An Economic Theory of Clubs

The Suntory and Toyota International Centres for Economics and Related Disciplines An Economic Theory of Clubs Author(s): James M. Buchanan Reviewed work(s): Source: Economica, New Series, Vol. 32, No. 125 (Feb., 1965), pp. 1-14 Published by: Blackwell Publishing on behalf of The London School of Economics and Political Science and The Suntory and Toyota International Centres for Economics and Related Disciplines Stable URL: http://www.jstor.org/stable/2552442 . Accessed: 03/02/2012 20:00 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org. Blackwell Publishing, The London School of Economics and Political Science, The Suntory and Toyota International Centres for Economics and Related Disciplines are collaborating with JSTOR to digitize, preserve and extend access to Economica. http://www.jstor.org An Economic Theory of Clubs' By JAMES M. BUCHANAN The implied institutional setting for neo-classical economic theory, including theoretical welfare economics, is a regime of private property, in which all goods and services are privately (individually) utilized or consumed. Only within the last two decades have serious attempts been made to extend the formal theoretical structure to include com- munal or collective ownership-consumption arrangements.2 The " pure theory of public goods" remains in its infancy, and the few models that have been most rigorously developed apply only to polar or extreme cases. For example, in the fundamental papers by Paul A. Samuelson, a sharp conceptual distinction is made between those goods and services that are " purely private " and those that are " purely public ". No general theory has been developed which covers the whole spectrum of ownership-consumption possibilities, ranging from the purely private or individualized activity on the one hand to purely public or col- lectivized activity on the other. One of the missing links here is " a theory of clubs ", a theory of co-operative membership, a theory that will include as a variable to be determined the extension of ownership- consumption rights over differing numbers of persons. Everyday experience reveals that there exists some most preferred or " optimal" membership for almost any activity in which we engage, and that this membership varies in some relation to economic factors. European hotels have more communally shared bathrooms than their American counterparts. Middle and low income communities organize swimming-bathing facilities; high income communities are observed to enjoy privately owned swimming pools. In this paper I shall develop a general theory of clubs, or consumption ownership-membership arrangements. This construction allows us to move one step forward in closing the awesome Samuelson gap between the purely private and the purely public good. For the former, the optimal sharing arrangement, the preferred club membership, is clearly one person (or one family unit), whereas the optimal sharing group 1 I am indebted to graduate students and colleagues for many helpful suggestions. Specific acknowledgement should be made for the critical assistance of Emilio Giardina of the University of Catania and W. Craig Stubblebine of the University of Delaware. 2 It is interesting that none of the theories of Socialist economic organization seems to be based on explicit co-operation among individuals. These theories have conceived the economy either in the Lange-Lerner sense as an analogue to a purely private, individually oriented social order or, alternatively, as one that is centrally directed. 3 See Paul A. Samuelson, " The Pure Theory of Public Expenditure ", Review of Economics and Statistics, vol. xxxvi (1954), pp. 387-89; " Diagrammatic Exposition of a Theory of Public Expenditure ", Review of Economics and Statistics, vol. XXXVII (1955), pp. 350-55. A 2 ECONOMICA [FEBRUARY for the purely public good, as defined in the polar sense, includes an infinitely large number of members. That is to say, for any genuinely collective good defined in the Samuelson way, a club that has an infinitely large membership is preferred to all arrangements of finite size. While it is evident that some goods and services may be reason- ably classified as purely private, even in the extreme sense, it is clear that few, if any, goods satisfy the conditions of extreme collectiveness. The interesting cases are those goods and services, the consumption of which involves some " publicness ", where the optimal sharing group is more than one person or family but smaller than an infinitely large number. The range of " publicness " is finite. The central question in a theory of clubs is that of determining the membership margin, so to speak, the size of the most desirable cost and consumption sharing arrangement.1 In traditional neo-classical models that assume the existence of purely private goods and services only, the utility function of an individual is written, (1) Ui_ Ui (Xii, X2L . . .X Xni) where each of the X's represents the amount of a purely private good available during a specified time period, to the reference individual designated by the superscript. Samuelson extended this function to include purely collective or public goods, which he denoted by the subscripts, n+l, . . ., n+m, so that (1) is changed to read, (2) U ,=U (Xj,X2, . . v Xi; Xvi 1, X42, . . . os X This approach requires that all goods be initially classified into the two sets, private and public. Private goods, defined to be wholly divisible among the persons, i=l, 2, . . ., s, satisfy the relation s Xi - Xi' i=1 while public goods, defined to be wholly indivisible as among persons, satisfy the relation, Xn +j= X'+Jo I propose to drop any attempt at an initial classification or differen- tiation of goods into fully divisible and fully indivisible sets, and to incorporate in the utility function goods falling between these two extremes. What the theory of clubs provides is, in one sense, a " theory of classification", but this emerges as an output of the analysis. The first step is that of modifying the utility function. 1 Note that an economic theory of clubs can strictly apply only to the extent that the motivation for joining in sharing arrangements is itself economic; that is, only if choices are made on the basis of costs and benefits of particular goods and services as these are confronted by the individual. In so far as individuals join clubs for camaraderie, as such, the theory does not apply. 1965] AN ECONOMIC THEORY OF CLUBS 3 Note that, in neither (1) nor (2) is it necessary to make a distinction between " goods available to the ownership unit of which the reference individual is a member " and " goods finally available to the individual for consumption ". With purely private goods, consumption by one individual automatically reduces potential consumption of other individuals by an equal amount. With purely public goods, consumption by any one individual implies equal consumption by all others. For goods falling between such extremes, such a distinction must be made. This is because for such goods there is no unique translation possible between the " goods available to the membership unit " and " goods finally consumed ". In the construction which follows, therefore, the " goods " entering the individual's utility function, the Xj's, should be interpreted 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 be included in the utility function along with arguments representing goods and services. For any good or service, regardless of its ultimate place along the conceptual public-private spectrum, the utility that an individual receives from its consumption depends upon the ntumber of other persons with whom he must share its benefits. This is obvious, but its acceptance does require breaking out of the private property straitjacket within which most of economic theory has developed. As an extreme example, take a good normally considered to be purely private, say, a pair of shoes. Clearly your own utility from a single pair of shoes, per unit of time, depends on the number of other persons who share them with you. Simultaneous physical sharing may not, of course, 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 that are consumed in the moment of acquisition the extension is somewhat more difficult, but it can be made none the less. Sharing here simply means that the individual receives a smaller quantity of the service. Sharing a " haircut per month " with a second person is the same as consuming " one-half haircut per month ". Given any quantity of final good, as defined in terms of the physical units of some standard quality, the utility that the individual receives from this quantity will be related functionally to the number of others with whom he shares.' Variables for club size are not normally included in the utility function of an individual since, in the private-goods world, the optimal club size is unity. However, for our purposes, these variables must be explicitly included, and, for completeness, a club-size variable should be included for each and every good. Alongside each Xi there must be placed an Nj, which we define as the number of persons who are to participate as " members " in the sharing of good, Xj, including the 1 Physical attributes of a good or service may, of course, affect the structure of the sharing arrangements that are preferred. Although the analysis below assumes symmetrical sharing, this assumption is not necessary, and the analysis in its general form can be extended to cover all possible schemes. 4 ECONOMICA [FEBRUARY ith person whose utility function is examined. That is to say, the club-size variable, N1, measures the number of persons who are to join in the consumption-utilization arrangements for good, Xj, over the relevant time period. The sharing arrangements may or may not call for equal consumption on the part of each member, and the peculiar manner of sharing will clearly affect the way in which the variable enters the utility function. For simplicity we may assume equal sharing, although this is not necessary for the analysis. The rewritten utility function now becomes, (3) Ui- U[(X1, Ni), (XI, N), ... *, (Xni+m, Nni?m)] We may designate a numeraire good, X7, which can simply be thought of as money, possessing value only as a medium of exchange. By employing the convention whereby the lower case u's represent the partial derivatives, we get u;/zu, defined as the marginal rate of substitution in consumption between Xj and XA for the ith individual. Since, in our construction, the size of the group is also a variable, we must also examine, u,/4u, defined as the marginal rate of sub- stitution " in consumption " between the size of the sharing group and the numeraire. That is to say, this ratio represents the rate (which may be negative) at which the individual is willing to give up (accept) money in exchange for additional members in the sharing group. We now define a cost or production function as this confronts the individual, and this will include the same set of variables, (4) F =F'[(Xj', Ni), (X2, ND, . . ., (Xn'+m, Nn+m)]* Why do the club-size variables, the Nj's, appear in this cost function ? The addition of members to a sharing group may, and normally will, affect the cost of the good to any one member. The larger is the mem- bership of the golf club the lower the dues to any single member, given a specific quantity of club facilities available per unit time. It now becomes possible to derive, from the utility and cost functions, statements for the necessary marginal conditions for Pareto optimality in respect to consumption of each good. In the usual manner we get, (5)ul ui/U = A Ifr Condition (5) states that, for the ith individual, the marginal rate of substitution between goods Xj and XA, in consumption, must be equal to the marginal rate of substitution between these same two goods in " production " or exchange. To this acknowledged necessary con- dition, we now add, (6) u Nj1u'=fNj1f,' 1 Note that this construction of the individual's utility function differs from that introduced in an earlier paper, where " activities " rather than " goods " were included as the basic arguments. (See James M. Buchanan and Wm. Craig Stubblebine, " Extemality," Economica, vol. xxxi (1962), pp. 371-84.) In the altemative construction, the " activities " of other persons enter directly into the utility function of the reference individual with respect to the consumption of all other than purely private goods. The construction here incorporates the same inter- dependence through the inclusion of the Nj's although in a more general manner. 1965] AN ECONOMIC THEORY OF CLUBS 5 Condition (6) is not normally stated, since the variables relating to club size are not normally included in utility functions. Implicitly, the size for sharing arrangements is assumed to be determined exogenously to individual choices. Club size is presumed to be a part of the environ- ment. Condition (6) states that the marginal rate of substitution " in consumption" between the size of the group sharing in the use of good Xi, and the numeraire good, X7, must be equal to the marginal rate of substitution " in production ". In other words, the individual attains full equilibrium in club size only when the marginal benefits that he secures from having an additional member (which may, and probably will normally be, negative) are just equal to the marginal costs that he incurs from adding a member (which will also normally be negative). Combining (5) and (6) we get, (7) uz/fs = ur/flr=4 u /fi . Only when (7) is satisfied will the necessary marginal conditions with respect to the consumption-utilization of Xj be met. The individual will have available to his membership unit an optimal quantity of Xj, measured in physical units and, also, he will be sharing this quantity " optimally" 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 the one-person and the two-person club, and since discrete changes in membership may be all that is possible, we may get, (7A) uj u4 > uz j u u l I-- I fI fi fN Nj=l j ft v Nj=2 which incorporates the recognition that, with a club size of unity, the right-hand term may be relatively too small, whereas, with a club size of two, it may be too large. If partial sharing arrangements can be worked out, this qualification need not, of course, be made. If, on the other hand, the size of a co-operative or collective sharing group is exogenously determined, we may get, (7 B) uj' u' UNJ f{i f,' f>' i i ff j Nj=k Note that (7B) actually characterizes the situation of an individual with respect to the consumption of any purely public good of the type defined in the Samuelson polar model. Any group of finite size, k, is smaller than optimal here, and the full set of necessary marginal conditions cannot possibly be met. Since additional persons can, by definition, be added to the group without in any way reducing the availability of the good to other members, and since additional members could they be found, would presumably place some positive value on the good and hence be willing to share in its costs, the group always remains below optimal size. The all-inclusive club remains too small. 6 ECONOMICA [FEBRUARY Consider, now, the relation between the set of necessary marginal conditions defined in (7) and those presented by Samuelson in applica- tion to goods that were exogenously defined to be purely public. In the latter case, these conditions are, s (8) ? (z4+j/u') =fn+jlfr' where the marginal rates of substitution in consumption between the purely public good, X,,+j, and the numeraire good, Xr, summed over all individuals in the group of determined size, s, equals the marginal cost of X1+j also defined in terms of units of Xr. Note that when (7) is satisfied, (8) is necessarily satisfied, provided only that the collectivity is making neither profit nor loss on providing the marginal unit of the public good. That is to say, provided that, a (9) fn+jlfr = E (fn+JIfr) i=1 The reverse does not necessarily hold, however, since the satisfaction of (8) does not require that each and every individual in the group be in a position where his own marginal benefits are equal to his marginal costs (taxes).' And, of course, (8) says nothing at all about group size. The necessary marginal conditions in (7) allow us to classify all goods only after the solution is attained. Whether or not a particular good is purely private, purely public, or somewhere between these extremes is determined only after the equilibrium values for the Nj's are known. A good for which the equilibrium value for Nj is large can be classified as containing much " publicness ". By contrast, a good for which the equilibrium value of N1 is small can be classified as largely private. II The formal statement of the theory of clubs presented in Section I can be supplemented and clarified by geometrical analysis, although the nature of the construction implies somewhat more restrictive models. Consider a good that is known to contain, under some conditions, a degree of " publicness ". For simplicity, think of a swimming pool. We want to examine the choice calculus of a single person, and we shall assume that other persons about him, with whom he may or may not choose to join in some club-like arrangement, are identical in all respects with him. As a first step, take a facility of one-unit size, which we define in terms of physical output supplied. On the ordinate of Fig. 1, we measure total cost and total benefit per person, the latter derived from the individual's own evaluation of the facility in terms of the numeraire, dollars. On the abscissa, we measure the number of persons in possible sharing arrangements. Define the full cost of the one-unit facility to be Y1, and the reference individual's 1 In Samuelson's diagrammatic presentation, these individual marginal conditions are satisfied, but the diagrammatic construction is more restricted than that con- tained in his earlier more general model. 1965] AN ECONOMIC THEORY OF CLUBS 7 evaluation of this facility as a purely private consumption good to be El. As is clear from the construction as drawn, he will not choose to purchase the good. If the single person is required to meet the full cost, he will not be able to enjoy the benefits of the good. Any enjoyment of the facility requires the organization of some co-operative-collective sharing arrangement.' Two functions may now be traced in Fig. 1, remaining within the one-unit restriction on the size of the facility. A total benefit function and a total cost function confronting the single individual may be : 0 L. Figure 1 Co 0 4-i- -o N5Bh oEl 0 0 Bi 0 N I Si S h Number of persons deri