Je tiens en premier lieu à remercier mon directeur de thèse, Stéphane Gauthier, qui m'a encadré depuis ma deuxième année de master et sans qui cette thèse n'aurait pu voir le jour. Sa direction et son exigence m'ont appris à mieux construire et developper mes reflexions. Je le remercie également pour sa bienveillance durant ces quatres années, et pour avoir été disponible chaque fois que j'en ai eu besoin.

Je souhaite également exprimer ma gratitude aux membres de mon jury, Florence Naegelen, Jerôme Wittwer, Ching-to Albert Ma, Philippe Choné et Jean-Philippe Tropéano. Je les remercie pour l'honneur qu'ils me font, d'accepter que je partage mon travail avec eux, pour le temps qu'ils m'ont accordé et pour leurs nombreuses remarques qui, j'en suis sûr, m'ont permis de grandement améliorer la qualité de cette thèse, et me permettrons de mieux aborder mes travaux futurs.

Pendant toute la durée de ma thèse, j'ai pu bénéficier des meilleures conditions pour mener à bien ma recherche, tant au Centre d'Économie de la Sorbonne, qu'à l'École d'Économie de Paris. J'ai notamment pu participer à de nombreuses conférences toutes très instructives, et également lors de l'une d'elle, rencontrer un de mes coauteurs, Nikolaï Brandt. J'en profite pour le remercier, lui et mon second coauteur Robin Hege, pour m'avoir donné à voir d'autres horizons de recherche, et pour leurs amitiés. Une pensée va également aux participants du regretté atelier santé de Paris 1 qui a rendu l'entrée dans ma première année si enthousiaste, et en particulier à leurs initiateurs (Agnès Gramain et Stéphane Gauthier) pour nous avoir donnés accès à un tel environnement d'échange et retours sur nos travaux de doctorants. Je remercie également Lise Rochaix et Daniel Herrera du laboratoire Hospinomics, de m'avoir accueilli dans leur atelier de lecture, et aux ateliers Coopétition-Hospinomics qui m'ont permis d'écouter des professionnels de terrain du secteur hospitalier et ont nourri de nombreuses réflexions. Je remercie tout les chercheurs avec qui j'ai eu le plaisir d'échanger. Je remercie également ceux qui font que tout fonctionne à la Maison des Sciences Economiques, où j'ai passé la majeure partie de ces années de thèse, pour leur aide et leur bonne humeur. Ces quatre années de thèse ont évidemment aussi été l'opportunité de riches rencontres et de nouvelles amitiés. J'ai une pensée particulière pour ceux avec qui l'aventure à démarré dès le magistère, Francesco, Antoine (« H. et P. »), Remi, et bien sûr le trio Max-Zoe-Antoine, pour les bons moments passés ensembles, leurs amitiés et leur soutien. Je tiens à remercier ceux avec qui j'ai eu le plaisir de partager notre bureau, et ceux qui ont toléré mes régulières irruptions v dans le leur, la "team comportementaliste" du 228 et "épistémologique" du bureau 215. Merci à tous les participants des réunions « Intermède » pour les franches rigolades et les conversations, parfois vertigineuses, mais toujours divertissantes. Merci également aux toucanneurs pour leur bonne humeur, ils se reconnaitront. Mes plus chauds remerciements vont également à ceux que ces années parisiennes éloignées ne m'ont pas permis de voir autant que je l'aurais souhaité. Merci pour leurs visites parisiennes rafraichissantes, pour leur accueil festif et leurs écoutes patientes lors de mes retours à Mérignac. Merci à Thomas, et Thomas, à Jérémy, Marc et Nicolas pour leurs amitiés que je ne tenterais pas de décrire. Une pensée particulière va à Elise qui a accompagné mon installation à Paris et me supporte depuis toutes ces années et pendant nos joyeux moments de colocation avec Yannick et Silvestre. Merci à tous mes amis et mes proches. J'aurais enfin une pensée émue pour ma famille, mes parents Sylvie et Christian, et ma soeur Marion. Je ne sais comment les remercier du soutien, des conseils et de la confiance chaleureuse qui m'accompagnent depuis toutes ces annèes. A ma compagne également, un infini merci. Un grand merci à tous

INTRODUCTION -RÉSUMÉ SUBSTANTIEL EN FRANÇAIS

Le cabinet de conseil français Altense a lancé en 2010 le logiciel Hoscare qui permet de comparer différents territoires en termes de prestations de santé (consommation de soins pour 1 000 habitants, production de soins, taux de fuite des patients, notamment). Il combine des données d'activité hospitalière en médecine, chirurgie et obstétrique (MCO), afin d'accompagner les établissements de santé dans leurs choix de positionnement sur leur territoire (part de marché, poids de l'offre au sein du territoire, zones de recrutement...). En parallèle, la société de conseil française Heva a lancé en octobre 2012 sa solution ADAPT (Analyse Dynamique de l'Attractivité du Positionnement des Territoires de santé et des hôpitaux) pour répondre à une demande similaire de la part des hôpitaux.

La préoccupation des hôpitaux à l'égard de leur volume d'activité n'est pas spécifique au cas français et a été renforcée par les récentes réformes des systèmes de financement des hôpitaux observées dans la plupart des pays développés (cf. infra). En France, les rapports publics du [START_REF] Sénat | Rapport d'information sur l'enquête de la Cour des comptes relative aux maternités[END_REF] et de la Cour des comptes (2014), relatifs aux maternités, soulignent notamment le rôle joué par la concurrence accrue dans le secteur hospitalier sur la fragilisation de l'activité de certains acteurs dont la présence est jugée parfois déterminante pour le maintien de la couverture territoriale des soins. Ces problématiques semblent toucher la plupart des territoires français, mais restent peu documentées. À titre d'illustration, le niveau d'activité de la maternité de Grasse (sud de la France) qui demeure inférieure à l'activité que l'on peut attendre d'une maternité de sa catégorie (type IIB) 1 inquiète les pouvoirs publics. Le nombre de grossesses par an y est passé de 1836 en 2010 à 1574 en 2012, et le rapport [START_REF] Sénat | Rapport d'information sur l'enquête de la Cour des comptes relative aux maternités[END_REF] suggère la concurrence des maternités voisines de Cannes et de Nice (respectivement environ 20 et 40km) parmi les causes possibles. L'autorité publique est également préoccupée par le cas de Mantes-la-Jolie 1. Les maternités en France sont catégorisées en fonction du périmètre de complexité de la grossesse que leur service permet de prendre en charge: du niveau I pour une grossesse normale à III pour les unités incluant notamment un service de réanimation en cas de complications graves, avec deux niveaux intermédiaires IIA et IIB.

1

(nord de la France) où, à l'hôpital public François Quesnay, le nombre de grossesses prises en charge annuellement a connu une baisse de 6 % entre 2010 et 2012. La précarité accrue dans sa zone d'activité semble avoir conduit les populations les plus privilégiées à se reporter vers ses concurrents privés en cas de grossesse non risquée. La pénurie de médecins dans l'hôpital qui a conduit à limiter le nombre de visites au début de la grossesse est également mentionnée comme un facteur contribuant à sa baisse d'attractivité.

En outre, en ce qui concerne plus généralement les services de chirurgie, des exemples de fermeture ou de fusion de services sont régulièrement transmis par les médias. À leurs égards, les ARS (Agences Régionales de Santés) ont également envisagé la mise en place de planchers d'activité destinés a autoriser ou non le maintien des services. Pour les ARS, les préoccupations en termes de volume d'activité des établissements font notamment intervenir des considérations d'économies d'échelle et de maintien de niveaux de pratique des chirurgiens suffisants afin qu'ils restent efficaces. Ainsi, du point de vue de l'efficacité productive du système de santé et du maintien de la qualité des services, ces modifications du tissu hospitalier, exacerbées par les pressions concurrentielles dans le secteur, peuvent être justifiées et vertueuses dans la limite des objectifs de garantir l'accès aux soins de santé sur l'ensemble du territoire.

La mise en oeuvre de la Tarification à l'activité (T2A) a renforcé l'importance du volume d'activité des établissements non seulement en termes de pérennisation de leurs services, mais aussi en tant que principale source de revenus des établissements, en termes de latitude à investir et à proposer de nouveaux services. Favoriser la concurrence hospitalière en termes de volume d'activité a été une des motivations d'origine de l'adoption des systèmes de paiements prospectifs tels que le T2A dans la plupart des pays développés. Dans un contexte de fortes pressions budgétaires nationales et de préoccupations quant à la soutenabilité du système de santé, la concurrence a un rôle à jouer pour promouvoir la réduction des coûts de santé et l'amélioration de la qualité des services.

Bien que de très bonnes performances soient attribuées au système de santé français, comme mentionné par [START_REF] Askenazy | Pour un système de santé plus efficace[END_REF], il ne semble pas particulièrement efficace, compte tenu du niveau élevé de ses dépenses, au regard des performances d'autres économies avancées pour lesquelles les dépenses de santé sont relativement moins élevées.. Concernant la hausse général-isée des dépenses de santé, les auteurs estiment qu'il reste encore beaucoup à faire en termes de responsabilisation de l'offre afin de contrôler les dépenses et d'assurer la soutenabilité du système. Il existe en effet des craintes du fait que les dépenses de soins ne sont pas toutes efficaces du point de vue d'une analyse coût-bénéfice. Le cadre dans lequel ces soins sont dispensés demande donc à être étudié plus en avant, notamment du point de vue des incitations qu'il fournit, afin d'envisager des modes de régulation plus efficaces. À ce propos, [START_REF] Chandra | Who Ordered That? The Economics of Treatment Choices in Medical Care[END_REF] souligne que "dans les zones grises de la médecine, où il n'est pas clair quel traitement est dans le meilleur intérêt du patient, les incitations financières peuvent s'avérer décisives".

Toutefois, la recherche en économie n'est pas encore parvenue à trouver de consensus sur les réels effets des récentes politiques, participant à promouvoir la concurrence, en matière de qualité et de rationalisation des dépenses de santé. En termes d'efficacité allocative des soins, des effets de sur-utilisation et de sous-utilisation de certaines procédures de soins peuvent être attendus et sont observés. L'objectif de cette thèse est de décrire les formes que peut prendre la concurrence entre les prestataires de soins de santé, de discuter des conséquences possibles Cette thèse de doctorat propose une analyse théorique des effets de la concurrence entre prestataires de soins au regard de l'hétérogénéité des patients. Une attention particulière est portée sur l'efficacité allocative du système en termes d'utilisation des ressources dans ce contexte de concurrence accrue. Cette thèse identifie notamment des circonstances dans lesquelles les incitations des prestataires de soins à augmenter leur activité peuvent se traduire par une redéfinition de leurs pratiques préjudiciables au bien-être social, de sorte qu'une plus grande prudence devrait être observée quant à leur régulation.

Cette thèse est composée de trois chapitres. Les deux premiers chapitres sont consacrés à la description de l'impact d'une concurrence accrue sur les modalités de décentralisation des plans de traitements que définissent les autorités de santé. Tout d'abord, nous nous concentrons sur le cas des hôpitaux, dont l'activité et financée et régulée par des paiements fixes et prospectifs. Le second chapitre suit une approche plus normative pour mettre en lumière les défis de la régulation du processus de soins (du choix d'effectuer des examens supplémentaires au choix des méthodes de traitements appropriées) compte tenu de la nature incomplète de l'information sur la gravité réelle de la maladie des patients. Dans ces deux chapitres, nous traitons de la régulation des soins d'hôpitaux ne disposant pas d'obligations de service universel. Le dernier chapitre de cette thèse traite du rôle des obligations de services qui sont généralement asymétriques entre les prestataires de soins publics et privés. Dans une application au marché d'aide à domicile des personnes âgées, nous discutons de la concurrence public-privé sur le marché de la santé à la lumière de son asymétrie en termes d'obligation de service. Nous y interrogeons la définition de couverture efficace de deux fournisseurs verticalement différenciés et analysons sa décentralisation à un marché concurrentiel mixte (public-privé).

Contexte

L'inflation généralisée des dépenses de santé demeure une préoccupation majeure des politiques publiques dans les pays développés. Au cours des trois dernières décennies, les dépenses de santé ont augmenté à un rythme plus rapide que les autres postes de consommation dans la plupart de ces pays. En France, leur part dans le PIB est passée de 9% en 2010 à 11% en 2015 (INSEE -Comptes nationaux de la santé). Si cette inflation s'explique en grande partie par le vieillissement de la population et l'apparition continue d'innovations thérapeutiques généralement plus coûteuses, les inquiétudes quant à la soutenabilité de ces dépenses ont accentué l'attention des pouvoirs publics sur les autres leviers des dépenses de santé publique.

Depuis les travaux fondateurs de [START_REF] Arrow | Uncertainty and the welfare economics of medical care[END_REF] et [START_REF] Pauly | The Economics of Moral Hazard: Comment[END_REF], une littérature abondante s'est saisie de l'analyse des différentes défaillances de marché spécifiques au marché de la santé. Il y a évidemment plus d'éléments constitutifs des dépenses de santé et de leur régulation que nous ne pouvons présenter en détail ici. En particulier, les littératures spécifiques à la médecine de ville et aux systèmes d'assurance des patients seront éludées. Cette dissertation se concentre sur le problème de régulation des soins en présence d'une assurance complète des patients.

Le secteur hospitalier concentre une part importante des dépenses de santé (par exemple, en France, elles représentaient 46% du total des dépenses de santé en 2010) et à ce titre, ses modalités de régulation ont fait l'objet d'une attention particulière. Les autorités de santé publique ont un rôle important à jouer pour garantir l'efficacité des dépenses de santé qui sont, dans une large mesure, financées par des fonds publics.

Décentraliser la prestation de soins peut faire intervenir de nombreux arbitrage: encourager la maîtrise des coûts sans sacrifier leur qualité ni inciter à la sélection des patients; encourager une couverture étendue des soins, ou limiter le rationnement, sans induire que des soins injustifiés soient dispensés. En ce qui concerne la régulation des soins hospitaliers, des systèmes de paiements prospectifs (SPP), mieux connu sous le nom de T2A en France, ont été adoptés par la plupart des économies avancées afin de mieux faire face à l'inflation des dépenses. Ces systèmes sont avant tout conçus pour favoriser l'efficacité productive des systèmes de santé. Depuis leur introduction, une vaste littérature s'est construite autour de deux défaillances de marché majeures liées aux asymétries d'information réunies sur les marchés de la santé, à savoir: un aléa moral du côté de l'offre dont les efforts à maintenir les coûts sont inobservés (Laffont and Tirole, 1993) et dont certaines dimensions de la qualité sont inobservables de sorte que leur conjonction donne lieu à des problèmes multitâches [START_REF] Holmstrom | Multitask Principal-Agent Analyses: Incentive Contracts, Asset Ownership, and Job Design[END_REF]. Cette littérature a mis en lumière les possibles antagonismes entre objectifs de réduction des coûts et d'amélioration de la qualité et des moyens d'y remédier. Elle souligne notamment les facultés d'une demande plus avertie de la part des patients et d'une intensification de la concurrence à alléger ces antagonismes. Les principales leçons de cette littérature sont détaillées dans la seconde section de notre introduction en anglais. Cette thèse s'intéresse plus particulièrement aux problématiques d'anti-sélection liées à l'hétérogénéité des patients et au pouvoir discrétionnaire des médecins en matière de décision des prestations pertinente pour chaque patient (chapitres 1 et 2). [START_REF] Garber | Is American Health Care Uniquely Inefficient?[END_REF] ont souligné que la distinction entre efficacité productive et efficacité allocative du système de santé est essentielle pour mieux appréhender les conséquences des nouveaux modes de régulation des hôpitaux et leurs incidences sur les dépenses de santé. Ils suggèrent que, dans la pratique, les politiques publiques tendent parfois à s'attaquer à un type d'inefficacité en ayant des effets limités ou involontaires sur les autres. En particulier, les résultats des incitations à l'efficacité productive du système de soins sur les dépenses totales peuvent être atténués par une offre accrue de traitements généralement plus innovants et plus coûteux.

Dans ces termes, l'efficacité allocative des soins réfère à la question de savoir si les bénéfices retirés de la dépense de fonds publics dans un type de traitement dépassent systématiquement le coût d'opportunité que représentent les autres biens ou traitements qui auraient pu être fournis. C'est une dimension particulièrement importante quand la pertinence d'un soin peut-être discutée, et s'adresse donc plus naturellement aux soins non urgents. L'efficacité allocative peut également devenir déterminante si le système de santé devient limité dans sa propension à couvrir l'ensemble des coûts des soins. Dans ces circonstances, les autorités sanitaires sont contraintes de choisir quels soins devraient bénéficier d'une assurance complète ou non sur la base de ce même critère [START_REF] Askenazy | Pour un système de santé plus efficace[END_REF].

Une question centrale devient alors d'évaluer le rapport coût-efficacité des traitements novateurs et coûteux par rapport aux anciennes méthodes afin de déterminer dans quelle mesure ils devraient être dispensés de façon généralisée ou restreinte. La hausse continue des taux d'utilisation des examens d'imagerie innovants, tels que les IRM ou les scanners, est probablement l'exemple le plus représentatif de ces préoccupations. On dispose en effet, à notre connaissance, de peu d'évidence que la hausse de leur utilisation ces dernières années est associée à une amélioration comparable de la santé des patients de sorte que leur mode de régulation interroge. Les problématiques relatives à la décentralisation des décisions d'examen de diagnostics médicaux feront l'objet du second chapitre de cette thèse.

Outre le niveau général de la qualité des services de santé, l'efficacité allocative du système fait référence ici à des décisions spécifiques au traitement de chaque patient. Par conséquent, une seconde question centrale est de déterminer comment décentraliser une l'efficacité allocative si les objectifs des autorités de santé des médecins et des patients ne sont pas alignés concernant ce qui doit être dispensé, et qui plus est, dans la mesure où chaque décision de traitement est relative à des caractéristiques du patient que seuls les médecins peuvent observer. [START_REF] Pauly | The Economics of Moral Hazard: Comment[END_REF] suggère qu'un certain degré d'inefficacité allocative est inévitable dans tout système de santé du fait des problèmes d'aléa moraux que sous-tend une assurance complète des patients du côté de la demande, et qui peuvent conduire à une sur-utilisation des ressources. Dans un contexte de pression budgétaire forte, la sur-utilisation peut également nuire à la capacité du système à assurer une couverture étendue des soins de façon soutenable. Dans cette thèse, nous nous intéressons aux instruments de régulation de l'efficacité allocative relatifs au cadre incitatif du côté des offreurs de soins, et aux problématiques d'anti-sélection qui leur sont liées (chapitres 1 et 2).

Cette thèse porte sur les questions d'anti-sélections que soulève l'asymétrie d'information entre régulateur et offreurs de soins vis-à-vis de l'hétérogénéité des patients que seuls les offreurs observent. Les deux premiers chapitres abordent en particulier les défis de la décentralisation des plans de traitements en matière de sélection des patients et de sélection des procédures de traitement. Ces problématiques sont devenues particulièrement saillantes depuis les réformes du système de financement des hôpitaux, et en particulier les raffinements des classifications de l'activité des hôpitaux sur lesquelles s'appuient leurs financements. Le troisième chapitre de cette thèse porte sur l'efficacité allocative les prestataires en termes de standards de qualité lorsque ceux-ci diffèrent en termes d'objectif et d'obligations de services.

Les conséquences de la concurrence entre établissements de santé dont nous discutons dans cette thèse s'articulent autour de trois caractéristiques communes aux marchés de la santé et à chaque chapitre de cette thèse:

-L'hétérogénéité de la demande de soins de santé qui est composée de consommateurs aux caractéristiques variables et qui se traduit par une hétérogénéité du coût des soins.

-Le pouvoir discrétionnaire des praticiens sur l'offre de soin: ils sont les seuls observateurs des caractéristiques précises des patients et les seuls en mesure de décider du type de soins à dispenser.

-La présence ou l'absence d'obligation de service universelle qui font varier la capacité des prestataires à discriminer les patients ou à les référer à d'autres prestataires de soins.

Adoption généralisée des systèmes de paiements prospectifs dans les pays développés

Le système de prise en charge publique des soins de santé s'appuyait jusqu'à très récemment sur le coût de ces soins : des systèmes de paiements rétrospectifs critiqués pour l'insuffisance des incitations données aux producteurs de soins à réduire leurs coûts. Cette organisation a donc été remise en cause pour avoir participé au gonflement substantiel des dépenses de santé dans les pays développés. Pour cette raison, à partir du milieu des années 1990, les pouvoirs publics ont commencé à réduire la part accordée aux coûts des soins dans la détermination du montant de l'enveloppe laissée aux producteurs de soins. Après l'introduction des paiements à l'activité dans le système américain Medicare en 1983, des systèmes de paiements prospectifs similaires ont été adoptés dans la majorité des pays développés (Allemagne, Australie, Royaume-Uni, France, Pays-Bas, Portugal, Belgique...). En France, la Tarification à l'activité (T2A) est devenue, depuis son entrée en vigueur en 2004, le mode quasi unique de financement de l'activité médicale, chirurgicale, obstétrique et odontologique des hôpitaux. Dans leur globalité, les SPP ont été mis en oeuvre en réponse aux composantes d'aléa moral du côté de l'offre de soins, à savoir le manque d'incitations des hôpitaux à ralentir la croissance des coûts et des dépenses de santé: limiter l'utilisation des ressources, améliorer l'organisation de leurs services, réduire la durée moyenne des séjours hospitaliers ou investir en vue de réduire les coûts des soins. Contrairement aux systèmes de paiement rétrospectifs, les SPP visent à faire supporter les coûts de production des soins à ceux qui les produise. L'introduction des SPP vise également à promouvoir la concurrence entre les prestataires de soins et à les inciter à maintenir des standards de qualité élevés dans le but de rester attrayants.

Les SPP s'appuient sur la définition d'une grille distinguant un ensemble de types de soins et le montant auquel chacun de ces soins doit être financé. Ils reposent sur une classification de l'activité des hôpitaux en groupes homogènes de malades (GHM). L'autorité de régulation définit les remboursements forfaitaires pour un séjour hospitalier dans chaque GHM et, pour chaque patient soigné, les hôpitaux déclarent le GHM auquel le patient appartient et reçoivent le remboursement correspondant de la part du régulateur. La composante prospective des systèmes basés sur les GHM consiste à déterminer à l'avance les transferts aux hôpitaux, qui peuvent varier en fonction de la pathologie du patient, mais qui restent indépendants du coût réel encouru par les hôpitaux. Ils incitent donc les hôpitaux à internaliser pleinement les bénéfices issus d'efforts de réduction des coûts. Les hôpitaux subissent une perte si le coût réel est supérieur au remboursement, et un surplus autrement. Ainsi, les paiements prospectifs ont été promus comme un moyen de décentraliser l'efficacité productive dans le système de santé.

Ce principe fondateur des systèmes de paiement basés sur les GHM est un mécanisme désormais bien compris de la théorie économique, et appartient à la littérature sur le partage des coûts dans les relations d'agence (les paiements purement prospectifs consistant en l'absence totale de partage des coûts entre le payeur et le producteur). Toutefois, la conjonction des objectifs de maîtrise des coûts et d'amélioration de la qualité a été et demeure un sujet important de la littérature théorique traitant des contrats dans le domaine de la santé. Les principaux enjeux éclairés par cette littérature sont détaillés dans l'introduction générale en anglais de cette thèse.

Nous y soulignons notamment qu'une autre composante centrale des SPP est de promouvoir la concurrence sur le marché en reliant les recettes des hôpitaux à leurs activités et donc leur capacité à attirer les patients.

Le principe des paiements prospectifs de type T2A en pratique Le calcul des montants de remboursements associés à chaque GHM repose sur l'évaluation du coût moyen des soins codés dans le groupe. Plus précisément, l'établissement des prix des GHM suit les principes de la concurrence par comparaison. [START_REF] Shleifer | A Theory of Yardstick Competition[END_REF] souligne que les propriétés des paiements prospectifs peuvent être stratégiquement détériorées si l'autorité s'engage à fixer les prix des GHM aux coûts moyens réels des hôpitaux: pour un hôpital, les efforts de réduction des coûts afin d'augmenter sa marge seraient alors anticipés comme vains.

Le principe de la concurrence par comparaison consiste à fixer les tarifs en fonction, non pas du coût moyen dans chaque hôpital, mais de celui de l'ensemble des hôpitaux (en pratique un souséchantillon de l'ensemble des établissements), afin d' instituer artificiellement une concurrence pour réduire les coûts et retrouver les propriétés d'une concurrence par les prix à l'équilibre de Nash. Ce système présente cependant plusieurs lacunes, notamment en présence de patients entièrement assurés. Il met l'accent sur les préoccupations concernant l'aléa moral du côté de la demande et de l'exposition à des comportements de demande induite du côté de l'offre.

Quand les hôpitaux sont régulés par paiements prospectifs, leurs incitations à accroître leur activité peuvent affecter le niveau total de consommation de soins. Supposer une assurance complète des patients implique en effet que leur reste à charge est indépendant du montant total des soins qu'ils consomment. Les prestataires de soins peuvent s'en saisir pour accroître leur activité en prodiguant des soins moins ou non nécessaire. Par conséquent, la consommation globale en termes de nombre de séjours est susceptible de gonfler et accroître les montants versés par l'autorité. Cet inconvénient des SPP en présence d'assurance complète est désigné par les phénomènes de "demande induite par les médecins" et a fait l'objet d'une grande attention, en particulier en ce qui concerne la médecine de ville dispensée par les médecins généralistes et les soins hospitaliers non urgents (endoscopie ou chirurgie ou de la cataracte, par exemple).

Afin de contrôler les effets d'induction de la demande du côté de l'offre sur les dépenses nationales de santé, les systèmes de tarification à l'activité ont été complétés par l'introduction de contraintes budgétaires globales et d'un mécanisme dit de "point flottant". Il consiste à adapter rétrospectivement la valeur-prix des GHM pour satisfaire à l'objectif budgétaire national (en France, l'ONDAM, Objectif National des Dépenses d'Assurance Maladie, détermine annuellement l'enveloppe nationale consacrée aux soins de santé. Voir Or. (2014) pour une présentation plus exhaustive du cas français). Le coût moyen des GHM est utilisé pour les pondérer les uns relativement aux autres, et la valeur du point de référence "flotte" afin de faire correspondre le remboursement total avec l'objectif budgétaire. Par conséquent, le niveau global de l'activité des hôpitaux n'a pas d'incidence sur le niveau des dépenses nationales de santé. [START_REF] Mougeot | Hospital price regulation and expenditure cap policy[END_REF] montrent que si l'intensité de la concurrence est suffisamment grande, les politiques de plafonnement des dépenses peuvent être un instrument efficace, en particulier lorsque l'établissement de transferts forfaitaires négatifs vers les hôpitaux n'est pas envisageable et que les prix fixés par GHM sont les principaux instruments de l'autorité de régulation. Les auteurs mettent cependant en garde vis-à-vis de leur utilisation dans la mesure où la crédibilité des autorités à s'engager à ne pas réduire l'enveloppe si la consommation globale diminue l'année suivante est cruciale pour les vertus du mécanisme.

La présentation ci-dessus ne fournit qu'une représentation simplifiée du calcul des prix des GHM qui, en réalité, peuvent englober de nombreux autres facteurs de sorte que les valeurs relatives des GHM en pratique s'écartent de la stricte évaluation de leur coût moyen. Cette situation est souvent perçue comme une autre lacune des SPP en pratique, et s'explique notamment par l'intégration d'objectifs des politiques de santé publique (Milcent, 2016, Hafsteinsdottir and[START_REF] Hafsteinsdottir | DRG prospective payment systems: refine or not refine?[END_REF] et la volonté d'orienter les pratiques de soins au moyen d'incitations financières. Par exemple, la promotion de la chirurgie ambulatoire (séjour de moins de 24 heures) a été en partie mise en oeuvre par une convergence des tarifs avec les GHM non ambulatoires correspondants de sorte à rendre la pratique ambulatoire relativement plus attrayante.

De ce point de vue, un défi fondamental de la régulation des soins de santé consiste à promouvoir l'efficacité et la qualité de la production sans induire d'inefficacités en termes de sélection des patients ni de distorsions non souhaitées des pratiques de traitements. Ces enjeux sont au coeur de l'analyse du premier et du second chapitre de cette thèse, qui étudient les effets de sélection de procédures de soins et de performance de tests de diagnostics (typiquement d'imagerie médicale). Les préoccupations liées à la sélection des patients sont inhérentes à la mise en oeuvre de paiements prospectifs fondés sur des grilles de GHM. En effet, les hétérogénéités du coût de prise en charge des patients au sein d'une pathologie donnée font porter un risque financier aux hopitaux étant donné que les remboursement associés sont fixe. En revanche, nous allons voir que les préoccupations qui concernent les distorsions des pratiques de soins sont le résultat de leur design qui n'est, dans la pratique, que partiellement prospectif.

Le raffinement des classifications et les problématiques de sélection des procédures de soin L'adoption généralisée de systèmes de paiements prospectifs s'est accompagnée de l'élaboration de différentes classifications pour et coder les groupes homogènes de malades (GHM) et déterminer les remboursements des hôpitaux. Cependant, une tendance généralement observée consiste à affiner progressivement les différentes classifications [START_REF] Reinhard | Diagnosis-Related Groups In Europe: Moving Towards Transparency, Efficiency And Quality In Hospitals[END_REF], et à introduire la performance de certaines procédures comme indicateurs d'appartenance à certains GHM. Ces raffinements visent à réduire les risques financiers que les prestataires supportent dans le cadre des systèmes de paiement prospectifs au moyen d'une diminution de l'hétérogénéité de coûts au sein des GHM. Toutefois, les indicateurs qui ont été utilisés pour ces raffinements ont renforcé les problématiques liées à l'asymétrie de l'information sur les caractéristiques précises des patients et au pouvoir discrétionnaire des établissements de santé. Dans la mesure où les fournisseurs de soins de santé décident de ce qui devrait être fourni, le choix des niveaux de remboursement des GHM peut sensiblement faire varier leur choix lorsque plusieurs alternatives de soin existent et sont codées différemment. À ce sujet, [START_REF] Chandra | Who Ordered That? The Economics of Treatment Choices in Medical Care[END_REF] note que "dans les zones grises de la médecine où il n'est pas clair quel traitement est dans le meilleur intérêt du patient, les incitations financières pourraient être décisives" .

Incitations financières et décisions de traitement

De nombreuses études empiriques confirment que les pratiques de traitements sont sensibles aux incitations économiques telles que des variations du remboursement des GHM. La plupart des études existantes, cependant, portent sur le système de santé des États-Unis.

Dans une étude portant sur le choix des méthodes d'accouchement des femmes enceintes, [START_REF] Gruber | Physician Financial Incentives and Cesarean Section Delivery[END_REF] teste les prédictions des modèles de demande induite des médecins et confirment la présence d'effets de revenu sur les pratiques médicales. La stratégie d'identification repose sur la baisse des taux de fécondité et l'existence d'un écart de tarifs entre un accouchement par césarienne, relativement mieux remboursé, et les accouchements par voie basse au moment de l'étude. Ils étudient si la diminution du nombre d'actes d'obstétriques, et donc des revenus des médecins, consécutive la baisse de la fécondité, à conduit les médecins à pratiquer davantage d'accouchements par césarienne. Leurs résultats confirment que des chocs de revenu peuvent conduire les médecins à dispenser relativement plus souvent l'alternative de traitement la mieux remboursée. Les auteurs précisent que les effets de revenu qu'ils discutent n'expliquent que 16 à 32 % de la croissance au cours de la période, le reste étant attribuable à d'autres facteurs tels que l'amélioration de la pratique des césariennes qui, devenue moins risquée, est devenue la pratique recommandée pour un plus grand nombre de patients. Gruber, [START_REF] Gruber | Physician fees and procedure intensity: the case of cesarean delivery[END_REF] ont effectué une autre étude sur la substitution des accouchements par césarienne aux accouchements par voie basse, cette fois-ci consécutivement à une variation de l'écart des remboursements des GHM associés à chaque procédure. Les auteurs estiment que la hausse relative du remboursement de la césarienne expliquerait plus de la moitié de l'écart de taux d'utilisation. L'étendue de la relation a été tempérée dans une réplication par [START_REF] Grant | Physician financial incentives and cesarean delivery: new conclusions from the healthcare cost and utilization project[END_REF] qui toutefois confirme qu'elle est statistiquement significative.

En ce qui concerne le soin des crises cardiaques, [START_REF] Coey | Physicians' financial incentives and treatment choices in heart attack management[END_REF] estime qu'en l'absence de distinction des paiements, des méthodes de soin plus ou moins invasives conduiraient à un changement de pratique pour 20 % des patients de leur échantillon et une baisse globale du coût du traitement de 27 % vis-à-vis du scénario où les paiements sont raffinés. Les résultats de l'auteur confirment la relation entre volumes relatifs des procédures de soins dispensés et niveaux relatifs des paiements des praticiens, et confirme les prédictions théoriques de Hafsteinsdottir and Siciliani (2010) en termes de distorsion attendue des pratiques (cf. infra). Cependant, l'étude ne permet pas d'estimer si l'absence de raffinement des paiements générerait au contraire une sousdistribution relative des traitements les plus intenses (et donc ici relativement mieux remboursés quand les paiements discriminent entre les méthodes de soins dispensés), comme les prédictions théoriques de Hafsteinsdottir and Siciliani (2010) le suggèrent. [START_REF] Jacobson | Physician agency and patient survival[END_REF] observent également des résultats conformes aux prédictions théoriques suite à une réduction du remboursement des traitements recourant à certains types de médicaments de chimiothérapie, et constatent par ailleurs que le changement de pratique a été associé à une amélioration de la santé des patients.

Les problématiques liée à la sélection de procedure que nous venons brièvement de documenter sont le fruit de la définition des grilles d'activité des hôpitaux qui en pratique ne sont pas purement prospectifs. En effets la classification des soins aux patients dans les GHM ne dépend pas seulement des diagnostics primaires des patients, mais inclut souvent l'exécution de procédures de traitement spécifiques ou la performance de tests diagnostiques. Cette adaptation des SPP a introduit une part significative de partage rétrospectif des coûts dans les systèmes de paiements basés sur les grilles de GHM, c'est-à-dire, des variations de prix du GHM qui ne sont pas fondées sur des variations prévisibles des coûts attribuables à des caractéristiques, exogènes, vérifiables des patients (typiquement, la pathologie du patient), mais sur les décisions de traitement dont on sait qu'elles conduisent à des variations de coûts. Cette dérogation au principe des SPP en pratique a d'abord été documentée par [START_REF] Mcclellan | Hospital Reimbursement Incentives: An Empirical Analysis[END_REF] concernant le système de tarification par GHM américain. L'auteur explique cette augmentation du partage rétrospectif des coûts par le manque de précision des informations diagnostiques observables, et le fait que la pertinence de certains traitements n'est pas claire pour de nombreux patients, de sorte qu'elles soient difficilement imputables à des caractéristiques exogènes vérifiables. Mc-Clellan (1997) estime que plus dans plus de 40 % des cas le codage d'un soin dans un GHM (Diagnostic Related Groups ou DGRs, dans la classification américaine) était tributaire de la performance d'une procédure de soins particulière en 1993. Selon [START_REF] Gilman | Hospital response to DRG refinements: the impact of multiple reimbursement incentives on inpatient length of stay[END_REF], cette proportion a augmenté avec les raffinements progressifs, atteignant près de la moitié des quelques 500 DRG du système américain de Medicare en 2000. En France, la classification est passée de 600 groupes en 2004 à près de 2300 en 2009, en incluant notamment 4 sous-niveaux de gravité dans la plupart des GHM préexistant. Le niveau de raffinement observé en France est ainsi nettement supérieur à la moyenne européenne [START_REF] Reinhard | Diagnosis-Related Groups In Europe: Moving Towards Transparency, Efficiency And Quality In Hospitals[END_REF], sans qu'il puisse être associé avec certitude à un degré supérieur de partages de coûts rétrospectifs. À notre connaissance, le degré de rétroactivité du partage des coûts n'a pas été documenté avec précision pour le cas français.

Approche théorique des thématiques de sélection des procédures de soins

Le partage rétrospectif des coûts a suscité des préoccupations vis-à-vis des distorsions dans les pratiques de traitements des hôpitaux. Du point de vue théorique la question des modes de décentralisation des plans de traitements fait intervenir les problématiques liées au pouvoir discrétionnaire des médecins concernant le choix des soins à dispenser et l'hétérogénéité inobservée des patients.

En ce qui concerne l'affinement de la classification, l'autorité doit faire un compromis entre (i) la définition d'une grille se basant uniquement sur les diagnostics primaires vérifiables et laisser subsister de fortes hétérogénéités de coût au sein des GHM, ou (ii) raffiner la classification, souvent en s'appuyant sur des indicateurs non exogènes, afin de réduire l'hétérogénéité des coûts entre les GHM et d'ainsi réduire les risques financiers subits par les hôpitaux, mais augmenter celle liée aux sélections de procédures. [START_REF] Hafsteinsdottir | DRG prospective payment systems: refine or not refine?[END_REF] analysent cet arbitrage et montre que la régulation par prix fixe peut entrainer des distorsions des pratiques de traitements qu'un raffinement de la classification ait eu lieu ou non. En revanche, les auteurs suggèrent que la procédure qui est sujette à des incitations de sur-utilisation varie en fonction du scénario envisagé. [START_REF] Hafsteinsdottir | DRG prospective payment systems: refine or not refine?[END_REF] supposent l'absence de risque de sélection des patients et l'établissement des tarifs des GHM au stricte coût moyen du service, les auteurs montrent que le raffinement induit une sur-utilisation des traitements les plus intensifs (coût-efficaces et donc optimaux en particulier pour les patients aux sévérités élevés), alors que leur sous-provision serait à prévoir si le raffinement de la classification entre traitements de base et avancé n'avait pas lieu.

Concernant uniquement l'étude de pathologies dont la classification des modes de prises en charge ont été raffinés, [START_REF] Malcomson | Supplier Discretion Over Provision: Theory and an Application to Medical Care[END_REF] montre qu'à l'optimum de second rang, une modulation des remboursements par GHM, en s'écartant de la règle du coût moyen, permettent de réduire l'étendue de la sur-utilisation, et de se rapprocher d'une allocation efficace. L'auteur fournit une typologie du problème de régulation d'un hôpital en situation de monopole en fonction de différents scénarios d'évolution des coûts avec la sévérité des patients, et en tenant compte des risques de sélection des patients que font encourir le tarif prospectif et la contrainte que cela fait porter sur leurs calculs. Si, en outre, les patients préfèrent le traitement "avancé" plus coûteux, la concurrence hospitalière peut alors renforcer le problème de la régulation des décisions de traitements par prix fixe. C'est ce qui est discuté dans le premier chapitre de cette thèse où nous soutenons que, dans ce cas, la concurrence pour les patients peut diminuer l'efficacité de la modulation du différentiel des tarifs à induire les pratiques de traitement souhaitées. Comme dans les travaux de Hafsteinsdottir and Siciliani (2010) et [START_REF] Malcomson | Supplier Discretion Over Provision: Theory and an Application to Medical Care[END_REF], notre modèle reflète que dans un scénario de classification raffinée, les traitements avancés devraient en effet être mieux remboursés que les traitements de base pour permettre qu'ils soient dispensés aux patients les plus sévères (pour lesquels ils sont relativement plus coûteux à dispenser, mais également plus efficaces du point de vue d'une analyse coût-bénéfice). Dans ces circonstances, afin de limiter l'étendue d'utilisation des traitements avancés aux patients les moins sévères pour lesquels ils sont moins efficaces en termes de coût-bénéfice, l'autorité de régulation peut donc soit augmenter le remboursement des traitements de base et les rendre plus attractifs pour le prestataire, soit accepter un certain degré de sous-utilisation des traitements standards. Dans chaque cas, vis-à-vis d'une configuration non concurrentielle, nous observons une diminution du bien-être social, issue soit d'une perte d'efficacité allocative, soit d'une hausse du coût de décentralisation du plan de traitement efficace.

Le premier chapitre interroge donc les problématiques de sélection des procédures de soins lorsque les hôpitaux sont réglementés par des prix fixes, et en ce sens, suit l'approche de [START_REF] Siciliani | Selection of treatment under prospective payment systems in the hospital sector[END_REF], [START_REF] Hafsteinsdottir | DRG prospective payment systems: refine or not refine?[END_REF], [START_REF] Miraldo | Price adjustment in the hospital sector[END_REF]Street (2011) et Pflum (2015). Comme [START_REF] Pflum | Physician Incentives and Treatment Choice[END_REF], nous y introduisons les problématiques liées à la concurrence entre établissements de santé, mais notre analyse porte plus particulièrement sur l'analyse du bien-être social de second rang lorsque la concurrence s'intensifie, en tenant compte des contraintes que les risques de sélection des patients font peser sur l'établissement des prix. Notre approche de la sélection des patients suit [START_REF] Ma | Health Care Payment Systems: Cost and Quality Incentives[END_REF], [START_REF] Ellis | Creaming, skimping and dumping: provider competition on the intensive and extensive margins1This is a substantially rewritten version of a paper entitled 'Creaming, Skimping, and Dumping: Provider Competition for Patients'.1[END_REF] [START_REF] Malcomson | Supplier Discretion Over Provision: Theory and an Application to Medical Care[END_REF], dans la mesure où nous considérons la possibilité que des tarifs de GHM trop bas puissent inciter les hôpitaux à éviter la prise en charge des patients les plus coûteux.

Le second chapitre de cette thèse propose d'établir un cadre d'analyse d'un autre aspect des décisions de soins qui concerne le choix de recourir ou non à des examens de diagnostiques tels que des méthodes d'imageries médicales. Du point de vue de l'autorité de régulation, la réalisation d'examens supplémentaires avant de décider de la méthode de traitement à appliquer peut, malgré leurs facultés à produire des informations, ne pas être justifiée pour tous les patients. La décision dépend évidemment des caractéristiques du patient, des symptômes particuliers et de la prévalence de comorbidité spécifique, mais aussi du coût des tests. Ainsi, la réalisation d'un scanner ou d'une IRM se justifie que dans certains cas. Cet aspect des plans de traitement appartient aux problématiques évoquées précédemment dans la mesure où les tests peuvent affecter le paiement de l'hôpital, et devraient être déterminés par les types de patients qui sont également observés par l'hôpital seulement. Afin de capturer l'idée que la valeur informative d'un examen dépend des croyances ex ante sur la sévérité de la maladie du patient, nous nous appuyons sur une nouvelle définition d'informativité construite par Bradley et al. (2014), Brandt, Eckwert and[START_REF] Brandt | Weak Informativeness: An Information Order with Variable Prior[END_REF], à partir d'un concept d'information introduit par [START_REF] Ganuza | Signal Orderings Based on Dispersion and the Supply of Private Information in Auctions[END_REF].

Nous examinons le problème d'une autorité de régulation qui peut observer les tests effectués, les résultats des tests et les méthodes de traitement mises en oeuvre (ou du moins nous supposons qu'ils sont vérifiables et peuvent être contractés), mais pas les croyances ex ante sur l'affection des patients qui appartiennent au savoir non objectivable des médecins et dépendent de caractéristiques non observées du patients. Ce chapitre traite donc de l'impact du cadre incitatif sur la décentralisation des procédures de soins (depuis le choix des examens de diagnostics jusqu'au choix des méthodes de traitement), à la fois pour des hôpitaux en situation monopolistiques et concurrentiels. Le cadre de notre modèle fait intervenir une autorité de régulation, un secteur hospitalier et deux types de patients pour lesquels l'information sur la sévérité de la maladie est incomplète. La décision d'acquérir de l'information au moyen d'examens de diagnostique doit donc dépendre des types de patients et est laissée à la discrétion des médecins qui seuls les observent. Nous considérons deux types de patients ex ante: un type dit élevé ou "asymptomatique", dont le degré d'affection est plus incertain, ou du moins les chances pour qu'il soit élevé sont plus grandes que pour le second type de patient, bas. Malgré la production d'informations, la réalisation d'examens diagnostiques supplémentaires coûteux n'est pas nécessairement efficace pour chaque type de patients. Pour le type de patient élevé, la valeur informative de l'examen ainsi que les coûts espérés de prise en charge sont plus élevés. En second rang, les coûts d'implémentation du plan de traitement issue de l'asymétrie d'information sur les types de patients ex ante dépendent fortement du fait qu'aucun type de patient, tous ou seulement un type élevé ne devrait être testé. Nous trouvons qu'il peut être optimal de tester plus de patients que ce qui est efficient en premier rang. En outre, si l'utilisation efficiente des tests de diagnostiques est clivante et consiste à le mettre en oeuvre pour un seul des deux types de patients ex ante, c'est toujours la pratique optimale pour un hôpital monopolistique en second rang également. La structure du secteur hospitalier peut cependant affecter le bienêtre social et les pratiques de traitement optimales. Notre modèle suggère que la concurrence peut renforcer le problème de régulation et de définition du cadre incitatif pour prévenir une sur-provision d'examens. En effet, tester les patients et afin de mieux adapter la méthode de traitement choisie à leur sévérité de maladie sous-jacente (inobservée), même imparfaitement, peut améliorer le statut de santé moyen des patients dans l'hôpital, en particulier si l'on venait à découvrit que la gravité est beaucoup plus sévère qu'attendu. Dans ces circonstances, les préoccupations liées aux phénomènes de sur-traitement ou de sous-traitement dans la cascade diagnostic-thérapeutique se retrouveraient amplifiées lorsque les hôpitaux se font concurrence pour attirer des patients.

Obligations de service universel et statuts des prestataires de soins concurrents

Nous avons souligné qu'une particularité des services de santé réside dans l'hétérogénéité des caractéristiques des consommateurs pouvant se traduire par une hétérogénéités de leurs coûts de prise en charge. Cette spécificité soulève des préoccupations d'accès des populations aux services de santé, dans la mesure où les prestataires qui font face à des tarifs fixes peuvent être incités à éviter ou refuser certains types de patientèles. Les analyses proposées dans les deux premiers chapitres de cette thèse tiennent compte des contraintes que les possibilités de sélection des patients peuvent faire peser sur la modalité de financement des soins de santé. Il apparait au regard de nos résultats et de ceux des travaux supposant l'absence de ces mécanismes de sélection que la présence d'obligations de services peut influer de façon critique sur les contrats établis à l'optimum et sur la gestion des effets de la concurrence.

La mise en oeuvre d'Obligations de Service Universel (OSU) à une partie des prestataires constitue une des modalités d'intervention des politiques publiques visant à garantir l'accès au service à toute la population. C'est le cas en France où des établissements privés coexistent avec des établissements publics qui sont tenus d'accepter toutes les demandes de soins qui leur sont adressées. Le secteur privé y dispose alors d'une plus grande latitude dans le choix des services qu'il offre et dans sa capacité à accéder aux demandes de soins qui lui sont adressées.

Le troisième chapitre de cette thèse a pour motivation d'analyser les conséquences possibles de cette asymétrie d'obligation de service.

Dans le secteur hospitalier, les différences d'obligation universelle de service ont fait l'objet de débats sur l'hétérogénéité des tarifs des GHM entre les secteurs public et privé et la question de leur convergence. En France, le secteur privé, qui représente environ 44 % des séjours hospitaliers, est financé sur la base de la même classification GHM que les hôpitaux publics, mais à des prix différents. Une partie de cette différence s'explique par un différent système de rémunération des médecins, qui, dans le secteur privé; n'est pas inclus dans les tarifs. Dans l'esprit de la concurrence par comparaison théorisée pas [START_REF] Shleifer | A Theory of Yardstick Competition[END_REF], l'introduction de la T2A en France a d'abord prévu une convergence progressive des tarifs entre les deux secteurs.

La convergence des tarifs devait alors refléter la convergence de la productivité des hôpitaux de secteurs concurrents. Elle a cependant été abandonnée, en raison de la persistance d'écarts de coûts de production et de la réticence des acteurs. Un argument principal qui a été avancé reposait sur l'idée que, en raison de leur obligation de service, les établissements du secteur public étaient amenés à traiter des affections en moyenne plus graves que le secteur privé. La littérature traitant des modulations des paiements lorsque l'étendue des sévérités de maladies prises en charge peut varier d'un hôpital à l'autre est présentée dans la version anglaise, plus exhaustive, de l'introduction de cette thèse, et souligne que des écarts de paiements peuvent alors se justifier. Si, en revanche, elles ne s'expliquaient pas par le traitement de patients plus coûteux, les différences de coûts entre secteurs seraient plutôt le reflet de différences de productivité et la convergence des tarifs devrait alors favoriser la convergence de la productivité entre les deux secteurs. À cet égard, l'étude de Dormont and Milcent (2012) montre que, si l'on tient compte des différences dans les caractéristiques des patients pris en charge entre les secteurs public et privé, les différences dans les productivités hospitalières semblent être gommées, voire même sont meilleures dans le secteur public. Les résultats des auteurs suggèrent donc que l'hétérogénéité des coûts moyens des séjours entre les deux secteurs reflète principalement la variation des caractéristiques des patients qu'ils traitent. En ce qui concerne la sélection des patients, cela pourrait également être interprété comme le reflet de mécanismes de sélection des patients en l'absence d'obligation de services.

Les tenants théoriques des OSU et leurs implications pour des marchés concurrentiels ont surtout été abordés par la littérature sur les économies de réseau (télécommunications, gaz, électricité, services postaux...). Dans cette littérature, l'hétérogénéité des coûts de fourniture de service aux consommateurs repose sur une composante géographique: les coûts de connexion/coûts d'accès aux consommateurs varient en fonction ce leur localisation. La littérature a mis l'accent sur les différents rôles joués par les interdictions de discrimination tarifaires et les contraintes de couverture universelle qui sont les deux exigences traditionnelles des OSU. Les fondements des contraintes d'ubiquité, c'est-à-dire l'obligation d'une couverture universelle à un des fournisseurs du marché, peuvent être divers et ne seront pas discutés dans cette thèse.

Nous vous référons par exemple aux travaux de Poudou and Roland (2017) pour une discussion sur la contrainte de couverture optimale d'une autorité averse aux iniquités.

Le troisième chapitre de cette thèse s'interroge plutôt sur la question de savoir si l'asymétrie de l'obligation de couverture entre les prestataires peut avoir un fondement théorique du point de vue de l'organisation des services. Notre analyse soutient qu'une asymétrie de couverture peut être socialement optimale pour limiter la multiplication des "coûts d'accès" encourus par les fournisseurs. En ce qui concerne le secteur hospitalier, on peut interpréter cela comme la possibilité que l'ampleur des coûts d'investissement et de spécialisation, nécessaires pour pouvoir prendre en charge n'importe quel degré de sévérité des patients au sein d'une pathologie, justifie qu'ils ne soient encourus que par certains prestataires. Toutefois, le troisième chapitre ne traite pas du cas de l'asymétrie des hôpitaux publics et privés en matière d'obligation de service.

Bien que le problème pourrait cependant faire l'objet de travaux futurs en nous appuyant sur nos premiers travaux, ceux-ci portent en particulier sur le cas des services de soins à domicile pour les personnes âgées.

Ce chapitre analyse les enjeux de l'asymétrie d'obligations de services et d'ubiquité entre les secteurs public et privé de la prestation d'aide à domicile. Nous discutons en particulier du degré efficace de couverture territorial des prestataires privés ne disposant pas d'obligation d'ubiquité et étudions les résultats de sa décentralisation au moyen d'une concurrence publicprivé d'acteurs différenciés en termes de qualité. Notre modèle suit une représentation du problème de différenciation verticale des entreprises en termes de standard de qualités proposées par [START_REF] Laine | Quality and competition between public and private firms[END_REF], et est étroitement lié à la littérature sur les obligations de service universel dans les économies de réseaux [START_REF] Bourguignon | Skimming the other's cream: Competitive effects of an asymmetric universal service obligation[END_REF]Ferrando (2007) et Calzada (2009) en particulier).

Nous portons notre attention au cas du marché des soins d'aides à domicile où des entreprises publiques, réglementées et soumises à une obligation de service universel sont en concurrence avec des entreprises privées qui ne font l'objet que d'une interdiction de discrimination tarifaire entre ses consommateurs. Comme dans le cas des économies de réseau, notamment postaux, les OSU sur le marché des soins d'aide à domicile sont motivés par sa composante géographique. Le coût de l'approvisionnement des consommateurs varie selon l'éloignement et la densité de la zone du territoire où ils vivent. De plus, la couverture de zones du territoire données par plusieurs entreprises multiplie les coûts de transport totaux encourus sur le marché.

Cependant, la présence de plusieurs fournisseurs différenciés en termes d'offre qualité-prix est valorisable socialement en raison de l'hétérogénéité des besoins des consommateurs et, par conséquent, socialement optimale dans les zones associées à des coûts d'accès relativement faibles.

Dans le problème décentralisé que nous étudions, l'autorité de régulation fixe le prix et la qualité de l'entreprise publique qui encourt une obligation de services, tandis que son concurrent privé choisit librement sa couverture territoriale, son prix et les standards de qualités de ses services.

La discrimination par les prix est interdite pour les fournisseurs des deux secteurs public et privé de sorte que l'asymétrie porte uniquement sur l'obligation d'accéder à toutes les demandes sur tout le territoire (ubiquité) du prestataire public. Nous constatons que la politique de couverture de l'entreprise privée est sans ambiguïté excessive à l'équilibre du marché, c'est-à-dire qu'elle consiste à fournir des zones plus éloignées que ce que le critère d'efficacité prescrit. En l'absence de contraintes de couverture pour l'entreprise privée, l'entreprise publique est alors amenée à renforcer l'intensité de la concurrence en prix pour limiter le pouvoir de marché de la firme privée et sa propension à couvrir des zones éloignées. À l'équilibre, ces tensions se traduisent par une limitation de la différentiation en qualité des acteurs et un alignement de la qualité du prestataire public vers les standards de qualité de son concurrent privée (qu'ils soient en premiers lieux plus élevé ou plus bas) au détriment de l'efficacité allocative sur le marché.

GENERAL INTRODUCTION

Launched in France in 2010 by the consulting firm Altense, the software Hoscare allows to compare the areas in terms of health-care provision (consumption of care per 1,000 inhabitants, production of care, flight rate of patients, etc.). It combines data from the hospitals' activity in medicine, surgery, obstetrics (Médecine, Chirurgie, Obstétrique, "MCO") , in order to help the strategic positioning of health institutions on their territory (market shares, weight within the territory supply, zones of recruitment ...). In parallel, the French council firm Heva launched its ADAPT solution (Dynamic analysis of the attractiveness of the positioning of health regions and hospitals) in October 2012 to answer similar demand from the hospitals.

The preoccupation of hospitals about their activity is not specific to France and has been strengthened by the recent changes of the hospitals' care-funding system and the adoption of prospective payment systems in many developed countries (cf.infra). In France, the public reports from the [START_REF] Sénat | Rapport d'information sur l'enquête de la Cour des comptes relative aux maternités[END_REF] and the Cour des comptes (2014), relative to the maternity services, have emphasized the role of an increased competition in jeopardizing the sustainability of some actors' activity whose presence plays a role in maintaining the territorial health coverage. Those concerns seem to affect most of the French territories but remain poorly documented. For instance, activity levels of the maternity ward in Grasse (South of France) which is below the activity one can expect from a maternity of its category (type IIB)2 worries the public authorities. The number of pregnancies per year has decreased from 1836 in 2010 to 1574 in 2012, and the report suggests a competition effect from the maternity ward of Cannes and Nice nearby (respectively around 20 and 40km) [START_REF] Sénat | Rapport d'information sur l'enquête de la Cour des comptes relative aux maternités[END_REF]. The public authority is also concerned by the case of Mantes-la-Jolie (North of France) where the number of pregnancies in the public hospital François Quesnay experienced a 6% decrease between 2010 and 2012. The increased precariousness of those areas of activity seems to have led the more privileged population to register to private competitors in case of normal pregnancy. The fact that we observe a short-age of physicians in this hospital has led to limit the number of visits in the beginning of the pregnancy is mentioned as a factor contributing to its decreased attractiveness.

News media also provide examples of closing or merging of hospital's surgery services. Concerning hospital's surgery services, the ARS (Agences Régionales de Santés) has also considered the implementation of a lower limit bound of activity to authorize the maintain of the services. These preoccupations also includes economies of scale concerns and that the surgeon's level of practice is sufficient so that they remain efficient. In terms of the productive efficiency of health-care system and the maintain of high quality standards these consequences of the increased role of hospital volume of activity can be justified and virtuous, in the limit of the concerns of guaranteeing health-care access on the whole territory.

The implementation of the "Tarification à l'activité" (T2A) has srentghtened the role of the hospitals' activity in maintaining their sustainability and more generally, as the hospitals' main source revenue, in determining their latitude to invest and engage in the provision of new services. Fostering hospitals' competition in terms of volume activity has actually been an important motivation for the implementation of the prospective payment systems such as the T2A in most developed countries. In a context of strong national budget pressure and concerns about the sustainability of the health-care system, competition can play a role toward the promotion of both cost-reduction and improving the quality standards. In the past few years, however, one has also observed an increased tendency of mergers between competitive services in order to avoid closure of services and alleviate the competitive pressures. Although very good performance is attributed to the French health system, as mentioned by [START_REF] Askenazy | Pour un système de santé plus efficace[END_REF], it does not seem particularly effective, given the high level of its spending, when compared to the performance of other advanced economies that spend relatively less. The authors argue in this context that there is still much to be done in terms of supply-side accountability in order to control expenditure and ensure the sustainability of the system. There are indeed concerns that health-care expenditures are not systematically cost-effective. The framework in which these cares are provided therefore needs to be studied further, particularly from the point of view of the incentives it provides, in order to consider more effective modes of regulation. In these respects, [START_REF] Chandra | Who Ordered That? The Economics of Treatment Choices in Medical Care[END_REF] emphasize the fact that "in the gray area of medicine where it is not clear what treatment is in the patient's best interest, financial incentives might prove decisive".

However, economic research hasn't reach consensus on the real effects of the recent policies promoting competition in terms of quality and rationalization of health care spending. In terms of allocative efficiency and rationalization of health-care spending, over-provision and underprovision of treatment can be expected and are observed. With respect to these concerns, the motivation of this Ph.D thesis is to describe the shapes that competition among health providers may take, to discuss the possible consequences of an increased competition in this sector in terms of allocative efficiency, and study the modalities of public interventions to regulate the interactions between health-care providers.

This dissertation thesis provides a theoretical analysis of competition effects in health-care with regards to the heterogeneity of care consumers. A particular attention is carried out on the efficiency of resources' allocation in health-care in the context of increased competition.

Along this thesis, we notably identify circumstances so that the health-care providers' incentive to increase their activity can also translate into redefinitions of their praxis that are detrimental to social welfare.

The thesis is composed of three chapters. The first two chapters are devoted to describe the impact of an increased competition on the modalities of treatment plans' decentralization by health authorities. In a first place, we focus on the case of hospitals regulated by fixed, prospective, repayments. The second chapter follows a more normative approach to highlight the challenges of regulating the care process when the incomplete nature of information about patients' actual severity of illness is taken into account. In these two chapters we discuss hospital-care regulation in the absence of universal services obligation. The optimal health coverage is endogenously determined with respect to cost-effectiveness criteria and takes into account the patients selection concerns in health-care. The last chapter of this dissertation addresses the role of services' obligation that are generally asymmetric between the public and the private sectors of health care provision. In an application to the home-care market, we discuss the public-private competition in health care at the light of its asymmetry in terms of obligation of services. We question the definition of an efficient coverage of two vertically differentiated providers and analyze its decentralization to a mixed-duopoly market.

In what follows we will first present the economic stakes of health care and hospital regulation and the founding principles of the current system for funding hospital care. Next, we will introduce the modalities of competition in the hospital sector and discuss its virtues for promoting quality of care under a fixed price regime of regulation. Further, we present the concerns about allocative efficiency of care that will be at the heart of this thesis. More precisely, we will discuss in more details the implementation of the prospective payment systems through payments based on Diagnostic Related Groups (DGR), and the role that classifications design and payments' levels may play into shaping the evolution of hospital-care practices. A last section will be devoted to present our approach of universal coverage of care and the public-private competition component for health-care regulation.

Context

The general inflation of health expenditures remains a major preoccupation of public policies in developed countries. During the past three decades, health expenditures have grown at a faster rate than the other consumption items in most developed countries. In France, their shares in the GDP grew from 9% in 2010 to 11% in 2015 (INSEE -Comptes nationaux de la santé). Although this inflation can largely be explained by the ageing of the population and therapeutic innovations that are generally more expensive, the concerns about the sustainability of these expenses have stressed the attention of public authorities on the other drivers of the growth of health spending. Since the seminal papers of [START_REF] Arrow | Uncertainty and the welfare economics of medical care[END_REF] and [START_REF] Pauly | The Economics of Moral Hazard: Comment[END_REF], a large literature seizes the analysis of the different market failures of health care. There is obviously more components of health-care spending and health-care regulation than we can thoroughly present here. In particular, the rich literature specific to the ambulatory medicine and patients' insurance system will be eluded in our presentation. We instead focus on the problem of hospital care regulation with fully insured patients.

The hospital sector concentrates a large share of health expenditures (for instance in France they represented 46% of total health expenses in 2010) and as such, the modalities of its regulation have been subject to a specific attention. The public health authorities have an important role to play in guaranteeing the efficiency of health-care expenditures that are, in a large extent, financed by public funds.

The decentralization of health care provision must deal with many trade-offs: providing incentives to costs' containment without sacrificing their quality nor induce patients' selection ; and also encouraging the health coverage or limiting rationing without inducing unjustified care. Concerning hospital care regulation, we have observed a broad adoption of prospective payment systems (PPS) in advanced economies' hospital sector to better deal with the inflation of hospital-care expenditures. The design of these systems have mostly been introduced to foster the productive efficiency in health-care. Since the introduction of PPS, a large literature has built upon two critical market failures due to information asymmetry in health care that are supply-side moral hazard in regulation problem (Laffont and Tirole, 1993) and multitasking issues [START_REF] Holmstrom | Multitask Principal-Agent Analyses: Incentive Contracts, Asset Ownership, and Job Design[END_REF]. This literature has highlighted the main trade-offs between cost-containment and quality enhancement in health-care, and the role of competition to alleviate this problem. We will review the main issues and lessons from this health-care literature in the second section of this introduction. This dissertation however will be mostly concerned by the issues of adverse-selection in health care regulation. In particular we will focus on the regulation problems due to patients' heterogeneity and physicians' discretion over provision (Chapters 1 and 2). [START_REF] Garber | Is American Health Care Uniquely Inefficient?[END_REF] have emphasized that distinguishing between productive and allocative efficiency is key to better understand the different consequences of health care reforms and their impact on health care expenditures. They suggest that, in practice, public policies tend to address one type of inefficiency but have limited or unintentional effects on the others. The effects of productive efficiency's improvement on health expenditures can indeed be deterred by an increased provision of, typically, innovative and costlier treatments. In these terms, allocative efficiency of care questions whether the benefits from public funds spent into a treatment consistently exceed the opportunity cost of other goods or treatments that might be provided. It GENERAL INTRODUCTION is particularly relevant for elective (non-emergency) treatments for which the relevance can be sometimes discussed. It has also been underlined as a fundamental matter of public policy if the health-care funding system becomes limited in its propensity to cover the cost of all cares [START_REF] Askenazy | Pour un système de santé plus efficace[END_REF]. In this case, the health authorities become constrained into choosing which care should be provided with full insurance or not on the basis of a same criterion.

A central question is therefore to evaluate the cost-effectiveness of these innovative treatments with respect to former methods in order to determine to what extent their provision should be generalized or restricted. The continuous expansion of innovative imaging tests' performance such as MRI or CT-scan is probably the most representative example of these concerns. Their provision has not stopped increasing in the past years, in particular in the U.S. while the evidence of improvement in the patients' health outcomes are limited. Testing decision will be the specific topic of Chapter 3. Aside general health care standards of quality, allocative efficiency of care refers here to patients-specific decisions. Hence, a second matter straightly becomes how to decentralize such allocative efficiency taking into account that the treatment strategy should depend on patient types that only the physicians can observe before deciding about the treatment that should be provided. In a seminal paper, [START_REF] Pauly | The Economics of Moral Hazard: Comment[END_REF], argue that allocative inefficiency is inevitable in any healthcare system because insurance for medical care causes over-utilization due to demand-side moral hazard. This dissertation will instead discuss these issues with regard to supply-side adverse selection and discretion over-provision in health care.

The dissertation will focus on adverse-selection issues related to patient heterogeneity and physicians' discretion over provision. In particular, the first two chapters will discuss the challenges to decentralize allocative efficiency dealing with patients' selection and the concerns about care procedures' selection that are presented in section 3 of this introduction. We will argue that these topics have become particularly salient with the reforms of the health-care funding system, especially the refinements of the Diagnostic Related Group classifications.The third chapter of this thesis will study allocative efficiency in terms of quality standards when providers are differentiated and differ in their obligation of services i.e. the patients' selection concern is relevant for only one provider.

Broad adoption of Prospective Payment Systems in developed countries

Since the introduction of payment per case schedule in the U.S. Medicare system in 1983, similar Prospective Payment Systems (PPS) have been widely adopted in developed countries (Australia, United-Kingdom, France, Germany, the Netherlands, Portugal, Belgium, ...). In France, the "Tarification à l'activité (T2A)" has become the quasi-unique way of financing medical, surgical, obstetric and odontology hospitals' activity since its implementation in 2004. PPS have been implemented to better deal with the supply-side moral hazard component of hospitals' regulation problem, namely, the hospitals' lack of incentives to slow down the growth of health-care expenditures. The contracts that were formerly predominating were mostly retrospective and determined the hospitals' revenue based on the costs they declared. Retrospective payments were blamed for not giving hospitals enough incentives to improve their services' organization, to reduce the average length of stays or to engage in investments to reduce the costs of care, neither to limit the use of resources. In opposition to retrospective payment systems, PPS aim at bearing the cares' production costs on their providers. The introduction of PPS, in addition, intends to promote competition between care providers and give them incentives to maintain their high standards of services to remain attractive.

The PPS rest on a classification of hospitals' activities into diagnostic groups (hereafter termed as DRGs, Diagnostic Related Groups, following the terminology of the leading U.S.

Medicare system). The authority of regulation defines fixed repayments for a hospital stay in each diagnostic group. For every patient they treat, hospitals report the diagnostic group they belong to and receive the corresponding repayment.

The prospective component of DRG-based systems lies into predetermining the repayment to hospitals that may vary according to the patient's pathology but are independent from the actual cost that hospitals incur. It therefore induces hospitals to fully internalize the benefits from undertaking efforts to reduce costs. Hospitals realize a loss if the actual cost exceeds the repayment, and a surplus otherwise. To the extent that it is also hospitals which support the disutility from undertaking these efforts, prospective payments have been promoted as a way to decentralize productive efficiency in the health care system (with capitation as its counterpart in GENERAL INTRODUCTION ambulatory medicine). This founding principle of DRG-based payment systems is a now a well-understood mechanism in economic theory, and belongs to the literature on the cost-sharing (with purely prospective payments consisting in the particular case of complete absence of cost-sharing between the payer and the producer). However, the conjunction of cost containment and quality enhancements' objectives have been and remains a major topic of the theoretical literature addressing contracting in health care. In this respect, another central component of PPS is to promote competition on the market by linking hospitals' revenues to their activity and thereby their ability to attract patients (c.f. infra).

Implementation of DRG-based systems in practice

The determination of DGR prices rests on the evaluation of the average cost of care of the patients that have been coded in the group. More precisely, DRG-pricing follows the principles of yardstick competition. [START_REF] Shleifer | A Theory of Yardstick Competition[END_REF] emphasizes that the property of prospective payment may be strategically undermined if the authority commits to set the DRG-prices at the hospitals' actual average costs: for the hospital, the efforts to reduce costs in order to increase its margin would be in vain. The principle of yardstick competition is to set prices according to not each but all hospitals' average cost to virtually institute a competition to reduce costs and retrieve the properties of competitive prices at the Nash equilibrium. This system, however, presents several shortcomings in particular in presence of fully insured patients. It stresses the concerns of the demand-side moral hazard [START_REF] Pauly | The Economics of Moral Hazard: Comment[END_REF]) and exposure to the supply-side induced demand.

Under PPS, the hospitals' incentive to increase their activity may not be neutral on the overall consumption of care. Full insurance of patients, indeed, implies that the patients' outof-pocket is independent from the total amount of care they consume. This can be grasped by physicians as an opportunity to increase their activity by delivering unnecessary care. Therefore, the overall consumption of number of stays could inflate. This drawback of PPS under full insurance has been named as "physicians induced demand" and has carried a lot of attention, in particular concerning ambulatory medicine provided by general practitioners and hospital elective care (e.g. cataract surgery or endoscopy).

In order to control for supply-side demand effects on national health expenditures, the DRGprice system is complemented by the introduction of global budget constraints and a "floating point" mechanism. It consists in retrospectively adapting the price-value of DGRs to satisfy the national budget target (in France the ONDAM (Objectif National des Dépenses d'Assurance Maladie) annually determines the national envelop dedicated to health care. See Or. (2014) for a complete presentation of the French case). The average cost is used to weight DRGs relatively to each other, and the value of the point "floats" in order to match the total repayment with the budget target. As a result, the overall level of the hospitals' activity does not impact the level of national health expenditures. [START_REF] Mougeot | Hospital price regulation and expenditure cap policy[END_REF] show that if the intensity of competition is large enough, expenditures cap policy can be a useful instrument when fining hospitals is not an option and fixed prices are the main instruments of the regulation authority.

They show, however, that without the possibility to impose negative lump-sum, the first-best cannot be retrieved.

The floating point mechanism has given rise to other concerns. In particular, it penalizes virtuous hospitals that would remain sober in terms of treatment decisions since all hospitals would suffer from a decrease in the point value [START_REF] Dormont | Comment évaluer la productivité et l'efficacité des hôpitaux publics et privés ? Les enjeux de la convergence tarifaire[END_REF]. Moreover, it does not address inequity between heterogeneous hospitals in heterogeneous territories and other critical caveats to PPS such as selection or discrimination among patients. As any payment systems, the PPS in practice has received many critics on several bases and many issues remain to be dealt with.

Shortcomings of DRG-based systems implementation

In practice, the classification of patients into DRGs does not only rely on observable characteristics of patients, such as their primary diagnoses, but integrate the performance of specific treatment procedures or tests in their definition. This derogation to the theoretical principle of PPS reflects the health authorities' will to decrease the heterogeneity of treatments' costs within DGRs and address patients selection and discrimination concerns. In France, the classification went from 600 groups in 2004 to nearly 2300 in 2009, in particular by including 4 sub-levels of severity in most former DRG groups. The level of refinement observed in France is far above the European average. For instance, Spain reached the second position with fewer than 1500 groups (see Reinhard, Alexander and Wilm (2011) p. 54). At this level, the prospective dimension of its DRG-based system can hence be discussed. The classification's refinements have stressed the adverse selection problem in the hospitals regulation by encouraging the provision of more intense treatments. The ins and outs of classification refinements and their implications are central aspects of this dissertation (Chapters 1 and 2) and discussed in more details further. However, we will not discuss up-coding behaviors, i.e. fraudulent coding practices which lead to unjustified payments and for which room or occurrence has also increased with classification refinements. See [START_REF] Milcent | Upcoding and heterogeneity in hospitals' response: A Natural Experiment[END_REF] for an estimation of up-coding after the French classification refinements and the reference therein, and [START_REF] Kuhn | Upcoding and Optimal Auditing in Health Care (or The economics of DRG creep)[END_REF] for a theoretical approach of the issue.

The above presentation provides in fact only a simplified representation of the calculus of DGR-prices, which in reality may embody many other factors and the relative values of the DRG's weight often depart from the strict evaluation of their average cost. This is often perceived as another shortcoming in practice, but it can find many explanations such as the integration of public health policy targets (Milcent, 2016, Hafsteinsdottir and[START_REF] Hafsteinsdottir | DRG prospective payment systems: refine or not refine?[END_REF]. For instance, the promotion of ambulatory surgery (less than 24 hours) has been in part implemented by a convergence of tariffs with the corresponding non-ambulatory DGRs in order to make the ambulatory surgery relatively more attractive. Along this introductory presentation, we will see that prices departing from the average cost may have founded theoretical justifications in terms of public policies. However, there are also concerns that DRGs system, as they are now, are more used as a way to control health-care expenditures at the expense of productive or allocative efficiency targets. See [START_REF] Mougeot | La tarification à l'activité : une réforme dénaturée du financement des hôpitaux, The French Prospective Hospital Pricing Policy : A Perverted Reform[END_REF] for a more exhaustive discussion of the shifts between theory and practice of the implementation of DRG-based payments.

Patient's demand and quality standards of health care

A major challenge of health care contracting consists in giving the proper incentives to hospitals to contain costs without sacrificing their quality. For the health authority payer, the problem consists in decentralizing two types of, possibly antagonists, efforts and both difficult to verify and enforce by contract. More precisely, the sequel will consider that the monitoring of standards is too expensive, and even impossible for some aspects of services. This section will discuss the potential virtues of prospective payment in promoting hospitals' competition to maintain standards of services and therefore alleviate this trade-off.

To be consistent with the restriction of our presentation to the health-care regulation problem with fully insured patients, we will consider hospital competition under administered prices only, i.e. non-price competition. In this context, the role of competition to maintain quality depends mostly on whether the demand of patients reflects quality, and if so, which dimension of the hospitals' quality.

Cost-reduction vs. quality trade-off when patients' demand does not reflect quality

In absence of quality competition, the impact of prospective payments depends on the hospital's own valuation of quality. In principle, under fully prospective payments, a purely for-profit hospital would systematically skimp on non-verifiable quality. Hospitals might achieve cost containment efficiently, but not engage in quality enhancement efforts that would increase their production costs thereby decrease their margin (or at least not beyond the minimum requirement to avoid for instance being sued for malpractice, or activity ban). However, it is now well recognized and admitted that health-care providers' objectives include some extent of both benevolence or altruism components, and self-interest.

Despite a certain degree of altruism, [START_REF] Ellis | Provider behavior under prospective reimbursement: Cost sharing and supply[END_REF] have emphasized that if the physician agency remains imperfect, i.e. if their own relative weighting of patients' benefit is less than those of the (principal) social planner, some degree of cost-sharing should be implemented. By partially relieving health-care providers of the weight from increasing quality in terms of costs, efficiency can be refined and the gap between the physician and the social planner's valuations of quality be filled. This can typically be achieved by supplementing the prospective payments to some degree of retrospective cost-based repayment. In practice, we observe the authorities' lump-sum transfers to complement DRG-based system and share with hospitals the cost of undertaking particular activities (Or., 2014). For instance, in France, the MIGAC (Missions d'Intérêt Général et d'Aide à la Contractualisation) program provides financial support to hospitals for undertaking several public interest missions (the "MIG"), or help hospitals to adapt their offer notably with acquiring new technology (the "AC" component) and can be interpreted in this view.

The former reasoning implicitly assumed a one-to-one relationship between cost and quality, but a multi-tasking agency arises if there is no guaranty that any cost's increase is the reflection of quality improvement. While stressing the duality of both targets, Chalkley and Malcomson (1998b) show evidence that choosing the degree of cost-sharing must solve a second-best tradeoff between the promotion of standards of services and limiting the cost of health care. [START_REF] Jack | Purchasing health care services from providers with unknown altruism[END_REF] furthermore extends this problem by addressing that the degree of benevolence may vary upon hospitals and is also unobserved.

Patients' demand in health care

There are many reasons to think that the volume of demand to health care practitioners reflects at least some dimensions of quality. However, having a very precise and descriptive model of how patients choose their practitioner or hospital is a complex matter. The parameters involved are numerous and include observable quality, recommendation by general practitioners or relatives, referral and distance to the provider. Moreover, in some health sectors such as hospital-care, patients do not receive frequent treatments so that the quality they observe might not affect their own demand in the future, but instead the demand of other patients. Due to these complexities, most theoretical approaches use reduced forms of demand functions that are increasing in quality in order to capture the relationship between quality and patients' demand [START_REF] Ma | Health Care Payment Systems: Cost and Quality Incentives[END_REF][START_REF] Ma | Optimal Health Insurance and Provider Payment[END_REF], Chalkley and Malcomson, 1998a[START_REF] Ellis | Creaming, skimping and dumping: provider competition on the intensive and extensive margins1This is a substantially rewritten version of a paper entitled 'Creaming, Skimping, and Dumping: Provider Competition for Patients'.1[END_REF]. Concerning hospital care, it is admitted that the demand effect should occur through reputation effects or recommendations of the, better informed by experience, general practitioners. Our approach of patients' response to hospitals' decisions will follow the same underlying interpretation in the first two chapters of this dissertation.

The use of reduced-demand form has been first introduced in the view of physicians' induced demand, i.e. the propensity of physicians to increase the volume of care they provide, possibly without affecting these of the other physicians. This view therefore also raises concerns that if the demand increase is not the result of being more attractive than competitors, it can consist in the delivery of unnecessary care. [START_REF] Allard | Provider Competition in a Dynamic Setting[END_REF] reconcile the result of many static models with reduced demand forms with a more comprehensive/descriptive model of physicians competition in a dynamic setting. [START_REF] Gravelle | Imperfect information in a quality-competitive hospital market[END_REF] approach the demand to hospitals in response to their quality by accounting for the imperfection of patients' information who actually observe noisy signals about qualities. However, this approach from which results are detailed further, introduces tractability issues for other applications such as ours. We should also mention other modelling of hospital competition oriented toward horizontal differentiation [START_REF] Calem | Competition and Specialization in the Hospital Industry: An Application of Hotelling's Location Model[END_REF][START_REF] Beitia | Hospital quality choice and market structure in a regulated duopoly[END_REF][START_REF] Brekke | Quality and Location Choices under Price Regulation[END_REF][START_REF] Brekke | Hospital Competition and Quality with Regulated Prices*[END_REF]. These works analyze the hospitals' case-mix differentiation or specialization degrees and we will not detail these aspects here.

Benefits of prospective payments for promoting competition

By relying on the hospital's revenue on their activity, the implementation of DRG-based system has substantially increased profit-related incentives in the whole hospital sector. In particular, increasing their demand by attracting more patients has become their main source of surpluses' increases. We should note here that these incentives are admitted to affect both forprofit and not-for-profit hospitals and because the surplus can be spent on perks for staff or on improving facilities3 .

To the extent that higher repayments increase the incentive for hospitals to attract patients, [START_REF] Ma | Health Care Payment Systems: Cost and Quality Incentives[END_REF] shows that the relationship between the hospital's demand and quality of services gives a supplementary leverage to the authority when setting the fixed repayments. Competition may therefore play a role, through the patients' demand for quality, to palliate the hospital's lack of altruism in a measure that depends on the marginal gain to attract patients i.e. the level of prospective payment. In this view, while by design fixed prospective payments allow to decentralize the productive efficiency, an appropriate choice of their level may induce hospitals to internalize the social benefits of quality and alleviate the issue of multitasking. Chalkley and Malcomson (1998a) extend the problem by considering the multiple dimensions of hospital care quality (medical services, nursing, hotel services...). The authors stress that the fact that the patient perceives every aspect of quality correctly (or similarly to the payer) is critical for assessing the role of competition and maintaining the appropriate standards of services. They also show that if patients lack of information about qualities then more complex contracts should be considered.

Information disclosure on measures of hospitals' quality has been pointed out as the most natural way to address this issue. For instance, the 2003 NHS reform, that introduced patients' possibility to choose their hospital in the U.K., has been accompanied by the provision of publicly assessable information on the provider's quality to inform the patients' choices. [START_REF] Ma | Information disclosure and the equivalence of prospective payment and cost reimbursement[END_REF] provide a theoretical analysis about how the way information is disclosed may impact the outcome and show that a well-designed quality index can also decentralize efficiency under cost-reimbursement. [START_REF] Gravelle | Imperfect information in a quality-competitive hospital market[END_REF] analyze a problem with heterogeneous hospitals in duopoly competition and argue that a condition so that a more accurate information effectively promotes quality enhancement is that hospitals' "endowment" in technology should be similar. The authors explain that, otherwise, better information only reveals the gap between hospitals that, therefore, have less expected gains in improving quality whether it is to confirm their leadership or to try to maintain the confusion for the provider with the lowest structural quality.

The results of empirical studies on the relationship between hospital competition and quality have not led to a consensus. However, the most recent studies tend to find a positive relationship [START_REF] Bloom | The Impact of Competition on Management Quality: Evidence from Public Hospitals[END_REF], [START_REF] Bloom | In defence of our research on competition in England's National Health Service[END_REF][START_REF] Gaynor | Free to Choose? Reform and Demand Response in the English National Health Service[END_REF]) with the exception of [START_REF] Gobillon | Competition and Hospital Quality: Evidence from a French Natural Experiment[END_REF] who finds heterogeneous effects depending on the health-care practitioners' status (distinguishing for-profit, non-for-profit, and public) in France.

Those studies generally use indicators of mortality rates of emergency care (generally heartattack surgery), which have been shown to be a good instrument for hospitals' standards of quality [START_REF] Bloom | In defence of our research on competition in England's National Health Service[END_REF], Cooper et al., 2011)). How to define the relevant hospitals' market, in particular with regard to the data that are available, remains a difficulty for this literature that we will not examine here. Using different methodologies, the recent studies we mentioned tend to confirm that promoting hospital competition may affect the decisions of patients in the sense of quality enhancement.

The development of new incentive devices

Though competition can be a powerful instrument to maintain standards of quality, it may reveal itself insufficient to induce the efficient levels of all the quality dimensions and has many shortcomings that must be dealt with. On this matter, countries such as the U.S and the U.K have supplemented DGR-pricing systems by value-based-purchasing programs such as pay-forperformance (P4P). They consist in rewarding physicians for meeting certain targets considered as good practices to improve quality indicators. This is the case for instance of the rate of hemoglobin testing for diabetic patients or reduced blood pressure and cholesterol measurements in patients with ischemic heart disease [START_REF] Kaarboe | Multi-tasking, quality and pay for performance[END_REF]. However, due to the numerous dimensions of qualities, for which many are not verifiable and thereby not contractible, those systems must also deal with the teaching to the test concerns, i.e. the concerns that targeting some verifiable dimensions of quality may only induce hospitals to focus all their effort on these contracted dimensions at the expense of other dimensions. In a model with partially altruistic health providers Eggleston (2005)'s finding suggests that altruism may alleviate teaching to the test issues if some degree of cost-sharing is implemented. In an extension of Eggleston's work, [START_REF] Kaarboe | Multi-tasking, quality and pay for performance[END_REF] characterize whether incentive designs, such as P4P, should be implemented for given services in function of the degree of substitutability between the verifiable and non-verifiable dimensions of these services. [START_REF] Glaeser | Not-for-profit entrepreneurs[END_REF] show that effects of teaching to the test concern both for-profit and non-for-profit hospitals, but explain that its main driver may differ between each status. They furthermore explain why health care providers may prefer to choose non-for-profit status since it might have positive connotations for patients and affect their perceptions of quality.

We have briefly reviewed the main issues and conflicts for decentralizing efficient quality standards and costs containment. The potential virtues of competition on this matter have fostered its promotion in the actual DRG-based system for funding health care. So far, we have mostly discussed quality dimensions that are unconditional to patients and have a public good flavor, i.e. patients may benefit differently from the quality, but the quality standards are the same for all patients. Vertical differentiation of providers will be considered in the last chapter of this thesis only. One should also observe that the terms of allocative efficiency could also have been used to discuss the objectives of the authority in terms of quality enhancement of care. In this case, allocative efficiency would have referred to the property that the average marginal benefit of quality equals the marginal cost of its production.

In an application to home-care services, the third chapter of this thesis will discuss the decentralization of quality standards to a public-private duopoly, where only the public firm has a ubiquity constraint, i.e. must provide services to the consumers that demand it, whatever the location of the consumers on the territory. We will show that an asymmetry in terms of coverage constraint between public and private competitors can be detrimental to the allocative efficiency in terms of quality standards. In particular, it discourages the quality differentiation between providers that, otherwise, should be efficient given the heterogeneity of consumers' valuations/marginal benefit from quality.

We have not referred to efficient quality in terms of allocative efficiency so far to avoid any confusion with the dimension of allocative efficiency that we will discuss in the sequel and will be central in the first two chapters of this dissertation. In this view, we will still consider that patients may differently benefit from quality, but now also discuss the allocative efficiency of care by taking into account that the care intensity can differ across patients. The effect of hospital competition in terms of allocative efficiency when evaluated at the level of patient types have been theoretically analyzed first by Ellis (1998) (c.f. infra).

Patients heterogeneity and allocative efficiency of care

This section addresses another fundamental challenge of health-care regulation that is to promote productive efficiency and quality without inducing inefficiency from patients' selection neither distortions in treatment praxis. We will argue that if concerns of patient selection are inherent to the implementation of DRG-based prospective payments, those from the distortions of treatments' praxis are the result of their design that is only partially prospective in practice.

Allocative efficiency of care questions whether the benefits from public funds spent into a treatment consistently exceed the opportunity cost of other goods or treatment that might be provided. The central criterion is therefore the cost-effectiveness of treatment provision.

In practice, the criteria of cost-effectiveness play an increasing role in determining whether patients should receive a treatment or not. In this respect, how to measure and value the patient's benefits is a constant source of debate and will not be addressed here. The most established measure is the QALYs (Quality-Adjusted Life-Year) [START_REF] Garber | Cost Effectiveness and Payment Policy[END_REF]. In terms of implementation, the National Institute for Health and Care Excellence (NICE) in the U.K. remains the best example of the application of costs-effectiveness criteria. The agency explicitly gives a threshold of between £20000 and £30000 per QALY gain so that a new technology treatment gets funded by the National Health System [START_REF] Garber | Evidence-Based Guidelines As a Foundation For Performance Incentives[END_REF]Sculpher, 2011, NICE, 2013).

One should, however, notice that the evaluations are conducted on average, and acceptance does not account for heterogeneous gains and costs across patient types as the following discussion will suggest.

Over-provision, under-provision, and excessive rationing will be the main concerns here.

The problem of health care regulation they address mostly concerns the asymmetry of information on patients' illness and physicians' discretion over provision, and the problem of decentralizing an appropriate use of resources in this context. These concerns will be shared by all chapters of this dissertation. We will first present the different issues related to patient selection in the terms of [START_REF] Ellis | Creaming, skimping and dumping: provider competition on the intensive and extensive margins1This is a substantially rewritten version of a paper entitled 'Creaming, Skimping, and Dumping: Provider Competition for Patients'.1[END_REF], and then present the issues of procedure selection that relies on the classifications designs.

Allocative efficiency at the intensive and extensive margins

The heterogeneity of patients within DGRs is inherent to the principle of PPS that aim at dissociating the repayment to the actual cost of care. We will follow the terminology of Ellis (1998) who proposes to decompose the payments system's impact on hospital's use of resources among patients into its intensive margin and extensive margin. The authors describe that due to the heterogeneity of patients within a DRG, hospitals may be led to undertake 'creaming', the over-provision of services to low-cost patients, 'skimping', the under-provision of services to high cost patients (the intensive margin) and 'dumping', the explicit avoidance of high cost patients (extensive margin).

Payments effects at the intensive margin

On the intensive margin effect, [START_REF] Ellis | Provider behavior under prospective reimbursement: Cost sharing and supply[END_REF], [START_REF] Ellis | Creaming, skimping and dumping: provider competition on the intensive and extensive margins1This is a substantially rewritten version of a paper entitled 'Creaming, Skimping, and Dumping: Provider Competition for Patients'.1[END_REF] and [START_REF] Ma | Health Care Payment Systems: Cost and Quality Incentives[END_REF] argue that cost-based retrospective payments may lead to cream all patients. This is due to both the demand-side moral hazard in presence of fully insured patients and to supply-side moral hazard to the extent that physicians also value benefits from treatment that otherwise is incumbent to patients only (altruism) and has discretion over what should be provided. [START_REF] Ellis | Creaming, skimping and dumping: provider competition on the intensive and extensive margins1This is a substantially rewritten version of a paper entitled 'Creaming, Skimping, and Dumping: Provider Competition for Patients'.1[END_REF] moreover suggests that prospective payment systems may lead to skimping on the costlier patients (high severity patients) or dumping. Reducing the treatment intensity of high patient types below its optimal level allows the provider to refine a positive margin but still treat the patient, while dumping occurs if skimping is insufficient to fulfill the physicians' sustainability constraints.

Prospective payment systems here are a fixed price for all patients' severity types, and can be interpreted as patients belonging to a given DGR.

To avoid any confusion, this dissertation will mainly focus on adverse-selection issues and put hidden actions aside, i.e. we will consider treatments' intensities that are observable or at least verifiable and contractible. In this view, [START_REF] Choné | Optimal Health Care Contract under Physician Agency[END_REF] consider a problem of health care contracting when over-utilization, creaming, may arise on observable quantities of care. They analyze the problem of contracting given that both physician intrinsic motivations (or altruism) and the type of illness of the patient are heterogeneous and unobserved. The authors consider that there is a positive relationship between the physicians' altruism and the average severity of illness of their patients and a possible antagonism between the physicians' incentive (that are also motivated by their altruism), and liability constraint (that is not). As a result, they find that the optimal contracting issue can lead to some degree of pooling between physicians with different altruism, who should all provide the same quantity of care despite the fact they have different patients case-mix. More precisely, they suggest that maximal quantity of care at the intensive margin can arise at the second best optimum.

Payments effects at the extensive margin

The effects of extensive margin rely on the risk that prospective payments bear on the health providers: excessively costly patients may be associated with significant losses. Therefore, it can induce hospitals to undertake risk selection/dumping i.e. avoiding types of patients for whom the treatment cost exceeds the DRG-price (see for instance [START_REF] Ma | Health Care Payment Systems: Cost and Quality Incentives[END_REF], [START_REF] Ellis | Creaming, skimping and dumping: provider competition on the intensive and extensive margins1This is a substantially rewritten version of a paper entitled 'Creaming, Skimping, and Dumping: Provider Competition for Patients'.1[END_REF], [START_REF] Eggleston | Risk Selection and Optimal Health Insurance-Provider Payment Systems[END_REF], [START_REF] Fraja | Contracts for health care and asymmetric information[END_REF], [START_REF] Makris | Optimal Incentive Schemes for Altruistic Providers[END_REF]). In addition, to the extent that DRG-prices do not necessarily reflect the average cost of treatments, they may encourage hospitals to specialize into the provision of treatments for which expected costs are lower than DGR prices [START_REF] Dranove | Rate-Setting by Diagnosis Related Groups and Hospital Specialization[END_REF]. In the view of Ellis (1998)'s terminology, one can also interpret specialization as an extensive margin effect.

In practice, dumping mechanisms are not necessarily restricted to strict admission refusals of some patient types. This can actually be contrary to the institutional framework. It can, however, consist in more indirect mechanisms, from the influence of the General Practitioner (GP) referral system, use of waiting lists, choice of the size of the services in terms of beds, or simply closure of loss-making services. Due to its various forms and the difficulty to obtain data on patient selection, there are few empirical assessments of the phenomenon. [START_REF] Newhouse | Do unprofitable patients face access problems?[END_REF] finds that less profitable patients are more likely to be found in last resort (typically public) hospitals, which suggests a dumping practice of the other hospitals.

The regulation issue lies on the fact that hospitals observe patient types and can anticipate their cost of treatment while the purchaser authority cannot. [START_REF] Chalkley | Cost sharing in health service provision: an empirical assessment of cost savings[END_REF] shows that introducing some degree of cost-sharing could again be a privileged instrument to deal with this issue. The principle is the following: under fully prospective payment, DRGprices must be high enough to avoid dumping on the most costly patients, but implies to leave hospitals a rent when patients' treatment costs are low. Some degree of cost-sharing would reduce hospitals' cost-containment motivations, but limit incentives for dumping when cost is high and allow the authority to pay less when cost is low, thereby also decreasing the rent left.

However, such form of cost-sharing is poorly used in health care in practice, mostly because they involve large monitoring costs. In this view, the problem of risk selection has also been addressed by the implementation of outliers payment policies which implement supply-side cost-sharing if the hospital's loss exceeds a given threshold. Outliers payment policy design raises other issues of implementation, that won't be detailed in our presentation (see [START_REF] Sappington | Using Subjective Risk Adjusting to Prevent Patient Dumping in the Health Care Industry[END_REF], or [START_REF] Mougeot | Supply-side risk adjustment and outlier payment policy[END_REF] for theoretical analyses).

Over-provision of treatment at the extensive margin and rationing in health care

The concerns of patient selection also relates to the question of over-provision at the extensive margin. This kind of over-provision addresses the possibility that some degree of patient selection at the extensive margin may actually be efficient. To the extent that the costs of treatment may not necessarily exceed its value in terms of patient benefits, or the opportunity cost of providing other goods or treatments to other patients, their coverage could be limited at the optimum and lead to the rationing of treatment provision. Patients would therefore need to pay if they want to access the treatment. Theoretically, rationing can generally be interpreted as the consequence of the costs of some advanced technologies that can be too high relatively to the taxpayer's willingness to pay [START_REF] Fraja | Contracts for health care and asymmetric information[END_REF].

Concerning over-provision of treatment at the extensive margin, it is mostly block contracts and cost-volume contracts that are considered and discussed (Chalkley and Malcomson, 1998b).

Block contracts consist in fixed prospective payment (lump sum) for the care of all patients within a pathology, independently from their number. Cost-volume contracts generally specify a lump sum and a cap to the number of patients to treat. Given the heterogeneity of patients, a critical theoretical concern for the implementation of block contracts lies into the asymmetry of information between the providers and the regulation authorities about the patient's characteristics. Given this asymmetry, the first issue to implement cost-volume contracts is to determine what are the patient types that each actor aims at seeing treated in priority. This is a critical aspect that will determine whether hospitals and health authorities are aligned in their sorting of patients and whether volume control can be effective. For the authority, the question is whether the net social benefit from treatment is higher for the more severe or less severe patients. The literature addressing rationing generally considers that, within a pathology, the net social benefit from treatment is decreasing in patient's severity of illness i.e. the cost increases at a faster rate than benefits (Chalkley and Malcomson, 1998b[START_REF] Fraja | Contracts for health care and asymmetric information[END_REF][START_REF] Makris | Optimal Incentive Schemes for Altruistic Providers[END_REF][START_REF] Choné | Optimal rationing within a heterogeneous population[END_REF]. As a result, the for-profit motivations are aligned with the social planner's objectives: the hospital would decide to treat first the patients for whom the treatment is the most valuable socially and contracts specifying a number patients to treat can be effective. When altruism is considered it is also necessary that patients with high benefits are also having lower costs, so that the altruistic and profit motives are aligned. When these necessary conditions are met, the use of block contracts may no more be effective. See [START_REF] Mougeot | Asymmetric Information and Pooling Contracts in the Hospital Sector[END_REF] for an analysis of the problem when these criteria are not met and a study of the non-responsiveness issues that may arise.

Due to specialization or hospital size and economy of scale for instance, some hospitals may be more productive in the provision of certain health services than others. The way payment contracts should account for hospital differences in productivity therefore becomes an additional question. Indeed, the optimal degree of rationing may vary across hospitals, and the most severe cases should be treated in the most productive hospitals only.A key aspect of this matter is that if more productive hospitals must treat a larger number of patients, including costlier patients, they might actually end up with larger total costs which might complicate the use of block contracts with lump-sum payments only. De [START_REF] Fraja | Contracts for health care and asymmetric information[END_REF] analyzes the problem of optimal rationing when hospitals' productive efficiency varies and is unobserved while [START_REF] Makris | Optimal Incentive Schemes for Altruistic Providers[END_REF] and [START_REF] Mougeot | Asymmetric Information and Pooling Contracts in the Hospital Sector[END_REF] extend the analysis to unobserved physicians' degree of altruism.

In practice, block contracts may also be difficult to implement, as beyond the requirement of responsiveness, it requires that the number of patients to be treated is known in advance. Chalkley and [START_REF] Chalkley | Contracts and competition in the NHS[END_REF] therefore argue that their design is better suited to elective/nonemergency care where the capacity for treatments is, in general, limited and fully used. Moreover, optimal block contracts depend on the patients' distribution in each hospital. The distribution of patients' characteristics/severity of illness may vary across hospitals and can be difficult to observe by the authorities. On the difficulty to assess the proper number of patients to treat in each hospital, [START_REF] Choné | Optimal rationing within a heterogeneous population[END_REF] consider a problem where the authority does not know the relevant distribution of patient types. The authors characterize conditions on the dispersion across possible condition so that if the efficient number of patients to treat is large enough, the optimal policy can consist in increasing the number of patients treated.

Another caveat of the use of cost-volume contracts in theory is that their counterpart in praxis generally consists in specifying fixed prices and quantities associated to lump sum transfers (see [START_REF] Fraja | Contracts for health care and asymmetric information[END_REF] for a discussion of cost-volume contracts' implementation). In particular, it generally requires that the authority can set negative lump sum which is not observed in practice. [START_REF] Miraldo | Price adjustment in the hospital sector[END_REF] have addressed the problem of rationing where authorities are limited to use fixed prices as their instrument. The authors argue that the analysis is more constrained but more realistic. Following also [START_REF] Malcomson | Supplier Discretion Over Provision: Theory and an Application to Medical Care[END_REF], our approach of rationing in Chapter 1 will be restricted to a fixed price instrument. As the former, we will not discuss the hospitals' heterogeneity but the heterogeneity of treatment procedures and procedure selection issues that are presented below.

Refinement of DRG classification and health-provider's procedure selection

The broad adoption of PPS has come together with the development of different classifications for coding the patients' diagnostic groups and determine the hospitals' repayments. However, a common observed trend consists in the progressive refinement of the different classifications [START_REF] Reinhard | Diagnosis-Related Groups In Europe: Moving Towards Transparency, Efficiency And Quality In Hospitals[END_REF], and the introduction of procedures performed into DGRs. The refinements of the classifications are intended to diminish the financial risks for providers under prospective payment systems by diminishing the within DGRs heterogeneity. However, the indicators that have been used for refinements have strengthened the issues of asymmetry of information on patients and the discretion over the provision problem of hospital regulation. To the extent that health providers have the discretion on what should be provided, the choice of DRG-repayments could involve strong distortions in what is provided. On this matter, [START_REF] Chandra | Who Ordered That? The Economics of Treatment Choices in Medical Care[END_REF] note that "in the gray area of medicine where it is not clear what treatment is in the patient's best interest, financial incentives might be decisive".

Refinement of DRG classification and health-provider's incentive for "procedure selection"

In practice, the classification of patient care into DGRs does not only depend on their primary diagnosis, but often includes the performance of specific treatment procedures or diagnostic tests. This adaptation of the PPS introduced a significant amount of retrospective costsharing in the DRG-based systems i.e. variations in the DRG-price that are not founded on predictable variations of expected costs based on, exogenous, verifiable characteristics of patients (typically the patient's pathology), but on the treatment decisions that are known to lead to variation in costs. This limit of DRG-pricing in practice has been first documented by [START_REF] Mcclellan | Hospital Reimbursement Incentives: An Empirical Analysis[END_REF] who explains the rise of retrospective cost-sharing by the lack of precision of observable diagnosis information, and that whether or not certain treatments are appropriate is unclear for many patients. The author estimates that more than 40% of DGRs were procedure-dependent in 1993. According to [START_REF] Gilman | Hospital response to DRG refinements: the impact of multiple reimbursement incentives on inpatient length of stay[END_REF], this proportion has increased with the next refinements attaining nearly half of the roughly 500 DRGs in the US Medicare system in 2000. We have also already emphasized the evolution from 600 groups in 2004 to nearly 2300 groups in 2009 in France, but the degree of retrospective cost-sharing has not been precisely documented.

Retrospective cost-sharing has raised concerns on distortions in treatment praxis. They rely on physicians and discretion over provision [START_REF] Malcomson | Supplier Discretion Over Provision: Theory and an Application to Medical Care[END_REF] and can be particularly salient in the presence of demand-side moral hazards associated with full insurance. If the alternative procedure for a given disease leads to a different repayment, hospitals may have stronger incentives to use one over the other.

Back to the design of the classification, the authority faces a trade-off between setting the tariffs based on primary diagnostics only and refining the classification in order to decrease costs heterogeneity between DRGs groups and decrease the incentives of patient or procedures selection. [START_REF] Hafsteinsdottir | DRG prospective payment systems: refine or not refine?[END_REF] analyze this trade-off and show that a selection effect may take place both with and without the refinement but the procedure that is subject to over-provision changes. Assuming the absence of patient selection issues and the possibility of pricing DRG at the average cost, the authors show that refinement induces an over-provision of the most intense treatments (socially optimal for high patient types), while its under-provision should be expected if refinement does not occur. On refined systems only, [START_REF] Malcomson | Supplier Discretion Over Provision: Theory and an Application to Medical Care[END_REF] shows that DGR prices, departing from the average cost rule, allow to reduce the extent of overprovision at the optimum, in particular when patient selection at the extensive margin should be expected.

If, moreover, patients have a preference for the costlier "advanced" treatment, hospital competition may strengthen the problem of regulating procedure selection. This is what is shown and discussed in the first chapter of this thesis where we argue that, in this case, competition for patients can decrease the effectiveness of repayment differential to decentralize an efficient allocation of treatments. Our results suggest that in a refined scenario, advanced treatments should indeed be better repaid than basic treatments to allow their provision to the most severe patients for whom they are costlier but cost-effective, the regulation authority can either increase the repayment of basic treatment and make them more attractive for the provider or accept their under-provision.

Empirical evidence

There are many empirical evidences supporting that treatment practices react to economic incentives such as changes in DRGs prices (the fixed amounts repaid to hospitals for each treatment). Most of the studies, however, relate to the U-S Medicare system. [START_REF] Gruber | Physician Financial Incentives and Cesarean Section Delivery[END_REF] test the predictions of "physicians induced demand" models and find evidence that physicians praxis are sensitive to income effects. The identification strategy uses the decrease in fertility rates and the repayment differential between Caesarean birth deliveries that were relatively better reimbursed, and natural deliveries, at the time of the study. They test how the consecutive decrease in the number of obstetric acts, and thereby the revenue, has led the physicians to perform more caesarean birth deliveries. They confirm that income shocks may lead physicians to substitute toward better reimbursed alternatives. Moreover, when testing whether the rapid growth of caesarean was only due to improved skills on their performance that diminished the risk of complications, the authors' findings suggest that income pressure only explained from 16% to 32% of the growth during the period. Gruber, [START_REF] Gruber | Physician fees and procedure intensity: the case of cesarean delivery[END_REF] conducted another analysis on the substitution from natural v.s. cesarean childbirth delivery based on variations in the repayment difference of the two procedures. They find that relatively higher repayment of cesarean could explain more than half of the utilization rate differential, though the extent of the relationship has been tempered in a latter replication by [START_REF] Grant | Physician financial incentives and cesarean delivery: new conclusions from the healthcare cost and utilization project[END_REF].

On heart attacks management, [START_REF] Coey | Physicians' financial incentives and treatment choices in heart attack management[END_REF] estimates that the refinement of the DRG involves a change in praxis for 20% of the patients of their sample and an increase in the treatment cost of 27% compared to an unrefined scenario (termed as a bundle payment in the article).

The author's result confirms the relationship between repayment levels and hospitals' treatment practices which confirms the prediction of [START_REF] Hafsteinsdottir | DRG prospective payment systems: refine or not refine?[END_REF], and argue in favor of bundle payments (which, with respect to their approach, can be associated to no refinements). However, their study does not allow to estimate whether the absence of refinement would generate instead under-provision of the most intense treatments. [START_REF] Jacobson | Physician agency and patient survival[END_REF] also observe results in line with the theoretical predictions after a repayment cuts in some available alternative drugs of chemotherapy, and find it was associated with an increase in patients' health outcomes.

Procedure selection in theory

In this context, one observes a growing theoretical literature analyzing how payments should be designed to account for procedure selection [START_REF] Malcomson | Supplier Discretion Over Provision: Theory and an Application to Medical Care[END_REF], Siciliani (2006), Hafsteinsdottir and[START_REF] Hafsteinsdottir | DRG prospective payment systems: refine or not refine?[END_REF], [START_REF] Pflum | Physician Incentives and Treatment Choice[END_REF]). The health authority's problem is to decentralize a treatment plan, i.e. what type of treatment should be provided to what type of patient in order to maximize welfare. Similarly to rationing issues, the efficiency will be determined by cost-benefit analyses that may vary with patient types. The regulation problem lies into the physician's discretion over provision, i.e. the asymmetry of information between health care practitioners who observe the patients' characteristics and the authority. The regulator only observes the treatment procedures performed through the DRG the patients have been coded into. The concerns of procedure selection will also be the major topic of Chapters 1 and 2 of this dissertation, which will address respectively treatment and test selection regulation in a competitive framework.

The work of [START_REF] Malcomson | Supplier Discretion Over Provision: Theory and an Application to Medical Care[END_REF] provides a theoretical typology of the consequences of hospitals' response to DRGs prices for the decentralization of treatment plan in the case of a monopoly hospital. The author describes the second-best trade-off between an efficient provision of three alternative treatments decisions (including a basic treatment, an advanced treatment and rationing of patients, i.e. no treatment), and the rent extraction when a monopoly hospital is regulated through fixed prices.

Taking into account the possibility of a hospital dumping behavior plays a crucial role in the results. In particular, it imposes a constraint on DGR-price levels that must exceed the average cost of providing each treatment and explains the hospital rent. More precisely, the author takes a reservation surplus normalized to zero, but more generally it is sufficient that the variance of the costs within the DGR exceeds the hospital's willingness to incur losses (reservation surplus) to foster the hospital to engage in a dumping process on high cost patients and prevent averagecost pricing.

Our baseline framework in Chapter 1 will follow Malcomson's approach of the problem to analyze the effect of competition on fixed price regulation. As in [START_REF] Ellis | Creaming, skimping and dumping: provider competition on the intensive and extensive margins1This is a substantially rewritten version of a paper entitled 'Creaming, Skimping, and Dumping: Provider Competition for Patients'.1[END_REF], the model exhibits hospital incentives to cream low-severity patients. In our model this translates into an expansion of the provision of advanced treatment to low severity patients (at given DRG prices) when hospitals compete. Skimping, however, does not increase with competition in terms or treatment procedure effects. Our problem will also be related to [START_REF] Pflum | Physician Incentives and Treatment Choice[END_REF], which, to our knowledge, provides the first analysis of procedure selection problems in theory. However we will depart from his analysis by excluding that prices are neutral on patient selection. In the problem he studies, the absence of threat of dumping allows for cross-subsidization across treatments, i.e. the hospital on average makes losses on the provision of advanced treatment, but are compensated by profits on the basic treatment provision. As in [START_REF] Malcomson | Supplier Discretion Over Provision: Theory and an Application to Medical Care[END_REF], and suggested by the empirical evidence, it is the repayment differential that determines the relative provision of each treatment. [START_REF] Pflum | Physician Incentives and Treatment Choice[END_REF] shows that, in these circumstances, the efficiency can be decentralized under competition. We will, however, take into account limits to the cross-subsidization possibility, i.e. taking into account patient selection and focusing on the second-best welfare analysis. Our problem shows that although the efficient allocation can be decentralized, it is no more the optimal allocation in second best, and in fact competition can be detrimental to welfare if patients prefer the advanced treatment.

In our sense, this issue is particularly relevant regarding the increased amount of innovative technologies that are available to treat patients, for which provision can significantly raise the hospitals' attractiveness, but also health care expenditures if they are not efficiently allocated.

For similar reasons, the increase in utilization rates of new testing technologies such as imaging methods worries public authorities for being significant drivers of health expenditures. The second chapter of this thesis intends to provide a framework to address the issue of decentralizing test decisions and their consequences for treatments' provision in the diagnostic-therapeutic cascade. This is closely related to procedure selection issues but requires to take into account the incomplete nature of information in health care.

Universal obligation of services and providers status in health care

Universal coverage and hospital care

We have well emphasized that a particularity of health care services lies into the translation of the heterogeneity of the consumers' characteristics into heterogeneity in the cost of their coverage. This may raise concerns of access to the health services since providers may have incentives to dump some consumer types. We have detailed the selection at the extensive margin in sections 3.2 and 3.3 of this introduction. Besides, the first two chapters of this dissertation will account for the constraints that the threat of dumping may bear on the way health care is funded. A lesson of our first two works is that the possibility for dumping critically affects the optimal pricing policies and the effect of competition. In particular, the first chapter of this thesis explored the effect of competition when the threat of hospital' dumping prevents the authority to strictly follow the average cost DRG-price rule without inducing the hospital to engage in risk selection. In our second chapter, we show how the regulation problem of decentralizing care procedures (diagnostic and treatment) is relaxed if the hospitals are committed to cure the patient they accept and cannot refer them to other hospitals if they learn that they have a very costly affection. In practice the constraints of services exist and generally vary across public and private providers' status.

The implementation of universal services obligations (USO) to part of the providers can be found as a solution when the public policies aim at guaranteeing the access to the service. This is the case in France where private firms coexist with public firms which must accept to provide all demands of care they are addressed. The private sector has more latitude in choosing the services it provides and to accept or refuse to answer demands. The consequences of this difference in obligation of services are the topic of the third chapter of this thesis.

In the hospital sector, the differences in universal obligation of services have been the topic of debates on the pricing heterogeneity between the public and the private sectors. In France, the private sector, which accounts for about 44% of the hospital stays [START_REF] Dormont | Comment évaluer la productivité et l'efficacité des hôpitaux publics et privés ? Les enjeux de la convergence tarifaire[END_REF], is funded based on the same DRG-classification as the public hospitals but at different prices levels. A part of this difference is explained by the different waging system of physicians in the private sector that is not included in the tariffs. In the spirit of yardstick competition, the introduction of T2A in France has first planned a progressive convergence of tariffs between the two sectors. The convergence should have therefore reflected the convergence in productivity of the competing hospitals. It has, however, been abandoned, due to the persistence of differences in production costs, and the reticence of the actors. A main argument held on the idea that, due to its obligation of services, the public sector was led to treat more severe affections while in parallel the private sector would have specialized in more profitable types of services.

Following our discussion of payments design when the patient heterogeneity can vary across hospitals in section 3.3, different payment levels can have theoretical justifications if the scope of patient types that are treated does vary across hospitals. If not explained by the treatment of costlier patients, however, cost differences would only reflect differences in productivity and the convergence of tariffs would foster the convergence of productivity between the two sectors. In this respect, the study of [START_REF] Dormont | Comment évaluer la productivité et l'efficacité des hôpitaux publics et privés ? Les enjeux de la convergence tarifaire[END_REF] shows that, when accounting for differences in patients characteristics between the public and private sectors, differences in hospital productivities were no more significant or even better in the public sector. The authors' findings therefore suggest that the heterogeneity in average costs of stays between both sectors mostly reflects the variation in the characteristics of the patients they treat. Concerning patient selection at the extensive margin, this could also be interpreted as the reflection that risk-selection does occur in absence of obligation of services.

Theoretically, the implications of universal obligation of services (USOs) and their justifications have mostly been addressed in the literature on network utilities (telecommunications, gas, electricity, postal services,...). In this literature, the heterogeneity of consumer costs of provision relies on a geographical component: the connection costs or costs to access consumers to provide them with services vary in function of their location. The literature has emphasized the different roles played by non-discrimination and coverage constraints that are the two traditional requirements of USOs. The foundation for ubiquity, i.e. the obligation of universal coverage to one provider, can be various, and won't be discussed in this dissertation. See for instance [START_REF] Poudou | Equity justifications for universal service obligations[END_REF] for a discussion of the optimal coverage constraint for an inequity-averse authority.

The third chapter of this thesis will instead interrogate whether the asymmetry of coverage obligation between providers can have theoretical foundation from a social welfare point of view. We will argue that asymmetric coverage can be socially optimal to limit the multiplication of "access costs" incurred by the providers. Concerning the hospital sector, this can be interpreted as the possibility that the magnitude of investment and specialization costs, required to be able to care any degree of severity, implies that they should not be incurred by all providers. The third chapter will, however, not address the case of hospitals' asymmetry in terms of obligation of services. We will instead discuss an application to home-care services to elderly where the foundation of coverage constraints also relies on the geographical component of the sector.

The particular case of home care services

In home care, the concerns of universal services obligation are raised by its geographical components. The firms must incur transportation costs for every care they deliver and may not be willing to incur the levels of access costs of each area in the territory. More precisely, the firm's employees spend unproductive hours reaching the consumers during their daily rounds, and the firm must pay wages for these hours (e.g. in France, they are repaid for all transport time exceeding 15 minutes). The transportation costs are composed of the reimbursements of the employees' transportation costs (gas or public transport fees) and the employees' remuneration during their traveling time (unproductive hours). The total transportation costs of a firm depends on both how much remote the consumers are from the firms' employees' locations and the density of the area the consumers live into. The public report of Bruant-Bisson, de Reboul and Aube-Martin (2010) suggests that the unproductive remuneration accounts for 20% to 25% of the total employee's remuneration which is the main component of the firms' costs. In addition, the reimbursement of the employee's travel expenditures (car usage) represents 1% to 7% of the firms' total expenditures.

The home care market in France is organized according to two regimes, "l'autorisation" (authorized), and "l'agréement" (approval or non-authorized) that vary the flexibility of firms in terms of acceptance policy of consumers, price and quality setting. The authorized firms are regulated by the local authorities and submitted to a universal service obligation (USOs). In contrast, the "agréées" firms can freely set their hourly tariffs, acceptance policy and the quality of their services. While the per-hour tariff differs across firms, both regimes of formal home care services are subject to uniform pricing constraint. In parallel, the APA (Allocation Personnalisée d'Autonomie, literally: personalized helps to autonomy) program aims at guaranteeing the financial access to home care so that the elderly supports from 0 to 90% of the firms' tariffs according to their income (see for instance [START_REF] Roquebert | Pay less, consume more? The price elasticity of home care for the disabled elderly in France[END_REF] or Bourreau-Dubois, [START_REF] Bourreau-Dubois | Les disparités départementales dans la mise en oeuvre de l'APA à domicile, reflets des choix de politiques locales ?[END_REF] for a more detailed presentation of the APA program).

Outline of the dissertation

The consequences of health-care competition that we discuss in this dissertation revolve around three characteristics common to the health-care markets and to each chapter of this thesis:

-The heterogeneity of health-care demand which is composed of consumers with variable characteristics and translate into a heterogeneity of costs of cares.

-The discretion over provision of health-care practitioners: they are the only observers of patients' precise characteristics and are able to decide the kind of care that should be dispensed.

-The presence or absence of universal obligation of services that varies the providers' ability to discriminate among patients or to refer them to other care providers.

The first two chapters of the dissertation will address procedure selection issues that we have discussed earlier, in terms of treatment and in terms of testing decision respectively. The last chapter of this thesis will be dedicated to explore the consequences of asymmetric obligation of services between public and private competitors.

The first chapter of this thesis provides an analysis of the procedure selection problem when competing hospitals are regulated through fixed prices, and in this sense follows [START_REF] Siciliani | Selection of treatment under prospective payment systems in the hospital sector[END_REF], [START_REF] Miraldo | Price adjustment in the hospital sector[END_REF], and more particularly [START_REF] Malcomson | Supplier Discretion Over Provision: Theory and an Application to Medical Care[END_REF] and [START_REF] Pflum | Physician Incentives and Treatment Choice[END_REF]. Our approach of patient selection follows [START_REF] Ma | Health Care Payment Systems: Cost and Quality Incentives[END_REF], [START_REF] Ellis | Creaming, skimping and dumping: provider competition on the intensive and extensive margins1This is a substantially rewritten version of a paper entitled 'Creaming, Skimping, and Dumping: Provider Competition for Patients'.1[END_REF][START_REF] Malcomson | Supplier Discretion Over Provision: Theory and an Application to Medical Care[END_REF], in the sense that we consider the possibility that too low DRG-prices can induce hospitals to dump the costlier patients.

We consider the problem of a regulation authority which cannot infer the population faced by a given hospital but only the treatments provided. In our model, the total demand of patients of a hospital reacts to their treatments' practices. Hospital competition aligns practitioners' praxis with patients' preferences of treatment. By assuming that patients prefer advanced treatments we show that, compared to a monopoly market structure, competition increases the social cost to decentralize any allocation of treatments and decreases social welfare. Furthermore, we find that hospital competition leads to adopt a repayment scheme that increases the dumping incentives of hospitals. We extend the model by allowing out-of-pocket participation of patients as a regulation tool and we find that it allows to retrieve positive effects of competition if patients' preferences over treatments are slack. More precisely, a competitive provision of treatments can improve the social outcome if treatments do not differ too much in terms of benefits. Otherwise, the provision of the advanced treatment by a monopoly hospital remains the optimal market structure in terms of the social surplus outcome.

The second chapter focuses on the problem of testing decisions. From a policy maker's point of view, the performance of additional diagnostic tests before deciding of the treatment method which should be applied may not be justified for all patients, despite the fact that valuable information is produced. The decision obviously depends on the patient's characteristics, particular symptoms and prevalence of specific co-morbidity, but also on the cost of the tests. This belongs to the procedure selection issues in the sense that testing can affect the hospital's payment, and should be determined by the patient types that are also observed by the hospital only. In order to capture the idea that value of the test depends on ex-ante beliefs about the patient's severity of illness, we will rely on a novel definition of informativeness built by Bradley et al. (2014), Brandt, Eckwert and[START_REF] Brandt | Weak Informativeness: An Information Order with Variable Prior[END_REF], from an information concept introduced by [START_REF] Ganuza | Signal Orderings Based on Dispersion and the Supply of Private Information in Auctions[END_REF].

We will consider the problem of a regulation authority that can observe the tests that have been performed, the test results, and the treatment methods that are provided to all patients (or at least we assume that they are verifiable and contractible). The authority does not observe, however, the patients ex-ante types and thereby, neither their expected costs of care. This article addresses the use of economic incentives to decentralize care procedures (from the choice of diagnostic testing to the choice of treatment methods), to both monopoly and competitive hospitals. Our framework interacts a regulation authority, a hospital sector, and two types of patients for whom the information about illness is incomplete. The decision of acquiring information through diagnostic testing will depend on the patient types and is at the discretion of physicians. The prior belief on a patient's illness depends on the patient's characteristics which are observed by the physician only. We consider two ex-ante patient types: a high-type such that its severity of illness is more uncertain, or expected to be higher, than for the other patient type, the low-type. Despite the production of valuable information, the performance of costly additional diagnostic tests may not be efficient for all patients. For the high patient type, both the value of the diagnostic test and the expected costs are higher. In second best, the implementation costs arising from the information asymmetry on ex-ante patient types heavily depends on whether none, all or only high patient type should be tested. We find that it can be optimal to test more patients than what is efficient in first best. However, since making the tests procedures separating between patient types is actually the most effective way to reduce informational rent, we find that testing all patients may not be optimal even though it is efficient. Moreover, if the efficient use of diagnostic tests is separating between the two patient ex-ante types, then it is the optimal practice for a monopoly hospital in second-best as well. The structure of the hospital sector may, however, affect social welfare, and the optimal treatment praxis. We show that competition may strengthen the problem of preventing over-testing. Indeed, testing patients and adapting the treatment method to their underlying severity of illness, even imperfectly, can improve their expected health outcomes, in particular if one discovers that the severity is much more serious than expected. In these circumstances, the concerns of over-testing should be higher when hospitals compete to attract patients.

The last chapter of this dissertation, Chapter 3, analyzes the impact of asymmetric universal service obligation between the public and the private sectors of health care provision in an application to health-care services. In particular, we discuss the relevance of this asymmetry at efficiency and study its decentralization to a public and private competitors that are differentiated in terms of quality. Our model follows a representation of vertical differentiation of firms in terms of quality standards proposed by [START_REF] Laine | Quality and competition between public and private firms[END_REF], and is related to the literature on universal service obligation in network utilities. In particular, our model is related to the analysis of [START_REF] Calzada | Universal service obligations in the postal sector: The relationship between quality and coverage[END_REF] on postal services.

We focus our attention on the home-care market where public, regulated, firms with universal service obligation (USO) compete with private firms that are subject to non-price discrimination only. As in network utilities, USOs in the home-care market is motivated by its geographical component. The cost to provide the consumers varies with both the remoteness and the density of the area they live in. Moreover, the coverage of given areas by several firms multiplies the total transportation costs incurred on the market. Yet, the presence of several vertically differentiated suppliers is socially valuable due to the heterogeneity of the home-care consumers' needs and, thereby, socially optimal in areas associated with relatively low transportation costs. In the decentralized problem, the authority sets the price and quality of the public firm with USO while its private competitor freely chooses its coverage, price and quality standards. Price discrimination is forbidden for both providers so that the obligation of service GENERAL INTRODUCTION is asymmetric in terms of ubiquity obligation only. We find that the private firm's equilibrium coverage policy is unambiguously excessive i.e. consists in providing more remote areas than what the efficiency criterion prescribes. In absence of coverage constraints to the private firm, the public firm is led to strengthen the competition in prices to limit the excessive coverage of the private firm. At the equilibrium, it translates into limiting the firm's differentiation at the expense of allocative efficiency firms' quality standards. Until the 2000s, most of the national health systems were determining the hospitals' revenues on the costs they declared, a retrospective system that has been blamed for the important deficits in health care. The general inflation of health care expenditures has raised the attention of policy makers to the introduction of productivity incentives in hospitals, and led to a broad adoption of Prospective Payment Systems in developed countries (Australia, England, France, Germany, the Netherlands, Portugal, Belgium, etc.). The Prospective Payment Systems (PPS) rests on a classification of hospitals' activities into diagnostic groups (for instance Diagnostics Related Groups, DRG's in the US Medicare system). The authority of regulation defines predetermined, fixed, repayments for hospital stay in each diagnostic group. For every patient they treat, hospitals report the diagnostic group they belong to and receive the corresponding repayment. One main motivation for introducing this mechanism is to develop incentives to contain the costs of health care by bearing cares' production costs on their providers. It also ties the hospitals' revenues to their inpatients case mix i.e, the number and variety of diseases that a hospital takes charge of, as well as the severity of those diseases. Prospective Payment Systems therefore tends to strengthened strategic behaviors in the definition of treatment plan by hospitals.

The heterogeneity of patients and hospitals discretion over provision are fundamental aspects of hospital care regulation. The regulation authority only observes which treatment is provided and the procedures performed, but not the patients type, and therefore its actual cost production. Hospitals' practitioners are the only ones to observe the patients' characteristics to decide what should be provided. The hospitals' response, in terms of treatment decisions, to repayments' levels have therefore become a central preoccupation since the introduction of PPS. It is also the main topic of this paper.

The relationship between economic incentives and hospitals' practices have been widely addressed by the empirical literature. The most classical case study is the choice between natural childbirth and cesarean section that both involve similar average costs [START_REF] Grant | Physician financial incentives and cesarean delivery: new conclusions from the healthcare cost and utilization project[END_REF], Gruber, [START_REF] Gruber | Physician fees and procedure intensity: the case of cesarean delivery[END_REF], [START_REF] Stafford | Cesarean section use and source of payment: an analysis of California hospital discharge abstracts[END_REF]). This literature shows evidence that hospitals' praxis reacts to economic incentives such as changes in DRGs' prices (the fixed amounts repaid to hospitals for each treatment). When defining the classification, the authority face a trade-off between setting the tariffs based on primary diagnostics only, and refining the classification in order to decrease cost heterogeneity between DRGs groups and decrease the incentives of patients or procedures' selection. In practice, since their introduction, the classifications has been subject to many refinement: splitting DRGs relating to a given primary diagnosis into several DRGs, mainly based on the type of treatment the patient receives 1 . If the alternative procedures for a given disease leads to a different repayments, hospitals may still have stronger incentives to use one over the other. [START_REF] Hafsteinsdottir | DRG prospective payment systems: refine or not refine?[END_REF] show that a selection effect may take place both with and without refinement but the praxis that is subject to overprovision changes. This is due to the subsistence of cost heterogeneity within diagnostic groups after refinement (cost heterogeneity also is one of the leitmotiv of Prospective Payment System in order to makes hospital internalize cost reduction objective). The work of [START_REF] Malcomson | Supplier Discretion Over Provision: Theory and an Application to Medical Care[END_REF] provides a theoretical typology of the consequences of hospitals' response to DRGs' prices for the decentralization of treatment plan in the case of a monopoly hospital. Our baseline framework will follow Malcomson's approach of the problem and its results will serve as a benchmark to draw the effects of an increase in competition.

The development of competition in hospital care is twofold. The introduction of PPS that links the hospitals' revenue to their activity rises the importance for hospitals to be attractive.

In parallel, one observes an increase in the information available to patients for their choice of a health care provider. A large literature documents the effect of hospitals' competition on hospitals' global quality (in the sense that its impact on patients is homogeneous, see Gravelle and Sivey (2010), [START_REF] Allard | Provider Competition in a Dynamic Setting[END_REF], [START_REF] Lyon | Quality Competition, Insurance, and Consumer Choice in Health Care Markets[END_REF], [START_REF] Ma | Health Care Payment Systems: Cost and Quality Incentives[END_REF]). The discussion of competition effect on treatment decisions which typically depend on the patients' characteristics is introduced by the seminal work of [START_REF] Ellis | Creaming, skimping and dumping: provider competition on the intensive and extensive margins1This is a substantially rewritten version of a paper entitled 'Creaming, Skimping, and Dumping: Provider Competition for Patients'.1[END_REF].

They theoretically highlights the patients' cream-skimming and dumping, and discusses 1. There is, in our knowledge, no recent evaluation of the role played by classifying procedures in determining the coding of patient into DRGs. Concerning the Medicare program in the U-S, [START_REF] Mcclellan | Hospital Reimbursement Incentives: An Empirical Analysis[END_REF] has assessed that over 40% of the diagnostic groups where determined by the performance of specific procedure but additional refinement has occurred since this evaluation. their theoretical implications for the provision of one treatment varying in intensity. Yet, the consequences of an increased competition in treatment decisions that may vary, not in intensity, but in kind (i.e: possibly leading to different DRG-prices), have not been thoroughly explored and is the topic of this work. With the increasing extent that patients' choice of hospitals relies on their treatment practices while many new-technology treatments are developed and become available, the propensity of care providers to internalize the patients' preferences may become a critical aspect for the regulation of treatment plans and the control of health expenditures. This article addresses the problem of treatment plans' decentralization to competitive hospitals. We study the use of fixed repayments to decentralize an allocation of services by competitive providers to a heterogeneous population in presence of both discretion over provision and the possibility for consumers' dumping. The patients choose a hospital based on their treatment practices. What matters for our analysis is the presence of a demand reactions to hospitals' treatment practices. It can be interpreted as patients getting direct information or that practitioners whom patients consult for their decision are substituting to the patient's decision in their best interest. We will focus the presentation on the decentralization problem of treatments when patients have a preference for the advanced over the basic treatment which leads to more instructive results.

The definition of the efficient allocation takes into account, for each type of treatment, its respective benefits and costs depending on the patient's severity, as well as the opportunity costs of their provision in terms of public funds' allocation 2 . Of course, the objectives of the hospitals and the planner can diverge whatever the degree of competition, and hospitals may be willing to provide treatments that may not correspond to cost-efficient practices defined by the treatment plan. In parallel some patients may be rationed in terms of treatment provision compared to what they would prefer either because of cost-effectiveness concerns or limited coverage of treatment costs. In line with the literature, we find that an appropriate choice of repayment contracts can, in most cases, align the objective of the hospitals and the health authority, but implementation costs vary between competitive and monopoly hospitals. Competition tend to align hospitals 2. Typically, those costs are important if one considers a health planner that cannot support all treatments' costs and thereby must arbitrate between the coverage of some treatments over others.

incentives with patient preferences in particular with the patients for whom the treatments' costs is covered by the health planner's repayment contract (somehow, a creaming effect). Overall we find that if the efficient allocation of treatment consists in rationing the provision of the advanced treatment compared to the patients' demand, competition is detrimental to welfare and reduce the global health-coverage. It inflates the social costs to decentralize the provision of the more costly patients which end up suffering from the consecutive downward change in coverage.

The most related paper is the recent work of [START_REF] Pflum | Physician Incentives and Treatment Choice[END_REF] which, to our knowledge, is the first theoretical study of hospital competition effects on treatments' decisions that can be associated with different DRG-prices. [START_REF] Pflum | Physician Incentives and Treatment Choice[END_REF] shows that, as soon as hospitals compete, the insurer's control over the physicians practice critically depends on the degree of patient informedness. When hospitals compete, a very low elasticity of demand (relatively to the number of physicians in the market) can prevent the insurer from decentralizing the efficient allocation.

The current paper, however, focus on the impact of competition on social welfare at the second best optimum, compared to a monopoly situation. It shares features with the work of Pflum in terms of hospitals' response to the insurer contract. Our approach is also complementary by addressing treatment's rationing or incomplete coverage of treatments' costs. Theoretically, those issues are generally interpreted as the consequence of the costs of some advanced technologies that can be too high relatively to the taxpayer's willingness to pay (see [START_REF] Fraja | Contracts for health care and asymmetric information[END_REF]).

The paper is organized as follows. Section 1 introduces the baseline model of hospital discretion over provision and the competitive framework. In section 2, we present our central result before detailing hospitals' responses to fixed repayment. Section 3 characterizes the effects of competition on the optimal treatment allocation and presents an extension of the model to discuss possibilities to retrieve positive effects of competition by charging patients with different out-of-pocket amounts. Section 4 concludes.

Framework

The model is composed of a regulator, H hospitals h i , i = 1, ..H, and a population of patients suffering from a same disease. In the sequel the monopoly case (H = 1) studied by [START_REF] Malcomson | Supplier Discretion Over Provision: Theory and an Application to Medical Care[END_REF] will be our benchmark, and we will focus our analyze on the duopoly case (H = 2).

The patients differ are their severity s ∈ [s ¯, s], distributed over F, a log-concave c.d.f. The hospitals observe the severities of patients who ask for a treatment in their hospital and choose among three alternatives: they can provide either a basic treatment, t = 1, an advanced treatment t = 2, or dump the patient t = / 0, i.e. not treating the patient at all.

The patients' utility is their benefit from the treatment and they choose the hospital providing the treatment that maximizes this benefit. The patients' benefit from treatment t ∈ {1, 2} is b t

, and normalized to zero otherwise. One can therefore interpret b t as the additional benefit from being treated with treatment type t ∈ {1, 2} rather than not. For a matter of simplicity we consider that benefits are independent from the patients' characteristics 3 . Also, we assume that the advanced treatment leads to a higher benefit:

A.1 -b 2 > b 1
If a hospital treats a patient of severity s with a treatment, t ∈ {1, 2}, it incurs the cost c t (s) > 0. Following the earlier literature [START_REF] Malcomson | Supplier Discretion Over Provision: Theory and an Application to Medical Care[END_REF], [START_REF] Siciliani | Selection of treatment under prospective payment systems in the hospital sector[END_REF], [START_REF] Hafsteinsdottir | DRG prospective payment systems: refine or not refine?[END_REF], [START_REF] Pflum | Physician Incentives and Treatment Choice[END_REF]) we assume that the costs verify A.2, A.3 and are convex.

Hospitals can dump patients 4 , and the hospitals' reservation surplus to treat a patient is normalized to zero 5 .

3. Our results can be extended to non-constant benefits, such that expected benefits depend on the patients' severities. More precisely, a sufficient and, in our sense, reasonable condition to extend the results to non-constant benefits is that the difference in expected benefits b 1 (s) -b 2 (s) is decreasing in severity, i.e. the higher the severity the higher the relative benefit from the advanced treatment. The remaining question is whether the efficient allocation will consist in rationing patients in the provision of advanced treatment (our case of study under b 1 < b 2 , or the reverse. We discuss the alternative case corresponding to b 1 > b 2 through the presentation of our extension section 4.2 4. Patients' dumping reflects one aspect of selection incentives that heterogeneity of costs within DRG groups may introduce. Our representation of dumping as a basic refusal follow the literature [START_REF] Ellis | Creaming, skimping and dumping: provider competition on the intensive and extensive margins1This is a substantially rewritten version of a paper entitled 'Creaming, Skimping, and Dumping: Provider Competition for Patients'.1[END_REF], [START_REF] Malcomson | Supplier Discretion Over Provision: Theory and an Application to Medical Care[END_REF]) but in practice the hospitals means to avoid too costly patients may goes from influencing the referencing system to pleading for example limited specialization in this area, no available capacity or inflated waiting times (OECD (2012) p.33) 5. The literature has shown that hospitals may undertake negative profits, for instance when accounting for the A.2 -For every treatment, the treatment costs are increasing with the patients severities:

c � t (s) > 0, ∀t ∈ {1, 2}.
A.3 -The difference in treatments' costs c 2 (s) -c 1 (s) decreases with patient severity:

c � 1 (s) > c � 2 (s).
To interpret A.3 one can think for instance of the basic treatment as being a medical procedure that is more time and drugs consuming when the severity increases, and the advanced treatment as being a surgical procedure, which possibly requires high-tech equipment, such that more of the costs are fixed (equipment, operating room, ect).

Finally, we define w t (s) = b t -(1 + λ )c t (s) the net social benefit from providing treatment t to the patient type s, where λ > 0 accounts for the cost of public funds (it costs (1 + λ )

USD to transfer 1 USD to the hospitals). To focus the analyses on the treatment decision we also assume interior allocation with A.4 and A.5, i.e: no treatment dominates the other for all levels of severity. Otherwise, the provision of the dominated treatment should simply be forbidden and hospitals' competition in treatment decision could not occur, implying dropping any analysis. The first, A.4, is a boundary assumption and the second, A.5, insures that the advanced treatment leads to sufficient increase in welfare so that the regulator implements the provision of both treatments at the second best optimal allocation of a monopoly hospital, see Appendix B. for details.

Assumption for interior allocations of treatment:

A.4 -w 1 (s ¯) > w 2 (s ¯) > 0 and w 1 ( s) < w 2 ( s) ≤ 0 A.5 - w 2 ( s) c � 2 ( s) > w 1 ( s) c � 1 ( s)
, where s would be the highest severity of patient which receives a treatment at the second best optimum if the advanced treatment did not exist.

Efficient allocation of treatment

The efficient allocation is the treatment t ∈ {1, 2, / 0} that maximizes the net social surplus from treatment, w t (s), for each patient's type s. The basic and the advanced treatments are labeled with corresponding efficiency, respectively low-s and high-s.

altruism of the practitioners. One can think of a negative reservation surplus without impacting the results.

The assumption on benefits and costs, A.2, implies that the net social surplus from being treated decreases with severity, w � t (s) < 0, t ∈ {1, 2}. Assuming further A.3 guarantees that the net social surplus from being treated with the basic versus the advanced treatment crosses at most once: w � 1 (s) -w � 2 (s) < 0. Therefore, if it is efficient to provide a patient of type s with the basic treatment then it is also efficient for every patient whose severity is less than s. Similarly, if it is efficient to provide a patient of type s with the advanced treatment then it is also efficient for patients whose severity is less than s, as long as the basic treatment does not dominate i.e: s remains above the crossing severity of w � 1 (s) and w � 2 (s). The assumption of interior allocation A.4 finally ensures that each possibility of treatment, t ∈ {1, 2}, is efficient for some types of patients, so that the efficient provision can be characterized by two severity cut-offs s * * 1 , s * * 2 . The basic treatment 1 is efficient for low-s (s ≤ s * * 1 ) the advanced treatment, 2, is efficient for high-s (s * * 1 < s ≤ s * * 2 ), while none of them might be efficient for the upper -very high -severities.

Proposition 1. Under A.1 to A.3 there are two unique severity cut-offs, s * * 1 , s * * 2 , such that:

w 1 (s) > max{0, w 2 (s)} for s ∈ [s ¯, s * * 1 ) (2.1) w 2 (s) > max{0, w 1 (s)} for s ∈ (s * * 1 , s * * 2 ) (2.2) w 1 (s) < w 2 (s) < 0 for s ∈ (s * * 2 , s] (2.3) If A.4 is satisfied, then the efficient allocation is interior: s ¯< s * * 1 < s * * 2 ≤ s.

Second best allocation of treatment

When the patients' severities are publicly observed, a set of contracts such that the repayment of each treatment depends on the patient type receiving it, (r 1 (s), r 2 (s)), can decentralize the efficient allocation, but this is not feasible in our second best problem.

The second best problem is characterized by the hospitals' discretion over provision. The regulator observes the treatment provided but not the types of the patients treated, and therefore, neither the actual costs the hospitals incur. The regulator sets a fixed repayment contract to hospitals, (r 1 , r 2 ), where r t is the repayment attached to treatment t ∈ {1, 2}. The hospitals receive nothing when they do not implement any treatments.

If one allows for negative repayments for not treated patients, (r / 0 < 0), or a non-linear tariff with a negative lump sum, one might still find feasible to decentralize the efficient allocation in this framework. We do not observe such systems in reality, and we aim at studying the effects of hospital competition when managed with the Prospective Payment Systems that are currently used. Besides, the previous literature on hospitals' optimal contracting has dismissed negative repayment or lump sum for the decentralization of treatment decision. Relying on the taxation principle, [START_REF] Malcomson | Supplier Discretion Over Provision: Theory and an Application to Medical Care[END_REF] restricts the procurement problem to the use of one fixed price per treatment (r 1 , r 2 , r / 0 ), and argues that a regulator that cannot monitor potential patients who are not treated is constrained to set r / 0 ≤ 0. In a framework that also allows for both non-treatment and hospitals dumping, but focuses on the heterogeneity in hospital's cost effectiveness, De [START_REF] Fraja | Contracts for health care and asymmetric information[END_REF], concludes on an optimal pricing with the feature of a positive lump sum (set at the least efficient hospital total costs, so in our case, none of the hospital), and a per-treatment-provided tariff.

Monopoly hospital benchmark

Our monopoly problem fits with a particular case analyzed in [START_REF] Malcomson | Supplier Discretion Over Provision: Theory and an Application to Medical Care[END_REF] so that we only detail the aspects that are essential for our discussion. The monopoly hospital chooses

t(s) = argmax t � r t -c t (s)
� for all s, and the monopoly allocation can be characterized by two severity cut-offs s m 1 ≤ s m 2 . Under the assumptions on costs A.2 and A.3, the basic treatment is provided to patients types s ≤ s m 1 , the advanced treatment to patients types s m 1 < s ≤ s m 2 and none are provided to patients types s > s m 2 . Under A.5 we can restrict attention to repayment implying interior allocation of treatment such that s ¯< s m * 1 < s m * 2 < s. The cut-off type s 1 , is the patient type such that hospital is indifferent between the provision of the basic treatment and the advanced treatment (r 1 -c 1 (s 1 ) = r 2 -c 2 (s 1 )), while the cut-off type s 2 is the highest severity the hospital is willing to treat (r 2 -c 2 (s 2 ) = 0). Given those relation between repayments and allocations, it is equivalent, and more convenient, to consider the problem of setting the treatment cut-off (s 1 , s 2 ) that maximize the welfare: Lemma 1. Under A.2 to A.5, the monopoly problem is equivalent to the choice of cut-off types s 1 , s 2 that maximize the reduced social welfare function:

V m (s 1 , s 2 ) = � s 1 s ¯�b 1 -c 1 (s) � d F(s) + � s 2 s 1 � b 2 -c 2 (s) � d F(s) -λ r m 1 (s 1 , s 2 )F(s 1 ) -λ r m 2 (s 2 )[F(s 2 ) -F(s 1 )] (2.4) Where r m 1 (s 1 , s 2 ) = c 1 (s 1 ) + c 2 (s 2 ) -c 2 (s 1 ), and r m 2 (s 2 ) = c 2 (s 2 ).
The Duopoly framework a -Patients' behavior

The patients choose the hospital which will provide the treatment that maximizes their benefits. Denoting t h (s) the treatment provided to patients type s in hospital h: the patients of type s choose the hospital h(s) = argmax h {b t h (s) } for all s, and choose randomly with equal probabilities if they are indifferent, i.e. if the hospitals provide the same treatments 6 . The Demand function from a patient of type s to hospital h according to its treatment practice for type s, t h (s) is:

D h (s,t h (s),t -h (s)) =          1 if b t h (s) > b t -h (s) 1 2 if b t h (s) = b t -h (s) 0 otherwise (2.5)
To illustrate, suppose that only one hospital has generalized the use of innovative and less invasive advanced treatments, such as laser ablation or robotic surgery, to patients with low severities while the other hospital continues providing them with the basic treatment. The demand function states that patients with low severities, or those whom they consult for making health care choices, should choose the hospital that has generalized the use of the advanced treatment to their type.

6. What matters for our analysis is the introduction of patients' reaction to the hospitals' practice in terms of treatment. There is no loss of generality when using such a simple demand, as long as the patients are not too captive in the market. For instance, following [START_REF] Pflum | Physician Incentives and Treatment Choice[END_REF], one can consider that a given proportion of patients are uninformed and always chooses hospitals randomly, implying only weaker incentives to internalize the patients' preferences but it would lead to the same results.

b -Hospitals' behavior

Hospitals maximize their profit and compete through their treatment decisions for each patient type, t h (s). The surplus realized by the hospital h when choosing practice t for the patients of type s is:

π h (s,t h (s),t -h (s)) = D h (s,t h (s),t -h (s)) � r t h (s) -c t h (s) (s) � (2.6) c -Equilibria treatments' strategies
We only consider pure strategies. A treatment practices (t h 1 * (s),t h 2 * (s)), with t h * (s) ∈ {1, 2, / 0}, is an equilibrium if and only if :

∀s, π 1 (s,t h 1 * (s),t h 2 * (s)) ≥ π 1 (s,t h 1 ,t h 2 * (s)) t h 1 ∈ {1, 2, / 0} and π 2 (s,t h 2 * (s),t h 1 * (s)) ≥ π 2 (s,t h 2 ,t h 1 * (s)) t h 2 ∈ {1, 2, / 0} (2.7)
We pick the pareto undominated nash equilibria in case of multiplicity. Such situation may arise because all the patients of a given type s have the same preferences order across treatments, which implies a discontinuity in their demand function.

d -The regulation problem

The regulator maximizes a utilitarian social surplus function defined as the sum, for every type of patient, of the patients' benefits and the hospitals' profits net of the regulator's transfers.

The regulator chooses the repayment contract (r 1 , r 2 ) that maximizes the social surplus subject to the hospitals' participation constraints and strategic interaction on treatment decisions.

Max

r 1 ,r 2 � s s ¯∑ h=1,2 D h (s,t h * (s),t -h * (s)) � b t h * (s) + (r t h * (s) -c t h * (s) (s)) -(1 + λ )r t h * (s) � dF(s) (2.8)
s.to:

� s s ¯�D h (s,t h * (s),t -h * (s)) � r t h * (s) -c t h * (s) �� dF(s) ≥ 0 h = 1, 2 (2.9) π h (s,t h * (s),t -h * (s)) ≥ π h (s,t h ,t -h * (s)) t h ∈ {1, 2, / 0} , h = 1, 2
(2.10)

The impact of hospital competition on welfare

The form of the social welfare in the reduced problem differs according to the competition intensity: for a given interior allocation

(s 1 , s 2 ), V c (s 1 , s 2 ) � = V m (s 1 , s 2 )
, where the superscripts c and m denote the hospital market structure, competitive and monopolistic respectively, and V the welfare function.

Hospitals under competitive pressure try to attract patients by providing them with their preferred treatments, here expanding the provision of the advanced treatment, t = 2. This implies that some low-severity patients who should receive the basic treatment are in fact treated with the advanced one. To recover the threshold s 1 , the regulator must raise the (relative) repayment of the basic treatment. Hence, it is more costly under competition to decentralize the allocation (s 1 , s 2 ) of treatments. This loss is the heart of the argument that implies that competition may be detrimental to welfare.

Proposition 2. Under assumption A.1 to A.4 we

find that V c (s c * 1 , s c * 2 ) < V m (s m * 1 , s m * 2 )
, hospitals' competition leads to a decrease in the social surplus compared to a monopolistic provision of treatments.

The following provides the essential intuitions for the proof of this result. Let's assume for the moment that interior allocation (s 1 , s 2 ) aligned with efficiency (s ¯< s 1 < s 2 ≤ s) can be decentralized when the hospitals compete. For such a given allocation (s 1 , s 2 ), the total benefit of patients and total hospitals' cost are identical across market structures so that the difference in social surplus relies only on the volume of public transfers involved and their social cost, λ .

(r 1 (s 1 , s 2 ), r 2 (s 1 , s 2 )) denote the repayment contract that decentralizes the given allocation.

We decompose the difference in public transfers involved in each market structure into, for each treatment, the repayment differential to decentralize the allocation, weighted by the number of patients that receive the given treatment:

V c (s 1 , s 2 ) -V m (s 1 , s 2 ) = λ [(r m 1 (s 1 , s 2 ) -r c 1 (s 1 , s 2 ))F(s 1 ) + (r m 2 (s 2 ) -r c 2 (s 2 ))(F(s 2 ) -F(s 1 ))] (3.1)
Lemma 1 and the forthcoming lemma 2 (i), states that the repayment differential for the basic treatment is r c

1 (s 1 , s 2 ) -r m 1 (s 1 , s 2 ) = c 2 (s 2 ) -c 2 (s 1 ) while the repayments of the advanced treatment are identical r m 2 (s 2 ) = r c 2 (s 2 ) = c 2 (s 2 )
. Indeed, since the possibility of patients' dumping constrains the repayment of the advanced treatment to cover the cost of treatment of high severity patients, s 2 , it must be that r c 2 (s 2 ) = c 2 (s 2 ). Besides, the hospitals' gain from each additional patient of type s 1 attracted by the expansion of the advanced treatment's provision is:

r c 2 (s 2 ) -c 2 (s 1 ) = c 2 (s 2 ) -c 2 (s 1 ).
To discourage a change in praxis, the relative repayment of the basic treatment must increase of this same amount.

The difference in welfare is therefore

V c (s 1 , s 2 ) -V m (s 1 , s 2 ) = -λ � c 2 (s 2 ) -c 2 (s 1 ) � F(s 1 ).
Since s 1 < s 2 and the costs are increasing in the severity of patients by assumption A.2, the difference is negative, i.e. and r m 1 (s 1 , s 2 ) < r c 1 (s 1 , s 2 ), and we have shown in particular that

V c (s c * 1 , s c * 2 ) < V m (s c * 1 , s c * 2 ). Of course, (s c * 1 , s c * 2 ) � = (s m * 1 , s m * 2 )
, but the definition of the monopoly's optimal allocation implies that

V m (s c * 1 , s c * 2 ) < V m (s m * 1 , s m * 2 ), and it is straightforward to conclude that V c (s c * 1 , s c * 2 ) < V m (s c * 1 , s c * 2 ) < V m (s m * 1 , s m *
2 ), i.e. the optimal welfare decreases when hospitals compete with each other.

Competitive equilibrium sorting of patients

The following details hospital equilibrium decision of treatment given a repayment contract that are characterized in Lemma 2 and complete the proof of Proposition 2. From equilibria conditions in (2.7), the competition over treatment can be analyzed for every type s, independently of other types s � � = s. The payoff matrix associated with the game relative to type s is:

t h 1 (s) = 1 t h 1 (s) = 2 t h 1 (s) = / 0 t h 2 (s) = 1 1 2 � r 1 -c 1 (s) � ; 1 2 � r 1 -c 1 (s) � 0 ; r 2 -c 2 (s) r 1 -c 1 (s) ; 0 t h 2 (s) = 2 r 2 -c 2 (s) ; 0 1 2 � r 2 -c 2 (s) � ; 1 2 � r 2 -c 2 (s) � r 2 -c 2 (s) ; 0 t h 2 (s) = / 0 0 ; r 1 -c 1 (s) 0 ; r 2 -c 2 (s) 0 ; 0 Table 1.
1 -Hospitals' strategic interaction for the choice of treatment of a given patient s

The game has three pure strategy Nash equilibria (1, 1), (2, 2), ( / 0, / 0). Let S c t (r 1 , r 2 ) define the set of patients types treated with treatment t ∈ {1, 2, / 0} at the equilibrium for a given repayment contract (r 1 , r 2 ).

S c 1 (r 1 , r 2 ) = � 1 2 (r 1 -c 1 (s)) ≥ r 2 -c 2 (s) and r 1 -c 1 (s) ≥ 0 � (3.2) S c 2 (r 1 , r 2 ) = {(r 2 -c 2 (s)) ≥ r 1 -c 1 (s) and r 2 -c 2 (s) ≥ 0} (3.3)
To characterize the segmentation of the treatments' provision we distinguish between two different configurations. The configuration (i) that is used to present Proposition 2, and the configuration (ii) such that equilibrium segmentation of patients is not aligned with efficiency,

i.e. it might not be possible to decentralize an allocation such that only low-s patients receive the basic treatment, and high-s the advanced treatment.

The configuration

(i) is such that r 2 -c 2 (s) -1 2 (r 1 -c 1 (s)) is increasing in s.
The equilibrium treatment practice is a segmentation between two cut-offs types of patients such that s ¯≤ s 1 ≤ s 2 ≤ s. Under A.2 there is a unique patient cut-off type s 2 such that is the hospital chooses treatment 2 for such type at the equilibrium, then all types s > s 2 are dumped by the hospitals, i.e. receives no treatment at the equilibrium. Besides, c � 2 (s) < 1 2 c � 1 (s) for all s implies that there is a unique patient type s 1 such that if hospitals provide such patient type with treatment 1 at the equilibrium, then all patients s < s 1 receive treatment 1 at the equilibrium.

Corollary, all patient types s ∈ (s 1 , s 2 ] receive the advanced treatment, 2, at the equilibrium.

If allocations of treatment are interior (s ¯< s 1 < s 2 ≤ s), then s 2 is such that r 2 -c 2 (s 2 ) = 0, and s 1 is the patient type such that hospitals are indifferent between providing the basic treatment or deviate toward the provision of the advanced treatment to attract more patients, i.e.

1 2 (r 1 -c 1 (s 1 )) ≥ r 2 -c 2 (s 1 ). The configuration (ii) is such that r 2 -c 2 (s) -1 2 (r 1 -c 1 (s)) is not monotonic or decreasing in s. First, observe that whenever c � 2 (s) > 1 2 c � 1 (s) for all s, if s ∈ S c 2 (r 1 , r 2
) then all patient types s � ≤ s receive the advanced treatment, 2. Corollary, if the allocation is interior, the patients who receive the basic treatment are high-s patients, which is not aligned with the efficient provision and is obviously highly inefficient. Then, note that when

r 2 -c 2 (s) -1 2 (r 1 -c 1 (s))
is not monotonic in s it might be several patients types such that hospitals are indifferent between the provision of treatments 1 and 2. In this case, different treatments are provided between each patient cut-off type such that 1 2 (r 1 -c 1 (s)) = r 2 -c 2 (s), and the equilibrium praxis is neither aligned with efficiency.

Lemma 2. (i) If c 2 (s) -1 2 c 1 (s)
is decreasing in the patient severity, then hospitals' equilibrium treatment practice is a segmentation between two cut-offs types of patients such that s ¯≤ s 1 ≤ s 2 ≤ s. Furthermore, there is a bijective mapping between the repayments contract (r 1 , r 2 ) and the interior treatment allocations s ¯< s 1 < s 2 ≤ s:

r c 1 (s 1 , s 2 ) = c 1 (s 1 ) + 2(c 2 (s 2 ) -c 2 (s 1 )) (3.4) r c 2 (s 2 ) = c 2 (s 2 ) (3.5)
The reduced social welfare function is:

V c (s 1 , s 2 ) = � s 1 s ¯�b 1 -c 1 (s)) � d F(s) + � s 2 s 1 � b 2 -c 2 (s) � d F(s) -λ [[c 1 (s 1 ) + 2(c 2 (s 2 ) -c 2 (s 1 ))]F(s 1 ) + c 2 (s 2 )F(s 2 )] (3.6)
(ii) Otherwise, repayment contract (r 1 , r 2 ) may not be sufficient to decentralize allocations aligned with efficiency, i.e. both interior, and such that only low-s types receive the basic treatment and high-s the advanced treatment.

The interim result of Proposition 2 in equation (3.1) is obtained from statements (2.4) in lemma 1 and (3.6)in lemma 2, which close the proof of the proposition for configuration (i).

The argument must be adapted in configuration

(ii).The argument is short if c � 2 (s) > 1 2 c � 1 (s) for all s ∈ [s ¯, s].
It consists in establishing that due to the high inefficiency of interior allocations the optimum consist in providing the advanced treatment only. Whatever the optimal allocation of the advanced treatment, however, it can also be achieved with a monopoly hospital with the same transfer, and thereby welfare. Since such allocation of a unique treatment is never optimal in the monopoly case 7 , one can again conclude that monopoly allows to achieve a higher welfare.

Finally, when r 2 -c 2 (s) -1 2 (r 1 -c 1 (s)) is not monotonic in s, one cannot establish a general relationship between repayments and the equilibrium segmentation of treatment. However we can show that for any interior allocation there is an allocation that at least weakly dominates in terms of allocative efficiency, and can be decentralized to a monopoly hospital at a lower social cost. The argument relies on the incentive of competitive hospitals to expand the provision of the basic treatment which implies that at given repayments (r 1 , r 2 ), the highest patient type receiving the basic treatment is always lower in competition (the upper bound of S c 1 (r 1 , r 2 ) is lower than the upper bound S m 1 (r 1 , r 2 )). The complete proof is detailed in Appendix B and appeals both to the argument of configuration (i) and to the special case c � 2 (s)

> 1 2 c � 1 (s) for all s ∈ [s ¯, s].
7. More precisely assumption A.5 is a necessary and sufficient condition so that both advanced and basic treatment are provided at the monopoly optimum, while it is necessary but not sufficient in the competitive framework 1 to s c 1 , following the introduction of competition when the repayment is r 1 . To induce competitive hospitals to adopt the "before-entry" monopolistic allocation, the regulator must increase the basic treatment's repayment (and therefore the public spending for each patient receiving it) from r 1 to r � 1 . The change in repayment counters the hospitals' incentive to expand the provision of the advanced treatment to low-s patients.

Finally, one can observe that competition involves a prisoner's dilemma situation for the patients with severity between the cut-off s c 1 and its counterpart for monopoly hospital s m 1 . The patients in this interval are the types the competitive hospitals try to attract by expanding the provision of the advanced treatment. Yet, both hospitals should deviate at the equilibrium.

Therefore, each hospital ends up conceding the use of a less profitable treatment to attract patients but fails to increase their activity given that their competitor will also adopt this treatment practice.

Extensions and empirical predictions

Hospital competition and global coverage

From now we restrict the attention to configuration (i) of lemma 2, such that competition does not prevent from decentralizing an allocation aligned with efficiency. In our sense, the alternative configuration is secondary and mostly stem from our simplifying assumption of perfectly elastic demand 8 .

Assumption 6: c 2 (s) -1 2 c 1 (s) is decreasing in s
Following Lemma 2, under A.6, the optimal competitive allocation of treatment must verify:

� b 1 -(1 + λ )c 1 (s 1 ) -(b 2 -(1 + λ )c 2 (s 1 )) � = λ (c � 1 (s 1 ) -2c � 2 (s 1 )) F(s 1 ) f (s 1 ) + λ (c 2 (s 2 ) -c 2 (s 1 )) (4.1) b 2 -(1 + λ )c 2 (s 2 ) = λ c � 2 (s 2 ) [F(s 2 ) + F(s 1 )] f (s 2 ) (4.2)
The LHS of (4.1) and (4.2) are the differences in the net social benefit, respectively, providing the cut-off patients' type s 1 with the basic treatment instead of the advanced treatment ; and providing the patients' type s 2 with the advanced treatment instead of leaving the hospitals with the incentive to dump those patients. The RHS of both FOC are the increases in the social cost of transfers consecutive to increases in allocation cut-offs.

One can show that the standard rent-extraction v.s. efficiency trade-offs lead to downward distortion of each allocation cut-off s 1 , s 2 , in both the competitive and the monopoly cases.

Proposition 3 below states this result and characterizes the effect of the market structure on the optimal decentralized allocation.

8. Precisely, in the duopoly case, perfect elasticity implies that a hospital can double the volume of its demand. It would be more likely to be in configuration (i) with a less elastic demand without changing the mechanisms. Indeed the factor in A.6 would be larger than 1 2 and tends toward 1, so that the room for encountering configuration (ii) diminishes.

Proposition 3. Under assumption A.1 to A.5 when hospitals compete, the optimal range of patients treated with 1 ( s

∈ [s ¯, s c * 1 ]), 2 (s ∈ (s c * 1 , s c * 2 ]
) and not treated (s > s c * 2 ) are such that s c * 1 < s * * 1 and s c * 2 < s * * 2 and:

(i) If c 2 (s) -1 2 c 1 (s)
is decreasing in patient severity, then it is feasible to decentralize an allocation such that s ¯< s 1 < s 2 ≤ s, and :

s c * 2 < s m * 2 < s * * 2 and s * * 1 > s c * 1 > s m * 1 iff : F(s m * 1 ) f (s m * 1 ) > � s2 s m * 1 c � 2 (s) c � 2 (s m * 1 ) ds (4.3) otherwise s c * 1 ≤ s m * 1 < s * * 1 9 . (ii) If c 2 (s)-1 2 c 1 (s)
is increasing in patient severity, competition prevents from decentralizing an efficient allocation of the basic treatment and:

s c * 1 = s ¯< s m * 1 and s * * 2 > s m * 2 = s c * 2 .
When hospitals compete, not only fewer patients receive treatment than efficiency would prescribe but further less than at the monopoly optimal allocation. The social cost of decentralizing a large provision of the advanced treatment to high-s increases under competition. The higher the repayment of the advanced treatment the stronger the hospitals' incentive to deviate toward providing low-s with the advanced treatment and increase their activity. Therefore, decentralizing a high coverage also requires to increases the "extra-repayment" of the basic treatment to prevent this deviation. Formally it translates into the RHS of condition (4.2) being larger than its counterpart in the monopoly case 10 when both are evaluated at s m * 2 . Moreover, although we found that the competitive pressure leads hospitals to expand the provision of advanced treatments to low-s patients at given repayments levels, its effect on the optimal level of the provision of the basic treatment to low-s patients is ambiguous. The final effect on the basic treatment provision depends on the relative volume of provision of treatments at efficiency, which competition strengthen making it more uniform. For instance, suppose that the advanced treatment is efficient for a large range of the population or, for instance, that the population has grown older, raising the proportion of high severity patients. A decrease in F(s 1 ) involves a decrease of the RHS of (4.2), so the global coverage is likely to be closer to the 9. Where s2 is the solution of (4.2) when s 1 is held equal to s m * 1 . 10. Which is equal to λ c � 2 (s 2 ) F(s 2 ) f (s 2 ) , see Appendix.A for the complete formulas.

monopoly optimum and thereby to the efficient level. Though an increase in r 2 to induce more coverage still requires increases in r 1 to keep the provision of basic treatment unchanged, its provision concerns fewer patients and has a lower social cost. The RHS (4.3) is unchanged, the LHS decreases and the condition is less likely to be satisfied. In that case, we can conclude to an enlargement of the segment receiving the advanced treatment toward both more high-s, and low-s.

Different patients' out-of-pocket participation

So far we have made the implicit assumption that all patients pay the same premium to access the health-care system, independently of the care they receive. In this section we relax this assumption and allow for the patients' out-of-pocket to depend on the type of treatment they receive, and to be controlled by the regulation authority. One can think for instance of the Top-up system in the National-Health-System in UK.

Apart from few adaptations of the baseline model, the main changes concern the patients' preferences over treatment that, now, are also a function of the out-of-pockets associated with each treatment.

Let the patients' income y be identical for all patients, and p t denoting the patient outof-pocket for treatment t. For convenience, we define v(y, p t ) = u(yp t ) -u(y), the patient dis-utility from out-of-pocket charges, with u(•) the utility of income and u � > 0, u �� < 0. The patient net surplus from receiving treatment is now v(y, p t ) + b t , for t ∈ {1, 2, / 0}, and the social benefit from the patients that are dumped by the two hospitals remain normalized to zero. The efficient out-of-pocket and treatment provision are such that it maximizes the net social benefit

from treatment v(y, p t (s)) + b t -(1 + λ )(c t (s) -p t (s)) for all s.
Assuming that the disutility to pay is the same for all patients that receive a treatment they are all charged with the same out-of-pocket. Under A.2 to A.4, the efficient allocation of treatment is a segmentation between two cut-offs patient types with the same feature as in Proposition 1. Lemme 4 establishes that at the efficient and monopoly optimal level, the former baseline model is just a particular case of this extension. The monopoly hospital still makes treatments' decisions regardless of the preferences of the patients so that, distorting the patients' out-of-pocket away from the efficient level would not change the decentralized problem. Therefore, in monopoly we find that the optimal out-ofpocket is equal to its efficient level.

We find that differentiating patients' out-of-pocket depending on the care they receive may allow to refine positive effects of strategic interactions in the competitive case. This is true in particular when the benefits from each treatment are close to each other.

If patients were to prefer the basic treatment, compared to a monopoly, competing hospitals tend to expand the provision of basic treatment to high-s patients compared to a monopoly11 .

In this case, a decrease in the repayment level of the basic treatment would discourage this deviation, with no other allocative effect (while decreasing the advanced treatment's repayment would increase dumping in the symmetric case). Therefore the total transfer to hospitals to decentralize any given allocations decreases when hospitals compete, implying an increase in social welfare.

Intuitively, a regulator could enforce this situation by simply setting the out-of-pocket p t , t = 1, 2, such that the patients' surplus is larger when receiving the basic treatment. This shift in patients' preferences over the treatments is obtained by distorting the participation of the patients away from efficiency and is not necessarily optimal. In particular, if the difference in treatments' benefits is high, the distortion in patients' out-of-pocket that is required to change their preferences leads to an important loss in welfare and the provision by a monopoly should be preferred.

Proposition 4. The control of the patients' out-of-pocket allows to retrieve a positive effect of competition, so that V c (s

c * 1 , s c * 2 , p c * 1 , p c * 2 ) > V m (s m * 1 , s m * 2 , p m * ), if and only if: � v(y, p m * ) -v(y, p c 1 ) + (1 + λ )(p m * -p c 1 ) � F(s m * 1 ) < 1 2 λ [c 2 (s m * 2 ) -c 2 (s m * 1 )] (4.4)
Where, p c * 1 < p c * 2 = p m * = p * * and then the optimal decentralization of treatment is such that,

s * * 2 > s c * 2 > s m * 2 and, s * * 1 > s c * 1 > s m * 1 .
Otherwise, the results of Propositions 2 and 3 still holds.

The condition (4.4) intuitively states that the total effect of out-of-pocket differentiation is positive if the decrease in the transfers to hospitals more than compensate the loss of distorting the patients' direct participation to the funding of their treatment costs.

Finally, Proposition 4 states that when competition allows to retrieve positive effect of competition, it also increases the global coverage at the optimum, while the difference in the optimal composition of treatments compared to its provision by a monopoly remains unclear.

Testable predictions and concluding remarks

Our results balance the former theoretical prediction on hospitals' competition. Most of the theoretical literature on hospital competition studies its effects on the hospitals' global quality and finds a welfare improving relationship. In this paper we explore this relationship with respect to quality dimensions that are specific to the patients' characteristics such that the nature of the treatment patients are provided with. Most of this work therefore relies on taking into account the heterogeneity of patients within the diagnostic groups that are defined by Prospective Payment Systems. We show that when hospitals internalize the patients' preferences to take treatment decisions, a preference of patients for the advanced treatment may be at the expense of social welfare and health coverage.

Testing the mechanism supporting these results meet with two difficulties due to the nature of patients heterogeneity that is observed by physicians only, and the prerequisite that the authority adapt the repayment levels to the hospital sector on the different territories it regulates.

Therefore the predictions on the impact of the hospital sector on optimal treatment praxis in section 4.1 may not be relevant to test. The central prediction that more intense competition should lead to expand the provision of the advanced treatment to patients with lower severities, however, suppose two hospital sector confronted to the same repayment contract and might be more convenient. In addition we find that an increase in competition should increase the sensitivity of hospitals' treatment practices consecutive to given changes in repayment levels. This is actually quite standard but in this model it also translates into larger variations in the patient segmentation for competitive hospitals instead of the speed of the change in practice. Also note that as mentioned in the presentation section 4.2, these predictions are symmetric when there are reasons to think that patients are rationed in the provision of the basic treatment, i.e. if one reverse the assumption 1.

The model also predicts that hospitals' competition is to make the treatment practices uniform according to two different scenarios. The first scenario refers to the competition effect on optimal allocation discussed in section 4.1 and seem hardly testable. The second scenario predict completely uniform praxis is likely to arise when the marginal costs of severity are similar for the two treatments (configuration (ii) of lemma 2), and only holds under assumption 1.

This case suggests that an increase in competition could lead to give up on the provision of the basic treatment. In the absence of competition there is no obstacle to decentralize the provision of the basic treatment to low-s patients since it is more cost-effective for those types, thereby implying the highest surplus for the hospital. But we find that, in competition, the opportunity for hospitals to increase their demand may always dominate the possibility to achieve larger costs reduction by providing the basic treatment to low-s patients. This is particularly true if the effect of decreases in severity on the costs of both treatments is similar.

Finally, our extension also shows that if one can change patients' preferences over treatments, for instance by setting different out-of-pockets for each treatment, positive effects of competition can be retrieved. In particular, it is more likely to restore positive effects of competition when the benefits of treatments are not too different. This result can be interpreted in terms of hospitals' specialization. When the difference in the benefits attached to the alternative treatments is substantial, our results suggest that one should prefer that the provision of the ad-vanced treatment is dedicated to a unique hospital on each territory. Otherwise, if treatments' benefits are similar, the differentiation of patients' out-of-pocket over treatments may restore positive effects of competition. In our sense, those predictions may participate to provide a reading grid for the use of Top-up system similar than this of the National-Health-System in the UK.

In the monopoly case, the regulator maximizes the social welfare subject to the hospital treatment decision rule in t * (s) = argmax t {r t -c t (s)} with t ∈ {1, 2, / 0} for all s, and the hospital participation constraint.

Lemma 1:

(i) Under the costs assumption A.2 and A.3, the monopoly segmentation of treatment is necessary such that the patients that receive the basic treatment have lower types than those receiving the advanced treatment. Given a repayment contract (r 1 , r 2 ), there is a unique patient cut-off type s 2 such that if hospitals choose treatment t ∈ {1, 2} for this patient then all s > s 2 receive no treatment t = / 0. Besides, there is a unique cut-off type s 1 such that is if hospitals choose treatment 1 for this patient then all s ≤ s 1 receive treatment 1. Corollary, all patients types s ∈ (s 1 , s 2 ] receive treatment 2. For convenience, denote s 1 = s ¯and s 2 = s 1 , when the hospital provides neither treatments 1 nor 2, respectively, given the repayments r 1 , r 2 .

(ii) By definition of the hospital decision rule, the participation constraint of the regulation problem is always satisfied. By Lemma 1 (i) there is a bijective relation between the repayment couple (r 1 , r 2 ) and the monopoly cut-offs couple (s 1 , s 2 ) of all interior allocations, such that:

r m 1 (s 1 , s 2 ) = c 1 (s 1 ) + c 2 (s 2 ) -c 2 (s 1 ), and r m 2 (s 1 , s 2 ) = c 2 (s 2 ).
Hence for convenience we reduce the monopoly problem to the choice of the cut-offs s 1 and s 2 that maximizes the welfare function in equation (2.4).

Taking the derivatives of the reduced program with respect to s 1 , s 2 and rearranging we obtain the following FOC: To insure that the solution of the problem is a maximum and is unique it is sufficient to impose log concave distribution of severity and that the treatments' costs are convex and that c �� 2 (s) is not too large compared to c �� 1 (s).

� b 1 -(1 + λ )c 1 (s 1 ) -(b 2 -(1 + λ )c 2 (s 1 )) � =λ � c � 1 (s 1 ) + c � 1 (s 1 ) � F(s 1 ) f (s 1 ) (4.5) b 2 -(1 + λ )c 2 (s 2 ) =λ c � 2 (s 2 ) F(s 2 ) f (s 2 ) (4.6) When λ > 0,

Appendix B.

Proposition 2: The proof corresponding to the configuration (i) of lemma 2 is provided in the main text, and it only remains to show that both monopoly and competitive optimal allocation are interior under assumption A.5. Observes that A.5 it is necessary and sufficient in the monopoly case, while it is not sufficient in the competitive case (in particular in configuration (ii)).

Proof: Suppose that only treatment 1 is available or, the corresponding welfare is equivalent with holding s 2 = s 1 in (3.6). Then the FOC for its allocation is this in (4.7). Define s ∈ [s ¯, s] ,

the cut-off type that maximizes the social welfare, i.e. verifies:

b 1 -(1 + λ )c 1 (s) = λ c � 1 (s) F(s) f (s) (4.7)
Now consider that s 2 is raised above s. The change in social surplus is:

b 2 -(1 + λ )c 2 ( s) f ( s) -λ c � 2 ( s)F( s) - � b 1 -(1 + λ )c 1 ( s) f ( s) -λ c � 1 ( s)F( s)

�

When the above term is positive then it is optimal to provide the advanced treatment to some patient types. A sufficient condition is that:

w 2 ( s) c � 2 ( s) > w 1 ( s) c � 1 ( s) . Since c � 1 ( s) > c � 2 ( s) another sufficient condition is that s > s * *
1 , such that w 2 ( s) > w 1 ( s). The same path can be followed in the monopoly case, though under A.2 it is a necessary and sufficient to conclude that the optimal allocation is interior.

To establish the proof corresponding to configuration (ii) of lemma 2 we must distinguish between two cases depending on whether 1 2 (r

1 -c 1 (s)) -(r 2 -c 2 (s)) is increasing in s or non monotonic. If 1 2 (r 1 -c 1 (s)) -(r 2 -c 2 (s)) increasing in s : When (c � 1 (s) < 2c � 2 (s))
no interior allocation of each treatment can be optimal as it involves high-s to receive the basic treatment, and low-s the advanced treatment. Either the problem becomes convex in the cut-offs s 2 < s 1 , and so that either s c * 2 = s, or s c * 2 = s 1 , i.e. only one treatment is provided. If the problem becomes convex in s 2 , it is straightforward to conclude since allocations of a unique treatment are feasible but optimal in Monopoly. More precisely,

V c * (s t ) = V m (s t ) < V m * (s m * 1 , s m * 2 )
. If the problem is not convex in s 2 , it exists a maximum such that s c * 2 < s 1 c * = s, where s verify (4.7). This maximum is global iff the economy of rent from treating low-s with the treatment 2 is larger than the loss of allocative inefficiency implied by (i) treating low-s with the treatment 2, and more importantly from not treating patients type s ∈ [ s, s m * 2 ]. Indeed, interior allocation in monopoly under assumption 5 implies that s < s m * 2 . Since competition does not prevent allocation consisting in providing only one treatment type, especially the treatment 2, a maximum such that only the treatment 2 is provided, with s 1 c * = s and s c * 2 = s m * 2 exists. By definition, if the optimal provision of the treatment 2 only is such that s < s m * 2 , it implies not treating patients with severity higher than s consists in a larger welfare loss than the loss due to increasing the repayment of treatment 2 and hospital rent.

If 1 2 (r 1 -c 1 (s)) -(r 2 -c 2 (s))
is not monotonic in s :

In this case, we cannot directly compare the welfare corresponding to given repayment neither to a given segmentation since the correspondence between repayment and segmentation of patients is unclear. However the deduction we can make on the upper bound of S c t (r 1 , r 2 ) are sufficient to establish the proof.

V c (r 1 , r 2 ) < V m (r 1 (s * * 1 , s * * 2 ), r 2 (s * * 2 ), there is V c (r 1 , r 2 ) ≤ � Sup{S c 1 (r 1 ,r 2 )} s ¯b1 -c 1 (s) -λ r 1 d F(s) + � Sup{S c 2 (r 1 ,r 2 )} Sup{S c 1 (r 1 ,r 2 )} b 2 -c 2 (s) -λ r 2 d F(s). V c (r 1 , r 2 ) = � S c 1 (r 1 ,r 2 ) b 1 -c 1 (s) -λ r 1 d F(s) + � S c 1 (r 1 ,r 2 ) b 2 -c 2 (s) -λ r 2 d F(s) (i) Suppose (r 1 , r 2 ) is such that S c t (r 1 , r 2 ) = / 0, for at least one treatment t ∈ {1, 2}.
We already argue that the same allocation can therefore be achieved with a monopoly with the same transfers, but is not optimal under A.5 in the monopoly case.

(ii) Suppose (r 1 , r 2 ) is such that S c t (r 1 , r 2 ) � = / 0 t ∈ {1, 2}
, and SupS c 1 < SupS c 2 . This directly implies that, r 2 > r 1 and, r m

1 (SupS c 1 , SupS c 2 ) < r 1 and r m 2 (SupS c 1 , SupS c 2 ) = r 2 . Moreover, it is clear that for all s ∈ [SupS c 1 , SupS c 2 ], s ∈ S c 2 (r 1 , r 2 ). Either SupS c
1 ≤ s * * 1 and we show that

V c (r 1 , r 2 ) < V m (r m 1 (SupS c 1 , SupS c 2 ), r m 2 (SupS c 1 , SupS c 2 )) ≤ V m * . Indeed, V c (r 1 , r 2 ) = � S c 1 (r 1 ,r 2 ) b 1 -c 1 (s) -λ r 1 d F(s) + � S c 1 (r 1 ,r 2 ) b 2 -c 2 (s) - λ r 2 d F(s) ≤ Proposition 1 and r 2 >r 1 � SupS c 1 s ¯b1 -c 1 (s) -λ r 1 d F(s) + � SupS c 2 SupS c 1 b 2 -c 2 (s) - λ r 2 d F(s) < r m 1 (SupS c 1 ,SupS c 2 )<r 1 � SupS c 1 s ¯b1 -c 1 (s) -λ r 1 d F(s) + � SupS c 2 SupS c 1 b 2 -c 2 (s) - λ r 2 d F(s) = r m 2 (SupS c 1 ,SupS c 2 )=r 2 V m (r m 1 (SupS c 1 , SupS c 2 ), r m 2 (SupS c 1 , SupS c 2 )). Or SupS c 1 > s * * 1 V c (r 1 , r 2 ) = � S c 1 (r 1 ,r 2 ) b 1 -c 1 (s)-λ r 1 d F(s)+ � S c 1 (r 1 ,r 2 ) b 2 -c 2 (s)-λ r 2 d F(s) < Proposition 1 and r 2 >r 1 � s * * 1 s ¯b1 -c 1 (s) -λ r 1 d F(s) + � SupS c 2 s * * 1 b 2 -c 2 (s) -λ r 2 d F(s) ≤ r m 1 (r 1 >SupS c 1 ,SupS c 2 )>r m 1 (s * * 1 ,SupS c 2 ) � s * * 1 s ¯b1 -c 1 (s) -λ r m 1 (s * * 1 , SupS c 2 ) d F(s) + � SupS c 2 s * * 1 b 2 -c 2 (s) -λ r 2 d F(s) = r m 2 (s * * 1 ,SupS c 2 )=r 2 V m (r m 1 (s * * 1 , SupS c 2 ), r m 2 (s * * 1 , SupS c 2 )) ≤ V m * (iii) Suppose (r 1 , r 2 ) is such that S c t (r 1 , r 2 ) � = / 0 t ∈ {1, 2}, and SupS c 1 > SupS c 2 .
Then it might be optimal only if SupS c 1 ≤ s where s verifies (4.7). And the same argument as previously when 1 2 (r 1 -c 1 (s)) -(r 2 -c 2 (s)) increasing in s applies. Any such allocations implied by such repayments are dominated by setting r 1 < c 1 (s), and r 2 = c 2 (s m *

2 ) so that the advanced treatment is provided. Therefore

V c (r 1 , r 2 ) ≤ V c ( 1 (s), c 2 (s m * 2 )) = V c (s 1 = s, s m * 2 ) = V m (s 1 = s, s m * 2 ) < V m * (s m * 1 , s m * 2 ). V c (r 1 , r 2 ) < Proposition 1 and r 2 >r 1 � s c 1 s ¯b1 -c 1 (s) -λ r 1 d F(s)
Proposition 3: The LHS of (4.1), (4.2) are equal to zero respectively at s * * 1 , s * * 2 by definition of the first best efficient allocation (2.1)- (2.3) . Furthermore c � 1 (s) > 2c � 1 (s) > 0 imply that interior allocation is such that s 1 < s 2 so that both RHS of FOC are positive. With the LHS being decreasing in s it follows that competition optimal solutions s c 1 , s c 2 are satisfied iff s c 1 ≤ s * * 1 and s c 2 ≤ s * * 2 . The condition so that the optimum is an interior allocation is given by A.5 and the proof follow the same path as in the monopoly case (see Appendix A.)

To compare competitive and monopoly optimal allocations first observe that the LHS of their respective FOC are identical (see Appendix A. for the monopoly FOC). Since by lemma 2 (i), interior allocations are such that s ¯< s 1 < s 2 ≤ s one find that the competitive FOC in s 2 when evaluated at the monopoly optimum s m * 2 is negative, and the concavity of the problem implies that s c 2 < s m 2 . Furthermore, denote s2 , the solution of 4.2 for s 1 = s m 1 and taking the derivative of V c (s 1 , s 2 ) with respect to s 1 at s m 1 , s2 we get:

λ � c � 1 (s m 1 ) F(s m 1 ) f (s m 1 ) -λ (c 2 ( s2 ) -c 2 (s m 1 )) = 0
Rearranging we obtain the inequality in (4.3) as the condition for

∂V c (s m 1 , s2 ) ∂ s 1 to be positive.
When it is positive it implies that at a given coverage level s2 it is welfare improving to increase

s c 1 beyond s m * 1 .
To finish observe that the LHS of FOC is decreasing by definition of efficiency, so it is sufficient that the RHS is non-decreasing for the problem to admit a unique solution. A set of sufficient conditions for the RHS to be non-decreasing is that the distribution of patients satisfy monotone hazard rate property and c �� 2 (s) is not substantially larger than c �� 1 (s) > 0.

Proposition 4

The regulation problem for competitive hospitals is:

Max r 1 ,r 2 ,p 1 ,p 2 � s s ¯∑ i=1,2 D i (s,t h i * (s),t h -i * (s)) � (v(y, p t h i (s) ) + b t h i * (s) ) + (r t h i * (s) -c t h i * (s) (s)) -(1 + λ )(r t h i * (s) -p t h i * (
s.to : Therefore the definition of allocation in function of repayment in lemme 2 (i) holds, and so the reduced form of the regulation problem is similar to (2.4) except for the additional effect of out-pocket level on the social surplus.

D i � s, p 1 , p 2 t h i (s),t -i (s) � =                        1 if v(y, p t h i (s) ) + b t h i (s) > v(y, p t -i (s) ) + b t -i (s) 1 2 if v(y, p t h i (s) ) + b t h i (s) = v(y, p t -i (s) ) + b t -i (s) 0 otherwise � s s ¯∑ h i =1,2 D h i � s, p 1 , p 2 t h i (s),t h -i (s) � r t h i * (s) -c t h i * (s) dF(s) ≥ 0 (PC h i ), i = 1, 2 π i (s,t h i * (s),t h -i * (s)) ≥ π i (s,t h i ,t h -i * (s)) t h i ∈ {1, 2, / 0} , i = 1, 2 (
In case (c): v(yp 1 ) + b 1 > v(y, p 2 ) + b 2 which will be the case of interest, the hospital strategic interaction changes compared to the baseline model. Hospitals now have incentives to provide the basic treatment to low-s patient in order to attract them. Under A.2, the equilibrium treatment praxis is a segementation between two cut-off type of patient

s 1 , s 2 such that r 1 -c 1 (s 1 ) = 1 2 � r 2 -c 2 (s 1 ) � and r 2 -c 2 (s 2 ) = 0 such that s ¯≤ s 1 ≤ s 2 ≤ s. Under A.
2 the provision of basic treatment is always relatively more profitable for low-s patient so that interior allocation is always aligned with efficiency. The regulation problem can be reduced to the choice of the cut-off s 1 , s 2 which maximizes :

V c (s 1 , s 2 ) = � s 1 s ¯�v(y, p 1 ) + b 1 -c 1 (s)) � d F(s) + � s 2 s 1 � v(y, p 2 ) + b 2 -c 2 (s) � d F(s) -λ � c 1 (s 1 ) + 1 2 (c 2 (s 2 ) -c 2 (s 1 )) � F(s 1 ) -λ c 2 (s 2 )F(s 2 ) (4.9)
Proposition 4: The proof consists in, first argue that the optimum of case (b) is never a global maximum, then to exhibit the condition such that case (c) leads to a higher welfare than case (a)

First we show that if it is optimal to First observe that by the concavity of patients' utility of income, the distortions in out-ofpocket levels that leads to the lowest social loss are obtained by decreasing the out-of-pocket of the basic treatment. It follows that if it is optimal to set different out-of-pockets, they are such that p 1 < p 2 = p * * . Now, assume the prices are such that b 1 + v(y, p 1 ) = b 2 + v(y, p 2 ), and consider a marginal decrease in p 1 so that b

1 + v(y, p � 1 ) > b 2 + v(y, p 2 ), with p � 1 = p 1 -ε < p 1 .
The change in social in surplus associated to the decentralization of a given interior allocation

(s 1 , s 2 ) is : V c (b) � s 1 , s 2 , p 1 , p 2 ) -V c (c) (s 1 , s 2 , p � 1 , p 2 ) ≈ - 1 2 � c 2 (s 2 ) -c 2 (s 1 ) � F(s 1 ) (4.10)
The difference is always negative for interior allocations such that s 1 < s 2 . Furthermore it is enough to observe that A.5 guaranty that optimal allocation is interior in each case (the argument used in the baseline model holds). Therefore we have shown that

V c (b) � s 1 , s 2 , p 1 , p 2 ) < V c (c) (s 1 , s 2 , p � 1 , p 2 ) ≤ V c (c) (s (c) * 1 , s (c) * 2 , p (c) * 1 , p (c)
*

2 ) so that the solution to case (b) is never qualified for the global extrema. Now one can show that if it is optimal to set the patients' out-of-pocket at a different level than p * * they must be set so that the solution belongs to (c). Indeed if the solution belongs to case (a), i.e. b 1 + v(y, p c� 1 ) < b 2 + v(y, p c 2 ), then the problem of treatment decentralization if the same as in the baseline model, and difference in out-pocket only generates inefficiency in the funding of care without relaxing the decentralization problem which can't be optimal. More precisely, one can show that the local optimum of case (a) is such that p A sufficient condition so that the global extrema belong to case (c) and it is therefore optimal to set different out-pocket for each treatment is:

V c (c) ( s1 , s2 , p 1 , p 2 ) -V c (a) ( s1 , s2 , p * * ) (4.11)
To finish the proof one must observe that proposition 2 still holds for the comparison of competitive optimum in case (a) and the monopoly optimum. Indeed, since the optimal outof-pocket is equal in monopoly and in the competitive case (a) (equals to p * * ), one can extend the proof by simply considering the benefits from treatment b � t = v(y, p * * ) + b t , instead of b t only. Therefore the condition such that the social welfare when out-of-pocket are distorted in competition, i.e we are in case (c) is a sufficient to conclude that the global optimum of the competitive problem belongs to case (c). In other words is the condition that follows is verified, then it is optimal to distort the patients out-of-pocket so that patient prefer the provision of the basic treatment and competition lead to an increase in welfare compared to monopoly. The necessary condition so that

The difference in surplus to decentralize any interior condition in monopoly and competition when optimal out-of-pocket is different is:

V m (s m * 1 , s m * 2 , p m * ) -V c (c) (s m * 1 , s m * 2 , p (c) * 1 , p (c) * 2 ) = � v(y, p m * ) -v(y, p (c) * 1 ) + (1 + λ )(p m * -p (c) * 1 ) � F(s m * 1 ) -1 2 λ [c 2 (s m * 2 ) -c 2 (s m * 1 )] (4.12)
Rearranging this condition gives (4.4).

Introduction

Between 2000 and 2006 the spending for imaging services, including MRI, CT and PET scanner, has more than doubled in the U-S Medicare system. With an average annual growth rate twice as large as the overall rate of growth in physician fees schedule services, they are pointed out as a significant driver of the health-care expenditures' inflation in the past decades [START_REF] Smith-Bindman | Rising Use Of Diagnostic Medical Imaging In A Large Integrated Health System[END_REF]Larson, 2008, Lehnert and[START_REF] Lehnert | Analysis of Appropriateness of Outpatient CT and MRI Referred From Primary Care Clinics at an Academic Medical Center: How Critical Is the Need for Improved Decision Support?[END_REF]. On these respects, over a 6-month period in 2007, the share of inappropriate CT-scan and MRI performed in a large urban university hospital in the U-S has been estimated at 26% (118 out of 459 acts, see [START_REF] Lehnert | Analysis of Appropriateness of Outpatient CT and MRI Referred From Primary Care Clinics at an Academic Medical Center: How Critical Is the Need for Improved Decision Support?[END_REF] ). Another recent national study suggests that higher rates of performance of coronary angiography on asymptomatic patients were associated with more frequent inappropriate treatment decisions due to diagnostic-therapeutic cascade [START_REF] Bradley | Patient selection for diagnostic coronary angiography and hospital-level percutaneous coronary intervention appropriateness: insights from the National Cardiovascular Data Registry[END_REF]. In their article, asymptomatic patients refer to patients with no ischemic symptoms, and the inappropriate treatment decisions to inappropriate Percutaneous Coronary Intervention. At the aggregate level we are, however, unaware of clear evidence on whether over-testing leads to better health outcomes.

The studies assessing appropriateness of care rely on evidence-based guidelines that are increasingly developed to support care decision and complement physician experiences. They are designed to help physicians in deciding which diagnostic tests to order, which treatments to use for specific conditions, when to discharge patients from the hospital, and many other aspects of clinical practice 1 . For instance, in the U.K. the National Institute of Health and Care Excellence (NICE) publish guidelines based on efficacy and cost-effectiveness to orientate hospital care practices. As suggested by [START_REF] Garber | Evidence-Based Guidelines As a Foundation For Performance Incentives[END_REF], compliance, however, remains difficult to measure, especially at a general level. Different factors are mentioned explaining departures from guidelines and include medical liability fears, economically motivated self-referral, patient demand, regional differences practice style, and physician experience and training in the appropriate use of newer imaging modalities (Smith-Bindman, Miglioretti and Larson, 2008, Lehnert and Bree, 1. see for instance [START_REF] Berg | Detection of Breast Cancer With Addition of Annual Screening Ultrasound or a Single Screening MRI to Mammography in Women With Elevated Breast Cancer Risk[END_REF] or [START_REF] Mainiero | ACR Appropriateness Criteria Breast Cancer Screening[END_REF] for recent examples of their construction for imaging use. 2010). To our knowledge no empirical economic studies have been conducted to determine the salience of these possible factors. One should also mention that the physicians' own judgment still play a crucial role for adapting treatment to each patient's particular condition, and can also explain departures from guidelines when it remains uncertainty about what is the best praxis.

A purpose of this work is to provide a theoretical framework to highlight the role of economic incentives and hospital competition on the care procedures hospitals might follow. We will consider the problem of a regulation authority that can observe the tests that have been performed, the test results, and treatment method that are provided to all patients (or at least we assume that they are verifiable and contractible). The authority does not observe, however, the patients ex-ante types and thereby, neither their expected cost of care. The verifiability of tests, signals and treatment can reflect the idea that the physicians cannot lie about the patient's pathology, neither about the treatment method it has used to cure it (i.e. we do not discuss upcoding). As evidence-based guidelines suggests, though the signal reveals a complication for a patient, how exactly its treatment should be managed, what treatment intensity is appropriate can depends on its characteristics. This is the underlying idea that our model intends to reflect.

In particular, in our problem, the decision to perform a test is at the discretion of the physicians that are the only ones to observe the patient. Decentralizing the performance of tests, as much as the treatment praxis, therefore, falls into the problem of discretion of over-provision of health-care practitioners. Understanding its implication is key for designing economic incentives in health care and apprehending its impact on the care procedure that hospitals follow. Our model notably reflects the prediction that an over-performance of tests may turn into overtreatment (if for instance, evidence-based guidelines are the support of treatment decision but are designed from the analyses of symptomatic patients types only). Although it is associated with the possibility of over or under-treatment effects, we believe that our approach could highlight in what circumstances an over-performance of tests might actually improve the patient's health outcomes in expectations. In this view, a hospital competition related to hospital reputation or average health outcomes indicators would be consistent with the rise of utilization rates of diagnostic tests. The discussion will use the generic term of hospital to refer to the health-care provider. Our model could, however, be applied to other sectors of health provision such as am-bulatory medicine which is also an important source of tests orders from general or specialist practitioners.

From a policy maker's point of view, the performance of additional diagnostic tests before deciding of the treatment method which should be applied may not be justified for all patients, despite the fact that valuable information is produced. The decision obviously depends on the patient's characteristics, particular symptoms and prevalence of specific co-morbidity, but also on the cost of the tests. We will address this aspect of health-care practice by relying on an information concept which allows to characterize the informativeness of a test in function of the prior on patient illness/health status [START_REF] Brandt | Weak Informativeness: An Information Order with Variable Prior[END_REF]. In the economic literature on information, usually the prior belief is fixed. Therefore, the classical notions of informativeness only compare information structures (see e.g. [START_REF] Blackwell | The comparison of experiments[END_REF], [START_REF] Kim | Efficiency on an information system in an agency model[END_REF], [START_REF] Lehmann | Comparing location experiments[END_REF]).

Consequently, those notions neglect the impact of the prior on informativeness of information structures. Building on another information concept related to posterior beliefs dispersion introduced by Ganuza and Penalva (2010), [START_REF] Brandt | Information and the dispersion of posterior expectations[END_REF] has built a definition of informativeness that can be related to informative value orders of agents in a quasi-linear decision problem. [START_REF] Brandt | Weak Informativeness: An Information Order with Variable Prior[END_REF] generalizes the concept and show if an information structure is more informative for one of two agents, then information is also more valuable for this agent the than others. To our knowledge, this paper provides the first application of this informativeness criterion to discuss contracting and agency issues.

This paper provides a framework to analyze the role of information acquisition in the regulation of health care and the consequences of hospital competition on this relation. By care procedures we mean the process of care from diagnostic testing to the choice of treatment methods. We consider that the information on patients' illness is incomplete and the prior's beliefs about their illness is heterogeneous. For instance, finding a high blood pressure leads to suspect the same degree of illness depending on the other symptoms the patient presents or on the patient's medical history. Based on the patient's characteristics and its experience, the physician will attach a different prevalence on patients severity of illness or co-morbidity. These ex-ante beliefs will first affect the decision to conduct further tests or not and therefore the decision of treatment method to apply to each patient will be updated through Bayesian updating.The incentive problem will also depend on whether the hospital sector is competitive or not and will be detailed in each respective section separately.

We consider two ex-ante patient types: a low and high-type patients such that the illness of the high-type patient is either expected to be higher or more uncertain. We also consider that the cost to take care of the patient depends on their degree of illness, that is unknown before the treatment method is chosen. Due to the stochastic orders between the prior beliefs on patients' illness, one patient type has a higher expected cost of care than the other. Finally, our results are restricted to the case such that the information value of the test is higher for costlier patients i.e. those with the higher probability of being affected by a severe illness.

We will analyze the optimal payment scheme to hospitals when patients are fully insured.

We find that the optimal contracts share features with bundle contracts in the sense that payments are conditioned to the whole episode of care and are not separable in diagnostics test and treatment as soon as only some patients should be tested. The main discussion will take into account the possibility that hospitals undertake risk selection at every stage of the care procedure.

More precisely, we will focus on a case such that hospitals can dump patients or refer them to other providers if they learn that their severity is too high after a test result. The alternative case such that hospitals are committed to take charges of all the care process if they accept a patient will be discussed in a separate section.

First, we analyze the optimal decentralization of tests for a monopoly hospital. Beyond information value for treatment per se, our model exhibits two other possible values of diagnostic testing in the regulation problem: (i) testing has a monitoring value as it allows to reduce the uncertainty on the costs that hospitals will incur depending on the patient type they treat, (ii) testing bears a risk on the hospital when it is not behaving truthfully, and contracts are separating in terms of test procedures, i.e. in our case if the hospital should only test one type of patient. The second effect relies on incompleteness of information at the interim stage, after the observation of a non-fully informative test. In our model it implies that at the optimum, the authority could choose contract separating in terms of test-procedures more often than it is efficient. Hence, testing high patient types might be optimal even though tests are inefficient for both patient types, and not testing low-types might be optimal in second best when it is however efficient. More clearly, the efficient testing procedure is always optimal in second best if and only if efficiency consists in only testing the high-type patients.

We further analyze the effect of hospital competition on the decentralization and optimal design of contracts and care procedures. We find that despite the risk of over or under-treatment, over-testing low patient types can increase hospitals' attractiveness for patients. In that case it can be a solution for hospitals to increase its surplus is to increase its activity. Suppose that after the preliminary consultation the expected illness of the low-type patient is subject to low uncertainty, so that the treatment for this patient can be decided with no further test. The performance of a test keeps a value for the patient in the sense that it could, with high probability, exclude a more severe condition, and if it reveals (less likely) a more severe condition, allows for adapting the treatment to the updated beliefs. However, if the test is intended for high, more symptomatic, patient types only, diagnostic-therapeutic cascades following the test of low types can consist in over or under treatment of these patients. Our model of information value suggests that it can still be valuable for patients to be treated. Establishing clear and tractable sufficient conditions is, however, still a work in progress. Yet, the intuition is simple: given that low/asymptomatic patient-types can actually suffer from a very serious affection, testing would limit the "risk" that this is not discovered, and that treatment is not adapted to the patient's actual illness. Even-though including the possibility of over or under treatment, over-testing can actually increase the average patient's health outcome, and engaging in such praxis can increase hospital attractiveness. We, moreover, discuss how such issue could lead to distortion in treatment praxis of each patient, and cream low-patient type at the optimum.

We are aware of very few works that highlight the use of economic incentive to decentralize care procedures with taking into account the incompleteness of information in health care. In [START_REF] Grassi | Information acquisition, referral, and organization[END_REF], the acquisition of information by exerting effort allows the physician to better infer the unobserved type of patients before deciding whether they should refer the patient to another physician or not. Physicians differ in their cost-effectiveness to treat the different patient types so that referral to other providers can be socially optimal. In our model, however, the hospitals are not differentiated in terms of costs, and the decision to perform tests relies on the possibility to better fit the treatment of the patient's unobserved illness, i.e. information has value in terms of patients' health outcomes. Hence, our problem is more related to [START_REF] Pflum | Physician Incentives and Treatment Choice[END_REF], that analyzes hospital treatment decision after the performance of a fully informative/revealing test under fixed-price regulation. In this respect, we show that the decentralization problem is highly dependent on the test's informativeness. Our generalization also considers non-fully informative tests and can reflect the gray areas of medicine that we discussed above. Regarding the effects of hospital competition, Ellis (1998) describes creaming, skimping and dumping as hospitals behaviors, which provide a meaningful prism for the analysis of our results.

The paper is organized as follows. Section 2 presents the framework, characterizes an efficient care procedure and details the role of prior on information value with an illustration. Section 3 discusses the optimal care procedures with respect to efficiency for a monopoly hospital, while section 4 is dedicated to presenting the effect of competition on optimal procedures. Section 5 presents the result of a relaxed problem when a hospital must commit to take charge of all the care procedures of the patient they accept. Section 6 concludes.

General framework

Setting : general structure

The model consists of a regulator, a hospital sector, and a continuum of patients. In the sequel, we will separately analyze the monopoly and the competitive hospital market cases. Each patient's severity of disease is (ex-ante) unknown and described by a random variable d that has realizations

d ∈ D := [d ¯, d].
The regulation problem is to decentralize the provision care procedure, i.e. the choice of diagnostic methods m, and the choice of treatment method t to cure the patient. Hospitals only can observe the patient's type θ , and the problem will be characterized by the hospitals' discretion over provision. Before treating a patient θ ∈ {l, h}, the hospital can choose to test the patient m = m, or not m = m. If a patient is tested, an observable signal is produced, s ∈ S := [0, 1], which is correlated with the (not directly observable) severity of disease of the patient and, hence, contains information about the patient's actual disease. After the observation of a signal s the physician updates their belief about the disease according to Bayes's rule and chooses a treatment t ∈ [t ¯, t]. 2 In contrast, the regulator can only verify and contract the applied diagnostic method m, the test result/signal s, and the type of treatment provided t. In second best, the regulator specifies a repayment T (t, s, m) for each possible list (t, s, m) ∈ [t ¯, t] × { / 0 ∪ S, {m, m}} but does not observe θ nor the patient's (expected) severity of disease. We will study revealing contract about whom details are provided at the end of this section. Figure 1 bellow summarizes the problem. We adopt a representation of signals and treatment as a continuum for technical reasons.

Moreover, one could interpret the treatment dimension t as the "classifying" treatment practices that already exist in the payments schedules based on the Diagnostic Related Groups 3 . The non-contractible dimensions of patients' care that are not directly linked to the performance of the treatment t belongs to the part of the care driven by the patient's (ex-ante and unknown) severity of disease d which we consider reveals step by step along the treatment process.

m θ (s, d) = f m (s|d)g θ (d) � J m θ (s, d) = d � d ¯s � s ¯jθ (s � , d � )ds � dd � � .
The (marginal) pdf of the test's result, s, of a patient of type θ when the diagnostic test m is applied is:

f m θ (s) = � D θ � f m (s| d) � = d � d f m (s|d)g θ (d)dd
, and the corresponding c.d.f is

F θ (s) = � s 0 f m θ (s � )ds � .
Then, by Bayes' rule, g θ (d|s, m) = j θ (s,d)

f m θ (s)
3. [START_REF] Mcclellan | Hospital Reimbursement Incentives: An Empirical Analysis[END_REF] and [START_REF] Gilman | Hospital response to DRG refinements: the impact of multiple reimbursement incentives on inpatient length of stay[END_REF] have studied and documented the use of verifiable medical acts to define the diagnostic-related groups that determine payments in current hospitals funding systems.

Prior beliefs and testing technology :

Ex-ante, the beliefs of the physician on the illness of the patient vary upon the patient's characteristics that are embodied by the patient's type θ ∈ {l, h}. For the patient type θ , the diseases are distributed according to the p.d.f. g θ (d) (and c.d.f. G θ ). The draw of ex-ante beliefs G θ from the observation of patients characteristics θ reflects the non-objectivable knowledge of physicians, and θ is non-verifiable and non-contractible.

Assumption 1. G h � FOSD G l or G h � MPS G l
We will focus on two interpretations of patient difference in terms of prior: mean preserving spread (G h � MPS G l ) and first order stochastic dominance (G h � FOSD G l ) so that respectively, ex-ante, the severity of the disease of the h-type patient is more uncertain, or higher than for the l-type. With respect to a given pathology, as in [START_REF] Bradley | Patient selection for diagnostic coronary angiography and hospital-level percutaneous coronary intervention appropriateness: insights from the National Cardiovascular Data Registry[END_REF], h-type patient can be interpreted as the symptomatic patient for whom the prevalence of serious diseases is higher, and the type l as asymptomatic in the sense that her expected disease is lower (FOSD), or at least not extreme (MPS). 4 Moreover, under Assumption 3 we restrict our attention to the tests technology such that higher signals are associated with higher probability of high degree of illness d. It notably implies that � Dθ

� d|s

� is increasing in s, i.e. higher signals consist in worse news.

Assumption 2. We restrict our attention to diagnostic tests with Monotone Likelihood Ratio Property (MLRP), i.e. for s

� ≥ s, d � ≥ d we have f (s|d) f (s � |d � ) ≥ f (s|d � ) f (s � |d) for all m ∈ {m, m}.
4. When it is not relative to a given pathology, asymptomatic patients refers to patients with no symptoms at all and has a contrary interpretation of complete uncertainty about the patient pathology. In order to avoid confusion, we won't make an extensive use of the word in the main discussion.

Patients :

The patients' utility from being treated depends on both, the treatment they receive , t ∈ [t ¯, t],

and their, unknown, severity of disease d ∈ [d, d]:

u(t, d) = ϕ 1 (t)h(d) -ϕ 2 (t)
We need to assume that u(t, d) is quasi-linear and monotonic in d, and additionally assume that the patients' surplus from being treated is increasing in treatment intensity and disease:

ϕ � 1 (t) > 0 and h � (d) > 0.
In particular, with this functional form the marginal gain from increasing the treatment intensity is increasing in the severity of the disease: ∂ u(t,d) ∂t∂ d > 0. It amounts to consider that more intense treatments give more benefits, and should be provided to patients with more severe diseases. The part ϕ 1 (t) could be interpreted as the positive effect of the treatment while ϕ 2 (t) could be interpreted as the potentially negative side effects of the treatment, increasing in their intensity if for instance ϕ 2 ≥ 0 and ϕ � 2 > 0.

From the patient benefits' point of view the informative value of a test will therefore rely on the possibility to adapt the treatment method to the signal that will be observed after the test. Typically, the patient would receive a heavier treatment if the test result s indicates a higher severity of illness and the marginal benefit from increasing the treatment intensity is high compared to the negative effect. Similarly the patient would receive and a less intensive treatment otherwise, if the signal indicates a low severity.

In principle, ϕ 2 (t) doesn't need to be strictly positive and increase. For instance, ϕ 2 (t) = 0, however, has the unsatisfying implication of patients' non-satiety in terms of treatment intensity.

Hence, the test would have no value from the strict point of view of the patients: all patients would prefer higher treatment whatever the test results. 5

5. The test could, however, has value from a social welfare point of view since it would take into account treatments provision costs, but considering ϕ 2 (t) > 0 and ϕ � 2 > 0, and that patients may not always prefer more intense treatment remains more appealing in terms of interpretation, and information value.

Hospitals :

The profit of the hospital that receive the transfer T (m,t, d) for performing the diagnostic method m ∈ {m, m} and [t ¯, t] is:

π(m,t, d) = T (m,t, d) -c(t, d) -c m
When the hospital decides to perform a test to the patient it incurs the cost c m > 0, and the cost is c m = 0 otherwise. After updating her beliefs, the physician chooses the treatment t ∈ [t ¯, t] to cure the patient and incurs the cost c(t, d). We assume that costs are additively separable in treatment and illness of disease t and d, i.e. c(t, d) = c 1 (t) + c 2 ( d), and increasing and convex in d. This reflects the simplifying idea that the overall costs to cure a patient can separately be imputed to the type of treatment performed, and to the patient-specific illness that may affect the overall costs by driving the additional performance, non-coding (or non-contractible) acts or efforts. Moreover, the idea that the impact of the patient severity on the treatment costs is convex in severity is a common assumption in the health economics literature (see for instances [START_REF] Siciliani | Selection of treatment under prospective payment systems in the hospital sector[END_REF], [START_REF] Malcomson | Supplier Discretion Over Provision: Theory and an Application to Medical Care[END_REF] and [START_REF] Pflum | Physician Incentives and Treatment Choice[END_REF]). Under assumption 1 it will imply that, ex-ante, the expected cost of curing h-type patients will be larger than for the patient type l.

The separability of both dimensions may, however, seem less satisfying since, in principle, the choice of the treatment method could affect the impact of severity on the overall costs. In this view, information may therefore also have value for hospitals, in the sense that it would increase the likelihood to directly adopt the best treatment strategy if the test is accurate. More precisely, this simplification on costs implies that we will discuss information value from the patient's point of view, as we described earlier, and from a social welfare point of view only. It will affect social welfare both through patients' benefits and by allowing to better adapt treatment intensity to the patients' underlying disease and hence undertaking costly incentive treatments only when the signal suggests it is cost-effective.

Regulation authority :

The regulatory authority decentralizes the care of patients to profit maximizing hospitals, and in return transfers T to hospitals for each patient they treat, at cost (1 + λ )T, where λ accounts for the cost of public funds. The regulator's objective is to maximize the social welfare defined by the sum of the patients' payoffs and hospitals' profits minus the transfer costs:

w(m,t, d) = u(t, d) � �� � patient's utility + π(m,t, d) � �� � hospital's profit -(1 + λ )T (m,t, d)
We also need to assume that the total negative effects of treatments on social welfare ϕ 2 (t) + c 1 (t) are sufficiently increasing in treatments guaranteeing that the treatment optimization problem is well-behaved and has a unique interior solution: arg max

t {w(m,t, d)} ∈ [t ¯, t] for all d ∈ � d ¯, d� .
Now, we should observe that for a given severity of diseases and treatment method, the net social surplus is independent of the patient's characteristics θ i.e. no matter if the exante patients' characteristics indicated that the disease d was more likely to be associated with one patient type, if d is the actual patient's degree of illness for both patient types l and h, the performance of a given treatment t would result in the same utility u(t, d) for the patient, We call a hospital's treatment praxis {t θ (m, s)} s∈S the set of treatment decisions that a hospital follows for each possible result s from the test of patients type θ with a diagnostic method m.

With this, one can respectively define the ex-ante expected utility EU θ , profit Π θ and social welfare W θ when the care procedure of the patient type θ is {(m,t θ (m, s))} s∈S :

EU θ = � S θ � � D θ � u(t θ (m, s), d)|m, s �� (2.1) Π θ = � S θ � T (m, s,t θ (m, s)) -c 1 (t θ (m, s)) -� D θ � c 2 ( d)|m, s �� -c m (2.
2)

W θ = EU θ + Π θ -(1 + λ )� S θ [T (m, s,t)] (2.3)
Determinants of the second-best contracting problem :

Relying on the revelation principle, we will restrict our attention to the choice of a menu of contracts for each possible ex-ante patient type and interim signal � m θ ,t θ (s), T θ (s) � that maximize the welfare subject to the hospitals' participation and incentive constraints for each patient type. The formal regulation problem depends on the structure of the hospital market and are detailed in each section separately.

The following details the determinants each problem will have in common. The contracts are designed ex-ante. For each type of patient, the hospital follows the care procedure that maximizes its profit given the associate repayment. Whether the incentive problem will hold ex-ante or at the interim stage will depend on the type of care procedure that is decentralized.

There are three possible types of care procedure to consider in terms of testing: a separating contract (m l � = m h ) and two pooling contracts m l = m h . The separating contract implements diagnostics for a patient of type h only. The pooling contracts implement the testing of either all patients, or no patient. The incentive problem will also depend on whether the hospital sector is competitive or not and will be detailed in each respective section separately.

Moreover, in the main discussion we assume that hospitals are not committed to treat patients when they engage in testing procedures which involves both an ex-ante and interim participation constraint. In particular, in order to account for the threat of patient selection or patient dumping (hospital triage or refusal of patients) in the problem, we consider a hospital reservation surplus from accepting to cure the patients that we normalize to zero.One might alternatively consider that hospitals are committed to support the whole care procedure and of the patients they start to cure ex-ante. The reality certainly lies in between and we believe that normalizing the hospital's reservation surplus to zero is without loss of generality for the analyzes of second-best distortions.

Finally, assumption 2 bellows ensure that the incentive problem we discuss is not dual when

G h � MPS G l .
In the subsequent we will observe that it is always satisfied if

G h � FOSD G l . Assumption 3. � S l � � D h � c 2 ( d)|m, s �� -� D h [c 2 ( d)] ≤ 0 2.

The efficient care procedures

The efficient care procedure is defined by the solution of the regulator's first best problem.

In the full information, efficiency benchmark, the regulator observes θ , and decides the care procedure to follow, i.e. the diagnostic test to perform m, and the treatment t after observing the signal s, in order to maximize the social welfare, and chooses the transfers T (m,t, s) under the hospital's participation constraint (PC θ ,s ): 6

� D θ � π(m,t, T (m,t, s), d)|m, s � = T (m,t, s) -c 1 (t(s)) -� D θ � c 2 ( d)|m, s � -c m ≥ 0
for all s ∈ S, and θ

(PC θ ,s )
For each patient type θ , the regulator's problem is:

             max m,t(s),T (m,t,s) � S θ � � D θ � u � t(s), d� + π � m,t(s), T (m,t, s), d� -(1 + λ )T (m,t, s))|m, s �� -(1 + λ )c m s.to : (PC θ ,s ) for all s ∈ S (2.4) 6.
Where the notation uses the representation of no test as a fully uninformative that systematically returns the ex-ante prior.

Since the public funds are costly (λ > 0), the regulator will bind the hospital's participation constraint (PC θ , s) at the optimum, and reimburse the hospital with the total expected costs of care for each patient type (θ , s).

With this argument we easily get a reduced form of the problem. Formally, for θ ∈ {l, h} the efficient care procedure solves:

max m∈{m,m},t(s)∈{τ:[0,1]→[t ¯,t]} � S θ � � D θ � u(t(s), d) -(1 + λ )c(t(s), d))|m, s �� -(1 + λ )c m (2.5)
This optimization problem can be solved as follows. First, we determine the set {t * θ (m, s), s ∈ S} of efficient treatment praxis conditional on having performed the diagnostic test m. Then, given the expected social surplus associated with the optimal treatment praxis for each test we determine the efficient testing procedure m * * θ . Since the net social benefit is quasi-linear in the disease, the efficient treatment after observing a signal s from diagnostic test m purely depends on the conditional expected disease

� D θ � d|m, s � . In particular, t * θ (m, s) := arg max t∈[t ¯,t] � D θ � u(t, d) -(1 + λ )c(t, d)|m, s � (2.6)
For each type θ , the efficient test procedure m * * θ is the one that maximizes the ex-ante expected payoff from performing the test m:

m * * θ = arg max m � S θ � � D θ � u(t * θ (m, s), d) -(1 + λ )c(t * θ (m, s), d)|m, s

��

To determine whether it is efficient to perform a test or not, one must verify if the information value of the test exceeds its costs. We should therefore distinguish between informativeness and information value which also depends on the surplus function form and the decision made after the observation of the signal. Basically, an informative test may have zero information value if the same decision is made regardless of the signal received (although such decision is hardly efficient).

Definition 1. For the authority, the information value of performing the diagnostic test m on the patient type θ is the difference between the ex-ante expected payoff when the test is done and that without the test. In particular, in first best, the information value is:

V * θ := � S θ � � D θ � u(t * * θ (m, s), d) -(1 + λ )c(t * * θ (m, s), d)|s m �� -� D θ � u(t * * θ , d) -(1 + λ )c(t * * θ , d) �
More precisely, this is a definition of "gross" information value which makes abstraction of tests' costs, and not of net information value which would account for the social cost of implementing the test. Providing the test m to a patient type θ is therefore efficient if and only

if V * θ > c m .
The next section will detail the relation between informativeness and information and provide an illustration so that the following assumption hold. Assumption 4 below amounts to restrict attention to the configurations such that the test is more valuable for the patient type h, i.e. the more "symptomatic" patients or for whom the prevalence of serious severity of disease is higher.

Assumption 4. V * l < V * h
Under assumption 4 , if performing the additional test is efficient for l, it is efficient for h also.

Proposition 1 below characterizes the efficient care procedure (from tests to treatment method) depending on the value of information from the test and its cost.

Proposition 1. , under Assumptions 4,

If V * θ < (1 + λ )c m , then m E (θ ) = m, and t * * θ = arg max t∈T E θ � u(t, d) -(1 + λ )c(t, d) � . If V * θ ≥ (1+λ )c m , then m E (θ ) = m, and t * * θ (m, s) = arg max t∈T = � D θ � u(t, d) -(1 + λ )c(t, d)|m, s � ∀ s ∈ S.

Informational structure of diagnostic testing

The informativeness of a diagnostic test heavily depends on the prior beliefs. For instance, the diagnostic of injuries can require to perform an X-ray (typically to detect bones infections), and ultrasound (typically to detect tendon infections) and sometimes to perform both or additional, costlier, tomographic methods such as a CT-scan of MRI. The decision to perform either or several of these tests depends on the physician's primary observation of the patients.

Similarly, CT-scan of MRI may be used to confirm the presence of osteoarthritis when it is suspected since it would affect the treatment process, but when exactly to perform the supplementary exam relies on the physician's observations of the patient. One can also think about the example provided in introduction.

We call ( f m , g θ ) the diagnostic system, i.e. a diagnostic system that consists of both a diagnostic test and a prior belief about the disease. With these notations, we can define informativeness. In order to capture the impact of the prior on informativeness. Informativeness is defined as in Brandt et al. ( 2014): a diagnostic system is more informative than another one if and only if the conditional expected disease is more spread out (in the sense of an MPS) when the patient type is a high-type than when it is a low-type.

Definition 2. ( f m , g θ ) is more informative than ( f m, g θ ), denoted by ( f m , g θ ) � in f ( f m, g θ ), if and only if � D θ � γ( d)|m, s � -� D θ � γ( d) � is a MPS of � θ � d| m, s � -� θ � d� , ∀γ : D � → �,γ � > 0
The definition must hold in particular for γ(d) = d. Intuitively, the more informative the test, the higher the sensitivity of conditional expectations � � d|m, s � to the received signal, and the higher the dispersion of conditional expectations. Typically, a fully uninformative test would return the same conditional expectation whatever the signal s received. Hence, the test being more informative for the patient type h means that, overall, the variation's magnitude of the posterior beliefs about the h-type disease is larger than for the patient l, see figure 2.3 for an illustration.

Information value :

Intuitively, the value of information of the test stems from the opportunity to better fit the decision with the actual (unknown) state of nature, in our case the patient's severity of illness.

Hence, for a given patient type θ , a more informative test has a higher informative value whenever the treatment decision consecutive to the signal is optimal. Building on [START_REF] Brandt | Information and the dispersion of posterior expectations[END_REF] we can show that it the test m is not fully uninformative then, according to the intuition, it has a positive value (if

( f m , g θ ) � in f ( f m , g θ ), then V * θ > 0).
Moreover, if the ex-ante expected disease � D θ � d� are the same for both patients so that the main difference between patients is the degree of uncertainty (e.g. if G h � MPS G l ) we can show from [START_REF] Brandt | Weak Informativeness: An Information Order with Variable Prior[END_REF] that if the test is more informative for the patient

type h ( f m , g h ) � in f ( f m , g l ) then it is also more valuable: V * h > V * l .
It is, however, not sure whether such prior order implies the informative order in definition 1. We have taken a step toward characterizing class of prior dispersion and test technologies that are sufficient to imply the weak informativeness order from definition 2, and satisfy the information value orders we assumed under the assumption 4.

It is, however, not the main topic of this work that focus on the regulation problem. The following provides an illustration for the reader to understand the main intuitions, while details are provided in Appendix C.

Illustration :

This section presents a restrictive case such that assumption 3 and 4 are satisfied, and gives the main intuition of the relation between informativeness and information value. The property of the restriction are details in Appendix C and has three components: the priors beliefs are mean preserving spread, their c.d.f cross only once at the mean, and the test technology is linear.

d G h ( d) G l ( d) d E θ   d  = d e Figure 2.3 -conditional expected disease 1 0 E h [ d|m, s ] E l [ d|m, s ] Figure 3
illustrates that posterior expectation on h types patients' severity is more dispersed across the set of possible test results s than these of the l-type. In line with the definition 2, this implies that the informativeness of the test is higher for the patient type h: (where the correction for ex-ante mean is not necessary since they are the same for both patients).

The following give the intuitive argument that explains why assumption 4 is satisfied, and the test has a higher value for the h-type of patient.

First, observe that increasing the intensity of the treatment has a marginal social benefit that is higher if the patient suffers from high severity of disease. This is related to the assumption

∂ u(t,d)
∂t∂ d > 0. Therefore, according to the treatment decision problem in (2.6), in efficiency if the patient is tested, more intense treatment is provided when the posterior expected severity of disease, � D θ � d|m, s � , are high.

As a result, at the efficient treatment praxis, the social welfare gain from care is convex in the patients' expected disease � D θ � d|m, s � , i.e. the higher the severity, the higher the net gain from providing this patient a given treatment and the higher the treatment the patient will receive.

Let denote the posterior expectation by

x θ = � D θ � h( d)|m, s �
, formally the optimality condition of treatment implicitly defines a function of t * θ (x θ ), such that

∂t * θ (x θ ) ∂ x θ > 0 since ϕ 1 (t) � > 0. The crucial point is therefore, ∂ w(t * θ (x θ ),x θ ) ∂ x θ = ϕ 1 (t * θ (x θ ))
, and follow by the envelop theorem 7

7. With definition 2 holding for all increasing transformation, in particular h(d) the definition MPS implies

The convexity reflects that there is more social gain from adapting the treatment to the severity disease when it is expected to be high. Hence, from ex-ante perspective, the h-type of patient for whom the expectations are more dispersed can have large gains for being tested.

3. Decentralization of care to a monopoly hospital

Monopoly regulation problem

A monopoly hospital decides the care procedures to follow independently from the patient's preferences. According to the second-best contracting problem, the hospital accepts to take charge of the care of a patient of type θ , (θ , s), at the ex-ante and interim stage respectively, if and only if the contract satisfies the hospital's participation constraints:

(PC θ ) Π ex (θ ) = � S θ [� D θ [π(m θ , T θ (s),t θ (s), d)|m θ , s]] ≥ 0 θ ∈ {l, h} (3.1) (PC θ ,s ) Π int (θ , s) = � D θ [π(m θ , T θ (s),t θ (s), d)|m θ , s] ≥ 0
for all s ∈ S, and θ ∈ {l, h}

(3.2)
The incentive problem varies upon the type of care procedure that is decentralized. If the contract is separating in terms of test procedures (m l � = m h ), the decision of hospitals to perform tests or not will entirely determine the care procedure the patient falls into, and the treatment method resulting from the test results. In this case, the incentive problem holds ex-ante.

If, however, the testing decision is the same in the care procedure of both types of patient (m l = m h ), it does not determine the treatment procedure to follow and the incentive constraints must hold at the interim stage i.e. after observing the patient type. The two types of incentive constraints that the contract must verify are the following (Appendix A. gives more extensive [START_REF] Brandt | Information and the dispersion of posterior expectations[END_REF] for a complete and general proof of the fixed priors case, and [START_REF] Brandt | Weak Informativeness: An Information Order with Variable Prior[END_REF] for the case with different priors. See appendix C. for the proof or the informative order in a first place.

� S h � w(t * h (x h ), x h ) � > � S l � w(t * l (x l ), x l ) � . See
formulations and presentation of the incentive problem in each possible case):

max θ ∈{l,h} Π θ ( θ ) = � S θ � � D θ � π(m θ , T θ (s),t θ (s), d)|m θ , s �� i f i f m l � = m h max θ (s)∈{l,h} Π θ ( θ , s) = � D θ � π(m θ , T θ (s),t θ (s), d)|m θ , s � ∀ s ∈ S, ∀ i f m l = m h (3.3)
Hence, the regulator chooses the contract that maximizes the social welfare subject to the participation and the incentive constraints that are described above:

max m θ ,t θ (s),T θ (s) ∑ θ ∈{l,h} µ(θ ) � � S θ � � D θ [u(t θ (s), d)|m θ , s] � + � S θ � T θ (s) -� D θ [c(t θ (s), d)|m θ , s] -c m θ � -(1 + λ )� S θ � T θ (s) �� (3.4)
sub ject to : (3.1) to (3.3), and where µ θ denotes the share of patients types θ ,(with µ l = 1µ h ).

(3.5)

We will show that the efficient care procedure described in Proposition 1 can always be decentralized under assumption 1 to 4. Compared to the first best problem, the information asymmetry, however, typically increases the implementation costs of the care procedures, and can lead to adopt different procedures at the second best optimum than these that are efficient.

The solution of the monopoly's second-best problem is found by following a similar path than our first best problem. First, we determine the second-best optimal tariffs T θ (m θ , s) conditional on a given treatment procedure {m θ ,t θ (s)} s∈S for all θ , and m θ ∈ {m, m}. This equivalently gives the hospital rent or second best implementation costs at the optimum for each possible test procedure.

Second, we find the optimal treatment praxis conditional on each possible testing procedure t * θ (m θ , s) for all θ , and m θ ∈ {m, m}. Finally, we can determine the optimal testing procedure m SB θ , associated with the treatment praxis t SB θ (s) = t θ (m SB θ , s), and tariffs T SB θ (s) = T θ (m SB θ , s) for all patients types θ and signals s.

However, to characterizes the hospitals informational rents it is equivalent and more convenient to consider the choice of the contract � m θ ,t θ (s), Π θ (s) � .

Monopoly optimal payment schemes and implementation costs

Before we detail the results and intuitions for the monopoly case, we should observe that the incentive problems only arise through expected cost of severity d. Indeed, apart the patient severity of disease, the cost of care due to treatment c 1 (t) is known by the authority and t is assumed to be observable. Hence if the patients' severity of disease did not impact the costs, the authority could simply pay the hospital back for the treatment cost and eventually diagnostic costs. This would make the hospitals indifferent between all contracts, and the incentive problem would vanish, just as the need to leave some rent to hospitals.

Given the previous observation in absence of effects of patients' demand, the incentive problem will entirely rely on the ex-ante and interim expected costs from patients severity of disease 8. We choose linearity and single crossing case or our restriction for simplicity, but we should observe for the discussion that the figure does not hold in general. 1 0

E h [ c 2 ( d)|m, s ] E l [ c 2 ( d)|m, s ] Π l = � D h � c 2 ( d) � -� D l � c 2 ( d) � , and, Π h = 0 if m l = m h = m Π l = max{� S l � � D h [c 2 ( d)|m, s] � -� D l [c 2 ( d)], 0}
, and,

Π h = 0 if m l = m, m h = m Π θ = � S θ � max{� D -θ [c 2 ( d)|m, s] -� D θ [c 2 ( d)|m, s] , 0} � , for θ = l, h if m l = m h = m
For the following discussion, one can have in mind that the efficient treatment decision conditional on tests will be decentralized (c.f. argument for lemma 1 in the sequel), so that the idea that the treatment will be higher when the expected severity of disease is higher holds.

If m l = m h = m: no test is performed to neither ex-ante patient-type θ , and the treatment method is chosen according to the ex-ante expected illness of both θ -types. In this case, the total cost of curing a patient varies only upon their ex-ante expected severity of illness. Due to the convexity of costs in patients severity of illness d, and the higher prevalence of high severity for the h-type, under the assumption 1, the cost of care of a symptomatic patient is expected to be larger than these of a asymptomatic l-type, i.e.

� D h � c 2 ( d) � > � D l � c 2 ( d) �
. Therefore, the hospital could take advantage of this difference in costs by applying the treatment procedure of high-types, typically a more intense treatment, when the patient is a low-type. This standard observation results in the first item of Proposition 2. Moreover the optimal transfers are such that

T θ = c 1 (t θ ) + � D h � c 2 ( d) �
, for θ ∈ {l, h}. Hence, if the same treatment t θ is applied to both patients in absence of the test (which can arise with a mean preserving spread order of prior beliefs), the optimal contract would consist in a unique payment level to hospitals whatever the type of patients they treat.

If m l = m, m h = m: the contract must be designed such that only the patients of types h should be tested. In that case, the expected disease and associated costs are updated after the observation of the test results s at the interim stage, and transfers must account for the updated expected costs from illness. There will still be no relevant incentive problems preventing hospitals to apply the l-type procedure to h-types since transfers in the l-type contract should not cover the higher expected costs of the severity of disease of h-types. Hence, the incentive problem still exists if and only if hospitals may find advantageous to provide tests to l-types also choosing the contract of h-types when the patient is l. A necessary condition so that it can be

advantageous is that � S l � � D h [c 2 ( d)|m, s] -� D l [c 2 ( d)|m, s] � > 0.
There is indeed not necessarily an incentive problem in this case. Suppose that both patient types get the test result s, one can possibly infer that the low-type patient has actually a larger illness than the high-type, and actually be more costly to treat. If such signals are the most likely result from testing the low patients' types, it might even be that the incentive problem vanishes.

If indeed we verify that

� D h � c 2 ( d)|m, s � -� D l � c 2 ( d)|m, s � > 0
, hospitals can still take advantage from applying the care procedure of high-types to the patients of type l (assuming the participation constraint of high-type holds), and the second item of Proposition 2 gives the minimal rent to leave to hospitals for the care of a low-type, l, in order to discourage it. Assumption 3 gives the necessary and sufficient condition such that this contract is truthful, so that the efficient solution can always be decentralized in second best. The case so that Assumption 3 does not hold is discussed in Appendix A, and implies that the incentive problem is dual. We just observe now that it is systematically satisfied if G h � FOSD G l , under the MLRP assumption, and also verify with equality in our illustration in section 2.3.

Under assumption 3, at the optimum, the transfers verify

T l = c 1 (t l ) + � S l � � D h � c 2 ( d)|m, s �� and, T h (s) = c 1 (t h (s)) + � S h � � D h � c 2 ( d)|m, s �� + c m
Hence, when the h-type patients should be the only one to be tested, the optimal contract should not separate the repayment of test performance and costs of curing the patient. Relying on the transfers on the whole episode of care allows that, apart from test paybacks, if t h (s) = t l , at the optimum the hospital would therefore be repaid less if the treatment does not follow the performance of the test, and more otherwise since

� S l � � D h � c 2 ( d)|m, s �� ≤ � S h � � D h � c 2 ( d)|m, s �� = � D h � c 2 ( d)|
� under Assumption 3. Typically, t h (s) = t l will occur if the two priors are mean preserving spreads and the test only confirms the ex-ante beliefs, i.e.

� D h � d|m, s � = � D h � d| � = � D l � d| � .
Such test results can be particularly when likely if the h-type severity of disease is not too uncertain and the distribution g h (d) is centered around the mean as in our illustration.

Note that the consequences of unintended over-testing if the payments fail to induce trustful care procedures will be discussed in detail in the competition case section 4.2.

If m l = m h = m: all patients are tested, and hospitals can choose whether they follow the procedure of one or another contract at the interim stage, after they update their beliefs. To highlight the lemma on this case, consider the care of low-type patients. The hospitals will have an incentive to apply the treatment method for patient h if, after updating its beliefs, the expected cost of treating h-types

is larger i.e. if � D h � c 2 ( d)|m, s � -� D l � c 2 ( d)|m, s � > 0.
Otherwise for this given signal, it is when the patient type is high that the hospitals might have an advantage if they follow the l-type procedure. Hence, the average rent left to hospitals can be positive for each patient type and its magnitude depends on what signals are received and their likelihood for each patient ex-ante type. At the optimum the transfers verify

T θ (s) = c 1 (t θ (s)) + max θ ∈{l,h} {� D θ � c 2 ( d)|m, s � } + c m
for all θ and s.

A more detailed presentation of the different configurations and the optimal contracts asso-ciated is available in Appendix A.

Monopoly optimal care procedures

We have observed that following Proposition 2, the incentive problem holds on the costs of care that are specific to patients' underlying, unobserved, severity of disease. The consequence is that hospitals' rent cannot be reduced by distortions of the treatment procedures, and yield:

Lemma 1. Under the assumption of observable treatment and separability between treatment choice and the patient's severity of disease which impact the cost of care: c(t, d) = c 1 (t) + c 2 (d), the second-best optimal treatment praxis conditional on testing procedure equals the conditional efficient treatment praxis:

t SB * θ (m, s) = t * θ (m, s)
Hence, only variations in the testing procedure, i.e. the decision to perform tests or not to each patient type, is subject to change the treatment praxis for a given patient type in second best compared to efficiency. Another important implication of lemma 1 is that the gross information values are unchanged w.r.t to the first best benchmark and only the implementation costs of test procedures vary. Our most robust result is that efficient care procedure is always decentralized at the second best optimum if and only if they consist in testing one patient type only. Moreover, we find that testing more patients than in first best can be optimal to reduce implementation costs, in particular if the information value net of the test costs is not too low. We also find that distortions consisting in less testing can be optimal but this result seems to be more sensitive to the information structure, and therefore less general. Proposition 3 characterizes the optimal testing procedure m SB w.r.t the efficiency benchmark, and its consequences for treatments.

Proposition 3. Under assumptions 1 to 4, the optimal second best care procedure is such that:

(i) If V * θ < (1 + λ )c m , θ ∈ Θ,
then it is optimal to decentralize the efficient procedure and m SB θ = m * * θ , and t SB θ (m) = t * * θ (m) for all θ ∈ Θ, if and only if:

µλ � S l � � D h � c 2 ( d) � -� D h � c 2 ( d)|m, s �� ≤ (1 -µ)(V * h -(1 + λ )c m )
Otherwise, it is optimal to perform a test on the high-type, so that for the low-type

(m SB l ,t SB l (m l , s)) = (m * * l ,t * * l (m))
, and for the high-type

m SB h = m � = m * * h , t SB h (s) = t * h (m, s) (as defined in (2.6)). (ii) If V * l < (1 + λ )c m < V * h
, it is always optimal to decentralize the efficient procedure m SB θ = m * * θ , and t SB θ (m, s) = t * * θ (m, s) for all θ ∈ Θ, and s ∈ S.

(iii) If V * θ > (1 + λ )c m , θ ∈ Θ,
then it is optimal to decentralize the efficient procedure iff,

µ(V * l -(1 + λ )c m ) > λ µ� S l � max θ ∈Θ {� D θ � c 2 ( d)|m, s � } -� D h � c 2 ( d)|m, s � � +λ (1 -µ)� S h � max θ ∈Θ {� D θ � c 2 ( d)|m, s � } -� D h � c 2 ( d) � �
Otherwise, it is optimal to give up on testing low-types and implement the efficient praxis for high-types:

� m SB l ,t SB l � = � m,t * l (m � , and � m SB h ,t SB h (s) � = � m * * h ,t * * h (s) � for all s ∈ S.
Proof. See Appendix A.

We find that incompleteness of information at the interim stage may lead the authority to choose separating test procedures more often than it is efficient. This is due to both the fact that information reduces cost heterogeneity compared to the ex-ante problem and that in presence of non-fully informative tests, contracts that are separating in tests can lead the hospital to bear a risk when its praxis is not truthful. What follows decomposes the results of Proposition 3 and details the main intuitions concerning distortions in testing procedures.

(i) First, we find that the second-best optimal care procedure may consist of testing more patients than what is efficient. Intuitively, information reduces the heterogeneity of the expected cost of care between patient types l, and h at the interim stage. To see this, suppose the implementation of a fully informative test for one type of patients, say θ . Thanks to a fully revealing signal, the ex-ante type of patient becomes irrelevant. Indeed, whatever the ex-ante patient type θ ∈ Θ, a given test result s leads to the same expected costs of care, and therefore to the optimal treatment t. A similar intuition, with weaker magnitudes, applies for non-fully informative tests.

Somehow, testing gets a monitoring value in second best in addition to the information value stemming from the possibility to better fit the treatment method to the disease.

We should also observe that, in our illustration, the restrictions imply that the condition so that testing more is optimal is never met. It is due to the combination of MPS priors and the linearity of the test technology. Indeed, together they imply that the marginal distributions of signals are the same for both patient types. As a result, the difference in conditional expected costs between l and h-type may vary from the ex-ante difference, but is, on average, the same.

This observation is more linked to the risk introduced in the hospital decisions when the testing procedure is separating (which is what we discuss below) than to the strict monitoring effect of the test. This risk component will also explain why the monitoring effect from the test never leads to test both l and h-type patients at the optimum.

(ii) We formerly discuss the monitoring value of testing for the regulation problem. Our second result reflects that, however, the costs to implement a care procedure is the lowest when only some patients are tested and, in our case, the symptomatic patient type h. The previous observation on Proposition 2, gives most of the intuitions for these results. When all patients are tested, the care procedure is not separating ex-ante but only at the interim stage when treatment methods are chosen, for all possible patient types (θ , s). We argued that the incentive problem is relaxed when it holds ex-ante since it bears on hospitals the risk that the signal's realization reveals a higher expected disease and therefore a higher expected cost of care for an asymptomatic patient l than for the ex-ante symptomatic patient h.

The consequence of this observation is that it is never optimal from the authority's point of view to distort the care procedure in second best if it consists in separating test praxis in first best. Moreover, efficiency can be decentralized even in terms of implementation costs in

the case that � Sθ l � � D h [c 2 ( d)|m, s] -� D l [c 2 ( d)|m, s] � ≤ 0.
As we explained in the discussion of proposition 2, this might occur if signals revealing higher disease to l-type relatively to h-types are particularly likely.

(iii) The last result of proposition 2 is related to the discussion of the second item. Since contracts that are separating in tests can lower the implementation cost, we find that the secondbest optimal procedure may consist in testing fewer patients than it is efficient. The necessary condition is that the reduction in implementation costs balances the loss in net information value from testing. For instance, if the illness is of low-types, it is quite certain and the information value for this patient is close to the cost of the test.

We should emphasize that increase in testing at the optimum described in (i), and the current effect we discuss of decreasing testing is highly related to the informativeness of the test. The result in (i) would never occur with fully uninformative test, while the effect in (iii) on the contrary never occurs if the test is fully informative. However the possibility of decreasing testing seem the less robust to changes in the patient priors.

Consider for instance our illustration, and in particular figure 3. While crossing between the

normalized variation � D θ � c( d)|m, s � -� D θ � c( d)
� is implied by definition of the informativeness order, as figure 3 suggests is it might not be the case for the non-normalized posterior cost expectations. Indeed, the larger the dispersion (in our particular graph), the lower the signal where the posteriors cross, and possibly does not cross any more. In such case, the incentive problem will always hold on the low-type, and given the rents in proposition 2, the implementation cost between testing only h and testing both patients would be the same. As a result, the condition so that decreasing testing might be optimal in (iii) could never be verified.

Decentralization of care to competitive hospitals

The competitive regulation problem

To present the effect of competition on optimal care procedures, we will restrict our attention, under Assumption 5 to the particular configuration such that it is efficient that the test is performed for only one patient. In a first approach, it prevents to duplicate the configuration to the analysis when evaluating the impact of hospital competition.

Assumption 5. V l < (1 + λ )c m < V h
We consider now a hospital sector composed of two competitive hospitals H = 1, 2.

Let assume for simplicity a reduced demand form such that the hospital H's respective market of θ -type patients is a function of these patients' average utility in each hospital:

D H θ (EU H θ + , EU -H θ -
).Our view here is that, if the hospitals' activity is not too low, the average utility of θ -types patients in hospital H must tend toward the ex-ante expected utility from the care procedure that this hospital H applies to these patients:

EU H θ = � S θ � � D θ � u(t(s), d)|m, s �� .
Furthermore, we assume that the demand must verify:

Σ H=1,2 D H θ = 1, and D H θ (EU H , EU -H θ ) = 1 2 if EU H θ = EU -H θ
One can interpret this simple demand as representing the reputation effect of hospitals' praxis on their demand (or the one from whom they consult for the choice of their health-care provider, typically their general practitioners). We consider non-differentiated hospitals, i.e.

they are equally attractive if they follow the same treatment praxis.

In terms of the observable, the regulation problem remains unchanged and is as described in section 2.

( θ H , θ -H ) = D H θ ( θ H , θ -H )π θ ( θ H
) denotes the hospital H surplus associated with applying the care procedure θ H to patient types θ given that its competitor applies the procedure θ -H to these types (where

π θ ( θ H ) = � S θ � � D θ � π θ (m( θ H )),t( θ H , s), T ( θ H , s)|s, m
�� is defined as previously).

For hospitals to accept to take care of both patient types θ ∈ {l, h}, and each patient type (θ , s), the contract must verify the ex-ante and interim participation's constraints, respectively:

(PC H θ ) D H θ � θ 1 , θ 2 � π H θ (θ ) ≥ 0 θ ∈ Θ H = 1, 2 (4.1) (PC H θ ,s ) π H θ (θ , s) = T θ (s) -� D θ [c 2 ( d)|m θ , s] ≥ 0 for all s ∈ S θ ∈ Θ (4.2)
Moreover, the hospitals compete on the basis of the care procedures they apply to each patient type. A care procedure strategy

� ( θ 1 * l , θ 1 * h ), ( θ 2 * l , θ 2 * h ) � with ( θ H l , θ H h ) ∈ {l, h} 2
, is an equilibrium if and only if:

for all θ ∈ Θ, Π 1 θ ( θ 1 * , θ 2 ) ≥ Π H θ ( θ 1 , θ 2 ) θ 1 ∈ Θ Π 2 θ ( θ 2 * , θ 1 ) ≥ Π H θ ( θ 2 , θ 1 ) θ 2 ∈ Θ (4.3)
We only consider a pure Nash equilibrium and in case of multiple equilibria, we assume that the hospital equilibrium care procedure is a trustful contract if it is an undominated equilibrium.

The regulator's problem is to choose the menu of contracts, { � m θ ,t θ (s), T θ (s) � } s∈S , that maximizes the social welfare under hospitals' equilibrium constraint and the, ex-ante and interim hospitals' participation constraints.

Max

m θ ,t θ (s),T θ (s) ∑ H=1,2 � ∑ θ ∈Θ � µ(θ )EU H θ (θ ) + D H θ � θ H , θ -H � π H θ (θ ) -(1 + λ )� S θ [T θ (s)]

Competitive optimal payments

We show that for any contract that verifies the participation constraint for all patients and signals, hospitals' equilibrium care procedure is symmetric for each patient-type.

Lemma 2. For all contracts separating in tests, such that EU H (l) � = EU H h (h) and compatible with the participation constraints, the hospitals' equilibrium choice of care procedures is symmetric:

θ H * = θ -H *
The intuition for the lemma is straightforward since the hospitals are not differentiated in costs, and the proof is detailed in appendix B. One can therefore restrict the attention to the conditions such that the contract is revealing for one of the two hospitals. The per-patient sur-plus that the hospital can gain by choosing non-revealing strategies is unchanged compared to monopoly. However, competing hospitals account for the effect of their choice of care procedures on the patients' surplus since it affects their attractiveness and hence their final demand.

When hospitals compete, the impact of applying care procedure of h-types to l-patient types and testing them can improve their health in average and is determinant for designing the contract. When engaging in such praxis, the hospitals can expect to make a surplus for similar reasons than a monopoly: low patient types are expected to be less costly than the transfers associated to this procedure. If, in addition, this praxis consists in improving the patients' surplus, then it can also increase their attractiveness and demand: D l (h, l) > 1 2 . Hence, competitive hospitals would have further reasons to over-test than a monopoly. If this view, competition might deter the social welfare by strengthening the revelation problem and increase the rent to leave for l-types treatment Π l compared to the monopoly case. Lemma 3. Under Assumptions 1 to 4, the optimal contract such that m h = m and m l = m must verify:

Π h = 0 Π l = D H l (EU H l ( θ = h), EU -H l ) max{E Sl � � D h [c 2 ( d)|m, s] -� D l [c 2 ( d)] � , 0} . Max m θ ,t θ (s) ∑ θ ∈Θ � µ(θ )EU H (θ ) -(1 + λ )E θ [c(t θ (s), d)] � -2λ µD l (h, l)π l ( θ = h) (4.5)
What follows discusses why one can expect over-testing concerns to increase or not under competition with regard to our approach of information value.

We have emphasized in our illustration that, contrary to the informativeness of the test, the information value depends on the form of the utility function and on the treatment praxis that are made. Suppose the contract is designed to implement separating test procedures as it should be the case under the assumption 5. In that case, the diagnostic-therapeutic cascade following an unintended test of a l-type patients would correspond to these of h-types for whom the treatment praxis is designed for. Following such praxis can result in over or under-treatment to the extent that the prevalence of high (and low) severities differs among the patients. As a result, the conditional exceptions should vary, and for instance if, given a signal, one should expect a less severe affection of the low-type patient (if

� D l � d|m, s � < � D h � d|m, s � ), then a treatment t h (s)
being chosen based on the h-type patients' prior might be excessively intense for a low-type.

Hence, it is not obvious that the performance of the test, though informative, can improve the hospital's average health outcome of l-types patients.

However, it remains possible that such excessive adaptation of patients l treatment is still better than no adaptation with no test. This is particularly the case when the patient's (unobserved) true illness d if effectively high (actually possibly, though unlikely, equal to

� D h � d|m, s � ).
In the spirit of our presentation of informative value in our illustration, a sufficient condition is that the patients' utility u(t For the moment, we are restricted to conjectures to determine whether competition increases the welfare or not.

(� D h [h(d)|m, s]), � D l [h(d)|m, s]) keeps being convex in � D l [h(d)|m, s], (with t(� D h [h(d)|m, s]) = t h (s)).
To finish this discussion we should clarify that our claim if not that is the hospital is not truthful and over-test it would therefore over/under treat the patients on purpose. An interpretation of these aspects of our results can be to consider that evidence-based guidelines which physicians follow after testing low types are these of high types simply because no other guidelines are available, or because of the limited knowledge about good praxis for low types. In particular, these concerns have been pointed out with insistence in the discussion of evidence-based guidelines. For instance, [START_REF] Garber | Evidence-Based Guidelines As a Foundation For Performance Incentives[END_REF] interrogates whether tests for colon cancer in high-risk patients apply to low-risk groups, in a discussion about the use of guidelines to found incentive devices. In a review of the guidelines that the National Institute for Health and Care Excellence (NICE) provides to support care decisions in the U.K. NHS, [START_REF] Steel | A review of clinical practice guidelines found that they were often based on evidence of uncertain relevance to primary care patients[END_REF] suggest about the guidelines that "nearly two thirds were of uncertain relevance to patients in primary care" and recommend clearer identification of which guidelines were relevant for primary care patients to avoid potential over-treatment. In this view, primary care patients could correspond to l-type patients in our problem. Although we do not directly aim at discussing the transfers conditional on guidelines that raise many issues that we do not take into account, they play an increasing role in treatment decision, and give a clear interpretation of the former discussion.

Competitive optimal treatment praxis

The former discussion emphasized whether hospitals may be more attractive by expanding the performance of tests to low patients type depends a lot on the treatment procedures praxis that is recommended for high types. Hence, contrary to a monopoly and lemma 2, the implementation costs of care procedures to competitive hospitals also depend on the treatment procedures. We find that competition leads to cream patients that are not tested in terms of treatments in order to discourage the hospital to instead cream the l-type in terms of the test by engaging into an over-provision.

The result of proposition 4 becomes difficult to interpret in a general sense for high patient types since it is highly dependent on the configuration one considers: in terms of cost and utility function and in terms of the priors beliefs dispersions. The main intuition is that when hospitals compete, the care procedures can be used to discourage them from engaging into another strategy than revealing because their reputation has become determinant. Since the regulation problem lies in preventing the hospital to cream low-type patients by unintended provision of tests, improving the standards of treatments for low-types would decrease the net gain from being tested. Similarly, making the treatment procedure of high patient types more extreme makes it less attractive for low patients types Proposition 4. Suppose that m h = m � = m l = m is optimal, under assumptions 1 to 5, the optimal second best contract is therefore such that:

(i) t SB l > t * l (ii) and t SB h (s) =                        t * h (m, s) + d t if h(E l [ d|s, m]) < h(E h [d|s, m]) - c � 1 (t h (s) * ) ϕ � 1 (t h (s) * ) t * h (m, s) if h(E l [ d|s, m]) = h(E h [d|s, m]) - c � 1 (t h (s) * ) ϕ � 1 (t h (s) * ) t * h (m, s) -d t otherwise
with d t > 0, and t * h (m, s) defined in (2.6)

For the most straightforward interpretation one can consider a case such that c � 1 (t) ≈ 0. 9 The heterogeneity of the costs of care is, therefore, almost completely driven by the illness's severity of the patients. Then, the efficient treatment praxis maximizes the patient utility (e.i. there is no rationing in terms of treatment methods due to the cost of their intensity). In that case the result can be directly interpreted with respect to the posterior expectations of patients' severity of disease.

Suppose for instance that the choice of the treatment method has a low impact on the final cost of care, i.e. c � 1 (t) ≈ 0. In that case, the proposition suggests that competition should lead to intensify the treatment praxis for every signal that indicates a more serious illness to h-types (i.e. If however, a more intensive treatment leads to a significant increase in the treatment costs, c � 1 (t) >> 0, the efficient treatment procedure may actually consist in rationing with respect to treatments that would maximize the patient's utility. In this case, an increase in the treatments of h-types would consist in creaming h-types, and is optimal if it is, furthermore, too intensive for an l-type. In our illustration this can happen in particular for high signals, so for high (h, s)types. Otherwise, aside information value, over-testing can be only motivated by justifying 9. Such supposition, however, strengthens the condition on the negative impact of treatment ϕ 2 (t) to ensure that the treatment decision problem is well-behaved and admits an interior unique solution. more costly treatment for the l-type if the efficient praxis rations this patient. In that case,

such that � D l � d|m, s � < � D h � d|m, s � ),
∂ u(t h (s) ast ,E l [ d|s,m]) ∂t
> 0, and both patient-types are actually rationed if receiving t h (s) ast when the signal is s. When treatment costs justify such strong rationing, Proposition 4 suggests that the treatment praxis is distorted downward, at the expense of the h-type patients' health outcomes.

Figure 2.5 -Conditional expected disease-U-shaped g h (d)

1 0 E h [ d|m, s ] E l [ d|m, s ]
The figure 2.5 gives another illustration, falling into our restricted case section 2.3, so that the prior p.d.f of the h-type is U-shaped while this of the l-type is as before. This can be interpreted as a h-type such that the lack of symptoms gives to think that the patient either fall into a rare kind of affection of the pathology of concerns or might actually just not be affected and not need treatment for this disease (MPS is convenient but, non-MPS, dispersion should better suit this idea). In this interpretation h is the asymptomatic patient, and the test has value for being able to exclude the rare condition, so low value for the l-type for whom it is somehow excluded. The l-type in this case is the symptomatic patient with low Suppose again for simplicity that c � 1 (t) ≈ 0, following proposition 4, the distortion would consist in increasing the treatment's adaptation to expected disease in the first ellipse and decrease it in the second, w.r.t. to the efficient adaptation, i.e. the distortion suggests flattening the treatments for intermediate signals where the signal does not learn much for the h-type since, w.r.t. to ex-ante believes it does not allow to settle the case, but does for the l-type.

We interpret such distortion as going toward three types of treatment: no treatments for low signals, an intermediate fixed treatment for intermediate signals, and a higher for high signals.

The direction of the distortions is actually towards killing the value of information of interim signals where is has the most significant impact for l-types, but the marginal gains from efforts from learning how to perfectly adapt the treatment are low.

Discussion of hospitals' competition effect on optimal testing : We cannot assure that assumption 5 is a sufficient condition so that separating test praxis is optimal by the same path as in the monopoly case. We are sure that there is a condition such that it is true, but discussing the optimal testing becomes more complicated, and requires a better understanding of the link between prior, testing technology, and informativeness orders, which is still work in progress. If assumption 5 is a sufficient condition so that contract separating in test is optimal, the effect of competition on social welfare would depend on whether the unintended over-testing would lead to increase the average l-type patient's outcome or not. If it does increase the patient's outcome in average as we suggest is likely, implementations cost would rise compared to the monopoly case due to a demand effect. Suppose under assumption 5, that a contract is designed so that all patients should be tested, then the unit rent would increase as suggested in proposition 3 (iii) for similar reasons. However, suppose, furthermore, that the treatment praxis is efficient given the test (hence omitting possible treatment distortions in a first place). If hospital care procedures are not trustful for l-types, their expected surplus should decrease (strictly if c � (t) = 0). In these circumstances, designing a contract such that all patients are tested might discourage hospital to non-trustful choices of care procedures and the total implementation costs might be lower than with separating tests due to the volume of patients' demand effects. We would hence interpret such results as a tension towards establishing guidelines for the test of each category of patients and to test them all at the optimum.

All those considerations depend on the patients' prior and the information technology considered that, both, affect the propensity that unintended over-testing increases or not the l-type surplus. Better characterization of this property is necessary to determine the cost of the separating contract and comparing it with these we just discuss. Those issues are left to further research.

Hospitals' commitment to cure patients

In this section we will assume that the hospitals cannot defer the treatment of a patient, or dump patients at the interim stage after learning that the patient's illness, is for instance very severe and costly. In practice this type of referral can arise if the hospital lacks of specialization. We will not yet address heterogeneity in hospital here, and only consider the problem for, somehow, fully specialized hospitals for which such referral praxis would not be understood and therefore not authorized.

We show that, if it is efficient to test a patient type, it is a sufficient condition to decentralize efficiency in terms of care procedures and social costs of health care. Otherwise, the secondbest problem may still lead to test more patients than it is efficient. Somehow, this section underlines the role of coverage's prerogative in the decentralization of health care. However, in our opinion, this not realistic in general since the mechanisms at stakes may require to bear a high risk on hospitals, so that an interim participation constraint may arise to ensure that the contract is acceptable. The regulation problem corresponds to (3.4), but with making abstraction of participation constraints in (3.2).

Proposition 5. Under assumptions 1 to 5 if V l < (1 +λ )c m < V h , then there exist a contract such that (PC θ ) are binding and (IC θ ) holds, so that the efficient praxis is implemented: m SB θ = m * * θ , t θ (s) * * = t θ (s) SB , for all θ ∈ Θ and s ∈ S, and NSB * * = NSB SB , where the superscript SB denotes the second best optimum.

Otherwise, if v l < V h < (1 + λ )c m , it might be optimal to increase testing compared to efficiency:

m SB h = m � = m * * h = m iff: µλ � D h � c 2 ( d) � -� D l � c 2 ( d) � � �� �
expected di f f erence in socialcosts f or l,i.e. decrease in socialcosts f or l

> (1 -µ)|V h -(1 + λ )c m | � �� � expected social loss through testing h
The proof is intuitive when one supposes that the test verifies MLRP and G h � FOSD G l .

Suppose the regulator sets two transfers, one for each possible type of patients θ , so that in average the hospital makes no loss for the taking care of each type. In that case, applying the care procedure of a high-type to a low-type patient allows the hospital to make a surplus.

The hospital would be repaid for the care of a patient type that, on average, is costlier due to the higher uncertainty about his/her health status (for instance if comorbidities are suspected), while incurring the same costs from treatment and diagnostic test. However, under the assumption 2, the authority can rely on the simple idea that high signals are less likely for low-type patients than high-types10 . Suppose that the authority segments the signal set around the cut-off s � such that F l (s � ) < F l (s � ), and sets two fixed transfers for the care of high-type patients conditional on the test result so that: if s < s � the tariff is below the conditional expected cost, and above if s > s � . This standard mechanism implies that if the patient is a low-type, the probability to incur the loss is higher. Hence, a simple manipulation of the levels of expected losses and surplus associated with each segment allows the health authority to both: ensure the hospital makes no loss when treating high-types, and discourage hospitals to follow this procedure when the patient's type is low.

Actually, neither assumption 1 nor 4 need to hold here since it is sufficient that the c.d.f. of signal differs across types for at least one signal realization s � . The proof is clear as long as

V l < (1 + λ )c m .
If, however, is if efficient to test all patients type θ the argument may be slightly different and the authority may have to segment the diagnostic group of high-types in as many parts as there are signals such that

� D l � c 2 ( d)|m, s � = � D h � c 2 ( d)|m, s � .

Concluding remarks

In this paper we have proposed a framework to analyzes agency issues when the decision of information acquisition does not only depends on the information structure, in our case the testing technology, but can also vary upon the ex-ante priors, that together define the information system. Relying on this novel information concept allows us to highlight the impact of priors about patients' severity of disease in diagnostic testing problems. Furthermore the relation between informativeness and dispersion of posteriors' expectation that this emphasized by this concept, highlights how the efficient treatment praxis can vary with the patient's characteristics and allows to discuss it formally. The notion of dispersions of posteriors also gives a clearer idea of the value of testing and separating testing procedures in a second-best problem in terms of implementation costs and the risk induced by unintended information acquisition for the hospital agent. Our model also reflects the concerns of treatment's appropriateness that have emerged in the medical literature since the development of evidence-based guidelines. In particular, it integers the prediction that if the treatment decision makes use of evidence-based guidelines designed on asymptomatic patients (h), and the payment scheme fails to discourage over-testing, then further inefficient use of resources in terms of over/under-treatments are to be expected. The continuation of our research will first focus on characterizing the conditions so that the arise of over-testing should be expected to improve the patient health outcome in order to give better interpretation of when the over-testing concerns should be particularly expected and relevant in praxis if hospitals compete for patients. A first, but not very satisfying, approximation would concern the case so that patients are close in terms of priors so that the treats of over/under treating is low. However if one considered a continuum of patients ex-ante types, this suggests that the problem would arise for sure at the patient cut-off type under which test should not be provided. The state of our research remains preliminary but we believe it can be a fecund approach to highlights many aspects of health care that are related to testing. For instance, one could consider the case of a physician differentiated in terms of training, such that one can have a better clue than another about the patient's severity of disease, and discuss the problem of decentralizing referral among physicians. This view can somehow be reflected by the current model if one considers that the patient is unique and the difference in priors reflects the difference in physician specialization. The interpretation of over and under treatment that may arise if the specialized physician over-test would, however, be more puzzling. Another work could therefore consider specifying a referral system in the spirit of [START_REF] Grassi | Information acquisition, referral, and organization[END_REF], and analyses how the specialized physician gains from being refereed patients by the non-specialized hospital or general practitioner, and thereby increases its activity, can affect the problem and its predictions. We think that being refereed more patients could relax the problem so that the specialized hospital might spontaneously reveals itself while the problem might lie into given proper incentive to the non-specialize hospital to refer enough patients. Besides, we believe that following a similar approach could allow to highlight the implementation issues of screening programs, in particular non-resort, since the approach integer the idea that the valuation of information can be deterred by a confusion of patient about their relevant distribution.

Concerning the problem we study in this paper, we also think about many extensions that could contribute to further improve the understanding of test performance decentralization. For instance, we haven't address the implementation of cost-volume or block contracts to control the number of tests provided. Due to incompleteness of information it is difficult to assess the conditions implying that volume control could be effective to decentralize test provision in our problem. Indeed, beyond forecasting the appropriate number of tests to provide, implementing volume-based contracts requires to discuss the conditions so that the hospitals and the authority are aligned in terms of whom patients to provide with the restricted number of tests. It is not clear whether volume control could be implemented for tests as long as the authority cannot infer the ex-ante patient types to whom the test has been provided to, based on the signal received.

Moreover, in our problem, the patient that should be provided with tests is also the one that is expected to be the costliest to cure, so that fully for-profit hospitals might not provide this patient with tests in priority.

One could also want to relax the separability of the two sources of costs that are treatments and severity of illness. In that sense information would have value for diminishing the impact of severity on the costs of care. It, however, becomes difficult to discuss with keeping patient-side information value since the problem would no more be quasi-linear in severity disease. We have focus in a first place on health benefits-side of information value because it appeared to us more fecund. As far as we have explored cost-side information value, it would mostly imply that the authority can pay less for the diagnostic test than it costs. More precisely, it could pay for its cost discounted of its information value in terms of expected cost of care reductions as soon as the contract would make the hospitals internalize the value of cost reduction(which is straightly achieved by the prospective nature of the contracting problem we analyze).

Last, we have also thought about accounting for physician altruism in the model. In this view the over-testing incentives motivated by patient benefits could arise for other reasons than competition. This will be an interesting direction to gather the different aspect of the discussion that would therefore all occurs together in the monopoly problem. We would, however, be confronted with the same issues in terms of determining whether unintended over-testing can be relevant for the patient, but this will be explored in later research. Moreover, a moral-hazard problem in terms of unobserved treatment intensity could be analyzed in the monopoly problem.

We, however, think it would be puzzling with the view that hospitals does whatever it takes to adapt to the underlying severity when it reveals during the treatment process (i.e. does support the cost sources by the severity), that is in our sense appealing. We have therefore focus on the view that physicians can engage in creaming in terms of patients' treatment method/technology, by relying on the information asymmetry, but does adapt to their actual severity that reveals step by step in the treatment process. We think that, in this view, accounting for partial altruism, and possibly duality of physician motives, could make the story behind the model smoother by limiting the possibly disturbing implications of purely for-profit motivations in terms of healthcare interpretations.

Appendix A : Monopoly case Proposition 2

We will successively consider the decentralization of procedures such that both ex-ante patient types are tested, only one type, and no type. The argument being quite standard we only give details when its application may seem ambiguous.

Pooling testing procedure:

m l = m h If m l = m h = m
When the test is performed to both patient's type, the authority can observe the test results

s, but does not know whether the patient type is (h, s) or (l, s) so that hospital can choose both contract (m,t θ (s), T θ (s)) θ ∈ Θ.

For each possible signal s ∈ S, the incentive constraints specific to this case are:

(IC h,s ) T h (m, s) -c 1 (t h (m, s) -� D h � c 2 ( d)|m, s � ≥ T l (m, s) -c 1 (t l (m, s)) -� D h � c 2 ( d)|m, s � and (IC l,s ) T l (m, s) -c 1 (t l (m, s) -� D l � c 2 ( d)|m, s � ≥ T h (m, s) -c 1 (t h (m, s)) -� D l � c 2 ( d)|m, s �
The incentive problem depends on whether, given the signal s, the expected costs from patients severities d, is higher for the patient-type h,

� D h � c 2 ( d)|m, s � > � D l � c 2 ( d)|m, s � or for the patient-type l: � D h � c 2 ( d)|m, s � < � D l � c 2 ( d)|m, s �
. Indeed, if the transfers are adapted to the difference in treatment costs c 1 (t), then following another procedure than these designed for the test can only be advantageous for hospitals if they expect lower costs due to lower patient severity of illness d. Similarly, observe that if

� D h � c 2 ( d)|m, s � = � D l � c 2 ( d)|m, s � the
ex-ante patient type is not relevant to determine neither the treatment nor the transfer there is no incentive problem.

Finally observe that for each ex-ante type θ , if the interim participation constraints hold (PC θ ,s ) for all signals s, then the ex-ante hospitals' participation constraints hold (PC θ ), so that one can ignore they are irrelevant.

Suppose, the signal is such that the hospital expect large costs from severity if it is the result of a high patient type test, i.e.

� D h � c 2 ( d)|m, s � > � D l � c 2 ( d)|m, s � . A standard argument concludes that if (PC h,s
) and (IC l,s ) holds then (PC l,s ), holds. One can therefore show that (PC h,s ) must be bind at the optimum since hospital's rent is socially costly (λ > 0), so that:

T h (s) = c 1 (t h (s)) + � D h � c 2 ( d)|m, s � + c m .
Otherwise, one can modify the transfers with T � h (s) = T h (s) -dT , dT such that (PC h,s ) still hold, and a T � l (s) = T l (s) -dT such that (IC h,s ) still hold and (IC h,s ) is not impacted, while the ex-ante change of social welfare is:

λ [µ dT f l (s) + (1 -µ) dT f h (s)] > 0.
A similar argument concludes that (IC l,s ) should also be binding at the optimum, and:

T l (s) = c 1 (t l (s)) + � D h � c 2 ( d)|m, s � + c m .
The symmetric reasoning applies in the alternative, i.e.

if � D h � c 2 ( d)|m, s � < � D l � c 2 ( d)|m, s �
, so that at the optimum:

T θ (s) = c 1 (t θ (s)) + max θ ∈Θ {� D θ � c 2 ( d)|m, s � } + c m .
The last item of Proposition 2, gives the implication of these remarks on the hospital ex-ante surplus in function of the patient ex-ante type.

If m l = m h = m
In this case the problem is simple, there is no interim participation constraint and our labeling of the respective patient types implies that the incentive problem lie in the possibility of hospital to make surplus by not being trustful in the report of low patient type l. Since ex-ante low-patient types are expected to be less costly to care:

� D h � c 2 ( d) � > � D l � c 2 ( d)
� will imply:

T θ = c 1 (t θ ) + � D h � c 2 ( d) � , θ ∈ {l, h} .
Finally observes that he consequence of this configuration as the same interpretation as in Chapter 1: if the hospital is willing to provide a treatment to patients with high severity, then it is also willing to do so for lower severities, and a rent must be left if the authority aim at preventing the hospital to provide intense treatment to all patients types.

Separating test procedures:

m l � = m h Case 1 m l = m, m h = m
In this case, if the test is performed and the result is s the authority still does not know whether the patient type is (h, s), or (l, s), but only one set of procedures is consistent with testing {(m h ,t h (s)} s∈S . Hence, the fact that hospital does not know what the test result is going to be is critical.

The incentive constraints specific to this case are:

(IC h,s ) � S h � T h (m, s) -c 1 (t h (m, s)) -� D h � c 2 ( d)|m, s �� ≥ T l (m) -c 1 (t l (m)) -� D h � c 2 ( d) � and (IC l,s ) T l (m) -c 1 (t l (m) -� D l � c 2 ( d) � ≥ � S l � T h (m, s) -c 1 (t h (m, s)) -� D l � c 2 ( d)|m, s ��
It is straightforward that, due to larger expected cost from the severity, if the contract bind

(PC l,s ), i.e. if T l (m) = c 1 (t l (m) -� D l � c 2 ( d)
� , then the hospital would make a loss if caring high patient types as low patient types. Hence, if the incentive problem exists it holds on preventing that hospitals want to test and treat low-patient has high-types.

The hospital may have incentive to perform test to low patient types i.f.f :

Π l ( θ = h) = � S l � T h (s) -c 1 (t h (s)) -E l � c 2 ( d)|m, s �� > T l -c 1 (l) -E l � c 2 ( d) � = Π l
We can show that there is an incentive problem i.f.f.

� S l � � D h � c 2 ( d)|m, s � -� 1 D l � c 2 ( d)|m, s �� = � S l � � D h � c 2 ( d)|m, s �� -� D l � c 2 ( d) � > 0 (6.1)
Otherwise, the authority can also bind the participation constraint for all high types (h, s), without breaking the incentive constraint for low types. This case can be interpreted as the hospital expecting to receive mostly signals that indicates higher illness for a low-type than a high type at the interim stage. Hence testing low-types and receiving transfer designed for hightypes would involve loss-making in expectation. Hence with the former argument there is no incentive problem in the separating case.

If 6.1 holds, we can show that if (PC h,s ) holds for all s and (IC l ) holds implies that (PC l ) strictly hold.

However, we cannot directly argue that the participation constraint for high types will binds.

Indeed, supposing that (PC h,s ) and (IC l ) binds, one can verify that (IC h ) would now be violated

if assumption 3 is not verified, if � S l � � D h � c 2 ( d)|m, s �� -� D h [c 2 ( d)] ≤ 0 does not hold.
If assumption 3 is indeed verified the replication of the former arguments lead to conclude that both (PC h,s ), for all s, and (IC l ) bind at the optimum iff

� S l � � D h � c 2 ( d)|m, s �� -� D h [c(d)] < 0.
Then one can conclude that transfers verify:

T l = c 1 (t l ) + � S l � � D h � c 2 ( d)|m, s �� and, T h (s) = c 1 (t h (s)) + � S h � � D h � c 2 ( d)|m, s �� + c m
On assumption 3: If it does not hold then the incentive problem becomes dual, and truthful contract can bind the ex-ante participation constraint for neither patient type if assumption 3. This problem will be studied in more details in further works. More precisely, we suspect that transfers such that

T h (s) = c 1 (t h (s)) + � D h � c 2 ( d)|m, s � + R(s), and T l = c 1 (t l ) + � S l � � D h � c 2 ( d) + R(s)|m, s
�� could be incentive compatible iff, R(s) ≥ 0 for all s, and verify

� S h [R(s)] -� S l [R(s)] ≥ � S l [� D h [c 2 |m, s]] -� D h [c 2 ( d)]
. This could imply that separating con- tract involves higher implementation costs than testing all patients' types and should be explored with interests.

Observes, however, that this assumption should hold in many cases. For instance, it is the case in our illustration in section 3.1 where it holds with equality since if implies F h (s|d) =

F l (s|d) and � S l � � D h � c 2 ( d)|m, s �� = � S h � � D h � c 2 ( d)|m, s �� = � D h [c 2 ( d)]. It is also always veri- fied by definition of FOSD, if G h � FOSD G l since under MLRP this implies F h � FOSD F l , and � D h � c 2 ( d)|m, s � is increasing in s. Case 2 m l = m, m h = m
Under assumption 4 we exclude the possibility that testing only low-type is efficient. Hence, for the proposition 3 to still holds, it is sufficient to show that testing only low-types cannot reduce the implementation costs.

In that case at least one incentive problems occur, since by following the high-type procedure, the hospital can make a surplus when the patient type is low for the same reason as in the no test configuration. Hence there the hospital must be left with a unit rent equal to at least

� D h � c 2 ( d) � -� D l � c 2 ( d)
� for the treatment of low-types. Furthermore, an additional incentive

problem may arise if � S h [E l [c(d)|m, s] -� D h [c(d)|m, s]] > 0.
This is the counterpart of the con- figuration such that the contract specifies to treat only high-type. Therefore the implementation costs are at least as high as when no test is performed, which is all what we needed to show.

Proposition 3

Proposition 2 implies that findings the optimal care procedures is equivalent with finding the set of couples (m θ ,t θ (s)) that maximizes

∑ θ ∈Θ [ NBS θ (m θ ,t θ (s)) -λ Π θ (θ )]
, for all θ and s.

Also following Proposition 2, at the optimum the hospital surplus in independent from the choice of treatment practices t θ (s). Therefore, given a testing procedure, the optimal treatment in conditional on the signal s is the solution of (2.6). Therefore if the optimal test provision if the efficient one, treatment also is.

It remains to show that hospital's informational rent is a the lowest when m l = m and m h = m, and the optimality conditions follows from comparing the maximand of each possible testing procedure.

Consider the contracts with no test versus the contract with the performance of a test for h only. By Proposition 2,

T l = c 1 (t l ) + � S l � � D h [c 2 ( d)|m, s]
� , and

T h (s) = c 1 (t l (s)) + � D h [c 2 ( d)|m, s] + c m
for all s is the transfer binding (PC h,s ), and (established in Milgrom 1981), so that:

(IC l ) if m l = m � = m h = m. It is compatible with (IC h ) if and only if Π h ( θ = l) = � S l � � D h [c 2 ( d)|m, s] -� D h � c 2 ( d) �� ≤ 0. First observe that � S θ � � D θ [c 2 ( d)|m, s] � = � D θ � c 2 ( d) � , then that MLRP and G h � FOSD G l implies than F h � FOSD F l
� S l � � D h [c 2 ( d)|m, s] -� D h � c 2 ( d) �� F h � FOSD F l < � S h � � D h [c 2 ( d)|m, s] -� D h � c 2 ( d) �� = 0
The above inequality also appears to be the different between the implementation costs of testing only high-types, and testing no types i.e the first and second item of Proposition 2.

Hence, the first inequality in the proposition, gives the condition such that the net loss from testing h is larger than the economy of public funds it allows. If the inequality does not hold, it is therefore optimal to test high patient types. The same reasoning holds when G h � MPS G l , as long as a revealing contract exists, i.e. the condition in assumption 3 is verified.

The second inequality gives the condition such that the net gain from testing l is larger than the increases in public expenditures its performance implies. The term on the RHS of the inequality is necessary positive by definition of the maximum function, implies that testing all is more costly to implement.

Corollary the hospital rent can't be decreased when only h types are tested, so that a sufficient condition for this procedure to be optimal is that it is efficient, i.e. maximizing ∑ θ ∈Θ NBS θ (m θ ,t θ (s)), which leads to the first item of the proposition.

Appendix B : Competition

Lemma 2

Proof. Suppose tests are separating and the equilibrium care procedure are different among between the two hospitals, i.e. is θ 1 * � = θ 2 * . Following (4.3) equilibrium can be analyses for each patient type separately. Moreover, participation constraints must hold for revealing strategies so that π θ (θ ) ≥ 0 , then ( θ 1 * , θ 2 * ) can be equilibrium only if each strategy leads to non-negative profit in each hospital: π 1 ( θ 1 * ), π 2 ( θ 2 * ) ≥ 0. Otherwise they both choose the revealing contract.

Besides, if the hospitals choose a different praxis, in particular hospitals must prefer their praxis than aligning with their competitor:

D 1 θ ( θ 1 * , θ 2 * )π( θ 1 * ) ≥ 1 2 π( θ 2 * ) (6.2) D 2 θ ( θ 2 * , θ 1 * )π( θ 2 * ) ≥ 1 2 π( θ 1 * ) (6.3)
Since they must be simultaneously verifying (6.2) and (6.3) implies

D 2 θ ( θ 2 * , θ 1 * ) � π( θ 2 * ) -π( θ 1 * ) � ≥ 1 2 � π( θ 2 * ) -π( θ 1 * ) � (6.4)
Then either, by definition,

EU θ ( θ 1 * ) > EU θ ( θ 2 * ), implies that D 1 θ ( θ 1 * , θ 2 * ) > 1/2 > D 2
θ ( θ 2 * , θ 1 * ), and (6.3) implies π( θ 2 * ) ≥ π( θ 1 * ) ≥ 0 with at least one inequality being strict. Or EU θ ( θ 1 * ) < EU θ ( θ 2 * ), with (6.2) implies π( θ 1 * ) ≥ π( θ 2 * ) ≥ 0 by a similar path. In either case the condition (6.4) leads to a contradiction.

Finally, in the particular case that EU θ ( θ 1 * ) = EU θ ( θ 2 * ), condition (6.4) and

D 1 θ ( θ 1 * , θ 2 * ) = D 2 θ ( θ 2 *
, θ 1 * ) = 1/2 imply that non-symmetrical strategies are equilibrium iff π( θ 1 * ) = π( θ 2 * ), so that the hospitals are actually indifferent between either strategy, and we consider they both choose the revealing procedure in such case.

Lemma 3

Proof. Suppose the regulator aim at decentralizing (m θ ,t θ (s), T (Θ, s)), with m l = m, m h = m.

A standard argument establishes if that (PC h,s ) is verified for all s then (PC h ) holds. Therefore revealing contracts are such that (PC l ) holds.

Indeed trustful contract must verify :

D H l ( θ 1 = h, θ 2 = l)π l ≥ D H l ( θ 1 = h, θ 2 = l)π l ( θ = h) = D H l ( θ 1 = h, θ 2 = l) � π h + � Sl � � D h � c 2 ( d)|m, s � -� D l � c
Where the last inequality is implied by definition of D H θ , (PC h ) holding, and

� D h � c 2 ( d)|m, s � -� D l � c 2 ( d)|m, s � > 0.
One can therefore restrict attention to (PC h,s ), and equilibrium condition, and show that at the optimum the participation constraint of high-types are binding for all s. This follows the standard argument we used to establish Proposition 2.

The revealing strategy being equilibrium for low-types requires in particular that:

D H (l, l)π l ≥ D H (h, l)π l ( θ = h) = D H (h, l)π l ( θ = h) max{E Sl � � D h [c 2 ( d)|m h , s] -� D l [c 2 ( d)] � , 0}
otherwise, the equilibrium is: θ 1 l = θ 1 l = h. More precisely it is a necessary and sufficient condition when hospital participation constraints hold. Lemma 3 follow by showing that the above constraint is binding.

Proposition 4

Proof. Suppose that the efficient testing is optimal so that m SB l = 0, and m SB h = 1. We have shown in the monopoly case that under the assumption 3, the implementation cost is minimal when the procedure specify to test high patient types only, so that it is always optimal under 5.

Suppose that m SB l = 0, and m SB h = 1 is optimal (condition to be clarified by further work).

Lemma 3 gives the implementation costs of the test procedures are such that m h = m � = m l = m. This costs depends on the treatment praxis through the patient expected utility that determines whether increasing test for low patient types makes the hospital more attractive.

Indeed, consider first the treatment of low patients:

∂ SW ∂t l = ∂ NSB ∂t l -2λ ∂ EU l ∂t l D H l ( θ 1 = h, θ 2 = l) ∂ EU l π l
When evaluated at the efficient praxis t m * l , the first term is zero by definition of efficiency, and the second term is positive since

∂ EU l ∂t l = ∂ NSB ∂t l + c � 1 (t l ), positive if c � 1 (t l ) > 0. And ∂ EU l ∂t l < 0,
by definition of hospitals' demand.

Hence at the competitive second best t SB l ≥ t * l , and strictly greater if c � 1 (t) > 0.

Now consider the treatment of high-types h. The treatment depends on the signal s consecutive to the test. For a given signal s:

∂ SW ∂t h (s) = ∂ NSB ∂t h (s) -2λ ∂ EU l ( θ = h) ∂t h (s) D H l ( θ 1 = h, θ 2 = l) ∂ EU l ( θ = h) π l
If one evaluates ∂ SW ∂t h (s) , at t * h (m, s), the first term is zero by definition of t * h (m, s) in (2.6), and the sign of the second term depend on the sign of :

∂ EU l ( θ = h) ∂t h (s) = ϕ � 1 (t * h (m, s))h(� D l � c 2 ( d)|m, s � ) -ϕ � 2 (t * h (m, s))
. If it is positive then ∂ SW ∂t h (s) is positive, and t SB h (s) > t * h (m, s)

To determine its sign we know that t * h (m, s)

verify ϕ � 1 (t * h (m, s))h(� D h � c 2 ( d)|m, s � ) - ϕ � 2 (t * h (m, s)) -c � 1 (t * h (m, s)) = 0 Hence, ∂ EU l ( θ =h) ∂t h (s) > 0 ⇐⇒ h(� D l � c 2 ( d)|m, s � ) > ϕ � 2 (t * h (m,s)) ϕ � 1 (t * h (m,s)) = h(� D h � c 2 ( d)|m, s � ) - c � 1 (t * h (m,s)) ϕ � 1 (t * h (m,s))
The distortions we have claimed in the proposition stem from the above inequality.

Appendix C.

The restriction : Consider that patient priors are ordered so that G h is a mean preserving

spread of G l with G l (d) (≥) ≤ G h (d) ⇔ d (
f θ (s|d) = � θ � f (s| d) � = f (s|� θ � d� ) =: f (s) ∀ s ∈ S, θ = 1, 2
and , hence,

F θ (s|d) = � θ � F(s| d) � = F(s|� θ � d� ) = F(s|d e ) =: F(s) ∀ s ∈ S, θ = 1, 2.
I.e. under the Assumptions 1 and 2, the marginal signal distribution of the diagnostic test is the same no matter to which patient the test is applied to. Consequently (former Assumption 2)

� S l � � D h � c 2 ( d)| s�� = � S h � � D h � c 2 ( d)| s�� = � D h � c 2 ( d) � . (6.5)
Moreover, strict MLRP implies that F(s|d) is strictly decreasing in d. Therefore, for all s ∈ (s ¯, s):

F(s|d) > = < F(s) ⇔ d < = > d e .
(6.6)

Last, [START_REF] Brandt | Information and the dispersion of posterior expectations[END_REF] have shown that the following definition of informativeness is equivalent with the definition in 2. The proofs make use of the version below:

Definition 2.2.bis Consider f m , f m � ∈ F. Information system ( f m , g h ) is weakly more infor- mative than ( f m � , g l ) -denoted by ( f m , g h ) � w-in f ( f m � , g l ) -iff J m h (F m -1 h (p), d) -pG h (d) ≥ J m � l (F m -1 l (p), d) -pG l (d), ∀ p, d.
The next Proposition shows that for a fixed test f ∈ F, then the diagnostic system becomes more informative when the prior is more spread out in the sense of SC -MPS-dispersion.

Our Claim:

Consider a diagnostic technology f ∈ F and two (ex-ante) distributions G h (d) and G l (d).

If G h � SC-MPS G l then ( f , g h ) � w-in f ( f , g l ).
Proof.

Define W I θ (s, d) := J θ (s, d) -F(s)G θ (d), θ = l, h. Since, F m l (s) = F m h (s) =: F(s), we get that ( f , g h ) � w-in f ( f , g l ) ⇔ W I h (s, d) ≥ W I l (s, d)∀ d, s. (6.7) Next define, T hl (d) := G -1 l (G h (d)) and T lh (d) := G -1 h (G l (d)). Then, G h (d) = G l (T hl (d))
and G l (d) = G h (T lh (d)) for all d ∈ D. Hence, the transformation theorem implies

W I h (s, d) = d � d ¯F (s|d � )g h (d � )dd � -pG h (d) = T hl (d) � d ¯F (s|T lh (d � ))g 2 (d � )dd � -pG h (d) (6.8)
and

W I l (s, d) = d � d ¯F (s|d � )g l (d � )dd � -pG l (d) = T lh (d) � d ¯F (s|T hl (d � ))g h (d � )dd � -pG l (d).
(6.9)

Moreover, equation (6.6) implies W I l (s, d) (6.12),(6.10)

d dd W I θ (s, d) = g θ (d) [F(s|d) -p] > = < 0 ⇔ d < = > d e . ( 6
≤ W I l (s, T hl (d)) = T hl (d) � d ¯F (s|d � )g l (d � )dd � -p G l (T hl (d)) � �� � =G h (d)
MLRP,(6.11)

≤ T hl (d) � d ¯F (s|T lh (d � ))g l (d � )dd � -pG h (p) = W I h (s, d).
(6.13)

d > d e :
W I h (s, d) (6.10),(6.12) Next, we related informativeness with its value. The proof makes use of lemmata [START_REF] Brandt | Weak Informativeness: An Information Order with Variable Prior[END_REF], with only an adaptation of the second lemma.

≥ W I h (s, T lh (d)) (6.8) = T hl (T lh (d)) � d ¯F (s|T lh (d � ))g l (d � )dd � -pG h (T lh (d)) = d � d ¯F (s|T lh (d � ))g l (d � )dd � -pG l (d) =W I h (
Observations: Observes that this aspect of the restriction may be unsatisfying regarding our interpretation of hospital rent in separating testing. Indeed due to the combination linearity of the test technology and MPS, it is never optimal to test more patients than it is efficient, since separating introduce, in average not risk. This is provided by the implication in (6.5) above.

Because of this restrictive view we have kept the discussion general about the test technology and priors.

Also note that we can provide a relaxed version of this restriction with non MPS property so that the risk dimension of testing would remains. The proofs can be provided on requests and belongs to a separate work more dedicated to the information concept. This current state instead is dedicated to its applications.

Appendix D.

Proposition 5 Considering the following design of transfers, one can verify that it verifies both incentive and participation constraint, 3.3 and (3.1), and more precisely binds both hospital participation constraints. If, m h = m, m l = m, the contract only need to verify:

T l = c 1 (t l ) + � D l � c 2 ( d)|m, s �
, and with s � such that F m h (s � ) � = F m l (s � ):

T h =                        F m h (s � )EC l -F m l (s � )EC h F m h (s � ) -F m l (s � ) if s < s � (1 -F m l (s � ))EC h -(1 -F m h (s � ))EC l F m h (s � ) -F m l (s � )
Otherwise where, EC θ denotes the expected cost of following the procedure, (m h , (h, s)) for all s, if the patient type is θ ∈ Θ, and m h = m.

EC h = � S h [c 1 (t h (s))|m, s] + � D h � c 2 ( d) � -c m , EC l = � S l [c 1 (t h (s))|m, s] + � D l � c 2 ( d) � -c m ,
Since participation constraints are binding, no other contract can increase the welfare, and the solution of the regulation problem is this of efficiency (if V h > (1 + λ )), otherwise the proposition gives the condition such that the efficient testing is optimal if V h < (1 + λ ).

CHAPTER 3

Mixed Market and Asymmetric Universal Service

Obligations: an Application to Home Care

This chapter is a joint work with Robin Hege. 151

Introduction

The home care market in France is organized according to two regimes, "l'autorisation" ( authorization), and "l'agréement" (approval or non-authorized) that vary the flexibility of firms in terms of acceptance policy of consumers, price and quality setting. The authorized firms are regulated by the local authorities and submitted to a universal service obligation (USOs).

In contrast, the "agréées" (approved) firms can freely set their hourly tariffs, acceptance policy and the quality of their services 1 . While the per-hour tariff differs across firms, both regimes of formal home care services are subject to uniform pricing constraint. 2 In parallel, the APA (Allocation Personnalisée d'Autonomie, literally: personalized helps to autonomy) program aims at guaranteeing the financial access to home care so that the elderly supports from 0 to 90% of the firms' tariffs according to their income (see for instance [START_REF] Roquebert | Pay less, consume more? The price elasticity of home care for the disabled elderly in France[END_REF] or Bourreau-Dubois, [START_REF] Bourreau-Dubois | Les disparités départementales dans la mise en oeuvre de l'APA à domicile, reflets des choix de politiques locales ?[END_REF] for a more detailed presentation of the APA program). In order to focus on the geographical component of access we will not address financial access concerns in this work.

The coexistence of firms producing differentiated services at different prices is a usual topic of economic theory (see for instance [START_REF] Laine | Quality and competition between public and private firms[END_REF] and the reference therein), and relies on heterogeneity in consumers' valuation of the firms' quality of services. As far as home care is concerned, the consumers' needs vary a lot upon their degree of dependence: it can mostly consist in housekeeping, support for feeding and administrative tasks, or be supplemented by psychological, hygienic and up to physical autonomy helps. The vertical differentiation among firms could correspond to the differences in qualification requirements set by each firm. Theoretically, the diversity of suppliers is necessarily welfare-improving since their coexistence requires that some consumers are better-off with choosing their price-quality offer, for each firm. Since the seminal work of Chamberlin, the literature has identified the market failures inherent to such oligopolies and suggested solutions to palliate them. In particular, Cremer, 1. Although minimum quality standards must be met for their activity to be allowed.

2. An anonymity chart prevents us from providing a very precise example, but we can report for instance a difference of 19 versus 22 euros per hour respectively in an "agréée" and "authorisée" firm from the same French department in 2012. [START_REF] Cremer | Mixed oligopoly with differentiated products[END_REF] show that the regulation of one of the competing firms makes it possible to alleviate those market failures and increase the social welfare. We will address this aspect of our problem by following a "non-addressed" branch approach and in particular the work of [START_REF] Grilo | Mixed Duopoly under Vertical Differentiation[END_REF] and [START_REF] Laine | Quality and competition between public and private firms[END_REF]. [START_REF] Laine | Quality and competition between public and private firms[END_REF] characterize the outcome of a public-private interaction in terms of consumers' allocation and equilibrium qualities. They show that efficiency can be decentralized by the mixed duopoly interaction. We will discuss, moreover, the introduction of a geographical component and the presence of asymmetric ubiquity constraints, i.e. the constraint to be active and answer demands everywhere on the territory.

In this work we are interested in the conjunction of asymmetric USOs and the price-quality competition of a public-private duopoly. In home care, the concerns of universal services obligation are raised by its geographical components. The firms must incur transportation costs for every care they deliver and may not be willing to incur the levels of access costs of each area in the territory. More precisely, the firm's employees spend unproductive hours reaching the consumers during their daily rounds, and the firm must pay wages for these hours (e.g. in France, they are paid for all transport time exceeding 15 minutes). The transportation costs are composed of reimbursements of the employees' transportation costs (gas or public transport fees) and the employees' remuneration during their traveling time (unproductive hours).

The total transportation costs of a firm depends on both how much remote the consumers are from the firms' employees' locations and the density of the area the consumers live into. The public report of Bruant-Bisson, de Reboul and Aube-Martin (2010) suggests that the unproductive remuneration accounts for 20% to 25% of the total employee remuneration which is the main component of the firms' costs. In addition, the reimbursement of the employee's travel expenditures (car usage) represents 1% to 7% of the firms' total expenditures.

The literature has emphasized the different roles played by non-discrimination and ubiquity constraints that are the two traditional requirements of USOs. The foundation for ubiquity can be various, see for instance [START_REF] Poudou | Equity justifications for universal service obligations[END_REF] for a discussion of the optimal coverage constraint for an inequity-averse authority. [START_REF] Valletti | Universal Service and Entry: The Role of Uniform Pricing and Coverage Constraints[END_REF] and [START_REF] Poudou | Efficiency of uniform pricing in universal service obligations[END_REF] characterize the impact of coverage obligation when uniform pricing is imposed to part or all of the firms in problems without vertical differentiation. Following [START_REF] Bourguignon | Skimming the other's cream: Competitive effects of an asymmetric universal service obligation[END_REF] and [START_REF] Calzada | Universal service obligations in the postal sector: The relationship between quality and coverage[END_REF], who also discusses competition between differentiated providers, we consider that only one firm, the public, has an ubiquity constraint and we will not discuss whether ubiquity is the optimal coverage constraint. In particular, we consider that both firms can possibly be active on each area of the territory, while only one firm, the public, has an ubiquity constraint and must answer all requests of service on the whole territory. We also consider non-discrimination constraints of uniform pricing form: the prices can vary across firms but not across the consumers of a given firm.

The work of [START_REF] Calzada | Universal service obligations in the postal sector: The relationship between quality and coverage[END_REF] provides an analysis of the impact of asymmetric obligation of ubiquity on vertical differentiation in a mixed duopoly following the representation of [START_REF] Cremer | Service Quality, Competition, and Regulatory Policies in the Postal Sector[END_REF]. Their work is orientated towards an analysis of postal services that, in terms of USOs, present similar features with the home-care market. [START_REF] Calzada | Universal service obligations in the postal sector: The relationship between quality and coverage[END_REF] compares equilibrium coverages of private-private interaction with these of a mixed-duopoly. They find that the presence of the public firm neutralizes strategic dimensions of coverage for the private entrant and leads the entrant to a higher coverage than a private-private competition. We will not address the private-private game in this paper. Our work instead aims at characterizing the mixed-duopoly equilibrium in terms of coverage with respect to the social optimum and its impact on equilibrium qualities. We use a more general representation and slightly different timing that retrieve many of their prediction. Our main result, however, suggests that, not only the private firm coverage is possibly higher than in private-private competition it is, furthermore, excessive in the mixed-duopoly framework. [START_REF] Bourguignon | Skimming the other's cream: Competitive effects of an asymmetric universal service obligation[END_REF] discuss the competition between a provider of an essential good (electricity) with ubiquity constraints and a provider of non-essential good (gas) entrant. The ubiquity constraint of the electricity provider is justified by the essential nature of the good it provides, the consumers choose between electricity only, and gas plus electricity. They show that ubiquity, if justified by the essential nature of the good, can actually give advantage to its provider, that compete more aggressively than the entrant. Our problem departs from [START_REF] Bourguignon | Skimming the other's cream: Competitive effects of an asymmetric universal service obligation[END_REF], in the sense that we will consider that the firm with the ubiquity constraint is public and maximizes welfare while competing with a private firm that provides an essential good also. More importantly, the somehow, substitutability of each providers' product is endogenously determined by the firm's choice of the quality standards.

In network utilities, such as postal services, transports or telecommunication, an obvious drawback from the diversity of supply on a given area stems from the multiplication of access costs that are supported by each firm. Concerning home care, one can also observe that if a countryside area is covered by two firms, it implies that both firms have to incur the transportation cost to access the town 3 . Hence, it is efficient that both firms supply a given market area only if the net welfare gain from diversity of supply is larger than the additional transportation that the entry of a second supplier implies. The main topic of the paper concerns the decentralization of such trade-off to a mixed duopoly.

The decentralized problem is a three-stage game: both of the firms simultaneously choose their quality, then simultaneously choose their price and the private firm chooses its coverage while the public firm is constrained to access all requests whatever the consumers' locations 4 . Simultaneous move is a common assumption in vertical differentiation of mixed duopoly (see for instance [START_REF] Valletti | Universal Service and Entry: The Role of Uniform Pricing and Coverage Constraints[END_REF], [START_REF] Grilo | Mixed Duopoly under Vertical Differentiation[END_REF], [START_REF] Cremer | Service Quality, Competition, and Regulatory Policies in the Postal Sector[END_REF], [START_REF] Laine | Quality and competition between public and private firms[END_REF]) and will be discussed in more details in the main discussion.

Our main result is that the equilibrium surplus of the private firm (which enters the market and competes with the public firm with USOs) overcomes the net welfare gain from its entry. This result first suggests a transfer of surplus between firms and consumers in favor of the private firm. In our model, the regulation authority has no social preference across consumers and firms so that we focus our attention on the consequences of this property in terms of equilibrium coverage. We, however, believe that this result could have other implications and fields of interpretations since it only requires that the consumers' distributions are log-concave, which is a commonly assumed to guaranty the uniqueness of equilibrium.

Concerning the impact of asymmetric USOs, we intuitively find that the private firm is 3. In practice the multiplication of access costs does not only stem from the need for both firms to access the area. It also constrain the organization of rounds sine the consumer of each area are segmented between the two providers. As descriptive representation of the last effect bringing too much complexity, in a first approximation we will consider that, for either of these two reason (dividing the density of the area, and requiring that both firms access the area), the presence of both firms multiplies the access cost incurred.

4. In terms of timing [START_REF] Calzada | Universal service obligations in the postal sector: The relationship between quality and coverage[END_REF] considers that coverage is decided first, quality in second and price at last, also simultaneously. Both can be discussed and we believe that our approach better suits the home care and captures the idea that the private firm policy in terms of coverage is flexible.

willing to cover a market area only if the expected surplus gain from providing the consumers of this area compensates the transport costs to access it. Hence, our main result predicts that the private firm equilibrium coverage will be excessive since its expected surplus on an area is larger than the associate gain in welfare. Indeed, from the public firm's point of view, the private firm's coverage is optimal only if the welfare gain from the diversity of suppliers out-passes the multiplication of transportation costs it induces. In other words, despite the possibility that a consumer would prefer the private firm's price-quality offer, it might not be optimal that the private firm competes with the public firm in a remote area as it implies a too large increase in total transportation costs.

We, furthermore, show that in order to limit the inefficient multiplication of costs and the private firms' coverage, the public firm's best response is to decrease the degree of differentiation between the two firms. By aligning its quality on the private firm's standards, the public firm decreases the private firm's market power due to the intensification of price competition.

The expected surplus by the area of the private firm decreases, as wells as its coverage of remote areas. For instance, if the public firm is the high quality provider, we can conclude on a deficient level of quality at the equilibrium. In terms of policy recommendations, our results tend to argue for the application of a coverage license to the private competitor on the market. We can also characterize the expected difference in qualities between a public-public market (that would correspond to our benchmark) and the mixed-duopoly equilibrium.

We should last observe that our problem does not address the problem of how of the USOs are funded. See [START_REF] Choné | Allocating and funding universal service obligations in a competitive market[END_REF], for a study of the effects of alternative funding mechanisms of both ubiquity and non-discrimination constraint on welfare, including non-linear pricing. In particular, our problem will implicitly assume that the regulation authority payback the public firm if it incurs losses due to its obligation of service, and only impose that the public firm's tariffs exceed its marginal cost i.e. is not structurally in deficit. We will see that the participation constraint of the public firm does not necessarily bind at the equilibrium. Hence, the main restriction from our simplifying approach is that we do not discuss the distortions that arise due to the cost of public funding if the public firm break-even constraint does bind. The impact of positive cost of public funding of the sustainability of the ubiquity constraint on the public firm price-quality trade-offs should be explored in extension of our problem.

The paper is organized as follows. Section 1 presents the framework and the efficient solution to the trade-off between the social gain from the supplier diversity, and its cost in terms of transportation. In section 2, we present the mixed-duopoly and the equilibrium coverage of the private firm. Section 3 characterizes the impact of the private firm's excessive coverage on the equilibrium quality, and section 4 concludes.

Framework

Model settings

The model is composed of a continuum of consumers, and two firms i ∈ {u, pr}: a public, regulated, firm with ubiquity constraints, u, and a private firm pr. The consumers are differentiated by their valuations for quality v ∈ [v ¯, v], and their geographical location on the territory, θ ∈ � 0, θ � (city center, town, village..). Our representation of the territory and consumer distribution on the territory follow the literature ( [START_REF] Valletti | Universal Service and Entry: The Role of Uniform Pricing and Coverage Constraints[END_REF][START_REF] Bourguignon | Skimming the other's cream: Competitive effects of an asymmetric universal service obligation[END_REF][START_REF] Calzada | Universal service obligations in the postal sector: The relationship between quality and coverage[END_REF][START_REF] Poudou | Efficiency of uniform pricing in universal service obligations[END_REF]). The territory is represented as a line with the two firms being located at the same point on the line so that locations are ordered by increasing fixed access costs, C(θ ) = θ , with the same order at each firm. For interpretation, zero is the cost to serve the city center, and θ is the cost to serve the remotest/less dense areas.

Moreover, there is a unit mass of consumers in each area, which are differentiated in terms of their valuation of quality v distributed over the c.d.f F(v). In home care, the heterogeneity of valuation reflects the different needs and requirements of consumers. We assume under A.1 that the consumers' valuation and location are independently distributed 5 and that their respective c.d.f F(v), G(θ ) are log-concave to ensure that the problems of each firm are quasi-concave.

Assumption 1. F(v) ⊥ G(θ )

5. We make this assumption, common in the litterature on USOs, for technical reason. Going toward joint distributions could be the motivation of further works but, seemingly would lead to conduct a similar analysis conditional on consumers' valuation distributions.

Both firms produce a good differentiated in quality, q i , and the price at which they sell it, p i .

In the sequel, we will consider both the case of the coexistence of a "low-cost" and of a more "luxurious" private provider with a public firm providing higher and lower quality respectively (both cases can be observed on the home-care market).

The unitary production cost of a good of quality q is c(q) increasing and convex in quality, with c(0) = 0 (in home care services, differences in quality standards can be interpreted as differences in qualification of the firms' employees that directly impact the hourly wage of the workers). The firm's costs also depend on the location of the consumers they provide. We characterize areas by their cost of access so that it costs θ to the firms to provide the consumers of a given area of type θ . The marginal profit of providing an additional consumer in the area is:

π(p i , q i ) = p i -c(q i ) (2.1)
All consumers buy one unit of the good 6 , and choose the provider with the price-quality offer that maximizes their utility. The surplus of a consumer of type v, that buys from firm i is :

u(q i , p i , v) = vq i -p i (2.2)
The social welfare is defined as the sum of the consumers' surplus and the firms' profits over all market areas. Consider an area of a given type θ and denote by S i (θ ) ⊆ [v ¯, v], the subset of consumers' types of this given area that are provided by the firm i ∈ {u, pr}. The net social welfare from the provision of this area by the firm i is:

W i (S i (θ ), q i ) -θ = � S i (θ ) � vq i -c(q i ) � dF(v) -θ , if S i (θ ) � = / 0, zero otherwise. (2.3)
When both firms compete in a given area, i.e: S i (θ ) � = / 0, for i = u, pr, each firm incurs the 6. We do not consider the financial exclusion. Concerning home care in France, the concerns of financial access are dealt within the APA program, that we do not take into account in this model. Moreover, for essential goods and services one can consider that the utility from no consumption tends to -∞ or, following [START_REF] Cremer | Service Quality, Competition, and Regulatory Policies in the Postal Sector[END_REF], one could also assume a utility u(q i , p i , v) = vq ip i + k with k large enough. cost to access the area. Yet the presence of the two firms on a given area may be justified by their differentiation in quality (vertical differentiation) which is always strictly welfare improving for consumers.

Universal obligation of services' constraints: In the sequel, we will consider two firms constrained to uniform pricing and quality i.e. quality and price can differ across the two providers but each of them must offer the same service at the same price to all of their respective consumers. In addition, one of the two firms is regulated and has a ubiquity constraint, i.e. has the obligation to answer the demand of any consumer on the whole territory.

Before presenting the decentralized three-stage game and its equilibria, we need to define the socially optimal allocation and coverage of each firm.

Social optimum and limited coverage a -Efficient segmentation of consumers and firms' coverage

This section defines the socially optimal allocation of consumers between two vertically differentiated firms when one firm has a universal service obligation, denoted by the superscript u. Hence, only the coverage of the firm without ubiquity constraints, needs to be defined. To avoid confusion with the public-private problem, here we denote by nu the firm that has no ubiquity constraint. The efficient segmentation of consumers and firms on the market is the set of S i (θ ) that maximizes the total social surplus for each location type θ . The sequel only formally explicits the result for the case such that the firm with USOs constraint is the high quality provider, i.e. q u ≥ q nu . The alternative case admits symmetric results and interpretation and is presented in Appendix A.

At given qualities and prices, the activity of both firms on a given market area is efficient (i.e. S i * (θ ) � = / 0, i = u, nu) if and only if firms are differentiated in terms of quality and each firm should be allocated some consumers' types. Namely, given the qualities q u , q nu there must exist a unique consumer type ṽ ∈]v ¯, v[ such that the social planner is indifferent between its provision by either of the two firms: ṽ(q nu , q u ) = c(q u ) -c(q nu ) q uq nu (2.4)

The efficient segmentation of consumers is such that consumers with low-valuation are provided by the low quality firm, nu, i.e. S nu * (θ ) = [v ¯, ṽ), and the high-valuation consumers by the high quality firm: S u * (θ ) = ( ṽ, v]. Moreover, since the coverage of the market area θ by the two firms implies that the transportation's costs are incurred twice, the activity of the two firms is efficient only if the welfare gain from the diversity of supply exceeds the loss from increasing the total transportation costs, i.e. if:

W nu ([v ¯, ṽ) , q nu ) +W u ((v ¯, ṽ] , q u ) -2θ ≥ W u ([v ¯, v] , q u ) -θ (2.5)
If the above condition is not satisfied at θ then it is not socially optimal that both firms cover the whole territory and this will be the configuration of interest in the sequel. Hence, all types of areas such that (2.5) doesn't hold are provided by the firm u. The net social welfare by area being a decreasing function of the transportation costs, there is a unique cut-off location type θ (q nu , q u ), such that S nu * (θ ) = [v ¯, ṽ), S u * (θ ) = ( ṽ, v] for all θ ≤ θ while, S nu * (θ ) = / 0 and

S u * (θ ) = [v ¯, v] otherwise.
The larger the social gain from the diversity of supplier, the larger the coverage of the firm nu, θ (q nu , q u ). Hence, a sufficient condition so that ubiquity is not optimal for the firm nu, is that the maximal access cost θ , or the territory, is large enough. Following [START_REF] Bourguignon | Skimming the other's cream: Competitive effects of an asymmetric universal service obligation[END_REF], we will restrict attention to asymmetric coverage and we will need the following condition to hold:

θ < θ (2.6) b -Efficient qualities
The characterization of the social optimum depends upon whether it is the firm that provides high or low quality to which universal coverage obligation is incumbent to. If we suppose first that q u ≥ q nu , the socially optimal qualities are characterized by the solution of the following program:

Max q u ≥q nu � θ 0 [W nu ([v ¯, ṽ) , q nu ) +W u (( ṽ, v] , q u ) -2θ ] dG(θ ) + � θ θ [W u ([v ¯, v] , q u ) dF(v)] -θ dG(θ ) (2.7)
with ṽ being defined by equation ( 2.4) and, θ (q nu , q u ) =

� ṽ v ¯[(vq nu -c(q nu )) -(vq u -c(q u ))] dF(v) (2.8)
Proposition 1. If q u ≥ q nu the efficient qualities, q i * must verify:

c � (q nu ) = � ṽ v ¯v dF(v) F( ṽ) (2.9) c � (q u ) = G( θ ) � v ṽ v dF(v) + (1 -G( θ )) � v v ¯v dF(v) 1 -F( ṽ)G( θ ) (2.10)
Otherwise, the characterization of the efficient qualities such that q u < q nu is a close mirror, and is presented in Appendix A.

The efficient level of quality of the two respective firms are such that their marginal costs equate the average valuation of their consumers. The differences in (2.9) and (2.10) below result from the difference in coverage between the two firms. In the case where q u ≥ q nu , as opposed to the low-quality firm, the firm with the ubiquity constraint is not fully specialized on one segment of the market and provides consumers with both high and low valuations for quality in all area types larger than θ .

The problem admits an interior solution in each case q u ≥ q nu , and q u < q nu . Moreover the limit case such that q u = q nu is never optimal due to convexity of unitary costs c(q). There is a unique optimal quality for each consumers' type, so that there is always a diversification of quality that can be welfare improving for some consumers. In other words, c �� (q) > 0 implies that diversity of suppliers is welfare improving and that some degree of differentiation should be exerted at the optimum, at least at the "city center" θ = 0. Also for this reason, ṽ(q n , q nu ) ∈]v ¯, v[ is always verified at the optimum.

It is, however, difficult to establish a clear condition such that the social planner should impose ubiquity on the high or on low quality firm, i.e. whether one of the two cases q u ≥ q nu , and q u < q nu globally dominates. In the sequel, we will consider the two possible positioning of the public firm with the ubiquity constraints and analyses the equilibrium condition of each case.

Decentralization to a mixed duopoly

In the mixed duopoly decentralized problems, the firm without USOs constraint is private and freely competes with the public firm to maximize its profit. We consider the firms play a simultaneous three-stage game: the public and private firm simultaneously choose qualities then simultaneously choose prices and the private firm chooses its coverage. If the private firm does not cover some areas, the consumers of these areas can all turn to the public firm.

The simultaneity assumes an absence of leadership from the public firm and is obviously a questionable assumption. It can also be interpreted as the absence of a commitment of the public firm that is often assumed in the mixed-oligopoly literature and reflects that the public firm is likely to adapt its offer in response to the private firm's actions. In our opinion, this consideration suits well the description of home care market in France, where, de facto, price and quality norms seem to be set by the authorities depending on the evolution of the market and negotiations with the private firms, and thereby, subject to regular changes.

-Stage 1: The public firm, u, and the private firm, pr, simultaneously choose their respective quality q u , q pr .

-Stage 2: Qualities in Stage 2 are common knowledge. The public and the private firms simultaneously choose their respective prices, and the private firm decides which consumers in which areas it is willing to provide.

-Stage 3: Consumers observe the offers (p i , q i ), and address their demand to their preferred provider. The consumers that are rejected by the private firm can pick the public firm as a second choice.

Given the prices and qualities, at stage 3, the consumers choose to buy from firm i if and only if vq ip i > vq -ip -i . Since the consumers' utility from a given price-quality offer is increasing in their valuation, there is at most one type of consumer that is indifferent between the two firms (in which case their demand is shared by half between the two firms). Denote by v the consumer's type cut-offs such that:

v(p u , p pr , q u , q pr ) = p up pr q uq pr (2.11)

It will be convenient to denote by ϕ( v) the potential market share of the private firm on each area θ . This will allow to present the results in a most general way when it is possible, and accounts for both possible equilibrium positioning of the public and private firm. Hence, if q pr < q u then vq prp pr > vq up u , for all v < v such that the potential demand to the private firm is ϕ( v) = F( v) and conversely ϕ( v) = 1 -F( v) if q pr > q u (firms equally share the demand by half if q pr = q u ). Also for convenience, we follow the notations of the previous literature and define h(v) and k(v) the inverse hazard rate and the inverse reverse hazard rate respectively.

They respectively measure the importance of the marginal demand of the private firm relatively to its total demand when q u < q pr and q u ≥ q pr , and their rate of growth follows by log-concavity of the c.d.f. F :

h(v) = 1 -F(v) f (v) , k(v) = F(v) f (v)
, with h � < 0, and k � > 0 3. Mixed-duopoly equilibrium prices and coverage

Equilibrium prices and consumers' segmentation

This section presents the firm equilibrium prices and coverage that are chosen in stage 2, and illustrate the consecutive segmentation of consumers. Given the qualities q u , q pr chosen in stage 1, the private firm sets its price and coverage to maximize its profit and the public firm sets its price to maximize the social welfare.

Profit maximization implies that the private firm offers services only if its unit margin is positive, and its surplus on the market area covers the access cost. Given that the share of the consumer that prefers to buy at the private firm is ϕ( v) in each area θ (by assumption 1) the condition so that private firms will accede to their demand is ϕ( v) [p prc(q pr )] -θ ≥ 0. The private firm will therefore cover all areas θ with access costs lower than θ (p u , p pr , q u , q pr ) such that:

θ (p u , p pr , q u , q pr ) = ϕ( v)( v(p u , p pr , q u , q pr )) [p prc(q pr )]

Hence, the total surplus of the private firm is given by the product of its demand and its unit margin in all areas it serves, net of the costs θ to access each area:

Π(p pr , p u , q pr , q u , θ ) =

� θ 0 ϕ( v) [p pr -c(q pr )] -θ dG(θ ) (3.2)
The objective of the public firm is to maximize social welfare defined in the social planner problem in (2.7) if q u ≥ q pr (or in (6.1) otherwise, see Appendix A). Only the set of the variables it chooses and the status of the firm without ubiquity changes, where nu is now private firm and its choices are take as given. Hence, in the decentralized problem, at stage 2 the public firm takes the price, quality and coverage of the firm without ubiquity constraints, p pr , q pr , θ as given.

If the private firm covers the area θ , the private firm's demand is ϕ( v) and the public firm's demand is 1 -ϕ( v). In this case, whether one firm serves high or low valuation consumers depends on its positioning on the market, that is, whether they are the high or low quality providers. If the private firm does not cover an area θ , all consumers buy from the public firm. Lemma 1 characterizes the respective best response in price of the private and the public firm. The public firm sets its price so that the equilibrium segmentation of consumers between Figure 3.1 -Consumers segmentation with q u ≥ q pr : Private

Firm Public Firm θ v = p u -p pr q u -q pr v v θ θ consumers' valuation for quality, v consumers' location type, θ
with USO the two firms is efficient, v = ṽ(q u , q pr ). More precisely, the public pricing rule insures that the difference in firms' prices reflects their difference in costs at the equilibrium, i.e. p prp u = c(q pr ) -c(q u ). The pricing strategy of the private firm exhibits the classic relationship between the firm's margin rate and the inverse price elasticity of demand.

Lemma 1. At the equilibrium, the private firm margin and the consumer segmentation respectively verify:

p pr -c(q pr ) = |q u -q pr | ϕ( v) f ( v) (3.3)
p uc(q u ) = p prc(q pr ) , and v = c(q pr ) -c(q u ) q prq u (3.4)

One should observe that both the private and the public firms have a positive margin at the equilibrium. However, for the public firm that cannot control its coverage, it does not guaranty that its global participation constraint is binding. This depends on whether the surplus it achieves in the least remote areas compensates for its losses in the most remote areas (or whether the equilibrium price achieves to mutualize the transportation costs across the public firms con-sumers). We do not address this issue in this work. We will consider that the authority payback the public firm if it incurs losses due to transportation costs at the equilibrium. Although in line with what is observed on the French home-care market, this simplification omits to discuss the trade-off that may be involved due to the cost of public funds attached to this transfer. Further works should take a step into this direction, but this work will only consider the constraint that the public firm's pricing does not involve a structural deficit at the equilibrium, i.e. that p u ≥ c(q u ).

Given the previous remarks, the equilibrium prices in (3.3) and (3.4) are obtained by rearranging the respective FOC in price of each firm and where the coverage verifies (3.1). Those conditions hold when both firms are active at the equilibrium, i.e. ṽ(q u , q pr ) ∈ [v ¯, v], but one can show that it is the only possible equilibrium of the game.

The convexity of unit costs implies that remaining out of the market is never a best response for the public firm. Whatever the quality of the private firm q pr , there is a price-quality offer that is welfare improving for at least some consumers, and that is compatible with the public firm participation constraint as long at the private firm neither makes losses. Hence, there is no equilibrium such that vq prc(q pr ) > vq uc(q u ) for all v ∈ [v ¯, v].

Besides, there is no equilibrium with the private firm providing a quality such that vq prc(q pr ) < vq uc(q u ) for all v ∈ [v ¯, v] and makes non-negative profit. Indeed, if the private firm chooses such quality, the public firm's best response in price is to exclude the private firm by setting a price such that p u < p pr -v|q uq pr |, for all v ∈ [v ¯, v], and this without breaking the participation constraint p u ≥ c(q u ) since it holds as long as the private firm makes no losses. 7 .

As a result, the private firm would make no profit, and this is strictly dominated by the private firm choosing a quality level such that ṽ ∈ [v ¯, v]. Therefore the equilibrium must be such that the two firms are active, and the equilibrium price verifies lemma 1.

Also observes that the closer the two firms' qualities, the lower the coverage of the private firm. However, the extreme case q u = q pr , which implies a zero profit for both firms and that the private firm covers only areas without access costs (e.g. city centers) is never an equilibrium 7. If vq prc(q pr ) < vq uc(q u ) for all v and p pr ≥ c(q pr ), then setting p u = c(q u ) is sufficient to achieve that vq prp pr < vq up u ) for all v so that no consumer chooses the private firm (multiple prices level are possible as soon as the private firm unit margin is strictly positive).

Proposition 2. At any equilibrium quality q u , q pr , the private firm surplus is larger than the net welfare gain from product differentiation which results in an excessive coverage of the private at the equilibrium: Π(q pr , q u , θ (q u , q pr )) = [p prc(q pr )] ϕ( v(q pr , q u )) -θ (q u , q pr ) > 0 ⇐⇒ θ > θ (3.5)

Proof. The result of excessive coverage of the private firm follows by successive integration by parts (IBP). One must, however, distinguish two cases depending on whether q pr < q u , or on the contrary, q pr > q u (there is no equilibrium such that q pr = q u ). Let suppose q pr > q u , the proof of the opposite case is detailed in appendix A . By rearranging the difference θ (q pr , q u ) -θ (q pr , q u ) where the efficient and effective equilibrium coverage given the qualities q pr and q u are characterized in equations (2.8) and (3.1) respectively, we obtain θ (q u , q pr ) -θ (q u , q pr ) = (q uq pr )F( v)

[ � v v ¯[ v(q u , q pr ) -v] dF(v) F( v) -k( v) ]
A first IBP gives a correspondence between the average distance to the pivotal consumer the c.d.f of valuation distribution:

� v v ¯(v -x) f (x) dx = � v v ¯F (x) dx.
Now, by observing that F(x) = k(x) f (x), a second IBP gives:

� v v ¯F (x) dx = F(v)k(v) - � v v ¯k� (x)F(x) dx
The combination of the two gives: θ nuθ nu = (q uq pr )F( v)

� - � v v ¯k� (x)F(x) dx F( v) � . Hence, under
the assumption of log-concavity of F the last term is negative and we have shown that entry of the private firm is excessive: θ nuθ nu < 0.

To understand the result of Proposition 2, one can first observe that a higher differentiation in quality proportionally increases both, the welfare gain from the diversity of supply, and the equilibrium margin of the private firm in (3.3), through the change in the price elasticity of demand. The degree of product differentiation does not affect the sign of the difference between effective and efficient coverage, but only its extent. The sign of the difference between the efficient and the effective coverage of the private firm therefore only depends on structural properties of the distributions of the consumers' quality valuation.

Suppose for instance that q u ≥ q pr and the low-quality consumers are concentrated toward the indifferent pivotal consumers. Then most consumers of the private firm are not far from being indifferent to choosing the public provider and the diversity of supply gain from the private firm covering an area is obviously low. Therefore it would not be optimal that the private firm covers remote locations where the additional costs that this implies is high. The private firm market power and propensity to cover areas, however, does not depend on the distribution of the consumer's valuation . It depends on ϕ( v) f ( v) , the relative weight of the marginal consumers cut-off types v in its total demand. Besides, a necessary condition for the private firm to serve an interior market share at the equilibrium is that F( v) f ( v) is increasing, and 1-F( v) f ( v) decreasing ( and insured by log-concavity of F). We show that this condition also implies that the private firm should be able to extract a share of social surplus that is larger than the net welfare gains consecutive to the entry of a differentiated firm.

Though this property translates into excessive coverage in our application, it does not rely on the asymmetric obligation of ubiquity, but seems to embody a more general property on the equilibrium distribution of the social surplus in a mixed-duopoly problem with vertical differ-entiation.

The condition in (2.6) gave a condition on the size of the territory so that the discussion of asymmetric coverage is relevant. Following a similar path, the sequel we will only discuss the cases so that the size of the territory implies θ > θ . We could obviously discuss the alternative case so that the private firm chooses full-coverage. However, it appears that we would quickly be led to the conclusion that the coverage of the private firm becomes irrelevant for the the quality game if the public firm choice of quality cannot discourage the private firm choice to cover all areas.

Impact of asymmetric USOs on equilibrium qualities

4.1. The firm with USOs provides the high quality q u ≥ q pr

The firms choose the quality that maximizes respectively (2.7), and (3.2) taking into account the impact the qualities have on the equilibrium prices and coverage in (3.3), (3.4), (3.1) that implicitly define pi (q pr , q u ) and θ (q pr , q u ).

Lemma 2. If the equilibrium qualities are such that q u < q pr they must satisfy:

c � (q pr ) = v - k( v) 1 + k � ( v) (4.1) c � (q u ) = G( θ ) � v v v dF(v) + (1 -G( θ )) � v v ¯v dF(v) 1 -F( v)G( θ ) + ∂ θ ∂ q u g( θ ) θ (q u , q pr ) -θ (q u , q pr ) 1 -F( v)G( θ ) (4.2)
We find that the public equilibrium quality is left at a deficient level to limit its differentiation in quality with the private firm and maintain an intense price competition. Indeed, compared to a problem coverage concerns, the public firm tries to contain the market power which, following Proposition 2, translates into an excessive coverage that is detrimental to welfare. The second term in the RHS of (4.2) measures the impact of the public firm quality q u on the equilibrium coverage of the territory with:

∂ θ ∂ q u = F( v) � k( v) + (1 + k � ( v))(c � (q u ) -v) � (4.3)
An increase in the public firm quality increases the private firm coverage for two reasons:

it diminishes the price elasticity of demand, and increases the private firm equilibrium market share. Indeed, the public firm quality becomes inefficiently high for the cut-off consumer types v that should now consume from the private firm (the convexity of costs implies that ∂ v ∂ q u = c � (q u )-v q u -q pr > 0). The first term in the RHS of (4.2) corresponds to the average valuation of the public firm's consumers. It is enough to observe that the second term is negative to conclude that the public firm equilibrium quality is below the average evaluation of its consumers from the concavity of the public firm problem: q u < qu .

Before characterizing the result, one should make an observation on the socially optimal qualities in (2.9) and (2.10). They are defined with respect to the average valuation of each firm's consumers given the optimal coverage θ * . To the extent that the private firm coverage would be excessive even at the efficient qualities, the efficient qualities q i * , would not more be the reflect of the firms consumers valuations. Indeed θ (q u * , q pr * ) > θ * implies that the public firm serves both low and high valuations consumers in fewer areas. Therefore in the decentralized problem, the average valuation of the public firm consumers would be higher than in the efficiency benchmark at q u * , q pr * . The same remark applies concerning v. Though chosen following the efficiency criterion at the price equilibrium following lemma 1, it suffices that one of the two firm qualities differs from the efficiency level for the cut-off to depart from ṽ * = ṽ(q u * , q pr * ).

To consistently evaluate qualities w.r.t the efficiency criterion, we must therefore account for the change in consumer valuation. The characterization with respect to the efficient quality levels per se will be presented further. We define qi ( v, θ ), i = u, pr, as the qualities that satisfy the efficiency rule given the mixed-duopoly effective coverage, θ , (with qi ( ṽ * , θ * ) = q i * ).

should be welfare improving. Two contradictory forces also intervene. The choice of the quality is affected by these of the private firm, and must take into account its impact on limiting or worsening the excessive coverage of the private sector. This will imply that even if increased, the quality standard will remains deficient at the equilibrium following proposition 3. In fact, we find that the welfare gains from maintaining an intense price competition may be larger than from adapting the quality to the consumers' valuation. In this case the coverage inefficiency strictly discourages quality improvements of the public firm, and leads to a quality lower than in the efficiency benchmark even though it serves fewer areas alone. This result is summarized in Proposition 4 below.

Proposition 4. Under A.1 to A.4, the mixed-duopoly equilibrium qualities with respect to the efficiency benchmark depends on properties of distributions F and G, and are such that:

-If k(•) is concave or linear, and g(θ ) not too substantially decreasing, the mixed market equilibrium, q pr , q u is such that: q pr < q pr * < q u < q u * -Otherwise the mixed market equilibrium w.r.t the efficiency in terms of qualities is ambiguous.

The characterization of the equilibrium with respect to the efficient quality levels depends both on valuation and areas distribution and becomes difficult. Our attempts on the characterization relies on the decomposition of the equilibrium condition of the public firm in (4.2) in three effects. Two effects on the public firm's consumers' average valuation: (i) a strict quality effect of the private firm on the public firm's consumers' valuation and (ii) the excessive coverage. And (iii) one effect from the public firm's motives to contain the excessive coverage and consecutive multiplication of transportation costs (see appendix B (6.13) for the formal rearrangements of the FOC). The following details these effects for the only case we can globally conclude on. Suppose the quality of the public firm is at the efficient level q u * :

(i) if k is non-convex the private firm quality is weakly deficient qpr (q u * ) < q pr * following proposition 3. Following lemma 1 it should therefore serve fewer high-valuation consumers so its market share should be decreased such that v < ṽ * . Moreover, the social gain from its entry being lower it should cover fewer areas θ (q u < q pr ) < θ * . Hence we can conclude that, in absence of coverage issues (for instance with coverage licenses), the public firm's average valuation of consumers would still be lower, and there is a first force toward the reduction of its quality below q pr * .

(ii) The second effect now accounts for the effective coverage of the private firm that is excessive following Proposition 2. We have emphasize that it implies an increases in the average consumers' valuations of the public firm, and supports improvement of its quality above the efficient level q u * . The welfare gains from a quality improvement are larger by

F( v) � c � (q u ) -� � v � � v ≤ v�� g(θ )
on each location's type where entry is excessive, i.e. by the decreased mismatch of providing low valuation consumers with high quality in these areas.

(iii) The last effect from the moderation of the private firm's coverage, however, is larger than the change in average valuation effect from (ii) if the conditions of the first item of the proposition 4 are reunited. Indeed, following lemmas 1 and 2, the effect from the private firm gain in surplus following an increase in the public quality is F( v)(1 + k � ( v))(c � (q u ) -c � (q pr )) in each area it serves ( and follows from (4.3), see the proof in Appendix B). Increasing the private firm surplus amounts to increase its coverage, and affects g( θ ) locations. Moreover, this effect can be directly compared to the effect in (ii) because the coverage effect on the social welfare is also proportional to the extent of the excessive coverage: θθ (following equation (4.2)).

Suppose that g is not too decreasing. 8 To conclude, it is sufficient to observe that the private firm's coverage growth is larger than welfare gains from quality improvement when the private

firm quality is deficient: c � (q u ) -E � v � � v ≤ v� < c � (q u ) -c � (q pr
) (due to the weak concavity of k following Proposition 3).

In that case, the three effects have the same direction and we find that adapting the highquality to effective equilibrium coverage deters more the coverage efficiency (by benefiting more the private firm) than it benefits the public firm's pool of consumers.

8. Otherwise, if the population is too strongly concentrated in the center of the territory (if g(θ ) is too substantially decreasing) the containment of the private firm's coverage, though more effective, may only have an impact that is marginal compared to the increase average valuation of the public firm's consumers because of the "number" of areas concerned. See the decomposition (6.13) in appendix B for more details.

4.2. The firm with USOs provides the low quality q u < q pr This section briefly details the implications of our main result when the public firm with ubiquity constraint is positioned on the segment of low-valuations' consumers.

When the public firm provides the low-quality the distortion is still sourced by the prediction of excessive coverage in Proposition 2 but, quite intuitively, leads to symmetric effects in terms of equilibrium qualities. The public firm best response in quality is now such that the marginal cost of quality must equal the average valuation of its consumers augmented from its positive effect on limiting the private firm coverage:

c � (q u ) = G( θ ) � v v ¯v dF(v) + (1 -G( θ )) � v v ¯v dF(v) 1 - � 1 -F( v)G( θ ) � + ∂ θ ∂ q u g( θ ) θ (q u , q pr ) -θ (q u , q pr ) 1 -(1 -F( v))G( θ r ) (4.4)
When it is the low quality provider, an increase of the public firm quality increases the intensity of price competition. This is quite intuitive to the extent that reducing the degree differentiation between the two firms, |q prq u |, now consists in improving quality for the public firm now that it provides the low-quality. It implies that quality improvement at the public firm now has a positive effect in terms of containment of the private firm's degree of coverage.It narrows the degree of differentiation between the two firms so that consumers are more sensitive to prices' variations. The private firm's market power is therefore decreased, as well as its surplus by areas and its propensity to cover remote markets areas.

The observation below summarize this effect, and is the counterpart (4.3), and where h(•) is the inverse hazard rate defined in section 2.3:

∂ θ ∂ q u = (1 -F( v)) � (c � (q u ) -c � (q pr ))(1 -h � ( v)) � < 0 (4.5)
Proposition 3 bis below characterizes the equilibrium qualities of the two firms when q u < q pr . Proposition 3 bis The public firm's best response in quality is always excessive w.r.t its consumers' valuations of quality: qu > qu . Moreover, the private firm's best response in quality is such that:

(i) If h(•) is concave then v + h( v) 1-h � ( v) < E [v|v ≤ v],
implies that the private firm's best response in quality is deficient: qpr (q u ) < qu

(ii) If h(•) is linear then v + h( v) 1-h � ( v) = E [v|v ≤ v],
implies that the private firm's best response in quality is efficient: qpr (q u ) = qu

(iii) If h(•) is convex then v+ h( v) 1-h � ( v) > E [v|v ≤ v],
implies that the private firm's best response in quality is excessive: qpr (q u ) > qu For similar motives than in the previous section, at the equilibrium the public firm chooses an excessive quality level, given its consumers valuations, to limit the excessive coverage of the private firm. Also similarly the private firm equilibrium quality depends on the consumer distribution (see Appensix B for the private firm best response in quality). Whether the private firm provides higher or lower quality than the efficiency criterion prescribes now rely on the curvature of the inverse hazard rate. It determines how basing the choice of the quality level on relative demand h( v) of the pivotal consumers types, drives the quality away from the average valuation of the private firms' consumers.

Discussion and robustness

Existence of equilibrium qualities :

We have omitted so far to discuss whether the equilibrium qualities within sections 4.1 and 4.2 are mutual best responses in terms of firm positioning. For instance, one should consider the possibility that at the "constrained" equilibrium in section 4.1, the private firm's actual best response is to change its positioning by choosing a quality higher than the public firm's and provides the high-valuation consumers (and conversely in section 4.2). We can show that there is a unique cut-off quality of the public firm qu such that the private firm has a maximal surplus when choosing a quality below the public firm's quality if q u > qu and above otherwise. More precisely, the maximum of the private firm's surplus in its constrained problems q pr ≤ q u (q pr > q u ) is increasing (decreasing) in the public firm's quality by the Envelop Theorem. This gives a clear-cut condition to determine whether the private firm's quality in Proposition 3, and 3 bis, is a global best response.

However, such characterization becomes more difficult concerning the public firm's objective, i.e. the maximal social welfare. The reasons are similar than in our efficiency benchmark where it is difficult to determine whether the welfare is higher if the ubiquity constraint is imposed to the high quality or the low quality firm. It must exist intervals where both qualities are mutual best-responses, and they should be characterized by further works. We should however emphasize that our main result in proposition 2 does depend on the firms' equilibrium positioning. Moreover, concerning the public firm, we believe that the constrained view of positioning can be satisfying to the extent that positioning could relate other aspects such as political concerns of the local authorities. In this view, the choice of the public positioning only needs to respect that its quality is above qu if it wants to provide consumers with high valuations (in our home care problem, the highly dependent elderly) and conversely if it aims at providing the low-valuations' consumers.

Simultaneity and multi-stage timing:

We also have emphasised that an important aspect of the model is the simultaneity of public and private decision. We should here be more precise about this view. As the discussion by [START_REF] Laine | Quality and competition between public and private firms[END_REF] suggests that, holding the sequentiality of the game, the simultaneity has actually only a relative importance as long as the first best outcome can be decentralized so that commitment is unnecessary. It is not the case in our problem that the first best is decentralized and it might be a room for improvement from Stackelberg moves. The main trade-off would however remain unchanged if the sequentiality of the game remains unchanged. We have already argued in the presentation of the timing section 2.3 why the simultaneous view may seem relevant for an application to the home-care market. Its precise impact on the equilibrium distortions compared to a Stackelberg view could be assessed by further research.

It appears in fact that it should be the sequential nature of the game more than the simultaneity that matters for our discussion. In the sequential game, the private firm's expected gain from an increase in its quality is actually tempered by the presence of the public firm that adapts its price to guaranty an efficient segmentation of consumers. Due to the public firm's price reaction, the private firm's propensity to increase its price after a quality improvement is linked to the net social gain of the cut-off consumers vc � (q pr ), instead of the consumers types v gains solely.

If one supposes a Stackelberg game in which the authority commits both to price and quality standards at first, without sequentiality, the private firm would therefore not internalize that its quality decisions might induce changes in the public firm prices. More precisely, it would choose its price and quality to maximize its margin on the cut-off consumer v. Hence, in line with the seminal predictions of [START_REF] Spence | Monopoly, Quality, and Regulation[END_REF], the private firm would tend to set a very excessive quality and price regarding the consumers with low-valuations (if q u ≥ q pr and conversely), and the segmentation of consumers would be very inefficient.

We argued just above that, holding the sequentiality of the game, we believe that the main trade-offs discussed in our model would remain unchanged, and mostly the magnitudes of the distortions they implies should vary. This, however, may not be accurate concerning commitment in stage 2 since the public firm would be able to internalize the effect of its pricing on the private firm's coverage. In order to take a first step toward assessing the impact of such difference we now relax that coverage and prices are jointly chosen by the private firm. Below we present the results of an alternative timing allowing the public firm to internalize the effect of its pricing on the private firm's coverage in our simultaneous game. More precisely, we now consider that the private firm's coverage is chosen in stage 3, and is observed when the firms choose their price in stage 2. For similar reasons than before, the private firm's pricing is independent of its coverage policy so that the change of timing will not affect the private firm's pricing-strategies.

Proposition 5. If q u ≥ q pr , and the private firm's coverage is decided in stage 3, and is observed in stage 2 when prices are chosen:

(i) The private firm's best response in price is unchanged and verifies (3.3) in lemma 1 at the equilibrium. The public firm's equilibrium price verifies:

F( v)g( θ ) � θ - θ � = G( θ ) f ( v)
[ v(q pr , q u ) -ṽ(q pr , q u )]

(5.1)

(ii) At the equilibrium in price, the private (public) firm has a lower (larger) market share than what is efficient, but its coverage remains excessive: v < ṽ(q pr , q u ) and θ > θ

To highlight this complementary result, we observe that the public firm now internalizes the effects of its pricing on the private firm's coverage so the equilibrium prices and coverage change. We show that our result of excessive coverage in Proposition 2 is robust to this change.

We also find that, in order to limit the coverage of the private firm, the public firm will adopt a more aggressive pricing strategy, and end up with a larger segment of the market than it is efficient. For instance, in the case where q u ≥ q pr , it is optimal that the public firm provides high quality to consumers with relatively low valuations in each area. Despite the fact that it uses its price instrument, we find that the public firm will not completely deter the private excessive coverage because its reduction is in balance with increasing the inefficiency of consumers' segmentation. The public firm will therefore use both prices and quality to contain the private firm's coverage.

Optimal coverage of the public firm:

Finally, our main discussion has not been discussing whether the ubiquity constraint of the public firm where socially optimal, and whether a weaker coverage constraint should be imposed instead. The answer depends on the social cost to access the most remote location θ with respect to the social welfare gain to provide the consumers of these locations. In the spirit of [START_REF] Poudou | Equity justifications for universal service obligations[END_REF], too large access costs can also be balanced by inequity-aversion of the authority, but no such analysis has been conducted yet in a problem of price-quality competition. In vertical differentiation problems, the optimality of ubiquity also relies on the quality of the public firm. When it is the sole provider of given market areas, the social welfare is higher if the public firm quality is intermediate given that it serves both low and high-valuation consumers. In that sense, the excessive coverage of the private firm, which we believe would hold, by leading the public firm to reduce vertical differentiation with the private firm, could also imply that higher coverage should be optimal. Such expected effects should be explored and validated by further works.

Another interesting aspect to explore would consist in allowing a non-continuous coverage of the public firm, i.e. the possibility that the public firm could pull out some market areas that are covered by its private competitor. Such discontinuous coverage could particularly be optimal in answer to the excessive entry of the private firm in locations, i.e. where the presence of the two firms is detrimental to welfare in terms of organization of services and transportation costs. To palliate the absence of control of the private firm's coverage, the welfare could be increased if the public firm did not cover these intermediate locations, and only the private firm serves consumers. To our knowledge no work on the USOs has address the possibility of discontinuous coverage yet. The way the segment of areas that the public firm should not cover would vary with the change in equilibrium consumers segmentation in each location and firms and qualities would, however, makes the computation and characterization of the equilibrium complicated.

Concluding remarks

We have shown that the asymmetry of USOs can lead to significant distortions in terms of territorial coverage and quality standards with respect to what would be socially optimal. More precisely, we have shown that excessive coverage would systematically arise at the expense of an efficient provision of quality in the two providers. The most natural response could therefore be to introduce a coverage license on the private provider in the spirit of [START_REF] Poudou | Efficiency of uniform pricing in universal service obligations[END_REF]. If the private firms' coverage would be constrained to efficiency, the public-private interaction can achieve to decentralize efficient quality, as [START_REF] Laine | Quality and competition between public and private firms[END_REF] emphasized in a problem without coverage concerns. This, however, may be difficult to efficiently implement due to constant evolution of demand in the home-care market for the elderly, and may be more relevant for other sectors such as postal services. Moreover, in our discussion of proposition 4

we have seen that an efficient setting of coverage licenses might be insufficient. This is due to the fact that the quality equilibrium outcome under efficient coverage licenses still highly depends on the distribution of consumers' types as it is emphasized by our Proposition 3 echoing Laine and [START_REF] Laine | Quality and competition between public and private firms[END_REF]. Concerning quality per se, imposing minimum quality standard has been argued to be the most relevant way to increase average quality [START_REF] Cremer | Service Quality, Competition, and Regulatory Policies in the Postal Sector[END_REF]Grimaud, 1997, Valletti, Hoernig and[START_REF] Valletti | Universal Service and Entry: The Role of Uniform Pricing and Coverage Constraints[END_REF]). We do observe such standards in the French home-care market. Theoretically they can be particularly relevant for the unregulated firm or in privateprivate markets where neither quality is regulated. It could also play a role in our problem when the private firm tends to provide deficient qualities, but we have seen that this might not be the case and regulation of the private firm's quality could also be required to achieve efficiency.

There is however other supports for the presence of a private section competing with the public regulated sector that could balance the benefits of such complete regulation of the sector (see for instance [START_REF] Caillaud | Regulation, competition, and asymmetric information[END_REF]). Such benefits from the presence of a private sector mostly relies on agency issues in the regulation problem that are not being discussed here.

We believe that our main result could highlight aspects of public-private competition other than coverage. In particular, it would be interesting to verify how the number of firms could affect our results to both better understand the problems of coverage, and more generally, the underlying distribution of social surplus. However, the case of more than two firms is more difficult to solve. We also believe that the approach we follow could be used to highlight some features of competition in hospital-care. However, such straight extrapolation is made difficult by the variety of specificities of the hospital sector. Although vertical differentiation takes place on quality dimensions such as nursing, hotel services or physicians' skills, many other aspects are patient-type dependent. However, the cost of specialization that the hospital must incur strongly depends on the range of patients' severities the hospital intends to be able to treat. To that extent, it might be inefficient that all hospitals are able to cover the most severe patients, and a similar mechanism and stakes could take place in terms of over-specialization. In the French hospital system for instance, public hospitals have the obligation to be able to deal with all severities of patients, while private hospitals can decide to refer the most severe pathologies to public hospitals. In particular, if one had modeled the hospital specializations' cost as a function of the maximum severity of illness it can take charge of, our results in proposition 2 should hold. We think this should be investigated by further works, but this requires to modify important aspects of our problem. A main caveat would concern price-competition that, in most countries, can only occur in the private sector which is often the only sector able to vary patient's out-of pocket (hospitals prices are otherwise widely administered). The second term is simplified by using the definition of θ (q pr , q u ), and rearrangements of the first term lead to the expression (4.2) of lemma 2.

The FOC for the private firm is:

∂ Π(q pr ,q u ) ∂ q pr = ∂ θ ∂ q pr � � v v ¯( ppr -c(q pr )) dF(v) - θ � g( θ ) +G( θ ) ∂ v ∂ q pr [ ppr -c(q pr )] f ( v) + G( θ )F( v) � ∂ ppr ∂ q pr -c � (q pr ) �
Here, similarly, by the Envelop Theorem or direct use of lemme 1, the effect through the variation of θ in the first term is zero. Hence, rearranging the second term with the expression of ∂ v ∂ q pr and ∂ ppr ∂ q pr , that we detail below, yields the expression in (4.1).

The two following statements are absent from the main text but useful for the resolution, and allow to derive the static comparatives of the prices neither presented in the discussion. See [START_REF] Laine | Quality and competition between public and private firms[END_REF] for a more exhaustive presentation of those aspects of the problem that are not specific to the coverage dimension.

∂ v ∂ q u = 1 q uq pr � c � (q u ) -c(q u ) -c(q pr ) q uq pr � (6.8)

∂ v ∂ q pr = 1 q uq pr � c(q u ) -c(q pr ) q uq pr -c � (q pr ) � (6.9)

Since lemma 1 implies p uc(q u ) = p prc(q pr ), the differentiation of the firms' best responses in price 9 and the above observation leads to the comparative statics of equilibrium prices w.r.t qualities, and in particular:

∂ ppr ∂ q pr = c � (q pr ) -k( v) -k � ( v)(c � (q pr ) -v) (6.10) Proposition 3: First, concerning the equilibrium quality of the public firm, we only need to show that the second term of (4.2) is negative. Then the conclusion is straightforward, given that the first term corresponds to the definition of qu : the counterpart of the efficiency rule in Proposition 1 w.r.t the effective coverage of the private firm. The final formula of ∂ θ ∂ q pr is obtained by rearrangements of the derivative in (3.1). We use the equilibrium price definition in lemma 1 and the following observation on equilibrium segmentation :

∂ v ∂ q u = 1
q u -q pr � c � (q u ) -c(q u )-c(q pr ) q u -q pr � .

Second, concerning the characterization of the private equilibrium quality, it follows by IBP and few rearrangements. The two following statements are required for each item:

9. d puc � (q u )dq u = -(dq prdq u )k( v) -(q prq u )k � ( v) � ∂ v ∂ q pr dq pr + ∂ v ∂ q u dq u � ;

d pprc � (q pr )dq pr = -(dq prdq u )k( v) -(q prq u )k � ( v) � Then observing that F(x) = k(x) f (x) and integrating by parts we obtain:

� v v ¯F (x) dx ≡ � v v ¯k(x) f (x) dx = F(v)k(v) - � v v ¯F (x)k � (x) dx (6.12)
We give the essential steps for the concave case only, the proof of the successive item of the lemma follows the same path, except that the first inequality changes. The concavity of k(•) implies:

� v v ¯k� (v)F(x) dx < � v v ¯k� (x)F(x) dx. Rearranging we obtain =⇒ F(v)k(v) -k � (v) � v v ¯F (x) dx > F(v)k(v) - � v v ¯k� (x)F(x) dx =
F(v)k(v) 1 + k � (v) > � v v ¯F (x) dx
A last rearrangement and (6.11), gives:

v -k(v) 1+k � (v) < v - � v v ¯F (x) dx F(v) = � v v ¯v dF(v) F(v)
The conclusion on the private firm's best responses in quality rests on the convexity of the costs functions when comparing the private firms' best response to the definition of q pr in (2.9)

Proposition 4: The decomposition we use to support the discussion of the proposition and is the following:

∂W ( qu )
∂ qu (q u , q pr ) = (i)

� �� � � �[v|v ≥ v] -c � ( qu ) � G( θ )(1 -F( v)) + � �[v] -c � ( qu ) � (1 -G( θ )) + � θ θ    g(z)F( v) � c � (q u ) -�[v|v ≤ v] � � �� � (ii) - � g( θ )(1 + k � ( v))F( v)(c � (q u ) -c � (q pr )) � � �� � (iii)    dz (6.13)
The proof consists in showing that if k( v) is non-convex, so that the best response of the private firm in quality is deficient, then if the best response in quality is below the efficient level. We show that, given the private firm best response in quality, if the public firm chooses the efficient quality, then the welfare could be increased by a decrease in quality of the public ∂W ( qu )

∂ q u (q u * , qpr (q u * )) < 0. Moreover we observe that, since ∂ Π( qu ) ∂ q pr ∂ q u > 0, then qpr (q u ) is increasing in the public firm quality q u . Therefore, a decrease of q u below the efficient level q u * implies that and the private firm quality is further below the efficient level q pr < q pr * . Hence, in that case we can conclude that the equilibrium such that q u ≥ q pr must verify that q pr < q pr * < q u < q pu * Proof. We now need to prove our first statement that if k() in non-convex and g(θ ) not too decreasing then ∂W ( qu ) ∂ q u (q u * , qpr (q u * )) < 0 More generally, our aim is to drop the ambiguity of the sign of , an to do so we need to decompose the partial derivative of welfare w.r.t to the public firm quality q u and see under what conditions we can drop the ambiguity of its sign. It appears that it relies strongly on the properties of the distributions of valuation, F, and territory, G. The only situation where is not ambiguous is when k() in non-convex and g(θ ) not too decreasing in θ .

The form of the partial derivative ∂W ( qu ) ∂ q u (q u * , qpr (q u * )) can be found in appendix B equation (6.7). Our decomposition below is obtained by rearranging the first term of (4.2) with entering the efficient cut-off θ (q u , q pr ), and the second term with using (4.1). First observe that

G( θ ) � v v x -c � ( qu ) dF(v) + (1 -G( θ )) � v v ¯x -c � ( qu ) dF(v) = G( θ ) � v ṽ * x -c � ( qu ) dF(v) + (1 -G( θ )) � v v ¯x -c � ( qu ) dF(v) - � θ θ g(z) � � v v ¯x -c � ( qu ) dF(v) � dz
Then observe that, from formula (4.3) and the private firm best response in (4.1), in an equilibrium one must verify that ∂ θ ( qu , qpr ) ∂ qu = (1 + k � ( v))F( v)(c � (q u ) -c � (q pr ))

Therefore, the partial derivative can be decomposed as follows:

∂W ( qu )

∂ qu (q u , q pr ) = G( θ ) (6.14) This decomposition allows to distinguish between the effects of the distortion in qualities of the private firm w.r.t to the efficiency criterion in the first line, and the effect of the distortion in coverage w.r.t. the efficiency criterion ( θ ) in the second line. The proof therefore consists in evaluating the expression at q u * , given the private firm best response in quality is qpr (q u * ).

� v ṽ * x -c � ( qu ) dF(v) + (1 -G( θ )) � v v ¯x -c � ( qu ) dF(v) + � θ θ g(z) � � v v ¯c� (q u ) -x dF(v) � dz - � θ θ � g( θ )(1 + k � ( v)) � v v ¯(c � (q u ) -c � (q pr )) dF(v) � d
We first observe that the sign of the second line is negative when g(θ ) is not too decreasing since by log concavity we have k � > 0, and k �� ( v) ≤ 0 implies c � (q u ) -E � v � � v ≤ v� < c � (q u ) -c � (q pr ) by proposition 3. If one supposes that g is uniform or increasing the observation is made easier since, g(z) ≤ g( θ ) for all z ∈ [ θ , θ ] (with θ < θ by proposition 2). In that case, it is straightforward that since g is a p.d.f. if, moreover, we have g � > 0, then:

g(z)(c � (q u )-E � v � � v ≤ v� ) ≤ g( θ )(c � (q u )-c � (q pr )) ≤≤ g( θ )(1+k � ( v))(c � (q u )-c � (q pr )), for allz ∈ [ θ , θ ]
This last observation shows that the second line of our decomposition is negative when the condition of the first item of the proposition is concomitant. Now, to sign the first line of our decomposition in (6.14), we show that the efficient coverage of the private firm is lower if it provides a lower quality. Hence, if k() is non-convex, so that the private firm quality is weakly deficient ( qpr (q u * ) ≤ q pr * ), then θ (q u * , qpr (q u * )) ≤ θ (q u * , q pr * ).

GENERAL CONCLUSION

The ambition of this thesis is to shed light on the way competition among health-care providers can affect the modalities of treatment plans' decentralization (Chapters 1 and 2), and how the asymmetry among providers in terms of obligation of services can have an impact on the outcomes of competition between the public and private sectors (Chapter 3). This general conclusion provides an overarching view of the contributions of this thesis and presents some perspectives of research.

The first two chapters address the procedure selection issues in the problem of treatment plan's decentralization, i.e. we take into account that payment modalities to health care providers can affect their treatment praxis. We focus on the problem of an authority that aims at decentralizing the allocative efficiency of care, in terms of both treatments and testing decisions. In the view of guaranteeing an efficient use of publicly funded resources, the authority must take into account both the benefits and costs from the treatments. A central consequence is that the efficient treatment plan may actually consist in rationing the provision of an advanced costlier treatment procedure to some type of patients in particular if their exists effective and less costly treatment alternatives for these types (typically less severe patients). We find that in these circumstances, hospital competition for patients can deteriorate the effectiveness of fixed payments to decentralize the allocative efficiency. In line with the existing theoretical and empirical literature, our model reflects that payments' differentials between the alternative procedures can affect their relative level of provision. We furthermore describe how competition affects this relation and should be taken into account when setting the payments levels of each procedure. We

show that an increased competition tends to decrease the effectiveness of payments differential to discourage an over-provision of advanced treatment procedures and may lead to inflate health care spending. Somehow, the more intense the competition, the more health providers internalize that creaming patients that would be rationed at efficiency can increase their activity. We believe that these predictions could be empirically tested by using mergers of hospitals' services as an identification strategy of variations of competition intensities, and should be the topic of further works. Moreover, in our sense, these theoretical effects of competition should be looked into with a particular attention when the global health-care budget constraint becomes tight.

Our model does account for the cost of public spending, but we believe that further work should address the issues related to a limited budget in more details. In a note for public deciders, [START_REF] Askenazy | Pour un système de santé plus efficace[END_REF] emphasises the importance to take into account the opportunity costs of health-care public spending when, due to a limited budget, the authority becomes forced to decide which care provision should be fully insured or not. In this view, the opportunity cost of public spending should be endogenously determined with respect to the strength of the budget constraint, and an inflation of health care spending on some care provision could be at the expense of the coverage extent of others. How to precisely address these aspects and discuss the difficult trade-offs that limited budged implies is unclear and should carry our attention in the future.

In the first two chapters we focus on the treatment plan decentralization problem when hospitals have no obligation of service. More precisely, we have considered the possibility that payments levels are not neutral on the incentives that hospitals have to engage in patients selection or dumping. Though dumping effects can embody other ways than strict refusal, in France, this view may seem puzzling concerning the public sector that generally has universal obligation of services and, in our sens, is better suited to the analysis of the regulation of the private sector. In addition, due to the presence of a public sector with obligation of service, further works should account for the impact of USO's asymmetry on the segmentation of patients between public and private competitors. In principle, the patients' types that might be dumped into private services should actually be referred to public hospitals. This is the approach adopted by [START_REF] Ma | Health Care Payment Systems: Cost and Quality Incentives[END_REF], where it is considered that patients who are dumped end up treated at a fixed cost in a public hospital. If instead, one supposes homogeneous costs of care in both sectors, we could expect the results of such referral to be neutral in terms of allocative efficiency.It seem however, that a strict homogeneity would be a too strong assumption and that accounting for heterogeneity in hospitals in terms of productivity should be considered to conduct such analyses. Moreover, an analysis of patients' discharge should account for the hospitals' capacity constraints that are not taken into account in our current work. In terms of capacity there might be concerns such that a too large degree of discharge from the private toward the public sector could generate bulks if the public sector is not able to absorb all the demand so that rationing by queuing could furthermore arise. Further work should address this question, and explore to what extent the keeping service units dedicated to curing serious illnesses in the private sector might be optimal, and how it could be regulated by a public intervention.

The second chapter of this thesis has proposed a framework to analyze the agency issues when the decision of information acquisition does not only depend on the information structure, in our case the testing technology, but can also vary upon the ex-ante priors. We analyze how the prior beliefs about the patient's severity of illness, which may vary across patients, can affect the decision to perform tests. Relying on a novel information concept allows us to highlight the impact of priors about patients' severity of disease in a problem of decentralizing care procedures, from testing to the treatment decision. Our model notably reflects the concerns of treatment's appropriateness that have emerged in the medical literature since the development of evidence-based guidelines. In particular, it integers the prediction that if the treatment decision use the evidence-based guidelines on the sample of patients that are intended to be tested, a failure of the payment scheme to discourage over-testing can lead to a further inefficient use of resources in terms of over/under-treatments if clear guidelines are not available for the corresponding type of patient. We believe that the implications of these predictions need to be further investigated at the light of providers' heterogeneity in terms of specialization or training. Indeed, prior beliefs may not only change across patients types, but also across physicians regarding a given patient type. In this view, one should consider the possibility that some providers, due to specialization for instance, have a better evaluation patients' specific degree of affection than others. Such differentiation across providers also emphasize the critical role that the referral system should play in order to decentralize the efficient segmentation of patients across providers, and an efficient use of diagnostic tests.

The last chapter of this thesis contributes to better understand the stakes of asymmetric obligation of service between public and private providers. With the particular case of home-care services to the elderly, we emphasise that an asymmetry of coverage constraints between com-peting providers can be efficient in terms of organization of health consumers' coverage. We however show that its decentralization to the public-private competition can lead to significant distortions in terms of territorial coverage and quality standards with respect to what would be socially optimal. More precisely, we show that excessive coverage of the provider without obligation of service would systematically arise at the expense of an efficient provision of quality by both providers. A limit of this work lies into the simplified representation of the organizational costs issued from the presence of several providers on a market segment. Following the literature, we consider that transportation costs are incurred twice if the two competitors provide a given market area. We believe that a different approach wouldn't change the tradeoff that we highlight. However, we think it would be important to better understand the actual weight of diversity of suppliers in terms of the organization of coverage and have an idea of the extent to which it can inflate the total costs incurred in the market. In this view, one direction for further work would be to estimate the constraint that the consumers' segmentation between competitor providers bear on the optimization problem of home-care providers' daily rounds.

A possible approach would be to combine data on the segmentation of consumers between the providers of a particular French department, and use micro simulation to estimate how the total transportation costs incurred on the market could be decreased if the number of firms was reduced.

The approach we follow to describe the problem of home-care could be used to highlight some features of competition in hospital-care that we formerly discussed. In particular, the question that both or only one type of provider should invest in service units sufficiently specialized to take care of any degree of patients' severities. However, a straight extrapolation is made difficult because of the variety of specificities of the hospital sector that should be accounted for. Although vertical differentiation takes place in quality dimensions such as nursing, hotel services or physicians' skills, many other dimensions of quality are patient-type dependent as underlined in the two first chapters. However, the cost of specialization that the hospital must incur should strongly depend on the range of patients' severities the hospital intends to be able to treat. To that extent, it might be inefficient that all hospitals are able to cover the most severe patients, and a similar mechanism and stakes could take place in terms of over-specialization.

A propos de l'Optimalité de la Concurrence en Economie de la Santé Résumé: Cette thèse de doctorat a pour objet d'évaluer les effets potentiels d'une concurrence accrue sur le marché de la santé. Elle porte une attention particulière aux effets de la concurrence sur l'efficacité allocative du système de santé en termes de soins et de dépenses de santé. Dans leur ensemble, nos résultats suggèrent que les effets canoniques de la concurrence ne s'appliquent pas nécessairement au marché de la santé, et détaillent des circonstances dans lesquelles une hausse de la concurrence pourrait nuire au bien-être social. Cette thèse comporte une introduction et trois chapitres (articles académiques), chacun portant sur un aspect diffèrent de l'efficacité du système de santé. Le premier chapitre analyse l'impact de la concurrence sur les pratiques de soins hospitaliers et leur régulation par tarification prospective. Le second chapitre détaille les enjeux de régulation liés à la nature incomplète de l'information sur les patients au moment de choisir la procédure de soins à adopter, à commencer par la décision de mettre en oeuvre des tests de diagnostic supplémentaires. Le dernier chapitre de cette thèse discute les conséquences possibles de l'asymétrie qui peut exister entre fournisseur de soins public et privé en termes d'obligation de couverture et pouvant être appliqué au marché de l'aide à domicile des personnes âgées.

Mots-clefs: concurrence entre les hôpitaux -tarification à l'activité -efficience allocative du système de santé -obligations de service universel About the Optimality of Competition among Health-Care Providers Abstract: The purpose of this thesis is to study the potential effects of an increased competition between health-care providers on the allocative efficiency of the health-care system. In a theoretical framework it discusses the effect of competitive pressure considering the decentralization of treatment decisions, diagnostic tests' performance, and the organization of care coverage. It is composed of an introduction and three chapters (essays), each of them focusing on a different aspect of the health-care system efficiency. Our findings globally suggest that the canonical effect of competition does not necessarily applies to the health-care market, and detail circumstances in which competition could degrade social welfare. The first chapter analyses the impact of competition on hospitals' treatment praxis and their regulation through fixed-repayments. The second chapter details the regulation issues related to the incomplete nature of the information on patients' illness before the decision to perform diagnostic tests. The last chapter of this thesis discusses the possible outcomes of the asymmetry of obligation of services between public and private competitors in an application to the home-care market for the elderly.

Keywords: hospital competition -prospective payment systems -allocative efficiency of health-care systems -universal obligation of services

  d'une concurrence accrue dans ce secteur en termes d'allocation des ressources et d'étudier les modalités d'interventions des pouvoirs publics pour réguler les interactions entre prestataires de soins.

Figure 1

 1 Figure 1.1 -Implementing a given allocation: competition vs. monopoly

Lemma 3 .

 3 (i) Under A.1 to A.4, the efficient out-pockets verify p 1 (s) = p 2 (s) = p * * and v � (y, p) = 1 + λ for all s ∈ [s ¯, s], and the efficient allocation cut-offs are s * * 1 , and s * * 2 such that s ¯< s * * 1 < s * * 2 ≤ s. (ii) In monopoly case, the optimal out-of-pockets are equal, p m * 1 = p m * 2 = p * * and the results of lemma 2 hold with b ext t = v(y, p t (s)) + b t denoting the new patient net benefit from treatment in(2.4).

  4.8)For the resolution one must distinguish between three cases. The out-of-pockets can be such that (a) patients prefer the provision of the advanced treatment, (b) patients are indifferent among the two treatments, or (c) patients prefer the provision of the basic treatment. Each of the three cases appeal to a different reduced welfare function and contain a unique optimum.In case (a): v(yp 1 ) + b 1 < v(y, p 2 ) + b 2 the hospital strategic interaction is unchanged compared to the baseline model. Therefore the definition of allocation in function of repayment in lemma 2 holds, and so the reduced form of the regulation problem is similar to(3.6) except that it takes into account the effect of out-pocket level on the social surplus.In case (b): v(yp 1 ) + b 1 = v(y, p 2 ) + b 2 the hospitals' treatment decision does not affect demand, and hospitals' treatment decision rules are the same as for a monopoly .

Figure 2 .

 2 Figure 2.1 -Baseline problem

2 .

 2 More precisely, we denote the posterior p.d.f (c.d.f) for a patient of type θ after observing a signal from diagnostic test m by g θ (d|s, m) (G θ (d|s, m)). For every disease d ∈ D,a diagnostic test { f m (s|d)} s∈S,d∈D defines a conditional signal distribution on the set of signals S = [0, 1]. When the test m is applied to a patient of type θ ∈ {l, h}, the joint p.d.f (c.d.f) of signals and diseases is j

  costs for the hospital c(t, d) and transfer T . The patients' characteristics θ only play a role in assessing the ex-ante patients' degree of illness d and in updating this belief after the observation of the test result s. In other words, prior beliefs would have no impact on the problem if the information becomes complete and the d are observed before the care. The definition of the ex-ante expected surplus functions requires to specify the contracts and define care procedures. Let denote the expectation operator of the patient's severity of disease d conditional on the test results s from a diagnostic method m by � D θ [•|m, s], and by � S θ [•|m] the expectation operator for the signal (test result) of a patient type θ .

Figures 2

 2 Figures 2 and 3 gives one possible representation the restrictions. What is important to observe in this restriction is that only the degree of uncertainty about the illness varies between both patient types (d e � l � d� = � h � d�), and the probability/ prevalence of high severity of illness is higher for the h-type patients. To interpret figure 3, one could also simply think of a completely uniform distribution of disease for the h-type patient, while the uncertainty would be lower for the l type that has an inverse U-shaped function (for another illustration and interpretation see Figure2.5 in section 4.3).

Figure 2 . 2 -

 22 Figure 2.2 -ex-ante priors c.d.f

Figure 2 .Proposition 2 .

 22 Figure 2.4 gives an illustration of the impact of the signals on conditional expected costs that corresponding to the earlier restriction and Figures 1 and 2.8

Figure 2 .

 2 Figure 2.4 -Conditional expected costs from patient's severity of disease c 2 (d):

  Characterizing such property is still work in progress and requires an analysis on how the difference in the conditional expectation of both types varies with the signals, i.e. the relationship between � D l [h(d)|m, s] and � D h [h(d)|m, s].

  and reduces its intensity otherwise. With respect to our illustration in figure 2.3, such distortion would consist in limiting the provision of intermediate treatments. In order words, it would lead to more extreme variations in the recommendations for the h-types treatment.

2 �

 2 e . We denote this special case as single crossing-mean preserving spread -formally G h � SC-MPS G l . Moreover a diagnostic technology { f (s|d)} s∈S, D∈D with strict MLRP (which satisfies Assumption 2) and which is linear in d i.e. f (s|ad + b) = a f (s|d) + b. Denote the set of such diagnostic tests by F. An example of such a test is a technology where the conditional signal distribution is equal to a distorted observation of the disease, i.e. s|d = d + ε. d� = d e , linearity in d implies

  lh (d � ))g l (d � )dd � -p(G l (d) -G h (d e )) lh (d � ))g l (d � )dd � -p(G l (d) -G h (d e )) G h (d e )=G l (d e ),MLRP, T lh (d)≥d for d≥d e ≥ W I l (s, d e ) + d � d e F(s|d � )g l (d � )dd � -p(G l (d) -G l (d e )) = d � d ¯F (s|d � )g l (d � )dd � -pG l (d) = W I l (s, d)Hence W I h (s, d) ≥ W I l (s, d) for all s ∈ (s ¯, s) and d ∈ D. From (6.7) follows ( f , g h ) � w-in f ( f , g l ).

Figure 3 .

 3 Figure 3.1 describes the allocation in the case where the public firm is the high-quality provider.

Lemma 2 :

 2 Under A.1 and the assumption of costs' convexity, the programs of the private and public firms are concave in quality and respectively: v) ( pprc(q pr ))dθ ] dG(θ ) uc(q u )) dF(v) -θ dG(θ ) (6.6)Observing that the quality effect on welfare through the consumers' segmentation is zero by lemma 1, one obtains the following FOC for the public firm :∂W ( qpr , qu ) ∂ qu = G( θ ) � v v xc � ( qu ) dF(v) + (1 -G( θ )) � v v ¯x -c � ( qu ) dF(v) + ∂ θ ( qu , qpr ) ∂ qu g( θ ) � � v vx qprc( qpr ) -(x quc( qu )) dF(v) -

  the c.d.f F, is assumed to be log-concave, k � (v) > 0 we obtain:

Table of Contents

 of

	Introduction -Résumé Substantiel en Français
	General Introduction

  by definition of efficiency (2.1), (2.2),(2.3) the LHS of (4.5), (4.6), evaluated respectively at s * * 1 , s * * 2 are zero, while the RHS are negative under A.2 and A.3. Therefore, with the concavity of the regulator problem it implies that the cut-off s m * 1 , s m *

	2
	satisfying the FOC m are such that : s m * 1 < s * * 1 and s m * 2 < s * * 2 .

Maternity wards in France are categorized in function of the perimeter of pregnancy complexity their service allows to handle: from level I for normal pregnancy to III for services including notably reanimation unit in case of serious complications.23

We will not discuss in detail the implication of their differences in terms of objectives (see[START_REF] Sloan | Chapter 21 Not-for-profit ownership and hospital behavior[END_REF] and the references therein for more details).

We do not provide a detailed analysis because of this symmetry. The main intuition in this section and the optimal segmentation in the baseline model correspond to proposition 4 (ii) with the difference that out-of-pocket and disutility to participate are zero.

We already referred to this property in the proof of proposition 5.

Remerciements

We think this should be investigated by further works, but this requires to modify important aspects of our problem. A main caveat would concern price-competition that, in most countries, can only occur in the private sector which is often the only sector that able to vary the patients' out-of pocket (hospitals prices are otherwise widely administered).

Appendix A.

Appendix A. provides the minimum insights for the results of the monopoly case under assumption A.5 which implies an interior allocation of treatment at the monopoly optimum (s ¯< s m * 1 < s m * 2 < s). It also details the role of the condition in A.5 which implies an interior allocation of treatment at the monopoly optimum.

Appendix C. Lemme 3: (i) In first best, the regulator can condition the out-of-pocket on treatment and patient type, s, and chooses the treatment decision that maximizes the net social benefit from treatment, v(y, p t (s)) + b t -(1 + λ )(c t (s) -p t ) for every patient type. The concavity of patients' surplus in net income implies that the efficient participation of patients are such that: p * * 1 (s) = p * * 2 (s) = p * * and v � (y) = 1 + λ . Under A.2 to A.4, the efficient allocation of treatment is a segmentation between two cut-offs patient types with the same feature as in Proposition 1.

(ii) Monopoly hospital's treatment praxis is not affected by patients' preferences. Therefore, the authority should not distort the patients out-of pocket and one again find p m * = p * * . Furthermore, the relationship between the repayment contract and interior allocations of treatment s m 1 , s m 2 is unchanged, so as the FOC w.r.t s 1 since a change between treatments 1 and 2 do not involve a change in patients' disutility to pay and only affect welfare through cost-benefit concerns just as in the baseline model. Only the FOC with respect to the cut-off s 2 changes, but it does not affect the baseline results.

Incomplete Information and Economic Incentives in Health Care

This chapter is a joint work with Nikolai M. Brandt. Proposition 3. Under A.1 to A.4, the public firm's best response in quality is always deficient w.r.t its consumers' valuations of quality: qu < qu . Moreover, the private firm's best response in quality is such that:

implies that the private firm's best response in quality is deficient: qpr (q u ) < qpr .

implies that the private firm's best response in quality is efficient : qpr (q u ) = qpr .

implies that the private firm's best response in quality is excessive qpr (q u ) > qpr .

The characterization of the private firm quality in (i), (ii), (iii) in function of the curvature of the inverse hazard rate is obtained by successive integration by parts and detailed in Appendix B. Following Lemma 1, the presence of the public firm guarantees that the variations in the firm's market shares strictly reflect the changes in their production costs. It bounds the private firm's ability to instruments the differentiation in quality to increase its market power. As a result, the inefficiency in terms of quality of the private firm is limited compared to the standard results of private-private monopolistic competition. In particular [START_REF] Laine | Quality and competition between public and private firms[END_REF] have shown that the consumers' internalization of firms' production costs can be sufficient to imply efficient qualities at the mixed-duopoly equilibrium. It turns out that this result is retrieved when introducing coverage concerns in term of private firm quality, but the coverage inefficiency implies that it is no longer an equilibrium.

Beyond the characterization of equilibrium qualities w.r.t the efficiency criterion, one can be interested in the prediction with respect to the efficient quality levels q u * , q pr * . Due to the absence of agency issues, the efficiency in our model indeed corresponds to a fully regulated market with two firms. This configuration is also observed in the French home-care market so that such predictions could give interesting insights.

At the mixed-duopoly equilibrium, the public firm that covers fewer locations alone, and therefore has a lower proportion of low-valuation consumers compared to the efficiency benchmark in Proposition 1. This observation suggests that an increase in the public firm's quality

Appendix A

This appendix details the proof for the resolution of our benchmark problem, and provides the complementary details for the proof equilibrium price and our main result in Proposition 2.

Proposition 1: If one supposes that q nu > q u , for every area covered by both firms, lowvaluation consumers are provided by the firm u with ubiquity, and the high-valuation consumers are provided by the firm with partial coverage only. The problem is:

with ṽ being defined by equation ( 2.4) and, θ (q nu , q u ) =

Then the efficient qualities, q i * must verify:

Lemma 1 : Given that the equilibrium coverage of the private firm is θ (q pr , q u ) in (3.1), the respective objectives of the two firms are the following:

We have already argued that both firms are always active at the equilibrium so that the equilibrium qualities necessarily verify the FOC of the above objective. Besides, the log-concavity of the distribution of consumers' valuation F(v) by A.1, guaranties that the problem of both firms are concave in prices and the FOC are sufficient. Rearrangement of the firms' respective FOC gives the firm's best response in Lemma 1.

Proposition 2: The proof for the case such that q u ≥ q pr has been provided in the main text.

Now consider the alternative case such that q u < q pr .

Proof. If q u < q pr , one can rearrange the difference θθ , respectively defined in equations (6.2) and (3.1), with ϕ( v) = (1 -F( v)), as: θ (q u , q pr ) -θ (q u , q pr ) = (q uq pr )(1 -F( v)

A first IBP gives a correspondence between the average distance of the private firm's consumers' valuation and the pivotal consumer v:

The combination of the two, together with A.1 which implies h � < 0, allows to conclude directly that the structure of the distribution implies an excessive coverage of the private firm θθ < 0.

Appendix B

This appendix details our characterizations concerning the impact of our main result in proposition 2 on the equilibrium qualities. This implies that if the private firm coverage where efficient despite its quality is qpr (q u * ) < q pr * , the public firm would have more low valuation consumers than at efficiency when it chooses q u * and the private firm choose q pr * , and therefore should choose a lower quality. This is actually the first line being negative at (q u * , qpr (q u * )) when qpr (q u * ) < q pr * . In order to show this, we first observe that:

By lemma 1, the first effect through the segmentation is zero, and a deficient quality at the private firm implies that the second term is negative by following Proposition 3 (i) and (ii). Therefore, we find that ∂ θ (q pr ,q u ) ∂ q pr > 0, i.e. when its quality is deficient the efficient coverage of the private firm is decreased θ ( qpr (q u * ), q u * ) < θ * . Together with ṽ(( qpr (q u * )) < ṽ(( qpr (q u * ), q u * ) = ṽ * we can conclude that the average valuation of the public firm consumers is lower than the marginal cost of quality when its quality is efficient but these of the private firm deficient, since

Hence, the above observation directly yields that the first term in our decomposition of welfare is negative.

Finally one can observe that q u > q u * at the equilibrium, under the condition of Proposition 4 (i) so that the second term of our decomposition is negative, necessarily requires that the first term is positive (and compensates for the second term effect.) This however, given the form of the first term of our decomposition that corresponds to the partial derivative of our benchmark problem, would imply that there is another critical point (q pr , q u ) in the first best problem and is in contradiction with the quasi-concavity of the problem. Indeed, following proposition 3, if k is non-convex, ∂W ∂ q pr (q u , qpr (q u )) is also positive. Therefore, if, moreover, the first term of the decomposition in (6.14) is positive when q u > q u * , the welfare could be increased by both increasing q u and q pr , suggesting another critical point than (q u * , q pr * ) which is impossible.

Lemma 2 bis: Suppose now the case such that q u < q pr , the programs of the private and public firms are concave in quality and respectively:

Similar computations than in the alternative q u ≥ q pr yield that the equilibrium qualities from the public-private interaction are such that:

Proposition 3 bis: The second term in the RHS of (6.18) measures the impact of the public firm quality q u on the equilibrium coverage of the territory. The first statement that characterizes the public firm quality is straightforward from the efficiency criterion in (6.3) in appendix A., and the following rearrangement of the partial derivative of the private firm equilibrium coverage in (3.1) with respect to q u (with ϕ( v) = 1 -F( v)). The different steps are more details and we follow the same here.

∂ θ

The characterization of the private firm's best response in quality follows the same reasoning as in Proposition 3.

Appendix C

Proposition 5

Proof. The surplus of the private firm, if it covers the area θ , is:

Suppose the pricing rule of the public firm is such that ṽ(q pr , q u ) = v(q pr , q u ), i.e. it follows lemma 1 with the previous timing. Then, according to proposition 2 Π(q pr , q u , θ (q u , q pr )) > 0 which is equivalent to θ (q pr , q u ) > θ (q pr , q u ). Therefore, if the price of the public firm decentralizes an efficient segmentation of consumers, the LHS of (5.1) is negative and the RHS is zero so it is no more its best response. The concavity of the public firm's problem implies that the public firm's best response should be to set a lower price. Taking into account that the decrease in of the public firm prices implies a welfare loss by distorting the consumers segmentation between firms, it is not optimal that the public firm sets its price so low that the coverage of the private firm would no more be excessive.

To complete the proof, one should ensure that (5.1) never violates the participation constraint of the public firm at the equilibrium. Suppose p u = c(q u ), then v = c(q u )-p q u -q = ṽk( v) since p = c(q)+(q u -q)k( v) following lemma 1. Hence, p u = c(q u ) implies θ (q pr , q u )-θ (q pr , q u ) = � � v v ¯ṽ(q u ,q pr )-v dF(v)

� which is positive by definition of v, and we have argued that this is never the case at the equilibrium. Therefore the constraint p u ≥ c(q u ) is never violated when the FOC in (5.1) is satisfied.

List of Tables

1.1 Hospitals' strategic interaction for the choice of treatment of a given patient s . 70