which there has been large increases in direct
government expenditure (D in our set up). The fact that the Government has responded
to an increased demand for these services by increasing the direct grant, rather than by
increasing the subsidy to private giving, is consistent with our explanation.
4.2.2
Optimal public policy at the new equilibrium
Substantial …eld, experimental and neuroeconomic evidence supports the assertion that
private charitable giving is a source of warm glow but direct government grants are not
(see the discussion in Remark 2). Hence, once the economy is established on the G+
locus in Figure 3, it would be welfare improving to replace government grants by an
equivalent amount of private giving, if possible. Propositions 13, 14, 15 in Section 5,
below, characterize the optimal policy when we are on the G+ locus.
21
5
Welfare analysis
We now ask what should be the optimal policy parameters s; t of the government when
its objective is to maximize the social welfare function (27), below. This also allows us to
directly answer the question that we posed in subsection 4.2.2 above. Namely, what should
be the optimal mix of policy parameters once the high equilibrium gets established and
(using the stability of beliefs) the economy moves along the G+ locus. g (s; t; G (s; t) ; x)
Substituting the optimal level of giving of consumer x, g (s; t; G (s; t) ; x), and the
equilibrium aggregate level of giving, G (s; t) (where G lies either on the G or the G+
locus) in the individual utility function (14) gives the consumer x’s indirect utility function
v (s; t; x) = u ((1
t) m (x) + (x) G
(1
s) g ; g ; G ; x) .
(24)
where g = g (s; t; G (s; t) ; x) and G = G (s; t). Di¤erentiating (24), using Lemma 4b
or using the envelope theorem, gives
vs = g u1 + Gs ( u1 + u3 ) ,
vt =
mu1 + Gt ( u1 + u3 ) ,
(25)
(26)
where, the partial derivatives Gs ; Gt are given by Proposition 8a,b, respectively.22
The government chooses, s and t, so as to maximize the social welfare function,
Z 1
W (s; t) =
! (x) v (s; t; x) dx,
(27)
x=0
where v (s; t; x) is given by (24). We assume that the individual-speci…c weights, ! (x), are
measurable and integrable functions from [0; 1] into R such that ! (x) > 0 (so that each
R1
individual is socially important) and x=0 ! (x) dx = 1.
Di¤erentiating (27), using the chain rule, and (25), (26), we get
Z 1
Z 1
!g u1 dx + Gs
! ( u1 + u3 ) dx,
(28)
Ws =
Wt =
x=0
x=0
Z
Z
1
!mu1 dx + Gt
x=0
1
! ( u1 + u3 ) dx,
(29)
x=0
where subscripts denote appropriate partial derivatives.
Propositions 13, 14 and 15, below, derive the optimal mix between private and public
giving to charity in di¤erent cases. These propositions are used extensively in Section 6
and are crucial in determining the optimal public policy at di¤erent equilibria.
Proposition 13, below, implies that in a social optimum where Gt
0 no government
intervention is needed, warm glow and/or altruism su¢ ce to maximize social welfare.
22
To facilitate readability, and to keep formulas to a manageable length, we have ommitted the arguments of g ; G from (25)-(29).
22
0 and g > 0 on a subset
Proposition 13 : Let (s ; t ) be a social optimum where Gt
of [0; 1] of positive measure. Then s = t = D = 0. So all giving to charity is private
giving.
We now show, in Proposition 14, that if subsidies are e¤ective then no direct government
grant is needed. This is because when private donations replace an identical amount of
public donations, D, welfare improves on account of the warm glow received by private
givers.
Proposition 14 : Let (s ; t ) is a social optimum where Gs
of [0; 1] of positive measure. Then D = 0.
0 and g > 0 on a subset
Proposition 15 : Let G be a social optimum with g > 0 on a subset of [0; 1] of positive
measure. Suppose (i) Fs 0 and FG < 1 at G or (ii) Fs 0 and FG > 1 at G . Then
D = 0.
The intuition behind Proposition 15 can be seen from the results in Propositions 10,
11. In Proposition 15i, for instance, the comparative statics are normal (and not perverse),
hence, private giving can be encouraged through the use of subsidies. Since private giving
leads to warm glow, and an improvement in welfare, it is optimal to generate all charitable
giving through private giving, rather than by direct government grants. A similar intuition
explains Proposition 15ii.
We can now use these propositions to revisit the question that we posed in subsection
4.2.2, above. Namely, the question of the optimal policy once the economy gets established
on the G+ locus. Once established on the locus G+ comparative statics are normal, i.e.,
G+
s > 0. From Proposition 14 we know that, in this case, s attains its maximum value and
the government makes no direct contributions to charity, i.e., D = 0. All tax revenues are
then used to …nance subsidies to private giving. The reason is that private giving confers
warm glow on individuals and, hence, raises welfare relative to the same level of giving
being undertaken directly by the government. Hence, it is optimal to set D = 0.
6
Two examples
We now present two examples of the general model. In Example 1 (subsection 6.1) charitable contributions provide income to consumers who, otherwise, have no income. In
Example 2 (subsection 6.2) charitable contributions …nance public good provision.
23
Figure 7: Illustration of the main features of Example 1.
6.1
Example 1: Charitable contributions as public redistribution
We consider an economy where some consumers have no income. Their consumption
expenditure is …nanced entirely by either charitable donations, g, made by other ‘caring’
consumers with positive income and/or by tax …nanced direct government grants, D (which
now have the interpretation of social welfare payments). This is illustrated in Figure 7.
The assumptions are as follows.
1. m (x) > 0 for 0
x
p < 1, m (x) = 0 for p < x
1.
2. The aggregate of all donations to charity (private and public), G, is divided among
R1
the consumers with no income. Hence, (x) = 0 for x 2 [0; p] and x=p dx = G.
3. If x 2 [0; k], where 0 < k p, then consumer x cares about the plight of those with
no income. Each of these caring consumers has the utility function
u (c; g; G; x) = ln c + a (x) gG, a (x) > 0, x 2 [0; k] ,
(30)
so there is strategic complementarity between g and G (see De…nition 1). The
following technical condition holds:
1
1
<
a (x) G
1
t
m (x) , x 2 [0; k] .
s
(31)
4. Some individuals with positive incomes do not care about individuals with zero
incomes; these are individuals x 2 (k; p].23 The utility function of all individuals
x 2 (k; 1] are given by
u = ln c, x 2 (k; 1].
(32)
23
In the special case of k = p all consumers with postive incomes are caring. We allow for both cases,
k = p and k < p.
24
Let m be the aggregate income of the caring consumers (see Figure 7). Then
m=
Z
k
m (x) dx
x=0
Z
p
m (x) dx =
x=0
Let
A=
Z
k
x=0
Z
1
m (x) dx = M .
(33)
x=0
1
dx > 0.
a (x)
(34)
Proposition 16 : (a) (Multiple equilibria) The only economically interesting cases occur
when [m + t (M m)]2 > 4(1 s)A: In this case, we have two distinct, real, positive,
equilibria 0 < G (s; t) < G+ (s; t). These are given by
q
1
G (s; t) =
m + t (M m)
[m + t (M m)]2 4(1 s)A ,
2
and, correspondingly,
g
s; t; G (s; t) ; x =
1 t
m (x)
1 s
1
a(x)G (s;t)
0
if x 2 [0; k]
:
if x 2 (k; 1]
(b) (Increasing and concave desire to contribute) For all G, the aggregate desire to give,
F , (i) responds positively to subsides i.e., Fs > 0 and, (ii) it is increasing and concave,
i.e., FG > 0; FGG < 0. So we have the case depicted in Figures 4, 5, 6. Furthermore,
G=
G+ ) FG < 1
G ) FG > 1
(c) (Perverse and normal comparative statics) The comparative statics with respect to the
subsidy are perverse at the low equilibrium and normal at the high equilibrium, i.e., Gs < 0;
G+
s > 0. For m < M (and so k < p) the same holds for the comparative static result with
respect to the tax rate, i.e., Gt < 0; G+
t > 0. For k = p, Gt = 0.
The equilibria are as in Figures 4, 5, 6. Suppose that an economy is at the low
equilibrium, G , and also suppose that it is socially desirable to move it to the high
equilibrium, G+ . How can this be done? The situation is identical to the one presented in
Section 4.2, and so we follow the same solution method.
From Proposition 16b, due to perverse comparative statics at the low equilibrium,
Gt
0: Hence, from Proposition 13, it would appear that the best policy is no intervention
i.e., s = t = 0; leaving the economy at the low equilibrium, G . However, an alternative
policy is possible, as we shall now describe; see Figure 8.
Set s; D as follows: s = 0 and D = G (0; 0). The government budget constraint (4)
implies that the tax rate t required to …nance D is t = G (0; 0)=M . Since G (0; 0) is an
25
Figure 8: Multiple equilibria when Ft > 0; FG > 0; FGG < 0.
equilibrium, and there is an interior solution to consumption (from (9)), so 0 < G (0; 0) <
M . It follows that 0 < t < 1 and, hence, it is feasible.
Thus, the government gives a direct grant equal to G (0; 0) …nanced with an income
tax (any grant D > G (0; 0) will also work). Since g > 0 on some subset of [0; k]
of positive measure, it follows, from (5), that G > D = G (0; 0). Hence, because we
are now on the F (0; t; G) locus (see Figure 8) and the equilibrium aggregate donation
G > G (0; 0) > G (0; t), the only possible candidate for equilibrium (guaranteed by the
existence result in Proposition 6) is
G = G+ (0; t).
(35)
Once the economy establishes itself at the high equilibrium, G+ (0; t), the stability of
beliefs (see subsection 4.1 above) ensures that the economy is on the G+ locus. The …scal
parameters, s; t can then be adjusted to their socially optimum values, assuming that the
economy is on the G+ locus, using the results of Section 5. We now address this issue.
6.1.1
Socially optimal public policy at the new equilibrium
At the low equilibrium, G (0; 0), private individuals cannot be induced to make additional
contributions due to the perverse comparative statics, Gs (0; 0) < 0, Gt (0; 0) < 0. In
contrast, the comparative statics at the high equilibrium, G+ (s; t), for any values of s; t
are normal. Allow the economy to establish itself at the G+ locus. We now illustrate the
insights of subsection 4.2 for Example 1 (subsection 6.1). Consider two cases.
1. Let k < p, so that not all consumers with positive income are caring. From Propo+
sition 16c, G+
s > 0, Gt > 0, i.e., the comparative static e¤ects are normal at the
high equilibrium. Depending on the parameter values, the optimal tax rate may be
positive, in which case it can be found by setting Vt = 0 in (29). From Proposition
16b, at the high equilibrium G+ , Fs > 0, and FG < 1, so, Proposition 15 implies that
26
D = 0. Thus, once the economy has moved from the G locus to the G+ locus, the
direct grant from the government to the charity is phased out. In the new, socially
optimal equilibrium, all contributions to charity are private (because only private
contributions are associated with warm glow) and all income tax revenue is used to
subsidize private donations to charity.
2. Let k = p, so that all consumers with positive income are caring. Hence, from
Proposition 16c, G+
t = 0. It follows, from Proposition 13, that once the economy has established itself on the G+ locus, s = t = 0. Thus, once the (one-o¤)
direct government grant to charity (…nanced by an income tax) has shifted the
economy from the bad equilibrium, G , to the good equilibrium, G+ , no further
government intervention is needed and the entire charitable contributions are voluntary private contributions. At the new optimum, equilibrium is described by
R1
G+ (0; 0) = x=0 g + (0; 0; G+ (0; 0); x)dx.
6.1.2
A numerical illustration
9
,
We now illustrate the above results by a numerical simulation. Speci…cally, let p = 19
1
9
9
9
9
k = 38 , a (x) = 190000 for x 2 0; 38 , a (x) = 0 for x 2 ( 38 ; 1], m (x) = 1900 for x 2 0; 19 ,
R1
9
9
m (x) = 0 for x 2 ( 19
; 1], (x) = 0 for x 2 0; 19
, x= 900 (x) Gdx = G. Thus,
1900
m =
Z
9
38
m (x) dx = 450, M =
x=0
A =
Z
9
38
x=0
Z
1
x=0
1
dx = 45000.
a (x)
m (x) dx =
Z
9
19
m (x) dx = 900,
x=0
(36)
Initially, assume that s = t = 0. Then, from (36) and Proposition 16a, G (0; 0) = 150,
9
, g + = 3800
for x 2 0; 38
and g = 0 for
G+ (0; 0) = 300. Correspondingly, g = 1900
3
3
9
x 2 ( 38 ; 1].
Suppose that the locus passing through G is considered to be socially inferior to that
passing through G+ . Since there are perverse comparative statics at the low equilibrium,
i.e., Gs < 0; Gt < 0, the best policy would appear to be no intervention: s = t = 0.
However, there is an alternative. The government sets s = 0; t = 1=6. This raises a total
tax revenue equal to tM = 900=6 = 150. Since Gt < 0, it follows that G (0; 1=6) < 150 at
t = 1=6. The government makes a direct grant D = G (0; 0) = 150 to the charity. Since
R 9
9
38
, we must have G(0; 1=6) = D + x=0
g(0; 1=6; G; x) > D = 150 =
g > 0 for x 2 0; 38
G (0; 0) > G (0; 1=6). Hence, the only possible equilibrium is G = G+ (0; 1=6). Once the
economy is on the G+ locus and the new equilibrium gets established (see subsection 4.1
above) s; t can be given their optimal values as shown in Propositions 13, 14, 15 in Section
5.
27
6.2
Example 2: Voluntary contributions to a public good
Individuals often voluntarily contribute to, and directly use, several kinds of public goods
such as health services and education24 . Let the utility function of consumer x be
u (c; g; G; x) = [1
a (x)] ln c
b (x)
+ a (x) ln g,
G
(37)
where
b (x)
< (1 t)m (x) .
(38)
G
Condition (38) ensures that consumer x has su¢ cient disposable income, (1 t)m (x), to
and also a positive level of donation to
sustain a level of consumption, c, greater than b(x)
G
charity, g (x). It is straightforward to check that u1 > 0, u2 > 0, u3 > 0.
This example can be given the following interpretation. Private (voluntary) contribuR1
tions to public goods, x=0 gdx, plus public contribution, D, …nanced from income taxation,
provide the necessary infrastructure for private consumption, c. An increase in aggregate
R1
expenditure on infrastructure, G = D + x=0 gdx, leads to a higher level of utility for a
given level of consumption.25
De…ne the constants B; C as:
Z 1
Z 1
Z 1
B=
a (x) m (x) dx + t
[1 a (x)] m (x) dx, C =
a (x) b (x) dx.
(39)
0 < a (x) < 1, b (x) > 0,
x=0
x=0
x=0
The main results for this example are listed in Proposition 17, below.
Proposition 17 : (a) (Multiple equilibria) The only economically interesting cases occur
when B 2 > 4C. In this case, we have two distinct real positive equilibria 0 < G (s; t) <
G+ (s; t). These are given by
G (s; t) =
p
1
B
2
B2
4C ,
(40)
and, correspondingly, the levels of private giving are
g (s; t; G (s; t) ; x) =
a (x)
(1
1 s
t) m (x)
b (x)
, x 2 [0; 1] .
G (s; t)
(b) (Increasing and concave desire to contribute) The aggregate desire to give, F , (i) responds positively to taxes, i.e., Ft > 0, (ii) is unresponsive to subsidies, i.e., Fs = 0,
24
For the US, education, health and human services account for the greatest proportion of private giving
after religion; see Table 3 in Andreoni (2006).
25
For example, one would derive a higher utility from one’s own car if a higher quality of public roads
were provided.
28
and, (iii) it is increasing and concave, i.e., FG > 0; FGG < 0 (as in Figures 4, 5, 6).
Furthermore,
G+ ) FG < 1
G=
G ) FG > 1
(c) (Neutral comparative statics) Comparative statics are neutral, i.e., Gs = 0. Thus,
subsidies are ine¤ective in in‡uencing aggregate giving. Furthermore, Gt < 0, G+
t > 0.
From Proposition 17, we know that the economy has two equilibria, as in Figure 8.
1. The low equilibrium is characterized by low voluntary contributions to the public
good, causing low aggregate spending on the public good infrastructure, G . From
(37), to achieve any speci…c utility level, high private consumption expenditure is
needed. From the budget constraint, (7), we see that, as a consequence, less income can be contributed to the public good, which is a strategic complement, thus,
perpetuating the low expenditure on infrastructure.
2. The high equilibrium is characterized by high contributions to the public good, causing high aggregate expenditure on infrastructure, G+ . In turn, this implies that relatively less private consumption expenditure is needed to reach any speci…c utility
level. Hence, relatively more income is left over to donate to charity, perpetuating
high expenditure on infrastructure.
Suppose that the economy is at the low equilibrium, G , and that it is socially desirable
to move the economy to the high equilibrium, G+ . How can this transition be achieved?
From Proposition 17c, we know that incentives in the form of a subsidy will not work
because Gs = 0. From Proposition 17c, Gt < 0, so, from Proposition 13, it would appear
that, at the low equilibrium G , the best feasible policy is no intervention: s = t = 0,
leaving the economy at the low equilibrium G . However, an alternative policy is possible,
as we shall now describe; see Figure 8. Set s; D as follows: s = 0 and D = G (0; 0). The
government budget constraint (4) implies that the tax rate t required to …nance D is given
by
G (0; 0)
:
(41)
t=
M
Since G (0; 0) is an equilibrium, 0 < G (0; 0) < M . Hence, it follows, from Proposition 6a, that 0 < t < 1, which is a feasible tax rate. Since g > 0 it follows, from (5),
that G > D = G (0; 0). Since we are now on the F (0; t; G) locus (see Figure 8) and the
equilibrium aggregate donation G > G (0; 0) > G (0; t), the only possible equilibrium
(guaranteed by Proposition 6) is G = G+ (0; t). Once the high equilibrium is established,
the stability of beliefs ensures that the economy is on the G+ locus. The …scal parameters
s, t can then be adjusted to their socially optimal values as suggested in Section 5. We
now turn to this issue.
29
6.2.1
Socially optimal public policy at the new equilibrium
From Proposition 17b the relevant graph is as in Figure 8. Once the economy has moved
to the new equilibrium, G+ , the direct grant from the government, D = G (0; 0), towards
the public good can be phased out. It is welfare improving to do so because of the normal
comparative statics at the high equilibrium, and the fact that private giving confers warm
glow, while an equivalent amount of direct grants does not.
In the low equilibrium, G , we have seen above that the optimal policy solution is
s = t = 0. However, from Proposition 17c, we know that at the high equilibrium, G+
t > 0,
and so, the comparative static results are reversed from the low equilibrium. The optimal
tax rate, which can be found from (29), balances the loss in private consumption against
the gain arising from the additional amount of the public good. Also, from (28), Vs > 0,
hence, it is welfare improving to provide additional subsidies. Thus, all tax revenues are
used to …nance subsidies on charitable donations. In the socially optimal solution at the
high equilibrium, G+ , therefore, s > 0; t > 0 while D = 0 (see Proposition 14).
6.2.2
A numerical illustration
Consider an economy of identical consumers, each with income m (x) = 50, x 2 [0; 1]
(hence, M = 50). Choose a (x) = 0:1, b (x) = 0:8, x 2 [0; 1] (the relevant utility function
in given in (37)). Suppose that initially, s = t = 0. Then (39), (40) give G (0; 0) = 1 and
G+ (0; 0) = 4, and the feasibility condition (38) is satis…ed (see Figure 8). Correspondingly,
g (x) = 1, g + (x) = 4, x 2 [0; 1].
Suppose that the economy is, initially, at the low equilibrium G (0; 0) = 1. If the move
to a high equilibrium is considered desirable, then, from (41) we get that the required tax
rate, t, is
1
G (0; 0)
=
= 0:02;
(42)
t=
M
50
and the feasibility condition (38) still holds at this tax rate. The government uses its entire
tax revenue tM = 1 = G (0; 0) to make a contribution D = G (0; 0) = 1 to the public
R1
good. Since g > 0; G (0; 0:02) = D + x=0 g (x) > 1 = G (0; 0). In terms of Figure 8,
the economy is on the F (0; 0:02; G) locus. Hence, the only candidate for equilibrium is
G = G+ (0; 0:02). Once the economy is on the high equilibrium, G+ (0; 0:02), the policy
parameters s; t can be adjusted to their socially optimal values. This will involve the
phasing out of the direct grant. Once the …nal position of the, new, socially optimal,
equilibrium is established, the government uses all tax revenues to subsidize voluntary
contributions to the public good. The direct grant, here, is only a temporary measure to
shift the economy from the low equilibrium locus G to the high equilibrium locus G+ .
30
7
Dynamics
In this paper we have concentrated exclusively on equilibrium analysis. For certain purposes, a rigorous analysis of the time path of adjustment may be required. This, however,
will involve specifying a precise list of the information sets of each of the players, the
updating rules, and a learning process26 . This lies beyond the scope of our paper.
The reader of this paper might have wondered about the stability properties of the
equilibria. To study the stability of equilibria, an adjustment process needs to be speci…ed.
A popular adjustment process in economics is the partial adjustment scheme. Let G
be an equilibrium, G ( ) the value of G at time, , and a positive constant. Then the
partial adjustment scheme is given by
G( ) =
[G
G ( )] ,
(43)
from which we can see that G is a stable equilibrium.27 Although a plausibility argument
can be given for (43), it cannot be derived from the behavior of the decision makers in our
model. The reason is that there are no dynamic constraints in our model. So if all agents
believe that the resulting equilibrium will be G , then they will jump straight to G . It is
not clear why they should follow (43).
Another popular adjustment process is
G( ) =
[F (s; t; G ( ))
G ( )] .
(44)
For > 0, (44) says that aggregate giving moves in the direction of the aggregate desire
to give, which appears plausible. It is similar to the tatonnement process in general
competitive equilibrium and the Cournot adjustment process in oligopoly theory. Both
have been extensively studied but are no more regarded as true dynamic processes.
For the moment, let us take (44) seriously. Suppose > 0. If FG < 1 at an equilibrium,
G , then G is stable. This is compatible with either normal comparative statics (if Fs > 0)
or perverse comparative statics (if Fs < 0). If FG > 1, then G is unstable; and this is also
compatible with normal comparative statics (if Fs < 0) or perverse comparative statics
(if Fs > 0). The exact reverse happens if < 0. Therefore, we cannot assume that an
equilibrium with perverse comparative statics is unstable and, therefore, uninteresting.
26
At the moment there is a lack of consensus in the profession about the appropriate model of learning and there is ‘too large’ a variety of models to choose from. These include reinforcement learning,
learning through …ctitious play, learning direction theory, Bayesian learning, imitation learning, experience weighted attraction learning, etc. For a survey of the experimental evidence on these theories, see
Camerer (2003)
27
The solution to the di¤erential equation (43) is given by G ( ) = G + [G(0) G ]e
. Stability
follows because lim !1 G ( ) = G .
31
Moreover, it can be argued that the correct stability notion is saddle path stability.
This requires that the number of stable eigenvalues of a dynamic system (calculated at
equilibrium) be equal to the number of predetermined variables and the number of unstable eigenvalues be equal to the number of jump variables. Our system (44) is one
dimensional (though non-linear). We have no predetermined variables (our model lacks
dynamic constraints). The one variable, G, of (44) is a jump variable (our model allows
agents to choose any g and, hence, any G). Thus saddle path stability (which is arguably
the correct notion of stability here) actually requires that (44) be unstable (as in simple
macro models of exchange rate determination).
To summarize, the only natural dynamics for our model is the very simple one where
agents jump immediately to whatever equilibrium is in line with their expectations.
8
Conclusions
Private philanthropic activity is much studied in economics and is of immense economic
importance. The empirical evidence suggests that the vast bulk of giving activity is undertaken by very large numbers of diverse, dispersed and small givers. For such givers,
the competitive equilibrium model in economics would seem to be a natural …t. However,
the existing literature has typically described the activity of ‘giving’within the ambit of
a Nash equilibrium in strategic giving (often taking the form of a symmetric equilibrium).
Furthermore, the literature has, by and large, restricted itself to a unique equilibrium in
giving. We relax both these features of the existing literature.
Coordination problems and multiple equilibria are the norm in many situations of
economic interest, but they are often ruled out by assumption. It is not surprising that
uncoordinated private giving might lead to the possibility of multiple equilibria. We show
that a necessary condition for multiple equilibria is our proposed condition of strong aggregate complementarity between own-giving and aggregate-giving to a charity.
Once we allow for multiple equilibria that can potentially be ranked according to some
social criterion, the following question arises naturally. If society is stuck at a low equilibrium, characterized by low levels of giving, can public policy help it to attain a high
equilibrium? In the context of private philanthropic activity (and public good contexts),
we show that temporary direct government grants to charities allow for such engineering
of moves between equilibria.28 The signi…cance of this result, to our mind, goes beyond a
charity or a public goods context.
We also perform a welfare analysis and examine the optimality of alternative mixes
of private and public contributions to charity. We show, for some parameter values, that
28
This result, in our model, also has the potential to explain the e¢ cacy of seed money, leadership
contributions, and direct grants by large donors, e.g., the National Lottery (as in the UK).
32
additional incentives to giving can reduce the aggregate of private contributions in equilibrium (perverse comparative statics). In such cases, it might be best to …nance charitable
giving, if required, by direct government grants …nanced through taxation. For other parameter values, however, giving to charity responds well to incentives (normal comparative
statics). In this case, charitable giving should be entirely funded by private individual
contributions, possibly subsidized through taxation. This is welfare improving because
private giving leads to warm glow, while direct government grants do not. The aggregate
desire to give function has played a key role in this paper. It may also be useful in other
contexts.
Throughout we focus on equilibrium analysis. The dynamics of time paths from one
equilibrium to another involve fundamental questions about the precise learning mechanisms to be used. Although progress on learning mechanisms is being made and in due
course such mechanisms may enrich our model, currently such issues lie beyond the scope
of this paper.
9
Appendix: Proofs.
R1
0 we get, from (12), that 0
c (x) dx < M
Proof of Lemma 1: Since c (x)
x=0
m(x) R 1
and, hence, 0 < t
1. From (6), (13), (5) and (5) we get c (x) < M y=0 c (y) dy +
h
i
R1
(x) M
c
(y)
dy
= (1 t) m (x)+tM (x) = (1 t) m (x)+ (x) D (1 t) m (x)+
y=0
(x) G. .
Proof of Lemma 2: Part (a) follows from (2) and (6), respectively. Part (b) follows
from part (a), (14) and the fact that u (c; g; G; x) is a measurable function of c; g; G; x and
an integrable function of x. Part (c) follows from Cohn (1980), exercise 3, p57. .
Proof of Proposition 3:
(a) We have that U , as a function of g, is a continuous strictly concave function de…ned
on a compact interval; recall (14), (15). Hence, a maximum exists and is unique.
(b) Since U is measurable in x and continuous in g, the Carathéodory conditions hold.
Hence, the result follows. See Kharazishvili (2006), Chapter 15 and, in particular, Exercises
2 and 3 pages 312 and 313.
(c) Denote the optimal values by superscript stars. From the restriction (9) we know
that c (x) > c (x) and so g < 1 1 s [(1 t) m (x) + (x) G c (x)].
R1
R1
R1
R1
(d) From (c) we get (1 s) x=0 g dx < (1 t) x=0 m (x) dx+G x=0 (x) dx x=0 c (x) dx.
R1
Hence, from (2), (3), (6) and recalling that c (x) 0, we get (1 s) x=0 g dx < 2M tM ,
from which the required result follows.
(e) follows from (d).
(f) (11) implies that the marginal utility of giving is in…nite as giving approaches zero,
33
hence, g > 0. .
Proof of Lemma 4: If g > 0, then, in the light of Proposition 3c, g is an interior
maximum and part (a) follows. Part (b) follows from (14). Appealing to the implicit
function theorem, or di¤erentiating the identity, (1 s) u1 = u2 , establishes parts (c),
(d) and (e). Part (f) follows from parts (c), (d) and (e), Lemma 2c and Cohn (1980,
proposition 2.1.6, p52) .
Proof of Lemma 5: Follows from (21) and De…nition 2. .
Proof of Proposition 6: In the text. .
Proof of Lemma 7: From De…nitions 3 and 4 we get
R1
G (s; t) = tM + (1 s) x=0 g (s; t; G (s; t) ; x) dx. Implicitly di¤erentiate this identity
R1
R1
@g
1
dx
=
+
g dx . .
+
G
G
and rearrange to get x=0 @g
s
s
@s
@G
1 s
x=0
Proof of Proposition 8: Let G be an equilibrium at which FG 6= 1. Then, from
Proposition 6c, G is isolated. By De…nition 4, G = F (s; t; G ). Di¤erentiating this
implicitly, and rearranging, gives the required results. .
Proof of Corollary 9: Immediate from Proposition 8 and De…nition 6. .
Proof of Propositions 10 and 11: Obvious from Proposition 8a,b, respectively. .
Proof of Lemma 12: From the government budget constraint, (18), evaluated at equiR1
librium, we get D (s; t; G (s; t)) = tM s x=0 g (s; t; G (s; t) ; x) dx. Rewrite this with
D and t as the independent instruments, to get
R1
D = tM s (D; t) x=0 g fs (D; t) ; t; G (s (D; t) ; t)g dx. Di¤erentiate implicitly with reR1
@g @G
@g
@s
spect to D and rearrange
to get @D
= 1= x=0 g +
i s @s + @G @s dx. Use Lemma 7
hR
R
1
1
@s
= 1= x=0 g dx + 1 s s Gs + x=0 g dx . Simplify to get the required result.
to get @D
.
Proof of Proposition 13: Let (s ; t ) maximize social welfare (27). By assumption,
u1 > 0, u3 0, ! (x) > 0, (x) 0 and m (x) 0 with m (x) > 0 on a subset of positive
0 then, from (29), it follows that Wt < 0. Hence, necessarily, t = 0.
measure. If Gt
Since D
0, s
0 and g > 0 on a subset of positive measure, it follows from the
government budget constraint (4) that s = D = 0. .
Proof of Proposition 14: Let (s ; t ) maximize social welfare (27). By assumption,
u1 > 0, u3 0, ! (x) > 0, (x) 0 and g
0 with g > 0 on a subset of positive measure.
Hence, if Gs
0 then (28) implies that Ws > 0. Viewing D and t as the independent
government instruments we get, @W (s(D;t);t)
= Ws (s (D; t) ; t) @s(D;t)
. From this and Lemma
@D
@D
R
1
12 we get, @W
= (1 s) Ws = sGs + x=0 g dx . Since Ws > 0, Gs
0, g
0 with
@D
< 0. Suppose D > 0. Since
g > 0 on a subset of positive measure, it follows that @W
@D
@W
< 0, it follows that W can be increased further by reducing D, which contradicts the
@D
assumption that (s ; t ) maximizes social welfare. Hence, D = 0. .
Proof of Proposition 15: From Proposition 8a, Gs (s; t) = 1 FFs G . So when Fs 0 and
34
FG < 1 at G , or if Fs 0 and FG > 1 at G , we get Gs (s; t) 0. Proposition 14 then
implies that D = 0. .
Proof of Proposition 16: Maximizing (30) subject to (7) gives
g (G; s; t; x) =
1
1
t
m (x)
s
c (G; s; t; x) =
1
, x 2 [0; k] ,
a (x) G
(45)
1 s
, x 2 [0; k] .
a (x) G
(46)
From (31) and (45) we see that g (G; s; t; x) > 0 and from (46) c (G; s; t; x) > 0. On the
other hand, from (32) we get that
g = 0, x 2 (k; 1].
(47)
From (19), (33), (34), (45), the aggregate desire to give to charity is
F (s; t; G) = m + t (M
1
m)
s
G
(48)
A,
and, hence,
Ft = M
m
(49)
0.
The inequality in (49) is strict for m < M , i.e., k < p. Direct di¤erentiation of (48) proves
part (b), i.e., Fs > 0, FG > 0, FGG < 0. Di¤erentiating (48) we get
A
A
A
> 0, FG = (1 s) 2 > 0, FGG = 2 (1 s) 3 < 0.
G
G
G
From De…nition 4 and (48), an equilibrium G, must satisfy the quadratic equation
Fs =
G2
[m + t (M
m)] G + (1
(51)
s)A = 0.
The quadratic equation (51) has the solutions
q
1
m + t (M m)
G (s; t) =
[m + t (M
2
m)]2
(50)
4(1
s)A .
(52)
For real roots we need [m + t (M m)]2
4(1 s)A. If [m + t (M m)]2 = 4(1 s)A
then, from (50), FG = 1. From Proposition 8a,b it would follow that Gs , Gt are unde…ned.
Hence, the only economically interesting cases occur when,
[m + t (M
m)]2 > 4(1
s)A,
(53)
in which case we have two distinct real positive equilibria
0 < G (s; t) < G+ (s; t) ,
35
(54)
and, correspondingly,
g
s; t; G (s; t) =
1
1
t
m (x)
s
1
, x 2 [0; k] ,
a (x) G (s; t)
(55)
g = 0, x 2 (k; 1].
(56)
p
p
p
Using the fact that for real numbers a; b, a > b: a b > a
b, as well as (50) and
(52)-(54), we get
G = G+ ) FG < 1, G = G ) FG > 1.
(57)
From Proposition 8a,b, (49), (50), (57) we get
G+
s > 0, Gs < 0,
(58)
G+
t > 0, Gt < 0 (for m < M , i.e., k < p),
(59)
Gt = 0 (for m = M , i.e., k = p).
(60)
Proof of Proposition 17: Applying Lemma 4b to the utility function (37), and using
the budget constraint (7), which must hold with equality, gives
g (s; t; G; x) =
c (s; t; G; x) = [1
a (x)
(1
1 s
a (x)] (1
t) m (x)
t) m (x)
b (x)
,
G
b (x)
b (x)
.
+
G
G
(61)
(62)
From (38) and (61), (62), we see that g (s; t; G; x) > 0 and c (s; t; G; x) > b(x)
. FurG
thermore, it is straightforward to verify that the second order conditions also hold. Hence,
given s; t; G; g (s; t; G; x), c (s; t; G; x) maximize utility (37) subject to the budget constraint (7), and are unique.
Substituting from (61) into (19), the aggregate desire to give, F (s; t; G) is:
Z 1
Z 1
Z
1 1
F (s; t; G) =
a (x) m (x) dx + t
[1 a (x)] m (x) dx
a (x) b (x) dx. (63)
G x=0
x=0
x=0
From (63) we get:
Z
1
1
a (x)] m (x) dx > 0; FG = 2
G
Fs = 0, Ft =
[1
x=0
Z 1
2
FGG =
a (x) b (x) dx < 0.
G3 x=0
Z
1
a (x) b (x) dx > 0;
x=0
(64)
From (64) and Proposition 8a, we get
Gs =
Fs
= 0.
1 FG
36
(65)
From (64) and Proposition 15, it follows that, at a social optimum, D = 0, i.e., no direct
grant from the government to the charity is involved. Giving to charity is entirely funded
by private donations, which are subsidized from taxation if s > 0, t > 0.
To make further progress, we need to determine the equilibrium values of G. From
(19), (63) and De…nition 4, the equilibrium values of G are the solutions to the equation
Z 1
Z
Z 1
1 1
a (x) m (x) dx + t
[1 a (x)] m (x) dx
a (x) b (x) dx.
(66)
G=
G x=0
x=0
x=0
Substituting (39) in (66) we get
G2
(67)
BG + C = 0,
with solutions
i
p
1h
B
B 2 4C .
(68)
2
If B 2 < 4C, then no equilibrium exists. If B 2 = 4C then a unique equilibrium exists, and
p
is G = B2 = C: But then, from (64), (39), FG = 1. In this case, neither Gs nor Gt are
de…ned, see Proposition 8a,b. Hence, the only interesting case is when B 2 > 4C.
In this case, (67) has two distinct real positive roots:
G (s; t) =
0 < G (s; t) < G+ (s; t) ,
(69)
and, correspondingly,
a (x)
(1
1 s
b (x)
, x 2 [0; 1] .
(70)
G (s; t)
p
p
p
Using the fact that for real numbers a > b > 0: a b > a
b, as well as (64),
(39) and (68) - (69), we get
g (s; t; G (s; t) ; x) =
t) m (x)
G = G+ ) FG < 1, G = G ) FG > 1.
From (64), (71) and Proposition 8b, we get G+
t > 0, Gt < 0.
(71)
.
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