We study the optimal provision of public goods in the context of a special class of altruistically linked utility functions. We show that the usual Samuelson condition holds as if the utility functions were independent.Samuelson condition, Pareto efficiency

Eduardo Ley

English

Research Papers in Economics

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Forthcoming: Economics Letters
Optimal Provision of Public Goods
with Altruistic Individuals
Eduardo Ley
Abstract. We study the optimal provision of public goods in the context of a special class of altruistically linked utility
functions. We show that the usual Samuelson condition holds as if the utility functions were independent.
Keywords. Samuelson condition, Pareto e±ciency
JEL Classi¯cation System. D6, H0, H4
1. Modelling Altruism
Suppose that we have n agents with altruistically interrelated utility functions. Denote by zi agent i's
consumption bundle, and let z =( z1;:::;z n) represent an allocation. Each agent is assumed to have
preferences over allocations, z, which are additively separable over individual bundles, zi. Thus, we
assume that agent i's utility index, Vi, can be represented by
Vi = Ãi(z)=
X
j
¯ijUj(zj)=¯iiUi(zi)
| {z }
ego
+
X
j6=i
¯ijUj(zj)
| {z }
alter
; (1)
where Ui(¢) is a twice-di®erentiable, strictly quasi-concave, and monotonically increasing function. We
will always assume that ¯ii > 0. Utility results from an ego part, ¯iiUi, and an alter part,
P
j6=i ¯ijUj.
If ¯ij = 0 for all i 6= j, so that there is no alter, then we have the usual egoistic preferences. Otherwise,
we shall say that the system is altruistic. While some of the results below also hold for malevolent
systems, when dealing with altruistic systems we will always assume that they are benevolent systems
so that we have ¯ij ¸ 0. Systems like (1) have been used to represent altruism by Becker (1974), and
Abel and Bernheim (1991), among others. Becker (1976) uses a more general formulation |i.e.,h e
uses a utility function not necessarily separable.
As discussed, e.g., in Bergstrom (1990), there is an alternative way to model interrelated utility.
Instead of using a system like (1), it is sometimes more natural to specify i's preferences over his own
consumption bundle and everybody else's `happiness':
Vi = Ái(zi;V »i)=°iUi(zi)
| {z }
ego
+
X
j6=i
±ijVj
| {z }
alter
= Ui(zi)+
X
j
®ijVj (2)
where utility, Vi, is provided by the ego part, °iUi(zi), and the alter part,
P
j6=i ±ijVj; and V»i represents
the vector of Vj's excluding Vi. We also have ®ii =( °i ¡ 1)=°i and ®ij = ±ij=°i for i 6= j. This
formulation is used, e.g., in Barro (1974), Bernheim and Stark (1988), Bergstrom (1989).
Stacking the Ui's and the Vi's in column vectors U and V , a system like (2) can be expressed in
matrix form as
V = U + AV (20)
I thank Dallas Burtraw and Tim Brennan for useful comments.
Email: edley@km0.com Version: May 23, 1997where ®ij is the ijth element of A.I f( I ¡ A)¡1 exists, we can write (20)a s
V =( I ¡ A)¡1U = BU (10)
where the ijth element of B corresponds to ¯ij in (1). Conversely, if we start from V = BU, and B¡1
exists we can use A = I ¡ B¡1 to transform (10) into a system like (20).
There are two types of issues when going from one representation to the other. There is a technical
issue dealing the existence of the inverse matrices (I ¡ A)¡1 or B¡1. But there is an additional issue
dealing with the `consistency' of the utility representations. It is reasonable to expect that a benevolent
system like (2) with all the ±ij ¸ 0, should have all the ¯ij ¸ 0 when transformed into a system like
(1). Bergstrom (1990) establishes the conditions under which a well-behaved system like (1) can be
represented by a well-behaved system like (2) and viceversa.1 We will call those well-behaved systems
felicitous.2
In what follows we only need to assume that a utility representation like (1) exists. Provided that
this utility representation is also felicitous, then there exists an associated representation like (2) to
which the results apply as well.
2. Optimal Public Good Provision
For the ease of exposition, suppose that there is only one private good, xi, and one pure public good,
Y ; so that zi =( xi;Y).3 We shall assume that the public good can be produced at a constant marginal
cost. Choosing units suitably, we can make the (constant) marginal rate of transformation between the
private good and public good equal to one.
Let wi represent i's endowment of the private good and let w =( w1;:::;w n). We will assume in
what follows that B always has strictly positive diagonal elements and positive o®-diagonal elements, so
that it can be used to represent an altruistic system. We shall use Ehw;BUi to represent an economy
with altruistic individuals (whose preferences can be represented by a system like (1), or in matrix
form as V = BU). We shall use Ehw;Ui to represent the same economy with the egoistic individuals
that would be obtained by making the ¯ij's equal to zero in the altruistic system. That is, for every
altruistic system we obtain an egositic system by simply dropping the alter part in (1).
Denote by W =
P
i wi aggregate resources, and let X =
P
i xi; then Y = W ¡ X. Pareto optimal
allocations of Ehw;BUi are maxima of
L =
X
i
¸iVi =
X
i
¸i
X
j
¯ijUj(xj;W¡ X); (3)
for any row vector ¸ =( ¸1;:::;¸ n) > 0 |see,e.g., Cornwall (1984).
Proposition 1. Pareto e±cient allocations of the altruistic economy Ehw;BUi are also Pareto e±cient
allocations of the egoistic economy Ehw;Ui.
Proof. We can rewrite (3) as:
L =
X
j
Uj(xj;W¡ X)
X
i
¸i¯ij =
X
j
¹jUj(xj;W¡ X): (4)
1 To get a sense of the perversities that can occur, take n = 3 and start out from Vi = Ui(zi)+
P
j6=i Vj. This system
transforms into Vi = ¡0:5
P
j6=i Uj(zj). Thus, in a system of apparent benevolence, when we obtain the representation
of the preferences over allocations we ¯nd that agent i: (1) does not care about his own consumption bundle and (2)
cares negatively about other peoples' ego-happiness. A planner concerned with maximizing welfare would just need to
destroy the economy's resources!
2 A system like (10), with A>0, will be called felicitous if there exists a non-negative row vector ´ such that ´>´ A ;
and we shall then say that A is a felicitous matrix |in linear models, a consumption matrix A which has this property
is called productive, see, e.g., Gale (1960) or Cornwall (1984). A key property of a felicitous system is that (I ¡ A)¡1
exists and it is non-negative so that for any U>0w eh a v eV =( I ¡ A)¡1U>0 |see, e.g., Gale (1960). We shall say
that B>0 is felicitous when B¡1 = I ¡ A and A is felicitous.
3 This private-public terminology is valid in an egoistic economy. In an altruistic system, `private' goods generate
consumption externalities so they are not properly private.
2where ¹j =
P
i ¸i¯ij > 0, since ¸i > 0, ¯ii > 0, and ¯ij ¸ 0. Maxima of (4) correspond to Pareto
e±cient allocations of an economy where Vi = Ui(xi;Y) with welfare weights ¹ =( ¹1;:::;¹ n).
As a corollary, e±cient allocations of an altruistic system like (1) must satisfy an `unaltered' Samuel-
sonian condition:4
X
i
@Ui(xi;Y )
@Y
@Ui(xi;Y )
@x
= 1; (5)
instead of
P
i
@Vi=@Y
@Vi=@x = 1, as one might have expected.
Proposition 2. If Ui(xi;Y)=v(Y )xi +ui(Y ), then the optimal level of the public good in Ehw;BUi
is the same in all Pareto e±cient allocations and it does not depend on the values of the ¯ij's.
Proof. If Ui(xi;Y)=v(Y )xi + ui(Y ) then
P
i
@Ui=@Y
@Ui=@x only depends on
P
i xi which implies that the
optimal provision of Y in an egoistic system is independent of the distribution of the private good
among the agents |see Bergstrom and Cornes (1981). Therefore, by proposition 1, the e±cient level
of Y is determined independently of B in an altruistic system.
We can rewrite (3) and (4) in matrix form as L = ¸BU = ¹U. Since, for every ¹>0, you can
always ¯nd a regular B>0, that guarantees that ¹B¡1 > 0.5 Then, for each Pareto e±cient allocation
of an egoistic system U you can always ¯nd an altruistic system BU for which that allocation is also
Pareto e±cient.
However, what about the reverse statement to proposition 1? Are all Pareto optima of the egoistic
system Ehw;Ui also Pareto optima of the altruistic system Ehw;BUi for any altruistic B? The answer
is no. To establish the reverse proposition, we would have to show that for any ¹>0 we can ¯nd ¸>0
such that ¹ = ¸B. Postmultiply both sides by (I ¡ A)=B¡1 and we obtain ¹(I ¡ A)=¸. It should
be clear that, given a felicitous A>0, we cannot always guarantee that ¹(I ¡A) > 0 for any arbitrary
¹>0.
Proposition 3. If ¹B¡1 > 0, a Pareto e±cient allocation of the egoistic economy Ehw;Ui with
welfare weights ¹, is a Pareto e±cient allocation of the felicitous altruistic economy Ehw;BUi with
welfare weights ¸ = ¹B¡1.
However, if preferences are of the form Ui(xi;Y)=v(Y )xi + ui(Y ), it follows from proposition 2
that Pareto optima of Ehw;Ui are Pareto optima of Ehw;BUi for any B>0.
3. Concluding Remarks
Bernheim and Stark (1988) show that altruism can alter the utility possibilities frontier in most sur-
prising ways. Here we derive the conditions for optimal provision of a public good and we ¯nd the
intriguing result that an unaltered Samuelson condition must hold. That is, the sum of ego-marginal
rates of substitution must equal the marginal rate of transformation. As a result, Pareto optima of an
altruistic economy are also Pareto optima of the egoistic economy obtained by eliminating all altruistic
links.
4 We shall only deal with interior solutions. Corner solutions only add complication to the exposition without providing
additional insights.
5 For example, make ¯ij =0i fj ¡ 1 6=0 ;1 and ¯ii = 1. Write ¹ = ¸B, start with ¸1 = ¹1. Then recursively choose
¯i¡1;i <¹ i=¸i¡1.
3References
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Optimal Provision of Public Goods with Altruistic Individuals

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