A competitive equilibrium for a warm-glow
economy∗
Nizar Allouch
Queen Mary, University of London
School of Economics and Finance
[email protected]
Abstract
The warm-glow model [Andreoni (1989,1990)] of public goods provision has received widespread interest, yet surprisingly most attention
has focused on the voluntary contribution equilibrium of the model,
and only very little attention has been devoted to the competitive
equilibrium. In this paper we introduce the concept of competitive
equilibrium for a warm-glow economy [Henceforth, warm-glow equilibrium] and establish both existence and welfare properties. The
warm-glow equilibrium concept may prove to be very useful to the
normative and positive theory of public goods provision. First, it is a
price based mechanism achieving efficient outcomes. Second, not only
could the warm-glow equilibria outcomes serve as a point of reference
to measure free-riding and welfare loss but also, as suggested by Andreoni (2006) and Bernheim and Rangel (2007), they are more likely
to be achieved.
Keywords: warm-glow, public goods provision, competitive equilibrium, voluntary contribution.
JEL Classification Numbers: H41, D64, C62.
∗
I thank Ted Bergstrom, Richard Cornes, Clive Fraser, Herakles Polemarchakis, Chris
Tyson, and seminar participants at Warwick University, Cambridge University, East Anglia University, CORE, Vanderbilt University, PET 2008, and SAET 2009 meetings for
insightful comments that substantially improved the paper. The hospitality of the University of Cambridge, Department of Economics is gratefully acknowledged.
1
1
Introduction
Altruistic behavior in societies has intrigued economists ever since the “homo
economicus”model advocated by Adam Smith and David Ricardo. The puzzle is to explain how a palpable altruistic behavior may emerge amongst
supposedly selfish members of the society. Prosocial behaviors are a pervasive feature of many economic activities and, not surprisingly, they are the
subject of ongoing interest in many fields of economics ranging from experimental economics to development economics. In public economics, the standard neoclassical model of public goods provision assumes that consumers
care only about the magnitude of their public goods provisions insofar as
these provisions affect the aggregate level of provision. The lucidity of the
standard model compounded with the straightforwardness of its policy suggestions, have made it of paramount significance to the study of all areas of
public economics such as taxation, pensions, and charity giving.
The work of Sugden (1982), Warr (1982), Bergstrom, Blume, and Varian
(1986), and Andreoni (1988) amongst others has revealed some testable implications of the standard model of public goods provision. Amongst these
implications, an almost complete “Dollar to Dollar ”crowding out between
internal and external funding, the neutrality of the level of provision to a significant class of income redistributions and a feature where only the wealthy
contributes as the size of the population grows large. These predictions,
which are the consequence of the fact that public goods are an additive externality of the various provisions where the identity of the contributor is
immaterial, are in conflict with empirical tests and laboratory experiments.
For instance, Kingma (1989) in studying the charitable contributions to public radio stations has shown crowding out has much smaller effects than the
theoretical Dollar to Dollar prediction and that contributors were not significantly wealthier than average. These observations amongst others have led
economists to question the ability of the standard model to explain some of
the empirical and experimental evidence.
Andreoni (1989,1990), inspired by earlier works of Becker (1974) and
Cornes and Sandler (1984), has proposed a “warm-glow ”model of public
goods provision. In the warm-glow model, consumers do not solely benefit
from the aggregate level of public goods provision but they may also benefit
from the warm-glow effects of their own public goods provision. Thus, unlike
the standard public goods model, one consumer’s public goods provision does
not constitute a perfect substitute for other consumers’ provision. For exam2
ple, when someone paints his house (we assume that his choice of the color
is to the taste of his neighbors), his action not only benefits himself but also
his neighbors. At the same time, having one of his neighbors houses painted
will not substitute for having his own house painted.1 In the warm-glow
model, free riding is shown to be less severe, crowding out is not complete,
and neutrality of income redistribution does not hold. The above theoretical predictions are broadly in line with both empirical and experimental
evidence.
Despite the prominence of the warm-glow model in subsequent research,
surprisingly no competitive equilibrium for this model has been developed
yet. In this paper we introduce the warm-glow equilibrium concept which
follows the tradition of a decentralized competitive equilibrium. The notion
of competitive equilibrium in public goods economies originates in the early
work of Lindahl (1919) who introduced the concept of personalized prices
for public goods. With private goods, different people can consume different
bundles, but in equilibrium they all must pay the same prices. With public
goods, everyone is faced with an equal provision, but consumers may pay
different prices according to their preferences. The warm-glow equilibrium
resembles the Lindahl equilibrium in this regard by considering personalized
prices. However, due to the warm-glow effects, each consumer is endowed
with two sets of personalized prices, a personalized price to finance his own
public goods provision and another personalized price to finance other consumers’ provisions. In this paper we show that the warm-glow equilibrium
has interesting properties. First, the warm-glow equilibrium shows how efficiency can be achieved in a warm-glow economy via personalized prizes.
Second, a Pareto efficient allocation can be supported as a warm-glow equilibrium. In addition, under general hypothesis, a warm-glow equilibrium is
shown to exist. Not only is the price based mechanism which achieves efficient outcomes a highly desirable mechanism for its normative prescriptions,
but the equilibria outcomes could also serve as a point of reference to measure free-riding and welfare loss (see, for example, Cornes and Sandler (1984)
and Gaube (2006)).
In the public economics literature, the Lindahl equilibrium is often criticized for its lack of incentive compatibility since consumers have incentives
to report false preferences to get a better deal from the public goods. The
recent contributions of Andreoni (2006) and Bernheim and Rangel (2007)
1
This example was kindly suggested to us by Ted Bergstrom.
3
suggest that as the size of the population grows the in-part private good
aspect of each consumer’s public goods provision, at the margin, overwhelms
other concerns about the aggregate level of public goods provision.2 It would
be interesting to investigate in subsequent research whether this dominance
of warm-glow effects will reduce the vulnerability of the warm-glow equilibrium to strategic manipulations and in turn will result in less demanding
mechanisms than those required to implement the Lindahl equilibrium (see,
for example, Walker (1981) and Tian (1989)).3 It also remains to be seen
whether other solution concepts in the Lindahlian tradition such as the Ratio
Equilibrium of Kaneko (1977), and the Cost Share Equilibrium of Mas-Colell
and Silvestre (1989) could be defined for a warm-glow economy.4
The paper is organized as follows. In Section 2, we present the model
of a warm-glow economy, discuss alternative utility function formulations,
and define the concept of warm-glow equilibrium. In Section 3, we establish
the existence of warm-glow equilibrium, analyze its welfare properties, and
provide an example to distinguish it from the voluntary contribution equilibrium. In Section 4, we conclude the paper and in Section 5, we show our
results.
2
The Model
We consider a public goods economy E, with L private goods and K public
goods. There are N = {1, . . . , n} consumers (n ≥ 2) in the economy, each
L+K
of whom is characterized by his consumption set R+
. We will often write
L+K
consumption bundles for consumer i in the form (xi , gi ) ∈ R+
, where xi ∈
RL+ is consumer i’s private goods consumption and gi ∈ RK
refers
to consumer
+
i’s public goods provision. We consider an aggregate production technology
in order to concentrate on the consumer side of the economy. The aggregate
2
See also Ribar and Wilhelm (2002).
For instance, under a similar assumption Allouch (2010) shows in a warm-glow economy that eventually the core shrinks to the set of warm-glow equilbria. We note that
the core-equilibrium convergence of Debreu and Scarf (1963) is often considered as a justification for the emergence of the competitive behavior from social stability. However,
a similar result does not hold in the standard model of public goods provision since the
core is usually bigger than the set of Lindahl equilibria (see, for example, Muench (1972),
Champsaur, Roberts, and Rosenthal (1975), and Buchholz and Peters (2007)).
4
For recent contributions on these concepts, see, for example, Van den Nouweland,
Tijs, and Wooders (2002) and De Simone and Graziano (2004).
3
4
production technology is described by the production set Y ⊂ RL+K . For
simplicity, we assume that Y is a closed convex cone with vertex the origin,
satisfying the usual properties of irreversibility, no free production, and free
disposal. We also assume the possibility of producing public goods, that is,
Y ∩ (RL × RK
++ ) ̸= ∅.
Each consumer has an initial endowment of private goods wi ∈ RL++ . We
assume that there are no initial endowments of public goods. Each consumer
i’s preferences
can be represented by a utility function ui (xi , gi , G−i ), where
∑
G−i = j̸=i gj is the aggregate supply of public goods of other consumers.
It is worth noting that in the warm-glow model introduced by Andreoni
(1989,1990), each consumer’s public goods provision enters into the arguments of his utility function twice. That is, for consumer i with public goods
provision gi , ui (xi , gi , G) = ui (xi , gi , gi + G−i ). In the framework of voluntary
contribution, Andreoni’s formulation of utility functions leads to an easier
characterization of the theoretical results than the current paper’s utility
formulation due amongst other things to the straight forward expression of
convexity. Moreover, such a formulation has the methodological advantage
of being flexible enough to accommodate the two polar cases of standard
public goods provision model and pure warm-glow model. In our framework,
since we are interested in the competitive equilibrium the two formulations
of utility functions will eventually result in the same equilibrium concept.5
Arguably, this is due to the fact that we are interested in the underlying
preferences rather than their utility representations.
We assume that each utility function ui satisfies the following properties:
[A.1] Monotonicity: The utility function ui (·, ·, ·) is increasing.
[A.2] Continuity: The utility function ui (·, ·, ·) is continuous.
[A.3] Convexity: The utility function ui (·, ·, ·) is quasi-concave.
Assumptions [A.1]-[A.3] are standard for a public goods economy.
(L+K)N
An allocation of the economy is a list ((xi , gi ), i ∈ N ) ∈ R+
, specifying each consumer’s private good consumption and public goods provision.
5
In Remark 2, we show that the warm-glow equilibrium could be re-written for Andreoni’s utility functions formulation.
5
Definition: An allocation of the economy ((xi , gi ), i ∈ N ) is a feasible allocation if
∑
∑
( (xi − wi ),
gi ) ∈ Y.
i∈N
i∈N
Thus in a feasible allocation we require the net inputs of private goods and
outputs of public goods to be consistent with the aggregate production technology.
Definition: An allocation ((xi , gi ), i ∈ N ) is (weakly) Pareto efficient if it is
feasible and if there does not exist another feasible allocation ((x′i , gi′ ), i ∈ N )
such that for each consumer i
ui (x′i , gi′ , G′−i ) > ui (xi , gi , G−i ).
We note that the definition of Pareto optimality can be also stated for
(strongly) Pareto efficient allocations.
2.1
Warm-glow equilibrium
Definition: A warm-glow equilibrium is ((x∗i , gi∗ , πi , π−i )i∈N , p, pg ), where
((x∗i , gi∗ ), i ∈ N ) is a feasible allocation, p ∈ RL+ is a price system for private
K
goods, pg ∈ RK
+ is a price system for public goods, πi ∈ R+ is the personalized price of consumer i’s own public goods provision, and π−i ∈ RK
+ is
the personalized price of consumer i’s for the other consumers public goods
provision, such that
(i). for all (y, g) ∈ Y,
∑
∑
(p, pg ) · (y, g) ≤ (p, pg ) · ( (x∗i − wi ),
gi∗ )
i∈N
i∈N
(profit maximization);
(ii). for each consumer i ∈ N ,
p · x∗i + πi · gi∗ + π−i · G∗−i = p · wi ,
6
and if
ui (xi , gi , G−i ) > ui (x∗i , gi∗ , G∗−i )
then
p · xi + πi · gi + π−i · G−i > p · wi
(maximization of utility given personalized prices);
(iii). for each consumer i ∈ N ,
πi +
∑
π−j = pg
j̸=i
(for each consumer, personalized prices sum to the price of public
goods).
Condition (i) states profit maximization by the producer. Condition (ii)
requires that each consumer maximizes his utility subject to his budget constraint. Note that in the voluntary contribution equilibrium consumer i faces
the price pg to finance his own public goods provision gi . In the warm-glow
equilibrium, due to the externality effects of each consumer’s public goods
provision, consumer i will finance G−i at the personalized price π−i in addition to financing his own provision gi at the personalized price πi . Condition
(iii) is a restatement of Samuelson’s condition equating the sum of the personalized prices for each consumer’s public goods provision to the price of
the public goods.
Remark 1. We note that if we consider the standard public goods model,
that is ui (xi , gi , G−i ) = ui (xi , gi + G−i ), the perfect substitution between gi
and G−i for consumer i implies that πi = π−i . Thus, in the case of the
standard public goods provision model the warm-glow equilibrium coincides
with the Lindahl equilibrium.
Remark 2. To illustrate the observation made earlier in regard to the
two alternative formulations of utility functions, assume that instead of
7
ui (xi , gi , G−i ) representing the preferences of consumer i we have a utility
function of the form ui (xi , gi , G). Then the budget constraint of consumer i
could be re-written as
p · x∗i + (πi − π−i ) · gi∗ + π−i · G∗ = p · wi ;
and the utility maximization of consumer i could be re-written as
ui (xi , gi , G) > ui (x∗i , gi∗ , G∗ )
implies
p · xi + (πi − π−i ) · gi + π−i · G > p · wi .
3
Welfare theorems and existence of a warmglow equilibrium
We now state the first welfare theorem for a warm-glow economy: a warmglow equilibrium is Pareto efficient. Hence, amongst other things, the warmglow equilibrium provides a price based mechanism achieving efficient outcomes.
Theorem 1. A warm-glow equilibrium is Pareto efficient.
Proof of Theorem 1. See the Appendix.
In the following we provide a decentralization of Pareto efficient allocations as warm-glow equilibria. Our result is the analogue for a warm-glow
economy of the second welfare theorem for a private goods economy which
states that every Pareto efficient allocation can be supported, with an appropriate redistribution of initial endowments, as a competitive equilibrium.
Similar results for public goods economies were first introduced by Foley
(1970), where the concept of public competitive equilibrium corresponds to
a Lindahl equilibrium after tax redistribution. More recently, the work of
Boyd and Conley (1997), Conley and Smith (2005), and Murty (2010) consider a similar concept in economies with externalities. It is worth noting
8
that public goods could be accommodated in their model as a special case of
an additive externality.
Theorem 2. Assume that the economy satisfies [A.1]-[A.3]. If the allocation
(L+K)N
is Pareto efficient, then there exists a price system
((x∗i , gi∗ ), i ∈ N ) ∈ R++
g
((πi , π−i )i∈N , p, p ) ≥ 0 such that
(i). for all (y, g) ∈ Y,
(p, pg ) · (
∑
(x∗i − wi ),
∑
gi∗ ) ≥ (p, pg ) · (y, g);
i∈N
i∈N
(ii). for each consumer i ∈ N, if
ui (xi , gi , G−i ) > ui (x∗i , gi∗ , G∗−i ),
then
p · xi + πi · gi + π−i · G−i > p · x∗i + πi · gi∗ + π−i · G∗−i ;
(iii). for each consumer i ∈ N ,
πi +
∑
π−j = pg .
j̸=i
Proof of Theorem 2. See the Appendix.
As with any proposed equilibrium concept, it is of paramount importance to show existence of an equilibrium in a meaningful class of economies.
In public goods economies, the work of Foley (1967,1970), Milleron (1972),
and Bergstrom (1976) show the existence of the Lindahl equilibrium by appealing to standard existence results for a private goods economy.6 In our
framework, it turns out that just like the existence of the Lindahl equilibrium
6
See also Conley (1994), Wooders (1997), and Florenzano and del Mercato (2006) for
some specific core-equilibrium convergence.
9
resorts to standard existence results of competitive equilibrium for a private
goods economy, the existence of warm-glow equilibrium resorts to standard
existence results of the Lindahl equilibrium.
Theorem 3. Assume that the economy satisfies [A.1]-[A.3]. Then there
exists a warm-glow equilibrium.
Proof of Theorem 3. See the Appendix.
The following example shows that the warm-glow equilibrium is different
from the voluntary contribution equilibrium (and the Lindahl equilibrium).
Example. Consider an economy consisting of 2 consumers i = 1, 2 that all
have the same income wi = 1 and the same utility function
ui (xi , gi , g−i ) = xi (2gi + g−i ) = xi (gi + G),
where xi denotes consumer i’s private good consumption and gi denotes consumer i’s public good provision. The public good is produced by a unit-linear
production technology. That is, any non-negative quantity of the private
good can be converted into the same quantity of the public good. Then the
unique voluntary contribution equilibrium strategies are
3 2
(x1 , g 1 ) = (x2 , g 2 ) = ( , ).
5 5
The unique warm-glow equilibrium allocation is
1 1
(x∗1 , g1∗ ) = (x∗2 , g2∗ ) = ( , ),
2 2
and, if we normalize p = 1, the warm-glow equilibrium prices are
2 1
(π1 , π−1 ) = (π2 , π−2 ) = ( , ) and pg = 1.
3 3
10
4
Conclusion
The fact that some public goods are often assumed to suffer from congestion provides a motive for exclusion and the forming of clubs to supply these
goods. As a consequence, optimal club sizes may be bounded and less than
the total population. The pioneering contribution of Buchanan (1965) introduced club theory to highlight the fact that much consumption occurs in
clubs and is a collective activity. A related literature initiated by the seminal paper of Tiebout (1956) notes that many public goods are local rather
than pure and that the benefit of sharing the cost of the public goods will
eventually be offset by congestion. The club/local public goods literature
provides an interesting insight: market forces underlying the formation of
communities and clubs could potentially settle the economy in an efficient
equilibrium.
The concept of warm-glow equilibrium as developed in this paper does
not contain a partition of consumers into groups. Recent contributions to
the warm-glow literature suggest that individuals may also value the relative
position of their public goods provision. For instance, this could be motivated
either by the social status and prestige acquired from their public goods
provision or the desire to make an impact contribution (see, for example,
Harbaugh (1998) and Duncan (2004)).
Due to the common rivalry properties in provision, a promising avenue
for future research would be to explore whether the warm-glow equilibrium
concept could be extended to accommodate prestige or impact contributions,
similar to the way the clubs/local public goods literature accommodates congestion and crowding.
5
Appendix
Proof of Theorem 1. Suppose the Theorem is false. Then there is a
warm-glow equilibrium ((x∗i , gi∗ , πi , π−i )i∈N , p, pg ) with the property that the
feasible allocation ((x∗i , gi∗ ), i ∈ N ) is not Pareto efficient. This means that
there exists a feasible allocation ((xi , gi ), i ∈ N ) such that for each consumer
i, it holds that
ui (xi , gi , G−i ) > ui (x∗i , gi∗ , G∗−i ).
11
From utility maximization, it holds that
p · xi + πi · gi + π−i · G−i > p · wi .
∑
Given the fact that G−i = j̸=i gj , summing over the above inequalities, we
obtain
∑
∑
∑
(p · xi + πi · gi + π−i ·
gj ) >
p · wi .
i∈N
i∈N
j̸=i
Rearranging the terms with respect to each consumer’s public goods provision, it follows that
∑
∑
∑
(p · xi + (πi +
π−j ) · gi ) >
p · wi .
i∈N
i∈N
j̸=i
∑
g
π−j = p , it holds that
∑
∑
∑
p·
x i + pg ·
gi > p ·
wi
Given the fact that πi +
j̸=i
i∈N
and
i∈N
i∈N
∑
∑
(p, pg ) · ( (xi − wi ),
gi ) > 0.
i∈N
i∈N
∑
∑
Since the allocation ((xi , gi ), i ∈ N ) is feasible, we have ( i∈N (xi −wi ), i∈N gi ) ∈
Y. This yields a contradiction to condition (i) of the warm-glow equilibrium.
Proof of Theorem 2. The idea of the proof follows Foley (1970) in extending the commodity space for public goods. However, due to the warm-glow
effects our extension is quite different since each consumer distinguishes between his own public goods provision and others’ public goods provisions.
We first define the set
def
F = {(y, g1 , G−1 , . . . , gn , G−n ) | for each i, G−i =
∑
j̸=i
gj and (y,
∑
gj ) ∈ Y }.
j∈N
Since Y is a convex cone with vertex the origin, it follows that F is also a
convex cone with vertex the origin.
12
We next define the set
∑
def
D = {(
yi , g1 , G−1 , . . . , gn , G−n ) | for each i, ui (yi +wi , gi , G−i ) > ui (x∗i , gi∗ , G∗−i )}.
i∈N
The set D is nonempty and convex since the utility functions are increasing
and quasi-concave.
Since the allocation ((x∗i , gi∗ ), i ∈ N ) is Pareto efficient, it holds that
F ∩ D = ∅.
We now use the separation theorem to find prices (p, Π) = (p, π1 , π−1 , . . . , πn , π−n ) ̸=
0, where (p, Π) ∈ RL+KN , and a real number C such that
(i). for all (y, g1 , G−1 , . . . , gn , G−n ) ∈ F ,
(p, Π) · (y, g1 , G−1 , . . . , gn , G−n ) ≤ C;
(ii). for all (y, g1 , G−1 , . . . , gn , G−n ) ∈ D,
(p, Π) · (y, g1 , G−1 , . . . , gn , G−n ) ≥ C.
Since F is a closed convex cone with vertex zero, we have C = 0. Moreover,
since the allocation ((x∗i , gi∗ ), i ∈ N ) is Pareto efficient, we obtain
∑
( (x∗i − wi ), g1∗ , G∗−1 , . . . , gn∗ , G∗−n ) ∈ F ∩ clD,
i∈N
where clD denotes the closure of the set D. Hence, it follows from properties
(i) and (ii) in the separation theorem that
∑
(p, π1 , π−1 , . . . , πn , π−n ) · ( (x∗i − wi ), g1∗ , G∗−1 , . . . , gn∗ , G∗−n ) = 0.
(1)
i∈N
Now, for each public good k, we set7
∑ j
def
pgk = maxj
(Πi )k .
i∈N
7
For any a ∈ RK , (a)k is the k th component of a.
13
Let us consider the following set of consumers
∑ j
Jk = {j ∈ N |
(Πi )k = pgk }
i∈N
and the following public goods bundle (g1′ , . . . , gn′ ) defined as follows:
{∑
∗
i∈N gi
if j ∈ Jk ,
′
|Jk |
(gj )k =
0
otherwise.
∑
∑
Since ( i∈N (x∗i − wi ), i∈N gi∗ ) ∈ Y, it follows that
∑
( (x∗i − wi ), g1′ , G′−1 , . . . , gn′ , G′−n ) ∈ F.
i∈N
We claim that Jk = N , for each public good k. Suppose this were not
true. Since, by construction, the public goods bundle (g1′ , . . . , gn′ ) allocates
each public good to the consumers with the highest sum of personalized
prices, we obtain
∑
(p, π1 , π−1 , . . . , πn , π−n ) · ( (x∗i − wi ), g1′ , G′−1 , . . . , gn′ , G′−n ) > 0.
i∈N
This yields a contradiction to property (i) of the separation theorem.
From monotonicity of preferences it follows that
(p, π1 , π−1 , . . . , πn , π−n ) ≥ 0.
Suppose that p = 0. Then it holds that for some consumer i and public good
(L+K)N
k we have either (πi )k > 0 or (π−i )k > 0. Since ((x∗i , gi∗ ), i ∈ N ) ∈ R++
,
we obtain
∑
(p, π1 , π−1 , . . . , πn , π−n ) · ( (x∗i − wi ), g1∗ , G∗−1 , . . . , gn∗ , G∗−n ) > 0.
i∈N
This yields a contradiction to (1).
Now, suppose that for some consumer i and some consumption bundle
(xi , gi , G−i ), it holds that
ui (xi , gi , G−i ) > ui (x∗i , gi∗ , G∗−i ).
14
Then, it follows that
∑
(xi − wi +
(x∗j − wj ), g1∗ , G∗−1 , . . . , gi , G−i , . . . , gn∗ , G∗−n )
j̸=i
is in the closure of D. Property (ii) in the separation theorem entails
∑
(p, Π) · (xi − wi +
(x∗j − wj ), g1∗ , G∗−1 , . . . , gi , G−i , . . . , gn∗ , G∗−n )) ≥ 0
j̸=i
Subtracting (1), we obtain
p · xi + πi · gi + π−i · G−i ≥ p · x∗i + πi · gi∗ + πi · G∗−i
Suppose that this were an equality. Since ((x∗i , gi∗ ), i ∈ N ) ∈ R++
, by
continuity of preferences there is a point in the consumption set of consumer
i which, at the same time, is strictly preferred and costs less when compared
to (x∗i , gi∗ , G∗−i ). This yields a contradiction to property (ii) of the separation
theorem.
(L+K)N
Proof of Theorem 3. The idea of the proof is to consider each consumer’s
public goods provision as a bundle of public goods on its own merit. Thus,
understandably this construction will capture the warm-glow effects of each
consumer’s provision.
b with N consumers, L private
We first consider an auxiliary economy E,
goods, and KN public goods. A bundle of public goods for the auxiliary
b will be G
b = (b
economy E
g1 , . . . , gbn ) and list the each consumer’s public
goods provision.
b as follows:
We define the production set for the auxiliary economy E
∑
def
Yb = {(y, gb1 , . . . , gbn ) ∈ RKN +L | (y,
gbi ) ∈ Y }.
i∈N
It is clear that Yb is a closed convex cone with vertex the origin, satisfying
the usual conditions of irreversibility, no free production, and free disposal.
b as follows:
We define the utility functions u
bi for the auxiliary economy E
b = ui (xi , gbi , G
b−i ),
bi (xi , G)
u
15
b−i = ∑ gbj . The utility functions u
where G
bi are well defined and are inj̸=i
creasing, continuous, and quasi-concave.
Using standard existence theorems for the Lindahl equilibrium (see, for
example, Foley (1970), Milleron (1972), and Florenzano and del Mercato
b∗ , p) of the auxiliary
(2006)) there exists a Lindahl equilibrium ((x∗i , Πi )i∈N , G
b that is,
economy E,
b ∈ Yb ,
(i). for all (y, G)
∑
∑
∑
b ≤ (p,
b∗ ) = 0;
(p,
Πi ) · (y, G)
Πi ) · ( (x∗i − wi ), G
i∈N
i∈N
i∈N
(ii). for each consumer i ∈ N , it holds that
b ∗ = p · wi ,
p · x∗i + Πi · G
and if
then
b >u
b∗ )
u
bi (xi , G)
bi (x∗i , G
b > p · wi .
p · xi + Πi · G
j
For each consumer i ∈ N , let Πi = (Π1i , . . . , ΠN
i ), where Πi is the personalized price faced by consumer i to finance consumer j’s public goods
provision.
Because the construction of the warm-glow equilibrium prices from the
Lindahl equilibrium prices requires several intermediate constructions, the
proof is in several steps.
Step 1: For each public good k, we set
∑ j
def
pgk = maxj
(Πi )k
i∈N
In addition, we consider the following set
16
Hk = {j ∈ N | (b
gj∗ )k ̸= 0}
and once again the set
Jk = {j ∈ N |
∑
(Πji )k = pgk }.
i∈N
Claim 1. For each public good k, it holds that Hk ⊂ Jk .
Proof. Suppose this were not true. Then for some public good k, there
exists a consumer j1 ∈ Hk \ Jk . Hence, there exists a consumer j2 such that
∑ j
∑ j
(Πi 1 )k <
(Πi 2 )k .
(2)
i∈N
i∈N
Let ek denote the k th unit vector in RK . Let us consider the following public
b′ defined as follows:
goods bundle G
 ′
gj∗1 )k ek ,
gbj1 = gbj∗1 − (b
′
′
∗
b = gbj = gbj + (b
gj∗1 )k ek ,
G
2
 ′2
gbj = gbj∗ ,
if j ∈ N \ {j1 , j2 }.
∑ ∗ ∑ ′
∑
b∗ ) ∈ Yb , it follows that (∑ (x∗i − wi ), G
b′ ) ∈
Since j gbj = j gbj and ( i∈N (x∗i − wi ), G
i∈N
Yb . In addition, since
∑
∑
b∗ ) = 0
(p,
Πi ) · ( (x∗i − wi ), G
i∈N
i∈N
and (2), it follows that
(p,
∑
∑
b′ ) > 0.
Πi ) · ( (x∗i − wi ), G
i∈N
i∈N
This would contradict condition (i) of the Lindahl equilibrium of the auxiliary
b
economy E.
Now, we prove condition (i) of the warm-glow equilibrium. First, it follows
from Claim 1 that
∑ j
pg · gbj∗ = (
Πi ) · gbj∗ , for each consumer j.
(3)
i∈N
17
Moreover, for any bundle (y, g) ∈ Y, let us consider the following bundle
b = (b
G
g1 , . . . , gbn ) defined as follows:
{ gk
if j ∈ Jk ,
(b
gj )k = |Jk |
0
otherwise.
b ∈ Yb and
It is obvious that (y, G)
∑ j
Πi ) · gbj , for each consumer j.
pg · gbj = (
(4)
i∈N
Then it follows from (3), (4), and condition (ii) of the Lindahl equilibrium
that
∑
∑
(p, pg ) · (y, g) = (p, pg ) · (y,
gbj ) = p · y +
pg · gbj
j∈N
= p·y+
∑∑
j∈N
Πji
· gbj = p · y +
j∈N i∈N
= p·y+
= p·
∑
∑
b ≤p·
Πi · G
i∈N
(x∗i
= p·
i∈N j∈N
(x∗i − wi ) +
i∈N
− wi ) +
i∈N
∑
∑
∑∑
− wi ) +
i∈N
∑
p ·
g
Πji · gbj
∑
i∈N
gbj∗
=p·
∑
(x∗i − wi ) +
∑∑
·
gbj∗
∑
∑
= (p, p ) · ( (x∗i − wi ),
gbj∗ ).
i∈N
j∈N i∈N
g
j∈N
Step 2: For each consumer i and public good k, we set
def
(π−i )k = minj̸=i (Πji )k .
In addition, we consider the following sets
Hi,k = {j ∈ N \ {i} | (b
gj∗ )k ̸= 0}
and
Ji,k = {j ∈ N \ {i} | (π−i )k = (Πji )k }.
18
b∗
Πi · G
Πji
i∈N j∈N
(x∗i
∑∑
i∈N
j∈N
Πji · gbj∗
Claim 2. For each consumer i and public good k, it holds that Hi,k ⊂ Ji,k .
Proof. Suppose this were not the case. Then there exists a consumer i, a
public good k, and a consumer j1 ∈ Hi,k \Ji,k . Hence, there exists a consumer
j2 ∈ N \ {i} such that
(Πji 1 )k > (Πij2 )k .
Let us consider once again the following public goods bundle
 ′
gj∗1 )k ek ,
gbj1 = gbj∗1 − (b
′
′
∗
b = gbj = gbj + (b
gj∗1 )k ek ,
G
2
2
 ′
if j ∈ N \ {j1 , j2 }.
gbj = gbj∗ ,
Since,
we obtain
b ∗ = p · wi
p · x∗i + Πi · G
b ′ < p · wi .
p · x∗i + Πi · G
Moreover, it is easy to check that
b′ ) = ui (x∗ , gb′ , G
b′ ) = ui (x∗ , gb∗ , G
b∗ ) = u
b∗ ).
u
bi (x∗i , G
bi (x∗i , G
i
i
−i
i
i
−i
b
Hence, by monotonicity and continuity of preferences, there exists (xi , G)
such that
b >u
b∗ )
u
bi (xi , G)
bi (x∗i , G
and
b ≤ p · wi .
p · x i + πi · G
This yields a contradiction to condition (ii) of the Lindahl equilibrium of the
b
auxiliary economy E.
Step 3: For each consumer i ∈ N, we set
∑
def
π i = pg −
π−j .
(5)
j̸=i
Claim 3. For each consumer i ∈ N (resp. i ∈ Hk ) and public good k, it
holds that (πi )k ≥ (Πii )k (resp. (πi )k = (Πii )k ).
19
Proof. By construction, for each consumer i, public good k, and consumer
j ̸= i, it holds that
∑
(pg )k ≥
(Πij )k and (π−j )k ≤ (Πij )k .
j∈N
It then follows that
(πi )k = (pg )k −
∑
j̸=i
(π−j )k ≥
∑
(Πij )k −
j∈N
∑
(Πij )k = (Πii )k .
j̸=i
Moreover, if (gi )k ̸= 0, then it follows from Claim 1 and Claim 2 that the
above inequalities are equalities.
Now, we prove condition (ii) of the warm-glow equilibrium. First, we
show that the budget constraint of each consumer is binding. Indeed, for
each consumer i, from Claim 2 and Claim 3 we respectively get
π−i · gbj∗ = Πji · gbj∗ , for each consumer j ̸= i
and
πi · gbi∗ = Πii · gbi∗ .
Hence, it follows that
b∗−i = p · x∗i + πi · gbi∗ +
p · x∗i + πi · gbi∗ + π−i · G
∑
π−i · gbj∗
j̸=i
= p·
x∗i
+
= p · x∗i +
Πii
·
∑
gbi∗
+
∑
Πji · gbj∗
j̸=i
Πji
· gbj∗
j
b∗
= p · x∗i + Πi · G
= p · wi .
Suppose that for some consumer i and some consumption bundle (xi , gi , G−i ),
it holds that
b∗−i ).
ui (xi , gi , G−i ) > ui (x∗i , gbi∗ , G
20
Let us consider the following public goods bundle (b
gj )j̸=i defined as follows:
{
(G−i )k
if j ∈ Ji,k ,
|Ji,k |
(b
gj )k =
0
otherwise.
∑
Then, it is obvious that G−i = j̸=i gbj . Moreover, from Claim 2 and Claim
3 we respectively get
π−i · gbj = Πji · gbj , for each consumer j ̸= i
and
πi · gi ≥ Πii · gi .
Then, it holds that
b∗−i ) = u
b∗ )
u
bi (xi , gb1 , . . . , gi , . . . , gbn ) = ui (xi , gi , G−i ) > ui (x∗i , gbi∗ , G
bi (x∗i , G
Hence, it follows from the utility maximization of the Lindahl equilibrium
that
∑
π−i · gbj
p · xi + πi · gi + π−i · G−i = p · xi + πi · gi +
j̸=i
≥ p · xi +
Πii
· gi +
∑
Πji · gbj
j̸=i
b
= p · x i + Πi · G
> p · wi .
Finally, condition (iii) of the warm-glow equilibrium follows trivially from
definition (5). Hence, ((x∗i , gi∗ , πi , π−i )i∈N , p, pg ) is a warm-glow equilibrium.
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