Journal
of Public
Economics
52 (1993) 217-235.
North-Holland
Exclusion and moral hazard
The case of identical demand
Emilson CD. Silva and Charles M. Kahn*
Department of Economics, University
Received
September
of Illinois, Champaign-Urbana, IL 61820, USA
1991, tinal version received July 1992
This paper examines the problem of costly exclusion of individuals from a public good. Previous
analyses of exclusion have treated it as solely a question
of technologies;
in our analysis
exclusion depends on technology
and incentives. In this paper providers of the good design a
mechanism
to provide
an optimal
level of deterrence
to free riders. If individuals
are
heterogeneous
optimal deterrence may allow some free riders. We examine the effect of costs of
exclusion on the Samuelson condition for optimal provision, and see that the desire to deter free
riding leads to underprovision
of the good irrespective of the degree of rivalry of the good.
1. Introduction
Economics textbooks distinguish among goods according to levels of
excludability of consumption benefits. Private goods are perfectly excludable;
the seller of a private good can always prevent people who do not purchase
the good from enjoying its benefits. Pure public goods are perfectly nonexcludable; the seller of a pure public good is unable to prevent any agent
from enjoying the good.
However, almost all goods lie between these two extremes. It is neither
absolutely costless nor absolutely impossible to prohibit non-purchasers from
enjoying the benefits of most goods. Software computer packages are a
natural example of imperfect exclusion. Designers incorporate
copyprotection in their computer programs to prevent non-purchasers from using
them. Nevertheless, exclusion is not perfect because the more secure the
program is made the less valuable it is to legitimate users. The services of
satellite television provide another instance of imperfect exclusion, since it is
costly to employ scramblers that impede viewers’ reception of channels not
Correspondence to: CM. Kahn, Department
of Economics, University of Illinois, Box 111, 330
Commerce W. Building, 1206 South 6th Street, Champaign,
IL 61820, USA.
*We thank Jan Brueckner,
Myrna Wooders, two referees of this journal and participants
in
the Public Finance workshop of the University of Illinois for useful comments. We also thank
Christine M. Silva for typing assistance. Kahn’s research was funded by NSF grant SES 8821723
and Silva’s research was funded by CNPq (Conselho National
de Desenvolvimento
Cientifico e
Tecnolbgico-Brazil)
scholarship
200883/87.9.
0047-2727/93/$06.00
0
1993-Elsevier
Science Publishers
B.V. All rights reserved
218
E.C.D. Silua and C.M. Kahn. Exclusion and moral hazard
previously paid for. Thus, the satellite television industry also faces a
trade-off between levels of excludability and final prices.
Previous analyses of the choice of the degree of exclusion have treated the
problem as purely technological. But it is also a question of incentives.
Exclusion technologies make it more difficult for a consumer to use the good
without paying; the degree to which he attempts to do so will depend upon a
comparison of the benefits he can achieve by consuming licitly with the
benefits he can achieve by consuming illicitly.
In this paper we investigate choices of degree of exclusion made by the
producer of a public good. His choices are mediated both by the technology
and by the incentives of the consumers to free ride. When exclusion is
imperfect, provision depends on both marginal costs and marginal incentive
benefits from inspection, even when the good is non-rivalrous. Although
willingness to pay for exclusion devices increases with the degree of rivalry,
rivalry is not necessary for exclusion to be desirable.
Our investigation points out a natural economy of scale in free riding. If
the effectiveness of the deterrence of a particular degree of exclusion depends
on the number of other agents attempting to free ride (as it does in the case
of typical monitoring technologies), then there may be multiple equilibrium
responses to a given exclusion technology. We focus on stable equilibrium
responses by considering the incentives both of individuals and of groups of
individuals to free ride.’
First we consider a situation in which all agents are homogeneous and the
public good is non-rivalrous. In this case, costly exclusion reduces the quality
of the good provided. Next, we examine the provision of a rivalrous public
good. Here, costly exclusion also affects membership in the organization. As
long as agents are homogeneous, the optimal arrangement involves no free
riding. We conclude with an example of a case of heterogeneous agents
where some free riding may be part of an optimal arrangement.
2. Literature review
Most papers on local public goods and clubs take exclusion for granted.2
The theory of local public goods assumes that exclusion from benefits occurs
as soon as individuals select their localities of residence. By choosing a
location, the individual determines his pattern of taxes and consumption of
local public goods. Thus, as long as local governments can costlessly enforce
‘Formally,
our analysis applies coalitional
retinements
to a game which has multiple Nash
equilibria.
*The literatures on local public goods and clubs [introduced
by Tiebout (1956) and Buchanan
(1965), respectively]
study goods which exhibit excludability
and some sharing properties,
but
limited availability.
Current papers in these literatures
have focused on congestion
effects and
pricing mechanisms.
Well-known
examples are Littlechild (1975) Berglas (1976), Wooders (1978,
1989) Berglas and Pines (1981) and Scotchmer and Wooders (1987).
E.C.D. Silva and C.M. Kahn, Exclusion and moral hazard
219
taxation of residents, exclusion of benefits is costless. Club theory gives a
non-spatial approach to costless exclusion of non-contributors.
Since the
collective goods provided by clubs are generally limited in availability, club
members find it beneficial to exclude some individuals. The implicit assumption is that exclusion is costless and perfectly effective. Indeed, perfect
exclusion is sufficient for allocation of collective goods provided by clubs
through a market system. For example, Wooders (1978) develops a model in
which a local public good equilibrium has multiple jurisdictions, where
within each jurisdiction members consume the same amount of the local
public good, face the same amount of congestion in consumption and pay
lump-sum taxes which equal their Lindahl tax times the amount of public
good consumed minus their profit shares. An interpretation of the equilibrium in that model regards jurisdictions as competing with each other and
with potential jurisdictions to attract clientele. Such equilibria are in the core
of that economy.3
Oakland (1972) is a very early example of a paper which examines the
optima1 degree of exclusion. In that paper the marginal cost of provision of
the public good consists of a production cost component and an exclusion
cost component. The marginal benefit from exclusion is the increase in
consumers’ utility from the associated reduction in congestion. As exclusion
becomes costly, optima1 utilization rises. Since the exclusion technology is
treated as a ‘black box’, the mode1 can only be used to describe optima1
levels of provision of the good, not to consider whether any mechanism can
achieve the optimum.
According to Oakland, costly exclusion should be adopted if society is
better off by restricting consumption of rivalrous goods to high demanders.
Laux-Meiselbach (1988) formalizes this line of argument by describing social
gains from exclusion as internalization of consumption externalities. Public
provision should arise when the costs of exclusion exceed the social gains.
Since the framework only considers a choice between perfect exclusion and
no exclusion, the question of the optima1 degree of exclusion does not arise.
The optima1 degree of exclusion for rivalrous public goods is examined by
Helsley and Strange (1991). In their analysis, exclusion is a choice variable
under the control of providers of rivalrous public goods. Private providers
have two exclusion technologies available. With ‘coarse’ exclusion, providers
incur the costs of verifying membership. With ‘tine’ exclusion, they also incur
the costs of monitoring intensity of use. As the authors note, since their
mode1 does not consider exclusion of non-payers, it ignores the free-riding
problem.
Although our analysis also examines costly exclusion, its focus is quite
3Wooders (1989) shows for a quite general model that core-equilibrium
equivalence
obtains
for large economies
with local public goods or public goods provided by clubs where small
groups can realize almost all gains to collective consumption.
220
E.C.D. Silva and CM. Kahn, Exclusion and moral hazard
different. In our model, the degree of exclusion is a choice variable controlled
by the private company providing the public good. This company acts as a
principal, and consumers act as agents. Unlike Oakland and LauxMeiselbach, our model has agents determining their best responses to
exclusion levels chosen by the principal. Therefore it can be used to
investigate the feasibility of achieving efficient allocations. Unlike Helsley and
Strange, our exclusion technology is directed against agents who have not
paid for consumption.
Our framework builds on the literature on incentives. Most investigations
of public goods and incentives focus on preference elicitation.4 In contrast,
we assume agents’ preferences are known to the principal and focus on a
problem of moral hazard - an agent’s attempt to enjoy the benefit of
consumption without paying.
3. The models
There are two goods in our economy: a private good ‘money’ and a public
good ‘public transportation’. A private company is considering the provision
of public transportation. Since individuals will be tempted to free ride when
the good is provided, the owner (principal) must design a mechanism that
induces agents to pay for the public good. The mechanism will specify the
responsibilities of the company, the number of subscribers, and the enforcement rules.
The other agents in the economy have the option of purchasing or not
purchasing the good. Subscribers pay a fee to purchase the good. Those who
do not pay have a further decision, namely whether or not to attempt to free
ride. The principal may attempt to exclude free riders.
We deal with a good for which free riding can be detected, at a cost.
Subscribers are distinguished from free riders via a system of inspection. A
fine, exogenously determined by the legal system, is charged to anybody who
is caught free riding.5
The mechanism consists of four parameters, {n, z, k, s}, corresponding,
respectively, to the maximum number of subscribers the owner will admit,
the total amount of the public good provided, the degree of monitoring, and
the fee subscribers pay.
The actual number of subscribers is determined endogenously. The cost of
“See, for instance, Groves and
costly exclusion.
5We assume the tine is paid to
The results will continue to hold
the tine is not completely efficient
exceed the compensation
received
Ledyard
(1977). Novas
and Waldman
(1988) also investigate
the legal system; the provider of the good gets none of the tine.
in more general circumstances
as long as the administration
of
- that is, as long as the costs borne by those paying the tine
by the agency.
E.C.D. Silva and C.M. Kahn, Exclusion and moral hazard
221
the service depends on its degree of rivalry; for a fixed level of provision, z,
and fixed ridership the cost of provision increases with the degree of rivalry.
The level of monitoring
corresponds
to the number
of riders who are
inspected during the allocation of the public good. The fee in equilibrium
will
consist of the equal share of the total cost of provision
every subscriber
pays.
Although we focus on a single owner, we assume that he faces competition
from other (potential
or actual) providers
of the good. Throughout
the
paper, we will deal with situations
where agents have identical demands for
public goods. Therefore competition
with free entry is characterized
by the
assumption
that the owner chooses a mechanism which maximizes the utility
of a representative
subscriber.‘j
In designing the mechanism,
the principal takes into account his payoffs
from providing the good, agents’ willingness to subscribe, and their incentives
to free ride. The models in subsection 3.1 show how these incentives influence
the shape of the mechanism
designed
for a homogeneous
population.
Subsection
3.2 presents
an example where a mechanism
designed for a
heterogeneous
population’
allows some free riding.
3.1. Homogeneous population
3.1.1. Technology
The firm provides service of quality z to r riders at a cost T(z,r),* where
T is increasing
and convex in z and T(z, r)/r+oo as r-0.
In addition, the
firm pays an exogenous
cost of 4 per inspection
undertaken.
Our results
depend on the assumption
that the agent’s tine f, if caught free riding,
exceeds the unit cost 4 of inspection,
that is, f >q. Otherwise,
there will
typically be no feasible positive level of production.
The probability
of apprehending
a free rider is p(k,r), a function of the
number of inspections
k and the total ridership r. We assume p is concave
and weakly increasing
in k. A natural
example is the case of random
sampling without replacement:
p( k, r) = min (k/r, 1).
6The situation
where agents disagree about the trade-offs between quality and price of an
imperfectly
excludable
good presents a large number
of interesting
and difficult modeling
problems and must be deferred for future research.
‘As in Scotchmer and Wooders (1987) heterogeneity
of tastes can be consistent with identical
demand for the public good.
*That is, we focus on cases where the total cost of provision is affected by ‘service rivalry’
such as maintenance
cost.
222
E.C.D. Silva and C.M. Kahn, Exclusion and moral hazard
3.1.2. Tastes
An agent who consumes
has utility’
m units of money
and transportation
of quality
z
U(m, z, r) = m + v(z, r),
where v is weakly decreasing and concave in r, weakly increasing in z and
~(0, I) = 0. Thus, r indexes congestion within a firm.
Agents are initially endowed with m, units of the private good. The total
population
of agents is represented
by P. Ridership
on the public good
consists of n subscribers
and a free riders. Thus the total number of honest
non-subscribers
is P-n-a.
The utility of a subscriber
is U(m, -s, z, I) =
m, - s + v(z, r); the utility of an honest non-subscriber
is U(m,,, 0,O) = m,. The
utility of a non-subscriber
who attempts to free ride is stochastic because it
depends on the probability
of being caught during inspections.
His expected
utility is
pU(m,-f,z,r)+(l-p)U(m,,z,r)=m,-p(k,r)f+v(z,r).
3.1.3. The case of a non-rivalrous public good
For a non-rivalrous
public good crowding affects neither utility or cost.
Therefore, for this section we write T(z) for cost and v(z) for utility.
We begin by analyzing
the incentives
to free ride. A non-subscriber
attempts to free ride if free riding is both individually
rational and incentive
compatible.
Free riding is individually
rational if the expected utility of a free
rider is no less than the utility of an honest non-subscriber.
Free riding is
incentive compatible
if the expected utility of a free rider is no less than the
utility of a subscriber. Thus, whenever (1) and (2) are satisfied, free riding is
the best response of any non-subscriber
given inspection
levels k, and the
number of riders, r:
v(z) 2 p(k, r)f,
(1)
s 2 p(k, r)f.
(2)
Constraint
(1) states that free riding is individually
rational if the action’s
expected cost is not greater than the action’s benefit. Constraint
(2) states
that free riding is incentive compatible
if its expected cost is not greater than
the subscription
cost.
Note that the desirability
of free riding depends on the total ridership.
There is a kind of ‘economy of scale’ in free riding. The more non-subscribers
who free ride, the more difficult it is to catch a non-subscriber
free riding. It
is possible to have multiple equilibria such that, provided no non-subscriber
is free riding, no other non-subscriber
finds it desirable to do so, but if large
‘Analogous
results hold for general
additively
separable
utility functions.
E.C.D. Silva and C.M.
Kahn,
Exclusion
223
and moral hazard
numbers of non-subscribers are free riding, every non-subscriber finds it
desirable. For example, in the case of random sampling without replacement,
constraint (1) becomes
u(z) 2 fk/(n + a).
Define a, =fk/u(z) --n; that is, a,, corresponds to the minimum number of
free riders necessary to make free riding desirable. If a, lies between 1 and
P-n, then it is not rational for a non-subscriber to attempt free riding
provided no other non-subscriber is free riding:
u(z) <fkl(n + l),
but all non-subscribers will prefer to free ride whenever at least a,
non-subscribers are free riding. Such a mechanism will have two stable
equilibria: one with no free riding and one with complete free riding by
non-subscribers.” Of the two stable equilibria, the one with no free riding is
coalitionally unstable, since non-subscribers collectively prefer to free ride.”
In general then, some coalition of non-subscribers prefers to free ride unless
min Cs,WI 5 Ak W.
In deciding whether to purchase the public good an agent compares the
utility from subscribing with the utility from not consuming the good and
with the expected utility from free riding. Thus, for an agent to subscribe
both of the following conditions must hold:
u(z) 2 s,
(3)
p(k, P)f 2 s.
(4)
Condition (3) is the agent’s individual rationality constraint; it guarantees
that the benefits of subscribing exceed the fee. Condition (4) is the agent’s
incentive compatibility constraint; if (4) is satisfied, no agent prefers to free
ride.r2
The company provides the good if total revenue is at least as great as the
total cost:
“‘There will also be an unstable equilibrium with exactly a,, free riders.
“For coalitional
equilibrium
concepts see Bernheim et al. (1987) or Kahn
(1992).
‘*If we simply required that
and
Mookherjee
s4Ak,r)f,
then no agent would find it desirable to free ride, given that the firm is currently serving r riders.
Condition (4) is more stringent; it captures coalitional
incentives for free riding. It says no agent
finds it desirable to free ride even if everyoneelse decided to use the system at the same time.
224
E.C.D. Silva and CM. Kahn, Exclusion and moral hazard
ns 2 T(z) + kq.
(5)
The principal (owner) designs a mechanism {n, z, k, s} which is feasible - that
is, which satisfies (3), (4) and (5). With potential entry, the principal
maximizes the surplus accruing to subscribers. (If he did not, another
provider would offer a better arrangement and compete away his subscribers.) Thus, the principal solves this problem:
max u(z) --s
(n,Z,k,Sl
(Pl)
subject to (3), (4), (5) and
It is possible that the solution to problem (PI) is to provide no public
good at all - i.e. to set z, s, and k equal to 0. However, if the public good is
provided, the following theorem provides the solution to (Pl). The conditions
are stated in the form appropriate for the case of random sampling without
replacement. The appendix gives the proof for the general case as well as
necessary and sufficient conditions for a non-zero solution.
Theorem 1.
The following conditions characterize
the non-zero solution to
(Pl):
n=P.
PO,= TzflCf - d,
k = Wl(f
- 4,
s = (T(z) + kq)/P.
(10)
Note that with potential entry and identical customers the solution to the
principal’s mechanism design problem also maximizes the following utilitarian social welfare function:
W=n(m,-s+v(z))+a(mO-pf+o(z))f(P-n-a)m,
subject to constraints (3), (4), and (5) which make the mechanism feasible.
The theorem tells us the following information about the solution to the
problem. Condition (7) states that the optimal mechanism totally inhibits free
riding (a=O) and, thus, everybody prefers to purchase the public good. Given
the lack of rivalry, the entire population is served by a single provider.
E.C.D. Silva and C.M. Kahn, Exclusion and moral hazard
225
Condition (8) determines z, the level of public good. It is the Samuelson
condition for the optimal provision of the public good under free-riding
possibilities. The left-hand side represents the sum of the marginal rates of
substitution between public and private goods (marginal social benefit); the
right-hand side is the marginal social cost of provision. Observe that the
marginal social cost is greater than the marginal production cost, T,, as long
as the inspection cost 4 is positive. Since T, is the marginal social cost in the
conventional case where 4 =O, the public good is underprovided relative to
the situation of costless inspection.
Condition (9) is the optimal monitoring condition. It shows that the
optimal level of inspection consists of the total technological cost divided by
the difference between the free-riding tine and the marginal cost of inspection.
In other words, the provider of the public good must spend resources on
monitoring so as to equate free-riding’s expected cost, kf/P, to subscription
cost, s.
Condition (10) determines the subscription fee. It shows that total cost is
shared equally among subscribers. As an alternative description of the
pricing, we could imagine that each member is charged a personal Lindahl
price v, = T,f/P(f -q) per unit of quality consumed plus a ‘participation
price’ [see Wooders (1978, 1980)] equal to s-zuZ. As long as the technology
T( .) is strictly convex, this participation price is positive. If there is constant
returns to z, then the participation price is zero.
Subscribers pay a higher subscription fee for the good when exclusion is
costly. To see this, substitute (9) into (10) to obtain
s= W)flP(f - 4
’
W/f’.
In short, relative to the case of costless exclusion, subscribers
consumption of a good which is underprovided and costs more.
3.1.4.
share the
The case of a rivalrous public good
When the public good is rivalrous, subscribers will be sensitive to the
presence of free riders. Crowding could drive away some subscribers and
force remaining subscribers to consume a good whose quality is degraded.
A non-subscriber will attempt to free ride if free riding is both individually
rational and incentive compatible, that is
r(z, r) 2 p(k r)f,
(11)
s2 dk, r)f.
(12)
Observe that now the desirability of free riding depends upon a comparison
between the opposing effects increased ridership has on consumption rivalry
and the probability of being caught. In general, as before, free riding will be
226
E.C.D. Silua and C.M. Kahn, Exclusion and moral hazard
desirable if there exists an rs P such that (11) and (12) hold. For simplicity,
we assume that
is decreasing in r whenever u(z, r) 2 T(z, 1)/r. In other words, in the relevant
region the disutility from congestion rises faster than the reduction in costs
of free riding as ridership increases. This assumption allows us to keep the
constraints for free riding simple since it implies that if no non-subscriber
finds it desirable to free ride, no coalition of non-subscribers will find it
desirable to free ride either.
An agent will purchase the public good if subscribing is desirable:
v(z, r) 2 s,
(13)
p(k, r)f L r(z, r).
(14)
The principal is willing to provide the good if
ns 2 T(z, r) -t kq.
(15)
Given competitive entry, the principal’s problem is to maximize the surplus
accruing to subscribers subject to feasibility constraints (13), (14) and (15) as
follows:‘3
max u(z, r) -s
(n.2,k.s.r)
(P2)
subject to (13) (14), (15) and
P2rkn20,
--~20,
kz0,
The non-zero
below.14
Theorem 2.
(164
~20.
solution
WW
to this problem
is summarized
in Theorem
2
The non-zero solution to (P2) satisfies the following conditions:
n = r,
(17)
k = rub, 4/f,
(18)
‘%~ce no
behaves as a
non-members’
l4The proof
non-subscribers
will prefer to free ride if (13) and (14) are satisfied, the principal
club manager who maximizes total net benefits accruing to members and keeps
utilities unchanged.
of Theorem 2 is similar to the proof of Theorem 1 and is thus omitted.
E.C.D. Silua and C.M. Kahn, Exclusion and moral hazard
221
s= T(z, W + 4z, dqlf,
(19)
ru, = T,(fl(f
(20)
- q)),
T(z, r)/r = T, - ru,( 1 - q/f),
if r < P,
(214
T(z,r)/r>=T,-ru,(l-q/f),
if r=P.
@lb)
Condition (17) states that the optimal mechanism totally inhibits free
riding (a=O). Condition (18) is the optimal monitoring condition. It shows
that the optimal level of inspection equals the sum of all subscribers’ benefits
from consumption divided by the free-riding line. In other words, the
expected cost from free riding equals the benefit from consumption. Condition (19) indicates that the optimal fee is the sum of average production
cost, T(z, r)/r, and the average inspection cost, u(z, r)q/f. Since q/f >O, the
optimal fee here is greater, ceteris paribus, than in the conventional case
where q =O. Condition (20) is the Samuelson condition for the optimal
provision of the rivalrous public good. Note that, as in the case of a nonrivalrous public good, each agent’s personalized Lindahl price is increasing in
exclusion costs. Since the marginal social cost of provision equals the sum of
all members’ personalized Lindahl prices, the rivalrous public good is
underprovided, ceteris paribus, relative to the costless exclusion case.
Condition (21) is the optimal ‘membership’ condition. It should be
contrasted with the optimal membership condition in the case of costless
inspection:
T(z, r)/r = T, - rv,.
(22)
The benefit of taking on an extra member is the payment that member gives
in sharing the cost of the public good. Since each member pays average cost,
the left-hand side of eq. (22) is this benefit. There are two costs to taking on
an extra member. The first is the marginal social cost of providing the service
T,; the second is the social congestion cost, - ru,. Membership expands until
benefit and costs are equated.
One interpretation of eqs. (21) is to note that the social congestion cost is
reduced by an amount q/f once associated inspection costs are included.
Thus, since average costs are unchanged and the marginal social costs of
increasing ridership decrease with marginal inspection costs, the optimal
membership increases with costly inspection.
For the purpose of comparison, consider a social planner who maximizes a
utilitarian social welfare function:
W=n(m,+u(z,r)-s)+a(m,+v(z,r)-p(k,r)f)+(P-n-a)m,
subject to the feasibility constraints (13) and (14). As part of the process, he
228
E.C.D. Siloa and C.M. Kahn, Exclusion and moral hazard
is assumed to pick the number of firms to operate and is thereby able to
control per firm’s crowding. If r is the crowding level per firm, then (n+a)/r
firms are operating. Thus the production
constraint
of the planner is
ns 2 ((n + a)/r) ( qz,I) + kq) .
It is easily verified that the solution to Theorem 2 is identical to the solution
to this planner’s problem.
It is also useful to compare the solution
in this section with the local
public goods model of Wooders (1978), where equilibria
are shown to exist
provided that the economy is replicated in the proper way. For our example,
such replication
requires that the total population
grows as an integer
multiple of the optimal firm size r as calculated in Theorem 2. Each active
firm in such an economy
would offer the good and would find its offer
accepted by r subscribers. Thus, our solution corresponds
to the equilibrium
in Wooders’
model. But by the standard
definition
of efficiency, these
allocations
would not be efficient. Expenditure
on monitoring
is a pure social
cost; all agents in our economy would prefer an allocation conforming
to the
public goods equilibrium
under costless monitoring.
Of course, such an
allocation
is not incentive compatible.
Since the solution to problem (P2)
does solve a planner’s problem with corresponding
incentive constraints,
it is
clear that it will satisfy a notion
of constrained
Pareto optimality
and
correspond
to a suitable version of the Incentive Compatible
Core [see Kahn
and Mookherjkee
(1992)].
In sum, our results indicate that a company tends to provide less to more
subscribers and at a higher fee than it would were exclusion costless.’ 5
3.2. Heterogeneous population
An example where free riding is optimal
In the model of the previous subsection,
the firm found it optimal to
completely exclude all free riders. In this subsection we consider a modification where complete exclusion can be suboptimal.
Suppose now that different people in the population
vary in the intensity
with which they use a non-rivalrous
transportation
system. Since the good is
non-rivalrous,
this variation
has no effect on provision.
But we assume that
the likelihood of being caught by the monitoring
technology depends on the
frequency of usage. (If a rider rides daily without a ticket, he is much more
“An earlier version of this paper also considered
an example of a natural monopoly
with
potential entry. Results were similar, although the limitation on the number of firms meant that
it was possible for equals to be treated unequally in the allocation.
229
E.C.D. Siloa and C.M. Kahn, Exclusion and moral hazard
likely to be caught than if he rides once a month.) Thus, different portions of
the population
place differing valuations
on free riding.
We assume an agent of type t derives the following utility from consuming
x trips, each of quality z:
U(m, t, x, 2) = m + min (t, x)v(z)/t.
In other words, the marginal value of a trip of quality z is o(z)/t up to t trips;
after which it is zero.
Monitoring
occurs on a per ride basis; the probability
of any agent being
caught is assumed to depend on the number of rides he takes. If J is the
total number of rides taken on the system and j is the number of rides taken
by a free rider, we represent the probability
of being caught as p(j, J, k). For
example, if we again considered random sampling without replacement,
then
p(j,J,k)=l-(l-k/J)’
(as long as k<J).
Thus, there is an economy of scale in free riding. The
marginal probability
of being caught decreases with the total number of rides
taken. A similar phenomenon
will occur with many other monitoring
technologies.
Therefore, henceforward
we may simply assume that a rider of
type t will either not ride at all or ride exactly t times. In other words, the
utility of a subscriber is m,--s + u(z), the utility of an honest non-rider is m.
and the utility of a free rider of type t is mo-p(t,J,
k)f +u(z). Observe that
since p(t, J, k) is non-decreasing
in t, if some type t purchases the good, so do
all higher types.
As before, the only stable levels of free riding are 0 or P-n; only now the
free-riding population
is of a different type than the paying population.
Let
p(t) be the total population
of type t or higher. Thus, ~(0) = P. Let J, be the
total number of rides if all people ride. Potential competition
forces the firm
to choose an arrangement
which is most satisfying to its customers. (Recall
that in this example all customers
agree on the benefits from subscribing
although
they disagree on the benefits from free riding.) Thus, the firm’s
problem is16
max u(z)-(2,k,s,t)
subject
(P3)
to
44 2 s,
p(t,
(23)
J,, Qf = s,
16For simplicity
we state constraints
(24) and (25) as equalities.
constraints
are binding is identical to the previous section and, therefore,
(24)
The proof
omitted.
that
these
E.C.D. Silua and C.M. Kahn, Exclusion and moral hazard
230
A# = T(z)+ kq,
(25)
~20,
(26)
kz0,
~2.0.
We now provide an example where an arrangement
that permits free
riding is optimal. Suppose all members of the population
are of one of two
types; namely, low-frequency
riders, I, and high-frequency
riders, h.” We
take the monitoring
technology to have the functional form
P(C,J,, k) = kg(t).
where g is non-decreasing
in t.
There are two candidates for the optima1 arrangement.
Either the firm will
choose an allocation
which prevents both types from free riding or it will
choose an allocation
which prevents only high types from free riding. With
perfect exclusion, every rider pays a subscription
fee. With free riding by low
types, only high types pay for provision. In either case, the entire population
consumes the public good.18
Theorem 3 below shows that high types prefer (and therefore choose) the
mechanism which involves free riding riding by low types if either the relative
cost of catching
low types is high or the number
of low types in the
population,
P-p(h),
is small. The proof of Theorem 3 is in the appendix.
Theorem 3.19
AhMh)f
The optimal mechanism involves free riding by low types if
> 4,
l/g(l) - l/g(h) > CP -
(27)
,#lf
/q
(28)
Condition
(27) is the generalization
in this problem
of the sufficient
conditions
for a non-zero
optimum.
The left-hand side of condition
(28) is
large if g(1) is much smaller than g(h) - for instance, if low types ride much
less frequently than high types. On the other hand, if there is little difference
in the ease of catching them, then the arrangement
should be designed to
induce both types to subscribe.”
The two sides of this inequality are closely related to the costs and benefits
of increasing
the base of subscribers.
As the subscriber
base increases costs
“The two-point
distribution
is made only for expositional
convenience.
This example can
easily be extended to arbitrary distributions
on the space of types.
‘sConditions
identical to those described in the appendix are sufficient to guarantee
that the
optimum involves positive provision.
“For general distribution
of types, Theorem 3 extends as follows: There exists a critical level t
such that it is the optimum; all types higher than t subscribe and all types lower than t free ride.
“‘This is the natural generalization
of the previous section’s result that when agents are
identical the optimal arrangement
induces all of them to subscribe.
E.C.D. Silva and C.M. Kahn, Exclusion and moral hazard
231
can be shared among a larger number of customers. On the other hand, the
increase in the number of subscribers can only be achieved by more intensive
monitoring so as to eliminate more free riders. If additional population
gained is small [right-hand side of inequality (28)] it does not compensate
for increased monitoring costs.
The appendix also demonstrates that the first-order conditions describing
the optimal arrangement when the optimum allows partial free riding are as
follows:
0, = ~&(~)fl(P(M~)f
- 4).
It is a corollary of the theorem that whichever of the two possible
arrangements yields higher quality service (higher z) is the one that will be
observed in equilibrium.
Corollary.
Let (z,, k,,s,) be the optimum i‘n problem
Then, v(z,J -s,, > v(zJ - s1 if and only if z,, > z,.
(P3) for
t = h or t = 1.
Finally, it should be noted that, as in the previous sections, the optimum
arrangement involves positive levels of monitoring, lower quality and higher
per-member costs than would be optimal under costless monitoring.
4. Summary and concluding remarks
This paper has examined provision of a public good for which exclusion is
feasible but costly. Relative to the standard cases in the literature, exclusion
costs reduce provision and increase subscription cost. If the good is rivalrous,
exclusion costs tend to increase the sharing group.
We considered as a base case providing a good with non-rivalrous
consumption to a homogeneous population. In this case, optimal provision
leads to all individuals subscribing for the good. We then separately
considered two modifications: rivalrous consumption and heterogeneous
population. Rivalrous consumption (with a homogeneous population) led to
excluding some individuals but did not lead to positive levels of free riding;
perfect exclusion is always desirable where consumers are homogeneous. On
the other hand, imperfect exclusion can be desirable where consumers are
heterogeneous. Heterogeneous populations (with non-rivalrous provision) led
to positive levels of free riding and no exclusion. We conjecture that
combining the two modifications would lead to a situation in which all three
types of behavior are observed in the optimum: subscription, partial exclusion and free riding.
Although we focused on private incentives for providing a good, it should
232
E.C.D. Silva and C.M. Kahn, Exclusion and moral hazard
be clear from our discussion that a similar analysis would also be applicable
to the issue of optimal public provision in a world where public providers
are constrained
by consumers’ willingness
to pay. When agents are homogeneous, the outcome is identical to a constrained
social planner’s outcome.
When agents receive identical
benefits from consumption,
a government
could improve the allocation
by charging each individual
a lump-sum
tax
and avoiding exclusion altogether. This outcome would no longer be feasible
if there were heterogeneity
in the preferences for the good. For example,
suppose a small unidentifiable
segment of the population
were interested in
consuming
the good and the segment had no power to impose the provision
of the good for the entire population.
Optimal
public provision
would
require
that public providers
break even and the segment
receive full
provision.
Therefore, optimal provision
is feasible if public providers,
constrained by consumers’ incentives, designed mechanisms
similar to the ones
designed by the principal in this paper.
The analysis of this paper focused on cases in which the good provided
identical benefits to all who consume it. A priority for future research is the
examination
of cases where agents are heterogeneous
not only in ease of free
riding as considered here, but also in benefits received from consumption.
In
more general circumstances
the design problem will incorporate
both the
free-riding
analysis
of this paper and the preference
elicitation
analyses
examined by other authors.
Appendix
In this appendix we consider the general form of monitoring
technology.
Thus, random
sampling
without replacement
examined
in the paper is a
special case.
Proof of Theorem 1. Note that z =s=k=O
is a feasible solution. Either this
zero solution is optimal or s<min(v(z),
p(k, P)f) in the optimum.
The latter
implies z>O. Then, n>O and s>O by (5) and k>O by (4). Note that n < P is
not optimal,
since the objective
function
increases
in n and all other
constraints
become less binding as n increases. Then, in the optimum n=P.
Observe that the objective function
is decreasing
in s. This implies (5) is
binding in the optimum. Second, note that whereas (5) is less binding, (4) is
more binding as k decreases. Thus, (4) is also binding in the optimum. This
establishes
conditions
(9) and (10) of the text. The principal’s
problem
simplifies to the following:
max u(z) - T(z) + kq)/P
b,k)
(PI’)
233
E.C.D. Silva and C.M. Kahn, Exclusion and moral hazard
subject
to
Pfp(k, P) = T(z) + kq.
(A.1)
Letting I stand for the Lagrange multiplier
first-order
conditions
for maximizing
the
sufficient for a solution to (Pl):
Pv,-
on the principal’s problem,
Lagrangian
are necessary
T,-AT,=O,
(‘4.2)
64.3)
-4+w-qfp,)=O,
and, thus, manipulating
0, = TAfp,l(Pfp,
(A.2) and (A.3) to eliminate
1, we obtain
64.4)
- 4)).
that
(8)
and
Observe
p(k, P) = min (k/P, 1). Q.E.D.
Corollary.
the
and
(9)
result
from
(A.4)
and
(A.l)
if
The zero outcome is not optimal if and only if
(Pfp,(O> P) - qM0)
’ T,(O)fp,(O, P)
Proof. It can be verified that if this condition
holds, then there are feasible
outcomes in the neighborhood
of the zero outcome which dominate the zero
outcome. Since problem (Pl) is convex and the zero outcome is always a
feasible solution, these local conditions
are necessary and sufficient.
Q.E.D.
of marginal costs and
Comment. The condition is essentially the comparison
benefits of increasing
the level of provision
z, incorporating
the necessary
changes in the other parts of the contract to maintain
incentive compatibility. Note that
PfP,(OY P) ’ 4
(A.4’)
is necessary for the condition
to be satisfied; if (A.4’) does not hold, the zero
outcome
is optimal.
In the special case of random
sampling
without
replacement,
condition (A.4’) reduces to the requirement
f >q assumed in the
text.
Proof of Theorem 3. Given
technology, (24) becomes
the special
functional
form for the monitoring
kg(t)f=s.
Consider
problem
(24’)
(P3) with t held fixed. If t = 1, the problem
is the choice of
234
E.C.D. Silva and C.M. Kahn, Exclusion and moral hazard
the optimum arrangement
under which no individual
chooses to free ride. If
t=h, the problem is the choice of the optimum arrangement
under which no
high-type rider wishes to free ride. Either way, the problem has a non-zero
solution if and only if
P(&o)f’q.
(A.3
Since in the solution (24) and (25) are binding,
Thus, the objective function reduces to
u(z) - T(Mt)fl(P(r)s(r)f
and the first-order
0, =
condition
principle,
g(r)flMr)s(r)f
smaller.
- 4)
(A.6)
is
Tzg(t)fl(PMW-
By the envelope
s= T(z)g(t)f/(p(t)g(t)f-q).
4).
(A.6) is maximized
(A.7)
by choosing
whichever
t makes
(A.8)
- 4)
Thus, if
~(h)f-q/g(h)‘~(l)f-q/g(l),
the arrangement
that keeps only high-frequency
dominating
arrangement.
Q.E.D.
(A.9)
riders from free riding
is the
References
Berglas, E., 1976, On the theory of clubs, American Economic Review, Papers and Proceedings
66, 116121.
Berglas, E. and D. Pines, 1981, Clubs, local public goods, and transportation
models: A
synthesis, Journal of Public Economics
15, 141-162.
Bernheim,
D., B. Peleg and M. Whinston,
1987, Coalition-proof
Nash equilibrium,
Part I,
Journal of Economic Theory 42, l-12.
Buchanan, J., 1965, An economic theory of clubs, Economica
33, l-14.
Groves, T. and J. Ledyard,
1977, Optimal allocation
of public goods: A solution to the ‘freerider’ problem, Econometrica
45, 7833809.
Helsley, R.W. and W.C. Strange, 1991, Exclusion and the theory of clubs, Canadian
Journal of
Economics 24, no. 4, 888-899.
Kahn, C.M. and D. Mookherjee,
1992, The good, the bad, and the ugly, coalition
proof
equilibria in infinite games, Games and Economic Behavior 4, 101&121.
Laux-Meiselbach,
W., 1988, Impossibility
of exclusion
and characteristics
of public goods,
Journal of Public Economics 36, 127-137.
Littlechild,
SC., 1975, Common
costs, fixed charges, clubs and games, Review of Economic
Studies 42, 162-166.
Novas, I.E. and M. Waldman,
1988, Complementarity
and partial nonexcludability:
An analysis
of the software/comouter
market, Journal of Institutional
and Theoretical
Economics
144,
443461.
Oakland,
W.H., 1972, Congestion,
public goods and welfare, Journal of Public Economics
1,
339-357.
E.C.D. Silva and C.M. Kahn, Exclusion and moral hazard
235
Scotchmer, S. and M.H. Wooders, 1987, Competitive equilibrium and the core in club economies
with anonymous
crowding, Journal of Public Economics 34, 159-173.
Tiebout,
C., 1956, A pure theory of local expenditures,
Journal
of Political
Economy
64,
416424.
Wooders,
M.H., 1978, Equilibrium,
the core, and jurisdiction
structures
in economies
with a
local public good, Journal of Economic Theory 18,328-348.
Wooders, M.H., 1980, The Tiebout hypothesis: Near optimality
in local public good economies,
Econometrica
48, 1467-1485.
Wooders, M.H., 1989, A Tiebout theorem, Mathematical
Social Sciences 18, 33-35.