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Congestion function specification and the “publicness” of local public goods
JOURNAL
OF URBAN
ECONOMICS
27,80-96 (1990)
Congestion Function Specification and the “Publicness” of Local Public Goods* JOHN H. Y. EDWARDS Department of Economics, Tulane University, New Orleans, Louisiana 70118
Received November 25,1986; revised August 26,1987 This paper is about the measurement of congestion within the median voter demand framework. The model of congestion traditionally used in the literature and four others are compared and tested in a median voter framework. The results show that a more flexible functional form which imposes fewer a priori restrictions on the data does at least as good a job of explaining variations in local government expenditures. Furthermore, it is shown that measured “publicness” of local public goods depends entirely on the a priori restrictions imposed on the congestion fUnCtiOn
Q 1990 Academic
Press. Inc.
I. ON THE NON-PUBLIC NATURE OF LOCAL GOODS The central role of congestion in public finance theory is clear in a number of works. Samuelson [24] posited a two-good world and argued that the uncongestable good should be provided by the government. At the same time he raised the preference revelation conundrum and the specter of inefficiency from which Tiebout [27] subsequently sought to rescue the local public sector. By imposing a spatial range on local goods, Tiebout and later Buchanan [7] reintroduce congestion and thus eliminate the non-convexity associated with Samuelson’s pure public goods. Tiebout goods and club goods lie between Samuelson’s polar, or pure, cases. Empirical incarnations of the median voter model have also incorporated population size and the notion of a continuum between private and public goods. This is accomplished by means of a congestion function of the form Z* = ZN-7, where Z is the public good, Z* is the level or quality perceived by the median voter, and N is population size. A null value of y indicates Z is a pure public good, while y = 1 implies Z is like a private good in the sense that the median voter, at least, receives his share l/N. Though their work is seldom cited in this context, the economic rationale behind congestion functions for local public goods is clearly laid out in *I am indebted to Wally Oates for invaluable comments and encouragement in the early stages of this research. I also thank two anonymous referees and Nahid Aslanbegui, Melville McMillau, Bill Oakland, John Pritchet, and Ingmar Prucha for their help and suggestions. Computer time was graciously provided by the University of Maryland and Tulane University computer centers. 80 0094-1190/90 $3.00 Copyright All rights
0 1990 by Academic Press. Inc. of reproduction in any form reserved.
THE CONGESTION OF LOCAL PUBLIC GOODS
81
Bradford, Malt, and Oates [5]. They distinguish between D-goods directly produced by the government (police cars) and C-goods of direct concern to the citizenry (the level of safety). The level of C-good perceived depends jointly on the level of D-good provided by the government and on a vector of socio-economic variables such as income level, population size, and education.’ Median voter demand studies typically find that y > 1. Because their congestion function is admittedly arbitrary, Borcherding and Deacon [4] caution against drawing normative conclusions from their estimates. Yet one’s curiosity is piqued by results which seem to indicate that goods provided by local governments lack the publicness property we would expect. The Bergstrom and Goodman [2] study is perhaps the most widely known in this tradition, because of the size of its data base and their rigorous definition of the conditions and assumptions necessary for the validity of a median voter demand model. Their findings lead them to reject the hypothesis that y < 1, concluding that collective provision of goods and services such as general administration, police protection, and parks apparently are “largely a matter of indifference from an efficiency viewpoint” [2, p. 2941. This “privateness” result has found widespread acceptance in the literature. Two widely-cited texts, Mueller [19, pp. 107-1111 and Atkinson and Stiglitz [l, p. 3231,report the result. Niskanen [20] and Borcherding, Bush, and Spann [3] point to it as evidence of voter exploitation. Finally there are strains of the literature (McMillan, Wilson, and Arthur [18]; Epple, Filimon, and Romer [12]) in which theoretical analysis is founded on this empirical result. In stark contrast to the median voter demand results, Brueckner [6] found that fire protection services possessthe congestion properties of a pure public good. It would appear that the non-publicness result needs to be examined more closely. Indeed, a number of recent works by McMillan et al. [18], Edwards [lo, 111,McKinney [16], Craig [8], and Oates [21] have already begun this examination. The contribution of this paper is to examine the sensitivity of the non-publicness result to alternative specifications of the congestion function. This is done within the traditional median voter demand framework to facilitate comparison with earlier results. The next section will lay the foundation for an empirical study of the effect of congestion function specification on the measurement of publicness. In the third section I develop five congestion function specifications that have been tailored to reflect specific properties (such as increasing ‘As will be seen shortly, the procedure in the median voter demand literature has been to control for so&o-economic variables at the time of estimation.
a2
JOHN H. Y. EDWARDS
marginal congestion and camaraderie) that are described in the literature. These models are estimated for three publicly provided goods in Section IV. The comparison of congestion function specifications is based on these 15 estimated equations. Section V provides a summary and the conclusions. II. MEDIAN
VOTER DEMAND AND CONGESTION
Assume there are two goods: one provided by the local government, 2, one procured privately, X. In the median voter framework as developed by Bergstrom and Goodman [2], we might say that the government’s problem is to maximize median-voter utility, subject to the voter’s budget constraint, i.e., maxU=
U(X, Z, N),
s.t. x + tz = Y,
(0 J>
where N is persons sharing Z, t is median voter tax price, and Y is income. Rather than having N enter the utility function directly, we may introduce a “technology of consumption” in the senseof Sandmo [26]. Assuming private goods are not congestable allows us to rewrite the utility function as u = u(x,
z*),
(0 4
where Z* is the perceived amount of public good, and Z* =f(Z,
N).
(0.3)
The “congestion function” described by (0.3) may be seen as a special case of the transformation of a D-good into the C-good perceived by the median voter, as a function of the number of persons sharing it.* From (0.3) we obtain
Z = f-‘(Z*,
N).
(04
The local government’s problem may be reformulated as maxL=
U(X,Z*)+X[Y-X-tf-'(Z*,N)],
where A is a Lagrange multiplier. If population is not a choice variable of *“Congestion” is used here in the broadest sense.It should be understood to include all the effects of population size on the C-good. If the latter were police protection, then congestion, as used here, would encompassall such effects of population size as higher crime rates, longer police response times, and reduced willingness of bystanders to intercede at the scene of a crime. The formulation in (0.3) is analogous to that used in the literature on clubs and optimal community size.
83
THE CONGESTION OF LOCAL PUBLIC GOODS
the government, utility maximization leads to 8L ax=Ux-X=O
aL
-LYz* = up - xtf;: CYL ax=
= 0
Y-x-l“‘=o.
Certain specifications of the utility and congestion function allow us to solve for Z*,
z* = h(N, t, Y).
(0.5)
(0.4) and (0.5) provide the individual’s demand function for Z, z =f-l(h,
N) = H(N,
t, Y)
(04
which will provide population, price, and income elasticities of demand for the public good. The research problem becomes one of specifying (0.1) and (0.3) in a manner flexible enough to provide a solution as in (0.6) without sacrificing the ability to describe behavior which economic theory leads us to expect. The main issue to be resolved here is the form of (0.3) the congestion function. It will be helpful at this point to set four properties that a reasonable specification of (0.3) should have; this will restrict our choice of congestion models to those that make economic sense. Property 1. When there is a single user, he perceives what there is-as though Z were purchased privately: Z*( Z, 1) = Z. Property 2.
Z* is predicated on the existence of Z: Z*(O, N) = 0.
Property 3. If the good is congestable, then there is some value of N > N* such that 8 Z*/aN < 0. Property 4.
Z* must be increasing in Z; that is aZ*/aZ
> 0.
III. FIVE MODELS OF CONGESTION This section describes five congestion models. The first four are based on descriptions of congestion found in the public goods literature, while the last is a flexible functional form that imposes no a priori restrictions on the shape of the congestion function and yet can pick up properties described by the other models if these are present in the data. Four of the models satisfy the four congestion function properties described above. While the
84
JOHN H. Y. EDWARDS
GCF model does not, it was included nonetheless because Inman [14] has shown it to be the general form from which many of the highway congestion models can be derived. (1) Decreasing Marginal
Congestion
The (DMC) model of congestion developed by [2] and [4] is the one that has been used in median voter studies of demand for the past decade. The specification corresponding to (0.3) is Z* = ZN-“.
(1.1)
For y > 0 we have i3Z*/aN -z 0 and a2Z*/aN2 > 0; congestion decreases at the margin. (2) Increasing
Marginal
Congestion
The IMC model is designed to satisfy the more usual assumption that congestion accelerateswith the intensity of use (Litvack and Oates [15] and Walters [30]). Eventually the good has no value, or even negative value to consumers, as when highway trafhc comes to a standstill. From (1.2),3 Z* = Z(2 - exp[y(N - I)]}, we obtain -y2Zexp[y(N
az*/aN =
-yZexp[y(N
- l)] < 0 and
(1.2)
a2z*/aiv2 =
- l)] < 0.
(3) Camaraderie Function
Economics of clubs literature suggeststhat there may be a range where sharing congestable goods actually increases benefits to each user until these are eroded by crowding. The increased benefit has been called a “camaraderie effect” [25, p, 14821. It can be modeled with a modified Gamma function as Z* = ZNYexp[a(l
- N)]
$!$ = Z*($ - a).
(1.3)
(1.4)
Marginal congestion is shown in (1.4). For y, a > 0, there is benefit from sharing until N = y/a. For y, a < 0 the model conveniently reverts to decreasing marginal congestion with the traditional DMC model (a = 0) as a special case. 3The apparently curious appearance of 2 in (1.2) is needed to satisfy Property 1.
THE CONGESTION
OF LOCAL
PUBLIC
85
GOODS
(4) Generalized Congestion Function
The GCF developed by Inman [14] is specified in terms of 2 as z =
N(7)
+ pz*w
O-5)
where N(Y) = (NY - 1)/y for y # 0 and N(Y) = In(N) for y = 0, while z*(w) = [(z* - ~7~)~:- 11/w,, = ln(Z* - w2),
for wi Z 0
for wi = 0.
0 *6)
Thus, we have for y, wi # 0 z = (NY - 1)/y + p[(z*
- w*)W1- 11/w,
z* = [l + w,(yz - NY - l)/py]l’wl
+ w,.
(1.7) (1.8)
My attempts to estimate the demand equation derived from (1.8) were unsuccessful. However, I was able to estimate a restricted version which, -without affecting marginal congestion-assumed wz = 0. Dropping the subscript on wi, we obtain z = (NY - 1)/y + p(z*w - 1)/w,
(1.7a)
and z* = [l + w(yz - NY - l)/yp]““. (5) Exponential
(1.8a)
Function Model
Given the diversity of functional forms that can be drawn from the literature, the inclusion of a flexible functional form seemedappealing. The simple exponential polynomial function of (1.9), Z* = Zexp[aN + bN2 + dN3 - (a + b + d)],
0.9)
has two desirable properties. Most important is that it imposes no a priori restrictions on the shape of the congestion function. The signs of the first and second partial derivatives of (1.9) depend on the sign and relative magnitude of the population parameters. Second, the demand equation derived from it is log-linear and can be estimated using OLS. IV. ESTIMATION The DMC demand equation shown in Table 1 was derived from (1.1) in the following manner. Solving (1.1) for Z, the equivalent of (0.4) is Z = Z*N7.
(2-l)
86
JOHN H. Y. EDWARDS TABLE I Price Per Unit of Z*, by Model DMC fNY IMC r(2 - expy(N - 1)1-t Camaraderie rNmYexp[a( N - l)] @z*(w-‘) GCF Exponential texp[(a + b + d) - aN - bN2 - dN3] Demand Functions for Z Estimated, by Model
- exp y(N - l)]-(‘+a)fiS;~
IMC
ctsY[2
Camaraderie
cts~N-Y(‘+a)exp[a(l
GCF*
K, + K,t’*V*
Exponential
ctarexp(l
*K,= -P/w;
B
+ (NY - 1)/y + &L,S, + ih,SF
+ @[(a + b + d) - ai - bN2 i dN3]
c
K2=;[l+S~1-w~l
+ S)(N - l)]fi,Sp
=; and $I = w/6(2 - w).
The budget constraint (recall “t ” is tax price) thus becomes Y = x + tZ*NY,
(2.2)
so that price per unit of Z* is
P,* = NY.
(24
Assuming a multiplicative demand function, demand for Z* is
By substitution in (2.1), demand for Z is Z, = ,t8y~NY(l+8)
(2.6)
where the Si are vectors of socio-economic variables described below.4
4Hulten [13] has noted that using the multiplicative form of (2.5) and (2.6) is tantamount to assuming that the technology of consumption is Hicks-neutral with respect to the S,.
THE CONGESTION OF LOCAL PUBLIC GOODS
87
The other four specifications in Table 1 were derived in the same way from the respective congestion functions. The DMC, Camaraderie, and Exponential Function models were estimated using ordinary least squares. The GCF and IMC models yield demand equations that are not log-linear and were estimated using the non-linear maximum likelihood procedure developed by K. B. White [31]. The dependent variable is in logarithmic form in all but the GCF model.5 Each of the demand equations was estimated for general administrative expenditures (GE), parks and recreation (PR), and police protection (PP). The five socio-economic variables (Si) included as control variables are % owner-occupied housing, % non-white, employment/residential ratio, % population change since the last census, and 4%over 65 years of age. Data used are for 78 New York State municipalities of between 10,000 and 150,000 inhabitants. Median income (Y) and median house values (V) as well as the demographic data (the Si) come from [29]. PR and PP expenditures were drawn from [28], while GE expenditures were obtained from [22]. Property tax rates (T) and market price equilization rates (E) came from [23]. Median voter tax price is computed as t = (V X E) X T/R, where R is total tax revenue. Observations were omitted when some of the needed data were not available. Full results for four of the models are not presented because of space limitations. Table 2 compares my results for the DMC model, based mostly on 1977 data, to the original Bergstrom and Goodman results for New York with 1962 data. The price, income, and population elasticities are of the expected signs. With the exception of the employment/residential ratio, parameter estimates for the control variables from the two data sets are of the same sign when they are both sign&ant. Significant economic parameters have a similar range of values in both sets of data, though in the latter period the range is somewhat narrower and population elasticity estimates are slightly higher. The congestion parameter y is in all caseslower than the 1962 estimates, though significantly greater than one at the 95% level of confidence. This implies non-publicness and “over-crowding” of goods examined in all six cases. Table 3 shows the arithmetic means of elasticity estimates for the 15 equations estimated. Price and income elasticity estimates are of the expected sign and for the most part reassuringly similar. The estimated congestion functions for police protection are shown in Figs. (l)-(5) as the percentage of Z, perceived by the median voter at each ‘1 believe this to be in keeping with the spirit of the original Inman [14] formulation of the GCF. Implicit in (1.5) is a test of the logarithmic form, since lim,,,(NY - 1)/y = log N. Further statistical discrimination is provided in the non-nested test developed by MacKinnon [17] that is used in this paper.
88
JOHN H. Y. EDWARDS TABLE 2 Bergstrom and Goodman Model Estimates Parks and recreation
Police protection
General administration expenditures
Parameters
1962
1977
1962
1977
1962
1977
Tax share
-0.81 (1.53) 1.74 (2.71) 0.69 (1.19) 3.63 1.42 (2.21) -3.48 (2.02) 14.27 (2.53) 1.66 (0.60) 0.35 (1.84) - 8.03 0.74
- 0.23 (2.78) 1.25 (3.93) 1.05 (7.97) 1.36 - 1.05 (1.33) - 1.39 (3.53) 0.33 (1.19) - 0.07 (1.11) - 1.43 (1.99) - 3.83 0.75
-0.31 (0.48) 1.78 (2.31) 1.02 (1.45) 1.47 -2.73 (3.54) - 1.16 (0.56) 10.38 (1.53) 2.32 (0.70) -0.14 (0.61) - 8.58 0.69
-0.31 (1.64) 1.08 (1.39) 1.08 (3.45) 1.56 0.06 (0.W -1.85 (1.96) 1.44 (2.21) 0.01 (0.08) - 3.26 (1.89) - 10.07 0.49
-0.50 (3.57) 1.03 (6.05) 0.75 (5.00) 1.50 - 0.03 (0.17) - 1.68 (3.73) 3.70 (2.52) - 0.19 (0.26) 0.10
- 0.12 (1.94) 0.91 (3.82) 1.10 (11.55) 1.16 -0.13 (0.24) - 0.69 (2.42) 0.83 (4.14) 0.05 (1.16) - 1.71 (3.20) -7.32 0.86
Income Population Crowding parameter Population change Owner occup. % over 65 % nonwhite Employee/resident ratio Intercept R2
WV - 1.68 0.94
Note. Absolute value of “t” statistics in parentheses,variables logged.
value of N.6 In the first four figures, Z,, can be any arbitrary amount of expenditures, since the congestion functions specify log-separable roles for Z and N. The last graph represents the GCF model where congestion varies not only with population size, but also with the level of expenditures. The IMC function explicitly restricts the rate of congestion and hence the shape of the congestion function. Estimates based on the IMC show congestion having a small effect on the amount of public good provided in the towns at the lower limit of the sample. In a town of 10,000 persons, congestion would reduce perceived services by about 7%, falling off to zero when N = 106,638. While the DMC model also restricts the rate of congestion, the Camaraderie, Exponential, and GCF models do not. There are thus nine instances of free-form estimation here, namely the last three models estimated for three goods. The general shape in all nine caseswas essentially the same, providing strong evidence in support of the traditional median voter de 61t is important to note, however, that the vertical scale varies from figure to figure. The intention here is to examine the forms of the various congestion functions.
89
THE CONGESTION OF LOCAL PUBLIC GOODS TABLE 3 Representative Values of Elasticities by Model Expenditure category Model DMC IMC Camaraderie Exponential GCF* DMC IMC Camaraderie Exponential GCF* DMC IMC* Camaraderie* Exponential* GCF*
PR Price elasticities - 0.23 - 0.43 - 0.23 - 0.22 -0.31 Income elasticities 1.25 1.13 1.25 1.27 0.81 Population elasticities 1.05 0.21 1.08 1.13 1.23
PP
GE
- 0.32 - 0.65 - 0.32 - 0.32 -0.51
-0.12 -0.57 -0.12 -0.12 -0.12
1.08 1.34 1.08 1.09 2.51
0.91 2.34 0.91 0.92 0.85
1.08 0.16 1.10 1.12 3.32
1.10 0.62 1.08 1.34 1.38
*Arithmetic means of estimated values.
mand assumption that congestion costs for local public goods decrease at the margin. If we write the utility function as U( X, 2, R), where R = l/N, this result can be interpreted to mean that (given X and Z) R is subject to decreasing marginal utility. The Measurement of Publicness Traditionally, median voter demand studies have reached their “nonpublic” conclusion by relying on the DMC model where the test for privateness from (1.1) involves the null hypothesis Ha: y = 1. The other four models used in this paper imply a non-constant rate of congestion. Parametric tests of publicness are therefore not possible; we must develop a more general one which preserves the spirit of the traditional criterion. In the DMC model, y 2 1 is said to imply privateness because the median voter receives at most “his aliquot share.” From (1.1) we have Z* = Z/N. Of course, not rejecting the null hypothesis y < 1 implies public good characteristics given that for all observed values Zj and Nj,
where Fj is the estimated value of Z* at observation “j.”
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JOHN H. Y. EDWARDS
Camaraderie
GCF (Z. = $512,000)
FIGS. 1-5. Police protection: Estimated congestion functions (N in thousands, Z* as percentage of Z,).
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THE CONGESTION OF LOCAL PUBLIC GOODS
As the basis for an alternative test of publicness, consider the function Gij = 4,/Z,
- l/Nj,
(3.1)
where Fij is the estimated value of congestion function “i ” at observation “j.” The functions G, have an upper bound of unity for pure public goods. Also, 0 < Gi < 1 implies the publicness of Z in the sense that each person receives more than l/N. Finally, G, I 0 implies the privateness of Z, as each person receives at most his share of the publicly provided good. (3.1) thus preserves the spirit of the parametric test traditionally used for the DMC model, but can be applied to other functional forms. Of all the models examined, only the DMC and the closely related Camaraderie models allow us to state unequivocally that none of the three goods are public goods, since the estimated value of Gi in Table 4 is negative for all of the New York State municipalities sampled. At the opposite extreme, the Exponential function strongly indicates publicness. For the other two models the degree of publicness varies across goods. The IMC function indicates publicness of PR and of PP, but implies that congestion costs accelerate to the point where in the largest 23% of the municipalities, GE appears to be overly congested. Finally, the GCF model shows police and 97% of observations for parks as private and yet that in only 3% of the municipalities are administrative services like private goods. TABLE 4 Estimated Publicness (Representative Values of G, = E;,/Z - l/N) Minimum
DMC Camaraderie IMC Exponential GCF
- 0.0001 - o.ooo1 0.063 0.017 - 0.002
loo 100 0 0 97
DMC Camaraderie IMC Exponential GCF
-
100 100 0 0 100
DMC Camaraderie IMC Exponential GCF
-
Mean
Maximum
S for which GsO
Model
Parks and recreation - O.OOOOOl - o.oooo5 - 0.00005 - O.OOOOOl 0.80 0.932 0.097 0.231 - 0.009 0.0001 Police protection - 0.00001 0.0001 - 0.00005 0.0001 0.00005 - 0.00001 0.023 0.80 0.93 0.004 0.047 0.13 0.009 - 0.00024 - 0.004 General administrative expenditures 0.00009 - O.cWOl - 0.00005 0.00009 - 0.00001 - 0.00005 8.37 -0.06 0.74 0.03 0.17 0.35 0.003 0.0004 0.003
100 100 23 0 3
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JOHN
H. Y. EDWARDS
Above all, Figs. (l)-(5) and the results in Table 4 should serve as a warning. Within the median voter demand framework descriptions of congestion properties and-more important for policy prescription-conclusions on the publicness of local public goods are highly sensitive to the a priori congestion assumptions which underlie demand models. There are two possible criteria for choosing congestion models. The first -conformance to economic theory-has already been used in the preliminary selection of models. Given the conflicting nature of this theoretical guidance, my preference is for flexible functional forms-like the exponential function-which let the data speak for themselves.The second criterion that can be used is the statistical performance of conflicting models in explaining variations in municipal expenditures on local public goods. Statistical Comparison The DMC model of (1.1) can be seen as a special (nested) case of (1.3) where the coefficient “a ” is assumed equal to zero. Using the standard t-test, the null hypothesis H,: a = 0 cannot be rejected for any of the three goods.7 This means that we cannot reject the hypothesis that “Camaraderie” adds no explanatory power to the DMC over the sample range. In order to test the other models against one another, I have used the “J-test,” a method of non-nested hypothesis testing developed in Davidson and MacKinnon [9] and MacKinnon [17]. The procedure is called “artificial embedding” and, as the name suggests, consists of creating an artificial grand model as the convex combination of competing models. Predicted values from alternative specifications “j” are added as explanatory variables to the model “i ” being tested. For the parameters thus created, the joint test HOi: Xjj = 0 (for i # j) is the null hypothesis test for model “i.” The likelihood ratio test, based on the statistic -2log L, where L is the ratio of restricted to unrestricted likelihood function values, is distributed as x2 with degrees of freedom equal to the number of restrictions in each Z& (in our case, three). A clear choice between models occurs only if three null hypotheses of no significant enhancement are rejected and one Ho: is not. In the present context, this could be the case in four of 16 possible outcomes. To reduce the probability of type 2 error, a 95% confidence interval was chosen. Table 5 presents the results of tests where three Xjj are held equal to zero, for a critical x2 value of 7.81. We reject the hypothesis of no ‘The absolute critical value at the 95% level of significance with 75 degrees of freedom is 2.00. The absolute values of the calculated r-statistics were Parks and Recreation: t = 0.99 Police Protection: r = 1.13 General Administrative Expenditures: t = 0.30
93
THE CONGESTION OF LOCAL PUBLIC GOODS TABLE 5 Non-nested Hypothesis Tests for Four Congestion Models Four models Critical x2 values Values of - 2 log L for
7.81
Two models 3.84
Parks and recreation DMC IMC Exponential GCF
4.46
13.86 3.89 52.59
4.00
1.76 -
Police protection DMC IMC Exponential GCF DMC IMC Exponential GCF
2.58
142.90 5.68 9.68
General administrative expenditures 1.00 126.86
1.00
2.26 0.74
-
0.32
0.00
10.00
-
significant enhancement in the IMC and GCF cases. It is also true, however, that we cannot reject the equivalent hypothesis for the DMC and the Exponential models. They appear to outperform the other models for all three goods, PR, PP, and GE. The statistical test results are consistent with the observation that elasticity estimates from the two models in Table 3 are altogether quite similar. Distinguishing between these two models is vital for, although both appear to do a good job of explaining inter-municipal variations in expenditures, Table 4 shows they have opposite implications for publicness. In repeating the non-nested tests with only the DMC and Exponential models, our null-hypotheses restrict one parameter, for a critical x2 value of 3.84. Expenditures on PR are clearly best explained by the Exponential model (see Table 5) whose predicted values significantly add to the performance of the DMC model, while the converse is not true. Unfortunately, the two-model test sheds no more light on the other two goods. For PP and GE alike, there is no statistically significant difference between the models. The Exponential function model explains inter-municipal variations in public expenditures at least as well as the DMC model traditionally used in the literature. Since the theoretical guidelines for restricting the shape of the congestion function are not clear, this statistical comparison again indicates that the Exponential function model is probably preferable for its more general form. It is therefore by no means certain that local governments are providing goods that behave like private goods.
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JOHN
H. Y. EDWARDS
V. CONCLUSIONS AND FINAL COMMENTS This paper defined the congestion function in terms of the Bradford, Malt, and Oates distinction between C-goods and D-goods, thus developing the theoretical basis for including population and other socio-economic variables in studies of demand for local public goods. Four properties that should be satisfied by reasonable specifications were then defined and five models of congestion developed and incorporated in median voter demand equations. First examined was the DMC model, which has become entrenched in the median voter demand literature. Three others were drawn from descriptions of congestion provided elsewhere in the literature and the fifth is a flexible functional form which imposes no a priori restrictions on marginal congestion. Four main results emerged from this experimentation. (1) Variation of functional forms was found to have a sizable impact on estimates of population elasticity of demand, a parameter of paramount importance to the local public goods literature. (2) Unexpectedly, estimation of free-form congestion functions provided substantial evidence supporting the traditional assumption that congestion of local goods decreasesat the margin. This may be interpreted as evidence of decreasing marginal utility for congestion alleviation. (3) The results in Table 4 imply that the degree of publicness is not constant, but varies indirectly with population size. This result was foreseen by McMillan et al. [18]. There may be discontinuities, they suggest, in the provision of “optimally sized facilities and service areas” so that “empirical investigation should reveal a high degree of publicness.. . for services in communities which are small enough that service units need not be replicated” (p. 598).8An alternative theory was recently put forth in Oates [21] arguing that the privateness result in the median voter demand literature may be due to what he calls the “zoo-effect.” Larger communities provide a greater diversity of goods so we are really not observing non-publicness but an aggregation problem that results in an upward bias in the measurement of congestion. The observed inverse relation between population size and the degree of publicness may be interpreted as supporting both hypotheses. The relation to the theory of McMillan et al. is obvious. The Zoo-effect bias would be present in all five models considered in this paper, but the severity of the bias should be less in smaller municipalities-and the measured degree of publicness should be correspondingly larger.
‘Note, however, that as originally formulated the DMC model does not allow the congestion parameter “y” of (1.1) to vary across municipalities (see Edwards [ll]).
THE CONGESTION OF LOCAL PUBLIC GOODS
95
(4) My most important result concerns the public nature of local goods. Economic theories of shared goods, such as the Tiebout and Buchanan models, depend on the notion of a trade-off between cost savings and congestion, a trade-off which implies the existence of an optimal group size. In relying exclusively on what now appears to have been an ad hoc specification of congestion, the median voter literature had concluded that the services of municipalities have private good, not public good, characteristics. The DMC specification was a convenient device a decade ago, but until now, it had not been tested against other models. The DMC function used in these early median voter demand studies has essentially the right shape and-in two out of three goods examined here-did as good a job of explaining inter-municipal variations in expenditures as the more flexible Exponential function model. However, imposing constant elasticity of congestion apparently results in an upward bias in measuring the level of congestion. When the Exponential function is used we find that services provided by local governments have the public good characteristics that the theory of shared goods leads us to expect. REFERENCES 1. A. B. Atkinson and J. E. Stiglitz, “Lectures on Public Economics,” McGraw-Hill, New York (1980). 2. T. C. Bergstrom and R. P. Goodman, Private demands for public goods, Amer. Econ. Rev. 63, 280-296 (1973). 3. T. E. Borcherding, W. C. Bush, and R. M. Spamr, The Effects on public spending of the divisibility of public outputs in consumption, bureaucratic power, and the size of the tax-sharing group, in “Budgets and Bureaucrats” (T. E. Borcherding, Ed.), pp. 211-228, Duke Univ., Durham, NC (1977). 4. T. E. Borcherding and R. T. Deacon, The demand for the services of non-federal governments, Amer. Econ. Reu. 62, 842-853 (1972). 5. D. F. Bradford, R. A. Malt, and W. E. Oates, The rising cost of local public services, Nut. Tax J. 22, 185-202 (1969). 6. J. K. Brueckner, Congested public goods: The case of fire protection, J. Public Econ. 15, 45-58 (1981). 7. J. M. Buchanan, An economic theory of clubs, Economicu 32, 1-14 (1965). 8. Steven G. Craig, The impact of congestion on local public good production, J. Public Econ. 32, 331-353 (1987). 9. R. Davidson and J. MacKinnon, Several tests for model specification in the presence of alternative hypotheses, Econometrica 49, 781-793 (1981). 10. John H. Y. Edwards, “Congestion Function Specification and the Publicness of Goods Provided by Governments,” Dissertation, University of Maryland (1985). 11. John H. Y. Edwards, A note on the publicness of local goods: Evidence from New York state municipalities, Cunud. J. Econ. 19, 568-573 (1986). 12. D. Epple, R. Filimon, and T. Romer, Housing, voting, and moving, in “Research in Urban Economics” (J. Henderson, Ed.), Vol. 3 J.A.I., London (1983). 13. C. R. Hulten, Productivity change in state and local governments, Reu. Econ. Statist. 66, 256-266 (1984). 14. R. P. Inman, A generalized congestion function for highway travel, .I. Urban Econ. 5, 21-34 (1978).
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JOHN H. Y. EDWARDS
15. James M. Litvack and W. E. Oates, Group size and the output of public goods, Public Finance 25, 41-58 (1970). 16. S. McKinney, “The membership margin and the median voter,” mimeo (1985). 17. J. MacKinnon, Model specification tests against non-nested alternatives, Econometric Reu. 2, 85-100 (1983). 18. M. L. McMiBan, W. Wilson, and L. M. Arthur, The pubhcness of local public goods: Evidence from Ontario municipalities, Cunad. J. Econ. 14, 596-608 (1981). 19. D. C. Mueller, “Public Choice,” Cambridge Univ. Press, London (1979). 20. W. A. Niskanen, Jr., Bureaucrats and politicians, J. Law Econ. l&617-643 (1975). 21. W. E.Oates, On the measurement of congestion in the provision of local public goods, J. Urban Econ. 24, 85-94 (1988). 22. Office of the State Comptroller, “Special Report on Municipal Affairs,” Albany, New York (1977). 23. Office of the State Comptroller, “Overall Real Property Tax Rates: Local Governments in New York State,” Albany, New York (1980). 24. P. A. Samuelson, The pure theory of public expenditure, Rev. Econ. Statist. 36, 387-389 (1954). 25. Todd Sander and J. T. Tschirhart, The economic theory of clubs: An evaluation survey, J. Econ. Lit. 18, 1481-1519 (1980). 26. A. Sandmo, Public goods and the technology of consumption, Rev. Econ. Stud. 40, 517-528 (1973). 27. C. M. Tiebout, A pure theory of local expenditures, J. Polit. Econ. 64, 416-424 (1956). 28. U.S. Department of Commerce, Bureau of the Census, “Census of Governments: New York,” Washington, D.C. (1977). 29. U.S. Department of Commerce, Bureau of the Census, “Census of Population and Housing: New York,” Washington, D.C. (1980). 30. A. A. Walters, “The Economics of Road User Charges,” Johns Hopkins Press, Baltimore (1968). 31. K. White, A general computer program for econometric methods-Shazam, Econometrica 46,239-240 (1978).
OF URBAN
ECONOMICS
27,80-96 (1990)
Congestion Function Specification and the “Publicness” of Local Public Goods* JOHN H. Y. EDWARDS Department of Economics, Tulane University, New Orleans, Louisiana 70118
Received November 25,1986; revised August 26,1987 This paper is about the measurement of congestion within the median voter demand framework. The model of congestion traditionally used in the literature and four others are compared and tested in a median voter framework. The results show that a more flexible functional form which imposes fewer a priori restrictions on the data does at least as good a job of explaining variations in local government expenditures. Furthermore, it is shown that measured “publicness” of local public goods depends entirely on the a priori restrictions imposed on the congestion fUnCtiOn
Q 1990 Academic
Press. Inc.
I. ON THE NON-PUBLIC NATURE OF LOCAL GOODS The central role of congestion in public finance theory is clear in a number of works. Samuelson [24] posited a two-good world and argued that the uncongestable good should be provided by the government. At the same time he raised the preference revelation conundrum and the specter of inefficiency from which Tiebout [27] subsequently sought to rescue the local public sector. By imposing a spatial range on local goods, Tiebout and later Buchanan [7] reintroduce congestion and thus eliminate the non-convexity associated with Samuelson’s pure public goods. Tiebout goods and club goods lie between Samuelson’s polar, or pure, cases. Empirical incarnations of the median voter model have also incorporated population size and the notion of a continuum between private and public goods. This is accomplished by means of a congestion function of the form Z* = ZN-7, where Z is the public good, Z* is the level or quality perceived by the median voter, and N is population size. A null value of y indicates Z is a pure public good, while y = 1 implies Z is like a private good in the sense that the median voter, at least, receives his share l/N. Though their work is seldom cited in this context, the economic rationale behind congestion functions for local public goods is clearly laid out in *I am indebted to Wally Oates for invaluable comments and encouragement in the early stages of this research. I also thank two anonymous referees and Nahid Aslanbegui, Melville McMillau, Bill Oakland, John Pritchet, and Ingmar Prucha for their help and suggestions. Computer time was graciously provided by the University of Maryland and Tulane University computer centers. 80 0094-1190/90 $3.00 Copyright All rights
0 1990 by Academic Press. Inc. of reproduction in any form reserved.
THE CONGESTION OF LOCAL PUBLIC GOODS
81
Bradford, Malt, and Oates [5]. They distinguish between D-goods directly produced by the government (police cars) and C-goods of direct concern to the citizenry (the level of safety). The level of C-good perceived depends jointly on the level of D-good provided by the government and on a vector of socio-economic variables such as income level, population size, and education.’ Median voter demand studies typically find that y > 1. Because their congestion function is admittedly arbitrary, Borcherding and Deacon [4] caution against drawing normative conclusions from their estimates. Yet one’s curiosity is piqued by results which seem to indicate that goods provided by local governments lack the publicness property we would expect. The Bergstrom and Goodman [2] study is perhaps the most widely known in this tradition, because of the size of its data base and their rigorous definition of the conditions and assumptions necessary for the validity of a median voter demand model. Their findings lead them to reject the hypothesis that y < 1, concluding that collective provision of goods and services such as general administration, police protection, and parks apparently are “largely a matter of indifference from an efficiency viewpoint” [2, p. 2941. This “privateness” result has found widespread acceptance in the literature. Two widely-cited texts, Mueller [19, pp. 107-1111 and Atkinson and Stiglitz [l, p. 3231,report the result. Niskanen [20] and Borcherding, Bush, and Spann [3] point to it as evidence of voter exploitation. Finally there are strains of the literature (McMillan, Wilson, and Arthur [18]; Epple, Filimon, and Romer [12]) in which theoretical analysis is founded on this empirical result. In stark contrast to the median voter demand results, Brueckner [6] found that fire protection services possessthe congestion properties of a pure public good. It would appear that the non-publicness result needs to be examined more closely. Indeed, a number of recent works by McMillan et al. [18], Edwards [lo, 111,McKinney [16], Craig [8], and Oates [21] have already begun this examination. The contribution of this paper is to examine the sensitivity of the non-publicness result to alternative specifications of the congestion function. This is done within the traditional median voter demand framework to facilitate comparison with earlier results. The next section will lay the foundation for an empirical study of the effect of congestion function specification on the measurement of publicness. In the third section I develop five congestion function specifications that have been tailored to reflect specific properties (such as increasing ‘As will be seen shortly, the procedure in the median voter demand literature has been to control for so&o-economic variables at the time of estimation.
a2
JOHN H. Y. EDWARDS
marginal congestion and camaraderie) that are described in the literature. These models are estimated for three publicly provided goods in Section IV. The comparison of congestion function specifications is based on these 15 estimated equations. Section V provides a summary and the conclusions. II. MEDIAN
VOTER DEMAND AND CONGESTION
Assume there are two goods: one provided by the local government, 2, one procured privately, X. In the median voter framework as developed by Bergstrom and Goodman [2], we might say that the government’s problem is to maximize median-voter utility, subject to the voter’s budget constraint, i.e., maxU=
U(X, Z, N),
s.t. x + tz = Y,
(0 J>
where N is persons sharing Z, t is median voter tax price, and Y is income. Rather than having N enter the utility function directly, we may introduce a “technology of consumption” in the senseof Sandmo [26]. Assuming private goods are not congestable allows us to rewrite the utility function as u = u(x,
z*),
(0 4
where Z* is the perceived amount of public good, and Z* =f(Z,
N).
(0.3)
The “congestion function” described by (0.3) may be seen as a special case of the transformation of a D-good into the C-good perceived by the median voter, as a function of the number of persons sharing it.* From (0.3) we obtain
Z = f-‘(Z*,
N).
(04
The local government’s problem may be reformulated as maxL=
U(X,Z*)+X[Y-X-tf-'(Z*,N)],
where A is a Lagrange multiplier. If population is not a choice variable of *“Congestion” is used here in the broadest sense.It should be understood to include all the effects of population size on the C-good. If the latter were police protection, then congestion, as used here, would encompassall such effects of population size as higher crime rates, longer police response times, and reduced willingness of bystanders to intercede at the scene of a crime. The formulation in (0.3) is analogous to that used in the literature on clubs and optimal community size.
83
THE CONGESTION OF LOCAL PUBLIC GOODS
the government, utility maximization leads to 8L ax=Ux-X=O
aL
-LYz* = up - xtf;: CYL ax=
= 0
Y-x-l“‘=o.
Certain specifications of the utility and congestion function allow us to solve for Z*,
z* = h(N, t, Y).
(0.5)
(0.4) and (0.5) provide the individual’s demand function for Z, z =f-l(h,
N) = H(N,
t, Y)
(04
which will provide population, price, and income elasticities of demand for the public good. The research problem becomes one of specifying (0.1) and (0.3) in a manner flexible enough to provide a solution as in (0.6) without sacrificing the ability to describe behavior which economic theory leads us to expect. The main issue to be resolved here is the form of (0.3) the congestion function. It will be helpful at this point to set four properties that a reasonable specification of (0.3) should have; this will restrict our choice of congestion models to those that make economic sense. Property 1. When there is a single user, he perceives what there is-as though Z were purchased privately: Z*( Z, 1) = Z. Property 2.
Z* is predicated on the existence of Z: Z*(O, N) = 0.
Property 3. If the good is congestable, then there is some value of N > N* such that 8 Z*/aN < 0. Property 4.
Z* must be increasing in Z; that is aZ*/aZ
> 0.
III. FIVE MODELS OF CONGESTION This section describes five congestion models. The first four are based on descriptions of congestion found in the public goods literature, while the last is a flexible functional form that imposes no a priori restrictions on the shape of the congestion function and yet can pick up properties described by the other models if these are present in the data. Four of the models satisfy the four congestion function properties described above. While the
84
JOHN H. Y. EDWARDS
GCF model does not, it was included nonetheless because Inman [14] has shown it to be the general form from which many of the highway congestion models can be derived. (1) Decreasing Marginal
Congestion
The (DMC) model of congestion developed by [2] and [4] is the one that has been used in median voter studies of demand for the past decade. The specification corresponding to (0.3) is Z* = ZN-“.
(1.1)
For y > 0 we have i3Z*/aN -z 0 and a2Z*/aN2 > 0; congestion decreases at the margin. (2) Increasing
Marginal
Congestion
The IMC model is designed to satisfy the more usual assumption that congestion accelerateswith the intensity of use (Litvack and Oates [15] and Walters [30]). Eventually the good has no value, or even negative value to consumers, as when highway trafhc comes to a standstill. From (1.2),3 Z* = Z(2 - exp[y(N - I)]}, we obtain -y2Zexp[y(N
az*/aN =
-yZexp[y(N
- l)] < 0 and
(1.2)
a2z*/aiv2 =
- l)] < 0.
(3) Camaraderie Function
Economics of clubs literature suggeststhat there may be a range where sharing congestable goods actually increases benefits to each user until these are eroded by crowding. The increased benefit has been called a “camaraderie effect” [25, p, 14821. It can be modeled with a modified Gamma function as Z* = ZNYexp[a(l
- N)]
$!$ = Z*($ - a).
(1.3)
(1.4)
Marginal congestion is shown in (1.4). For y, a > 0, there is benefit from sharing until N = y/a. For y, a < 0 the model conveniently reverts to decreasing marginal congestion with the traditional DMC model (a = 0) as a special case. 3The apparently curious appearance of 2 in (1.2) is needed to satisfy Property 1.
THE CONGESTION
OF LOCAL
PUBLIC
85
GOODS
(4) Generalized Congestion Function
The GCF developed by Inman [14] is specified in terms of 2 as z =
N(7)
+ pz*w
O-5)
where N(Y) = (NY - 1)/y for y # 0 and N(Y) = In(N) for y = 0, while z*(w) = [(z* - ~7~)~:- 11/w,, = ln(Z* - w2),
for wi Z 0
for wi = 0.
0 *6)
Thus, we have for y, wi # 0 z = (NY - 1)/y + p[(z*
- w*)W1- 11/w,
z* = [l + w,(yz - NY - l)/py]l’wl
+ w,.
(1.7) (1.8)
My attempts to estimate the demand equation derived from (1.8) were unsuccessful. However, I was able to estimate a restricted version which, -without affecting marginal congestion-assumed wz = 0. Dropping the subscript on wi, we obtain z = (NY - 1)/y + p(z*w - 1)/w,
(1.7a)
and z* = [l + w(yz - NY - l)/yp]““. (5) Exponential
(1.8a)
Function Model
Given the diversity of functional forms that can be drawn from the literature, the inclusion of a flexible functional form seemedappealing. The simple exponential polynomial function of (1.9), Z* = Zexp[aN + bN2 + dN3 - (a + b + d)],
0.9)
has two desirable properties. Most important is that it imposes no a priori restrictions on the shape of the congestion function. The signs of the first and second partial derivatives of (1.9) depend on the sign and relative magnitude of the population parameters. Second, the demand equation derived from it is log-linear and can be estimated using OLS. IV. ESTIMATION The DMC demand equation shown in Table 1 was derived from (1.1) in the following manner. Solving (1.1) for Z, the equivalent of (0.4) is Z = Z*N7.
(2-l)
86
JOHN H. Y. EDWARDS TABLE I Price Per Unit of Z*, by Model DMC fNY IMC r(2 - expy(N - 1)1-t Camaraderie rNmYexp[a( N - l)] @z*(w-‘) GCF Exponential texp[(a + b + d) - aN - bN2 - dN3] Demand Functions for Z Estimated, by Model
- exp y(N - l)]-(‘+a)fiS;~
IMC
ctsY[2
Camaraderie
cts~N-Y(‘+a)exp[a(l
GCF*
K, + K,t’*V*
Exponential
ctarexp(l
*K,= -P/w;
B
+ (NY - 1)/y + &L,S, + ih,SF
+ @[(a + b + d) - ai - bN2 i dN3]
c
K2=;[l+S~1-w~l
+ S)(N - l)]fi,Sp
=; and $I = w/6(2 - w).
The budget constraint (recall “t ” is tax price) thus becomes Y = x + tZ*NY,
(2.2)
so that price per unit of Z* is
P,* = NY.
(24
Assuming a multiplicative demand function, demand for Z* is
By substitution in (2.1), demand for Z is Z, = ,t8y~NY(l+8)
(2.6)
where the Si are vectors of socio-economic variables described below.4
4Hulten [13] has noted that using the multiplicative form of (2.5) and (2.6) is tantamount to assuming that the technology of consumption is Hicks-neutral with respect to the S,.
THE CONGESTION OF LOCAL PUBLIC GOODS
87
The other four specifications in Table 1 were derived in the same way from the respective congestion functions. The DMC, Camaraderie, and Exponential Function models were estimated using ordinary least squares. The GCF and IMC models yield demand equations that are not log-linear and were estimated using the non-linear maximum likelihood procedure developed by K. B. White [31]. The dependent variable is in logarithmic form in all but the GCF model.5 Each of the demand equations was estimated for general administrative expenditures (GE), parks and recreation (PR), and police protection (PP). The five socio-economic variables (Si) included as control variables are % owner-occupied housing, % non-white, employment/residential ratio, % population change since the last census, and 4%over 65 years of age. Data used are for 78 New York State municipalities of between 10,000 and 150,000 inhabitants. Median income (Y) and median house values (V) as well as the demographic data (the Si) come from [29]. PR and PP expenditures were drawn from [28], while GE expenditures were obtained from [22]. Property tax rates (T) and market price equilization rates (E) came from [23]. Median voter tax price is computed as t = (V X E) X T/R, where R is total tax revenue. Observations were omitted when some of the needed data were not available. Full results for four of the models are not presented because of space limitations. Table 2 compares my results for the DMC model, based mostly on 1977 data, to the original Bergstrom and Goodman results for New York with 1962 data. The price, income, and population elasticities are of the expected signs. With the exception of the employment/residential ratio, parameter estimates for the control variables from the two data sets are of the same sign when they are both sign&ant. Significant economic parameters have a similar range of values in both sets of data, though in the latter period the range is somewhat narrower and population elasticity estimates are slightly higher. The congestion parameter y is in all caseslower than the 1962 estimates, though significantly greater than one at the 95% level of confidence. This implies non-publicness and “over-crowding” of goods examined in all six cases. Table 3 shows the arithmetic means of elasticity estimates for the 15 equations estimated. Price and income elasticity estimates are of the expected sign and for the most part reassuringly similar. The estimated congestion functions for police protection are shown in Figs. (l)-(5) as the percentage of Z, perceived by the median voter at each ‘1 believe this to be in keeping with the spirit of the original Inman [14] formulation of the GCF. Implicit in (1.5) is a test of the logarithmic form, since lim,,,(NY - 1)/y = log N. Further statistical discrimination is provided in the non-nested test developed by MacKinnon [17] that is used in this paper.
88
JOHN H. Y. EDWARDS TABLE 2 Bergstrom and Goodman Model Estimates Parks and recreation
Police protection
General administration expenditures
Parameters
1962
1977
1962
1977
1962
1977
Tax share
-0.81 (1.53) 1.74 (2.71) 0.69 (1.19) 3.63 1.42 (2.21) -3.48 (2.02) 14.27 (2.53) 1.66 (0.60) 0.35 (1.84) - 8.03 0.74
- 0.23 (2.78) 1.25 (3.93) 1.05 (7.97) 1.36 - 1.05 (1.33) - 1.39 (3.53) 0.33 (1.19) - 0.07 (1.11) - 1.43 (1.99) - 3.83 0.75
-0.31 (0.48) 1.78 (2.31) 1.02 (1.45) 1.47 -2.73 (3.54) - 1.16 (0.56) 10.38 (1.53) 2.32 (0.70) -0.14 (0.61) - 8.58 0.69
-0.31 (1.64) 1.08 (1.39) 1.08 (3.45) 1.56 0.06 (0.W -1.85 (1.96) 1.44 (2.21) 0.01 (0.08) - 3.26 (1.89) - 10.07 0.49
-0.50 (3.57) 1.03 (6.05) 0.75 (5.00) 1.50 - 0.03 (0.17) - 1.68 (3.73) 3.70 (2.52) - 0.19 (0.26) 0.10
- 0.12 (1.94) 0.91 (3.82) 1.10 (11.55) 1.16 -0.13 (0.24) - 0.69 (2.42) 0.83 (4.14) 0.05 (1.16) - 1.71 (3.20) -7.32 0.86
Income Population Crowding parameter Population change Owner occup. % over 65 % nonwhite Employee/resident ratio Intercept R2
WV - 1.68 0.94
Note. Absolute value of “t” statistics in parentheses,variables logged.
value of N.6 In the first four figures, Z,, can be any arbitrary amount of expenditures, since the congestion functions specify log-separable roles for Z and N. The last graph represents the GCF model where congestion varies not only with population size, but also with the level of expenditures. The IMC function explicitly restricts the rate of congestion and hence the shape of the congestion function. Estimates based on the IMC show congestion having a small effect on the amount of public good provided in the towns at the lower limit of the sample. In a town of 10,000 persons, congestion would reduce perceived services by about 7%, falling off to zero when N = 106,638. While the DMC model also restricts the rate of congestion, the Camaraderie, Exponential, and GCF models do not. There are thus nine instances of free-form estimation here, namely the last three models estimated for three goods. The general shape in all nine caseswas essentially the same, providing strong evidence in support of the traditional median voter de 61t is important to note, however, that the vertical scale varies from figure to figure. The intention here is to examine the forms of the various congestion functions.
89
THE CONGESTION OF LOCAL PUBLIC GOODS TABLE 3 Representative Values of Elasticities by Model Expenditure category Model DMC IMC Camaraderie Exponential GCF* DMC IMC Camaraderie Exponential GCF* DMC IMC* Camaraderie* Exponential* GCF*
PR Price elasticities - 0.23 - 0.43 - 0.23 - 0.22 -0.31 Income elasticities 1.25 1.13 1.25 1.27 0.81 Population elasticities 1.05 0.21 1.08 1.13 1.23
PP
GE
- 0.32 - 0.65 - 0.32 - 0.32 -0.51
-0.12 -0.57 -0.12 -0.12 -0.12
1.08 1.34 1.08 1.09 2.51
0.91 2.34 0.91 0.92 0.85
1.08 0.16 1.10 1.12 3.32
1.10 0.62 1.08 1.34 1.38
*Arithmetic means of estimated values.
mand assumption that congestion costs for local public goods decrease at the margin. If we write the utility function as U( X, 2, R), where R = l/N, this result can be interpreted to mean that (given X and Z) R is subject to decreasing marginal utility. The Measurement of Publicness Traditionally, median voter demand studies have reached their “nonpublic” conclusion by relying on the DMC model where the test for privateness from (1.1) involves the null hypothesis Ha: y = 1. The other four models used in this paper imply a non-constant rate of congestion. Parametric tests of publicness are therefore not possible; we must develop a more general one which preserves the spirit of the traditional criterion. In the DMC model, y 2 1 is said to imply privateness because the median voter receives at most “his aliquot share.” From (1.1) we have Z* = Z/N. Of course, not rejecting the null hypothesis y < 1 implies public good characteristics given that for all observed values Zj and Nj,
where Fj is the estimated value of Z* at observation “j.”
90
JOHN H. Y. EDWARDS
Camaraderie
GCF (Z. = $512,000)
FIGS. 1-5. Police protection: Estimated congestion functions (N in thousands, Z* as percentage of Z,).
91
THE CONGESTION OF LOCAL PUBLIC GOODS
As the basis for an alternative test of publicness, consider the function Gij = 4,/Z,
- l/Nj,
(3.1)
where Fij is the estimated value of congestion function “i ” at observation “j.” The functions G, have an upper bound of unity for pure public goods. Also, 0 < Gi < 1 implies the publicness of Z in the sense that each person receives more than l/N. Finally, G, I 0 implies the privateness of Z, as each person receives at most his share of the publicly provided good. (3.1) thus preserves the spirit of the parametric test traditionally used for the DMC model, but can be applied to other functional forms. Of all the models examined, only the DMC and the closely related Camaraderie models allow us to state unequivocally that none of the three goods are public goods, since the estimated value of Gi in Table 4 is negative for all of the New York State municipalities sampled. At the opposite extreme, the Exponential function strongly indicates publicness. For the other two models the degree of publicness varies across goods. The IMC function indicates publicness of PR and of PP, but implies that congestion costs accelerate to the point where in the largest 23% of the municipalities, GE appears to be overly congested. Finally, the GCF model shows police and 97% of observations for parks as private and yet that in only 3% of the municipalities are administrative services like private goods. TABLE 4 Estimated Publicness (Representative Values of G, = E;,/Z - l/N) Minimum
DMC Camaraderie IMC Exponential GCF
- 0.0001 - o.ooo1 0.063 0.017 - 0.002
loo 100 0 0 97
DMC Camaraderie IMC Exponential GCF
-
100 100 0 0 100
DMC Camaraderie IMC Exponential GCF
-
Mean
Maximum
S for which GsO
Model
Parks and recreation - O.OOOOOl - o.oooo5 - 0.00005 - O.OOOOOl 0.80 0.932 0.097 0.231 - 0.009 0.0001 Police protection - 0.00001 0.0001 - 0.00005 0.0001 0.00005 - 0.00001 0.023 0.80 0.93 0.004 0.047 0.13 0.009 - 0.00024 - 0.004 General administrative expenditures 0.00009 - O.cWOl - 0.00005 0.00009 - 0.00001 - 0.00005 8.37 -0.06 0.74 0.03 0.17 0.35 0.003 0.0004 0.003
100 100 23 0 3
92
JOHN
H. Y. EDWARDS
Above all, Figs. (l)-(5) and the results in Table 4 should serve as a warning. Within the median voter demand framework descriptions of congestion properties and-more important for policy prescription-conclusions on the publicness of local public goods are highly sensitive to the a priori congestion assumptions which underlie demand models. There are two possible criteria for choosing congestion models. The first -conformance to economic theory-has already been used in the preliminary selection of models. Given the conflicting nature of this theoretical guidance, my preference is for flexible functional forms-like the exponential function-which let the data speak for themselves.The second criterion that can be used is the statistical performance of conflicting models in explaining variations in municipal expenditures on local public goods. Statistical Comparison The DMC model of (1.1) can be seen as a special (nested) case of (1.3) where the coefficient “a ” is assumed equal to zero. Using the standard t-test, the null hypothesis H,: a = 0 cannot be rejected for any of the three goods.7 This means that we cannot reject the hypothesis that “Camaraderie” adds no explanatory power to the DMC over the sample range. In order to test the other models against one another, I have used the “J-test,” a method of non-nested hypothesis testing developed in Davidson and MacKinnon [9] and MacKinnon [17]. The procedure is called “artificial embedding” and, as the name suggests, consists of creating an artificial grand model as the convex combination of competing models. Predicted values from alternative specifications “j” are added as explanatory variables to the model “i ” being tested. For the parameters thus created, the joint test HOi: Xjj = 0 (for i # j) is the null hypothesis test for model “i.” The likelihood ratio test, based on the statistic -2log L, where L is the ratio of restricted to unrestricted likelihood function values, is distributed as x2 with degrees of freedom equal to the number of restrictions in each Z& (in our case, three). A clear choice between models occurs only if three null hypotheses of no significant enhancement are rejected and one Ho: is not. In the present context, this could be the case in four of 16 possible outcomes. To reduce the probability of type 2 error, a 95% confidence interval was chosen. Table 5 presents the results of tests where three Xjj are held equal to zero, for a critical x2 value of 7.81. We reject the hypothesis of no ‘The absolute critical value at the 95% level of significance with 75 degrees of freedom is 2.00. The absolute values of the calculated r-statistics were Parks and Recreation: t = 0.99 Police Protection: r = 1.13 General Administrative Expenditures: t = 0.30
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THE CONGESTION OF LOCAL PUBLIC GOODS TABLE 5 Non-nested Hypothesis Tests for Four Congestion Models Four models Critical x2 values Values of - 2 log L for
7.81
Two models 3.84
Parks and recreation DMC IMC Exponential GCF
4.46
13.86 3.89 52.59
4.00
1.76 -
Police protection DMC IMC Exponential GCF DMC IMC Exponential GCF
2.58
142.90 5.68 9.68
General administrative expenditures 1.00 126.86
1.00
2.26 0.74
-
0.32
0.00
10.00
-
significant enhancement in the IMC and GCF cases. It is also true, however, that we cannot reject the equivalent hypothesis for the DMC and the Exponential models. They appear to outperform the other models for all three goods, PR, PP, and GE. The statistical test results are consistent with the observation that elasticity estimates from the two models in Table 3 are altogether quite similar. Distinguishing between these two models is vital for, although both appear to do a good job of explaining inter-municipal variations in expenditures, Table 4 shows they have opposite implications for publicness. In repeating the non-nested tests with only the DMC and Exponential models, our null-hypotheses restrict one parameter, for a critical x2 value of 3.84. Expenditures on PR are clearly best explained by the Exponential model (see Table 5) whose predicted values significantly add to the performance of the DMC model, while the converse is not true. Unfortunately, the two-model test sheds no more light on the other two goods. For PP and GE alike, there is no statistically significant difference between the models. The Exponential function model explains inter-municipal variations in public expenditures at least as well as the DMC model traditionally used in the literature. Since the theoretical guidelines for restricting the shape of the congestion function are not clear, this statistical comparison again indicates that the Exponential function model is probably preferable for its more general form. It is therefore by no means certain that local governments are providing goods that behave like private goods.
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V. CONCLUSIONS AND FINAL COMMENTS This paper defined the congestion function in terms of the Bradford, Malt, and Oates distinction between C-goods and D-goods, thus developing the theoretical basis for including population and other socio-economic variables in studies of demand for local public goods. Four properties that should be satisfied by reasonable specifications were then defined and five models of congestion developed and incorporated in median voter demand equations. First examined was the DMC model, which has become entrenched in the median voter demand literature. Three others were drawn from descriptions of congestion provided elsewhere in the literature and the fifth is a flexible functional form which imposes no a priori restrictions on marginal congestion. Four main results emerged from this experimentation. (1) Variation of functional forms was found to have a sizable impact on estimates of population elasticity of demand, a parameter of paramount importance to the local public goods literature. (2) Unexpectedly, estimation of free-form congestion functions provided substantial evidence supporting the traditional assumption that congestion of local goods decreasesat the margin. This may be interpreted as evidence of decreasing marginal utility for congestion alleviation. (3) The results in Table 4 imply that the degree of publicness is not constant, but varies indirectly with population size. This result was foreseen by McMillan et al. [18]. There may be discontinuities, they suggest, in the provision of “optimally sized facilities and service areas” so that “empirical investigation should reveal a high degree of publicness.. . for services in communities which are small enough that service units need not be replicated” (p. 598).8An alternative theory was recently put forth in Oates [21] arguing that the privateness result in the median voter demand literature may be due to what he calls the “zoo-effect.” Larger communities provide a greater diversity of goods so we are really not observing non-publicness but an aggregation problem that results in an upward bias in the measurement of congestion. The observed inverse relation between population size and the degree of publicness may be interpreted as supporting both hypotheses. The relation to the theory of McMillan et al. is obvious. The Zoo-effect bias would be present in all five models considered in this paper, but the severity of the bias should be less in smaller municipalities-and the measured degree of publicness should be correspondingly larger.
‘Note, however, that as originally formulated the DMC model does not allow the congestion parameter “y” of (1.1) to vary across municipalities (see Edwards [ll]).
THE CONGESTION OF LOCAL PUBLIC GOODS
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(4) My most important result concerns the public nature of local goods. Economic theories of shared goods, such as the Tiebout and Buchanan models, depend on the notion of a trade-off between cost savings and congestion, a trade-off which implies the existence of an optimal group size. In relying exclusively on what now appears to have been an ad hoc specification of congestion, the median voter literature had concluded that the services of municipalities have private good, not public good, characteristics. The DMC specification was a convenient device a decade ago, but until now, it had not been tested against other models. The DMC function used in these early median voter demand studies has essentially the right shape and-in two out of three goods examined here-did as good a job of explaining inter-municipal variations in expenditures as the more flexible Exponential function model. However, imposing constant elasticity of congestion apparently results in an upward bias in measuring the level of congestion. When the Exponential function is used we find that services provided by local governments have the public good characteristics that the theory of shared goods leads us to expect. REFERENCES 1. A. B. Atkinson and J. E. Stiglitz, “Lectures on Public Economics,” McGraw-Hill, New York (1980). 2. T. C. Bergstrom and R. P. Goodman, Private demands for public goods, Amer. Econ. Rev. 63, 280-296 (1973). 3. T. E. Borcherding, W. C. Bush, and R. M. Spamr, The Effects on public spending of the divisibility of public outputs in consumption, bureaucratic power, and the size of the tax-sharing group, in “Budgets and Bureaucrats” (T. E. Borcherding, Ed.), pp. 211-228, Duke Univ., Durham, NC (1977). 4. T. E. Borcherding and R. T. Deacon, The demand for the services of non-federal governments, Amer. Econ. Reu. 62, 842-853 (1972). 5. D. F. Bradford, R. A. Malt, and W. E. Oates, The rising cost of local public services, Nut. Tax J. 22, 185-202 (1969). 6. J. K. Brueckner, Congested public goods: The case of fire protection, J. Public Econ. 15, 45-58 (1981). 7. J. M. Buchanan, An economic theory of clubs, Economicu 32, 1-14 (1965). 8. Steven G. Craig, The impact of congestion on local public good production, J. Public Econ. 32, 331-353 (1987). 9. R. Davidson and J. MacKinnon, Several tests for model specification in the presence of alternative hypotheses, Econometrica 49, 781-793 (1981). 10. John H. Y. Edwards, “Congestion Function Specification and the Publicness of Goods Provided by Governments,” Dissertation, University of Maryland (1985). 11. John H. Y. Edwards, A note on the publicness of local goods: Evidence from New York state municipalities, Cunud. J. Econ. 19, 568-573 (1986). 12. D. Epple, R. Filimon, and T. Romer, Housing, voting, and moving, in “Research in Urban Economics” (J. Henderson, Ed.), Vol. 3 J.A.I., London (1983). 13. C. R. Hulten, Productivity change in state and local governments, Reu. Econ. Statist. 66, 256-266 (1984). 14. R. P. Inman, A generalized congestion function for highway travel, .I. Urban Econ. 5, 21-34 (1978).
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15. James M. Litvack and W. E. Oates, Group size and the output of public goods, Public Finance 25, 41-58 (1970). 16. S. McKinney, “The membership margin and the median voter,” mimeo (1985). 17. J. MacKinnon, Model specification tests against non-nested alternatives, Econometric Reu. 2, 85-100 (1983). 18. M. L. McMiBan, W. Wilson, and L. M. Arthur, The pubhcness of local public goods: Evidence from Ontario municipalities, Cunad. J. Econ. 14, 596-608 (1981). 19. D. C. Mueller, “Public Choice,” Cambridge Univ. Press, London (1979). 20. W. A. Niskanen, Jr., Bureaucrats and politicians, J. Law Econ. l&617-643 (1975). 21. W. E.Oates, On the measurement of congestion in the provision of local public goods, J. Urban Econ. 24, 85-94 (1988). 22. Office of the State Comptroller, “Special Report on Municipal Affairs,” Albany, New York (1977). 23. Office of the State Comptroller, “Overall Real Property Tax Rates: Local Governments in New York State,” Albany, New York (1980). 24. P. A. Samuelson, The pure theory of public expenditure, Rev. Econ. Statist. 36, 387-389 (1954). 25. Todd Sander and J. T. Tschirhart, The economic theory of clubs: An evaluation survey, J. Econ. Lit. 18, 1481-1519 (1980). 26. A. Sandmo, Public goods and the technology of consumption, Rev. Econ. Stud. 40, 517-528 (1973). 27. C. M. Tiebout, A pure theory of local expenditures, J. Polit. Econ. 64, 416-424 (1956). 28. U.S. Department of Commerce, Bureau of the Census, “Census of Governments: New York,” Washington, D.C. (1977). 29. U.S. Department of Commerce, Bureau of the Census, “Census of Population and Housing: New York,” Washington, D.C. (1980). 30. A. A. Walters, “The Economics of Road User Charges,” Johns Hopkins Press, Baltimore (1968). 31. K. White, A general computer program for econometric methods-Shazam, Econometrica 46,239-240 (1978).