The instrumentation of public subsidies for private schools: Different regulatory models with concurrent equity implications

The availability of public funding for private schools, in both primary and secondary education, has become a common feature in a number of OECD countries. The expansion of public subsidies for privately owned schools has consequences that go far beyond the involvement of private actors in the provision of education. These include deepening forms of regulatory governance in educational systems and the blurring of frontiers between public and private education. Public subsidies for privately owned schools have been adopted following diverse rationalities and in pursuit of different goals. In light of the diversity, this research examines the regulatory configurations of private subsidized education provision across OECD countries, from a policy instruments’ perspective. Based on a systematic review of the literature, the article identifies four models of regulation of private subsidized education, and analyses why and how these models have been problematized and have evolved accordingly. The paper pays particular attention to recent educational reforms adopted, in most cases, to tackle the equity challenges posed by publicly subsidized private provision. Finally, the paper elaborates on the implications that this form of provision has for public education and the achievement of equity goals, and reflects on the potential and limits of regulatory reforms when confronting these issues.


Introduction
Over the past decades, publicly funded private schooling has become a central feature in numerous educational systems. The presence of private subsidized schools in educational systems impacts public education in numerous ways. Firstly, by design or by default, private actors' involvement in education provision is frequently accompanied by the extended school choice available to families, triggering competitive dynamics between schools. Secondly, the fact that private subsidized education directly competes with public education, in particular, to attract middle-and middle-high-class students, has resulted in a significant decrease in public education enrolment and has encouraged the white flight from public to private schools 1 . Thirdly, the competitive pressure from publicly funded private schools has encouraged the public school sector to gradually adopt management principles, values and techniques that are more typical of the private sector and, in doing so, has blurred the frontier between public and private education. Fourthly, in terms of educational governance, public subsidies for private schools render more evident the decoupling of the funding and provision roles, traditionally assumed by the state, which is something that contributes to the state managing publicly and privately owned schools as independent operators, through resource allocation and accountability instruments. Fifthly, the presence of private subsidized education also poses certain risks in terms of the conception of education as an equitable public good. Numerous studies show that the presence of private subsidized education impacts negatively on school segregation and social stratification between schools (Alegre and Ferrer, 2010; Dumay and Dupriez, 2014;Eurydice, 2020;OECD, 2012). Existing research challenges the idea that public funding might be a sufficient mechanism to equate public and private schools in terms of their social commitment, or to make private actors comply with equity goals or align with a more public ethos.
The effects of publicly funded private provision in public education have generated a large debate in both political and academic circles. Over the past few years, public regulation has acquired a central position in this debate. International organizations, such as the Organisation for Economic Co-operation and Development (OECD), consider that effective regulations should allow the combination of expected benefits of educational providers' diversity and inhibit their potential negative impact in social dimensions, such as equity (OECD, 2017a). This approach assumes that the diversity of institutional settings regulating subsidized private schools is at the root of the heterogeneous impact of public funding schemes for private education providers (Boeskens, 2016: 57). Scholars such as Rizvi (2016: 2) consider that the recent expansion of private provision in public education systems obliges both scholars and policymakers to reflect further on 'to what extent and how should [the] activities [of the private sector] be regulated, and to what end'.
Nonetheless, the configurations of public subsidies for privately owned schools' arrangements and the resulting hybrid models of educational provision -models that combine a significant share of publicly subsidized private provision with conventional public provision -are not univocal. Countries' decisions to fund private schools respond to different policy goals and can be enacted through different policy instruments, which, in turn, can be assembled in different ways. These decisions need to be understood as the product of specific political and institutional circumstances which, in many cases, date back many decades. At the same time, the different models of public funding for private schooling schemes are not static but are revisited, transformed and calibrated over time. In this paper, we analyse the public subsidies for private schools (PSPS) as a polymodal policy instrument that has been adopted, enacted and regulated differently, according to various political objectives and circumstances. We argue that in the OECD context, at least four regulatory models of private subsidized education can be differentiated, namely, freedom of instruction, equivalence, pro-market competition and autonomy for diversification. Nonetheless, despite their multiple origins, policy goals and regulatory configurations, PSPS have been problematized by different stakeholders and policy actors in numerous countries, usually due to their negative implications in terms of educational equity.
To develop this argument, the paper is structured as follows. The first section starts by presenting the notion of private subsidized education, and the prominence it has acquired in OECD countries. The second section presents the research framework. This first shows how a policy instruments perspective can contribute to the analysis of trajectories, modes of regulation and calibrations of PSPS in educational systems, and also presents the methods of the research. The following sections present the main results regarding the instrumentation of PSPS. Specifically, the third section identifies and presents four different models of regulation in which PSPS have crystallized, whereas the fourth section analyses the evolution of PSPS, with a focus on their problematization over time (especially from the perspective of educational equity) and on the most recent regulatory challenges adopted to face these issues. Finally, we discuss our main results and elaborate on the policy implications of PSPS for public education systems.

Private subsidized education: definition and acquired dimension
Most OECD countries have adopted policies that allow private schools to receive public subsidies at some level (Boeskens, 2016). However, the prevalence and relevance of the private subsidized school sector vary significantly across countries. The private subsidized school sector encompasses any educational institution managed by a non-governmental organization (e.g. faith-based institution, private company or foundation) that receives more than 50% of its funding from public sources (UOE, 2019), which is why schools in this sector are often denominated in international statistics as government-dependent private schools (OECD, 2017b).
On the basis of this notion, Figure 1 shows the evolution of the percentage of secondary education students enrolled in private subsidized schools in OECD countries between 2000 and 2018. While in countries such as Italy, Finland or Norway, this sector does not represent a significant share of the total enrolment, in countries such as Belgium, the Netherlands, the United Kingdom and Spain, it plays an important role in the provision of secondary education (similar figures also apply to primary education). In terms of the evolution of PSPS over the last two decades, it is possible to observe how the share of private subsidized education in terms of the total enrolment has remained relatively stable in most of the countries. The exceptions to this trend are Belgium, where the enrolment in private subsidized education has decreased slightly, and the United Kingdom and Chile, where it is possible to observe a significant increase in the weight of this type of provision between 2000 and 2018.

Analysing PSPS from a policy instruments perspective
Over the past few decades, the regulation of many education systems has evolved from a bureaucratic to a post-bureaucratic model of educational governance, in which school providers are more autonomous and diverse, families' capacity for choice has been expanded and performance-based management encouraged (Maroy, 2009). Overall, this new form of regulatory governance implies a substantive transformation in the role of the state in education, which moves away from playing a leading role in educational provision and acts primarily as a regulator of education through legal, financial and accountability instruments (Majone, 1994;Yeung, 2010). In increasingly fragmented educational systems, the regulation of the preferences and behaviour of autonomous educational actors becomes a significant feature of public action in education. Making public funding available for private providers only adds complexity, from a regulatory perspective, to this situation.
As shown in the previous section, PSPS have been key in the expansion of private schooling in many OECD countries. Nonetheless, PSPS are a polymodal instrument that can crystallize in different policies and funding schemes (e.g. vouchers, contracted schools, charter schools), all of which embody different values, rationales and goals. Analysing PSPS from a policy instruments perspective involves assuming first that instruments' choice and their continuous adaptation do not always respond to pragmatic reasons or follow a linear process, but rather result from the complex and ever-evolving interaction between institutions, political and economic interests, and ideas (Capano and Lippi, 2017;Peters, 2002). From this perspective, the social construction of both the problems and the policy objectives behind various instruments' choice is a key moment in the analysis of any policy process. This implies analysing the set of problems associated with the choice or use of a given policy instrument, as well as the representations and problematizations involved in the reformulation and evolution of the instrument in question (Lascoumes and Le Galès, 2007).
The adoption of PSPS can be motivated by a wide range of rationales and objectives. As highlighted by West and Nikolai (2017), educational authorities can pursue very different policy goals and objectives when adopting PSPS. The authors note that, while certain education systems have adopted them as a means of promoting competition and other market dynamics, others use PSPS to support freedom of instruction, promote pedagogic diversification or achieve educational expansion. Thus, while in certain countries the emergence of publicly funded schools was totally aligned to the neoliberal agenda in the public sector reform of the 1980s, in other countries, the state has been funding private providers for many decades, as a means of supporting faith-based educational institutions and promoting freedom of instruction (Verger et al., 2016). It is precisely this diversity of origins and policy goals that influences -and to a great extent explains -the diversity of regulatory configurations that can be observed among those countries that have a relevant publicly subsidized private education sector.
As will be analysed in this paper, governments opt to publicly fund private schools in response to a diverse set of objectives and factors, that vary according to each country's political, social, educational and historical context. However, the arguments used by private providers to exert pressure on political powers and to advocate public funding, seem to be more similar across national contexts. Usually, private actors emphasize the need for PSPS as a means of guaranteeing freedom of instruction and families' right to choose; they also allege that private provision is more efficient than direct public provision.
Policy instruments do not operate as isolated devices, but as a part of broader regulatory regimes (cf. Levi-Faur, 2011). The way in which policies are instrumented and operate is contingent on their interaction with other instruments and regulations, and with the specific norms, rules and mechanisms that are established to implement them effectively. To some extent, instrumentation should be viewed as a process of policy sedimentation and layering in which one instrument might alter the enactment and effects of the other. PSPS are not absent from these interactive and relational dynamics. Expectedly, the effects of PSPS are conditioned by their assemblage with existing regulations in areas such as school choice, the entrance of new private providers in the system, school accountability or the level of autonomy that schools can enjoy, to name the most relevant. At the same time, regulatory regimes and the policy instruments that configure them are not static. They evolve over time and tend to be calibrated and re-regulated, especially when they do not achieve the expected effects or are problematized by key stakeholders.
Relatedly, another core premise of the policy instruments perspective is that, even when instruments are adopted to achieve specific behavioural or social effects, they tend to generate unexpected and even undesired effects. Policy instruments evolve in ways that are unforeseen when originally adopted; indeed, it is difficult to predict the form that any instrument will end up assuming, let alone their effects (Bezes, 2007). To a great extent, the enactment of PSPS policies has policy implications that go far beyond the mere involvement of non-state actors in the provision of education. Even when PSPS are not designed to promote market competition, their use, with the passage of time, frequently triggers market logics and mechanisms. PSPS frequently advance in parallel with higher levels of freedom of school choice and school autonomy, two features conducive to competitive attitudes and values among educational demand and supply. It follows from this that policies promoting the involvement of private schools in public education systems, end up promoting a certain level of market competition, even when they do not intend to do so, or even if this is not their primary objective. This is the case, as such arrangements alter the subjectivities and logics of action developed by schools (Jabbar, 2015;van Zanten, 2009), as well as the way in which families and students understand, engage with and 'consume' education (Cribb and Ball, 2005;Rizvi, 2016).
The increasing role of private subsidized education entails the coexistence of different types of provision and forms of ownership and makes educational systems more difficult to govern. The autonomy that private subsidized providers tend to enjoy becomes a challenge from a public accountability perspective, especially when educational authorities aim at aligning all types of providers with the education quality and equity goals, set at national level (Hooge et al., 2012). In fact, one of the most well documented, unintended effects of PSPS is found in the area of educational equity. Available evidence on the topic consistently shows that the involvement of private actors in educational provision tends to lead to an increase in student performance inequalities, school segregation or social stratification among schools (Alegre and Ferrer, 2010;OECD, 2012;Waslander et al., 2010). A recent report from Eurydice, comparing European education systems concludes: When the level of public spending per pupil is controlled for, academic segregation (i.e., where students of different levels of academic ability are concentrated in particular schools) in primary education is higher in education systems with a larger government-dependent private sector (Eurydice, 2020: 15).
Over the past few years, the regulation of the private subsidized sector has been at the centre of a debate of both a technical and political nature regarding the potential and limits of public-private mixes for educational provision. A key question in the context of this debate is whether hybrid systems of educational provision can be regulated and calibrated in such a way as to make these systems compatible with the achievement of equity goals and the fulfilment of the right to inclusive education. As we argue in this paper, the equity challenges that PSPS generate in different institutional settings are key to understanding the recent wave of educational reforms that numerous countries with hybrid systems of school provision have adopted.

Methods and data
The evidence presented in this article relies on the analysis of literature, focusing on the instrumentation and regulation of private subsidized education in OECD countries. Primary studies were gathered applying a systematic review approach. Searches were conducted in the main international academic repositories (Scopus and Web of Science), and complemented with searches in other repositories (Dialnet, HAL, Revues/OpenEditions Search) that include literature in French and Spanish. Searches were carried out using a common syntaxis, based on three groups of keywords: (a) denominations of private subsidized education; (b) dimensions of regulation and regulatory instruments; and (c) selected cases 2 (i.e. countries or education systems in which private subsidized education represents more than 10% of the total enrolment in primary or lower secondary education). The primary studies, gathered through these searches, were screened in two different rounds to assess their relevance for the purposes of the study. Table 1 summarizes the search and screening process of the primary studies.
The academic literature finally selected (n = 112) was complemented with hand searching of grey literature (i.e. reports and policy documents), as well as different studies considered relevant but not indexed in the academic literature repositories, such as working papers and articles that are not published in high-impact journals (n = 141).
All the primary studies and complementary literature selected were reviewed and systematized using extraction sheets specific to each country or education system of interest. Each one of the sheets synthesized the normative and legal provisions that characterize the PSPS' regulatory framework in place in each country. More specifically, the review was guided by the following codes, each of which refers to a regulatory dimension of private subsidized education provision: 1. Authorization of providers. These are the legal provisions that establish the requirements with which educational providers need to comply in order to be eligible to receive public funding.

2.
Funding. This dimension includes the regulation of the level of public funding received by private providers, the criteria followed to distribute public resources and which expenses are covered by these funds. 3. School choice and admissions. These constitute the legal provisions as to how the process of school choice should be organized and the admission processes schools should follow. 4. Autonomy. Relates to the level of autonomy of private subsidized providers regarding curricular, pedagogical and managerial aspects. 5. Evaluation and accountability. This category includes school inspections and evaluations, accountability measures and the participation of schools in large-scale assessments. 6. Workforce (teachers). This dimension includes certification requirements and the working conditions of teachers employed in private subsidized schools.
Additionally, and together with information in relation to the different regulatory dimensions, information regarding the recent policy developments in private subsidized education and evidence of reform were also systematized for each country. Finally, for each of the countries, we collected information relating to the historical origins of the PSPS arrangements in place, with particular attention focused on the policy goals that education administrators and decision-makers were pursuing with the establishment of such arrangements.
On the basis of this information collected and synthesized through country-specific extraction sheets, we identified, in an inductive manner, four different regulatory models of private subsidized education. The construction of the typology was primarily informed by the main policy goal that had originally motivated the adoption of a system of public funding for private schools. This was in keeping with West and Nikolai's (2017) observations that the diversity of policy goals, pursued by PSPS, is at the root of the heterogeneity of regulatory frameworks. Thus, we grouped together those education systems in which the institutionalization of PSPS had been driven by similar or comparable motivations and values. For each of these clusters, we identified a number of regulatory patterns, i.e. regulations and normative arrangements shared by most of the countries included in the model.
It should be noted, however, that while each of the models is characterized by a number of regularities, they also exhibit a considerable degree of internal heterogeneity. Thus, while the different countries grouped into a given model share a common dominant policy goal, they might present certain inconsistencies in terms of regulatory provisions. This is the case as the same policy goal can be served by different policy instruments and also because certain PSPS are driven by a combination of policy goals and regulatory frameworks that are in a process of constant evolution. Therefore, the models represent ideal types that synthesize the main regulatory orientations and characteristics of existing educational systems, and there is not necessarily an exact correspondence between the empirical models and the referent cases. Thus, they serve analytical purposes, rather than being oriented towards accurately capturing empirical reality.

Models of private subsidized education: policy goals and regulatory configurations
This section presents four different regulatory models of private subsidized education identified in the OECD context, namely, freedom of instruction, equivalence, market competition and autonomy for educational diversification. As noted above, the models need to be understood as ideal types, and countries only exhibit some of the constitutive elements considered to be specific to each model. Nonetheless, we illustrate the models with numerous examples of the regulatory features of the national cases that fit better within each model.

Freedom of instruction
One of the possible goals of PSPS programmes is to guarantee freedom of instruction. 3 The freedom of instruction model prevails in countries such as Australia, France, Denmark and Canada, in which PSPS schemes were originally adopted as a means of ensuring the financial viability of an already existing private provision, usually managed by faith-based institutions (Barman and Edwards, 2009;Caldwell, 2010;Olsen, 2015). Within these education systems, the regulation of the private subsidized sector is oriented towards guaranteeing that private providers comply with basic quality standards in exchange for public funding, without undermining their autonomy as private educational institutions. In Australia, France and Canada, PSPS were adopted as a response to the precarious economic situation experienced by faith-based private schools in the second half of the 20th century. The decline of religious education was often linked to the creation of a free public education system in the context of the process of educational expansion (Angus, 2003;Fowler, 1992). In this context, religious schools mobilized themselves to force the government to institutionalize a system of public subsidies for private providers and to protect freedom of instruction and the presence in the system of an already established private school sector (Caldwell, 2010;Meuret, 2004;Teyssier, 2011). An exception is Denmark, where PSPS were adopted by the end of the 19th century as a means of avoiding state and church domination in educational provision and guaranteeing a certain level of diversity within the educational offer (Olsen, 2015).
Within this model, the authorization of new private subsidized providers relies primarily on social demand criteria. Indeed, in Australia and France, attempts to reduce the centrality of social demand in the authorization of new private providers (for instance by including certain education planning criteria) have been dismissed because of the strong opposition they generated among private providers, some political parties and a percentage of the public (Guardia Hernández, 2019;Vickers, 2005). Nonetheless, since the main objective of this model is to guarantee the continuity of an existing private supply, rather than foster its expansion, the entry barriers for new private subsidized providers are frequently high. In some cases, this means that private schools are required to demonstrate an initial period of economic self-sufficiency as a prerequisite, before applying for public subsidies (Meuret, 2004;Patrinos, 2001;Van Pelt et al., 2017).
The public funding scheme for private providers only covers the operational expenses of the schools in part, either though block or earmarked grants. In line with the partial nature of public funding, subsidized private schools are allowed to charge families add-on tuition fees. In many cases, these fees represent a substantial share of the total funding of schools (Donnelly, 2012;Nusche et al., 2016;Van Pelt et al., 2017). In the case of Denmark, a country with high levels of public spending in education, school fees in the subsidized private sector are mandatory since they are considered as guaranteeing a minimum parental implication in the school (Schindler-Ragvin, 2008).
The process of school choice in the private subsidized sector is managed independently of the regulation that applies to the public sector. In most cases, the choice of private subsidized schools is not guided by catchment areas or other controlled choice policies that might apply to public schools. Private subsidized schools are allowed to establish their own prioritization criteria and are granted full autonomy to manage the process of admissions (Bosetti and Gereluk, 2016;Caldwell, 2010;Olsen, 2015).
Another feature of this model is that private providers enjoy higher levels of pedagogical and managerial autonomy than public schools. Although in most cases, private subsidized schools must follow basic national curricular standards or learning goals, they can adapt the curriculum to their specific pedagogical or philosophical approaches (Olsen, 2015). France is an exception in relation to this general trend. In France, known for its highly centralized educational system, private subsidized schools must follow exactly the same national curriculum as public schools. In order to guarantee the non-denominational nature of French education, religious private subsidized schools are allowed to offer religious courses as long as they remain optional subjects (Meuret, 2004;Pons et al., 2015). In relation to managerial and staffing decisions, private subsidized schools are also highly autonomous. This usually means that they are in charge of hiring and firing teachers, and in certain countries, they even establish their own salary scales (Caldwell, 2010;Eurydice, 2016;Olsen, 2015).
Finally, regarding evaluation and accountability, the integration of the private subsidized sector into the policies and procedures applied to the public sector is only partial. In most of the countries, private subsidized schools are required to participate in national external tests; however, inspection and evaluation procedures and requirements differ significantly between private subsidized and public schools. In most cases, the evaluation of private subsidized schools is developed internally, without the intervention of public authorities (OECD, 2013).

Equivalence
The second model refers to educational systems in which public funding for private subsidized schools has been adopted as a means of guaranteeing freedom of instruction, but also seeking equivalence between private subsidized and public sectors. Broadly speaking, the regulatory approach of this model can be characterized as an 'exchange logic', according to which the equalization of public funding received by private subsidized and public schools, is paralleled by the loss of autonomy of private providers in areas such as school admissions, the curriculum or evaluation and accountability.
Most of the education systems grouped in this model can be considered as 'old quasi-markets' (Vandenberghe, 1999) or historical public-private partnerships (Verger et al., 2016), in which education provision managed by faith-based institutions is a long-standing tradition. In most cases, the institutionalization of private subsidized education took place in a context in which the state needed to expand provision to respond to an increasing educational demand. This was the case in England 4 and Belgium in the 1950s (De Rynck, 2005;Walford, 2003) and Spain in the 1980s, after the end of the military dictatorship (Calero and Bonal, 1999;Olmedo, 2013). In most of these national contexts, the historical role of faith-based education provision played an important role as an argument and source of legitimization of private providers advocating the adoption of PSPS. In the Netherlands, Belgium and Spain, private providers also argued that national constitutions guaranteed freedom of instruction and that private education institutions should be supported by state funding (Bonal, 2000;Dupriez and Maroy, 2003;James, 1984).
The equivalence logic has clear consequences from a regulatory point of view. Firstly, private subsidized and public schools are funded through the same scheme, and the level of public funding for both sectors is essentially the same (Danhier and Friant, 2019;De Groof, 2004;Patrinos, 2013). In the majority of cases, the public funding that schools receive depends partially on the socioeconomic characteristics of their student body, in order to compensate for the social stratification between private subsidized and public schools (for instance, through formula funding arrangements). Secondly, and in exchange for the equalization of the public funding received, private subsidized schools are prohibited from charging add-on tuition fees, or families' financial contributions are considerably restricted, only representing a small percentage of the total funding received by schools (De Groof, 2004;Nusche et al., 2015;Patrinos, 2013;Vogels, 2002, as cited by Levin et al., 2013). Ireland and Spain are exceptions to this pattern. Despite the equivalence goal, the public funding received by private subsidized schools in both countries is significantly lower than in public schools. In Ireland, this is the case as public funding to cover all operational expenses is only provided for private subsidized schools that do not charge fees to families (Rougier and Honohan, 2015). In the case of Spain, private subsidized and public schools are funded through different schemes and arrangements. This has led to a significant difference in the public funding received by the private and the public sector over the years, in favour of the latter, thereby encouraging private subsidized schools to charge fees to families (Pérez-Díaz and Rodríguez, 2011; Rogero-García and Andrés-Candelas, 2014).
Within this model, the authorization of new private subsidized providers is mainly based on the social demand of schools. However, such considerations are frequently combined with certain educational planning criteria. For example, in the Flemish and French-speaking communities of Belgium and the Netherlands, the potential contribution of the new school to the pedagogical diversification of the school supply in the local area is taken into account when authorizing new providers (Maussen and Vermeulen, 2015; Ministère de la Fédération Wallonie-Bruxelles -Direction des Relations Internationales, 2016; Vlaams Ministerie van Onderwijs en Vorming, 2020). Similarly, in England, education authorities consider the existing supply of religious education in the local area as a criterion for the authorization of a new denominational subsidized school (Walford, 1995). In Spain, private providers need to demonstrate the existence of an unmet demand in the area where they are located as a requirement for receiving public funding (Villarroya, 2000).
School choice in private subsidized education tends to be regulated through controlled school choice systems or common admissions and prioritization criteria. In Spain and England, catchment areas are equivalent for private subsidized and public schools, and admissions are regulated by the same procedures and prioritization criteria (Roberts and Danechi, 2019;Vega-Bayo and Mariel, 2018;Walford, 2003). In Belgium (Flemish and French-speaking communities), the assignment of school places is managed in a centralized way and relies on the common criteria of students' prioritization in the case of overdemand (Cantillon, 2011;Lambrechts and Geurts, 2008). Nevertheless, in the Netherlands and Ireland, both private subsidized schools and public schools enjoy autonomy to manage the process of admission, although only private subsidized schools are legally permitted to consider students' religious (and philosophical in the case of the Netherlands) beliefs as prioritization criteria in relation to applications (Ledwith and Reilly, 2013;Patrinos, 2013;Vermeulen, 2004).
The level of school autonomy enjoyed by private subsidized schools tends to be lower than in the previous model. In terms of curricular aspects, private subsidized and public schools have to follow the national curriculum and achieve the learning standards established by educational authorities (Darmody et al., 2012;Eyles and Machin, 2019;Rouw et al., 2016;Walford, 2003). Regarding the management of human resources, while private subsidized schools have a greater level of discretion with regard to the staff hiring process than public schools, certification requirements and salary scales tend to be harmonized between public and private subsidized schools (Eurydice, 2016;Nusche et al., 2015;Rougier and Honohan, 2015 as cited by Skerrit and Salokangas, 2020).
Finally, regarding evaluation and accountability, and in line with the equivalence objective, there are, formally speaking, no significant differences between private subsidized and public providers in terms of inspection procedures and external assessments (OECD, 2013).

Market competition
Certain countries adopted PSPS with the explicit goal of promoting market competition within the educational system. Although most PSPS programmes tend to generate a certain level of competition between schools, the distinctive feature of this model is that competition is a central programmatic goal, and that competition is promoted at system level.
The market competition model of education regulation is part of the broader pro-market approach to public sector reform, which materialized with the emergence of neoliberalism in the 1980s. According to the neoliberal doctrine, the public funding of private providers constitutes a means of actively encouraging private sector participation in educational provision, increasing the ability of families to choose and fostering competition between schools through funding schemes focused on demand. Competition is conceived as a mechanism undermining the 'monopolistic' situation of the state as a provider, incentivizing schools to perform and respond better to 'clients' demands and, overall, improving both the quality and the efficiency of the educational system. Uncoincidentally, the funding modality in which this model tends to crystallize is the so-called voucher scheme, a per-capita funding mechanism that encourages schools to compete in order to attract students.
The most emblematic national cases representing this model are Chile and Sweden. In both countries, the market competition approach is characterized by the adoption of a voucher scheme and high levels of decentralization. In Chile, market reform in education was adopted at the beginning of the 1980s, in the context of the military dictatorship led by General Augusto Pinochet (Bellei and Vanni, 2015), whereas in Sweden it was adopted at the beginning of the 1990s by a conservative government. The market reform in Sweden surprised the world in the sense that it became a counter-intuitive case of a market approach in a highly developed social-democratic welfare state (Klitgaard, 2008;Wiborg, 2013). In both countries, the adoption of PSPS was justified as a means of promoting competition in the education system, but also because of the supposed superiority of the private sector in terms of efficiency and effectiveness (Chubb and Moe, 1990).
In both Chile and Sweden, private subsidized schools, in addition to funding from vouchers, have been allowed to charge families add-on fees. The possibility of schools charging fees was accompanied by a reduction in the value of the public subsidy. These add-ons have encouraged and exacerbated the social segmentation of the school system generated by market competition. For this reason, recent educational reforms have banned the schools from charging add-on tuition fees and have raised the value of public subsidies accordingly (Böhlmark et al., 2016;Centro de Estudios Mineduc, 2017).
Within this model and in line with market principles, the authorization of new providers originally relied exclusively on social demand criteria (Böhlmark et al., 2016;Grau et al., 2018) and had few restrictions on the type of providers allowed to operate and receive public funding. Thus, in both countries, private schools were initially entitled to public funding as long as they had enough demand, and for-profit providers were eligible for public subsidies. Nonetheless, over the past few years, regulatory reforms adopted in Chile and Sweden have placed more significance on educational planning criteria. In the case of Chile, these changes have been paralleled by the establishment of more discriminant entry criteria. The Inclusion Law, adopted in 2015, impedes for-profit providers from receiving public funding, and prohibits the authorization of new publicly funded schools in areas with oversupply (Muñoz and Weinstein, 2019).
School choice procedures also follow a market logic; families frequently have the right to choose any private subsidized school, independently of their place of residence. In the case of Chile, oversubscribed schools have enjoyed a certain level of discretion in selecting students or in defining admission criteria. Nonetheless, this situation has also changed as a result of the 2015 Inclusion Law. Currently the admission procedures in all publicly funded schools are centralized in a unique platform, the main objective being to reduce the opportunities for schools to select students according to academic ability or socioeconomic status (Carrasco and Honey, 2019).
In terms of school autonomy, private subsidized schools must follow the national curriculum but can adapt it according to their philosophical or educational orientation (OECD, 2017c;Wennström, 2020). Schools also enjoy freedom to choose their pedagogic approach and have the authority to make important decisions in relation to budgets, school staff and organization (for instance, in terms of the school timetable). In contrast, public schools are more strictly subordinated to the prerogatives of local education authorities in many of these domains (Rönnberg, 2011;Santiago et al., 2017). All publicly subsidized schools, regardless of their ownership, take part in the performance-based accountability system; however, there are schools' and teachers' evaluation procedures in place that are different in the case of public and private schools. The results of external evaluations are published in school rankings, browsers or similar artefacts to promote and inform school choice.

Autonomy for educational diversification
The last regulatory model identified covers education systems that have encouraged the private management of public schools and the expansion of a private network of school providers with high levels of pedagogical, curricular and managerial autonomy. Here, the main goal is to subsidize private schools, which are exempt from following certain public regulations, as a means of diversifying the educational offer and favouring the generation of school improvement dynamics within the system. The line between this model and the market competition model is thin and porous, since school autonomy is a necessary condition for competition between schools. In fact, the autonomy model has advanced in certain countries that attempted to promote more dramatic market reforms in the past, which, however, did not progress. Accordingly, market advocates had to be content with 'second best' pro-private sector solutions, such as charter schools in the US and Academies in England.
The main argument behind the educational reforms that have given way to this model is that public school provision is excessively uniform, rigid and is somehow captured by corporatist interests. In addition, PSPS have the potential to unlock this situation, foster innovation and diversify the school offer accordingly. These reforms also emphasize equity values since they have often been adopted under the guise of giving more opportunities of choice to socially disadvantaged populations, particularly since social justice arguments have played a central role in the legitimation of such schemes (Gorard et al., 2019;Vergari, 2007).
The autonomy model has tended towards equating the funding allocated to public and private schools. When first adopted in the 1990s, public subsidies did not cover the total expenses of Academies and charter schools, but at the same time, these schools were not allowed to charge fees to families. It was expected that the school owners would resort to sponsors, philanthropic organizations or competitive funding calls to complement the public baseline funding. Nonetheless, the English PSPS system has evolved towards equalization with public schools' funding (West and Nikolai, 2017). In most US states, charter schools' managers can apply for different sources of public funding, which are usually available to support them in providing for students with special educational needs.
Initially, the Academies programme targeted socially disadvantaged areas; however, a reform adopted in 2010, allowed public schools to convert to Academies in all types of contexts; the same reform also allowed the creation of new Academies, known as Free Schools (Wiborg, 2015). Academies require the authorization of the Department for Education and have a non-profit owner, either a private foundation or a trust. Charter schools, in turn, need to be authorized by an external agency, which can be a school district, an NGO or a university. In certain US states, the charter school sector is liberalized, whereas in others (22 out of 51) there is a cap on the number or proportion of charter schools. 5 Regarding school choice, families seeking a school place in an Academy in England need to follow the established procedure for accessing public schools. Nonetheless, Academies can establish prioritization criteria in cases of overdemand (Roberts and Danechi, 2019). In the US, admission criteria to charter schools have evolved in parallel to broader school choice policies, with a tendency towards more freedom of choice. Accordingly, in the vast majority of states, charter schools can admit students from outside their district and, in fact, from any location within the state. Charter schools typically manage their own admission process, but in the case of overdemand, they are obliged to select their students' intake randomly, for instance, applying a lottery system (Skinner, 2014).
School autonomy is the cornerstone of this regulatory model. Both charter schools and Academies enjoy high levels of autonomy in terms of curricular, pedagogical, budgetary and organizational aspects. This high level of autonomy is intended to favour pedagogic innovation, as well as the customization of the educational project of schools to contextual particularities. These schools also have autonomy in relation to staffing decisions, and in defining the school calendar and timetables (Podgursky, 2006;West and Currie, 2008). Especially in the US context, charter schools tend to be more intense with regard to classroom hours than public schools, which, in fact, has become one of the main 'selling points' of this type of school offer.
Academies and charter schools also have a great deal of authority as regards defining the school curriculum (Bulkley and Fisler, 2003;Exley, 2017;Roberts and Danechi, 2019). However, in practice, this freedom is restricted by the strong presence of external evaluations and accountability measures in the educational system. While the quality-assurance mechanisms are the same for public and private subsidized schools, evaluation and accountability instruments are particularly consequential in the subsidized private school sector, since underachievement in literacy, numeracy and/or other subject areas could lead to the discontinuity of the public funding contract. Thus, accountability pressure is especially tangible in these schools, significantly affecting their curricular decisions and instructional strategies.

Equity issues in private subsidized education: key problematizations
The expansion and consolidation of private subsidized education have posed several equity challenges and promoted profound educational debates, especially in those countries where PSPS have a longer trajectory. At least four main equity-related concerns generated by PSPS can be identified, namely, school segregation, student selection and discrimination, for-profit schools and faith-based schools.
School segregation. PSPS have been frequently problematized due to their contribution to school segregation and social stratification between schools. Although these phenomena are not exclusive to education systems that combine public and private subsidized forms of provision, comparative evidence shows that the presence of private subsidized providers and pro-market policies tends to intensify ethnic and socioeconomic school segregation (Boeskens, 2016;Eurydice, 2020;OECD, 2012). In countries as diverse as Chile, the Netherlands, England, Belgium and Denmark, an important body of evidence shows how the involvement of private providers and the adoption of pro-market policies (i.e. school choice and competition) have negatively affected school segregation (see, for instance, Alegre and Ferrer, 2010;Bonal and Bellei, 2018;Demeuse and Friant, 2011;Dumay and Dupriez, 2014;Dupriez et al., 2018;Elacqua, 2012;Vandenberghe, 1999).
In the Netherlands and the Flemish community of Belgium, the dramatic growth of schools with a substantial percentage of immigrant or ethnic minority students fostered the political debate around private subsidized education and school segregation (Peters and Walraven, 2011;Sierens et al., 2011). In both countries, the debate surrounding school segregation led to the adoption of regulatory reforms to reduce the concentration of ethnic minority students in certain schools (Ladd et al., 2010;Nusche et al., 2015;Sierens et al., 2011). In the particular case of the Netherlands, in 2010, an OECD report warned of the role of school choice triggering immigrant students' segregation and recommended adopting regulatory reforms to make school choice compatible with a more balanced distribution of this group of students across public and private subsidized schools (OECD, 2010;Peters and Walraven, 2011).
In numerous countries, PISA results and data became determinant in problematizing the relationship between private subsidized education and school segregation. This has been the case in countries such as Sweden, Denmark and the French-speaking community of Belgium (Dupriez et al., 2018;Olsen, 2015;Swedish Ministry of Education and Research, 2016). In the Frenchspeaking community of Belgium, this type of evidence was actively used by civil society organizations to denounce market dynamics in the educational system as a key driver of socioeconomic segregation between schools (Dupriez et al., 2018). As Danhier and Friant (2019) point out, the public and political debate around the relationship between private subsidized education and school segregation is essential in explaining the centrality that the distribution of students among schools acquired, in the reforms adopted during the 2000s. In Chile, the high level of school segregation triggered a large political and public debate around the role and impact of private subsidized education that resulted in profound regulatory reforms being adopted over the past decade. Indeed, the important student protests that took place in 2006 and 2011 drew attention to the social stratification of the Chilean education system as one of the main effects of the pro-market policies inherited from the military dictatorship that ruled the country between 1973 and 1989 (Bellei and Cabalin, 2013). An OECD report, released in 2004, pointed out that the high level of school segregation in Chile was mainly explained by the lack of effective regulation of private subsidized education (OECD, 2004). Indeed, the relationship between private subsidized education, regulation and school segregation has been one of the key drivers of the compensatory policies adopted in 2008, and the more ambitious educational reforms passed between 2014 and 2017 (Zancajo, 2019).
Student selection and discrimination. The autonomy in admission procedures that private subsidized schools enjoy favours the discrimination of certain social or minority groups. In certain contexts, student admissions have become a specific area of debate and problematization of private subsidized education.
In Chile, student selection and exclusion has been part of the educational debate for many years. Academic research has shown repeatedly that cream-skimming is one of the main factors explaining the higher academic performance of private subsidized schools, compared with public schools, as well as high levels of social stratification between schools (Contreras et al., 2010;Hsieh and Urquiola, 2006). In 2009, student selection, based on socioeconomic or academic criteria, was legally prohibited in primary education (Santa Cruz, 2016). However, the fact that the admission process continued to be managed by the schools and that the criteria to prioritize applications in the case of overdemand was unclear, led to the continuation of student selection practices, particularly among private subsidized schools (Carrasco et al., 2017). In England, the autonomy of private subsidized schools (Academies and Free Schools) regarding school admissions has also been problematized as it has facilitated these schools enrolling more socially advantaged students (Allen et al., 2010;West et al., 2004).
In the US, there has also been a public debate regarding the selective nature of charter schools. Although this type of school is not legally permitted to select new entrants, marketing strategies, interviews with families and other practices have been identified as promoting the self-exclusion of certain social groups (Welner, 2013). Similarly, and although the existing evidence is scarcer, the prevalence of discriminatory selective practices by private subsidized schools has also been reported in Sweden or Spain. In both countries, there is evidence that private subsidized schools develop student selection practices oriented towards excluding applicants based on their socioeconomic or ethnic background (Villarroya, 2003;West, 2017).
For-profit schools. The management of private subsidized schools by for-profit organizations has also been problematized in the countries where this form of provision is allowed. The debate around this type of educational provider has an equity component, since for-profit privately subsidized schools tend to be more selective than not-for-profit privately subsidized schools (Verger et al., 2020), but the main source of concern relates to the very fact that schools receiving public funding can make a profit from their educational activity. Nonetheless, the number of OECD countries that allow for-profit providers to receive public funding is limited. Only in Chile 6 , Sweden and certain US states, do for-profit providers represent a relevant share among private subsidized schools (Boeskens, 2016).
In Chile, the prohibition of for-profit subsidized providers has been one of the key demands of students and civil society mobilizations. At a political level, one of the main arguments to support the prohibition of for-profit providers was the need to guarantee that all public funding for education is effectively used to improve quality and reduce inequalities, particularly in a context in which public funding for education was expected to increase (Muñoz and Weinstein, 2019). In Sweden, for-profit subsidized providers have also been a source of controversy (West, 2017). The high margin of benefits obtained by these providers (Wenglén, 2016), the bankruptcy of emblematic school chains (Rönnberg, 2017) and the strong opposition of the most important teacher union (Arreman and Holm, 2011) has fostered the public and political debate around the presence of forprofit providers in the system.
Faith-based schools. Another recurrent source of controversy stems from the fact that, in certain contexts, a significant share of publicly funded private providers has a religious orientation and/or is affiliated with a religious denomination. This is particularly the case in the freedom of instruction and the equivalence models, in which, as noted above, the institutionalization of the PSPS sector owes much to the pressure exerted by faith-based institutions and cannot be dissociated from the historical centrality of the church in the provision of education. In many of these contexts, the confessional orientation of PSPS has hardly been an object of controversy; however, a public debate has emerged, when private schools with a 'minority' (i.e. non-Christian) religious orientation have attempted to access the system of public subsidies. This is, for instance, the case in the Netherlands and Denmark, where the recent expansion of Islamic schools has generated a heated debate in which concerns over religious extremism and the limited 'integration' of the migrant population feature prominently (Olsen, 2015;Peters and Walraven, 2011).
Conversely, in other contexts in which religious provision has a long-established tradition, the centrality of faith-based schools has been problematized on equity grounds and in response to the challenges associated with the quasi-monopolistic position enjoyed by a given confession. This is the case in Ireland, where the increasing ethnic and religious diversity of the country has resulted in a debate regarding the capacity of private subsidized schools to prioritize the admission of Catholic students (Rougier and Honohan, 2014). A government report, released in 2011, warned that the regulation of admission processes had not been able to address the recent needs of a more diverse society and has led to the exclusion of certain ethnic groups (Ledwith and Reilly, 2013). Similarly, in England, the selection (and exclusion) of students on the basis of their religious beliefs, has also been a recurrent theme of debate, and has contributed directly to the problematization of school autonomy with regard to student admission (West et al., 2009).

Recent educational reforms: converging towards a pro-equity approach?
The debates and controversies around PSPS have resulted, on many occasions, in initiatives to recalibrate this policy instrument, with the objective of reversing or compensating for its negative equity effects. The identified reforms have focused on three main areas of regulation: the authorization of new providers, the funding of private subsidized schools and the school choice system and admissions.
The authorization of new providers: from social demand to planning criteria. The first identified intervention constitutes the revision of the authorization of new private providers receiving public funding. While originally, the authorization of new publicly funded providers in the context of most of the regulatory models identified was based on their social demand, more recently, educational planning criteria have acquired a more prominent role in several education systems.
Broadly speaking, the regulatory changes oriented towards increasing the role of educational planning criteria in the authorization of private providers can follow three main approaches. The first approach focuses on avoiding situations of oversupply. This has been the case in Sweden, where currently, the process of authorization needs to take into account the 'no harm requirement' (Sahlgren, 2016), meaning that the opening of a new private subsidized school should not negatively affect the level of enrolment of public schools located in the same area. This requirement is mainly oriented towards avoiding oversupply, but also the instability of private supply (Meyland-Smith and Evans, 2009;Miron, 1996;Sahlgren, 2016).
The second approach considers how new publicly funded providers contribute to the diversification of the existing school supply. In the Netherlands and the Flemish and French-speaking communities of Belgium, the authorization of a new provider takes into account, among other factors, to what extent the new provider offers an educational project or pedagogical approach, which is not already covered by other schools in the same area (Ministère de la Fédération Wallonie-Bruxelles -Direction des Relations Internationales, 2016; Vlaams Ministerie van Onderwijs en Vorming, 2020).
The third approach combines the two aforementioned criteria. This is the case in Chile, where a recent regulatory reform has established two main requirements to authorize the opening of a new private subsidized school: firstly, the existence of demand in the local area that is not satisfied by the available school offer (whether it is private subsidized or public) and secondly, the absence of a similar educational project in the local area. New private providers, seeking to obtain public funding, need to justify how they accomplish both requirements (Ministry of Education, 2015).
The funding of private subsidized schools: towards needs-based formulas. Different education systems have redesigned their PSPS schemes along the lines of formula funding or targeted schemes. The logic behind these funding formulas is to make socially disadvantaged students more 'attractive' for those schools that previously excluded them, due to the fact that these students were more 'costly' to educate. Providing schools with additional funding, depending on the number of socially disadvantaged students that they enrol, is supposed to disincentivize cream-skimming practices and accordingly, reduce school segregation. Furthermore, these funding schemes are a means of compensating for the uneven distribution of students across schools, which is a common issue in those education systems that combine public and private subsidized forms of provision (Elacqua et al., 2018;Levin, 2002;OECD, 2019).
Reforms oriented towards establishing need-based funding mechanisms have been adopted in several countries, although they can follow at least two different approaches. A first approach is the adoption of needs-based formula funding, which comprises allocating all or a significant percentage of public funding depending on the socioeconomic or educational characteristics of the school's student body, among other possible factors. This is the funding mechanism that has been in place in the Netherlands since 1985, and the approach adopted in Flemish and French-speaking communities of Belgium during the 2000s.
A second approach consists of the adoption of targeted funding schemes that provide additional resources to schools based on the number of socially disadvantaged students that they enrol. In contrast to formula funding, these schemes are additional to the main allocation of school funds (OECD, 2017c). This approach was adopted in England (Pupil Premium) and Chile (Preferential School Subsidy). In both cases, schools (public and private subsidized) receive additional resources for each student enrolled who is identified as socially disadvantaged. In both countries, targeted funding schemes were introduced to disincentivize cream-skimming practices and counter school segregation, but also to reduce the achievement gap between different social groups (Elacqua and Santos, 2013;West, 2015).
School choice and admissions: recentralizing procedures and criteria. The main objectives of reforms in this third area are to reduce the capacity of schools to develop explicit or subtle student selection processes, and to guarantee that socially disadvantaged families have the same opportunities to make use of the greater capacity of choice that accompanies PSPS (Musset, 2012;OECD, 2019). In order to accomplish these objectives, countries have followed two main regulatory approaches. The first approach consists of the establishment of controlled school choice systems. The Frenchspeaking community of Belgium and Chile have recently adopted a controlled school choice system as a means, among other objectives, of reducing school segregation. In both countries, families express their schooling preferences using a common and centralized process. Applications are prioritized using common criteria which provide access preference to students from socially disadvantaged backgrounds (Cantillon, 2013;Carrasco and Honey, 2019). In both cases, regulatory changes have been highly controversial from a political point of view. Certain private providers and middle-class families organized strong opposition campaigns, as they considered that the centralization of the process of school choice and the prioritization of socially disadvantaged students, was undermining the 'right' of families to choose the education for their children (Bellei, 2016;Delvaux and Maroy, 2009).
The second policy approach to the regulation of school admissions constitutes establishing common criteria of student prioritization, while maintaining the management of the process at school level. This approach was adopted in the Flemish community of Belgium and England. In the case of the Flemish community of Belgium, in 2011, local school boards were allowed to establish mandatory quotas for socially disadvantaged students in all schools in their area (Cantillon, 2011;OECD, 2015). In England, the reforms of the Admission Code in 2007 and 2009 established, among other requirements, the mandatory prioritization of socially disadvantaged students and students with special needs in cases of overdemand (West et al., 2011).

Conclusions and discussion
PSPS have spread across numerous OECD countries over the last few decades. However, the reasons why and how this policy instrument has been adopted, designed and regulated vary significantly across national education systems. While in some countries PSPS were adopted and designed to promote competition between schools and market-like dynamics within national education systems, in other contexts, they have sought to guarantee freedom of instruction, support educational expansion or promote the diversification of the educational system. This diversity of policy objectives is clearly linked to the historical junctures and to the institutional, social and economic contexts in which PSPS have been instrumented.
PSPS configure different regulatory models of hybrid educational provision, according to their design characteristics, but also according to the objectives and principles that they aim to realize. The regulatory models that we have identified in the OECD context are freedom of instruction, equivalence, pro-market competition and autonomy for educational diversification (see Table 2). These models embody different values and notions of educational governance and are rather complex from a regulatory perspective. Within each of them, PSPS interact with a broad range of policies and procedures, such as those regarding school choice, students' admission, the entrance of new providers, accountability and school autonomy.
Nonetheless, regulatory models, and especially the instruments that configure them, are far from static, and are frequently calibrated as a means of adapting to changing environmental circumstances. With the passage of time, policy instruments might generate unintended effects and opportunistic behaviours within the policy system. When these effects are harming specific social groups, public policies frequently come under pressure, especially when evidence relating to these detrimental effects accumulates and a public debate emerges, with different stakeholders pressuring decision-makers into addressing them. In the case of PSPS, their deployment has had welldocumented equity implications, which usually manifest themselves in the form of school segregation and segmentation of the school system, which are triggered by student selection and exclusion practices, among other mechanisms. Although these effects are stronger when PSPS are combined with market-like policies, they cut across all the regulatory models identified. Thus, within a broad range of educational settings, the main source of PSPS problematization is their negative impact on equity, particularly regarding school segregation and the discrimination of socially disadvantaged students and minority groups.
As a result of these equity concerns, and the individual and collective action that has raised them, education systems with a long-standing tradition of private subsidized education have tended to converge towards the adoption of reforms that increase state control and central planning in the regulation of private subsidized providers. The most frequent measures identified in this study include the establishment of controlled school choice systems, needs-based funding schemes and authorization processes that place more importance on education planning criteria. These reforms can be identified in the context of different models of regulation. Specifically, educational systems that are part of the equivalence model are strengthening their regulations as a means of better aligning private provision to public goals and realizing the equivalence goal formally established in the legislation, with measures oriented towards correcting implementation problems and preventing irregular behaviours, etc. For their part, educational systems that are part of the market and autonomy models are adopting new public regulations in the direction of the equivalence model with measures aimed at reducing the discretional practices of private providers and harmonizing the

Freedom of instruction
The aim is to guarantee the freedom of instruction appears as the main driver of public funding of private schools. Frequently, the original aim was to ensure a certain level of state support to denominational schools.

Equivalence
The main policy goal is to guarantee freedom of instruction but ensuring equivalence between private subsidized and public schools. In many cases, the original aim was to incorporate faith-based providers into the 'public system'.

Market competition approach
Private subsidized schools emerged as a consequence of pro-market (ideological) reforms. Although this type combines policy goals of other types, the distinctive characteristic is the (initial) aim of promoting competition as a means to improve effectiveness and equity.

Autonomy for educational diversification
Autonomy as the main driver of effectiveness, innovation and equity improvement New modalities of school are envisaged as the means to overcome state schools' lack of autonomy. Frequently, this type can be considered as a second generation (or second best) of pro-market policies.

US, England (Academies and Free Schools)
Source: Own elaboration.
regulatory frameworks of the public and private subsidized sectors. These regulatory trends are less prevalent (or at least less well-documented) in those countries that integrate the freedom of instruction model. Nonetheless, these regulatory reforms tend to calibrate existing policy instruments rather than altering the policy goals and objectives that these instruments aim to serve. Even the emblematic case of Chile, which has undergone profound pro-equity reforms recently, after a long period of problematization of the market model, has focused on reforming existing instruments and adopting new related tools and techniques. However, these changes have not significantly altered the market approach to educational regulation (instrumented in a voucher funding scheme with total freedom of school choice) that has been enacted in the country since the beginning of the 1980s.
Although it is still early to assess the effectiveness of this recent wave of pro-equity reforms, documented in most OECD education systems with PSPS, emerging evidence of their impact indicates that it might be challenging to inhibit opportunistic behaviours or to compensate for the unintended negative effects of private providers' involvement in educational delivery. The possibility of calibrating PSPS and related instruments to guarantee the achievement of certain equity standards or goals is put into question in light of the available evidence. For instance, the choice and funding reforms adopted in Belgium, both in the French-speaking and Flemish communities have had little or no impact on reducing school segregation (Danhier and Friant, 2019;Sierens et al., 2011). Similarly, the legal prohibition of selective admission in Chile in 2009 and the adoption of a targeted voucher scheme in 2008 did not prevent schools from continuing to select students based on their socioeconomic or educational characteristics (Carrasco et al., 2017;Elacqua and Santos, 2013).
Over the past few years, various international and national stakeholders have begun to warn of the equity challenges posed by the expansion of PSPS. At the same time, they consider that appropriate and well-designed regulatory reforms can allow national education systems to enhance the stated goals of PSPS (i.e. freedom of instruction, school supply diversification, innovation and responsiveness, or educational effectiveness) without undermining equity (OECD, 2017a;World Bank, 2018). However, further research on this topic is necessary. So far, national experiences demonstrate that regulatory reforms are not necessarily retained and properly implemented. One of the reasons for this is related to the powerful interest groups generated, precisely, by public funding of the private sector. Associations of private schools, like-minded private foundations and middleclass families tend to oppose, on occasions vehemently and in a well-coordinated manner, any regulatory change that is perceived as limiting the autonomy of private schools or the freedom of choice for families (Delvaux and Maroy, 2009;Zancajo, 2019). Furthermore, once implemented, instrumental changes might have limited potential in effectively tackling the equity challenges posed by the well-entrenched market forces and logics that, by design or by default, emerge and sediment within hybrid systems of school provision. The logic of public action is frequently circumscribed by existing policy instruments. It may be time for governments to rethink their public funding priorities, if educational equity and social cohesion are to be the primary goals.

Declaration of conflicting interests
The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding
The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the British Academy through the British Academy Postdoctoral Fellowship (PF2\180006).