Chapter 6: Club goods and local public goods

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Chapter summary

The issues that this chapter addresses are similar to those involved with pure public goods. It begins by defining club goods and public goods and investigating the relationship between them. The efficiency question is then addressed for single product clubs when all consumers are identical. The role of two-part tariffs in achieving efficiency is stressed. The role of clubs in achieving efficiency for the economy as a whole is then considered. The extension is then made to consider heterogeneous consumers. Heterogeneity leads naturally into a discussion of the influential Tiebout hypothesis concerning local public goods.

6.1 Definitions

One of the defining features of the public good of Chapter 5 was non-rivalry: once the good was provided, its use by one consumer did not affect the quantity available for any other. This is clearly an extreme assumption. Many commodities, such as parks, roads and sports facilities, satisfy non-rivalry to a point but eventually are subject to congestion. Although not pure public goods, these goods cannot be classed as private goods either.

There are also public goods whose benefits are restricted to given geographical areas. For instance, radio and television signals can only be received within the range of the transmitter and a police service may only patrol a limited jurisdiction. Provided a consumer is located within the relevant area, they can then benefit from the public good, otherwise the public good is unavailable to them. These are again not pure public goods. The geographical restriction on availability can also be accompanied by congestion within the region.

The first class of goods are called club goods to reflect the fact that there are benefits to groups of consumers forming to coordinate provision and that group size chosen may be less than the total population. The second class are local public goods, with the name capturing the idea of geographical restriction.

6.2 Single product clubs

Efficiency with a pure public good involves determining how much of it should be provided. With a club good, it is not just the quantity of the good that needs to be decided but also the number of club members. The latter is especially important when there is congestion. Adding a new member allows the cost of providing a given quantity of public good to be spread among more members but reduces the benefit obtained by each existing member. With a club good there is a second efficiency condition involved.

The natural assumption to make on the method of charging is that the cost of the club is divided equally among the members. This will make the club just break even.

Let each consumer have the utility function U(x, G, n), where x is consumption of a private good, G provision of the club good and n the number of club members. If the cost of providing G units of the club good is C(G), then the budget constraint of a member with income M will be:

M = x + (C(G)/n) (6.1)

The decision problem for those in charge of the club involves choosing G and n to maximise the welfare of a typical member. Putting together the budget constraint and the utility function, this can be expressed as

max {G,n} U(M –C(G)/n ,G,n).

This gives the necessary conditions for efficiency

MRT G,x ≡ UG/Ux = C/n ' ,

and

MRTn,x ≡ Un/Ux = − C/n2 .

The first of these conditions is a version of the Samuelson rule. It states that the marginal rate of substitution between the public good and the private good should be equated to the marginal rate of transformation (or the marginal cost) of the club good.

6.3 Two-part tariffs

The utility function is modified to make the number of visits, v, a variable. Assume that utility has the form

U(x, G, v, nv), (6.5)

so utility is now obtained from visits, v, but disutility comes from the total of visits, nv , made by all members – this is the congestion effect. The cost of providing the club is

C(G, nv). (6.6)

The necessary conditions for the choice of G, v and n are

n(UG/Ux) = CG

n(Unv/Ux_ = - C/n2+vCv/n and Uv/Ux = Cnv-n(Unv/Ux)

Now notice that if a fixed fee is paid – a share of the total cost – then visits will not be correctly chosen. The individual incentive is to consume visits until the marginal benefit is equal to the marginal cost of zero. This does not meet the efficiency condition. Furthermore, if visits are priced at marginal cost then the fixed costs of supplying the club will not be covered.

6.4 Clubs and the economy

Two different settings are considered which give very different perspectives. The first setting is to consider an economy with a very large number of individuals and an optimum membership for each club that is small relative to the total population. Then the outcome for the economy will be that a very large number of clubs will form, each with the correct number of members and each providing the efficient level of service. Hence efficiency will be attained for the economy as a whole. In this case, the efficiency of each individual club is reflected at the aggregate level.

The second, and analytically more interesting, case arises when the optimal membership of each club is relatively large compared to the total population i.e. there can at most be a finite number of optimally-sized clubs. A normal outcome is that there will be some remainder when total population is divided by optimum club size. To analyse this situation, assume that the total population is more than the optimum size of a club but less than twice the optimum.

The conclusion of this section has to be that the efficiency of the individual club does not translate into efficiency for the economy when there are small number problems. This should not be surprising since small numbers introduce problems akin to those found in oligopoly problems. What
occurs is that small groups of consumers are able to affect their own utility levels by choosing to form optimal size clubs. Therefore they possess market power and this is reflected in the inefficiency. To put this into context, observe that if the optimal club size is greater than (or equal to) the total population, then a single club will form and this will achieve the highest attainable level of welfare.

6.5 Approximate efficiency

There is one situation in which these arguments do not apply. The problems of dividing the population into optimal clubs resulted from the fact that there was a uniquely optimal level of club membership. However if there is a non-unique optimum club efficiency can be achieved. The interpretation of this
figure is that maximal attainable utility is given by the upper envelope of the utilities for each club size. As the figure reveals, as population size increases the gaps between the envelope and the utility attainable at optimum club size become smaller and the upper envelope converges towards a straight line at this utility value. This shows the result that the efficiency will be approximately achieved in a large economy and the deviation from efficiency will become close to zero.

6.6 The Tiebout hypothesis

The previous discussion has adopted the perspective of a club in the normal sense of the word: an organisation that individuals can join to enjoy shared facilities. The concept of a local public good, as a special case of a club good, has already been introduced. A local public good has the feature that its benefits are restricted to a particular geographical area and it cannot be enjoyed outside of that area. Relating this idea to the analysis of club goods, one can think of local communities as clubs which are formed to provide local public goods. To become a member of a local community, a consumer must move into the area (i.e. join the club) and pay whatever local taxes are levied in that community (i.e. pay the membership fee). Once they have done this, they can then enjoy the local public goods that are provided.

Tiebout observed that pure public goods lead to market failure because of the difficulties connected with information transmission. Since the true valuation by a consumer of a public good cannot be observed, and since a pure public good is non-excludable, free-riding occurs and private provision is inefficient.

Now assume that there are a number of alternative communities in which a consumer can choose to live and that these differ in their provision of local public goods. In contrast to the pure public good
case, a consumer’s choice of which location to live in provides a very clear signal of preference. The chosen location is obviously the one offering the provision of local public goods closest to the consumer’s ideal. Hence, through community choice, preference revelation takes place. Free riding
cannot benefit a consumer since the choice of a non-optimal location merely reduces their welfare level. The only rational choice is to act honestly.

Since preference revelation is taking place, it follows that if there are enough different types of community and enough consumers with each kind of preference then all consumers will allocate themselves to a community that is optimal for themselves and each community will be optimally sized. Thus, the market outcome will be fully efficient and the inefficiencies discussed in connection with pure public goods will not arise.

Where problems do arise is in the link between income and location. An assumption that can justify the previous analysis is that consumers obtain all their income from ‘rents’ (e.g. from the ownership of land, property or shares). In this case it does not matter where they choose to reside
since the rents will accrue regardless of location. Once some income is earned from employment, then the Tiebout hypothesis only holds if all employment opportunities are also replicated in all communities. Otherwise communities with better employment prospects will appear more attractive even if they offer a slightly less appealing set of local public goods. If the two issues become entangled in this way then the Tiebout hypothesis will naturally fail.

Further difficulties with the hypothesis arise when the numbers of communities and individuals is considered. When these are both finite the problems already discussed above with achieving efficiency through market behaviour arise again. These are compounded when individuals of different types are needed to make communities work.

The hypothesis depends upon the freedom of consumers to move to preferred locations. This is only possible if there are no transaction costs involved in changing location.

6.7 Empirical evidence

The Tiebout hypothesis provides the reassuring conclusion that efficiency will be attained by local communities providing public goods. If correct, the forces of economics and local politics can be left to work unrestricted by central government intervention.

It is necessary to note that these results do not confirm that the Tiebout hypothesis is completely operating, but only that some sorting of residents is occurring. It is supportive evidence for the hypothesis but not complete confirmation.

Reading: Jean Hindriks and Gareth D. Myles (2004), Intermediate Public Economics

Chapter 9 Club Goods

"One of the defining features of the public goods of Chapter 8 was non-rivalry: once the good was provided, its use by one consumer did not affect the quantity available for any other. This is clearly an extreme assumption. Many commodities, such as parks, roads and sports facilities, satisfy non-rivalry to a point but are eventually subject to congestion.

A good which has some degree of non-rivalry but for which excludability is possible is called a club good. The name is intended to reflect the fact that there are benefits to groups of consumers forming a club to coordinate provision and that the group size may be less than the total population."
p171

"[Geographically limited] goods are again not pure public goods ... and are termed local public goods, with the name capturing the idea of geographical restriction. The geographical restriction on availability can also be accompanied by congestion within the region." p171-172

"*Definition 2 (Club good) A club good is a good which is either non-rivalrous or partly rivalrous but for which exclusion by the providers is possible.*" p172

"*Definition 3 (Local public good) A local public good can only benefit those within a given geographical area. It may be non-rivalrous within that area or it may be partially rivalrous." p173

".. registration at schools can be restricted by policy choice to pupils in the local area and the size of the local population can be controlled by prohibition on new building." p173

Let each consumer have the utility function U (x, G, n), where x is the consumption of a private good, G provision of the club good and n the number of club members. Utility increases in x and G, and decreases in n if there is congestion. If the cost of providing G units of the club good is C (G), then the budget constraint of a member with income M when the cost of the club is shared equally between members will be

M = x + C(G)/n

The decision problem for those in charge of the club involves choosing G and n to maximize the welfare of a typical member. Putting together the budget constraint and the utility function, this can be expressed as

max_{G,n}U ((M - C(G)/n), G, n)

The first-order conditions for this optimization produce the following pair of equations that characterize efficiency:

nMRS_G,x := n (U_G/U_x) = CG

MRS_n,x ≡ Un/Ux = -C/n^2

The first of these conditions, is the version of the Samuelson rule and describes the level of public good, G, that the club should supply. It states that the sum of marginal rates of substitution between the public good and the private good for the n members of the club should be equated to the marginal rate of transformation (or the marginal cost), C_G , of another unit of the club good.

"A club is able to exclude non-members from consumption of the public good and can levy a charge on members. If all consumers are identical, then the club will achieve an efficient level of the club good and an efficient level of membership. If the club good suffers from congestion, then the optimum membership will be finite. Without congestion, the entire population will be members of the club. The collection of membership fees by the club will ensure that it breaks even in its financing of the provision of the club good. This fundamental insight that clubs can attain efficiency in the provision of public goods is attributed to the seminal work of Buchanan who was the first to develop the theory of clubs. In terms of the earlier discussion, Buchanan observed that joining a club constitutes an act of preference revelation which permits the attainment of efficiency." p175

"When this is considered further, it becomes apparent that it is not the number of club members that matters for congestion but how frequently the facilities of the club are used. Let v be the number of visits that each member makes to the club. An increase in the number of visits raises the utility of the member making those visits, but causes congestion through the total number of visits of all members. Letting the total number of visits be V = nv, the utility function is written U = U (x, G, v, V ) with the marginal utility to a visit, U v , positive and the marginal congestion effect, U V , negative. The cost function for providing the club is also modified to make it dependent upon the total number of visits,
nv. " p175

"However, there is a very important distinction between the cases of variable and fixed utilization. This analysis of variable utilization retained the assumption that there is a fixed charge for membership but no further charges for visits. Consequently, once someone has become a member of the club, the price for each additional visit is zero. In choosing visits, each member will only take account of the private cost of the increase in congestion and not the cost they impose on other members. Therefore, they will make an excessive number of visits to the club. In brief, the fixed charge does not impose the correct incentives on members to decentralize the efficient outcome. To implement the optimum defined, it is therefore necessary for a club charging a fixed fee to directly regulate the number of visits." p176

"if the efficient membership of each club is small relative to the total population, then the outcome for the economy will be that a very large number of clubs will form each with the correct number of members and each providing the efficient level of service. Hence efficiency will be attained for the economy as a whole. In this case, the efficiency of each individual club is reflected at the aggregate level." p179

"The second and more interesting case, from both a practical and an analytical perspective, arises when the optimal membership of each club is relatively large compared to the total population. In this case the population size can support only a limited number of optimally-sized clubs." p180

"Tiebout observed that pure public goods lead to market failure because of the difficulties connected with information transmission. Since the true valuation by a consumer of a public good cannot be observed, and since a pure public good is non-excludable, free-riding occurs and private provision is inefficient." p189

"... the Tiebout hypotheses has much the same foundations as the Two Theorems of Welfare Economics since both concern economies with no rigidities and large numbers of participants." p190

"Phrased more prosaically, consumers reveal their preferences by voting with their feet and this ensures the construction of optimal communities. This also shows why the analysis of the previous section failed to find efficiency. The existence of at most two localities violated the large-number assumption of the Tiebout hypothesis." p190

"To sum up, the Tiebout hypothesis provides support for allowing the market, by which is meant the free movement of consumers, to determine the provision of local public goods. By choosing communities, consumers reveal their tastes. They also abide to local tax law so free-riding is ruled out. Hence efficiency is achieved. Although apparently simple, there are a number of difficulties when the practical implementation of this hypothesis is considered. The population may not partition neatly into the communities envisaged and employment ties may bind consumers to localities whose local public good supply is not to their liking. Transactions costs in housing markets are significant and these will limit the freedom of movement that is key to the hypothesis. The hypothesis provides an interesting insight into the forces at work in the formation of communities but it does not guarantee efficiency." p191

Lev Lafayette

Chapter 6: Club goods and local public goods

Reading: Jean Hindriks and Gareth D. Myles (2004), Intermediate Public Economics

Further Reading

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Chapter 6: Club goods and local public goods

Reading: Jean Hindriks and Gareth D. Myles (2004), Intermediate Public Economics

Further Reading