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Encyclopedia of Law & Economics - 0750 PUBLIC GOODS & CLUB GOODS



Patrick McNutt
Chairperson, Competition Authority, Dublin and Research Associate,Department of Political Science, University of Dublin
© Copyright 1998 Patrick McNutt



1. Introduction  

2. Excludability & Non-Rivalry  

3. Public Goods Paradox  

4. The Coase Theorem & Property Rights  

5. Tiebout-Oates World  

6. A Marginal Decision Curve  

7. A Buchanan-Ng Framework  

8. Precis on General Models  

9. Homogeneity  

10. Future Research and Controversies  

11. Concluding Comment  

Bibliography on Public Goods and Club Goods (0750)  


Public goods contrast with private goods. Pure public goods have the uniquecharacteristics of non-excludability and non-rivalry in consumption while privategoods are sold to those who can afford to pay the market price. Theunder-supply equilibrium of a public goods provision is an important aspect ofthe provision of public goods. The economic theory of clubs represents anattempt to explain the under-supply equilibrium of a public goods provision. Itraises many different and controversial issues which impinge on governmentpolicy in the public sector. In many respects, a club provision proffers analternative to a central government provision of local public goods. The salientcharacteristc of a club, the excludabiity factor, may militate against an equal anddemocratic distribution of the club good. At the level of voluntary clubs, withwhich Buchanan was originally concerned, club theory can critically appraise theefforts at achieving optimal membership of the club and the maximum utility ofclub members. As the literature introduces increasing problems with cooperationthen it behoves law and economics scholars to research and develop non-marketand/or non-cooperative solutions to an optimal provision of public goods.

JEL classification: D60, D71, K00.

Keywords: Free Rider, Pareto Optimality, Club Goods, Excludability &Non-rivalry, Coase Theorem, Homogeneity.

1. Introduction  

Pure public goods as originally defined by Samuelson (1954) have the uniquecharacteristics of non-excludability and non-rivalry in consumption. Publicgoods contrast with private goods; public goods are non-excludable andnon-rivalrous in consumption while private goods are sold to those who canafford to pay the market price. The market price excludes some consumers whilethe property of rivalrous consumption ensures that not all consumers who canafford to pay the price, actually purchase the private good. The public goodsproperty of non-rivalry ensures that a provision of the good for consumer Aentails a provision for consumer B. Likewise, the property of non-excludabilityensures that one cannot exclude consumer B from securing the benefits of thepublic good, consequently there is no incentive for consumer B to pay the costsof providing the public good. Therefore a consumer may 'free ride' Kim andWalker (1984) on the provision of the public good, securing the benefits but notpaying the costs of provision.

A lighthouse signal is a classic example of a pure public good, where theprovision is both non-rival and non-excludable. Local radio or community radio,theatre performances and untelevised sports events are interesting examples ofa local public good, where the provision in non-rival but excludable. The marketis not the only mechanism through which goods and services are provided in amodern economy Coase (1974) ; public goods and club goods are characterisedby their provision wholly through a political process since by their very naturethey are unmarketable.

A primary reason why market failure persists is reflected in the inability ofcitizens to act co-operatively and it is this lack of co-operation which mandatesan allocative role for government in the economy. A public good that becomesexcludable is a club good McNutt (1996) . The economic analysis of clubspioneered by Buchanan (1965) can be applied to the provision of local publicgoods, ranging from the supply of decentralised regional public goods [localhealth boards] to community projects and neighborhood schemes, such ascommunity sports clubs and residents associations.

In the theory of clubs, however, there is collective consumption but with anexclusion principle, for example, a membership fee. One can think of club goodsas public goods sans non-excludability. There are economies of scale in thatadditional members reduce the average cost of the club good. But additionalmembers also lead to crowding which in the long run could be regarded as theintroduction of rivalrous consumption. Indeed the clubs goods has polarextremes as noted by Mueller (1989, p. 131) : "for a pure public good the additionof one more member to the club never detracts from benefits of clubmembership........[for] a pure private good, say an apple, crowding begins to takeplace on the first unit".

2. Excludability & Non-Rivalry  

There are therefore two salient properties pertaining to the provision of publicgoods, namely, non-excludability in supply and non-rivalry in consumption. Thelatter implies that inter-citizen consumption is mutually exclusive, that is, theconsumption by one citizen of the public good will not affect the consumptionlevel of any other citizen. Radio broadcasts, clean air or defence spring to mindas examples of a non-rivalrous public good. Non-excludability is the hallmark ofa political system where the central government funding emanates directly fromcitizen taxation. However, in the provision of some public goods, either localpublic goods or club goods, the citizens often prefer to act independently ofgovernment. The property of excludability in the supply of the public good isthe sine qua non of club goods.

A Prisoner's dilemma characterisarion of the market failure problem wouldindicate a Pareto inferior outcome as long as a dominant strategy existed for theindividual citizen. The incentive to cheat on collective decisions, otherwiseknown as the free rider problem, illustrates one dominant strategy whichundermines the optimal provision of public goods. In the classic tradition ofpublic choice, government intervention per se would represent an externality. Itis the increasing trend towards local public goods in the provision of publicsector output that has facilitated the application of club theory which exhibitsa co-operative response to the resolution of a local or regional issue.

Buchanan (1965) , who was one of the first scholars to consider the efficiencyproperties of voluntary clubs, derived the economic conditions under which anoptimal provision of a local public good could be attained. This early workoutlined a justification for club analysis in the explanation of why clubs wouldorganize. Both Buchanan and Olson (1965) recognised independently that clubsenable members to exploit economies of scale in the provision of the public goodand to share in the cost of its provision. They each addressed the issue ofmembership restrictions, with Olson distinguishing between exclusive clubs andinclusive clubs with no membership constraints.

Likewise, Tiebout (1956) had much earlier addressed a club related issue inhis work on population mobility and size of local government. His 'voting withthe feet' hypothesis has many direct applications in the area of local publicgoods. Other scholars, notably Schelling (1969) and McGuire (1974) justifiedclub formation on the basis of 'a taste for association'. This has since beentranslated in the club literature as the assumption of homogeneity [identicaltastes], an assumption which has raised the policy issue as to whether or notmixed clubs are optimal. For example, if mixed clubs are not optimal then thepolicy of group segregation is optimal whereas the policy of busing, as practisedin some US states, is sub-optimal. The issue of optimality, however, is notcompletely resolved across the club literature.

3. Public Goods Paradox  

To what extent the theory of clubs enables policy makers to escape theunder-supply equilibrium in the optimal provision of public goods remains achallenging issue. In other words, the optimal provision of public goodsgenerally is constrained by what can broadly be defined as the public goodsparadox, that is, unless the spoils of the public good are divisible there is noincentive for the individual to participate in its provision. Club theory overcomesthe problem of non-excludability in so far as members of the club use the clubgood. The non-excludability characteristic of a pure public good may constrainthe realisation of economies of scale in any interest-group provision of the goodunless the gains are divisible.

Table 1. An Economics Typology
Excludable Non-Excludable
Rival Private good Public good
Non-Rival Local public good Pure public good

The public good in Table 1 is characterised as non-excludable and rival. Inother words, rivalness in consumption is the distinguishing feature between apublic good and a pure public good. The good could be described as a commongood in the absence of any rival behaviour between citizens; some examplesinclude air quality, frontier land and outer space. Rivalrous behaviour, however,converts the common good into a public good as frontier land is zoned, airquality control becomes necessary and space stations are constructed.

Once property rights are established the good eventually becomes anexcludable and rival private good. For example, if a toll-free congested bridge, arival and non-excludable good, becomes a congested bridge with Pigou-Knighttolls, the good therefore becomes a rival and excludable private good. There areincreasingly few examples remaining Hummel (1990) of a pure public goodotherwise defined as a public externality. Medical knowledge is one example butthe classic examples of national defense, the environment, outer space andunpolluted air are no longer regarded as pure public goods.

Table 2. A Law & Economics Typology
Excludable Non-Excludable
Rival Private good Private externality
Non-Rival Club good Public externality

To what extent they represent McNutt's (1996) 'collective good' thuswarranting a citizen tax, depends upon how acceptable the good is to the citizensand the citizens' effective demand for that good. For example, should peaceniks who may regard defence as an unacceptable public good or Gaelic speakers whomay regard the English-language public radio broadcasts as an unacceptablepublic good, be obliged to pay the requisite fee or charge to have the goodsupplied? While pollution represents the classic example of an externality, maywe suggest pollution control as a modern example of a pure public good. Thiswould include anti-smoking legislation, catalytic converters in car exhausts andCFC legislation. Albeit, the classic lesson from the literature Van Zandt (1993) isthat an optimal provision of pure public goods may escape the policy maker.

The property of excludability, as noted in Table 2, is the essence of a clubtheory approach to the provision of public goods. If consumption of the publicgood is not contingent on payment, individuals have no incentive to reveal theirtrue preferences. The individual becomes a free rider and if all individuals behavelikewise the net result is an absence of effective demand for the good. Whereconsumption is non-rival, for example, exclusion could be easily applied.However, because the marginal cost to previous consumers of adding one extraconsumer is zero, the price should be zero. In this case there is no need toexclude. However the administrative costs of the public good provision must becovered somehow and with non-rival consumption in the absence of exclusion,the usual market method cannot determine price.

Musgrave and Musgrave (1980) have argued in favour of thenon-excludability characteristic; they have argued that with excludability,non-rivalrous goods can be effectively provided by private production. In adifferent context Ng (1979, p. 190) emphasised the non-rivalrous characteristic,particularly if we do not regard public production as a necessary and sufficientcondition for a public good. Since free riders impact on these conditions it israther difficult to compute exactly the individual's valuation of a public good.And this is particularly difficult if payment is not contingent to a particularpreference revealation. Preference revelation mechanisms Kormendi (1980) forexample, where individuals pay a price that equates will their revealed preferencefor the good, are presented as experimental attempts to minimise the problem.Another alternative to the market failure result in the provision of public goodsis to be found in the general theory of clubs. Tanzi (1972) had shown that welfarecosts may be involved in providing public goods which differ with respect tohow individuals are excluded from consuming the good.

4. The Coase Theorem & Property Rights  

In standard public goods analysis it is assumed that consumption of the publicgood can be extended to all consumers at a zero marginal cost. It is also assumedthat a free rider problem exists or that individuals Cohen (1991) can only beexcluded at some positive cost. Loehr and Sandler (1978, p. 27) consider theissue of a "forced rider" in which people "are forced to consume, whether theylike them or not" a range of public goods, for example defence. They furthercomment that "it is entirely possible that the welfare of some individuals mightfall when a marginal unit of the public good is provided". The Pareto optimalityconditions would have to allow for subsidies for these individuals to ensure thatthe marginal utility to tax price ratios for all individuals are equal. The forcedrider may influence the provision of the public good. This could be extended tolocal goods and services where forced riders may be involved in decisionmaking.

Pigou (1920) had suggested that government intervention was necessary inorder to abate the externality problem. The transactions costs of groupingconcerned citizens together in order to resolve the externality problem wasprohibitive. Coase (1960) argued that in the absence of transaction costs,concerned citizens could resolve the problem, independent of government.Theorem 1, the Coase theorem, and the liability rules amend the public choiceanalysis of the externality problem.

Theorem 1: In the absence of transactions costs and bargaining costs, concerned citizens willagree to resolve an externality problem and arrive at a Pareto optimal allocation of resources,independent of government.

The apportionment of blame and the allocation of property rights, that is, theright to clean air, the right to pollute, proffer an alternative, indeed a complementto the introduction of Pigovian taxes. The idea behind liability rules was toapportion blame; an alternative to this procedure in tort law is to establishoptimal conditions which may prevent the accident or property rights disputeoccuring. The traditional response in public finance was either to compensatethe offended party or tax the offending party. This required an apportionmentof blame which may have induced unnecessary government expenditure andrent-seeking activity. The costs incurred must be weighted against aninter-citizen or club resolution of the initial dispute.

The costs of providing the public good must include the bargaining costsattributable to the resolution of the ensuing debate on the amount of publicgood supplied, if at all. The treatment of these bargaining costs are a centalfeature in Buchanan and Tullock (1962) whose framework was used by Loehrand Sandler (1978) in considering the impact of bargaining costs in the provisionof public goods. They illustrate the net indirect costs imposed on forced ridersand the number of individuals required to reach agreement on public provision.They further represents costs imposed upon a person who bears some burdenunder all decision rules with the exception of unanimity".

In this case if the individual was a forced rider he would agree to the decisiononly when adequately compensated, that is when net costs are zero where theentire population is in agreement. Loehr and Sandler further comment that theircost function is "downward sloping since the greater the proportion of thepopulation needed for agreement, the more likely persons similiar to himself (butnot identical to him) will be wooed by the early proponents of the public action".A point may be reached where the need to form larger and larger coalitionswould force bargains between free riders and forced riders. A particularlyinteresting point in Loehr and Sandler (p. 31) is their comment that the cost curveneed not end at zero when unanimity is reached.

In other words, some free riders, they argue, may still exist, even whereeveryone is in agreement on the policy". Summation of all individual cost curvesin their presentation creates a community cost curve which indicates that moreand more decisive groups would imply a higher cost in terms of effort andbargaining. If the decisions have to be made at the point where community costsare at a minimum then we are abandoning Pareto optimality. The solutionpresented represents a second best solution. McNutt (1996) considered aninter-citizen resolution by adapting an earlier argument in Turvey (1968, p. 310) who had argued that the traditional interpretation of an externality is ratherrestrictive. How much group B suffers from A's externality depends not only on"the scale of A's diseconomy but also on the precise nature of A's activity andB's reaction to it". For example, the victim in Pigou's chimney example couldreduce the disutility by installing an indoor clothes-line.

The Pigouvian solution of reducing the amount of smoke contrasts with thealternative solution of either building a higher chimney or using differentsmokeless fuel. McNutt (1996) shows that by allowing an inter-citizen resolutionto a dispute, the cost may be less than the government cost. If citizens can agreeon the resolution of an externality problem, the cost to the government offinancing the inter-citizen solution may be less than a central governmentsolution. An inter-citizen resolution like the Coase theorem offers an alternativeto government action in the resolution of an externality problem. One policyimplication of this result applies to traffic congestion in large cities. Rather thanimpose a tax on car owners who persist in driving to the city at rush hour,car-users should be encouraged to resolve the externalities of long tailbacks, caremissions and queues by acting collectively. Car pools with special motorwaylane access, such as the HOV (heavy occupancy lanes with at least threepassengers per vehicle) lanes in the US, would be socially more efficient thanallowing as many fee paying cars to enter the city limits; citizens would preferto incur the lower garage parking fee for the pooled car.

5. Tiebout-Oates World  

It is useful to re-examine the conditions which independently underpin the Tiebout (1956) and Oates (1972) models of local public goods and adapt theLoehr-Sandler model in a search for some common ground in a Tiebout-Oatestype world. Forced riders, can leave the local neighboorhood; this assumes norelocation constraints; crucial to the question posed here is the failure ofindividuals to reveal their true preference for local public goods. In his analysis,Tiebout recognised the efficiency in the supply of public goods and furtheracknowledged that voting process was the only recourse to reveal thepreferences of the sharing group. The optimal allocation is determined by a'voting with the feet' exercise.

Tiebout had presented an earlier framework for the theory of clubs inassuming an infinite number of individuals who form themselves into manyclubs of different sizes. Under certain conditions the infinity assumption allowseach club to maximise its own benefit without violating Pareto optimality. TheBuchanan-Ng framework may be preferable to the Tiebout framework in the casewhere location of consumers is exogenous, transport is costly and where thereare few clubs. In the Tiebout model individuals can vote with their feet, movingto regions according to their preferences for public goods.

Nevertheless, in order to examine this model further we note twoassumptions of the Tiebout model, viz [i] consumer-voters are fully mobile and[ii] they have full information on the differences on revenue and expenditure inthe local areas. These two assumptions depend on the absence of relocationconstraints such as employment, house purchase and school availability. It alsopresupposes a large number of alternative communities with which the consumercan effectively rank order each community. The remaining assumptions includethe following: [iii] there are no external economies or diseconomies of scale in thesupply of the public services [iv] there is an optimal community size for everycommunity service and finally [v] communities below the optimal size attract thenew residents.

This set of assumptions establish the classic Tiebout model and ensure theglobal optimality of excludable public goods provision. Mueller (1989, p. 157) outlines an illustrative proof of this global property. However the new residentscan produce congestion in the new area and the resulting congestion costs andpossible negative externalities if the community has grown beyond the optimalsize, forces Mueller to conclude that in general the Tiebout model will notproduce a Pareto optimal outcome. In his illustration he shows quite clearly how

a non-Parero though stable equilibrium can emerge. Empirical evidence tosupport the hypothesis has been forthcoming, for example, Cebula (1979) showed that inter area differences in welfare benefits influenced migrationdecisions while Aronson and Schwartz (1973) in an earlier and original analysisshowed that those towns likely to gain in relative population are those that offerresidents equal or better services at an equal or lower tax rate.

6. A Marginal Decision Curve  

McNutt (1996) offered an alternative interpretation to the global condition in aTiebout-Oates world by considering the idea of a marginal decision [MD] curve.This differs from the average benefit curve employed initially by Mueller (1979) ;while both curves represent benefit, Mueller's curve assumes that benefit is afunction of community size whereas McNutt's curve is a function of the numberof internal members (who form an internal group) in the sharing group. Theconcept of an internal group is used to explain the formation of alliances in theprovision of public goods. In many instances, for example, the alliance mayexpressly form to prohibit the supply of public goods as with defence orenvironmental quality. As illustrated by McNutt (pp198-199) , the group MDschedules are mirror images of each other which reinforces the point that utilityin the club is maximised by dividing the club good equally between each group.

Let us take the example of tulips in a public square; tulips represent a publicgood, planted in the public square by the local authority. Assume that the tulips,for whatever reason, offend a sub-group of the individuals who spend the dayin the square. For this sub-group the tulips represent an externality. The squareitself is a public good, but the presence of tulips reduces the utility of thissub-group. Next we introduce the concept of internal member:

Definition 1: define the sub-group S of citizens such that there is an issue i which at least onemember j of the group regards as an externality, then j S is defined as an internal member ofthe set S. The set S is a proper subset of the set, C, of all individuals in the square.

If the committee responsible for planting tulips decides against plantingtulips in the square, the internal group is defined as decisive. The significanceof an internal group is in its ability to rank local public goods in descendingorder of preference. The important characteristic of an alliance supplied publicgood is jointness in supply, that is, the supply includes private benefits as wellas public goods. The private good may include cultural or educational benefitsbut may also include private externalities as with the tulips example.

Club theorists may have underestimated how members of a sharing groupbecome associated. Apart from similiar tastes, there is the possibility of an'association by alliance', that is an alliance of internal citizens who expresslyobject to the supply of a public good. How this manifests itself in theory, is asfollows: the 'sharing group', that is the group of all citizens who consume thegood, is subdivided into group A which derives exactly half as much utility asgroup B, the internal group, in any provision of a local public good. Group B, aninternal group, has a negative impact on the remaining members, [MDA] =1/2.[MDB].

If the rule is to maximise the utility of the sharing group then emphasis willbe in the directon of group B. Ironically the utility of the A group will decrease.The dominance of the internal group secures a reduction in the amount of localpublic good in order to maximise the utility of the sharing group, B. McNutt(1996, p. 198-199) called this 'the tulips paradox', that is, in the local provision ofa public good the presence of a decisive internal heterogeneous group withidentical tastes may reduce the supply of the local public good in order tomaximise the utility of the larger citizenry group.

7. A Buchanan-Ng Framework  

There are two basic models across the literature on club theory, the Buchanan(1965) within-club model and the more general Oakland (1972) total economymodel which will be developed in a later section. Buchanan's model is the classictreatment of clubs while the Oakland model is more general in extending clubtheory to include heterogenous members, discrimination, variations in theutilization of the public good and exclusion costs. Neither model however,guarantees Pareto optimality in the provision of local goods, which ironically isthe raison d'etre of club theory as a methodoligical study of the allocativeefficiency of [impure] public goods.

The assumptions underpining the Buchanan model include the following: [i]individuals have identical tastes for both private and public goods [ii] the sizeof the club good [a swimming pool], hence its total cost, is fixed, and [iii] equalsharing of costs. Mueller (1979) has argued that [iii] follows as an assumptionfrom [i]. In a simple model Buchanan determines the optimal size of the clubmembership. Mueller shows that with some algebraic manipulation, by deductingeach individuals share [equal shares] of the cost of providing the good fromprivate income to obtain "net of public good income", and substituting this intoan objective function with the amount of public good and club size asexplanatory variables, the Buchanan model obtains the Samuelson condition forthe efficient consumption of a public good.

The crucial assumption in the Buchanan model and in club theory generally,is the assumption of identical tastes and incomes. The Tiebout model shows thatit is inefficient to have individuals of differing tastes in the same club.Intuitively, think of ten women golfers in a golf club of 25 players. The resulthere is akin to Pauly's (1967) result, obtained much earlier, that no stableequilibrium will exist, if the women golfers form a winning majority. This isparticularlly the case if the number of women golfers increased and the threat ofexit by the male golfers becomes credible - they could leave and form analternative club. The dynamics of the situation would suggest that a smallmembership size is optimal - in other words, there has to be a limited degree ofpublicness (an excludability factor) as additional members beyond the optimalmembership size will impose a cost on existing members. Congestion may ariseon the golf course, reducing the utility of existing members.

According to Ng the relevant Pareto optimality condition requires that anyindividual in the club must derive a total benefit in excess of the aggregatemarginal cost imposed on all other consumers in the club. So the Buchanan-Ngtheory is to optimise the membership; alternatively Oakland considers the degreeof congestion or overcrowding to be important. Club theory has manyinteresting applications in the analysis of congestion and in establishing theoptimal group size for [say] a local golf club to a local community. Buchanan'seconomic theory of clubs builds on three rather important assumptions [i] thatthe benefits and costs are divisible amongst the club members. As moremembers join, average costs for the provision of the club declines, but marginalbenefits begin to fall as more members contribute to congested levels ofmembership; [ii] it is costless to the club to exclude members. This conveneintlyremoves any distortion should exclusion be deemed necessary in order to attainan optimal [MC = MB] membership. Finally it is assumed that [iii] there is nodiscrimination across members. This is a rather difficult assumption to defend inpractice, as in the case of golf clubs and swimming pools where there is evidenceof sex discrimination, still discrimination respectively. However with these threefundamental assumptions, an individual quasi-concave utility function ismaximised in order to find the optimal club size and the optimal quantity of thegood.

The public good is not a pure public good, but rather there is an element ofcongestion as individuals consume the good up to its capacity constraint. Whatarises then is some exclusion mechanism in order to charge consumers a pricefor the provision and use of the good. Brown and Jackson (1990, p. 80) hadcommented that the purpose of a club "is to exploit economies of scale, to sharethe costs of providing an indivisible commodity, to satisfy a taste for associationwith other individuals who have similar preference orderings". For Buchanan-Ngthe main club characteristic is membership or numbers of consumers, and it isthis variable that has to be optimised. For Tiebout an assumption of infinity ofindividual consumers presupposes costless exit from one region to another andthe formation of many clubs. Oakland considered the degree of congestion asan important characteristic in the provision of a club good. There is room for allof the characteristics in a general theory of clubs that seeks to determine aPareto optimal distribution of public goods.

What appears not to have been examined in this context is the interpretationof an individual's income elasticity of demand as a proxy for tastes for a publicgood. In the Tiebout world high income individuals may migrate to the same areawhich leaves relatively poorer individuals consuming only the public goodswhich they themselves can afford to provide. No one really objects to clubmembership when the public good is tennis courts, squash courts or golf clubs.To avoid congestion in the club and to achieve economies of scale, a Paretoefficient outcome is arrived at by introducing an exclusion principle. But in aTiebout world of clubs, right handed golfers exiting to form an alternative club,is quite different to the world in which high income individuals migrate to onearea and low income individuals to another area.

As Mueller (1979, p. 144) pointed out "the voluntary association approachis likely to affect the distribution of income". If individuals can vote with theirfeet and have positive income elasticities of demand for public goods they canbenefit from living in a community with incomes higher on average than theirown. But for the poorer individuals transport and mobility is costly and for thehigher income individuals the formation of interest groups [for example, regionalor local environmental lobby] is a concomitant to the provision of the publicgood. Each militate against an egalitarian distribution of the public good. Anyattempt to transfer across from rich to poor "runs directly into the issue of theproper bounds of the polity, and the rights of citizenship" according to Mueller.However, in order to reach levels of efficient voluntary provision in Paretianterms, cooperation is necessary.

8. Precis on General Models  

The presumption is that a voluntary provision of the public good will lead to asuboptimal outcome. The general model further assumes the existence of aprivate good and an impure public good, with the private good acting as anumeraire. The members are heterogenous, non-members are costlesslyexcluded, and club members determine their utilization rate of the club good byvarying the number of visits [to the public park] and time spent at the club.Optimal provision in this general model, within which both members andnon-members are considered in deriving the optimal conditions for a single club,requires according to Sandler and Tschirhart (1980, p. 1489) "that the marginalbenefits from crowding reduction, resulting from increased provision, equal themarginal costs of provision [MRT]". This is analogous to the earlier Paretooptimal condition [MRS = MRT] for public goods provision, and not unlike theconclusion extracted by Buchanan. The utilization condition in the generalOakland model requires an equal rate of utilization for all members, although totaltoll payments [for utilization] vary between heterogenous members.

Oakland's model is identical to the Buchanan model under the followingconditions [i] all members are homogenous and each consumes the availablequantity (say) X of the public good, such that Xi = Xj; [ii] for the members S thecrowding function must be an identity mapping, i.e. C(S) = S, this reduces thegeneral Oakland utility function to the Buchanan function U[Y1,X1,S], where Ssubstitutes for C(S). The insertion of a crowding function into the utilityfunction is one major difference between the models in club theory. Sandler(1978) argued that by including a crowding function, crowding externalities suchas poor view can be considered, (a) increases in the provision of the public goodreduces crowding [dc/dc < 0] and (b) increases in member use of the goodincreases crowding, that is [dc/dxi > 0]. It has been argued that the general modelimplicitly assumes cardinality of the utility function. Sandler and Tschirhant(1980, p. 1490) in their review of club theory comment that since "the generalmodel requires an ordering of the population based upon club preferences",cardinality is implicit.

Cardinality may rule out particular functional forms of the utility function,that may be otherwise appropriate for club analysis, for example thetransformation W = LogU. In practice however populations cannot be ordered;this applied weakness in the Oakland model has been overcome by Hillman andSwan (1979) who proposed an ordinal representation that does not require anordering of the population. Their model, a ceteris paribus type model, maximisesan arbitrary members utility subject to the constancy of other members utilitylevels. Recall that Buchanan's model maximised individual utility U(Y,X,S)subject to a production:cost constraint F(Y,X,S) = O. The Hillman and Swan(1979) result is akin to this basic Buchanan model when (i) C(S) = S and (ii) F =U(Y,X,C(S)). The (ii) condition is the Buchanan constraint in the optimizationprocedure; an analogy requires that the Hillman and Swan constraint berewritten as F = U(Y1,X,C(S)) = O. This may be unlikely but worthy of furtherresearch.

Both Tiebout (1956) and Oakland (1972) represent alternative frameworks tothe approach adopted by Buchanan (1965) in accounting for the under-supplyof public goods. Oakland looked at the degree of congestion while the Tieboutmodel is an application of club theory to community size. A Tiebout-Oaklandpublic goods problem would manifest itself for those public goods for whichcongestion begins at a certain size of community. As the community gets larger,residential density increases [community congestion], reducing the utility ofeveryone living in the community. Two factors which are important in thecontext are [i] that the total number of people may not be an integral multiple ofN, the number of workers, i.e. there may be a fixed population as identified by Pauly (1967) and [ii] the number of communities may be fixed, the one exception,alluded to by Atkinson and Stiglitz (1980) , is a frontier society.

If the communities are fixed [say] to two, an optimal provision of the publicgood may involve an equal treatment, a result which in Atkinson & Stiglitz (1980) yields a local minimum (maximum) solution with population shortage (excess),hence social welfare could be increased by moving to an unequal treatment. Asimiliar point was alluded to earlier in the discussion of the marginal decisioncurve. However, the general theory of clubs with the property of nodiscrimination of members assumes a group of homogenous individuals. TheTiebout world has heterogenous individuals sorting themselves out intohomogenous populations with homogenous tastes. Hence doctors and lawyerslive in the same neighbourhood, and there are golfers in the golf club andswimmers in the swimming club. Health and sports clubs have to acquire anoptimal mix of members in order to minimise crowding and queues. A sortingmechanism has to be introduced such as a rota or a time schedule based onmembershp age. But is the sorting optimal? In answering this question we haveto refer to the concept of homogeneity.

9. Homogeneity  

In the literature there are at least two interpretations of homogeneity in the clubliterature, namely [i] Tiebout (1956, p. 419) homogeneity as captured in his workwhere he commented on "restrictions due to employment opportunities are notconsidered". In mixed communities doctors and lawyers do not have equalincomes since the respective income depends on labour supply. Consequentlythey are not perfect substitutes, and the community needs both; the communityis better off if they have the same tastes. And secondly [ii] an Atkinson-Stiglitz(1980, p. 531) type homogeneity, which is a weaker version of the Tiebouthomogeneity, and argues "that individuals are [not] always better off forminghomogeneous communities with people of identical tastes". In their argument,they consider a third public good produced as a compromise to a mergedcommunity forming from the separate communities. In the merged case theindividual can enjoy the benefits of the economies of scale associated with threepublic goods [equivalent to our average cost reductions in the Buchananmodel], but when these benefits are weighted against diminishing returns tolabour N [equivalent to the declining benefits in a Buchanan model], theindividual is better off.

An interesting dimension arises in the context of a heterogenous populationwhich can be translated into different marginal valuations. If for example the localauthority does not tax the individuals according to their respective valuations,by imposing an equal tax, there may not be an optimal provision of the localpublic good in the merged community. Those who value the public good less,are essentially subsidised by the high-value individuals and receive a windfallgain in the provision of the good. The movement from separate communities toa merged community is not a Pareto improvement. Atkinson and Stiglitz (1980) arrive at a similar result, assuming no diminishing returns to labour, in lookingat positive benefits, that is "everyones taxes [are] cut". Whether the sorting isoptimal or not depends clearly on the assumptions of diminishing returns tolabour, the existence of a windfall provision to individuals with lower valuationsand on the assumption of homogeneity.

Pauly (1970b) and McGuire (1974) in their generalisation of the earlier workof Tiebout assume an indefinitely large number of individuals, forming clubs ofdifferent sizes. Pareto optimality is not violated with the assumption of infinity[uncountable infinity according to Ng (1979) as each individual can join a clubthat suits his or her preference, thus maximising the individual (average) benefitor the benefit of the club. The applicability of this infinity framework isaccording to Ng (1979, p. 212) suitable for the cases where the number of clubsfor the same good is large and the population is mobile; he suggests groupsegregation in housing the nomadic life and sports clubs.

In the typology of public goods presented in Table 2 earlier,, the club goodis defined as a non-rival excludable public good. A different usage of rivalry hasbeen discussed in the literature by Starrett (1988, p. 58) in the context of clubtheory and local communities. The spatial element in local communities, withcompeting use for a limited [same] space, generates "club rivalry that isindependent from the rivalries we have been discussing". In what he refers to asa bare bones model, Starrett concludes with an optimality condition whichsuggests that efficient size will require that average provision cost equal the sumof the various marginal rivalry costs. In the model transport costs play the roleof rivalry costs, as Starrett (1988, p. 59) argues "transportation has no value tothe members per se but must be incurred if they want to share the collectivegood".

That each individual in the club is equal distance from Starrett's collectivegood, the assumption of radical symmetry, is dropped in an alternative modelwhich allows for choice in the number of trips to the collective good [forexample, the public park] and in the amount of residential land held by eachindividual. The first best solution is an unequal division of land as individualscloser to the public good represent an externality to these further out in theresidential area. The latter residents have larger tracks of land. Starrett'sunsurprising conclusion is a formulation "that treats equals equally [p60] "; thereason, apart from the formal rigor of his model, is that in the real world thepolitical system will impose this constraint on society. Of the Lagrangeanoptimization results presented by him the one that is of interest is the conditionfor optimal club size.

Theorem 2: The Henry George Theorem states that if public expenditure is fixed and populationvaries, the population that maximises consumption per capita is such that rents equal the publicgood expenditure.

The Starrett (1988, p. 62) result which states that the supply of the publicgood should equal the pseudo-land-rent in the optimal spatial club is in manyrespects similiar to the Henry George Theorem as derived by Atkinson andStiglitz (1980,p525) . Optimization on club size leads to the Starrett result. In aHenry George world, each citizen had identical tastes, an assumption which isimported by Buchanan into his original club model. Since club rivalry involvesspatial separation the marginal cost of rivalry is reflected in the marginal premiaon limited space. Starrett concludes "that in our bare-bones model this premiacould be measured in terms of transport costs, [but] differential land rents turnsout to be the right measure in broader contexts [p62] ". The measure is right,relatively speaking, in that it secures an optimal club size. The differentapproaches within the general theory of clubs highlight the many differentcharacteristics of a club and of a club good. The general theory of clubs offer asolution to the optimal provision of public goods.

10. Future Research and Controversies  

In this final section we look at some of the more interesting areas of researchwithin the public and club goods literature, areas of recent controversies indeedwhich have arisen across the literature. Many of the issues have an importantbearing on the optimal provision of local public goods and consequently onlocal public finance.

10.1 Membership Homogeneity  

Membership homogeneity has to be one of the more controversial issues withinthe club literature, particularly from a public policy perspective. For example, ifmixed clubs with heterogenous membership are found to be non-optimal, asoutlined in our earlier discussion, serious policy implications for group housingor education schemes may arise. The literature is divided on the optimality ofmixed clubs, with Ng (1973b) and Oakland (1972) arguing for the optimality ofclubs and Berglas and Pines (1978)   Helpman (1979) , McGuire (1974) and Stiglitz(1977) arguing in favour of homogenous clubs. The latter group, according to Sandler and Tschirhart (1980, p. 1492) "have recognised that mixed clubs may bedesirable when strong scale economies require a larger membership thanpossible with homogeneity".

Mixed clubs however are not Pareto optimal due to an important assumption:the equal cost sharing assumption which states that in a mixed club, albeit allmembers pay the same membership fee, those members with higher valuationsof the public good have a higher total payment as they use [visit the park] thegood more frequently. Conversely mixed clubs are shown to be efficient whenthere are no second best constraints imposed. Hence by invoking second bestconstraints requiring all members to share club costs equally, as alluded to in ourargument on windfall gains or requiring all members to use the club equallyirrespective of tastes as in McGuire (1974) and Porter (1977) , mixed clubs canalways be shown to be less desirable than homogenous clubs. It is the set ofsecond best constraints that relegates the mixed clubs to second place in theefficiency comparisons. A scale of membership fees may (paradoxically)encourage the intense user of the good to use it less and while her less frequentuser revisits frequently.

10.2 Pareto Optimality  

Neither the within-club Buchanan model nor the Oakland economy model,ensure Pareto optimality. As Sandler and Tschirhart (1980.p1493) conclude"[within-club] may fail when the membership size is large relative to the entirepopulation, [general model] will fail when multiple clubs are desirable". Themultiple clubs translates into a variable number of clubs and this requires thatboth the optimal number and optimal size of clubs be determined simultaneously.A rather different slant on the optimality controversy is whether or notBuchanan in his original article, failed to consider Pareto optimality. Ng (1973,

p. 294) has argued that Buchanan did fail to give Pareto optimal conditions inmaximising the "average net benefits instead of total net benefits"; Ng (1979, p.212) in defending his position has reiterated that his analysis aims "at Paretooptimality or maximising total benefits of the whole population". Both Berglass(1976) and Helpman and Hillman (1977) criticised Ng's (1973b) attack onBuchanan and questioned whether or not Ng had maximised total benefits of oneclub, which in general, is non-Pareto optimal.

The Buchanan-Ng framework on clubs which concentrates on each particularclub, is preferable, according to Ng (1979, p. 212) to "the more general model[wherein] these conditions are not satisfied (our italics)". The conditions referredto are generally the infinity conditions outlined in our discussion. In contrast Berglass (1976) defended Buchanan on optimality and Helpman and Hillman(1977, p. 295) suggested that the issue is very much dependent "on a recognitionof the different types of club problems analysed" and a realisation of thedifference between maximizing average net benefits [for the members] andmaximising total net benefits for the club. Buchanan proceeded with the former,whereas Ng proceeded with the latter "in maximising total net benefits for theentire economy [p1493]" according to Sandler and Tschirhart (1980) . Otherscholars have considered the issues arising from exclusion costs, memberdiscrimination and the analysis of an efficient membership fee or toll for optimalclub provisions. The interested reader is directed to the review by Sandler andTschirhart (1980) and Mueller (1989) and the bibliographies contained therein.

Game theory has helped to shed some light on the issues raised in the clubliterature and in particular Pauly (1967) to whom we referred earlier, defined theoptimum club size as that size for which average net benefits are maximised. Thisis at variance with the non-game arguments by Ng (1973b)   Helpman and Hillman(1977) and the Oakland general model. A direct comparison between the gameand non-game outcomes is complicated by the different assumptions used. Inparticular the game approach does not admit the interdependency between themembership and the provisions which characterises the classic Buchanan typemodel; nor does it consider a simultaneous solution to membership, provisionof the good and finance. In many cases the club fee decided ex-post. Theapproaches do converge on the optimum number of clubs in the homogenouscase.

Pauly (1970, p. 60) divided a mixed population into homogenous groups, witheach group divided into multiple clubs where average net benefits are maximised.He proved that the core was non-empty and existed "if the clubs consist ofidentical members with equal payoffs and that clubs with higher averagepay-offs have fewer members". There has been an increase in game theoreticcontributions, for example, Cornes and Sandler (1986)   Sandler and Posnett (1991) and notaby Sugden (1981, p. 118) who has argued that where there is ''aconsistent theory of non-Nash, utility-maximising behaviours, even less of thepublic good would be supplied than in a Nash equilibrium". The conclusion isthat public goods would never be supplied at all.

10.3 Profit Maximising Clubs  

However there are two more recent controversial developments to which wewould like to turn our attention. The first concerns the issue of profit maximisingclubs, alluded to in the classic survey by Sandler and Tschirhant (1980)   Berglasand Pines (1978) have demonstrated that a perfectly competitive industry withidentical firms [each frim acts as a club] supplying the shared club good wouldachieve the same efficiency conditions as those of a private co-operative.

Hillman (1978) found that the non-discriminating monopolist provided smalleroutput and charged a higher price and operated more crowded facilities than thenon-profit co-operative. In contrast Hillman and Swan (1979) have shown thata discriminating monopolist will always achieve an effecient outcome. Ng (1974) argued that a government was necessary in order to achieve the efficientoutcome, defined as maximising total benefits. He continued to argue, in thespirit of our earlier discussion that since members under a monopolist willmaximise net benefit rather than total benefit, an efficient outcome is not attainedin the absence of a centralised government.

Ng apparently underestimated the impact of short-run political objectives inguiding a government-run club, as later outlined by Sandler (1978) . Scotchmer(1985, p. 39) has argued that with a homogenous population, profit maximisingclubs will achieve an equilibrium that is "within epsilon" of being efficient. Thereis entry in response to profits, and with incumbent clubs making a conjecturalvariation on "the price and facility response in other clubs when it changes itsstrategy", the numnber of clubs will be too large. The strategy space is definedby facility X and price P, not facility X and the member N. With the strategyspace (X,P) each club believes that it can get more clients at the expense of otherclubs. The set of strategies is a Nash equilibrium if no club can charge (X,P)such as to make more profit, with the zero conjectural variation assumptions. Thestrategy space (X,N) is abandoned because the Nash equilibrium requires theassumption, deemed unlikely by Scotchmer (1985, p. 27) , that "the other [clubs]will change their prices in whatever manner necessary to maintain the clientiele".

The earlier profit-maximising club literature explored by Berglass (1976) and Wooders (1980) had assumed that there was an efficient size sharing group, andthe conclusion has been that provided entry forces profits to zero, a clubequilibrium will be efficient. However these firms are competitive in the sense ofbeing a "utility-taker", whereas Scotchmer (1985) departs from this in arguingthat firms take as fixed the strategies of other firms. It is essentially a nonco-operative game and the equilibrium is cast as a Nash equilibrium. Formembers the utility available in other clubs will change as membership changes.

10.4 Multi-Product Clubs  

A further area of research which was introduced in the wake of new material oncontestability theory is the idea of a multi-product club, footnoted initially by Sandler and Tschirhart (1980, p. 1513) In particular, they had suggested a role forthe concept of economies of scope defined simply as complementarity inproduction. Within the literature however, some scholars have considered thisissue already, although the joint products include a private good and an impure(or indeed pure) public good. Examples would include the Samuelson constraintand the Henry George Theorem. However in the area of local government, wherecommunities and cities share multiple club goods, this application may prove tobe useful. Berglas and Pines (1978) did however present a multiproduct clubmodel, but did not consider the concept of economies of scope.

The essence of this assumption in any industry-type analysis is that the twoproducts cannot independently be provided at a cheaper cost than jointproduction. It is important to recall that the relationship in the club literaturebetween the average cost curve and the number of clubs is related to thedefinition of a single product monopoly. The condition of sub-additivity in thecost function had already been used already in the club literature by Pauly (1970,p. 55) in his argument that "club characteristic functions may be sub-additive".The many variants to the economic analysis implicit in Buchanan's originalmodel have advanced our understanding of club theory and have helped toincorporate club theory into the economic analysis of local public finance.

11. Concluding Comment  

The economic theory of clubs represents an attempt to explain the under-supplyequilibrium of a public goods provision. It raises many different andcontroversial issues which impinge on government policy in the public sector.In many respects, a club provision proffers an alternative to a centralgovernment provision of local public goods. The salient characteristc of a club,the excludabiity factor, may militate against an equal and democratic distributionof the club good. At the level of voluntary clubs, with which Buchanan wasoriginally concerned, club theory can critically appraise the efforts at achievingoptimal membership of the club and the maximum utility of club members.

Game theoretic approaches to public goods provision may give scholars thelatitude within which they could abandon the conventional postulate ofindividual utility maximisation and critically evaluate how rational behaviour canbe encouraged in the individual for the voluntary provision of the public good.Arguably, it is in the arena of an interchange between club provision and aninterest group provision of a local public good that the contestable issue ofsub-additivity may arise. The externalities, both private and public, to a certaindegree, may discourage rational individuals from contributing more in order toattain a Paretian outcome.

If the literature identifies increasing problems with cooperation then itbehoves law and economics scholars to adopt an approach which will researchand develop non-market and/or non-cooperative solutions to an optimalprovision of public goods. This approach will contribute positively to anevaluation of the economics of the provision of excludable club goods. Theapproach will also precipitate a much wider debate on the policy issues of localneighborhood supply and provision of public services; it may also impact on thetheory of public goods provision generally by focusing more on the(intra-interest group) economies of organisation per se in an attempt to explainthe under-supply equilibrium of a public goods provision.

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© Copyright 1998 Patrick McNutt

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