Keywords: Sociology, European Union, Mainstreaming, EU studies, Political sociology, Historical sociology

In the 1950s and 1960s, the first scholars who studied the post-war institutions that would develop into the EU were influenced by the dominant social science paradigm of their times and sociologists such as Talcott Parsons. They were interested in European integration, the complex process whereby supranational political institutions would generate strong crossborder economic cooperation, thus leading to social interactions between people in the member states and creating new allegiances towards a new political centre and a sense of shared identity beyond the nation-state. Sociologists then disappeared from the field until the 1990s for reasons that we will examine. The core of the chapter then focuses on the "return" of a research agenda inspired by major sociological traditions. "Bringing sociology back in" first meant focusing on the social bases of political integration at a time on the one hand and the effects (or lack thereof) of EU rules. This agenda has developed as many observed the end of the permissive consensus underpinning the European project, what Neil Fligstein labelled the "Euroclash". Beyond extending the realm of EU studies beyond the study of EU institutions and laws, sociologists brought a number of theoretical perspectives and tools of empirical analysis. There are thus sociologies of the EU and the chapter will provide examples of this diversity.

In his classic exposition of the principles of his new discipline, the French sociologist Emile Durkheim emphasized that any truly scientific study requires the application of a specific method to a specific object [START_REF] Durkheim | The Rules of Sociological Method: And Selected Texts on Sociology and Its Method[END_REF](Durkheim [1895]]). This implies, first, to clearly define the phenomena under scrutiny by distinguishing them from others; this also suggests that sociological investigation, to be scientific, needs to stay clear of other types of discourses, be they political, or other scientific approaches to the same phenomenon. Accordingly, the use of rigorous sociological methods to investigate the social transformations brought about by the emergence of the European Union (EU) constitutes the core of the scientific subfield known as 'sociology of the EU'. It revolves around concepts such as European identity, transnational fields, Europeanization or social integration. The sociology of the EU is practiced by a number of professionals using qualitative or quantitative methods, writing in journals and book series specializing in EU studies as well as sociology, and it is routinely taught in academic curricula introducing to European integration and its politics. In both regards, however, the sociology of the EU faces dilemmas. The aim of this chapter is to offer anecessarily selectiveoverview of the achievements of sociological studies of the EU in facing these dilemmas, as well as to point at some avenues where it can be developed by future research. We will mostly focus on empirical sociology, based on qualitative or quantitative studies, rather than on a social theory which, given its broad ambition, exceeds the field of EU studies strictly speaking (Outhwaite 2022; G. [START_REF] Delanty | Social Theory and European Transformation: Is There a European Society?[END_REF]. The chapter first presents the historical development of the sociology of the EU, and the early debates it faced. Then, it delves into the main issues with which sociologists of the EU are busy today, and reflects on the new directions the discipline has been following in recent years.

WHAT IS THE SOCIOLOGY OF THE EU?

On the one hand, the very object of the sociology of the EU remains questionable: Is it about the slow formation of a 'European society' whose boundaries and very existence remain bitterly disputedor is it mostly concerned with an institutional process, the organization of that society by a set of relatively stabilized institutionsspecifically here, that of the EU? Each process raises distinct questionsfor instance, do members of this emerging society actually share an identity, and if so how is it (re)produced? Or, how do transnational elites manage to use the supranational institutions in order to achieve certain goals? In this chapter, we will focus on sociological studies directly approaching the construction of European political, economic and legal institutions. In this perspective, the sociology of the European Union is concerned with the study of the social processes linked to European integration, and more precisely, to the construction of the European Communities (EC) and later the EU. However, both dimensions and processes can hardly be completely separated: The interactions between the social changes at work in European societies and the building of new institutions also beg for clarifications. For instance, are these institutions the mere reflection of a solidifying pre-existing social order, or do they trigger the development of such an order? That is, what is the relation of social integration and of political integration? Of course, sociologists have long faced similarly intricate issues in analysing the social processes at work within nation-states. But attempting to equate the sociology of the EU and the sociology of national societies might well be methodologically and theoretically problematic, too. Indeed, on the other hand, the methodological and theoretical apparatus specifically defining the 'jurisdiction' [START_REF] Abbott | The System of Professions: An Essay on the Division of Expert Labor[END_REF] of the sociology of the EU as a scientific discipline remains contested. This is not only because of the general situation of sociologythat is, the entanglement of sociologists with their object of studybut also because the study of an object reaching beyond national boundaries confronts different national traditions. Each carries with it a distinct representation of what it means to practice sociology for sociology has professionalized along different national lines. It results in a diversity of approaches to the EU by national sociologistsfrom wide-ranging conceptualizations dear to social theory, to indepth qualitative studies, through quantitative researches bringing together an immense amount of data. As a consequence, the sociology of the EU often overlaps with cognate disciplines, in particular political science. Thus, one has to pay attention to these different, and at times conflicting, national disciplinary practices. But this is only part of the issue. It should also be clear that applying concepts developed to account for national societies is normatively loaded, as it inadvertently prescribes a familiar horizon to European integrationeventually bound to turn into something similar to a nationstate. This certainly reflects the entanglement of European studies with the political construction of Europe [START_REF] Rosamond | Field of Dreams: The Discursive Construction of EU Studies, Intellectual Dissidence and the Practice of "Normal Science": Discursive Construction of EU Studies[END_REF]. More generally, though, it might well be that the concepts and methods applied to the study of the EU are 'hopelessly confused by borrowing the conceptual vocabulary of the nation-state' (Delanty 1998, 106) that is, that the 'methodological nationalism' built-into most of the social sciences leads to misrepresenting the actual processes unfolding in the EU [START_REF] Beck | Cosmopolitan Vision[END_REF]. This is not to say that the sociology of the EU should ignore national sociologies and instead seek salvation by borrowing from 'globalization studies'. Rather, this means that the concepts and methods it uses need a thorough examination before they can be meaningfully put to work. To that aim, a short historical detour will prove useful.

A BRIEF HISTORY

Sociological studies have accompanied European integration, as well as its uncertainties and crises. The formal beginnings of the European Communities (EC) in the 1950s prompted dedicated sociological research on the construction of Europe. It benefited from the growing institutionalization of sociology in universities [START_REF] Calhoun | Sociology in America: A History[END_REF]Ruegg 2011, 375-86). Furthermore, in the post-war context, studies of European integration were also shaped by political motivations [START_REF] Rosamond | Field of Dreams: The Discursive Construction of EU Studies, Intellectual Dissidence and the Practice of "Normal Science": Discursive Construction of EU Studies[END_REF][START_REF] Canihac | La Fabrique Savante de l'Europe: Une Archéologie Des Savoirs de l'Europe Communautaire[END_REF]: Understanding Europeans and their endless struggles for poweras well as the way to overcome them. In this context, two approaches stood out during the early years: Transactionalism and neo-functionalism. Both regarded the first Communities as extraordinary attempts to consciously build a new type of supranational polity. But, in line with the functionalist theory developed notably by Talcott Parsons, both sought to analyze the EC as social systems. In particular, they set out to sociologically understand the process of 'integration' allegedly taking place in Europe. Transactionalism was mostly developed from the late 1950s around Karl Deutsch. An eclectic scholar, he tried to move beyond the study of political institutions only. To that aim, he proposed to start from the distinction between 'community' and 'society' inherited from the German sociologist Ferdinand Tönnies. While society is united by division of labour and mutual interest, a community is a function of the communication taking place between its members: It is not so much a certain type of institutional organization, as 'a group of persons united by their ability to exchange information' (Deutsch 1951, 243;[START_REF]Nationalism and Social Communication -An Inquiry into the Foundations of Nationality[END_REF]. Accordingly, he submitted that these various 'transactions', i.e. communications, are key in understanding the construction of national, as well as supranational, communities. He hypothesized that an intensification of communications would lead to the construction of shared values, habits and, eventually, a common 'way of life' (Deutsch 1957). In turn, this may result in a shift of 'loyalties' towards the new community. From this perspective, integration is the social process of constructing a 'sense of community' among the members of the integrated units. Neofunctionalism, arguably the most famous early approach to European integration, elaborated on these ideas while slightly changing their empirical focus (see Saurugger's chapter). It was initially theorized by Ernst B. Haas, a German-born scholar emigrated to the USA, whose book on the ECSC published in 1958 was to become a classic [START_REF] Haas | The Uniting of Europe[END_REF]. For E. Haas, too, integration was both an institutional and a social process: 'Political integration is the process whereby political actors in several distinct national settings are persuaded to shift their loyalties, expectations and political activities toward a new centre, whose institutions possess or demand jurisdiction over the pre-existing national states' (Haas 1958, 16). This is to say that the Communities are institutional systems following an expansive logic. The concept capturing this logic is the famous 'spill-over', i.e. the mechanism by which institutional integration in a limited area requires an extension to neighbouring areas by the force of practical and economic necessities set in motion by the actors. When looking at the EC as social systems, on the other hand, E. Haas proposed to focus less on individuals in general, and more on the activity of certain groupsthe political, economic and social elites. These groups were expected to redirect their 'allegiances' towards the new centre being built, and thus to increasingly shift the focus of political action from national polities to the new European organizations. It is thus at the intersection of the institutional dynamics of overcoming the state and the social dynamics of recomposing national communities that neo-functionalism located the sociological originality of the first European Communities. In spite of these early developments, sociological studies of the European Communities would stall from the late 1960s. This relative eclipse was driven by multiple factors. Changes in the main paradigms in the social sciences led to the domination of modes of explanation less sensitive to issues of community-building, shifting loyalties and socialization (Guiraudon and Favell 2011, 6); at the same time, the institutional study of the European Communities started to be taken over by the small but dynamic field of Community law, which remained largely impervious to sociology (Vauchez 2015). Besides, sociologically-oriented researchers experienced an 'existential crisis', as their theories seemed to be refuted by the political developments of European integration. K. Deutsch observed early on that it was hardly possible to empirically confirm the emergence of a true European 'community' (Deutsch 1963). Most importantly, the 'empty chair crisis'when the French President de Gaulle stopped French participation in the institutions of the Community -seemed to confirm the 'obstinate' force of nationstates [START_REF] Hoffmann | Obstinate or Obsolete? The Fate of the Nation-State and the Case of Western Europe[END_REF]. Neofunctionalists thus questioned their own ideas, and E. Haas himself wrote about the 'obsolescence' of his theory in the mid-1970s (Haas 1975). The preparation and signing of the Maastricht Treaty (1992), however, brought the European Communities, now EU, back on the agenda of sociology. Indeed, the Maastricht reforms not only marked a significant extension of EU competences. It was also meant to complement the economic achievements of European integration by fostering the development of a 'citizens' Europe': Placing concepts such as citizenship, legitimacy or identity centre stage, the heated debates surrounding its ratification raised questions familiar to sociologists (Trenz 2016, 6). Both dimensions of the 'Maastricht moment' triggered numerous studies seeking to illuminate the Europeanization of the categories and practices usually attached to nation-states [START_REF] Caporaso | Transforming Europe: Europeanization and Domestic Change[END_REF]C. Radaelli 2004). This new research, that unfolded throughout the 1990s and early 2000s, schematically followed three main paths (for a slightly different presentation, Guiraudon and Favell 2011). The first one tended to apprehend the EU as a special type of international organization (IO). Much work here attempted to account for the way the EU is governed. Updating neofunctionalist insights, the institutionalization of distinct European 'social fields' was carefully scrutinized [START_REF] Fligstein | Constructing Polities and Markets: An Institutionalist Account of European Integration[END_REF]. But in line with the 'constructivist turn' in the study of IOs, many researchers displaced the focus away from formal institutions and legal rules. Instead, the 'social construction of Europe' [START_REF] Christiansen | The Social Construction of Europe[END_REF] was explored through the emergence of shared norms, ideas and strategies (Parsons 2003;McNamara 1998). This led to empirical studies of the actors populating EU institutions, such as MEPs [START_REF] Abélès | Political Anthropology of a Transnational Institution: The European Parliament[END_REF] and European Commissioners [START_REF] Ross | Jacques Delors and European Integration. Europe and the International Order[END_REF][START_REF] Smith | Politics and the European Commission: Actors, Interdependence, Legitimacy[END_REF]), or of policy sectors usually closely related to national sovereignty, such as defence or immigration [START_REF] Guiraudon | The Constitution of a European Immigration Policy Domain: A Political Sociology Approach[END_REF]Mérand 2008a). Crucially, on the other hand, the increase of available data, for instance the Eurobarometer and the European Social Survey, also opened a path to analyse the EU from the vantage point of its citizens. It led to discussing the production of a European public sphere [START_REF] Koopmans | Towards a European Public Sphere?: Vertical and Horizontal Dimensions of Europeanized Political Communication[END_REF], of European social movements [START_REF] Imig | Contentious Europeans: Protest and Politics in an Emerging Polity[END_REF], or of a European identity driven by the institutional development of the EU, in particular the new European citizenship (Risse 2003; [START_REF] Duchesne | Is There a European Identity?[END_REF][START_REF] Inglehart | Long Term Trends in Mass Support for European Unification[END_REF]. Finally, others still used the tools of comparative sociology to move the inquiry from the EU itself, in order to explore its differentiated interactions with national societies. It was, for instance, illustrated by ambitious comparative social histories of Europe [START_REF] Kaelble | The Social History of Europe, 1945-2000: Recovery and Transformation after Two World Wars[END_REF][START_REF] Crouch | Social Change in Western Europe[END_REF] or by studies of everyday practices transformed by European integration, such as transnational mobility (Favell 2008a;Mau and Mewes 2012). While echoing some older concerns of transactionalism and neofunctionalism, these works significantly broadened the scope of sociological research on the EU. They contributed to 'normalizing' it by demonstrating that the EU could be apprehended with the usual concepts and methods of sociology. However, they largely failed to approach it as an object deserving interest in itself: Instead, it was analysed either as a (special) case of IO, or as a 'secondary variable' explaining other processes. It is only slowly, in the wake of the intense controversies that surrounded the failure of the projected European constitutional treaty (2005), that the process of 'mainstreaming sociology' in EU studies [START_REF] Saurugger | Does European Integration Theory Need Sociology?[END_REF], that is, of treating it as a standard object of sociology, was further taken up. This still ongoing process means that the sociology of the EU is now confronted with standard debates about sociological theories and methods; conversely, it also led to the fruitful re-evaluation of general issues of sociology through the lenses of the EU.

SOCIOLOGY'S CONTRIBUTION TO EU STUDIES

Even though sociological approaches have gained importance over the last decade in EU studies, students or researchers will at some point face the question of disciplinary boundaries: What place is there actually for sociology in 'EU studies'? Indeed, more than ten years after [START_REF] Favell | The Sociology of the European Union: An Agenda[END_REF] agenda for a sociology of the EU, the field remains dominated by integration theories (see Saurugger's chapter) and neo-institutionalist approaches (see Bulmer's chapter). Dialoguing with this literature still is one of the big challenges for sociologists of the EU. In this part, we try to contribute to this dialogue by highlighting the new theoretical and methodological paths opened by sociology for studying the EU.

Sociology of the EU and European integration theories

EU studies gained autonomy as a disciplinary field through the a posteriori rebuilding of theoretical debate between functionalists and intergovernmentalists to explain European integration [START_REF] Rosamond | Field of Dreams: The Discursive Construction of EU Studies, Intellectual Dissidence and the Practice of "Normal Science": Discursive Construction of EU Studies[END_REF]). The 1990s-2000s have thus seen the multiplication of 'integration theories', still discussed, developed and updated today (see for example [START_REF] Schmidt | Rethinking EU Governance: From "Old" to "New" Approaches to Who Steers Integration*[END_REF]. Sociology remained rather at the margins of the field (Ross 2011), diverging from integration theories on at least three main aspects. First of all, they share the same object but differ in their purpose. One of the 'big questions' integration theories try to answer is 'who got the power' between the intergovernmental, supranational and parliamentarian institutions of the EU as sui generis object. On the other hand, the objective of sociology is to understand objects such as public policies, identities, or power struggles through the EU as a case among others.

It thus aims to understand the broader social dynamics the EU is embedded in [START_REF] Georgakakis | The Historical and Political Sociology of the European Union: A Uniquely French Methodological Approach?[END_REF]. Second, sociologists adopt an actor-based rather than institutions-based approach [START_REF] Saurugger | Les approches sociologiques de l'intégration européenne[END_REF]2020). EU integration scholars tend to reify EU institutions to understand the distribution of formal powers and to a lesser extent informal powers between them, as an explanation for integration dynamics. Sociology of the EU, where constructivist structuralism (Kauppi 2018) tends to dominate, is on the other hand more interested in both the internal and the more structural power struggles. The first means opening those 'institutional black boxes' and looking at agency within and between the EU institutions and their representatives. The second implies to analyze actors by replacing them in larger social structures, such as education (Michon 2019) or religion [START_REF] Foret | Religion and Politics in the European Union: The Secular Canopy[END_REF] The attention paid to structures depends on one's epistemological posture. That being said, if not all sociologists are structuralists, they all share an interest for social relations. By doing so, they often offer nuanced and refined understanding that not only contribute to the analysis of processes such as institutionalization, professionalization or politicization, but also more indirectly of where power lies within the European space. In doing so, they often reject the hypothesis that power can be assigned to monolithic institutions (Smith 2010). The third divergence lies in the theoretical ambitions of sociological approaches. Integration theories aim at producing a general theory of European integration, prioritizing theoretical discussions and deductive reasoning. On the contrary, sociologists value empirical-based analysis and more inductive or abductive reasonings. Thus, a contribution of sociology to EU studies is bringing in 'new' methods. Micro-sociology for instance brought fieldwork into the picture (Adler-Nissen 2016). Sociologists study their objects notably by interacting with them, through semi and unstructured interviews, as well as participant and non-participant observations (see below for examples). Macro-sociology also contributed greatly to the renewal of EU studies by bringing in tried-and-tested methods to explore new dimensions of EU integration. For example, it opened a research agenda on European social classes through the use of aggregated data-basis (Dìez Medrano 2011; Lebaron and Blavier 2017). These rather fundamental divides often lead to a mutual disdain between scholars sharing the same object, even though sociology of the EU progressively gained space in EU studies. For example, it is now more common to find dedicated chapters to sociological approaches in handbooks on EU studies (for example, see Jones and Menon 2012; Brack and Gürkan 2020; this handbook).

Sociology of the EU and neo-institutionnalism(s)

Among the calls for more sociological approaches within EU studies that flourished in the late 2000s, Jenson and Mérand (2010, 75) defended the stimulating stand that sociology of the EU actually shares common roots with (neo-)institutionalism, especially its historical and constructivist variations, that tend to dominate EU studies since the mid-1980s. They argue that bridges can be built between the two literatures. The common ground between institutionalism(s) and sociological approaches lies in a shared acknowledgement of the importance of actors' strategies, ideas, identities, norms and interests. The main difference then is that where institutionalism(s) treat them as separate independent variables, sociology doesn't distinguish between them [START_REF] Jenson | Sociology, Institutionalism and the European Union[END_REF]. Indeed, institutionalists distinguish between a logic of consequences (interest-driven) and a logic of appropriateness (idea-driven), where sociologists consider them to be interwind (Mérand 2008b;[START_REF] Jenson | Sociology, Institutionalism and the European Union[END_REF]. Interestingly, this debate was taken a step further in another close but distinct field: international relations (IR). For instance, Samuel B.H. [START_REF] Faure | La « logique du praticable » à l'épreuve de la pratique scientifique[END_REF] critically discussed the relevance of distinguishing a third logic, that is "of practicality" (Pouliot 2008)creating an analytical separation between actors' interests, ideas and practices. Following this critical perspective that first arouse in IR [START_REF] Bigo | Pierre Bourdieu and International Relations: Power of Practices, Practices of Power[END_REF]), Didier Georgakakis (2021) underlined how political sociology, and especially the classical field theory [START_REF] Bourdieu | Microcosmes. Théorie des champs[END_REF], applied to the EU, provides a theoretical framework that encompasses all the preoccupations of the different variants of institutionalism. Indeed, like historical neoinstitutionalism, it pays particular attention to the historical patterns that structured the field; like constructivist neo-institutionalism, it looks at the norms that sustain actors' representations; like rational neo-institutionalism, it seeks to objectify the resources that drive actors' strategies; and finally, like discursive neo-institutionalism (V. [START_REF] Schmidt | Discursive Institutionalism: The Explanatory Power of Ideas and Discourse[END_REF][START_REF] Crespy | When "Bolkestein" Is Trapped by the French Anti-Liberal Discourse: A Discursive-Institutionalist Account of Preference Formation in the Realm of European Union Multi-Level Politics[END_REF], it includes actors' stances in the analysis. Further, applying the concept of field to the EU, both nourishes the debates around its heuristic value, and contributes to EU studies by covering understudied actors of EU integration (Kauppi 2018). For instance, Antoine Vauchez's (2015) analysis of the EU legal and judicial arena as a 'weak field' both offers a take on how to apply field theory to a transnational object, and highlights the key role of lawyers in the building of the European community. Interestingly, US-based sociologists Doug McAdam et Niel Fligstein (2015) took a side-step from Bourdieu's understanding of the concept. They suggest that social life is composed of 'strategic action fields', where actors develop specific 'social skills'. The difference lies in the fact that power relations are not part of the construction of their object, as such, they are closer to constructivist neo-institutionalism [START_REF] Jenson | Sociology, Institutionalism and the European Union[END_REF].

EU STUDIES' CONTRIBUTION TO SOCIOLOGY

The 'mainstreaming' of sociology not only brought traditional sociological debates about concepts and methods back in EU studies. It also produced works that, while empirically analysing the EU, contributed in their own right to more general sociological debates. Here, we briefly illustrate some of these contributions to the debates about historical sociology; identity; elites; and public policies.

Historical sociology

One area in which the study of European integration has lent insights of major sociological interest is historical sociology. Historical sociology is concerned with the long-term historical social processes that have shaped the presente.g. at the macro level, state formation and nation building, or, at the micro level, behavioural norms of politeness and citizenship (Gerard Delanty and Isin 2003). Similarly, a number of works (especially in the French-speaking tradition of 'socio-histoire') have relied on in-depth analyses of shorter episodes of European integration to bring to light the social and historical processes at work in the EU [START_REF] Déloye | Introduction : Éléments Pour Une Approche Socio-Historique de La Construction Européenne[END_REF][START_REF] Cohen | Le "Père de l'Europe[END_REF].Applying socio-historical lenses to the study of the EU has both contributed to normalizing the EU, and has allowed to shed new light on the processes of polity-formation (McNamara 2010;Vauchez 2015;S. Faure 2019). For instance, following early attempts by sociologists [START_REF] Elias | The Society of Individuals[END_REF]Marks 1997) or historians (Milward 1992), Stefano Bartolini has forcefully argued that it represents the sixth major transformation of European polities since the 16 th century (following state building, capitalist development, nation building, democratisation, and the emergence of the welfare-state). However, as an essentially economic project, the EU has not been built through the usual processes of monopolization of legitimate violence and cultural homogenization: It is a 'disjointed' polity in which economic, military or cultural boundaries do not overlap [START_REF] Bartolini | Restructuring Europe: Centre Formation, System Building, and Political Structuring between the Nation State and the European Union[END_REF]. By showing how it departs from to the standard account of nation-state building, this thus allows to conceptualize more systematically, in a nonstate-centric manner, the general processes at work in the building of a polity .

Identity and citizenship

More recently, other studies have sought to complement this perspective by exploring the micro-foundations of such macro processes. This has for instance been applied to the (non-) emergence of a shared feeling of belonging about European citizenship, and to the issues it raises for the legitimacy of the EU [START_REF] Delmotte | Belonging and Indifference to Europe: A Study of Young People in Brussels[END_REF], or to the possibility of an EU-wide sense of solidarity in relation to social policies [START_REF] Börner | European Integration, Processes of Change and the National Experience[END_REF]. Moreover, EU-building opens new avenues to study the effect of increased transnational mobility on identities (Favell 2008b), which leads to new multi-layered identities and, eventually, social conflicts (Fligstein 2009). In this perspective, Kristine Mitchell ( 2015) has demonstrated the existence of an "Erasmus effect" positively impacting the European identification of its participants. European integration also constitutes an opportunity to conduct transnational qualitative inquiries on citizens' attitudes toward a shared object, the EU [START_REF] Gaxie | L'Europe des Européens. Enquête comparative sur les perceptions de l'Europe -Daniel Gaxie[END_REF]. Thus, questioning identities and citizenship in relation to the EU also allows to cast a new light on preexisting (national, local) groups (see Van Ingelgom's chapter). For instance, Juan Dìez [START_REF] Medrano | Framing Europe: Attitudes to European Integration in Germany, Spain, and the United Kingdom[END_REF] showed how differentiated European integration from one member state to another can be explained by national histories.

Sociology of (EU) elites

The EU as a sociological object is a great 'laboratory' for those interested in studying 'elites' in a broad sense. Indeed, the EU and its myriad of actors constitute a timely case to study the emergence of transnational elites. Sociology of the EU notably participated in the understanding of the constitution and evolution of bureaucratic and political elites beyond states' structures. In that sense, it takes up with Ernst Haas early works on elites as social groups, leaving aside the neofunctionalist assumption of a shift of loyalty. It points out both common trends and peculiarities, testing state sociology concepts by applying them to the EU. For example, following Yves Dezalay's work on global elites, Antoine Vauchez and Lola Avril thoroughly documented the early formation of a group of European lawyers [START_REF] Vauchez | How to Become a Transnational Elite. Lawyers' Politics at the Genesis of the European Communities (1950-1970)[END_REF]Avril 2020a), showing how the constitution of an elite is deeply intertwined with policy-and polity-building. In a different vein, Willy [START_REF] Beauvallet | L'institutionnalisation inachevée du Parlement européen. Hétérogénéité nationale, spécialisation du recrutement et autonomisation[END_REF] studied the socialization of eurodeputies and demonstrated the unstable nature of the European Parliament's institutionalization. Further exploring the 'field of Eurocracy' through the study of EU bureaucrats [START_REF] Georgakakis | The Historical and Political Sociology of the European Union: A Uniquely French Methodological Approach?[END_REF]Rowell 2013), Didier Georgakakis (2017) highlighted how new public management overflowed national state administrations and affected the European commission. In her study of diplomats, Rebecca Adler-Nissen (2014) pointed to the struggle for symbolic power between state and European diplomatic corpses, underlining the tensions between national and transnational elites. Recent works have paid special attention to the practices (Adler-Nissen 2016) of the latter within the institutions, showing how they concretely participate in the building of non-state polity. For instance, Laura Landorff ( 2019) has looked at the informal practices of parliamentary actors. Andy Smith (2019) and Frédéric [START_REF] Mérand | The Political Commissioner: A European Ethnography[END_REF] have both highlighted the heuristic value of the concept of political work to study the policy and polity making practices of political elites. Aside from the usual suspects, sociologists have also paid attention to rather unexpected actors within EU politics, such as football players [START_REF] Gasparini | The European treatment of young football players migrations. New issue of power at a European level[END_REF], writers [START_REF] Bonnamy | L'Écrivain et le Marché unique. La mobilisation des représentants d'écrivains français face au projet de réforme européenne du droit d'auteur (2014-2016)[END_REF], wine producers [START_REF] Smith | Vin et politique[END_REF] or even surfers (Weisbein 2015). As such, the sociology of the EU paid great attention to lobbying practices and actors (Morival 2019;[START_REF] Beauvallet | EU affairs[END_REF], extending the scope of the sociology of elites, and building bridges with the sociology of mobilization [START_REF] Balme | L'action collective en Europe[END_REF].

Sociology of (EU) public action

Sociology of public action also paid greater attention to the development of European public policies [START_REF] Balme | Chapitre 11 / Analyse des politiques publiques et intégration européenne. Apports et points aveugles[END_REF]. One of its entry points had been the conceptualization of a Europeanization of public policies process (Radaelli 2002), which both contributed to public policies analysis and theories of integration (see Bartenstein and Wessels' chapter). Studying European public policies participates in the understanding of the transfer of powers among sovereign states, especially through cases extremely closely related to core state powers, such as defence policy [START_REF] Irondelle | Europeanization without the European Union? French Military Reforms 1991-96[END_REF][START_REF] Hoeffler | L'européanisation sans l'Union européenne[END_REF][START_REF] Faure | Défense Européenne. Émergence d'une Culture Stratégique Commune[END_REF] or immigration policy [START_REF] Guiraudon | The Constitution of a European Immigration Policy Domain: A Political Sociology Approach[END_REF]. The EU also constitutes an interesting case for the sociology of instruments of public policy (Lascoumes and Le Galès 2005). For example, through her analysis of the EU gender equality policy, Sophie [START_REF] Jacquot | The Paradox of Gender Mainstreaming: Unanticipated Effects of New Modes of Governance in the Gender Equality Domain[END_REF] shows how gender mainstreaming constitutes a tool of public action, along with others such as benchmarking and the open method of coordination [START_REF] Bruno | Europeanization through Its Instrumentation: Benchmarking, Mainstreaming and the Open Method of Co-Ordination … Toolbox or Pandora's Box?[END_REF]. In her work on the Schengen visas policy, Juliette [START_REF] Dupont | Verrou Ou Vitrine ? Politiques Du Visa Schengen En Algérie et En Chine[END_REF] showed how short-term visas can play both as a security and a marketing tool, going beyond border control. Analysis of European public policies contributed greatly to the literature studying regulation policies and more generally State-Market relations and the intertwining of polity-and market-building. Considering the EU as a regulatory state (Avril 2020b), it notably pointed to the inner tensions between liberalism and protectionism that irrigate European industrial public policies, from defence [START_REF] Hoeffler | European Armament Co-Operation and the Renewal of Industrial Policy Motives[END_REF] to agri-food (Smith 2021). Finally, recent works in sociology of the EU applied a programmatic action framework [START_REF] Hassenteufel | The Programmatic Action Framework: An Empirical Assessment[END_REF] to EU public policy cases, as well as in a comparative perspective between member states bridging further sociology of elites and sociology of public action [START_REF] Genieys | Bringing the state elites back in ?: Les gardiens des politiques de l'État en Europe et aux États-Unis[END_REF]S. B. H. Faure 2020). As such, this emerging literature contributes to a double dynamic of mainstreaming of sociology in EU studies and to EU as an timely case for sociologists.

CONCLUSION

Today, sociology constitutes one of the most fruitful and dynamic disciplinary approach to the EU. This is the result of a long process of 'mainstreaming' sociology in EU studiesthat is, of turning it into a normal way of studying the EU. Conversely, this has produced insights of interest for sociologists more generally, especially for those studying identity, elites, state formation and public action. It also has proven to be a timely case for sociological history. This process of normalization is still ongoing, though, on at least three respects. First, the sociology of the EU has hardly touched on some traditional areas of sociology. For instance, questions and methods of the old field of sociology of knowledge have only recently been applied to the EU (Adler-Nissen and Kropp 2017). Likewise, despite isolated research in the 1990s, such as George [START_REF] Ross | Jacques Delors and European Integration. Europe and the International Order[END_REF] work on the Delors Commission, ethnographic methods remained rather unexplored territory. In this regard, following previous works such as Kathleen McNamara's work (2015) on the role of everyday social practices in EU-building, Rebecca Adler-Nissen (2016) called for a 'practice turn' in EU studies. She seems to have been heard both by experienced and young researchers that recently conducted ethnographic fieldwork in various EU spaces such as the European Commission (Mérand 2021a), far-right groups within the European Parliament [START_REF] Delaine | A l'extrême Droite de l'hémicycle : Une Sociologie Politique Des Nationalistes Dans Le Champ de l'Eurocratie[END_REF] or the College of Europe [START_REF] Behar | Faire Le Collège d'Europe. Une Sociologie de La Formation Des[END_REF]. Second, the EU itself is an evolving object. Sociological studies have to integrate this inherent historical dimension to specify how, exactly, they can contribute to sociology at large. The numerous crises of European integration, the study of the 'resistances' it triggers [START_REF] Crespy | From Euroscepticism to Resistance to European Integration: An Interdisciplinary Perspective[END_REF][START_REF] Hamm | Les Conditions de l'Europe[END_REF], as well as the prospects of 'disintegration' or of'differentiated integration' they bring about, have become major objects of study for sociologists of the EU, opening new paths for further research. Finally, the debates on the place of sociology in EU studies seem to have set aside its pluralistic dimension, with its own internal debates and controversies [START_REF] Weisbein | Vers une sociologie pragmatique de l'Union européenne ?[END_REF]. While sociology of the EU tend to be dominated by approaches inspired by Pierre Bourdieu's and Norbert Elias' fields and configurations' theories, other major authors such as Bruno Latour or Luc Boltanski remain at the margins. For instance, one can wonder why, whereas pragmatic sociology has proven heuristic to study national public policies [START_REF] Zittoun | La fabrique politique des politiques publiques[END_REF][START_REF] Zittoun | L'État sous pression[END_REF], ten years after Julien Weisbein's call (2011), it remains anecdotic when studying the EU. It opens new paths for mainstreaming of sociology in EU studies : bringing long going ontological and epistemological controversies of the first into the latter.

Journal of Social Science Research 17 (2): 97-118. https://doi.org/10.1080/1351161042000238643. Landorff, Laura. 2019. Inside European Parliament Politics: Informality, Information and Intergroups. Springer. Lascoumes, Pierre, and Patrick Le Galès, eds. 2005. Gouverner par les instruments. Académique. Paris: Presses de Sciences Po. http://www.cairn.info/gouverner-par-les-instruments--9782724609492.htm. Lebaron, Frédéric, and Pierre Blavier. 2017. 'Classes et nations : quelle articulation à l'échelle européenne ?' Actes de la recherche en sciences sociales 219 (4): 80-97. Marks, Gary. 1997. 'A Third Lens: Comparing European Integration and State Building'. In European Integration in Social and Historical Perspective -1850 to the Present, edited by Louise A. Tilly and Jytte Klausen. New York & Oxford: Rowman&Littlefield Publishers. Mau, Steffen, and Jan Mewes. 2012. 'Horizontal Europeanization in Contextual Perspective: What Drives Cross-Border Activities within the European Union?' European Societies 14 (1): 7-34. https://doi.org/10.1080/14616696.2011.638083. McNAMARA, Kathleen R. 1998. The Currency of Ideas : Monetary Politics in the European Union. Cornell University Press. Ithaca. McNamara, Kathleen R. 2010. 'Constructing Europe: Insights from Historical Sociology'. Comparative European Politics 8 (1): 127-42. https://doi.org/10.1057/cep.2010.8. McNamara, Kathleen R. 2015. The Politics of Everyday Europe: Constructing Authority in the European Union. Oxford: Oxford University Press. https://doi.org/10.1093/acprof:oso/9780198716235.001.0001. Mérand, Frédéric. 2008a. European Defence Policy beyond the Nation State. Oxford; New York: Oxford University Press. ---. 2008b. 'Les institutionnalistes (américains) devraient-ils lire les sociologues (français) ?' Politique europeenne 25 (2): 23-51. ---. 2021a. The Political Commissioner: A European Ethnography. Transformations in Governance. Oxford: Oxford University Press. https://doi.org/10.1093/oso/9780192893970.001.0001. ---. 2021b. 'Political Work in the Stability and Growth Pact'. Journal of European Public Policy 0 (0): 1-19. https://doi.org/10.1080/13501763.2021.1901961. Michon, Sébastien. 2019. A l'école des eurocrates. La genèse de la vocation européenne. Res Publica. Rennes: Presses Universitaires de Rennes. Milward, Alan. 1992. The European Rescue of the Nation State. Routledge. London. Mitchell, Kristine. 2015. 'Rethinking the "Erasmus Effect" on European Identity'. Journal of Common Market Studies 53 (2): 330-48. Morival, Yohann. 2019. Les Europes du patronat français depuis 1948. https://www.peterlang.com/document/1057814. Outhwaite, William. 2022. 'Social Theory and European Integration'. In Routledge International Handbook of Contemporary Social and Political Theory, edited by Gerard Delanty and Stephen P. Turner, Second edition. Routledge International Handbooks. London New York: Routledge. https://doi.org/10.4324/9781003111399. Parsons, Craig. 2003. A Certain Idea of Europe. Cornell University Press. Ithaca & London. Pouliot, Vincent. 2008. 'The Logic of Practicality: A Theory of Practice of Security Communities'. International Organization 62 (2): 257-88. https://doi.org/10.1017/S0020818308080090. Radaelli, Claudio. 2004. 'Europeanisation: Solution or Problem?' European Integration Online Papers 8 (16). Radaelli, Claudio M. 2002. 'The Domestic Impact of European Union Public Policy: Notes on Concepts, Methods, and the Challenge of Empirical Research'. Politique Europeenne 5 (1): 105-36. Risse, Thomas. 2003. 'The Euro between National and European Identity'. Journal of European Public Policy 10 (4): 487-505. https://doi.org/10.1080/1350176032000101235.


Jean-Baptiste Hiriart-Urruty

Lassère Patrice

When is L r (R) contained in L p (R) + L q (R) ?

We prove a necessary and sufficient condition on the exponents p, q, r ≥ 1 such that L r (R) ⊂ L p (R)+L q (R). In doing so, we explore the structure of L p (R) + L q (R) as a normed vector space.

1 Introduction.

In a recent mathematical note aimed at undergraduate students and their teachers ([3]), J.-B. Hiriart-Urruty and M. Pradel proposed a way to extend the Fourier transformation to all the spaces L r (R) with 1 ≤ r ≤ 2 in the following manner.

-First, they classically define the Fourier transformation on L 1 (R). Then they define it on L 2 (R) using in that case the much less known Wiener's approach which relies on a specific Hilbertian basis made of the so-called Bernstein functions.

-After having checked the coherence of both definitions on L 1 (R) ∩ L 2 (R), they extend the definition of the Fourier transformation to the space L 1 (R) + L 2 (R).

-Finally, and this is the key-point, they prove the inclusion L r (R) ⊂ L 1 (R) + L 2 (R) for all 1 ≤ r ≤ 2, so that the Fourier transformation can be extended to all the Lebesgue spaces L r (R).

An obvious question which arises is, what happens if r ∈ [1, 2] ? For example, is the inclusion L 3 (R) ⊂ L 1 (R) + L 2 (R) true or not ? The objective of the present mathematical note is to answer this question. We provide a necessary and sufficient condition for the inclusion L r (R) ⊂ L p (R) + L q (R) to hold true. For that purpose, concentrate on the sum L p (R) + L q (R) and see how it is structured as a normed vector space.

2 How to norm a sum of normed vector spaces ?

Let V and W be two vector subspaces of a "holdall" vector space, E. If V is equipped with a norm • 1 and W with a norm • 2 , is there a natural way to define a norm on the vector subspace V + W ? Of course, we do not assume that V ∩ W = {O E }. If this was the case, a natural way to define a norm N on V + W would be N (u) = v 1 + w 2 , whenever u ∈ V + W is (uniquely) decomposed as u = v + w, with v ∈ V and w ∈ W . What we have in mind is indeed V = L p (R), W = L q (R) and E = L(R) as the "holdall" vector space (L(R) stands for the set of all Lebesgue classes of mesurable functions on R).

The theorem below answers the question posed above. It does not seem to be well-known, except by people who have to deal with the interpolation of functional spaces (like in [START_REF] Bergh | Interpolation Spaces, Grundlehren der mathematischen Wissenschaften[END_REF]).

Theorem 1 Let N be defined on V + W as follows N (u) := inf{ v 1 + w 2 ; u = v + w with v ∈ V and w ∈ W }.

(1)

Then, N is a semi-norm on V + W . It is a norm under the following "compatibility" assumption :

(T )

z k ∈ V ∩ W, z k -→ a in (V, ||.|| 1 ) z k -→ b in (W, ||.|| 2 )    =⇒ a = b. Proof : To check that N (O E ) = 0, N (λu) = |λ|N (u) for all λ ∈ R, u ∈ V +W ,
and N (u 1 + u 2 ) ≤ N (u 1 ) + N (u 2 ) for all u 1 and u 2 in V + W , does not raise any difficulty. It suffices to use the definition of the lower bound (or infimum) of a set of real numbers.

To prove that N (u) = 0 implies that u = O E is a bit more tricky. Our experience with that question with undergraduate students shows that they usually fail to answer it correctly. Their common mistake is to deduce that a sequence (v k + w k ) k converges to O E using the fact that (v k ) k converges to O E in V and (w k ) k converges to O E in W . We therefore provide a proof here.

We first begin by observing that

ν : V ∩ W u → ν(u) := max( u 1 , u 2 ) ( 2
)
is a norm on V ∩ W ; this is an easy result to prove. Consider therefore, u ∈ V + W such that N (u) = 0. We take for example

u = v + w, with v ∈ V and w ∈ W. (3)
Due to the definition (1) of N (u), for all positive integers k, there exists

v k ∈ V and w k ∈ W such that u = v k + w k , and (4)
v k 1 + w k 2 ≤ N (u) + 1 k = 1 k . ( 5
) Thus, (v k ) k converges to O E in V and (w k ) k converges to O E in w. But what about (v k + w k ) k ?
Recall that no norm, hence, no topology, has yet been defined on V + W . We infer from (3) and (4) that

v + w = v k + w k , and thus v -v k = w k -w. (6)
As a consequence, this common vector

z k := v -v k = w k -w lies in V ∩ W and, since ν(z k ) = v -v k 1 or w k -w 2 , z k → v in (V, • 1 ) and z k → -w in (W, • 2 ).
The assumption (T ) then ensures that

v = -w, that is u = v + w = O E .
Note that the technical "compatibility" assumption (T ) is satisfied -trivially if V ∩ W = {O E } (in that case it amounts to 0 = 0).

-in the cases where

V = L p (R), W = L q (R) (indeed, convergence of (f k ) k towards f in L p (R) implies convergence almost everywhere of a subsequence of (f k ) k towards f ).
-when the "holdall" vector space E is a Hausdorff topological vector space in which V and W are continuously imbedded.

We suppose that (T ) is in force for the rest of the section.

The vector space V +W , equipped with the norm N as defined in (1), inherits some properties of (V, • 1 ) and (W, • 2 ). Here is one.

Theorem 2 If (V, • 1 ) and (W, • 2 ) are Banach spaces, then so is (V +W, N ).
Proof : The proof of this theorem offers the opportunity to use a caracterization of completeness of normed vector spaces which is not well-known. Let (X, • ) be a normed space. We have indeed

((X, • ) is complete ) ⇐⇒   Every series in X, of general term a k , for which ∞ k=0 a k < +∞ does converges in X (toward a sum denoted as ∞ k=0 a k )   . (7)
The implication (⇒) is classical, it is the most often used. The converse implication (⇐) is not often used, but we have an opportunity to do that here. For a proof of the equivalence [START_REF] Sonntag | Topologie et Analyse Fonctionnelle[END_REF], see for example ([6], Theorems 2-XIV-2.1 and 2.2, pages 164-165), ( [START_REF] Megginson | An Introduction to Banach Space Theory[END_REF], page 20) or ( [START_REF] Sonntag | Topologie et Analyse Fonctionnelle[END_REF], pages 262 and 270); it is also sketched in ( [START_REF] Bergh | Interpolation Spaces, Grundlehren der mathematischen Wissenschaften[END_REF], page 24). The proof is not very difficult however, but readers are encouraged to work through it themselves. We now proceed in this manner to a proof of Theorem 2.

Consider a series in V + W , of general term u k for which

∞ k=0 N (u k ) < ∞.
We have to prove that n k=0 u k converges as n → +∞, to some element u ∈ V + W . In view of the definition (1) of N , for all positive integer k, there exist v k ∈ V and w k ∈ W satisfying

u k = v k + w k , and (8)
v k 1 + w k 2 ≤ N (u k ) + 1 k 2 . (9)
Thus,

∞ k=0 v k 1 < +∞ and ∞ k=0 w k 2 < +∞. Since both (V, • 1
) and (W, • 2 ) have been assumed to be complete, there exist v ∈ V and w ∈ W such that

n k=0 v k -→ n→+∞ v in V, and n k=0 w k -→ n→+∞ w in W. ( 10
) Let u := v + w ∈ V + W . Let us check that, as expected, n k=0 u k converges to u in (V + W, N ). Indeed, N u - n k=0 u k = N u + v - n k=0 (v k + w k ) ≤ v - n k=0 v k 1 + w - n k=0 w k 2 .
It remains to apply (10) to get the desired result, N (u -

n k=0 u k ) -→ n→+∞ 0.
Comments : The construction of the norm ν on V ∩ W and N on V + W deserves some geometrical interpretation. Even if • 1 (resp. • 2 ) is only defined on V ⊂ E (resp. on W ⊂ E), we can extend it to the whole of E by setting u 1 = +∞ if u ∈ V (resp. u 2 = +∞ if u ∈ W ). We still denote by

• 1 and • 2 the extended functions.

Clearly, • 1 and • 2 are convex positively homogeneous functions on E. Modern convex analysis accepts and can handle convex functions possibly taking the value +∞ ( [START_REF] Rockafellar | Convex Analysis[END_REF]). An important geometrical object associated with a convex function

f : E → R ∪ {+∞} is its so-called strict epigraph epi s f := {(x, r) ∈ E × R : f (x) < r}
epi s N = (epi s • 1 ) + (epi s • 2 ). ( 11
)
The sets where ν (resp. N ) is finite, called the domain of ν (resp. of N ) in convex analysis, is just V ∩ W (resp. V + W ).

The binary operation which builds a convex function f from two other ones f 1 and f 2 , via the geometric construction [START_REF] Rockafellar | Convex Analysis[END_REF]). This operation enjoys properties similar to the usual (integral) convolution in classical analysis.

epi s f = epi s f 1 + epi s f 2 is called the infimal convolution of f 1 and f 2 ([2],
In brief, the norm N has been designed as an infimal convolution of the norms • 1 and • 2 .

Returning to our particular setting E = L(R), V = L p (R) and W = L q (R) with 1 ≤ p, q < +∞, the vector space L p (R) + L q (R) can be equipped with a norm that we denote • p,q as follows,

f p,q = inf{ g p + h q ; f = g + h with g ∈ L p (R) and h ∈ L q (R)}. (12)
As proved in Theorem 2, (L p (R) + L q (R), • p,q ) is a Banach space.

3 Comparing L r (R) with L p (R) + L q (R)

We know that the Lebesgue spaces L r (R) and L s (R) (for 1 ≤ r = s < +∞) cannot be compared. Neither L r (R) is contained in L s (R) nor the converse. A direct comparison is however possible if we deal with the sum of these spaces. Here is the main result of this section.

Theorem 3 If 1 ≤ p < q < +∞, then we have the following :

1. L r (R) is contained in L p (R) + L q (R) whenever r ∈ [p, q], 2. if 1 ≤ r < +∞ does not lie in [p, q], then L r (R) is not contained in L p (R) + L q (R).
Proof : (1) Let f ∈ L r (R) and consider X := {x ∈ R : |f (x)| > 1} (a measurable set defined within a set of null measure) as well as X c = R \ X (the complementary set of X in R). We decompose f as follows :

f = f 1 + f 2 , with f 1 = f • 1 X and f 2 = f • 1 X c . ( 13
)
We claim that (13) provides an explicit decomposition of f in L p (R) + L q (R), that is to say f 1 ∈ L p (R) and f 2 ∈ L q (R). We first prove that,

R |f 1 (x)| p dλ(x) = X |f (x)| p dλ(x) = X |f (x)| p-r • |f (x)| r dλ(x). (14)
For x ∈ X, |f (x)| > 1 and, since the exponent p-r is nonpositive, |f (x)| p-r ≤ 1.

Consequently, the last integral in the string of equalities ( 14) is bounded above by X |f (x)| r dλ(x). Finally,

(14 ) R |f 1 (x)| p dλ(x) ≤ X |f (x)| r dλ(x) ≤ R |f (x)| r dλ(x) < +∞.
We thus have proved that

f 1 ∈ L p (R).
Second, we prove that

f 2 ∈ L q (R). Indeed, R |f 2 (x)| q dλ(x) = X c |f (x)| q dλ(x) = X c |f (x)| q-r • |f (x)| r dλ(x). (15) For x ∈ X c , |f (x)| ≤ 1 and, since the exponent q-r is nonegative, |f (x)| q-r ≤ 1.
Again, the last integral in the string of equalities (15) is bounded above by

X c |f (x)| r dλ(x). As a result, R |f 2 (x)| q dλ(x) = X c |f (x)| r dλ(x) = R |f (x)| r dλ(x) < +∞. ( 16
)
We therefore, have proved that f 2 ∈ L q (R).

(2) The second part of Theorem 3 is a bit harder to prove (like most of the negative results in mathematics). We actually have to distinguish two cases for r in the segment [p, q] : r < p and r > q.

Case 1 : r < p. Choose α satisfying 1/p < α < 1/r and let f be defined on R by f (x) = x -α 1 (0,1] (x). Since |f (x)| r = x -αr for x ∈ (0, 1] and 0 elsewhere, the choice of α implies that f ∈ L r (R) (since αr < 1). The same argument shows that f ∈ L p (R) (since αp > 1).

Suppose now that L r (R) ⊂ L p (R) + L q (R). Then

f ∈ L r (R) ⊂ L p (R) + L q (R) ⊂ L p ([0, 1]) + L q ([0, 1]). (17)
But since [0, 1] is of Lebesgue finite measure and p < q, L q ([0, 1]) is contained in L p ([0, 1]), so that (17) yields that f ∈ L p ([0, 1]). This is not the case. Thus we have proved that L r (R) is not contained in L p (R) + L q (R).

Case 2 : r > q. Our proof in this case relies on a technical lemma that we present separately.

Lemma 1 Let 1 ≤ p < q < +∞, let Ω be a mesurable subset of R, and let f ∈ L p (Ω) + L q (Ω). Then f ∈ L q (Ω) whenever it is essentially bounded on Ω.

Proof of the Lemma : Let f be decomposed as f = f p + f q , with f p ∈ L p (Ω) and f q ∈ L q (Ω). So, to prove that f ∈ L q (Ω) amounts to proving that f p ∈ L q (Ω).

Let X := {x ∈ R : |f p (x)| > 1}. To show that Ω |f p (x)| q dλ(x) is finite, we cut it into two pieces : Ω∩X c |f p (x)| q dλ(x) and Ω∩X |f p (x)| q dλ(x).

Consider the first piece. Since f p ∈ L p (R), the set X is of finite (Lebesgue) measure. Now with the definition of X and the fact that q -p > 0, we obtain

Ω∩X c |f p (x)| q dλ(x) = Ω∩X c |f p (x)| q-p • |f p (x)| p dλ(x) ≤ Ω∩X c |f p (x)| p dλ(x) ≤ R |f p (x)| p dλ(x) < +∞.
This concludes the argument for the first piece.

Consider now the second piece. Since f has been assumed essentially bounded on Ω,

|f p (x)| ≤ |f (x) -f q (x)| ≤ f ∞ + |f q (x)| for almost all x in Ω. Consequently, Ω∩X |f p (x)| q dλ(x) ≤ Ω∩X ( f ∞ + |f q (x)|) q dλ(x). (18)
The convexity of the function t → t q on [0, +∞) implies that (

f ∞ + |f q (x)|) q ≤ 2 q-1 ( f q ∞ + |f q (x)| q )
. So, we pursue the string of inequalites (18) with

Ω∩X |f p (x)| q dλ(x) ≤ 2 q-1 Ω∩X ( f q ∞ + |f q (x)| q ) dλ(x) ≤ 2 q-1 [ λ(X) f q ∞ + ( f q q ) q ] .
To summarize, we have proved that

Ω |f p (x)| q dλ(x) = Ω∩X c |f p (x)| q dλ(x) + Ω∩X |f p (x)| q dλ(x) < +∞.
Thus, f p ∈ L p (Ω), which was our objective. That concludes the proof of the technical lemma.

Let us go back to the second part of the proof of Theorem 3, the case where r > q. Choose α satisfying 1/r < α < 1/q and let f be defined on R by f

(x) = x -α 1 [1,+∞) (x).
Since |f (x)| r = x -αr for x ∈ [1, +∞) and 0 elsewhere, the choice of α implies that f ∈ L r (Ω) (since αr > 1). But also f is essentially bounded on R. If f were in L p (R) + L q (R), the technical lemma would imply that f ∈ L q (R). But, this is not the case since |f (x)| q = x -αq for x ∈ [1, +∞) and 0 elsewhere, the choice of α implies that αq < 1.

Thus, again in the case where r > q, L r (R) is not contained in L p (R) + L q (R).

We end this note with the following observation, which links sections 2 and 3. In the first part of Theorem 3, we have proved that L r (R) ⊂ L p (R) + L q (R) whenever r ∈ [p, q]. In the course of its proof, a simple explicit decomposition of f ∈ L r (R) as f = f 1 + f 2 , with f 1 ∈ L p (R) and f 2 ∈ L q (R) has been provided (see (13) and the upper bounds (14') and ( 16)). Indeed, as a consequence of ( 14) and (16), f p,q ≤ f 1 p + f 2 q ≤ f r/p r + f r/q r .

Hence, the injection of L r (R) into L p (R) + L q (R) is continuous ; therefore, there exists C > 0 such that

f p,q ≤ C f r . (20)
Indeed, using the inequality (19), we can get an upper bound for the norm of this injection (a somewhat complicated expression in terms of p, q, r). This result complements a more classical one which says that, when r ∈ [p, q], L p (R)∩L q (R) is contained in L r (R), then the injection is continuous.

(

  literally, what is strictly above the graph of f ). In our situation, K 1 := epi s • 1 and K 2 := epi s • 2 are open convex cones of E. So, what are the strict epigraphs of the norm functions ν and N ? One easily checks the following epi s ν = (epi s • 1 ) ∩ (epi s • 2 ), and

PATRICE LASS ÈRE received his Doctorat ès sciences mathématiques from the University Paul Sabatier, Toulouse, France, in 1991. He currently teaches at the University Paul Sabatier. His research interests includes complex analysis in one or several variables and functional analysis. Institute of Mathematics, Paul Sabatier University, TOULOUSE, FRANCE lassere@math.ups-tlse.fr http://www.math.univ-toulouse.fr/ lassere/


Bastien Drut

Richard Duhautois

Assortative Matching Using Soccer Data: Evidence of Mobility Bias

Keywords: assortative matching, soccer, mobility

 methodology (an estimation of a two-way fixed effects model) do not provide clear evidence of the existence of assortative matching between workers and firms, i.e., there is no clear evidence that good employees work in good firms. In fact, negative or very small correlations are found in the literature using this approach. Andrews et al. (2008, 2012) suggest that positive assortative matching has not been observed in the empirical literature because of a limited mobility bias. This paper uses Italian soccer championship data to test positive assortative matching.

Because job turnover is significantly high in professional soccer, we use numerous movers to detect sorting. Our results indicate that estimating a standard Mincerian wage equation with numerous movers enables us to find a positive correlation between players' and firms' fixed effects, meaning that good players tend to move to good teams. This suggests that superstar effects are at work as the unobserved component of the player fixed effect is far more correlated with the wage than the observed component.

Introduction

In each labor market, the remuneration earned by workers and the wages paid by firms vary significantly. As suggested by [START_REF] Torres | The Sources of Wage Variation: A Three-Way High-Dimensional Fixed Effects Regression Model[END_REF], explanations for this heterogeneity can be found in both supply-side and demand-side determinants, although the more interesting explanations focus on a combination of the two approaches. The determinants of job mobility traditionally were analyzed by focusing only on workers' characteristics; only since the 1990s have these factors also been considered in relation to firms' characteristics, thanks to the development of linked employee-employer data (LEED). Therefore, the following key question emerged in the literature: Do good workers move to good firms? Using his theory on the economics of marriage, [START_REF] Becker | A theory of marriage: Part I[END_REF] introduced the idea of assortative matching, through which spouses tend to choose each other by sorting potential candidates as a function of their observable characteristics (IQ, education, attractiveness, height, etc.). Similarly, in labor economics, the theory of assortative matching focuses on the mutual selection of workers and firms. Several authors have studied this topic from a theoretical point of view (e.g., [START_REF] Sattinger | Assignment models of the distribution of earnings[END_REF][START_REF] Mortensen | Job creation and job destruction in the theory of unemployment[END_REF].

Empirical studies 1 do not provide clear evidence regarding the existence of assortative matching between workers and firms, i.e., there is no clear evidence that good employees work in good firms. The first and most influential study on assortative matching in the labor market is attributed to Abowd, Kramarz and Margolis (AKM) (1999). Using a linked employer-employee dataset from France, these researchers estimate assortative matching by decomposing wages into worker and firm fixed effects. According to these authors, the correlation between these two effects is an indicator of sorting because the firms' fixed effects are a proxy of their wage policy and the workers' fixed effects are a proxy of their "quality". This methodology has been applied to different developed countries with similar results 1 Abowd, Kramarz and Margolis, 1999; Abowd, Creecy and Kramarz, 2002; Abowd, Kramarz, Lengeman and Perez-Duarte, 2004; Andrews, Gill, Shank and Upward, 2008; Torres, Portugal, Addison and Guimaraes, 2013. [START_REF] Abowd | Computing person and firm effects using linked employer-employee data[END_REF], for the United States; Andrews, Gill, Shank andUpward, 2008, 2012, for Germany;Iranzi, Schivardi and Tosetti, 2008, for Italy;[START_REF] Lopes De Melo | Firm Wage Differentials and Labor Market Sorting: Reconciling Theory and Evidence[END_REF] provides a summary of evidence based on AKM). Using this approach, negative or very small correlations are found in the literature: good workers appear to move to bad firms.

Because the negative correlations are not intuitive, recent studies have focused on possible statistical biases and have developed different methodologies. The primary concern is the detection of sorting using pioneering methods. [START_REF] Eeckhout | Sorting versus screening: Search frictions and competing mechanisms[END_REF] indicate that using unobserved workers' and firms' effects does not enable a direct test of the existence of assortative matching. These authors propose the construction of an indicator of the strength of sorting; unfortunately, however, this indicator does not determine the direction. Lopes de [START_REF] Lopes De Melo | Firm Wage Differentials and Labor Market Sorting: Reconciling Theory and Evidence[END_REF] proposes constructing an indicator to determine both the strength of sorting and the direction; his results yield strong evidence of positive assortative matching. In a recent paper, [START_REF] Bartolucci | Better Workers Move to Better Firms: A Simple Test to Identify Sorting[END_REF] propose a test that uses information on workers' mobility, wages and firm profits to identify the sign and strength of assortative matching.

Overall, these three recent studies suggest that positive assortative matching is a pervasive feature of the labor market, i.e., good workers appear to move to good firms.

From a statistical point of view, [START_REF] Andrews | High wage workers and low wage firms: negative assortative matching or limited mobility bias?[END_REF][START_REF] Andrews | High wage workers match with high wage firms: Clear evidence of the effects of limited mobility bias[END_REF] suggest that positive assortative matching has not been observed in the empirical literature using two-way fixed effects models because of limited mobility bias. These researchers indicate that the correlation between workers' and firms' fixed effects is negative when the number of movers is small and positive when the number of movers is large. In other words, empirical literature fails to find positive assortative matching because the data often include only a small number of movers.

In this paper, we use Italian soccer championship data to test positive assortative matching by estimating a two-way fixed effects model. First, because job turnover is significantly high in professional soccer, we use numerous movers to detect sorting. Second, we note that labor economists have difficulty ranking workers and firms by performance. In our study, however, we have extensive information on players' and teams' performance because soccer is the most popular sport worldwide. Because of this information, we are able to calculate "naïve" correlations between the subjects. Moreover, in the literature related to the AKM methodology, the issue of inter-industry differential in terms of productivity and wages has been highly debated (see for instance in [START_REF] Abowd | Persistent InterIndustry Wage Differences: Rent Sharing and Opportunity Costs[END_REF][START_REF] Abowd | Sorting between and within industries: a testable model of assortative matching[END_REF] as worker may be highly productive in a given sector and not in another one. We did not face this issue in our study as football players very good in a given team are very likely to be also very good in another one. [START_REF] Gandelman | Mobility Among Employers and Assortative Matching: Field Evidence From Soccer Data[END_REF] has already shown positive assortative matching using soccer data from the Uruguayan National League. Examining the determinants of soccer players' promotions and demotions, [START_REF] Gandelman | Mobility Among Employers and Assortative Matching: Field Evidence From Soccer Data[END_REF] finds that players' performance on the pitch correlates to the probability of being promoted; however, because the author has no information on wages, he does not estimate a wage variation equation.

The soccer sector is also particularly interesting as there are superstar effects [START_REF] Lucifora | Superstar effects in Sport: Evidence from Italian soccer[END_REF] linked to the popularity of "top" soccer players: earnings are found to be highly convex in performance measures, other things been equal. Franck and Nüesch (2008) insist on the fact that the exact talent of a star player is difficult to determine and find that the market values of German soccer stars are better predicted by expert evaluations revealing hidden talent characteristics than by firsthand observable talent measures. Our objective is also to contribute to this vein of the literature by estimating whether unobservable talent contributes more or not than observable talent to assortative matching in soccer.

Our results suggest the following: (1) using observable player and club performance, there is evidence of positive assortative matching, i.e., good players tend to move to good teams; (2) estimating a standard Mincerian wage equation, we find a positive correlation between players' and firms' fixed effects, also indicating that good players tend to move to good teams; (3) introducing information about workers' and firms' performance only marginally affects the estimation of this two-way fixed effect model; the contributions of wage variation and the correlation between the observed and unobserved components are virtually identical with and without performance; and (4) that superstar effects are at work as the unobserved component of the player fixed effect is more correlated with wage than the observed component. Overall, our work suggests that these models are appropriate for detecting assortative matching when mobility is not limited.

The paper proceeds as follows: Section 2 presents the data and the concept of the players' and teams' performance. Section 3 presents the descriptive evidence of assortative matching. We focus on empirical strategy in Section 4 and present our results in Section 5.

Section 6 provides concluding remarks.

The Data

We use data from the Italian Serie A (first division) for the seasons 2009[START_REF] Eeckhout | Sorting versus screening: Search frictions and competing mechanisms[END_REF][START_REF] Eeckhout | Sorting versus screening: Search frictions and competing mechanisms[END_REF]-2011,2011-2012, 2012[START_REF] Lopes De Melo | Firm Wage Differentials and Labor Market Sorting: Reconciling Theory and Evidence[END_REF]and 2013[START_REF] Abowd | Sorting between and within industries: a testable model of assortative matching[END_REF]. According to Deloitte (2012), the Serie A is the fourth most important European football league in terms of revenues, i.e., the average revenue per club and per season accounts for €77 million. A total of 20 clubs participate each year in the Serie A, the first division of the professional football system in Italy. At the end of the season, the three clubs with the worst performance in the Serie A are demoted to the Serie B, the second division of the professional football system in Italy, whereas the three clubs with the best performance in the Serie B2 are upgraded to the Serie A; therefore, 26 different clubs are considered in this study.

We retain only the 872 distinct soccer players who played at least two seasons and three games during the same season with the same team. For each player, we have the following variables: field position, team, wages, nationality, birth date (age), a dummy indicating whether the player plays for his country, his performance and his team's performance. All the data are obtained from the newspaper La Gazzetta dello Sport.

The players' performance

As previously noted, the framework of professional sports is particularly suitable for studying assortative matching in the labor market because information about players' skills and performance is easily accessible. In particular, some specialized newspapers evaluate each player after each game by rating the athletes. In this study, we use the ratings produced by the Italian newspaper La Gazzetta dello Sport, ranging from the worst (0) to the best (10).

For the four first seasons of our sample (i.e., 2009-2010 to 2012-2013), we consider the average rating for each player during the season. We only consider the four first seasons because we use ratings of the previous year to analyze the player mobility.

As noted by [START_REF] Gandelman | Mobility Among Employers and Assortative Matching: Field Evidence From Soccer Data[END_REF], these ratings can be subjective and are imperfect for the following reasons: first, the evaluation varies according to the writer; second, the writer may be influenced by personal preferences, and some players may receive better or worse evaluations than deserved. Overall, however, we consider the ratings to be a good proxy of each player's average performance over the entire season by offering a broad measure of performance on numerous aspects of the game.

Table 1 presents the distributions of the average ratings. The distributions are very similar for the four seasons: the ratings mean varies from 5.84 to almost 5.89. Individual player ratings range from 3.88 to 7.06 for the four seasons.

The clubs' performance

Similar to [START_REF] Gandelman | Mobility Among Employers and Assortative Matching: Field Evidence From Soccer Data[END_REF] approach, we construct three different rankings to test the robustness of our results. Table 2 presents 3 The ranking does not change significantly when using other weights.

Stylized facts

Each season, about 43% of soccer players move from one club to another (42.7% to 44.1% according to the seasons). Among those players, less than 50% move within the first division. Approximately 30% of the players move to the second division or retire and approximately 20% move to other foreign leagues. Table 3 represents the share of players moving within the first division who shift to a better-ranked or a worse-ranked club based on the three rankings. On average, 30% move to a better club and 70% move to a worse one.

To obtain a preliminary idea regarding assortative matching, Figure 1 shows the distributions of ratings of players who move to a better club and of those who move to a worse club. Unsurprisingly, on average, players moving to better clubs have better ratings than players moving to worse clubs. Although the average rating of players who move to a better club is only 3.5% higher than the average of players who move to a worse club, this difference is statistically significant. [START_REF] Gandelman | Mobility Among Employers and Assortative Matching: Field Evidence From Soccer Data[END_REF] finds a 5% difference between promoted and demoted players.

Figure 1: Distribution of ratings according to movements

Descriptive evidence shows signs of positive assortative matching. We now turn to the descriptive correlations between players' and teams' performance. Table 4 presents the correlations between rankings and ratings for players who change teams. The correlation between the different rankings is very high: the correlation between CL1 and CL2 is 0.84, between CL1 and CL3 is 0.95 and between CL2 and CL3 is 0.95. These results demonstrate that rankings are homogeneous.

We find that the correlations between clubs' rankings and players' ratings (for movers) are positive and range between 0.29 and 0.30. Of course, these correlations are only descriptive. Using the AKM methodology, the magnitude of the estimated correlation is approximately 0.2 to 0.3 in the literature. [START_REF] Torres | The Sources of Wage Variation: A Three-Way High-Dimensional Fixed Effects Regression Model[END_REF] find a correlation of 0.27 and [START_REF] Andrews | High wage workers match with high wage firms: Clear evidence of the effects of limited mobility bias[END_REF] find correlations ranging from 0.2 to 0.3. The aim of the next section is to estimate a two-way fixed effects model to obtain correlations between the fixed effects components.

Methodology

In pioneering methods, the evidence of assortative matching comes from the correlation between the workers' and firms' fixed effects ( i  and j  in equation (1) below).

We focus on this correlation to compare our results with other studies, although we use the methodology developed by [START_REF] Torres | The Sources of Wage Variation: A Three-Way High-Dimensional Fixed Effects Regression Model[END_REF], which allows workers' and firms' fixed effects4 to be simultaneously estimated by decomposing them into observed and unobserved components. This methodology enables us to directly introduce the indicators for team and player performance into the two-way fixed effects model to see whether this approach improves estimations.

Using the same notations as [START_REF] Torres | The Sources of Wage Variation: A Three-Way High-Dimensional Fixed Effects Regression Model[END_REF], we estimate the following linear wage equation:

ijt j i ijt ijt X W         ln (1)
where ijt W ln is the natural logarithm of the annual wage of player i (i=1,…,N) in team j (j=1,….,J) at year t (t=1,…,T). For each player, T is equal to 2 to 5. ijt X is a vector of the time-varying characteristics of the player and team.

i  is the player fixed effect and j  is the team fixed effect.

ijt

 is an error term following the conventional assumptions.

We first estimate equation ( 1) using pooled OLS with and without player and club performance (Table 6). We include the following (time varying and non-time varying) variables: age, age squared, seniority (years in a team), seniority squared, a dummy for the forward players, nationality (Italian=1; 0 otherwise), a dummy indicating whether the player is an international player (i.e. playing for his country), the share of Italian players in each team, the share of international players in each team, the years dummy, his performance and his team's performance. Results of these two estimations are presented in appendix (Table A1).

We then estimate equation (1) using fixed-effects OLS. In the first step, we use the time-varying variables (age, age squared seniority and seniority squared). Results are presented in appendix (Table A2). In the second step, we decompose the two estimated player ( i  ˆ) and team ( j  ˆ) fixed effects into an observed and an unobserved component.

For the players, we estimate the following:

i i i V ercept       int (2)
where i V is a of non-time-varying player characteristics (a dummy for the forward players, nationality and a dummy indicating whether the player is an international player),  is the associated vector of coefficients and the product of the two is the observed player heterogeneity component. To obtain the unobserved heterogeneity player component, we calculate the residuals

   i i i V  
. Results are presented in appendix (Table A3).

For the teams, we estimate the following:

j j j Z ercept       int (3)
where i Z is a vector of non-time-varying team characteristics (the share of Italian players in each team and the share of international players in each team),  is the associated vector of coefficients and the product of the two is the observed team heterogeneity component. To obtain the unobserved heterogeneity team component, we calculate the residuals

   j j j Z  
. Results are presented in appendix (Table A4).

We have three different components (and the residuals): The correlation between i  and j  and the correlation between  i V and  j Z may be taken as an estimate of the degree of sorting.

Results

Table 5 presents the correlations between the fixed effects components. Despite different datasets, our results are broadly in line with those of [START_REF] Torres | The Sources of Wage Variation: A Three-Way High-Dimensional Fixed Effects Regression Model[END_REF] and those of j  (remember that the team fixed effect is a proxy of the club's wage policy and the player fixed effect is a proxy of the player's "quality") suggests evidence of positive assortative matching (0.46). As previously mentioned, [START_REF] Torres | The Sources of Wage Variation: A Three-Way High-Dimensional Fixed Effects Regression Model[END_REF] find a correlation of 0.27 and [START_REF] Andrews | High wage workers match with high wage firms: Clear evidence of the effects of limited mobility bias[END_REF] find correlations ranging from 0.2 to 0.3. When focusing on observed components only, the result is the same: the correlation between the components is 0.40 (0.34

for [START_REF] Torres | The Sources of Wage Variation: A Three-Way High-Dimensional Fixed Effects Regression Model[END_REF]).

Of the two primary components, the player fixed effect component exhibits the highest correlation with the log wage (0.83; 0.74 for [START_REF] Torres | The Sources of Wage Variation: A Three-Way High-Dimensional Fixed Effects Regression Model[END_REF]); the correlation between the club fixed effect component and the log wage 0.71 (0.67 for [START_REF] Torres | The Sources of Wage Variation: A Three-Way High-Dimensional Fixed Effects Regression Model[END_REF]). Both the observed and the unobserved components of the player fixed effect are highly correlated with the log wage (respectively 0.50 and 0.65) but the correlation with the unobserved component is far stronger, which differs slightly from the results of Torres et al. ( 2013) 5 . Concerning the components of the team fixed effect, both the observed and unobserved components are also correlated with the log wage (respectively 0.65 and 0.32). Our results differ slightly from those of [START_REF] Torres | The Sources of Wage Variation: A Three-Way High-Dimensional Fixed Effects Regression Model[END_REF] on one particular way: this is not the observable part of the fixed effects that is the most correlated with the log wage but rather the unobserved component of the player fixed effect. The opposite is true for the team fixed effect, for which the observed component is the most correlated with the log wage. One explanation may be that a significant share of soccer players' wages depends on an "extra-sportive" component and in particular on the players' popularity. This explanation is in line with the works of Franck andNüesch (2008, 2012) who show that soccer players' wages and market values are higher when players are popular their quality. Overall, our results indicate that the relationship between teams' wage policies and the quality of the players selected is positive and that superstar effects may be at work in our sample as the unobserved component of the player fixed effect is more correlated with the wage than the observed component.

To test the idea of the correlation between the firm and the worker fixed effects is biased downwards when there are few movers, we drop randomly movers from our sample. while the share of movers remains above 20%. The correlation between player and club fixed effects is smaller when the share of movers is below 20% and becomes even negative when the share of movers is below 7%. As previously noted in [START_REF] Abowd | Are Good Workers Employed by Good Firms? A Test of a Simple Assortative Matching Model for France And The United States[END_REF] and Andrew et al. (2008Andrew et al. ( , 2012)), the correlation between the firm and the worker fixed effects is biased downwards when there are few movers in the data. The comparison between the descriptive correlation (in Section 3) and the correlations estimated with the fixed effects and the observed portion of the fixed effects show that when the number of movers is large, the twoway fixed effects model is stronger.

        p p ijt ijt j j i i ijt ijt C Z W X W       ln
where p ijt C is the pth component. To measure the contribution of each component to wage variation, we calculate the following:

    ijt p ijt ijt W Var C W Cov ln , ln where     1 ln , ln   ijt p ijt ijt p W Var C W Cov
In Table 6, we report the contributions to the wage variation for pooled and fixed effects estimations. Both of these estimations are run with and without player and club performance (the of the equations are shown in the Appendix).

For pooled estimations, adding player and club performance (player ratings and a club ranking) into the estimation increases the R² by 2 points; player performance explains 1.9% and club performance explains 8.8% of total wage variation. Focusing on FE estimations enables us to decompose the player and firm fixed effects. The largest contribution comes from player fixed effects (54.7%), followed by club fixed effects (27.6%) and time-varying explanatory variables (10.9%). Among the player fixed effects, the unobserved component accounts for more than two-thirds of the total when player performance is not included; which is another indication of superstar effects, i.e. that "extra-sportive" criteria explain a big share of wage. For instance, Franck and Nüesch (2008) show that having a personal website and the media presence increase the players' market value, all other things being equal. However, contrary to the player fixed effects, the unobserved component accounts for slightly less than one-third of the total among the club fixed effects. Including both player and firm performance in the FE estimations increases the observed components by 1.5 and 0.7 points, respectively, which is very small.

Conclusion

Empirical studies on assortative matching using two-way fixed effect models have often failed to show that good workers move to good firms. One explanation of this result is the mobility bias: there are too few movers in firms to identify positive assortative matching.

We therefore use soccer data because job turnover is very high in soccer, providing us with numerous movers to detect sorting. [START_REF] Gandelman | Mobility Among Employers and Assortative Matching: Field Evidence From Soccer Data[END_REF] has already demonstrated matching between better workers and better employers but without estimating a wage equation.

In this paper, we show evidence of positive assortative matching through the use of both descriptive statistics and estimations of firm and worker fixed effects, i.e., good players tend to move to good teams. We also suggest that introducing information about worker and firm performance marginally affects the estimation of the two-way fixed effect model; the contribution of wage variations and the correlation between the observed and unobserved components are virtually identical with and without performance. Furthermore, our results are consistent with the recent literature and with [START_REF] Torres | The Sources of Wage Variation: A Three-Way High-Dimensional Fixed Effects Regression Model[END_REF]: worker heterogeneity is the most important source of wage variation, followed by firm heterogeneity. However, a significant difference emerges if we compare our results with other studies; the unobservable portion of the player fixed effects is more closely correlated with log wage than the observable portion, which could be explained by the fact that soccer players' wages may depend on an "extra-sportive" component, as suggested by Franck and Nüesch (2012).

  these rankings of the 26 clubs that played at least one season in the Serie A division during the five seasons of our study. The first ranking (CL1) is contemporaneous because it is based on the final position of the club for the past five seasons in the first division Italian Soccer championship(from 2004-2005 to 2008-2009). We use seasons before 2009 to avoid reversed causality between wages and performances. For instance, because Internazionale Milano (a Milan team) won the championship four times during this period and finished second once, the team is ranked first. Juventus FC (a Turin team) is only ranked four in the CL1 because the club was downgraded to the Serie B in 2007.Among CL1 four teams were in Serie B during the five seasons, therefore they have the same ranking (23). The second ranking (CL2) is historical, based on the titles won in the first division championship (Serie A), the European cups (Champions League and Europa League), the Italian National Cup and the second division championship (Serie B). For each team, we construct an index of the number of titles won. To account for the prestige of each competition, we weight the number of titles as follows: 0.5 for the European cups, 0.3 for the first division championship and 0.1 each for the Italian Cup and the second division championship 3 . Finally, the third ranking (CL3) is the average of CL1 and CL2. For CL2, we tried to use UEFA rankings of club but only 12 clubs out of 26 were referenced. We used the 12 clubs rankings and for the others we used CL2. The results are not affected by this ranking.

Figure 2

 2 Figure 2 represents estimated correlations between player and club fixed effects (the

Figure 2 :

 2 Figure 2: estimated correlation by average number of movers

1

  Because we have indicators for firm and worker performance, we estimate the following equation with and without performance to measure the contribution of each component to wage variation:

Table 1 : Average player performance

 1

				Standard	Minimum Maximum
	Season Observations	Mean	D.
	2009-2010	425	5.888	0.320	4.00	7.06
	2010-2011	445	5.874	0.264	4.75	6.73
	2011-2012	472	5.828	0.304	4.50	7.00
	2012-2013	479	5.841	0.322	3.88	7.00

Source: La Gazzetta dello Sport

Table 2 : Average club performance (Rankings)

 2 Rankings are based on the performance of clubs in Italian championships (Serie A and B), in the Italian National Cup and in the European cups.

	Club	Ranking 1 Ranking 2 Ranking 3
	Inter	1	3	2
	Milan	2	1	1
	Roma	3	10	4
	Juventus	4	2	3
	Fiorentina	5	9	5
	Palermo	6	15	8
	Udinese	7	23	16
	Lazio	8	7	6
	Sampdoria	9	12	9
	Cagliari	10	14	11
	Atalanta	11	13	12
	Siena	12	25	18
	Livorno	13	18	17
	Chievo	14	24	19
	Parma	15	8	10
	Genoa	16	4	7
	Catania	17	22	21
	Napoli	18	11	15
	Torino	19	5	13
	Lecce	20	21	23
	Bologna	21	6	14
	Brescia	22	16	20
	Bari	23	17	22
	Novara	23	19	24
	Pescarra	23	20	25
	Cesena	23	26	26
	Note:

Table 3 : Share of movements

 3

		Share of	Share of
		players	players
		going to a	going to a
		better club	worse club
	Ranking 1	28.5%	72.5%
	Ranking 2	31.4%	68.6%
	Ranking 3	29.1%	70.9%
	Notes: rankings are based on the performance of clubs in the Italian championships (Serie A and B), in
	the Italian National Cup and in the European cups.

Table 4 : Correlations between players and firm performance (movers)

 4

		Ranking 1 Ranking 2 Ranking 3	Ratings
	Ranking 1	1	0.84	0.95	0.29
			<.0001	<.0001	<.0001
	Ranking 2		1	0.95	0.30
				<.0001	<.0001
	Ranking 3			1	0.30
					<.0001
	Notations				1
	Notes: correlations between players' ratings and clubs rankings are based on clubs post-moving
	rankings.

Table 5 : Correlations between components

 5

			Player fixed	Player fixed		Team fixed	Team fixed
			effect:	effect:		effect:	effect:
		Player fixed	unobserved	observed	Team fixed	unobserved	observed
	Log(wage)	effect	component	component	effect	component	component

Table 6 : Contributions to wage variation

 6

	Contributions
	Pooled	FE
	Without performance With performance Without performance With performance

The two first teams are directly promoted while the teams ranked between the third and the sixth position take part to a play-off to determine the third promoted club.

These researchers estimate a three-way fixed effects model. The third fixed effect is a job title effect; if we had more observations, we could introduce this effect as the field position.

This slight difference may be due to the estimated decomposition into observed and unobserved components in the second step, i.e. the potential correlation between Z with ε.

Appendix