clubs as status symbol: would you belong to a club that accepts you as a member?

Socio-Economic Planning Sciences 36 (2002) 93–107

Clubs as status symbol: would you belong to a club thataccepts you as a member?

Ann van Ackerea,*, Christian Haxholdtb

aHEC Lausanne, BFSH1, CH 1015 Lausanne-Dorigny, SwitzerlandbDepartment of Management Science and Statistics, Copenhagen Business School, Solbjerg Plads 3,

2000 Frederiksberg C, Denmark

Abstract

We present a stylised model that goes beyond traditional analyses involving crowding and exclusiveness,and addresses the status issue by asking ‘Do I want to be associated with those individuals?’ rather than ‘Do Iwant to be associated with that many individuals?’. As the population cares more about status, exclusion fromwell-defined groups/clubs occurs: less desirable individuals are refused. Inability to exclude induces the mostdesirable individuals to leave, and the club collapses. Offering honorary membership to the most desirablepotential members is not only a commercially optimal strategy when exclusion is not allowed, it evenoutperforms exclusion as a revenue maximisation strategy. r 2002 Elsevier Science Ltd. All rights reserved.

Please accept my resignation. I don’t want to belong to a club that will accept me as a memberGroucho Marx

1. Introduction

The literature on clubs goes back to Buchanan’s [1] seminal paper, where he goes beyond thesharp distinction between purely public and purely private goods, arguing that in between thesetwo extremes there is a continuum of ownership–consumption possibilities. He introduces theterms clubs and club membership to refer to specific ownership–consumption agreements.For a purely public good, the optimal club membership size is infinity, as an additional

individual can enjoy the good without infringing on the enjoyment of others; i.e. there are noexternalities. While goods satisfying this extreme condition are fairly rare, one could think oftuning into one’s favourite broadcast: this does not affect other people’s ability to tune in, nor thepleasure they derive from doing so (assuming one keeps the volume down to a reasonable level!).

*Corresponding author.

E-mail address: [email protected] (A. van Ackere).

0038-0121/02/$ - see front matter r 2002 Elsevier Science Ltd. All rights reserved.

PII: S 0 0 3 8 - 0 1 2 1 ( 0 1 ) 0 0 0 1 4 - 3

For a purely private good, the favoured membership is one individual or a family unit. Examplesinclude clothes and a housing unit.Examples of goods falling in between these extremes include public libraries and private sports

clubs. In both cases, as the good is shared by an increasing number of people, the utility to eachmember starts to decrease due to externalities, which can be interpreted as congestion in itsbroadest sense.Buchanan therefore redefines an individual’s utility function as depending not only on the

amount of each good an individual acquires, but also on the number of other people who sharethe benefits of each good. For instance, the utility of membership in a given sports club dependson the total number of members (labelled, the club size variable). Buchanan notes that anindividual’s cost function will also depend on these club size variables, as they will affect the costborne by an individual (e.g. more members may imply lower fees). This theory requires thatexclusion be enforceable. Thus, if it is impossible to exclude non-members from enjoying thegood, the notion of club membership looses its relevance, creating a free-rider problem:individuals will no longer be willing to pay for membership. Buchanan summarises this concisely(p. 13): ‘‘Hence, the theory of clubs is, in one sense, a theory of optimal exclusion, as well as one ofinclusion’’.Since Buchanan, a great deal has been written about clubs. For a survey of work up to 1980, see

Sandler and Tschirhart [2]. An overview of more recent work can be found in Atkinson [3] and inChapter 4 of Cornes and Sandler [4]. Most of this literature has in common that individuals’utility depends on (i) the absolute amount of each good, and (ii) the number of people who shareeach good.The first of these assumptions is relaxed in the literature dealing with positional versus non-

positional goods. Hirsh [5] defines positional goods as those which (p. 27) ‘‘are either (1) scarce insome absolute or socially imposed sense or (2) subject to congestion or crowding through moreextensive use’’. In this paper, we focus on goods falling in the first category. Frank [6] rephrasesthis definition as (p. 101) ‘‘y those things whose value depends relatively strongly on how theycompare with things owned by others’’; i.e. he emphasises the relative aspect. Educationalqualifications are one example of such goods: the value of your qualifications depends on howthese compare to the qualifications of other individuals in the job market. This may lead to over-investment in positional goods by rational individuals, as ownership of such goods becomes asignal of one’s ability. Frank argues that there is a significant tendency to overspend on positionalgoods across the population, and that the rate of overspend is higher for people near thebottom of the positional goods’ hierarchy. Overspend is socially sub-optimal as one’s gainis another one’s loss; i.e. the total ‘status’ pie remains unchanged, but resource allocation issub-optimal.Congleton [7] takes this line of thought a step further and talks about status seeking games,

which are (p. 175) ‘‘games in which an individual’s utility is determined by his relative expenditureon status seeking activities rather than on his absolute consumption’’. This is closely related to thepositional goods described in Franks [6]. Congleton argues that these games are not necessarily aswasteful as implied by the literature on positional goods, as they may involve benefits forindividuals not actively involved in these games. For instance, expenditure on attempts to breakcertain records (flying around the world in a balloon, landing the first man on the moon) yieldsbenefits to people not directly involved, such as looking on, and technical spin-offs.

A. van Ackere, C. Haxholdt / Socio-Economic Planning Sciences 36 (2002) 93–10794

Congleton also points out that status seekers may take co-ordinated actions to limit the costs ofstatus seeking activities by making it difficult, if not impossible, for new status seekers to join. Forinstance (p. 184), ‘‘Most exclusive clubs restrict eligibility for membership in arbitrary ways.Many clubs require new members to be acceptable to current members and/or limit membershipto persons of a particular genology, creed, or sex’’.H.aackner and Nyberg [8] extend the above arguments to the concepts of vanity and congestion.

They focus on negative reciprocal externalities, with congestion as a special case (e.g. thecongestion caused by an extra car is felt by all drivers, including the driver of the extra car).H.aackner and Nyberg mention the interesting example of prestigious brand-name goods, where thevalue of the brand is negatively related to the total number of purchasers, as it is the exclusivenessof the product that creates its value. Owners of these exclusive goods can be thought of as a clubsharing an amount of status: the larger the number of people sharing, the lower each person’sutility.H.aackner and Nyberg present a model where identical consumers choose between a number

of brand goods of identical intrinsic quality, and a composite good. The utility of the brandgood is increasing in exclusiveness (i.e. decreasing in the sales volume of that brand). Theyconclude that in equilibrium, exclusive high price firms may coexist with cheaper mass marketbrands.

1.1. Structure of the game

The various papers discussed above address the issue that the relative, rather than the absolute,amount of a good may be important. Still, all assume that what matters, is the total number ofusers with whom a good is shared. No attention is thus given to who these individuals are, so thepopulation is generally assumed to be homogenous.In this paper, we extend the notion of clubs to specifically include the issue of a heterogeneous

population, where some individuals are more desirable members than others. We interpret theconcept of a club in a very broad sense, ranging from citizens of a town having access to their localpublic library, to belonging to an exclusive sports club, or belonging to the club of ‘Rolex watchowners’.This definition is related to what Schelling [9] calls open models of sorting and mixing. His focus

is mainly on the process by which people join or leave a group, and on how individually rationalchoices (micromotives) can lead to an outcome most individuals find undesirable (macro-behaviour). Schelling also considers situations where a fraction of the population needs to beinduced to remain in order that the rest stay voluntarily. This is in the same spirit as our ‘honorarymembers’. The key difference between Schelling’s approach and ours lies in the absence of a profitmaximising club manager.To focus on the aspect of a population that is heterogeneous with respect to desirability, we will

ignore the issue of congestion in its narrow sense. Our aim is to focus on the impact of whobelongs, rather than on the impact of how many belong. Note that in many examples, congestionin its narrow sense is not an issue (e.g. Rolex watch owners).Cowan et al. [10] address some of these issues. Their model disregards the price issue, and

assumes that individuals’ preferences for the objective qualities of a good are correlated with theirsocial status. Additionally, consumption decisions are affected by the consumption decisions of

A. van Ackere, C. Haxholdt / Socio-Economic Planning Sciences 36 (2002) 93–107 95

other agents. A key difference with our work is that we consider a join/do not join decision, whilethey consider a probability of consumption for each individual. Also, they assume that consumersare most influenced by the consumption behaviour of individuals with similar social status, andlittle or not by socially more distant people. On the other hand, in our model, the joining decisionof a small number of socially very different people would have a significant impact on the value ofbelonging to the club.The clubs we consider are characterised by two parameters: the intrinsic value of the facilities

and services they offer (sports facilities, knowing the time), and the average desirability of itsmembers, which we approximate in this paper by their average wealth level. The population ischaracterised by the importance individuals attach to ‘who’ belongs.The structure of the game is as follows: the club selects a membership fee and, in the case of

exclusion, also decides which individuals are eligible to join. Given this information, individualssimultaneously elect whether or not to join the club. They base their decision on the assumptionthat all individuals are rational, and that eligible individuals will join if and only if they benefitfrom doing so.Our main results can be summarised as follows. As the population attaches an increasing

weight to status, it becomes optimal to exclude an increasing number of less desirable potentialmembers. The inability to exclude causes the most desirable members to leave, and the clubcollapses. Honorary memberships enable one to alleviate this problem.The paper is structured as follows. In Section 2 we describe the model. In Section 3 we interpret

the results. In Section 4 we discuss how sensitive our results are to the various assumptions, whileSection 5 contains some concluding remarks.

2. The model

As mentioned above, clubs are characterised by two parameters: the intrinsic value of thefacilities and services offered by the club, denoted by c; and the average wealth level of members,denoted by %ww: The club sets a fee level (which influences its membership) so as to maximise itsrevenue.The population is characterised by its sensitivity to status, or exclusiveness, denoted by k: A

value k ¼ 0 implies that exclusiveness is irrelevant, while k ¼ 1 implies a very high weight toexclusiveness. Each individual is characterised by his wealth level w: Wealth levels are distributeduniformly on the interval ½0; 1�: For simplicity, we will refer to the individual with wealth level w asindividual w: In this paper, we assume that individuals favour being associated with high wealthindividuals, i.e. we equate wealth and desirability. An interesting extension would be to consider atwo-dimensional array of individuals, characterised by separate wealth and desirabilityparameters.The utility an individual derives from joining the club depends on the service provided by the

club ðcÞ; the importance of status ðkÞ; how this individual compares to other club members, i.e.does he benefit from being associated with them ð %ww� wÞ; the fee charged by the club ð f Þ; and hisown wealth level (wÞ: The utility from joining the club is defined as

Uðw; %ww; k; c; f Þ ¼ cþ kð %ww� wÞ � f =ð1þ wÞ2: ð1Þ


Note that this expression captures the change in utility resulting from joining the club, i.e. for anyindividual, we focus on the change in his utility rather than on his total utility. The first term is theintrinsic value of the services offered by the club, the second term represents the ‘status’ aspect(note that this term is positive (negative) if the member ranks below (above) average), and thethird term is the cost aspect (the higher an individual’s wealth, the less price sensitive he is). Notethat, as mentioned in the introduction, we ignore congestion effects.It is straightforward to show that this function is concave in w; implying that the set of

individuals interested in joining will be an interval of ½0; 1�: This, together with the continuity of Uimplies the existence of a unique Nash Equilibrium.As will be shown in the next section, in some cases it is optimal for the club to exclude a range

of lower wealth individuals who wish to join. Note that we use the term exclusion in the sense ofrefusing membership to potential members wishing to join, rather than in the sense of preventingindividuals who did not join from using the services of the club.It is intuitively obvious that the club will never wish to exclude high wealth individuals. Let wu

denote the wealth level of the highest ranked individual wishing to join, and wl the wealth level ofthe lowest ranked individual interested in joining and allowed to do so. Then, the assumption ofuniform distribution of wealth implies that the average wealth level of club members will equal%ww ¼ ðwu þ wlÞ=2:

2.1. The case with exclusion

We first assume that the club is able to exclude potential members if this is optimal. In this case,the club’s problem can be formulated as follows:

maxf ;wl;wu

f * ðwu � wlÞ

s:t:

Uðwl; %ww; k; c; f ÞX0;

Uðwu; %ww; k; c; f ÞX0; ð2Þ

0pwlpwup1:

The first and second equation (together with the concavity of Uðw; %ww; k; c; f Þ) imply that onlyindividuals who wish to join do so, but some may be prevented from joining. This will be the casewhenever the first constraint is not binding.We ignore costs incurred by the club to provide service level c in order to avoid confusing the

results of the model with the impact of an arbitrarily selected cost function; i.e. we consider thelevel of service as given, and maximise contribution (revenue-variable cost). An interestingextension would be to optimise the service level c for a given population (as characterised by k).We use a superscript e to denote the case with exclusion. Appendix A provides closed form

solutions for f e;weu and wel : Let Uðw

el Þ and Uðw

euÞ; respectively, denote the utility of individuals w

eu

and wel : The highest wealth individual always joins, i.e. weu ¼ 1 for all values of k and c:As illustrated in Fig. 1, the c–k space is divided into two large regions, separated by a very small

transition region. In the self-selection region, it is optimal for the club to select a fee and allow anyone who wishes to join to do so: Uðwel Þ ¼ 0 and UðweuÞ > 0: The same strategy applies to the


transition region, but here Uðwel Þ ¼ UðweuÞ ¼ 0: In the exclusion region, it is optimal for the club to

refuse entry to a range of individuals who would benefit from joining: Uðwel Þ > 0 and UðweuÞ ¼ 0:Figs. 2 and 3, respectively, show total contribution and Uðwel Þ for k ranging from 0 to 1, and

selected values of c: Table 1 provides comparative statics with respect to k and c:Note that these comparative statics only apply within each region. For instance, for k ¼ 0:7; wel

decreases in c up to approximately c ¼ 0:6 (column C) and increases thereafter (columns A andB). This is due to the switch from the self-selection region to the exclusion region.It is not surprising that individuals are willing to pay more for better facilities, i.e. f e and the

contribution increase in c: UðweuÞ also increases despite the fee increase. Let us first focus on lowvalues of k; i.e. kp1:176c:For k ¼ 0; we have wel ¼ 1=3; implying that two-thirds of the population joins the club.

Individual wel has zero gain from joining, while the gain for the top ranking individual is strictlypositive. As k increases, the club attracts more members (wel decreases), despite an increase in fees,due to the increasing status-value of membership. Revenue increases. Uðwel Þ remains at zero, butUðweuÞ decreases due to the combination of an increased fee, a decrease in %ww and the increasedweight to status (recall that individuals with wealth above %ww loose from being pooled with lowerranked individuals). UðweuÞ reaches zero as k reaches 1:176c: In this region, an increase in c leads toan increase in wel ; i.e. the club size is reduced.For k in the region ½1:176c; 1:183c�; we have Uðwel Þ ¼ Uðw

euÞ ¼ 0: Contribution reaches its

maximum in this region, for k ¼ 1:178c: Club membership increases steeply. This is paired with asharp drop in fees.When k > 1:183c; it becomes optimal for the club to exclude lower wealth individuals, so as to

increase %ww; thereby retaining the higher wealth individuals without need to decrease fees anyfurther. This results in lower contributions and a smaller club. Fees remain constant. Contrary tothe self-selection region, wel decreases in c; i.e. a higher intrinsic value of the facilities leads to alarger club size.The utility of the individual with wealth wel deserves a closer look (Fig. 3). For small c; exclusion

starts for fairly low values of k: As k increases, the loss to the marginally excluded individualremains relatively low. For larger c; e.g. c ¼ 0:7; exclusion only takes place for larger values of k;

Fig. 1. Overview of the Exclusion case.


and on a smaller scale (i.e. a larger club size); but, the loss to the marginally excluded individual ismuch larger.

2.2. The no-exclusion case

Next, let us consider the situation where a club is unable to refuse potential members wishing tojoin. One example is the limited control a brand owner can exert on stores electing to sell products

Fig. 2. Exclusion Case, Contribution. F Self-selection region. - - - - - Exclusion region. The transition region (thatcontains the maximum) is too small to be visible.

Fig. 3. Exclusion Case, Utility of lowest wealth individual. F Self-selection and transition region (0-value). - - - - -Exclusion region.


below the recommended retail price, thereby making them available to individuals unable toacquire them at the regular price.In this case, the club’s problem can be formulated as follows:

maxf ;wl;wu

f * ðwu � wlÞ

s:t:

wl *Uðwl; %ww; k; c; f Þ ¼ 0;

Uðwl; %ww; k; c; f ÞX0; ð3Þ

Uðwu; %ww; k; c; f ÞX0;

0pwlpwup1:

The first constraint requires that either wl ¼ 0 (the lowest wealth individual joins) orUðwl; %ww; k; c; f Þ ¼ 0 (i.e. any individuals with wealth level below wl do not wish to join). Theremaining constraints are as in the previous case. Note that the concavity of Uðw; %ww; k; c; f Þ in wimplies that all individuals in the interval ½wl;wu� benefit from joining.We use a superscript n to denote the no-exclusion case. Appendix B provides closed form

solutions for f n;wnu and wnl : Let Uðw

nl Þ and Uðw

nuÞ; respectively, denote the utility of individuals w

nu

and wnl :As illustrated in Fig. 4, the c–k space is now divided into three main regions, labelled self-

selection, all-belong, and collapse. The self-selection region is identical to the case with exclusion,since, in this region, it is optimal not to refuse any potential members. In the transition regionbetween self-selection and collapse, we have Uðwnl Þ ¼ Uðw

nuÞ ¼ 0: Note that this region

encompasses the transition region of the exclusion case. At the boundary between the transitionarea and the all-belong area, we have wnl ¼ 0 and wnu ¼ 1; i.e. the entire population joins the club.This holds throughout the all-belong area. From this point onwards, Uðwnl Þ > 0 and UðwnuÞ ¼ 0;i.e. the lowest wealth individual gets a net benefit from joining. As we enter the collapse area, thehigh-wealth members start to leave, i.e. wnuo1: This area is largest for large k; thus defeating theinitial purpose of the club (benefiting from being associated with the more desirable individuals)when status matters most.

Fig. 4. Overview of the No Exclusion case.


Table 1 provides comparative statics with respect to k and c: As before, these comparativestatics only apply within each region.For values of ko1:183c; the results are identical to the exclusion case. As k exceeds 1:183c; wnl

decreases further, and so does the fee, until wnl reaches zero for k ¼ 1:2c; and wnu remains at 1. Fork in the interval ½1:2c; 1:33c�; everyone joins the club. As k increases further, the highest wealthindividuals leave the club. As in the exclusion case, contribution decreases in k as k exceeds 1:178c:Not surprisingly, contribution is lower than in the exclusion case as soon as the non-exclusionconstraint becomes binding ðk > 1:183cÞ:The fee drops sharply in the transition area, and continues to drop at a decreasing rate

throughout the all-belong and collapse areas. For high k; this results in a significantly lower feethan in the exclusion case.The utility of the highest wealth member, UðwnuÞ; is as in the exclusion case: it drops to 0 as k

increases, and stays there. Uðwnl Þ is also fairly similar to the exclusion case as far as the generalshape is concerned. There is a significant difference in interpretation, though: the high positivevalue when k is large is no longer a result of exclusion. Rather, it is the benefit the lowest wealthmember of society obtains from being associated with higher wealth individuals (recall that in thisrange, wnl ¼ 0).

2.3. Honorary members

It is common practice for many clubs to offer honorary memberships (implying a fee waiver orreduced fees) to particularly desirable individuals who, by their presence, will provide others withan incentive to join. In this section, we provide several numerical examples illustrating that this

Table 1Comparative staticsa

A B C D E

(a) Impact of an increase in c for a fixed value of k

Contribution m m m m mwu 1 1 1 1 mwl m m k m mFee m m m m mUðwlÞ 0 0 Non-monotone k Non-monotoneUðwuÞ m m m 0 0

(b) Impact of an increase in k for a fixed value of cContribution m Non-monotone k k kwu 1 1 1 1 kwl k k m 0 0Fee m k Constant k kUðwlÞ 0 0 m m mUðwuÞ k 0 0 0 0

aNote: Exclusion case: A: Self-selection region; B: Transition region; C: Exclusion region.

No-exclusion case: A: Self-selection region; B: Transition region; D: All-belong region; E: Collapse region.


approach can significantly improve the club’s contribution, outperforming not only the no-exclusion case, but also the exclusion case (i.e. the use of honorary members is more effective thanexclusion from a contribution maximising point of view). We consider introducing honorarymembers in a situation where exclusion is not allowed.Table 2 provides optimal policies with respect to honorary membership for three numerical

examples ðc ¼ 0:5 and k ¼ 0:6; 0:8; 1Þ: Referring to the no exclusion case, k ¼ 0:6 is the limitbetween the transition and all belong regions, while k ¼ 0:8 and 1 are in the collapse region.Regular members pay the fee, while honorary members pay the discounted fee.In all three examples, the honorary members scenario outperforms not only the no-exclusion

case, but also the exclusion case. The larger the k; the more significant the improvement over theno-exclusion case. This is not surprising as, the higher the k; the higher the degree of exclusion.For k ¼ 1; contribution is more than doubled compared to the no-exclusion case, and increases bymore than 30% compared to the exclusion case. Allowing exclusion in the honorary members casecan lead to a further small improvement. For instance, the first column of Table 3 shows that forc ¼ 0:5 and k ¼ 1; contribution increases from 0.66 to 0.70.It is interesting to compare the no exclusion case for k ¼ 1 (Table 2) to the scenario presented in

the second column of Table 3: by offering free membership (discounted fee ¼ 0Þ to thoseindividuals not interested in joining, the club can double the fee charged to other members, andthe resulting contribution!To illustrate this idea, think, for instance, about companies signing up top athletes to wear their

brand of sports clothes, thereby being able to command a premium price in the market. In thiscase, it is the brand name of the clothes that is a club good (the actual use of the clothes is clearly aprivate good).This example actually goes a step further, as athletes are paid (i.e. the discounted fee is negative)

to wear these brands. This idea is illustrated in the third column of Table 3, where animprovement of about 35% is achieved compared to the no-exclusion case by subsidising the top

Table 2Honorary members: overview of numerical examples ðc ¼ 0:5Þ

Case k ¼ 0:6 k ¼ 0:8 k ¼ 1:0

Honorary Regular members 0.20–0.98 0.17–0.74 0.17–0.59

Honorary members 0.98–1.0 0.74–1.0 0.59–1.0Fee 1.07 1.14 1.25Discounted fee 1.04 0.67 0.34

Contribution 0.85 0.82 0.66

No-Exclusion Members 0–1.0 0–0.82 0–0.64

Fee 0.8 0.57 0.48Contribution 0.8 0.47 0.31

Exclusion Members 0.17–1.0 0.37–1.0 0.5–1.0

Fee 1.0 1.0 1.0Contribution 0.83 0.63 0.5


10% of the population to induce them to join. Note that this scenario exhibits a ‘gap’, i.e.individuals in the range ½0:64; 0:9� elect not to join, while both lower and higher wealth individualsdo join.

3. Discussion

Let us consider four extreme parameter combinations as described in Table 4:Case I is not particularly interesting: the club offers limited intrinsic value and the population

attaches little importance to status. Case II describes situations where the club offers a valuableservice, but status is of little or no importance. This implies higher fees, and club size decreases inc: As an example, consider outdoor public sports facilities with little or no comfort (case I)compared to a modern sports complex (case II), or an omnibus (I) compared to a high speed train(II) linking the same cities.Case III combines high intrinsic value and high importance to status. This results in high fees

wherein the value of the service reduces the impact of the status aspect. This category includes, forinstance, high quality private sports clubs and health care facilities. If one considers ‘knowing thetime’ as a valuable service, then owning ‘a’ watch would fall in category II (high c; low k), whileowning a Rolex would fall in category III. There is no exclusion: no one is prevented fromacquiring a watch, nor a Rolex watch.Next, consider case IV. This is the interesting combination of low intrinsic value, with a high

weight to status. This results in a highly exclusive club (high wel ; Uðwel Þ > 0).

This category encompasses some of the more exclusive social clubs that offer limited amenities,but have strict membership conditions (e.g. maximum membership size, new members must berecommended by existing members, etc.) and ownership of limited edition products. It is thiscategory of clubs that is non-sustainable when exclusion is not allowed (see Fig. 4: case IV falls inthe collapse region). It is in this region that offering honorary membership to selected individualsis most profitable. One of the most striking examples in this context is sportswear: the brandimage, determined by which professional players endorse that specific brand, has a crucial impacton the retail price. The danger of collapse of a brand if the exclusivity is not maintained helpsexplain why producers of famous brands are willing to invest in tightly controlled distributionchannels.

Table 3Variations on honorary membership: numerical examples ðk ¼ 1; c ¼ 0:5Þ

Hon. with exclusion Discounted fee ¼ 0 Discounted fee o0

Regular members 0.27–0.72 0.0–0.64 0.0–0.64

Honorary members 0.72–1.0 0.64–1.0 0.9–1.0Fee 1.23 0.97 0.71Discounted fee 0.54 0.0 �0.34Contribution 0.70 0.62 0.42


Let us consider the point of view of those individuals who are excluded, but would benefitfrom joining, given the present structure of the club, i.e. %wwe and f e: If one individual withwowel was allowed in, he would benefit considerably, on the assumption that one additionalindividual does not affect %ww in any significant way. But, if a group of individuals, say,with wealth levels in the interval ½w;wel � were allowed to join, %ww would drop significantly,resulting in these individuals no longer being interested in joining; and many of the existingmembers electing to leave! This is exactly what happens in the no-exclusion case (collapseregion).

4. Sensitivity: other utility functions

The individual’s utility function contains two arbitrary components: (i) the impact of anindividual’s relative position in the club (captured by ð %ww� wÞ in our example), and (ii) thediffering value of money for different wealth levels (captured by the term f =ð1þ wÞ2 in ourexample). In this section, we carry out a sensitivity analysis with respect to these two aspects,focusing on the case where exclusion is allowed.First consider the more general utility function

Uðw; %ww; k; c; f Þ ¼ cþ kð %ww� wÞ � f =ð1þ wÞm: ð4Þ

Table 4Overview of results

k (Importance of status)

Low High

Case I Case IVLow fees Exclusion optimal if allowed.

e.g. Basic sports facilities Collapse in No Exclusion caseLow provided by local authorities. (not sustainable without exclusion).

Basic public transport facilities. Honorary memberships most valuablee.g. Branded goods endorsed by stars.

c Exclusive sports clubs where fee(intrinsic) exceeds the value of the services.value)

Case II Case IIISelf-selection (optimal solution does Self-selection (optimal solution does

not require exclusion). not require exclusion).High Higher fees and smaller club size Less exclusive and higher fees than Case IV

compared to Case I e.g. High quality private sports facilities.

e.g. High standard sports facilities. High quality private healthcare.High speed trains.


The parameter m (which equals 2 in the base case) captures the extent to which increasing wealthdiminishes the impact of the financial expenditure f : As m increases, higher wealth individualsbecome increasingly less affected by the fee, implying that the fee becomes more effective as aselection mechanism. Consequently, the no exclusion region increases.Decreasing m leads to (a) the gradual disappearance of the no-exclusion region as it exists in the

base case, and (b) its replacement by a region where the whole population belongs to the clubðwl ¼ 0;wu ¼ 1Þ: In this region we still have UðwlÞ ¼ 0; as was the case in the no exclusion region.Next, consider an alternative formulation for the status term, for instance

Uðw; %ww; k; c; f Þ ¼ cþ %ww� k*w� f =ð1þ wÞ2: ð5Þ

So, rather than penalising the difference between the club’s average wealth and an individual’sown wealth level, we assume that the utility of membership increases in average wealth, anddecreases in the individual’s own wealth. This decrease is stronger, the larger k: This equationcaptures the idea that high wealth individuals have less to gain from joining, but the financial costof doing so is also less. The qualitative results are similar, but there are a few differences in thecomparative statics. Each individual’s utility is now decreasing in k: Profits are monotonicallydecreasing in k and the optimal fee is decreasing in the exclusion region. Club membership reacheszero for k ¼ 1þ c; while no profits can be achieved when k exceeds 1þ c:

5. Concluding remarks

In this paper, we presented a model that went beyond traditional analyses involving crowding(congestion in a narrow sense) and exclusiveness (e.g. how many individuals own a specific good,congestion in a broad sense) by adding the dimension of status: who are the individuals who alsoown this good, are members of that club, go to that theatre, etc. Is it desirable to associate withthem? To isolate this issue, we presented a model that ignored crowding issues. We showed that,when the population attaches a high value to status, it is optimal from the club manager’sperspective to exclude ‘undesirable’ individuals, even though these are willing to pay the joiningfee and would gain from joining. If exclusion is not an option, increasing the weight of statusresults initially in the whole population joining, while further increases cause the most desirablemembers to leave.We also illustrated that, from a commercial point of view, the introduction of honorary

members (i.e. inducing some highly desirable individuals to join by offering them a reduced fee, oreven pay them to join) could lead to strongly enhanced revenue. This practice is an intrinsic partof the promotion of fashion goods (sports equipment, beauty products, etc.), enablingmanufacturers of these brands to charge premium prices to the general public by sponsoring anumber of ‘stars’ to use their product.It would be interesting to extend the work presented in this paper by considering a two-

dimensional array of individuals, characterised by separate wealth and desirability parameters. Itwould also be interesting to investigate the optimal service level c for a given population. Thiswould require making assumptions with respect to the cost of providing service, and explicitlyaddressing the issue of crowding.


Appendix A. Closed form solution for the exclusion case

f e ¼

ð32kÞ�2ð9c3 þ c2ð26k� 3ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi9c2 þ 4ckþ 4k2

pÞÞ

þ 4ckð13k� 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi9c2 þ 4ckþ 4k2

pÞ if kpa1c;

þ 4k2ð2kþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi9c2 þ 4ckþ 4k2

pÞ

2ð3c� 2kþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffic2 þ 4ck� 4k2

pÞ if a1cpkpa2c;

2c if a2cpk;

8>>>>>>>><>>>>>>>>:

wel ¼

ð3cþ 2k�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi9c2 þ 4ckþ 4k2

pÞð4kÞ�1 if kpa1c;

ðc� kþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffic2 þ 4ckþ 4k2

pÞk�1 if a1cpkpa2c;

1� ck�1 if a2cpk;

8>><>>:

weu ¼

1 if kpa1c;

1 if a1cpkpa2c;

1 if a2cpk;

8><>:

where a1 solves

� 9� 40a31 þ 3ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi9þ 4a1 þ 4a21

qþ a21ð124� 20

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi9þ 4a1 þ 4a21

qÞ

þ a1ð�26þ 8ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi9þ 4a1 þ 4a21

qÞ ¼ 0; i:e: a1 ¼ 1:176

and

a2 ¼ 4�1ð3þffiffiffi3

pÞ; i:e: a2 ¼ 1:183:

Appendix B. Closed form solution for the no-exclusion case

f n ¼

ð32kÞ�2ð9c3 þ c2ð26k� 3ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi9c2 þ 4ckþ 4k2

pÞÞ

þ 4ckð13k� 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi9c2 þ 4ckþ 4k2

pÞ if kpa1c;

þ 4k2ð2kþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi9c2 þ 4ckþ 4k2

pÞ

2ð3c� 2kþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffic2 þ 4ck� 4k2

pÞ if a1cpkpð6=5Þc;

4c� 2k if ð6=5Þcpkpð4=3Þc;

ð9c3 � k2ðk�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi9c2 þ 2ckþ k2

pÞÞð32kÞ�2

þ ðc2ð13kþ 3ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi9cþ2ckþ k2

pÞÞð32kÞ�2 if ð4=3Þcpk;

þ ðckð13kþ 4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi9c2 þ 2ckþ k2

pÞÞð32kÞ�2

8>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>:


wnl ¼

ð3cþ 2k�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi9c2 þ 4ckþ 4k2

pÞð4kÞ�1 if kpa1c;

ðc� kþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffic2 þ 4ckþ 4k2

pÞk�1 if a1cpkpð6=5Þc;

0 if ð6=5Þcpkpð4=3Þc;

0 if ð4=3Þcpk;

8>>>><>>>>:

wnu ¼ l

1 if kpa1c;

1 if a1cpkpð6=5Þc;

1 if ð6=5Þcpkpð4=3Þc;

ð3c� kþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi9c2 þ 2ckþ k2

pÞð4kÞ�1 if ð4=3Þcpk;

8>>>><>>>>:

where a1 is as defined in Appendix A:

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clubs as status symbol: would you belong to a club that accepts you as a member?

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