Is there an optimal size for local governments? A spatial panel data model approach

ABSTRACT

The paper presents a framework for determining the optimal size of local jurisdictions and whether it varies depending on the geographical heterogeneity of the territory. To that aim, it first develops a theoretical model of cost efficiency that takes into account spatial interactions and spillover effects among neighbouring jurisdictions. The model solution leads to a spatial Durbin panel data specification of local spending as a non-linear function of population size. The model is tested using a large local data set over the period 2003–11 for an aggregate measure of public spending. The empirical findings suggest a ‘U’-shaped relationship between population size and the costs of providing public services. A second step investigates the role of geographical characteristics such as elevation and terrain ruggedness in the determination of the optimal jurisdiction size. The results reveal that optimal city size decreases with elevation and increases with ruggedness.

KEYWORDS:

• optimal government size
• spatial panels
• Spanish municipalities
• topography

JEL:

• C11
• C23
• H4
• H7

ACKNOWLEDGEMENTS

The authors gratefully acknowledge the associate editor and three anonymous referees for their numerous insightful comments and constructive suggestions that helped to improve the work significantly. They also thank the participants at the 3rd International Conference on ‘Decentralization after the Great Recession: Fine-tuning or Paradigm Change?’, Santiago de Compostela, Spain, October 2017, for very helpful comments.

DISCLOSURE STATEMENT

No potential conflict of interest was reported by the authors.

Notes

1. See Slack and Bird (2012) and Tavares (2018) for a review of the empirical literature on the existence of economies of scale, the efficiency gains and the democratic outcomes of consolidated local governments.

2. For papers focusing on the optimal size of a coalition and its characteristics, see, for example, Alesina and Spolaore (1997, 2005), Blume and Blume (2007), Bolton and Roland (1997), Gordon and Knight (2009), Saarimaa and Tukiainen (2010) and Hanes (2015).

3. Unlike Solé-Ollé and Bosch (2005), where the omission of relevant spatial interaction terms could lead to bias/inconsistent and inefficient estimates (Elhorst, 2014; LeSage & Pace, 2009), we estimate a panel data version of the SDM that allows one to quantify with accuracy the magnitude of spatial spillovers, thus minimizing the possibilities of overstating or underestimating the optimal municipal size in this context. On the other hand, the analysis displays two key advantages when compared with the cross-sectional analysis based on the general nesting spatial model (GNSM) of Hortas-Rico and Salinas (2014). First, unlike the GNSM, the SDM does not suffer from parameter identification issues (Elhorst, 2014).

4. The empirical analysis carried out here presents some similarities with respect to Bastida et al. (2013) and Rios et al. (2017), as they also employ spatial econometric techniques to model local government spending. However, they do not focus on the link between population and spending. Rios et al. (2017) control for population density but do not include population; Bastida et al. (2013) do not consider the possibility of a non-linear effect.

5. This function F is assumed to satisfy the following assumptions: (1) F is continuous, twice differentiable, positive, displays diminishing marginal products and constant returns to scale (i.e., $\alpha +\text{β}=1$); and (2) F satisfies the Inada conditions.

6. Note that:

$[I_N-\psi W{]}^{-1}={\sum }_r^{\mathrm{\infty }}{\psi }^r{W}^r=1/1-\psi$given that ${\sum }_r^{\mathrm{\infty }}{W}^r\psi =\psi \mathrm{a}\mathrm{n}\mathrm{d}{\sum }_r^{\mathrm{\infty }}{\psi }^r=1/1-\psi \mathrm{i}\mathrm{f}\left|\psi \right|<1.$

7. $r_{it}$ is assumed to be the constant across municipalities owing to the perfect mobility of capital.

8. The signs of the first and second partial derivatives of $f\left(N_{it}\right)$ depend on the sign and relative magnitude of the population parameters.

9. For an alternative approach to investigate non-linear effects in panels with large T while accounting for cross-sectional heterogeneity, cross-sectional dependence and feedback effects among the dependent and independent variables, see Chudik, Mohaddes, Pesaran, and Raissi (2017).

10. Accordingly, the recent reform of the local administration (27/2013 Act of Rationalization and Sustainability of Local Administration) establishes measures to encourage the voluntary merger of municipalities and shifting services of municipalities of fewer than 20,000 inhabitants upwards to the provincial councils. Recent examples of territorial reforms in Europe with a similar spirit can be found in Denmark, Greece, Germany and Switzerland. For a more detailed review, see Steiner, Kaiser, and Eythorsson (2016).

11. Given that some of these control variables are not the main objective of the present study, they are discussed here only in brief. See Ladd and Yinger (1989) and Ladd (1992, 1994) for a review of arguments that justify their inclusion in the local spending model.

12. Specifically, public provision is compulsory for all municipalities in services such as rubbish collection, street cleaning, water supply, sewerage and street lighting, among others. Municipalities with a population greater than 5000 inhabitants additionally have to provide parks, public libraries and solid waste treatment. Municipalities with a population greater than 20,000 have to provide local police and social services. Finally, municipalities with a population greater than 50,000 inhabitants also have to provide public transport and environmental protection.

13. However, at least in Spain, this does not necessarily translate into different spending levels because local government tends to provide services even without explicit official responsibility. It is ultimately citizens’ demands (and lack of intervention by other layers of government) that help to explain why a service is provided. Thus, the relationship between the level of responsibility and local spending might not be as evident as it might seem at first (Solé-Ollé & Bosch, 2005).

14. This implies assuming perfect mobility of labour among the municipalities within the same province.

15. These variables have been provided by Goerlich and Cantarino (2010). According to them, the inclusion of both variables is justified on the grounds that they are almost completely unrelated and both exhibit a very different spatial distribution.

16. These estimated parameters do not refer to the cost function but to the expenditure function. Nonetheless, cost effects can be recovered by dividing the total effects by $1/\left(1+\eta \right)$, where $\eta$ is the estimated total effect of the tax share.

17. The equation implied by our estimates is given by the total effects: $-0.832+0.045\ast 2N\ast =0$.

18. This procedure makes inference slightly more complex than that of linear models with interaction effects or with spatial effects, but without interaction terms. The reason is that in a spatial model with interaction terms, the researcher cannot directly draw from the variance–covariance matrix of the parameters and the estimated parameters $\mathrm{V}\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{o}\mathrm{v}\left(\mu ,\text{β},\theta ,\sigma \right)$ but, instead, the draws have to be taken from ${\mathrm{\Sigma }}_{\mathrm{T}\mathrm{E}}=\mathrm{V}\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{o}\mathrm{v}\left(\mathrm{\partial }{Y}_t/\mathrm{\partial }{X}_t\right)$.

19. To calculate the size of the flypaper effect from our log–log regression and make it comparable with previous studies, elasticities are transformed using:

$\left(Y_t/T{r}_t\right)\mathrm{T}{\mathrm{E}}_{Tr}-\left(Y_t/I_t\right)\mathrm{T}{\mathrm{E}}_I,$

where $\mathrm{T}{\mathrm{E}}_{Tr}$ and $\mathrm{T}{\mathrm{E}}_I$ denote the total effects obtained with the log–log specification for transfers and income, respectively; and $Y_t/T{r}_t$ and $c{Y}_t{I}_t$ are the ratio of spending to grants and income, respectively.

20. According to our data, about 70% of the municipalities with fewer than 1000 inhabitants were included in the high-altitude subsample.

21. Available data show that the number of population clusters is higher in rugged locations, but, at the same time, the average distance between the population clusters within those municipalities is lower.

22. The quality of the results obtained when estimating the pooled model is less than that obtained by the fixed-effects model (R² = 50.8% versus 89.5%), and, therefore, they should only be taken as indicative. In fact, the likelihood ratio tests on the joint significance of the fixed effects is LR = 76,824.60 (d.f. = 5521), with p = 0.00, suggesting that the simplified model should not be employed to perform inference. Obviously, the fact that in this setting we are not including spatial fixed effects precludes one from performing any comparison with the implied N* reported in Table 1.

23. In particular, we estimate alternative specifications of the following augmented SDM version:$\begin{array}{rl}{Y}_t& =\alpha +\lambda Crisi{s}_t+{\gamma }_1{N}_t+{\gamma }_2{N}_t^2+{\gamma }_3{G}_t+{\gamma }_4{G}_t^2+{\gamma }_5{N}_t\\ & \times G_t^2+{\gamma }_6{N}_t^2\times G_t^2\\ & +{\delta }_{1}W{N}_t+{\delta }_{2}W{N}_t^2+{\delta }_{3}W{G}_t+{\delta }_{4}W{G}_t^2\\ & +{\delta }_5{N}_t\times G_t^2+{\delta }_{6}W{N}_t\times G_t^2\\ & +\sum _k{X}_{t,k}{\text{β}}_k+\sum _{k}W{X}_{t,k}{\theta }_k+{ϵ}_t\end{array}$where N denotes the logarithm of the population; G is the key geographical variable of the regression model interacting with N; and X includes the set of previously defined demand and cost factors (including population density and the other geographical variable). Solving for N in $\left(\mathrm{\partial }{\mathbf{Y}}_{\mathit{t}}/\mathrm{\partial }{\mathbf{N}}_{\mathit{t}}\right)=0$ allows one to obtain the optimal population size N* as a function of G. The simulation of the effects for elevation and terrain ruggedness are reported in Tables A6 and A7 in Appendix A in the supplemental data online, whereas the median forecasts of N* stemming from the effects of models (1) to (6) in Tables A6 and A7 are shown in Figures A2 and A3 (online), respectively.

Additional information

Funding

This work was supported by the Ministerio de Economía y Competitividad (Spain) [grant number ECO2016-76681-R].