The Linear Expenditure System and the Demand for Municipal Public Services: The Median Voter Specification Revisited

Abstract

Following Bergstrom and Goodman (1973), this paper contributes to the set of studies estimating demand for publicly provided goods and services. The main innovation in this paper is methodological as it uses a Stone–Geary utility function to describe median voter preferences. Unlike in previous studies, the local public expenditure function, in a (simplified) linear expenditure system, can be derived directly from the theoretical framework. The translated expenditure function provides an estimate of the minimum required local public services in each municipality. Using cross-sectional municipal data from France, it is found to be, at the most, 30 per cent of the total per capita expenditure. Compared with the specification conventionally used in the literature, linear and non-linear expenditure systems show a greater sensitivity to price and income variations and reveal a greater number of significant variables.

1. Introduction

The estimation of household demand for publicly provided goods and services has been a concern of public finance economists for many years. The well-known median voter model is usually used to understand local per capita expenditures because these expenditures are assumed to satisfy the demand of the median voter. The seminal works of Borcherding and Deacon (1972) and Bergstrom and Goodman (1973) are examples in which this model is used. In the literature, as the functional form of the median voter’s utility function is not specified, empirical studies assume that there is a linear relation (or a log–log transformation) between local public spending (per capita) and the median income and tax price, in addition to other control variables. Such an approach has been used by Colburn and Horowitz (2003), who analyse spending on education, and by Turnbull and Djoundourian (1994) to explain the behaviour of municipal governments.

However, as noted by Reiter and Weichenrieder (1997) in their critical survey of this empirical literature, this approach has some methodological drawbacks. Atkinson and Stiglitz focus in particular on one issue

The results of estimating median voter models provide insight into a number of aspects, but are open to a number of criticisms … for example the specification of the functional form (Atkinson and Stiglitz, 1980, p. 326).

In this article, we examine this question and provide a methodological innovation. More precisely, using a Stone–Geary utility function to describe median voter preferences, we are able to derive the public spending function in a simplified linear expenditure system (with only two goods, private consumption and local publicly provided goods and services). The public spending function consequently is derived directly from the median voter model.

The linear expenditure system (LES) is a system of demand functions proposed by Stone (1954) by imposing theoretical restrictions (additivity, homogeneity and symmetry) on a general linear formulation of demand. The non-linear expenditure system (NLES) is an extension of the LES, proposed and tested by Carlevaro (1976, 1977, 1982), which includes non-linear terms for income and price effects. In both specifications, the translating method, inspired by Pollak and Wales (1981), allows the introduction of variables to capture production costs and demographic differences across municipalities.

Another attractive feature of the LES is that parameters have economic meaning. LES consequently allows total public expenditure to be broken down into two constituent parts. The first is the incompressible part, which may be interpreted as required expenditure in the sense that there is a minimum level of services that must be provided by the local authority. It can also be interpreted as a measure of expenditure needs. The second component of total public expenditure is the variable part which depends on income and tax share levels.

Finally, to demonstrate that the linear expenditure system has something useful to contribute to existing empirical literature, our specification is tested against the specification that is generally used in other studies.

This article consequently is an original contribution to empirical public finance literature as only two other recent articles have used a linear expenditure system on a similar topic, Aaberge and Langorgen (2003) and Allers and Elhorst (2011). The modelling approach of both pairs of authors is, however, very different because they do not include a price variable. They also consider a system of municipal public services that are supposed to be substitutes for each other.

Estimating the expenditure systems with non-linear least squares on the 109 French municipalities with more than 50 000 inhabitants in 2005 (excluding Paris) allows us to obtain original results. First, we are able to assess the minimum local spending value, which is on average 13 per cent (non-linear expenditure system) or 30 per cent (linear expenditure system) of the total municipal per capita spending level. We consequently conclude that a dominant part of municipal public spending fluctuates with income and tax share variations. The economic climate and tax policy are thus important factors explaining public expenditure levels.

Secondly, the translating method shows that two opposite forces contribute to explaining the incompressible value. On one side, an increase in the size of the population contributes to reducing the per capita expenditure due to economies of scale in consumption. On the other, expenditures linked to investment and the maintenance of municipal capital stock increase per capita expenditure. Finally, values of elasticities derived from the LES reveal a greater sensitivity to price and income than values obtained with the standard method used in the literature described later.

This article is organised as follows: the next section presents the alternative demand specifications. Section 3 contains a description of the data and presents the translating method. In section 4, the three demand specifications (two alternative and one traditional) are applied in a model explaining per capita municipal public spending for French municipalities with over 50 000 inhabitants and the results are compared. Section 5 concludes.

2. Demand Estimates for Local Public Services: Alternative Functional Forms

First, we describe the method traditionally used in the literature. We then present the linear expenditure system and describe its non-linear extension.

2.1 Traditional Approach

The traditional specification used to explain cross-section differences in public spending between municipalities is based on the median voter model. In this model, the economy is composed of different citizens who derive utility from publicly provided goods and services and from private consumption. Given the median voter’s utility function $U_{m} (x, g)$ , the municipal commodity supplied matches with the quantity demanded by the median voter if

Max U_{m} (x, g) u . c {\begin{matrix} y_{m} = t b_{m} + x & (1) \\ C_{G} G = tB & (2) \\ g = \frac{G}{N^{λ}} & (3) \end{matrix}

Private consumption, measured in quantities x, serves as numeraire, while g is the volume of publicly provided goods and services available for each citizen. The first equation is the budget constraint of the median voter where $y_{m}$ is the median income within the local jurisdiction and $b_{m}$ is the median voter’s tax-base. The second equation describes the budget constraint of the jurisdiction where t is the uniform municipal tax rate. B gathers all the local tax-bases, $C_{G} G$ is total public spending and G denotes the public output. $C_{G}$ is the unit cost of providing public services; g differs from the aggregated level G of local public services only when there is congestion, see (3) where $λ$ denotes the congestion parameter and N is the size of the population.

We substitute (3) in (2) to obtain an expression of the tax rate t to be substituted in (1). Differentiating by g and x, we obtain the marginal rate of substitution between private consumption and local publicly provided goods and services

MR S_{x, g} = \frac{\partial U_{m} / \partial g}{\partial U_{m} / \partial x} = \frac{b_{m}}{b} N^{λ - 1} C_{G} = p_{g}

(4)

where, b is the average tax-base. In the median voter model, tax price $p_{g}$ measures the price that the median voter considers to formulate her demand for publicly provided goods and services. Tax price includes the tax share $\frac{b_{m}}{b}$ , unit costs of public services $C_{G}$ and the size of the population, taking into account congestion effects.

As Reiter and Weichenrieder (1997) have pointed out, the tax price is certainly the most difficult variable to measure. Existing studies generally retain a simplified specification to estimate the price elasticity of demand. Bergstrom and Goodman (1973) assume that the tax price is proportional to the median voter’s tax share. Unit costs of public services then are supposed to be the same in each locality. In contrast, Borcherding and Deacon (1972) suppose that tax shares are identical across all states in the US and use public wages as the price variable. In this article, the tax price $p_{g}$ in equation (4) is supposed to be measured by the tax share as in Bergstrom and Goodman (1973). Therefore, we examine whether publicly provided goods and services are elastic or not with respect to the median voter’s tax share.

Moreover, the measurement of the median’s voter tax share poses practical problems. For example, using the 1960 US census of housing data, Bergstrom and Goodman (1973) compute the tax bill on the house of median value. In our article, lacking data on median house value, tax share is measured by the average residential municipal tax-base divided by the total average tax-base (including the local business tax). This variable therefore denotes the share of tax-bases between firms and households. The ratio is near 1 in residential areas, while it tends towards 0 in industrial areas.

In the literature, as the functional form of the decisive voter’s utility function is not generally specified, the estimable function of local public expenditures is supposed to be a linear function of the tax price and the median income, with additional control variables. Therefore, the level of public expenditure can be expressed as a linear regression of the form

Exp = α_{1} + α_{2} N + α_{3} \frac{b_{m}}{b} + α_{4} y_{m} + \sum_{i = 1}^{k} β_{i} X_{i} + ε

(5)

where, Exp measures the total per capita expenditure. The $X_{i}$ s include the demographic characteristics of the population and variables describing cost differentials in supplying local public services; $ε$ is the error term.

We then are able to deduce price and income elasticities of demand using parameters $α_{3}$ and $α_{4}$ respectively and mean values of tax share and income. Alternatively, if we apply a log transformation of all the variables, we obtain constant price and income elasticities. In the literature, typical estimates of price elasticity are negative and greater than −1, whereas income elasticity is generally positive and smaller than unity.

However, as mentioned previously, this approach is open to criticism. In particular, we assume in equation (5) that voters are fully aware of the costs and benefits of publicly provided goods and services.¹ Fiscal illusion is ruled out since it would mean a lack of awareness on the part of the voters of the cost of providing any given service. In such circumstances, voters have an incentive to increase demand for publicly provided goods and services to the point where their marginal costs and benefits are equivalent. Oates (1988) and Dollery and Worthington (1996) provide a survey of the literature on this issue.

At least three fiscal illusion hypotheses can be tested through public expenditure functions by adding explanatory variables intended to capture such a phenomenon. First, one source of fiscal illusion is the complexity of the revenue system. According to this hypothesis, the more complex the revenue system, the larger the level of public expenditure will be, ceteris paribus. Many studies have employed the Herfindahl index as a measure of revenue complexity—for example, Dollery and Worthington (1996). The expected sign of this variable is negative.

Secondly, the renter illusion hypothesis argues that renters, who bear property tax only indirectly, underestimate the tax price of public services and may therefore vote for higher public expenditures. Property taxes levied on owners of rental dwellings may be passed forward in the form of higher rents to tenants. The renter illusion hypothesis contends that renters do not perceive this to be the case. This can be tested by adding a variable indicating the proportion of renters in the municipality to the econometric specification.

These two sources of fiscal illusion in local taxation are relevant in the French case. For instance, the complexities of the local French taxation system are well known. However, these issues are disregarded because of the lack of appropriate data.

A third potential form of fiscal illusion is the flypaper effect. The flypaper effect occurs when an unconditional lump-sum grant to a local government increases spending in a greater proportion than an equivalent rise in local income. The flypaper effect is also relevant in the French local public sector as grants from the central government are the second most important revenue stream. However, by definition, the linear expenditure system cannot include easily more than one income variable ( $y_{m}$ in equation (6)) which is equal to the median income plus the national grant per capita. We consequently assume here that there is no flypaper effect.

Overall, the detection and measurement of fiscal illusion are difficult tasks (Oates, 1988) and should be explored in a separate work. Therefore, the issue of fiscal illusion is beyond the scope of this article.

Another difficulty is the problem of identifying the median voter. According to Bergstrom and Goodman (1973), the median voter earns the median income if we observe a monotonic relation between income and desired public spending. However, they show that, without the assumption of proportionality of income distributions, the expenditure would depend on the detailed structure of income distribution. For example, Colburn and Horowitz (2003) have added the percentage of families with income greater than $75 000 and the percentage of families with income less than $15 000 to address the role of income distribution. However, by definition, the linear expenditure specification again cannot include more than one income variable as shown in (6). Indeed, we suppose here that all residents benefit equally from publicly provided goods and services. However, according to Le Grand (1987), this is not generally the case as there is a large amount of evidence suggesting that most services often benefit the middle class more than the poor. Nevertheless, at the municipal level in France, this hypothesis is probably not too restrictive. For example, French municipalities are in charge of primary schools (students between the ages of 3 and 10) which are relatively equally distributed as all education expenditure prior to the school leaving age.

2.2 Linear Expenditure System

The main originality of this paper is to specify the median voter’s utility function in the following Stone–Geary form

U_{m} = β \ln (g - m_{g}) + (1 - β) \ln (x - m_{x})

The assumptions of this function are strong separability, the adding-up restriction and positive marginal budget share $β$ . $m_{x}$ is often considered as a subsistence level of consumption; $m_{g}$ can be regarded as a minimum required for local public services.

Then, in our study, the local public expenditure function is derived in a straightforward manner from $MAX U_{m} (x, g)$ subject to $y_{m} = p_{g} g + p_{x} x$

p_{g} g = p_{g} m_{g} + β (y_{m} - p_{g} m_{g} - p_{x} m_{x})

(6)

where, $p_{g} g$ denotes per capita expenditure in each municipality; and $p_{x}$ is the price of the private consumption.

This specification is a (simplified) linear expenditure system (LES) if we consider only two goods (publicly provided goods and services and aggregate private consumption). This specification has the advantage of being compatible with theory while explicitly modelling a minimum level of consumption, irrespective of the price of the consumed good or the consumer’s income. The median voter first purchases the minimum level of each good and the left-over income is then allocated in a fixed proportion $β$ to the demand for local public services. See Deaton and Muellbauer (1980) for a complete presentation.

Since public spending is usually characterised by inertia, the LES specification is particularly well suited to account for these features of the fiscal process. However, in concrete terms, the minimum expenditure level is difficult to define as it can be interpreted in many ways. As suggested by Le Grand (1975a, p. 538) “There are certain minimum levels of most services which are required by statutory or moral pressures”. Indeed, each French municipality must finance compulsory tasks corresponding to the payment of wages, loan repayment, general administration and the maintenance costs of the public capital stock, in particular primary school buildings and the transport network. As unit costs of public services vary across municipalities, $p_{g} m_{g}$ can also be considered as an indicator of expenditure needs. In France, as in most countries, central government grants to local authorities are distributed to correct two sources of inequality, needs and resources.

In the general case (with n substitute goods), the LES is a system in which expenditures on individual commodities can be expressed as n linear functions of income and prices. However, at the empirical level, we estimate a system of $n - 1$ equations. Then, as we only consider two goods here, and considering the adding-up restriction, we simply have to estimate the public expenditure function. As we are only interested in the demand for municipal public services, and without spatial variation for $p_{x}$ , we suppose that the minimal private consumption value can be incorporated in parameter $γ$

p_{g} g = p_{g} m_{g} + β (y_{m} - p_{g} m_{g} - γ)

(7)

The income elasticity of demand may be written as

E (g / y_{m}) = \frac{β}{ω} with ω = \frac{p_{g} g}{y_{m}}

In addition, price elasticity is

E (g / p_{g}) = - 1 + (1 - β) \frac{m_{g}}{g}

Thus, the income elasticity is always positive since the marginal budget share $β$ is positive. Furthermore, as $0 < β < 1$ , price elasticity is greater than −1 and the demand is inelastic.

In our study, there are at least two advantages in measuring the price elasticity of demand. Firstly, measuring the tax share elasticity of spending provides implications for policy analysis. One of the robust findings of the literature is a very low price elasticity of demand for publicly provided goods and services. If we obtain larger values here (in absolute values), this consequently would mean that the literature has underestimated the decrease in public expenditures that follows an increase in tax share. In our study, variations in tax shares can stem from variations in the amount of commercial and industrial tax-bases. For example, following the 2010 local business tax reform in France, the base of the local business tax (or taxe professionelle) has been reduced in many municipalities. Ceteris paribus, this reform increases the tax share and this has potential expenditure-reducing effects approximated by our tax price elasticity.

Secondly, as Reiter and Weichenrieder (1997) have pointed out, many empirical studies offer little evidence that local governments provide a pure public good in the Samuelson sense as the crowding parameter estimate $\hat{λ}$ is near 1 (see equation (3)). Using total public per capita expenditure as a dependent variable, Borcherding and Deacon (1972) derived the degree of crowding jointly from the population elasticity and from the price elasticity of demand (respectively $α_{2}$ and $α_{3}$ in our equation (5) if we apply a log transformation of all the variables): $λ = \frac{α_{2}}{1 + α_{3}} + 1$ . As the population elasticity is often near zero (it equals −0.024 in the LES specification in Table 2), the crowding parameter is near 1 if we consider the estimated price elasticities obtained with the standard method (from −0.36 to −0.064 in Table 4). However, this result changes if the estimated price elasticity switches from inelastic to elastic. Indeed, the crowding parameter is equal to 0.76 when the price elasticity is −0.90 (mean value for NLES), indicating quasi publicness, where crowding effects are present but there are still economies of scale in consumption (Reiter and Weichenrieder, 1997).

Indeed, all households in a municipality may not be sensitive to tax share variations. In France, around 20 per cent of buildings occupants have low income levels and are tax exempt. They consume municipal public services at zero prices. Nevertheless, it is useful to discuss tax share elasticity. In our theoretical framework, the median voter is assumed to be in charge of expenditure decisions. And, the median voter, who earns the median income, does really pay the municipal taxes.

2.3 The Non-linear Expenditure System

The non-linear expenditure system (NLES) to which we refer is an extension of the LES proposed and tested by Carlevaro (1976, 1977, 1982). If we consider the general case (with n goods), the NLES can be written as

q_{i} = m_{i} + β_{i} \frac{y - \sum_{i = 1}^{n} p_{i} m_{i}}{p_{i}}, i = 1, \dots, n

(8)

with

β_{i} = β_{i}^{0} + β_{i}^{1} Φ (\frac{y - \sum_{i = 1}^{n} p_{i} m_{i}}{P}), i = 1, \dots, n

where, $q_{i}$ denotes the demand for good i, $p_{i}$ its unit price and $m_{i}$ the minimum value. P is an index price.

The coefficients $β_{i}^{0}$ and $β_{i}^{1}$ have to check the two following linear restrictions

\sum_{i = 1}^{n} β_{i}^{0} = 1

and

\sum_{i = 1}^{n} β_{i}^{1} = 0

Thus, if we formulate the municipal expenditure function in a quadratic form, we obtain

p_{g} g = p_{g} m_{g} + β_{1} (y_{m} - p_{g} m_{g} - γ) + β_{2} (y_{m} - p_{g} m_{g} - γ)^{2}

(9)

Then, the income elasticity can be written as

E (g / y_{m}) = \frac{β_{1} + β_{2} (y_{m} - p_{g} m_{g} - γ)}{ω}

with

ω = \frac{p_{g} g}{y_{m}}

Therefore, the income elasticity is positive only if $β_{1} + β_{2} (y_{m} - p_{g} m_{g} - γ)$ is positive.

And the price elasticity is

E (g / p_{g}) = - 1 + [1 - β_{1} - β_{2} (y_{m} - p_{g} m_{g} - γ)] \frac{m_{g}}{g}

This specification completes the LES in two ways. First, the expenditure function is quadratic in income, which can be a more suitable formulation of a demand equation according to Aronsson et al. (2000). Secondly, if $1 - β_{1} - β_{2} (y_{m} - p_{g} m_{g} - γ) < 1 - β$ , the price elasticity derived from the NLES is lower than the one derived from the LES and then closer to −1 (which is the case if we consider the estimates obtained in Table 2 and the elasticity values in Table 4).

3. Application to Municipal Public Expenditure

Following the translating method developed by Pollak and Wales (1981), one way to propose a better empirical specification and to reduce unexplained variation in expenditure behaviour is to postulate that the parameter measuring the minimum required local public services is a function of the characteristics of the municipality. Following the description of the data, we thus develop the translating method in the second sub-section.

3.1 Overview of the French Local Public Sector and Data Description

We first offer a brief description of the local public sector in France and then introduce the data selected for empirical analysis.

Institutional background

The French local public sector includes four overlapping administrative divisions in France. In order, from the lowest level up, there are 36 680 municipalities, 2599 groups of municipalities (referred to in the remainder of the paper as intercommunity groups), 100 departments and 22 metropolitan regions. Municipalities form the lowest level of local government in France and finance about 60 per cent of total local spending.

French decentralisation is based on a unitary model as opposed to a federal model. All municipal governments therefore have the same responsibilities and source of funding even though they vary in size and in particular population structure. French municipalities provide a wide range of major public services. General administration, town planning, running water, garbage collection, road maintenance and public transport in the municipal area are services intended for the entire population. Primary education (from 3 to 10 years old) and expenditures related to poverty are services intended for specific users. In France, the majority of public expenditures associated with poverty are financed by the national government and the departments. Departments are indeed specialised in providing minimum benefits paid to those concerned by the loss of independence. Therefore, municipalities do not provide specific services for the elderly. The poverty-related expenditures at the municipal level consequently are relatively small—provision of public services to low-income groups at reduced (social housing units) or zero prices (public transport, museums). According to Le Grand (1975b), this kind of aid can be viewed as a form of price discrimination. However, especially in large cities, a part of poverty-related expenditure burdens also comes from indirect poverty-related expenditures as mentioned by Pack (1998).

Each municipality has its own budget that mainly is financed by local tax resources (about 50 per cent) and by grants from the national governments (about 40 per cent) and borrowing (about 10 per cent). Under the period under study, French municipalities had three main tax instruments at their disposal: a business tax (taxe professionnelle) and two local taxes on households: a residence tax paid by occupants of rental housing (taxe d’habitation) and a building property tax paid by homeowners (taxe foncière bâtie).

Data description

As the median voter model has been found to apply to the lowest tier of localities and is appropriate for explaining the aggregate behaviour of governments, we decided to explain the total expenditure per capita at the municipal level. A discussion on this subject may be found in Turnbull and Djoundourian (1994). To deal with heterogeneity in terms of population size, we concentrate here on the 109 French municipalities with over 50 000 inhabitants in 2005 (excluding Paris). The year 2005 is representative of the current situation of French municipalities, posterior to the latest decentralisation laws of 2003. Data are provided by the Ministries of the Economy and Interior.

The dependent variable is total municipal expenditure per capita and is measured by the operating expenditure plus gross savings which cover the repayment of loans for past investments. Such an approach seems desirable with cross-section data because investment expenditure is discontinued in the time dimension. Running water and garbage collection spending are not in those figures.

The set of available variables, measured in 2005, and summary statistics for all variables are presented in Table 1.

Table 1.

Dataset description: 109 French municipalities of more than 50 000 inhabitants in 2005

Description of variables	Mean	Minimum	Maximum
Total municipal expenditures per capita (€)	1 141	731	3 137
Yearly median municipal income (€)	9 314	5 184	36 286
Tax share (per cent)	0.24	0.12	0.41
Municipal population	106 107	50 070	807 070
Number of social benefit beneficiaries (for 1000 habitants)	1 670 (14)	72	19 679
Number of secondary residences	1 798	90	23 560
Number of school pupils	16 874	7 010	132 000
Number of social houses (for 1000 inhabitants)	12 005 (120)	600	69 200
Road length (metres)	224 357	27 000	1 250 000

Among the 109 municipalities with over 50 000 inhabitants, 86 belong to an intercommunity group (80 per cent) and 50 (46 per cent) have a left-wing government. As explained previously, we consider the median income increased by the per capita national grants received by the municipality.

3.2 Determinants of the Incompressible Public Expenditure

Pollak and Wales (1981) developed the translating method in a demand specification to allow subsistence parameters to depend on demographic variables. In recent cross-section analyses, Gaudin et al. (2001), Aaberge and Langorgen (2003) and Allers and Elhorst (2011) have also implemented the translating method to estimate the demand function for public services in a LES specification.

In the LES, the minimum level of consumption is by definition independent of price and income. Therefore, it depends here on variables measuring the unit cost of the public services and the number of public users (as the price variable is measured here by the tax share). The choice of these variables inevitably is constrained by the availability of data.

If we consider the municipal general services intended for the whole of the population, we expect that the per capita public expenditure will decrease with an increase in population size (economies of scale in consumption). However, for more specific services, an increase in each category of users is likely to have an opposite, positive effect on expenditure needs. Therefore, the municipal population is differentiated by distinguishing the main categories of specific users of municipal public services in France. We then differentiate the number of school pupils (variable pupils) and the number of recipients of social benefits to deal with the provision of public services to a low-income group at reduced or zero price (variable social beneficiaries) from the rest of the municipal population (variable pop) in equation (11).

The cost of providing public services is supposed to depend on the size of the public capital stock. Taking into account the main investment expenditures in French municipalities, several variables such as the number of municipal social housing units $Y_{1}$ , the number of secondary residences $Y_{2}$ or the road network length $Y_{3}$ may be used. The introduction of the number of secondary residences is a proxy used to capture costs of public investment in terms of leisure or tourism.

Therefore, $m_{g}$ can be defined by the following system of equations (10–14)

m_{g} = α_{1} N_{c} + α_{2} Y_{1} + α_{3} Y_{2} + α_{4} Y_{3}

(10)

where, $N_{c}$ measures the size of the population using the municipal public services, according to

N_{c} = pop + δ_{1} pupils + δ_{2} social beneficiaries

(11)

Equations (12)–(14) enable us to include the effect of the intercommunity group on the unit costs of the public services since membership in an intercommunity group is supposed to create economies of scale (in production). $D (Intercommunity)$ denotes a dummy variable that takes the value of 1 if the municipality does not belong to an intercommunity group, zero otherwise. We also try to identify partisan effects with the introduction of another dummy variable $D (Left)$ , which takes the value of 1 when the municipality has a left-wing incumbent, zero otherwise

α_{2} = α_{20} + α_{21} D (Intercommunity) + α_{22} D (Left)

(12)

α_{3} = α_{30} + α_{31} D (Intercommunity) + α_{32} D (Left)

(13)

α_{4} = α_{40} + α_{41} D (Intercommunity) + α_{42} D (Left)

(14)

where, $α_{1}$ , $α_{20}$ , $α_{21}$ , $α_{22}$ , $α_{30}$ , $α_{30}$ , $α_{31}$ , $α_{32}$ , $α_{40}$ , $α_{41}$ , $α_{42}$ , $δ_{1}$ and $δ_{2}$ are the parameters to be estimated. Equations (10)–(14) are substituted in the municipal expenditure function to obtain one equation with non-linearity in parameters.

4. Empirical Results

The estimation results for the linear and non-linear expenditure systems are reported (4.1) and compared with the results derived from the method generally used in the literature (4.2).

4.1 Expenditure Systems

We present the results from estimating the linear expenditure system and its quadratic extension. Insignificant values of parameters obtained in preliminary estimations finally have been set equal to 0. In the same way, the value of the subsistence private consumption is fixed to be 0 to obtain consistent estimates.² As the Breush–Pagan test reveals the presence of heteroscedasticity, robust standard errors are computed in a heteroscedastic-consistent matrix. Empirical results obtained using a non-linear least squares estimate to deal with the non-linear coefficients are listed in Table 2.

Table 2.

Estimation results of the linear and the non-linear expenditure systems (N = 109)

Specification	LES	NLES
Coefficient estimates	Value (probability)	Value (probability)
$β_{1}$ (marginal budget share)	0.081 (0.000)***	0.136 (0.000)***
$β_{2}$ (marginal budget share, quadratic form)		−0.2810⁻⁵ (0.000)***
$α_{1}$ (population)	−0.024 (0.000)***	−0.013 (0.000)***
$α_{20}$ (social housing, common effect)	0.198 (0.00)***	0.097 (0.003)***
$α_{21}$ (social housing, intercommunity effect)	0.246 (0.00)***	0.198 (0.00)***
$α_{22}$ (social housing, left-wing effect)	0	0
$α_{30}$ (secondary residences, common effect)	0.198 (0.00)***	0.155 (0.00)***
$α_{31}$ (secondary residences, intercommunity effect)	0	0
$α_{32}$ (secondary residences, left-wing effect)	0	0
$α_{40}$ (road length, common effect)	0.0039(0.005)**	0.0009 (0.005)**
$α_{41}$ (road length, intercommunity effect)	0	0
$α_{42}$ (road length, left-wing effect)	0	0
$δ_{2}$ (social beneficiaries)	0	0
$δ_{1}$ (pupils)	0	0
Lm het p-value	0.00***	0.00***
Schwarz BIC	799	776
MAPE (mean absolute percentage error)	19.22	15
Mean value of expenditure needs	361(31 per cent)	149(13 per cent)

Notes: significance level: *** for 1 per cent; ** for 5 per cent and * for 10 per cent.

For each municipality, the minimum required local public services ${\hat{m}}_{g}$ can be computed using estimates parameters and equations (10)–(14).³ The corresponding spending per capita values then are obtained by multiplying ${\hat{m}}_{g}$ by the corresponding tax share. Finally, average values change substantially across the specifications: the average is around 13 per cent of total per capita municipal expenditure with the non-linear expenditure system, yet around one-third if we consider the linear expenditure system.

The NLES has the smallest SBIC and MAPE (mean absolute percentage error) and therefore appears to be the best predictive model. As $β_{2}$ is significant, left-over income has a non-linear effect on municipal per capita expenditures (see equation (9)). This explains the divergence between the findings of the linear and the non-linear approaches. One implication is the greater values of price elasticities obtained with the NLES, as shown in Table 4.

To delve further into the interpretation of the ‘incompressible’ per capita municipal public consumption, we can derive marginal effects according to

\begin{matrix} LES : {\hat{m}}_{g} = - 0.024 pop \\ + [0.198 + 0.246 D (Intercommunity)] \\ Social housing + 0.198 Secondary residences \\ + 0.0039 road length \end{matrix}

(15)

\begin{matrix} NLES : {\hat{m}}_{g} = - 0.013 pop \\ + [0.097 + 0.198 D (Intercommunity)] \\ Social housing + 0.155 Secondary residences \\ + 0.0009 road length \end{matrix}

(16)

Results reveal a negative and significant impact of population size on total per capita local public expenditure

\frac{\partial (p_{g} g)}{\partial pop} = \frac{\partial (p_{g} g)}{\partial m_{g}} \frac{\partial m_{g}}{\partial pop}

This effect is stronger in the LES specification: an increase of 1000 inhabitants will result in a decrease of 5 euros in per capita local expenditure for the LES (respectively -2.8 euros for NLES).

Conversely, whatever the specification, a greater number of social housing, secondary residences and road length significantly increase the incompressible per capita expenditure. However, these effects are weaker if we consider the NLES specification, which explains differences in mean values of expenditure needs. Estimates obtained with the LES show that an increase of 100 social housing units will generate an increase of around 4 euros in per capita total municipal spending (respectively +2 euros with the NLES). Furthermore, this positive effect is larger when the municipality does not belong to an intercommunity group (+9.7 euros in the LES case and +6.5 in the NLES case).

The number of secondary residences also has a positive impact on per capita municipal spending: 100 additional secondary residences will generate an increase of around 4 euros in per capita spending in the LES case (respectively +3 euros in the NLES specification). Whatever the specification considered, an increase in road length also has a positive impact upon municipal spending, with 1000 additional metres generating an increase of less than 1 euro per capita.

Finally, the numbers of social beneficiaries and of pupils have no significant impact on the per capita public spending level ( $δ_{1} = δ_{2} = 0$ ) whatever the specification, nor does the partisan effect ( $α_{22} = α_{32} = α_{42} = 0) .$

We now develop the results obtained with the standard method in the second sub-section. Price and income elasticities derived from the three specifications will then be presented and compared.

4.2 Traditional Method and Comparison of Alternative Specifications

We initially used the same variables as in the expenditure systems, including specific effects (multiplying explanatory variables by the two dummy variables), but insignificant values of parameters were obtained in preliminary estimations and finally were removed. Table 3 reports final OLS estimates obtained with a heteroscedastic-consistent matrix.

Table 3.

Standard linear specification

Explanatory variables	Coefficient estimates (probability)
Constant	1012 (0.000)***
Income	0.022 (0.173)
Tax share	−980 (0.027)***
Population	−0.0021 (0.161)
Social housing (without intercommunity group)	0.065 (0.000)***
Secondary residence	0.054 (0.02)**
Road length	0.00019 (0.381)
Lm het p-value	0.00***
Schwarz BIC	757
MAPE	13 per cent
Adjusted R²;	0.66

Notes: significance level: *** for 1 per cent; ** for 5 per cent and * for 10 per cent.

This specification offers the lowest SBIC value, but results reveal insignificant values relating to the influence of road length, population and income. The coefficients for tax share and income also have the predicted sign but are statistically different from zero for tax share only. The coefficients of the other variables are similar to those obtained with the NLES.

We turn now to the distributions of income and price elasticities obtained for the three different specifications which are described in Table 4.

Table 4.

Comparison of income and price elasticities of demand estimated with the different specifications

	Specification	LES	NLES	Standard
Price elasticity	Mean	−0.72	−0.90	−0.22
	Minimum	−1.14	−1.44	−0.36
	Maximum	−0.10	−0.42	−0.064
Income elasticity	Mean	0.69	0.68	0.18
	Minimum	0.32	−1.92	0.08
	Maximum	2.32	1.01	0.62

Price elasticities obtained with the standard method (last row) range from −0.36 to −0.064. Taken as a whole, they are lower (in absolute values) than their expenditure system counterparts. Indeed, price elasticities vary from −1.14 to −0.10 for the LES (respectively −1.44 to −0.42 for the NLES). The income elasticity derived from the expenditure systems also reveal greater values than values obtained with the standard approach, with mean values equal to 0.69 for LES, 0.68 for NLES and 0.18 for the standard specification if we retain the coefficient 0.022. Furthermore, the estimated income elasticity is always positive except for one municipality (−1.92 with the NLES), which therefore can be considered as an outlier. The price elasticities derived from the expenditure systems must be greater than −1, by construction. However, 10 estimated values lower than −1 are obtained. All other estimated values suit this constraint.

5. Concluding Remarks

The estimation of demand function for publicly provided goods and services hypothesises that the level of expenditures is explained in an ad hoc linear regression form. The main innovation in this paper is methodological as the median voter’s satisfaction is described by a Stone–Geary utility function. We are then able to specify the public spending function in a straightforward manner from the median voter model.

Such a specification provides a richer diagnostic of local expenditure determinants than the one conventionally used. First, by separating the variables influencing the municipal expenditure from price and income effects, we are able to measure and define the minimum required per capita public spending. The main results show that this is between 13 and 33 per cent of the total public spending. It decreases with population level and membership in an intercommunity group. As French municipalities are required to cover the costs of equipment maintenance, results confirm that minimum required per capital public spending increases with the size of the public capital stock.

Secondly, we are able to propose a non-constant formulation of the price and income elasticities. Thus, we obtain higher values, in absolute terms, than those obtained with the traditional method. This is compatible with the existence of economies of scales in consumption.

Therefore, the linear expenditure system retained in this article provides initial results which are thought to be quite promising. Indeed it calls for further investigations to revisit the numerous topics previously discussed in the literature (fiscal illusion, crowding parameter estimates, etc.), using estimation of public expenditure functions.

Footnotes

Acknowledgements

I am grateful to an anonymous referee and to the Editor Kenneth Gibb for their helpful comments and suggestions on the earlier versions of this paper.

Notes

Funding Statement

Part of this research has been funded by the French Àgence Nationale de la Recherché (ANR), under the grant ANR–08–GOUV–054.

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