Individual Responsibility and the Funding of Collective Goods

Abstract

When a deficit occurs in the funding of collective goods, it is usually covered by raising the amount of taxes or by rationing the supply of the goods. This article compares the efficiency of these institutions. We report the results of a 2x2 experiment based on a game in the first stage of which subjects can voluntarily contribute to the funding of a collective good that is being used to compensate the victims of a disaster. In the second stage of the game, in case of a deficit, we introduce either taxation or rationing. Each treatment is subjected to two conditions: the burden of the deficit is either uniform for all the subjects, or individualized according to the first-stage contribution. We show that the individualized treatments favor the provision of the collective good through voluntary cooperation whereas the uniform treatments encourage free-riding. Individualized taxation brings the voluntary contributions closer to the optimum while uniform rationing appears to be the worst system since free-riding restrains the provision of the good.

Full text

Page 1

IZA DP No. 3041
Individual Responsibility and the Funding of
Collective Goods
Louis Lévy-Garboua
Claude Montmarquette
Marie-Claire Villeval
D I S C U S S I O N P A P E R S E R I E S
Forschungsinstitut
zur Zukunft der Arbeit
Institute for the Study
of Labor
September 2007

Page 2

Individual Responsibility and the
Funding of Collective Goods


Louis Lévy-Garboua
University of Paris I and CIRANO

Claude Montmarquette
CIRANO and University of Montréal

Marie-Claire Villeval
CNRS-GATE, University of Lyon,
and IZA


Discussion Paper No. 3041
September 2007



IZA

P.O. Box 7240
53072 Bonn
Germany

Phone: +49-228-3894-0
Fax: +49-228-3894-180
E-mail: iza@iza.org


This paper can be downloaded without charge at:
http://ssrn.com/abstract=1014382

An index to IZA Discussion Papers is located at:
http://www.iza.org/publications/dps/



Any opinions expressed here are those of the author(s) and not those of the institute. Research
disseminated by IZA may include views on policy, but the institute itself takes no institutional policy
positions.

The Institute for the Study of Labor (IZA) in Bonn is a local and virtual international research center
and a place of communication between science, politics and business. IZA is an independent nonprofit
company supported by Deutsche Post World Net. The center is associated with the University of Bonn
and offers a stimulating research environment through its research networks, research support, and
visitors and doctoral programs. IZA engages in (i) original and internationally competitive research in
all fields of labor economics, (ii) development of policy concepts, and (iii) dissemination of research
results and concepts to the interested public.

IZA Discussion Papers often represent preliminary work and are circulated to encourage discussion.
Citation of such a paper should account for its provisional character. A revised version may be
available directly from the author.

Page 3

IZA Discussion Paper No. 3041
September 2007









ABSTRACT

Individual Responsibility and the Funding of Collective Goods*

When a deficit occurs in the funding of collective goods, it is usually covered by raising the
amount of taxes or by rationing the supply of the goods. This article compares the efficiency
of these institutions. We report the results of a 2x2 experiment based on a game in the first
stage of which subjects can voluntarily contribute to the funding of a collective good that is
being used to compensate the victims of a disaster. In the second stage of the game, in case
of a deficit, we introduce either taxation or rationing. Each treatment is subjected to two
conditions: the burden of the deficit is either uniform for all the subjects, or individualized
according to the first-stage contribution. We show that the individualized treatments favor the
provision of the collective good through voluntary cooperation whereas the uniform
treatments encourage free-riding. Individualized taxation brings the voluntary contributions
closer to the optimum while uniform rationing appears to be the worst system since free-
riding restrains the provision of the good.


JEL Classification: H41, H21, H30, H50, C91

Keywords: collective goods, taxation, rationing, responsibility, interior optimum, experiment


Corresponding author:

Marie-Claire Villeval
CNRS-GATE
93, Chemin des Mouilles
69130 Ecully
France
E-mail: villeval@gate.cnrs.fr





* We are grateful to Mark Isaac and participants at several conferences and seminars for helpful
comments. We also thank Romain Zeiliger for programming the experiment presented in this article
and Nathalie Viennot for research assistance. Financial support from the French Ministry of Research
(ACI Inter-SHS) and the Government of Québec is gratefully acknowledged.

Page 4

1. INTRODUCTION
There are instances in which members of a group or society are confronted to the same
basic need in their lifetime but do not face it simultaneously. For example, all families will need
to send their young children at school and all persons suffering from diseases will need health
care. Over a great number of periods, such needs will be felt with near certainty. In a given
period, however, the distribution of needs in a large population might be considered random.
Consequently, privately consumed goods and services of this sort can be mutually funded like an
insurance policy in order to take advantage of risk diversification. Redistributive taxation and
rationing are two widely used policies for financing such collective goods. In the first case, the
optimal provision of the good is guaranteed but the deficit generated by the insufficient level of
voluntary contribution is funded by compulsory taxes at an additional cost for society. In the
second case, the collective good is provided only to the extent that it has been voluntarily funded
by group members. The budget constraint is always met at the cost of possibly providing the
collective good at a suboptimal level. Taxation adjusts aggregate contributions to the socially
desirable consumption level, while rationing adjusts aggregate consumption to the sum of
voluntary contributions. In any case, however, redistribution generally occurs. This has the
effect of blurring the relation between the price paid by each individual to satisfy his need and the
marginal utility of the collective good, at the expense of efficiency.
The goal of this paper is to compare the efficiency of various modes of redistributive taxation and
rationing in the provision of collective goods. Specifically, we introduce uniform and
individualized modes of both taxation and rationing policies. Uniform and individualized modes
reflect different conceptions of equity: the former value equal treatment a priori whereas the latter
value equal outcomes a posteriori. By crossing two policies (taxation, rationing) with two modes

2

Page 5

(uniform, individualized), we are able to compare four different institutions that are widely used
for funding collective goods. We question here the ability of these institutions to give individuals
a sense of “responsibility” by letting them spontaneously come closer to efficiency.
For this purpose, we have designed a laboratory experiment based on a two-stage collective good
(insurance) game. We observe individual contribution behavior when private needs are met
collectively and prices are not expressed on a market.1 In the first stage, subjects receive an
endowment and voluntarily contribute to a common pool intended to compensate for the losses
suffered by those members of the group who happen to be randomly hit at the beginning of the
second stage. In the second stage, two situations may occur. If the losses can be fully or more
than fully covered by voluntary contributions in the aggregate, each victim can be totally
compensated and any surplus is burnt. On the other hand, if the voluntary contributions fall
short of the losses to be covered in the aggregate, four different solutions for equalizing the
supply of funds and the demand for compensation are being considered. Uniform policies offer
equal treatment of group members in the second stage (ex post), irrespective of their individual
effort in the first stage (ex ante). In contrast, individualized policies offer a differentiated
treatment of group members in the second stage, specifically favoring those who contributed
most in the first stage. Thus, four institutions are considered in the second stage in a 2x2
experimental design. We describe them respectively as uniform taxation, individualized taxation,
uniform rationing and individualized rationing.
The four treatments share the same social optimum of “equal contribution for all”. However,
they lead to very different Nash equilibria: no contribution for uniform policies, an interior

1 Olson (1965), Dawes and Thaler (1988) and Ostrom (1990) have shown how, in this context of collective action, it
is tempting for the actors to free-ride on the others’ contribution.

3

Page 6

contribution for individualized rationing, and both symmetric and asymmetric contributions
including a number of over-contributions for individualized taxation. Repeating the game fifty
times allows us to observe the convergence to the predicted Nash equilibria and its speed, and the
selection of equilibrium when many equilibria exist. Thus, the experimental design enables a
clean comparison of four widely used policies which are normatively equivalent but may lead to
much contrasted equilibria
The specificity of our paper with respect to the experimental literature on the provision of public
goods is fourfold. The first specificity has to do with the nature of the good. Our game considers
collective goods and not public goods, in the sense that only the victims of the disaster can share
the good whereas a pure public good is non-excludable. In public good games, each subject
generally benefits from a unique marginal return from the good (Davis and Holt 1993; Ledyard
1995). In our game, since only the victims of the damage share the collective good, an
individual’s contribution can have a null internal return and a positive external return if he does
not become a victim himself.2 This design allows us to study behavior in a framework where the
subjects are not assured to get a return from their contribution and to compare the contributions as
a function of whether there is a guarantee to compensate the victims or not.
The second novelty of our experiment resides in the comparison of the efficiency of sanctions on
voluntary contributions when various combinations of private and public funding of the collective
good are used. Here, the sanction is targeted only to the victims in the case of purely private
financing of the collective good (rationing), whereas it is inflicted to the entire group in the case

2 Other experiments allow a contribution to the public good to have an interior return (for the person who
contributes) different from its public return (for the other members) to isolate altruism (Carter, Drainville, and Poulin
1992; Goeree, Holt, and Laury 2002). In contrast, in our experiment, when an individual decides of his contribution,
he ignores whether or not he will be a victim of the damage.

4

Page 7

of mixed financing (taxation). Since taxation triggers an additional monetary cost, a crowding-
out effect of the voluntary contributions by taxation (Bergstrom, Blume, and Varian 1986;
Roberts 1987; Andreoni 1993) is less likely in our experiment.
The third novelty of our experiment is the comparison of the efficiency of uniform versus
individualized mechanisms. Individualization is here related to a difference in the individuals’
contribution behavior, not to a difference of their ex ante situation (in terms of endowment, Isaac
and Walker, 1988, or of marginal return of the good, Chan, Mestelman, Moir, and Muller, 1999).
It becomes feasible to test whether individualizing the consequences of a deficit gives the agents
a sense of responsibility towards the voluntary funding of the collective good, whereas uniform
treatments reduce the feeling of responsibility.3
A last specificity of our experiment lies in the non linearity of the collective good’s return, which
allows us to compare various treatments having a common interior optimum but different Nash
equilibria. Introducing an interior equilibrium in public good games, Andreoni (1993), Sefton
and Steinberg (1996), Keser (1996), Isaac and Walker (1988), Willinger and Ziegelmeyer (1999),
Holt and Laury (2000) show that the subjects still over-contribute, but over-contribution is
reduced as the equilibrium draws closer to the optimum. In our game the optimum is interior and
similar across treatments, and we vary its distance to the equilibria. Thus, we can study how
various cooperation risks, i.e. the gap between the equilibrium and the optimum, influence
behavior.
The predictions of our theoretical model are confirmed by the results of the experiment. Uniform
rationing is the least efficient system in terms of the contribution level in that it does not

3 From a theoretical point of view, Kirchsteiger and Puppe (1997) show that a linear uniform tax-subsidy policy,
where the individual tax depends solely on the sum of the other members’ voluntary contributions, can implement
the optimal contribution only under very strict conditions.

5

Page 8

discourage free-riding and the service is not provided to the victims. Uniform taxation does not
perform much better since the collective good is provided but at an extra cost for the community,
which puts a strain on efficiency. Individualized rationing enhances the sense of responsibility,
but cooperation is not complete and the service is not provided in full. The most efficient
institution turns out to be individualized taxation. With an average contribution in the
neighborhood of the optimum, cooperation is more important and is stable over time. This
mechanism confers to individuals the highest sense of responsibility towards the funding of the
collective good. It cancels most of the crowding-out effect of taxes on the voluntary
contributions, by protecting cooperative individuals against being exploited by the free riders.
The rest of the article is organized as follows. Section 2 presents the theoretical model. Section 3
details the experimental design. Section 4 analyzes the results. Finally, section 5 concludes and
discusses the implications of these observations.
2. THEORY
The game consists of a two-stage collective good (insurance) game. For each of the four
treatments that we consider, we identify in this section the social optimum and Nash equilibria.
Every individual decides whether to spend part of his income Y on a mutual fund, which has the
purpose of covering losses for randomly hit subjects.
Let Y 100 tokens be the income perceived by subject i
≡
),..., 1 (
N
∈
in a given period;
ig : the voluntary contribution of individual i to the collective good (); 0
≥
ig
%
di: the random loss suffered by individual i, which can only take two values: 0 or . This
loss is an i.i.d. Bernoulli variable.
0
>
d

6

Page 9

i
π : the profit or net payoff of i (
i π~ if the individual profit is random).
Each of the S victims may experience one disaster at most in any period of game.4 Thus,
N
S
p ≡

represents the probability that any individual in a group of N become a victim following S draws
without replacement. In our experiment,
; 4, 12
==
SN
hence,
3
1
=
p
.
We consider successively 2 x 2 policies that can be implemented in case of a budget deficit:
(taxation, rationing) x (uniform, individualized). In case total contributions exceed the total
losses experienced by the victims, the surplus is burnt.5
2.1. Uniform taxation
The payoff function is:
2
1
11
1
if
1
N
100
2
if
100
⎪ ⎩

N
NN
j
j
ijj
jj
i
N
j
i
j
Lg
gLgLg
N
Lg
g
α
β
⎢
⎢
⎣
π
=
==
=
⎧
⎪
⎪
⎨
⎪
⎡⎤
⎥
⎥
⎦
>
⎡
⎢
⎣
⎤
⎥
⎦
⎛
⎜
⎝
⎞
⎟
⎠
−−−−+−
=
≤
−
∑
∑∑
∑
(1)
where
0
and
αβ >
.
The individual i’s final payoff is represented by the second row of Equation (1) when the budget
is sufficient ex ante to cover losses in the aggregate. It is then simply obtained by subtracting i’s
voluntary contribution to the collective good from the period income. There is an obvious

4 This hypothesis makes the number of disasters S and the total amount of losses () fixed values,
thus shortening the gap between the situation of a small experimental group and what the law of large numbers
would naturally achieve in a large group.
5 We justify this assumption by its experimental simplicity, and for reflecting the inflationary nature of public
expenditure through the principle that most taxes are not targeted, so that a local surplus will be redistributed to other
public uses within the same period.
LSdd
N
i
i
=≡
∑
=1
~

7

Page 10

incentive for each player to free ride if losses can be covered by the voluntary contributions of
other players. However, irresponsible behavior tends to create a budget deficit ex ante; and, if
this occurs, individual i’s final payoff will be defined by the first row of Equation (1). The
income net of own contribution is now diminished by the cost of financing the deficit imputed to
i. The third term describes the individualized contribution to covering the deficit. The last term
between brackets with ,0
α β > represents the extra cost of taxation. The latter is a quadratic
function of the deficit because the deadweight loss of taxation is supposed to increase in the size
of the deficit. Under uniform taxation, all members of the pool share the same burden
irrespective of their initial contribution.
The complete coverage of victims removes any risk. The planner simply maximizes the sum of
payoffs which is symmetric in all individual contributions. Hence, everybody must contribute by
the same amount:

()
1,...,
ji
ggjiN
≡ ∀ ≠ ∈
.
The normative payoff per capita (1) is described as follows:
[]
2
*
i
100
⎪
⎨
⎪
⎩
2
100
i
i
L
N
L Ng
N
g
α
β
⎨
⎩
π
⎧
⎧⎫
⎬
⎭
−−+−
=
−

ig
i
L
N
L
N
ifg
if
<
≥
(2)
Function
*
i π reaches its maximum when
,
i
L
N
g
=
and the budget is balanced ex ante. This
defines the social optimum.
The Nash equilibrium is attained when each player i maximizes his payoff with respect to the
others’ contributions. If g denotes the average contribution of all the other players, i’s payoff is:

8

Page 11

πi=
100− gi−1
N
L− (N −1)g − gi
⎡⎣⎤⎦− β +α
⎣⎢
2N
L− (N −1)g − gi
()
2
⎡⎤
⎦⎥
if gi+ (N −1)g < L
100− gi
⎩
if gi+ (N −1)g ≥ L
⎧
⎪
⎨
⎪

(3)
In the presence of a deficit, the payoff is given by the first row of Equation (3). Its maximum is
reached for a null contribution if the following condition is met:
1
−≤ NL
α
, which is the case for
reasonable values of α and, in particular, in our experiment. By symmetry, all the players then
have a null contribution in equilibrium. In the absence of a deficit, the maximum payoff is
obtained for:
0) 1
−
(
≥≡ −=
Dg NL gi
, whereD is the ex ante deficit before i’s contribution
and takes an integer value. For Nash equilibrium to exist without an ex ante deficit, we must
have: ) 0 (
i
π
) (
iD
π
≥
. Comparison of the first and second rows of Equation (3) yields after
simplification:
0
1
2
2
> +
−
−β
α
N
D
N
N
D
.
With the numerical values given to the constant parameters α and β in our experiment
(
2,005.
==
βα
), the only admissible values are:
=
D
0, 1, 2, that is,
1
−
N
subjects almost cover
total expected losses all together and the
th player gives almost nothing. However, such
equilibria seem highly implausible and should not be selected if players have rational
expectations. Indeed, they imply that a player believes that some other players have over-
contributed (above L/N), which is not a rational expectation in our design in which any budget
surplus is burnt, unless the fixed and marginal deadweight cost of a deficit grow large enough.
Thus, when uniform taxation is used to finance a budget deficit, zero contribution is the likely
Nash equilibrium if
N
1
−≤ NL
α
.


9

Page 12

2.2. Individualized taxation
In contrast with the previous policy, the tax levied on individual i for financing a budget
deficit now offsets the voluntary contribution. Player i’s payoff is therefore defined as follows:

πi=
100− gi−
⎨
⎪
⎪
⎪
L
N− gi
∑
L−
gj
j=1
N
L−
gj
J =1
N
∑
⎡
⎣⎢
⎤
⎦⎥− β +α
⎩⎪
2N
L−
gj
j=1
N
∑
⎡
⎢
⎣
⎤
⎥
⎦
2
⎧
⎨
⎪
⎫
⎬
⎭⎪
⎪
100− gi
⎧
⎪
⎪⎪
⎩
if L >
gj
j=1
∑
N
∑
if L ≤
gj
j=1
N

The term between brackets with ,0
α β > represents the extra cost of financing the deficit. The
third term describes the individualized contribution to covering the deficit.
After simplification,
2
1
1
1
if
100
⎪
=⎨
⎪
⎩
2
if
100
N
N
j
j
j
i
j
N
j
i
j
Lg
L
N
−
Lg
N
Lg
g
α
β
π
=
=
=
⎧
>
⎡
⎢
⎣
⎤
⎥
⎦
−−−−
≤
∑
∑
∑
(4)
As in the previous case, the complete coverage of victims removes any risk. As a consequence,
the normative payoff function has the same expression as in (2). Thus, the social optimum is
unchanged: all subjects must equally contribute L
N to reach a balanced budget ex ante. Although
the social optimum does not depend on the nature of taxation, individualization has the effect of
making the social optimum a Nash equilibrium as well. If agent i is looking for the maximum
payoff given the others’ contributions (equal tog on average), i’s payoff is expressed as:
()2
100
⎪
=⎨
⎪
⎩
(1)(1)

2
100(1)
ii
i
ii
L
N
LNgg if gNgL
N
gif gNgL
α
β
π
⎧
−−−−−−+−<
−+−≥
(5)

10

Page 13

It reaches its maximum when the budget is balanced ex ante, if
β+≤
NLD
, i.e. if (
1
−
N
)
members of the group have made on average an almost optimal contribution. One of these
balanced budget-equilibria is symmetric and coincides with the social optimum. Intuitively, an
individual who believes that other members have paid their share of expected total losses and
knows that he will have to pay his own share anyway will prefer to do it ex ante rather than ex
post in order to avoid the payment of the additional cost.
There are also asymmetric equilibria with an ex ante budget deficit. In these, the individualized
taxation guarantees that the more generous players will not be penalized ex post, since all players
eventually will get the same profit. Furthermore, as their payoff depends on the ex ante deficit
and rapidly decreases in this deficit (given the quadratic term), some players may over contribute
relative to the social optimum in order to curb the deficit and the resulting efficiency cost.
2.3. Uniform rationing
Rationing the contributors is another way of financing a publicly-funded insurance. Now,
there can be no deficit to be covered ex post because the amount of indemnities will be
automatically adjusted to the sum of voluntary contributions if the latter are not high enough to
cover all losses. Pool members collectively bear the risk of being only partly covered if they
become victims, so that an agent’s net payoff becomes uncertain even though the sum of
individual losses is certain. We first consider uniform rationing of the victims: each victim’s loss
is limited by the sum of all contributions to the fund relative to losses. This is expressed as
follows:

11

Page 14

N
j
1
L
j=1
∑
N
j
j=1
if g
100
⎪
⎪
⎨
⎪
⎪
⎪⎩
1
if g
100
N
j
j
ii
i
i
g
L
gd
%
L
g
π
%
=
⎧
⎪
⎛
⎜
⎜
⎜
⎜
⎝
⎞
⎟
⎟
⎟
⎟
⎠
<
−−−
=
≥
−
∑
∑
(6)
The presence of an individual risk makes the definition of the social optimum problematic
because the social planner must account for the players’ risk attitudes. A psychologically-
founded solution (see Lévy-Garboua et al., 2006) is to assume that the social planner behaves like
an impartial judge who, by lack of knowledge of others, attributes them his own characteristics in
terms of initial wealth and risk aversion. Since all members of the group enter symmetrically in
this game, the social planner i would choose the same contribution gi for all and eventually
maximize his own expected utility, )
~
π
(
∗+
iiiwEU
, with:
*
i
if
100
⎪
⎪
⎨
⎪
⎪ ⎩
1
if
100
i
i
ii
i
i
L
N
L
N
Ng
L
g
gd
%
g
g
π
%
⎧
⎛
⎜
⎝
⎞
⎟
⎠
<
−−−
=
≥
−
(7)
If the sum of contributions covers the sum of losses, all subjects are fully insured and receive a
sure payoff. The optimum is then obtained for
i
L
N
g
=
. However, payoffs are uncertain if
contributions fall short of the amount of losses. Consequently, the social planner maximizes the
following expected utility:
() (
=
)()
*
i
11001001
i
iiiiiiii
Ng
L
EU wp U wgpUwgd
π
%
⎛
⎜
⎝
⎞
⎠
⎛
⎜⎝
+−+−++−−−
⎠
⎞⎟ ⎟ (8)
where
designates his concave utility function of wealth (
i
U
0, 0
<
′ ′
i
>
′
i
UU
). It is shown in
Appendix A that the social optimum still corresponds to
N
L
gi=
, as in the case of taxation. A

12

Page 15

remarkable feature of our definition of the social optimum is that it is robust to variations of
endowed wealth and risk aversion.
To determine the Nash equilibrium, we consider that player i maximizes his expected utility
given the other players’ contributions
. The payoff writes as:
),...,,,...,(
111
Nii
gggg
+−
100
⎪
=⎨
⎪
⎪
⎪
⎩
[1]
100
ij
j i
L
iiij
j i
≠
i
iij
j i
≠
gg
gd
%
ifggL
g ifggL
π
%
≠
⎧
⎪
⎪
+
−−− + <
−+ ≥
∑
∑
∑
(9)
If there exists a Nash equilibrium without a budget deficit, it must be
and
Dgi=
D
i
−=100
π
.
In case of a budget deficit, the random payoff is described by the first line of equation (9) and the
expected utility writes as follows:
)
1
100 ( )100 ()1 ( )
~
π
(
i
i
S
j
j
iiiiiii
g
S
S
g
S
L
wpUgwUpEU
−
−+−++−+−=
∑
≠


)
1
100()100()1 (
iiiiii
g
S
S
S
D
wpUgwUp
−
−−++−+−=

It reaches its maximum when the player’s contribution is null regardless of the other players’
contributions. The null contribution represents the only equilibrium since the expected utility is
then greater than the balanced budget utility )100(
DwU
ii
−+
for all D.
2.4. Individualized rationing
The rationing is individualized by having it depend on each individual contribution to the
fund relative to the contributions of the other victims. We impose here the two following

13