This study proposes a sequence of methods to obtain geolocated estimates of primary school provision, costs, and access. This sequence entails: (1) location-allocation, an approach that mimics school location patterns in case of decentralised governance, such as exists in the EU and UK; (2) balanced floating catchment areas, an approach to assign pupils to schools assuming free school choice; and (3) school costs estimates, which are induced from pupil counts and the distributional properties of observed school costs. The method is fine-tuned using observed school locations and school-level costs data. It is developed to assess how much local population densities and demography affects school access and schooling costs across Europe. Its results can be aggregated by degree of urbanisation to quantify the differences across human settlements ranging from mostly uninhabited areas to densely populated cities.
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1 Introduction
Consider that the location of primary schools can be a balancing act between on the one hand, school costs, or the operational financial costs of providing education; and on the other hand school access, or the costs of travelling to school. Local demand for schools comes from the amount of children in a community that are eligible for primary education. Finding a satisfactory balance between costs and access is much harder in areas with low densities of demand. Thus, due to the scale economies that primary schools are subject to Andrews et al. (2002), Duncombe (2007) and Zimmer et al. (2009), areas with low demand densities often face either poor school access or excessive school costs. School provision is therefore under pressure in areas where a steady reduction in pupil numbers is at odds with the goal to maintain schools close to families (Barakat 2015; Christiaanse 2020; OECD 2021). As schools in areas with low demand densities cannot benefit from scale economies, demographic change can magnify the impact of economic rationalisation, which has already led to school closures in many countries, in particular after the 2008 financial crisis (Stockdale 1993; Witten et al. 2007; Agasisti 2014; López-Torres and Prior 2020). Such school consolidation is a sensitive topic because closures can have a broader impact on communities that are already in decline (Salant and Waller 1998; Lyson 2002; Elshof et al. 2014).
School costs and access depend on school provision, which in turn depends on local demand and governance. However, general effects of public intervention, low demand densities and foreseen demographic changes on school access and costs can be difficult to establish from factual data because sufficiently fine-grained data is often unavailable. In addition, population densities as well as political responses to demographic challenges differ widely across countries and regions. In the EU, such responses have included incentives to form larger school communities; colocation of multiple school levels; and school consolidation (OECD 2021). The lack of harmonised data calls for an approach that identifies the effects of population densities on school provision, and subsequently on school access and cost, in an ex-ante manner. Such studies are rare. This paper therefore proposes a method to obtain internationally comparable estimates of school access and costs by first modelling school locations and their sizes. The aim of the introduced method is to provide an internationally comparable reference of how population densities affects the balance between access and cost efficiency in education provision, and to allow counterfactual exploration of how changes in demand densities or governance would affect that balance.
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The proposed method entails a sequence of solutions to problems that are, individually, well-documented in the economy and geography literature. It is inspired by the Swedish system for municipal finance equalisation (Tillväxtanalys 2011) which takes population distribution, school locations and transport costs into account. The method is based on three principles: (1) costs arise in facilities (e.g. schools) and not in areas (e.g. school districts or municipalities); (2) public services are consumed locally, and are provided close to places of residency; and (3) additional costs arise as a result of transport costs, lack of economies of scale, and/or scope of small facilities. The method relies on fine-grained geographical data and yields results for individual schools. This geographic granularity is necessary because aggregate statistics can exaggerate equity in accessibility and efficiency, as the demand densities and demographic compositions that affect local school provision differ substantially within and across regions [see also Franklin (2021)].
All components in the modelling sequence have been calibrated using observed school locations, travel and costs statistics on primary education in Europe. The aim of calibration was to verify whether structural differences in school provision, access and cost between rural and urban geographies can be reproduced on aggregate, without attempting to reproduce local results with high accuracy, as country-specific governance rules and locally relevant factors are ignored for the sake of international comparability. We expect the calibrated model to reproduce empirically observed annual and transport costs, which in turn are assumed to represent an implicit societally accepted equitable balance between school efficiency and proximity. The methods to simulate school locations, estimate school sizes and assess accessibility were developed for GeoDMS. Cost estimations were done in R. All used software is open source, and all scripts and data are available upon request.
The remainder of this paper proceeds as follows. In the next section, we review the literature relevant to the presented method. We then present the methods to simulate school locations and estimate their pupil populations and costs, followed by considerations regarding the application of our approach in empirical case studies. We subsequently compare the simulated school placement, access, size and costs of primary schools with observed distributions that are described in comprehensive, publicly available databases from Portugal, England and France. Comparisons are given for England and Portugal, the countries on which separate parts of the modelling sequence were calibrated; and for France, a country that was not included in model calibration and serves as a test case here. The paper closes with some general observations and a discussion of possible further work.
2 Literature review
Analyses that link demography, school provision, cost and access in an ex-ante framework are infrequent in the literature. Few studies even consider demand as an endogenous factor [see for instance Andrews et al. (2002)], and only a handful of studies consider the link between changes in local demand and supply in education provision (Elshof et al. 2014; Christiaanse 2020). Holland and Baritelle (1975) suggest a way to redesign school districts in rural areas in order to balance current and expected future operating and transport costs for schools in the USA. To include local demand in our access and cost estimates, we resort to a location-allocation heuristic. Location-allocation methods compute optimal facility distributions, in which the number and capacity of those facilities may be unknown a priori, while taking into account-specific constraints and predefined distributions of demand (Cooper 1963). They have, among others, been relevant to search locations that minimise user distances to public and private facilities, for instance in healthcare (Pacheco and Casado 2005; Tao et al. 2014) and retail (Kohsaka 1989).
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Multiple studies have developed location-allocation models for education. Fortney (1996) introduces a school location model to establish an efficient balance between the costs and benefits of proximate and high-quality education through consumer and producer surpluses. Unfortunately, equity considerations cannot be taken into account in this framework and would need to be addressed after an efficient distribution is obtained. Chen et al. (2018) introduce a spatial decision support system in which school locations and school attendance are optimised, given an a prior set number of schools and while taking into account capacity constraints. Others employ genetic algorithmic heuristics to optimise locations that meet physical requirements, and balance between minimising travel time for the majority of students and minimising travel time for students that are farther away (Doerner et al. 2009; Lotfi et al. 2021).
The previous contributions to allocating school locations, and in fact most works in the location-allocation literature, assume top-down coordination [for an overview, see Wang (2012)], aiming for a distribution that is universally optimal in terms of facility or travel costs (Pacheco and Casado 2005; Tao et al. 2014; Xu et al. 2020). However, school governance is decentralised in most countries. States, acting as the main collector of tax revenues, only impose minimum bounding conditions for grants to schools, for instance by setting minimum school sizes (De Haan et al. 2011; Tillväxtanalys 2011; DfE 2019). Presumably the majority of basic education pupils receive education in institutions that rely on public grants. In Greece, Italy and the UK, where the share of pupils that are not educated in institutions that receive public grants is the largest among European countries, only 5%-6% of pupils go to privately funded facilities (Eurydice 2000). Because of decentralised governance, the number of schools in a country cannot be identified a priori, as the conditions for public grants make the presence of schools a result of local demand densities and bounding conditions. This may drastically affect local access and cost outcomes, and we therefore consider centralised optimisation methods a poor fit for the goals of this paper.
Due to the bounding conditions imposed through decentralised governance, resources for schools are scarce. Local communities, in turn, have a stake in obtaining and maintaining a school because of considerations related to access, population attraction (Elshof et al. 2014; Barakat 2015), broader social benefits (Salant and Waller 1998; Lyson 2002) and existing capital stock (Delmelle et al. 2014). The tension between scarce resources and localised benefits implies that communities compete for school locations, which likely does not lead to a central optimum distribution. This makes centralised location-allocation approaches that rely on an a priori target facility countless suitable to mimic school locations. In this paper, we therefore adopt a bottom-up allocation method in which local communities compete for the allocation of a school, given centralised rules but without top-down coordination.
The adopted location-allocation procedure leads to a simulated spatial distribution of school locations, which may contain or more school facilities. This can be used to capture the distance to closest locations, a classical location-based accessibility measure (Geurs and van Wee 2004). However, assigning all pupils to the closest location may lead to implausible results, as pupils likely experience some freedom in school choice and education boards may seek to balance pupils over the schools in their district. Thus, travelled distances are presumably higher than closest distances. Assignment to closest school locations may also lead to implausibly small and large facilities, which is problematic as the nonlinear nature of school costs may exaggerate estimated costs. Demand and supply therefore need to be assessed and balanced locally for school locations, as is done in accessibility measures that take competition into account (Merlin and Hu 2017; Kelobonye et al. 2020). To assess initial demand, an origin-constrained spatial interaction model (Fotheringham and O’Kelly 1989) is applied, in which users may be assigned to the closest school location or others. Such spatial interaction models share roots with potential accessibility approaches (Hansen 1959; Geurs and Van Wee 2004; Stępniak and Jacobs-Crisioni 2017) and catchment area approaches, with the main difference being in how degrees of separation are expected to affect interactions such as service uptake (Luo and Qi 2009). Given initial demand, number of schools in a location is modelled using models fitted on observed links between number of attending pupils in a grid cell and the number of schools. Subsequently, pupils are assigned to schools in a balanced floating catchment area approach (Paez et al. 2019; Pereira et al. 2021). That approach is preferred over other floating catchment approaches such as E2SFCA (Luo and Qi 2009) as it ensures that the aggregate student population and estimated number of pupils are identical, which is key for the later cost estimates. The total number of pupils in a school, and the summed distance they travel, are used to capture the average distance that pupils travel to school. That average travelled distance is considered as a measure of school access for the remainder of this paper.
Finally, variable annual school operating costs are estimated by deriving staffing costs from estimated pupil counts and adding plausible additional residual costs. School costs are thus estimated based solely on the estimated number of pupils in each school, without relying on often unavailable country-specific information or other school information. We consider running costs that are variable with school size only and do not attempt to model other costs, including capital costs. Our inductive approach differs considerably from most studies, which tend to deduce efficiency from total costs and student counts from the top-down. It must be noted from the literature that there is no definitive say on optimum size. On the one hand, schools exhibit considerable returns to scale (Kenny 1982; Zimmer et al. 2009). However, ex-post econometric analyses of spending following school consolidation, which are available mostly for the USA [see Andrews et al. (2002) for an overview], focus on cost effects but often oversee the opportunity costs of longer travel, in many cases leading to severe underestimation of the welfare implications of school consolidation for pupils and their families (Kenny 1982). In fact, Andrews et al. (2002) argues that optimal school sizes may be much lower than suggested in studies that do not account for impacts of increased travel time. Moreover, larger school areas have been associated with worse pupil attendance and performance, lesser parent involvement, higher transport costs and higher teacher wages (Talen 2001; Zimmer et al. 2009; Williams and Wang 2014).
3 Methods
3.1 Simulating school locations
Our approach consists of three steps (see Fig. 1). First, school locations are allocated to most likely locations across the geography of a country. The modelling of locations is based solely on universal location simulation rules, and local pupil counts and road distances. Second, pupils are assigned to allocated schools using a balanced floating catchment area approach, and distances travelled to schools are estimated to proxy the transport costs of travel to school. Transport costs are expressed in road distances travelled, while their monetary value remains unknown. We assume that longer travel distances are linear with transport costs, as network distances and travel times tend to be correlated (Rietveld et al. 1999). Monetary values of travel to school are ignored as the means to travel to schools, as well as the organisation of school transport, likely differs substantially between contexts and may be endogenous to distance to schools. An assessment of travel time or costs is therefore too challenging for the scope of this paper. Finally, school costs are estimated using the assigned school sizes. Only annual operational school provision costs are estimated, and they are expressed in monetary values.
Fig. 1
Input data, steps and applied methodology in the adopted approach
×
The first step of our methodology is to establish the likely locations of schools, a question that we treat as a decentralised location-allocation problem. To operationalise our approach, we adapt the facility allocation algorithm of Kompil et al. (2019b). The iterative location-allocation procedure, in every step, selects the highest utility location in a predefined region1 as an additional discrete location for schools. There is no central optimisation process, and the number of locations is not defined a priori. Instead, the number and size of facilities are obtained endogenously from the allocation procedure. The outcome of our procedure is an equilibrium that arises when, universally, demand is completely exhausted, or when no more potential locations meet the imposed bounding conditions.
The competing communities are represented as network nodes that are distributed on a 1km2 regular lattice covering all land in a country. On the onset, all nodes are considered potential school locations. We assume that the selected 1km2 spatial resolution is sufficiently fine to avoid misrepresentation errors due to spatial aggregation (Goodchild 1979; Stępniak and Jacobs-Crisioni 2017). For every considered node, road travel distances are obtained to all other nodes in the maximum allowed catchment area. We assume that the pupil populations in every node in the catchment area contribute to local school demand. Nodes can be flagged as having either satisfied or unsatisfied demand. On the onset, all nodes have unsatisfied demand.
3.1.1 Grid search calibration
The applied algorithm approximates centrally imposed boundary conditions through three threshold values, which were obtained through calibration. These thresholds are: (T1) the distance in kilometres that defines a potential school’s largest allowed catchment area; (T2) the minimum number of pupils that a potential school needs in its catchment area; and (T3) the target number of pupils for a school. The calibration of T1, T2 and T3 was done through a grid search that aimed at most accurately reproducing observed school distributions in Portugal. Accuracy was measured with a composite measure aiming to minimise discrepancies in total number of facilities and differences in allocated shares of schools in urban, rural and intermediate locations. Portugal was preferred to England as the latter has a relatively small dispersed rural population of pupils (7% of pupils in England and 20.6% in Portugal), so that rural school locations in Portugal are presumably more representative for location rules. Portugal was preferred to France to limit computational burden, as the requisite repeated modelling of school distributions in the much larger and more populous France proved very cumbersome. The results comparisons presented later in this paper show that school patterns in other countries can be reproduced fairly well. Table 1 indicates the thresholds that gave the most accurate results in Portugal. The grid search and its results are discussed in more detail in Annex A; a key takeaway is that T1 and T3 have a much bigger impact on model outcomes than T2. The established target school sizes are relatively small, which is instrumental to ensure that facilities allocated in early iterations do not cannibalise the market for later iterations, which in turn typically offsets the balance between allocated rural and urban facilities. These target sizes should thus not be confused with the much larger sizes that are considered optimal from a cost–benefit point of view (Andrews et al. 2002; Zimmer et al. 2009).
Table 1
Threshold values for modelling primary school locations, obtained through calibration
(T1) Maximum catchment area (km)
15
(T2) Minimum size (nr pupils)
7
(T3) Target size (nr pupils)
280
3.1.2 Allocation approach
Figure 2 outlines the placement procedure schematically. In each model iteration, eligible school locations are sought by summing unsatisfied pupils in the maximum catchment area. We express this as ‘market’ (1):
where \(P_{i}^{iter}\) describes unsatisfied demand at each origin node in the iteration (iter) at hand. In the example, the student population in N3 (which is at least 16 km removed from the other nodes) is not considered for the market sized of the other nodes. All nodes that have enough unsatisfied market size (T2) within the distance range (T1) are considered eligible for a school location; thus the nodes N1, N2 and N4 remain eligible until the last school is allocated. In the first iteration, locational utility U is measured as a potential accessibility measure (Geurs and van Wee 2004) at the destination to unsatisfied demand at the origins. This measure is specified as (2):
in which \(d_{ij}\) indicates travel distance between origin node i and destination node j, while 100 m is kept as the minimum distance between population and facilities relevant for the special case that j = i. In later iterations, potential accessibility is weighted by relative inaccessibility to account for equity considerations, as will be explained in following sections.
Fig. 2
Placement simulation procedure in a hypothetical case with four nodes on a regular lattice. Nodes are expressed as circles. Road distances between nodes are expressed in km; numbers between brackets at each node express number of pupils
×
In each separate region, the eligible destination with the highest locational utility is selected as additional school location, so that a school is allocated to N1 first, N3 second, and N4 last. When selecting a location, we ignore site-specific factors such as ownership, land use and zoning (Koomen et al. 2015; Lotfi et al. 2021), and assume that in all cases, a suitable site can be found in the 1km2 of land that the selected node represents. To flag the nodes for which the newly placed location satisfies demand, we assess which pupils would presumably frequent that location. This is necessary for the next placement cycle, which only needs to consider nodes with unsatisfied demand. We assign all demand at nodes in the vicinity, in order of distance, to the newly placed school, either until the location has reached or exceeded its target size threshold (T3), or until the demand in all nodes in the maximum catchment area is assigned to the new facility.
In the example, in the first iteration, pupils from N1 and N2 are therefore assumed to visit the school location in N1, causing the location in N1 to have 300 students, which is slightly more students than the target size. Subsequently, we take all the nodes that have been assigned to the newly placed school out of the pool of nodes with unsatisfied demand, so that the 300 students in N1 and N2 are removed from the market at the end of the first iteration. The reduced pool of nodes with unsatisfied demand is subsequently passed to the next model iteration, reducing eligibility and accessibility scores for competing nodes that partially share the same catchment. Finally, the placement routine model stops when no more nodes are found that meet the eligibility criterion; in the example, this occurs in the fourth iteration.
3.1.3 Counterweighting mechanism
We can readily verify that the node with the best access to potential pupils would minimise the average travel distance, at least when all available pupils in the catchment area would visit the additional school. Thus, this location would offer the lowest travel cost compromise in the region and may therefore be considered highest utility. However, the choice to use potential accessibility as a measure of utility tends to play out as a preference for central nodes. To simulate top-down incentives to improve access to schools in peripheral nodes, we introduce a counterweighting mechanism in the utility measure for intermediate iterations. The results of this counter weighting are indicated as ‘Ctr weight’.
To do this, we weigh pupils by their relative inaccessibility to schools. If few schools are allocated in the pupils’ vicinity, those pupils obtain a boost in their contribution to the utility score, so that potential locations that would serve pupils with poor access to schools are given a slightly higher preference score. The pupils’ iteration-specific potential access to facilities as (3):
in which \(D0_{j}^{iter}\) is a vector of dichotomous values that indicate whether facilities have been allocated in prior iterations in the destination nodes in j. Subsequently, through iteration-specific weighting values \({W}\), population is weighted by their access to services in A, relative to the average of the collection of nodes in a region in I, so that (4):
in which \(P_{i}^{{\left( {iter - 1} \right)}}\) contains all population, passed on from the previous iteration, that is not yet assigned to an already allocated facility. The function \(f\left( {A, w} \right)\) rescales the relative facility accessibility to prevent extreme outcomes. In this exercise, this weight is scaled between 0.1 and 2, so that pupils with the lowest or no accessibility to schools count double in the allocation mechanism. The applied scales were set on an ad hoc basis for this introduction and may be adjusted to account for other equity considerations. Broader scale ranges have been tested and provide somewhat similar results. Finally, the locational utility of a node in intermediate iterations is computed as (5):
The pupil-to-location results that are obtained during the location placement procedure are only a rough approximation of final school sizes, as the number of facilities at a selected location may be larger than one, and free school choice is not considered. We apply a two-stage spatial interaction modelling approach to adjust the number of schools at a location and distribute pupil populations over available facilities.
3.2.1 First-stage assignment
In the first stage we use an origin-constrained spatial interaction model (Alonso 1978; Fotheringham and O’Kelly 1989) to distribute all pupils from every grid cell of residence to the five closest locations. We believe that a choice set of five locations per origin gives an acceptable balance between computational complexity and a comprehensive choice set. We must acknowledge nevertheless that in practice, pupils from well-served areas may perceive a much larger choice set, while in underserved areas, their choice may be more limited. This model takes the form (6):
The number of pupils in each origin i is described by O. The initial attractiveness of a location for pupils is described by \(D1_{j} = D0_{j}^{last iter}\). This variable is set to 1 for all locations, so that all are considered equally attractive from the point of view of pupils, and only travel distance to the locations in the pupil’s choice set is a relevant factor in location preferences. A1 is a balancing factor computed as \(A1_{i} = \mathop \sum \limits_{j = 1}^{n = 5} D1_{j} C_{ij}^{prim}\). Finally, \(C_{ij}^{prim}\) is a decayed measure of distance. As they have a sizeable impact on accessibility results, there is a vast literature on the selection of distance decay parameters (Stępniak and Rosik 2018). We select a relatively steep distance decay function as we expect that travel minimisation is particularly important in primary school selection. This helps minimising overall travel cost. The distance decay function takes the form \(C_{ij}^{prim} = \acute{d}{^{-2}_{ij}}\), in which \(\acute{d}_{ij}\) is a transformation of \(d_{ij}\), which in turn contains road distances from pupils to schools. A comparison with primary school pupils’ stated travelled distances (DfT 2021), presented in the Results section, shows that the imposed assignment model yields differences in distances travelled in high and low demand density environments with realistic orders of magnitude.
Distance transformation Distances are transformed to ensure aggregate travel cost reductions. It can easily be verified that with power law distance decay, the spatial distribution of schools matters for aggregate travel costs. Take the case that distance decay is defined as \(t_{ij}^{ - 1.5}\). If a pupil can choose between schools at 1 km and 2 km away, the schools at 1 km would be a factor 2.8 more attractive than the school at 2 kms. In the case of schools only at 20 and 21 km, the school at 20 km would be only 1.07 times as attractive as the school at 21 km, implying much more indifference in choice in the latter case, even though the absolute difference in distance between the two schools is equal between cases.
While in human choice behaviour, there is presumably more indifference to destinations in the latter case, the 1 km increase in travel distance would have the same effect on total system costs in both cases. As we attempt to minimise system costs, the straightforward inclusion of \(d_{ij}^{ - \alpha }\) is not ideal. The distance decay function in \(C_{ij}\) therefore has an unorthodox specification through \(\acute{d}_{ij}\), which describes the additional travel distance to facilities, compared to the facility nearest to the origin. This transformation takes the form \(\acute{d}_{ij} = \left[ {d_{ij} - \min d_{i} } \right] \ge 1\), so that \(\acute{d}_{ij}\) effectively describes the additional distance to reach a school location compared with the closest school location. Using \(\acute{d}_{ij}\) rather than the actual travel distances in \(d_{ij}\) imposes that the distance-decayed travel distances retain high sensitivity to farther destinations even if the closest facility is relatively far. On a side note, the distance decay computation may behave capriciously with small changes in travel distances smaller than one, while the precise distances cannot be known accurately given the 1km2 spatial resolution. Therefore, if any travel distance observed in \(\acute{d}_{ij}\) is lower than 1 min, the system uses 1 km as travel distance.
3.2.2 Estimating school sizes
The origin-constrained spatial interaction model leads to a first-stage number of pupils attending per location, which we derive as (7):
Every location in j represents a 1km2 area, which may contain multiple school facilities, especially in areas with a high demand density. In a second step, we therefore use \(s1_{j}\) to estimate the number of schools per selected location. This is done using the results of a model which fits the (natural log of) number of facilities in a 1 km2 grid cell on the (natural log of) number of pupils in that grid cell, so that the continuous expected number of facilities are computed as (8):
which has been fitted using data from England.2 We need to allocate a discrete number of facilities in every location, so that we round the number of facilities in a location, with a minimum of one (9):
Given the discrete number of school facilities allocated to every location, we subsequently compute expected and allocated school supply by dividing the assigned number of pupils with both the continuous and rounded expected number of facilities in a location, which yields expected school size (10a) and first-stage allocated school sizes (10b):
First-stage allocated school sizes (\(AS_{j}\)) will under- and overshoot the expected school size (\(ES_{j} )\) in most locations, signalling empty capacity and overcrowding. In order to balance school sizes over the choice set that is presented to pupils, we subsequently modify the attractiveness of schools for every origin-to-location relation separately with the ratio of the maximum expected (i.e. \(\max ES_{j} | i)\)) over the first-stage allocated school sizes, so that \(D2_{ij} = \left[ {\max ES_{j} | i} \right]/AS_{j} .\) We then replace D1 by D2 in the applied origin–destination model and its balancing factor, yielding final school allocation (11a), balancing factor (11b), school size (11c) and mean distances travelled (11d):
This modification of attractiveness in D2 is imposed to simulate that a pupil faced with the choice of multiple proximate schools, of which some have few pupils and others threaten to be crowded, will give preference to any the smaller schools. With this rebalancing extremely small schools are avoided if there are sufficient pupils in the neighbourhood, while simulated schools typically do not exceed a size of roughly 600–700 pupils (except in rare cases where all proximate schools are crowded, in which case school sizes can reach to over 1,000). Finally, we use the average distance travelled per school as indicator of access to schools; while we use school sizes to estimate school costs.
The results of assignment given the hypothetical case from Sect. 3.1 are given in Table 2. In the second stage of pupil allocation, all locations have more expected than actual (rounded) locations, indicating that all three locations are somewhat crowded. The location in N1 is by far the most crowded. As a consequence, some of the pupils assigned to N1 in the first stage are diverted to N4. N3 does not gain many pupils in the second stage, despite being of a size roughly similar to N4. This is the consequence of the relatively long travel needed to reach N3 from any of the other nodes.
Table 2
Results of two-stage pupil assignment and school size estimation procedures
Variable
Description
N1
N2
N3
N4
First stage of pupil allocation
\(O_{i}\)
Pupils at origin
180
120
150
110
\(D1_{j}\)
First-stage nr of facilities
1
0
1
1
\(A1_{i}\)
First-stage balancing factor
0.98
17.31
0.99
0.98
\(s1_{j}\)
First-stage school size
264.97
0
157.21
137.82
Second stage of pupil allocation
\(FAC_{j}^{cont}\)
Estimated nr of facilities (continuous)
1.18
0
1.04
1.01
\(FAC_{j}^{^{\prime}round}\)
Estimated nr of facilities (rounded)
1
0
1
1
\(ES_{j}\)
Expected facility size, continuous fac
225.33
0
150.99
136.48
\(AS_{j}\)
Expected facility size, rounded fac
264.97
0
157.21
137.82
\(A2_{i}\)
Second-stage balancing factor
1.14
19.50
1.03
0.99
Results
\(s2_{j}\)
Second-stage facility size
260.28
0.00
157.93
141.79
\(\overline{{d2_{j} }}\)
Mean distance travelled to facility
1.7
0.0
1.1
2.2
3.3 Estimating school location costs
Given estimated pupil numbers in s2, the tool proceeds with estimating annual school location costs. Table 3 summarises the assumed parameters for primary school costs estimation. The first step is to derive teaching staff costs, which according to school-level data for England (DfE 2020) constitutes more than half of total costs in primary schools. Teaching staff costs is the product of the number of teaching staff, multiplied by their corresponding salaries. To estimate the number of teaching staff in each school, we take at least one full-time staff member per primary school, and additional staff from an ordered probability distribution of pupil-to-teacher ratios. Assigning teaching staff in this way ensures that schools have teaching staff counts that are proportional to their size while allowing for some variation in the number of staff across schools in the same size range.
Table 3
Assumed parameters for school costs estimation
\(PT_{j}\)
Mean pupil-to-teacher ratio (SD)
13 (1)
\(\overline{{TS_{j} }}\)
Mean costs on teaching staff per head (SD)
EUR 43,000 (1 000)
Ft
Fixed full-time staff paid at average costs levels
1
%TA
Percentage TAs in total teaching staff (paid at half the average costs on teaching staff)
40%
-
Proportion of non-teaching staff to teaching staff
1/4
\(NTS_{j}\)
Costs of non-teaching staff per head
EUR 34,000
Sources: Author’s elaboration based on Eurostat (2021) and UK Department of education (DfE 2020). Parameters based on EU-average values and actual school costs data
Pupil-to-teacher ratios PT for the primary schools in j are drawn randomly based on a normal distribution with mean of 13 and standard deviation of 1, so that (12):
$$PT_{j} \sim N\left( {13,1} \right),$$
(12)
Values of pupil-to-teacher ratios across primary schools in England follow a normal distribution (see Fig. 10 in Annex B). The imputed mean and standard deviation values are based on average ratios across the EU in 2017 (which is the latest year for which data are available). Subsequently, teaching staff in schools \(T_{j}\) is computed as (13):
$$T_{j} = s2_{j} / PT_{j}$$
(13)
where s2 describes the estimated number of pupils from [see Eq. (11c)].
To obtain costs on teaching staff in each school, we make two adjustments. The first adjustment follows the assumption that larger schools pay higher salaries per teaching staff (Zimmer et al. 2009). Besides the argument that a larger organised workforce entails greater negotiation power, larger schools may pay higher wages if the composition of larger schools favours a teaching workforce with more skills/experience; if larger schools retain teachers for longer time; and/or if larger schools are more competitive than smaller schools in attracting more qualified teachers. Teachers in larger schools may be paid more to teach larger classes, and in some cases, they may receive bonuses for teaching very large classes. In modelling terms, we impose that teaching staff salaries per head follow a normal distribution, for which we find support in our reference data (see Fig. 9 in Annex B). Similarly as before, costs in teaching staff are assigned to each school by drawing random values for a normal distribution of teaching staff with mean of 43,000 and standard deviation of 1000, so that (14):
As reference, the mean starting salary for primary schools across OECD countries stands around EUR 32,912, while top salaries average EUR 54,622 (OECD.stat 2020, conversion rate July 2023).
The second adjustment is made because it is unrealistic to assume that each teaching staff is paid at average salary levels when the level of qualification and associated pay is known to vary across teachers of the same school. To obtain teaching costs, each school is assumed to have one base full-time staff paid at mean school salary levels. To these fix costs we add a percentage of the teaching staff paid at half the mean school salaries and the remaining share paid at mean school salaries. This is equivalent to assuming that a share of the teaching workforce in each school has low qualifications and/or experience.
We can then estimate total teaching costs specified in Eq. (4) based on mean school-teacher salaries \(\overline{{TS_{j} }}\) (15):
where Ft is the number of fixed full-time teaching staff, and %TA is the share of teaching assistants in the teaching staff,3 with \(Ft\) = 1 and \(\% TA\) = 0.4. For reference, in the financial records we obtained for primary schools in England, teachers make up 56% of the school teaching staff, and teaching assistants make up the remaining 44%.
To estimate non-teaching staff costs, we assume that every four teaching staff require one non-teaching staff. These proportions follow those observed in the actual data for England. We set median salaries per non-teaching staff in \(NTS_{j}\) at EUR 34,000, a lower value than those of teaching staff, under the assumption that non-teaching staff require lower qualifications. Total costs in non-teaching staff in each school is then expressed in \(NTC_{j}\) (16):
So that non-teaching staff costs are equal to one fixed non-teaching staff (plus the count of non-teaching staff times the mean salary). Finally, we can compute total school costs \(C_{j}\) (17):
$$C_{j} = TC_{j} + NTC_{j}$$
(17)
Table 4 shows the way that total costs are estimated from assigned school sizes in the same hypothetical case study used before. The pupil-to-teacher ratios were set to 13 for all nodes, and teacher salaries were kept at 43,000. The results indicate some non-linearity in total costs and costs per pupil, with larger schools being relatively less costly.
Table 4
Cost estimations for school locations in example nodes
Variable
N1
N2
N3
N4
Number of pupils
260.28
0
157.93
141.79
Number of teaching staff
20.02
0
12.15
10.91
Staff cost estimates
Teaching assistants costs
172,185
0
104,479
93,797
Teacher costs
516,556
0
313,438
281,391
Teaching staff costs
935,924
0
598,182
544,894
Non-teaching staff costs
204,183
0
137,265
126,706
Results
Total costs
935,924
0
598,182
544,894
Costs per pupil
3596
0
3788
3843
The same mean pupil-to-teacher ratios (13) and teacher costs (43,000) were assigned to all school locations for simplicity
3.4 Application of methods to empirical case studies
To verify the performance of the simulations with respect to observed geographical differences in school provision, access and cost, we run the proposed method for England, France and Portugal. England and Portugal were selected as they represent data on which parts of the model chain were calibrated. France is selected to assess our model outside the calibration process because it is a sizeable country with reliable information on school locations. We compare the simulation results with actual school locations, school sizes, stated travelled distances and financial records. Data on school locations were provided by the English, French and Portuguese departments of education (INE 2011; MEN 2017; DfE 2020). They describe school locations and pupil attendance for: 14,963 schools with 4,200,779 pupils in England; 35,302 schools with 5,040,089 pupils in France; and 4620 schools with 529,376 pupils in Portugal. For the goals of this paper, these school location databases are considered actual and comprehensive sources of school provision. In addition, the English department of education has provided a sample of school locations including financial records, covering 60% of primary schools with 2,727,656 pupils; and the UK department of transport has provided stated distances travelled to school, averaged by rural–urban classification, collected in the UK National Travel Survey (DfT 2021).
To apply the proposed methods, road travel distances were obtained from road network data (TomTom, 2018). To measure demand, pupil population distributions were obtained from 1 km2 LUISA population age grids for 2011 (Goujon et al. 2021; Jacobs-Crisioni et al. 2020). These describe total local population per 5-year age group and are based on the GEOSTAT census-based, 1km2 population grid (EUROSTAT 2021) and regional population projections prepared for the 2015 ageing report (EC, 2015). We set age ranges at 6–11 year olds for primary school pupils. This range roughly aligns with International Standard Classification of Education (ISCED) level 1. The age range is also close to the English system on which our approach is mostly calibrated (Eurydice 2020). As pupil populations are composed of multiple age groups, fixed percentages were taken of each group, so that the primary school population consists of 80% of the 5–9-year-old population and 40% of 10–14 year olds (covering ages 6, 7, 8 and 9, and 10 and 11, respectively).
We acknowledge however that school attendance ages differ from the selected age ranges across countries. In fact, education demand was underestimated in the presented simulations, and this may have affected the allocation considerably. Schools were allocated in England assuming a total pupil population of 3,710,593 in England and 4,604,098 and in France, roughly half a million less pupils than observed. The total assumed pupil population Portugal was 604,790, which is a roughly 70,000 pupil overestimation. The regionally aggregate sizes of every population group in the used population grids are consistent with official population statistics (provided to Eurostat by the UK). It is therefore unlikely that the discrepancies between aggregate observed and simulated populations differed substantially. Thus, the primary reason for the found discrepancy between primary school populations must be that the modelled school ages tend to undershoot or overshoot the real length of pupils’ school attendance. We will show in the results section that the proposed method can reproduce observed urban–rural differences in costs and access fairly accurately despite demand differences.
To verify whether the adopted method reproduces observed access and cost differences between urban and rural areas, we mostly compare results that are aggregated by the Degree of Urbanisation classification. This classification entails a harmonised definition that was recently endorsed by the United Nations’ Statistical commission (European Commission 2020). It is based on 1km2 population density grids and designates these grid cells as cities, towns, suburbs, villages, dispersed rural or mostly uninhabited. The results presented in this paper encompass statistics based on estimated distances, and statistics based on school characteristics and efficiency. Distance-based statistics are reasoned from the pupils’ perspective, and thus aggregated based on the degree of urbanisation of pupils’ place of residence. Unfortunately, comparable statistics on actual facility costs and attended facilities by pupil’s residences could not be obtained; so that distance statistics are an exception in this paper, and all other statistics are aggregated using the degree of urbanisation of the grid cell in which a school is placed.4
4 Results
Results are split into four sections, subsequently discussing the accuracy of location-allocation, pupil assignment, school size and school cost simulation. We use four sources of data in this section (they are marked by single letters for brevity), representing data from England and Portugal on which parts of the method were fitted, and data on school locations from France, which is used as a partial validation set. Where relevant, we present results for: (1) simulated locations (marked by an s), (2) actual locations (a), (3) the subset of English schools with financial record data (f), and (4) the UK national travel survey (t), see DfT (2021). In most material, the focus will be on England, as it provides the most comprehensive dataset including locations, partial costs and aggregate distances travelled. The aim of the simulated placement is not to reproduce actual numbers of schools and pupils perfectly, as observed school distributions are without doubt dependent on many factors that are left unaccounted for in our approach. Factors that are not accounted for include the historical distribution of schools as well as local or regional political decisions governing school density and school size. We nevertheless consider it useful to compare observed and simulated placements for a number of reasons, for instance to test whether the applied approach yields geographically biased results and to obtain a baseline set of access and cost values.
4.1 School locations
Figure 3 gives an overview of simulation results for England. Table 5 reports numbers of school locations and pupil’s average distances to closest schools per degree of urbanisation. While we consider travelled distances our central access indicator, distance to the closest school is used here to verify how similar simulated locations are compared with observed locations (Kompil et al. 2019a). The simulations tend to allocate less schools than observed, except for mostly uninhabited areas in France and Portugal and dispersed rural areas in Portugal where the simulation approach slightly overestimates school provision. This may be related to underestimated pupil populations or can be linked to factors other than population distribution, such as historical locations. In relative terms, the simulated allocation results are in line with actual locations. In England and France, absolute and simulated results per degree of urbanisation are almost perfectly correlated (both with a 0.98 correlation coefficient) and in Portugal correlation between simulated and actual provision is considerable (with a correlation of 0.76).
Fig. 3
Simulated school locations and road distances to closest school locations in England
Table 5
Number of school locations, and population-weighted average distances to closest school locations, per degree of urbanisation
Mostly uninhabited
Dispersed rural
Villages
Suburbs
Towns
Cities
Number of schools
England
a
297 (2.0%)
1964 (12.9%)
1555 (10.2%)
1535 (10.1%)
2456 (16.1%)
7410 (48.7%)
s
278 (2.4%)
1471 (12.6%)
1249 (10.7%)
1022 (8.7%)
2048 (17.5%)
5635 (48.2%)
f
214 (2.1%)
1370 (13.7%)
1023 (10.2%)
1036 (10.4%)
1550 (15.5%)
4789 (48.0%)
France
a
1201 (3.4%)
12,618 (35.7%)
7076 (20.0%)
2979 (8.4%)
4316 (12.2%)
7112 (20.1%)
s
1561 (7.7%)
7228 (35.6%)
4121 (20.3%)
1568 (7.7%)
2304 (11.3%)
3519 (17.3%)
Portugal
a
63 (1.4%)
976 (21.1%)
726 (15.7%)
916 (19.8%)
751 (16.3%)
1188 (25.7%)
s
75 (2.6%)
1021 (36.0%)
455 (16.0%)
366 (12.9%)
403 (14.2%)
515 (18.2%)
Distance to closest school (km)
England
a
3.58
2.47
1.17
1.32
0.75
0.53
s
3.69
2.84
1.49
1.45
0.61
0.47
France
a
3.65
2.09
0.86
1.08
0.58
0.35
s
3.92
2.85
1.42
1.41
0.69
0.37
Portugal
a
4.98
2.94
1.05
1.07
0.61
0.37
s
4.17
2.94
1.48
1.53
0.68
0.38
Table shows results from actual locations (a), simulated locations (s) and a subset of schools in England for which financial records are available (f)
×
The expected structural effects of demand density on distances to school are also represented well by the model. Distances statistics are similar along the various degrees of urbanisation, with the largest difference being a roughly 760 m overestimation of distance to schools for pupils from dispersed rural areas in France. Overall, average distances to school decline similarly with increased with the degree of urbanisation for both the actual and simulated school locations. Distances to simulated and actual locations being are almost perfectly correlated when averaged by degree of urbanisation (0.98 in France and Portugal, 0.99 in England), and considerable correlation when comparing these distances at the grid level (0.73 in England, 0.74 in France and 0.67 in Portugal).
4.2 Pupil assignment
Table 6 shows simulated or, for actual locations, reported numbers of pupils by the degree of urbanisation of school locations. Inaccuracies in simulated pupil numbers may arise from differences in estimated pupil populations, misses in the school location model, or erroneous assumptions in the floating catchment and school sizes models by which pupils are assigned to school locations. In Portugal, absolute differences play out chiefly in an overestimation of students in schools in dispersed rural locations, seemingly at the cost of schools in towns. This is consistent with discrepancies in school allocation results. In France and England, the relative pupil distributions are fairly similar, while total pupil numbers per degree of urbanisation differ substantially, so the source of discrepancy must be the underestimation of total pupil population. In fact, correlations between observed and simulated pupils per degree of urbanisation indicate that the simulation approach captures differences between urban and rural areas almost perfectly, with correlation coefficients of 0.99 for England and France, and 0.97 for Portugal.
Table 6
Number of pupils by school degree of urbanisation, and distances travelled by degree of urbanisation of pupil residence
Mostly uninhabited
Dispersed rural
Villages
Suburbs
Towns
Cities
Assigned pupils (thousands)
England
a
33.4 (0.8%)
208.9 (4.9%)
260.7 (6.1%)
372.9 (8.7%)
700.2 (16.4%)
2694.0 (63.1%)
s
25.5 (0.7%)
180.7 (4.9%)
227.4 (6.1%)
276.6 (7.5%)
671.8 (18.1%)
2328.4 (62.8%)
f
23.7 (0.9%)
147.7 (5.4%)
169.2 (6.2%)
244.5 (9.0%)
431.2 (15.8%)
1711.4 (62.7%)
France
a
65.6 (1.3%)
867.1 (17.2%)
1021.6 (20.3%)
533.6 (10.6%)
849.2 (16.8%)
1702.9 (33.8%)
s
122.7 (2.7%)
861.9 (18.7%)
839.3 (18.2%)
449.2 (9.8%)
750.7 (16.3%)
1580.2 (34.3%)
Portugal
a
4.8 (0.9%)
47.5 (9.0%)
62.2 (11.7%)
91.4 (17.3%)
124.5 (23.5%)
199.0 (37.6%)
s
3.3 (0.5%)
91.9 (15.2%)
74.2 (12.3%)
102.0 (16.9%)
125.0 (20.7%)
208.5 (34.5%)
Distance travelled (km)
England
a
4.30
3.17
2.06
2.09
1.59
1.20
s
4.41
3.56
2.35
2.23
1.57
1.32
t
2.24
1.62
0.86
France
a
4.38
2.83
1.74
1.78
1.28
0.83
s
4.67
3.62
2.35
2.21
1.69
1.24
Portugal
a
5.64
3.64
1.79
1.78
1.33
0.90
s
4.93
3.71
2.46
2.32
1.73
1.27
Table shows results from actual locations (a), simulated locations (s), a subset of schools in England for which financial records are available (f), and stated distances by urban–rural classification according to the UK National Travel Survey (t), which are paired with similar degree of urbanisation classes by the authors for the sake of comparison
Table 6 also shows distances travelled by pupils to attend school. For England, the table also includes actual travelled distances. In general, differences in travelled distances between actual and simulated school locations repeat the differences found in the closest location analysis. Because of the imposed balancing mechanism, a limited number of pupils are assigned to farther schools, causing estimated travelled distances that are monotonously larger than distances to the closest school. In the case of simulated school locations, the differences between travelled distances and distances to the closest facility tend to be more sizeable than is the case for the actual school locations; this may be due to a smaller number of simulated schools making up for more dispersed school locations, and hitherto longer distances to the other schools considered in the balancing mechanism. The method is capable of capturing general urban–rural differences but produces substantial estimation differences between simulated and actual locations locally. Similar to the distance to closest school measures, actual and simulated travelled distances are highly correlated, with correlation coefficients of 0.99 for all countries when comparing values at the aggregate level; and 0.54, 0.49 and 0.55 for England, Portugal and France at the grid level. Figure 4 shows in detail how these relatively limited correlations at grid level play out on the map, as limited differences in school allocation cause checkerboard patterns where travelled distances are alternatively under- and overestimated locally.
Fig. 4
Simulated and actual school locations, estimated distances travelled to simulated school locations, and differences in estimated travelled distances between actual and simulated locations, around the North York Moors national park in England
×
Importantly, distances travelled to actual locations cannot be used as an empirical validation of that procedure because travelled distances to both actual and simulated locations are based on the floating catchment procedure outlined in the Methods section. As the distances travelled to both actual and simulated locations depend on the balancing method discussed before, only comparisons with stated distances can be used as an empirical validation here. The obtained data describing travelled distances is classified using the UK urban–rural classification, which is not identical, but somewhat similar to the Degree of Urbanisation. We link the UK ‘urban conurbation’ class with ‘cities’; UK ‘urban city and town’ with ‘suburbs and towns’; and UK ‘rural villages, hamlets and isolated’ with the ‘dispersed rural’ and ‘villages’ classes. The UK ‘rural town and fringe’ category is ignored here as it seems to have no clear counterpart in the degrees of urbanisation classification. The discrepancies in the territorial classifications and the inherent differences between stated distances and the results of network analyses (Rietveld et al. 1999) pose a challenge to comparing our simulations and stated distances. Nevertheless, the stated travelled distances seem reasonably in line with our allocation results, although stated distances are all slightly lower than our travelled distances estimates. Most importantly, the stated travelled distances corroborate that travel distances tend to be longer in rural settings.
4.3 School sizes
Table 7 compares actual and simulated school sizes, showing the number of pupils per school and the number of small schools. In line with the UK national block funding formula of schools eligible for sparsity funding (DfE 2019), the threshold to consider a school small depends on average age group sizes. This threshold is set to 21.4 for primary schools, so that schools are considered small if they have less than 128.4 (6 × 21.4) pupils.
Table 7
Average school sizes and number of small schools by school degree of urbanisation
Mostly uninhabited
Dispersed rural
Villages
Suburbs
Towns
Cities
Average pupils per school
UK
a
112.5
106.4
167.7
242.9
285.1
363.6
s
91.6
122.8
182.1
270.7
328.0
413.2
f
110.8
107.8
165.4
236.0
278.2
357.4
FR
a
54.6
68.7
144.4
179.1
196.8
239.4
s
78.6
119.3
203.7
286.5
325.8
449.1
PT
a
75.8
48.7
85.6
99.8
165.8
167.5
s
43.5
90.0
163.0
278.7
310.2
404.8
Small schools
UK
a
213 (71.7%)
1478 (75.3%)
584 (37.6%)
226 (14.7%)
136 (5.5%)
123 (1.7%)
s
223 (80.2%)
861 (58.5%)
198 (15.9%)
20 (2.0%)
2 (0.1%)
11 (0.2%)
f
156 (72.9%)
1024 (74.7%)
387 (37.8%)
157 (15.2%)
89 (5.7%)
70 (1.5%)
FR
a
1097 (91.3%)
11,273 (89.3%)
3471 (49.1%)
953 (32.0%)
1093 (25.3%)
898 (12.6%)
s
1337 (85.7%)
4096 (56.7%)
413 (10.0%)
20 (1.3%)
2 (0.1%)
7 (0.2%)
PT
a
49 (77.8%)
907 (92.9%)
582 (80.2%)
688 (75.1%)
353 (47.0%)
566 (47.6%)
s
72 (96.0%)
740 (72.5%)
154 (33.8%)
2 (0.5%)
5 (1.2%)
1 (0.2%)
Table shows results from actual locations (a), simulated locations (s), and a subset of schools in England for which financial records are available (f). Percentages of small schools are relative to total number of schools in the respective degree of urbanisation and country
In general, the simulations seem to overshoot school size, with larger schools in most countries in most degrees of urbanisation. This overshoot is fairly surprising, as the method reproduces distances to closest facilities fairly well despite substantially lower simulated pupil populations. It is possible that the imposed allocation rules overemphasise the dispersion of primary schools at the cost of efficiency. This does not seem to be the case here, so that the simulation yields school distributions with similar access characteristics at a lower cost. Possibly, excluded factors contributing to inertia (for instance capital costs) limit cost efficiency in actual school distributions. Despite overestimations in school size, the simulations reproduce school size discrepancies between degrees of urbanisation fairly well, with correlations between actual and simulated school sizes at the aggregate level measuring 0.99 (England), 0.98 (France) and 0.89 (Portugal).
The overestimation of school sizes is evident in the number of simulated small schools. The method underestimates the presence of small schools in all degrees of urbanisation, both in absolute and relative terms, except in mostly uninhabited areas. The much higher number of small schools for example in the west part of the mapped area in Fig. 5. Differences in small schools are especially sizeable in France and Portugal, where even a substantial share of city schools would be considered small by UK criteria. The surprising actual presence of small schools even in sizeable cities can thus not be reproduced by the adopted method. In Portugal, school sizes tend to be smaller across the country, reflecting different national criteria in the allocation of students to schools (OECD 2022). In France, schools are smaller as well, although the fraction of small schools is much smaller than in Portugal. This may be due to omitted factors such as school quality and whether a school is privately or publicly funded. In mostly uninhabited areas, the method tends to overestimate the absolute and relative number of small schools, which we believe is the combined effect of two factors that lead to structurally smaller schools in mostly uninhabited areas. First, there are slightly too many simulated schools in those areas, and second there is a general underestimation of student populations. Despite the considerable differences in simulated numbers of small schools, the simulation method is still able to reproduce structural differences between degrees of urbanisation, with considerable correlation between actual and simulated small schools (0.97 in England, 0.93 in France and 0.80 in Portugal) at the aggregate level.
Fig. 5
Actual and simulated schools considered small or not small around York, England
×
4.4 School costs
School costs have three components: costs on teaching staff, costs on non-teaching staff, and remaining costs. Key estimated costs findings will be presented here; the intermediate steps through estimating costs on teaching staff, costs on non-teaching staff, and remaining costs, are documented in Annex B for England, which is the only country for which facility financial records were available. Other results are available upon request. Key take away from Annex B is that the observed costs can be estimated fairly accurately using the proposed intermediate staff estimates, with a slight underestimation of primary school teaching staff. For France and Portugal, financial records were not found, but some country-wide statistics could be collected for France. Estimating French primary school teaching staff using pupil counts from actual school placements yields a total teaching staff of 369,508. As a benchmark, the number of teaching staff in public schools in France in 2015 was 340,500 in preschool and primary schools (MEN 2015), so that aggregate teaching staff is estimated fairly accurately with the adopted method even using parameters calibrated on data from England. Thus, given observed pupil counts, teaching staff can be estimated fairly accurately. However, when applying the costs estimation method to simulated facilities and pupil numbers, the simulation discrepancies discussed in the previous sections are exacerbated, leading to more sizeable differences in estimated costs between observed and estimated placements.
This work is based on the assertion that schools enjoy returns to scale, so that schools with more pupils can operate more efficiently (Kenny 1982; Andrews et al. 2002; Zimmer et al. 2009). When juxtaposed with pupil counts, actual costs in f (top in Fig. 6) clearly confirm the presence of returns to scale in the set of primary schools for which financial information is available. This figure also includes costs per pupil based on the estimation method proposed in this paper, and actual (a) and simulated (s) school locations, with observed and assigned pupil counts, respectively. Through the imposed relation between pupil numbers and staff required, these offer a stylised representation of the observable scale benefits that schools enjoy with more pupils. Although these distributions are fairly identical, it is noteworthy that compared with actual placements, the simulated placements distribution has a shorter tail and contains some extremely small schools, leading to higher costs peaks. Such small schools may not be viable in England given that such extremely small schools are not recorded in the full dataset of actual school locations.
Fig. 6
Comparison of actual and estimated costs per pupil in England
×
Figure 7 shows box plots of the distribution of estimated costs per pupil for actual and simulated school locations in England. Costs distributions for primary schools again confirm the accuracy of estimates for those schools. In addition, for both the actual and simulated distributions, it is evident that there are significant geographical differences in school costs, with high demand density in cities enabling substantially higher cost efficiency both for the actual schools in England and the simulated placements.
Fig. 7
Box plots of the distribution of estimated costs per pupil, in actual and simulated placements
×
Table 8 shows the estimated costs per pupil for the full actual placement and pupil counts, as well as for simulated school locations, by degree of urbanisation. For England, cost estimates do not vary considerably between actual and simulated placements, in line with the fairly accurate assignment of pupils for those facilities. Nevertheless, despite generally larger schools, the simulated placement procedure tends to overestimate costs per pupil, which may be due to the shorter tail in simulated school sizes. An example of the spatial pattern of costs per student is shown in Fig. 8, clearly showing the discrepancies between central and peripheral locations, and reflecting the patterns of small school sizes seen in the previous section.
Table 8
Estimated absolute and normalised costs per pupil, for actual (a) and simulated (s) primary school placements
Mostly uninhabited
Dispersed rural
Villages
Suburbs
Towns
Cities
Absolute estimated costs (EUR)
UK
a
4350
4284
3863
3632
3528
3412
s
4700
4449
4190
3977
3872
3718
FR
a
4536
4268
3658
3533
3486
3392
s
4784
4380
4029
3850
3785
3627
PT
a
4092
4646
3992
3861
3533
3525
s
5596
4587
4146
3874
3812
3663
Indexed costs (country harmonic mean = 0)
UK
a
538.1
472.1
51.1
− 179.9
− 283.9
− 399.9
s
575.6
324.7
66.2
− 146.8
− 252.4
− 405.9
FR
a
1862.9
944.9
− 280.1
− 444.1
− 546.1
− 616.1
s
1112.5
523.5
− 71.5
− 292.5
− 369.5
− 487.5
PT
a
185
739
85
− 46
− 374
− 382
s
1404.7
395.0
− 45.9
− 318.0
− 380.2
− 528.3
Source: UK Department for Education (2020) and authors’ simulations. Normalised values are normalised to the country mean (country harmonic mean = 0)
Fig. 8
Estimated costs per student in simulated school placements around York, England
×
In France, schools are much smaller on average, and consequently the estimated costs per pupil are considerably higher. Nevertheless for both simulated and actual placements, our estimated costs per pupil are lower than the 2015 value of EUR 6200 per pupil quoted by the French department of education (MEN 2015). There is a sizeable gap in estimated costs between simulated and actual placement of French schools, especially in low-density contexts. For France, simulated schools tend to be more cost-efficient. Although absolute costs per pupil differ, the estimation method clearly captures structural cost differences between degrees of urbanisation in England and France. When aggregated by degree of urbanisation, cost estimates from actual and simulated placements there are very correlated (0.98 for England, 0.99 for France). Portugal is a surprising exception as, in comparison with the other countries, cost efficiency results for mostly uninhabited and dispersed rural schools seem inverted. Portuguese governing rules seem to favour cost efficiency for mostly uninhabited areas, leading to a limited number of relatively cost-efficient schools at the expense of longer travel, while those rules seem to allow relatively cost-inefficient schools in dispersed rural location. Our simulations are unable to capture this nuance, leading to a relatively modest correlation of 0.59 when comparing aggregated costs per student in actual and simulated school placements.
5 Conclusions
This paper introduces a tool to simulate the allocation of school facilities, and estimate likely pupil enrolment and the costs of providing education to the schools’ pupils. The main goal of this tool to produce internationally comparable estimates of access to schools and costs of school provision at a fine spatial scale. Whereas actual access and costs data will be influenced by a wide range of factors, the estimates produced by this tool only take into account the spatial distribution of pupils and the road network. As a result, it can reveal whether the population distribution in one country leads to higher costs or longer access distance as compared to another country. In contrast with many contributions to the location-allocation literature (Wang 2012; Xu et al. 2020), school allocation is not governed by a central optimisation objective, but instead is done in a bottom-up manner, which we assume is more consistent with the governing of school locations in Europe. The school distribution simulation is calibrated on observed spatial school distributions, so that it reflects a balance between access and efficiency that presumably is societally acceptable. Pupil enrolment is simulated through an adapted spatial interaction model that is calibrated to yield general observed distances to schools. Consistent with English school costs and staff records, costs are estimated by deducing staffing and additional costs from school size. The adopted method is validated by comparing modelled and observed results in England, Portugal and France.
In this study, the geography of school demand was described by fine resolution population grids by age class (Jacobs-Crisioni et al. 2020). However, the method is not constrained to the selected data sources. The method, however, can also be used with other data sources. Another attractive feature of the proposed method is that the approach can be repeated using spatial data that describes future or counterfactual pupil distributions (Goujon et al. 2021). It can be used to compare a scenario where the number and location of schools responds to changes in demand to a scenario where the number of schools and their location is kept constant. This method can indicate the likely impact on travel time, class sizes and school costs of these different scenarios. The proposed method can also help to explore the equity effects of expected demographic change and/or changes in the number and location of schools (Proietti et al. 2022).
Validation results show that with the used approach, the spatial distribution of schools, school sizes, school access and school provision costs can be reproduced fairly accurately in England, and similarly, relative model results over rural and urban areas in France and Portugal can be reproduced without salient errors. When comparing statistics aggregated by degrees of urbanisation, model results correlate very well with actual placements. Thus, structural differences between distances to the closest school correlate almost perfectly in England and France, and considerably in Portugal. The amounts of actual and simulated pupil numbers in schools per degree of urbanisation correlate almost perfectly, and so do the actual and simulated distributions of small schools. Aggregate cost comparisons are unfortunately harder to draw, as geographically detailed comprehensive costs data is unavailable.
Across the studied countries, school patterns seem to respond fairly similar to spatial differences in demand, confirming findings for generic services in Europe (Kompil et al. 2019b). The obtained results indicate that with regard to school provision, territories face a trilemma, in which demand density, distance to schools and costs of school provision are interconnected, and it is seemingly impossible to score low values in all three. Given this trilemma, it stands to reason that territories do well to prepare for demographic change, if their demand densities are under pressure due to ageing and population decline (OECD 2021).
The proposed method, and its findings, are not without limitations. These results pertain only to primary schools. Model outcomes depend on a well-specified and granular spatial representation of demand. Given the sizeable differences in estimated and observed pupil populations that we find, such demand is difficult to establish even for primary schools, even though those constitute the least specialised tier of education, and therefore arguably are the most straightforward to model. The method to allocate school locations assumes decentralised governance affecting all school supply. It is trained to reproduce differences between urban and rural school distributions in Portugal at a specific moment in time, and further builds on a model of how staffing costs respond to school size. When comparing with actual school placements, access and costs, we find that our simulation yields considerable absolute differences between model results and actual school data. These differences may, amongst others, be the consequence of differences in imposed demand, local governance rules, unobserved local factors, and even inertia in school supply adapting to changes in demand. For this paper, local accuracy was sacrificed deliberately in order to obtain internationally comparable results of how school provision, access and costs respond to local demand densities. However, through recalibration, the proposed method can in principle be adjusted to better reflect local conditions. School provision costs are limited to annual running costs. School access is assumed linear with pupils’ distances travelled to school, which may ignore considerable heterogeneity in how distances and school travel policies affect actual travel costs. In addition, this method cannot take into account the considerable benefits to active travel (Moodie et al. 2009; Masoumi et al. 2020), nor the opportunity costs for travel to school (Kenny 1982).
Finally, we identify ample opportunities to further extend this research. The method is setup to estimate access and costs to primary schools across the EU and UK; and follow-up publications are planned to present these in full, given current and projected demand. Ideally, the results from this method, assuming decentralised governance, are complemented with access and cost results from methods that optimise access (Tao et al. 2014) or cost efficiency (Fortney 1996; Pacheco and Casado 2005). Such complementary results would be useful to explore the viable space of outcomes, given the trilemma that low densities pose on school provision. To the extent that our results for England, Portugal and France are representative, structural differences between rural and urban areas, in terms of accessibility and costs of provision, can be meaningfully estimated using only information on the spatial distribution of demand and a limited number of governing principles. In Europe’s near future, ageing and depopulation are expected to have a considerable impact on demand for education (European Commission 2023), jeopardising access to schools in particular in remote and rural areas. We consider the introduced methods a useful addition to point where intervention is likely needed as a result of demographic changes, and to assess the likely results of such interventions.
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The adopted location-allocation approach is meant to reproduce observed school placement patterns accurately, under the assumption that the real-world placement patterns yield a societally acceptable balance between school costs (as a function of the size) and travel costs. A grid search has been performed in which parameters and thresholds were structurally changed, and allocation results were compared with observed facility distributions. It is important to repeat here that the applied method is not per se meant to reproduce observed facility distributions. The method is put in place to represent, in a stylised and general fashion, the consequences of population distribution on service provision. Observed school distribution is no doubt dependent on many factors that are left unaccounted for in our approach, such as the historical distribution of schools and political decisions governing school density and school size. We nevertheless believe a comparison is useful, in order to learn how much the allocation procedure deviates from the observed distribution; to test whether the applied approach yields geographically biased results; and to obtain a baseline set of values.
The grid search was performed by adapting values related to maximum distance, minimum size, target size and accessibility weighting. A composite objective function was computed to measure model accuracy given the imputed values. That function was composed of three criteria, namely percentage difference between modelled and observed nationwide number of facilities; the difference between modelled and observed rates of number of urban versus rural facilities; and the mean squared error of percentage points for shares of number of schools per level 2 degree of urbanisation (European Commission 2020), thus discerning school provision in cities, towns, suburbs, villages, dispersed rural areas and mostly uninhabited areas. The grid search has been performed for primary schools in Portugal. Due to its relatively small size and the implications of country size for computational burden, Portugal was found a better fit for this grid search than the other countries for which observed school distributions were readily available (France and England). The results follow that with the adopted threshold values, English school distributions and costs can be reproduced accurately.
One key finding from this grid search is that some parameters have a much more substantial impact on allocation outcomes than others. In particular, the maximum catchment area distance and the school’s target size, which both come into play in the school placement stage of the modelling procedure, have a considerable impact on facility distribution. A grid search of Portuguese primary school allocation yielded that the allocation procedure performs best with a maximum distance of 15 km, and a target school size of 280 (see Fig. 11). These threshold values have therefore been selected as baseline values for allocation of primary schools throughout Europe.
Annex B: Additional detail on cost estimation using pupil counts only
We assess school costs on the assumption that costs vary primarily through number of teaching staff, which in turn depends on pupil counts. As the cost estimations hinge on those assumptions, we go through alternative explanations and provide evidence that supports our assumptions in this annex.
To start, it is worth verifying that the distribution of costs per teacher follows a normal distribution; see Fig. 9. Note that costs are given in GBP in this annex, and EUR in the paper; for the paper we have centred our estimates around European averages, while we discuss the results from the UK DfE here. The primary school data shows that the mean gross salary for teachers is GBP 38 716, and that costs are fairly normally distributed.
Fig. 9
Distribution of costs of teaching staff per teacher—primary schools (actual costs)
×
No salient geographical differences in school cost structures
In the actual data for England, teaching staff costs makes up 57.6% of total school costs. However, there are no noticeable differences in average costs shares by degree of urbanisation (Table
Table 9
Costs structure, actual placement
Degree of urbanisation
Exp. in teaching staff (%)
Exp. in non-teaching staff (%)
Exp. on premises
Exp. on teaching resources
Exp. on catering
Mostly uninhabited
56.7
17.9
8.9
7.4
5.8
Dispersed rural
57.0
18.0
8.9
7.1
5.7
Villages
57.5
17.0
8.8
6.8
6.0
Suburbs
57.5
16.8
9.1
6.5
6.3
Towns
57.9
17.0
8.9
6.2
6.5
Cities
57.4
17.9
9.2
6.0
6.1
Source: UK DfE (2020). Five costs categories aggregated from more disaggregated categories in the original data for presentation purposes
9). This suggest that the average establishment located in a lower density area has a similar costs structure than one located in an urban area. The implication for the costs estimation is that it suffices to estimate establishment-level costs from first principles, i.e. the number of teaching staff required for the school size, instead of explicitly modelling costs structure differences arising from purely geographical factors.
The lack of difference in the costs structure of schools does not mean however that there are no differences in average costs per pupil across geographical areas. While costs in teaching staff per teaching staff does not vary significantly across settlements (in line with nationally set wages), teaching staff costs per pupil is higher in lower density areas—e.g. it is about GBP 700 higher per pupil in dispersed rural areas compared to towns. These differences are reflected in differences in total costs per pupil in rural areas compared to the national mean, which are as high as GBP 921 per pupil in mostly uninhabited areas.
Both total costs per pupil and costs on teaching staff per pupil are higher in rural (mostly uninhabited, dispersed rural, and villages) versus urban areas (suburbs, towns, and cities), and are the lowest in towns (Table
Table 10
Teaching costs statistics, actual placement
Degree of urbanisation
Costs per pupil
Normalised value
Costs per teacher
Normalised value
Mostly uninhabited
5369
921
3016
474
Dispersed rural
5333
886
3031
489
Villages
4497
50
2572
29
Suburbs
4145
-303
2374
-168
Towns
4045
-403
2335
-208
Cities
4339
-109
2479
-64
Source: UK DfE (2020). Values normalised to the country mean (country mean = 0)
10). Differences in costs per pupil across types of settlements are to a large extent driven by differences in teaching staff costs per pupil. This confirms that costs differences in urban versus rural schools are not driven by geographical wage differences or different costs structures between rural and urban schools.
Differences in pupils per teaching staff seem to drive geographical cost differences
As expected, the number of pupils per teacher is also smaller in lower density areas than in more dense areas. For instance, the number of pupils per teacher in a school in dispersed rural areas is 2.5 pupils smaller than in the schools located in a town. However, pupils per teaching staff (which includes both teachers and teaching assistants, all measured in full-time equivalent) differ less across settlement types. This is because the ratio of teaching assistants to teachers is higher in suburbs, towns and cities compared to rural areas (i.e. in villages, dispersed rural areas and mostly uninhabited areas). The variation among schools in rural areas in the number of pupils per teacher is larger than in towns or cities. This suggests that the teaching staff in each school has a bottom limit, so the number of teachers cannot adjust fully to smaller class sizes (Table
Table 11
Schools, pupils and teaching staff, actual placement
The actual school data for England reveals that the distribution of pupil-to-teacher ratio follows a normal distribution with mean 11.9 and standard deviation 2.3 (Fig. 10). In addition, this figure also reveals that pupil-to-teacher ratios vary between degrees of urbanisation, with the most variance in rural areas, and the least in suburbs (Fig. 11).
Fig. 10
Distribution of pupil-to-teacher ratio in primary schools by degree of urbanisation
Fig. 11
Results of adapting maximum distance and target size in allocation procedure for primary schools in Portugal
×
×
Simulating staff using the proposed staff estimation method
Applying the estimation approach to the actual data leads to 202 077 estimated teaching staff, 28 404 teaching staff lower than the actual count (Table
Table 12
Comparison actual and estimated school and teaching staff counts
Degree of urbanisation
Number of schools (actual placement)
Number of teaching staff (actual placement)
Number of teaching staff—estimated (actual placement)
Average teaching staff (actual placement)
Number of schools (simulated placement)
Number of teaching staff—estimated (simulated placement)
12). This result makes sense given that the assumed pupil-to-teacher ratio (13) is higher than the one derived from the actual distribution (11.9). Despite the absolute differences in the number of teaching staff, the approach mimics well the geographic distribution of the number of teachers and the average number of teachers per school by degree of urbanisation, capturing higher averages in towns and cities in simulated placement schools as observed in the actual data.
We apply the same procedure to the simulated placement of schools with their corresponding number of pupils. The results show that as in the actual data, the average number of teachers per school also increases with density, but at a slower rate (Table 12). This is because the simulated placement generates more schools in urban areas than those observed in the actual data.
As Table
Table 13
Comparison of estimated versus actual costs on teaching and non-teaching staff
Degree of urbanisation
Exp. teaching staff per teacher (actual placement)
Exp. teaching staff per teacher—estimated (actual placement)
Exp. teaching staff per teacher—simulated (actual placement)
Exp. non-teaching staff per pupil (actual placement)
Exp. non-teaching staff per pupil—estimated (actual placement)
Exp. non-teaching staff per pupil—simulated (actual placement)
13 shows, the estimated value of teaching costs is 11.3% higher than the actual value. Despite the differences in the average teachers per school between the actual and simulated placement, applying the costs estimation approach to the simulated placement reproduces the size and geographical variation of actual teaching staff costs.
After estimating each of the three costs types, the comparison by degree of urbanisation using actual and simulated placements to actual total costs shows that the proposed approach captures well the levels and geographical variation of costs per pupil (Table
Table 14
Comparison of estimated versus actual costs per pupil
Degree of urbanisation
Exp. per pupil—estimated (actual placement)
Normalised value
Exp. per pupil—simulated (actual placement)
Normalised value
Mostly uninhabited
4350
712
4541
969
Dispersed rural
4284
646
4239
667
Villages
3863
225
3779
207
Suburbs
3632
− 6
3613
41
Towns
3528
− 110
3463
− 109
Cities
3412
− 226
3347
− 225
Source: UK DfE (2020) and authors’ simulations. Values normalised to the country mean (country mean = 0)
This number roughly corresponds to observed share of teaching assistants in total teaching staff. We performed sensitivity tests (not shown but available upon request) of the final costs results to the assumption of fixed staff and conclude that it does not affect the geographical distribution of costs, only minimally its scale.
Through the spatial interaction model used for pupil assignment, school costs for the simulated school placements can technically be linked to the origins of pupils; given the absence of comparable observed data, a discussion of costs brought back to the origins of pupils are left out of this paper.
