# Assessing French departments' spending efficiency over time

Kassoum Ayouba, Marie-Line Duboz, Julie Le Gallo

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Kassoum Ayouba, Marie-Line Duboz, Julie Le Gallo. Assessing French departments' spending efficiency over time. Applied Economics, 2023, online, pp.1-26. 10.1080/00036846.2023.2167925. hal-04166497

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Assessing French departments’ spending efficiency over time

Kassoum AYOUBA<fnref n="1" />, Marie-Line DUBOZ<fnref n="2" /> and Julie LE GALLO<fnref n="3" />

**Abstract**

We analyze way in which spending efficiency of French departments evolved from 2007 to 2019. We use the time-dependent conditional order-$m$ approach used allows to account for the heterogeneity among departments, characterized by contextual variables, and to evaluate the effect of these variables. The results reveal a continuous improvement in their performance (management of total expenditures) over time and show that departments have much more room for improvement in the management of investment expenditures than in the management of operating costs. Moreover, the time variable has a positive effect on performance (shape of the frontier and catch-up) for both the operating and the investment specific models.

**JEL codes:** C14, C61, H21, H50

**Key words:** Efficiency, Local governments, Expenditures, Conditional model, Contextual variables

<footnote n="1">¹ UMR Territoires, Université Clermont Auvergne, AgroParisTech, INRAE, VetAgroSup, kassoum.ayouba@inrae.</footnote>

<footnote n="2">² CRESE EA3190, Université de Bourgogne Franche-Comté, marie-line.duboz@univ-fcomte.fr</footnote>

<footnote n="3">³ CESAER UMR0141, Institut Agro, INRAE, Université de Bourgogne Franche-Comté, julie.le-gallo@agrosupdijon.fr, corresponding author.</footnote>
# 1 Introduction

As in many other developed countries, local governments play an important role in providing public goods and services to citizens in France. In 2020, they accounted for 19% of French public spending.<fnref n="1" /> There are three subnational levels of French government: regions (régions), departments (départements) and municipalities (communes). One specific feature of departments is that the French decentralization process made them responsible for providing social services.<fnref n="2" /> The evolution of their finances over time should therefore be analyzed in the light of this major responsibility.

The level of their revenues depends on the type of revenue source (own taxes and grants) and on the sensitivity of these revenue sources to the economic cycle, while the level of their expenditures, a large proportion of which is devoted to social services, essentially depends on economic fluctuations, that is, the economic cycle. This means that departments regularly face a “scissors effect”: when activity drops, their revenues fall, while spending increases due to the need for social welfare programs. In times of crisis, like the subprimes crisis in 2008 or the COVID-19 crisis in 2020, the “scissors effect” is even stronger, since tax revenue losses are larger whereas the crisis leads to higher spending on subsistence wage and social protection.

Alongside the effect of a crisis or, more generally, a slowdown in economic activity, a structural effect also occurs. In 2021, continuing the downward trend in the weight of direct taxes in departmental operating revenue, the property tax (Taxe foncière sur les propriétés bâties, TFPB) was transferred to the municipalities and was replaced by an indirect tax on

<footnote n="1">¹ This figure and those in the rest of the introduction come from the report by the Finance and Local Public Management Observatory (Observatoire des Finances et de la Gestion Publique Locales, 2021).</footnote>

<footnote n="2">² The decentralization process in France is characterized by several stages known as “Act I”, “Act II” and “Act III”. Competencies regarding social aid were mainly transferred from central government to departments’ responsibilities after the law of July 22, 1983. The competencies of departments were gradually extended later with other laws, such as the law of December 18, 2003 and the law of December 1, 2008.</footnote>
added value (*Taxe sur la valeur ajoutée*, TVA). This reform brought about a new loss of fiscal power for the departments, calling into questions their funding model. Revenues increasingly correlated with economic growth are matched by largely counter-cyclical expenditures.

As has already happened in the past, with increasing difficulty, departmental authorities face a complex situation: satisfying demand for public goods and services in terms of both quantity and quality, which implies growing expenditure, and, simultaneously, complying with increasing constraints on resources. More than ever, the question of improving their public spending performance (i.e., their efficiency) has become crucial.

Over the three last decades, local governments' spending efficiency has been extensively studied by economists with a focus on the evaluation of public policies: Narbón-Perpiñá & De Witte (2018a, b), in their systematic review, identified 84 empirical studies on local government efficiency analysis from 1990 to August 2016. Spending efficiency refers to the ability of local governments to provide a maximum amount of output for a given level of inputs, or to use a minimum level of inputs for a given output. Local government efficiency has been analyzed in multiple contexts and countries: for example, Belgian municipalities by De Borger & Kerstens (1996); German municipalities by Geys et al. (2013), Tuscan municipalities by D'Inverno et al. (2018); Spanish local governments by Narbón-Perpiñá et al. (2019) and Spanish municipalities by Cordero et al. (2020); Portuguese municipalities by Cordero et al. (2017); Norwegian municipalities by Bruns & Himmler (2011) and US municipalities by O'Loughlin & Wilson (2021).

The efficiency literature can be divided into two main strands of research. On the one hand, some studies focus on efficiency in specific areas of public good provision such as waste collection, street cleaning, road maintenance and water services. On the other hand, other studies evaluate local performance from a global perspective considering that local authorities
provide a wide variety of services and facilities. While the former studies are of interest and
usually motivated by data availability, they do not allow an assessment of the overall
performance of local governments. Moreover, according to Cordero et al. (2017), this type of
study has a major drawback: it is difficult to know which parts of the local inputs are assigned
to each public service. This observation explains why many studies, including ours, prefer the
second approach, which addresses the global efficiency of local government.

Among the pioneering works on local government efficiency, De Borger & Kerstens (1996) point out that estimated mean efficiency scores are highly dependent on the reference technology considered. Consequently, to measure public sector performance accurately, more research to identify the “best” reference technology is necessary. Narbón-Perpiñá et al. (2019) emphasized the fact that there is no commonly accepted methodology to measure efficiency and Aiello & Bonanno (2019) show, through their meta-analysis, that methodological choices lead to heterogeneity in results.

As summarized by D'Inverno et al. (2018), the available methods to infer the unknown and unobservable “efficiency frontier” from data can be classified into two distinct approaches: the econometric approach (“stochastic frontier analysis” – SFA) and the optimization nonparametric approach (“data envelopment analysis” – DEA). This non-parametric methodology has been often used in the local government efficiency literature because of “its conceptual simplicity and its versatility” (Balaguer-Coll & Prior, 2009). With DEA, only the convexity assumption is required to estimate the production function converting inputs into outputs. No assumption on the functional form of the production function or on the distribution of efficiencies is needed. An extension of the DEA has also been proposed, called the “free disposal hull” (FDH): its main difference from DEA is that it drops the convexity assumption.
In this analysis, we opt not to maintain the convexity assumption and therefore to consider the
FDH framework. This is in line with O'Loughlin and Wilson (2021) whose results strongly suggest non-convexity of the United States local government production set.<fnref n="3" />

As a deterministic approach, FDH is sensitive to outliers and extreme values: since it includes all observed input-output combinations, the efficient frontier might be affected by the observations that are extreme points. Consequently, any outliers may influence the estimated frontier as well as the estimated efficiency scores. To mitigate this problem, we follow the “partial” frontiers framework suggested by Cazals et al. (2002). We use their order-$m$ approach involving the concept of partial frontier, opposed to the traditional full frontier, which reduces the influence of outliers and extreme values. For local government efficiency assessment, the order-$m$ approach has been used, among others, by Narbón-Perpiñá et al. (2019).

Moreover, Cordero et al. (2017) emphasize the need to consider that local authorities usually operate in highly heterogeneous contexts in terms of demographic, economic or geographical characteristics. It is therefore necessary to account for the effect of contextual variables when measuring the spending efficiency of these local authorities, as it is done in a growing literature exploring this question of heterogeneity among local entities, reviewed in Narbón-Perpiñá and De Witte (2018b).

Our paper addresses the measurement of the spending efficiency of French departments and the way it evolved from 2007 to 2019. We chose to focus on departments since, although they represent nearly 30% of local authority expenditure (28.3% in 2019), little work has been devoted to them. Indeed, while the literature on public expenditure in France is relatively abundant for municipalities (Guengant, 1999; Breunig & Rocaboy, 2008; Frère et al., 2014), it has paid little attention to departments, even less so on the question of the efficiency of public spending. To the best of our knowledge, the question of the efficiency of public spending in

<footnote n="3">³ The appendix contains a discussion on convexity.</footnote>
French departments has only been empirically addressed by Seifert & Nieswand (2014) for one year, 2008, using DEA.

We approach the question of the French departments’ efficiency differently, such that our contribution is twofold. First, taking advantage on recent developments in the literature on conditional efficiency measures (Bădin et al., 2012, Mastromarco & Simar, 2015), we use a time-dependent conditional order-m model. This allows us to consider heterogeneity among departments, characterized by a set of contextual variables, and to evaluate the effect of these variables. Second, we use the available decomposition of total expenditures into operating expenditures and investment expenditure over a long period (2007-2019), thanks to the database of the General Direction on Local Jurisdictions (Direction Générale des Collectivités Locales, DGCL).

The paper is organized as follows. Section 2 introduces the methodology. Section 3 provides a short description of the institutional context and describes the data. Section 4 presents the results. Section 5 highlights our main conclusion.

# 2 Methodology

## 2.1 From unconditional to conditional robust efficiency scores

We consider a set of n departments using a set of inputs $x$ ($x \in \mathbb{R}_+^p$) to provide services, denoted $y$, to the population ($y \in \mathbb{R}_+^q$). The production technology $\Psi$ used to convert inputs into services (i.e. outputs) is the set of all feasible input-output combinations:

$$ \Psi = \{(x, y) \in \mathbb{R}_+^{p+q} | x \text{ can produce } y\} \quad (1) $$

Following Cazals et al. (2002) and Daraio & Simar (2005), under the free disposability assumption of inputs and outputs, the unconditional feasible set $\Psi$ can be characterized by:
$$ \Psi = \{(x, y) \in \mathbb{R}_+^{p+q} | H_{XY}(x, y) > 0\} \quad (2) $$

with $H_{XY}(x, y) = \text{Prob}(X \le x, Y \ge y)$ representing the probability of dominating a unit operating at level $(x, y)$. It provides the probability of finding another unit that produces as much output while using no more input. Therefore, $\Psi$ is the support of $H_{XY}$. Throughout this paper, we denote elements of the observed sample at hand by lowercase letters $(x, y)$ whereas uppercase letters $(X, Y)$ denote the unknown random variables which can take any value.

The (unconditional) input-oriented technical efficiency of a production plan $(x, y)$, defined as the proportionate decrease in inputs (holding output levels fixed) required to ensure that there is zero probability of being dominated, is given by:

$$ \theta(x, y) = \inf\{\theta|(\theta x, y) \in \Psi\} = \inf\{\theta:S_{X|Y}(\theta x, |y)>0\} \quad (3) $$

where $S_{X|Y}(x, y) = \text{Prob}(X \le x|Y \ge y) = \frac{\text{Prob}(X \le x, Y \ge y)}{\text{Prob}(Y \ge y)} = H_{XY}(x, y)/F_Y(y)$ is the conditional survival function of $X$ given that $Y \ge y$. $S_{X|Y}(x, |y)$ is a nonstandard conditional survival function as the condition is on $Y \ge y$ and not $Y = y$. $\theta(x, y) = 1$ indicates that the evaluated production plan is fully efficient, whereas with $\theta(x, y) < 1$ the evaluated production plan is inefficient.

The Free Disposal Hull (FDH) estimator of $\theta$ (denoted $\hat{\theta}_{\text{FDH}}$) is obtained by replacing $S_{X|Y}$ with its empirical analogue in equation (3), using the observed sample: $\chi_n = \{(x_i, y_i)|i = 1, \dots, n\}$:

$$ \hat{S}_{X|Y}(x|y) = \frac{\sum_{i=1}^{n} \mathbb{P}(x_i \le x, y_i \ge y)}{\sum_{i=1}^{n} \mathbb{P}(y_i \ge y)} \quad (4) $$

where $\mathbb{P}$ is the indicator function.
The estimator $\hat{\theta}_{\text{FDH}}$ may be sensitive to outliers and atypical observations since it is based on the full observed sample. Therefore, to obtain robust measures, we follow Cazals et al. (2002) and make use of the “partial frontier”, which no longer envelops all production plans. Specifically, we consider an order-$m$ frontier, which is defined as the expected frontier when considering a random set $\Psi^m$ of $m$ random production plan with $Y \ge y$. Since atypical observations are not part of the subsample $\Psi^m$ in every draw, the potential detrimental impact of these observations on the $\theta_{\text{FDH}}$ is mitigated. The order-$m$ efficiency score $\theta_{\text{FDH}}^m$ has the following expression:

$$ \theta_{\text{FDH}}^{m}(x, y) = \int_{0}^{\infty} [1 - (1 - S_{X|Y}(ux|y))^{m}] du \quad (5) $$

As for $\hat{\theta}_{\text{FDH}}$ we can obtain the estimator for the order-$m$ efficiency $\hat{\theta}_{\text{FDH}}^m$ by replacing $S_{X|Y}$ with $\hat{S}_{X|Y}$ in equation (5). When $m$ increases and converges to $\infty$, $\hat{\theta}_{\text{FDH}}^m \to \hat{\theta}_{\text{FDH}}$. Also, the share of the super-efficient production plan (i.e with $\hat{\theta}_{\text{FDH}}^m > 1$) is inversely related to the value of $m$.

The estimation of departments' performance might be affected by contextual factors $z \in \mathbb{R}_+^r$ that are beyond their control. To ensure a fair comparison among units, the efficiency analysis should account for these factors. To this end, following the idea of Cazals et al. (2002) and Daraio & Simar (2007), extended by Mastromarco & Simar (2015) who introduce time as an additional conditioning variable, we define the attainable set $\Psi_t^z \in \mathbb{R}_+^{p+q}$ as the support of the conditional probability $H_{X,Y|Z}^t(x,y|z) = \text{Prob}(X \le x, |Y \ge y|Z = z, T = t)$. The conditional input-oriented technical efficiency of a production plan $(x, y) \in \Psi_t^z$, at time $t$ and facing conditions $z$, is defined as:

$$ \theta^t(x, y|z) = \inf\{\theta | (\theta x, y) \in \Psi_t^z\} = \inf\{\theta : S_{X|Y,Z}^t(\theta x|y, z) > 0\} \quad (6) $$
where $S_X^t(x|y, z) = \text{Prob}(X \le x, |Y \ge y, Z = z, T = t)$.

The effects of atypical observations on conditional input-oriented efficiency can also be mitigated, replacing $S_{X|Y}(ux|y)$ in equation (5) by $S_{X|Y,Z}^t(ux|y, z)$. Moreover, as above the conditional robust FDH estimator of $\theta^t$, denoted $\hat{\theta}_{FDH,m}^t(x, y|z)$ is obtained by replacing $S_{X|Y,Z}^t$ with its empirical analogue in equation (5) computed with the observed sample: $\chi_n^t = \{(x_{i,t}, y_{i,t}, yz_{i,t})|i = 1, \dots, n \text{ and } t = 1, \dots, s\}$. However, unlike the unconditional case, the estimation of $S_{X|Y,Z}^t$ in the conditional case is less straightforward. It requires the use of smoothing techniques in z and t:

$$ \hat{S}_{X|Y}^t(x|y, z) = \frac{\sum_{j=(i,v)} \P(x_i \le x, y_i \ge y) K_{h_z}(z, z_i) K_{h_t}(v, t)}{\sum_{j=(i,v)} \P(y_i \ge y) K_{h_z}(z, z_i) K_{h_t}(v, t)} \quad (7) $$

where $j = (i, v)|z - h_z < z_{i,v} < z + h_z; t - h_t < v < t + h_t$. $K_{h_t}(\cdot)$ and $K_{h_z}(\cdot)$ are respectively, a univariate kernel function with compact support and an r-variate kernel (with compact support) product. The time variable has a discrete nature so that discrete kernels could be used for it. However, as pointed out in Li & Racine (2007), the usual continuous kernels are more appropriate when the discrete variables can take many different values, as is the case in our application. Also, the environmental variables z can be both discrete and continuous. This implies the use of both discrete and continuous kernels. Bandwidths $h_t$ and $h_z = (h_1, \dots, h_r)$ of appropriate size are selected by the data-driven procedure suggested by Bădin et al. (2012), based on the least squares cross-validation (LSCV) approach.

## 2.2 Influence of conditioning variables on efficiency

Following Bădin et al. (2012), we examine the influence of conditioning variables $(z, t)$ both (i) on technological change (shift of the frontier) and (ii) on technological catching-up
(distribution of efficiency scores). The first effect can be investigated by considering the ratios of conditional to unconditional efficiency measures, which are measures relative to the full frontier of the conditional and the unconditional attainable sets, respectively. Since some atypical production plans may hide the real effect of (z, t), we follow the suggestion in Daraio and Simar (2007) and do the analysis with order-m frontier, with large values of m to get robust estimates of the full frontier. The ratio to be analyzed is:

$$R_{I,m}(x, y|z, t) = \frac{\theta_m^t(x, y|z)}{\theta_m(x, y)} \qquad (8)$$

In our input-oriented framework, a tendency of $R_{I,m}(x, y|z, t)$ to decrease (resp. increase) with the conditioning variables indicates a favorable (resp. unfavorable) effect of these variables on local governments technological change. The analysis of the ratio in equation (8) for small values of m enables us to investigate the second effect, i.e the influence of conditioning variables on the distribution of efficiency scores (technological catching-up). The interpretation of this ratio is the same as above. To perform this analysis and investigate the statistical significance of conditioning variables on $R_{I,m}(x, y|z, t)$, one can replace $R_{I,m}(x, y|z, t)$ by its estimate using the sample of observed production plan and specify the following model:

$$\hat{R}_{I,m}(x, y|z, t) = g(z, t) + \varepsilon \qquad (9)$$

where $\varepsilon$ is the error component with $E(\varepsilon|z, t)$. Following Racine & Li (2004) and Daraio & Simar (2014), we use kernel weighted local-linear least squares. The following local-linear least squares minimization problem has to be solved:

$$\min_{(\hat{\alpha}, \hat{\beta})} \sum_{i=1}^{n} \{\hat{R}_{I,m} - \hat{\alpha} - \hat{\beta}(w_i - w)\}^2 K_{h_w}(w, w_i) \qquad (10)$$
with $w = (z, t)$. $\hat{\alpha}$ and $\hat{\beta}$ are the solutions of the above minimization problem such that $\hat{\alpha} = \hat{\beta}(w)$ and $\hat{\beta} = \hat{\beta}(w)$ are consistent estimators of the true conditional mean function $g(w) = E(R_{I,m}|w)$ and the gradients $\hat{\beta}(w) = \partial E(R_{I,m}|w)/\partial w$. $K_{h_w}$ is a generalized product kernel function with properties as in equation (7). The bandwidths used are again estimated by LSCV. The statistical significance of the conditioning variables is analyzed by employing procedures described in Racine et al. (2006). These are implemented in the R package np (Hayfield & Racine, 2008).

In this paper, we go a step further and investigate input-specific radial efficiency and its
determinants. In the first stage, this means that we optimize one input ($x_1$) and the other inputs
($x_2$) are inactive in the process: $x = x_1 + x_2$. This is common in the standard DEA framework
and input-specific radial efficiency are often called sub-vector input efficiency (Färe et al.,
1983). Theoretically, decisions makers often control some inputs more than others, optimizing
radially on all inputs may not be appropriate. Specifically, when $x_1$ is considered active (i.e.,
controllable), we look for the maximum possible reduction in its level relative to the technology
given the level of inactive input $x_2$ (i.e., noncontrollable) and given the level of the outputs.
We extend the sub-vector input approach in a radial order-$m$ conditional framework. Some
straightforward adjustments are necessary to perform this estimation. For example,
$S_{X|Y}(x|y)$ should be replaced by $S_{X_1|X_2,Y}(x_1|x_2, y) = \text{Prob}(X_1 \le x_1|X_2 \le x_2, Y \ge y)$. The
logic of the second stage remains the same.
# 3 Institutional context and data

## 3.1 The departments: the French intermediate level of local government

The French local public sector consists of three layers of decentralized governments. Since 2020, in mainland France, the upper layer embodies 13 regions, the intermediate one includes 96 departments and the lower tier contains 34 839 municipalities. Local government entities must respect a golden rule: they must vote in a budget that balances revenues and expenditures. This equilibrium rule is required for each section of the budget (Frouté, 2021): both the operating and the investment sections must be balanced. Equilibrium in the investment section can be reached thanks via borrowing, but interest payments are considered as operating expenditure.

In 2015, the last major reform of local governments, named *loi NOTRe*, for a new territorial organization of the French Republic, certainly led to a reduction in the competencies of the departments to the benefit of the regions, especially in terms of economic development. However, the departments remain the competent authority to promote solidarity and territorial coherence. Therefore, expenditures on welfare represent the largest part of departments' operating expenditure. On average, social services absorb more than half of this expenditure and fall into four main categories: child welfare (*Aide sociale à l'enfance*), benefits for the disabled (*Prestation de compensation du handicap*), aid to the elderly (*Allocation personnalisée d'autonomie*) and minimal subsistence wage (*Revenu de solidarité active*). On average, local roads and, to a lesser extent, secondary schools represent half of the departments' investment expenditure (excluding repayment of departmental debt).

Two further points are worth emphasizing. First, they are very heterogeneous, in particular in terms of population. Indeed, the most populated is *Nord* (with 2,641,081 inhabitants) and the
least is *Lozère* (with 76,800 inhabitants). Second, like regions before them, in 2021 departments
lost their last direct tax, the property tax, when it was transferred to the municipalities and
replaced by an indirect tax on added value. As they have no power over governmental grants,
this raises the question of the departments' fiscal autonomy and more importantly, the need to
monitor and improve their performance in terms of spending.

## 3.2 Data and variables

We focus on departments located in mainland France from 2007 to 2019. We remove four
departments from our initial sample. *Corse-du-Sud, Haute-Corse* (the two departments of the
*Corsica* island), and *Paris* have a specific tax system with implications in the expenditure
measurement. We also remove *Rhône* as its delimitation, competencies and hence tax system
changed in 2015 when it became a metropolis (*metropole*), implying a rupture in the series.
Our final sample consists of 1196 observations: 92 departments observed from 2007 to 2019.

As in O'Loughlin & Wilson (2021), the variable used as an input is the total expenditure (TOT EXPEN). Also, O'Loughlin & Wilson (2021, p. 2685) mentioned that since this input measures costs, their input-oriented efficiency estimates can be interpreted as estimates of cost efficiency or input overall efficiency. In our analysis, following De Borger and Kerstens (1996), we assume that departments face identical input prices, meaning that there are no allocative inefficiencies in the analysis and that technical efficiency and cost efficiency coincide. Then, our input-oriented technical efficiency estimates give an idea of how much departments might feasibly reduce costs while holding outputs constant. Finally, when departments face different prices, inference on allocative, technical and cost efficiencies is possible following Simar and Wilson (2020).
The total expenditure is expressed in constant prices (2015) and contains operating (OPE
EXPEN) and investment (INV EXPEN) expenditures. Operating expenditures, those that occur
each year, include the remuneration of employees and electives and also other expenditures
such as social aid (which represented 62% of operating expenditures in 2019), interest
payments, general operating costs, etc. This category is the most widely used input indicator
(Narbón-Perpiña and De Witte, 2018a). Moreover, as other papers reviewed in the same
reference, we also analyze investment expenditures, which contain all investments in education
and infrastructures (departments build and maintain secondary schools and operate certain local
roads), debt amortization, etc. We report in Table 1 descriptive statistics on these variables for
the whole sample.

<caption>**Table 1:** Descriptive statistics of expenditure variables (in 2015 millions euros)</caption>

The departments' total expenditures mainly consist of operating expenditures (on average, the ratio of investment to total expenditures is around 20%). Regarding the evolution over time of these variables (Table A1 in the appendix), we observe that the average ratio of investment to total expenditures decreased continuously from 28% in 2007 to 16% in 2019.<fnref n="4" /> Since the average total expenditures is relatively stable over the years, this suggests that French departments are reducing their investment expenditures in order to provide the increasing number of services needed by citizens (see the evolution of outputs averages in Tables A2 to A4). As noted by Blochliger et al. (2010) in a study on sub-central governments' spending, investment is easier to reduce in the face of budget constraints, while "current spending is often politically sensitive or mandated and thus difficult to change". From these observations, it is

<footnote n="4">⁴ This characteristic in the evolution of French departments’ spending over these years is extensively commented on and analyzed in Duboz et al. (2021).</footnote>
worth analyzing whether there are any differences in the way i) departments manage their operating and investment expenditures, and ii) contextual variables affect the management of these two types of expenditures. We pursue both these aims in the empirical section.

Regarding outputs, we select seven variables proxying the level of services provided by departments. The choice is both driven by the adequacy to the departments' specific competences and the availability of data. Hence, we first use the length (in kilometers) of the road network (ROAD) as a proxy for the construction and maintenance of roads. The education services provided are approximated by the number of pupils in public secondary schools (PUB PUPILS). The main competences of departments are social services, we therefore include three variables capturing this dimension: the number of beneficiaries of the Personalized Autonomy Allocation (*Allocation personnalisée d'autonomie*, BENEF APA), the number of beneficiaries of Disability Compensation Prestation (*Prestation de compensation du handicap*, BENEF PCH), and the number of beneficiaries of Childhood Social Aid (*Aide sociale à l'enfance*, BENEF ASE). Social welfare payments are represented by the number of beneficiaries of minimal subsistence support (*Revenu de Solidarité Active* and *Revenu Minimum d'Insertion*, BENEF RSA/RMI). Finally, we use the total population (TOT POP) as a proxy for general administrative services. As pointed out by Narbón-Perpiñá and De Witte (2018a), this is the indicator most frequently used in local governments efficiency analysis as it can act as a good proxy for all the needed administrative tasks provided by local governments in the absence of more direct outputs. Descriptive statistics corresponding to these variables are presented in the first 7 rows of Table 2 (whole sample). Tables A2 to A4 present the evolution of these variables over time from 2007 to 2019, confirming the high increase of the number of beneficiaries of the various social services over the period (+68% for BENEF RSA/RMI, +23% for BENEF APA, +35% for BENEF ASE and +707% for BENEF PCH).
<caption>**Table 2:** Descriptive statistics of original outputs and composite indicator</caption>

Given that the correlations between all output variables, but ROAD, are high and range from 0.68 and 0.98 (Table A5), we follow Wilson (2018) and reduce the outputs' space dimension (i.e., aggregate outputs) through Principal Component Analysis (PCA) to avoid the curse of dimensionality of non-parametric estimation. This approach has been followed among others by Cordero et al. (2020). As exact aggregation of micro-macro relations is impossible (Van Daal & Merkies, 1984), PCA may lead to efficiencies in the reduced space that are different from the efficiencies in the original space. Furthermore, converting the original dimension into a reduced dimension may partially lose the original physical characteristics of the output variables. Consequently, it may prevent from learning about the production process transforming inputs to outputs (e.g., marginal rate of transformation between outputs, Lee & Cai, 2020). With PCA, we identify one component that represents 85.15% of the information provided by the six original correlated output variables (Table A6). All six original variables are positively and strongly correlated with this component, with correlations ranging from 83% to 96% (Table A7) and well represented (Table A8). Therefore, we include this component (named composite index, COMP INDEX) as second output along with the variable ROAD. As in Cordero et al. (2020), the value of the composite index (COMP INDEX) is rescaled by adding the maximum negative value to all the values to ensure that the COMP INDEX value is positive for all departments. The descriptive statistics (whole sample) and dynamics over the years for COMP INDEX are presented in the last row of Table 2 and in the last rows of Tables A2 to A4 respectively, showing an increase of 52% over 2007-2019.

We are interested in analyzing how the demographic, financial, economic and political context in which the departments operate affect their performance in providing services to
citizens. To this end, in our paper we consider four continuous variables, namely population
density (DENSITY) for the demographic context, level of debt (DEBT) for the financial
context, income level of the living population (INCOME) and unemployment level
(UNEMPLY) for the economic context. We also add one discrete variable, political color (POL
COL), for the political context, taking three modalities related to the political majority in the
department council: left wing, center (reference) or right wing. Descriptive statistics on these
variables are presented in Tables 3 and 4 (whole sample). Their evolution over the period is
presented in Table A9 for the continuous variables and A10 for the political color categorical
variable. In particular, debt per inhabitant has increased by 19% over the period.

**Table 3:** Descriptive statistics of continuous contextual variables included in the analysis

**Table 4:** Distribution of the discrete variable (POL COL) in %

As departments are heterogeneous, we analyze the average of the inputs, outputs, and
contextual variables according to the population size of departments (Table A11). We split the
departments into four groups of equal number of observations: Extra-small, Small, Large and
Extra-large. All inputs and original outputs but ROAD increase with population size. The
absence of clear relation between ROAD and population size may be explained by the
existence, in big departments, of a dense public transport network, which complement roads.
Regarding contextual variables, on average, population size decreases with debt but increases
with income and population density. The unemployment level seems unrelated to the
population size. Globally, it is therefore relevant to explore the distribution of efficiencies
according to department size.
# 4 Results

## 4.1 Unconditional scores

To estimate the unconditional scores, we use a robust order-$m$ model with two outputs (ROAD and COMP INDEX) and one input (TOT EXPEN or OPE EXPE or INV EXPEN) to characterize the partial technology used by departments. In line with Daraio & Simar (2005), we determined the value of $m$ (i.e., the size of the partial frontier) as the value for which the change in percentage of super-efficient observations with $m$ decreases smoothly with $m$. In our analysis, $m = 200$. The unconditional robust input-oriented efficiency scores obtained are displayed in Table 5.

<caption>**Table 5:** Distribution of unconditional robust efficiency estimates (2007 – 2019)</caption>

The average efficiency score is 0.8663 for total expenditure: on average, there exists room for improvement in departments' management of their total expenditure. The five best performing departments are *Territoire de Belfort* in 2018 (1.3962), *Lozère* in 2019 (1.3179) and in 2007 (1.3068), *Territoire de Belfort* in 2019 (1.2807) and *Lozère* in 2016 (1.2749). Conversely, the five worst performing are *Hauts-de-Seine* in 2011 (0.4078), in 2008 (0.4216), in 2007 (0.4217), in 2010 (0.4321) and in 2009 (0.4371). Figure A1 maps the unconditional efficiency scores for total expenditure for 2007, 2011, 2015 and 2019. The spatial distribution remains overall quite stable. Using a DEA estimation on total expenditures as a single input for only one year, 2008, Seifert & Nieswand (2014) also find significant room to improve spending efficiency: French departments could save a significant amount of inputs while providing the same level of output. In addition, as we do, they find that *Territoire de Belfort* and *Lozère* are very efficient departments.
A look at the disaggregation between operating and investment expenditures shows that French departments have much more room for improvement in the management of investment expenditures than in the management of operating ones. Indeed, the average score for investment expenditures is 0.6349 compared to 0.9015 for operational expenditures. A possible interpretation of these results is that departments' operating expenditures are somewhat constrained (no matter what, they must pay employees) whereas investment expenditures are more discretionary. Their level depends on the departments' medium- and long-term strategies which are likely to be very heterogeneous (context-specific).

Figure 1 shows an almost continuous improvement in the average unconditional efficiency of departments over the period 2007-2019.<fnref n="5" /> In addition, they become much less heterogeneous in terms of performance over the period, as shown by the continuous decrease in the coefficient of variation. This is likely to indicate, from 2007 to 2019, an improvement in the departments' technology, and also much less dispersed (technological catch-up) units. We pursue these exploratory observations in detail later in this paper.

<caption>**Figure 1.** Evolution of the mean and variation coefficient of unconditional efficiencies over time (top panel: mean, bottom panel: standard deviation)</caption>

Table 6 reports the distribution of the average unconditional efficiency (across the whole sample and over time) by departments' population size groups. Contrary to Seifert & Nieswand (2014) who catch hold of heterogeneity between departments with their territorial size and find no impact of this variable on the departments' spending efficiency, we observe that small units in terms of population seem to outperform large ones. This result suggests that departments do not face similar environments and that unconditional efficiencies may not produce a fair

<footnote n="5">⁵ This is confirmed by average growth rates of efficiency scores (Table A12).</footnote>
ranking of the units under analysis. The corresponding tables for operating expenditures and investment expenditures are Tables A13 and A14, respectively, with similar findings.

<caption>**Table 6:** Mean unconditional efficiency per department size and its evolution over time for total expenditures</caption>

## 4.2 Conditional scores

### 4.2.1 Conditional efficiency estimates

Table 7 displays the estimated conditional robust efficiency scores considering the five contextual variables presented in section 3 and the year. On average, conditional efficiency estimates are higher than unconditional ones. Indeed, departments are now benchmarked against similar ones in term of context (i.e., smaller subsamples in the evaluation process). The overall mean conditional efficiency value is 0.9851 for total expenditure: on average, when comparing “like for like”, there is little room for departments to reduce their expenditure given the level of services. However, as the minimum and the first quartile values indicate, some departments can reduce expenditures even when compared to their likes. This is the case of Seine-Saint-Denis (0.7691) Aude (0.8277) and Haute-Garonne (0.8315). The high level of average conditional efficiencies estimated also means that our exogenous variables explain most of the efficiency differentials detected with the unconditional model and are therefore relevant.<fnref n="6" /> D’Inverno et al. (2022) also estimated high conditional municipal service provision scores for 307 Flemish municipalities in the period 2006–2011.

<footnote n="6">⁶ Descriptive statistics on optimal estimated bandwidth confirm this observation (Table A15). Indeed, the median values of bandwidth estimates have reasonable (narrow) values, predicting significant influence on the production process. High maximum values can be a result of smoothing out, for some departments, of an insignificant variable.</footnote>
Looking at the disaggregation between operating and investment expenditures, we observe, as for the unconditional efficiency scores, that there is much more room for improvement in the management of investment expenditures compared to operating expenditures when efficiency scores are adjusted (i.e., when contextual variables considered). This reflects the fact that investment is a risky enterprise. Finally, following Bădin et al. (2012), we calculate “pure efficiency”, i.e., the efficiency scores cleaned by the effect of the contextual variables which are not under the control of local decision makers. Table A16 reports descriptive statistics on operational, investment and global “pure” efficiencies. Results indicate that, on average, the operational expenditures are better managed than investment ones.<fnref n="7" />

<caption>**Table 7:** Distribution of conditional robust efficiency estimates (2007 – 2019)</caption>

### 4.2.2 Influence of contextual variables on performance for the total expenditure model

We now examine more formally the influence of contextual variables on the boundary of the frontier and on the distribution of efficiency scores. To this end, as explained in Section 2, we non parametrically regress the ratio of the conditional to unconditional efficiency measures for total expenditures (and for full and partial frontiers) on the contextual variables. Figures 2 and 3 show the behavior of the full frontier ratio and partial frontier ratio respectively as marginal functions of contextual variables and help to assess the effects on the boundary and on the distribution of efficiencies.

<caption>**Figure 2:** Marginal effects of exogenous variables on the conditional-to-unconditional efficiency ratio (full frontier)</caption>

**Figure 3:** Marginal effects of exogenous variables on the conditional-to-unconditional efficiency ratio (partial frontier, $m = 1$)

<footnote n="7">⁷ Large (positive) value of “pure efficiency” indicates a department which has poor performance, and small (negative) value indicates very good performance.</footnote>
The level of debt (DEBT) has no influence on the shape of the technology used by French departments (Figure 2) and almost no influence on the distribution of their efficiency scores distribution (Figure 3). This is in line with Pérez-López et al. (2015) on a sample of Spanish municipalities. Population density (DENSITY), while not affecting the technology used in our analysis, has a negative impact on the distribution of efficiencies. This would mean, in line with Doumpos & Cohen (2014) among others, that with an increasing density, the provision of public services is slightly more complex, thus less efficient (i.e., there are agglomeration diseconomies). Also, the ratio of conditional to unconditional efficiency measures for the full frontier increases with INCOME, which then has a negative effect on the technology used to provide services. This would mean that high income departments are less motivated to monitor their expenditures and/or that some services/characteristics of services provided by these departments to citizens are missing from our analysis (e.g., quality of services). The negative association is less pronounced with values of INCOME from 12,000€ to 14,000€ per inhabitant. The effect of INCOME on technological catch-up (distribution of efficiencies) is similar, although more linear and less pronounced than the effect found on the shape of the frontier. It can be also noted that lower (from 4% to 8%) and higher (from 13% to 16%) unemployment rates affect in a very slight positive way the estimated technology, while for moderate values, there is no discernible effect. Indeed, a good socio-economic departments' situation (i.e., a low unemployment rate) implies lower spending on social services and therefore tends to increase the estimated efficiency. For higher rates of unemployment, the positive effect could be explained by the fact that a poor socio-economic situation could imply lower demand for high-cost or high-quality public services. This variable has almost no effect on the distribution of efficiencies inside the frontier of the estimated technology.
We observe that the estimated ratio of conditional to unconditional efficiencies decreases over the years (Figure 2), highlighting an improvement over time in the technology used by French departments (i.e., with the same level of expenditures, they are able to deliver more services). This could be explained by the decrease in the share of investment expenditures in total expenditures (from 28% in 2007 to 14% in 2019). Indeed, as investment expenditures translate less in direct services for citizens than operational ones, with a constant budget, reducing investment expenditures in favor of operational ones enable to provide more services. This may also mean that the quality of services provided (missing characteristic) has declined over years. The effect of time is also positive on the distribution of efficiencies (technological catch-up), although less pronounced than the effect on the technology (Figure 3). To sum up, it should be noted that while it has a positive impact on both efficiency (technological catch-up) and technological change (shifts in the frontier), the role of time is more important in accelerating technological change for departments. Finally, the political color of the departments shows no significant association with the shape of the technology or the distribution of efficiencies.

### 4.2.3 Influence of contextual variables on performance for operating and investment expenditure models

We also investigate the influence of contextual variables on the boundary of the estimated frontier and on the distribution of estimated efficiency scores, distinguishing between operating and investment expenditures. Figures 4 and 5 show the behaviors of the full frontier ratio as marginal functions of contextual variables for the operating expenditures and the investment expenditures models respectively.
**Figure 4:** Marginal effects of exogenous variables on the conditional-to-unconditional efficiency ratio (full frontier)– operating expenditure model

**Figure 5:** Marginal effects of exogenous variables on the conditional-to-unconditional efficiency ratio (full frontier)– investment expenditure model

Regarding variables INCOME and YEAR, in terms of direction, their effect on the estimated operating expenditures technology and investment expenditure technology are almost the same as the effect found for the total expenditure technology. However, in terms of magnitude, these variables have a much higher impact on the investment expenditure technology than on that of operating expenditures. For example, consider the time variable. While it has a positive effect on technological change for both models, its estimated effect is greater on the investment expenditure technology. This would suggest that as investment expenditures are decreasing over time, French departments are discarding the riskier investments. Variables DEBT and DENSITY, while having no influence on the shape of the operating expenditure technology used by French departments, do appear to affect the investment expenditure technology. For the variable DEBT, the estimated effect is nonlinear (positive trend until 800, then negative) whereas for the variable DENSITY, it is almost linear with an increasing trend. This would mean that the management of investment expenditures is more complicated with an increasing population density. The effects of unemployment rates on the estimated operating expenditure technology and investment expenditure technology are relatively different. Up to 10%, this variable appears to have no association with the operating expenditure technology. From 10% to 13%, the association is negative, then it turns positive (from 13%). Regarding the investment expenditure technology, it should be noted that lower (from 4% to 10%) and higher (from 13% to 16%) unemployment rates positively affect the estimated technology, while for moderate values, there is a slight negative effect. The effects
of unemployment rates on the investment technology and on the total expenditure technology
are similar in terms of direction, but different in terms of magnitude. Lastly, political color has
no effect on either technology.

Figures 6 and 7 illustrate the effect of our contextual variables on the distribution of operating expenditures and investment expenditures efficiencies. For the operating expenditure model, the effects of contextual variables on the distribution of efficiencies are similar to those observed on the technology. However, the effect of INCOME, while linear on the expenditure technology, is nonlinear on the distribution of expenditure efficiencies. Regarding the investment model, differences exist for certain contextual variables. The variable DEBT has a nonlinear effect on the investment technology, while it has almost no effect on the distribution of efficiencies. The unemployment rate is positively associated with the distribution of investment efficiencies on its support as a whole. This contrasts with the effect found on the technology, which is negative from 10% to 13% and positive on the remaining support of the variable. Finally, for the investment expenditure model, the estimated effects on the catching-up process are lower (in terms of magnitude) than those on the technology.

**Figure 6:** Marginal effects of exogenous variables on the conditional-to-unconditional efficiency ratio (partial frontier)—operating expenditure model

**Figure 7:** Marginal effects of exogenous variables on the conditional-to-unconditional efficiency ratio (partial frontier)—investment expenditure model

# 5 Conclusion

Against a backdrop of increasing constraints on the resources of French departments, we
propose an analysis of their spending efficiency and its dynamics. We use a time-dependent
conditional order-$m$ approach which allows to account for outliers and heterogeneity among departments (characterized by a set of contextual variables) in the estimation of efficiency. Using a sample of 1196 observations (92 departments observed from 2007 to 2019), the results reveal a continuous improvement in the performance of French departments (management of total expenditures) over time, mainly thanks to technology enhancement. In addition, we use the decomposition of total expenditures into operating expenditures and investment expenditures. We find several interesting results.

First, the average ratio of investment to total expenditures decreased continuously, from 28% in 2007 to 14% in 2019. Since the average total expenditures are relatively stable over the years, this would mean that French departments reduced their investment expenditures to cater to the increasing needs of citizens. Second, departments have much more room for improvement in the management of investment expenditures than in the management of operating costs, probably because operating expenditures are somewhat constrained, whereas investment expenditures are more discretionary. Third, the time variable has a positive effect on performance for both the operating and investment specific models. However, the estimated effects are greater in the investment model, meaning that, as investment expenditures decreased over time, French departments discard the riskier investments.

Our results clearly underline the specificity of investment expenditures compared to operating expenditures when studying the efficiency of local government. Since the ability of the departments to foster growth depends on their ability to invest, the need to increase efficiency in their investment choice should not be underestimated, as there is room for improvement. Rather than ruling out investments that might seem risky, departments should perhaps favor greater cooperation with each other to finance their investments when they reach
a critical size. Cooperation between departments should help to fund projects and yield
economies of scale in a context of limited resources.

Indeed, several departments now seem to have chosen this solution, as highlighted by a
recent survey made by the French Department Assembly (Assemblée des Départements de
France, 2020). The modalities of the cooperation actions vary from one department to another
(e.g., creation of an investment fund for the environment by Sarthe, Mayenne and Maine-et-
Loire, gradual pooling of road networks for Loire and Haute-Loire, joint development of
economic and tourist attractiveness for Cher and Indre), but all express an aspiration for a
renewal of decentralization, based on a possible pooling of resources, as a mean to improve
their efficiency. The next phase of our work may consist in evaluating the effects of these ties
in terms of expenditure incurred and public goods provided.
# 6 Declarations and ethics statements

## Ethical Approval

This article does not contain any studies with human participants or animals performed by any of the authors.

## Informed consent from participants

All authors of this paper have consented to its submission and its publication.

## Data availability statement for basic sharing policy

Data will be made available: https://github.com/ayoubakassoum/local_gvt

## Conflict of Interest

Authors declare that they have no conflict of interest.

## Funding

Authors have no funding to declare.

## Acknowledgements

We would like to thank Kristoff De Witte for providing his R code. We adapt and use parts of this code in our setting.
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Tables and figures

<table><caption>Table 1: Descriptive statistics of expenditure variables (in 2015 millions euros)</caption><thead><tr><th></th><th>Min</th><th>Max</th><th>Mean</th><th>V</th></tr></thead><tbody><tr><td>OPE_EXPEN</td><td>74.3918</td><td>2625.3659</td><td>548.6013</td><td>0.7532</td></tr><tr><td>INV_EXPEN</td><td>13.6132</td><td>581.5011</td><td>133.2378</td><td>0.7783</td></tr><tr><td>TOT_EXPEN</td><td>113.5290</td><td>2997.8573</td><td>681.8392</td><td>0.7317</td></tr></tbody></table>

<table><caption>Table 2: Descriptive statistics of original outputs and composite indicator</caption><thead><tr><th></th><th>Min.</th><th>Max.</th><th>Mean</th><th>CV</th></tr></thead><tbody><tr><td>ROAD (kilometers)</td><td>221.6670</td><td>7764.0000</td><td>4014.9128</td><td>0.3709</td></tr><tr><td>PUB PUPILS (No. of pupils)</td><td>2295.0000</td><td>96115.0000</td><td>24726.9415</td><td>0.7639</td></tr><tr><td>BENEF RSA/RMI (No. of benef.)</td><td>718.0000</td><td>111101.0000</td><td>13807.5485</td><td>1.0839</td></tr><tr><td>BENEF APA (No. of beneficiaries)</td><td>1919.0000</td><td>49371.0000</td><td>12278.7734</td><td>0.6345</td></tr><tr><td>BENEF ASE (No. of beneficiaries)</td><td>88.0000</td><td>12632.0000</td><td>1613.6338</td><td>0.9174</td></tr><tr><td>BENEF PCH (No. of beneficiaries)</td><td>69.0000</td><td>15623.0000</td><td>2116.0769</td><td>0.8570</td></tr><tr><td>TOT POP (No. of inhabitant)</td><td>76800.0000</td><td>2641081.0000</td><td>649812.2667</td><td>0.7161</td></tr><tr><td>COMP INDEX (Unitless)</td><td>0.0000</td><td>16.0296</td><td>2.7298</td><td>0.8284</td></tr></tbody></table>

<table><caption>Table 3: Descriptive statistics of continuous contextual variables included in the analysis</caption><thead><tr><th></th><th>Min</th><th>Max</th><th>Mean</th><th>CV</th></tr></thead><tbody><tr><td>DEBT (2005 euro/inhabitant)</td><td>0.0000</td><td>1488.9845</td><td>522.1124</td><td>0.4808</td></tr><tr><td>DENSITY (inhabitant/km2)</td><td>14.7689</td><td>9129.7079</td><td>346.1336</td><td>3.6231</td></tr><tr><td>INCOME (2005 euro/inhabitant)</td><td>10441.1374</td><td>24775.1032</td><td>13238.1559</td><td>0.1353</td></tr><tr><td>UNEMPLY (%)</td><td>4.1000</td><td>16.0000</td><td>8.9526</td><td>0.2238</td></tr></tbody></table>

<table><caption>Table 4: Distribution of the discrete variable (POL COL) in %</caption><thead><tr><th>right</th><th>left</th><th>center</th></tr></thead><tbody><tr><td>50.1672</td><td>47.2408</td><td>2.5920</td></tr></tbody></table>
<table><caption>Table 5: Distribution of unconditional robust efficiency estimates (2007 - 2019)</caption><thead><tr><th rowspan="2"></th><th colspan="3">Input</th></tr><tr><th>TOT EXPEN</th><th>OPE EXPEN</th><th>INV EXPEN</th></tr></thead><tbody><tr><td>Min.</td><td>0.4078</td><td>0.4706</td><td>0.0803</td></tr><tr><td>1st Qu.</td><td>0.7744</td><td>0.8339</td><td>0.3589</td></tr><tr><td>Median</td><td>0.8861</td><td>0.9362</td><td>0.5899</td></tr><tr><td>Mean</td><td>0.8663</td><td>0.9015</td><td>0.6349</td></tr><tr><td>3rd Qu.</td><td>0.9784</td><td>1.0000</td><td>1.0000</td></tr><tr><td>Max.</td><td>1.0531</td><td>1.0000</td><td>1.3962</td></tr><tr><td>SD</td><td>0.1415</td><td>0.1071</td><td>0.3019</td></tr></tbody></table>
<table><caption>Table 6: Mean unconditional efficiency per department size and its evolution over time for total expenditures</caption><thead><tr><th></th><th>Extra small</th><th>Small</th><th>Large</th><th>Extra large</th></tr></thead><tbody><tr><td>2007</td><td>0.8620</td><td>0.7674</td><td>0.7251</td><td>0.7212</td></tr><tr><td>2008</td><td>0.8398</td><td>0.7540</td><td>0.7199</td><td>0.7346</td></tr><tr><td>2009</td><td>0.8345</td><td>0.7485</td><td>0.7381</td><td>0.7371</td></tr><tr><td>2010</td><td>0.8618</td><td>0.8015</td><td>0.7694</td><td>0.7589</td></tr><tr><td>2011</td><td>0.8784</td><td>0.8328</td><td>0.7913</td><td>0.7664</td></tr><tr><td>2012</td><td>0.9002</td><td>0.8494</td><td>0.8191</td><td>0.7984</td></tr><tr><td>2013</td><td>0.9040</td><td>0.8620</td><td>0.8441</td><td>0.8073</td></tr><tr><td>2014</td><td>0.9462</td><td>0.9024</td><td>0.8912</td><td>0.8448</td></tr><tr><td>2015</td><td>0.9580</td><td>0.9094</td><td>0.8963</td><td>0.8605</td></tr><tr><td>2016</td><td>0.9833</td><td>0.9222</td><td>0.9085</td><td>0.8624</td></tr><tr><td>2017</td><td>1.0119</td><td>0.9404</td><td>0.9297</td><td>0.9049</td></tr><tr><td>2018</td><td>1.0755</td><td>0.9742</td><td>0.9783</td><td>0.9414</td></tr><tr><td>2019</td><td>1.0683</td><td>0.9777</td><td>0.9767</td><td>0.9550</td></tr><tr><td>Total</td><td>0.9298</td><td>0.8651</td><td>0.8450</td><td>0.8252</td></tr></tbody></table>

<table><caption>Table 7: Distribution of conditional robust efficiency estimates (2007 - 2019)</caption><thead><tr><th rowspan="2"></th><th colspan="3">Input</th></tr><tr><th>TOT EXPEN</th><th>OPE EXPEN</th><th>INV EXPEN</th></tr></thead><tbody><tr><td>Min.</td><td>0.7691</td><td>0.8369</td><td>0.2321</td></tr><tr><td>1st Qu.</td><td>0.9776</td><td>1.0000</td><td>0.9199</td></tr><tr><td>Median</td><td>1.0000</td><td>1.0000</td><td>1.0000</td></tr><tr><td>Mean</td><td>0.9851</td><td>0.9953</td><td>0.9270</td></tr><tr><td>3rd Qu.</td><td>1.0000</td><td>1.0000</td><td>1.0000</td></tr><tr><td>Max.</td><td>1.0000</td><td>1.0000</td><td>1.0000</td></tr></tbody></table>
<figure><img src="image_2.png" /><figcaption>Figure 1. Evolution of the mean and variation coefficient of unconditional efficiencies over time (top panel: mean, bottom panel: standard deviation)</figcaption></figure>
<figure><img src="image_3.png" /><figcaption>Figure 2: Marginal effects of exogenous variables on the conditional-to-unconditional efficiency ratio (full frontier)</figcaption></figure>
<figure><img src="image_4.png" /><figcaption>Figure 3: Marginal effects of exogenous variables on the conditional-to-unconditional efficiency ratio (partial frontier, m = 1)</figcaption></figure>
<figure><img src="image_5.png" /><figcaption>Figure 4: Marginal effects of exogenous variables on the conditional-to-unconditional efficiency ratio (full frontier)—operating expenditure model</figcaption></figure>
<figure><img src="image_6.png" /><figcaption>Figure 5: Marginal effects of exogenous variables on the conditional-to-unconditional efficiency ratio (full frontier)—investment expenditure model</figcaption></figure>
<figure><img src="image_7.png" /><figcaption>Figure 6: Marginal effects of exogenous variables on the conditional-to-unconditional efficiency ratio (partial frontier)– operating expenditure model</figcaption></figure>
<figure><img src="image_8.png" /><figcaption>Figure 7: Marginal effects of exogenous variables on the conditional-to-unconditional efficiency ratio (partial frontier)– investment expenditure model</figcaption></figure>
# A Appendix

## Convexity in efficiency analyses

The use of the convexity assumption in efficiency analyses may be questionable and that there exists some explicit test for convexity (Kneip et al., 2016). From a theoretical perspective, a variety of reasons may explain the existence of non-convexities in production technology. One can mention indivisibilities, i.e., the fact that inputs and outputs are not necessarily perfectly divisible. This limits the up- and downscaling of production processes (see, e.g., Kerstens & De Woestyne, 2021). Other well-known factors in the literature that are likely to generate non convexities are economies of scale and economies of specialization, as well as the existence of production externalities (see, e.g., Briec et al., 2020). However, most researchers tend to ignore these factors and maintain the convexity assumption either using a time divisibility argument (Shephard, 1970; Hackman, 2008) or simply because of analytical convenience. The argument of time divisibility is questionable as it ignores setup and lead times that make switching between production activities costly in terms of time (Kerstens & De Woestyne, 2018).

Briec, W., Kerstens, K., & Van de Woestyne, I. (2020). Nonconvexity in production and cost functions: An exploratory and selective review. *Handbook of Production Economics*, 1-34.

Hackman, S. T. (2008). Nonconvex models of technology. *Production Economics: Integrating the Microeconomic and Engineering Perspectives*, 125-145.

Kerstens, K., & Van de Woestyne, I. (2021). Cost functions are nonconvex in the outputs when the technology is nonconvex: convexification is not harmless. *Annals of Operations Research*, 305(1), 81-106.

Kerstens, K., & Van de Woestyne, I. (2018). Enumeration algorithms for FDH directional distance functions under different returns to scale assumptions. *Annals of Operations Research*, 271(2), 1067-1078.

Kneip, A., Simar, L., & Wilson, P. W. (2016). Testing hypotheses in nonparametric models of production. *Journal of Business & Economic Statistics*, 34(3), 435-456.

Shephard, R. W. (1970). *Theory of cost and production functions*. Princeton University Press.
Tables and figures

<table><caption>**Table A1:** Averages of expenditure variables from 2007 to 2019 in 2015 millions euros</caption><thead><tr><th></th><th>OPE EXPEN</th><th>INV EXPEN</th><th>TOT EXPEN</th><th>INV EXPEN/TOT EXPEN</th></tr></thead><tbody><tr><td>2007</td><td>480.8898</td><td>177.5773</td><td>658.4671</td><td>0.2806</td></tr><tr><td>2008</td><td>503.6643</td><td>177.8271</td><td>681.4911</td><td>0.2708</td></tr><tr><td>2009</td><td>530.7811</td><td>180.3348</td><td>711.1155</td><td>0.2601</td></tr><tr><td>2010</td><td>544.6266</td><td>163.8905</td><td>708.5168</td><td>0.2344</td></tr><tr><td>2011</td><td>551.6002</td><td>156.6115</td><td>708.2113</td><td>0.2242</td></tr><tr><td>2012</td><td>550.6972</td><td>148.6932</td><td>699.3903</td><td>0.2161</td></tr><tr><td>2013</td><td>560.9952</td><td>140.3684</td><td>701.3640</td><td>0.2056</td></tr><tr><td>2014</td><td>568.8738</td><td>107.7313</td><td>676.6056</td><td>0.1631</td></tr><tr><td>2015</td><td>577.4477</td><td>98.8454</td><td>676.2934</td><td>0.1504</td></tr><tr><td>2016</td><td>577.4853</td><td>92.6929</td><td>670.1779</td><td>0.1396</td></tr><tr><td>2017</td><td>570.2833</td><td>90.8594</td><td>661.1427</td><td>0.1410</td></tr><tr><td>2018</td><td>556.6449</td><td>92.2993</td><td>648.9453</td><td>0.1467</td></tr><tr><td>2019</td><td>557.8277</td><td>104.3603</td><td>662.1881</td><td>0.1606</td></tr></tbody></table>

Note: OPE_EXPEN, INV_EXPEN and TOT_EXPEN correspond respectively to the departments' operating, investment and total expenditures.
<table><caption>**Table A2:** Averages of original outputs and composite index from 2007 to 2009</caption><thead><tr><th></th><th>2007</th><th>2008</th><th>2009</th><th>2010</th></tr></thead><tbody><tr><td>ROAD (kilometers)</td><td>4017.44</td><td>4019.19</td><td>4025.80</td><td>4023.41</td></tr><tr><td>PUB PUPILS (No. of pupils)</td><td>24376.20</td><td>24412.93</td><td>24561.34</td><td>24675.59</td></tr><tr><td>BENEF RSA/RMI (No. of benef.)</td><td>10254.98</td><td>10061.92</td><td>11207.63</td><td>11848.73</td></tr><tr><td>BENEF APA (No. of beneficiaries)</td><td>10796.41</td><td>11193.05</td><td>11537.33</td><td>11779.88</td></tr><tr><td>BENEF ASE (No. of beneficiaries)</td><td>1447.03</td><td>1457.43</td><td>1475.85</td><td>1488.03</td></tr><tr><td>BENEF PCH (No. of beneficiaries)</td><td>405.03</td><td>800.87</td><td>1205.05</td><td>1551.50</td></tr><tr><td>TOT POP (No. of inhabitant)</td><td>622332.49</td><td>637101.23</td><td>637101.23</td><td>641032.62</td></tr><tr><td>COMP INDEX (Unitless)</td><td>2.1261</td><td>2.2391</td><td>2.3796</td><td>2.4901</td></tr></tbody></table>

<table><caption>**Table A3:** Averages of original outputs and composite index from 2011 to 2014</caption><thead><tr><th></th><th>2011</th><th>2012</th><th>2013</th><th>2014</th></tr></thead><tbody><tr><td>ROAD (kilometers)</td><td>4024.33</td><td>4027.15</td><td>4020.41</td><td>4018.67</td></tr><tr><td>PUB PUPILS (No. of pupils)</td><td>24939.77</td><td>24847.57</td><td>25337.14</td><td>25328.05</td></tr><tr><td>BENEF RSA/RMI (No. of benef.)</td><td>12268.71</td><td>13087.57</td><td>14175.12</td><td>14929.33</td></tr><tr><td>BENEF APA (No. of beneficiaries)</td><td>12038.83</td><td>12223.98</td><td>12413.66</td><td>12436.35</td></tr><tr><td>BENEF ASE (No. of beneficiaries)</td><td>1512.60</td><td>1536.02</td><td>1572.76</td><td>1595.80</td></tr><tr><td>BENEF PCH (No. of beneficiaries)</td><td>1849.12</td><td>2080.33</td><td>2291.02</td><td>2498.86</td></tr><tr><td>TOT POP (No. of inhabitant)</td><td>644362.83</td><td>647670.63</td><td>650737.49</td><td>653854.04</td></tr><tr><td>COMP INDEX (Unitless)</td><td>2.5915</td><td>2.6786</td><td>2.7855</td><td>2.8590</td></tr></tbody></table>

<table><caption>**Table A4:** Averages of original outputs and composite index from 2015 to 2019</caption><thead><tr><th></th><th>2015</th><th>2016</th><th>2017</th><th>2018</th><th>2019</th></tr></thead><tbody><tr><td>ROAD (kilometers)</td><td>4018.67</td><td>4021.34</td><td>4004.84</td><td>4004.77</td><td>3967.84</td></tr><tr><td>PUB PUPILS (No. of pupils)</td><td>24386.01</td><td>24305.65</td><td>24505.62</td><td>24739.64</td><td>25034.73</td></tr><tr><td>BENEF RSA/RMI (No. of benef.)</td><td>15383.10</td><td>14819.13</td><td>17006.61</td><td>17168.82</td><td>17286.50</td></tr><tr><td>BENEF APA (No. of beneficiaries)</td><td>12659.73</td><td>12863.89</td><td>13086.47</td><td>13232.93</td><td>13361.54</td></tr><tr><td>BENEF ASE (No. of beneficiaries)</td><td>1625.67</td><td>1679.16</td><td>1766.62</td><td>1862.85</td><td>1957.40</td></tr><tr><td>BENEF PCH (No. of beneficiaries)</td><td>2691.51</td><td>2801.70</td><td>2952.85</td><td>3113.05</td><td>3268.11</td></tr><tr><td>TOT POP (No. of inhabitant)</td><td>657193.41</td><td>660416.27</td><td>663556.49</td><td>666100.37</td><td>666100.37</td></tr><tr><td>COMP INDEX (Unitless)</td><td>2.9133</td><td>2.9467</td><td>3.0819</td><td>3.1609</td><td>3.2351</td></tr></tbody></table>

Note Tables A2-A4: ROAD is the length in kilometers of the road network, PUB PUPILS is the number of pupils in public secondary schools, BENEF RSA/RMI is the number of beneficiaries of minimal subsistence support, BENEF APA is the number of beneficiaries of the Personalized Autonomy Allocation, BENEF ASE is the number of beneficiaries of minimal subsistence support, BENEF_PCF is the number of beneficiaries of Disability Compensation Prestation and TOT POP is the total population. All variables are measured at the department level.
<table><caption>**Table A5:** Correlation matrix between original output variables except ROAD</caption><thead><tr><th></th><th>ROAD</th><th>PUB-PUPILS</th><th>BENEF-RSA/RMI</th><th>BENEF-APA</th><th>BENEF-ASE</th><th>BENEF-PCH</th><th>TOT-POP</th></tr></thead><tbody><tr><td>ROAD</td><td>1.00</td><td>-0.09</td><td>-0.06</td><td>0.20</td><td>0.12</td><td>0.05</td><td>-0.04</td></tr><tr><td>PUB PUPILS</td><td>-0.09</td><td>1.00</td><td>0.86</td><td>0.81</td><td>0.83</td><td>0.68</td><td>0.97</td></tr><tr><td>BENEF RSA/RMI</td><td>-0.06</td><td>0.86</td><td>1.00</td><td>0.85</td><td>0.89</td><td>0.76</td><td>0.88</td></tr><tr><td>BENEF APA</td><td>0.20</td><td>0.81</td><td>0.85</td><td>1.00</td><td>0.84</td><td>0.75</td><td>0.87</td></tr><tr><td>BENEF ASE</td><td>0.12</td><td>0.83</td><td>0.89</td><td>0.84</td><td>1.00</td><td>0.71</td><td>0.87</td></tr><tr><td>BENEF PCH</td><td>0.05</td><td>0.68</td><td>0.76</td><td>0.75</td><td>0.71</td><td>1.00</td><td>0.73</td></tr><tr><td>TOT POP</td><td>-0.04</td><td>0.97</td><td>0.88</td><td>0.87</td><td>0.87</td><td>0.73</td><td>1.00</td></tr></tbody></table>

Notes: see notes for Tables A2-A4
<table><caption>**Table A6:** Principal component analysis – explained variance</caption><thead><tr><th></th><th>Total</th><th>Variance (%)</th><th>Cumulative (%)</th></tr></thead><tbody><tr><td>Component 1</td><td>5.11</td><td>85.15</td><td>85.15</td></tr><tr><td>Component 2</td><td>0.39</td><td>6.42</td><td>91.57</td></tr><tr><td>Component 3</td><td>0.21</td><td>3.56</td><td>95.12</td></tr><tr><td>Component 4</td><td>0.17</td><td>2.84</td><td>97.97</td></tr><tr><td>Component 5</td><td>0.10</td><td>1.73</td><td>99.70</td></tr></tbody></table>

*Notes*: The second column provides the variance accounted by each component and the last column provides the cumulative variance. The first component represents 85.15% of the total variance.

<table><caption>**Table A7:** Correlation between original outputs and components</caption><thead><tr><th></th><th>Comp.1</th><th>Comp.2</th><th>Comp.3</th><th>Comp.4</th><th>Comp.5</th></tr></thead><tbody><tr><td>PUB_PUPILS</td><td>0.9328</td><td>-0.2472</td><td>0.2457</td><td>0.0413</td><td>0.0026</td></tr><tr><td>BENEF_RSA/RMI</td><td>0.9476</td><td>-0.0073</td><td>-0.1351</td><td>0.1383</td><td>-0.254</td></tr><tr><td>BENEF_APA</td><td>0.926</td><td>0.0332</td><td>-0.14</td><td>-0.3481</td><td>-0.0144</td></tr><tr><td>BENEF_ASE</td><td>0.9302</td><td>-0.0716</td><td>-0.2628</td><td>0.157</td><td>0.1892</td></tr><tr><td>BENEF_PCH</td><td>0.8309</td><td>0.5349</td><td>0.1394</td><td>0.0512</td><td>0.0388</td></tr><tr><td>TOT_POP</td><td>0.9633</td><td>-0.1775</td><td>0.1631</td><td>-0.0372</td><td>0.0449</td></tr></tbody></table>

<table><caption>**Table A8:** Quality of the representation of variables components (cos2)</caption><thead><tr><th></th><th>Comp.1</th><th>Comp.2</th><th>Comp.3</th><th>Comp.4</th><th>Comp.5</th></tr></thead><tbody><tr><td>PUB_PUPILS</td><td>0.87</td><td>0.0611</td><td>0.0604</td><td>0.0017</td><td>0</td></tr><tr><td>BENEF_RSA/RMI</td><td>0.8979</td><td>1,00E-04</td><td>0.0183</td><td>0.0191</td><td>0.0645</td></tr><tr><td>BENEF_APA</td><td>0.8575</td><td>0.0011</td><td>0.0196</td><td>0.1212</td><td>2,00E-04</td></tr><tr><td>BENEF_ASE</td><td>0.8653</td><td>0.0051</td><td>0.069</td><td>0.0246</td><td>0.0358</td></tr><tr><td>BENEF_PCH</td><td>0.6904</td><td>0.2861</td><td>0.0194</td><td>0.0026</td><td>0.0015</td></tr><tr><td>TOT_POP</td><td>0.928</td><td>0.0315</td><td>0.0266</td><td>0.0014</td><td>0.002</td></tr></tbody></table>

*Notes for Tables A7 and A8:* Comp1-5 refer to the first five components of the Principal Factor Analysis. Table 7 provides the correlation between the original outputs and the five components Table 8 provides the quality of representation of the variables in each component, measured by the squared cos2 of the orthogonal projection of each observation on each component.
<table><caption>**Table A9:** Averages of contextual variables from 2007 to 2019</caption><thead><tr><th></th><th>DEBT<br>(euro/inhabitant)</th><th>DENSITY<br>(inhabitant/km2)</th><th>INCOME<br>(2005 euro/inhabitant)</th><th>UNEMPLY<br>(%)</th></tr></thead><tbody><tr><td>2007</td><td>422.52</td><td>343.39</td><td>12678.68</td><td>7.27</td></tr><tr><td>2008</td><td>444.29</td><td>341.47</td><td>12533.17</td><td>7.76</td></tr><tr><td>2009</td><td>517.63</td><td>341.49</td><td>12973.83</td><td>9.46</td></tr><tr><td>2010</td><td>543.67</td><td>343.39</td><td>12808.95</td><td>9.10</td></tr><tr><td>2011</td><td>535.19</td><td>340.09</td><td>12873.76</td><td>9.36</td></tr><tr><td>2012</td><td>535.07</td><td>342.13</td><td>13060.46</td><td>10.18</td></tr><tr><td>2013</td><td>541.37</td><td>343.97</td><td>13258.12</td><td>9.67</td></tr><tr><td>2014</td><td>553.39</td><td>345.75</td><td>13383.78</td><td>10.25</td></tr><tr><td>2015</td><td>563.88</td><td>347.39</td><td>13517.23</td><td>10.04</td></tr><tr><td>2016</td><td>560.45</td><td>349.52</td><td>13634.97</td><td>9.14</td></tr><tr><td>2017</td><td>543.26</td><td>351.90</td><td>13655.25</td><td>8.70</td></tr><tr><td>2018</td><td>522.72</td><td>353.98</td><td>13700.47</td><td>8.40</td></tr><tr><td>2019</td><td>504.03</td><td>355.34</td><td>14017.358</td><td>7.05</td></tr></tbody></table>

<table><caption>**Table A10:** Evolution of the distribution of POL COL from 2007 to 2019</caption><thead><tr><th></th><th>right</th><th>left</th><th>center</th></tr></thead><tbody><tr><td>2007</td><td>48.9130</td><td>51.0870</td><td>0.0000</td></tr><tr><td>2008</td><td>38.0435</td><td>58.6957</td><td>3.2609</td></tr><tr><td>2009</td><td>38.0435</td><td>58.6957</td><td>3.2609</td></tr><tr><td>2010</td><td>38.0435</td><td>58.6957</td><td>3.2609</td></tr><tr><td>2011</td><td>38.0435</td><td>58.6957</td><td>3.2609</td></tr><tr><td>2012</td><td>38.0435</td><td>58.6957</td><td>3.2609</td></tr><tr><td>2013</td><td>38.0435</td><td>58.6957</td><td>3.2609</td></tr><tr><td>2014</td><td>38.0435</td><td>58.6957</td><td>3.2609</td></tr><tr><td>2015</td><td>67.3913</td><td>30.4348</td><td>2.1739</td></tr><tr><td>2016</td><td>67.3913</td><td>30.4348</td><td>2.1739</td></tr><tr><td>2017</td><td>67.3913</td><td>30.4348</td><td>2.1739</td></tr><tr><td>2018</td><td>67.3913</td><td>30.4348</td><td>2.1739</td></tr><tr><td>2019</td><td>67.3913</td><td>30.4348</td><td>2.1739</td></tr></tbody></table>
<table><caption>**Table A11:** Averages of considered variables per department size</caption><thead><tr><th></th><th>Extra small</th><th>Small</th><th>Large</th><th>Extra large</th></tr></thead><tbody><tr><td>OPE_EXPEN</td><td>199.9262</td><td>352.6696</td><td>515.7202</td><td>1126.0892</td></tr><tr><td>INV_EXPEN</td><td>54.4752</td><td>88.4694</td><td>129.7601</td><td>260.2465</td></tr><tr><td>TOT_EXPEN</td><td>254.4017</td><td>441.1383</td><td>645.4806</td><td>1386.3361</td></tr><tr><td>ROAD</td><td>3587.7092</td><td>4422.0631</td><td>4395.2728</td><td>3654.606</td></tr><tr><td>PUB_PUPILS</td><td>7600.6923</td><td>15102.1037</td><td>23920.9431</td><td>52284.0268</td></tr><tr><td>BENEF_RSA/RMI</td><td>3736.4348</td><td>8031.5117</td><td>11883.505</td><td>31578.7425</td></tr><tr><td>BENEF_APA</td><td>5607.8763</td><td>8995.1906</td><td>12492.5318</td><td>22019.495</td></tr><tr><td>BENEF_ASE</td><td>554.0736</td><td>1060.5084</td><td>1571.0067</td><td>3268.9465</td></tr><tr><td>BENEF_PCH</td><td>764.0268</td><td>1530.9732</td><td>2200.5686</td><td>3968.7391</td></tr><tr><td>DEBT</td><td>619.7713</td><td>510.1715</td><td>468.0032</td><td>490.5037</td></tr><tr><td>DENSITY</td><td>46.1509</td><td>64.2749</td><td>110.0199</td><td>1164.0888</td></tr><tr><td>INCOME</td><td>12119.1369</td><td>12734.7958</td><td>13457.1615</td><td>14641.5295</td></tr><tr><td>UNEMPLOY</td><td>8.5632</td><td>9.1365</td><td>8.9712</td><td>9.1395</td></tr></tbody></table>

Notes: The departments are split into four groups of equal number of observations. The extra small departments have between 76,800.0 and 312,784.0 inhabitants. The small departments have between 312,784.0 and 547,390.5 inhabitants. The large departments have between 547,390.5 and 819,234.0 inhabitants and the extra large departments have between 819,234.0 and 2,641,081.0 inhabitants.
<figure><img src="image_9.png" /><figcaption>**Figure A1:** Maps of unconditional efficiency scores of French departments for each year</figcaption></figure>
<table><caption>**Table A12:** Average growth rates for unconditional efficiency scores</caption><thead><tr><th></th><th>TOT EXPEN</th><th>OPE EXPEN</th><th>INV EPEN</th></tr></thead><tbody><tr><td>2007-2008</td><td>-0.6243</td><td>-0.6543</td><td>3.9228</td></tr><tr><td>2008-2009</td><td>0.8879</td><td>-0.406</td><td>13.0963</td></tr><tr><td>2009-2010</td><td>4.6935</td><td>1.8463</td><td>11.3128</td></tr><tr><td>2010-2011</td><td>2.6901</td><td>1.6796</td><td>12.336</td></tr><tr><td>2011-2012</td><td>3.2758</td><td>2.2563</td><td>10.0814</td></tr><tr><td>2012-2013</td><td>1.7896</td><td>0.6513</td><td>14.3473</td></tr><tr><td>2013-2014</td><td>5.0891</td><td>1.5125</td><td>29.2563</td></tr><tr><td>2014-2015</td><td>1.0587</td><td>0.1063</td><td>10.0538</td></tr><tr><td>2015-2016</td><td>1.5244</td><td>1.2359</td><td>9.2884</td></tr><tr><td>2016-2017</td><td>3.6026</td><td>2.8019</td><td>11.8664</td></tr><tr><td>2017-2018</td><td>4.9377</td><td>3.6191</td><td>16.3094</td></tr><tr><td>2018-2019</td><td>0.4035</td><td>0.8798</td><td>-0.0553</td></tr></tbody></table>

Notes: see notes for Table A1
<table><caption>**Table A13:** Mean unconditional efficiency per department size and its evolution over time for operating expenditures</caption><thead><tr><th></th><th>Extra small</th><th>Small</th><th>Large</th><th>Extra large</th></tr></thead><tbody><tr><td>2007</td><td>0.9171</td><td>0.8728</td><td>0.8490</td><td>0.8064</td></tr><tr><td>2008</td><td>0.9042</td><td>0.8476</td><td>0.8458</td><td>0.8207</td></tr><tr><td>2009</td><td>0.8866</td><td>0.8347</td><td>0.8408</td><td>0.8342</td></tr><tr><td>2010</td><td>0.8971</td><td>0.8662</td><td>0.8469</td><td>0.8395</td></tr><tr><td>2011</td><td>0.9048</td><td>0.8919</td><td>0.8670</td><td>0.8393</td></tr><tr><td>2012</td><td>0.9223</td><td>0.9031</td><td>0.8776</td><td>0.8689</td></tr><tr><td>2013</td><td>0.9176</td><td>0.9086</td><td>0.8910</td><td>0.8704</td></tr><tr><td>2014</td><td>0.9430</td><td>0.9148</td><td>0.9025</td><td>0.8741</td></tr><tr><td>2015</td><td>0.9278</td><td>0.9150</td><td>0.9106</td><td>0.8844</td></tr><tr><td>2016</td><td>0.9538</td><td>0.9194</td><td>0.9224</td><td>0.8802</td></tr><tr><td>2017</td><td>0.9678</td><td>0.9395</td><td>0.9354</td><td>0.9185</td></tr><tr><td>2018</td><td>0.9866</td><td>0.9686</td><td>0.9824</td><td>0.9443</td></tr><tr><td>2019</td><td>0.9903</td><td>0.9680</td><td>0.9812</td><td>0.9691</td></tr><tr><td>Total</td><td>0.9312</td><td>0.9040</td><td>0.8962</td><td>0.8746</td></tr></tbody></table>

Notes: see notes for Table A11

<table><caption>**Table A14:** Mean unconditional efficiency per department size and its evolution over time for investment expenditures</caption><thead><tr><th></th><th>Extra small</th><th>Small</th><th>Large</th><th>Extra large</th></tr></thead><tbody><tr><td>2007</td><td>0.6070</td><td>0.4019</td><td>0.3865</td><td>0.4631</td></tr><tr><td>2008</td><td>0.6050</td><td>0.3671</td><td>0.3624</td><td>0.4828</td></tr><tr><td>2009</td><td>0.5642</td><td>0.3901</td><td>0.3878</td><td>0.4747</td></tr><tr><td>2010</td><td>0.5118</td><td>0.4387</td><td>0.3972</td><td>0.4850</td></tr><tr><td>2011</td><td>0.5923</td><td>0.4751</td><td>0.4204</td><td>0.5239</td></tr><tr><td>2012</td><td>0.5932</td><td>0.4751</td><td>0.4955</td><td>0.5529</td></tr><tr><td>2013</td><td>0.6210</td><td>0.5366</td><td>0.5504</td><td>0.5343</td></tr><tr><td>2014</td><td>0.7351</td><td>0.6199</td><td>0.6822</td><td>0.6766</td></tr><tr><td>2015</td><td>0.7625</td><td>0.6974</td><td>0.7261</td><td>0.7467</td></tr><tr><td>2016</td><td>0.8631</td><td>0.7419</td><td>0.7620</td><td>0.7507</td></tr><tr><td>2017</td><td>0.9047</td><td>0.7919</td><td>0.7862</td><td>0.8205</td></tr><tr><td>2018</td><td>0.9577</td><td>0.8958</td><td>0.9057</td><td>0.8766</td></tr><tr><td>2019</td><td>0.9502</td><td>0.8637</td><td>0.8942</td><td>0.9041</td></tr><tr><td>Total</td><td>0.7077</td><td>0.5928</td><td>0.5957</td><td>0.6434</td></tr></tbody></table>

Notes: see notes for Table A11
<table><caption>**Table A15:** Estimated bandwidth by quintiles</caption><thead><tr><th></th><th>0%</th><th>20%</th><th>40%</th><th>60%</th><th>80%</th><th>100%</th></tr></thead><tbody><tr><th>POL COL</th><td>0</td><td>0.5</td><td>0.5</td><td>0.6667</td><td>0.6667</td><td>0.6667</td></tr><tr><th>DEBT</th><td>7.5676</td><td>45.0041</td><td>46.2354</td><td>46.6298</td><td>65.1748</td><td>200.95e+07</td></tr><tr><th>DENSITY</th><td>0.3244</td><td>1.343</td><td>1.9385</td><td>2.3083</td><td>4.8018</td><td>34.31e+07</td></tr><tr><th>INCOME</th><td>106.2765</td><td>225.2537</td><td>233.411</td><td>233.415</td><td>277.0492</td><td>951.48e+07</td></tr><tr><th>UNEMPLY</th><td>0.0894</td><td>1.1628</td><td>1.3416</td><td>2.3699</td><td>1709183.5392</td><td>10.44e+07</td></tr><tr><th>YEAR</th><td>0.4472</td><td>0.8857</td><td>0.8944</td><td>0.9265</td><td>1.0445</td><td>3.34e+07</td></tr></tbody></table>

*Notes*: The table reports the estimated bandwiths by quintiles for each variable

<table><caption>**Table A16:** « Pure » efficiencies</caption><thead><tr><th></th><th>OPE EXPEN</th><th>INV EPEN</th><th>TOT EXPEN</th></tr></thead><tbody><tr><th>Min.</th><td>-6.2221</td><td>-3.5875</td><td>-3.3272</td></tr><tr><th>1st Qu.</th><td>0.0742</td><td>-0.2595</td><td>-0.3186</td></tr><tr><th>Median</th><td>0.2304</td><td>0.4095</td><td>0.4564</td></tr><tr><th>Mean</th><td>0.0476</td><td>0.0613</td><td>0.1358</td></tr><tr><th>3rd Qu.</th><td>0.386</td><td>0.6219</td><td>0.7756</td></tr><tr><th>Max.</th><td>2.2479</td><td>1.5754</td><td>1.8162</td></tr><tr><th>SD</th><td>0.8083</td><td>0.8897</td><td>0.9322</td></tr></tbody></table>

*Notes*: The table reports the estimated bandwiths by quintiles for each variable