Dirk Holste

Ivana Roche

Marianne Hörlesberger

Dominique Besagni

Thomas Scherngell

Claire Francois

Pascal Cuxac

Edgar Schiebel

A concept for Inferring 'Frontier Research' in Research Project Proposals

This paper discusses a concept for inferring attributes of 'frontier research' in peer-reviewed research project proposals under the European Research Council (ERC) scheme. The concept serves two purposes: 1) to conceptualize and define, automatically extract, and comparatively assess attributes of frontier research in proposals; and 2) to build and compare outcomes of a statistical model with the review decision in order to obtain further insight and reflect upon the influence of frontier research in the peer-review process. To this end, indicators (including scientific 'novelty', 'risk', or interdisciplinarity') across scientific disciplines and in accord with the strategic definition of frontier research by the Council are elaborated, exploiting textual proposal information and other bibliometric data of applicants. Subsequently, a concept is discussed to measure ex-post the influence of indicators on the decision probability (or, alternatively, the odds) of a proposal to be accepted. The final analysis of the classification and decision probabilities compares and contrasts review decisions in order to, e.g., statistically explain congruence between frontier research and review decision or reveal differential representation of attributes. Ultimately, the concept is aiming at a methodology that monitors the effectiveness and efficiency of peer-review processes. * dirk.holste

Background and Objectives

Scientific disciplines use peer review as an essential mechanism for resource allocation and quality control (Bornmann, 2010). It serves either to determine what research deserves to be published (ex-post review) or what proposed research deserves to be funded through national or regionally operating agencies (ex-ante review). Reviewers face the challenge to find out what is the "best" scientific research according to a journal's/agency's strategy. Typically, journal and grant schemes are sufficiently different that there are no "best practices". Peer-review systems are rooted in critical rationalism, widely accepted and actively supported by the scientific community. Currently they are considered the most effective functioning quality control instrument at hand. Still they are not free form criticism on a number of issues, including poor reliability (congruent opinions); fairness (biased opinions of non-scientific merit); predictive validity (decisions in accord with merit to the scientific community); or inefficiency (e.g., delay, resources spent, opportunity cost). Several issues are 'systemic' in that biases and quality filtering are not necessarily incommensurable. E.g. making scientific progress is fundamentally a conservative act building incrementally upon previous work and requiring evidence resp. balance of risk/reward before publishing (funding) radical papers (ideas); or social system dynamics of subjective human interaction giving rise to lower reliability but offset by higher predictive validity (Bornmann, 2010;[START_REF] Powell | Making the Cut[END_REF]. Because of its central role for not only the scientific community but also publishers/editors and funding agencies, monitoring peer review processes is essential to shed light on to what extent set goals are actually accomplished by review decisions. The need for monitoring effects and the implicit reorientation of peer review processes is subject to current research activities [START_REF] Hojat | Impartial judgment by the "gatekeepers" of science: Fallibility & accountability in the peer review process[END_REF], Sweizer & Collen 1994[START_REF] Bornmann | The effectiveness of the peer review process: Inter-referee agreement and predictive validity of manuscript refereeing at Angewandte Chemie[END_REF][START_REF] Marsh | Improving the peer-review process for grant applications -Reliability, validity, bias, and generalizability[END_REF].

Where the scope and specialization of science has become too complex and sophisticated to be coped with by any one reviever, scientometric methods offer a 'helping hand' to either support the decision-making process or evaluate its outcome. In fact, scientometric evaluation has been witnessing a significant attention in the rising need to get a grip on science output and efficiency (Besselaar & Leydesdorff 2009;[START_REF] Norden | A profusion of measures[END_REF]). On the one hand, these methods carry strengths in that they are precisely defined and reliable, objective, efficient, and need no intervention; on the other hand, their weakness comes mainly in terms of limits of interpretation, applicability, confounding factors, and predictive validity -all of which are more or less debatable. Naturally a numerical model cannot be expected to substitute for expert peer review/process and the merit of the scientific community, each of which being hard to quantify. More faithfully models of this kind may serve to verify decisions, deliver support data efficiently, or hint at biases of the review process [START_REF] Juznic | Scientometric indicators: peerreview, bibliometric methods and conflict of interest[END_REF] ERC 2008). Due to the single evaluation criteria (scientific excellence), ERC grants provide a suitable test-bed for content analysis/text-mining and modelling in the field of bibliometric evaluation [START_REF] Yoon | Development and application of a keyword-based knowledge map for effective research[END_REF]. The primary interest is the extent to which research proposal comply with attributes of frontier research and the influence of these attributes on the selection of awarded grants. The remainder of the paper is structured as follows. After introducing the ERC peer review system and the definition of frontier research, the outline a concept of the review process is presented. Subsequently scientometric and text-analytic methods [START_REF] Roche | Identification and characterisation of technological topics in the field of Molecular Biology[END_REF][START_REF] Schiebel | An advanced diffusion model to identify emergent research issues: the case of optoelectronic devices[END_REF]) capturing desired attributes of frontier research are laid out. Then, a discrete choice model is adopted to approximate the selection function and how indicators influence the decision probability for a proposal to be accepted. Finally, a discussion of the concept closes the paper by elaborating a comparison of peer review process and model outputs and the assessment of the "influencing power" of the individual indicators.

A concept based on joint scientometric and content analysis

Peer review process set up by the ERC Proposal reviews are carried out in two steps. 1) Individual panel member assess both the qualification of PI (Criteria-1) and originality of the proposal (Criteria-2), discuss jointly in panel meetings, score 5 PI and proposal by at least three reviewers based on a scoring system 6 , and preliminary rank and eliminate low-ranking proposals. 2) Both panel members and remote referees individually assess Criteria-1,2 and the research environment 6 (Criteria-3), and interview SG applicants; subsequently proposals are further discussed, scored and ranked, upon which chair meeting consolidate results across different panels and compose a final rank order. The ultimate outcome is a list of top ranks inside the year's budget, next-in-rank (~50% of budget) pending and lower ranks above threshold with no chance of funding.

Defining frontier research and corresponding indicators

In response to the insight that the classical distinction between 'basic' and 'applied' research has ceased to exists, and that emerging areas of science and technology often embrace elements of both, a High Level Expert Group used the term "frontier research" to denote research that reaches beyond horizons of existing knowledge by being intrinsically risky endeavours without regard for established disciplinary boundaries.

Overall concept

The identification of frontier research through a combination of scientometric and text-mining methods is ambitious. The concept introduced here builds upon the understanding of the High Level Expert Group's notion of frontier research. The relation between a sought quantitative model and the above definition is made transparent through the correspondence between each identified key attribute and its indicator as well as a selection function (Figure 1).

Although each indicator has a clear measurement function and subjective interpretation, it is insufficient to represent frontier research. A faithful representation is achieved in combination, which is implemented in form of a selection function. Clearly the notion of 'revolutionary breakthrough' (cf. Table 2) is practically inaccessible by scientometric and textual methods alone. Here two indicators capture different albeit related aspects of the research activity in question: the "timeliness" (one aspect of novelty) of the knowledge-base explicitly used by the author and the "proximity to emerging research topics" (another aspect of novelty) of the proposed research project inferred through the dynamic change of the scientific research landscape pertinent to this discipline.

In computing indicators, an initial step identifies from a corpus of grant application relevant scientometric (e.g., publications, citations, patents) and content data (e.g. text-strings, keywords) bearing relevance to frontier research, extracts and subjects them to data mining. This is essentially a filtering step to pre-process raw data of high quantity into input data of lower quantity but higher quality. In a subsequent step, actual indicators are automatically and robustly computed and subjected to a selection function for comparison between empirical and model parameter. Finally, cross-validation and iterative variation of thresholds, classification criteria, and metrics as well as expert feedback from panel members and chairs, refine in a last step the performance of the model to sufficiently high usability. The following sections describe indicators and selection function in more detail. For various reasons the comparison of proposals is only meaningful within one scientific discipline. Thus we obtain indicator values and apply the selection function for each discipline (panel) individually.

Indicators of frontier research

The indicator TIMELINESS illustrated in Figure 2 is used as a proxy to infer the "innovative degree" of the proposal through bibliographic references; somewhat similarly, it can be applied to the top-N listed self-citations of the author. The underlying hypothesis is that the more recent references are, the more likely the work is at the frontier of science. Therefore, the bibliometric concept focuses is on references and the time elapsed since the publication listed to validating a proposal, and TIMELINESS is obtained from the difference between the times of proposal submission referenced publications. While there are alternative data sources, references of the proposal are considered appropriate because they constitute published documentation of highest velocity and directly relate to the research project. After identifying references and extracting publication dates in actual texts, the indicator can be calculated from the set of values: The simplest way to obtain TIMELINESS is to use the arithmetic mean. In addition, because it may be influenced by statistical outliers, other statistics (e.g. mode or median of the distribution) are considered to quantify the distribution and re-sampling methods like bootstrapping are applied. Finally, it is tested whether TIMELINESS follows a known theoretical distribution (e.g. Poisson or negative binomial distribution).

The indicator PROXIMITY illustrated in Figure 3 is used to infer the "innovativeness" of a proposal. The more sophisticated bibliometric concept employs a "publication landscape" (represented by a cluster map of scientific resp. technological information and characterized by the level of change over time), in which it positions and assigns to each proposal a PROXIMITY value in dependence to the distance and ranking of nearest clusters. The underlying hypothesis is that the closer a proposal is to clusters of positive dynamic change, the more innovative it is. To this end, raw data are obtained from external bibliographic reference databases (for international scientific and technological literature) and ERC panel descriptions to identify and extract discriminating terminological information. A clustering algorithm is employed to obtain a cluster map that groups similar references (through related keywords) and represents the publication landscape corresponding to the considered panel. A diachronic analysis (Roche et al. 2008) is used to study the evolution of the landscape across two time windows, by considering the content of each cluster and its relative location in the network of clusters.

Structural alterations of the network of clusters between the two time periods are identified and described: splitting or disappearing of clusters as well as persistence and emergence of clusters are investigated; as well cluster status changes (e.g. the evolution of a cluster the periphery toward a central position). In addition, techniques of association rule extraction are applied to complete the cluster evolution analysis [START_REF] Mahgoub | A text mining technique using association rules extraction[END_REF]). The result is a ranking of clusters of the more recent map with respect to its dynamics and for each proposal the distance of a proposal to nearest clusters. The indicator PASTEURESQUENESS illustrated in Figure 5 is used to infer the applicability of expected results of each proposal, by considering evidence for immediate or intended application. Input data are obtained from bibliographic and patent databases and proposals.

The term PASTEURESQUENESS originates from the definition of Pasteur's Quadrant [START_REF] Stokes | Pasteur's Quadrant. Basic Science and Technological Innovation[END_REF], which describes scientific research/methods that seek both fundamental understanding and at the same time social benefit (cf. Figure 4). PASTEURESQUENESS consists of in all three indicators: i) patents granted to the PI; ii) information related to business involvement; and iii) self-citations published in journals with prescribed 'applicability' (scoring function). The indicator RISK illustrated in Figure 6 is used to infer the "personal risk" of the PI in executing the proposed research under the assumption that references (of proposals, papers) serve as one informative source for constructing a 'bibliometric research profile'. In comparing the profile of a proposal to both past research by the PI and relevant research originating inside the same field, observed overlaps are used to categorize anticipated constant resp. aligned, modified or changed research directions.

On the first hand, one hypothesis is that the lower the overlap between two reference profiles (past vs. proposed research), the more risk-affine is the proposal for the PI, because it is indicative of a change from previous directions. On the other hand, another hypothesis is that the lower the overlap to all other '(field) profiles', the more unique is the proposed research. The computation of RISK can be refined by "exotic terms" (rare keywords) in textual information of the proposal; exotic in the sense that the terms are less frequently used than established terms in the same research area [START_REF] Roche | Identification and characterisation of technological topics in the field of Molecular Biology[END_REF][START_REF] Schiebel | An advanced diffusion model to identify emergent research issues: the case of optoelectronic devices[END_REF]. The indicator INTERDISCIPLINARITY illustrated in Figure 7 is used to infer self-consistently the presence and proportions of characteristic terminology or textual traits, which "belong" to different ERC '(home) panels', and thereby reveal the intra or inter-panel/disciplinary character of each proposal. It is build upon the assumption and previously successfully tested concept [START_REF] Schiebel | An advanced diffusion model to identify emergent research issues: the case of optoelectronic devices[END_REF]) that the frequency of occurrence and distribution of discipline specific keywords in scientific documents can be used to classify and characterize disciplines. To this end, each keyword is labeled according to its statistical frequency of occurrence across all PE or LS panels, filter are applied to distinguish relevant from irrelevant (i.e. panel unspecific) keywords, and the concentration of keywords from different home panels is assessed to classify proposals as very high, medium-high, low-medium, or very low in concentration. In addition to free keywords extracted from textual information of proposals, the analysis is supplemented by using fixed keywords from the description of ERC panels.

Modelling the selection function

The indicators described above are expected to have a positive effect on the decision probability of a grant application, which has in principle three possible outcomes: Type-A) above threshold and funded, Type-B) above threshold and not funded, and Type-C) below threshold. Yet as there are a number of other possible factors influencing the peer review process, a statistical analysis (discrete choice model) determining the actual association between indicators and the funding decisions is carried out separately for SG and AG applications.

A discrete choice model is used to estimate how various exogenous factors (captured by the indicators, cf. Table 3) influence the probability for a project proposal to get accepted. Discrete choice models have come into wide-spread use in econometrics to represent discrete alternatives rather than a continuous measure of activity [START_REF] Greene | Econometric analysis. Fifth Edition[END_REF]. In a first attempt, the selection model assumes a binary choice (types A/B vs. C) as they are the two central outcomes of the dependent variable: the rejection or acceptance of a project proposal. To this end, the dependent variable takes the form (empirical data on the i = 1,2,…,n proposals):

1 0o t h e r w i s e . i proposal is accepted Y ⎧ = ⎨ ⎩
The decision probability of a proposal to be accepted, Pr(Y i = 1), depends on a set of independent, exogenous factors (indicators) summarized in a matrix X such that , , Pr( 1) ( )

Pr( 0) 1 ( ) ii ii YF YF == == - X X β β , (2)
where X i is a set of k observed factors for proposal i, β is the estimated k-by-1 parameter vector reflecting the impact of changes in X i on the probability Pr(Y i = 1). It is common practice to define F by the logistic distribution function, leading to binary Logit model:

, , , exp Pr( 1) ( ) 1-exp i ii i Y == Λ = X X X β β β . (3)
Technically, the parameter estimation is based on Maximum-Likelihood (cf. [START_REF] Greene | Econometric analysis. Fifth Edition[END_REF], and the interpretation in the context of the research question can be comfortably framed in terms of odds-ratios, which are given by exp(β) (cf. Johnston and DiNardo 2007).

Discussion

The above concept aims at developing quantitative methods for determining and examining the relationship between peer review and decisions on grant allocation in terms of attributes of frontier research:

• Can attributes of frontier research be represented and quantitatively to evaluate the grant allocation decision by bibliometric approaches?

The detailed development has focused on the ERC grant scheme (with data in 2007-2009), but the concept might be applicable more generally, depending on mission, review process and guidelines, attributes and correspondence of indicators for grant schemes. The implemented concept is intended to yield a bibliometric model in which indicators are expected to have a positive effect on the decision probability for ERC grant applications. Thus a follow up and specific question is:

• How well do bibliometric indicators and the decision probability discriminate between grant applicants accepted resp. rejected for funding?

In particular, instead of relying on performance measures the concept builds on a combination of content analysis and data-mining in combination with scientometric methods. The concept presents a new approach in that it explicitly utilizes information present in research proposals submitted to a grant agency and relates this information to the bulk of information drawn from activities of the larger research community in a specific field. To this end, textual information (e.g. keywords and longer strings extracted from proposals) as well as references (e.g. papers or patents) are combined using, e.g., human expert systems, clustering techniques, relational mapping, networks, bibliometric functions, various indices and data filtering techniques.

Described indicators and selection function are currently implemented. First ex post analyses between model and review process can be expected a mixture of similarity between peer reviewer selection (i.e. Types-A/B) and dissimilarity (i.e. type C). Depending on the found correlation between the discrimination of types A/B vs. C obtained from the bibliometric model and selection by peer review, numerical algorithms for the computation of indicators might need refinement, resulting in a modification of the model, or improvements of the future operation of the peer review process can be envisioned. In any case, it requires careful investigation (statistical independence, positive vs. negative correlation, outliers), explicit differentiation between measurement concept and interpretation, and careful feedback on parameters on a discipline-specific basis. In order to serve its purpose, the development and refinement will be hand-in-hand with experts involved in the review process to determine what the metrics could be used for and how it is affecting the review process.

Ultimately the concept shall result in a methodology that allows the grant agency to monitor the operation of the peer review process from a bibliometric perspective and thereby provide a basis for the further refinement of the peer review process, including the ex ante bibliometric evaluation of future grant applications to support reviewers with orientation knowledge for the review assessment.

Figure 1 .

 1 Figure 1. Core review process and corresponding indicators and the selection function.

Figure 2 .

 2 Figure 2. The core bibliometric concept of the TIMELINESS indicator.

Figure 3 .

 3 Figure 3. The core bibliometric concept of the PROXIMITY indicator.

Figure 4 .

 4 Figure 4. Illustration of the Pasteur Quadrant.

Figure 5 .

 5 Figure 5. The core bibliometric concept of the PASTEURESQUENESS indicator.

Figure 6 .

 6 Figure 6. The core bibliometric concept of the RISK indicator.

Figure 7 .

 7 Figure 7. The core bibliometric concept of the INTERDISCIPLINARITY indicator.

  Table 1 lists selected examples of scientometric indicators that are performance-centred and use easy-to-measure (in principle) volume data. Other indicators incorporate data extracted from networks of co-authorships, citations or bibliometric coupling; curriculum vitae, e.g., age, gender, institution (teaching, research); referee scores; or socioeconomic relevance (previously attracted, requested funds).

Table 1 . Selected classical scientometric indicators for measuring scientific performance.

 1

	Indicator	Measurement	Interpretation options
	Authorship	Number of publications or co-publications	Research output, productivity
	Citation	Number of citations	Research influence, international impact
	Publication-citation h-index	Research productivity and impact
	Journal impact
	factor
			Global spread, attention in scientific
			community and beyond
	Network properties Social network parameters	Beyond stand-alone, reflect system properties
			and influence in network interconnectedness
			and speed of information exchange

Total number of journal citations in year n of papers published in years n-1 and n-2 over the number of articles published in n-1 and n-2

Reputation of scientific journals

Online access

Number of times a paper is accessed online in some time period T

Table 2 . Number of proposals submitted and grants awarded by the ERC in 2007-2009. ERC grant (year) Total budget (m €) Number of proposals submitted Total number of grants awarded Number of grants awarded (in PE and LS)

 2

	SG (2007)	335	9,167	299	242
	AG (2008)	553	2,167	282	198
	SG (2009)	325	2,503	244	187
	AG (2009)	515	1,584	244	202

Source:

ERC (2011); since 2009, SG and AG grants are awarded annually.

  The first European research funding body targets research at the highest level of excellence in any scientific discipline. It supports investigator-driven projects aiming at broadening the scientific and technological knowledge without regard for established disciplinary boundaries (frontier research 3 ) through open and direct competition. The selection of proposals for is based strictly on peer review. The ERC has established a process which is to identify scientific excellence of frontier research as the sole evaluation criterion for funding decisions(ERC 2010). Internationally renowned scientists and scholars constitute two sets of review panels (for the SGs and AGs), each of which subdivided into 25 individual panels that cover the entire range of disciplines and fall into in the domains PE, LS, and SH 4 . Each panel is composed of 10-12 members and headed by a chair. If further expertise is required, external reviewers may be consulted by providing assessments on a proposal-by-proposal basis.

Table 3 . Relation between the definition of frontier research and correspondence of indicators.

 3

	Indicator	Definition (Frontier research…)

NOVELTY

"… stands at the forefront of creating new knowledge and developing new understanding. Those involved are responsible for fundamental discoveries and advances in theoretical and empirical understanding, and even achieving the occasional revolutionary breakthrough that completely changes our knowledge of the world." RISK "… is an intrinsically risky endeavour. In the new and most exciting research areas, the approach or trajectory that may prove most fruitful for developing the field is often not clear. Researchers must be bold and take risks. Indeed, only researchers are generally in a position to identify the opportunities of greatest promise. The task of funding agencies is confined to supporting the best researchers with the most exciting ideas, rather than trying to identify priorities." PASTEURESQUENESS "… may well be concerned with both new knowledge about the world and with generating potentially useful knowledge at the same time. Therefore, there is a much closer and more intimate connection between the resulting science and technology, with few of the barriers that arise when basic research and applied research are carried out separately."

INTERDISCIPLINARITY "… pursues questions irrespective of established disciplinary boundaries. It may well involve multi-, inter-or trans-disciplinary research that brings together researchers from different disciplinary backgrounds, with different theoretical and conceptual approaches, techniques, methodologies and instrumentation, perhaps even different goals and motivations.

Source:

EC (2005)

.

  Table 3 cites key attributes of frontier research according to the Group's report (European Commission 2005).

The clear distinction between basic and applied science has become blurred because new areas of science and technology often display substantial elements of both. Frontier research reflects that reality and calls for concentrated efforts on a more receptive attitude to unconventional thinking (ERC 2010).

In addition, a fourth domain 'Interdisciplinary' has been defined for cross-panel and cross-domain proposals.

SG, intellectual capacity to reach beyond the state of the art and creativity (achievements, publications); grant as contribution to establish independence or its consolidation; commitment (time in-residence, on project). AG, ground breaking nature of research, potential impact; interdisciplinarity; novel concepts; highly novel and/or unconventional methodologies justified by high gain/high reword balance); feasibility; recourses claimed justified added-value of team members.

Categories and marks used: "non-competitive" (1), "very good" (2), "excellent" (3), or "outstanding" (4).

Acknowledgments

The authors acknowledge the support that this work was partially funded by the Ideas specific programme of the EU's FP7 Framework Programme for Research and Technological Development (project reference no. 240765). The authors thank Helga Nowotny and Jens Hemmelskamp of the ERC for stimulating discussions, and Manuela Kienegger for time and energy in drafting the manuscript.


Laurent Granier

Marion Podesta
email: mpodesta@univ-tlse1.fr

Bundling and Mergers in Energy Markets

Keywords: JEL classi…cation: D43, L13, L22, L41 Product Bundling, Endogenous Mergers, Energy Markets

published or not. The documents may come

Introduction

A trend towards deregulation of utilities industries, such as energy or telecommunications, is observed worldwide. This has an impact on market structures and pricing strategies. In particular, market structures shift from monopolies1 to oligopolies. Moreover, in energy industries, we observe multi-market mergers between …rms belonging to various oligopolies. Such mergers enable …rms to bundle several energies. 2 For instance, they provide packages of two di¤erent energies like gas and electricity. A signi…cant example is the merger case between E.ON and Ruhrgas on the German market. Although the merger proposal is rejected in 2002 by the competition authority, the Federal Minister of Economics and Technology even so approves the merger in 2003 [START_REF] Marsden | Competition Law and the Consumer: Results of the Legislative Survey on Fourteen European Competition Law Regimes[END_REF]. Before this acquisition, Ruhrgas was the …rst gas producer in Germany while E.ON, the …rst electricity one. E.ON only owned holdings in a few local subsidiaries of gas supply. Thus, the new leader of the German energy market now supplies bi-energy bundles. We also remark that other mergers become e¤ective on this market straight after. Indeed, electricity supplier EWE merges with gas suppliers Cuxhaven and SWB in 2003. This substantiates the merger wave phenomenon. The following question therefore becomes important: do bundling strategies trigger mergers in energy markets? This type of incentive could better explain the convergence phenomenon in energy industries. In this paper, we study the emergence of these mergers. In order to carry out our analysis, we use a horizontally di¤erentiated model derived from Reisinger (2006). It allows to study bundle competition. This analysis can be interpreted as a modelling of a competition between two electricity …rms and two gas …rms. 3 We build a merger game allowing to underline a merger wave phenomenon. This phenomenon is due to the ability to supply bi-energy bundles4 once a merger is achieved. Bundling entails two e¤ects. The …rst is a price discrimination one. The second is a competition one. The trade-o¤ between these e¤ects and merger choices causes an increase in pro…ts.

The results which we have just evoked allow to better assess a relevant phenomenon in the energy markets: the convergence phenomenon. Usually, convergence refers to a process that reduces di¤erences between activities. It corresponds to a gradual integration of formerly separate industries. To describe convergence in the energy industry, we analyze a speci…c trend: the convergence between gas and electricity. 5 This trend is widely observed during the 1990s in the US and is now described in Europe too [START_REF] Verde | Everybody merges with somebody-The wave of M&As in the energy industry and the EU merger policy[END_REF]. Multi-market mergers in energy industry participate in this convergence phenomenon. Indeed, downstream mergers allow the diversi…cation of energy supplies and clearly participate in the convergence phenomenon. For instance, the inter-industry merger Dong/Elsam/EnergiE2 (European Commission, 2006a) refers to the integration between the Danish gas incumbent and Danish companies active in the electricity sec-tor. The …rm could now exploit their complementarities and supply bi-energy bundles. Another example is Gaz de France/Suez merger proposition submitted to the European Commission in 2005. This corresponds to a national merger but concerning both midstream and downstream markets. Colette Lewiner (a senior vice president at Capgemini in Paris) says that this merger could have been "a plus for competition if Suez and GDF bundle their o¤erings to give customers like industry better o¤erings, perhaps in the form of a single bill for electricity, gas and water" (International Herald Tribune, 2006). As the proposition was declined by the authorities, Gaz de France has to purchase electricity to Electricité de France in order to supply bi-energy bundles. In 2008, the Gaz de France/Suez merger becomes e¤ective because European Commission approves it. This example suggests that bundling strategies may incite to merge. Empirical studies show that a lot of consumers6 use several types of energies [START_REF] Bernard | Quebec residential electricity demand: a microeconomic approach[END_REF][START_REF] Nesbakken | Energy consumption for space heating: a discrete-continuous approach[END_REF]. So bundling strategies may be a fundamental reason for merger decisions. 7 Despite the prevalence of this particular type of merger, to our knowledge they are not analyzed by the theoretical literature. The aim of this paper is to …ll this gap.

Before modelling the competition with bundles, we give more details about the bundling literature. Bundling refers to the practice of selling two or more goods at a unique price. 8The economic literature on bundling isolates several e¤ects. One of the main e¤ects is price discrimination. Bundling allows to sort consumers according to their willingness to pay. This characteristic is analyzed by [START_REF] Adams | Commodity bundling and the burden of monopoly[END_REF] for a monopoly producing two goods. In analysis dealing with speci…c cases, they show that mixed bundling is generally the optimal strategy9 since the correlation between the goods is negative. Whinston (1990), Nalebu¤ (2004) and [START_REF] Peitz | Bundling may blockade entry[END_REF] underscore the fact that a two-market monopolistic …rm can deter the entry of competitors by bundling10 if the potential entrant can enter only one market. In this framework, Nalebu¤ shows that pure bundling is optimal. A second e¤ect of bundling to consider is, in competitive environments, a competition e¤ect. [START_REF] Anderson | Why …rms prefer not to price discriminate via mixed bundling[END_REF] analyze bundling in a complementary-goods duopoly. In their view, independent pricing is a dominant strategy in the commitment case. [START_REF] Economides | .U Eckbo BE Horizontal mergers, collusion and stockholder wealth[END_REF], in the same framework, shows that …rms follow mixed bundling strategies in the Nash equilibrium. Firms, however, make lower pro…ts than they do when adopting an independent pricing strategy. [START_REF] Armstrong | Competitive nonlinear pricing and bundling[END_REF] examine principally a unit-demand model where consumers may buy one product from one …rm and another product from another …rm under nonlinear pricing. They show that bundling generally acts to reduce pro…t and welfare and to boost consumer surplus.11 However, they consider an intrinsic extra shopping cost when consumers purchase each good at different locations. [START_REF] Thanassoulis | Competitive mixed bundling and consumer surplus[END_REF] …nds that if buyers incur …rm speci…c costs or have shop speci…c tastes then competitive mixed bundling lowers consumer surplus overall and raises pro…ts. Reisinger (2006) also studies a duopoly that produces two types of horizontally di¤erentiated goods. He analyzes a framework for which consumers buy one unit of each good with neither substitutability, nor complementarity e¤ects created by variants choices for each type of goods. The correlation of the reservation prices is expressed by the correlation of consumers' location on each market. He shows that there are two e¤ects created by bundling: the well-known "sorting e¤ect" and the "business-stealing e¤ect," which results from bundle competition. Reisinger shows that …rms have an incentive to adopt a mixed bundling strategy. Nonetheless, the e¤ect on pro…ts is ambiguous. If the correlation of reservation prices is negative, then the competition e¤ect dominates and the bundling strategy lowers pro…ts. Such …rms are in a prisoner's dilemma situation. On the other hand, if the correlation of reservation prices is positive, then the sorting e¤ect allows …rms to make higher pro…ts.

We use the model of Reisinger (2006) in order to analyze the impact of bundling on merger incentives. We therefore consider two horizontally di¤erentiated markets, that are electricity and natural gas markets. As Reisinger (2006), the link between these two markets is the correlation of consumers' locations. Nevertheless, four …rms are present. Two …rms produce electricity, and the two others supply natural gas. In their respective markets, …rms compete in prices. We build an endogenous merger game and assume that monopolization was illegal. First, we exclude the post-merger bundling strategy. Second, we remove this assumption in order to analyze the e¤ect of bundling strategy on merger incentives. In a basic model in which bundling is not considered, we …nd that there is no incentive to merge. Once a merger is achieved, however, as we show, there is an incentive to adopt a mixed bundling strategy. Otherwise, the bundling strategy triggers a merger wave. Moreover, we show that relative to the correlation of reservation prices, two types of mergers are achieved. Furthermore, while Reisinger (2006) shows that there is a prisoner's dilemma, we show that the di¤erent types of mergers allow this dilemma to be removed. Finally, from a welfare point of view, we show that bundling is less harmful than Reisinger suggests (2006).

In order not to neglect merger interactions in our model, we endogenize merger decisions. In this sense, our study is closely linked to the endogenous merger literature, some of which seeks to explain mechanisms preventing mergers as the "insider's dilemma 12 " previously evoked in the exogenous merger model of [START_REF] Stigler | Monopoly and oligopoly by merger[END_REF]. For instance, Kamien andZang (1990, 1993) and Fridolfsson and Stennek (2005b) also consider the "insider's dilemma". Moreover, Kamien andZang (1990, 1993) add auction mechanisms to take into account …rms' acquisitions processes. We care about the "insider's dilemma" but without any auction mechanism. Indeed, we are not interested in surplus sharing rule. On the other hand, we did deal with other characteristics found in the endogenous merger literature, such as taking all …rms' combinations into consideration. For instance, some endogenous merger models allow merger interactions to be revealed [START_REF] Nilssen | Sequential horizontal mergers[END_REF]. More particularly, some models attempt to emphasize the phenomenon of preemp-12 "The insiders' dilemma means that a pro…table merger does not occur, because it is even more pro…table for each …rm to unilaterally stands as an outsider" (Lindqvist and Stennek, 2005). [START_REF] Salant | Losses from horizontal merger: the e¤ects of an exogenous change in industry structure on Cournot-Nash equilibrium[END_REF] validate the result of [START_REF] Stigler | Monopoly and oligopoly by merger[END_REF] when …rms compete in a Cournot fashion. Indeed, they show that if a takeover does not merge more than 80 per cent of an industry, such a takeover is not carried out because outsiders earn more than insiders. Going further, [START_REF] Inderst | The incentives for takeover in oligopoly[END_REF] focus on probability of hold-up (respectively hold-out) in a merger game that includes cases for which outsiders bene…t from mergers.

tive mergers (Fridolfsson and Stennek, 2005a[START_REF] Brito | Preemptive mergers under spatial competition[END_REF][START_REF] Matsushima | Horizontal mergers and merger waves in a location model[END_REF]. Finally, other models, such as those of [START_REF] Fauli-Oller | Takeover waves[END_REF] or [START_REF] Nilssen | Sequential horizontal mergers[END_REF], focus on merger waves phenomena. As the same type of merger interactions are possible in our framework, we build a merger game based upon [START_REF] Nilssen | Sequential horizontal mergers[END_REF]. Contrary to [START_REF] Nilssen | Sequential horizontal mergers[END_REF], we do not restrict merger possibilities in an ad hoc fashion. Indeed, the only restriction concerning merger choices is due to the prohibition of the monopolization. Some merger choices are then mutually exclusive but this is not determined in an ad hoc way. This is due to the fact that a homogeneous merger is de facto incompatible with a heterogeneous merger.

The following section presents the basic model. The section 3 introduces the bundling strategy on energy markets. Section 4 gives the equilibrium of the game and the social welfare analysis. The …nal section presents some concluding remarks.

Basic model

Throughout this section, we exclude bundling strategies. We start with the assumptions of the competition game. Next, a merging game is de…ned. Finally, we solve this game in order to establish the benchmark, before introducing the bundling strategy in the next section.

Assumptions

We will consider a four-…rm industry. Two …rms produce the electricity A at the marginal cost c A and two others produce the gas B at the marginal cost c B . In order not to introduce bias13 in our bundling analysis, we will assume that production costs are linear. Each type of energy is horizontally di¤erentiated. For each energy, the product variants are the locations of the …rms on a circle whose circumference is normalized to 1. According to the type and the location of their output, …rms are named either A i or B j with i; j = 1; 2. The …rm A 1 (respectively B 1 ) produces the good A, electricity (respectively B; gas) and is located at 0 on circle A (respectively B) while the …rm A 2 (respectively B 2 ) produces the same energy 14 but is located at 1 2 . There is a continuum of consumers, and, without loss of generality, we normalize its total mass to 1. Consumers' locations on both circles are x = (x A ; x B ). Each consumer wishes to buy one and only one unit of each type of energy. 15 This allows us to focus on the pure strategic e¤ect of bundling. Firms compete in prices in each energy market. Their prices are denoted by p i A and p i B . Thus, consumers can choose between four product combinations. They can buy either the electricity from …rm A 1 and the gas from …rm B 1 , i.e., (A1B1), or the electricity from …rm A 2 and the gas from …rm B 2 , i.e., (A2B2), or the electricity from …rm A 1 and the gas from …rm B 2 , i.e., (A1B2), or the electricity from …rm A 2 and the gas from …rm B 1 , i.e., (A2B1).

For instance, a customer located at 0 x A ; x B 1=2, buying the electricity from …rm A 1 and the gas from …rm B 2 has an indirect utility of:

V (x A ; x B ) = K A p 1 A t A (x A ) 2 + K B p 2 B t B (1=2 x B ) 2 . (1)
Utilities from consumption (gross of prices and transportation costs) of electricity, A and gas, B are given by K A and K B . The two markets (or the two energies) are denoted by k = A; B, and we note t k the transportation cost associated with circle k. Without loss of generality16 , we assume t A > t B > 0. The consumer reservation price R i k , for the variant i of the energy k, is given by K k t k (d i ) 2 , where d i is the shortest arc length between the consumer's location and …rm i on circle k. In order that all consumers buy both types of energies in each price equilibrium, we assume that K k is su¢ciently high. Reservation prices can be linked to the locations of consumers. Indeed, the joint distribution function of reservation values G(R i A j R i B ), and so, the correlation between reservation prices for the energy from location i on the two markets can be deduced from the joint distribution function of consumer location F (x A j x B ). Like Reisinger (2006), we give a structure to this distribution function. It is a simple function expressing all correlations of reservation prices. Thus, if a consumer is located at x A on circle A, then this consumer is located at where 0 1=2. This means a -shift of all consumers on circle B. If = 0, the reservation price correlation is one. Through adopting this simple structure, correlations of reservation values can be obtained easily by altering . The correlation coe¢cient

x B = x A + x A + 1 if x A + 1 if x A + > 1 on circle B,
[R A ; R B ]( ) = Cov[R A ;R B ]( ) (R A ) (R B )
is given17 by 1 30 2 + 60 3 30 4 . By way of illustration, note that for small values, if a consumer has a high reservation price for the electricity of …rm A 1 , then s/he has a high reservation value for the gas of …rm B 1 . If a consumer has a high reservation price for the electricity of …rm A 2 , then s/he has a high reservation value for the gas of …rm B 2 . Conversely, a high value of implies that consumers have very di¤erent reservation prices for the two energies sold at the same location on each circle. Now, we de…ne the merger game in which the four …rms are involved.

Merger game

We assume that monopolization is illegal. 18 Thus, potential mergers necessarily involve …rms from two di¤erent markets. We build a two-stage game. At the …rst stage, …rms choose either to merge or not to merge. At the second stage, …rms choose independently to follow either a bundling strategy or an independent pricing one. Then they compete in prices. The previous section already presents these stage assumptions. We now describe the …rst stage. Each electric …rm A i can choose to merge with the gas …rm with the same location in the gas market. We call this type of merger a "homogeneous merger". Each electric …rm A i can also choose to merge with the gas …rm located at the opposite in the gas market. In this case, the merger is called a "heterogeneous merger". 19 Finally, each electric …rm A i can choose not to merge (see Figure 1

below).

Insert Figure 1 Firms take sequential and non-cooperative decisions. The sequential nature of the game is similar to [START_REF] Nilssen | Sequential horizontal mergers[END_REF]. By this way, we focus on merger interactions. Moreover, we avoid coordination problems between …rms. 20 Without loss of generality, we assume decisions are made by electric …rms of the market A. Results are exactly the same if decisions are made by gas …rms of market B. We also assume that electric …rm A 1 makes its merger decision …rst, but the results are the same if electric …rm A 2 chooses …rst. In this paper, we look for subgame perfect Nash equilibria (SPNE). Therefore, we solve the game backward. Thus, if we solve competition subgames, then we solve the merger game. Concerning mergers, we do not de…ne any pro…t sharing rule. We focus exclusively on the following question: "Is the merger pro…t higher than the pre-merger pro…t sum of …rms involving in the merger?" Indeed, if such is the case, there is necessarily a pro…t-sharing rule that gives an incentive to merge. On the other hand, in parallel with the "strategic motives" consideration [START_REF] Nilssen | Sequential horizontal mergers[END_REF], we take into account interactions between merger decisions. To illustrate, two-…rm merger expectation can either incite or not incite another two-…rm merger. Therefore, merger decisions are endogenous in this model. At the …rst stage, both electric …rms A 1 and A 2 choose to merge or not to merge with gas …rms of market B. Thus, we present this game in Figure 2 below.

Insert Figure 2

Now, we solve the game without considering bundling strategy. This constitutes the benchmark. 19 To better illustrate these merger types, consider an example in energy. The homogeneous merger could be a merger between the …rm A 1 , which supplies electricity, and the …rm B 1 , which provides gas. The …rm A 1 has the reputation of providing a vaster supply network than A 2 , and the …rm B 1 has the reputation of supplying a more performant technical aid than B 2 does. In the same way, the heterogeneous merger could be a merger between the …rm A 2 and the …rm B 1 . The …rm A 2 has a much wider range of services than A 1 does, and regarding the gas storage the …rm B 2 has a better reputation than B 1 does. 20 In a simultaneous game, a coordination problem can arise by two ways. First, the potential buyers bid for the same target. So we could use a method for selecting a buyer …rm and the repetition of the simultaneous game would provide the same result. Second, …rms coordinate on the wrong continuation equilibrium for intermediate values of delta. But whatever are the type of the …rst merger and the value of delta, there is an incentive to achieve a second merger by repeating the game because of mixed bundling (see proof of proposition 1 in Reisinger (2006) for the homogeneous case). If the simultaneous game is repeated, …rms anticipate this and bid for the good continuous equilibrium since the wrong gives less pro…t.

Benchmark: mergers and independent pricing

By symmetry, we deduce from the game tree that there are …ve possible outcomes. There may be either two homogeneous mergers, two heterogeneous mergers, only one homogeneous merger, only one heterogeneous merger, or, …nally, no merger. We determine prices and pro…ts of the di¤erent game's outcomes in Appendix 6.1. We note that z A i B j is the pro…t of the merged …rm A i B j where z = 1; 2 is the number of mergers. Thus, we can establish the following equation:

z A i B j = A i + B j , 8i; j; z = 1; 2. ( 2
)
We therefore deduce the following proposition:

Proposition 1 If bundling strategy is not considered, then there is no incentive to merge.

When they cannot provide bundles, the monoproduct …rms are indi¤erent as to whether to merge or not to merge. Since markets are independent, there is no competition e¤ect due to mergers. Therefore, there is no incentive to merge. 21 After a merger, prices are unchanged and the global pro…t of a merger is merely the pro…t sum of the merging …rms. In this case of indi¤erence, we assume that …rms choose not to merge. After the benchmark analysis, the following section considers a case where merged …rms are able to follow a mixed bundling strategy. Thus, we focus on pure e¤ects of bundling in the competition game and on their impacts on incentives to merge.

Mergers and mixed bundling

In this section, we introduce mixed-bundling strategy. 22 Indeed, if two …rms merge, they can supply a bundle composed of both electricity and gas. A merged …rm A i B j 8i; j = 1; 2 can o¤er a package composed of electricity A i and gas B j . This bundle is denoted by (ABij) and its price is p ij AB . Mixed bundling may enable …rms to attract marginal consumers by lowering the bundle price. 23 A consumer who buys the energies from di¤erent …rms in the independent pricing case can then prefer to buy the two energies from the same …rm. The desutility when purchasing its non-preferred variant of a energy is balanced by a lower price. We show that results depend on correlation of reservation prices. In this section, we analyze game outcomes. Here, a mixed bundling strategy is possible. Thus, we study competition in each outcome. We exclude, however, the non-merged outcome, because bundling is not possible in this case, and so, pro…ts are already computed in the previous section. For simplicity, we analyze only the merger waves outcomes, i.e., the two-merger cases. We will explain why it is su¢cient to solve the game in section 4. First, we study the case of homogeneous two-…rm merger. This corresponds to Reisinger's model (2006). Next, a sub-section is devoted to the case of heterogeneous two-…rm mergers.

Bundling and homogeneous mergers

We assume electric …rms A i merged with gas …rms B j , 8i = j, with i; j = 1; 2. In this con…guration, customers choose between four product combinations. They can either buy the bundle of …rm A 1 B 1 , i.e., (AB11), at price p 11 AB , or buy the bundle of …rm A 2 B 2 , i.e., (AB22), at price p 22 AB . They can also purchase either the electricity A from …rm A 1 B 1 and the gas B from …rm A 2 B 2 , i.e., the product combination (A1B2), at price p 1 A + p 2 B , or the electricity A from …rm A 2 B 2 and the gas B from …rm A 1 B 1 , i.e., the consumption option (A2B1), at price p 2 A + p 1 B . We want to determine prices and pro…ts when …rms adopt a mixed bundling strategy, but this is interesting only if …rms have an incentive to bundle. Thus, we use the lemma of Reisinger (2006):

Lemma 1 If > 0, i.e < 1, then in equilibrium, homogeneously-merged …rms follow a mixed-bundling strategy.

Proof. See Reisinger (2006) In the benchmark, there is no merger incentive. Merger incentives are only due to bundling strategies. Lemma 1 shows that if there are two mergers, each …rm follows a mixed bundling strategy. In the same manner, every …rm is incited to merge because merger incentives and bundling incentives are exactly the same here. In this case, two homogeneous mergers are achieved. The proof is exactly the same as the proof of Lemma 1 given by Reisinger (2006). For this reason, we only studied the two-homogeneous merger outcome.

Lemma 2 Respectively for

3 2 ( t A +t B 5t A +t B ), 3 2 ( t A +t B 5t A +t B ) < 1 3 t B 6t A , and > 1 3 t B
6t A , equilibrium pro…ts are given by:

2 A i B j = 1 8 (t A + t B ) + 2 9 2 t A t B t A + t B , (3)
2

A i B j = 1 8 (t A + t B ) + t A t B (4(t A + t B ) 2 (6t A + t B ) 4 2 t A ) 2(t A t B ) 2 , ( 4
) 2 A i B j = 1 8 t A 7 72 t B . (5)
Proof. See Reisinger (2006) First, we assume that is small, i.e the correlation of reservation prices is positive. In this case, there are four consumption combinations: (AB11), (A1B2), (AB22) and (A2B1). Figure 3 below presents this demand con…guration (i):

Insert Figure 3 Second, we assumed that is high. In this case, there are six consumption options: (A1B2); (AB11), (AB22), (A2B1), (AB22) and (AB11). Figure 4 below presents this demand con…guration (ii):

Insert Figure 4 For the homogeneous mergers, the pro…t level explanation is similar to Reisinger (2006): in every case, …rms compete, à la Bertrand, on each horizontally di¤erentiated energy. But here, they can use bundling. Thus, bundle demands appear. When the correlation is high ( small), …rms compete with their own separate energies by lowering their bundle prices. Therefore, they have no incentive to decrease their bundle prices relative to the prices sum of the separate energies in the independent pricing strategy. Furthermore, bundling allows for consumers to be sorted and pro…ts to be raised. Moreover, this e¤ect increases when correlation decreases. Indeed, each …rm can raise the prices of its separate energies. Thus, pro…ts increase. On the other hand, the correlation decrease makes consumers more and more indi¤erent to the two bundles. Consequently, for a su¢ciently weak correlation ( > HM 1 ), each bundle competes directly with the rival's bundle. Firms must lower their bundle prices in order to maintain their market shares. However, when this new competition for this level of correlation ( HM 1 ), is integrated, the pro…t maximization does not give an equilibrium. Hence the necessity of computing a touching equilibrium [START_REF] Economides | The principle of minimum di¤erentiation revisited[END_REF] between this correlation threshold and the one for which the equilibrium becomes stable ( HM

2

). This implies a linear drop in prices to keep the initial demand structure (i). This new e¤ect dominates the positive sorting e¤ect. The decrease in prices entails both a drop in pro…ts and, for a su¢ciently high correlation, a prisoner's dilemma concerning the bundling strategy decision. We compute the equilibrium of the competiton game following a homogeneous merger wave. Results, however, can di¤er when mergers are heterogeneous. Subsequently, we study this game outcome.

Bundling and heterogeneous mergers

We assume electric …rms A i merge with gas …rms B j , 8i 6 = j, with i; j = 1; 2. In this con…guration, customers choose between four product combinations. They can either buy the bundle of …rm A 1 B 2 , i.e., (AB12) at price p 12 AB , or buy the bundle of …rm A 2 B 1 , i.e., (AB21) at price p 21 AB . They also can either purchase the electricity A from …rm A 1 B 2 and the gas B from …rm A 2 B 1 , i.e., the product combination (A1B1) at price p 1 A + p 1 B , or purchase the electricity A from …rm A 2 B 1 and the gas B from …rm A 1 B 2 , i.e., the product combination (A2B2) at price p 2 A + p 2 B . We want to determine prices and pro…ts when …rms follow a mixed-bundling strategy but this is interesting only if …rms have an incentive to bundle. Thus, we establish this lemma:

Lemma 3 If < 1
2 , i.e > 1, then, in equilibrium, heterogeneously-merged …rms follow a mixed-bundling strategy.

Proof. see Appendix 6.2

In the benchmark, there is no merger incentive. Merger incentives are only due to bundling strategies. Lemma 3 shows that if there are two mergers, each …rm adopts a mixed bundling strategy. In the same manner, every …rm is incited to merge because merger incentives and bundling ones are exactly the same here. In this case, two heterogeneous mergers are achieved. The proof is exactly the same as the proof of Lemma 3. For this reason, we only study the two-heterogeneous merger outcome.

First, equilibrium demand con…gurations will be described. 24 These depend on . We assume is small, that is, is high, i.e the correlation is positive. In this case, there are six consumption options: (A1B1), (AB12), (AB21), (A2B2), (AB21), and (AB12). Below, Figure 5 presents this demand con…guration (iii):

Insert Figure 5 Second, we assume is high. In this case, there are no longer four consumption options left: (AB12), (A2B2), (AB21), and (A1B1). Figure 6 below presents this demand con…guration (iv):

Insert Figure 6 Equilibrium prices and pro…ts are computed in Appendix 6.2. Thus, we just have to calculate for which value of the demand con…guration is changing. If both …rms set equilibrium prices as in Appendix 6.2.3, then there exists a threshold from which bundle (AB12) is no longer followed by (AB21) but by (A2B2). Calculating this threshold leads to = HT 1 = 1 6 tb 6ta . On the other hand, if …rms set equilibrium prices as in Appendix 6.2.4, then we …nd the threshold HT 2 beyond which the second demand con…guration arises. This threshold is given by = HT 2 = ta tb 5ta+tb . For more details, refer to Appendix 6.2.5. For between 1 6 tb 6ta and ta tb 5ta+tb , …rms set prices in such a way that the last consumer purchasing (AB12) is indi¤erent in the choice between (AB12), (AB21) and (A2B2). Therefore, the …rst demand con…guration still holds. The determination of the equilibrium prices in this region25 is similar to the one in a standard Hotelling model when we move from local monopoly to competition (Economides, 1984, Gabszewicz and[START_REF] Gabszewicz | Spatial competition and the location of …rms[END_REF]. We therefore establish the following lemma: 5ta+tb , equilibrium pro…ts are given by:

2 A i B j = 1 8 t A 7 72 t B , (6)
2

A i B j = 1 8 (t A + t B ) + t A t B 2(t A t B ) 2 (3(t B t A ) + 2 (8t A + t B ) 4 2 t A ), (7)
2

A i B j = 1 8 (t A + t B ) (1 4( 2 )) t A t B 18(t A + t B ) . ( 8
)
Proof. See Appendix 6.2 Intuitions about pro…t levels are the same as in the case of homogeneous mergers, but reversed relative to the correlation. In the heterogeneous merger case, bundles are composed of goods with opposite locations. Subsequently, and contrary to the homogeneous merger case, the bundle competition e¤ect appears for low correlation values. In the same way, the sorting e¤ect is also reversed relative to the correlation. Indeed, by sorting consumers, …rms can make more pro…ts than they can in cases of independent pricing, when the correlation is weak. Furthermore, when the correlation is high ( < HT 2 ), the competition e¤ect between bundles appears and dominates the sorting e¤ect. This is due to the opposite locations of merged …rms on each energy market. We …nd a prisoner's dilemma, but now for high correlation values. And now that competition outcomes corresponding to merger waves are studied, in the next section we compute and analyze the SPNE of the game.

Equilibrium of the game

In this section, we solve the whole game. Next, we analyze the welfare and competition policy implications.

Equilibrium computation

At the equilibrium of the merger game, we can write the following proposition:

Proposition 2 In equilibrium, a merger wave occurs and …rms choose a mixed bundling strategy.

Proof. The merger game presented in Figure 2 establishes …ve possible outcomes. If there are two homogeneous mergers (respectively heterogeneous) and according to the Lemma 1 (respectively Lemma 3), then …rms follow a mixed bundling strategy and pro…ts are given in section 3.1 (respectively 3.2). If there is no merger, then pro…ts are presented in section 2.3. At last, if there is one homogeneous merger (respectively heterogeneous), the proof of Lemma 1 (respectively Lemma 3) allows us to assert that the merged …rm follows a mixed bundling strategy. 26 By comparing the di¤erent payo¤s associated with the various outcomes, we …nd that in equilibrium, it occurs either a homogeneous merger wave or a heterogeneous one27 (according to the correlation of reservation prices).

The case where the …rst …rms merge with the wrong partner in order to prevent another merger is not feasible. Indeed, when a merger is already achieved, there always exists an incentive to achieve a second merger whatever is the value of delta (see proof of proposition 1 in Reisinger (2006) for the homogeneous case). As …rms anticipate this, the …rst merger will be those for which the competition e¤ect created by the second merger is minimized. Moreover, this …rst merger is achieved because, if is not the case, the second mover tries to bene…t from the discrimination e¤ect by merging and hurts the pro…t of the …rst mover. Thus, we are interested in outcomes which trigger a merger wave. As there can be only one type of merger at the same time, that is two homogeneous or two heterogeneous mergers, we compare the pro…ts associated with these outcomes according to . Because the game is symmetric, it is su¢cient to compare the pro…t of …rm A 1 B 1 further to a homogeneous merger wave and the pro…t of …rm A 1 B 2 further to a heterogeneous wave.

In order to rank these equilibria pro…ts according to , we must order the thresholds for which pro…t functions are modi…ed. These thresholds are given by HM

= t A t B 5t A +t B < HM 1 = 3 2 ( t A +t B 5t A +t B ).
We deduce the following ranking:

HT 1 < HT 2 < HM 1 < HM 2
. There is a turnover between the two types of merger according to the correlation of consumers' reservation values for the two energies. This turnover occurs around three thresholds, i.e., 1 , 2 and 3 . The pro…ts comparison according to allows the equilibrium and the associated pro…ts to be computed. The following Proposition28 presents this equilibrium and Figure 7, below, illustrates the turnover between homogeneous and heterogeneous merger pro…ts: Proposition 3 In equilibrium, …rms always make more pro…ts than they do in the independent pricing case. Moreover, …rms choose to merge either in a homogeneous way or in a heterogeneous one. This depends on the reservation prices correlation. They merge in a homogeneous way for 0 1 and 2 < 3 and they merge in a heterogeneous one for 1 < < 2 and for 3 1 2 .

Insert Figure 7 The intuitions behind the homogeneous merger wave and the heterogeneous merger wave have already been explained. There is a trade-o¤ between a sorting e¤ect due to bundling which is positive from the …rms' point of view and a competition e¤ect passing through energy bundles ("business-stealing e¤ect", Reisinger 2006). In the homogeneous merger case, this competition e¤ect exists only if the correlation is weak. In the heterogeneous merger case, this competition e¤ect exists only if the correlation is strong. Thus, when the correlation is su¢ciently high ( 1), …rms avoid this competition effect by merging in a homogeneous way. Conversely, if the correlation is su¢ciently weak (

3 ), …rms avoid this competition e¤ect by merging in a heterogeneous way. The sorting e¤ect is at a maximum for a weaker correlation with heterogeneous mergers than it is with homogeneous ones. Therefore, …rms choose to alternate between these two types of mergers when the correlation is intermediate ( 1 < < 3 ). This alternation is due to the following fact: according to the type of merger, the maximum pro…t values occur at di¤erent levels of correlation. These maximum values correspond to a strong sorting e¤ect without triggering the business-stealing e¤ect. Indeed, the sorting e¤ect is even stronger than product bundles, since the two …rms are similar from consumers' point of view. This lack of di¤erentiation between bundles …nally creates this new competition e¤ect.

Concerning the heterogeneous merger, and therefore the bundles of energies with opposite locations, the maximum value of pro…t is reached at HT 2 1 4 . Concerning the homogeneous merger, and therefore the bundles of goods with the same location, the maximum value of pro…t is reached at HM 1 1 4 . In both cases, the di¤erent opportunities of merger allow …rms to bene…t better from the sorting e¤ect by positioning their sales on the two energy markets according to the correlation. Because of these merger opportunities, …rms avoid the competition e¤ect between bundles. At the equilibrium, the competition e¤ect exists only for the two following ranges of parameters corresponding to "touching" equilibria: 1 < < HT . Such is the case when t B ! 0. Indeed, transportation costs vary the intensity of the two e¤ects and the thresholds de…ning the di¤erent types of equilibria. Intuitively, the weaker the transportation cost t B , the weaker the sorting and the competition e¤ects. The two maximum pro…t values tend to get closer. Therefore, the minimum and maximum ranges of correlation become larger. To an extreme degree, when t B ! 0, bundling has no e¤ect and the price of the gas B is equal to its marginal cost of production. We …nd the Bertrand paradox, since there is no horizontal di¤erentiation of the gas B and the sorting e¤ect does not exist any more. Conversely, when t B ! t A , the e¤ects are intensi…ed, and for the two merger cases, the sorting e¤ect is stronger and the competition e¤ect between energy bundles (business-stealing e¤ect) does not take place as easily. Indeed, the trade-o¤ between the low price of the bundle and the additional distance to cover for the gas B tends to favor separate energies consumption. Figure 8, below, illustrates these two limit cases:

Insert Figure 8 The possibility of the four energy …rms merging with …rms, either at the same or at the opposite location, eliminates the prisoner's dilemma, which is already underlined in the two merger types. Indeed, as …rms can choose their merger partner in relation to the correlation of reservation prices, …rms can bene…t better from the sorting e¤ect without being a¤ected by the competiton e¤ect. In order to evaluate the scope of this study, we will focus on welfare implications in the following section.

Welfare analysis

First, we focus on social welfare. As a benchmark, we calculated the maximum welfare. The welfare is maximized when transportation costs are minimized. Indeed, price levels do not a¤ect social welfare for the reason that the volume of consumption is unchanged in this model. Maximum welfare is achieved when consumers who are located at x k with 0

x k 1 4 and 3 4

x k 1 for k = A (respectively k = B) buy electricity A from …rm A 1 (respectively gas B from …rm B 1 ) and when consumers who are located at 1 4

x k 3 4 for k = A (respectively k = B) buy electricity A from …rm A 2 (respectively gas B from …rm B 2 ). This situation corresponds to the independent pricing case for which covered distances are minimal. We note W IP social welfare when energy …rms follow an independent pricing strategy:

W IP = W 0 1 48 (t A + t B ), (9)
with

W 0 = K A + K B c A c B .
In this case, consumers always buy the energies that are near their locations. Therefore, welfare is maximal. Now, we will focus on social welfare in cases of merger waves. We describe social welfare at the game equilibrium in the following proposition29 and in Figure 9:

Proposition 4 In equilibrium, social welfare is always lower with bundling strategy than in the independent pricing case or than without merger. Moreover, by noting W r the welfare obtained by Reisinger and W the welfare at the equilibrium of our model, we …nd that

W = W r if 0 1 or if 2 < 3 , that W < W r if 1 < 2 , and that W > W r if 3 < 1=2.
The fact that social welfare is always lower with bundling strategy is due to the sorting e¤ect created by this strategy. Moreover, the social welfare di¤erences with the model of Reisinger are explained by the opportunity given to energy …rms to merge in di¤erent ways according to the correlation between the two energy markets. The main intuition is the following. Firms avoid the competition e¤ect by selecting the right merger partners according to the correlation of consumers' reservation values for the two goods.

Insert Figure 9 When …rms do not merge, or when …rms sell their goods independently, social welfare is higer than in the equilibrium of our game. The …rst recommendation is as follows: European Commission should prevent mergers from being allowed to bundle on energy markets, if one relies on our model assumptions. For instance, one might assume e¢ciency gains which could make merger advantageous from a social welfare point of view. Another way for competition authorities might be to forbid bundling. It is interesting to note that the possibility for …rms to choose a type of merger, homogeneous or heterogeneous, makes our results concerning welfare more balanced than those in Reisinger (2006). But, in accordance with Reisinger (2006), we note that bundling is harmful to social welfare. Indeed, the prisoner's dilemma disappears, which increases …rms' pro…ts when correlation is low. That increases social welfare. When the correlation is low ( > 3 ), consumers choose packages with energies from faraway locations, thus there are lower transportation costs than in the model of Reisinger (2006). On the other hand, for the intermediate values of correlation, the possibility for …rms to merge in homogeneous or heterogeneous ways can make pro…ts higher but the transportation costs are also considerably higher. This accounts for social welfare jumps in 1 and 3 . Finally, for high correlation values ( < 1 ), we …nd the same equilibrium as Reisinger (2006) and the same level of social welfare. Moreover, we observe that the competition authorities generally prefer to take consumers' surplus into account rather than social welfare. Thus, we establish the following corollary30 to Proposition 4:

Corollary 1 In equilibrium, the consumers' surplus is always lower than in the independent pricing case or without merger.

In energy markets, several …rms merge and some of these mergers are presented in Table 1 in the Appendix 6.7. We can think that energy …rms merge in order to supply bundles. In Table 1, we make list of dowstream market mergers in energy markets. We leave out other types of mergers in energy markets in order to limit other e¤ects than bundling in incentives to merge, as synergy e¤ects due to a vertical integration for example. So, we can provide merger cases which appear mainly because of bundling strategies created by mergers. In Italy, the electric …rm Enel merged in 2004 with the gas retailer Italgestioni in the downstream energy market. This example among others given in Table 1 can be analyzed through the homogeneous and heterogeneous mergers concepts. This depends on merged …rms' characteristics with regard to each energy sold in packages.

Thus, we calculate the consumers' surplus. We note S IP , the consumers' surplus for the independent pricing case, corresponding to the maximum surplus case. This value is easily given by social welfare minus two times the merged …rm pro…t in the independent pricing strategy:

S IP = W 0 13 48 (t A + t B ), (10)
with

W 0 = K A + K B c A c B .
We can conclude that the analysis of consumers' surplus gives the same results as the social welfare analysis. The …rms' gains from bundling are insigni…cant in comparison with the bundling's e¤ect on social welfare. The graphical representation of equilibrium consumers' surplus is the same as it is with social welfare, with a lower level and the slopes more pronounced. The analysis of consumers' surplus as social welfare comes to the same conclusion.

Conclusion

We observe an increasing number of mergers in the energy markets. A lot of them concern …rms from di¤erent markets such as gas and electricity. Whatever the merger, it allows …rms to supply bi-energy bundles. Our paper studies the bundling e¤ects on merger incentives in energy markets. We show that bundling strategies always create merger incentives for specialized …rms (electricity or gas …rms). The intuitive explanation is the following. Although competition e¤ects of a merger involving …rms from two independent markets are non-existent with an independent pricing, competition e¤ects appear if the merger allows product bundling.

First, we show that there is always an incentive to follow a bundling strategy once the merger is achieved. When bundling is not possible, we …nd that there is no incentive to merge. From these two results, we deduce that bundling strategy generates not only a merger incentive but also a merger wave. This incentive comes from the sorting e¤ect of the bundling strategy. However, a competition e¤ect, which is negative for …rms' pro…ts, is also generated by the bundling strategy. We …nd that, in order to take better advantage of the sorting e¤ect and to avoid the competition e¤ect, energy …rms choose between two merger types. This choice is function of the correlation of consumers' reservation prices for the two energies. An electric …rm can merge with a gas …rm at the same location on the other circle. An electric …rm can also merge with a gas …rm at opposite location on the other circle. We call homogeneous and heterogeneous mergers, respectively. These merger opportunities remove the prisoner's dilemma created by the dominance of the competition e¤ect which is emphasized by Reisinger (2006).

Our model has important implications concerning competition policy. We show that bundling strategies have a negative e¤ect on social welfare, but in our model this e¤ect is weaker than in Reisinger (2006). Competition authorities should pay more attention to mergers in domestic markets, such mergers may be authorized by the governments in order to promote national champions. However, we must note that our analysis don't take into consideration potential e¢ciency gains following a merger. A direction for future research could be the introduction of these e¤ects. The introduction of other e¤ects like e¢ciency gains could a¤ect the results.

Appendices

Independent pricing case

In the independent pricing case, the computation of the equilibrium is the same as in the standard model of Salop (1979). We model two independent and horizontally di¤erentiated markets. It should be noted that A i (respectively B j ) is the pro…t of the single …rm A i (respectively B j ). In every case, equilibrium prices are given by:

p A = c A + 1 4 t A , (11)
p B = c B + 1 4 t B .
First, we consider the case where there is no merger. Thus, there are two monoproduct …rms in each energy market, A and B, competing in prices. The equilibrium pro…ts are given by:

A i = 1 8 t A , ( 12
) B j = 1 8 t B .
We note that z A i B j is the pro…t of the merged …rm A i B j where z = 1; 2 is the number of mergers. Whether there are one or two homogeneous mergers, the equilibrium pro…ts are given by: 8i = j,

2 A i B j = 1 8 t A + 1 8 t B , or (13)
8i = j, 1 A i B j = 1 8 t A + 1 8 t B , A i = 1 8 t A , B j = 1 8 t B .
Whether there are one or two heterogeneous mergers, the equilibrium pro…ts are given by:

8i 6 = j, 2 A i B j = 1 8 t A + 1 8 t B , or (14)
8i 6 = j,

1 A i B j = 1 8 t A + 1 8 t B , A i = 1 8 t A , B j = 1 8 t B .
6.2 Heterogeneous mergers 6.2.1 Proof of the Lemma 3

In the case of an heterogeneous merger wave, the …rms A i and B j are merged 8i 6 = j with i; j = 1; 2: Let us analyze if there is an incentive for the merged …rm A 1 B 2 to introduce a bundle. First, we consider the case where both …rms do not bundle. Since the equilibrium is symmetric, both …rms charge the same independent prices p IP A and p IP B , and earn pro…ts of

A i B j = 1 2 (p IP A c A + p IP B c B ) 8i
6 = j with i; j = 1; 2. Now, if …rm A 1 B 2 introduces a bundle, that means selling both goods together at a price p 12 AB < p 1 A + p 2 B . We analyze the case where p 12 AB = p IP A + p IP B , but where p 1 A = p IP A + " 1 and p 2 B = p IP B + " 1 , with " 1 > 0 but small. So, the …rm A 1 B 2 increases its pro…ts raising its independent prices by " 1 and sets the bundle price equal to the sum of the independent prices. or

A 2 B 1 = A 2 B 1 + " 2 (1 2 ) (2" 1 " 2 ) 1 t A + 1 t B (2" 2 ) 2 1 t A + 1 t B . (21)
If " 1 and " 2 are small, then A 2 B 1 > A 2 B 1 , provided that < 1 2 . Thus, …rm A 2 B 1 also has an incentive to bundle.

Proof of the consumption combinations:

In order to study the di¤erent equilibria of heterogeneous merger waves, we have to establish several claims concerning consumption combinations:

Claim 1 There cannot exist direct rivalry between product combination (A1B1) and (A2B2).

Proof. The method of proof is the same than for claim 1 of Reisinger (2006) Claim 2 .

(?) Take x A and x 0 A with 0 x A ; x 0 A 1 2 and x 0 A < x A . If (AB12) is optimal at x A , then (AB21) can never be optimal at x 0 A . (??) Take x A and x 0 A with 1 2

x A ; x 0 A 1 and x 0 A < x A . If (AB21) is optimal at x A , then (AB12) can never be optimal at x 0 A .

Proof. The method of proof is the same than for claim 2 of Reisinger (2006) Claim 3 . (?) Take x A and x 0 A with 0 x A ; x 0 A 1 2 and x 0 A < x A . If (A1B1) is optimal at x A , then (A2B2) can never be optimal at x 0 A . (??) Take x A and x 0 A with 1 2

x A ; x 0 A 1 and x 0 A < x A . If (A2B2) is optimal at x A , then (A1B1) can never be optimal at x 0 A .

Proof. The method of proof is the same than for claim 3 of Reisinger (2006) Claim 4 . (?) Take x A and x 0 A with 0 x A ; x 0 A 1 2 and x 0 A < x A . If (AB12) is optimal at x A , then (A2B2) can never be optimal at x 0 A . (??) Take x A and x 0 A with 1 2

x A ; x 0 A 1 and x 0 A < x A . If (A2B2) is optimal at x A , then (AB12) can never be optimal at x 0 A .

Proof. The method of proof is the same than for claim 4 of Reisinger (2006) Claim 5 .

(?) Take x A and x 0 A with 0 x A ; x 0 A 1 2 and x 0 A < x A . If (A1B1) is optimal at x A , then (AB21) can never be optimal at x 0 A . (??) Take x A and x 0 A with 1 2

x A ; x 0 A 1 and x 0 A < x A . If (AB21) is optimal at x A , then (A1B1) can never be optimal at x 0 A .

Proof. The method of proof is the same than for claim 5 of Reisinger (2006) 6.2.3 Proof of the equilibrium: heterogeneous mergers and strong correlation

Assuming small, we consider a consumer at location x A = 0. If we move clockwise on circle A, then the consumer who is indi¤erent between (A1B1) and (AB12) is located at:

x A = 1 4 + p 12 AB p 1 A p 1 B t B . ( 22
)
The product combination which is bought to the right of (AB12) is (AB21). The marginal consumer between to buy the bundle (AB12) and the bundle (AB21) is located at:

x A = 1 t A t B (p 21 AB p 12 AB + 1 4 (t A t B ) + t B ). (23)
Moving further to the right the next combination which is bought is (A2B2) and the marginal consumer is located at:

x A = 1 4 + p 2 A + p 2 B p 21 AB t B . ( 24
)
If we pass the point 1=2 and move upward on the left side of the circle, we get the same product structure as on the right side, because of symmetry, only with …rm A i and B j reversed. The pro…t function of …rm A 1 B 2 is therefore: Because of symmetry we get a similar function for …rm A 2 B 1 . Calculating prices and pro…ts, for both …rms, we get:

A 1 B 2 = (p 1 A c A )( 1 4 + p 12 AB p 1 A p 1 B t B + 1 4 + p 12 AB p 1 A p 1 B t B ) (25)
p A = c A + 1 4 t A 1 6 t B , (26)
p B = c B + 1 12 t B , 8i 6 = j p ij AB = c A + c B + 1 4 (t A t B ), 8i 6 = j z=2 A i Bj = 1 8 t A 7 72 t B .
with i; j = 1; 2: When the value of increases, there is a disappearance of bundle options. It exist a value of where the bundle (AB12) is not following by the bundle (AB21): At this threshold demand structure changes and there is only one option of bundle consumption.

Proof of the equilibrium: heterogeneous mergers and weak correlation

Assuming high, we consider a consumer at location x A = 0. If we move clockwise on circle A, there is a consumer who is indi¤erent between (AB12) and (A2B2) located at:

x A = 1 4 + p 2 A + p 2 B p 12 AB t A . ( 27
)
If we move clockwise on circle A, the next product combination is (AB21). So, The marginal consumer between (A2B2) and (AB21) is located at:

x A = 3 4 + p 21 AB p 2 A p 2 B t B . ( 28
)
If we pass the point 1=2 and move upward on the left side of the circle A, we get the same product structure as on the right side, because of symmetry, only with …rm A i and B j reversed. Therefore, the pro…t function of …rm A 1 B 2 is: ).

A 1 B 2 = (
Because of symmetry we get a similar function for …rm A 2 B 1 . Calculating prices and pro…ts, for both …rms, we get:

p A = c A + 1 4 t A + ( 1

Proof of the intermediate thresholds: heterogeneous mergers

For the pro…t function (26) arises, (A1B1) must be followed by (AB12) and not by (AB21). The frontier between (A1B1) and (AB12) at the equilibrium prices is given by:

x A = 1 12 . ( 31
)
The frontier between (A1B1) and (AB21) at the equilibrium prices is given by:

x A = 1 4 + t B t A t B . ( 32
)
If 2 < 3 , in equilibrium, …rms choose to merge in a homogeneous way. Prices and pro…ts are given by (3) for 2 < HM 1 and by (4) for HM 1 < < 3 . If 3 1 2 , in equilibrium, …rms choose to merge in a heterogeneous way. Prices and pro…ts are given by (30).

6.5 Proof of the social welfare 6.5.1 Proof of the social welfare: homogeneous mergers See Reisinger (2006).

Proof of the social welfare: heterogeneous mergers

As in the previous appendix, we compute the welfare by integrating equilibrium prices in every frontiers of consumption combinations and by calculating total transportation costs. It is not necessary to compute the welfare for < 1 6 t B 6t A since it is never the equilibrium welfare. Here, we compute the welfare when 3 2 ( t A +t B 5t A +t B ) < After some manipulations, we get:

1 3 + t B 6t A and when > 1 3 + t B 6t A . If 3 2 ( t A +t B 5t A +t B ) < 1 3 + t B 6t A : W HT 2 = K A + K B c A c B (37) t A 0 B B @ 1 4 t B 4t B t A t A t B Z 0 (x) 2 dx + 1 2 Z 1 4 t B 4t B t A t A t B ( 1 2 x) 2 dx 1 C C A t A 0 B B @ 1 4 3t B 4t B 3t A t A t B Z 1 2 (x 1
W HT 2 = K A + K B c A c

  . Moreover, these ranges are very restricted since, respectively, they correspond at the most to 1

Moreover, administrative "principle of specialization" that formerly assigned public monopoly operators to produce only a single specialized good, is removed.

2 For instance in France, the dominant operators propose energy and service packages to professionals such as "Provalys" for Gaz de France and "Essentiel Pro" for Electricité de

France. 3 As illustration, we can lean on the competition which existed in the German energy market before the mergers which we have

just quoted.4 We show that bundling creates an incentive to merge. Nevertheless, there exist other merger motives in

energy markets.[START_REF] Linqvist | The insider's dilemma: an experiment on merger formation[END_REF] For more explanations about convergence between gas and electricity, see[START_REF] Toh | The impact of convergence of the gas and electricity industries: Trends and policy implications[END_REF] andBazart (2007).

Note that professionals also consume several energies at the same time.

This reason is evoked byJacobsen et al. (2006).

When a …rm sells its goods both separately and bundled in a package, this …rm follows a mixed bundling strategy. When a …rm commits to supply only the bundle, it follows a pure bundling strategy.

[START_REF] Schmalensee | Gaussian demand and commodity bundling[END_REF] shows their results are robust to a bivariate normal distribution. As for them, McAfee et al. (1989) generalize these results to almost all distributions.

InNalebu¤ (2004), bundling is optimal even without any commitment.

By contrast, when consumers buy all their products from one …rm (the one-stop shopping model), nonlinear pricing leads to higher pro…t and welfare but often lower consumers surplus, than linear pricing.

In this model, bundling must not be incited by e¢ciency gains, for instance.

, but without loss of generality. Indeed, if we placed …rms more closely, results would be qualitatively the same. They would just be shifted relative to .15 Electricity and gas can be used for lighting and heating needs, for instance.

The limit cases t B ! t A and t B ! 0 are studied in section 4.

The proof is given byReisinger (2006).

There is always an incentive to monopolize the electricity market A or the gas market B as long as reservation prices of consumers are su¢ciently high. But this is detrimental for consumers and generally, authorities forbid monopolization.

Here, we model no other mechanism able to create a merger incentive. For instance, considering synergies could create an incentive to merge.

We excluded the pure bundling strategy, because …rms have no incentive to manage without an additional discrimination tool, given that …rms are in competition.

Firms supplying bi-energy packages o¤er discounts to consumers buying packages instead of independent energies. For instance, Gaz de France/Suez proposes the bi-energy package at the price of the two separate energies minus 36e (respectively 66e) for one year (respectively for two years).

See Appendix 6.2.

For more details, see appendix 6.2.6.

For clarity reasons, we do not present all asymmetric cases because they are dominated. As an illustration, we present a case in Appendix 6.3. For more details, please contact the authors.

Reisinger (2006) shows, in the extension of his model, that an independent pricing equilibrium appears when the game is sequential. This can occur if …rms can commit not to practice bundling. Here, the merger game is sequential. But, as …rms can choose in the …rst stage to merge either in a homogeneous way or in a heterogeneous one, it is never pro…table for …rms to commit to an independent pricing strategy or to commit not to merge. Indeed, it always exists one type of merger (homogeneous or heterogeneous) which allows …rms to earn higher pro…ts than in the separate sales con…guration. This is valid when there is only one merger or two sequential mergers.

The proof is given in Appendix 6.4.

The proof of the Proposition 4 is given in Appendix 6.5.

The proof for Corollary 1 is given in Appendix 6.6.

We have to distinguish between two cases, either if is "near" 1 2 or not, because this changes the demand structure on the circles. First, look at the case where is not near 1 2 . If …rms do not bundle there are four demand regions on the circles, namely (A1B1); (A1B2); (A2B2), and (A2B1). The frontiers between these regions (or the marginal consumers) are given by 1 4 for the frontier between (A1B1) and (A1B2), by 1 4 for (A1B2) and (A2B2), by 3 4 for the frontier between (A2B2) and (A1B2) and …nally by 3 4

for (A1B2) and (A1B1). If the …rm A 1 B 2 introduces the bundle (AB12), the frontiers are changed to 1 4

for the frontier between (A1B1) and (AB12), to 1 4 + " 1 t A for (AB12) and (A2B2), to 3 4

for the frontier between (A2B2) and (AB12) and …nally to 3 4 " 1 t B for (AB12) and (A1B1). The new pro…t function of …rm A 1 B 2 is given by:

or

This pro…t is always higher than the previous pro…t A 1 B 2 as long as > 0 because " 1 can made arbitrary small and so (" 1 ) 2 tends faster towards 0 than " 1 . We made the proof than the merged …rm A 1 B 2 has an incentive to introduce its bundle. Let us focus on …rm A 2 B 1 to introduce its bundle if the …rm A 1 B 2 is already bundling. The pro…t of …rm A 2 B 1 if …rm A 1 B 2 bundles while …rm A 2 B 1 not is given by:

If the …rm A 2 B 1 chooses to bundle (AB21) and set p 21 AB = p IP A + p IP B , and sells their goods independently at the price p 2 A = p IP A + " 2 and p 1 B = p IP B + " 2 , with " 2 > 0 but small, the frontiers are given by 1 4 (" 1 +" 2 ) t B for the frontier between (A1B1) and (AB12), by

t B for the frontier between (AB21) and (A2B2), by 3 4 (" 1 +" 2 ) tB for (A2B2) and (AB21), by 1

for the frontier between (AB21) and (AB12) and …nally by 3 4

for (AB12) and (A1B1). The new pro…t function of …rm A 2 B 1 ; when it proposes bundles, is then:

Thus, for " 1 and " 2 small, bundling is pro…table since (" 2 ) 2 and (" 1 " 2 ) tend faster towards 0 than " 2 . We shown that bundling is always a pro…table strategy for small. Now let us turn the case where is near 1 2 . First, we analyze the incentive of …rm A 1 B 2 to introduce its bundle while the other …rm practices independent pricing. If …rm A 1 B 2 does not bundle, product combinations are: (A1B2); (A2B2), (A2B1) and (A1B1). The frontiers are given by 1 4 for the frontier between (A1B2) and (A2B2), by 3 4 for (A2B2) and (A2B1), by 3 4 for the frontier between (A2B1) and (A1B1)and …nally by 5 4

for (A1B1) and (A1B2). If merged …rm A 1 B 2 bundles, then the frontiers are given by 1 4 + " 1 t A for the frontier between (AB12) and (A2B2), by 3 4 " 1 t B for (A2B2) and (A2B1), by 3 4 + " 1 t A for the frontier between (A2B1) and (A1B1) and …nally by 5

4

for (A1B1) and (AB12). The pro…t of …rm A 1 B 2 if it bundles is:

or

A 1 B 2 is always higher than A 1 B 2 if < 1 2 since (" 1 ) 2 tends faster towards 0 than " 1 . Therefore, A 1 B 2 has an incentive to bundle. Now, let us analyze the pro…t of the …rm A 2 B 1 if the …rm A 1 B 2 is already bundling. If …rm A 2 B 1 chooses not to bundle, its pro…t is:

If …rm A 2 B 1 also bundles, it sets prices p 21 AB = p IP A + p IP B , sells its goods separately at p 2 A = p IP A + " 2 and p 1 B = p IP B + " 2 , with " 2 > 0 but small. Now, the frontiers are given by

for the frontier between (AB12) and (A2B2), by 3 4

and (AB21), by 3 4

for the frontier between (AB21) and (A1B1) and …nally by

for (A1B1) and (AB12). Pro…t of …rm A 2 B 1 if both …rms bundle is then:

For the demand structure (iii) arises, then (31) must be smaller than (32). This gives the …rst threshold:

For the pro…t pro…t function (29) to arise, the option consumption (A1B1) must be followed by (AB21) and not by (AB12). Calculating in the same way as before by inserting the equilibrium prices corresponding to the pro…t function ( 29) in ( 31) and ( 32) gives that demand structure (iv) arises only if:

)

Proof of the intermediate equilibrium: heterogeneous mergers

In the region such as

5ta+tb , …rms set their prices in such a way that demand structure (iv) arises. Prices are determined in order to satisfy the following constraint:

This means that (A2B2) is followed by (AB12) and not by (AB21), and that thresholds

and HT 2 are given by ( 26) and ( 30) respectively. Because of the instability of the demands for these parameter values, we use a linear interpolation of the equilibrium prices given by ( 26) for HT 1 and by (30) for HT 2 . We check that prices and pro…ts given below constitute an equilibrium in the competition game:

Proof of the Proposition 2

Consider a homogeneous merger case, where A 1 and B 1 merge and the two other …rms A 2 and B 2 are independent. This appendix is still valid if A 2 and B 2 merge but sell their products separately. The merger pro…t of A 2 and B 2 is merely the sum of A 2 and B2 . As an illustration, we focus on the case where correlation of reservation values is weak, that is for 0 < < 3(ta+tb) 2(5ta+tb) . If the merged …rm follows an independent pricing strategy, then pro…ts are given by:

Prices are given by:

If the merged …rm practices mixed bundling, pro…ts are the following:

,

Prices are given by:

> 1 8 t A + 1 8 t B , the merged …rm A 1 B 1 has an incentive to practice mixed bundling. There is a trade-o¤ between two e¤ects. First, the bundle of the merged …rm decreases the market shares of independent …rms. Then, as prices of goods sold independently of merged …rm are higher than those of independent …rms (e.g.

, there is a positive e¤ect on the market shares of independent …rms. In this con…guration, these two e¤ects perfectly o¤set, what explains that the pro…ts of independent …rms are not a¤ected by the rival's bundle. By increasing its prices of separate sales, with unchanged market share, the competitor earns more pro…ts by bundling its goods together.

Proof of the Proposition 3

Pro…ts corresponding to homogeneous and heterogeneous merger waves are equal for three values of parameter.

, with HT 1 < 1 < HT 2 , the heterogeneous merger pro…t corresponding to the equilibrium ( 7) is equal to the homogeneous merger pro…t corresponding to the equilibrium (3). After 1 , the equilibrium pro…t (7) becomes greater than the equilibrium pro…t (3).

Next, for 2 = 1 4 , with HT 2 < 2 < HM 1 , the heterogeneous merger pro…t corresponding to the equilibrium (8) is equal to the homogeneous merger pro…t corresponding to the equilibrium (3). After 2 , the equilibrium pro…t (3) becomes greater than to the equilibrium pro…t (8).

, with HM 1 < 3 < HM

2

, the heterogeneous merger pro…t corresponding to the equilibrium ( 8) is equal to the homogeneous merger pro…t corresponding to the equilibrium (4). After 3 , the equilibrium pro…t (8) becomes greater than the equilibrium pro…t (4).

It is easy to prove these di¤erent pro…t functions are monotonic for considered parameter values. Thus, pro…t di¤erences increase or decrease as speci…ed here. It is also easy to prove that there is only for these three values of , that are 1 , 2 , and 3 , that pro…ts of heterogeneous and homogeneous merger wavesbecome level. Moreover, for t A t B 0; the following ranking is still valid: 0

HM 2 1 2 : We can deduce the following lemma:

Lemma 5 If 0 1 , in equilibrium, …rms choose to merge in a homogeneous way. Prices and pro…ts are given by (3).

If 1 < < 2 , in equilibrium, …rms choose to merge in a heterogeneous way. Prices and pro…ts are given by (36) for 1 < < HT 2 and by (30) for HT 2 < 2 .

Finally, for > 1 3 + t B 6t A , the welfare is given by:

After some manipulations, we get :

Proof of the social welfare at game equilibrium with bundling

The appendices 6.5.1 and 6.5.2, as the Lemma 5 allow to determine the social welfare at equilibrium game.

Lemma 6 In equilibrium, the social welfare is given by:

6.5.4 Proof of the welfare comparaison with that of Reisinger (2006) For 0

We obtain, for these parameter values, that

. But, for these parameter values,

At last, for HM 2 < 1 2 , we obtain

. But, for these parameter values, W R = W IP 1 36 (t A +t B ) t B t A . Thus, we obtain that W HT

6.6 Proof of the consumers' surplus 6.6.1 Proof of the consumers' surplus: homogeneous mergers

We just subtract two equilibrium pro…ts from equilibrium welfare to …nd the consumers' surplus. For a homogeneous merger wave and for < 3 2 ( t A +t B 5t A +t B ), the consumers' surplus is given by: For a homogeneous merger wave and for 3 2 ( t A +t B 5t A +t B ) < < 1 3 + t B 6t A , the consumers' surplus is given by:

Proof of the consumers' surplus: heterogeneous mergers

We just subtract two equilibrium pro…ts from equilibrium welfare to …nd the consumers' surplus. For a heterogeneous merger wave and for 1 6 t B 6t A < < ta tb 5ta+tb , the consumers' surplus is given by:

For a heterogeneous merger wave and for > ta tb 5ta+tb the consumers' surplus is given by:

.

Proof of the consumers' surplus at equilibrium game with bundling

The Appendices 6.6.1 and 6.6.2, as the Lemma 5, allow to determine the consumers' surplus at the equilibrium.

Lemma 7 In equilibrium, the consumers' surplus is given by: