Efficient Public Good Contributions by Impurely Altruistic Individuals
By
Emilson C. D. Silva*
School of Business, University of Alberta, Edmonton, Alberta, Canada
E-mail: [email protected]
Xie Zhu
Department of Economics, Oakland University, Rochester, MI 48309-4401
E-mail: [email protected]
April, 2017
Abstract: We show that Becker’s Rotten Kid Theorem applies to situations involving impurely altruistic individuals who
voluntarily contribute to a public good-individuals who we call “reluctant contributors”. Reluctant contributors care about
prestige levels derived from their contributions. Reluctant contributors behave as “Good Samaritans” whenever they
make contributions that add up to the socially optimal level of the public good and marginal rates of substitution between
prestige and the numeraire good are equalized across contributors. There are two recipes for turnovers. In the first, the
government redistributes income and prestige levels across contributors after contributions are made. In the second, the
government redistributes income after contributions and exchanges of prestige units take place. Prestige units can be
exchanged via brokerage services provided by a charity. We demonstrate that brokerage services in the market for
prestige provide a rationale for the existence of charities. This paper also shows that a policy of subsidizing contributions
is neutral in the presence of income redistribution.
*
Corresponding author.
Efficient Public Good Contributions by Impurely Altruistic Individuals
1.
Introduction
Becker’s infamous Rotten Kid Theorem is as follows: “Each beneficiary, no matter how selfish, maximizes the family
income of his benefactor and thereby internalizes all the effects of his actions on other beneficiaries” [Becker (1991), pp.
288]. Bergstrom (1989) formally demonstrates that the Rotten Kid Theorem holds as long as a benevolent parent
considers his children - the rotten kids - as normal goods and the rotten kids’ utilities are transferable.1 More recently,
Cornes and Silva (1999) extended Bergstrom’s analysis by showing that rotten kids will behave as Becker proposed
whenever they care about the sum of their individual contributions to a pure public good (bad) and each rotten kid makes
a strictly positive (negative) contribution.2
In this paper, we examine the circumstances under which voluntary contributions to a public good by individuals
who care not only about the sum of their individual contributions, but also about their contributions per se - individuals
who we shall call “reluctant contributors” - yield a socially optimal level of the public good. In other words, we
investigate the circumstances under which reluctant contributors behave as “Good Samaritans”.3
1
See also Bernheim, Schleifer and Summers (1985) for circumstances where the Rotten Kid Theorem does not apply.
Chiappori and Werning (1999) identify circumstances leading to interior and non-interior solutions.
3
One should not confuse our terminology - “Good Samaritans” - with the well known “Samaritan’s Dilemma” of
Buchanan (1975). This dilemma says that an individual may become impoverished if he anticipates that his benevolent
government is willing to help him out upon hard times. Bruce and Waldman (1990) show that in a two-period model
parental altruism can result in the Samaritan’s Dilemma. This possibility is not present in our analysis because we
consider a one-period model. In future work, we plan to extend our model to two periods and examine whether or not the
“Rotten Kid Theorem Meets the Samaritan’s Dilemma”, as in Bruce and Waldman.
2
1
By postulating that contributors care about both the public good and the act of contributing per se, our approach
builds on the impure public good model developed by Cornes and Sandler (1984) 4 and is closely related to the impure
altruism model of Andreoni (1989, 1990)5. As Harbaugh (1998a) points out, contributors may derive positive utility from
the act of contributing per se because publicly announced contributions endow contributors with “prestige”. 6 Indeed,
Harbaugh (1998b) finds empirical support while Andreoni and Petrie (2004) providing experimental evidence for the
hypothesis that donors derive utility from prestige.7 As in Harbaugh (1998a, 1998b), we shall adopt the convention that
contributors obtain prestige from having their contributions publicly known. Society or private charitable organizations
announce the levels of contributions after they are made. To simplify exposition, we shall refer to a contributor’s private
utility from making a contribution as utility from prestige.
This paper makes three main contributions to the literature, two of which are positive findings and one is
negative. Starting with the good news, we show that reluctant contributors behave as “Good Samaritans” under two
situations. In the first situation, reluctant contributors make strictly positive contributions to the public good in
anticipation that their benevolent and egalitarian government will redistribute income and levels of prestige. The
government redistributes levels of prestige by matching contributions and endowing contributors with prestige levels
4
The joint product impure public good model established by Cornes and Sandler (1984) has been developed to
analyze a variety of impure public good provision problems ranging from climate protection to fighting terrorism.
In the case of climate protection, reduction of greenhouse gas emissions generates not only global public benefits of
mitigating global warming, but also local ancillary benefits such as improved health from reduction in conventional air
pollutants - particulates, nitrogen dioxides and sulfur dioxides. See Caplan and Silva (2005), Loschel and Rübbelke
(2005), Rübbelke (2005a), and Silva and Zhu (2005, 2011, 2015) for climate policy analyses in an impure public good
framework. In fields other than climate protection, recent applications of the impure public good model include Kotchen
(2005), who studies environmental friendly consumption; Mazzanti, Cainelli, and Mancinelli(2005), who study firms’
R&D decisions and social capital; Rübbelke (2005b), who studies terrorism. For more examples, see Sandler (1997) and
Sander and Hartley (2001). For provision of impure public goods in monopolistic competitive markets, see Cavalletti,
and Levaggi (2002).
5
Andreoni (1989, 1990)’s warm glow model made significant progress to the charitable giving literature by treating
individual donations as contributing to an impure public good - individuals are motivated to give by an intrinsic feeling of
warm glow from their individual gifts as well as the sum of their donations. See Andreoni (2004) for a review on the
theoretical foundations, as well as the empirical and policy studies on charitable giving over the past 25 years. Some
important questions investigated by the literature as reviewed in Andreoni (2004) are, e.g., optimal subsidy to giving,
charitable contributions of time and money, fund-raising activities of charities as strategic players, measuring price and
income elasticity of giving, etc.
6
Similarly, donors may want others to know about their donation due to the desire to demonstrate wealth and status
(Glazer and Konrad (1996)), or to receive social approval (van de Van (2002)). Romano and Yildirim (2001) formulate a
general utility function which could be used to model warm-glow or prestige and reputation motives for donation.
7
See, e.g., Rege and Telle (2004), Masclet et al. (2003), among others, for how public recognition of contributions
and social approval can affect voluntary private contributions to public goods in experimental settings.
2
associated with their overall contributions to the public good. The matching grant changes the amount of prestige
received by the grantee by an amount exactly equal to the grant amount. Therefore, such grants, which can be interpreted
as grants in kind, have the key property of endowing grantees with levels of prestige that surpass (or fall short) of levels
of prestige that they can obtain on their own through their original contributions to the public good.
In the second situation, a market for prestige is created. Reluctant contributors make strictly positive
contributions to the public good and trade prestige levels in anticipation that their benevolent and egalitarian government
will redistribute income. The market for prestige is created by a non-profitable organization - a charity - whose
responsibilities are to provide the public good and act as a “broker” in the market for prestige. Reluctant contributors
utilize the charity to purchase and sell units of prestige, in excess of their direct contributions to the public good. Since
we show that the resulting allocation of resources in this setting is socially optimal, we obtain a true Rotten Kid Theorem
for the impure altruism model.
Our bad news is the neutrality of a governmental policy of subsidizing contributions to the public good. Such a
policy is shown to have no real effect on the allocation of resources whenever reluctant contributors anticipate that, after
they contribute to the public good, the government will redistribute income. This result is novel for two main reasons.
First, the policy neutrality results available in the literature are derived with the assumption that the center or the altruistic
head of a household - i.e., a benevolent or altruistic agent - is able to precommit to lump-sum or distortionary income
redistribution. This is a key assumption underlying the Ricardian Equivalence Theorem (see, e.g., Barro (1974) and
Bernheim and Bagwell (1988)). It is also an important assumption - even if not explicitly stated - underlying the
neutrality of income redistribution in economies with pure public goods (see, e.g., Warr (1982), Andreoni (1989),
Bernheim (1986) and Boadway, Pestieau and Wildasin (1989)). In our model, the government plays a three-stage game
with the reluctant contributors. This game is built after a three-stage game developed by Andreoni and Bergstrom (1996).
The government announces subsidies and tax rates in the first stage, the reluctant contributors make their contributions to
the public good in the second stage, and the government redistributes income amongst the reluctant contributors in the
third stage. Therefore, the timing of the game played between the government and the reluctant contributors in our
analysis is essentially different from the games examined by the policy neutrality literature. Indeed, our neutrality result
comes from the fact that the government’s income-redistribution policy takes effect after subsidies, tax rates and
3
contributions to the public good have been made. The second reason for the novelty of our neutrality result is that it is
derived in a setting where the public good is impure. Warr’s well known neutrality result, for example, depends crucially
on the assumption that the privately provided public good is pure.
As Bergstrom (1989) alerted us, the income-redistribution policy of a benevolent and egalitarian government,
which is unable to precommit to incentive schemes, will not generally yield the Rotten Kid Theorem. But, as we shall
demonstrate, this policy - being generally non-neutral - is an essential ingredient of a successful recipe for turnovers.
Income redistribution when mixed with either prestige redistribution or prestige exchange is capable of transforming
reluctant contributors into Good Samaritans!
We organize the paper as follows. Section 2 presents the socially optimal allocation. Section 3 shows us the
conditions that characterize the decentralized allocation, where the government does not intervene. In section 4, the
government is endowed with a policy instrument to redistribute income. Section 5 examines the effectiveness of a
governmental policy that subsidizes contributions to the public good in the presence of income redistribution. In section
6, the government is endowed with policy instruments to redistribute income and prestige levels. Section 7 introduces a
market for prestige and analyzes the allocation of resources where contributors trade prestige levels and provide
contributions to a charity in anticipation of the income-redistribution policy of the government. Section 8 concludes the
paper.
2.
Social optimum: the government’s most preferred allocation
Consider an economy with n individuals, indexed by j, j = 1,..., n and a benevolent and egalitarian government. There are
two goods. A private good can be used for either consumption or as an input in the production of a public good. For
simplicity, we assume that it takes one unit of the private good to produce one unit of the public good. The public good
level, G, equals ∑jgj.
Contributor j is assumed to derive the following utility from contributing gj units of the private good to the
public good and consuming xj units of the private good and G units of the public good:

u j x j , pj , G
  u  x , p  g , G  ,
j
j
j
j = 1 , ... , n ,
where, for each j, u j is increasing in all arguments and strictly quasiconcave. The function p transforms contributor j’s
contribution to the public good into units of prestige, pj. For simplicity, we assume that
4
 
pj  p g j = g j ,
j = 1 , ... , n .
Substituting this into contributor j’s utility function, enables us to write his utility function as follows:
u
j
x , g , G  ,
j
j
j = 1 , ... , n .
The government’s objective function is a strictly quasiconcave transformation W of individual utilities:
W
u 
1
n
x1 , g1 , G  , ... , u

xn , gn , G
 ,
(1)
where Wj=∂W/∂u j > 0 for all j. In words, the government cares about the welfare of each individual in society.
Let us suppose that the government is persuasive enough so that it is able to convince all contributors how they
should allocate their resources. The government determines its most preferred allocation - the social optimum - and
instructs all contributors to make their choices so that the socially optimal conditions are satisfied. The contributors
implement the socially optimal allocation by religiously following the instructions dictated by the government. All
contributors will obtain prestige from their contributions because they will contribute to the public good on their own.
The social optimum can be obtained by choosing {xj, gj}j
= 1,...,n
to maximize (1) subject to the resource
constraint:
j x j + j g j = j Ij ,
or
j x j + G = I ,
(2)
where Ij > 0 denotes individual j’s income level and I =∑j Ij represents the aggregate income level. Let λ be the
Lagrangian multiplier associated with (2). Assuming the solution is interior, the first order conditions yield (2) and the
following equations:
j
j = 1 , .... , n ,
W j ux =  ,
j
k
n
j = 1 , ... , n .
Wj up + k = 1 Wk uG =  ,
(3)
(4)
Equations (3) and (4) yield
j
k
up
uG
+ nk = 1
=1,
j
ux
ux supk
j = 1 , ... , n .
(5)
Equation (3) tells us that the marginal social utilities of income should be equalized. Income redistribution is socially
desirable because social preferences are strictly convex. Equations (5) show that the typical Samuelson condition for
optimal provision of a pure public good should be modified to account for the private benefits associated with
5
contributions to the public good. For each j, the first term of the left-hand side of (5) is individual j’s marginal rate of
substitution between prestige and the private good. Hence, the marginal rates of substitution between prestige and the
private good should be equalized across contributors:
j
i
up
up
= i ,
j
ux
ux
3.
i , j = 1 , ... , n .
(6)
Reluctant contributors on their own
Let us now examine a situation where the contributors voluntarily decide how much to contribute to the public good
without governmental intervention. Our objective in this section is to illustrate the shortcomings generated with
decentralized behavior. We wish to clearly identify which incentives governmental policy should promote in order to
induce reluctant contributors to voluntarily contribute to the public good at the socially optimal level.
All contributors play a simultaneous Nash game, whereby contributor j chooses { xj, gj } to maximize
u
subject to:
j
x,g,g+G 
j
x j + g j = Ij ,
j
-j
j
xj  0 ,
gj  0 ,
(7)
taking the choices of all other contributors as given. The contributions of all contributors other than j sum up to yield G-j.
Assuming interior solutions,8 the Nash equilibrium is characterized by (7), for all j, and the following equations:
j
j
u p uG
+ j =1,
j
ux ux
j = 1 , ... , n .
(8)
Equations (8) show us that each contributor neglects the positive effect of his contribution on each other. The
decentralized allocation deviates from the socially optimal allocation in three ways: (1) it violates the modified
Samuelson condition (5); (2) it does not yield equalization of marginal social utilities of income; and (3) it does not imply
equalization of marginal rates of substitution between prestige and the private good. Governmental policy should,
therefore, induce reluctant contributors to voluntarily choose levels of contributions that satisfy (5) while equalizing
marginal social utilities of income and marginal rates of substitution between prestige and the private good.
8
As in Andreoni (1990), we shall only consider equilibria with non-zero contributions. This restriction is
not without loss of generality, since the efficiency properties of the equilibria studied are sensitive to whether or not
the equilibria are characterized by corner solutions. By restricting our attention to equilibria with positive
contributions, we wish to illustrate the set of circumstances under which voluntary contributions and governmental
policy implement the socially optimal allocation described by (2) - (5).
6
4.
Reluctant contributors anticipate income redistribution
As discussed in the introduction, Cornes and Silva (1999) show that voluntary contributions to a pure public good yield
an efficient aggregate level of provision provided that each contributor makes a positive contribution to the pure public
good and a benevolent agent, who cares about all contributors, redistributes the incomes of contributors after
contributions are made. Such a finding motivates us to study the implications of income redistribution in the current
setting, with an impure public good. As we shall see, the fact that all contributors are impurely altruistic implies that
income redistribution will generally fall short of providing contributors with the right incentives to voluntarily provide a
socially optimal level of the public good. However, as we shall also demonstrate, income redistribution will be an
essential policy tool in the arsenal of policy instruments that the government should use to induce reluctant contributors
to behave as Good Samaritans.
Assume that the government’s sole economic role is to redistribute income across all individuals. Income
redistribution takes place after the reluctant contributors choose their contributions to the public good. When contributors
make their choices, they know that the government will subsequently redistribute income in order to maximize its
egalitarian objective function. That is, all contributors correctly anticipate that income transfers will be effected to
equalize marginal social utilities of income, as in (3). In game theoretical terms, the contributors and the government play
a two-stage game, whereby the contributors choose their contributions to the public good in the first stage and the
government chooses income transfers in the second stage, after having observed the levels of contributions. The
equilibrium concept utilized is subgame perfect equilibrium.
Let τj denote the income transfer received (if positive) or paid (if negative) by contributor j. This individual’s
budget constraint is
x j + g j = I j + j ,
j = 1 , ... , n .
(9)
Since ∑jτj = 0, we obtain the economy’s resource constraint (2) by summing up equations (9) over all j.
Let g-j denote the (n-1) - tuple of contributions which does not include gj. We are now able to formally describe
the two-stage game C which is called Game 1 C as follows:
Game 1
7
Stage 1: Taking g-j as given, contributor j chooses gj > 0 to maximize
u
j
 x g
j
j
, g- j , g j + G- j , I
, g
j
, g j + G- j
,
j = 1 , ... , n .
(10)
Stage 2: The government chooses {xj }j = 1,...,n to maximize (1) subject to (2) after observing {gj, g-j, G}.
It is easy to explain the crucial features of this game. The government, in the second stage, knows both the
prestige levels reached by contributors, {pj }j = 1,...,n, and the aggregate level of the public good provided, G. Since pj = gj
for all j, the government, therefore, knows {gj, g-j, G}. As we mentioned above, we obtain the economy’s resource
constraint (2) by summing up equations (9) over all j. The government chooses nonnegative {xj }j = 1,...,n to maximize (1)
subject to (2). Let λ be the Lagrangian multiplier associated with constraint (2). The first order conditions for this
problem yield the constraint (2) and equations (3); that is, a system of n + 1 equations, with n + 1 endogenous variables
{λ , xj }j = 1,...,n and n + 2 exogenous variables {G, I, gj, g-j }j = 1,...,n. Since the objective function is strictly quasiconcave
and constraint (2) is linear, the solution to the government’s problem - assumed to be interior - is a unique constrained
global maximum. Hence, we may apply the Implicit Function Theorem to obtain the n + 1 implicit functions λ (gj, g-j, G,
I), xj(gj, g-j, G, I ), j = 1,...,n. The functions xj(gj, g-j, G , I) tell us how the government’s transfers respond to changes in
the levels of prestige, aggregate public good and aggregate income. Substituting contributor j’s transfer function into his
utility function yields his objective function (10). The problem this contributor faces in the first stage is now clear and
well defined. He chooses his most desired contribution level anticipating that there will be transfers of the numéraire
good amongst contributors in order to satisfy the government’s income redistribution objective.
The following result is immediate.
Proposition 1. The subgame perfect equilibrium for Game 1 is given by ∑jτj = 0, (2), (3) and the following conditions:
j
i
k

 xk 
up
up
u
+ nk = 1 Gk = 1 -  in_ j i + nk = 1
 ,
j

psubk
ux
ux
ux


j = 1 , ... , n .
(11)
Proof. Consider the second stage. Maximization of (1) subject to (2) yields the constraint and equations (3) as the set of
first order conditions. As we explained above, this set of first order conditions enable us to derive the implicit functions
xj(gj, g-j, G , I). Substituting these functions into constraint (2), we obtain:

 j x j g j , g- j , G , I
Differentiating (12) with respect to G yields:
8
+ G  I.
(12)
j

 x j g j , g- j , G , I
G
 = -1.
(13)
Consider the first stage. Assuming interior solutions, the first order conditions can be written as follows:
j
j
  xj  xj 
u p uG
+
= - 
+
 ,
j
j
ux ux
  pj G 
j = 1 , ... , n .
(14)
Summing up equations (14) over all j, acknowledging equation (13) and rearranging, we obtain equations (11).
The subgame perfect equilibrium for Game 1 deviates from the socially optimal allocation in that the modified
Samuelson condition (5) is not satisfied and marginal rates of substitution between prestige and the numéraire good are
not equalized. As we shall demonstrate below, equalization of marginal rates of substitution between prestige and the
numeraire good is a necessary condition for the efficiency of an equilibrium of any game where reluctant contributors
voluntarily decide how much to contribute to an impure public good. If, as in the setting considered by Cornes and Silva
(1999), contributors did not care about prestige at all, the income redistribution policy of the government would suffice
for contributors to contribute the socially optimal amount of the public good. The question we investigate in the next
section is whether the government may induce reluctant contributors to behave as Good Samaritans with a policy of
subsidizing contributions, in addition to its income redistribution policy.
5.
Policy neutrality: subsidizing contributions in the presence of income redistribution
Andreoni (1990) finds that contributions of impurely altruistic individuals - i.e., reluctant contributors - increases with the
rate at which contributions are subsidized by the government. This result is derived from straightforward comparative
statics applied to the equilibrium conditions for a simultaneous Nash noncooperative game played by contributors.
Because the government is not a strategic player in his analysis, taxes and subsidies are exogenously given and income
redistribution is not considered.
We extend his analysis in two significant ways. First, the government is given authority to endogenously
determine: (i) the subsidy rate applied to contributions, (ii) the taxes levied to finance the subsidy scheme and (iii) the
income transfers needed to promote income redistribution. Second, we assume that the government is able to pre-commit
to the subsidy scheme and taxes. That is, the government announces both subsidies and taxes before contributors decide
how much to contribute to the public good. To be precise, we consider a three-stage game as follows. In the first stage,
the government announces subsidies and taxes. All contributors observe these quantities and, in the second stage, choose
9
their contributions. In the third stage, the government chooses income transfers.
Let s  [0, 1] denote the standard subsidy rate and tj denote contributor j’s tax rate, j = 1,...,n. This contributor’s
budget constraint is as follows:
x j + (1 - s) g j = I j + j - t j s G ,
j = 1 , ... , n .
(15)
Contributor j is taxed for a share tj of the cost of the subsidy scheme. Hence, tj> 0 and ∑j tj = 1. Since ∑jτj = 0, we obtain
(2) by summing up equations (15) over all j. We are now ready to present the formal description of the three-stage game.
Game 2
Stage 1: The government chooses {s, t }, where t = (t1,...,tn ), to maximize
W  u1  x1  g1  s, t , g-1  s, t , G  s, t  , I  , g1  s, t  , G  s, t   ,.. , un  xn  gn  s, t , g-n  s, t , G  s, t  , I  , gn  s, t  , G  s, t
(16)
subject to: 0  s  1 , j t j = 1 , t j  0 , j = 1 , ... , n .
Stage 2: Having observed {s, t } and taking g-j as given, contributor j chooses nonnegative gj to maximize (10).
Stage 3: The government chooses {xj }j = 1,...,n to maximize (1) subject to (2), knowing {s, t, gj, g-j, G}.
Stages 2 and 3 of this game are identical to stages 1 and 2 of Game 1, respectively. The additional set of
exogenous variables in this game - i.e., the n+1 exogenous variables, (s, t) - are only relevant in the first stage of the
game. This follows from the fact that the government is able to redistribute the numéraire good across individuals in the
third stage of the game, irrespective of subsidies and tax rates. Applying the same logic of the previous section, the first
order conditions for the third stage will enable us to define the functions xj (gj, g-j, G, I), j = 1,...,n. Inserting these
functions into the contributors’ utility functions yields their objective functions (10) in the second stage of the game.
Assuming interior solutions to the maximization problems of the second stage, as in the previous section, we obtain
equations (14). Provided the sufficient second order conditions are satisfied and the Nash equilibrium of the second stage
is (locally) unique, we will again be able to apply the Implicit Function Theorem to define the functions gj (s, t). Hence,
the function G(s, t)=∑j gj (s, t) is well defined. Substituting these functions into the contributors’ utility functions and the
implied utility functions into the social welfare function (1), we obtain (16). The problem facing the government in the
first stage is now also well defined.
10

Game 2 is purposely built very similar to a three-stage game considered by Andreoni and Bergstrom (1996). Our
construction of the subsidy scheme and tax rates mirrors theirs. Our precommitment and timing assumptions for the first
and second stages of the game are also identical to their own. Unlike us, they assume that contributors are purely
altruistic and do not consider income redistribution. The third stage of their game involves no strategic choice, but simply
the collection of taxes and subsequent payment of subsidies. As they acknowledge, the third stage of their game can be
collapsed into the second stage without altering the Nash equilibrium for the second stage. Their analysis focuses on the
Nash equilibrium for the second stage.
Before we characterize the subgame perfect equilibria for Game 2 - as we will see there is a continuum of
equilibria - as well as some of their key features, it is important to note that, because the government makes interpersonal
income transfers after all the other strategic variables have already been determined, there is no loss in generality in
letting t1 = 1, ti = 0 for I = 2,...,n, and considering the equilibria for Game 2 given these tax rates. All contributors know
that the government’s income redistribution policy will equalize marginal social utilities of income in spite of the values
set for the tax rates.
We are now ready to state our negative result:
Proposition 2. (Policy Neutrality) A policy of subsidizing contributions is completely neutral in the presence of income
redistribution.
Proof. Consider the third stage of the game. As we mentioned above, the conditions that characterize the equilibrium in
this stage are exactly the same as the conditions that characterized the equilibrium for stage 2 of Game 1. It is important
to note, however, that the equilibrium in stage 3 also satisfies the budget constraints (15). Since we assume that t1 = 1, ti
= 0 for i = 2,...,n, we can write:
1  s , g1 , g- 1 , G , I , I1
i  s , gi , g- i , G , I , Ii


x1  g1 , g- 1 , G , I
xi  gi , g- i , G , I
Note that:
11
-
-
I1 + s G + (1 - s) g1 ,
(17)
Ii + (1 - s) gsubi ,
i = 2 , ... , n .
 1
 1
 x1
 x1
+
=
+
+1,
 p1
G
 p1
G
(18)
 i
 i
 xi
 xi
+
=
+
+1- s ,
 pi
partialG
 pi
G
i = 2 , ... , n .
Differentiating equations (17) with respect to s yields:
 1
= G - g1  G- 1 ,
s
(19)
 i
= - gsubi ,
i = 2 , ... , n .
s
Consider the second stage. Assume that the Nash equilibrium for the second stage is (locally) unique. Equations
(14) can be utilized to implicitly define the functions gj(s). Hence, the function G(s) =∑j gj (s, t) is also well defined.
Inserting these functions into equations (17) yields:
1  s , g1 (s) , g- 1 (s) , G(s) , I , I1
i  s , gi (s) , gsub- i(s) , G(s) , I , Ii


x1  g1 (s) , gsub- 1(s) , G(s) , I
xi  gi (s) , g- i (s) , G(s) , I
 - I1 + s G(s)
+ (1 - s) g1 (s) ,
 - Isubi + (1 - s) gi (s) ,
(20)
i = 2 , ... , n .
Differentiating equations (20) with respect to s and utilizing (18) and (19), we obtain:
  x1
 x1
 1  1
n
+
+ s i = 2 
G
 pi G
  pi
  d gi 
 1
 
- G- 1 = 0 ,
 =

s
  ds 
(21)
  xi
xsubi
 i
 i
+
k _ i 


G


G
p
p
k
k

  d gsubk 
 i
 
+ gi = 0 ,
 =
 
ds
s


i = 2 , ... , n .
It should be clear that the solution to the system of equations (21) is given by:
d gj
ds
= 0,
j = 1 , ... , n .
(22)
Hence,
d gj
dG
= nj = 1
= 0.
ds
ds
(23)
In the first stage of the game, we have the following implied social welfare function:
W  u1  x1  g1 ( s ), g-1 ( s ), G ( s ) , I , g1 ( s ), G ( s ) ,.. , un  xn  gn ( s ), g-n ( s ), G ( s ) , I , gn ( s ), G ( s )  
The Lagrangian for the maximization problem in the first stage is as follows:
12
(24)
L = W  u1  x1  g1 ( s ), g-1 ( s ), G ( s ) , I , g1 ( s ), G ( s ) ,.. , un  xn  gn ( s ), g-n ( s ), G ( s ) , I , gn ( s ), G ( s )   +  s +  (1 - s)
The first order conditions are:
(25)
-=0,
s = 0 ,
  0 ,
s  0 ,
(26)
(27)
 (1 - s) = 0 ,
 0,
1- s  0 .
In writing (25), we made use of equations (22) and (23). Conditions (25) - (27) imply that any s  [0, 1] solves the
maximization problem.
Proposition 2 tells us that subsidies have no real effect on the allocation of resources. It follows that the level of
public good in any equilibrium with a positive subsidy rate will correspond to the level of public good in the equilibrium
where this rate is zero. Since the latter equilibrium allocation is equivalent to the equilibrium allocation described by
Proposition 1, the ability of the government of pre-committing to the subsidy scheme is of no real value. The equilibria
for Game 2 are as inefficient as the equilibrium for Game 1.
This begs the question: Is there some governmental policy other than subsidizing contributions which, when
implemented together with the income redistribution policy, induces reluctant contributors to voluntarily provide the
socially optimal level of the public good? As we shall demonstrate in the following section, the answer is an affirmative
one. We show that a governmental policy of actively providing contributors with prestige C a commodity whose market
is missing C promotes the right incentives for reluctant contributors to behave as Good Samaritans.
6.
A winner recipe’s key ingredients: income redistribution and prestige grants
Consider now a setting where the government takes two actions after contributions are made. First, as before, it
redistributes income amongst the reluctant contributors. Second, it allocates additional prestige units to contributors with
matching grants. The sole purpose of the “prestige policy” is to endow contributors with more (or less) prestige than the
prestige levels they can achieve on their own through their contributions to the public good.
Let mj denote the matching grant amount received by contributor j, j = 1, ..., n. Contributor j’s utility function is
now written as follows:
u
j
x , p , G 
j
j

= uj xj , p
 g , m , G  ,
j
j
j = 1 , ... , n ,
where, for simplicity, we assume that each contributor j derives prestige from his actual contribution to the public good,
defined as the sum of his own contribution and the matching grant received from the government:
13
pj = p
 g ,m  = g
j
j
j
+ mj ,
j = 1 , ... , n .
Substituting these expressions into the contributors= utility functions, enables us to write the utility functions as follows:
u
j
x , p , G 
j
j

= u j x j , g j + mj , G
,
j = 1 , ... , n .
Contributor j=s budget constraint is
x j + g j = I j + j + m j ,
j = 1 , ... , n .
(28)
Since the objective of the prestige policy is not to supplement private provision of the public good but to alter
prestige levels that can be achieved by contributors on their own, the prestige policy must satisfy the following constraint:
n
 j = 1 mj = 0 .
(29)
The matching grant policy of the government considered here implies that some contributors will receive negative grants.
This fully captures the notion that prestige redistribution is a zero-sum game - i.e., for some contributors to receive
additional prestige from the government, some others have to give up an equal amount of their initial endowment of
prestige. It may help the reader to think of the prestige redistribution policy of the government as policy of granting
medals to heroes - i.e., of granting medals to Good Samaritans! The analogy is a good one, since everyone should agree
that a policy of granting medals to heroes is a zero-sum game.9
Given equation (29) and ∑jτj = 0, we obtain the economy resource constraint (2) by adding up the budget
constraints (28) over all j. Now, note that when the government chooses mj, for all j, each individual’s contribution to the
pure public good is already predetermined. Since the government takes gj, for all j, as given and chooses mj, for all j, it, in
fact, chooses pj, for all j. Hence, the problem faced by the government in the second stage of the game it plays with the
reluctant contributors C formally described below C is to choose {xj , pj }j = 1,...,n to maximize:
W
subject to:
u 
1
n
x1 , p1 , G  , ... , u

j x j + G = I ,
j pj = G ,
9
xn , pn , G 
.
(30)
(2)
(31)
Our results do not depend on the restriction imposed by equation (29). The results go through even if the
prestige redistribution policy of the government granted additional prestige to everyone. This would require positive
public provision of the pure public good. It can be shown that the results remain the same as long as
*
n
j = 1 mj  G ,
where G* denotes the socially optimal amount of the pure public good.
14
where, given our definition of pj, we obtain constraint (31) by summing up all pj’s and noting equation (29).
It is important to note that the exogenous variables in the government’s maximization problem are G and I.
Since W is strictly quasiconcave and the constraint (2) is linear, the solution to the government’s problem - which will be
assumed to be interior - will be a unique and global constrained maximum. By applying the Implicit Function Theorem,
we will be able to implicitly define the following important functions:
xj = xj ( G , I ) ,
pj = pj ( G , I ) ,
(32a)
j = 1 , ... , n ,
j = 1 , ... , n .
(32b)
Functions (32) tell us how the government responds to both the aggregate private provision of the pure public
good and the aggregate income level. As we shall demonstrate below, the key difference between this situation, where the
government is able to redistribute prestige levels, relative to the situation where the government is not endowed with
instruments to redistribute prestige is that now information about the aggregate quantity of the pure public good suffices
for the implementation of a socially optimal allocation of resources.
For concreteness, we formally describe the two-stage game played by contributors and the government - called
Game 3. The game is as follows:
Game 3
Stage 1: Taking g-j as given, contributor j chooses gj > 0 to maximize
u
j
x
j
( g j + G- j , I ) , p j ( g j + G- j , I ) , g j + G- j
,
j = 1 , ... , n .
(33)
Stage 2: The government chooses {xj , pj }j = 1,...,n to maximize (30) subject to (2) and (31), after observing {G}.
We are now ready to present the first of our two main results:
Proposition 3. The subgame perfect equilibrium for Game 3 is socially optimal.
Proof. Consider the maximization problem in the second stage. Let λ and ψ be the Lagrangian multipliers associated with
(2) and (31), respectively. Assuming the solution is interior, the first-order conditions yield (2), (31) and the following
equations:
j
W j ux =  ,
j
W j ux =  ,
j = 1 , .... , n ,
j = 1 , .... , n .
15
(3)
(34)
Note that equations (3) and (34) imply equalization of marginal rates of substitution between prestige and the numéraire
good; that is, they imply equations (6). Equations (2), (3), (31) and (34) implicitly define functions (32) as well as
functions λ = λ (G, I ) and ψ = ψ (G, I ). Substituting functions (32) into equations (2) and (31) yields:
j x j ( G , I ) + G = I ,
j pj ( G , I ) = G .
(35a)
(35b)
Differentiating equations (35) with respect to G, we obtain:
 xj ( G , I )
= -1,
G
 pj ( G , I )
=1.
j
G
j
(36a)
(36b)
Now consider the first stage of the game. Substituting equations (32) into contributor j’s utility function, the
problem faced by contributor j is to choose gj > 0 to maximize (33), taking g-j as given. Assuming interior solutions, the
first order conditions for the contributors’ problems are:
or
  pj ( G , I ) 
 xj ( G , I ) 
j 
 + uGj = 0 ,
 + upj 
ux 


G

G




j
j
 xj ( G , I )
up   p j ( G , I ) 
uG
+ j 
+
= 0,
j

G
G
ux 
ux

j = 1 , ... , n ,
j = 1 , ... , n .
(37)
Summing up (37) over all j and noting (36a) enables us to write:
j
j
up
j
ux
  pj ( G , I ) 
j

 +  j uGsupj = 1 .


G
ux


(38)
Since the first order conditions of the second stage maximization problem imply that
j
i
up
up
= i ,
j
ux
ux
(6)
i , j = 1 , ... , n ,
equation (38) can be rewritten as follows:
i
  pj ( G , I ) 
u subGj
up


+
=1,


j
j
i
j


G
ux
ux


i = 1 , ... , n .
Utilizing equation (36b), we are able to rewrite (39) as follows:
i
j
up
u
+ j Gj = 1 ,
i
ux
ux
16
i = 1 , ... , n ,
(39)
which is the modified Samuelson condition (5).
Proposition 3 is a “sophisticated” Rotten Kid Theorem. Besides redistributing income, a benevolent agent - the
government - who cares about all his children - the reluctant contributors - promotes the right prestige incentives so that
the reluctant contributors, acting selfishly, internalize all externalities. Income redistribution is necessary but not
sufficient for our Rotten Kid Theorem because, in our setup, there is no market for prestige, where reluctant contributors
can trade prestige for the private good. The governmental prestige policy fulfills this market vacuum by equalizing
marginal rates of substitution between prestige and the numéraire good.
7.
Another winner recipe’s key ingredients: income redistribution and a market for prestige
17
As we observed in section 6, there are efficiency gains to be had from properly altering the levels of prestige that
contributors can obtain on their own through their contributions to the public good. We also noticed that, due to the
absence of a market for allocating prestige between contributors, governmental action is needed to induce rotten
contributors to behave as Good Samaritans. In this section, we investigate the efficiency properties associated with the
creation of a market for prestige. To simplify the exposition, we will assume that n = 2 and that the social welfare
function is utilitarian. The government’s sole policy instrument is the interpersonal income transfer used to redistribute
income between the two reluctant contributors.
Suppose that an entrepreneur who cares about the public good - say, contributor 1 - forms a non-profitable
organization - say, a charity - with the basic purpose of providing the public good. Contributor 1, however, knows that
both he and contributor 2 care not only about the public good, but also about the level of prestige that originates with the
act of contributing to the public good. Contributor 1 also knows that there are efficiency gains to be had from creating a
market mechanism whereby he and contributor 2 trade units of prestige for units of the private good, because their
marginal rates of substitution between these two commodities will generally be different in the absence of trade.
Contributor 1, therefore, decides that his organization will not only be a vehicle for the provision of the public good, but
will also function as a “broker” in the market for prestige to be created.
The charity works as follows. It sells gj+ zj-zk units of prestige to contributor j, j, k = 1, 2, j … k. The total
revenue collected by the charity is, thus, g1+ g2. This quantity is used to finance the provision of G units of the public
good. The brokerage function performed by the charity amounts to collecting zj units of the private good from contributor
j in order to purchase zj units of prestige from contributor k, j, k = 1, 2, j … k. Hence, zk, k = 1, 2, k … j, denotes either the
quantity of prestige sold by contributor j to the charity or the amount of private good offered, in excess of gk, by
contributor k.
Contributor 1 and the charity are assumed to be one and the same agent. The charity’s budget constraint is
x1 + g1 + G + z1 - z2 = w1 + G + 1 ,
which implies
x1 = I1 + 1 - g1 - z1 + z2 .
(40a)
Since τ1 + τ2 = 0, it follows that τ2 = - τ1. Hence, contributor 2's budget constraint yields
x2 = I2 - 1 - g2 - z2 + z1 .
18
(40b)
The charity, contributor 2 and the government play a two-stage game - called Game 4 - as follows.
Game 4
Stage 1: Taking {g2, z2} as given, the charity chooses {g1, z1} to maximize
1
u

I1 + 1  g1 , g2 , z1 , z2  - g1 - z1 + z2 , g1 + z1 - z2 , g1 + g2
.
(41a)
.
(41b)
Taking {g1, z1} as given, contributor 2 chooses {g2, z2} to maximize
u
2

I2 - 1  g1 , g2 , z1 , z2  - g2 - z2 + z1 , g2 + z2 - z1 , g1 + g2
Stage 2: Having observed (g1, g2, z1, z2), the government chooses τ1 to maximize
1
u

I1 + 1 - g1 - z1 + z2 , g1 + z1 - z2 , g1 + g2
+
u
2

I2 - 1 - g2 - z2 + z1 , g2 + z2 - z1 , g1 + gsub2
.
(42)
Since there are only two contributors, we find it easier to utilize the budget constraints to solve for the xj’s and
then substitute these quantities in the utility functions. Because in this section we assume that the social welfare function
is utilitarian, the government’s objective function in the second stage of the game is given by (42). Note that in the second
stage there is only one choice variable, τ1, but there are four exogenous variables, (g1, g2, z1, z2). The solution to the
government’s maximization problem - assumed to be interior - is a unique and global constrained maximum. The first
order condition that characterizes this solution enables us to implicitly define the function τ1(g1, g2, z1, z2). This function
gives the government’s responses to changes in the levels of contributions and prestige. Inserting this function into
equations (40) and the implied xj’s into the utility functions, we obtain the utility functions (41). The maximization
problems in the first stage are now well defined.
Our second main result is presented in Proposition 4.
Proposition 4: The subgame perfect equilibrium for Game 4 is socially optimal.
Proof. The first order condition for the second stage can be written as follows:
1
u x  I1 + 1  g1 , g2 , z1 , z2  - g1 - z1 + z2 , g1 + z1 - z2 , g1 + g2
- u
2
x
I
2

- 1  g1 , gsub2, z1 , z2  - g2 - z2 + z1 , g2 + z2 - z1 , g1 + g2
=
(43)
0.
Equation (43) yields the following response functions:
1
2
1
1
 1
ux x + ux G - ux G - ux p
=
,
1
2
 g1
ux x + ux x
2
2
1
2
 1
ux p + ux G - ux G - ux x
=
,
1
2
 g2
ux x + ux x
19
(44a)
(44b)
1
2
 1
ux p + ux p
= 1- 1
,
2
 z1
ux x + ux x
1
2
 1
ux p + ux p
= 1
-1.
2
 z2
ux x + ux x
Note that:
 1
 1
= ,
 z1
 z2
 1
 1
 1
=
.
 z1
partialg1  g2
(44c)
(44d)
(45)
Assuming interior solutions, the Nash equilibrium in the first stage is characterized by the following equations:
1
1
 1
uG u p
+ 1 = 1,
1
 g1
ux ux
2
2
 1
uG u p
+
= 1+
,
2
2
 g2
ux ux
1
up
= 11
ux
2
up
= 1+
2
ux
 1
,
 z1
 1
.
 z2
(46a)
(46b)
(46c)
(46d)
Then, the first equation in (45), (46c) and (46d) imply that
1
2
up
up
= 2 .
1
ux
ux
(47)
1
1
2
 1  1
uG uG up usubp
+ 2+ 1+
= 2+
.
1
2

g2  g1
ux
ux
ux
ux
(48)
Adding up (46a) and (46b) yields
2
Substituting (46d) into (48) and rearranging, we obtain
1
1
2
 1  1  1
uG uG u p
+
+
= 1+
.
1
2
1
 g2  g1  z2
ux
ux ux
(49)
Since the two equations in (45) imply that
 1  1  1
= 0,
 g2  g1  z2
it follows that equation (49) corresponds to the modified Samuelson condition (5) for n = 2. €
Proposition 4 is a “true” Rotten Kid Theorem. With the creation of a market for prestige, reluctant contributors
behave as Good Samaritans if they anticipate that their benevolent government will redistribute income after levels of
contributions and prestige have been allocated.
20
8.
Concluding remarks
We have, among other things, established a Rotten Kid Theorem in the impure public good model. An
important implication of our analysis is that the creation of a market for prestige provides a rationale for the existence of
charitable organizations.10 Charities should not only create the market for prestige, but also operate as brokers. Proper
brokerage encourages reluctant contributors to agree with society about how much of the public good should be provided.
To our knowledge, Harbaugh (1998a) is the only other study that formally examines prestige as a motivation for
making charitable contributions.11 Harbaugh (1998b) provides empirical evidence that charities advertise contributions
and allocate contributors into several categories according to the levels of their contributions. He also finds empirical
support for the hypothesis that contributors possess a taste for prestige. These pieces of evidence appear to indicate that
charities not only supply a market for prestige, but also operate as brokers in such a market. It is, therefore, quite possible
that the levels of public goods provided by charitable organizations may indeed be socially optimal.
9.
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10
In our simple model, we neglected the important fundraising function performed by charitable organizations.
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21
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See footnote 6 for studies on similar motives such as social approval, reputation, etc.
22
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