Yves Ducq
email: yves.ducq@ims-bordeaux.fr

David Chen
email: david.chen@ims-bordeaux.fr

Guy Doumeingts
email: guy.doumeingts@interop-vlab.eu

A contribution of System Theory to Sustainable Enterprise Interoperability Science Base

Keywords: System theory, organisational interoperability, enterprise modelling, evolution management

Even if the research domain related to interoperability has been developed for more than ten years and particularly for the last eight years, the different kinds of interoperability and the different problems to solve need to be consolidated in order to define a real science. Moreover, because of the continuous evolution of enterprises in supply chains, interoperability problems must continuously be considered and solved in order to reach a sustainable interoperability. The objective of this prospective research paper is to discuss how System Theory (ST),applied to System of Systems, is able to support the development of Sustainable Enterprise Interoperability Science Base. After an introduction which reminds the definition of Enterprise Interoperability and the development of this domain in Europe, the system theory conceptsare introduced. Then,the requirements are described to support the determination of the necessary concepts to develop a Science Base for Sustainable Enterprise Interoperability. This part also describes how the concepts of System Theory meet the defined requirements. The fourth part presents a specific approach based on system theory in order to manage the evolution of interoperability in enterprises and to reach sustainable interoperability. Then last part illustrates this workwith a concrete example showing how ST concepts are used in GRAI methodology for instance to represent business process and decision interoperability problems.

Introduction

Interoperability is defined as « the ability for a system or a product/service to work with other systems or products/services without special effort of the user » [1]. Enterprise Interoperability (EI) is defined as « the ability of an Enterprise to interact with other Enterprises, not only on an Information Technology point of view, but also on organisational and semantic points of view. This interaction must be flexible and developed at the lowest cost" [START_REF] Ec | Enterprise Interoperability Research Roadmap[END_REF]. This last definition was step by step elaborated through a serial of works and projects developed in the frame of the European Commission since 2000"s. Enterprise Interoperability in fact appears long time ago when the economic world starts to exist: the enterprises have to interact in order to develop business.

The use of IT applications has supported the development of EI with a strong acceleration in the last twenty years. But the economic environment has obliged Enterprise to develop EI solutions at a low cost and in a flexible way. It was recognised that 40% of the IT budget of enterpriseswas the consequence of the non interoperability of IT applications. This situation was the driver of an initiative launchin 2000"s by the European Commission to create a working group in order to develop several research works to meet the new economic constraints (cost, flexibility, security...). Based on the suggestions of this expert group composed of the main stakeholders, the thematic network "IDEAS" (Interoperability Development of Enterprise Applications and Software) was launched within FP5 (July 2002-June 2003). The objective of this network was to elaborate a roadmap to develop a research program in EI. Two main initiatives were launched within FP6: ATHENA Integrated Project (IP) (Advanced Technologies for Interoperability of Heterogeneous Enterprise Networks and their Applications) [START_REF] Athena | Advanced Technologies for Interoperability of Heterogeneous Enterprise Networks and their Applications[END_REF] and INTEROP Network of Excellence (NoE) (Interoperability Research for Networked Enterprise Applications and Software) [START_REF] Interop | Interoperability Research for Networked Enterprises Applications and Software, Network of Excellence, Proposal Part B[END_REF]. In the FP7, several projects were launched, among them the COIN FP7 Integrated Project (Collaboration and Interoperability for networked enterprises) which developed an advanced integrated solution, made of a secure Generic Service Platform providing the European Industry (and mostly SMEs) with EI and EC (Enterprise Interoperability and Collaboration) services, under innovative business models inspired by the SaaS-U paradigm (Software as a Service-Utility).

Through all these results developed since 2000"s, the concepts of Enterprise Interoperability have been established, the domain has been defined, the problematic identified and some solutions proposed. A new scientific domain is born which must be promoted in order to allow the recognition of this new discipline, its understanding and its use by the stakeholders of the domain.

Moreover, in order to take into account the rapid evolution of enterprises inside supply chains, EI domain must be extended towards sustainable EI. This last concept aims to manage continuously the evolution of interoperability between partners through the continuous modelling, the continuous performance measurement and the continuous implementation of interoperable solutions. This situation might be compared to the emergence of Enterprise Modelling (EM) at the beginning of 80"s, when it was necessary to create a Science Base for EM. System Theory was a good support to elaborate the theoretical concepts of EM.This research paper proposes to use a similar approach in order to contribute to the development of a Science Base in the domain of EI and sustainable EI.The European FP7 ENSEMBLE project which aims at developing a Science Base for Enterprise Interoperability has identified a list of relevant established sciences that potentially can contribute to Enterprise Interoperability development [START_REF] Ensemble | EISB (Enterprise Interoperability Science Base) State-of-the-Play report: Envisioning, Supporting and Promoting Future Internet Enterprise Systems Research through Scientific Collaboration[END_REF]. System science / General system theory is considered as one of the most relevant ones. In this paper, we start to investigate the basic concepts and principles of General System Theory and its use in Enterprise interoperability domain. Other system related approaches such as Complex Systems, System Dynamics or Systems Engineering could be studied in the next stage of the research. So, in the following part, the requirements to support the development of an EI science base will be detailed. Then, the concepts of system theory will be presented and their contribution to the EI science base will be discussed. In the fourth part of this paper, the evolution management approach based on system theory will be presented to reach a sustainable EI. Finally, a case study will be detailed to demonstrate the interest of system theory.

The System Theory

The system theory is the result of the research works done by many authors among whose one can cite L. Von Bertalanfy [START_REF] Bertalanfy | General System Theory, foundation, development, applications[END_REF], H. Simon [START_REF] Simon | The new science of management decision[END_REF], K.E. Boulding [START_REF] Boulding | General System Theory, the skeleton of science -Management Science[END_REF], Von Neuman [START_REF] Neuman | Theory of Games and Economic Behavior[END_REF], Jean Louis Le Moigne [START_REF] Le Moigne | La théorie du système général. Théorie de la modélisation[END_REF], Mesarovic [START_REF] Mesarovic | Theory of hierarchical, multilevel systems[END_REF] and many others. These research works applied originally the same concepts (System theory concepts) in various disciplines: biology, physics, economy, organisation, computer sciences, cybernetics. From all these works, several definitions are proposed below fora system and its related concepts.

A system is composed of a limited set of elements having attributes and relations between these elements. So, a system has a particular structure. It answers to the question WHAT? The elements composing a system have the particularity to contribute to reach one or several common objectives. These are the objectives of the system. These objectives answer to the question WHY? In order to reach these objectives, a system has several functions which are related to its structure. This answers to the question HOW? Moreover, a system has a boundary. Sometime it is easier to determine the elements inside the system by determining the elements outside the system. The elements outside the system composed the environment of the system and enable to also the borders of the system. This environment answers to the question IN WHAT? But this environment has the ability to modify the system properties and to influence its evolution. This capacity of evolution is the last property of a system. A modification of the borders and of the objectives of a system might lead to the modification of the different status of a system. So, a system can be represented by the figure 1

below:

Insert here figure 1 Figure 1: The concepts of the system But in the frame of new complex systems, the notion of system of systems is emphasised. Indeed, few systems are running independently to their environment and this environment plays a more and more importantrole. Moreover, few companies are able to manufacture a

STRUCTURE EVOLUTION FUNCTIONS O B J E C T I V E S E N V I R O N M E N T
product or a service in a whole (for economic reason they are focused on their core business) and they are obliged to look for partners in the frame of a network of companies. This network is in fact a network of systems which has the same properties than a single system, i.e. a structure, functionalities, objectives, an environment, and its own evolution. This leads to the concept of system of systems. The concept of system of systems could be represented in the figure 2 Based on these definitions, the system theory aims to represent (to model) the realities of a system, concrete or abstract, highlighting at the same time the global and the detailed representations of this system. For instance, GRAImethodology (Graph with Results and Activities Interrelated) is based on the system theory, allowing to represent the controlled system (often called the Operative System, including the added value activities of the enterprise) and the control system at the global (GRAI Grid, GRAI Nets and functional view) level and at a detailed level (GRAI Nets and business process views) and taking into account system objectives and environment. The explicit description of the control system enables to represent the elements which aim to reach the objectives [12] [13] [14]. This definition shows the importance to represent and to study the system at the global level (the level of the system of systems) and the detailed level, i.e. the level of each system. The first one allows to understand the whole system and to consider its whole objectives and structure and the second to understand each system separately in terms of practices and of control of these practices. One of the main problems, in the running of system of systems, is then the interoperability problem. This interoperability problem can be then defined at different levels, contributing to the E.I. science base definition:

 At the level of each system,  At the level of the system of systems, it means between the various systems of the network.

STRUCTURE EVOLUTION FUNCTIONS E N V I R O N M E N T STRUCTURE EVOLUTION FUNCTIONS E N V I R O N M E N T STRUCTURE EVOLUTION FUNCTIONS COMMON OBJECTIVES E N V I R O N M E N T STRUCTURE EVOLUTION FUNCTIONS E N V I R O N M E N T

Contribution of System Theory to the development of EI Science Base

There are mainly two main scientific approaches in the history of sciences: natural sciences and sciences of artificial. In the domain of natural sciences, the main objective of scientific investigations is to observe the real world phenomena, to explain it based on some hypotheses, and to verify those hypotheses are correct. In the domain of sciences of artificial, the objective is different. This domain is also called engineering sciences which aims at elaborating solutions to achieve a pre-defined engineering goal (control, design,…). Concerning the Enterprise Interoperability domain, how can science contribute to its development? As a science of explanation, the natural sciences can provide concepts and methods for observing the phenomena of non-interoperability and for explaining why systems are not interoperable. A better understanding of non-interoperability problems might lead to development of adequate solutions to solve these problems. On the other side, concerning the role of engineering sciences (i.e. science of artificial), it can contribute to elaborate scientific solutions to solve interoperability problems. The criteria to judge a scientific solution, is the repeatability and verifiability. In the future, based on the improved understanding of enterprise interoperability problems, repeatable and verifiable solutions can be built. Philosopher Karl Popper (1902Popper ( -1994) ) considered that a statement is only scientific if this is open to the logicalpossibility of being found false [START_REF] Popper | Realism and the aim of science -Routledge[END_REF]. This means that interoperability problems and solutions must be therefore tested in real systems and situations. To explain the phenomena of non-interoperability that can be observed in various situations, the following hypotheses can be made:

(1) Enterprise systems are not interoperable because there are barriers to interoperability that obstruct exchange of information and services.

(2) Barriers are different kinds of incompatibilities and can be found at different levels and sub-domains in an enterprise.

(3) Heterogeneity is the source of Incompatibilities. Whenever there is heterogeneity in two collaborating systems, there is a risk of non-interoperability (4) Barriers can be specifically linked to a particular application in a specific domain; however there are generic barriers which are common to all situations of non-interoperability. One of the requirements to develop a science base is to define an "instrument" to use for observing the phenomena of non-interoperability. However this is difficult because non interoperability problems are not always directly observable. They may be only observed through their consequences, such as for example, the impossibility to communicate, the impossibility to use or to understand the transmitted information etc. On the other hand, system theory can also provide concepts to represent interoperability problems through modelling approaches. One of the coming research subject identified is to develop formalisms (modeling constructs) in the frame of the enterprise modeling domain to model interoperation / interface problems and requirements for interoperability. Another requirement is to definean "instrument" for measuring the different degrees of interoperability [START_REF] Chen | An Approach for Enterprise Interoperability Measurement[END_REF] [START_REF] Kasunic | Measuring systems interoperability: challenges and opportuni-ties, Software engineering measurement and analysis initiative[END_REF]. As it is obvious that interoperability is not a binary state (all or nothing), metrics for measuring interoperability levelare needed to base any research in this area withan objective to keep a rigor basis.

In the Framework for Interoperability defined in the frame of Interop NoE, the phenomena of non-interoperability is approached and three categories of barriers (dimension of interoperability barriers) have been defined [START_REF] Chen | Framework for enterprise interoperability[END_REF], namely conceptual, technological and organizational barriers. These barriers can appear in the four interoperability aspects (dimension of interoperability concerns): data, service, process, business. These two first dimensions composed the problem space of enterprise interoperability (see figure 4). The intersection of an interoperability barrier and an interoperability concern is the set of interoperability problems having the same barrier and concern. A solution is considered as relevant to interoperability if it enable to remove at least one barrier for at least one interoperability concern. In such ways, the scientific interoperability research is problem driven. Some examples of interoperability barriers and problems are illustrated in the figure 3 below. For example, at the cross of conceptual barrier and process concern, one interoperability problem is the impossibility of exchanging process model information between IDEF3 and BPMN models because of the syntax incompatibility between these two models.

Figure 3: The problem space of enterprise interoperability The third dimension that must be added is named interoperability approach (integrated, unified and federate) (figure 4). The three dimensions together (also see figure4) composed the solution space of enterprise interoperability. The cross of an interoperability barrier, an interoperability concern and an interoperability approach includes the set of solutions to breakdown a same interoperability barrier for a same concern and using a same approach. components of systems that are concerned by interoperations). Therefore another requirement to develop a science base is to identify in all relevant existing sciences, the concepts, principles and methods that allow removing incompatibilities. This is defined as one of the grand challenges in the roadmap for enterprise interoperability published by the European Commission [START_REF] Ec | Enterprise Interoperability Research Roadmap[END_REF]. Among these sciences, system theory (or System of Systems) is considered as one of the most important ones. Among the three categories of barriers (conceptual, technological and organisational), the organisational barriers can be best dealt with system theory.

C O N C E P T U A L T E C H N O L O G IC A L O R G A N IS A T IO N A L B U S IN E S S P R O C E S S S E R V IC E D A
The organisational barriers are concerned with the incompatibilities of organisation structure, decision-making procedure and management techniques implemented in two enterprises. Indeed if two companies have different organisation structures (ex. hierarchical vs. networked) and management techniques, some necessary mappings may need to be done before the two systems become interoperable at the operational level. More precisely the three followingorganisational requirements (barriers) to meet are:

 Responsibility needs to be defined to allow the two parties knowing who is responsible for what (process, data, software, computer,…). If responsibility in an enterprise is not clearly and explicitly defined, interoperation between two systems is obstructed.

 Authority which defines who is authorised to do what. For example, it is necessary to define who is authorised to create, modify, maintain data, processes, services, etc. and who is authorised to take decisions in case of problems, etc.

 Organisation structure refers to the style by which responsibility, authority and decision making are organised. For example one can consider centralised vs. decentralised organisations, or hierarchical vs. matrix or networked organisation structures.

These organisational barriers (or problems) can concern four interoperability aspects, namely: data, service, process and business. Among these four aspects, the system theory can contribute in particular to solve problems of process interoperability and business interoperability thanks in particular to the global and local structure modelling and to the common (system of systems) and specific (system) functions and objectives identification:

 The interoperability of processes aims to make various processes work together. In the case of the networked enterprise, it is necessary to study how to connect internal processes of two companies to create a common process (collaborative process for example). In this sense, the modelling of the network processes and the definition of the systems objectives contributes to the process interoperability definition and solving.

 The interoperability of business refers to the ability to work together at the levels of organization and company in spite of the different modes of decision-making, methods of work, legislations, culture of the company and commercial approaches etc. In this sense, the definition of system functions and objectives and of system environment and evolution contributes to a first step in the business interoperability problems solving.

The system theory can also contribute to propose solutions for the three main approaches in order to relate systems together to establish interoperability, namely:

 Integrated Approach: itmeans that there is a common format for all models. Diverse models are built and interpreted using/against the common template. This format must be as detailed as the models themselves. The system structures are aligned in this case and common objectives are defined.

 Unified Approach: It means there is a common format but it only exists at meta-level. This format is not an executable entity as it is the case in the integrated approach.

Instead it provides a mean for semantic equivalence to allow mapping between models and applications. In this case, the system functions are coherent and common objectives are defined.

 Federated approach, there is no common format at all. To set up interoperability, parties must accommodate and adjust "on the fly". The use of the federated approach implies that no partner imposes their models, languages and methods of work. In this case, only common objectives are defined but each system keeps its own structure and its own functions. The environment of the systems must be accurately defined in order to adapt continuously the interoperability requirements "on the fly". This is also crucial to identify the potential of evolution of each system.

It has been considered that a scientific approach for Enterprise Interoperability research should be problem-driven. Considering interoperability as a problem is remarkable.

Interoperability is a requirement inside a system, whosematurity depends on the interactions or composition among itscomponents. This is the same for the system itself, when it needs to besufficiently flexible to interact with another system, or when itneeds to be open to new components. As soon as this ability is notachieved when systems or system"s elements need to operatetogether, interoperability becomes a problem that must be solved [START_REF] Naudet | Systems Science for Enterprise Interoperability[END_REF]. The basic concepts of the Framework for Enterprise Interoperability is modelled in relation to a system approach and integrated in OoI (Ontology of Interoperability) initially developed under INTEROP NoE and progressively enhanced in [START_REF] Ruokolainen | An Ontology of Interoperability in Inter-Enterprise Communities[END_REF] (see figure 5).

Figure 5. System theory based Ontology for Enterprise Interoperability

In figure 5, basic concepts of the enterprise interoperability (framework) are represented in "orange colour" around the three concepts: enterprise level, interoperability barrier and interoperability approach. System theory concepts are represented in blue colour and consist in two parts: system concepts (right part in figure 5 around system concept) and upper part (left) representing system theory solutions around the "solution" concept. Enterprise level (data, service, process and business) is referred to system concepts represented at the right side. Interoperability approach concept (integrated, unified and federated) is related to "relation" concept of the system theory. Interoperability barriers are removed by solutions provided from system theory perspective. For example (see figure 5), to solve interoperability problems (by removing the three kinds of interoperability barriers) using a federated approach, there are two types of solutions: a priori and a posteriori. A priori solution consists in negotiation and homogenization actions before the beginning of an interoperation. A posteriori solution takes place after collaboration starts. It consists in a domination, an adjustment or an exclusion.

According to the Ontology of Enterprise Interoperability presented in figure 5 [22] [START_REF] Guedria | Contribution of System Theory to develop Enterprise Interoperability, 8ème Congrès International de Génie Industriel[END_REF],

Enterprise interoperability problems are caused by the interoperability barriers (of three types) which exist at four different enterprise levels where system entities operate to fulfil their missions. A system has objective, function, structure, behaviour and operates in an environment. There are two types of relations that impact interoperability: structural relation and behaviour relation. A system can be represented by various models in which the mismatches of incompatible syntax and semantics are main problems of non-interoperability. So, based on these assumptions, the table 1 below summarizes and explain possible contributions of identified system theory concepts and principles to enterprise interoperability development. The 1 st column lists system concepts /principles presented in the ontology. In the 2 nd and 3 rd columns, the "x" indicates if the concept or principle contributes to EI problem space and solution space. Necessary explanations are given as remarks in the 4 th column. [START_REF] Ullberg | Barriers to Enterprise Interoperability[END_REF] and relationships to system concepts and solutions.

An ID is given to each barrier allowing categorizing the barriers according to the framework. This ID is constructed according to the following syntax: <type of barrier>"/"<type of concern>"-"< number within this category>, where the types are identified by the first letter in their respective names. For instance, O/P-2 is the second organizational barrier at the process level. The description of barrier is expressed as the heterogeneity (or difference of things) considered as the source of incompatibilities (barriers). In this table, the last column gives the relevant concept(s) from system theory and solution elements that is/are used to remove the barrier according to system theory concepts described in figure 5. However,the barriers listed in the table 2 are not exhaustive and need to be further identified and completed. For each barrier, the solution needs to be identified in the solution space. For example, PSL (Process Specification Language) [START_REF] Schelenoff | The Process Specification Language (PSL): Overview and version 1.0 Specification -NIST[END_REF] contributes to remove conceptual barrier (both syntax and semantics) concerning process through anunified approach. Figure6 shows the position of PSL solution in the framework. Figure 6. Position of PSL in the framework. Finally, the last problem to solve is the continuous management of enterprise interoperability to go towards sustainable interoperability. In this sense, system theory must enable to identify interoperability problems, to evaluate performance of interoperability and to facilitate the identification of decisions to make in order to implement new interoperable solutions.

Moreover, the contribution of system theory to interoperability, and in particular organisational interoperability problem solving and then science base, depends on the system life cycle phase which is considered. There are two types of solutions depending on the life cycle phase [START_REF] Naudet | A Systemic approach to Interoperability formalization[END_REF]: a priorii.e interoperability solutions during the system design and a posteriorii.e. during the execution of the system. For a priori solutions, system has a structure and behaviours, and most often, the structure determines the behaviour. Stating that some systems have higher interoperability potential than others, one can deduce that these interoperable systems might have some common structure characteristics can be defined. It means that some structure patterns supporting interoperability. System Theory allows toidentifysystem properties or attributes that must be amplified to improve interoperability potentials, such as openness, adaptability, flexibility, reconfigurability, modularity etc. These properties must be designed in the system as a priori solution. Fora posteriori solutions, three systemic solutions for interoperability can be considered: (1) Exclusion (rejection of a problematic subsystem), ( 2) Domination (modification of the system"s structure while keeping its original objective), and (3) Adjustment (limitation of the action field of a sub-system) [START_REF] Guedria | Contribution of System Theory to develop Enterprise Interoperability, 8ème Congrès International de Génie Industriel[END_REF].

System theory might also support the three main phases of interoperability problem solving during both phases of system life cycle:

-Identification of problem, through the consideration of the system structure and then of the structure of system of systems, -Evaluation of interoperability level in the system of systems through the consideration of system finality and objectives to reach in the frame of this finality, -Interoperability improvement through the modification of the system structure in the system of systems. This last point is linked to the management of system evolution in order to reach a sustainable interoperability all along the collaboration between systems of system as presented in the next part of the paper.

So, contribution of system theory concerns organisational interoperability characterisation, evaluation and evolution management. Indeed, as explained previously, organisational interoperability is mainly related to the coherence of decisions and practices inside each system and between systemsof system. Indeed, system theory will allow torepresent the decisional structure insisting on the hierarchicallinks between decisions,enabling to define also responsibilities and authorities at the global and the detailed levels of decision.

Moreover, system theory allows to consider and represent business processes and practices at different levels of granularities, from the global activity of the system of systems, to the detailed activities of each system, and allowing continuously to link all these practises. System theory, allowing to identify system objectives and structure also allows to characterise organisational interoperability authorities and responsibilities. The system theory is also very useful to represent and understand decision coordination and synchronisation and then improve the control of the system of systems and its evolution. Other contributions of system theory to sustainable enterprise interoperability have been studied in [START_REF] Guedria | Contribution of System Theory to develop Enterprise Interoperability, 8ème Congrès International de Génie Industriel[END_REF]. Several characteristics of systems can be observed to determine their potential forinteroperability. Based on [START_REF] Walliser | Systèmes et modèles, introduction critique à l"analyse des systèmes[END_REF],the main characteristics are considered as follows [START_REF] Naudet | Systems Science for Enterprise Interoperability[END_REF]:

-The openness of a system refers to relations between the system and its environment. A closed system, which does not or cannot interact with outside is itself not interoperable, as it cannot be connected to other systems. Exotropic systems, that can only send information to their environment, can be connected but have a poorinteroperability since they force other systems to be adaptable. Endotropic systems (that only support inputs from the environment), or mixed systems have a better interoperability since they are able to react to inputs coming to their environment.

-The stability of a system should be considered for interoperability. An unstable system will be prone to create interoperability problems due to its changing nature.

-The adaptability of a system is obviously an important factor for interoperability. A system that can self-react to changes and adapt its structure or behaviour accordingly while keeping its original objectives, has a greater interoperability potential with other systems.

-The reversibility is one of the properties that interoperable systems should have: Even if the implementation of the interoperability between two systems leads to their adaptation or modification, these systems have to be able to come back to their initial state (both from the point of view of structure and behaviour), when interoperation is over. The interoperability of systems that have predictable behaviours, can be better quantified since their inputs and outputs can be matched more easily with other systems to interoperate with. This is true for causal and deterministic systems. However this is much more difficult for stochastic or undetermined systems for whose interoperability is hardly predictable. However, in order to reach sustainable interoperability, systems must solve continuously interoperability problems and implement continuously new small projects to reach and to maintain interoperability. So, the following part proposes a specific method of evolution management to reach sustainable interoperability based on system theory concepts and in particular modelling and performance evaluation.

4 Towards sustainable interoperability management using evolution management method based on system theory

The aim of this part is to show how the concepts of ST are important to consider for the definition of a management evolution method to ensure sustainable EI. Moreover, this part also aims to show how ST concepts can be considered when modelling a set of companies in order to ensure the interoperability of practices and decision making.

The aim of the following method (figure 7) is to manage continuously interoperability performance in order to reach a sustainable interoperability. This method is based on system theory and on the GRAI Evolution Method (GEM) [START_REF] Malhene | Gestion du processus d'évolution des systèmes industriels -Conduite et méthode -PhD University Bordeaux 1[END_REF]developed at the origin to allow isolated enterprises to move towards the organisational interoperability and the collaboration inside a network. The principle of this method is to manage system evolution like a continuous process. In practice, the evolution process is composed of a sequence of steps representing the evolution of the system states. In figure 7, GEM was modified to take into account interoperability problems. Then, two different enterprises are considered, the Enterprise 1 (E1) and Enterprise 2 (E2). One considers that before the beginning of the evolution they do not collaborate because of a lack of organisational interoperability.

The AS IS represents the model of existing systems, insisting on the part dedicated to collaboration. The components of the system are here described and formalisedin a coherent way thanks to the GRAI Conceptual model which includes system theory concepts: it is possible to better understand how the system is running and also to detect the points to improve. The system theory, using to identify systems objectives, structure and evolution, as presented previously,is useful for this part of the evolution method. The TARGET corresponds to the strategic objectives of the network. The TARGET implementation corresponds to the "Effective Collaboration" between E1 and E2. The STEP is an intermediate stage between the AS IS and the TARGET. It corresponds to the future system which will be implemented.

To validate each step, a performance indicator system is established. In fact, in this method, there are two different measurement systems:  Performance Measurement System to manage the Evolution (PMSE),  Performance Measurement System to manage the NetworkExecution (PMSNE).

PMSE is readapted at the end of each step and is transformed in PMSNE when the collaboration becomes effective. The final goal is an effective collaboration between the two (or more) concerned enterprises: E1 and E2. The main objectives for the future Network are described at the beginning of the evolution project. They result from a Network Reference Model and are validated by the PMSNE.

It is possible to further detail the management evolution process described inthe figure 8.

Here, there are not only the AS IS, the STEP and the TARGET, but also a set of complementary steps which allow the definition of the three first. This reduces the field of the evolution, being focused on the interoperability problems. When the model of existing system (AS IS) is defined, points of interoperability can be identified thanks to previous method. The Users Specificationsare derived from the comparison between the AS IS and the TARGET. This comparison must be done for the business process and decision models and also at the global and local levels of running as proposed by ST. The Users Specifications concern flows, resources and activities modified and added to the system. These must indicate the technical and organisation solutions to integrate into the existing system. In most of cases, technical solutions refer to existing tools on the market. These tools have the advantage of being compliant to standard and of offering periodic updates. However, the greatest interest is an easy and controllableimplementation in term of time and cost. An intermediate step is characterized by the implementation of a project of change or an action allowing to establish an interoperable solution.

Figure 8: Method for sustainable interoperability management based on system theory

System theory

It is possible to identify various kinds of actions according to the priority of their implementation: strong, average and weak priority. The definition of the priorities associated with the intermediate states is done through the definition of an Action plan. When defining the action plan, it is necessary to valid the coherence of objectivesof actions in comparison to the objectives of the network, in order to ensure that the implementation of actions will contribute to the Network objective"s achievement. Performance measurement system must be defined, just before the action plan definition, in order to measureits effects after the implementation. Thus, all this evolution management approach is required to reach a sustainable interoperability. Moreover, this approach is obviously in line with ST in the sense that the modelling of the various concepts of a system is performed, the environment is taken into account, the functions and the objectives of each system are identified in the decision modelling and of course the evolution of each system is considered.

Example of application

The objective of this chapter is to illustrate the previous requirements and concepts through a real case study, showing how enterprise modelling technique based on system theory allows in particular the characterisation of organisational interoperability. The companies belong to the furniture sector. The furniture sector is a very complex industry regarding data sharing, process methodologies and business processes between organisations across the supply chain. The EU furniture industry accounts for about half the world furniture production. The furniture industry is one of the largest manufacturing industries in the EU. The furniture industry in the EU accounts for 8,800 enterprises with over 20 employees, employing 600,000 people, and more than 80,000 enterprises with under 20 employees (employing almost 300,000 people). The SMEs which are mostly owned by families as a labour-intensive industry provide employment for around one million people including sub-contracting. Those companies are using a wide range of information systems, many of them developed by small software companies, usually in a very strong competition in the market. Design, production and available services are the major aspects that SMEs have to deal with in order to achieve the success. The current usage of EI within the furniture industry is reduced to the use of internet for some email exchanges without a specific format.Everything is donemanually and in some cases the exchange of Information is done via postal mail or via fax. In this project, the objective of the furniture e-Procurement use case wasto developinteroperable solutionsin two domains:the selling which is oriented towards the Customer (Retailer), the procurement which is oriented towards the Provider. The diagraminfigure 9describes the e-Procurement scenario, as well as the flow of documents between the different actors involved in the scenario: Retailer, Manufacturer and Provider. These documents are marked as follows: R1, R2, R3, R4: these interactions are part of the Retailer"s side of the scenario. In this part, the Retailer asks for information onfurniture"s and receives the Manufacturer answers. M1, M2, M3, M4: on the other hand, these documents are part of the Manufacturer procurement side. In this part, the Manufacturer asks for raw material and the provider serves it. The figure 9shows the usual process between the different actors involved in the scenario including the Interior Decoration Project. The Interior Decoration Project is a draft performed by the Retailer according to the Consumer"s requirements. In a Deco Project, the pieces of furniture are placed in a room which takes into account the special configuration of the future room (dimensions, shape, walls, and painting …). ICT level interoperability issues -Repetitive manual process for regular bulk orders. Most of the manufactured products are generic and this involves repeated periodic processing of similar or identical orders.

Then, GRAI enterprise modelling method [START_REF] Doumeingts | Enterprise Modelling techniques to improve efficiency of enterprises[END_REF] [13], which is directly derived from System Theory, was used in order to represent the running of the scenario in detail in order to understand interoperability problems from a business and decisional points of view.

It was necessary to model first the relationships between the three actors: Retailer, Manufacturer and Provider: This is "the system of systems". This action is directly linked to the concept of "global model". The result is given in the next figures: The business process model above (figure 10) represents accurately the relationships between the retailer, the manufacturer and the supplier, as presented globally in figure 9.

Figure 10 shows that there are a lot of interactions between the three kinds of systems and then a lot of potential interoperability problems to solve in terms of software tools but also in terms of practices.This is the global modelling of the system proposed in the system theory in order to understand the global exchanges and the global interoperability problems between systems. The models highlight the importance to consider each system concept (structure, functions, and environment) in order to identify EIproblems.

Figure 11: The business process of manufacturing Figure 11shows the detailed manufacturing business process using the same concepts than the global one. This is an inclusive modelling as proposed by the system theory. This modelling allows to represent at which step of the process the various systems are in contact and then have to exchange information or products.This local/detailed representation, complementary to the previous one,is the second phase proposed by the system theory in order to identify and understand local problems. Indeed, the interoperability of tools and practice has a sense only in a certain context of use and this context is represented in this model. In the decision model presented in figure 12, the columns correspond to the functions of the various systems/actors. The retailer decisions in the column on the left, the manufacturer decisions in the four columns at the centre and the supplier/provider decisions in the column on the right.

Then this GRAI Grid represents the decisional interoperability problems between the various actors of the system of systems.For instance, the various decision frames (blue arrow) from the purchasing manufacturer function to the supplier, at the tactical level (level 20) shows potential problems in the collaborative decision making between partners. Indeed, the supplier may receive opposite objectives through these various decision frames.The decision frames include system objectives and decision variables as proposed in the ST.

Again, this representation at the global level aimsto consider the interoperability problems related to global objectives which must be coherent between systems, decisional structures which must be compatible between systems andenvironment.

Additionally to these interoperability issues, the following challenges have been identified:

-Media break: from paper and phone to Internet-based technology -Integration of orders/quotations directly into ERP software"s implemented in each system.

As the furniture is a traditional industry with small companies, in terms of technology use, most of the commercial transactions are performed in paper and by phone/fax. Therefore, there is a lack of integration among the different companies involved in the scenario. Solving these two issues would imply a clear change in their way of doing business, shifting from paper to e-documents and from phone/fax to Internet-based communication. Additionally, the solution would help in the integration of the commercial documents into their ERP systems, as the information would be easily treated.

Conclusion

In conclusion, organisational interoperability is one of the main interoperability requirements which are not very investigated in recent research works on interoperability.

System concepts and more particularly system theory which aims to represent these concepts in the real systems and system of systems, must strongly contribute to the organisation problem solving through: identification of decisions, responsibilities, authorities and organisational structure or decision system, identification of business process practices, all these systems being represented at the global and the detailed levels, with identifying links between the different levels.

The sustainable interoperability can be reached through the management of system evolution using modelling, objectives identification and performance evaluation all along the life of collaboration between systems. One perspective of this research work is also to consider interoperability problems between different kinds of systems: ecological, economic, research… included inside the same global system or ecosystem. The science of global system is emerging and is undertaken at European commission instigation. So, it is then necessary to reconsider the system of systems concepts, to valid their existence in the global systems and to define the new emerging concepts. Then, it will be necessary to undertake research on EI to represent these concepts. However, EC has identified that business process and decision modelling are key points to represent and control such kinds of global systems as this is coherent with the work presented in this paper.

Figure 2 :

 2 Figure 2: System composed of four systems with common objectives

Figure 4 .

 4 Figure 4. Problem space vs. solution space of enterprise interoperability domain Indeed based on the hypotheses made, the research in Enterprise Interoperability domain consists in elaborating solutions to remove barriers (i.e. incompatibilities between systems or

  T A Tw o basic dim ensions => P R O B LE M S pace E xam ple : C onceptual barrier x P rocess concern : tw o process des cription m odels (B P M N and ID E F 3) can't exchange inform ation X E xam ple : C onceptual barrier x P rocess concern : tw o process des cription m odels (B P M N and ID E F 3) can't exchange inform ation X X E xam ple : T w o infrastructures (W eb and U nisys) can not w ork together X E xam ple : T w o infrastructures (W eb and U nisys) can not w ork together X B arrier x C oncern S olution : A pproach x B arrier x C oncern

  te c no log ic a l o rg a nisa tion a l fe d e ra te d u te c no log ic a l o rg a nisa tion a l fe d e ra te d u n ifie d in teg ra ted p ro ce ss b u sin es s c o nc e ptu al

Figure 7 :

 7 Figure 7: Method to manage interoperability convergence

Figure 9 :

 9 Figure 9: Document flow including the Decoration Project Analysing the information already presented, it is possible to observe different interoperability issues at all three levels of the interoperability framework namely: Knowledge, Business and ICT levels. These are described below: Knowledge level interoperability issues -Confusion resulting from poor product descriptions (The clients often order the wrong products) -Missing information, both from supplier and buyer (the furniture company has 3 people employed on the client side and one person employed on the supplier side to ensure the integrity of received orders and RFQs)Business level interoperability issues-Lead time from product order to the delivery could be reduced (a shorter lead time from ordering to receiving raw materials from the supplier has a direct effect on the delivery date of the finished product) -Lead time spent to score suppliers (the furniture company performs tri-monthly reviews of their suppliers to ensure that standards are kept)

Figure 10 :

 10 Figure 10: Interactions between retailer, manufacturer and supplier

Figure 12 :

 12 Figure 12: GRAI Grid representing the control of business processes at the global level

Table 1 .

 1 Contribution of system theory concepts and principles to EI development The table 2 below gives non-exclusive examples of barriers to enterprise interoperability

Table 2 .

 2 List of the barriers with ID, name and a brief description

	Id	Name	Description				System concept
	C/D-1	Data content	Coverage, i.e. content, of the	Representation, Model,
			respective data representation	Information System structure
	C/D-2	Data syntax	Heterogeneous data format and	Representation, syntax,
			structure				Information System structure
								(homogenisation or mapping)
	C/D-3	Data semantics	Data meaning disagreements	Representation, semantics,
								Information System structure
								(Semantic annotation,
								mapping)
	C/S-1	Service content	Differences in the coverage, i.e.	Function concept,
			content, of the services offered	Representation, model
	C/S-2	Service syntax	Language/formalism syntax used to	Representation, syntax
			describe the services		(homogenisation, mapping)
	C/S-3	Service semantics	The meaning of services descriptions Representation, semantics
								(Semantic annotation)
	C/P-1	Process content	Coverage, i.e. content, of the	Behaviour
			processes				(Coordination)
	C/P-2	Process syntax	Process	description	language	Representation, syntax
			grammar	and	graphical	(homogenisation, mapping)
			representation
	C/P-3	Process semantics	The meaning of the processes	Representation, semantics
			description				(Semantic annotation)
	C/B-1	Visions, strategies &	Differences	in	the	respective	Objective, environment.
		Culture	companies goals, views, etc.	(Domination,	adjustment,
								negotiation)
	C/B-2	Business syntax	Format, template or model used for	Representation, syntax
			describing enterprise business	(homogenisation, common
								template, mapping)
	C/B-3	Business semantics	Meaning of terms used to express	Representation, semantics
			business issues			(Semantic annotation,
								mapping)
	T/D-1	Exchange format	Protocol or format available to	Interface, representation,
			exchange information		(Metamodel, common
								template)
	T/S-1	Service granularity	Definitions of what constitutes the	Interface, structure
			services, i.e. interface problems
	T/P-1	Process behaviour	Order of operations in the	Behaviour, function, objective
			computerized processes		(Model)
	T/B-1	Degree of	How much of data, services and	Objective, function
		computerization	processes that are automated in IT
	T/B-2	IT requirement	The ability of IT to support the	Objective, function
		fulfilment	requirements of the business
	O/D-1	Information ownership The structures for assigning rights to	(Organisation) structure
			data (different rights for different
			partners)
	O/D-2	Classified information Differences in which an information is	Organisation
			to be regarded as classified with	(Homogenisation)
			respect to the collaboration partner
	O/S-1	Service management Incompatible service management	Organisation (structure)
			rules and practices
	O/P-1	Business process	Order of operations in business	Behaviour, model
		behaviour	processes				(negotiation, homogenisation)
	O/B-1	Legislation	The legislative requirements that	Environment
			influence different actors.		(homogenisation, mapping)
	O/B-2	Organization structure How enterprises are organized on a	Relation, (organisation)
			high level				structure, model
	O/B-3	Methods of work	High level differences regarding how	(Organisation) structure
			work	is	performed	in	the	(homogenisation, adjustment)
			organizations


Porphyry deposits distribution along the western Tethyan suture : insights from a paleogeographic approach

Keywords: porphyry, deposits, geodynamics, kinematics, paleogeography, Tethyan suture

In order to better understand the relationships between ore deposit distribution and their tectonic context, and help identifying geodynamic-related criteria of favorability, we propose a paleogeographic approach. We use this approach to study the distribution of porphyry copper deposits along the western Tethyan suture. Spatial and temporal distribution of the deposits is not random and show that they were emplaced in five distinct clusters. Two clusters, in the Aegean-Balkan-Carpathian area, that were emplaced in Upper Cretaceous and Oligo-Miocene, are associated with a specific polyphased kinematic context, within the framework of the Africa-Eurasia convergence. This context is characterized by 1) a fast subduction rate, shortly followed by 2) a drastic decrease of this rate. Although preliminary, these results highlight the control of the geodynamic context, and especially the subduction kinematics, on the genesis of porphyry deposits. This study also confirms that the paleogeographic approach is a promising tool that could help identifying geodynamic and tectonic criteria favoring the genesis of various ore deposit types.

Introduction

The spatial approach of predictivity focuses on the geological context of mineral deposits and the parameters that control their distribution, from district to continental scales: geology, tectonic structures, geophysics and geochemistry, but also geodynamics and paleogeography. It is an upstream phase of prospection campaigns, which goal is to guide the exploration strategy by predicting the a priori most favorable areas. [START_REF] Loiselet | Spatial distribution of mineral deposit along easten Mediterranean subduction zone: a link with 3D mantle flow associated with slab rollback ?[END_REF] have shown the strong impact of the geometry and dynamics of the eastern Mediterranean subduction zone on the distribution of porphyry and epithermal deposits. More generally, it is widely accepted that the genesis of many types of mineralization is closely linked to the geodynamic context (e.g. [START_REF] Lips | Geodynamic causes of copper and gold concentrations in the Tethys belt[END_REF]. Therefore, a thorough knowledge of this context is a necessary condition to identify tectonic factors that control the genesis of ore deposits. A major problem, however, is to replace the mineralization within the geodynamic framework that prevailed at the time of its genesis. It is a necessary step to better understand the relationships between the mineralization and its environment (plate boundaries, tectonic structures, geology,…). This would, in turn, help identifying criteria that are favorable to their genesis. We believe that the paleogeographic approach, which has not been much used so far in the field of mineral resources predictivity, is appropriate to help solving this problem.

Porphyry copper deposits are associated to calc-alkaline magmatism in subduction zones. In the present study, we use the paleogeographic approach to better understand the geodynamic and kinematic context that controlled, at least in part, the genesis of porphyry copper deposits along the western Tethyan margin during Cretaceous and Cenozoic.

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Conclusion

The paleogeographic approach used in the present study allowed to evidence a specific kinematic context associated with the emplacement of two porphyry copper deposits clusters in the Aegean-Balkan-Carpathian area in Upper Cretaceous and Oligo-Miocene. This context is characterized by 1) first a relatively high velocity of subduction, shortly followed by 2) a drastic decrease of this velocity. Even though this result is preliminary, it confirms the control of the geodynamic context, and especially the subduction kinematics, on the genesis of porphyry deposits. It needs, however, to be confirmed by 1) studying other convergent margins, in order to check if other porphyry deposits clusters may also, or not, be associated with a similar kinematic context, 2) considering the magmatism associated with the porphyry deposits, in order to confirm the link between deep mantle processes and genesis of deposits in the upper crust, and 3) refining the kinematic models, that are global and lack the accuracy needed for such a regional studies. These points will be addressed in future developments of this work.

Nevertheless, the present study confirms that the paleogeographic approach is a promising tool that could help identifying geodynamic and tectonic criteria favoring the genesis of several mineral deposit types. As a corollary, concentrations of mineral deposits, according to their type, may also be interesting markers of past geodynamic contexts.

dist tial and t

	ine Worksh hop on Mine eral Resourc ces Potentia
		emporal	distributi ion of por rphyry co
	ve compiled a a list of porp phyry copper r deposits alo ong the weste
	data extract ted from 1) t the ProMine e Mineral De eposits datab
	sus" Mineral l Deposits da atabase of the e BRGM (un npublished) a
	ld" database	of the USGS S (Singer et	al., 2008). I It contains 11
	explicitly co ontain copper r. These depo osits are dist tributed alon
	°E and range	in age from	4.0 (Zanclea an) to 143.5	Ma (Berriasi
	ve studied	the spatial	and tempor ral distribut tion of thes
	is not rando om. On the c contrary, it s shows conce entrations of
	n Tethyan sut ture and duri ing distinct t time periods s. Plotting eit
	and their geo ographic dis stribution wi ith age-based d symbology
	nto five disti inct spatial an nd temporal	"clusters" (F Figure 1):
	"Older" dep posits of the C Caucasus are ea (Armenia a, Azerbaijan
	Balkan-Carp pathian area	deposits (Bu ulgaria, Serb bia, Romania
	(29 deposits s);
	Eastern Tur rkey-Caucasu us area depo osits (Georgi a, Armenia,
	(11 deposits s);
	Aegean-Bal lkan-Carpath hian area dep posits (Aeg gean Sea, Gr
	Slovakia) -	Oligocene a and Miocene	(47 deposits s);
	Middle-East t area deposi its (Iran, Afg ghanistan, Pa akistan) -Mio
	e 1 -Spatial a and tempora l distribution n of porphyry y copper dep
			suture in f five distinct	"clusters".
	tribution, a	paleogeogr raphic appro oach

Tatsumi Y., & Eggins S. (1995). Subduction zone magmatism. Cambridge, MA, Blackwell Science, 211 p.


Jean-Michel Courtault
email: courtault.jeanmichel@univ-paris13.fr.

Riccardo Magnani
email: riccardo.magnani@univ-paris13.fr.

How much can European governments squeeze out of their taxpayers?

Keywords: Distributable surplus, public decit, sovereign debt, debt sustainability, CGE models. JEL classication: D61, H61, H62, H63, E62, C68

 and Luenberger (1992), to evaluate the capacity of European countries to repay their debts. In our analysis, we use Computable General Equilibrium (CGE) models to simulate dierent policies that can be implemented to achieve debt sustainability. We rst evaluate the quantity of distributable surplus that can be extracted from policies aiming at increasing the quantity of labor and/or capital available in the economy. We show that the results are very sensitive whether we consider decits before and after the recent nancial and economic crises. Then, assuming that governments are able to capture all the distributable surpluses, we compute the date at which they are able to repay their debts. In particular, we nd that most EU countries, excepted Germany and to lesser extent France and the UK, cannot achieve debt sustainability. We nally discuss the usefulness of Eurobonds.

Introduction

Over the years European governments have built huge public debts. With the recent economic crisis the decits have reached unprecedented levels as it seemed necessary to support the economy with budgetary measures (consider for example, the help to the French automobile industry). Table 1 shows the level of public decits and debts in 2005, 2009 and 2010, i.e. before and after the nancial and economic crises. We can see that decits have worsen considerably for most countries except Germany and in particular for Greece, Spain, Ireland and the UK. Consequently the public debt has dramatically increased in this period. The question arises as to whether European governments will be able to repay their debt or whether they will have to resort to inationary measures, or explicit default. Governments need the cooperation of taxpayers to be able to levy enough taxes to repay their debts. However, if taxes are too high, people could modify their behavior in the labor market by reducing labor supply and in the capital market by transferring capital oshore, implying that the State will not be able to levy enough taxes. So, it is not possible as often advocated in the popular press, to simply take money from the relatively wealthy.

The idea of our paper is that only surpluses can be taxed away by the governments. Surplus can be loosely dened as the dierence between the maximum price a buyer is ready to pay and the minimum price a seller is ready to accept for any trade. In the absence of state intervention, the surplus is shared by the buyer and the seller according to their bargaining power. The government can potentially take all the surplus in the transaction but not more as the buyer and/or the seller would get less then their reservation price and would then withdraw from the trade. Because of its excessive greediness, the state would dry out the source of income on which it draws.

The surplus indicates the maximum amount that is taxable by the State. We will assume here that governments are able to tax all surpluses even if, in practice, the eective amount that can be extracted depends on the available scal tools as well as the information available to the State. The specic notion of surplus that is used in our paper is that of the distributable surplus proposed by [START_REF] Allais | A la recherche d'une discipline économique. 1e partie[END_REF][START_REF] Allais | La Théorie Générale des Surplus[END_REF] or, equivalently, of the benet function proposed by Luenberger (1992[START_REF] Luenberger | Microeconomic Theory[END_REF]. In particular, Allais and Luenberger dene surplus as the maximum quantity of a reference good that can be taken away from a consumer with a given level of utility.

Several methods permit the evaluation of distributable surplus. For example, methods based on contingent evaluation are particularly well suited for experimental microeconomic analysis. The method we follow in our paper is based on the use of Computable General Equilibrium (CGE) models as they seem more suitable for macroeconomic analysis. However, independently of the method used to compute the distributable surplus, it represents the upper limit of taxable output.

The aim of our paper is to use the concept of distributable surplus to evaluate how much surplus can be extracted from European economies in order to investigate whether European public debts are sustainable. Given that the distributable surplus represents the maximum taxable output that governments can extract from their respective taxpayers, we are able to compute the maximum level of debt that governments can aord and to determine the date at which governments are able to repay their debts. The result of our analysis is that most EU countries, excepted Germany and to lesser extent France and the UK, cannot achieve debt sustainability.

It is important to note that the methodology used in our paper diers from that of previous studies. Public debt sustainability has been empirically tested by assuming that past behavior of scal policies remains constant. [START_REF] Hamilton | On the Limitations of Government Borrowing: A Framework for Empirical Testing[END_REF] propose a framework for analyzing whether governments can run a Ponzi scheme or not and nd sustainability of US scal policy. A number of studies have tested the sustainability of public decits by analyzing the stationarity and the cointegration properties of total public expenditures and revenues as ratios of GDP. Concerning European countries, Santos [START_REF] Bravo | Intertemporal sustainability of scal policies: some tests for European countries[END_REF], assuming that cointegration of expenditures and revenues is a sucient condition for sustainability, nd sustainable scal policies in Germany, the UK, Austria, France and the Netherlands, but not in Belgium, Denmark, Ireland, Portugal, Italy and Finland. [START_REF] Greiner | Debt sustainability in the European Monetary Union: Theory and empirical evidence for selected countries[END_REF], nd that scal policies in some European countries are sustainable following the approach developed by [START_REF] Bohn | The Sustainability of Budget Decits in a Stochastic Economy[END_REF][START_REF] Bohn | The Behavior Of U.S. Public Debt And Decits[END_REF] implying that the intertemporal budget constraint of the government holds in the case in which the public debt to GDP ratio is a mean-reverting process.

The paper is organized as follows. In the following section we dene distributable surplus through the [START_REF] Allais | A la recherche d'une discipline économique. 1e partie[END_REF] and Luenberger (1992) benet function [START_REF] Boiteux | Le Revenu Distribuable et les Pertes Economiques[END_REF] distributable income function. Then we investigate whether European countries have generated sucient surplus during the period 2005-2009 to cover their public decits. In section 4, we analyze the sustainability of European public debts. We then present CGE models for seven European countries and evaluate, using dierent scal rules, whether various policies are able to generate sucient surplus. We conclude in the last section.

The distributable surplus

Let an economy be composed by two consumers j = 1, 2 and two goods i = 1, 2. They have the following utility function U j (x j ) = U j (x 1 j , x 2 j ) = i (x i j ) α i j with α 1 j + α 2 j = 1 and they are endowed with the bundle of goods ω j = {ω 1 j , ω 2 j }.

Let g ∈ 2 + be a reference bundle of goods, arbitrarily dened. Let u be a reference utility level which represents the minimum utility level that is acceptable by the individual. We can dene the distributable surplus relative to the reference utility u and the bundle of good x as :

b j (x j , u j ) = max β β s.t. u j (x j -βg) ≥ u j (1)
In words, the distributable surplus represents the maximum number of units of bundle g that the consumer j is ready to give up to obtain bundle x j when his initial utility level is u j . If g is a unit of gold, then b j (x j , u j ) can be interpreted as the maximum price that the agent will agree to pay in order to acquire x j knowing that he has a utility level of u j .

Hence, b j (x j , u j ) can be interpreted as the reservation price of x j for individual j as gold can be taken as the numéraire.

In the case of the above Cobb-Douglas utility function, for g = (1, 0), the distributable function is given by :

b j (x, u) = x 2 j - u j (x 1 j ) α 1 j 1 α 2 j
Alternatively, it is possible to use Boiteux' surplus function to evaluate total distributable surplus in the economy. [START_REF] Boiteux | Le Revenu Distribuable et les Pertes Economiques[END_REF] and [START_REF] Courtault | A note on Boiteux' surplus function and dual Pareto eciency[END_REF] present an analogue of the benet function in the dual space of price-income pairs, ranked with the agent's indirect utility functions v j . Agent j's Boiteux' surplus at utility level u j , relative to the price-income pair (p, R j ), is dened by:

d(p, R j , u j ) = min d d s.t. v j (p, R j + d) ≥ u j (2)
The Boiteux' distributable income function d(p, R j , u j ) measures the income that must be given to an individual to move from a reference utility level u j to an environment (p, R j ) which represents the new allocation. The Boiteux's distributable income function, being dened in terms of income, is more intuitive than the benet function which is dened in terms of an arbitrary bundle of goods. In the case of the Cobb-Douglas utility function, the distributable income function is given by :

d(p, R j , u j ) = R j -u j • i p i α i j α i j
In the Appendix we determine the distributable surplus in a 2x2 pure exchange economy.

In particular, we compute the equilibrium of this economy and we deduce the total maximal distributable surplus using both measures that can be extracted from this economy. This distributable surplus, that is also expressed as a percentage of GDP, is computed using as reference utility level for each consumer the utility of his initial endowment, as this represents the minimum utility level that an individual can achieve if he chooses not to trade.

CGE model for several European countries

In this section we use CGE models in order to evaluate the distributable surplus that European governments can extract from their taxpayers. The analysis is carried out by using CGE models for seven countries: France, Germany, Greece, Ireland, Italy, Spain and the United Kingdom.

It is important to note that computable general equilibrium models are built in such a way that observed situation represents the equilibrium of the economy. Hence, the initial allocation in CGE models is already Pareto-optimal as rst welfare theorem is always veried in the absence of externalities and market frictions. However, scal policies or other macroeconomic shocks can change the equilibrium of the economy. So, for each shock, we are able to compute the distributable surplus which can be generated from this shock. This will give us an idea as to which policy can be implemented to generate maximum possible surplus.

The structure of the model we used in our paper is fairly standard, along the lines of [START_REF] Devarajan | Policy Lessons from Trade Focused Two-Sector Models[END_REF] and [START_REF] Shoven | Applied General Equilibrium[END_REF]. CGE models are widely used to analyze the eects of macroeconomic shocks and policies in a coherent framework that takes into account the interrelations existing among economic agents (rms, households, government, and the rest of the world).

Description of the CGE models

In this paper, we use a CGE model for each of the seven countries considered. Each CGE model is multisectoral (we consider 16 sectors) and considers a representative household. The models are built by using 2005 input-ouput data provided by OECD and data concerning national accounts. In order to take into account the eects of the recent crisis on the possibility of surplus extraction, we use 2009 data from national accounts and we use the same structure for input-output.

In each sector, production depends on the quantity used of primary factors (labor and capital) and of intermediate goods. We use a two-stage CES production function where in the rst stage production depends on primary factors and total intermediate good, and in the second stage the total intermediate good depends on the intermediate goods produced by the other sectors. The production is sold in the domestic market or exported, where exports depend on the relative price, i.e. the ratio between the foreign price and the domestic price.

The production that is sold in the domestic market and the imports constitute a composite good that is sold in the market to the rms (as intermediate goods), the households, the government, or used as investment good. The production that is sold in the domestic market and the imports are assumed to be imperfect substitutes (Armington assumption).

Concerning the households, the disposable income is given by the dierence between the revenues (labor and capital incomes, transfers from the government and interests on the public debt) and direct taxation. An exogenous and constant fraction of the disposable income is saved and the complementary fraction is consumed. Households have CES preferences that allow them to determine the optimal quantity of goods demanded for each sector.

Concerning the budget constraint of the government, the dierence between the total expenditure (for goods demanded, transfers to households and interests on the public debt) and revenues (direct and indirect taxation) determines the government decit. We discuss the scal rule in the next section.

The equilibrium of the balance of payments is guaranteed by capital inows or outows that are endogenously determined by the net exports. This implies that the exchange rate is xed exogenously. Another possibility is to x the capital inows and determine the exchange rate endogenously in order to equilibrate the balance of payments. This is the case for the United Kingdom.

We use the neoclassical macro-closure implying that investments are determined by aggregate savings, i.e private and public savings and international capital ows.

All markets clear. For each sector, the domestic price adjusts in order to equalize the quantity produced and demanded (domestic and foreign). The real wage adjusts in order to equalize the total labor demanded by the sectors and the (exogenous) labor supplied by the households and the real capital remuneration adjusts in order to equalize the total capital demanded by the sectors and the (exogenous) capital supplied by the households. The models are solved by considering the producer price index as the numeraire.

Fiscal rules

The government budget constrain can be written as:

Def = G + Γ + r • B - i τ V A i • V A i + i τ prod i • Y i + τ Y • Y (3)
where G represents the total government expenditure for goods and services, Γ the transfers to families, r • B the interests paid on the public debt, τ V A i the VAT rate dierentiated by sector i, τ prod i the tax rate on products, τ Y the income tax rate. The government can set any of the following variables (except one): the decit, the total expenditure, transfers to households, the income tax rate, the VAT rates and the tax rates on products. We will not consider as instruments the transfers, the VAT rates and the tax rates on products and we consider to following ve scal rules.

In the rst scal rule, the government decit and the income tax rate are assumed to be exogenous and xed at the initial level (Def = Def 0 and τ Y = τ Y 0 ), and the total government expenditure (G) is endogenously determined in order to satisfy the budget constraint.

In the second scal rule, the government decit is assumed to be exogenous and xed at the initial level (Def = Def 0 ), the total government expenditure per worker is kept constant ( G L = α) and the income tax rate (τ Y ) is endogenously determined in order to satisfy the budget constraint.

In the third scal rule, the government decit is determined such that the ratio between the decit and GDP remains constant ( Def GDP = β), the income tax rate is assumed to be exogenous (τ Y = τ Y 0 ) and the total government expenditure (G) is endogenously determined in order to satisfy the budget constraint.

In the fourth scal rule, the government decit is determined such that the ratio between the decit and GDP remains constant ( Def GDP = β), the total government expenditure is kept constant (G = G 0 ) and the income tax rate (τ Y ) is endogenously determined in order to satisfy the budget constraint.

In the fth scal rule, the total government expenditure and the income tax rate are exogenous (G = G 0 and τ Y = τ Y 0 ) and the government decit (Def) is endogenously determined in order to satisfy the budget constraint.

Reference bundle

In our analysis we consider two reference bundles: in the rst case we consider the equilibrium consumption bundle of the representative household. In the second case, we will use instead the distributable income as the reference unit.

Two choices are possible: either we x the reservation utility at the 2005 utility level, or at 2009 utility level. In the rst case we are able to answer whether there was enough surplus in the economy to cover the 2009 decit, or whether it would have been necessary to cut spending. We will also be able to compute the reduction of well-being necessary to generate sucient surplus in the case in which diminution of spending is not politically feasible. In the second case, one considers implicitly that no reduction of well-being is acceptable by the community, hence we consider whether the government is able to generate surplus through the introduction of a policy aiming at increasing the quantity of labor and/or capital available in the economy.1 4 Have European economies generated sucient surpluses?

We examine now whether the economies have generated sucient surpluses in the period 2005-2009 to cover the public decits during the same period. Surpluses are sucient when the quantity of surplus generated by each economy is greater or equal than actual decit.

In order to compute the surplus it is necessary to x a reservation utility level dened as the minimum utility level that households will accept. We assume that the reference utility level is the level in 2005 i.e. before the crisis.

Table 2 shows that the surplus generated between 2005 and 2009 was not sucient to cover the 2009 decit for Greece, Ireland and Spain. For these countries in order to generate a surplus equal to the 2009 decit, it would be necessary to reduce the initial well-being (measured by the disposable income devoted to consumption) by 11% in Greece, 13% in Ireland and Spain, and 9% in the UK. Indeed to sustain such level of decit, Ireland for example, would have to accept a level of disposable income lower than 13% than the 2005 level, corresponding to the 2002 level. This table clearly shows that some countries (Ireland, Spain, the UK, and more particularly Greece) have increased their level of consumption between 2005 and 2009 even if they should have reduced it drastically. France has increased its consumption over and above the feasible level, while Italy has slightly decreased consumption as necessary. In contrast, Germany has increased consumption well below what they could have done.

Each government could have extracted more surplus each year instead of running decits. But they didn't. Hence, they have to repay past decits with current surplus. Here, we consider the case whether countries are able to repay past decits (cumulated in the period [2005][2006][2007][2008][2009] with the 2009 surplus. The only country that could have done that is Germany and could maintain the 2003 well-being level. All the other countries should accept a strong reduction in their well-being, comparable to their early 1990 level. 5 Simulation results of dierent macroeconomic shocks on distributable surpluses

In this section we consider the case in which people do not accept any reduction of their wellbeing and we analyze the policies that government could implement in order to generate sucient surplus. Indeed, we are entitled to consider that the reference utility level of agents is equal to the utility level that they achieved in the absence of the policies under consideration. In fact, the government is perfectly free to introduce or not these policies and, as a consequence, it can capture all the surpluses that can be generated by these policies.

The analysis is carried out by using the CGE models. The initial equilibrium used to calibrate our models gives us the reference utility level. Starting from this level, a macroeconomic shock or policy will move the economy out of the initial equilibrium and will allow us to compute the surplus generated by such a shock.

We consider three types of shocks under dierent scal rules: a shock on the labor supply and on the capital supply and on both. These shocks may be interpreted as government policies. An increase in the labor supply may be induced, for example, by an immigration policy, an increase in the retirement age, or an increase in the legal number of work hours per week. An increase in capital supply may be induced, for example, by policies stimulating foreign investments or the repatriation of national capitals invested abroad. Some of the scal rules considered in our simulations permits to consider the cost of these policies. For example, an immigration policy requires an increase in expenditures (health, education, infrastructures) which is explicitly taken into account by the second scal rule in which per capita public expenditure is kept constant.

In our analysis, we consider two distinct periods, before and after the 2009 economic crisis. This allows us to reach a more balanced view of the possibility to generate distributable surplus, since both periods of expansion and depression tend to give a biased evaluation of the eect of government policies.

Before the crisis

The objective of our simulations is to compute the increase in (i) labor supply, or (ii) capital supply, or (iii) both labor and capital supply, that is necessary to generate a surplus sucient to cover the public decit. In what follows, we dene the "eciency" of a production factor as the capacity of this factor to generate such a surplus. The higher is the necessary increase in a factor, the lower is the eciency of this factor.

The simulations are run only for the ve countries that display a decit (Greece, France, Germany, Italy, UK), and not for the two countries that have a surplus in 2005 (Ireland and Spain). For each country and the ve scal rules, we show in Table 3 the percentage variation of production factors necessary to generate a surplus equal to the budget decit.

First of all, the results shows that countries can be separated in two groups: France, Germany and UK on one side, and Greece and Italy on the other side. The dierence between the two groups is related to the share of labor incomes in total GDP that is quite low in the second group (lower that 50%). The two groups are also dierent in terms of public decits over GDP in 2005: countries in the rst group have decits close to the level xed by the Maastricht rule, while the countries in the second group have a much higher decit. By considering the rst scal rule, in which the government decit and the income tax rate are exogenous while the public expenditure is determined by the government budget constraint, we can see that, for countries in the rst group, it is necessary to increase capital supply by a much larger amount than labor supply while, for Greece and Italy, the dierence between the necessary increase in labor and capital supply is much lower. It is interesting to compare Germany and the UK as they have almost identical decit/GDP ratio. We can note that in order to generate a surplus equal to the decit, the UK has to increase labor supply by a smaller amount than Germany, but needs a greater increase in capital supply. Overall, the eciency is greater in the UK, in the sense that UK has to increase both factors by 5.1% vs. 5.5% for Germany. France needs to increase both factors by only 4.8% thanks to the lower decit/GDP ratio with respect to Germany and the UK. Interestingly, whereas Greece has a greater decit than Italy, it is necessary to increase labor and capital supply by a lower amount (6.7% versus 7.2% for Italy).

With the second scal rule the eciency of the increase in labor supply is much smaller for each country than in the rst scal rule. However, the necessary increase in capital supply is much smaller than in the rst scal rule. The results are explained by the fact that with this rule, per worker government expenditure is kept constant. This is justied by the fact that the increase in labor supply can be realized by an immigration policy. Consequently, an increase in labor supply produces an increase in public spending which has a negative inuence on the government nances. The overall eciency of the increase in both factors is lower than in the rst scal rule. Indeed, the rst scal rule is more overall ecient than any other scal rules. If we look at labor supply eciency, we can see that the most ecient scal rule is the rst one, whereas in terms of capital supply eciency, the most ecient scal rule is the fourth.

The results obtained with the third scal rule are very similar to the ones obtained with the rst scal rule, although the overall eciency of labor and capital supply is greater with respect to the rst scal rule.

After the crisis

After the crisis, the increase in only one production factor that would be necessary to generate a sucient surplus to cover the 2009 decit is too large to be considered as feasible. So, instead, we consider an increase in both factors. Table 4 show that only Germany appears to have maintained its capacity to generate enough surplus to cover the decit, whereas all the other countries have seen their situation drastically deteriorated. Indeed simple average across scal rules shows that for Germany an increase of 5.6% of both labor and capital is sucient to generate a surplus equal to the decit, whereas it was 5.9% before the crisis. For all the other countries the percentage shock over both labor and capital is much higher after the crisis, with a maximum average of 41.2% for Ireland. We can rank the countries by increasing order of diculty of getting out of the budget decit dilemma: Germany 5.6%, Italy with 9.9%, France with 15%, the UK with 22.7%, Spain with 23.5%, Greece with 24.6% and Ireland with 41.2%. The media have particularly stressed the diculty for Greece to repay its debt whereas its situation is not much worse than that of Spain and, surprisingly, the UK. The situation of Ireland seems catastrophic. This result is only partly explained by the value of the decits that are dierent in each country. The decit of Greece is more important that of Ireland. However the percentage increase in production factors to cover the decit is smaller.

The third scal rule seems to be the worst rule in order to generate a surplus sucient to cover the decit. For all countries, except Greece, the best scal rule is the fth, i.e. the case in which public decits are endogenous. It would seem that a Maastricht rule type is bad during crisis whereas it seems better to adopt a more permissive budgetary policy.

Equal decit

Here we make the assumption that the public decit, for each country, respects the Maastricht criterium, i.e. is equal to 3% of GDP. Table 5 shows that each European country is able to generate sucient surpluses. Surprisingly, Greece is now in a better position with respect to the other countries, while Ireland is again in the worst position. Except for the UK, where the best scal rule is now the second or the fourth, the ordering of the scal rules for the other countries has not changed. 6 Sustainability of public debts

In this section we compute for each country the year in which the actual public debt plus the present value of future public decits are compensated by the present value of future distributable surpluses. Public debt is dened as sustainable only in the case in which the date computed exists.

In our analysis we make the following assumptions: for each country, the ratio of public decit with respect to GDP decreases linearly over time towards zero between 2012 and 2020; real GDP and distributable surpluses grow at the constant rate of 1.5%.

In the computation of the present values, we consider four dierent interest rates. The rst one is the ten-year government bond rate observed on November 9 2011. 2 This interest rate is relevant in the case in which all the debt has to be renewed at that date. However, if the time to maturity of the debt is not immediate, this interest rate is not relevant. This is why, secondly, we consider the average ten-year government bond rate observed in 2011 and, thirdly, the current yield interest rate computed as the ratio between actual interests payments and the actual public debt. Finally, we consider the average ten-year government bond rate observed in the Euro zone weighted by the size of public debts. This rate is interesting because it could approximate the rate on Euro bonds in the case in which European governments decide to introduce this kind of instrument to nance the overall European debt.

Table 6 shows that, in any interest rate scenario, only France, Germany and the UK are able to repay the current public debt using future surpluses generated by their economies. This is precisely the case of the rst scenario in which all the other countries are not able to repay their debts within a nite horizon. If we consider the average 2011 interest bond rate (second scenario) debt is not sustainable for Greece and Ireland, while for Italy and Spain the date computed is so remote that we can have doubts on the sustainability of their debts. If we consider the current yield and the Euro bond scenarios (third and fourth scenarios) the debt is sustainable for all countries. However, the third scenario is not very realistic as nancial markets are not willing anymore to lend to most European countries on the basis of past interest rates at which the debt was contracted. It is interesting to note that in the Euro bond scenario the date computed is delayed, with respect to the second scenario, only for France and Germany by only two years. However, even this scenario is not very realistic since for Greece, Ireland, Italy and Spain the date is very remote implying a strong uncertainty concerning the validity of our hypotheses over such a long period. In this paper we use the concept of distributable surplus proposed by [START_REF] Allais | A la recherche d'une discipline économique. 1e partie[END_REF] and Luenberger (1992) to investigate about (i) the capacity of European governments to generate sucient surpluses to cover public decits and (ii) the sustainability of public debts.

After showing that European governments have not generated sucient surplus with their economic policies implemented over the recent period (2005)(2006)(2007)(2008)(2009), we investigate whether a policy aiming at increasing both labor and capital supply could be sucient to cover actual decits. Using CGE models for several European countries, we show that public decits observed before the crisis could be covered by a relatively small increase in labor and capital supply of about 5%. However, in order to cover the 2009 public decits would require for all countries, except Germany, a much larger increase in both production factors. Given that current public spending cannot be nanced by taxing the distributable surplus as the level of the increase in labor and capital necessary to achieve the budget equilibrium is not realistic, European governments will have to reduce their public spending, at the most, to the level observed before the crisis.

Concerning the sustainability of public debts in the Euro zone, that is necessary for the preservation of the Euro system, we found that the best solution is the introduction of Euro bonds. However, even this scenario is not very realistic since for Greece, Ireland, Italy and Spain the date at which the debt is completely repaid is so remote that it raises doubts on the eciency of this policy. In any case, this policy would imply the strict control of public expenditures of the dierent European countries by European institutions in order to maintain the debt under control.

Table 1 :

 1 Public decits and debts for several European countries Public deficit / GDP Public debt / GDP Public deficit / GDP Public debt / GDP Public deficit / GDP Public debt / GDP

		2005		2009		2010
	France	2.9	66.4	7.5	78.3	7.0	81.7
	Germany	3.3	68.0	3.0	73.5	3.3	83.2
	Greece	5.2	100.0	15.4	127.1	10.5	142.8
	Ireland	-1.6	27.4	14.3	65.6	32.4	96.2
	Italy	4.3	105.9	5.4	116.1	4.6	119.0
	Spain	-1.0	43.0	11.1	53.3	9.2	60.1
	UK	3.4	42.5	11.4	69.6	10.4	80.0
	Source: Eurostat

Table 2 :

 2 Analysis of the capacity of EU countries to cover public decits

	France	Germany	Greece	Ireland	Italy	Spain	Uk

Table 3 :

 3 Percentage increase in factor supply necessary to cover the 2005 decits

			1st	2nd	3rd	4th	5th	Average
		L	7.4%	10.9%	7.7%	10.9%	11.4%	9.7%
	France	K	16.6%	12.5%	18.2%	12.6%	12.2%	14.4%
		L and K	4.8%	5.4%	5.0%	5.4%	5.4%	5.2%
		L	9.3%	13.4%	9.8%	13.5%	14.5%	12.1%
	Germany	K	16.3%	12.8%	17.8%	13.0%	12.4%	14.5%
		L and K	5.5%	6.0%	5.9%	6.0%	6.1%	5.9%
		L	7.6%	9.0%	7.9%	9.0%	9.0%	8.5%
	UK	K	19.4%	15.7%	21.8%	15.7%	15.6%	17.6%
		L and K	5.1%	5.3%	5.4%	5.3%	5.3%	5.3%
		L	14.8%	22.9%	15.6%	23.0%	23.5%	20.0%
	Italy	K	16.9%	13.2%	19.3%	13.3%	13.1%	15.2%
		L and K	7.2%	7.5%	7.8%	7.5%	7.5%	7.5%
		L	14.8%	23.6%	15.5%	23.2%	21.7%	19.8%
	Greece	K	14.3%	12.2%	16.2%	12.0%	12.5%	13.4%
		L and K	6.7%	7.2%	7.3%	7.2%	7.2%	7.1%

Table 4 :

 4 Percentage increase in both factors necessary to cover the 2009 decits

		1st	2nd	3rd	4th	5th	Average
	Germany	6.2%	5.0%	6.6%	5.1%	4.9%	5.6%
	Italy	11.0%	8.7%	12.6%	8.8%	8.5%	9.9%
	France	15.4%	13.8%	18.3%	14.1%	13.6%	15.0%
	UK	23.5%	19.5%	31.3%	19.8%	19.2%	22.7%
	Spain	23.2%	20.9%	30.9%	22.0%	20.3%	23.5%
	Greece	21.3%	24.6%	27.6%	24.7%	24.7%	24.6%
	Ireland	36.8%	35.5%	58.9%	39.7%	35.0%	41.2%

Table 5 :

 5 Percentage increase in both factors necessary to cover a decit equal to 3% of GDP

		1st	2nd	3rd	4th	5th	Average
	France	5.8%	5.2%	6.2%	5.2%	5.1%	5.5%
	Spain	5.7%	5.0%	6.0%	5.1%	5.0%	5.4%
	Germany	6.1%	5.0%	6.5%	5.0%	4.9%	5.5%
	Ireland	6.7%	6.3%	7.2%	6.4%	6.3%	6.6%
	Italy	6.1%	4.8%	6.5%	4.8%	4.7%	5.4%
	Greece	3.5%	4.0%	3.7%	4.0%	3.9%	3.8%
	UK	5.7%	4.6%	6.0%	4.6%	4.7%	5.1%

Table 6 :

 6 Analysis of the sustainability of public debts

		November 09/2011 (1)	Average 2011 (2)	Current yield (3)	Euro average (4)
	Interest rate	3.16%	3.32%	2.76%	4.66%
	France
	Year	2026	2026	2025	2028
	Interest rate	1.81%	2.74%	3.52%	4.66%
	Germany
	Year	2021	2023	2023	2025
	Interest rate	30.69%	16.66%	4.20%	4.66%
	Greece
	Year	never	never	2049	2055
	Interest rate	8.24%	9.87%	2.90%	4.66%
	Ireland
	Year	never	never	2043	2061
	Interest rate	6.76%	5.14%	3.60%	4.66%
	Italy
	Year	never	2096	2049	2067
	Interest rate	5.71%	5.40%	2.80%	4.66%
	Spain
	Year	never	2136	2043	2064
	Interest rate	2.35%	3.13%	2.51%
	UK
	Year	2034	2037	2035
	Source: Ecowin (November 2011)
	(1) Bond yield on November 9 2011
	(2) Average bond yield in 2011
	(3) Interests / Public debt
	(4) Average bond yield in 2011 weighted by the size of public debts

This kind of policy is the only possible in order to increase GDP in a CGE model that is not demand- driven.

This date has been chosen in our analysis as it is one of the worst period in the history of the Euro zone in terms of high interest rates for most of highly indebted countries (Italy, Spain and Greece).

Appendix 1: An illustration with a 2x2 pure exchange economy Here, we present numerical simulations within a 2x2 pure exchange economy in which the economy is supposed to be composed by two consumers j = 1, 2 and two goods i = 1, 2. They have the following utility function U j (x j ) = U j (x 1 j , x 2 j ) = i (x i j ) α i j with α 1 j + α 2 j = 1 and they are endowed with the bundle of goods ω j = {ω 1 j , ω 2 j }. We also assume that there exists one unit of each type of good.

The following tables present, for a given initial distribution of endowments and for a specic reference bundle, the eect of varying elasticities of the Cobb-Douglas utility function with respect to each good and the eect of the initial distribution of endowments, on the value of the distributable surplus expressed as the percentage of the total equilibrium value of income.

The rst set of tables considers as reference bundle g = (0, 1). The second set of tables considers as reference bundle g = (1, 0). Firstly, we can see that the results are not aected by the choice of the reference bundle. Indeed, results are perfectly symmetrical when the reference bundle is g = (0, 1) instead of g = (1, 0). In addition, when we use income as the numéraire, the results are qualitatively similar. It is possible to note that when the elasticity with respect to one good is exactly equal to the initial endowment in that good for any agent, then the distributable surplus is nil however measured as the initial distribution is already Pareto-optimal (see for example Bewley, 2007, chapter 3). Moreover, the results show that the farther is the initial distribution from the Pareto-optimal allocation, the greater is the value of the distributable surplus that can be taxed by the government. Indeed, in the particular case where goods are equally distributed among consumers and preferences are identical, the distributable surplus is nil as the initial distribution is already Pareto-optimal, hence there is no incentives to trade.