Assessment of Public Services Distribution in African Urban Areas: a Cost-Benefit Analysis Based on Bid Rents from Spatial Interaction Models

The paper proposes a cost-benefit analysis of alternative school networks that are otimized to cover the territory of the central part of Huambo Municipality in Angola. Benefits comes from increases in housing prices associated with changes in employment generated by schools, evaluated by variations in values of real estate estimated by the bid-rents of a spatial interaction model using hedonic regression methods. Costs are associated with construction and functioning of schools. The more advisable school networks alternatives are the ones that optimize the coverage of the region with education services while securing individual viability of each one of the schools.


Introduction
In countries dependent on natural resources, as happens with many African countries, the spatial allocation of rents from natural resources is important for peace and development. If those funds go to the main urban areas then cumulative processes of unsustainable development may arise leading to congestion and slums in major cities and to abandonment, poverty and conflict in more detached regions. On the other hand with more equitative spatial allocation of funds, dependend development can spread accross the country soon to be degraded when the allocation of external support disapears. This study assumes that the spatial distribution of rents from natural resources can have a great impact on the economic modernization of the territories of developing countries. In this sense, suitable investments in population centres may be an important way to reduce spatial inequality and diversify the economy through nurturing the creation of viable social capital across the territory, namely social capital. The issue question is to know the spatial redistribution profile that can promote regional sustainable development.
The argument regarding the influence of social capital in the economic development has been widely discussed. Across European countries, Svendsen and Sørensen (2006) documented the social capital influence in Denmark's economic growth, and Woodhouse (2006) presented a case study that highlights the importance of the social capital in regional economic development. In Africa, Mubangizi (2003) demonstrated how social capital has been used as a strategy for economic development. Narayan and Pritchett (1999) reported the impact of social capital as well as the quality of services in rural areas and point out the importance of education services in the creation of social capital.
Investment in the social capital is also part of the agenda for developing countries (Pronyk et al., 2008;Rodríguez-Pose & Hardy, 2015;Theron & Theron, 2014;Tikly & Barrett, 2011), but there is not enough attention to the spatial distribution of those investments and there is a lack of information about their economic returns. In fact, the criteria for the allocation of public services is quite often the quantity of population served and not the spatial accessibility and modernization of the peripheral areas. This has led to the concentration of population in major towns and their detachment from their territories and resources, consequently promoting spatial and social inequalities. The analysis of the relation between public expenditure and regional development has been studied (Isserman et al., 2009;Wilson & Rahe, 2016) but they do not attend the network features of most public expenditures that we want to address in this paper. Furthermore, although the provision of public services is not profit maximization, spatial equity concerns that do not consider that public marginal benefits should equal provision marginal costs tend to inefficient (Justi, 1764, Samuelson, 1954) and unsustainable (Dentinho, 2011). The question is how to do a cost benefit analysis of a network of schools or any other public equipment that can be spatially distributed getting net benefits for all the schools or public equipments of the network.
There is some recent work on the Cost Benefit Analysis of Public Services in urban areas. How they evolve in shrinking cities (Louali et al., 2022); in sprawling 1 3 Assessment of Public Services Distribution in African Urban… cities (Lityński, 2022), and in the design and management of smart cities (Egin & Treleaven, 2019;Turečková & Nevina, 2020).
The specific question for Huambo is to evaluate alternative profiles of consistent education networks, defined by their distribution by zone (village in the rural area or urban district in town) and level (4th year, 9th year, 12th year and University level). To answer that question the paper presents a cost-benefit analysis of two alternative distributions of schools that minimize the cost subject to the minimum distance of access to school in Huambo municipality (Pakissi & Dentinho, 2016) assuming that the distribution of schools generates multiplier effects and economic benefits across space.
The choice of Huambo is justified because it used to be the second city of Angola befores the civil war that lasted for 28 years  with a lot of public infrastructures and equipments destroyed or abandoned. Also, because the authors were involved in the elaboration of the Master Plan of the Municipality which created the necessary database to estimate and calibrate the proposed models. The analysis can be an example for the African context where there is a growing demand of the decision support systems for spatial redistributive policies not only based on physical and environmental geography but also but also on social aspects considered synergistically (Viegas et al., 2013). Figure 1 presents the Conceptual Scheme for the Cost Benefit Analysis Spatial Networks of Public Services proposed in the paper. First (1) there is the design spatial distribtution of services that covers all the territory based on integer programming tools (Revelle et al., 2008) that minimizes the number of locations, subject to constraints that ensure that all nodes can reach any service location below a maximum distance allowed for service. Second (2) implicit to the spatial distribution of centrally allocated public services there is the spatial distribution of basic employment associated to theses services. Third (3) the bidrents of the Spatial Interaction Model are calibrated for the new situation because there is higher demand for space close to the zones where the service is allocated. Fourth (4) the new bid rents are related to real state prices -based on the hedonic regression between real estate prices and bidrents estimated for the status quo -and, weighted by the number of dwellings in each zone. The conceptual scheme allows the interaction with the decision maker that can chose other accessibility criteria if services in some zones are not viable in terms of cost benefit analysis; that is what we try to do moving from alternative 1 to 2 in the present paper.
Spatial interaction models are built to describe and predict flows of people; goods and information through space (Sen & Smith, 1995;Wilson, 1998) and their application to analyze spatial interactions are larlegy reported in established social sciences literature: showing how decentralization increases when land and capital gain importance (Carey, 1858); clarifying consumer's behaviour in space (Reilly, 1931; describing demographic phenomena (Stewart, 1948); relating human interaction with a gravity functions (Carrothers, 1956); or understandind the profile of trip within space (Schneider, 1959). Some studies have contributed to the creation of analytical tools that are regularly used in land use planning, geography and regional science (Anderson, 1979;Batty, 1976;Fotheringham & O'Kelly, 1989;Haynes & Fotheringham, 1984;Isard, 1974;Mikkonen & Luoma, 1999;Wilson, 1970). More specific fields involve transports and accessibility analysis (Evans, 1971(Evans, , 1976Hyman, 1969;Reggiani et al. 2011), commerce and marketing (Huff, 1964;Bergstrand, 1985;Deardorff, 1998;), migration (Plane, 1984) and, more recently, connectivity (Tranos et al. 2013;Reggiani et al. 2015). A broad review of the theoretical background of spatial interaction models is undertaken by (Roy & Thill, 2004). These authors as well as (Fotheringham & O'Kelly, 1989), had established, in their studies, a connection between the traditional SIMs and their geographical approach, which allows a much more consistent integration of the ongoing interpretations in space economic models.
The relationship between the central place theory (Lösch, 1940;Christaller, 1966) and Spatial Innteraction Models (SIMs) has been the center of research, and there are efforts made to transform the central place theory in SIMs. A set of studies made by Wilson suggested that variables usually used in SIMs can provide an implicit representation of various aspects of the central place system (Wilson & Senior, 1974;Wilson, 1970Wilson, , 1976Wilson, , 1979. Actually, central place theory is now regarded as being consolidated in SIMs, because informations about the number of centers in each hierarchical level as well as spacing of these centers are part of attraction term (Wilson, 1977).

Assessment of Public Services Distribution in African Urban…
When successfully developed, integrated and validated, SIMs can act as decision support systems that envision large populations behavior in relation to possible threats for local and regional economy, subsequently enabling ways of developing impact measuring methodologies in order to create local and regional sustainable plans (Couclelis, 2005). Moreover, improvements in the SIM field are expected due to hardware and software accelerated technological development over the years, which according to recent analysis, allows a broader and faster integration of social and ecological complexities as well (Irwin & Geoghegan, 2001).
Evaluation is an important element in a decision process, especially in public policies involving systemic phenomena as urban policies. The most common evaluation methods are Cost Benefit Analysis (CBA), which compares all benefits with costs associated with a policy. The hedonic price method is a preferably choosen method to estimate demand or amount normally used in CBA. Through the decomposition of what is being researched, it is possible to obtain estimates of the contribution for each characteristic (Ekeland et al., 2002).
The objective of this paper is to make a cost-benefit analysis of networks of public services provision in Africa by admitting that the distribution education services in population centres using a spatial equity criterion generates economic benefits through multiplier effects on the local economies estimated by the valuation of the real estate. This approach swiftly responds to the increasing demand for decision support systems that can evaluate the economic impact of alternative networks of public services in order to promote good decision-making in the application of public funds that affect the spatial profile of development Africa that depend very much on the spatial allocation of rents from natural resources.
The analysis focus the municipality of Huambo. The province with the same name is located in the central plateau of Angola and has 11 municipalities and an area of 38.271 km2 with a population of 1,896,147 (INE, 2014). Huambo municipality is the capital of the province and concentrates 35.1% of the population that lives in the urban center with rural conditions due to the inaccessibility to essential services (INE, 2011;Rocha, 2013). Figure 2 contains a geographical placement of the Huambo as well as the zone disaggregation within the central part of the municipality, closer to the city of Huambo. The purpose of the exercise is to analyse the costs and benefits associated to alternative allocations of education services in 100 zones of Huambo. The focus is to understand the impact that an increase in employment associated with the public services will have in the value of houses for each zone of Huambo. The second point of this paper discusses the methodology, which includes the SIM formulation developed in MATLAB, the data acquisition procedures and the basic concepts of hedonic modelling. The third point presents and discusses the benefits and costs of alternative allocations of education service in Huambo. Finally, in point four, taking the results into account, the conclusions come thought.

Methods
This point presents the methodology used in of this study. First, we describe the SIM formulation, development and programming with MATLAB software. Secondly, the process of data acquisition is explained. Finally, we briefly elucidate about the multiple linear regression methodology used for the construction of the hedonic model that relates the bid-rents of the SIM with the house prices.

SIM formulation
The formulated SIM is a gravity urban base model (Lowry, 1966) with residence and places to work constraints. The base model (Hoyt, 1939;North, 1955;Tiebout, 1956) consider exports or basic activities a strategic and driving role in regional and local development. The central idea underlying the base theory is that, as regional economies are open economies, the external demand for the products of a region will have a fundamental importance in its growth.

Assessment of Public Services Distribution in African Urban…
Starting with Basic Employment per zone the model distributes with a gravity function (Sen & Smith, 1995) population and employments, residences, and places to work in the various zones so that the average residence-employment and residence-services estimated travel distances are equal to the real ones. The dual variables on spatial constraints can be interpreted as bid-rents (Wilson, 2010). The model considers that exports (basic activities -E B ) are the propulsive factors of the economy; non-basic employment (E nb ) refers to employment that serves directly the local population. The sum of E b and E nb gives the total employment (E).
The model assumes that the spatial interaction T ij between one origin i and one destination j from a set of m zones, is positively related with the attraction V/W on destination j (V j /W j ) and negatively related with the distance between them (d ij ). A higher value W j on a specified zone signifies that the attraction (V j /W j ) must be reduced to refrain demand on that zone; W j reflects ultimately higher real estate values and is related to the bid-rent value. Notice that V is introduced to provide scale to zones with different dimensions.
Considering a specific zone i and an interaction with zone j from a set of m zones, the interaction between iand j is given by: For all zones i,j, the population comes: Whereas T ij refers to the commuters that work in zone i and live in zone j, E i is the employment of zone i, r is the inverse of the activity ratio (total population/total employment of the region), V j /W j is the attraction of zone j, α is the parameter that defines the friction by distance for the commuters, d ij is the distance between zone i and j, and P j is the population in zone j.
The activities generated for each zone i that serve the population living in all the other zones within a service range: For all zones I the Employment comes: Where S ji is the activity generated in zone i that serves the population in zone j, P j is all the residents in zone j, s is the non-basic activity ratio (E nb /Population), Vi/Wi is the attraction of zone i, β is the parameter that defines the friction produced by distance for To secure that the residence-employment and population-services average costs from the model are equal to the real average costs, the model goes through an iterated calibration for parameters α and β until the predicted average costs are similar to the real average costs. Finally, V i /W i values as well go through an iterated calibration so as to guarantee the accomplishment of constraints that the demand for space in each zone is lower or equal to the space available. Notice that, in this model, residents and employees compete for the same space in each zone.
The V i /W i calibrated attraction values can also be interpreted as bid rents (Roy &Thill, 2004). The bid-rent (ω i ) is complementary to the transportation costs and is given by the formula: Therefore, equations (7) and (8) expresses mathematically as (10) and (11) respectively:

SIM Implementation in MATLAB
The SIM described is coded and integrated in MATLAB 2013a (Mathworks, Natick, United States) that supports model calibration and simulation functions of alternative allocations of basic employmennt. For each zone, the user must insert the data for zone name, E b , space for population and space for employment. The distance matrix values and r as well as s parameters are also inserted. The r and s parameters are calculated by the following formulas: Other inputs include average distance costs for both jobs and services, maximum number of iterations (I max ) and required tolerance to stop the iterative cycle (error t ).
The attrition parameters α and β are adjusted by Hyman's calibration method (Hyman, 1969). For a hypothetical parameter γ and iteration I: With C real being the real average costs and C estimated being the model estimated average costs. The optimum stop condition is activated if the absolute value of both the differences of average costs is lower than the errort parameter previously defined: With EC e being the estimated average commuting cost, C e is the real average commuting cost, EC s is the estimated average cost for the population to access services in a specific zone, C s is the real average costs for the population to access to services in a specific zone and errort and maximum tolerance to end the iterative cycle.
When the maximum number of iterations (I max ) is lower than the current iteration, or when the Eq. (15).
The calibrated model serves to simulate alternative distributive policies related to modifications in E b , available space and distance matrix variables. The iterative process stops differently in the simulation procedure.
The program stops after a defined number of iterations.

Data Acquisition
The data related to population, employment, employment per sector of economic activity and housing data for each of the 100 neighborhoods, which were considered as zones for the model, were obtained from Huambo Municipality Master Plan (GPH-Huambo, 2012).
The basic employment was calculated according to (Haig, 1927), which stated that the minimal regional or national percentage of employments per sector could be deduced from the local percentage of jobs per sector in each zone. For instance, when the national norm of employment for agriculture is 5% and the local norm is 8%, then 3% of the employment for agricultureis considered as basic employment. On the other hand, when the minimal norm of employment per sector for supporting a zone is found, that value is equal to the E nb . After acquiring the values for employment per sector, it is transformed into percentages by dividing the number of jobs per sector per zone by the total amount of jobs for that zone. The minimum percentage of each sector is multiplied by the total amount of jobs for each zone, which will thus lead to the minimum number of jobs needed to support a population -Enb. Finally, Eb is calculated by subtracting Enb from the total employment value. The basic employment data is estimated according to the employment data obtained from the National Development Plan of Angola (Territorial, 2013). Data corresponding to basic employments of the education sector, the number of students to benefit from that education and the number of classrooms on each zone is estimated according to the percentages given by UNESCO (2014).
We consider an increase in the basic employment of the education sector according to two distribution of schools that minimize the number of schools per level subject to an accessibility threshold per level, which we will designate as alternative 1 and alternative 2 for the scope of this paper (Pakissi & Dentinho, 2016). The costs for each alternative are determined according to the sum of operational costs and infrastructuring cost that are obtained from the PND 2013-2017 (Lei n o 13/2013, 2013; da Rocha, 2013).
The total space available per zone is calculated through the sum of space available for population. Distance matrix is constructed through linear measurements calculated with the Geographic Information Systems (GIS). The average distances between residence -employment and population -services are estimated through the weighted sum of the average distance travelled by employees and population for their jobs or needed services inside or outside the municipality. These model estimates compare with the average distances travelled by commuters and students collected in a survey questionnaire done for the Governo Provincial do Huambo (2012) and serve to calibrate the attriction parameters of the model following formula (14). Finally, the (r) and (s) values were calculated according to equations (12) and (13), r = 3,408 and s = 0,155.

Hedonic modelling and impact estimation
For the hedonic modelling, housing data on each of the considered zones are required, so an inquiry is applied to the residents of these zones in order to gather data regarding house pricing, location (north-south, central-peripheral) and school level involved. Average house prices per zone were obtained by a survey questionnaire done in 2012 for the Huambo Municipality Master Plan (GPH-Huambo, 2012).
Average House Price for each zone is the dependent variable and bid rents and bidrents square are the independent variable. Control dummy variables test differences in land prices between the North and the South of the Benguela raiway line, differences between centre and periphery and for the existence of urban infrastructure in suburban zones (Table 1).
Afterwards, the economic impact of each distribution of schools comes from substituting the bidrents of each scenario in the regression estimated for the status. The total impact on each zone results from multiplying the changes in average house prices for each zone by the the number of houses on the zone.

Results
Four regressions were tested (Table 2). Results show that the 4th regression has the best significancy for all variables and a determination coefficient (R 2 = 0,498). The relation between the real and estimated prices from the hedonic model 4 (Fig. 3) indicate that, generally, bidrents simulated from the Spatial Interaction Model can be good estimates of house prices. The test would be better for land prices but there is no data for land process and the exercise for house prices works for the purpose of the paper (Data in Annex 1). Bid rents are in fact a measure of Spatial Interaction.
Actually, once we have the Spatial Interaction Model with bid rents calibrated for the status quo and the estimated hedonic equation able to relate calibrated bidrents with house prices then it is possible to go along the Conceptual Scheme of Fig. 1 to accomplish the Cost Benefit Analysis of Spatial Networks of Public Services.
First -Step 1 of the Conceptual Scheme in Fig. 1 -we obtain Alternative 1 by minimizing the number of locations, subject to constraints that ensure that all nodes can reach any service location below 8 Km to the 4th year school, 16 Km to the 9th year school, 28 Km to the secondary school and 49 Km to the University). Table 3 presents the costs and number of schools for each education level. Figure 4 displays the spatial distribution of different levels of schools and the implicit spatial distribution of public basic employment per zone as is proposed in Step 2 of the Conceptual Scheme of Fig. 1. Then, Step 3 of the Conceptual Scheme, we use the Spatial Interaction Model to estimate the new Bidrents for each zone and, Step 4, evaluation the   Benefits of Alternative 1 using the Hedonic Regression that relates the new bidrents with the new value of houses. Figure 5 shows the estimated Net Present Benefits of each school for Alternative 1. Finally, since some of the schools show negative Net Present Benefits (Fig. 5), we propose new threshold distances for Alternative 2 and do the process again. The process can continue until all the schools have positive Net Present Benefits. Doing that we not only secure that all the schools have a positive net present value but also that the whole area is covered with the education network. In the end: (1) the distribution of basic employment in education has a consistent distribution rule that minimizes the number of schools and secures accessibility; (2) the distribution of basic employment generates employment multiplier effects across space and related valuation of real estate; and (3) every schools school be viable for all levels. To achieve that the search should continue after alternative 2 testing, for example a threshold for the primary level between 8 Km and 3 Km.

Conclusion
The spacial distribution of employment has a major influence in the economic and demographic landscape (Alpkokin et al., 2008;Leeuwen, 2008;Mcarthur & Osland, 2014) The paper confirmes this while securing a viable spatial distribution of public services in developing countries. In fact, the distribution of education services according to sound feasibility principles creates social capital, covers the education needs of a population (Hu et al., 2014;Theron & Theron, 2014) and can generate sustainable income multiplier effects and modernization in the more remote areas of developing countries (Kandiero, 2009;Rocha, 2010;Strassburg et al., 2014).
Most studies that look at the impact of employment on regional developent frequently focus their attention on private investment and highlight the impact of the employment created by primary activities such as agriculture and tourism. This paper broaden out the analysis for the investment in the public networks namely education while securing their longterm viability. 1 3 The article presents results of cost-benefit analysis from two alternatives of investment in education networks in Africa, looking at the municipality of Huambo, in Angola. The use of the calibrated bid rents of a Spatial Interaction Model to the investment for all shcools. The tool is to adapt the minimum distance for each level of education so that each one and all schools of a consistent spatial distribution are viable.
Finally, one can conclude that the used methodology can be an important tool in decision support system in developing countries where the investment on public equipments and infrastructures is fundamental to structure cities efficiently and improve the economy through the creation of viable capital. In the case of Angola, the Provincial Government of Huambo should change the rule of school allocation based on the population of the zones that forces cumulative processes of congestion and slums in major towns and the abandonment, poverty and conflict in more detached regions. He should also avoid a voluntaristic attitude of investing on unsustainable schools in the more remote areas to where no teachers would like to go. On the contrary, the investment should go to where it is viable and the tools suggested in this paper helps to deal with the complexity of a spatial system and the capacity to influence it through external funding.