Public Goods, Club Goods, and the Measurement of Crowding

Ž .Journal of Urban Economics 46, 69]79 1999Article ID juec.1998.2112, available online at http:rrwww.idealibrary.com on

Public Goods, Club Goods, and theMeasurement of Crowding*

Michael Reiter†

Department of Economics, Uni ersitat Pompeu Fabra, Ramon Trias Fargas 25-27,E-08005 Barcelona, Spain

E-mail: [email protected]

and

Alfons J. Weichenrieder

Department of Economics, Uni ersity of Munich, Schackstrasse 4,D-80539 Munich, Germany

E-mail: [email protected]

Received December 27, 1997; revised August 20, 1998

This paper shows that some frequently used measures of the degree of public-ness of publicly provided goods and club goods are seriously affected by metriza-tion problems. The paper proposes measures that do not depend on arbitrarymetrization conventions, and discusses the relationship of these measures to thequestion of private provision and optimal club size. Q 1999 Academic Press

Key Words: Publicness, crowding, quasi public goods, club goods

The theoretical benchmark case of a pure public good has received wideattention in the public economics literature. Numerous empirical studies,however, have found that public services provided at the community levelare subject to heavy crowding effects.1 A central element of all thesestudies is a crowding function of the form

z s Z g , n . 1Ž . Ž .Here g denotes the quantity of a publicly provided good, and n is thenumber of users. g may be measured either in physical units or as

*We are grateful to the editor and to two anonymous referees for very helpful suggestionson an earlier version of the paper.

†Author to whom correspondence should be addressed.1 w x w xThe seminal papers by Borcherding and Deacon 2 and Bergstrom and Goodman 1

w xtriggered a huge literature; for a recent survey, cf. Reiter and Weichenrieder 10 .

69

0094-1190r99 $30.00Copyright Q 1999 by Academic Press

All rights of reproduction in any form reserved.

REITER AND WEICHENRIEDER70

expenditures. For the issues raised in this paper, the distinction is irrele-vant. Furthermore, the two measures are equivalent under constant re-turns to scale in the production of the good. The variable z is to beunderstood as the usefulness of the provided good to the individualŽ w x.Bergstrom and Goodman 1, p. 282 or the amount of the good captured

Ž w x.by the individual Borcherding and Deacon 2 . It is often called the‘‘service level’’ derived from the provision of g. If g is a private good,z s grn. Conversely, if g is a purely public good in the sense of Samuel-son, z s g. More interesting and more problematic is the case of impure,also called rivalrous, public goods, where users cause some degree ofcongestion. Little thought has been spent on the exact meaning andmeasurement of z in this case, and the failure to do so has led toconsiderable confusion in the literature when results of studies are com-pared which implicitly use different methods to define z. Closely related tothe measurement of z is the question of how to define the degree ofpublicness of g.

The contribution of this paper is twofold: first it presents three essen-tially different methods to define and measure z, and proposes measuresof publicness of g that do not depend on the way in which z is measured.Equipped with a solid definition of publicness, it then discusses interestingcharacteristics of crowding functions and how these are related to thequestion of whether a good can be efficiently provided by private compet-ing clubs. In particular, it investigates the role of the concept of increasingmarginal congestion, which has attracted considerable attention in theliterature.

1. METRIZATION

This paper abstracts from excludability issues and concentrates oncrowding effects. Therefore, we need not distinguish between impurepublic goods and club goods, which are defined as excludable impure

Ž w x.public goods Cornes and Sandler 4, p. 347 . In addition, we assume thatthere is only one publicly provided good, g, and that private goods can beaggregated and measured in monetary units.

In this framework, the utility of household i can be written as

ui s V i x i , g , n , 2Ž .Ž .

where x i is household i’s consumption of the private good in money units.Ž .The literature does not work with the general formulation 2 , but imposes

Ž .implicitly the assumption that the marginal rate of substitution betweeni Ž .g and n is independent of x , so that 2 can be decomposed into

ui s U i x i , z 3Ž . Ž .

MEASUREMENT OF CROWDING 71

Ž .and the crowding function 1 . We follow this tradition and make theŽ .assumptions necessary to justify representation 3 .

Ž .One should note that 2 is not a ‘‘pure’’ utility function, but is condi-tional on the technology available in the public sector. For example, if g isthe input of policemen, the utility of individuals depends on the technical

Ž .ability of the police to produce safety. If z is some measure of safety, 1can then be understood as a public sector production function. For a given

Ž .state of technology, however, specification 2 is more general than theŽ . Ž .combination of 1 and 3 , since it requires no separability assumptions.

Ž . Ž .The utility specification 3 and 1 leaves open the question of how zshould be defined or measured. Only an ordering is imposed on z. The

Ž .reason is that for any positive monotone transformation f . , the specifica-tion

i i i y1u s U x , f z , 39Ž . Ž .

z s f Z g , n , 19Ž . Ž .2 Ž . Ž .is equivalent to 3 and 1 and only amounts to a remetrization of z. The

choice of a metric for z is a matter of convention, and in this sensearbitrary. It has nothing to do with the real economic structure, i.e., theobservable behavior of households.

For most theoretical purposes, for example the characterization ofPareto-efficient allocations, an exact definition of z is unnecessary, be-cause the introduction of z is a superfluous intermediate step. In empiricalapplications, however, where z is either measured or, if not really mea-sured, at least interpreted, it is of crucial importance to be aware of

Ž .the intrinsic metrization problems that are posed by the formulation 3Ž .and 1 .

There are at least three different metrizations that appear plausible:

1. A natural metric. If, for example, the good in question is highwayservices, we may choose z as the speed at which highway travel can flowŽ w x.Inman 9 . One should note that a natural metric is not unique. Forexample, one might use the log of speed rather than speed itself.

2. Money metric. Here we define z as the willingness to pay for theuse of the public good. If we assume that U is of the quasi-linear form

U s x q z s x q Z g , n , 4Ž . Ž .

where the private good x is measured in currency units, money metric isunique. Otherwise, the willingness-to-pay will depend on the total income

2Quasi-concavity of U is also preserved. Requiring concavity of U would impose restric-tions on the transformation f.


of a household, and we get different money metrics for different incomelevels.

3. The proportional metric. Given n, we take z as being propor-tional to g :

z s g ? j n 5Ž . Ž .Ž .for some function j n , decreasing in n. Using the normalization

j 1 s 1 6Ž . Ž .Ž .the proportional metric is unique. It is important to note that 5 is not just

a metrization, but plays the dual role of making a substantial assumptionplus choosing a metric. The substantial assumption is that, for given n, themarginal rate of substitution yZ rZ is proportional to g. Otherwise, an g

Ž .representation of the form 5 is not possible. But even if this assumptionŽ .is met, a metric different from 5 could still be chosen. Metrization is a

matter of convention.

For a simple illustration, assume

U i x i , g , n s x i q g 0.5ny0 .4 . 7Ž .Ž .Using the money metric, we get

z m m s g 0.5ny0 .4 , ui s x i q z m m . 8Ž .

With the proportional metric we get

prop y0.8 i i prop'z s gn , u s x q z 9Ž .

Generally, we cannot expect any two of the three metrizations to be equalŽ .up to a multiplicative constant . Since money metric is equivalent towillingness to pay, and the proportional metric refers to physical quanti-ties, the two can be proportional to each other only if the marginalwillingness to pay for the physical quantity g is constant. This is clearly avery special and economically implausible case. A natural metric of z is

Ž .expressed in units speed, crime rates, etc. of a different category thanŽeither willingness to pay or the physical quantity of g highway length,

.number of policemen, etc. . There is therefore no presumption that anatural metric should be proportional to a money metric or a proportionalmetric. A simple example may clarify the point: doubling the length orwidth of a highway will normally not exactly double the speed of cars.

2. MEASURING PUBLICNESS

Different metrizations have in fact been used in the literature. Most ofw xthe papers following Borcherding and Deacon 2 and Bergstrom and


w xGoodman 1 use the proportional metric, with the special assumption

j n s nya , a ) 0. 10Ž . Ž .w xBrueckner 3, p. 48 uses the reduction in expected fire losses, which is a

w xform of money metric. Craig 5, fn. 5 uses a natural metric: safety,3 w xmeasured as a constant minus the crime rate. Edwards 7 states four

properties that should be fulfilled by reasonable congestion functions.Ž .These requirements are fulfilled by and are actually weaker than the

proportional metric. Edwards then investigates five different congestionfunctions, four of which use the proportional metric. He notes that the

Ž .fifth one Generalized Congestion Function, GCF does not meet the fourrequirements. However, this function may well make sense if used together

w xwith a natural metric, as Inman 9 originally did.The measure of congestion most widely used in the literature is the

elasticity of z with respect to n, keeping g constant:

z nzh s ? . 11Ž .n n zgsg

The discussion of the last section has made it clear that this is a problem-atic measure as it depends on the metrization of z. Elasticities areinvariant only to multiplicative changes of variables, but the differentmetrics we have discussed are not multiples of one another. This can be

Ž . Ž .seen from the example of the last section: while Eqs. 8 and 9 representz Ž .the same utility function, the elasticity h is y0.4 in the case of 8 andn

Ž .y0.8 in the case of 9 . It is therefore misleading to compare elasticityestimates in studies that use different metrics, as is sometimes done in the

w xliterature. For example, Brueckner 3, p. 53 uses a money metric and findsa congestion elasticity of y0.24, which is much lower than the values of

Ž .about y1 usually found. Transforming Brueckner’s congestion function 8Ž .into a proportional metric, the congestion elasticity would be gr a q b

Ž .s y0.60 in Brueckner’s notation . This still indicates considerably lesscrowding than is usually found, but not so dramatically less as Brueckner’s

4 w xinterpretation suggests. Craig 5, p. 346f. uses a natural metric andcompares his congestion elasticity to those found in the studies using a

Ž w x. w xproportional metric similar in Craig and Heikkila 6 . Edwards 7 com-pares the results of the GCF function, which uses none of the above threemetrics, to results of four different functional functions, all of which usethe proportional metric.

3Note that even in this simple case, choosing different constants leads to differentmetrizations of z which are not proportional to one another.

4 In his Footnote 15, Brueckner mentions the potential problem of measuring fire protec-tion services, but seems not to recognize the nature of the metrization problem.


If comparisons of results across different papers take care of themetrization problem, researchers may of course choose different metrics inthe measurement of crowding, depending on the specific aims of theinvestigation. However, in the measurement of publicness additional con-siderations come into play. While congestion is a technical concept,publicness is an economic concept which is intimately related to thequestion of efficient private provision. We therefore argue that a measureof publicness should tell us something about whether private or publicprovision is more desirable, and such a measure must not crucially dependon the arbitrary metrization of z. Dependence on the metrization canindeed be avoided by measuring the increase of the quantity of the publicgood which is necessary to keep consumers’ utility constant when popula-tion increases. The relevant elasticity is

g ngh s ? . 12Ž .n n gzsz

Since z is kept constant, the elasticity h g is independent of how z isnŽ .measured and reflects only the shape of the indifference curves in g, n

space.Ž .For example, a private good can be defined in terms of 12 by the

g Žcondition h s 1 more precise and comprehensive definitions are givenn.below . We will show in Section 3 that this definition has the property that

an excludable good is private if it can be efficiently provided by competi-Ž . Ž .tive private clubs. Definitions based on 11 and 12 are equivalent only in

g w Ž . Ž .x zthe case of a proportional metric, since h s y Nj 9 N rj N s yh .n nIf applied in conjunction with a metric other than the proportional one,definitions based on h z are inappropriate, a fact that seems not to ben

w xrecognized in the literature. For example, Brueckner 3, p. 47 uses amoney metric and defines a private good as a good where h z s y1. Withnthis definition, it is generally not true that a private good can be efficientlyprovided by competing clubs.

We will discuss the relationship between measures of publicness and theefficient provision of goods more systematically in Section 3. First, how-ever, we provide a comprehensive list of definitions, following the insightthat measures of publicness should be independent of z.

Definitions of Publicness

1. A good is purely public if

u x , g , n s u x , g , 1 , ; g , n , x . 13Ž . Ž . Ž .

2. A good is purely private if

u x , g , n s u x , grn , 1 , ; g , n , x . 14Ž . Ž . Ž .


For the impure cases, we distinguish between average and marginalpublicness. The average concept refers to the comparison between thesituation where n individuals consume g units of the public good, and thesituation where one individual consumes grn units. This leads to thefollowing definitions.

Ž .3. At point g, n , a good is a¨eragely

¡overcongested if u x , g , n - u x , grn , 1 ,Ž . Ž .private if u x , g , n s u x , grn , 1 ,Ž . Ž .~impure public if u x , grn , 1 - u x , g , n - u x , g , 1 ,Ž . Ž . Ž . 15Ž .public if u x , g , n s u x , g , 1 ,Ž . Ž .¢camaraderie if u x , g , n ) u x , g , 1 .Ž . Ž .

The marginal concept is concerned with local changes in n and g andtherefore makes use of the elasticity h g.n

Ž .4. At point g, n , a good is marginally

overcongested if h g ) 1,¡ ngprivate if h s 1,n

g~impure public if 0 - h - 1, 16Ž .ngpublic if h s 0,ng¢camaraderie if h - 0.n

Ž .Note that a purely public private good is also averagely and marginallyŽ .public private .

Next we define quantitative measures of average and marginal public-ness. They are normalized to have the value 1 for public and 0 for privategoods. The publicness measures can be understood as 1 minus a conges-

Ž .tion measure. Marginal publicness at point g, n is naturally measured by

MP g , n s 1 y h g . 17Ž . Ž .n

There are certainly several possible definitions of a¨erage publicness atŽ .point g, n . We propose the formulation

G 1; g , n nŽ .AP g , n s 1 y 1 y ? , n ) 1, 18Ž . Ž .ž /g n y 1

Ž .where G n9; g, n is the level of g necessary to achieve the same utilityŽ .level as at g, n with n9 users, formally

i iU x , G n9; g , n , n9 s U x , g , n . 19Ž . Ž .Ž .


Ž .Definition 18 has the required property that, for a pure public good,Ž .AP s 1 because G 1; g, n s g, while for a purely private good AP s 0

Ž .because G 1; g, n s grn. Of course, the proposed measures are invariantŽ .with respect to transformations of the utility function 2 .

If we adopt a proportional metric, we can easily express the abovepublicness concepts in terms of g and z. This is trivial for the marginal

g z Ž .concepts because h s yh . The average publicness measure 18 is givenn nby

n z 1AP s ? y . 20Ž .

n y 1 g n y 1

w xThis is similar to Edward’s 7, Eq. 3.1 measure zrg y 1rn. Both measuresassume the value 0 for purely private goods. For purely public goods, ourmeasure is exactly 1, contrary to Edward’s, which is bounded from aboveby 1.

3. IMPORTANT PROPERTIES OFCROWDING FUNCTIONS

In this section we discuss several properties of crowding functions thathave figured prominently in the literature. We are mainly interested inwhether these properties are independent of the metrization of z, and howthese properties relate to the economic question of public or privateprovision of goods. In the following, we assume that the public good g is

w xperfectly divisible. Edwards 8 analyzes the interesting case where g is notperfectly divisible. Indivisibility does not change the logic of the metriza-tion arguments presented above, but it makes the decision whether publicor private provision is desirable more complicated.

Iso-elasticity

As mentioned above, most of the literature uses the proportional metricŽ . Ž .5 with the specification 10 . This gives

h g s a . 21Ž .n

For a s 1, we have a purely private good. Club or city size is thenirrelevant. For a ) 1, the optimal club size would be infinitesimally small,for a - 1, the optimal club size would be infinite.

U-shape

Goods for which publicness varies with population size and quantity ofthe good provided are more interesting. Then there is often an optimal

w Ž .xfinite club size for given z* or utility level U x, g*, n* . This is the case if


Ž .the average cost function assume g is measured as expenditures

G n; g*, n*Ž .AC n s 22Ž . Ž .

n

has an interior minimum, or more specifically, is U-shaped. The minimumis at a point where5

ghngh s 1 and ) 0. 23Ž .nŽ . ln n zsZ g*, n*

Once this size has been sufficiently exceeded, more people are best servedby replicating clubs, and the good is private in this sense. Note that theoptimal club size, and more generally the question of whether the goodcan be efficiently supplied by private clubs, can be determined by measuresthat are independent of any metric of z. This provides a justification of thepublicness measures proposed in Section 2. Other measures of publicnessare appropriate only if they are compatible with those. For example,

w xEdwards 8, p. 566ff. provides conditions for U-shaped average costs whichuse a measure similar to h z, but they are equivalent to ours since hisnanalysis is in the framework of a proportional metric.

Increasing Marginal Congestion

A property of crowding function that has received considerable atten-( ) Ž .tion in the literature is increasing absolute marginal congestion IMC ,

defined as

2Z g , nŽ .- 0. 24Ž .2 n

w x w xCraig 5, p. 338f. and Edwards 7, p. 84 indicate that IMC is a morenatural assumption than decreasing marginal congestion. Note that thisproperty again depends crucially on the metrization of z. Most studies

Ždiscuss IMC in combination with the proportional metric in which case it.simply means j 0 - 0 . The literature seems to have overlooked that IMC

together with the proportional metric has implications that are extremelyimplausible from an economic point of view. Assume there is a n0 such

Ž 0. Žthat j 9 n - 0 this simply means that the good is not a camaraderie.good for all n . IMC implies, for a given g, that the service level z s 0 is

5 Ž .To see this most easily, note that minimizing 22 w.r.t. n is equivalent to minimizingŽ . Ž .ln g y ln n for given z w.r.t. ln n. The first order condition then is ln gr ln n y 1 s

g w 2 Ž .2 x Ž g .h y 1 s 0, and the second order condition is ln gr ln n s h r ln n ) 0.n n


reached at a finite population level n*, where

j n0Ž .0n* - n y . 25Ž .0j 9 nŽ .

z s 0 is the service level which is reached if g s 0, i.e., if the good is notprovided at all. This situation is called ‘‘gridlock’’. If z s 0 for n s n*,however, the proportional metric obviously implies that z is zero for allvalues of g. In other words, gridlock occurs at n* no matter how much of gis provided! Given the near inconsistency of IMC and proportional metric,

w x w xit is comforting that Edwards 7 is able to reject IMC. Craig 5 finds IMCin an empirical application using a natural metric, where it may well makesense.

4. SUMMARY

In the last three decades, great efforts have been made to measure thepublicness of publicly provided goods. The results provide potentiallyimportant information for privatization decisions. This paper has arguedthat some of the measures used to characterize publicness are flawed sincethey depend on arbitrary metrizations. The paper has described measuresthat are not affected by this problem, and has briefly discussed how theyrelate to the question of private provision and optimal club size. Finally,the paper argued that increasing marginal congestion is not an essentialcharacteristic of a local public good or a club good.

REFERENCES

1. T. C. Bergstrom and R. P. Goodman, Private demand for public goods, AmericanŽ .Economic Re¨iew, 63, 280]296 1973 .

2. T. E. Borcherding and R. T. Deacon, The demand for the services of non-federalŽ .governments, American Economic Re¨iew, 62, 891]901 1972 .

3. J. K. Brueckner, Congested public goods: The case of fire protection, Journal of PublicŽ .Economics, 15, 45]58 1981 .

4. R. Cornes and T. Sandler, The Theory of Externalities, Public Goods and Club Goods,2nd Ed., Cambridge University Press, 1996.

5. S. G. Craig, The impact of congestion on local public good production, Journal of PublicŽ .Economics, 32, 331]353 1987 .

6. S. G. Craig and E. J. Heikkila, Urban safety in Vancouver: Allocation and production of aŽ .congestible public good, Canadian Journal of Economics, 22, 867]884 1989 .

7. J. H. Edwards, Congestion function specification and the ‘publicness’ of local publicŽ .goods, Journal of Urban Economics, 27, 80]96 1990 .


8. J. H. Edwards, Indivisibility and preference for collective provision, Regional Science andŽ .Urban Economics, 22, 559]577 1992 .

9. R. P. Inman, A generalized congestion function for highway travel, Journal of UrbanŽ .Economics, 5, 21]34 1978 .

10. M. Reiter and A. Weichenrieder, Are public goods public? A critical survey of theŽ .demand estimates for local public services, Finanzarchi , 54, 374]408 1997 .

Public Goods, Club Goods, and the Measurement of Crowding

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