buchanan 1965 an economic theory of clubs

5/24/2018 Buchanan 1965 an Economic Theory of Clubs

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The Suntory and Toyota International Centres for Economics and Related Disciplines

An Economic Theory of ClubsAuthor(s): James M. BuchananSource: Economica, New Series, Vol. 32, No. 125 (Feb., 1965), pp. 1-14Published by: Blackwell Publishing on behalf of The London School of Economics and PoliticalScience and The Suntory and Toyota International Centres for Economics and Related

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An Economic Theory of Clubs'By JAMESM. BUCHANAN

The implied institutional setting for neo-classical economic theory,including theoreticalwelfareeconomics, is a regime of privateproperty,in which all goods and services are privately (individually) utilized orconsumed. Only within the last two decades have serious attemptsbeen made to extend the formal theoretical structure to include com-munal or collective ownership-consumptionarrangements.2The puretheory of public goods remains in its infancy, and the few modelsthat have been most rigorouslydeveloped apply only to polar or extremecases. For example, in the fundamental papers by Paul A. Samuelson,a sharp conceptual distinction is made between those goods andservicesthat are purelyprivate and those that are purelypublic .No generaltheory has been developed which covers the whole spectrumof ownership-consumption possibilities, ranging from the purelyprivateor individualizedactivityon the one hand to purelypublicor col-lectivized activity on the other. One of the missing links here is atheory of clubs , a theory of co-operative membership, a theory thatwill include as a variable to be determinedthe extension of ownership-consumption rights over differingnumbers of persons.Everydayexperience reveals that there exists some most preferredoroptimal membership for almost any activity in which we engage,and that this membershipvaries in some relation to economic factors.European hotels have more communally shared bathrooms than theirAmerican counterparts. Middle and low income communities organizeswimming-bathing facilities; high income communities are observedto enjoy privately owned swimming pools.In this paper I shall develop a generaltheory of clubs, or consumptionownership-membershiparrangements. This construction allows us tomove one step forwardin closing the awesome Samuelson gap betweenthe purely private and the purely public good. For the former, theoptimal sharingarrangement,the preferredclub membership, is clearlyone person (or one family unit), whereas the optimal sharing group

1 I amindebted o graduate tudentsand colleagues ormanyhelpfulsuggestions.Specificacknowledgement hould be made for the critical assistance of EmilioGiardinaof the Universityof Cataniaand W. CraigStubblebineof the Universityof Delaware.2 It is interesting hat none of the theoriesof Socialist economic organizationseemsto be basedon explicitco-operationamongindividuals.Thesetheorieshaveconceived he economyeitherin the Lange-Lernerenseas an analogueto a purelyprivate, ndividuallyorientedsocial orderor, alternatively,as one that is centrallydirected.3 See Paul A. Samuelson, The Pure Theoryof Public Expenditure , Reviewof Economicsand Statistics, vol. xxxvi (1954), pp. 387-89; DiagrammaticExpositionof a Theoryof PublicExpenditure , Reviewof Economics ndStatistics,

vol. XXXVII1955), pp. 350-55.A


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2 ECONOMICA [FEBRUARYfor the purelypublic good, as defined n the polarsense, includesaninfinitely argenumberof members.That is to say, for any genuinelycollective good definedin the Samuelsonway, a club that has aninfinitely argememberships preferred o all arrangementsf finitesize. Whileit is evidentthat some goods and servicesmay be reason-ably classifiedas purelyprivate,even in the extremesense,it is clearthatfew, if any, goods satisfythe conditionsof extreme ollectiveness.Theinteresting asesarethosegoodsand services, heconsumption fwhichinvolvessome publicness , wherethe optimalsharinggroupis morethanone personor familybut smaller hanan infinitely argenumber.The rangeof publicness is finite. Thecentralquestion na theoryof clubs is that of determininghe membershipmargin,soto speak,the size of the most desirable ost andconsumption haringarrangement.1

In traditionalneo-classicalmodels that assume the existence ofpurely private goods and servicesonly, the utility function of anindividual s written,(1) Ui_ Ui (Xii, X2L . . .X Xni)whereeach of the X's representshe amountof a purelyprivategoodavailableduringa specified ime period, to the reference ndividualdesignatedby the superscript.Samuelsonextendedthis function to include purelycollective orpublic goods, which he denoted by the subscripts, n+l, . . ., n+m,so that(1) is changed o read,(2) U ,=U (Xj,X2, . .Xi; Xvi 1, X42, . . . os XThis approachrequires hat all goods be initiallyclassified nto thetwo sets, privateand public. Privategoods, definedto be whollydivisible among the persons, i=l, 2, . . ., s, satisfy the relation

sXi - Xi'i=1whilepublic goods,defined o be whollyindivisible s amongpersons,satisfythe relation,

Xn +j= X'+JoI proposeto drop any attemptat an initialclassification r differen-tiation of goods into fully divisibleand fully indivisible ets, and toincorporate n the utility function goods falling betweenthese twoextremes.Whatthetheoryof clubsprovides s, in onesense,a theoryof classification ,but this emergesas an outputof the analysis.Thefirststepis thatof modifyingheutility unction.1 Note thatan economictheoryof clubscan strictlyapply only to the extentthatthe motivation for joining in sharing arrangements s itself economic; that is,only if choices are made on the basis of costs and benefitsof particulargoods andservicesas these are confronted by the individual. In so far as individualsjoinclubsfor camaraderie, s such,the theorydoesnot apply.


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1965] AN ECONOMIC THEORY OF CLUBS 3Note that, in neither (1) nor (2) is it necessary to make a distinctionbetween goods available to the ownership unit of which the referenceindividual is a member and goods finally available to the individualfor consumption . With purely private goods, consumption by oneindividual automatically reduces potential consumption of otherindividuals by an equal amount. Withpurely public goods, consumptionby any one individual implies equal consumption by all others. Forgoods falling between such extremes, such a distinction must be made.This is because for such goods there is no unique translation possiblebetween the goods available to the membership unit and goodsfinally consumed . In the construction which follows, therefore, thegoods entering the individual's utility function, the Xj's, should beinterpreted as goods available for consumption to the whole member-ship unit of which the reference individual is a member .Arguments that represent the size of the sharing group must beincluded in the utility function along with arguments representinggoods and services. For any good or service, regardlessof its ultimateplace along the conceptual public-private spectrum, the utility that anindividual receives from its consumption depends upon the ntumber fother persons with whom he must share its benefits. This is obvious,but its acceptance does require breaking out of the private propertystraitjacket within which most of economic theory has developed.As an extreme example, take a good normally considered to be purelyprivate, say, a pair of shoes. Clearly your own utility from a single pairof shoes, per unit of time, depends on the number of other personswho share them with you. Simultaneous physical sharing may not, ofcourse, be possible; only one person can wear the shoes at each par-ticular moment. However, for any finite period of time, sharing is

possible, even for such evidently private goods. For pure services thatare consumed in the moment of acquisition the extension is somewhatmore difficult, but it can be made none the less. Sharing here simplymeans that the individual receives a smaller quantity of the service.Sharing a haircut per month with a second person is the same asconsuming one-half haircut per month . Given any quantity offinal good, as defined in terms of the physical units of some standardquality, the utility that the individual receives from this quantity willbe related functionally to the number of others with whom he shares.'Variables for club size are not normally included in the utilityfunction of an individual since, in the private-goods world, the optimalclub size is unity. However, for our purposes, these variables must beexplicitly included, and, for completeness, a club-size variable shouldbe included for each and every good. Alongside each Xi there must beplaced an Nj, which we define as the number of persons who are toparticipate as members in the sharing of good, Xj, including the1 Physicalattributesof a good or servicemay, of course,affectthe structureofthe sharing arrangementshat are preferred.Althoughthe analysisbelow assumessymmetrical haring, his assumption s not necessary,and the analysis n its generalformcan be extendedto cover all possibleschemes.


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4 ECONOMICA [FEBRUARYith person whose utility function is examined.That is to say, theclub-sizevariable,N1, measuresthe numberof persons who are tojoin in the consumption-utilizationrrangementsor good, Xj, overthe relevant ime period.The sharingarrangementsmay or may notcall for equal consumptionon the part of each member,and thepeculiarmannerof sharingwill clearlyaffect the way in which thevariableenters the utility function. For simplicitywe may assumeequal sharing,althoughthis is not necessaryfor the analysis.Therewrittenutilityfunctionnow becomes,(3) Ui- U[(X1, Ni), (XI, N), ... *, (Xni+m, ni?m)]We may designatea numerairegood, X7, which can simply bethoughtof as money, possessingvalueonly as a mediumof exchange.By employingthe convention wherebythe lower case u's representthe partialderivatives,we get u;/zu,definedas the marginalrate ofsubstitutionn consumptionbetweenXj and XA or the ith individual.Since, in our construction, he size of the groupis also a variable,we must also examine,u,/4u, definedas the marginalrate of sub-stitution in consumption between he sizeof the sharinggroupandthe numeraire.Thatis to say, thisratiorepresentshe rate(whichmaybe negative)at which heindividuals willing o give up(accept)moneyin exchange or additionalmembersn the sharinggroup.We now definea cost or production unctionas this confronts heindividual,andthiswill include he same set of variables,(4) F =F'[(Xj', Ni), (X2,ND, . . ., (Xn'+m, n+m)]*Whydo the club-sizevariables, heNj's, appear n thiscost function?The additionof members o a sharinggroupmay, and normallywill,affectthecost of the good to anyone member.Thelarger s the mem-bershipof the golf club the lower the dues to any single member,givena specificquantityof club facilitiesavailableper unit time.It nowbecomespossible o derive, romtheutilityandcost functions,statementsor the necessarymarginal onditions or Paretooptimalityin respect o consumptionof eachgood. In the usualmannerwe get,(5)ul ui/UAIfrCondition(5) statesthat, for the ith individual,he marginalrate ofsubstitutionbetweengoodsXj andXA,n consumption,must be equalto the marginalrate of substitutionbetweenthese same two goodsin production or exchange.To this acknowledgednecessarycon-dition,we nowadd,(6) uNj1u'=fNj1f,'

1 Note that this constructionof the individual'sutility functiondiffers rom thatintroducedin an earlier paper, where activities rather than goods wereincluded as the basic arguments. (See James M. Buchanan and Wm. CraigStubblebine, Extemality, Economica,vol. xxxi (1962), pp. 371-84.) In thealtemativeconstruction,the activities of other personsenter directlyinto theutility function of the reference ndividualwith respectto the consumptionof allotherthanpurely privategoods. The constructionhereincorporates he sameinter-dependence hroughthe inclusionof the Nj's althoughin a more generalmanner.


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1965] AN ECONOMIC THEORY OF CLUBS 5Condition (6) is not normally stated, since the variablesrelating to clubsize are not normally included in utility functions. Implicitly, the sizefor sharing arrangements is assumed to be determined exogenouslyto individual choices. Club size is presumedto be a part of the environ-ment. Condition (6) states that the marginal rate of substitution inconsumption between the size of the group sharing in the use ofgood Xi, and the numeraire good, X7, must be equal to the marginalrate of substitution in production . In other words, the individualattains full equilibrium in club size only when the marginal benefitsthat he secures from having an additional member (which may, andprobably will normally be, negative) are just equal to the marginalcosts that he incurs from adding a member (which will also normallybe negative).Combining (5) and (6) we get,(7) uz/fs ur/flr=4fi .Only when (7) is satisfied will the necessary marginal conditions withrespect to the consumption-utilization of Xj be met. The individual willhave available to his membership unit an optimal quantity of Xj,measured in physical units and, also, he will be sharing this quantityoptimally over a group of determined size.The necessary condition for club size may not, of course, be met.Since for many goods there is a major change in utility between theone-person and the two-person club, and since discrete changes inmembershipmay be all that is possible, we may get,(7A) uj u4>z j u l I-- IfI fi fN Nj=l j ft v Nj=2which incorporates the recognition that, with a club size of unity,the right-hand term may be relatively too small, whereas, with a clubsize of two, it may be too large. If partial sharing arrangementscan beworked out, this qualification need not, of course, be made.If, on the other hand, the size of a co-operative or collective sharinggroup is exogenously determined, we may get,(7B) uj' u' UNJ

f{i f,' f>' i if Nj=kNote that (7B) actually characterizes the situation of an individualwith respect to the consumption of any purely public good of the typedefined in the Samuelson polar model. Any group of finite size, k,is smaller than optimal here, and the full set of necessary marginalconditions cannot possibly be met. Since additional persons can, bydefinition, be added to the group without in any way reducing theavailabilityof the good to other members,and since additionalmemberscould they be found, would presumably place some positive value onthe good and hence be willing to share in its costs, the group alwaysremains below optimal size. The all-inclusive club remains too small.


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6 ECONOMICA [FEBRUARYConsider,now, the relationbetweenthe set of necessarymarginalconditionsdefinedn (7) and those presentedby Samuelson n applica-tion to goods that wereexogenouslydefinedto be purely public. Inthe lattercase, theseconditionsare,

s(8) ? (z4+j/u')=fn+jlfr'wherethe marginalrates of substitutionn consumptionbetweenthepurelypublic good, X,,+j,and the numeraire ood, Xr, summedoverall individuals n the groupof determined ize, s, equalsthe marginalcost of X1+jalso defined n terms of unitsof Xr. Note that when (7)is satisfied, 8) is necessarily atisfied,providedonly that the collectivityis makingneitherprofitnorloss on providinghe marginalunit of thepublicgood.Thatis to say, provided hat,a(9) fn+jlfr = E (fn+JIfr)i=1The reversedoes not necessarilyhold, however,since the satisfactionof (8) does not require hateachandevery ndividualn the groupbein a positionwherehis ownmarginalbenefitsareequalto hismarginalcosts (taxes).' And, of course,(8) says nothingat all aboutgroup size.Thenecessarymarginal onditionsn (7)allowus to classifyall goodsonly afterthe solution is attained.Whetheror not a particular oodis purelyprivate,purelypublic,or somewherebetween heseextremesis determined nlyaftertheequilibrium alues or theNj'sare known.A goodfor which heequilibrium alueforNj is largecanbeclassifiedas containingmuch publicness . By contrast,a good for which theequilibrium alue of N1 is smallcan be classifiedas largelyprivate.

IIThe formalstatementof the theoryof clubspresentedn Section Ican be supplemented nd clarifiedby geometricalanalysis,althoughthenatureof the constructionmplies omewhatmorerestrictivemodels.Considera good that is known to contain,under some conditions,a degreeof publicness . For simplicity, hink of a swimmingpool.We want to examinethe choice calculusof a single person,and weshall assumethat other personsabout him, with whom he may ormay not choose to join in some club-likearrangement, re identicalin all respectswith him. As a firststep,takea facilityof one-unitsize,whichwe define n termsof physicaloutput supplied.Ontheordinateof Fig. 1,wemeasure otal cost andtotalbenefitperperson,the latter derived rom the individual'sown evaluationof thefacility n termsof thenumeraire, ollars.On theabscissa,we measurethe numberof personsin possible sharingarrangements.Define thefull cost of the one-unit acility o be Y1,andthereferencendividual's1 In Samuelson'sdiagrammatic resentation, hese individualmarginalconditionsare satisfied,but the diagrammatic onstruction s more restrictedthan that con-tainedin his earliermoregeneralmodel.


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1965] AN ECONOMICTHEORY OF CLUBS 7evaluationof this facility as a purelyprivateconsumptiongood to beEl. As is clearfrom the constructionas drawn,he will not choose topurchase he good. If the singlepersonis required o meet the fullcost,he will notbeableto enjoy hebenefitsof thegood.Anyenjoymentof the facilityrequires he organization f someco-operative-collectivesharing arrangement.'Two functionsmay now be tracedin Fig. 1, remainingwithintheone-unitrestriction n the size of the facility.A total benefit unctionand a total cost function confronting he single individualmay be

:0L. Figure 1

Co04-i--o N5Bh

oEl0

0 Bi0 N I Si S h

Number of personsderived.As morepersonsareallowed o share n the enjoymentof thefacility,of givensize, the benefitevaluation hat the individualplaceson the good will, aftersomepoint, decline.Theremay, of course,beboth an increasingand a constantrangeof the total benefitfunction,but at somepointcongestionwill set in, andhis evaluationof thegood

h1 e sharingarrangementneed not be either co-operative or governmental nform.Since profit opportunities xist in all such situations,the emergenceof profit-seekingfirms can be predicted n those settings where legal structurespermit,andwhere his organizationalorm possessesrelativeadvantages. Cf. R. H. Coase, TheNature of the Firm , Economica,vol. Iv (1937), pp. 386-405.) For purposesofthis paper, such firms are one form of club organization,with co-operativesandpublic arrangements epresentingother forms. Generally speaking,of course, thechoiceamong theseformsshould be largelydeterminedby efficiencyconsiderations.


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8 ECONOMICA [FEBRUARYwill fall. There seems little doubt that the total benefit curve, shownas B1, will exhibit the concavity property as drawn for goods thatinvolve some commonality in consumption.'The bringing of additional members into the club also serves toreduce the cost that the single person will face. Since, by our initialsimplifying assumption, all persons here are identical, symmetricalcost sharing is suggested. In any case, the total cost per person willfall as additional persons join the group, under any cost-sharingscheme. As drawn in Fig. 1, symmetrical sharing is assumed and thecurve, C1, traces the total cost function, given the one-unit restrictionon the size of the facility.2

For the given size of the facility, there will exist some optimal sizeof club. This is determined at the point where the derivatives of thetotal cost and total benefit functions are equal, shown as S1 in Fig. 1,for the one-unit facility. Consider now an increase in the size of thefacility. As before, a total cost curve and a total benefit curve maybe derived, and an optimal club size determined. One other suchoptimum is shown at Sh, for a quantity of goods upon which thecurves Ch and Bh are based. Similar constructions can be carried outfor every possible size of facility; that is, for each possible quantity ofgood.A similar construction may be used to determine optimal goodsquantity for each possible size of club; this is illustrated in Fig. 2.On the ordinate, we measure here total costs and total benefits con-fronting the individual, as in Fig. 1. On the abscissa, we measurephysical size of the facility, quantity of good, and for each assumedsize of club membership we may trace total cost and total benefitfunctions. If we first examine the single-member club, we may well findthat the optimal goods quantity is zero; the total cost function mayincrease more rapidly than the total benefit function from the outset.However, as more persons are added, the total costs to the single personfall; under our symmetrical sharing assumption, they will fall proport-ionately. The total benefit functions here will slope upward to the rightbut after some initial range they will be concave downward and atsome point will reach a maximum. As club size is increased, benefit

1 The geometricalmodel here applies only to such goods. Essentiallythe sameanalysismay, however,beextendedto applyto caseswhere congestion , as such,does not appear.For example, goods that areproducedat decreasingcosts, even iftheirconsumption s purelyprivate,may be shownto require omesharingarrange-mentsin an equilibriumor optimal organization.

2 For simplicity, we assume that an additional membership in the clubinvolves the addition of one separate person. The model applies equally well,however, for those cases where cost shares are allocated proportionatelywithpredicted usage. In this extension, an additional membership would reallyamount to an additional consumptionunit. Membership n the swimmingclubcould, for example,be defined as the right to visit the pool one time each week.Hence, the personwho plans to make two visits per week would, in this modifica-tion, hold two memberships.This qualification s not, of course, relevantunderthe strictworld-of-equalsassumption,but it indicates that the theoryneed not beso restrictiveas it might appear.


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1965] AN ECONOMICTHEORY OF CLUBS 9

c Cl Figure 2N=1

0~C ~ ~ ~ ~~~~~~~~~~~~~~raI

.4-A

U/

OUo4- //,

0

ol Q~~~~~kQuantity of good

functionswillshiftgenerally ownward eyond he nitialnon-congestionrange,and the point of maximumbenefitwill move to the right. Theconstructionof Fig. 2 allowsus to derive an optimal goods quantityfor each size of club Qk iS one such quantity or club size N=-K.Theresultsderived romFigs. I and2 are combined n Fig. 3. Herethe two variablesto be chosen, goods quantityand club size, aremeasuredon the ordinateandtheabscissa espectively.The valuesforoptimalclub size for each goods quantity,derived romFig. 1, allowus to plot the curve,Nopt,n Fig. 3. Similarly, he valuesfor optimalgoods quantity,for each club size, derivedfrom Fig. 2, allow us toplot the curve,Qpt.The intersection f thesetwo curves,N,ptand Qopt determines hepositionof full equilibrium,G. The individual s in equilibrium othwith respect o goods quantityandto groupsize, for the good underconsideration. uppose, or example, hatthe sharinggroup s limitedto size, Nk. The attainmentof equilibriumwith respect to goodsquantity,shown by Qk, would still leave the individualdesirousofshifting he size of themembershipo as to attainpositionL. However,once the groupincreases o this size, the individualprefersa largerquantityof thegood, and so on, untilGis attained.

Fig. 3 may be interpreted s a standardpreferencemap depictingthe tastes of the individual or the two components,goods quantity


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10 ECONOMICA [FEBRUARY

Figure 3

Nopt

%--0zQk L

a /

01 NkNumber of persons

and club size for the sharing of that good. The curves, Noptand Qopt,are lines of optima, and G is the highest attainable level for the in-dividual, the top of his ordinal utility mountain. Since these curvesare lines of optima within an individual preference system, successivechoices must converge in G.It should be noted that income-price constraints have already beenincorporated in the preference map through the specific sharingassumptions that are made. The tastes of the individual depicted inFig. 3 reflect the post-payment or net relative evaluations of the twocomponents of consumption at all levels. Unless additional constraintsare imposed on the model, he must move to the satiety point in thisconstruction.It seems clear that under normal conditions both of the curves inFig. 3 will slope upwardto the right, and that they will lie in approxim-ately the relation to each other as therein depicted. This reflects thefact that, normally for the type of good considered in this example,there will exist a complementary rather than a substitute relationshipbetween increasing the quantity of the good and increasing the sizeof the sharing group.

This geometrical model can be extended to cover goods falling atany point along the private-public spectrum. Take the purely public


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19651 AN ECONOMICTHEORY OF CLUBS 11good as the first extremecase. Since, by definition,congestiondoesnot occur, each total benefitcurve, in Fig. 1, becomes horizontal.Thus, optimalclub size,regardless f goods quantitys infinite.Hence,full equilibriums impossibleof attainment;equilibriumonly withrespectto goods quantitycan be reached,definedwithrespect o theall-inclusive inite group.In the constructionof Fig. 3, the N curvecannotbe drawn.A more realisticmodel may be that in which, atgoods quantityequilibrium,he limitationson groupsize imposeaninequality.For example,n Fig. 3, supposethatthe all-inclusive roupis of size, Nk. Congestions indicatedas beingpossibleoversmallsizesof facility, but, if an equilibriumquantityis provided,there is nocongestion,and, in fact, thereremaineconomies o scalein clubsize.The situationat the most favourableattainableposition is, therefore,in allrespects quivalento thatconfrontedn the caseof the goodthatis purelypublicunder he morerestricted efinition.

Figure 4

o | G| W Qopt

CS'00 0opt

G(0>,

0 N 1Number of persons

Considernow the purelyprivategood. The appropriateurvesheremay be shownin Fig. 4. The individual,with his income-price on-straints s able to attainthe peak of his ordinalpreferencemountainwithoutthe necessityof callingupon his fellowsto join himin sharing


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12 ECONOMICA [FEBRUARYarrangements. lso, the benefits hat he receives rom the good maybe so exclusivelyhis own that thesewouldlargelydisappearf otherswerebrought n to share hem.Hence,the full equilibrium osition,G,lies along the verticalfromthe N=1 memberpoint. Any attempttoexpandthe club beyondthis point will reduce the utility of the in-dividual.'

IIIThe geometricalconstruction mplies that the necessarymarginalconditionsare satisfiedat uniqueequilibrium alues for both goodsquantityand club size. This involves an oversimplificationhat is

made possibleonly throughthe assumptionsof specificcost-sharingschemesand identityamong individuals. n orderto generalize heresults,these restrictionsmustbe dropped.We know that, givenanygroupof individualswhoare ableto evaluatebothconsumptionharesand the costs of congestion, hereexists some set of marginalprices,goods quantity,and club size that will satisfy (7) above. However,thequantityof thegood,the sizeof the clubsharingn its consumption,andthecost-sharing rrangements ustbe determinedimultaneously.And, since there are always gains from trade to be realizedinmoving romnon-optimalo optimalpositions,distributionalonsider-ationsmustbeintroduced.Once heseareallowed o bepresent,hefinalsolution canbe locatedat anyone of a sub-infinity fpointsontheParetowelfaresurface.Only throughsome quite arbitrarily hosenconventions anstandard eometricalonstructions e madeto apply.The approachusedabove has been to imposeat the outset a set ofmarginalprices(tax-prices,f the goodis suppliedpublicly), ranslatedhereinto sharesor potentialshares n the costs of providing eparatequantitiesof a specificgood for groupsof varyingsizes. Hence, theindividual onfrontsa predictableetofmarginal rices or eachquan-tity of the good at everypossibleclub size, independently f his ownchoices on these variables.With this convention,and the world-of-equals assumption,the geometricalsolution becomes one that isrelevant oranyindividualn thegroup.If we droptheworld-of-equals

1 The constructionsuggestsclearlythat the optimalclub size, for any quantityof good, will tend to becomesmalleras the real incomeof an individual s increased.Goods thatexhibitsome publicness at low incomelevelswill, therefore, end tobecome private as income levels advance. This suggests that the number ofactivitiesthatareorganizedoptimallyunderco-operativecollectivesharingarrange-ments will tend to be somewhatlargerin low-incomecommunitiesthan in high-income communities, other things equal. There is, of course, ample empiricalsupportfor this ratherobvious conclusion drawnfrom the model. For example,in Americanagricultural ommunities hirty yearsago heavyequipmentwas com-munally shared among many farms, normally on some single owner-lease-rentalarrangement.Today, substantially he sameequipmentwill be found on each farm,even thoughit remains dle for much of its potentialworkingtime.The implicationof the analysisfor the size of governmentalunits is perhaps essevident. In so far as governmentsare organizedto provide communalfacilities,the size of suchunits measuredby the numberof citizens,should decline as incomeincreases.Thus, in the affluentsociety, the local school district may, optimally,-besmallerthanin the poor society.


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1965] AN ECONOMIC THEORY OF CLUBS 13assumption, the construction continues to hold without change for thechoice calculus of any particular individual in the group. The resultscannot, of course, be generalized for the group in this case, sincedifferent individuals will evaluate any given result differently. Themodel remains helpful even here, however, in that it suggests theprocess through which individual decisions may be made, and ittends to clarify some of the implicit content in the more formal state-ments of the necessary marginal conditions for optimality.1

IVThe theory of clubs developed in this paper applies in the strictsense only to the organization of membership or sharing arrangementswhere exclusion is possible. In so far as non-exclusion is a charac-teristic of public goods supply, as Musgrave has suggested,2the theoryof clubs is of limited relevance. Nevertheless, some implications ofthe theory for the whole excludability question may be indicated. Ifthe structureof property rights is variable, there would seem to be fewgoods the services of which are non-excludable, solely due to somephysical attributes. Hence, the theory of clubs is, in one sense, a theoryof optimal exclusion, as well as one of inclusion. Consider the classiclighthouse case. Variations in property rights, broadly conceived,could prohibit boat operatorswithout light licenses from approach-ing the channel guarded by the light. Physical exclusion is possible,given sufficient flexibility in property law, in almost all imaginablecases, including those in which the interdependence lies in the act ofconsuming itself. Take the single person who gets an inoculation,providing immunization against a communicable disease. In so far as

this action exerts external benefits on his fellows, the person takingthe action could be authorized to collect charges from all beneficiariesunder sanction of the collectivity.This is not, of course, to suggest that property rightswill, in practice,always be adjusted to allow for optimal exclusion. If they are not,the free rider problem arises. This prospect suggests one issue ofmajor importance that the analysis of this paper has neglected, thequestion of costs that may be involved in securing agreements amongmembers of sharing groups. If individuals think that exclusion willnot be fully possible, that they can expect to secure benefits as freeriders without really becoming full-fledged contributing members of

' A note concerningone implicit assumptionof the whole analysis is in orderat this point. The possibilityfor the individualto choose amongthe variousscalesof consumption sharing arrangementshas been incorporatedinto an orthodoxmodel of individual behaviour.The procedure mpliesthat the individualremainsindifferent s to which of his neighboursor fellowcitizens oin him in such arrange-ments. In other words,no attempthas been made to allow for personal selectivityor discrimination n the models. To incorporatethis element,which is no doubtimportantin many instances,would introducea wholly new dimensioninto theanalysis,and additionaltools to those employedherewould be required.2 See R. A. Musgrave,The Theoryof PublicFinance,New York, 1959.


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14 ECONOMICA [FEBRUARYthe club, they may be reluctant o entervoluntarilynto cost-sharingarrangements.This suggeststhat one importantmeans of reducingthe costs of securing voluntaryco-operativeagreements s that ofallowingfor moreflexiblepropertyarrangementsnd for introducingexcludingdevices.If the owner of a huntingpreserve s allowedtoprosecutepoachers,then prospectivepoachersare much more likelyto be willingto payfor thehuntingpermits n advance.University of Virginia, Charlottesville.

buchanan 1965 an economic theory of clubs

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