Nicolas Daclin

Sihem Mallek Daclin

Towards a sustainable implementation of interoperability solutions: bridging the gap between interoperability requirements and solutions

Keywords: Enterprise interoperability engineering, verification, interoperability requirements, interoperability solutions. Which solution? Which requirement? Interoperability requirements repository Interoperability solutions repository

The main objective of this communication is to present and discuss the need, for partners, to suitable interoperability solution according to their expectations. First, the problematic of the selection of a solution is presented and the stakeholders' needs to tackle this statement are highlighted. Then, existing works related to interoperability requirements and interoperability solutions are briefly presented and discussed. Finally, criteria -and associated examples -that guide stakeholders in their selection are presented.

Introduction

The concept of interoperability [START_REF] Iso | Advanced automation technologies and their applications -Part 1: Framework for enterprise interoperability[END_REF] is now considered as a crucial issue and a key factor of success for such enterprises that share and exchange processes, services data, enterprise applications [START_REF] Iso | ISO 15745 -1, Industrial automation systems and integration -Open systems application integration framework -Part 1 : Generic reference description[END_REF]… in a collaborative context. Numerous works has been led to characterize [START_REF] Chen | Architectures for enterprise integration and interoperability: Past, present and future[END_REF], implement [START_REF] Daclin | Developing enterprise collaboration: a methodology to implement and improve interoperability[END_REF] [START_REF] Bourey | Report on Model Driven Interoperability[END_REF] and improve [START_REF]C4ISR Architecture Working Group (AWG). Levels of Information Systems Interoperability (LISI)[END_REF] [START_REF] Ford | Interoperability measurement[END_REF] this aspect of a partnership. Originally coming from the computer science and Information and Communication Technologies (ICT) fields [START_REF]IEEE Standard Computer Dictionary: A Compilation of IEEE Standard Computer Glossaries[END_REF] and, exclusively considered as a technical problem, interoperability has expanded, since, and considers now other less technical aspects. Also, in addition to technical aspects, interoperability can includes organisational aspects (for instance, who does what, how, when and who is responsible for what?...) as well as conceptual aspects (for instance, does the data to be exchanged use a data model?...) [START_REF] Wang | DIMP: An Interoperable Solution for Software Integration and Product Data Exchange[END_REF] [10] [START_REF] Panetto | Information Systems for Enterprise Integration, Interoperability and Networking: Theory and Applications[END_REF]. Similarly, if interoperability was seen only as an ability that enables and improves the sharing and the exchange of data, it now considers other fields such as the interoperability of processes (synchronization / coordination of activities, mutual adjustment [START_REF] Vallespir | Enterprise modelling for interoperability[END_REF]). Finally, as a simple matter of interfacing issue, the development of interoperability takes also an interest in other aspects of a collaboration that can affect either a given partner or the partnership itself. In that sense, it takes, for instance, an interest in the preservation of the autonomy of each partner involved into the collaboration [START_REF] Chen | Enterprise interoperability framework and knowledge corpus -final report[END_REF].

As we may have noticed, interoperability has become an important expected capability -in other word a requirement (especially a non-functional requirement -NFR [START_REF] Willis | Systems Engineering and the forgotten '-Illities[END_REF]) -for enterprises that want to establish and maintain an effective partnership in terms of exchange and sharing of their information, products and resources (material as human)… but also that want to align and, further, if requested, to orchestrate their process commonly or else, that want to work together at an higher level in term of decision or policies. However, with a better understanding and definition of interoperability, also appeared a greater complexity in its implementation, monitoring and control and improvement. Naturally, interoperability solutions -that cover the set of interoperability aspects -are developed to satisfy the needs of interoperability for partners that collaborate. In this context it is important to allow partners to select solutions that meet -at best -all their expectations in term of interoperability. Although the space of problems of interoperability is clearly identified on one hand, and the space of interoperability solutions is clearly defined on the other hand, the link between both is not clearly established and formalized. This statement may lead partners to select their interoperability solutions in a hazardous way and without knowing the potential impacts of the implementation of a given interoperability solution onto other aspects (e.g. other requirements, performance of the collaboration…) of the partnership. As a consequence the here presented research attempts to clarify the link between interoperability requirements and interoperability solutions and to enable partners to select solutions fully adapted to their needs and in a proper way.

This paper is structured as follow. After this brief introduction, the problem and expected results of this research are presented. The research works related to the fields of this research is given and discussed in section 3. Section 4 presents the foundations of the proposed reference model to define and select accurate interoperability solutions regarding the dimensions that has to consider. To illustrate the first proposition to implement and to use the proposed model, a case study is made available to the readers and shows it into the overall context of interoperability within a collaborative process. The final section presents the conclusions and the prospects of this research.

Problem and expected outcomes

In enterprise collaboration engineering context as in any other domains (system engineering, Information System, mechatronic…), it is difficult for stakeholders to have and to select a solution that satisfies all their expectations initially expressed. Indeed, it is not enough to have a jumbled and unconnected list of solutions if it is not consistently and precisely related with a set of requirements and if the potential impact that can be caused by solution onto its environment is unknown (Figure 1) and vice and versa. Thus, "there is no interest to identify the problems if there are no adapted solutions just as there is no interest to give solutions if the problems are not clearly identified".

Fig. 1. The selection of solution dilemma

First, for a given expectation, several solutions can be available. However, all relevant solutions are not necessarily known and, naturally, the stakeholders favor the solution they control without further investigation and at the risk that such solution covers partially the problem or, in the worst case, is not adapted to the problem in the end (in operational phase for instance). To face up this problem, all solutions have to be (1) fully and clearly identified in order to avoid to stakeholders to miss out on a relevant solution and (2), mirrored from the (or the set of) need(s) they allow to satisfy. That means to make available a relevant structure in which requirements and solutions are properly related and in agreements with the considered problem namely the ability to interoperate whether conceptually, technically or else, organizationally and during all the life cycle of the partnership.

Second, a given solution, and this independently of the covered field, is developed to satisfy a need. However, it is often forgotten that a solution can fully match an (or a set of) expectation but at the same time, this one can have an impact on other expectation and further on the system to be developed. In that case, the problem is not reduced to the choice of "the" solution but to choice of "the best" solution that means the solution which will satisfy the need for which it is developed but also the solution for which its impacts on the context will be fully identified and defined to allow stakeholders to choose it in confidence. Thus, beyond to have a set of solutions related to a set of requirements as mentioned before, it is also required to guide the stakeholders in their process of selection according to identified criteria that characterize the solution precisely (for instance in term of impact on other requirements).

Interoperability requirements and solutions: a quick scan

Numerous works deal with interoperability requirements for collaborative enterprises on the one hand and with the development of interoperability solutions -either conceptual technical or organizational -on the other hand. However, few works are concerned with the definition of the relation that can exist between the world of interoperability requirements and the world of interoperability solutions.

First, the world of interoperability requirements engineering focuses on two aspects. On the one hand, it is about to identify, to define precisely [START_REF] Mallek | The application of interoperability requirement specification and verification to collaborative processes in industry[END_REF] [16] and writing requirements [START_REF]OMG): Semantics of Business Vocabulary and Business Rules (SBVR) -version 1.0[END_REF] in order to allow their verification -by stakeholders -either by the way of automatized and formal techniques (e.g. model checker) or not (e.g. test, expertise…). On the other hand, it is about to classify the interoperability requirements to ensure their management and traceability [START_REF] Daclin | Capturing and structuring interoperability rquirements: a framework for interoperability requirements[END_REF]. This aspect is strongly based on existing categorizations that are proposed by interoperability frameworks (Figure 2). These two aspects are fundamentals since the principle of requirements engineering is that, the more the requirements are accurately identified the more a solution fully adapted can be found. However, if the consideration of the interoperability as requirement is accepted and widely studied, the connection with existing interoperability solutions is weak and their choice is often at the discretion of the stakeholders without any guidelines or support.

Second, some approaches make interoperability solutions available and rely strongly on the structuration of interoperability through interoperability frameworks. Let's mention the enterprise interoperability framework [START_REF] Chen | Enterprise interoperability framework and knowledge corpus -final report[END_REF] that proposes no less than 66 interoperability solutions sorted onto its three axes such as interoperability concerns, interoperability approaches and interoperability categories [START_REF] Chen | Enterprise interoperability framework and knowledge corpus -final report -Annex: Knowledge Pieces[END_REF]. However, the exploitation of this framework is difficult and does not allow stakeholders to know if a given solution is fully adapted to their interoperability needs since (1), they don't know the possible impact (and its evaluation) on the environment, (2), several solutions can be proposed for a given intersection between axes, (3), they don't know if a selected solution does not go beyond their effective needs (some solutions cover several categories and concerns that are not necessarily interesting for partners) and ( 4), these solutions are not formally related to requirements since those latter are not clearly and explicitly formulated i.e. following a structured and accepted format [START_REF]INCOSE. Guide for writing requirements, v1.0, Requirements WG-INCOSE[END_REF]. Thus, although a large number of solutions is proposed, this remains too general in terms of guide in the choice of solutions (not related to specific needs) and without precision this large sample can lead stakeholders to select "a" solution and not "their" solution (Figure 3). In the same vein, let's mention also [START_REF] Elvesaeter | State-of-the-Art and Baseline EI Services Specifications[END_REF] that proposes a state of the art on enterprise interoperability services (business, semantic, data…).

Name of solution, author

Interoperability concern

To Other approaches propose a set of solutions -according interoperability aspectsbut without necessarily be positioned precisely into a framework. In this case, let's mention [START_REF] Balzert | State of the art solutions in enterprise interoperability. Enterprise information systems for business integration in SMEs: Technological, Organizational and Social dimensions[END_REF] that proposes interoperability solutions (strategic interoperability, process interoperability, technical interoperability and semantic interoperability) for SMEs. This kind of approach grasps the necessity to consider a solution not only as a simple answer to a problem but to also to position it into a context allowing to consider other aspects that can be linked to this one. Thus, it examines and proposes solution according to interoperability such as technical, process-based and semantic but also taking into account characteristics -allowing to evaluate their relevance and usability for an SME (from high relevance to not applicable) -such as:  Startup-costs: it represents the cost to implement technology or standard. This cost has to be as inexpensive as possible for SMEs.  Running costs: SMEs are not able to afford high running costs for their interoperability solutions. This cost has to be, also, as inexpensive as possible for SMEs.  Implementation effort: usually SMEs do not possess IT-system technically advanced as well as they do not employ IT-experts. Thus interoperability solution has to be easy to implement that means without strong efforts and advanced skills.  Direct use for an SME interoperability solution: this aspect is concerned by the evaluation of the use of an interoperability solution that fulfills the needs of SMEs.

 Conceptual input for an SME interoperability solution: interoperability solutions are also analyzed in term of conceptual input for further researches on interoperability for SMEs. Thus, this approach does not stop to the simple proposition of a set of interoperability solutions but, because of the specifics of SMEs, it has evaluated a set of interoperability solutions (BFC, BPMN, SAP…) according to defined criteria and it has provided an analysis allowing to guide SMES in their choice of interoperability solutions.

From these considerations, it appears that interoperability requirements on the one hand and solutions for interoperability on the other hand are clearly identified, but the link between them is not obvious or, at least, fully weaved. However the underlying difficulty remains the same for stakeholders: to select the better solution for their needs. In this way, it is required (1) to link solutions and requirements and (2), to highlight criteria that guide stakeholders in their choice of interoperability solution when one of them is identified as relevant to meet expectations.

Reference model for interoperability solutions

The reference model for interoperability solutions is a structure that embeds parameters aiming to make a set of solutions available (this set is fully related to those proposed in the enterprise interoperability framework [START_REF] Chen | Enterprise interoperability framework and knowledge corpus -final report[END_REF]) according to a (or a set of) requirement considered and positioned in agreements with the dimensions of the enterprise interoperability requirements framework (interoperability viewsinteroperability lifecycle -interoperability problems) [START_REF] Daclin | Capturing and structuring interoperability rquirements: a framework for interoperability requirements[END_REF]. However, as mentioned hereinbefore, the simple relation "requirements-solutions" is not sufficient to define accurately if a solution is really a well candidate solution regarding to interoperability project between partners and more broadly the partnership itself. Indeed, although a given solution allows to meet a requirement satisfaction, its implementation can have collateral effects on several parameters. As a consequence these parameters have to be integral part in the choice of an interoperability solution. To this purpose, the proposed model includes four parameters such as: interoperability problems resolution, solution overview, other requirements impact, and granularity impact.

Interoperability problems resolution. This parameter allows stakeholders to know precisely which problem(s) the proposed solution can solve in term of interoperability. These highlighted solutions depend not only on the resolved problem but also on fields covered in enterprises and the phase of the partnership. Thus the solution is positioned in agreements with the dimension of the framework for interoperability requirements that means according to (1) the interoperability views that represent the enterprises' domains that can be impacted by a problem of interoperability (business, process, service and data / resources / material), (2) the interoperability lifecycle levels that represent the moment of the partnership at which interoperability is requested by partners (e.g. the beginning of the partnership involves solutions that come under interfacing and thus, of compatibility) and (3), the abstraction levels that represent the categories of interoperability that can be developed in enterprise and thereby the problems of interoperability (conceptual, organizational ant technical). It is to note that a given solution can cover several interoperability views or else, interoperability life cycle levels. For instance solution such as semantic annotation can be used for each levels and can be implemented at the beginning of the partnership to solve existing -and avoid future -problems of misunderstanding (of models, data…). Solution such as Business Process Modeling and associated workflow engine are suitable for the process level to solve organizational and technical problems and can be implemented at the beginning of the partnership in order to align processes and useful for interoperation phase by the execution of the common process.

Solution overview. This parameter takes into consideration the general information about the proposed solution. It makes available a quick overview of the solution and allows stakeholders to identify easily relevant solutions from those that are not adapted (e.g. solutions previously implemented without success). To this purpose, it gives information about the name of the solution (not limited to its acronym, source of misunderstanding), applications that are possibly existing and that support the solution (e.g. BPMN modeler software to support processes modeling and execution), average cost (or range) for implementation, average time (or range) to implement the solution and, material and human means requested to implement the solution.

Other requirements impact. A solution is implemented to satisfy one (or a set of, for a solution that covers several aspects of interoperability) interoperability requirement. However, if a given solution allows to improve interoperability it can also induce a degradation by impacting requirements that are related to its application domain. For instance, let's consider the requirement defined such as "if task requires aptitude then resource has aptitude" that has to be satisfy when resources (human, material, software) are shared by partners. This requirement affects the domain of task, resources and aptitude. The non-satisfaction of this requirement can lead, at best, to a bad achievement of the considered task and, at worst, to its non-achievement. A solution could be the replacement of the allocated resource by another one that has the requested aptitude or to train the current resource for the task. However, the choice of a solution (e.g. changing of resource) can impact the satisfaction of other requirement, in this case, those ones that take an interest in the realization of other tasks and the availability of the shared resources (e.g. "It is possible that task is starting and resource is available"). It is to note that the impact is not limited to other interoperability requirements but includes also other ones (functional as well as nonfunctional, e.g. performance requirements). For instance, the possible solution previously proposed can impact the duration of the realization of the task. As a consequence, when an interoperability solution is developed or proposed, it has to highlight which other requirements can be impacted.

Granularity impact. It represents the level of detail of the object impacted by interoperability. The more precise the identification of the object impacted by interoperability is, the more the solutions selected and implemented by partners will be fully adapted and efficient. Indeed, interoperability can impact a partnership (mission, objectives…) as well as a given partner (mission, objectives but also component, resources…). For instance, the requirement "partners have necessary authorization to access shared data" has an impact on the good unwinding of the activities of the partnership (loss of time to access the information required to perform an activity). On the other side, the requirement "function f, performed by resource r involved in partnership, is even though performed" has a direct impact on a partner (precisely the execution of a function). From these knowledge, partners can select / adapt interoperability solutions or, possibly, relax requirements. As a consequence, the interoperability requirements (as other ones) are a part of a whole and their impacts must not be considered solely as a local matter but also as an overall matter. Finally, an important aspect of the model is the knowledge of the impact of a solution onto other requirements and stakeholders. Beyond the simple knowledge of the impacted elements, it is important for partners to know in which measure they are impacted. To this purpose, it would be interesting to quantify it or, at least, to define a trend showing the importance of the impact in order to guide more precisely stakeholders in their choice. Moreover, regarding the choice, itself, it would be interesting to rely on existing tools used in Systems Engineering to select a candidate solution [START_REF] Couturier | Tracking the consequences of design decisions in mechatronic Systems Engineering[END_REF]. Indeed, a solution is characterized in the proposed model with different parameters (impact granularity, cost/time to implement…) and the use of these kind of tools allow to weight them and implement mechanisms that partially automatize the selection.

As a case study, the drug circuit process, available online at goo.gl/chclk7, shows a possible first implementation of the proposed model of interoperability solutions (partially since the proposed model has to be validated). This illustration focuses on the verification of interoperability requirements -written or selected by stakeholders but the paper does not consider this problematic -and the proposition of adapted solution according to the unsatisfied requirement, the potential interoperability problem as well as the proposed solution and its possible impact on other requirements if it is implemented.

Conclusion and prospects

In a collaborative context, having mechanisms allowing to choose fully adapted interoperability solutions is a serious benefits, for enterprises, in term of costs for implementation and time to launch work. Furthermore, beyond the simple interoperability capabilities, benefits have an impact on the functioning of the whole partnership. The paper has presented a first proposition of a model that links interoperability requirements and interoperability solutions and, further, attempts to define a set of parameters allowing to guide stakeholders finely and accurately in their selection of a solution.

The advantage of such model is to guide precisely stakeholders in their choice rather than to have a sample list leading, too often, to select solutions unsuitable for their requirements. Obviously, parameters can be added depending on the special features of requirements (special needs in term of interoperability) or on the special features of partnership. However, having a link between requirements and solutions as well as general parameters characterizing them can be used as a base of selection.

Finally, it is to note this model -after its validation -will have to be integrated and technically implemented with existing tool and approach allowing to manage the lifecycle of an interoperability requirement from its elicitation and writing to the selection of a an adapted solution to satisfy it and via its verification [START_REF] Mallek | The application of interoperability requirement specification and verification to collaborative processes in industry[END_REF].

Fig. 2 .

 2 Fig. 2. The Framework for enterprise interoperability requirements

Fig. 3 .

 3 Fig.3. Template for interoperability solution proposal[START_REF] Chen | Enterprise interoperability framework and knowledge corpus -final report -Annex: Knowledge Pieces[END_REF]

Fig. 4 .

 4 Fig. 4. Reference model for interoperability solution.


Elena L Del Mercato
email: elena.delmercato@univ-paris1.fr

Vincenzo Platino
email: vin-cenzo.platino@gmail.com

Keywords: JEL classification: C62, D51, D62 externalities, private ownership economy, competitive equilibrium, homotopy approach

We consider a general equilibrium model of a private ownership economy with consumption and production externalities. Utility functions and production technologies may be affected by the consumption and production activities of all other agents in the economy. We use homotopy techniques to show that the set of competitive equilibria is non-empty and compact. Fixing the externalities, the assumptions on utility functions and production technologies are standard in a differentiable framework. Competitive equilibria are written in terms of first-order conditions associated with agents' behavior and market clearing conditions, following the seminal paper of Smale (1974). The work of adapting the homotopy approach to economies with externalities on the production side is non trivial and it requires some ingenious adjustments, because the production technologies are not required to be convex with respect to the consumption and production activities of all agents.

Introduction

We consider a general equilibrium model of a private ownership economy with consumption and production externalities. We use differential techniques to show that the set of competitive equilibria is non-empty and compact.

The general equilibrium model of Arrow and Debreu has been extended to economies with externalities. Our model of externalities is based on the seminal models given in [START_REF] Arrow | General Competitive Analysis[END_REF], [START_REF] Laffont | Effets externes et théorie de l'équilibre général[END_REF], and [START_REF] Laffont | Fundamentals of Public Economics[END_REF], where individual preferences and production technologies are affected by the consumption and production activities of all other agents in the economy. Consumption externalities have been widely recognized in the literature on network externalities and other-regarding preferences. Polluting production activities are a classical example of production externalities induced on individual preferences and technological processes of other firms. Furthermore, ICT (Information and Communication Technologies) firms also are in a between-firms externality relationship, because of the influence of the production activities of one firm on another's productivity.

We consider a private ownership economy with a finite number of commodities, households and firms. Each household's preferences are represented by a utility function. Each firm's production technology is represented by a transformation function. 3 Utility and transformation functions may be affected by the consumption and production activities of all other agents. Each agent (household or firm) maximizes his goal by taking as given both the commodity prices and the choices of every other agent in the economy. The associated concept of competitive equilibrium is an equilibrium à la Nash, the resulting allocation being feasible with the initial resources of the economy. This notion is given, for instance, in [START_REF] Arrow | General Competitive Analysis[END_REF], [START_REF] Laffont | Fundamentals of Public Economics[END_REF], and it includes as a particular case the classical equilibrium definition without externalities.

Fixing the externalities, the assumptions on utility and transformation functions are standard in a differentiable framework. In particular, preferences and production technologies are convex. However, we do not require preferences and production technologies to be convex with respect to the externalities. Our main theorem (Theorem 8) states that the set of competitive equilibria with strictly positive consumptions and strictly positive prices is non-empty and compact. We prove Theorem 8 following the seminal work by [START_REF] Smale | Global analysis and economics. II A: Extension of a theorem of Debreu[END_REF], and more recent contributions by [START_REF] Villanacci | Existence and regularity of equilibria in a general equilibrium model with private provision of a public good[END_REF], del [START_REF] Del Mercato | Existence of competitive equilibria with externalities: A differential viewpoint[END_REF] and [START_REF] Bonnisseau | General consumption constraints and regular economies[END_REF]. That is, we use:

(1) Smale's approach.

(2) Homotopy techniques.

(3) The topological degree modulo 2. In Smale's approach, the equilibrium is written as a solution of an extended system of equations consisting of the first-order conditions associated with agents' behavior and the market clearing conditions. No externalities are taken into account in the seminal work of [START_REF] Smale | Global analysis and economics. II A: Extension of a theorem of Debreu[END_REF]. However, this approach is used in many different settings such as incomplete markets, public goods and externalities. In the presence of externalities, Smale's approach overcomes the following difficulty: the individual demand and supply depend on the individual demand and supply of all other agents which, in turn, depend on the individual demand and supply of all other agents, and so on. Therefore, it is problematic to define an aggregate demand and an aggregate supply which depend only on prices and initial endowments.

The homotopy idea is that any economy is connected by an arc to some other economy that has a unique regular equilibrium. Along this arc, equilibrium moves in a continuous way without sliding off the boundary. It does not imply that at the end of the arc there is still a unique regular equilibrium. But, it implies that at the end of the arc the set of equilibria associated with the given economy is non-empty and compact. This is a kind of standard technique used in many contributions in general equilibrium theory, see for instance [START_REF] Villanacci | Existence and regularity of equilibria in a general equilibrium model with private provision of a public good[END_REF], del [START_REF] Del Mercato | Existence of competitive equilibria with externalities: A differential viewpoint[END_REF], [START_REF] Mandel | An index formula for production economies with externalities[END_REF], [START_REF] Bonnisseau | General consumption constraints and regular economies[END_REF], [START_REF] Kung | Voluntary contributions to multiple public goods in a production economy with widespread externalities[END_REF], and [START_REF] Ericson | Fundamental non-convexities and externalities: A differentiable approach, The B.E[END_REF]. However, the work of adapting this technique to economies with externalities on the production side is non-trivial and it requires some ingenious adjustments, because the production sets are not required to be convex with respect to the consumption and production activities of all the agents (for more details, see Subsection 4.2). Furthermore, our proof covers the case in which the economy is a standard private ownership economy without externalities. 4The technique described above is based on the homotopy invariance of the topological degree. Our proof is based on the topological degree modulo 2. 5 The degree modulo 2 is simpler than the Brouwer degree which is based on the notion of oriented manifold. 6 The reader can find in [START_REF] Geanakoplos | Solving systems of simultaneous equations in economics[END_REF] a brief review of both degree theories. In Section 6, we recall the definition and fundamental properties of the degree modulo 2.

We now compare our contribution with previous works. del [START_REF] Del Mercato | Existence of competitive equilibria with externalities: A differential viewpoint[END_REF] and [START_REF] Balasko | Wealth concerns and equilibrium[END_REF] study pure exchange general equilibrium models with externalities. In [START_REF] Villanacci | Existence and regularity of equilibria in a general equilibrium model with private provision of a public good[END_REF], and [START_REF] Kung | Voluntary contributions to multiple public goods in a production economy with widespread externalities[END_REF], there are no externalities on the production side. In [START_REF] Ericson | Fundamental non-convexities and externalities: A differentiable approach, The B.E[END_REF], utility and transformation functions are also affected by the price system. In order to get their existence result, the authors need to perturb utility and transformation functions. 7 Our contribution highlights that, in the presence of consumption and production externalities, there is no need for perturbing utility and transformation functions to establish the existence of an equilibrium. Furthermore, the contribution of [START_REF] Ericson | Fundamental non-convexities and externalities: A differentiable approach, The B.E[END_REF] presents some technical problems. 8 The existence results of [START_REF] Arrow | General Competitive Analysis[END_REF], [START_REF] Laffont | Effets externes et théorie de l'équilibre général[END_REF], [START_REF] Bonnisseau | Regular economies with non-ordered preferences[END_REF][START_REF] Bonnisseau | Existence of marginal pricing equilibria in economies with externalities and non-convexities[END_REF][START_REF] Mandel | An index formula for production economies with externalities[END_REF] are more general than ours. In [START_REF] Arrow | General Competitive Analysis[END_REF], [START_REF] Laffont | Fundamentals of Public Economics[END_REF][START_REF] Laffont | Effets externes et théorie de l'équilibre général[END_REF][START_REF] Bonnisseau | Regular economies with non-ordered preferences[END_REF][START_REF] Bonnisseau | Existence of marginal pricing equilibria in economies with externalities and non-convexities[END_REF], the authors use fixed point arguments, while we use differential techniques. The approach in [START_REF] Mandel | An index formula for production economies with externalities[END_REF] differs from ours for two main reasons: the author uses the excess demand approach and the topological Brouwer degree. In order to use the excess demand approach, the author has to enlarge the commodity space treating the externalities as additional variables. Moreover, in [START_REF] Bonnisseau | Regular economies with non-ordered preferences[END_REF][START_REF] Bonnisseau | Existence of marginal pricing equilibria in economies with externalities and non-convexities[END_REF][START_REF] Mandel | An index formula for production economies with externalities[END_REF], production technologies are also non-convex with respect to the individual firm's choice. For that reason, their equilibrium notion involves the concept of pricing rule and their existence proofs consist of techniques more sophisticated than those we use. In our simpler context, we provide an existence proof simpler than the ones provided in [START_REF] Bonnisseau | Regular economies with non-ordered preferences[END_REF][START_REF] Bonnisseau | Existence of marginal pricing equilibria in economies with externalities and non-convexities[END_REF][START_REF] Mandel | An index formula for production economies with externalities[END_REF].

The paper is organized as follows. In Section 2, we present the model and the assumptions. In Section 3, we provide the notions of competitive equilibrium and equilibrium function. The equilibrium function is built on the first-order conditions associated with households and firms maximization problems, and market clearing conditions. In Section 4, we first present our main result (Theorem 8) which states that the set of equilibria is non-empty and compact. Second, we provide the homotopy theorem (Theorem 9) used to prove Theorem 8. In order to apply Theorem 9, in Subsection 4.1 we build an appropriate private ownership economy that has a unique regular equilibrium. In Subsection 4.2, we provide our homotopy and its properties. All the lemmas are proved in Section 5. Finally, in Section 6, the reader can find a brief review on the topological degree modulo 2.

The model and the assumptions

There is a finite number C of physical commodities labeled by the superscript c ∈ C := {1, . . . , C}. The commodity space is R C . There are a finite number J of firms labeled by the subscript j ∈ J := {1, . . . , J} and a finite number H of households labeled by the subscript h ∈ H := {1, . . . , H}. Each firm is owned by households and it is characterized by a technology described by a transformation function. Each household is characterized by preferences described by a utility function, the shares on firms' profits and an endowment of commodities. Utility and transformation functions may be affected by the consumption and production activities of all other agents. The notations are summarized below.

• y j := (y 1 j , .., y c j , .., y C j ) ∈ R C is the production plan of firm j, as usual if y c j > 0 then commodity c is produced as an output, if y j < 0 then commodity is used as an input, y -j := (y f ) f =j denotes the production plan of firms other than j, y := (y j ) j∈J .

• x h := (x 1 h , .., x c h , .., x C h ) ∈ R C ++ denotes household h's consumption, x -h := (x k ) k =h denotes the consumption of households other than h, x := (x h ) h∈H .
• The production set of firm j is described by a transformation function. This is a convenient way to represent a production set using an inequality on a function called the transformation function, see for instance [START_REF] Mas-Colell | Microeconomic Theory[END_REF] and [START_REF] Villanacci | Differential Topology and General Equilibrium with Complete and Incomplete Markets[END_REF]. In the case of a single-output technology, the production set is commonly described by a production function. The transformation function is the counterpart of the production function in the case of production processes which involve several outputs.

The main innovation of this paper comes from the dependency of the transformation function t j on the production and consumption activities of all other agents. That is, t j describes both the technology of firm j and the way in which firm j's technology is affected by the activities of the other agents. More precisely, for any given (y -j , x), the production set of the firm j is given by the following set, Y j (y -j , x) := y j ∈ R C : t j (y j , y -j , x) ≤ 0 where the transformation function t j is a function from

R C × R C(J-1) × R CH ++ to R, t := (t j ) j∈J .
• Household h has preferences described by a utility function, • E := ((u, e, s), t) is a private ownership economy with externalities.

u h : (x h , x -h , y) ∈ R C ++ × R C(H-1) + × R CJ -→ u h (x h , x -h , y) ∈ R u h (x h , x -h ,
• p c is the price of one unit of commodity c, p := (p 1 , .., p c , .., p C ) ∈ R C ++ . • Given w = (w 1 , .., w c , .., w C ) ∈ R C , we denote w \ := (w 1 , .., w c , .., w C-1 ) ∈ R C-1 .

We make the following assumptions on the transformation functions.

Assumption 1 For all j ∈ J ,

(1) The function t j is continuous on its domain. For every (y -j , x) ∈ R C(J-1) × R CH ++ , the function t j (•, y -j , x) is differentiable and

D y j t j (•, •, •) is conti- nuous on R CJ × R CH ++ . (2) For every (y -j , x) ∈ R C(J-1) × R CH ++ , t j (0, y -j , x) = 0. (3) For every (y -j , x) ∈ R C(J-1) × R CH
++ , the function t j (•, y -j , x) is differentiably strictly quasi-convex, i.e., it is a C 2 function and for all y j ∈ R C , D 2 y j t j (y j , y -j , x) is positive definite on Ker D y j t j (y j , y -j , x).9 (4) For every (y -j , x) ∈ R C(J-1) ×R CH ++ , D y j t j (y j , y -j , x) 0 for all y j ∈ R C .

Fixing the externalities, the assumptions on t j are standard in a differentiable framework. From Points 1 and 4 of Assumption 1, the production set is a C 1 manifold with boundary of dimension C and its boundary is a C 1 manifold of dimension C -1. Point 2 of Assumption 1 states that inaction is possible. Consequently, using standard arguments from profit maximization, the individual wealth of household h derived from his endowment e h ∈ R C ++ and his profit shares is strictly positive for every price p ∈ R C ++ , from which one deduces the non-emptiness of the interior of the individual budget constraint. From Point 3 of Assumption 1, the production set is strictly convex. Furthermore, if the profit maximization problem has a solution then it is unique, because the function t j (•, y -j , x) is continuous and strictly quasi-convex. We remark that t j is not required to be quasi-convex with respect to all the variables, so we do not require the production set to be convex with respect to the externalities. From Point 4 of Assumption 1, the function t j (•, y -j , x) is strictly increasing and so the production set satisfies the classical "free disposal" property.

Remark 2 Our analysis holds true if some commodities are not involved in the technological process of firm j. In this case, for every firm j, one defines the set C j of all the commodities c ∈ C that are involved in the technological process of firm j, where C j denotes the cardinality of the set C j with 2 ≤ C j ≤ C. The production plan of firm j is then defined as y j := (y c j ) c∈C j ∈ R C j and the transformation function

t j is a function from R C j × f =j R C f × R CH ++ to R.
In this case, all the assumptions on the transformation functions are written just replacing

R C × R C(J-1) × R CH ++ by R C j × f =j R C f × R CH ++ .
Furthermore, in the definition of a competitive equilibrium, one also adapts the market clearing condition for every commodity c by considering only the firms that use commodity c in their technological process. That is, for every commodity c, the sum over j ∈ J is replaced by the sum over j ∈ J (c) := {j ∈ J : c ∈ C j }.

For any given externality (x, y) ∈ R CH ++ × R CJ , define the set of all the production allocations that are consistent with the externality (x, y), that is

Y (x, y) := {y ∈ R CJ : t j (y j , y -j , x) ≤ 0, ∀ j ∈ J } (1)
We remind that r = h∈H e h and we make the following assumption which is analogous to Assumption UB (Uniform Boundedness) in [START_REF] Bonnisseau | Existence of marginal pricing equilibria in economies with externalities and non-convexities[END_REF], and Assumption P(3) in [START_REF] Mandel | An index formula for production economies with externalities[END_REF].

Assumption 3 (Uniform Boundedness) There exists a bounded set C(r) ⊆ R CJ such that for every

(x, y) ∈ R CH ++ × R CJ , Y (x, y) ∩ {y ∈ R CJ : j∈J y j + r 0} ⊆ C(r)
The following lemma is an immediate consequence of Assumption 3.

Lemma 4 There exists a bounded set

K(r) ⊆ R CH ++ × R CJ such that for every (x, y) ∈ R CH ++ × R CJ , the following set is included in K(r). A(x, y; r) := {(x , y ) ∈ R CH ++ × R CJ : y ∈ Y (x, y) and h∈H x h - j∈J y j ≤ r}
It is well known that the boundedness of the set of feasible allocations is a crucial condition for the non-emptiness and the compactness of the set of equilibria. Fixing the externalities, from Assumption 3 one deduces that the set of feasible allocations is bounded. Therefore, in this sense, Assumption 3 is standard. Furthermore, Assumption 3 also guarantees that the set of feasible allocations A(x, y; r) is uniformly bounded with respect to the externalities (x, y). In particular, it implies that the set

F(r) = {(x, y) ∈ R CH ++ × R CJ | t j (y j , y -j , x) ≤ 0, ∀ j ∈ J and h∈H x h - j∈J y j ≤ r} of
feasible allocations that are mutually consistent is bounded. However, for a proof of the existence of an equilibrium it would not be sufficient to assume only the boundedness of the set F(r). 10 We use Lemma 4 to prove that the homotopy is compact once externalities move along the homotopy arc, see Step 2.1 in the proof of Proposition 15 in Section 5.

We make the following assumptions on the utility functions.

Assumption 5 For all h ∈ H,

(1) The function u h is continuous on its domain. For every

(x -h , y) ∈ R C(H-1) ++ × R CJ , the function u h (•, x -h , y) is differentiable and D x h u h (•, •, •) is con- tinuous on R CH ++ × R CJ . (2) For every (x -h , y) ∈ R C(H-1) ++ × R CJ , the function u h (•, x -h , y) is diffe- rentiably strictly increasing, i.e., D x h u h (x h , x -h , y) 0 for all x h ∈ R C ++ . (3) For every (x -h , y) ∈ R C(H-1) ++ × R CJ , the function u h (•, x -h , y) it is dif- ferentiably strictly quasi-concave, i.e., it is C 2 and for all x h ∈ R C ++ , D 2 x h u h (x h , x -h , y) is negative definite on Ker D x h u h (x h , x -h , y). (4) For every (x -h , y) ∈ R C(H-1) + × R CJ and for every u ∈ Im u h (•, x -h , y), cl R C {x h ∈ R C ++ : u h (x h , x -h , y) ≥ u} ⊆ R C

++

Fixing the externalities, the assumptions on u h are standard in a differentiable framework. From Point 3 of Assumption 5, the upper contour sets are strictly convex. Consequently, if the utility maximization problem has a solution then it is unique. Notice that u h is not required to be quasi-concave with respect to all the variables, that is, we do not require preferences to be convex with respect to the externalities. Point 4 of Assumption 5 is the classical Boundary Condition (BC), it means that the closure of the upper counter sets is included in R C ++ . We notice that in Point 1 and Point 4 of Assumption 5, we allow for consumption externalities x -h on the boundary of the set R C(H-1) ++ in order to look at the limit of the behavior of u h with respect to the consumption externalities. It means that BC is still valid whenever consumption externalities converge to zero for some commodities. This property is used in Step 2.2 of the proof of Proposition 15 in Section 5. 11 If u h does not satisfy BC whenever consumption externalities converge to zero for some commodities, we provide below an alternative assumption on u h from which one still gets Step 2.2 in the proof of Proposition 15, while maintaining Point 1 and Point 4 of Assumption 5 only for consumption externalities in R C(H-1) ++

. 12 (5) There exists δ > 0 such that for every

(x -h , y) ∈ R C(H-1) ++ × R CJ and for every (x h , x h ) ∈ R 2C ++ , if u h (x h , x -h , y) > u h (x h , x -h , y), then u h (x h , x -h + δ1, y) ≥ u h (x h , x -h + δ1, y) where 1 := (1, . . . , 1) ∈ R C ++ . 13
3 Competitive equilibrium and equilibrium function

In this section, we provide the definition of competitive equilibrium à la Nash and the notion of equilibrium function.

Without loss of generality, commodity C is the "numeraire good". Then, given p \ ∈ R C-1 ++ with innocuous abuse of notation, we denote p :

= (p \ , 1) ∈ R C ++ . Definition 6 (Competitive equilibrium) (x * , y * , p * \ ) ∈ R CH ++ ×R CJ ×R C-1
++ is a competitive equilibrium for the economy E if for all j ∈ J , y * j solves the following problem

max y j ∈R C p * • y j subject to t j (y j , y * -j , x * ) ≤ 0
(2)

11 A simple example of utility function that satisfies this property is given by any additively separable function

u h (x h , x -h ) = u h (x h ) + v h (x -h
) where u h is defined on R C ++ and satisfies the classical BC, and v h is defined on R C(H-1) +

. 12 For example, consider two commodities and the utility function

u h (x 1 h , x 2 h , x 1 k ) := x 1 h x 2 h x 1 k where (x 1 h , x 2 h ) ∈ R 2 ++ and x 1 k ∈ R + .
This function does not satisfy BC whenever the externality x 1 k converges to zero, but it satisfies Point 5. 13 A simple example of utility function that satisfies Point 5 is given by any multiplicatively separable function

u h (x h , x -h ) = u h (x h )m h (x -h ) where u h is defined on R C ++ and m h (x -h ) > 0 for every x -h ∈ R C(H-1) ++ .
for all h ∈ H, x * h solves the following problem

max x h ∈R C ++ u h (x h , x * -h , y * ) subject to p * • x h ≤ p * • (e h + j∈J s jh y * j ) (3)
and (x * , y * ) satisfies market clearing conditions, that is

h∈H x * h = h∈H e h + j∈J y * j (4)
Using first-order conditions, one easily characterizes the solutions of firms and households maximization problems. The proof of the following proposition is standard, because in problems ( 2) and ( 3) each agent takes as given both the price system and the choices of the other agents.

Proposition 7

(1) From Assumption 1, if y * j is a solution to problem (2), then it is unique and it is completely characterized by KKT conditions. 14 (2) From Point 2 of Assumption 1 and Assumption 5, there exists a unique solution x * h to problem (3) and it is completely characterized by KKT conditions.

(3) As usual, from Point 2 of Assumption 5, household h's budget constraint holds with an equality. Thus, at equilibrium, due to the Walras law, the market clearing condition for commodity C is "redundant". Then, one replaces condition (4) with

h∈H x * \ h = h∈H e \ h + j∈J y * \ j . Let Ξ := (R C ++ × R ++ ) H × (R C × R ++ ) J × R C-1
++ be the set of endogenous variables with generic element ξ := (x, λ, y, α, p \ ) := ((x h , λ h ) h∈H , (y j , α j ) j∈J , p \ ) where λ h denotes the Lagrange multiplier associated with household h's budget constraint, and α j denotes the Lagrange multiplier associated with firm j's technological constraint. We can now describe the competitive equilibria associated with the economy E using the equilibrium function

F : Ξ → R dim Ξ , F (ξ) := ((F h.1 (ξ) , F h.2 (ξ)) h∈H , (F j.1 (ξ) , F j.2 (ξ)) j∈J , F M (ξ)) (5)
where

F h.1 (ξ) := D x h u h (x h , x -h , y) -λ h p, F h.2 (ξ) := -p • (x h -e h - j∈J s jh y j ),
F j.1 (ξ) := p -α j D y j t j (y j , y -j , x), F j.2 (ξ) := -t j (y j , y -j , x), and

F M (ξ) := h∈H x \ h - j∈J y \ j - h∈H e \ h .
14 "KKT conditions" means Karush-Kuhn-Tucker conditions.

10 Documents de travail du Centre d'Economie de la Sorbonne -2015.34R (Version révisée)

ξ * = (x * , λ * , y * , α * , p * \ ) ∈ Ξ is an extended equilibrium for the economy E if and only if F (ξ * ) = 0. By Proposition 7, (x * , y * , p * \ ) is a competitive equilibrium for E if and only if there exists (λ * , α * ) such that ξ * is an extended equilibrium for E. We simply call ξ * an equilibrium.

Existence and compactness

In this section, we show that the set of competitive equilibria with strictly positive consumptions and strictly positive prices is compact and non-empty.

The result is provided in the following theorem.

Theorem 8 The equilibrium set F -1 (0) is compact and non-empty.

In order to prove Theorem 8, we use a homotopy approach following the seminal paper by [START_REF] Smale | Global analysis and economics. II A: Extension of a theorem of Debreu[END_REF]. The following theorem is a consequence of the homotopy invariance of the topological degree. Following Chapter 4 in [START_REF] Milnor | Topology from the Differentiable Viewpoint[END_REF], and more recent contributions by [START_REF] Villanacci | Existence and regularity of equilibria in a general equilibrium model with private provision of a public good[END_REF], del Mercato ( 2006) and [START_REF] Bonnisseau | General consumption constraints and regular economies[END_REF], our homotopy approach is based on the theory of degree modulo 2. The reader can find a survey of the degree theory, for example, in [START_REF] Geanakoplos | Solving systems of simultaneous equations in economics[END_REF] and in [START_REF] Villanacci | Differential Topology and General Equilibrium with Complete and Incomplete Markets[END_REF]. In Section 6, we recall the definition and fundamental properties of the degree modulo 2.

Theorem 9 (Homotopy Theorem) Let M and N be C 2 manifolds of the same dimension contained in euclidean spaces. Let y ∈ N and f, g : M → N be two functions such that f is continuous, g is C 1 , y is a regular value of g and #g -1 (y) is odd. Let L be a continuous homotopy from g to f such that

L -1 (y) is compact. Then, (1) g -1 (y) is compact and deg 2 (g, y) = 1, (2) f -1 (y) is compact and deg 2 (f, y) = 1.
The equilibrium function F plays the role of function f given in Theorem 9.

In order to construct the function playing the role of function g, we proceed as follows. In Subsection 4.1, we construct an appropriate economy called the "test economy". G is the equilibrium function associated with the test economy and it plays the role of function g in Theorem 9. The test economy has an unique regular equilibrium, i.e., #G -1 (0) = 1 and 0 is a regular value of G, see Proposition 12. In Subsection 4.2, we provide the homotopy H from G to F playing the role of the homotopy L. Proposition 15 states the compactness of H -1 (0). Using Propositions 12 and 15, all the assumptions of Theorem 9 are satisfied, and then one gets the following lemma.

Lemma 10 F -1 (0) is compact and deg 2 (F, 0) = 1.

Using Lemma 10 and the non-triviality property of the topological degree, one gets F -1 (0) = ∅, and then Theorem 8 is completely proved.

Finally, we remark that if, in addition, E is a regular economy (i.e., the equilibrium function F is C 1 and 0 is a regular value of F ), then using Lemma 10 and the computation of the degree modulo 2, one obviously obtains that, at a regular economy, the number of equilibria is finite and odd.15 However, this paper does not address any regularity issue. In the presence of consumption and production externalities, the analysis of regular economies is quite sensitive. For the case without externalities on the production side, see for example [START_REF] Bonnisseau | Regular economies with non-ordered preferences[END_REF], [START_REF] Kung | Voluntary contributions to multiple public goods in a production economy with widespread externalities[END_REF], and Bonnisseau and del Mercato (2010). In the presence of production externalities, the analysis of regular economies deserves a separate analysis, see del Mercato and Platino (forthcoming 2017).

The "test economy" and its properties

We construct the test economy in two steps. We first fix the externalities and we consider a Pareto optimal allocation of a standard production economy E without externalities. Second, using the Second Theorem of Welfare Economics, we construct an appropriate private ownership economy E that has a unique regular equilibrium. E is called the "test economy" and it is an economy without externalities.

Fix an allocation (x, y) ∈ R CH ++ × R CJ . Define u h (x h ) := u h (x h , x -h , y) for all h ∈ H, t j (y j ) := t j (y j , y -j , x) for all j ∈ J , and the production economy E := (u, t, r) where r = h∈H e h . Under Assumptions 1, 3 and 5, there exist a Pareto optimal allocation ( x, y) of the economy E and Lagrange multipliers ( θ, γ, β) such that ( x, y, θ, γ, β) is completely characterized by the first-order conditions for Pareto optimality. This result is provided in the following proposition.

Proposition 11 There exist a Pareto optimal allocation ( x, y

) ∈ R CH ++ × R CJ of the economy E and ( β, θ, γ) = (( β j ) j∈J , ( θ h ) h =1 , γ) ∈ R J ++ × R H-1 ++ × R C ++ such that ( x, y, β, θ, γ)
is the unique solution to the following system.

       (1) D x 1 u 1 (x 1 ) = γ (2) ∀ h = 1, θ h D x h u h (x h ) = γ (3) ∀ h = 1, u h (x h ) = u h ( x h ) (4) ∀ j ∈ J , γ = β j D y j t j (y j ) (5) ∀ j ∈ J , -t j (y j ) = 0 (6) h∈H x h - j∈J y j = r (6)
It is well known that a Pareto optimal allocation can be supported by some price system. 16 From system (6), one easily deduces a supporting price p, a redistribution of initial endowments e = ( e h ) h∈H and the equilibrium equations satisfied by ( x, y) for appropriate Lagrange multipliers ( λ, α) ∈ R H ++ × R J ++ . More precisely, first define

e h := x h - j∈J s jh y j (7)
and the economy E := ((u, e, s), t) which is a private ownership economy without externalities. Second, define the function

G : Ξ → R dim Ξ , G (ξ) := ((G h.1 (ξ) , G h.2 (ξ)) h∈H , (G j.1 (ξ) , G j.2 (ξ)) j∈J , G M (ξ)) (8)
where

G h.1 (ξ) := D x h u h (x h , x -h , y)-λ h p, G h.2 (ξ) := -p•(x h -e h - j∈J s jh y j ), G j.1 (ξ) := p -α j D y j t j (y j , y -j , x), G j.2 (ξ) := -t j (y j , y -j , x) and G M (ξ) := h∈H x \ h - j∈J y \ j - h∈H e \ h .
The function G is nothing else than the equilibrium function associated with the economy E. Finally, define ξ := ( x, λ, y, α, p \ ) with p :

= γ γ C , λ 1 := γ C , λ h := γ C θ h
for all h = 1 and α j := β j γ C for all j ∈ J . Using system (6), it is an easy matter to check that G( ξ) = 0. As stated in the following proposition, ξ is the unique regular equilibrium of the economy E.

Proposition 12 G -1 (0) = { ξ}, G is C 1 and 0 is a regular value for G.

Remark 13 Notice that the redistribution of endowments e = ( e h ) h∈H given in ( 7) is not necessarily strictly positive. However, at equilibrium, the individual wealth of household h is equal to p • x h which is strictly positive. One might wish to consider a different redistribution that gives rise to positive endowments. But, such a redistribution involves also some redistributions of the shares. 17 Our redistribution of endowments e = ( e h ) h∈H does not involve any redistribution of shares, so that we do not need to homotopize the shares, see the homotopy H in the next subsection. 18 16 Using Debreu's vocabulary, the Pareto optimal allocation ( x, y) is an equilibrium relative to some price system, see Section 6.4 in [START_REF] Debreu | Theory of Value[END_REF]. 17 For example, the redistribution e h := s jh h∈H e h with s jh :=

p• x h p• h∈H x h .
18 Even in the absence of externalities, homotopizing the shares complicates the homotopy proof, see for instance Section 2.2 in Chapter 9 of [START_REF] Villanacci | Differential Topology and General Equilibrium with Complete and Incomplete Markets[END_REF].

13 Documents de travail du Centre d'Economie de la Sorbonne -2015.34R (Version révisée)

The homotopy and its properties

The basic idea is to homotopize the endowments and the externalities by a segment in the two economies E and E. But, one finds the following difficulty. At equilibrium, the individual wealth is positive at the beginning as well as at the end of the homotopy arc. 19 Nevertheless, since the production sets are not required to be convex with respect to the choices of all agents, the equilibrium individual wealth may not be positive along the homotopy arc. Consequently, the individual budget constraints may be empty along the homotopy arc. We illustrate the details below.

First, homotopize the endowments by a segment. Then, for every τ ∈ [0, 1] the individual wealth is given by p

• [τ e h + (1 -τ ) e h ] + p • j∈J s jh y j which is equal to p • [τ e h + (1 -τ ) x h ] + p • j∈J s jh [y j -(1 -τ ) y j ]
Thus, the individual wealth is positive if p • y j ≥ p • (1 -τ ) y j for every j ∈ J . Using standard arguments from profit maximization, at equilibrium, this condition is satisfied if (1 -τ ) y j belongs to the production set of firm j. On the other hand if, at the same time, one homotopizes the externalities by a segment, the individual production set becomes the following set,

Y j (τ y -j + (1 -τ )y -j , τ x + (1 -τ )x)
But, one does not know whether or not the production plan (1-τ ) y j belongs to the production set given above, unless the transformation function t j is quasiconvex with respect to all the variables. Indeed, t j (0, y -j , x) = 0 because of the possibility of inaction, and t j ( y j , y -j , x) = 0 because G( ξ) = 0. If t j is quasi-convex with respect to all the variables, then t j ((1 -τ ) y j , τ y -j + (1τ )y -j , τ x + (1 -τ )x) ≤ 0, and so (1 -τ ) y j belongs to the production set given above. But, t j is not required to be quasi-convex with respect to all the variables.

In order to overcome the difficulty described above, we define below the homotopy H from the function G to the equilibrium function F in two times, by using the two homotopies Φ and Γ defined below. Namely,

• in the first homotopy Φ, we homotopize the endowments by a segment, without homotopizing the externalities,

• in the second homotopy Γ, we homotopize the externalities by a segment in utility and transformation functions, without homotopizing the endowments.

Remark 14 Notice that,

(1) If the externalities are fixed, then only one homotopy is needed, namely the homotopy Φ. Then, our homotopy proof covers the case in which the economy E is a standard private ownership economy without externalities.

(2) If the production sets are convex with respect to the production and consumption activities of all agents, then endowments and externalities can be homotopized at the same time.

We now provide the definition of all our homotopies. First, define the following convex combinations,

e h (τ ) := τ e h + (1 -τ ) e h , x(τ ) := τ x + (1 -τ )x, y(τ ) := τ y + (1 -τ )y (9)
and the two following homotopies Φ, Γ :

Ξ × [0, 1] → R dim Ξ , Φ (ξ, τ ) := ((Φ h.1 (ξ, τ ) , Φ h.2 (ξ, τ )) h∈H , (Φ j.1 (ξ, τ ) , Φ j.2 (ξ, τ )) j∈J , Φ M (ξ, τ )) (10) where Φ h.1 (ξ, τ ) = D x h u h (x h , y -j , x) -λ h p, Φ h.2 (ξ, τ ) = -p • [x h -e h (τ ) - j∈J
s jh y j ], Φ j.1 (ξ, τ ) = p -α j D y j t j (y j , y -j , x), Φ j.2 (ξ, τ ) = -t j (y j , y -j , x),

Φ M (ξ, τ ) = h∈H x \ h - j∈J y \ j - h∈H e h (τ ) \ . Γ (ξ, τ ) := ((Γ h.1 (ξ, τ ) , Γ h.2 (ξ, τ )) h∈H , (Γ j.2 (ξ, τ ) , Γ j.2 (ξ, τ )) j∈J , Γ M (ξ, τ )) (11) where Γ h.1 (ξ, τ ) = D x h u h (x h , x -h (τ ), y(τ )) -λ h p, Γ h.2 (ξ, τ ) = -p • [x h -e h - j∈J s jh y j ], Γ j.1 (ξ, τ ) = p-α j D y j t j (y j , y -j (τ ), x(τ )), Γ j.2 (ξ, τ ) = -t j (y j , y -j (τ ), x(τ )), Γ M (ξ, τ ) = h∈H x \ h - j∈J y \ j - h∈H e \ h .
Define then the homotopy

H : Ξ × [0, 1] → R dim Ξ , H(ξ, ψ) :=      Φ(ξ, 2ψ) if 0 ≤ ψ ≤ 1 2 Γ(ξ, 2ψ -1) if 1 2 ≤ ψ ≤ 1
The homotopy H is continuous since Φ and Γ are composed by continuous functions. Importantly, H ξ, 1 2 is well defined since Φ(ξ, 1) = Γ(ξ, 0). Furthermore, H (ξ, 0) = Φ (ξ, 0) = G(ξ) and H (ξ, 1) = Γ (ξ, 1) = F (ξ). We conclude the section by stating the following proposition.

Proposition 15 H -1 (0) is compact.

Proofs

Proof of Lemma 4. Let (x , y ) ∈ A(x, y; r). Since h∈H x h 0, y belongs to the bounded set C(r) given by Assumption 3. Furthermore, for every h ∈ H, 0 x h h∈H x h ≤ j∈J y j + r. Thus, there exists a bounded set K(r) ⊆

R CH ++ × R CJ such that for every (x, y) ∈ R CH ++ × R CJ , A(x, y; r) ⊆ K(r).
Proof of Proposition 11. Let E be the production economy defined in Section 4.1. We remind that A(x, y; r)

:= {(x , y ) ∈ R CH ++ × R CJ : t j (y j ) ≤ 0, ∀j ∈ J and h∈H x h - j∈J y j ≤ r}. Consider the set U (r) := {(u h ) h∈H ∈ h∈H Im u h | ∃(x , y ) ∈ A(x, y; r) : u h (x h ) ≥ u h , ∀ h ∈ H}. By Point 2 of
Assumption 1, the set U r is non-empty. Fix (u h ) h∈H ∈ U (r) and consider the maximization problem max

(x,y)∈R CH ++ ×R CJ u 1 (x 1 ) subject to                t j (y j ) ≤ 0, ∀j ∈ J u h (x h ) ≥ u h , ∀h ∈ H h∈H x h - j∈J y j ≤ r (12)
Step 1. Problem ( 12) has at least a solution. Let K be the set determined by the constraints of problem (12). K is non-empty since (u h ) h∈H ∈ U r . We claim that K is compact. Define the set N := {(x, y) ∈ R CH ++ × R CJ : u h (x h ) ≥ u h , ∀h ∈ H} and notice that K = N ∩ A(x, y; r). Then, K is bounded by Lemma 4. Furthermore, K is a closed set included in R CH ++ × R CJ by Point 4 of Assumption 5 and the continuity of the functions u h and t j . The theorem of Weierstrass implies that problem (12) has at least a solution.

Step 2. Let ( x, y) be a solution to problem (12). Then, ( x, y) solves the fol-lowing problem and it is a Pareto optimal allocation of the economy E.

max (x,y)∈R CH ++ ×R CJ u 1 (x 1 ) subject to                -t j (y j ) ≥ 0, ∀j ∈ J u h (x h ) -u h ( x h ) ≥ 0, ∀h = 1 r - h∈H x h + j∈J y j ≥ 0 (13)
Let K be the set determined by the constraints of problem ( 13), ( x, y) obviously belongs to K. Consider now (x, y) ∈ K and remind that K is the set determined by the constraints of problem ( 12). If u 1 (x 1 ) ≥ u 1 , then (x, y) ∈ K and so

u 1 ( x 1 ) ≥ u 1 (x 1 ). If u 1 (x 1 ) < u 1 , then u 1 ( x 1 ) > u 1 (x 1 ) since u 1 ( x 1 ) ≥ u 1 .
Thus, ( x, y) solves problem ( 13). Now, suppose by contradiction that ( x, y) is not a Pareto optimal allocation of E. Then, there is another allocation ( x, y) ∈ R CH ++ × R CJ such that t j ( y j ) ≤ 0 for all j,

h∈H x h ≤ r + j∈J y j , u h ( x h ) ≥ u h ( x h ) for all h, and u k ( x k ) > u k ( x k ) for some k ∈ H. If k = 1,
then we get a contradiction since ( x, y) solves problem (13). If k = 1, using the continuity of u k , there exists ε > 0 such that u k ( x k -ε1 c ) > u k ( x k ), where the vector 1 c ∈ R C + has all the components equal to 0 except the component c which is equal to 1. Consider the allocation (x, y) defined by x 1 := x 1 + ε1 c , x k := x k -ε1 c , x h := x h for all h = 1, h = k, and y j := y j for all j. We have that (x, y) ∈ K and u 1 (x 1 ) > u 1 ( x 1 ) since u 1 is strictly increasing. Then, once again we get a contradiction since ( x, y) solves problem (13).

Step 3. There exists ( β, θ, γ) := ((

β j ) j∈J , ( θ h ) h =1 , γ) ∈ R J ++ × R H-1 ++ × R C
++ such that ( x, y, β, θ, γ) is the unique solution to system (6).

We first prove the existence of ( β, θ, γ), afterwards we show the uniqueness of ( x, y, β, θ, γ).

Existence of ( β, θ, γ) 0. By Step 2, ( x, y) solves problem (13). The KKT conditions associated with problem (13) are given by

       D x 1 u 1 (x 1 ) = γ, ∀h = 1 : θ h D x h u h (x h ) = γ and θ h (u h (x h ) -u h ( x h )) = 0, ∀j ∈ J : γ = β j D y j t j (y j ) and β j (-t j (y j )) = 0, ∀c ∈ C : γ c (r c - h∈H x c h + j∈J y c j ) = 0 (14) where (β, θ, γ) := ((β j ) j∈J , (θ h ) h =1 , (γ c ) c∈C ) ∈ R J + × R H-1 + × R C
+ are the Lagrange multipliers associated with the constraint functions of problem (13). We first claim that KKT conditions are necessary conditions to solve problem (13). It is enough to verify that the Jacobian matrix associated with the constraint functions of problem (13) has full row rank equal to N := J + (H -1) + C.

For every firm j, fix a commodity c(j) ∈ C. The matrix given below is the N × N square sub-matrix which is obtained by considering the partial derivatives of the constraint functions with respect to ((y c(j) j ) j∈J , (x 1 h ) h =1 , x 1 ). Point 4 of Assumption 1 and Point 2 of Assumption 5 imply that the determinant of this square sub-matrix is different from zero, which complete the proof of the claim.20

             -D y c(1) 1
t 1 (y 1 ) . . .

. . . D x 1 H u H (x H ) 0 1 c(1) T . . . 1 c(J) T -1 1 T . . . -1 1 T -I C             
Therefore, there exists ( β, θ, γ) ≥ 0 such that ( x, y, β, θ, γ) solves system (14). Furthermore, Point 4 of Assumption 1 and Point 2 of Assumption 5 imply that all the Lagrange multipliers ( β, θ, γ) must be strictly positive. Consequently, all the constraints of problem ( 13) are binding, and then ( x, y, β, θ, γ) is a solution to system (6).

Uniqueness of ( x, y, β, θ, γ). Define θ 1 := 1, by equations ( 1) and ( 2) of system ( 6), for all h one gets

D x h u h ( x h ) = γ θ h
. Then, for every h, x h solves the maximization problem: max

x h ∈R C ++ u h (x h ) subject to γ θ h • x h ≤ γ θ h • x h because
KKT are sufficient conditions to solve this problem. Thus, the uniqueness of x h follows from the strict quasi-concavity of u h . Analogously, by equations ( 4) and (5) of system (6), y j solves the maximization problem: max

y j ∈R C γ β j
• y j subject to t j (y j ) ≤ 0 for every j. Thus, the uniqueness of y j follows from the continuity and the strict quasi-convexity of t j . Therefore, ( x, y) is unique and consequently, the uniqueness of ( β, θ, γ) follows from equations (1), ( 2) and (4) of system (6).

Proof of Proposition 12. We use the functions u h and t j defined in Subsection 4.1. We have already pointed out that G( ξ) = 0. Let ξ = (x , λ , y , α , p \ ) ∈ Ξ be such that G(ξ ) = 0, we show that ξ = ξ .

First, notice that

h∈H x h - j∈J y j = h∈H e h (15)
Indeed, summing G h.2 (ξ ) = 0 over h, one gets

h∈H x h - j∈J y j = h∈H e h by G M (ξ ) = 0.
Using the definition of e h given in (7) and Proposition 11, one deduces (15).

Second, we show that

u h (x h ) = u h ( x h ), ∀ h ∈ H (16)
Using the definition of e h given in ( 7) and G h.1 (ξ ) = G h.2 (ξ ) = 0, x h solves the following maximization problem max

x h ∈R C ++ u h (x h ) subject to p • x h ≤ p • x h + j∈J s jh p • (y j -y j ) (17)
because KKT are sufficient conditions to solve this problem. Analogously, from G j.1 (ξ ) = G j.2 (ξ ) = 0, y j solves the maximization problem: max y j ∈R C p •y j subject to t j (y j ) ≤ 0. Notice that y j satisfies the constraint of this problem because G j.2 ( ξ) = 0. Thus, p • (y j -y j ) ≥ 0 for all j, and consequently x h belongs to the budget constraint of problem (17). Then, u h (x h ) ≥ u h ( x h ) for all h. Now, suppose that u k (x k ) > u k ( x k ) for some k ∈ H. From (15) and G j.2 (ξ ) = 0 for all j, one deduces that (x , y ) is a feasible allocation of the production economy E, and then one gets a contradiction since ( x, y) is a Pareto optimal allocation of E by Proposition 11. Thus, ( 16) is completely proved. Now, define β := (β j ) j∈J where β j := λ 1 α j for all j, θ := (θ h ) h =1 where θ h := λ 1 λ h for all h = 1 and γ := λ 1 p . From G h.1 (ξ ) = 0 for all h, G j.1 (ξ ) = G j.2 (ξ ) = 0 for all j, (15) and ( 16), it is an easy matter to check that (x , y , β , θ , γ ) solves system (6). Then, Proposition 11 implies that ( x, y, β, θ, γ) = (x , y , β , θ , γ ), and consequently, one deduces that ξ = ξ .

We remark that G is C 1 by Point 1 of Assumptions 1 and 5. Finally, in order to show that 0 is a regular value for G, one proves that D ξ G( ξ) has full row rank. In this regard, we show that if ∆D ξ G( ξ) = 0, then ∆ = 0 where ∆ := (∆x h , ∆λ h ) h∈H , (∆y j , ∆α j ) j∈J , ∆p \ ∈ R dim Ξ . The system ∆D ξ G( ξ) = 0 is 19 Documents de travail du Centre de la Sorbonne -2015.34R (Version révisée)

given below.

                               (h.1) ∆x h D 2 x h u h ( x h ) -∆λ h p + ∆p \ [I C-1 |0] = 0, ∀ h ∈ H (h.2) -∆x h • p = 0, ∀ h ∈ H (j.1) h∈H ∆λ h s jh p -α j ∆y j D 2 y j t j ( y j ) -∆α j D y j t j ( y j ) -∆p \ [I C-1 |0] = 0, ∀ j ∈ J (j.2) -∆y j • D y j t j ( y j ) = 0, ∀ j ∈ J (M ) - h∈H λ h ∆x \ h + j∈J ∆y \ j = 0
We first prove that ∆x h = 0 for all h ∈ H. Otherwise, suppose that there is h ∈ H such that ∆x h = 0. The proof goes through the two following claims that contradict each others.

We first claim that ∆p

λ h ∆x h D 2 x h u h ( x h )(∆x h ) = -∆p \ • ( h∈H λ h ∆x \ h
). Multiplying G h.1 ( ξ) = 0 by ∆x h and using (h.2), we get ∆x h • D x h u h ( x h ) = 0 for all h. Therefore, Point 3 of Assumption 5 completes the proof of the claim since λ h > 0 for all h and ∆x h = 0.

Second, we claim that ∆p \ • ( h∈H λ h ∆x \ h ) ≤ 0. Multiplying both sides of G j.1 ( ξ) = 0 by ∆y j and using (j.2), we get ∆y j • p = 0. Then, multiplying (j.1) by ∆y j and summing over j, from (j.2) we get -j∈J α j ∆y j D 2 y j t j ( y j )(∆y j ) = ∆p \ • j∈J ∆y \ j . Since, α j > 0 for all j, Point 3 of Assumption 1 and (j.2) imply that ∆p \ • j∈J ∆y \ j ≤ 0. Using (M ), the claim is completely proved.

Since p C = 1 and ∆x h = 0 for all h ∈ H, from (h.1) we get ∆λ h = 0 for all h, and then ∆p \ = 0. Thus, multiplying (j.1) by ∆y j , Point 3 of Assumption 1 and (j.2) imply that ∆y j = 0. Therefore, using once again (j.1), we get ∆α j = 0 by Point 4 of Assumption 1. Thus, we get ∆ = 0.

Proof of Proposition 15. Observe that H -1 (0) = Φ -1 (0) ∪ Γ -1 (0). Since the union of a finite number of compact sets is compact, it is enough to show that Φ -1 (0) and Γ -1 (0) are compact.

Claim 1. Φ -1 (0) is compact.

We prove that, up to a subsequence, every sequence (ξ ν , τ ν ) ν∈N ⊆ Φ -1 (0)

converges to an element of Φ -1 (0), where ξ ν := (x ν , λ ν , y ν , α ν , p ν \ ) ν∈N . Since {τ ν : ν ∈ N} ⊆ [0, 1], up to a subsequence, (τ ν ) ν∈N converges to some τ * ∈ [0, 1]. From Steps 1.1, 1.2, 1.3 and 1.4 below, up to a subsequence, (ξ ν ) ν∈N converges to some ξ * := (x * , λ * , y * , α * , p * \ ) ∈ Ξ. Since Φ is continuous, taking the limit, one gets (ξ * , τ * ) ∈ Φ -1 (0).

We remind that for every τ ∈ [0, 1], e h (τ ) is given by (9).

Step 1.1. Up to a subsequence, (x ν , y ν ) ν∈N converges to some (x * , y * ) ∈ R CH + × R CJ . We show that for r = h∈H e h , the sequence (x ν , y ν ) ν∈N is included in the bounded set K(r) given by Lemma 4. By Φ j.2 (ξ ν , τ ν ) = 0, for every j we get t j (y ν j , y -j , x) = 0, ∀ ν ∈ N Thus, the sequence (y ν ) ν∈N is included in the set Y (x, y) given by (1). Summing Φ h.2 (ξ ν , τ ν ) = 0 over h, by Φ M (ξ ν , τ ν ) = 0 we have

h∈H x ν h - j∈J y ν j = h∈H e h (τ ν )
for every ν ∈ N. Using the definition of e h given by ( 7) and Proposition 11, one gets h∈H e h (τ ) = r for every τ ∈ [0, 1], and then

h∈H x ν h - j∈J y ν j = r for every ν ∈ N. Thus, (x ν , y ν ) ν∈N ⊆ A(x, y; r) ⊆ K(r). Consequently, (x ν , y ν ) ν∈N is included in cl K(r) which is a compact set. Therefore, up to a subsequence, (x ν , y ν ) ν∈N converges to some (x * , y * ) ∈ cl K(r) ⊆ R CH + × R CJ , and then (x * , y * ) ∈ R CH + × R CJ .
Step 1.2. The consumption allocation x * is strictly positive, i.e. x * h 0 for every h ∈ H. By Φ h.1 (ξ ν , τ ν ) = Φ h.2 (ξ ν , τ ν ) = 0 and KKT sufficient conditions, x ν h solves the following problem for every ν ∈ N.

max x h ∈R C ++ u h (x h , x -h , y) subject to p ν • x h ≤ p ν • [τ ν e h + (1 -τ ν ) x h ] + p ν • j∈J s jh (y ν j -(1 -τ ν ) y j )
(18) We first claim that for every ν ∈ N, the following vector

e h (τ ν ) := τ ν e h + (1 -τ ν ) x h (19)
belongs to the budget constraint of the problem above. By Φ j.1 (ξ ν , τ ν ) = Φ j.2 (ξ ν , τ ν ) = 0 and KKT sufficient conditions, y ν j solves the following problem for every ν ∈ N. max y j ∈R C p ν • y j subject to t j (y j , y -j , x) ≤ 0 (20) t j ( y j , y -j , x) = 0 since G j.2 ( ξ) = 0, see (8). By Point 2 of Assumption 1, t j (0, y -j , x) = 0. Then, we get t j ((1 -τ ν ) y j , y -j , x) < 0 since t j (•, y -j , x) is strictly quasi-convex, that is, the production plan (1 -τ ν ) y j belongs to the constraint set of problem (20). Thus, p ν • (y ν j -(1 -τ ν ) y j ) ≥ 0 for every j, and then p ν • j∈J s jh (y ν j -(1 -τ ν ) y j ) ≥ 0 which completes the proof of the claim.

Therefore, for every ν ∈ N, u h (x ν h , x -h , y) ≥ u h ( e h (τ ν ), x -h , y). By Point 2 of Assumption 5, for every ε > 0 we get u h (x ν h + ε1, x -h , y) > u h ( e h (τ ν ), x -h , y) where 1 := (1, . . . , 1) ∈ R C ++ . Taking the limit over ν and using Point 1 of Assumption 5, we get u h (x * h + ε1, x -h , y) ≥ u h ( e h (τ * ), x -h , y) for every ε > 0. Then, x * h belongs to the closure of the upper counter set of ( e h (τ * ), x -h , y), which is included in R C

++ by Point 4 of Assumption 5. Thus, x * h 0.

Step 1.3. Up to a subsequence, (λ ν , p ν \ ) ν∈N converges to some λ

* ∈ R H ++ × R C-1
++ . The proof is similar to the proof of Step 2.3.

Step 1.4. Up to a subsequence, (α ν ) ν∈N converges to some α * ∈ R J ++ . The proof is similar to the proof of Step 2.4.

Claim 2. Γ -1 (0) is compact.

Let (ξ ν , τ ν ) ν∈N be a sequences in Γ -1 (0). As in Claim 1, (τ ν ) ν∈N converges to τ * ∈ [0, 1]. From Steps 2.1, 2.2, 2.3 and 2.4 below, up to a subsequence, (ξ ν ) ν∈N converges to an element ξ * := (x * , λ * , y * , α * , p * \ ) ∈ Ξ. Since Γ is a continuous function, taking limit, we get (ξ * , τ * ) ∈ Γ -1 (0). We remind that for every τ ∈ [0, 1], x(τ ) and y(τ ) are given by (9).

Step 2.1. Up to a subsequence, (x ν , y ν ) ν∈N converges to some (x * , y * ) ∈ R CH + × R CJ . We show that for r = h∈H e h , the sequence (x ν , y ν ) ν∈N is included in the bounded set K(r) given by Lemma 4. Then, one completes the proof as in Step 1.1. By Γ j.2 (ξ ν , τ ν ) = 0, for every j we have that t j (y ν j , y ν -j (τ ν ), x ν (τ ν )) = 0, ∀ν ∈ N Thus, for every ν ∈ N, the production allocation y ν belongs to the set Y (x ν (τ ν ), y ν (τ ν ))

given by (1). Now, summing Γ h.2 (ξ ν , τ ν ) = 0 over h, by Γ M (ξ ν , τ ν ) = 0, we get h∈H x ν h -j∈J y ν j = r. Then, for every ν ∈ N, the allocation (x ν , y ν ) belongs to the set A(x ν (τ ν ), y ν (τ ν ); r) ⊆ K(r), and consequently, (x ν , y ν ) ν∈N ⊆ K(r).

Step 2.2. The consumption allocation x * is strictly positive, i.e. x * h 0 for every h ∈ H. The argument is similar to the one used in Step 1.2 except for the last part which is quite different due to the presence of consumption externalities in the utility functions. For this reason, at the end of this step, we need both the continuity of u h on R C ++ × R First, according to Γ h.1 (ξ ν , τ ν ) = Γ h.2 (ξ ν , τ ν ) = 0, replace problem (18) with the following problem max and replace e h (τ ν ) given by ( 19) with e h (τ ν ) := e h . Second, according to Γ j.1 (ξ ν , τ ν ) = Γ j.2 (ξ ν , τ ν ) = 0, replace problem (20) with the following problem max y j ∈R C p ν • y j subject to t j (y j , y ν -j (τ ν ), x ν (τ ν )) ≤ 0 Third, one follows the same strategy as for Step 1.2. For every ν ∈ N, u h (x ν h , x ν -h (τ ν ), y ν (τ ν )) ≥ u h (e h , x ν -h (τ ν ), y ν (τ ν )). Notice that x ν -h (τ ν ) belongs to R C(H-1) ++ , because x ν -h (τ ν ) = τ ν x ν -h + (1 -τ ν )x -h , and x ν -h and x -h are both strictly positive. Then, by Point 2 of Assumption 5, for every ν ∈ N and for every ε > 0 we get u h (x ν h + ε1, x ν -h (τ ν ), y ν (τ ν )) > u h (e h , x ν -h (τ ν ), y ν (τ ν )) (22)

Now, take the limit over ν. Differently from Step 1.2, we have the two following cases: x * -h (τ * ) 0 or x * -h (τ * ) ∈ Bd R C(H-1) ++

. The second case may happen here, but not in Step 1.2, because now there are consumption externalities in the utility functions. Indeed, if (τ ν ) ν∈N converges to τ * = 1, then x * -h (τ * ) = x * -h which a priori is not necessarily strictly positive.

Suppose that x * -h (τ * ) 0. Using ( 22) and Point 1 of Assumption 5, we get u h (x * h + ε1, x * -h (τ * ), y * (τ * )) ≥ u h (e h , x * -h (τ * ), y * (τ * )) for every ε > 0. Then, x * h is strictly positive, because it belongs to the closure of the upper counter set of (e h , x * -h (τ * ), y * (τ * )), which is included in R C ++ by Point 4 of Assumption 5. Suppose now that x * -h (τ * ) ∈ Bd R C(H-1) ++

. Under Point 1 and Point 4 of Assumption 5, the previous argument still holds true. Alternatively, if u h does not satisfy BC whenever consumption externalities converge to zero for some commodities, one uses Point 5 in Section 2 as follows. By Point 5 and ( 22), there exists δ > 0 such that for every ν ∈ N and for every ε > 0 one gets u h (x ν h + ε1, x ν -h (τ ν ) + δ1, y ν (τ ν )) ≥ u h (e h , x ν -h (τ ν ) + δ1, y ν (τ ν ))

Take the limit over ν and remark that x * -h (τ * ) + δ1 0. Consequently, one gets u h (x * h + ε1, x * -h (τ * ) + δ1, y * (τ * )) ≥ u h (e h , x * -h (τ * ) + δ1, y * (τ * )) for every ε > 0, because in this case u h is required to be continuous on R C ++ × R C(H-1) ++ × R CJ . Then, x * h is strictly positive since it belongs to the closure of the upper counter set of (e h , x * -h (τ * ) + δ1, y * (τ * )), which is included in R C ++ because x * -h (τ * ) + δ1 0 and because in this case Point 4 of Assumption 5 is required only for consumption externalities in R C(H-1) ++ .

Step 2.3. Up to a subsequence, (λ ν , p ν \ ) ν∈N converges to some (λ * , p * \ ) ∈ R H ++ × R C-1 ++ . By Γ h.1 (ξ ν , τ ν ) = 0, fixing commodity C, for every ν ∈ N we have λ ν h = D x C h u h (x ν h , x ν -h (τ ν ), y ν (τ ν )). Taking the limit, by Points 1 and 2 of Assumption 5, we get λ * h := D x C h u h (x * h , x * -h (τ * ), y * (τ * )) > 0.

By Γ h.1 (ξ ν , τ ν ) = 0, for all commodity c = C and for all ν ∈ N we have p ν c = D x c h u h (x ν h , x ν -h (τ ν ), y ν (τ ν )) λ ν h . Taking the limit, by Points 1 and 2 of Assumption 5, we get p * c := D x c h u h (x * h , x * -h (τ * ), y * (τ * )) λ * h > 0. Therefore, p * \ 0.

Step 2.4. Up to a subsequence, (α ν ) ν∈N converges to some α * ∈ R J ++ . For every firm j, fix a commodity c(j) ∈ C. By Γ j.1 (ξ ν , τ ν ) = 0, for every ν ∈ N we have that α ν j = p ν c(j) D y c(j) j t j (y ν j , y ν -j (τ ν ), x ν (τ ν ))

which is strictly positive by Point 4 of Assumption 1. Taking the limit, by

Points 1 and 4 of Assumption 1, we get α * j := p * c(j) D y c(j) j t j (y * j , y * -j (τ * ), x * (τ * ))

> 0.

Appendix

We introduce a definition of the degree modulo 2 for continuous functions, see Appendix B in [START_REF] Geanakoplos | Solving systems of simultaneous equations in economics[END_REF], and Chapter 7 in [START_REF] Villanacci | Differential Topology and General Equilibrium with Complete and Incomplete Markets[END_REF].

Let M and N be two C 2 manifolds of the same dimension contained in euclidean spaces. Let A be the set of triples (f, M, y) where

(1) f : M → N is a continuous function, (2) y ∈ N and f -1 (y) is compact.

Theorem 16 There exists a unique function, called degree modulo 2 and denoted by deg 2 : A → {0, 1} such that

C

  (H-1) + × R CJ and Point 4 of Assumption 5 for all consumption externalities in R C(H-1) + or, alternatively, the continuity of u h on R C ++ × R C(H-1) ++ × R CJ , Point 4 of Assumption 5 only for consumption externalities in R C(H-1) ++ , and Point 5 of Section 2.

  x h ∈R C ++ u h (x h , x ν -h (τ ν ), y ν (τ ν )) subject to p ν • x h ≤ p ν • e h + p ν •

  y) is the utility level of household h associated with (x h , x -h , y). That is, u h also describes the way in which household h's preferences are affected by the actions of the other agents. Denote u := (u h ) h∈H .• s jh ∈ [0, 1] is the share of firm j owned by household h; s h := (s jh ) j∈J ∈ [0, 1] J denotes the vector of the shares owned by household h; s := (s h ) h∈H ∈ [0, 1] JH . As usual,

	s jh = 1 for every firm j ∈ J .
	h∈H
	• e h := (e 1 h , .., e c h , .., e C h ) ∈ R C ++ denotes household h's endowment, e := (e h ) h∈H ∈
	R CH ++ and r :=

h∈H e h ∈ R C ++ .

The first version of this paper dates back to

and it has been available on the website: http://www.parisschoolofeconomics.eu/fr/del-mercato-elena/publications.

Documents de travail du Centre d'Economie de la Sorbonne -2015.34R (Version révisée)

In Chapter 9 of[START_REF] Villanacci | Differential Topology and General Equilibrium with Complete and Incomplete Markets[END_REF], the reader can find a homotopy proof for a standard private ownership economies without externalities. Our proof is simpler than the latter one, because we do not homotopize the shares.

See for instance, Chapter 4 in[START_REF] Milnor | Topology from the Differentiable Viewpoint[END_REF][START_REF] Villanacci | Differential Topology and General Equilibrium with Complete and Incomplete Markets[END_REF].

See for instance, Chapter 5 in[START_REF] Milnor | Topology from the Differentiable Viewpoint[END_REF][START_REF] Mas-Colell | The Theory of General Economic Equilibrium. A Differentiable Approach[END_REF].

See pages 53-55 in[START_REF] Ericson | Fundamental non-convexities and externalities: A differentiable approach, The B.E[END_REF].

Some basic assumptions for the existence of an equilibrium are missing, i.e., the possibility of inaction and the compactness of the set of feasible allocations, or any related assumptions. See for instance, Assumption 1(2) and Assumption 3 in our paper, or Assumption P(2) and Assumption UB in[START_REF] Bonnisseau | Existence of marginal pricing equilibria in economies with externalities and non-convexities[END_REF]. 4 Documents de travail du Centre d'Economie de la Sorbonne -2015.34R (Version révisée)

Let v and v be two vectors in R n , v • v denotes the scalar product of v and v . Let A be a real matrix with m rows and n columns, and B be a real matrix with n rows and l columns, AB denotes the matrix product of A and B. Without loss of generality, vectors are treated as row matrices and A denotes both the matrix and the following linear mapping A : v ∈ R n → A(v) := Av T ∈ R[m] where v T denotes the transpose of v and R[m] := {w T : w ∈ R m }. When m = 1, A(v) coincides with the scalar product A • v, treating A and v as vectors in R n .

See Section 2 in Chapter 6 of[START_REF] Arrow | General Competitive Analysis[END_REF],[START_REF] Bonnisseau | Existence of marginal pricing equilibria in economies with externalities and non-convexities[END_REF], and[START_REF] Mandel | An index formula for production economies with externalities[END_REF], where these authors also need uniform boundedness assumptions in order to prove the non-emptiness of the set of equilibria.

The computation of the degree modulo 2 for C 1 functions and regular values is given by Proposition 17 in Section 6.12 Documents de travail du Centre d'Economie de la Sorbonne -2015.34R (Version révisée)

Indeed, at the economy E, the equilibrium individual wealth is strictly positive because of the possibility of inaction (Point 2 of Assumption 1) and standard arguments from profit maximization.

We remind that for every commodity c, the vector 1 c ∈ R C + has all the components equal to 0 except the component c which is equal to 1. I C denotes the C ×C identity matrix.

Documents de travail du Centre d

(1) (Normalisation) deg 2 (id M , M, y) = 1

where y ∈ M and id M denotes the identity of M .

(2) (Non-triviality) If (f, M, y) ∈ A and deg 2 (f, M, y) = 1, then f -1 (y) = ∅.

(

where

If there is no possible confusion on the manifold M , we simply denote deg 2 (f, y) the degree modulo 2 of the triple (f, M, y).

As stated in the following proposition, in the case of regular values of a C 1 function, the degree modulo 2 is computed using the residue class modulo 2.

Proposition 17 If (g, M, y) ∈ A, g is a C 1 function and y is a regular value of g (i.e., for all z * ∈ g -1 (y), the differential mapping Dg(z * ) is onto), then g -1 (y) is finite (possibly empty) and the degree modulo 2 of g is given by