Karl-Johan Lundquist
email: karl-johan.lundquist@keg.lu.se

Michaela Trippl
email: michaela.trippl@wu.ac.at

Distance, Proximity and Types of Cross-Border Innovation Systems: A Conceptual Analysis

Keywords: O30, R10 innovation-driven integration, cross-border areas, regional innovation systems

Due to a lack of both theoretical and systematic empirical work, innovation-driven integration processes in cross-border areas are still poorly understood. Departing from the regional innovation system (RIS) approach, recent work on different forms of proximity, and the literature on cross-border areas, this paper seeks to identify in a conceptual way different stages in the development of cross-border innovation systems and to highlight the main characteristics and barriers characterizing each phase. The conceptual model proposed in this paper can serve as useful framework, guiding more systematic and comparative empirical work on cross-border innovation systems in different regional and national contexts.

 [START_REF]Scope, style, and theme of research on knowledge and learning societies[END_REF]. Simultaneously as these differences create the foundation for cross-border growth by offering potentials for new combinations and unexploited synergies, they also form barriers obstructing successful integration, especially when it comes to generate, transmit and share innovation-relevant knowledge [START_REF] Koschatzky | A River is a River -Cross-border Networking between Baden and Alsace[END_REF]HOECKMAN et al., 2009).

From this point of view there are probably 'easier' ways to go than following an innovationoriented growth path. At least in the short run and as a first step of cross-border integration other pathways might be more reasonable. These could include an integration and enlargement of local consumer and factor markets, increase of labor mobility, extended division of labor and specialization. Meanwhile, these more 'traditional' integration processes are well understood. Little is still known, however, about the nature of innovation-driven development processes in cross-border areas. Only a few scholars have so far explicitly used the RIS approach for empirically analyzing [START_REF] Coenen | Nodes, networks and proximities: on the knowledge dynamics of the Medicon Valley biotech cluster[END_REF][START_REF] Moodysson | Knowledge Collaboration and Proximity: The Spatial Organization of Biotech Innovation Projects[END_REF] or conceptualizing [START_REF] Trippl | Developing cross-border regional innovation systems: key factors and challenges[END_REF] integration processes in trans-frontier regions. Undoubtedly, their work has provided interesting insights into innovation activities taking place at the cross-border level. Recent conceptual work done by TRIPPL (2010) is particularly relevant for our article. Looking at five core dimensions of innovation systems, she elaborated on some key determinants of innovation-driven integration processes in crossborder areas. This analysis has to be acknowledged for shedding light on a set of factors that might favor or inhibit the development of a cross-border RIS and for suggesting some first ideas about the relevance of different types of distance in this regard. Although her conceptual framework constitutes an important starting point for our analysis, we go beyond this work in several crucial respects. First, [START_REF] Trippl | Developing cross-border regional innovation systems: key factors and challenges[END_REF] analysis is restricted to identifying hindering and favoring factors for the emergence of cross-border RIS. Our analysis is clearly different in nature and provides a dynamic perspective on the issue under consideration as our focus is on developing a typology of cross-border RIS that draws a clear distinction between different stages of innovation-driven integration processes. Second, our paper advances the existing body of work by discussing in a far more systematic way and in more detail the role of different kinds of distance and proximity in cross-border regions. Most importantly, we conceptualize various types of proximity and distance as both opportunities for and barriers to cross-border innovation activities. Third, the conceptual model suggested in this paper takes explicit account of the importance of innovation linkages to other spatial scales (including regional, national and international ones). In this paper it will be argued that cross-border RIS should be seen as the most advanced form of transnational integration, resting upon the success of previous incremental and less innovation-oriented modes of development. The key aim of this article is to develop a conceptual model that draws a clear distinction between these different types of integration. More specifically, by bringing together three strands of literature -the RIS approach, recent insights provided by the 'proximity school' and work done on cross-border regionalizationthe paper will deal with the following research questions:

• Which stages of cross-border RIS development can be identified in a conceptual way?

• What is the role of physical, cognitive, functional and institutional distance in shaping the opportunities and barriers for cross-border innovation activities and which kinds of proximity and distance are likely to prevail in each stage?

Departing from the RIS approach (see, for instance, [START_REF] Autio | Evaluation of RTD in Regional Systems of Innovation[END_REF]COOKE et al., 2004;[START_REF] Asheim | The Geography of Innovation[END_REF] [START_REF] Boschma R | Proximity and innovation. A critical assessment[END_REF][START_REF]Temporary Geographical Proximity in Knowledge Transmission[END_REF] we will elaborate on a characterization of each stage of cross-border RIS development. Drawing on the literature on cross-border areas, particularly on findings concerning barriers and driving forces connected to different integration processes, will further inform this discussion. This literature is also useful, as it provides some empirical evidence that underlines and supports our conceptual arguments.

The remainder of this paper is organized as follows. Section 2 provides a short literature review on cross-border areas and recapitulates recent insights provided by the 'proximity school'. Section 3 analyzes in a conceptual way different phases of innovation-driven integration in cross-border areas. Finally, in Section 4, the key arguments are summarized and some conclusions are drawn.

Cross-border areas and proximities

Basic features of heterogeneity

As a starting point for the conceptual discussion we will initially apply a very broad and simple definition of a cross-border region as an area consisting of adjacent territories belonging to different nation states. This broad definition covers all types of cross-border settings regardless of differences in terms of size, geographic conditions, history, culture and socio-economic conditions. The heterogeneity between different cross-border areas is, however, considerable. On the one extreme they can be very extensive in terms of space and population stretching over several nations and including a larger number of regions located in densely populated economic core areas. A prime example for such a setting is the Centrope region with a total population of more than six million inhabitants, consisting of the two the Czech Republic. On the other hand this definition also covers cross-border areas including only two neighboring regions with a small population distributed on a large area in the geographic as well as the economic periphery far away from core national and international markets. One example for such a cross-border region is the area around the twin cities of Haparanda and Torneå located along the northern part of the Swedish and Finish border.

Between these two extreme cases there is wide range of cross-border regions showing a considerable variation in basic geographical preconditions in terms of scale, size and location, engendering primary restriction of what kind of integration is likely to occur and what types of benefits might be related to different integration processes.

Furthermore, the magnitude and character of internal heterogeneity in cross-border areas will have a strong impact on the possibilities for and constraints to the emergence of an integrated socio-economic system. Many cross-border regions tend to show very dissimilar economic histories, technological trajectories and innovation capacities, institutional set-ups and positions in the regional system of their respective nations, as well as different social dynamics, political visions, governance structures, modes of regulation and cultural identities [START_REF] Dowd | Borders, Border Regions and Territoriality: Contradictory Meanings, Changing Significance[END_REF][START_REF] Trippl | Developing cross-border regional innovation systems: key factors and challenges[END_REF]. To a certain extent these differences create the foundation for cross-border growth. They might give rise to complementarities and synergies which could be capitalized on through interaction. Potential benefits might be related to an integration and enlargement of local consumer, labor and factor markets, enhanced competition, extended division of labor and increased specialization. Such processes could lead to shared growth effects and new opportunities for upgrading the competitive edge of the economy on both sides of the border. Simultaneously as some of these differences create the main driving forces for cross-border growth, they could also form barriers hindering successful integration. Thus, in addition to the political-administrative borders also economic, cultural and social borders tend to further divide these areas [START_REF] Dowd | Borders, Border Regions and Territoriality: Contradictory Meanings, Changing Significance[END_REF][START_REF] Smallbone | Challenges and prospects of cross border cooperation in the context of EU enlargement[END_REF][START_REF] Löfgren | Regionauts: The transformation of cross-border regions in Scandinavia[END_REF].

The complexity of integration processes in cross-border regions

Cross-border regionalization is a multi-faceted and complex phenomenon. It takes shape along nation state boundaries and requires contacts and partnership between public as well as private actors across borders. The process takes place in what could be labeled as a 'grey zone' between civil and public law in combination with the emergence of informal and formal networks between a wide spectrum of actors, ranging from the single citizen and firm to universities, industrial organizations, trade unions, political parties and cultural organizations [START_REF] Jönsson | Organizing European Space[END_REF]. The overall concern of the importance of increased cross-border integration and the different ways to achieve it can be assumed to vary among the actors, reflecting conflicting goals and asymmetry in power relations [START_REF] Johnson | Cross-border regions and territorial restructuring in Central Europe: Room for more transboundary space[END_REF]. Furthermore, all these stakeholders are more or less strongly embedded in or have relations to other spatial scales, from the local community to the global level. This historically rooted embeddedness shapes routines that will govern the actors' decisions in general, for instance a firm's decision about what to produce, when and how, investment decisions and search behavior for networking partners (NELSON and WINTER, 1982;[START_REF] Nelson R | Recent evolutionary theorizing about economic change[END_REF]2008). In many respects these routines will also frame the actors' potential 'cross-border behavior'. Furthermore, as noted above, the driving forces for cross-border integration processes, i.e., the differences in economic structure, innovation capabilities and cost structures giving rise to new complementarities and synergies, could also constitute major barriers to interaction between the different parts of a cross-border region. Consequently, to consider and analyze this tension and interplay between differences working as driving forces on the one hand and as barriers on the other, is essential for enhancing our knowledge about cross-border integration processes. A crucial point of departure for understanding cross-border integration in general and the formation of cross-border innovation systems in particular is to uncover the role of different types of proximity and distance and to explore how they have an influence on what kinds of linkages are likely to be established.

Distance and proximity

Recent research findings generated by the 'proximity school' [START_REF] Boschma R | Proximity and innovation. A critical assessment[END_REF][START_REF]Temporary Geographical Proximity in Knowledge Transmission[END_REF] provide a useful foundation for conceptualizing and categorizing different types of differences between the constituent parts of a cross-border area and to discuss their impact on potentials for and crucial impediments to cross-border knowledge interactions. Following the literature, three main categories of proximity, i.e. physical, functional and relational proximity, could be identified as central for this understanding [START_REF]On the Analytical Dimension of Proximity Dynamics[END_REF][START_REF] Moodysson | Knowledge Collaboration and Proximity: The Spatial Organization of Biotech Innovation Projects[END_REF][START_REF]Temporary Geographical Proximity in Knowledge Transmission[END_REF]. Physical proximity is closely related to the geographical dimension of agglomeration economies, transaction and transportation cost. When it comes to the exchange of knowledge (especially tacit one) which depends on face-to-face contacts the accessibility dimension could be assumed to be very important. As underlined in the literature it has less to do with pure distance measured in kilometers between different actors, but with the efforts it takes for them to interact in terms of time and costs. This could depend on several factors, for instance, the quality of the transport infrastructure and political-administrative set ups that facilitate or hinder mobility of goods and people. The construction of the fixed link between Sweden and Denmark is one example where investments in infrastructure have had a strong impact on reducing the physical distance between different parts of the cross-border area of the Oresund. In the case of new possibilities for the creation of a powerful international competitive cross-border region.

Functional distance [START_REF] Maggioni | Inter-regional knowledge flows in Europe: an econometric analysis[END_REF] refers to differences between regions in innovation performance. [START_REF] Maggioni | Inter-regional knowledge flows in Europe: an econometric analysis[END_REF] showed that knowledge does not flow easily between areas, if they differ strongly in their innovation capacity. Consequently, a strong asymmetry in performance and capability (i.e., too much functional distance) will limit the opportunities for mutual advantages of integration. Finally, the notion of relational proximity could be used as an umbrella term for a number of non-tangible dimensions discussed in the literature, for instance cognitive, organizational, social, institutional, cultural and technological proximity (see, e.g., [START_REF]On the Analytical Dimension of Proximity Dynamics[END_REF][START_REF] Boschma R | Proximity and innovation. A critical assessment[END_REF][START_REF] Moodysson | Knowledge Collaboration and Proximity: The Spatial Organization of Biotech Innovation Projects[END_REF]. Relational proximity is associated with the structures, relations and processes that originate, for instance, from the social dynamics, governance structures, regulation and cultural identities that together comprise the embeddedness of social action (GRANOVETTER, 1985). In a growing body of literature (GERTLER, 2003;[START_REF] Boschma R | Proximity and innovation. A critical assessment[END_REF][START_REF] Sternberg R | Entrepreneurship, Proximity and Regional Innovation Systems[END_REF] on relational proximity the discussion is centered around the importance of shared norms, institutions and regulation, mutual understanding, trust and codes of conduct and shared organizational and technological cultures for collaboration and knowledge exchange. A certain degree of relational proximity between key actors is a necessary condition for a fruitful knowledge exchange and collaboration in a crossborder area. In the conceptual model discussed in section 3, special attention will be given to the importance of the cognitive and institutional dimension of relational proximity. The cognitive dimension (NOOTEBOOM, 2000;NOOTEBOOM et al., 2007) is about the fine balance between being so close in terms of knowledge bases and technical know-how that the 11 partners are able to cooperate efficiently, but far away enough to learn something new through cross-fertilization and the exploitation of new complementarities. Good levels of 'related variety' (FRENKEN et al., 2007) defined in this sense can be regarded as key drivers of economic growth and innovation in cross-border regions. The institutional dimension of relational proximity reflects the significance of differences in both formal and informal institutions, laws, regulations and also differences in culture and language [START_REF] Boschma R | Proximity and innovation. A critical assessment[END_REF].

The relation between the three main types of proximity is complex (for a further discussion see. e.g., [START_REF] Coenen | Nodes, networks and proximities: on the knowledge dynamics of the Medicon Valley biotech cluster[END_REF][START_REF] Boschma R | Proximity and innovation. A critical assessment[END_REF][START_REF] Moodysson | Knowledge Collaboration and Proximity: The Spatial Organization of Biotech Innovation Projects[END_REF]. Physical proximity, for example, could in some cases be an important facilitator of relational proximity; in other cases relational proximity can emerge totally detached from physical proximity. The latter case is in line with GRANOVETTER's (1985) geographically unbounded view of the concept, where embeddedness can emerge regardless of physical proximity [START_REF] Moodysson | Principles and Practices of Knowledge Creation: On the Organization of "Buzz" and "Pipelines" in Life Science Communites[END_REF]. This aspect is important to remember in the discussion of cross-border areas. Being geographically close does not automatically mean that relational proximity abounds. More fruitful alternative relations might be found elsewhere. However, cross-border areas where physical, functional and relational proximity coincide might become major places of new knowledge generation and dynamic learning. It is this interplay between certain degrees of physical proximity and appropriate levels of functional and relational proximity that under certain circumstances shape a unique competitive advantage of crossborder regions compared to other spatial units. Furthermore, it is vital to understand crossborder regions in terms of their relations to and dependence on other spatial scales rather than concentrating only on their internal conditions. The embeddedness of actors and organizations in existing and historically evolved RISs and NISs as well as the importance of international

Stages of cross-border RIS development

This section departs from the RIS concept [START_REF] Autio | Evaluation of RTD in Regional Systems of Innovation[END_REF]COOKE et al., 2004;[START_REF] Asheim | The Geography of Innovation[END_REF] and identifies in a conceptual way different stages in the development of cross-border innovation systems. TRIPPL (2010) who applied the RIS approach to crossborder settings, suggested to analyze five different RIS elements and dimensions to disentangle those factors that influence a cross-border region's potential for joint innovation activities. RISs are considered to be made up of a subsystem of knowledge generation & diffusion including R&D organizations, educational bodies and technology transfer agencies (RIS dimension 'science base/knowledge infrastructure), and a subsystem of knowledge application & exploitation comprising the companies and clusters located in the region (RIS dimension 'economic structure/specialization pattern'). Intensive flows of knowledge, resources and human capital within and between these subsystems (RIS dimension 'nature of linkages') are regarded to be pivotal, constituting the foundations for systemic innovation.

The regional policy subsystem (RIS dimension 'policy structures') is often acknowledged to be another important RIS element. Finally, the specific institutional and socio-cultural setting prevailing in the region (RIS dimension 'institutional set-up) is ascribed to play a significant Focusing on these different RIS dimensions and on the role of various types of proximity, in the following we will discuss a conceptual framework that draws a distinction between different stages of cross-border RIS evolution.

[Figure 1 about here] To guide the discussion, three ideal types of cross-border settings are identified (see Figure 1) which represent different degrees of integration: weakly integrated systems (stage I), semiintegrated systems (stage II) and strongly integrated systems (stage III). In the following we will pinpoint basic conditions characterizing each stage regarding the main RIS dimensions identified above, i.e., economic structure, science and knowledge bases, nature of linkages, institutional set-ups, and policy structures. Furthermore, we add the dimension 'accessibility' to take into account the degree of physical proximity.

As shown in Figure 1, the linkages to other geographical scales as well as the effects of different barriers are crucial for understanding the character and magnitude of cross-border integration in each stage. Furthermore, two points should be underlined. Firstly, the three stages represent ideal types. In the real world the distinction between the three stages might not be as clear-cut as in our conceptual model. Cross-border areas which, for instance, display semi-integration in one dimension, can be more or less strongly integrated in other dimensions. Secondly, although we will discuss conditions for further development this does not imply that cross-border areas move necessarily or automatically from one stage to the next one. Table 1 summarizes the main features of the three ideal types of cross-border innovation systems.

[Table 1 about here] 3.1. Stage I: Weakly integrated systems Weakly integrated systems are characterized by low levels of cross-border economic relations in general and a lack of knowledge interactions and innovation linkages in particular. Various 'Euroregions' seem to represent a case in point in this respect (HASSINK et al., 1995;VAN GEENHUIZEN et al., 1996). Integration processes -if any exist or develop at all -show a dominance of asymmetric cost-driven linkages, mainly exploiting internal price and cost differences. Such an asymmetric pattern of integration has been reported in several analyses of cross-border regions (see, for instance, the work by [START_REF] Krätke | Regional integration or fragmentation? The German-Polish border region in a new Europe[END_REF] and [START_REF] Krätke | EU Eastern enlargement and the configuration of German-Polish inter-firm linkages[END_REF] on the German-Polish border area or the study done by TRIPPL (2008) on the Centrope region). There are two main reasons for such a situation, i.e. lack of synergies and under-exploitation of synergies.

Lack of synergies: Non-integration can be the outcome of missing synergies for cross-border linkages. On the one hand, too strong differences in the scientific specialization, knowledge bases, and economic structures (i.e. cognitive distance) make interaction useless, because 15 none of the involved regions can draw benefits from integration. On the other hand, if the regions are too similar in terms of their knowledge bases (reflecting too much cognitive proximity) also little can be learnt from interaction. Consequently, the potential benefits that result from establishing relations and investing in new cross-border linkages can be assumed to be rather low compared to those which are likely to result from further developing and capitalizing on already existing links with actors at other spatial scales. If sub-optimal levels of cognitive distance are the main reason for missing linkages, at least in the medium term, the prospects for the development of more advanced and interactive knowledge interactions are poor, regardless if it is possible to the remove other types of barriers. Absence of synergies might also be the consequence of too high levels of functional distance. Although there might be some similarities in the scientific or economic specialization pattern there is no interaction because the regions differ too strongly in their respective innovation capabilities, performances and receiver competences. For the strong region little can be learnt from the weak region, whilst the weak regions might potentially learn a lot but might miss the absorption capacity. Such a constellation has been observed for example in the US-Mexican border region [START_REF] Scott | European and North American Contexts for Cross-border Regionalism[END_REF] and in the Centrope area [START_REF] Trippl | Ökonomische Verflechtungen und Innovationsnetze im Wirtschaftsraum Centrope, (Economic linkages and innovation networks in the Centrope area)[END_REF]. Consequently, functional distance results in an unequal distribution of benefits from interaction and is closely related to the problem of absorptive capacity. Barriers related to functional distance can take decades to erode if ever. One key point in this respect is whether or not the form of asymmetric cost-driven integration can also generate a certain amount of learning for and knowledge spillovers to the weak side of the border. FDI activities, mobility of managers and key labor in general might be important mechanisms in this regard.

Under-exploitation of synergies: Some cross-border areas exhibit potentials for synergies in innovation, but several kinds of distance create borders and prohibit actors to establish the accessibility is restricted (resulting in high time costs to cross the border), the possibilities for exploiting potential synergies connected to enlarged and enhanced agglomeration advantages may be jeopardized. This type of distance could be one of many reasons for a low level of linkages and flows in general. Furthermore, a high degree of physical distance has a negative influence on knowledge linkages and learning processes which often contain a large proportion of tacit knowledge and therefore require frequent face-to-face contacts. The importance of physical proximity has been highlighted in many cross-border studies, especially in relation to debates about the benefits or drawbacks of new infrastructure investments in cross-border regions (see, for example, the case of the Oresund region documented in [START_REF] Matthiessen | The Öresund Area: Pre-and post-bridge corss border functional integration: the bi-national regional question[END_REF]. Arguably, barriers resulting from physical distance can be easier dealt with than those resulting from cognitive and functional distance. New investment in infrastructure or minimizing the cost to use it for crossing the border will for instance have a positive impact on accessibility.

Furthermore, various manifestations of institutional distance can lead to an under-exploitation of synergies. This could include differences in both hard institutions such as laws and regulations and soft institutions such as the lack of a common culture and language ( VAN HOUTUM, 1998;[START_REF] Klatt | Barrieren und Potenziale der grenzüberschreitenden Zusammenarbeit in der Fehmarnbeltregion[END_REF]. Weakly integrated systems are often characterized by institutional thinness depending on very few cross-border institutions and the absence of trustful cross-border 'leadership', reflecting a lack of legitimacy (HALL, 2008) and conflicting goals between different actors. One important aspect of this is, for instance, a low or asymmetric support and interest from the involved nation states (see [START_REF] Johnson | Cross-border regions and territorial restructuring in Central Europe: Room for more transboundary space[END_REF] insightful study on cross-border co-operation between Eastern Germany, Western Poland and the Northern Czech Republic). Strong forms of embeddedness of actors in their The combination of institutional thinness and strong forms of embeddedness on the one hand and a very low social acceptance of building a cross-border region among firms and citizens on the other hand will not only be a barrier to many forms of integration. It will also shape a very weak cross-border identity, and it has an influence on the positioning of the cross-border region in the global arena (HOSPERS, 2006). This means that it is more or less impossible to 'brand' the cross-border area as an attractive location for foreign direct investment.

Stage II: Semi-integrated systems

In semi-integrated settings we might observe what can be referred to as an emerging knowledge-driven system. Such systems feature decreasing levels of asymmetry and provide opportunities for new and more mutual beneficial linkages on both sides of the border. In addition to links drawing on pure internal price-cost differences resulting in increased economies of scales and the opening up of new markets, there are also windows of opportunities leading to economies of scope and increased cross-border learning processes in a few selective parts of the economy. These more interactive linkages can be observed in narrow segments of the science base and economic structure where good levels of cognitive and functional proximity prevail. Consequently, in semi-integrated systems we might find a few networked cross-border clusters exhibiting such sound degrees of cognitive and functional proximity (see, for instance, the case of the Canada-US cross-border region of [START_REF]Cascadia in comparative perspectives: Canada-U.S. relations and the emergence of cross-border regions[END_REF]. Innovation networking, however, is not a system-wide or region-wide phenomenon yet, but occurs in these few clusters only. In this stage of development, thus, one might identify innovative cross-border clusters but not a coherent trans-frontier innovation system. Apart from networking in a few clusters, cross-border knowledge links are still of subordinate importance when compared to linkages to other contexts.

The emerging knowledge-driven form of integration could also be partial in the sense that it only includes single steps in an innovation process as indicated by [START_REF] Moodysson | Knowledge Collaboration and Proximity: The Spatial Organization of Biotech Innovation Projects[END_REF] in their empirical analysis of the biotechnology sector in the Oresund area.

Furthermore in semi-integrated systems the linkages are also likely to be geographically concentrated to selective parts of the cross-border area and leaving out others, as indicated by an analysis of the internal heterogeneity of the Oresund region where only the core parts on both side of the border seem to have some degree of appropriate functional and cognitive proximity [START_REF] Lundquist | The interspace between Denmark and Sweden: The industrial dynamics of the Öresund cross-border region[END_REF].

These innovation activities of the cross-border area could be described as rather isolated 'islands of innovation' in an otherwise fragmented cross-border innovation system. However, these small segments of the economy could be important role models and drivers of change. Compared to stage I, physical accessibility is no longer a major obstacle for interaction. We might observe an increase of exchange of students, researchers, highly qualified people, scientific collaborations, university-industry partnerships, and enhanced institutional networking. Interaction at the cross-border level becomes essentially promoted by the emergence of bridging-organizations (often related to areas where good levels of functional and cognitive proximity exist), an increasing societal acceptance of cross-border projects and a growing consensus among different actors about potential benefits of increased integration.

Stage III: Strongly integrated systems

This stage of cross-border RIS development represents the most advanced form of innovationdriven integration. We look at an ideal type of integration, where the distinct RIS in the constituent regions of the cross-border area become more and more melted into a single one.

Strongly integrated trans-frontier RIS are characterized by a considerable flow of knowledge, expertise and skills across the border, brought about by a high intensity of mobility of students and labor, innovation related networking among firms, academic collaborations, university-industry partnerships, etc. These knowledge interactions have reached a level at which they form a central underpinning of the innovation performance of the cross-border region. This does not mean that innovation linkages at other spatial scales have lost in significance. The argument is that cross-border innovation linkages have grown in strategic importance and are no longer subordinate when compared to other innovation linkages. These linkages reflect the existence of substantial synergies for cross-border interaction and learning. Such synergies result from the co-existence of high levels of functional proximity (i.e. similar high levels of innovation abilities and knowledge generating capacities) and optimal levels of cognitive distance (related variety) in both the business systems and the knowledge bases. In contrast to stage II such synergies could not only be found in a few Strongly integrated cross-border RIS do not only provide good synergy potentials as they have been described above, but they offer also favorable conditions which enable and support actors to make effectively use of them. They exhibit an excellent transportation and communication infrastructure, thus, allowing for easy accessibility and interaction (physical proximity). Furthermore, other kinds of borders and barriers have been dismantled rather successfully, allowing for the establishment of cross-border interactions. Arguably, most critical and difficult to reach in this context is a good level of institutional proximity in the cross-border area. We might observe a high societal acceptance of building a strongly integrated cross-border RIS, probably even a gradual emergence of a common culture and identity and a high density of bridging institutions which help to deal with socio-institutional and cultural barriers (like differences in language) which might never vanish. A key feature of stage III is, indeed, that solving common problems at the cross-border level is becoming part of normal life, seeking and mobilizing knowledge that is available at the other side of the border are turning into routine activities and cross-border RIS development is widely accepted in business, academic and other societal spheres. Such processes are tightly connected to advanced forms of cross-border political governance (PERKMANN, 1999(PERKMANN, , 2003(PERKMANN, , 2007;;GUALINI, 2003). In strongly integrated RIS we might find fully working cross-border governance structures and high levels of institutional thickness. There are not only many organizations and policy actions geared towards innovation-driven integration, but they form a coherent whole and reflect a successful implementation of institutionalized and stable mechanisms for long-term policy coordination. Specialized organizations responsible for steering cross-border issues and a common innovation strategy and policy might be essential elements in this regard. In the best case we find transparent and democratic structures which allow for inclusive forms of governance and civic participation (see also the respective claims made by HALL, 2008) and which are vital for the emergence of societal acceptance and a common identity at the cross-border level. From a conceptual point of view, this final stage could be referred to as the 'utopia' of cross-border region building. Whilst some areas might succeed in reaching stage III in a few dimensions, in the real world it is probably very hard for most cross-border regions to meet the final stage in all dimensions outlined above.

However, the utopian world of stage III might play a powerful role in the development of visions, animating and guiding actors in search for enhancing innovation-driven integration processes.

Conclusions

Cross-border regionalization is a multi-faceted and complex phenomenon. Firstly, there is a wide range of cross-border regions showing a considerable variation in basic geographical preconditions in terms of scale, size and location causing primary restriction of what kind of integration is likely to occur and what types of benefits can result from different integration processes. Secondly, cross-border regionalization is a process taking place in a 'grey zone' between civil and public law in combination with the emergence of informal and formal interaction among diverse organizations and actors. The overall interest in increased crossborder integration and the ability to engage in such processes can be assumed to vary among the actors reflecting conflicting goals and asymmetry in power relations. Thirdly, these stakeholders are more or less strongly embedded in or have relations to other spatial scales, from the local community to the global level. This historically rooted embeddedness has an influence on the actors' decisions in general and on their 'cross-border behavior'. Fourthly, the driving forces for cross-border integration processes, e.g. the differences in economic structure, innovation capabilities and cost structure give rise to new complementarities and synergies, but often also generate the barriers that exist between the different parts of a crossborder region. Consequently, this tension and interplay between differences working as driving force on the one hand and as barriers on the other hand is crucial for understanding cross-border integration processes. As argued in this article, an essential point of departure to enhance our knowledge about cross-border integration in general and cross-border RIS development in particular is to uncover the role of different types of proximity and distance, and to explore how they influence driving forces and barriers and determine what linkages are likely to be established.

There is a consensus in both academic and policy circles that in the globalizing knowledge economy the long term competitive advantage of cross-border areas will be based on their capacity to create a common innovation system. As a general statement this is a rather undisputed one. However, a crucial question raised in this paper is how achievable this is for most cross-border settings as many of them are not characterized by the collective learning systems or the socio-cultural proximity that are assumed to be important prerequisites for successful innovation systems. In order to unleash this paradox, we suggested a conceptual model that distinguishes between three ideal stages of cross-border integration: weakly integrated systems (stage I), semi-integrated systems (stage II) and strongly integrated systems (stage III). To summarize, the three types are characterized by very different levels of physical, functional, cognitive and institutional proximity which in turn lead to very different possibilities for cross-border knowledge interactions to emerge. Cross-border RIS constitute the most advanced form of cross-border region building and will often be based on the success of previous incremental and less advanced modes of integration. Finally, the conceptual analysis provided in this paper might also have far reaching policy implications. There are at least two broad and general conclusions for policy which can be drawn immediately. First, the opportunities and prospects for successful policy interventions in cross-border RIS formation are subject to the types of barriers and sub-optimal levels of proximities prevailing in a specific cross-border setting. Whilst physical distance and some manifestations of institutional distance (particularly laws and regulations) can be dismantled rather easily, cognitive distance and differences in culture or language will probably never vanish or could be reduced only in the long term, requiring enormous efforts on the part of policy actors and other stakeholders. Second, our work on different stages of cross-border RIS development challenges any uniform policy models for promoting innovation-driven integration in cross-border regions and provides essential foundations for more tailor-made policy approaches. Arguably, the role of policy can be expected to differ strongly, depending on the respective phase of cross-border RIS evolution. In weakly integrated systems (stage I) which suffer from massive differences (particularly in terms of functional and cognitive distance) between adjacent regions, it might be more efficient for policy actors to address other issues than promoting cross-border innovation-driven integration. The lack of synergies 24 calls for a 'hands off' policy approach. The situation is clearly different, if poor integration results from unexploited synergies. Weakly integrated systems (stage I) of this type require a very proactive policy approach oriented on removing various forms of impediments in different RIS dimensions. Promoting a socio-cultural shift towards a stronger acceptance of cross-border integration processes, establishing cross-border governance structures, improving transport infrastructure and thus accessibility might be key policy tasks.

Furthermore, fostering knowledge interaction in a few fields which could serve as role models for other sectors should be on the policy agenda. To be sure, to support a transition of a weakly integrated system towards a more innovation-driven one is outstandingly challenging, calling for comprehensive measures and -given the long time span which such a transformation requires -a 'patient' policy system. In later phases of cross-border RIS evolution, there is a need for adjusting the innovation policy portfolio. A further strengthening of ongoing integration efforts in the economic, scientific, institutional and socio-cultural spheres, and broadening and deepening cross-border learning processes will become key policy tasks. Elaborating more on specific and adequate policy approaches required for different stages of cross-border RIS development and dealing explicitly with the question how crucial policy making is for the transition from one stage to the next one are worthy subjects for further research. Table 1:

  Vienna and Bratislava and adjacent regions in Austria, Hungary, Slovakia and

  fall of the iron curtain, followed by the EU-membership of the former Eastern countries and finally the Schengen Treaty have gradually decreased the physical distance between different parts of this area triggering the same kind of hope as in the Oresund region:

  taken into consideration. These already established linkages on both sides of the border have been created under similar constraints of physical, functional and relational proximity and will influence the character of new potential cross-border linkages. From this point of view it is important to take into consideration to what extent the exploitation of potential cross-border linkages will be affected by the quality and strength of existing linkages embedded in other geographical scales. It is still unclear in what ways cross-border integration reinforces, complements, changes, challenges or even substitutes existing relations and linkages.

  covers both formal institutions such as laws and regulations and informal institutions like routines, conventions and habits which have an impact on the behavior of and relations between actors and organizations.

  on the learning potential. Physical distance might have such effects. If

  or in other contexts and relations might also be a powerful impediment. Such well-established, long lasting and trust-based linkages have many advantages for actors and organizations, including lower search costs and a reduction of uncertainty due to strong social proximity. Consequently, actors who are inserted in such well-established links might be less prone to create new ones at the cross-border level.

  North West) as described by

  Actors who are involved in such processes might turn into what LÖFGREN (2008) has called 'regionauts'. Through their daily experience in working in a cross-border area they are able to identify the barriers in terms of accessibility, institutions, laws, regulation and also trustfully suggest how these could be transformed or removed in order to facilitate further cross-border integration.

  in many economic and scientific sectors. Consequently, we might observe a dynamic evolution of several fully developed and highly innovative cross-border clusters with strong collective learning mechanisms.

  suggested in this article could constitute a useful theoretical background, guiding systematic and comparative empirical work on the development of cross-border innovation systems in different national and regional contexts. So far, empirical studies have only looked at single dimensions of cross-border RIS development and neglected others. Our model could serve as a conceptual framework for empirical work that takes into account several dimensions and their interplay. Such empirical studies might also lead to conceptual adjustments.

  ,VAN OORT F. and VERBURG T. (2007) Related Variety, Unrelated Variety and Regional Economic Growth, Regional Studies 41, 685-697; GERTLER M. (2003) Tacit knowledge and the economic geography of context or The undefinable tacitness of being (there), Journal of Economic Geography 3, 75-99; GRANOVETTER M. (1985) Economic action and social structure: the problem of embeddedness, American Journal of Sociology 91, 481-510; GUALINI E. (2003) Cross-border Governance: Inventing Regions in a Trans-national Multilevel Polity, DISP 152, 43-52; HALL P. (2008) Opportunities for democracy in cross-border regions? Lessons from the Oresund Region, Regional Studies 42, 423-435; HASSINK R., DANKBAAR B. and CORVERS, F. (1995) Technology networking in border regions: Case study of the Euregion Maas-Rhine, European Planning Studies 3, 63-83; HOSPERS G-J. (2006) Borders, Bridges and Branding: The Transformation of the Öresund Region into an Imagined Space, European Planning Studies 14, 1015-1033;HOEKMAN J., FRENKEN K. andVAN OORT F. (2009) The geography of collaborative knowledge production in Europe, Annals of Regional Science 43, 721-738; .,VAN DER KNAAP B. and NIJKAMP P. (1996) Trans-border European networks: shifts in corporate strategy?, European Planning Studies 4, 671-682; VAN HOUTUM H. (1998) The development of cross-border economic relations, Center for Economic Research, Tilburg.

FigureFigure 1 :

 1 Figure

  and its application to cross-border contexts (TRIPPL, 2010) a conceptual framework that allows for identifying different types or stages of trans-frontier innovation systems. Focusing on different concepts of proximity

	we will suggest
	F o r
	P
	e
	e r
	R
	e
	v i e
	w
	O n l
	y
	6

29 NELSON R. R. (2008) "What enables rapid economic progress: What are the needed institutions?, Research Policy 37, 1-11; NELSON R. R. and WINTER S. (1982) An Evolutionary Theory of Economic Change, Belknap Press of Harvard University Press, Cambridge, Mass; NIEBUHR A. (2008) The impact of EU enlargement on European Border Regions, International Journal of Public Policy 3, 163-186; NOOTEBOOM B. (2000) Learning and Innovation in Organizations and Economies, Oxford University Press, Oxford; NOOTEBOOM B., VAN HAVERBEKE W., DUYSTERS G., GILSING V. and VAN DEN OORD A. (2007) Optimal cognitive distance and absorptive capacity, Research Policy 36, 1016-1034; PERKMANN M. (1999) Building Governance Institutions across European Borders, Regional Studies 33, 657-667; PERKMANN M. (2003) Cross-border regions in Europe. Significance and drivers of crossborder co-operation, European Urban and Regional Studies 10, 153-171; PERKMANN M. (2007) Policy entrepreneurship and multi-level governance: a comparative study of European cross-border regions, Environment and Planning C: Government and Policy 25, 861-879;

O3 -Technological Change|Research and Development < O -Economic Development, Technological Change, and Growth, R10 -General < R1 -General Regional Economics < R -Urban, Rural, and Regional Economics


Judith Przemeck

Christian Grund
email: ch.grund@uni-wuerzburg.dejudith.przemeck@uni-bonn.de

Oliver Gürtler

Matthias Kräkel

Subjective Performance Appraisal and Inequality Aversion

Keywords: Appraisals, Inequality aversion, Performance evaluation, Centrality bias, Leniency bias JEL Classification M5, D63 Appraisals, Inequality aversion, Performance evaluation, Centrality bias, Leniency bias

Making use of a subjective performance appraisal system, it is a well established fact that many supervisors tend to assess the employees too good (leniency bias) and that the appraisals hardly vary across employees of a certain supervisor (centrality bias). We explain these two biases in a simple theoretical model and discuss determinants of the size of the biases.

Introduction

Performance appraisal is a comprehensively discussed topic of economics and management research. [START_REF] Bretz | The Current State of Performance Appraisal Research and Practice: Concerns, Directions, and Implications[END_REF] as well as [START_REF] Levy | The Social Context of Performance Appraisal: A Review and Framework for the Future[END_REF] provide extensive surveys of the literature. Systematic performance appraisal systems can be implemented for several purposes including promotion and training decisions, performance based variable pay and personnel planning [START_REF] Cleveland | Multiple Uses of Performance Appraisal: Prevalence and Correlates[END_REF]. For some employees it might be possible to obtain objective performance indicators. Then, it is possible to analyze the advantageousness of certain measures (e.g. [START_REF] Reichelstein | Investment Decisions and Managerial Performance Evaluation[END_REF]). In many cases, however, it is not possible to obtain objective performance measures on an individual base. Besides, existing objective measures usually correspond only to part of employees' tasks, which may lead to distorted incentives [START_REF] Milgrom | An Economic Approach to Influence Activities in Organizations[END_REF][START_REF] Holmstrom | Multitask Principal-Agent Analyses: Incentive Contracts, Asset Ownership, and Job Design[END_REF]. Therefore, many firms implement some kind of subjective performance evaluation.

Subjective performance evaluation, however, is also fraught with some problems. One issue is that the accuracy of ratings is not given automatically. If the rater is a residual claimant (i.e. the owner of a firm) and the appraisal affects some kind of variable pay, she may underreport the performance of her subordinates in order to save costs. However, many supervisors are no residual claimants but themselves employed workers in multi layered firms. Therefore, supervisors can also be interpreted as agents with own utility functions, which may deviate from the principal's objectives. In this situation possible rater biases include the centrality bias and the leniency bias. 1 In many firms the majority of employees get ratings above the average mark. Therefore, the appraisals are by definition skewed to the top end of the scale and supervisors rate some kind of lenient (e.g. [START_REF] Jawahar | Where All The Children are Above Average: The Performance Appraisal Purpose Effect[END_REF]. Besides, several studies find that only a small fraction of possible performance levels is used by the supervisors when evaluating subordinates. They tend to differentiate only slightly between employees (e.g. [START_REF] Murphy | Performance Appraisal: An Organizational Perspective[END_REF]. These biases may not only be inefficient for firms but also lead to 2 the problem that the performance appraisal system is not accepted by several employees as [START_REF] Murphy | Performance Measurement and Appraisal: Motivating Managers to Identify and Reward Performance[END_REF] observes for the pharma company Merck & Co, Inc. More than 70 percent of employees are located in only three of 13 performance categories and only about 5 percent have got marks below the average one in this case (see Figure 1).

.2 0.3 1 1+ 2- 2 2+ 3- 3 3+ 4- 4 4+ 5- 5

Performance Rating

Note: 1 = unacceptable performance, 5 = exceptional performance.

Source: [START_REF] Murphy | Performance Measurement and Appraisal: Motivating Managers to Identify and Reward Performance[END_REF], p. 40.

Several reasons for these biases are discussed in the empirical management literature. If supervisors are not rewarded for accurate ratings, they may have insufficient motivation to invest time in gathering information [START_REF] Fox | The Halo Effect: Is it a Unitary Concept[END_REF]). Furthermore, they may also face cognitive limitations and tend to focus on some performance dimensions [START_REF] Ittner | Subjectivity and the Weighting of Performance Measures: Evidence from a Balanced Scorecard[END_REF] or arbitrarily favor certain employees e.g. to encourage loyalty or to serve their self-interest [START_REF] Ferris | Personnel/human Resources Management: A Political Influence Perspective[END_REF]. Besides, supervisors may have a preference for a pleasant relationship with their subordinates [START_REF] Varma | Interpersonal Affect and Performance Appraisals: A Field Study[END_REF]. Negative feedback may lead to undesirable discussions and is therefore avoided if possible. 2

The aim of this paper is to explain both the leniency bias and the centrality bias in a simple model.

2 See also [START_REF] Bol | Subjective Performance Evaluation[END_REF] for a more detailed overview of the literature. The contributions of [START_REF] Prendergast | Favoritism in Organizations[END_REF] as well as [START_REF] Prendergast | Uncertainty and Incentives[END_REF] are two of the few theoretical studies which provide explanations for the leniency bias by assuming that the well-being of subordinates is part of the supervisor's utility function.

In [START_REF] Prendergast | Favoritism in Organizations[END_REF] a supervisor can distort her reports but is monitored by the management. Since biased reports lead to inefficient job assignments, management punishes the supervisor if her report deviates from the management's own observation.

Similarly, [START_REF] Prendergast | Uncertainty and Incentives[END_REF] assumes that the supervisor faces some cost when deviating from the truth. Both contributions show that favoritism increases with the worker's incentives. Assessments are distorted upwards the more the supervisor likes a particular agent and downwards the more she dislikes a subordinate. [START_REF] Breuer | Social Ties and Subjective Performance Evaluations -An Empirical Investigation[END_REF] provide a similar model and indeed find in a quantitative case study that social ties between supervisor and appraised employee foster the leniency bias. Sliwka

(2007a) provides a model on performance appraisal with one loss averse agent and shows in a two period model that the leniency bias becomes more relevant over time. [START_REF] Giebe | Optimal Contracts for Lenient Supervisors[END_REF] examine optimal contracts and show that the leniency bias may hold if the supervisor cares for agents' well-being.

In this paper we build on Prendergast's argument and additionally assume that employees are inequality averse to each other. It is now more and more accepted that inequality aversion is an important driving force of human behavior in many situations.

Recent applications of inequality aversion in principal-agent-models include individual and team based incentive contracts, rank-order tournaments and adverse selection problems (see Englmaier 2005 for an overview). However, inequality aversion has not been applied to performance appraisal issues so far. Simultaneously and independently from this paper [START_REF] Sliwka | Accuracy, Social Preferences and Bias in Subjective Performance Evaluation[END_REF] gives another explanation for the centrality bias by incorporating two (not identical) signals of the agents' performances for a supervisor and a principal. The centrality bias is due to the supervisor's inequity aversion and a regression to the mean effect of the supervisor's report, because she wants to avoid own disadvantages from deviating from the principal's signal.

In the next section we proceed by providing a simple model. Section 3 concludes and discusses the results.

A simple model on performance appraisal with inequality averse agents

Suppose that a supervisor S has to assess the unverifiable performance of two agents i and j (i and j = 1, 2; i ≠ j) to determine their wages. Her verifiable report for agent i is characterized by the term r i > 0. We assume that the supervisor is not a residual claimant, but employed in a firm. She observes the true performance p i > 0 of both agents and then has to state a report r i for each agent. The situation of unverifiable performances of employees and the use of subjective performance appraisal systems may rather fit to the case of white-collar workers at lower and middle management levels than to blue-collar workers or executives. The wage W i of agent i consists of two components. He receives a performance appraisal contingent bonus br i (b > 0) in addition to his fixed wage w > 0, i.e. W i = w + br i . Linear incentive contracts are often used in practice and are also found to be optimal in many situations [START_REF] Holmstrom | Aggregation and Linearity in the Provision of Intertemporal Incentives[END_REF]. We focus on a given wage scheme and do not model some kind of principal.

We assume that agents may have social preferences and be inequality averse so that relative wages are important. This is in line with experimental evidence (see e.g. [START_REF] Fehr | Theories of Fairness and Reciprocity -Evidence and Economic Applications[END_REF] and findings of interviews with employees by [START_REF] Agell | Theories of Pay and Unemployment: Survey Evidence from Swedish Manufacturing Firms[END_REF]. For simplicity we assume that both agents are equally inequality averse. To model their utilities we use the utility function proposed by [START_REF] Fehr | A Theory of Fairness, Competition, and Cooperation[END_REF]. If the performance appraisal of i is lower than the performance appraisal of j, agent i is jealous of j. The resulting disutility increases in the difference of the performance ratings (r j -r i ), in the report dependent wage component b, and in the degree of disadvantageous inequality aversion α. If instead agent i is better assessed, he may feel some kind of compassion or guilt with agent j. The extent of such feelings is given by the parameter β. Thus, the utility function of an agent who is inequality averse is given by

F o r P e e r R e v i e w 5 U i (r i, r j ) = W i -α (max {W j -W i , 0}) -β (max {W i -W j , 0}) = w + b r i -α (max {w + b r j -(w + b r i ), 0}) -β (max {w + b r i -(w + b r j ), 0}) = w + b r i -α (max {b (r j -r i ), 0}) -β (max {b (r i -r j ), 0}) , i ≠ j and 0 ≤ β < 1, 0 ≤ α. 3
The special case of α = β = 0 pictures the utility of two purely self-interested agents who are only interested in their own wage. We assume β < 1. This implies that agent i cares more for his own utility than for the other agent's utility. In their seminal paper

Fehr and Schmidt assume that α ≥ β. We do not need this assumption for our model.

Furthermore, we assume that the supervisor is interested in the accuracy of her statement and in the utilities of the agents. On the one hand, the supervisor suffers from costs ( )

2 i i p r v - ⋅
, if her report deviates from her observation. Hereby, v > 0 measures the intensity of these costs from distorting evaluations. These costs may picture the supervisor's shame or her compunction because of lying. This means that the supervisor shows some kind of ethical behavior and has truth-telling preferences. This interpretation is independent of the effect of distorted reports on the firm, i.e. of (ex ante) gains or losses accruing to the firm. If distorted reports unambiguously hurt the firm, these costs may also represent the potential cost of being fired if found out to have stated extensively distorted evaluations. We therefore suppose increasing marginal costs with increasing deviations and use a quadratic cost function. On the other hand, S cares for the utilities of the agents. For example, she may want to avoid arguments with the agents and a negative working atmosphere in general [START_REF] Bernardin | Strategies in Rater Training[END_REF]. The strength of this factor is represented by the parameter µ. The supervisor's utility function is therefore given by

U S (r i , r j | p i , p j ) = µ (w + b r i -α (max {b (r j -r i ), 0}) -β (max{ b (r i -r j ), 0}) + w + b r j -α (max {b (r i -r j ), 0}) -β (max {b (r j -r i ), 0})) -v (r i -p i ) 2 -v (r j -p j ) 2 , µ ≥ 0.
3 Agents are assumed to be inequality averse with respect to outcomes and do not take inputs (e.g. efforts) into account. We think this is a reasonable assumption since there is evidence that the vast majority of employees consider themselves as top performers [START_REF] Meyer | The Pay for Performance Dilemma[END_REF]) so that differences in efforts will usually not be perceived equally across employees. Before deriving the optimal reports of the supervisor for inequality averse agents, we determine the performance appraisals for situations usually analyzed in more traditional economic models. The results can be taken as benchmarks compared to the results of our model:

1.) If the supervisor is merely self-oriented and does not care for her agents' utilities (µ = 0), it is easy to show that she will report the true performances

r i = p i (i = 1, 2)
by differentiating her utility function with respect to the reports, because there would only costs but no benefits from deviating from the observations.

2.) If the supervisor cares for agents' utilities (µ > 0) and the agents are not inequality averse but purely self-interested (α = β = 0), the supervisor maximizes µ (w + br i + w + br j ) -v (r i -p i ) 2 -v (r j -p j ) 2 . Solving the first order condition for r i leads to the optimal appraisals

i i p v b r + = 2 ~µ (i = 1, 2).
In this case the supervisor overstates the agents' performances (r > p). This result can be interpreted as the leniency bias. The size of this bias is increasing in the degree of the supervisor's preference for the agents' utilities and the monetary incentives of the agents. It is decreasing in the supervisor's costs of giving biased reports. Each employee is upgraded to the same degree so that there is no centrality bias in this case.

In our model the supervisor also has to choose r i and r j -given her observations p i and p j -to maximize her utility by taking the inequality aversion of the agents into account.

Note that the supervisor's utility function depends on her reports. Thus, we have to distinguish three different cases, because uneven reports lead to inequality between the agents. The case of equal reports can also be integrated into the other lines of the supervisor's utility function so that we can write this as Assume that S observes p i > p j . She wants to maximize her utility and has to decide whether it is optimal to choose different or equal reports for the agents. To determine the optimal reports r i and r j , we first look at the first line of her utility function. If S decides to choose a higher report for agent i or equal reports for both agents, she maximizes this line of her utility function subject to the constraint r i ≥ r j . Possible solutions include both different and equal reports (Appendix A provides an examination of the corresponding Lagrangian in more detail).

     < - - - - - - + - - + = - - - - + + > - - - - - - + - - + = . if ) ( ) ( )] ( ) ( 2 [ if ) ( ) ( ] 2 [ if ) ( ) ( )] ( ) ( 2 [ 2 2 2 2 2 2 j i j j i i i j j i j i j i j j i i j i j i j j i i j i j j i i S r r p r v p r v r
     ≤ - - - - - - + - - + ≥ - - - - - - + - - + = . if ) ( ) ( )] ( ) ( 2 [ if ) ( ) ( )] ( ) ( 2 [ 2 2 2 2 j i j j i i i j j i j i j i j j i i j i j j i i S r r p r v p r v r
The supervisor only provides different reports

i diff i p v b r + - - = 2 / ) 1 ( β α µ and j diff j p v b r + + + = 2 / ) 1 ( β α µ
, if the performance difference across agents is sufficiently large so that the creation of inequality is compensated. This is the case for

v b p p j i / ) ( β α µ + > -
. Note that the right-hand side this inequality is increasing in the inequality aversion parameters α and β. This is intuitive since it captures an agent's higher disutility when obtaining a report different from the one of his colleague. A similar reasoning can be applied to the maximization of the second line of the supervisor's utility function subject to the constraint r i ≤ r j.

4 It is never optimal for the supervisor to give agent j a better rating, while she observes that agent i is the top performer (see Appendix B). If S observes that agent i's performance is better than the performance of agent j, the only solution to the maximization problem of the second 4 The relevant Lagrange function and the resulting Kuhn-Tucker-conditions can be found in Appendix B. . Otherwise S sets equal reports.

The derivation of the optimal reports when agent j's performance is at least as good as the performance of agent i follows analogously. Our results are summarized in the following Proposition:

Let j i p p > .
If both agents are inequality averse and the performance difference between the better agent i and the worse agent j is sufficiently

large, i.e. v b p p j i / ) ( β α µ + > - , the supervisor optimally reports i diff i p v b r + - - = 2 / ) 1 ( β α µ and j diff j p v b r + + + = 2 / ) 1 ( β α µ . If the performance difference is rather small, i.e., /v ) ( β α µ + ≤ - b p p j i , the supervisor announces equal reports 2 / ) ( 2 / j i even j even i p p v b r r + + = = µ .
It is easy to see that the supervisor's reports picture both the centrality bias and the leniency bias. If there are large performance differences between the agents, the supervisor tries to diminish the wage gap between the agents. This effect represents the centrality bias. Note that the centrality bias occurs in our sense, if the difference between true performances exceeds the difference of performance ratings. The higher α and β , i.e. the more inequality averse the agents are, the larger is the range of equal reports and the larger the centrality bias.

The relevance of the leniency bias has to be illuminated separately for the cases of different and equal reports. First, we consider the case of different reports. As long as 1 < + β α holds, both agents' performance ratings are higher than their observed performances, which represent the leniency bias. Otherwise, only the report for the agent with the lower observed performance is adjusted upwards. Also in this case the sum of the upward bias for the weaker employee and the downward bias for the top performer is always positive, so that the leniency bias always exists in sum. This distortion increases in the preference of the supervisor for the agents' utilities (µ) and in the report dependent wage component (b). It is decreasing in her costs v of deviating from the observed performances.

Inspecting the case of equal reports even r , it is also straightforward that the supervisor generally rates the agent with the lower observed performance with leniency. However, similar to the case of different reports this does not always hold for the report for the agent with the higher observed performance. Again, the size of the leniency bias depends on the inequality aversion parameters α and β . Assume that

j i p p > , then i even i p r > if v b p p j i / µ < -
. Proposition 1 states that the supervisor announces even i r and

even j r if /v ) ( β α µ + ≤ - b p p j i
. Consequently the report for agent i is always adjusted upwards if 1 < + β α . In contrast the report for agent i is lower than his

observed performance, if 1 > + β α and v b p p v b j i / ) ( / β α µ µ + ≤ - <
. However, the upward bias for the weaker agent always outweighs the possible downward bias for the better one.

If the agents' performances are exactly the same (p i = p j = p), the supervisor reports

p v b r even i + = 2 /

µ

(for i = 1, 2). This is the only case, in which the performance appraisals for inequality averse agents correspond to the reports for self-interested agents simply because there is no inequality. The leniency bias is also present in the case of purely self-interested agents, but the centrality bias cannot be explained.

In our model we assume that the supervisor favors all agents. However, it may well be the case that a supervisor has only preferences for one of two subordinates (her favorite). For example, the supervisor knows one agent well and/or wants to reciprocate previous loyalty. It is quite obvious then that the leniency bias is particularly relevant for the favorite so that she may even receive a better report although having performed worse.

Conclusion

Many empirical studies have shown that subjective performance ratings of supervisors are subject to the centrality and leniency bias: Supervisors tend to differentiate only slightly between their subordinates so that ratings are compressed. Moreover, ratings are often skewed towards the top end of the rating scale. Our analysis offers a simple explanation for both biases: If supervisors care for the utility of their inequality averse subordinates, ratings may be distorted in both ways. The extent of the biases is influenced by the size of agents' inequality aversion and the difference in observed performances, for instance.

Biased appraisals may also affect agents' effort choices. This issue is not part of this paper. 5 Based on data of a financial service firm [START_REF] Bol | The Determinants and Performance Effects of Supervisor Bias[END_REF] empirically observes that biased performance appraisal outcomes influence future efforts. She differentiates between the leniency and centrality bias and shows that lenient ratings positively affect performance improvement. In contrast, the centrality bias has a negative effect on future performance. Further theoretical research may abstain from the typical ex ante perspective of principal agent models but also incorporate consequences for the future of employment relationships.

5 See an extended discussion paper version for this issue [START_REF] Grund | Subjective Performance Evaluation and Inequality Aversion[END_REF].

p r v b (K2) 0 ≥ λ , j i r r ≥ , ( ) 0 = -j i r r λ (K3)
From adding (K1) and (K2) we obtain

j i j i r r p p v b + = + + / µ . ( K4
)
Possible solutions include both different and equal reports. If S chooses j i r r > , it follows from (K3) that the Lagrangian multiplier λ has to be zero. From (K1) and (K2) we obtain the evaluations for agent i and j

i diff i p v b r + - - = 2 / ) 1 ( β α µ and j diff j p v b r + + + = 2 / ) 1 ( β α µ . 6
However, the first line of the supervisor's utility function is only valid for the range S may choose r i = r j in some cases, so that the agents cannot suffer from inequality. In

Figure 1 :

 1 Figure 1: Distribution of subjective performance appraisals at Merck & Co., Inc.

Furthermore

  , the right-hand side is increasing in µ and in the report dependent wage component b while decreasing in the supervisor's costs v of deviating from the performances do not differ that much.

  line of the supervisor's utility function corresponds to the second line of the utility function if reports are equal, different reports are only optimal if

  . Since the term on the right-hand side of inequality (1) is positive, the difference of the (observed) performances has to be sufficiently positive, too. Agent i 6 To capture the idea of different reports we will write diff i r and diff j r (i, j = 1, 2 and i ≠ j).

  his colleague to a sufficient amount to compensate for the creation of inequality if S announces

  If S sets r i = r j and we have 0

  the only solution to the maximization problem of the second part of the supervisor's utility function which satisfies all Kuhn-Tucker-conditions is to state

Page 11 of 17 Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK

		Submitted Manuscript
	µ	F o r F o r F r o α β j	j	λ
		P P P
		e e e
			e r e r e r
				R R R
				e e e
				v i e v i e v i e
				w w w

Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript

support by the Deutsche Forschungsgemeinschaft, grant SFB/TR 15, is gratefully acknowledged.

Appendix A

The supervisor will maximize the Lagrangian ( )

and S would decide to choose a lower report for agent i or equal reports for both agents, she maximizes the second part of her utility function subject to

The resulting Kuhn-Tucker-conditions are

From adding (K1') + (K2') we obtain

. This leads to a contradiction since we assume

from (K1') and (K2'). But

. With