domain: sde sdv

text:
Director, Philippe Grandcolas
email: pg@mnhn.fr

Biodiversity in France

Introduction

As a European country, France is at the crossroad of several biogeographic areas and pathways. Its terrestrial territory is limited by the North Sea, the Atlantic Ocean in the West and the Mediterranean Sea in the South, being therefore exposed to different marine influences. Mountains as well are part of this natural delimitation, with Vosges in the NorthEast, the Alps forming a mountainous belt in the South East and the Pyrenean chain in the Southwest separating France from the Iberian Peninsula. The resulting natural landscape is a very diverse mix of different influences, oceanic on the western side, continental in the East, alpine in South East and Mediterranean in the South [START_REF] Metzger | A climatic stratification of the environment of Europe[END_REF]. This landscape is located around 46°00'N latitude and 2°00'E longitude and the altitude peaks at more than 4000 meters in Alps and Pyrénées while more than 70% of the territory is below 500 m. Actually, among the biogeographic influences found in Europe, only the Macaronesian one is to be excluded and the boreal one is present in several remote and relict places of small size. The territory has been profoundly marked by past climatic events and the presence of human populations as well.

First a short account of biodiversity will be presented, providing numbers for large groups of organisms, signaling lacks or gaps in the knowledge as well and indicate resources to have access to primary data. Second, the different ecosystems comprising these species are detailed and are provided a few examples of organisms belonging to different groups and exemplifying different biogeographic trends. Third, emphasis will be put on the dynamics of the biota since the last quaternary climatic events, showing how they evolved and how they could be modified in the future according to climatic change.

Biodiversity numbers

The flora and fauna of France are remarkably rich for a temperate and human-modified landscape, because of the combination of different climatic and biogeographic areas. The total number of living species in France is more than 85000 of which more than 2000 are endemic and about 2500 are introduced (Gargominy et al., 2016). There about7000 species of Plants (Tracheophyta), which represent 40% of the European species, therefore less than a few other European Mediterranean countries (Tison and Foucault, 2004). If introduced and cultivated species are considered, the number of plants in France must be set up around 10000. The largest groups in this flora belong to the palearctic realm in the families Asteraceae (1165 species), Rosaceae (580 species), Poaceae (518 species), Fabaceae (436 species), Brassicaceae (275 species), Apiaceae(262 species) and Caryophyllacae (225 species). Other non-vascular plants (Bryophyta, Hepatica, Algae) totalize around 3000 species. Fungi comprise more than 3000 species.

Concerning animals, there are about 1500 vertebrate species (half of them are marine), among which 284 bird species nesting for 500 species observed in France [START_REF] Issa | Atlas des oiseaux de France métropolitaine[END_REF], 43 amphibians (Lescure andMassary, 2012), 135 mammals. Mammals belong to families Leporidae (4 species), Castoridae (1 species), Cricetidae (14 species), Muridae (5 species), Gliridae (3 species), Sciuridae (2 species), Erinaceidae (1 species), Soricidae (10 species), Talpidae (4 species), Bovidae (4 species), Cervidae (2 species), Suidae (1 species), orders Chiroptera (37 species), Carnivora (16 species) and Cetacea some of which are distributed and not erratically present (Savouré-Soubelet et al., 2016). It must also be mentioned that part of this biodiversity migrates in or from France (e.g., birds) and that conservation or population issues must be considered at a supra-national level.

Insects comprise more than 40000 species and the other large groups of arthropods are the Acari with several thousands of species and the Spiders with about 1500 species.Other groups like harvestmen (120 species), Pseudoscorpions (125 species) or Scorpions (5 species) are much less rich. These counts may be very accurate for vascular plants or vertebrates but they are still very incomplete for arthropods or other small animals. A general picture can be found on GBIF web site (GBIF, 2017) or on INPN web site, including the distribution of archeological organisms' remains [START_REF] Inpn | Inventaire National du Patrimoine Naturel[END_REF]. Even if the list of species is quite complete now for many groups or big organisms, their accurate distribution has still to be documented. A selection of all data on France biodiversity on the GBIF portal actually shows many sampling gaps that have been discussed in a recent report available on line (Witté andTourroult, 2017).

Regions and ecosystems

The ecosystems mainly of paleartic temperate origin are distributed in four different areas.

The Atlantic area with an oceanic climate is distributed along the western coast but includes also many lowland regions inside the country such as Ile de France The continental area is at the opposite on the eastern border from the North to the first and lower parts of the southeast mountains. As indicated by its name, the climate is continental, therefore colder and with a stronger seasonality than the previous one, more or less degraded in an oceanic one. The typical vegetation is oak and beech forest, with some elm stands in wet places.

Orobanche alsatica Kirschl., 1836: an example of continental distribution: This is a parasitic species of the family Orobanchaceae usually developing on the plant Peucedanum cervaria (L.) Lapeyr. (family Apiaceae). It occurs at the border of woodlands within high non-ligneous vegetation [START_REF] Inpn | Inventaire National du Patrimoine Naturel[END_REF][START_REF] Muller | Les plantes protégées de Lorraine: Distribution, écologie, conservation[END_REF]. Its distribution is mainly around central Europe and its occurrence in France is limited to the central eastern part of the country.

The mountainous area is obviously related to mountains, either in the Pyrénées or in the Alps, and at a lesser degree in Massif Central. Colder and with a shorter summer season, this area is divided in several successive altitudinal zones. From the lower to the higher, can be found the following levels: "collinéen"(<1300 m) mostly forested with oaks (Quercus), "montagnard" (between 1300m and 1800m) mostly forested with abies firs (Abies), beeches (Fagus) and some maples (Acer), with "subalpin"(1800-2400m) today occupied mainly by anthropogenic grasslands but originally with spruce (Picea sp.) and larch (Larix) and "alpin" levels (2400-3000) with alpine grasslands and small bushes (Juniperus). Elevation numbers are given for gross information but vary according to the mountain' side and to the latitude.

Soldanella alpina L. : an example of mountainous distribution

According to genetic analysis, this group of plants (family Primulaceae) originates in Central Asia and exemplifies the diversity of plants occurring in mountains of France. S. alpina L. is a short plant growing close to the soil, found at high elevation (>2000m) in Combes where snow is occurring until late in the season.

The Mediterranean area is limited to the Southern part of the country along the Mediterranean coast from Spain to Italy with the typical original corresponding climate, combining soft winter, rainy spring and hot summers. This area harboured olive tree (Olea) and oak (Q. suber or Q. ilex) forests, shrubby evergreen vegetation on acid or calcareous soils (maquis, garrigue).

Quercus suber L. : an example of Mediterranean distribution

The cork oak (Family Fagaceae) is distributed in the western part of the Mediterranean area in Spain, North Africa, southern France and Italy. This tree is typical of the Mediterranean area.

Evolution of the biota since Quaternary era and in the future

The landscape of France has been deeply modified during the last glaciation events. The ice belt has been moving far to the South and ecosystems have strongly shifted as well. Moving back to the North from more southern areas after glaciation events has been classically documented for many organisms with some presumptive roads of recolonization and diversification [START_REF] Hewitt | Post-glacial re-colonization of Europeanbiota[END_REF][START_REF] Taberlet | Comparative phylogeography and postglacialcolonization routes in Europe[END_REF][START_REF] Schmitt | Molecular biogeography of Europe: Pleistocene cycles and postglacial trends[END_REF]. Even 6000 years ago, the distribution of the main ecosystems was still quite different [START_REF] Prentice | Reconstructing biomes frompalaeoecological data: a generalmethod and its application to European pollen data at 0 and 6 ka[END_REF]. These changes have left some traces in term of genetic structure, endemism and more basically in term of ecosystem richness. A simple comparison with other areas in Europe or in other temperate areas in the world less directly affected by glaciations showed lower species numbers in ecosystems of France for many groups or organisms than in temperate areas less affected by glaciations. This situation has been especially mentioned for trees but is true for many groups.

Since ever after these dramatic events, Man has had a very strong influence on ecosystems and profoundly modified them. The first consequence in historical term, beginning thousands of years ago (Late Iron Age) and continuing until the 18th century, was the decrease in forested areas that have been replaced by cultivated areas and pastures, both in the lowlands and in the mountains. In addition to this big change in the ecosystems, many species have spread in a way that often has to be understood with archeological studies. For example, an animal as common as the rabbit (Oryctolagus cuniculus(Linnaeus, 1758)) was distributed mainly in the South East and has spread to the North with the practice of breeding and hunting in the Middle Ages [START_REF] Callou | De la garenne au clapier. Etude archéozoologique du lapin en Europe occidentale[END_REF]. Today, this species is distributed all over the landscape and is usually and misleadingly perceived as a common species that has always taken part of the natural communities. The same way, sheep and cowherds (Ovis aries and Bos taurus) have deeply modified the biota of the Alps mountainous area by grazing since the Late Iron Age and the Roman period [START_REF] Giguet-Covex | Long livestock farming history and human landscape shaping revealed by lake sediment DNA[END_REF].

Today, industrial agricultural and forestry practices significantly decrease the diversity of natural biota, by impacting natural communities and decreasing the species numbers. Primary functions of ecosystems are at risk of failure in several regions (especially pollination and water quality). Paradoxical situations begin to be observed with part of biodiversity higher in semi-urban areas than in intensive agricultural landscape [START_REF] Deguines | The whereabouts of flower visitors: Contrasting Land-Use preferences revealed by a country-wide survey based on Citizen Science[END_REF].

The effect of climate change is well documented, especially for some well-studied groups like birds or butterflies [START_REF] Devictor | Differences in the climaticdebts of birds and butterfliesat a continental scale[END_REF] and there is a shift for southern populations to go to the North.

Biodiversity in overseas territory

France includes as well many tropical overseas territories including French Guiana (Amazonia), les Antilles Françaises in the Caribbean region, New Caledonia in the Pacific, La Réunion and Mayotte in the Indian Ocean, and several other smaller or biologically less rich locations [START_REF] Gargominy | Biodiversité et conservation dans les collectivités françaises d'autre mer. Collection Planète Nature[END_REF]. Taking into account such places with very rich, endemic and diverse biota pushes up the estimates of species richness to twice the number cited higher (around 160000 species). The inventory of biodiversity in all these regions is still in progress (e.g., Guilbert et al., 2014) which means that the total number of species is actually much higher, including for endemic ones.

Natural Parks and Reserves

It exists10 national parks and 51 regional parks in France, which represent around 17% of area of the national territory. Several ones are located in overseas regions. These parks are tolerant to some agricultural, breeding and touristic activities in specific conditions.They welcome more than 5 millions of visitors per year.

  (around Paris), with a low elevation landscape originally harbouring either oak or hornbeam forests or open vegetation like moorland. Uromenus rugosicollis (Audinet-Serville, 1838): an example of Atlantic distribution: This species of katydid (Rough Saddle Bush-cricket) is distributed along the Atlantic coast; it goes north until the Cotentin and southwest until Marseille on the mediterranean coast (INPN, 2017). It is part of a group of katydids that shows a mediterranean distribution (Orthoptera Ensifera, subfamily Ephippigerinae).


Guillaume Touya

Carmen Brando-Escobar

Detecting Level of Detail Inconsistencies in VGI Datasets

Keywords: VGI, OpenStreetMap, level of detail, cartography

Whereas it was possible to define the level of detail (LoD) of authoritative datasets, it is not possible for Volunteered Geographic Information (VGI), often characterised by heterogeneous levels of details. This heterogeneity is a curb for mapmaking, particularly when using traditional map derivation processes such as generalisation. The paper proposes a method to infer the level of detail of VGI features. Then, inconsistencies between features with different levels of detail that get in the way of good mapmaking can be automatically identified. Some proposals are made to harmonise level of detail heterogeneities. The LoD inference is implemented and results are presented on OpenStreetMap data.

Introduction

Volunteered Geographic Information (VGI) is more and more used by the GIScience research community but also by all users of geospatial data. Indeed, as crowdsourcing communities, such as the OpenStreetMap community, continuously improve their processes the data quality increases [START_REF] Girres | An evaluation of the impact of cartographic generalisation on length measurement computed from linear vector databases[END_REF]Touya 2010, Haklay 2010), encouraging companies and administrations that rely on geospatial data to trust VGI. Even National Mapping Agencies (NMA) that produce authoritative and high quality data intend to integrate VGI in their databases (e.g. Canada). Beyond data quality, validating VGI for use by governments raises issues of integration into authoritative datasets (Du and others 2012) and trust in the volunteers [START_REF] Skarlatidou | Trust in web GIS: the role of the trustee attributes in the design of trustworthy web GIS applications[END_REF]). Moreover, it seems necessary to check if VGI can easily be used in the applications, based on the authoritative data it replaces or enriches. The main application of geospatial data is mapmaking, which can be a very complex task when scale varies and generalisation is required.

VGI and especially OpenStreetMap are successful because of their simplicity [START_REF] Brando | Edition collaborative d'un contenu géographique: une approche fondée sur des spécifications formelles et des relations spatiales explicites[END_REF]: almost anybody able to use a GPS device is able to contribute, as the specifications are not too complex to handle. The drawback of this simplicity is the heterogeneity of the produced data [START_REF] Girres | Quality assessment of the french OpenStreetMap dataset[END_REF]. The level of detail (LoD) of features is particularly heterogeneous. For instance, buildings with street display LoD coexist with land use parcels extracted from satellite imagery. Maps derived from VGI show that inconsistencies in LoD between spatially related features (e.g. a building and a containing built-up area) cause map conflicts [START_REF] Das | Conflicts in neogeography maps[END_REF], which are big obstacles to good quality mapping. The aim of the paper is to propose a method to identify LoD inconsistencies in VGI datasets. Indeed, if LoD inconsistencies are identified in the map, it is possible to further correct them to make more legible and less confusing maps. The key of our proposal to identify LoD inconsistencies is a two step process: first infer a LoD for each feature of the map and then identify spatial relations between conflicting LoD features.

Section 2 precisely defines the level of detail and its importance in mapmaking. Section 3 presents our proposal to infer the LoD of geographical features and detect inconsistencies. Section 4 shows some results on OpenStreetMap data. Section 5 briefly explores techniques to correct LoD inconsistencies and section 6 concludes and discusses further work.

The Importance of Level of Detail

Scale and Level of Detail

The scale of a map has a simple definition; it is the homothety factor between ground size and map size of features. In cartography, the word scale may also convey representation aspects: features cannot be mapped the same way at each mathematical scale, e.g. buildings can be mapped individually at large scales , e.g. 1:25, 000, but are represented as built-up areas at small scales, e.g. 1:200, 000.

However, scale cannot be used to characterise geographical databases, because data are not only intended for cartographic representation, and the term level of detail is preferred. [START_REF] Ruas | Echelle et niveau de detail[END_REF] define the level of detail in a geographical database as the conjunction of the conceptual schema of data, the semantic resolution, the geometrical resolution, the geometrical precision and the granularity.

The conceptual schema component is the way ground truth is represented in the geographical database: representing forests as polygonal features or with point features representing each tree are conceptual schemas that correspond to different levels of detail.

The semantic resolution is the quantity of details in the attribute data attached to geographic features. The issue is not related to mapping, so we will not focus on this component in the remaining of the paper.

By analogy with raster resolution, the geometric resolution of vector features is approximately the minimum distance between two vertices of the geometry.

Figure 1a shows that such a definition should be handled carefully as the distance between vertices is always shorter in sharp bends. Like [START_REF] Ruas | Echelle et niveau de detail[END_REF], we consider that the resolution value of a database should be used as a rough estimate of the features resolution. In the remainder of the paper, we consider LoD as the aggregation of all four components described above. We call high LoD a level of detail with high precision, resolution etc., and low LoD a level of detail with low precision, resolution etc. Despite this definition, LoD is a relative notion that is difficult to measure quantitatively or qualitatively. Section 3.1 of the paper describes of we propose to measure high and low LoD.

Case studies from OpenStreetMap

In order to illustrate the issue of level of detail inconsistencies and to test our propositions, three case studies were extracted from the French OpenStreetMap (OSM) dataset, as OpenStreetMap is a VGI source aiming at making maps. The case studies cover and share different kind of landscapes: urban, rural, suburban and mountainous areas (Figure 2). All three case studies contain a large amount of LoD inconsistencies, as OSM seeks to indifferently gather very detailed features such as cycle lanes and less detailed features such as sea routes or built-up areas. The inconsistencies are increased by the heterogeneity of capture tools and techniques used by OSM contributors. For instance, capturing a building automatically from cadastral image (as in French OSM), or from GPS field work do not yield similar LoDs.

The most common and obvious type of inconsistency in the case studies is the coexistence of detailed buildings and land use parcels. In Figure 3, buildings intersect a forest limit which is unlikely, or lie just outside the built-up area they should be part of. Even in rural areas where OSM datasets are less complete, the few existing features are LoD inconsistent. In Figure 4, we can see public bathrooms and an access path connected to less detailed footpaths, which coexist with very undetailed lake and river features. A footpath even crosses the lake because the lake is very imprecise at the footpath level of detail.

Figure 4.

Examples of LoD inconsistencies: a public bathrooms spot and its path coexist with less detailed footpaths (dashed lines) and even less detailed lake and river ©OpenStreetMap.

Mapmaking with Inconsistent LoDs

Inconsistent LoDs in the source data of a map may cause bad quality maps for two main reasons:  It causes legibility problems at multiple scales.  It may convey misleading information.

Inconsistent LoDs may make small scale maps, where high LoD features should not appear, not legible. Indeed, the knowledge of eye perception limits allows the definition of legibility thresholds (e.g. perception, separation, differentiation, etc.): below these thresholds, the eye only perceives noise that disturbs map reading, whatever the map legend is [START_REF] Duchêne | Generalisation, symbol specification and map evaluation: feedback from research done at COGIT laboratory, IGN france[END_REF]. Figure 5a shows an extract from OSM standard maps where most buildings are too small to be legible and disturb the reading. Here, generalisation is required, for instance by increasing building size. In the same time, too dense layers cause symbol overlapping problems, such as points of interest on a legible background map (Figure 5b). Although OSM is not designed to make maps with all its features, legibility problems may occur at most scales, even large ones. Inconsistent LoDs may also convey misleading information to the map reader that is not aware of each feature LoD and interprets the map as consistent. Figure 6 shows an example where inconsistent LoDs between buildings and forest in OpenStreetMap mislead the reader on the fact that buildings are in the forest. The forest LoD should restrict its use to small scale maps where buildings are not represented. Moreover, a map reader traditionally expects consistent LoD and his cognitive interpretation of the map may be biased by the inconsistency. Finally, automated mapping requires complex processes such as cartographic generalisation to make the geographical information legible at a given scale. Generalisation processes are mostly designed for NMA data that are homogeneous in LoD (Stanislawski and others 2012). Cartographic generalisation of VGI data has not been tackled yet: research projects are starting but do not focus on LoD heterogeneity (e.g. [START_REF] Klammer | Approaches for enhancing tile-based mapping with cartographic generalisation[END_REF] focus on generalising tile-based maps) and OSM based projects such as Mapnik 1 or CloudMade 2 only provide very simple generalisation operations, e.g. line filtering and feature selection. Thus, detecting LoD inconsistencies would greatly help to improve the way maps made with VGI data are perceived by a reader.

Detecting LoD Inconsistencies

This section describes our proposed approach to detect LoD inconsistencies: first, the LoD of features is inferred and then, spatial relations between features with different LoD are identified. The first subsection briefly presents the proposed LoD classification that is used to measure LoD. The second subsection deals with the inference of a LoD for individual features, based on criteria derived from LoD components (2.1). The last subsection deals with the detection of inconsistencies that relies on spatial relations.

Classification of Levels of Detail

LoD is a quite relative and fuzzy concept as its five components cannot be clearly and quantitatively defined, so we propose to measure it qualitatively with a Likert scale (Likert 1935) classification. We defined five LoD categories extracted from French geoportal 3 from the most detailed to the least: street, city, county, region and country. The street LoD contains features represented for parcel management or street orientation, e.g. the British OS MasterMap®. The city LoD contains features represented to describe what is visible on the ground (buildings, roads, rivers, forests, etc.). The county LoD contains features that represent a small region to allow displacements like the ones of a tourist in the area (i.e. visits, trekking, cycle rides). The regional LoD is related to the representation of a large region and the country LoD is even less detailed, for the representation of countries or big regions.

The LoD inference described in the next section classifies features in one of these five categories by the means of quantitative and qualitative criteria.

Assessing Features LoD

Criteria to Assess Geographic Features LoD

The components of LoD presented in 2.1 allow the definition of several criteria that are computed for each feature of the dataset to infer its LoD: Each criterion is designed to give a value normalised between 0 and 1, 1 being the lowest LoD, country. The empirical normalisation of each criterion implies some imprecision and fuzziness in the criteria values, which will have to be taken into account by the method using the criteria.

The feature type criterion analyses the feature type of features in relation to the data specifications and to a geographical ontology to infer LoD. The presence of some feature types in a map reveals the conceptual schema component of LoD. For instance, buildings or points of interest have higher LoD than built-up areas features.

The vertex density criterion analyses the number of vertices compared to feature length to give a clue on geometrical resolution, which is fuzzily defined. To empirically normalise the density, vertex density values have been studied for different types in datasets with different LoDs from the French NMA. The criterion value is computed according to (vertex_density_value = (1 -vertexDensity) 8 (Eq. 1). vertex_density_value = (1 -vertexDensity) 8 (Eq. 1)

The median edge length criterion analyses the median of edges length between two vertices to complete the assessment of resolution. [START_REF] Girres | An evaluation of the impact of cartographic generalisation on length measurement computed from linear vector databases[END_REF] states that it is the best criterion to assess geometrical resolution on roads and rivers. A piecewise function from  to [0,1] was empirically derived from values in the French NMA datasets and from the conclusion of [START_REF] Girres | An evaluation of the impact of cartographic generalisation on length measurement computed from linear vector databases[END_REF].

The capture source criterion analyses metadata on the capture source of features (e.g. GPS tracks, aerial images) to infer geometrical precision from knowledge on the sources. The values for the OpenStreetMap tag "source" were analysed in several French datasets, and a LoD value was assigned to each source value. For instance, features captured from the national cadastre have a 0.1 value for this criterion while features imported in OSM from CORINE Land Cover dataset (i.e. European land use dataset) have 0.7 value.

The shortest edge criterion analyses the length of the shortest edge between two vertices to infer granularity. It is a classical measure to assess building granularity in cartographic generalisation (Stöter and others 2010). It was also calibrated using OSM and French NMA datasets to obtain a piecewise function from  to [0,1].

The size criterion analyses the size of features to complete granularity inference. Indeed, the presence of small features indicates high granularity. We used the legibility thresholds of eye perception to relate features size to the LoD categories.

Finally, the coalescence criterion is dedicated to the inference of linear features granularity. It is based on the principle that if a linear feature symbol coalesces at a given symbol width (Figure 7), it means that the feature cannot be displayed at a scale that requires such a width for eye perception issues [START_REF] Girres | An evaluation of the impact of cartographic generalisation on length measurement computed from linear vector databases[END_REF]. To relate symbol width to scale and then to LoD categories, we studied the symbols used in the French NMA map series. In order to infer a feature LoD from the seven criteria, multiple criteria decision techniques are required. The next section presents the multiple criteria decision method we used, ELECTRE TRI [START_REF] Figueira | ELECTRE methods[END_REF].

Assessing LoD with ELECTRE TRI

Multiple criteria decision analysis is a field of computer science that develops techniques to make an appropriate decision based on multiple criteria. The decision can be: inferring the best element, sorting elements or classifying them. In our case, the decision we want to make is classify a feature into one of the LoD categories defined above. ELECTRE TRI is a multiple criteria decision method that classifies elements, using the principles of the ELECTRE multiple criteria decision methods [START_REF] Figueira | ELECTRE methods[END_REF]. ELECTRE methods only compare elements criterion by criterion and do not compare values from different criteria (i.e. it does not compare apples with oranges). It was previously used in map generalisation by [START_REF] Taillandier | Multi-criteria diagnosis of control knowledge for cartographic generalisation[END_REF].

The ELECTRE TRI method should be used when the following conditions are met [START_REF] Figueira | ELECTRE methods[END_REF]:

 Classification into ordered categories (e.g. from 'street' LoD to 'country' LoD.

 More than 3 criteria (7 here).  Heterogeneous criteria (e.g. it is hard to compare the granularity and source criteria).  For some criteria, a small value difference may be not significant, while the addition of several small differences may become significant. It compensates for the imprecision of our criteria.

ELECTRE TRI classifies vectors of values for each criterion (the dimension of the vector is the number of criteria), which correspond to the measured criteria for a given feature. For each category, two vectors are defined representing the lower bound of the category and its upper bound. For instance, for the category 'street LoD', the vertex density criterion value (vertex_density_value = (1 -vertexDensity) 8 (Eq. 1) of the lower bound vector is 0 (it is the first category) and the value of the upper bound vector is 0.2.

The ELECTRE TRI principle is to compare the vector for a given feature to the bound vectors of each category using an outranking relation: a feature is classified in a category if its vector outranks the lower bound vector of the category and is outranked by the upper bound vector. The comparison of two vectors, i.e. the computation of the outranking relation, is inferred from the comparison of the vector value for each criterion alone, two different criteria values are never compared (Figure 8). For one criterion, a value can be preferred to the other or both are considered equivalent for this criterion. Using this preference, each criterion votes for the assertion "vector u outranks vector v" [START_REF] Figueira | ELECTRE methods[END_REF]. In order to allow this comparison of values for a criterion despite their imprecision and fuzziness, four properties have to be added to criteria:

 Weight: it conveys the importance of this criterion compared to the others.  Preference threshold: if the difference between values for this criterion is bigger than preference, the vector with bigger value is preferred considering this criterion. It adjusts to the normalisation used for the criterion. For instance, vertex density has a 0.2 preference while size has a 0.5 preference.  Indifference threshold: if the difference between values for this criterion is less than indifference, both vectors are considered as equivalent for this criterion. For instance, for shortest edge, a difference less than 0.25 is not significant.  Veto threshold: if the difference between values for this criterion is bigger than veto, than the vector with bigger value will always be preferred to the other, whatever the other criteria. For instance, size has a high veto threshold as it is never possible to infer LoD only with this criterion.

The output of the ELECTRE TRI method for a given feature is its LoD category with a confidence rate on the classification, e.g. "this particular road feature has a topographic LoD with 80% confidence rate".

Spatial Relations Based LoD Inconsistencies Detection

Taking into account spatial relations is fundamental in automatic mapmaking and generalisation [START_REF] Duchêne | The CartACom model: transforming cartographic features into communicating agents for cartographic generalisation[END_REF]. [START_REF] Brando | Specifications for user generated spatial content[END_REF] state that the identification of implicit spatial relations helps to define good specifications for VGI contributors. [START_REF] Brando | Edition collaborative d'un contenu géographique: une approche fondée sur des spécifications formelles et des relations spatiales explicites[END_REF] follows the idea proposing to add integrity constraints checking on spatial relations in VGI contributing software (e.g. verify that roads do not cross buildings or that bus stops are along roads). We believe that the same approach could be used to identify places where LoD heterogeneity is a problem for mapmaking. If two features should respect a spatial relation for integrity reasons but do not, and if the features involved in the relation have a different LoD, there is a LoD inconsistency (Figure 9a). Conversely, if a forbidden spatial relation does exist with involved features having a different LoD, there is a LoD inconsistency (Figure 9d). Therefore, the proposed method uses as parameters a set integrity spatial relations and automatically process features to identify instances of these relations between LoD different features. For instance, if the set of integrity relations contains the four spatial relations depicted in Figure 9, the method searches the instances of buildings just outside a built-up area, etc. The reification of relations follows [START_REF] Duchêne | The CartACom model: transforming cartographic features into communicating agents for cartographic generalisation[END_REF] idea that defines spatial relations instances to guide generalisation.

Application on OpenStreetMap Case Studies

The criteria for individual LoD inference have been calibrated with features from zone 3 (see 2.2) and tested on both remaining zones. The method was implemented on CartAGen open source generalisation platform [START_REF] Renard | Capitalisation problem in researchexample of a new platform for generalisation: CartAGen[END_REF]. Figure 11 shows how buildings found in zone 2 can be classified into different LoDs, depending on their characteristics (only granularity, resolution and precision may vary here). Then, the four spatial relations from Figure 9 are selected as a set of relations to identify LoD inconsistencies: (i) building should be inside built-up area; (ii) no intersection between paths and lakes; (iii) bus stop should be along the road; (iv) a group of houses should be in a clearing. Instances of these relations between LoD inconsistent features were automatically computed in the case studies. Figure 12a shows a group of houses inside a forest whose LoD explains the lack of clearing around the group.

Figure 12b shows buildings identified as lying just outside a low detail built-up area.

Figure 12c shows with the red cross a bus stop too far to be along the road unlike the two at the bottom of the picture. Figure 12d shows a detailed footpath intersecting a low detail lake.

Resolving LoD Inconsistencies

Since LoD inconsistencies prevent good quality maps, strategies have to be developed to resolve the inconsistencies. When the aim is to make a map at a given scale from VGI, three strategies can be used. This section discusses the strategies but we have not implemented them yet.

The first strategy would be to build a multiple-resolutions database (MRDB) out of the VGI dataset as proposed in [START_REF] Müller | A framework for building multi-representation layers from OpenStreetMap data[END_REF]. It would allow adapting the displayed content to the representations corresponding to a given LoD. Building a MRDB would require matching techniques, maybe benefiting from linked data [START_REF] Hahmann | Connecting LinkedGeoData and geonames in the spatial semantic web[END_REF].

The second strategy would be the harmonisation of LoDs to the closest LoD to the map scale. When trying to make a map at a small scale, the harmonisation is the generalisation of the high LoD features, with specific constraints to correct the LoD inconsistencies. When trying to make a map at a large scale, the harmonisation is the enhancement of the low LoD features that have LoD inconsistencies. Figure 13 shows a simple automatic tool we developed on CartAGen open source generalisation platform [START_REF] Renard | Capitalisation problem in researchexample of a new platform for generalisation: CartAGen[END_REF], which enhances the built-up area LoD to put the buildings that should be in the area back into it: buffers are computed around the buildings belonging to the identified inconsistent relations and then merged with the built-up area. This tool is suited for harmonising undetailed areas that contain very detailed features and similar tools should be developed for the other types of LoD inconsistencies. The last strategy is to act when the VGI dataset is compiled, or updated, by automatically proposing a reconciliation if the contribution introduces LoD inconsistency. [START_REF] Brando | Edition collaborative d'un contenu géographique: une approche fondée sur des spécifications formelles et des relations spatiales explicites[END_REF] proposed a reconciliation tool to avoid integrity constraints violation during VGI contribution: for instance, it displaces the building off a road if a contribution introduces a building on a road. Harmonisation tools could reconcile contributions the same way for LoD inconsistencies, e.g. extend the built-up when a building is added near its limit.

Conclusion and Future Work

To conclude, the paper presented the problems raised by the LoD heterogeneity of VGI for mapmaking and proposed a method to infer the LoD of VGI features as a starting point for solving the problem. The method was successfully applied to French OpenStreetMap data.

As pointed all along the paper, there is still a lot to do to handle LoD heterogeneity problems in automatic mapmaking procedures. First, it would be interesting to integrate semantic resolution in LoD, using the ideas of [START_REF] Mooney | The annotation process in OpenStreetMap[END_REF] on heavily edited OSM objects. Then, it is necessary to adjust LoD inferences considering populations and not only individual features. Clutter (i.e. visual overload, [START_REF] Rosenholtz | Measuring visual clutter[END_REF]Nakano 2007, Stigmar andHarrie 2011) or high local density of features should increase the LoD of the concerned features. Moreover, the techniques presented for correcting LoD inconsistencies are just basic ideas that require further research. Finally, it would be interesting to study the sensitivity of existing automatic generalisation procedures to data with LoD inconsistencies. It would help to identify the weaknesses of these procedures regarding LoD homogeneity and correct them like Stanislawski and others (2012) did for river network pruning.

Figure 1 .

 1 Figure 1. (a) two lines with different resolutions. (b) the granularity of a building. The geometrical precision is simply the positional shift between ground truth and represented feature. Granularity describes the size of the smallest shapes of features, e.g. the smallest protrusion of buildings (

Figure 2 .

 2 Figure 2. Extracts from the three case studies: an urban area, a mountainous area, and a suburban one ©OpenStreetMap.

Figure 3 .

 3 Figure 3. Examples of LoD inconsistencies: the upper left buildings intersect the forest limit and buildings on the right lie outside the built-up area ©OpenStreetMap.

Figure 5 .

 5 Figure 5. (a) The building LoD is inconsistent with roads, rivers and land use LoD bringing noise to the map reader at this scale (©OpenStreetMap) (b) Point of interest symbols overlap at this scale.

Figure 6 .

 6 Figure 6. The forest LoD is inconsistent with the buildings in OSM misleading the reader because buildings are not in the forest but in a clearing.

  1 mapnik.org 2 cloudmade.com 3 geoportail.gouv.fr



  Feature type criterion (conceptual schema component)  Vertex density criterion (resolution component)  Median edge length criterion (resolution component)  Capture source criterion (geometrical precision component)  Shortest edge criterion (granularity component)  Size criterion (granularity component)  Coalescence criterion (granularity component)

Figure 7 .

 7 Figure 7. Linear granularity assessed by coalescence tests: (a) line has higher granularity than (b) as it coalesces quicker when symbol width is increasing.

Figure 8 .

 8 Figure 8. In ELECTRE TRI, the comparison between two vectors (i.e. the outranking relation) is inferred from the comparisons for each criterion.

Figure 9 .

 9 Figure 9. (a) building should be inside built-up area. (b) no intersection between paths and lakes. (c) bus stop should be along the road. (d) a group of houses should be in a clearing ©OpenStreetMap.

  Figure10shows some of the automatic inferences on road, building, path or land use features. As expected land use parcels are classified as LoD 1 (street LoD) as they are imported from European dataset CORINE Land Cover. Buildings, imported from cadastral maps, are mostly classified as LoD 5, but large buildings, that have fewer details, have a lower LoD. Roads are classified heterogeneously: some very detailed features such as the cycle lanes at the bottom right of Figure10are classified as LoD 4, while straight important roads are classified as LoD 2. After evaluation of the zone, results comply with a visual assessment of LoD.

Figure 10 .

 10 Figure 10. Some results of individual LoD inference in zone 2.

Figure 11 .

 11 Figure 11. Illustration of the individual LoD inference of some highlighted buildings in zone 2.

Figure 12 .

 12 Figure 12. Automatic detection of LoD inconsistencies for the four selected spatial relations (highlighted in red).

Figure 13 .

 13 Figure 13. Initial data (©OpenStreetMap) with several buildings outside a low LoD built-up area. The correction (automatically obtained) increases the built-up area LoD and includes the inconsistent buildings in it.


Denis Claude

Mabel Tidball

Managerial incentives and polluting inputs under imperfect competition *

Keywords: precommitment, externality, delegation, vertical relations, managerial incentives

This paper explores the link between upstream input pricing and downstream strategic delegation decisions. It complements earlier contributions by studying how environmental emissions and tax payments alter the incentives business owners have to divert their managers from prot maximization in favor of sales revenue generation. Two scenarios are compared depending on whether the upstream supplier precommits to a xed input price or adopts a exible price strategy. Corresponding Subgame-Perfect Nash-Equilibria are characterized and elements of comparative statics analysis are presented. The analysis conrms that previous results showing that a price precommitment makes the upstream supplier better o and downstream rms worse o carry over to situations in which production generates pollution.

Introduction

This chapter bridges two elds of research in which Georges Zaccour has been active: the analysis of vertical relations and environmental and resource economics. The former studies relations between rms that intervene successively along the value chain whereas the latter deals with the relations between the economy and the environment. We can trace back his interest for vertical relations to his early contributions on the analysis of energy markets [START_REF] Zaccour | Calcul des pseudo-équilibres sur des réseaux représentant des marchés séquentiels[END_REF][START_REF] Zaccour | Calcul des pseudo-équilibres sur des réseaux représentant des marchés séquentiels[END_REF][START_REF] Breton | Methods in the Analysis of Multistage Commodity Markets[END_REF]. Today this interest is mainly manifested in his work on marketing channels.

But, it also comes up tangentially in a va- riety of contributions ranging from environmental economics where sustainable tourism development may require tourism destinations to delegate expenditures in environmental remediation to a regional authority [START_REF] Claude | Investment in tourism market and reputation[END_REF] to institutional economics where good institutions are produced by the strategic precommitment of civil society to ght corruption [START_REF] Ngendakuriyo | Fighting corruption: To precommit or not?[END_REF].

Strategies of delegation and precommitment are at the heart of the literature on strategic delegation to which this chapter contributes. Starting with [START_REF] Vickers | Delegation and the Theory of the Firm[END_REF], Fershtman 1 Introduction (1987) and [START_REF] Sklivas | The strategic choice of managerial incentives[END_REF], this literature has examined the incentives owners have to delegate production decisions. By hiring a manager, the owner of a rm can credibly commit to pursue a goal that diers from maximizing prot. The managerial compensation contract will then be designed to convey the appropriate managerial incentives to guide the manager in his day-to-day decisions. Its terms will provide for a variable target-based bonus that rewards the manager's performance in achieving some alternative goal to (pure) prot maximization. This variable part may be based on any one or a combination of the following criteria : prot, sales volume or revenue, market share, and corporate social responsibilityor environmental objectives. Since rational managers respond to nancial incentives conveyed by the variable part of their remuneration, they will be encouraged to deviate from prot maximization.

2 & Judd
As is well known, by choosing to reward sales revenue rather than prot, the owner encourages the manager to adopt a more aggressive market behavior. Namely, the managerial rm will produce more (for any level of production of its competitors) than a prot-maximizing owner-managed rm. Financial disclosure rules usually ensure that the incentives embedded in managerial compensation contracts are common knowledge. ! Any change in the performance criteria presiding over managerial compensation will then affect the expectations of competing rms. This opens the door to a strategic manipulation of compensation contracts: each owner attempting to alter the expectations of rival rms to its own advantage.

But deviating from prot maximization is only protable when the deviation is unilateral.

And, since all owners face similar incentives to deviate, widespread deviations are to be expected and excessive output supply will result into lower prots for all. Hence, the opportunity to strategically delegate day-to-day production decisions to a manager closes as a trap on rm owners who actually nd themselves confronted with a Prisoner's Dilemma.

Strategic delegation provides a much needed rationale for observed deviations from prot maximization. This rationale, however, assumes that rms are vertically integrated and produce their own inputs. When this assumption is relaxed, the vertical externality linked to input pricing appears to have a disciplining role on downstream rms' behavior.

If duopolists buy in inputs from a common monopolist supplier, [START_REF] Park | Vertical externality and strategic delegation[END_REF] shows that strategic delegation becomes unprotable for business owners. [START_REF] Wang | Input pricing and market share delegation in a vertically related market: Is the timing order relevant[END_REF] reach the same conclusion by assuming that managerial incentive contract rewards a combination of prot and market share (rather than prot and sale revenues). On the contrary, [START_REF] Liao | Input price contracts and strategic delegation with downstream duopoly[END_REF]2010) shows that strategic delegation retains its strategic value if the input supplier precommits to a xed input price. This conclusion is backed by [START_REF] Wang | Delegation and Vertical Externalities[END_REF], who proves it true when compensation contracts reward the manager's performance relative to peers. " Finally, [START_REF] Claude | A reconsideration of the link between vertical externality and managerial incentives[END_REF] shows that [START_REF] Park | Vertical externality and strategic delegation[END_REF]'s main results no longer holds if we consider a downstream market consisting of more than two but a nite number of rms. This paper re-examines the link between vertical externalities and strategic delegation decisions. We consider an extended version of [START_REF] Park | Vertical externality and strategic delegation[END_REF]'s model in which downstream rms generate pollution emissions when they process the intermediate product into a nal good. Specically, we assume that the emission of a rm, per unit of output produced, is inversely related to that rm's productivity in processing inputs. Since pollution emissions are assumed to be taxed, downstream rms have an incentive to internalize them, at least in part.

Two papers have investigated the consequences of strategic delegation for environmental policy-making. In a Cournot duopoly with homogeneous goods and pollution emissions, ! See, [START_REF] Vural | Disclosure Practices by Family Firms: Evidence from Swedish Publicly Listed Firms[END_REF] for the example of the ball bearing compagny SKF " This form of strategic delegation was rst studied by [START_REF] Fumas | Relative performance evaluation of management[END_REF] and [START_REF] Miller | Relative performance as a strategic commitment mechanism[END_REF] Barcena- [START_REF] Barcena-Ruiz | Environmental taxes and strategic delegation[END_REF] nd that strategic delegation is protable. At the equilibrium, managerial rms produce and emit more than prot-maximizing owner-managed rms. Consequently, the optimal environmental tax is higher than that required to regulate a standard Cournot market. [START_REF] Pal | Delegation and Emission tax in a Dierentiated Oligopoly[END_REF] generalizes this result to dierentiated industries. However, none of these papers has investigated how factor market imperfections alter managerial incentives in downstream markets, which is the main purpose of our paper. Since our model encompasses those of [START_REF] Park | Vertical externality and strategic delegation[END_REF] and [START_REF] Claude | A reconsideration of the link between vertical externality and managerial incentives[END_REF], we check the robustness of their results.

The paper is organized as follows. Section 2 describes the model. Section 3 solves the managerial sub-game. Sections 4 and 5 characterize the sub game perfect Nash equilibrium (SPNE) depending on whether the upstream monopolist precommits to a price. SPNE outcomes are compared in Section 6. The last section concludes.

The Model

We extend the model of [START_REF] Park | Vertical externality and strategic delegation[END_REF] to account for the existence of pollution (or waste).

Consider a vertical market structure with upstream monopoly and downstream quantity competition. The single upstream monopolist (indexed by up) produces at no cost a homogeneous input x that it sells at a non-discriminatory price k>0. Let x i denote the input consumption of rm i (i = 1, 2, . . . , n) and X = ∑ n i=1 x i the aggregate input demand. The upstream supplier's prot function is simply π up = k X. Downstream rms rely on the same technology to turn the intermediate product x into a nal good y. Let y i denote the output of Firm i (i = 1, 2, . . . , n) and Y = ∑ n i=1 y i the aggregate downstream output. Firm i's production function is y i = ε x i , where ε ∈ (0, 1] is a parameter measuring Firm i's productivity.

Pollution emissions come as a by-product of production. More precisely, they are inversely related to Firm i's productivity and given by e i = (1 -ε) x i . # We assume that the government levies a tax on pollution emissions at a rate τ ≥ 0. Obviously, the more inecient the rm is (the lower the value of ε), the higher is the quantity of pollution emitted for each unit of the nal good y produced and the higher is the rm's tax bill for a given level of output. Conversely, if ε is assumed equal to 1 then the rm no longer emits pollution and its environmental tax bill is zero. Without loss of generality, we assume that downstream rms face no other production cost than that associated with input purchase.

Demand for the nal good y is represented by the inverse demand function P (Y ) = a-bY with a, b > 0. Under the above assumptions, Firm i's prot function is given by

π i = P (Y )y i -kx i -(1 -ε)τ x i .
(1) Using the production function, the above can be expressed in terms of y only:

π i = P (Y )y i -A (k, τ, ε) y i , where A (k, τ, ε) = k + τ ε -τ, ( 2
)
denotes rm i's eective marginal cost of production; namely, the sum of the rm's input expenditure (k/ε) and environmental tax bill (τ (1 -ε) /ε). Observe that ∂A/∂ε < 0, # We assume that the more ecient the rm is, the less input is used per unit of output and thus the lower the level of waste or emission generated. This assumption conforms to empirical ndings by [START_REF] Shadbegian | What Determines Environmental Performance at Paper Mills? The Roles of Abatement Spending, Regulation, and Eciency[END_REF]. Examining the determinants of environmental performance at paper mills, they found that high productivity plants pollute less. More precisely, a 10% higher productivity level is associated with a 2.5% lower emission per unit of output. Furthermore, they found that unexpectedly high productivity levels are associated with unexpectedly low levels of emissions per unit of output. [START_REF] Shadbegian | What Determines Environmental Performance at Paper Mills? The Roles of Abatement Spending, Regulation, and Eciency[END_REF] advance two main explanations for their results. On the one hand, newer production plants may be more ecient in production but also designed so as to reduce pollution emissions or waste. On the other hand, older, more inecient, rms may face less regulatory pressure and retrotting their facilities may be extremely dicult so that abatement possibilities are reduced.

∂A/∂τ > 0 and ∂A/∂k > 0, which is in accordance with intuition. Indeed, an increase in the price of factors (τ for emissions, and k for the intermediate product) leads to a corresponding increase in marginal cost, whereas an increase in the productivity parameter ε translates into a reduced (eective) marginal cost.

The owners of downstream rms may hire managers to run their rms on their behalf. The decision to hire a manager leads to the so-called divorce between ownership and control. Indeed, when management and control functions are divorced, agency problems arise if the manager's interests diverge from those of the owner [START_REF] Sengul | Strategic delegation a review, theoretical integration, and research agenda[END_REF]. In the remainder of this paper, we assume that all agency problems have been resolved

$ to focus our attention on the strategic value of delegation decisions. Following [START_REF] Vickers | Delegation and the Theory of the Firm[END_REF], [START_REF] Sklivas | The strategic choice of managerial incentives[END_REF] and [START_REF] Fershtman | Equilibrium incentives in oligopoly[END_REF], we assume that owners hire managers in order to credibly commit their businesses to objectives that dier from prot maximization. Here, we assume that this objective is the maximization of a weighted sum of prots and sales revenue; i.e.,

F i = α i π i + (1 -α i ) [P (Y )y i ] .
(3)

The prot weight α i is chosen strategically by the owner of rm i in order to manipulate the anticipations of rival rms to its own advantage (i.e., in a protable way).

The corresponding departure from prot maximization is credible since the inclusion of a target-based bonus in managerial compensation contracts ensures that appropriate incentives are conveyed to managers. % Depending on the value selected for α i , owners can encourage a wide range of behaviors. To see this, let us rewrite Equation (3) as:

F i = P (Y ) y i -α i A (k, τ, ε) y i . (4)
Then the performance measure has straightforward interpretation. By choosing α i = 1, rm owners encourage pure prot maximization. However, if α i is set lower than one, they direct their managers to pursue revenue generation at the expense of prots and if α i is set greater than one they direct their managers to pursue cost minimization at the expense of prot generation. We let ⃗ α = (α 1 , α 2 , . . . , α n ) denote the prole of prot weights chosen by downstream owners.

We consider two scenarios depending on whether the upstream monopolist is able to precommit to a xed input price. In the rst scenario, we assume that price is dicult to change and communicate to customers. In this case, price commitments are credible.

The order of moves is then as follows. The upstream monopolist moves rst and sets the input price. After observing this price, downstream owners design their managerial compensation contracts simultaneously and independently. Obviously, the latter simply means setting the value of the prot weight α i . At the last stage of the game, managers compete in quantities so as to maximize their compensation, taking into account their target-based bonus and the input price.

$ After delegation, the manager may not act in the owner's interest and engage in opportunistic practices and other self-serving behaviors. Managerial opportunism arises from two main sources [START_REF] Eisenhardt | Agency Theory: An Assessment and Review[END_REF]) : (a) the objectives of the owner and the manager may conict and (b) the owner may not be able to observe the behavior of the manager. The rst is of little relevance in our context. Indeed, we assume that the manager is oered a performance-based bonus which reduces conicts of interests and ensures that the manager will adhere to the owner's supply strategy. However, appropriate monitoring and governance mechanisms should be put in place in order to treat the second source of managerial opportunism. Indeed, unobservable behaviors may result in expropriation of the company funds [START_REF] Shleifer | A Survey of Corporate Governance[END_REF]. Common examples include the excessive consumption of perquisites such as luxurious oces and company jets. More broadly, managerial discretion may result in the allocation of company funds to the pursuit of pet projects or that of an irrational expansion of the rm.

% Consider a two-parts compensation contract w i = w F i + w V i F i where w F i ≥ 0 denotes the base salary and w V i > 0 denotes the bonus rate that rewards performance as measured by F i . Given this compensation contract, it is equivalent for manager i to maximize the compensation w i or the target F i . For more on this point see [START_REF] Kopel | Strategic Delegation in Oligopoly[END_REF]. In the second scenario, we assume that prices are easy to change and communicate and the upstream monopolist is unable (or unwilling) to make a price precommitment. Without price commitment the upstream monopolist retains the opportunity to adjust the input price to changes in the behavior of downstream rms. The order of moves is then as follows. In the rst stage of the game, Firm owners simultaneously and independently choose the compensation contracts that will be oered to hired managers (i.e., they set α i , i = 1, 2, . . . , n) . In the second stage, when the prole of managerial incentives ⃗ α is known and common knowledge, the upstream monopolist sets the input price k. Then, in the last stage, managers compete in quantities so as to maximize their compensation taking into account their target-based bonuses and the input price.

The dierence between the timing of moves in the above two scenarios is illustrated in Figure 1. Both three stage games are solved by backward induction. Since the last stage is common to both games, it will be analyzed separately in the next section.

Throughout the paper, we make the following assumption: Assumption 1. We assume that the eciency of rms is suciently high, 1 ≥ ε > τ a+τ , (or the price of energy is suciently low, aε

(1-ε) > τ > 0) to ensure that all rms are active at the equilibrium.

The managerial subgame

At stage 3, given ⃗ α and k, each manager i simultaneously and independently chooses a quantity y i so as to maximize F i , i = 1, 2, . . . , n. Assuming an interior solution, the resulting system of rst-order conditions is

P (Y ) + P ′ (Y )y i = α i A (k, τ, ε) , i = 1, 2, . . . , n. (5)
Obviously, these conditions repeat the standard provision that marginal revenue must equal the marginal cost of production. However, when the manager strikes the balance between marginal costs and revenue, he does not consider the rm's actual marginal cost of production A (k, τ, ε), but its depreciated (or inated) value by the weight factor α i . By solving the i-th equation in (5) for y i we obtain:

y i = α i A (k, τ, ε) -P (Y ) P ′ (Y ) , i = 1, 2, . . . , n. ( 6
)
If α i < 1, observe that the manager i is induced to produce more than a prot-maximizing rm owner (for each output choice of its competitors). By choosing prot weights that are lower than one, owners induce their managers to be more aggressive on the output market.

Summing Equations ( 6) over i results in:

Y = A (k, τ, ε) ∑ n i=1 α i P ′ (Y ) -n P (Y ) P ′ (Y ) , (7)
or, equivalently,

Y P ′ (Y ) + nP (Y ) = A (k, τ, ε) n ∑ i=1 α i . ( 8
)
This last equation implicitly denes the equilibrium industry output Y ⋆ as a function of a weighted sum of the marginal costs of production incurred by the rms making up the industry. This result repeats the observation by Bergstrom & Varian (1985a,b) that Cournot-Nash quantities depend solely on the sum of the rms' characteristics and, especially, are independent of how those characteristics are distributed. By substituting the inverse demand function P (Y ) = a -bY into the xed-point Equation ( 8) and solving for the equilibrium industry output level, we obtain:

Y ⋆ = na-A(k,τ,ε) ∑ n i=1 αi b(n+1) . (9)
Now, by plugging this quantity back into Firm i's (inclusive) reaction function (6), we obtain:

y ⋆ i = a+A(k,τ,ε)( ∑ n i=1 αi-(n+1)αi) (n+1)b . (10)
The implied equilibrium price is:

P ⋆ := P (Y ⋆ ) = a+A(k,τ,ε) ∑ n i=1 αi (n+1) . (11)

Price precommitment

In this rst scenario, the upstream monopolist precommits to an input price k. So, at the time when owners design their managers' compensation contracts, the pricing policy of the upstream monopolist is known and common knowledge. The timing of the game is depicted in Figure 1. At Stage 2, each owner simultaneously and independently chooses the incentives to provide to management (i.e., the prot weight α i ). At stage 1, the upstream supplier sets the input price k. Now, we resume the backward induction procedure starting with the design of managerial incentive contracts.

Choice of managerial incentive contracts

Each owner i sets the prot weight α i so as to maximize its prot taking the input price k as given; i.e., owner i solves

max αi π i = [P (Y ⋆ ) -A (k, τ, ε)] y ⋆ i , i = 1, 2, . . . , n. ( 12
)
Assuming an interior solution, we obtain the following system of n rst-order conditions for prot maximization:

P ′ (Y ⋆ ) ∂Y ⋆ ∂α i y ⋆ i + ∂y ⋆ i ∂α i [P (Y ⋆ ) -A (k, τ, ε)] = 0, i = 1, 2, . . . , n. ( 13
)
The rst term on the left-hand side corresponds to a gain in sales revenue linked to the price increase resulting from the reduction in total industry output. The second term corresponds to a prot loss linked to the reduction in rm i's supply. Conditions (13) indicate that managerial incentive contracts should be designed so as to balance these two countervailing eects.

From Equations ( 9) and ( 10), we obtain (See Appendix A.1):

∂Y ⋆ /∂α i = -A/ (b (n + 1)) , ∂y ⋆ i /∂α i = -nA/ (b (n + 1))) . ( 14
)
Replacing in Equation ( 13), P (Y ), P ′ (Y ), ∂Y /∂α i , ∂y i /∂α i and y i by their respective expressions, after straightforward calculations, we obtain:

-(n -1) (a + A ∑ n i=1 α i ) -(n + 1) A (α i -n) = 0, i = 1, 2, . . . , n. ( 15
)
Solving each of the above equations, we nd the expression of α i as a function of ∑ n i=1 α i :

α i = n -(n-1) (n+1)A (a + A ∑ n i=1 α i ) , i = 1, 2, . . . , n. ( 16
)
Summing Equations ( 15) over i = 1, 2, . . . , n, results in

-(n -1) na - ( 1 + n 2 ) A ∑ n i=1 α i + n 2 (n + 1) A = 0, (17)
from which we extract the expression of the weighted sum of managerial incentives

∑ n i=1 α i = n((n+1)nA-(n-1)a) (n 2 +1)A . ( 18
)
Plugging this sum back into Equation ( 16) we nd:

α c i = n -(n-1) (1+n 2 )A ( a + n 2 A ) , i = 1, 2, . . . , n. ( 19
)
Plugging α c i back into Equations ( 9), ( 10) and (11) yields:

P (Y c ) = a+n 2 A (1+n 2 ) , Y c = n 2 (a-A) b(1+n 2 ) , y c i = n(a-A) b(1+n 2 ) , π c i = n(a-A) 2 b(1+n 2 ) 2 , ( 20
)
i = 1, 2, . . . , n. For the time being, let us assume that k is given and xed. Then, equilibrium values are presented in (20). As expected the solution is symmetric. Furthermore, the solution is admissible (i.e., prices and quantities are strictly positive) provided that a > A. From Equation ( 19), we obtain:

α c i = α c = 1 -(n-1) (1+n 2 ) (a/A -1) , i = 1, 2, . . . , n. ( 21
)
Since admissibility implies a > A, it follows that ( a A -1) > 0 and α c < 1. In other words, equilibrium managerial compensation contracts provide that managers will be rewarded for deviations from prot maximization that favor sales revenue generation. Now, observe

that A(k, τ, ε) > A (k, τ, 1) = k. Since ∂α c /∂A = a (n -1) / ( A 2 ( 1 + n 2 ))
> 0, we nd that the equilibrium prot weight α c reaches a minimum when production generates no pollution (i.e., when ε = 1):

α c | ε=1 = 1 -(n-1) (1+n 2 ) (a/k -1) . ( 22
)
This is the same expression as found by [START_REF] Claude | A reconsideration of the link between vertical externality and managerial incentives[END_REF]. Finally, still assuming that k is given, we cannot exclude that α c might be negative for some parameter values. Indeed, this could be the case for a > (n (n + 1) A) / (n -1). However, in the next section it is shown that α c ranges in the interval [0, 1] when evaluated at the SPNE price k c .

Monopoly pricing

At stage 1, the upstream supplier sets the input price k so as to maximize its prot; i.e., the monopolist solves max k π c up = kn (y c /ε) .

Assuming an interior solution exists, the rst-order condition for prot maximization is

y c + k ( ∂y c ∂k ) = 0. ( 23
)
Solving the above equation for k, after straightforward computations, we obtain the following proposition.

Proposition 1. Suppose that the upstream monopolist precommits to a xed input price.

There exists a unique Subgame-Perfect Nash-Equilibrium. Equilibrium outcomes are, respectively,

k c = aε-τ (1-ε) 2 , α c = 1 - 2(n-1) k c (n 2 +1)(aε+τ (1-ε)) , y c = n k c b(n 2 +1)ε , ( 24
) P c = aε(n 2 +2)+n 2 (τ (1 -ε)) 2(n 2 +1)ε , π c up = n 2 (k c ) 2 b(n 2 +1)ε 2 , π c i = n(k c ) 2 b(n 2 +1) 2 ε 2 . ( 25
)
Assumption 1 ensures that all the quantities given in the above proposition are non negative at the SPNE.

Remark 1. If ε = 1, we obtain the managerial incentive as in [START_REF] Claude | A reconsideration of the link between vertical externality and managerial incentives[END_REF],

α c | ε=1 = 1 - (n -1) (1 + n 2 ) ,
and the same equilibrium values. Moreover, as ε tends to τ / (a + τ ) the input price k c tends to zero, α c tends to one and both upstream and downstream prots tends to zero.

We obtain the following comparative statics results:

∂k c ∂ε > 0, ∂y ∂ε c > 0, ∂α ∂ε c < 0, ∂π c i ∂ε > 0, ∂π c up ∂ε > 0. (26)
A reduction in pollution emissions or, equivalently in our model, an increase in the productivity of downstream rms, allows the upstream supplier to charge a higher input price k. Also, it provides rm owners with incentives to assign a lower weight to prot maximization in managerial compensation contracts. This lower weight encourages managers to adopt a more aggressive market behavior, which results in higher rm and industry output levels. As a result, both upstream and downstream prots rise. The exact inverse comparative static results obtains for changes in the tax rate τ . This should come as no surprise given the inverse relationship between productivity and pollution emissions in our model.

Finally, we have α c ∈ [0, 1], lim n→+∞ α c = 1, and ∂α c ∂n > 0, ∀n > 2. We conclude that α c is monotonically increasing in n and converges to one as the number of downstream rms become arbitrarily large. Indeed, as the number of rms increases, the downstream market becomes more competitive implying that aggregate output rises and market price falls. This in turn leads to a reduction in marginal revenue that must be compensated by lower production volumes. Firm owners achieve the required production cut by assigning a lower weight (1 -α) to sales in managerial compensation schemes.

Flexible pricing mechanism

We now turn to the second scenario in which the order of rms' moves is reversed. At the time when the upstream monopolist sets the input price, it is assumed that the terms of managerial compensation contracts are known and common knowledge. The timing of the game is depicted in Figure 1. In stage 1, rm owners choose managerial incentives. In stage 2, the upstream monopolist sets the input price. We now are in a position to resume the backward induction procedure starting with the resolution of the monopolist pricing problem.

Input pricing

In stage 2, the upstream monopolist sets the input price k so as to maximize its prot:

π ⋆ up = k ∑ n i=1 x ⋆ i . Recall that y ⋆ i = εx ⋆ i so that x ⋆ i = y ⋆ i /ε ⋆ .
The decision problem of the upstream monopolist then writes as

max k π ⋆ up = k n ∑ i=1 (y ⋆ i /ε ⋆ ) .
Assuming an interior solution exists, the optimal input price solves

n ∑ i=1 y ⋆ i + k [ ∂ ∂k ( n ∑ i=1 y ⋆ i )] = 0. ( 27
)
Replacing y ⋆ i by its value from ( 10) and solving for the optimal input price k gives:

k f = naε -τ (1 -ε) ∑ n i=1 α i 2 ∑ n i=1 α i . ( 28
)
As we shall see below, Assumption 1 ensures that k f is positive for the SPNE value of α i .

Note that

∂k f ∂α i = - naε 2 ( ∑ n i=1 α i ) 2 < 0. ( 29
)
An increase in the weight downstream rms place on prots results in a decrease in the output price. Alternatively, a greater sale orientation causes a reduction in the input price.

The economic intuition behind this result is straightforward. As the prot weight decreases, downstream managers are induced to behave more aggressively, producing more.

The increase in input demand then explains the increase in input price

Strategic delegation

In stage 1, each owner i simultaneously and independently sets the prot weight α i so as to maximize its prot. In other words, each owner i solves the following problem

max αi π f i = [ P (Y f ) -A ( k f , τ, ε )] y f i , i = 1, 2, . . . , n.
First-order conditions for prot maximization are:

( b ∂Y f ∂αi + 1 ε ∂k f ∂αi ) ( P (Y f ) -α f i A f ) + ( A f + α f i ε ∂k f ∂αi + b ∂Y f ∂αi ) ( P (Y f ) -A f ) = 0, (30) i = 1, 2, . . . , n. Assuming a symmetric equilibrium, it is shown in Appendix A.2 that condition (30) reduces to g (α) := ω 0 + ω 1 α + ω 2 α 2 + ω 3 α 3 = 0, (31)
where

ω 0 =a 2 n 2 (n + 1)ε 2 , ω 2 = -nτ (1 -ε)(n(n + 1)(aε + ϕ) -2aε), (32
)

ω 1 = -a ( n 2 -1 ) ε((n + 2)ϕ -nτ (1 -ϵ)), ω 3 = -n ( n 2 + 1 ) τ 2 (ε -1) 2 , ( 33
)
with ϕ = (aε -τ (1 -ε)).

The following proposition provides a characterization of equilibrium managerial incentives:

Proposition 2 (Equilibrium managerial incentives). Suppose that the upstream monopolist uses a exible pricing mechanism. Then, there exists a unique symmetric SPNE characterized as follows.

1. The equilibrium prot weight takes values on the interval (0, 1];

2. If production generates no pollution emission then it is given by α f ε=1 = n 2 (n-1)(n+2) ; 3. It is equal to oneimplying pure prot maximizationin two cases: (a) if the downstream market consists in a duopoly and production generates no pollution emission (n = 2, ε = 1) and, (b) in the limit case where downstream rms are so inecient that they prefer to be inactive at the equilibrium (ε = τ /(a + τ )).

Proof. See Appendix A.3.

We remark that Points 3(a) and (b) in Proposition 2 include as special cases results by [START_REF] Claude | A reconsideration of the link between vertical externality and managerial incentives[END_REF] which correspond to the model without pollution (ε = 1). Assuming that α i = α f for all i = 1, 2, . . . , n, and plugging (28) into Equations ( 9)-( 11)

gives the following proposition: Proposition 3. Equilibrium prices, quantities and prot levels are, respectively, given by

k f = aε -τ (1 -ε) α f 2α f , P f = aε(n + 2) + nτ (1 -ε) α f 2 (n + 1) ε , y f = α f k f b (n + 1) ε , (34) π f up = nα f ( k f ) 2 b (n + 1) ε 2 , π f = αk f ((n + 1)(aϵ + τ (ϵ -1)) -k f (α f n + n + 1)) b(n + 1) 2 ϵ 2 . ( 35
)
Under Assumption 1, it is straightforward to show that the prices k f and P f and the quantities y f and Y f are strictly positive.

Comparing equilibrium outcomes

This section attempts to compare equilibrium outcomes depending on whether the upstream monopolist makes a price precommitment. A major diculty in doing this is due to the complexity of the expression for α f . To begin with, we analyze how the decision to precommit alters managerial incentives in the downstream market. We are able to state the following proposition.

Proposition 4 (Comparison of managerial incentives). 1. The equilibrium prot weight is lower when the upstream monopolist precommits to a xed input price; i.e., α c < α f .

2. As the number of downstream rms becomes arbitrarily large and irrespective of the pricing strategy adopted by the upstream monopolist, the equilibrium prot weight converges to 1, implying that the behavior of managerial rms is eventually identical to that of prot-maximizing owner-managed rms; i.e., lim n→∞ α h = 1, h = c, f.

Proof. See Appendix A.4.

The above proposition extends previous results by [START_REF] Claude | A reconsideration of the link between vertical externality and managerial incentives[END_REF], that were established under the assumption that there is no pollution (ε = 1). It states that downstream owners oer compensation schemes that favor sale orientation over prot maximization if the upstream monopolist makes a price precommitment.

The intuition for this result can be traced back to the dierence in the timing of moves between the two games that we consider. In the precommitment scenario, downstream owners set managerial incentives when the input price is known and has become common knowledge. Accordingly, in their decision-making process, they take the input price as xed and given. By contrast, in the other scenario, the upstream monopolist adjusts the input price to the observed degree of competition on the downstream market (as proxied by managerial incentives). Then, downstream owners recognize that the price they pay for the input x depends on the managerial incentives they give to their managers. Specically, they know that a greater sale orientation (a lower value for α f ) results in higher total downstream production, which, in turn, implies an increase in both input consumption and input price. The implied surge in production costs reduces the extent to which rm owners nd it protable to divert managers away from prot maximization; i.e. the value of (1 -α f ).

Finally, as the number of rms rises, the downstream market becomes increasingly competitive so that the strategic value of delegation vanishes. Then, the behavior of managerial rms converges to that of prot-maximizing owner-managed rms.

Next, we compare equilibrium input prices between the two scenarios. When the downstream market structure is a duopoly and no environmental externality exists, [START_REF] Liao | Input price contracts and strategic delegation with downstream duopoly[END_REF]2010) showed that the upstream monopolist sets the same equilibrium price irrespective of whether a price commitment was made. However, [START_REF] Claude | A reconsideration of the link between vertical externality and managerial incentives[END_REF] proved that precommitment results in a lower input price if more than two rms operate on the downstream market. The following proposition extends this result to more general contexts in which production is polluting the environment:

Proposition 5 (Comparison of input prices). If the downstream industry (i) is a duopoly which generates no pollution emission or (ii) consists of innitely many rms, then the upstream monopolist sets the same price in both scenarios (commitment or no commitment).

Otherwise, precommitment results in a lower equilibrium input price.

Proof. From Equations ( 24) and ( 34), we obtain

k c -k f = - aε ( 1 -α f ) 2α f ≤ 0.
If n = 2 and ε = 1, then α f = 1 and k c = k f = a/2. Moreover, since α f tends to 1 as the number of downstream rms becomes arbitrarily large, it follows that lim n→∞ k ck f = 0.

Turning to the comparison of production levels, we are able to prove the following proposition:

Proposition 6 (Comparison of downstream production levels). Downstream managers are encouraged to choose higher output levels if the upstream monopolist precommits to a xed input price; i.e., y c -y f > 0. Consequently, precommitment results in more pollution.

Proof. We have

∆y = y c -y f = aε (n -1) - ( n 2 ( 1 -α f ) + n -α f ) (1 -ε) τ 2b (n + 1) (n 2 + 1) ε . (36)
It is easy to see that y c -y f > 0 if and only if α f > α where α = 1 -(n-1)(aε-τ (1-ε)) (n 2 +1)τ (1-ε) .

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From Proposition 4, recall that α f > α c . A little algebra shows that α c > α. Since α f > α, it follows that y c > y f .

The intuition for this result can be understood by considering propositions 4 and 5 above.

The former shows that, in the precommitment scenario, rm owners design compensation schemes so as to encourage a greater sale orientation (or, equivalently, a more aggressive market behavior). Under the same assumption, the latter proves that the upstream monopolist sets a lower input price. This, in turn, implies that the nal good y becomes less costly to produce. Then, the greater sale orientation combines with reduced production costs to encourage greater production and emissions of pollutant.

Next, we turn to the comparison of downstream and upstream prot levels. In the absence of environmental externalities, [START_REF] Claude | A reconsideration of the link between vertical externality and managerial incentives[END_REF] showed the upstream monopolist is strictly better o when committing to a xed input price. The exact opposite holds for downstream rms: if the upstream supplier engages in xed price contracts, they earn lower prots.

These result might seem surprising at rst sight, since the upstream monopolist charges a lower input price in the precommitment scenario. However, the basic intuition for this result is simple. If the upstream monopolist adopts a exible pricing mechanism, downstream owners anticipate that a more aggressive market behavior will result in a higher input price. Then, each owner provides his management with lower sales incentives. In other words, each owner assigns a lower weight (1 -α) to sales in managerial compensation schemes. This alleviates the problem of over-competition among rms arising from strategic delegation.

However, by waiving the right to adjust the input price to changes in managerial incentives, the upstream monopolist place rm owners back into their initial prisoner's dilemma situation. This strategic move creates the conditions for an overproduction that is unprotable only for downstream owners. Indeed, precommitment makes the upstream monopolist better o since the surge in input consumption implied by overproduction more than compensates for the loss in revenue due to a reduced input price.

Unfortunately, if we relax the assumption that production causes no pollution, it becomes dicult to sign the dierence between prots in the two scenarios for arbitrary parameter values. With that said, we are still able to shed some light on how prots compare and oer interesting insights on this issue. Let us recall that the admissible values of ε lie in the range τ /(a + τ ) < ε < 1. By evaluating upstream and downstream prots at both extremities, we obtain the following proposition.

Proposition 7 (Comparison of prots). 1. If production causes no pollution, then the upstream monopolist is better o committing to a xed input price:

( π c up -π f up ) | ε=1 > 0.
However, the precommittment decision of the upstream supplier is detrimental to downstream rms:

( π c i -π f i ) ε=1 < 0.
2. For a suciently high level of emission per unit of the nal good produced (or a suciently high environmental tax rate), downstream rms stop producing implying zero-prots for all:

π c up | ε= τ a+τ = π f up | ε= τ a+τ = 0 = π c i | ε= τ a+τ = π f i | ε= τ a+τ , i = 1, 2, . . . , n.
Proof. By plugging the value of ε into the corresponding expressions.

We retrieve the same results as in [START_REF] Claude | A reconsideration of the link between vertical externality and managerial incentives[END_REF]. As a work around for the diculty in signing dierences in upstream and downstream prots, we ran numerical simulations.

Despite extensive eorts, we found it impossible to nd even one numerical example that is admissible and reverses the ranking between prots in Proposition 7. If production generates few pollution emissions (ε is close to 1), numerical simulations conrm that the upstream rm is better-o when committing to a xed input price (i.e., π c up > π f up ).

The opposite (i.e., π c up < π f up ) was obtained only for so low values of ε that downstream rms produce nothing. Finally, we conrmed numerically that downstream rms makes lower prots in the precommitment scenario. We conclude that previous results by Claude (2018) are robust to the introduction of pollution emissions from productive activities.

Conclusion

Recent advances in the strategic delegation literature emphasize that factor market imperfections reduce the incentive business owners have to manipulate the structure of incentives embedded in managerial compensation contracts so as to encourage managers to deviate from prot maximization. The purpose of this paper was to re-examine this issue by allowing for pollution emissions and related environmental tax payments.

Two scenarios were considered depending on whether the upstream monopolist supplies the intermediate product through xed price contracts or relies on a exible pricing scheme. The corresponding two games were solved by backward induction. In both cases, we proved the existence of a unique Subgame-Perfect Nash-Equilibrium (SPNE).

Furthermore, we showed that the value of the equilibrium prot weight is restricted to the range from 0 to 1. It was shown that equilibrium managerial incentives encourage pure prot maximization only in limit cases. Hence, non-prot managerial incentives are expected to be the norm rather than the exception. Furthermore, except in limit cases, the following results hold. A price precommitment results in a lower input price which encourages greater production of the nal good and greater pollution emissions. Moreover, it makes the upstream supplier better o and downstream rms worse o. We conclude that upstream suppliers will choose to sign xed price contracts with their customers.

This paper has limitations. For tractability reason, we assumed identical rms. Relaxing this assumption oers interesting challenges for future research. Furthermore, our analysis has focused exclusively on how precommitment alters managerial incentives in downstream market. Accordingly, the rate of environmental taxation was regarded as exogenous and the welfare consequences of precommitment were not investigated. Future research could examine the normative question of optimal environmental policy.

A Appendices

A.1 Proof of Equation (14)

Recall that Firm i's equilibrium output level is (44)

A.2 Proof of Equation (31)

Let us consider the rst-order conditions for prot maximization which are given by :

-

( b ∂Y ∂αi + 1 ε ∂k ∂αi
) + ∂yi ∂αi (P (Y ) -A) = 0, i = 1, 2, . . . , n. ) .

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Let ∆(h) denote the discriminant of h(τ ). By direct computation, we nd that ∆(h) = -a 6 (n -1) 6 n(n + 1) 2 (ε -1) 6 ε 6 Γ, where

Γ = ( 8n 10 -n 9 + 6n 8 + 59n 7 -48n 6 + 156n 5 + 64n 4 + 28n 3 + 96n 2 -68n -64

) .

(

) 68
Since Γ is strictly positive, it follows that ∆(h) < 0. Hence, the cubic equation h (τ ) has a unique real solution. Since h(τ )| τ =0 = a 3 (n + 1)

( n 2 + 1 ) ( n 2 -2n + 2 ) ε 3 > 0,
and h ′ (t) > 0, it follows that h(τ ) > 0 for all τ ∈ [0, aε 1-ε ]. Then Φ>0 implies that g (α c ) > 0 so that α c < α f . 2. Since lim n→∞ α c (n) = 1 and α c (n) < α f (n) ≤ 1, it follows that lim n→∞ α f (n) = 1.

Fig. 1 :

 1 Fig. 1: Timing of the games: precommitment vs. exible pricing

  k,τ,ε)(∑ n i=1 αi-(n+1)αi)) (n+1)b , (38)whereas that of Firm j (j ̸ = i)

  y i and ∂y i /∂α i back into (45), results in [

-CE03-0005).

Plugging the following quantities

yields the cubic equation (31).

A.3 Proof of Proposition 2

Let us consider the existence problem rst. Let δ 0 = ω 2 2 -3ω 3 ω 1 and δ 1 = 2ω 3 2 -9ω 3 ω 2 ω 1 +27ω 2 3 ω 0 . The discriminant of the third-degree polynomial g(α) can be expressed as ∆ = -

3

. Let us show that ∆ < 0 so that Equation ( 31) has a unique real solution α f . To this end, it is convenient to express the discriminant as ∆ = η Γ where η = -4a 2 n(n + 1)τ 2 (ϵ -1) 2 ϵ 2 < 0 and (57)

Examining Equations (59-63), we nd that β j is positive (resp., negative) if j is even (resp., odd), for all j = 0, . . . , 4. Since z < 0, it follows that Γ > 0 and thus ∆ < 0.

We proceed by showing that α f takes values on the interval (0, 1]. Since g(0) = ω 0 > 0, it follows that α f cannot be equal to zero. Moreover, when τ a+τ < ε ≤ 1 we nd that

It follows that g(α) has a sign change on the interval (0, 1). Finally, if ε = τ /(a + τ ) recall that (aε -τ (1 -ε)) = 0, so that g(1) = 0. Hence, the unique real root α f takes values on (0, 1].

A.4 Proof of Proposition 4

1. We proved that g(α) has a sign change on the interval (0, 1). Observe that g(α) > 0 implies that α < α f . It follows that α c < α f if g(α c ) > 0. Then, the proof reduces to showing that g (α c ) > 0.

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