Estimating the publicness of local government services: alternative congestion function specifications.

• Author: Means, Tom S.

1. Introduction

   The standard approach to estimating median voter expenditure equations has incorporated the specification of a congestion function, [Alpha](N), to convert the total output (Q) of a government provided good into individual consumption ([q.sub.i]). In their classic papers, Borcherding and Deacon [2] and Bergstrom and Goodman [1] adopted a simple form, [Alpha](N) = [N.sup.-[Gamma]]. Using this form, estimated values of [Gamma] = 0 imply that the service in question is classified as a pure public good, while [Gamma] = 1 implies a private good. This approach is attractive from an empirical standpoint because it allows the data to determine the degree of publicness of the local government output. The empirical findings of Borcherding and Deacon, Bergstrom and Goodman, and numerous later studies that used the median voter approach,(1) indicate that most local government services do not exhibit a significant degree of publicness.(2)

   Recent studies have questioned whether estimates of publicness are sensitive to the particular specification of the congestion relationship [6; 9]. Researchers also have objected to the standard form because it holds the degree of publicness fixed, which implies a decreasing marginal rate of congestion [3; 6; 15]. Moreover, holding the degree of publicness fixed implies a single estimated value of [Gamma] is sufficient to classify the publicness of the good. Critics of the standard approach argue that the degree of publicness should vary with population size. For instance, a local park may exhibit no congestion for small population sizes, but eventually become congested as population increases.

   Unfortunately, no clear guidelines for specifying the congestion function are available. Edwards suggested that flexible functional forms are to be preferred because they "let the data speak for themselves" [6, 92]. He also argued that the preferred form is one that fits the data best. Based on these criteria, he concluded that a flexible form, such as an exponential form, is superior to the standard specification in the median voter literature. When he used the exponential form, Edwards found that, in contrast to prior studies, three local government services displayed substantial publicness. Hayes and Slottje [9], on the other hand, investigated alternative specifications, including an exponential form, and concluded they were not superior on statistical grounds to the conventional approach.

   Congestion functions that permit greater flexibility in measuring publicness offer clear advantages over the conventional form. Balanced against this, however, alternative and more complex congestion functions introduce several problems with respect to the estimation of publicness parameters in median voter demand functions. First, the alternative forms proposed in the recent literature do not always meet the simple criterion of "allowing the data to speak for themselves." Indeed, some of the proposed forms (in particular exponential-based forms) bias the estimated congestion parameters toward higher degrees of publicness (i.e., the amount of congestion is biased downward). Second, the alternative specifications of the congestion function involve more than one parameter to estimate. Gonzalez, Means, and Mehay [7] show that such specifications require joint tests on the parameters to test the pure public and private good hypotheses. Third, the proposed alternative forms yield non-nested models that require unnecessarily complicated procedures for testing whether an estimated equation provides a better fit of the data. Another related problem is that allowing the degree of publicness to vary with population size makes it difficult to apply conventional hypothesis testing methods. For the standard congestion function ([N.sup.-[Gamma]]) the estimated value of [Gamma] can be tested as a point hypothesis (e.g., [Gamma] = [Gamma] or [Gamma] = 1). When the degree of congestion varies with population size, the estimated degree of publicness must also vary. Some authors report the range (or frequency) of observations for which the good is less congested than a private good. This approach, however, cannot determine if these varying estimates of publicness are statistically different from a good that is purely private.(3) The clear advantage of holding the degree of congestion fixed is the ease in classifying the good as public, semi-public, or private. For the flexible functional forms, one cannot determine if a specific estimated value of the publicness parameter is different from unity.

   One goal of this paper is to demonstrate the potential bias toward publicness contained in some of the alternative functional forms. Section II also demonstrates that the pure public and private good hypotheses can be tested as restricted expenditure equations without prior specification of the congestion function. Section III shows that the exponential- and population-based forms can be constructed as nested models and tested using conventional methods for determining whether alternative forms improve the explanatory power of the model. The simple nested hypothesis tests of the publicness parameters are developed in section IV and empirically estimated in section V.

2. A Median Voter Demand Model

   In this section a reduced form expenditure equation is derived from a standard median voter model, in which the median voter's demand schedule is generated from a budget-constrained utility function. The flow of services to the local resident is defined by:

   [q.sub.i] = [Alpha](N)Q (1)

   where [q.sub.i] is individual i's consumption of the publicly provided output, Q. The budget constraint is written:

   [y.sub.i] = Z + [t.sub.i](PQ/N) (2) = Z + [[t.sub.i]P/([Alpha](N)N)][q.sub.i] = Z + [p.sub.i][q.sub.i]

   where [y.sub.i] is individual income, Z is the amount spent on private goods, [t.sub.i] is the individual's tax share (i.e., the taxpayer's share of the total budget) relative to per capita expenditures, P is the price of the output (Q), and [p.sub.i] is the tax-price to the individual resident for [q.sub.i]. Most median voter studies specify the budget constraint as [y.sub.i] = Z + [t.sub.i](PQ), where 0 [is less than or equal to] [t.sub.i] [is less than or equal to] 1. Following Dudley and Montmarquette [5], our tax share is defined relative to PQ/N, the per capita expenditure on the local public good.(4)

   Assuming a multiplicative demand function,

   [Mathematical Expression Omitted],

   the log-linear expenditure equation is derived by specifying [Mathematical Expression Omitted], substituting for [Mathematical Expression Omitted], and taking logs.(5)

   ln(E) = ln(A) + [[Beta].sub.1] ln([t.sub.i]/N) - (1 + [[Beta].sub.1])ln([Alpha](N)) + [[Beta].sub.2] ln([y.sub.i]). (4)

   Expression (4) represents the standard reduced form equation used in the publicness literature. The derivation of (4) assumes that [t.sub.i] may vary between voters but that the allocation of the good is the same for every voter.(6) An equal allocation of a pure public good requires [Alpha](N) = 1 in order to set [q.sub.i] = Q. For the pure private good case [Alpha](N) = 1/N in order to set [q.sub.i] = Q/N. Substituting these restrictions into (4) yields one reduced form expenditure equation for the pure public good case:

   ln(E) = ln(A) + [[Beta].sub.1] ln([t.sub.i]/N) + [[Beta].sub.2] 1n([y.sub.i]), (5)

   and one for the private good case:

   ln(E) = ln(A) + [[Beta].sub.1] ln([t.sub.i]) + ln(N) + [[Beta].sub.2] ln([y.sub.i]). (6)

   The pure public and private good hypotheses can be tested by estimating (4) subject to the restrictions implied by (5) and (6). In terms of elasticities. the restriction for the pure public good hypothesis is...