Competition in the health care market: a two-sided. approach

Transcription

1 Cometition in the health care market: a two-sie aroach Mario Pezzino Giacomo Pignataro Abstract Two ientical hositals comete for atients an octors choosing locations (e.g. secializations) on a Hotelling line, an selecting the quality of the treatment an the salary for the octors. Patients ay the rice chosen by a benevolent central lanner. Introucing the resence of cross-grou externalities for the atients (i.e. ceteris aribus atients refer the hosital with the highest number of octors), we show that in equilibrium hositals always maximally ifferentiate their services. The regulator, choosing the rice, can affect only the rovision of quality that, in equilibrium, may be rovie at the socially otimal level. Keywors: rice regulation, quality cometition, satial cometition, gatekeeing JEL classification: L13, L50, R30, R38, D8, I11, I18 School of Social Sciences, University of Manchester. Manchester, M13 9PL. Mario.Pezzino@manchester.ac.uk DEMQ, University of Catania. Catania, Giacomo.Pignataro@unict.it

2 1. Introuction The aer stuies the cometition between two ientical hositals that comete for atients an octors choosing secializations, quality an salaries, when rices are regulate by a benevolent central lanner. In articular, atients an octors are uniformly istribute on a segment of unitary length an incur quaratic mismatch costs to access a hosital locate at a ositive istance. Assuming in aition that atients exerience aositive cross-grou externality with resect to the number of octors emloye in a hosital, our moel connects two recent an growing branches of literature: contributions regaring imerfect hosital non rice cometition, an works concerning the stuy of cometition in two-sie markets. On one sie, recent contributions of health economics stuy the strategic behaviour of hositals that comete for atients choosing the secialization an the quality of their service when rices are in general regulate by a benevolent central lanner. In articular, the basic frameworks of the linear (H. Hotelling, 199) city an the circular (S.C. Salo, 1979) city have been use, following the extension roose by (N. Economies, 1993, 1989) in which a quality choice stage is introuce in the game. Examles of aers stuying (uoolistic) hosital cometition when locations are on the line are given by (P.S. Calem an J.A. Rizzo, 1995), (K. Brekke et al., 006), (K. Brekke et al., 007). While in the first two aers agents are assume to be erfectly informe, in the last one the authors introuce the ossibility that atients are not informe about the isease they suffer an about the quality an secialization choices of the hositals. The common result of these aers is that in equilibrium the level of rouct ifferentiation an the rovision of quality can be resectively excessive (insufficient) an insufficient (excessive) comare to the social otimum. In (P.S. Calem an J.A. Rizzo,

3 1995) it is assume that hositals share art of the transortation costs (mismatch costs) that atients incur to reach the chosen hosital; rices are regulate, but the analysis of otimal regulation is missing. (K. Brekke, R. Nuscheler an O.R. Straume, 006) stuy otimal rice regulation aoting a similar framework, removing, however, the assumtion that hositals share mismatch costs with their atients. The Hotelling framework is borrowe also in (P.P. Barros an X. Martinez-Giralt, 00); however, the authors are intereste in stuying the cometition (in rices an qualities) between two hositals when a thir-arty ayer bears art of the atient s treatment rice eening on ifferent coayment systems. Paers that stuy oligoolistic cometition in the health market using the framework of Salo s circular city are (H. Gravelle, 1999), (H. Gravelle an G. Masiero, 000), an (R. Nuscheler, 003). Assuming hositals (or in articular general ractitioners, GPs, for the first two aers) locate on the circumference of a circle, the interest of the authors if focuse to the entry ecisions rather than the secialization choice of health services. What in our oinion has not been fully exlore in this literature is the role laye by the inut sie of the market, i.e. the octors. Hositals comete on one sie for atients, but at the same time they comete to attract octors. Some ecision variables can be secific, such as salaries or rouctivity bonuses; others variables (such as quality an secialization) chosen by the hositals affect the ecisions of octors as well as atients. Doctors coul resent ifferent references towars the secialization chosen by the hositals; in aition, in our oinion it is reasonable to assume that ceteris aribus all octors woul receive a higher utility if hire by the highest quality hosital. 3

4 Moelling hositals cometition for octors introuces another asect that the literature mentione above has not stuie yet: inirect network externalities. Such a feature has been stuie in the literature concerning the so calle two-sie markets ((B. Caillau an B. Jullien, 003), (M. Armstrong, 005, M. Armstrong an J. Wright, 004), (J-C. Rochet an J. Tirole, 003)). In a two-sie market agents belonging to two ifferent grous receive utility only from interacting with the members of the other grou through the service of a latform. The number of agents of one grou that join a latform generates an inirect network externality to the member of the other grou joining the same latform. Tyical examles of such markets are the broacasting inustry, the ayment cars sector, software inustry. In our oinion, hositals can be thought as latforms facing atients an octors on the two sies of the market for health care, esecially for services that can be offere only with the facilities rovie in a hosital. (D. Barey an J-C. Rochet, 006) introuce such an iea stuying the cometition of health lans (instea of hositals) for atients an octors. In their moel atients have ifferent robabilities of being ill an they benefit from belonging to a health lan with a large number of octors, who in turn are istribute on a Hotelling line an are ai on a fee-for-service regime (imlying that octors too have a ositive cross-grou externality towars the number of olicy holers). Hosital lans comete in the level of octors remuneration an the remium. The authors show that less restrictive lans in equilibrium may obtain higher rofits ue to the resence of cross-grou externalities. The aim of our aer is instea to stuy hosital (uoolistic) cometition when both atients an octors are istribute on a line an hositals comete for both grous choosing qualities, salaries an locations. We shall assume that rices are regulate by a 4

5 benevolent central lanner. We want to stuy the case where ceteris aribus atients receive higher utility from being serve in a bigger (in terms of number of octors) hosital, while octors (being their salary not relate to the number of atients serve) o not have any externality regaring the number of atients who ecie to be serve by a hosital. Initially, we will stuy a moel of comlete information (as in (K. Brekke, R. Nuscheler an O.R. Straume, 006)), then we will stuy the case in which atients are imerfectly informe about hositals strategic ecisions an about the isease they suffer, an GPs woul lay a gatekeeing role in the system, (as in (K. Brekke, R. Nuscheler an O.R. Straume, 007)). The timing of the game (similar to (K. Brekke, R. Nuscheler an O.R. Straume, 007)) is given as follows: - In stage 1 the regulator sets rices an ecies whether all atients shoul visit a GP (at a cost) in orer to acquire the necessary information regaring the hositals (strict gatekeeing). - In stage hositals simultaneously choose secialisations. - In stage 3 hositals simultaneously choose qualities an salaries. - In stage 4 atients an octors choose a hosital. If the regulator can not imose a regime of strict gatekeeing, an aitional stage can be introuce in between stage 3 an 4, in which atients choose whether to visit a GP at a cost. We show that, exlicitly moelling the octors sie of the market, cometition increases to the oint that for any combination of arameters hositals will choose to maximally ifferentiate their services. In contrast to what has been shown by (K. Brekke, R. Nuscheler an O.R. Straume, 006) uner the assumtion of comlete information an by (K. Brekke, 5

6 R. Nuscheler an O.R. Straume, 007) uner the assumtion that only a ortion of the atients are informe regaring their illness an hositals characteristics, the regulator can now only inirectly affect the quality rovie through the rice selection an, inee, achieve in equilibrium the socially otimal quality rovision. Location choice, instea, is not affect by the rice chosen at the beginning of the game an being maximal ifferentiation subotimal, the equilibrium can only reach a secon best situation. The rest of the aer is structure as follows. Section two stuies the moel with comlete information an escribes the results. Section three introuces incomlete information an a gatekeeing role for GPs. Section four conclues.. The moel with informe atients Let us the market for hosital services be escribe by four grous of agents. Patients Patients eman inelastically one unit of hosital care. They are istribute uniformly with ensity equal to one on a segment of length equal to one. To access a hosital they nee to ay the regulate rice > 0. In aition, atients incur quaratic mismatch cost when they are treate in a hosital not secialize in their tye of illness. The inirect utility, net of the mismatch an access costs, of the generic atients locate at location z treate in hosital i locate at x i, i=1, is given by: ( ) u = v + γn + q t z x (1) i i i i 6

7 Where v > 0 i [ 0,1] is the utility that each atient obtains when recovere from her isease, n is the number of octors working for hosital i, γ 0 is the measure of the externality exerience by the atients when an extra octor works for hosital i; q i > 0 is the quality chosen by hosital i; t > 0 is the mismatch cost arameter. Doctors Every octor sulies inelastically one unit of labour. They are uniformly istribute with ensity equal to one on a segment of length equal to one. Working in a hosital is their only source of utility. The inirect utility of the generic octor locate at y emloye by hosital i locate at x i, i=1,, is given by: ( ) u = v + ϑq + w t y x () i i i i where v > 0 is the utility that octors obtain from their job, ϑ 0 is the measure of the benefit that octors obtain from working in a hosital roviing quality equal to 1; t > 0 is the arameter of the mismatch costs octors incur from working in a hosital not secialize i their referre treatment an wi 0 is the wage ai by hosital i. Hositals There are two hositals, inexe by i=1,. They choose their locations 1 (secialization) on a segment of length equal to one; in articular, let us efine x i, i=1,, the location of hosital i, an let us assume without loss of generality that x1 x an = x x1. Hositals can also invest in quality in orer to attract atients an octors. We assume 1 Symmetric by assumtion. 7

8 fixe costs for quality enhancement. In orer to attract octors, hositals choose also the salary wi 0. The rofit function for the generic hosital i is therefore given by: where [ 0,1] i qi Π i = ni wini k (3) n is the number of atients serve by hosital i, kq i / are the fixe costs of quality enhancement an k > 0 is the arameter measuring the intensity of costs for quality. Regulator The regulator is a benevolent central lanner who sets the rice of the hosital services in orer to maximize social welfare (given by the summation of atients, octors an hositals surlus), subject to the constraint that hositals o not earn negative rofits. We are therefore focusing in the stuy of a rosective ayment system such as the DRGricing system. We want to stuy the Subgame Perfect Nash Equilibrium, SPNE, of a three stage game in which, in stage 1 the regulator sets the rice > 0, in stage hositals choose secializations x i, i=1,, an in stage 3 hositals comete for atients an octors maximizing the rofit functions choosing qualities an wages. We solve the game by the metho of backwar inuction. This is a articular version (the regulator oes not nee to imose a strict gatekeeing regime since atients are assume to be informe) of the sequencing we escribe in the revious section. 8

9 Let us start from stage 3 of the game. For given rice an secializations, hositals choose simultaneously an not cooeratively quality an wages in orer to attract atients an octors an maximize rofits. The only source of revenues comes from the rice ai by atients, however hositals comete for octors since they generate a ositive network externality on atients. In orer to stuy hositals ayoff functions, we nee to efine k =,, i.e. the eman of atients an the suly of octors for each hosital. To o so, we look for the marginal atient an the marginal octor, i.e. the atient an the octor who are inifferent to access either of the two hositals. The marginal atient is locate at z [ 0,1] given by: k n i, 1 1 ( ) = ( ) = + γ ( n n ) + ( q q ) u z u z z t (4) an the marginal octor is locate at y [ 0,1] given by: 1 1 ( ) = ( ) = + ϑ ( q q ) + ( w w ) u y u y y t (5) Given the assumtions regaring atients an octors behaviour an istribution we know that: n = z n = 1 n 1 1 n = y n = 1 n 1 1 (6) Substituting (4) an (5) into (6), an solving the equalities exresse in (6), we obtain: n n ( w1 w + ϑ ( q1 q )) + t ( q1 q + t ) 1 tt 1 γ = ( q q ) + ( w w ) 1 ϑ = + t 1 1 (7) 9

10 Using the exressions of the market shares into (3), we can now exress the generic hosital i s rofits as function of qualities an wages. Assumtions 1 an ensure the existence of the equilibrium. Proosition 1 escribes the equilibrium of the last stage of the game. Assumtion 1 Let us assume that k is sufficiently high, i.e. k max { k, k, k, k } 1 3 4, where k = ϑ t, 1 / ( ϑ ) ( t t γ ) = + +, k = ( + ϑ )( t + γϑ ) t t, ( ) k t γϑ γ k 1 /4 3 1 /4 4 = + /4 tt. Assumtion Let us assume that atient s mismatch costs are sufficiently high, i.e. ( ( ( ( ) ) { γ ϑ γϑ γϑ ) ϑ ) ( γϑ ) γ ϑ } t > max / + t / + t + t + / /3, t + /4. Proosition 1 Assume that all agents are erfectly informe, that Assumtions 1 an hol. Then hositals equilibrium quality an salary at stage three of the game are escribe by: q = + t ϑ γ w = t t kt 1 γ Π = + t t 4kt ( + t ) ϑ (8) 10

11 if tt >, γ or by: q = w = 0 Π = t + γϑ kt t 4 ( 4 ( t + γϑ ) ) kt 8kt t 4 (9) otherwise. Proof See Aenix A. The equilibrium in the quality/salary subgame can be also escribe by the following comarative static results: (a) if > : q q q q q q q < 0 = 0 = 0 > 0 > 0 < 0 < 0 t t γ ϑ k w w w w w w w < 0 < 0 > 0 = 0 > 0 < 0 = 0 t t γ ϑ k (10) (b) otherwise: q q q q q q q t < t < γ > ϑ > > < k < (11) Results are intuitive. If >, an increase in the istance between the hositals,, or in the mismatch costs lowers the egree of cometition an in turn the incentive for hositals 11

12 to rovie high quality or ay high salaries (note in aition that the assumtion of inirect network externalities for the atients rouces the result that an increase of atients mismatch costs has a negative effect on salary: if cometition for atients is softene, hositals lose the incentive to comete fiercely for octors an the salary ecrease in equilibrium). If the regulator selects a higher rice, the incentive to comete for atients increases an consequently hositals ten to rovie higher quality an ay octors better. Clearly, an increase in the cost to rovie quality has a negative effect on such rovision, but not on salary selection. Quality (but not salary) is ositively affecte by an increase in ϑ, while the salary (but not the quality) is ositively affect by an increase in γ (again, ceteris aribus, the more atients benefit from an increase of the number of octors emloye in a hosital, the higher the incentive for hositals to ay higher salaries). In comarison to revious literature, now hositals have one more reason to increase quality, e.g. to attract an higher number of octors an, in turn, to be more cometitive on the atients sie of the market. At the same time, the resence of network cross-grou externalities exlains why hositals are willing to bear an extra cost (octors emloyment) in aition to investments in quality. If instea 0 <, results are mostly unchange regaring the equilibrium quality. However, now q/ t < 0 an q/ γ > 0. For such a range of rices hositals wish to ay a negative salary to octors, but we are not allowing negative salaries an in equilibrium salaries are equal to zero; it follows that if t ecreases (the octors sie of the market is more cometitive) or γ increases (atients benefit more from interacting with octors), hositals react increasing quality (rather than salaries) to attract octors. 1

13 We can now move to stage of the game. Proosition escribes the location ecisions of the two hositals. Proosition For any given > 0 an execting ay-offs given either by (8) or (9), if Assumtions 1 an 3 hol, then both hositals location best resonse is to locate at the extremes of the unitary segment, i.e. = 1. Proof See Aenix Proosition contrasts to what has been shown in revious literature, in which hositals might even ecie to minimally ifferentiate. Now, aing a new sie to the market, i.e. the octors, increase the egree of cometition to the oint that it is always a best resonse to maximally ifferentiate from the cometitor in orer to soften cometition. In orer to stuy stage 1 on the game, it is convenient to move to the analysis of the social otimum an, therefore, the rice ecision of the regulator. Let us escribe, then, the welfare function the regulator aims to maximize. Given the assumtions of the moel, the rice ai by the atients an the wage receive by the octors o not create any istortion of the eman an suly of care. It follows that welfare is affecte only by the location an quality ecisions of the hositals. Consequently, the welfare function is given by: ( ) γ t + t 1 W = v + v + + q ( 1+ ϑ ) kq + ( t + t ) (1)

14 where the first four elements of the summation are resectively the benefit for atients an octors to access a hosital, the ositive externality for the atients when they interact with octors, an the benefits for atients an octors from accessing hosital quality; the fifth element is the quality enhancement costs that hositals (symmetrically) incur; the last two terms are given by the mismatch costs atients an octors bear to access the hositals. If the regulator coul irectly choose quality an locations, the first best otimum woul be escribe by: 1+ϑ q = k 1 = (13) However, we are assuming that the regulator can not choose quality nor locations: he can at most affect quality through the rice, knowing that in the following stage hositals woul choose to maximally ifferentiate for any given rice. Proosition 3 escribes the equilibrium of the game. Proosition 3 If Assumtions 1 an hol an γ < t + t, in equilibrium the regulator chooses a rice that generates the socially otimal quality rovision, i.e. q = (1 + ϑ)/ k an maximal rouct ifferentiation, i.e. = 1. In articular, if t + t > γ > t (an quality/salary cometition is escribe by (8)), then the rice will be equal to t = ; if γ t (an quality/salary cometition is instea escribe by (9)), then the rice will be equal to tt ( ϑ ) ( t γϑ ) = ( 1 + )/ +. If, instea, γ t + t, then the rice will be equal to 14

15 ( ) ( ) ( )( ) = kt t γ + t k t + t γ kt kγ ϑ t ϑ > 0 an the level of quality rovie in equilibrium coul be below or above the socially otimal one. Proof See Aenix From a social oint of view, the moel rouces a clear result: the regulator is able to affect through rice selection only the rovision of quality; in articular, for values of γ sufficiently low, setting the rice aequately the regulator can achieve the socially otimal quality rovision. Introucing cometition for octors generates a fiercer egree of cometition comare to the one escribe in revious contributions that justifies hositals location ecision: no matter the rice selecte an for any combination of arameters, hositals ecie to maximally ifferentiate their services, as shown in roosition. When atients value relatively strongly the number of octors emloye in a hosital, i.e. high γ, the regulator can not inirectly regulate quality to its socially otimal value without the hositals earning negative rofits in equilibrium. The rice that woul ensure a socially otimal level of quality woul be so high that hositals woul comete more fiercely for octors to attract atients increasing salaries an quality (esecially for high ϑ ). It follows that for γ t + t the regulator can only choose the rice that inuces hositals to earn rofits equal to zero, with equilibrium quality excessive or insufficient from a social oint of view. Let us now briefly consier the case in which the regulator can commit to a rice only after the two hositals have chosen locations (artial commitment). 15

16 Corollary 1 Uner artial commitment hositals maximally ifferentiate their services an the equilibrium level of quality coincies with the socially otimal one if (a) γ > t an ˆ ˆ or (b) 0 < γ t an ˆ ˆ, where ˆ t, t 1 t ˆ t ( ϑ ) + + γϑ an ( ( )) ( ( ) ) ( ) ( ) ˆ t k t γ ϑ + t kt t + γ t k γ t + ϑ > 0. Otherwise, the regulator can at most select a rice that ensures zero rofits imlying that secializations an quality can be excessive or insufficient comare to the social otimum. Proof See Aenix 3. Hosital cometition with uninforme atients This section stuies the case in which atients have not comlete information regaring the characteristics of their illness (i.e. the location on the segment) nor hosital characteristics. In articular, atients know only that they nee hosital service an the istribution of the isease, say Z = F ( z ). To obtain information atients can go to a general ractitioner, GP, who instea is erfectly informe 3. We assume that ractitioners truthfully inform atients. Let us suose that a ortion λ ( 0,1] of atients visit a GP; consequently, a ortion ( 1 λ ) of atients is not informe. Let us assume that such an uninforme ortion of atients inelastically emans one unit of care with robability equal to 1/ from either hosital. Following (K. Brekke et al., 006), we assume cost heterogeneity with resect to 3 In Brekke et al (006) GPs iagnosis is assume not to be erfectly accurate. 16

17 GP consultations. Let r [ 0,1] be the GP cost tye of a atient an let r be uniformly istribute on the line. The cost to visit a GP is equal to ar where a > 0. The timing of the game with strict gatekeeing (i.e. λ = 1) as been escribe in section 1. If instea we introuce the ossibility that atients are free to choose whether to visit a GP (at a cost), i.e. inirect gatekeeing, we have to moify the first stage of the game not allowing the regulator to set λ, an introuce an aitional stage just after quality an salary choice in orer to let the atients to efine enogenously λ. By the metho of backwar inuction, let us first stuy the game uner strict gatekeeing, analysing atients an octors hosital choice an hositals quality an salary cometition, for given, λ an secializations x i, i = 1,. For those atients who visit a ractitioner an for octors the utility is again given resectively by (1) an (); the marginal agents inifferent to chose either hositals are again locate at z an y, resectively given by (4) an (5). The eman of treatment for hosital one will be now given by: n = λz + n 1 = 1 n 1 ( 1 λ ) (14) Hositals share the market again as shown in (6). Proosition 4 escribes the equilibrium at the final stage of the game. Proosition 4 Assume that a ortion λ ( 0,1] is informe an Assumtions 1 an hol. Assume in aition that there exist maximum feasible levels of quality an salary given by q an w. 17

18 Then the equilibrium level of salary an quality rovie at the final stage of the game in equilibrium are given by: t ϑ + λ q = min, q kt γλ w = min t, w t 1 γλ Π = + t t 4kt ( t ) ϑ + λ (15) if tt > =, or λ γλ ( + ) t γϑ λ q = min, q ktt w = 0 1 t Π = 4 ( + γϑ ) λ 4 kt t (16) if otherwise. Proof See Aenix. Easy comarative static exercises show that if > : q q q q q q q q > 0 > 0 < 0 = 0 = 0 > 0 < 0 < 0 λ t t γ ϑ k w w w w w w w w > 0 > 0 < 0 < 0 > 0 = 0 < 0 = 0 λ t t γ ϑ k (17) If 0 < : 18

19 q q q q q q q q > 0 > 0 < 0 < 0 > 0 > 0 < 0 < 0 λ t t γ ϑ k (18) Changes in the arameters resent qualitatively the same effects we have escribe for the case λ = 1 in section two. Now, however, there is an aitional arameter, λ, escribing the egree of information of atients. When λ increases, clearly the egree of cometition for atients increases as well, forcing hositals to choose in equilibrium higher quality an salary (for > ). Given quality an salary escribe in roosition 4, hositals in stage two choose their secialization. Proosition 5 escribes hositals secialization ecision when only a ortion λ > 0 of atients is informe. Proosition 5 For any given > 0 an 0 < λ 1, execting ay-offs given either by (15) or (16), holing Assumtions 1 an, both hositals location best resonse is to locate at the extremes of the unitary segment, i.e. = 1. Proof See Aenix In contrast to what has been showe by (K. Brekke et al., 006), hositals have a clear an ominant incentive to maximally ifferentiate their service in orer to soften cometition. Since 0 < λ 1, cometition now coul be miler than the one escribe in the revious section. However, since they have to comete for octors as well as atients, 19

20 hositals always maximally ifferentiate. We can now move to the first stage of the game in which a benevolent regulator sets the rice of the treatment,, an can imose a regime of strict gatekeeing, i.e. λ = 1, in orer to maximize the welfare function. Given the assumtion of the moel, the welfare function is: W γ = + q + ϑq kq TC TC GP (19) where TC reresents octors mismatch costs given by: TC ( 1 ) t t =. (0) 1 4 TC reresents instea atients mismatch costs an it is given by the weighte sum of the mismatch costs of informe an uninforme atients: t t t TC = ( 1 λ ) ( 1+ 3 ) + λ ( 1 ) (1) Finally, GP reresents the cost incurre by the λ ortion of atients who ecie to visit a GP: aλ GP = () The regulator s objective function is therefore given by: γ t 1 t W ( ϑ ) q kq ( λ ) aλ 1 1 ( ) ( ( )) = (3) As ointe out alreay by (K. Brekke et al., 006) assuming that hositals o not comete for octors, the λ that maximizes welfare is ientical to the one that maximizes the atients benefit to visit a GP net of the costs, i.e. social an rivate incentives to GP 0

21 t attenance coincie an λ =. Inee, if atients were free to ecie whether to visit a 4a GP or the regulator were able to set any λ ( 0,1], for a given rice, the equilibrium of the game woul be given therefore by: if >, or by: otherwise. λ ( ) = ( ) q ( ) w λ ( ) = ( ) q ( ) w t = 4a 1 + 4aϑ = 8ka γ = t 4a t = 4a 1 = = 0 ( + γϑ ) t 8kat (4) (5) Proosition 6 escribes the comarative static roerties of the secialization-qualityconsultation equilibrium. It coul be irectly comare to Proosition in (K. Brekke et al., 006). Proosition 6 The secialization-quality-consultation equilibrium resents the following comarative static roerties: 1

22 (i) if >, GP attenance is increasing in atients mismatch costs an ecreasing in GP consultation costs; quality is increasing in the treatment rice an the benefit that octors receive from hosital quality, ϑ, an ecreasing in the quality cost k an GP consultation costs; (ii) if 0 <, GP attenance is still escribe as in (i). However, an aitional negative effect is generate to equilibrium quality rovision if the octors mismatch costs increases; (iii) salary is increasing in treatment rice, he benefit that octors receive from hosital quality, ϑ, an ecreasing in the quality cost k, GP consultation costs an octors mismatch costs. While the escrition of equilibrium salary is quite intuitive, there are some clear ifferences between the results escribe in roosition 5 an those reorte in roosition in (K. Brekke et al., 006) an all of them are generate by the maximal ifferentiation result escribe in roosition 4 above. For examle, in (K. Brekke et al., 006) an increase in k, woul ecrease equilibrium quality rovision an consequently hositals incentive to ifferentiate; in turn, a ecrease in woul ecrease the benefits of consulting a GP, having a negative effect on λ. With a similar reasoning (K. Brekke et al., 006) exlain the ositive relationshi between λ an the rice or atients mismatch costs. However, since in our moel = 1 regarless the regulate rice an GP attenance, in equilibrium λ is not affecte by atients mismatch costs, quality costs nor treatment rice. Regaring equilibrium quality rovision, our results are in line with the literature. In aition, there is one more arameter that affect quality rovision: an increase in ϑ has a

23 irect ositive effect on quality rovision. When is sufficiently small, also octors mismatch costs have a (negative) effect on quality: if octors mismatch costs ecreases cometition for octors is fiercer an hositals have force to rovie higher quality. For a given rice, quality exresse in (4) an (5) are not necessarily otimal an if the regulator can at most imose a strict gatekeeing system, i.e. λ = 1, we have to consier when, for quality given by (15) or (16) an for = 1, the first erivative of the welfare function with resect to λ is non negative, imosing λ = 1. Such a erivative is given by: λ 1 ( ) W t + kt t 4a = λ = 4kt (6) if >, or ( 4 ) ( 1 ϑ )( γϑ ) ( γϑ ) W kt t t a + tt + t + t + λ = λ 1 = 4kt t (7) otherwise. Whether λ = 1 is a socially otimal strategy eens on the arameters of the moel (rice for the moment is assume to be exogenous). In articular, for a rice an GP consulting cost a sufficiently low an for octors mismatch costs t. Let us now introuce the ossibility that the regulator can set the rice to maximize W. Given the fact that the rivate incentive to visit a GP is equal to the socially otimal one, the regulator has only to set an let atients free to ecie whether to visit a GP. In other wors, as alreay argue by (K. Brekke et al., 006), it is not necessary to set u a strict 3

24 gatekeeing system if the regulator can set the treatment rice in orer to maximize welfare. Proosition 7 escribes the equilibrium of the game. Proosition 7 If Assumtion 1 hols, t ( ) γ > an a sufficiently large, i.e. ( γ ) ( ϑ) in equilibrium the regulator chooses a rice, a ( ) rovision of quality. If instea ( γ ) ( ϑ ) that hositals o not earn negative rofits, choose the rice: a 4k t k, = = 4 that ensures the otimal 0 < a < 4k t k the regulator, to ensures ( γ ) ϑ ( ( γ )( γ ϑ )) = 8ka 4a 4a + 8a k t + 4a 4ka k > 0 an quality rovision can excessive or insufficient from a social oint of view. If 0 < γ t an a is sufficiently high, i.e. a ( 1 + ϑ )( t + γϑ ) /16kt, in equilibrium the regulator again chooses a rice, 4at ( 1 ϑ ) / ( t γϑ ) rovision of quality. If instea ( )( ) hositals o not earn negative rofits, choose the rice: ( ) 64 / 0 = = + + that ensures the otimal 0 < a < 1 + ϑ t + γϑ /16kt the regulator, to ensures that = kt a t + γϑ > an quality rovision can excessive or insufficient from a social oint of view. For any combination of arameters the equilibrium level of horizontal ifferentiation is maximal an therefore excessive comare to the social otimum. Proof See Aenix. Results escribe in the revious section hol also for this version of the moel with imerfect information. The regulator can choose one variable, the rice, an consequently can at most affect the rovision of quality to the socially otimal level. In contrast with revious literature, the level of horizontal ifferentiation in the hosital market will always 4

25 be maximal, since now hositals comete simultaneously for atients (through quality an number of octors emloye) an for octors (through salaries). 4. Conclusions The aer stuie the cometition between two ientical hositals in a two-sie market comose on one sie by atients who require hosital treatment an on the other sie by octors who obtain ositive utility from working in a hosital. Assuming that both atients an octors resent a continuum of references with resect to the isease sace (the unit segment), hositals comete simultaneously for both grous of agents selecting the secialization first, the quality of the service an the salary for the octors then. The rice of the hosital treatment is centrally set by a benevolent regulator (a similar system can correson to the DRG-ricing system), an, if atients are not informe, a strict gatekeeing system can be set. In contrast to what has been emonstrate in relevant revious literature, the aer shows that for any ositive rice selecte by the regulator an for any level of information among atients, hositals maximally ifferentiate their services. Introucing salary cometition for octors increases the level of cometition between hositals an in turn justifies a centrifugal force in the market that generates a level of rouct ifferentiation always excessive comare to the social otimum. The regulator, however, can still inirectly affect the quality rovision that, when atients exerience sufficiently low cross-grou externalities with resect to octors emloye in a hosital, in equilibrium coincies with the socially otimal one. 5

26 Aenix Proof of Proosition 1 Substituting exressions (7) into (3), the rofits for hosital i, i = 1,, j = 1, i : Π = i ( ( i j + ) + γ ( i j + ϑ ( i j ))) i + i i j + + ϑ ( i j ) ( ( )) t q q t w w q q t kq t w w w t q q t t Secon orer conitions, SOCs, for rofit maximization are given by Πi/ wi = 1/ t, Det ( H ) ( 4 kt ϑ )/( 4t ) i i (8) Π / q = k, = +, where H is the Hessian matrix of the rofit function, an are satisfie if 4 >, that is true given the Assumtion 1. kt ϑ First orer conitions, Focs, are given by: ( + γϑ ) t ( kqt + w ϑ) Π t i = = 0 q i i i tt ( ) Π γ + t w i j wi t qiϑ + qjϑ = 0 = w t t i (9) Solving simultaneously the Focs, the caniate subgame symmetric (i.e. q1 = q = q an w1 = w = w ) equilibrium is escribe by (8). The salary in (8), for a range of rices sufficiently low, i.e. 0 <, coul be negative. In such a case, since we are imosing that hositals can not select a negative salary, hositals set a salary equal to zero an the caniate equilibrium in (9) follows. To rove that the quality an the wage in (8) an (9) are inee an equilibrium, we have to rove that no hosital has the incentive to unercut the cometitor on either sie of the market. Let us consier first the atients sie of the market. To treat all atients, either of the two hositals, say hosital 1, has to choose a combination of quality an wages such that even the atient locate at the oosite extreme of the segment is willing to be treate by the unercutting eviating hosital. Hosital 1 has therefore to choose: 6

27 ( 1 ) ( 1 ) γ ( 1 ) U x U x q q + t + n (30) where n 1 is given by (7). The eviating hosital has no incentive to choose a quality higher than the one exresse in (30). Suose >, substituting the exression of quality given by (30) into (7) an (3), an imosing that the quality an salary chosen by hosital are given by (8) first, we obtain hosital 1 s rofits as a function of the wage w 1. Such rofits are maximize for w 1 ( ( )) ( ) ( ) t ( t + γ ( kγ + ϑ )) ( ) kγ γ + t t t γ t + γϑ t t + t ϑ γ = (31) Substituting the wage given by (31) into (30), we have that the require quality to unercut is given by: q 1 ( γ ) 1 ϑ t t k = + t + + kt k t + γ ( kγ + ϑ ) (3) The rofits of eviation for hosital 1 are therefore given by: ( ( ) )( ) ( ( ) ) 1 t + kt γ t + γϑ 4kt ϑ k t ˆ γ + ϑ Π 1 = 8k t + γ ( kγ + ϑ ) t t (33) an they are less than the rofits given by (8). Let us now consier the case 0 < an substitute the exression of quality given by (30) into (7) an (3), an imose that the quality an salary chosen by hosital are given by (9). Following the revious reasoning, 7

28 we can show that the rofits of eviation are less than the rofits given by (9) as long as k k 1 ( γ ϑ ( ( ) ) ) γϑ γϑ ϑ an t t t ( t ) > / + / / /3 > 0, as assume in Assumtions 1 an. Since hosital is erfectly symmetric, no hosital has incentive to unercut the cometitor on the atients sie of the market. We have now to rove that the same is true for the octors sie too. In orer to attract all octors in the market the eviating hosital, say again hosital 1, has to offer a combination of salary an quality such that: ( 1 ) ( 1 ) U x U x w ϑq + w + t ϑq (34) Hosital 1 clearly has no incentive to choose a salary higher than the one exresse in (34) with equality. If > an we substitute the salary given by (34) into (7) an (3), an imosing that the quality an salary chosen by hosital are given by (8), we obtain the rofits of the eviating hosital as a function of its quality, i.e. q 1. The quality that maximizes the eviating hosital s rofits is: q 1 = + t ϑ kt (35) an the unercutting salary follows from (34): w 1 γ ϑ = t k (36) Assumtion 1 ensures that the salary given in (36) is ositive an rofits generate from eviation are less than rofits given by (8). Similarly, if 0 < an hosital 1 eviates from equilibrium escribe in (9), it can not obtain higher rofits. Hosital s behaviour is erfectly symmetrical an that roves that no hosital has incentive to unercut the cometitor. Finally, in orer to rove that the exressions in (8) an (9) reresent inee an 8

29 equilibrium uner the assumtion of comlete information we nee to rove that no hosital has the incentive to leave the market for octors an to attract atients through quality rovision. In orer to leave the market for octors, the eviating hosital, say again hosital 1, has to choose a salary such that: If ( ) ( ) U x U x w < ϑq + w t ϑq (37) <, eviating from equilibrium escribe by (8), for ( ) even for a eviating quality q 1 = 0 < t 4 kt ϑ /( k γ + ϑ), hosital 1 shoul choose a negative to leave the octors sie of the market, but given that we are not allowing such a ossibility in the moel, that is not feasible. For t ( 4 kt ϑ )/kγ t ( 4 kt ϑ )/( kγ ϑ) > > +, there is a non emty subset of quality an salary for which the eviating hosital can not attract any octor; however, for such a range of the quality that woul maximize the rofits of eviation woul again involve a negative salary. Therefore, for in such a case hosital 1 can at most choose w 1 = 0 an the corresoning quality that woul ensure not to emloy any octor woul be given by ( ) q1 = kγ 4 kt t + ϑ + t ϑ /kt ϑ. It can be shown that the rofits that follow from such kin of eviation are less than those reorte in (8) as long as k > k1. The same is also true for higher if ( ) > t 4 kt ϑ / k γ, that involves a ositive salary of eviation. If 0 < an hosital 1 is eviating from the equilibrium escribe by (9) (accoring to the eviation conition (37)), the salary that woul ensure rofits maximization in eviation woul be negative given that k > k1, requiring the hosital to choose a salary equal to zero. It follows that the rofits of eviation are again less than rofits when salary an quality are given by 9

30 (9). Symmetry ensures that hosital is not willing to eviate either, concluing this roof. Q.E.D. Proof of Proosition Given ay-offs in (8) or (9), Socs are easily verifie. The first erivative of rofits with resect to locations is given resectively by: 1 3 ( γ ϑ ) 3 Π Π + ktt + t k + = = > 0 x x 4kt (38) if rofits are given by (8), or by: 1 ( + γϑ )( t + γϑ ) t Π Π = = > 0 x x 4kt t 5 (39) It follows that the two hositals locate at the extremes of the segment at stage two, i.e. = 1. Q.E.D. Proof of Proosition 3 Suose first that t + t > γ > t. Substituting the exression for quality given by (8) into the welfare function (1), imosing = 1 an maximizing 4 with resect to, we obtain t = >. The rice that maximize the welfare function (1) when quality is given by (9) woul be given by ( ( 1 ϑ ))/( γϑ ) t t t t + t > γ > t the = + + >. It follows that for best feasible rice the regulator can choose to ensure that the equilibrium quality an salary are given by (9) is. Since W ( ) W ( ) <, it follows that for t + t > γ > t the 4 Socs are always satisfie. 30

31 caniate otimum rice is. Profits are concave in an equal to zero for ( ) ( ) ( )( ) 1, = kt t γ ± t k t + t γ kt kγ ϑ t ϑ > 0, 1. If k k, as escribe in Assumtion 1, then, an rofits are greater than zero in 1 equilibrium. For extreme values of γ, = can not be an equilibrium. If γ > t + t, >, imlying negative rofits in equilibrium. Therefore the regulator can at most choose = with equilibrium rofits equal to zero. Finally, for 0 < γ < t, we have that < <, imlying that the best feasible rice for the regulator to imose the regime escribe in (8) is given by. Since W ( ) W ( ) <, it follows that for t > γ > 0 the caniate otimum rice is. Profits are again concave (now since woul inuce an equilibrium with salaries equal to zero, rofits are equal to zero for { 0,(4 kt /( ) ) t t t } = + ). ensures ositive rofits in equilibrium for k k3 as efine in Assumtion 1. Q.E.D Proof of Corollary 1 Stage 3 cometition rouces again the exression (8) an (9) in Proosition 1. In stage the regulator selects the rice to maximize the welfare function. The welfare function is given by substituting into (1) the quality exresse (8) if >, an the quality exresse in (9) if 0 <. If γ > t, the welfare function is maximize (an stage 3 continuation rouces the socially otimal level of quality) for ˆ t =. In stage 1 rofits are strictly concave in > 0 an equal to zero for 31

32 1, ( ( )) ( ( ) ) ( ) ( ) = ˆ t k t γ ϑ ± t kt t + γ t k γ t + ϑ > 0. It follows that the best strategy for the regulator when γ > t, is to choose min { ˆ, ˆ } =. In stage 1 hositals, if execting = ˆ, woul choose to maximally secialize their services; in fact, the two hositals have to choose secializations to maximize rofits: ( ( ) ) ( ) + γ + ϑ 4 k t t Π = 1 (40) 8k obtaining Π = t + t an = 1. Otherwise, execting zero rofits in stage 3, the secialization choice oes not matter anymore: there is a continuum of equilibria, eening on [ 0,1] from a social oint of view., where the equilibrium quality can be excessive or insufficient If instea 0 < γ t, the welfare function is maximize (an stage 3 continuation rouces the socially otimal level of quality) for t 1 t ˆ = t ( ϑ ) + + γϑ. It follows that the best strategy for the regulator when γ > t, is to choose = min { ˆ, ˆ } In stage 1 hositals, if execting = ˆ, woul choose to maximally secialize their services; in fact, the two hositals have to choose secializations to maximize rofits: Π = ( ) ( 1+ ) t ( 4kt 1) ( 1+ ) ϑ ϑ ϑ γϑ 8k t ( + γϑ ) (41) 3

33 an it follows that Π = t + t, imlying that the two hositals will locate at the extremes of the segment again. If instea ˆ ˆ <, execting zero rofits in stage 3, there is a continuum of equilibria, eening on [ 0,1], where the equilibrium quality can be excessive or insufficient from a social oint of view. Q.E.D. Proof of Proosition 4 Given that λ ( 0,1] an given Assumtions 1 an it can be easily shown (following the same roceure we use for roof of Proosition 1) that no hosital has the incentive to eviate from the equilibria escribe in Proosition 4 unercutting the cometitor on either sie of the market, nor leaving the octors sie of the market choosing a salary sufficiently low. However, given the ossibility that a ortion ( 1 λ ) of atients is not informe, another ossibility of eviation arises: hositals might ecie to serve only the uniforme ortion of atients an ay a salary equal to zero an rovie quality level equal to zero. If a hosital ecies to o so, it will earn rofits equal to: ( 1 λ ) Π ˆ = (4) Let us first suose that > an efine the ifference between the rofits given by (15) an those given by (4) as: φ (, λ ) ( ( λ ) γλ ) ( ϑ λ ) 4kt t t + t + = Π Π ˆ = 8kt (43) It can be easily verifie that φ (, λ) / < 0, φ ( 0, λ ) > 0, lim φ (, λ ) { } there is a limit rice, say ( ) st φ ( ) =. So ˆ = 0,..,1 = 0, such that for < ˆ then 33

34 φ (,1) 0. Given the monotonic relationshi between q an w with resect to an λ, a unique salary an quality level will correson to such limit rice ˆ. In articular, ( ( )) ( ) ( ) ( ) ˆ = t kt ϑ kγ + kt t + γ t k γ t + ϑ (well efine an { } larger than if Assumtions 1 an hol). Let us efine min ˆ [ 0,1] ɶ an q ( ɶ λ ) an w arg max Π ( = ɶ, λ = 1) arg max Π =, = 1 (given the assumtion of the moel, the corner solutions are symmetric). The uer bouns on quality an wages ensure that for rices higher than ˆ, hositals have no incentive to eviate an serve only uninforme atients. It can be easily verifie that eviation from the corner solution (in which hositals select q1 = q = q an w1 = w = w ) is not rofitable. If instea, 0 < the caniate equilibrium is given by (16). The eviation rofits are again given by (4) an the ifference with rofits given by (16) is now given by: ( + γϑ ) ˆ (, 1 ) 4 λ t φ λ = λ 4 8 ktt (44) ˆ, / 0 ˆ 0, 0 ˆ =. It can be easily verifie again that φ ( λ ) <, φ ( λ ) =, lim φ (, λ ) ˆ { } ˆ = 0,..,1 = 0, such that for So there is again a limit rice, say ( ) st φ ( ) ˆ then ˆ,1 0 ( ) ˆ = 4 kt t / t + γϑ > it follows that for 0 < hositals 4 φ. Since, ( ) ( ) have no incentive eviate from the equilibrium an serve only uninforme atients. 34

35 Proof of Proosition 5 For any give rice an λ hositals choose secializations x i, i = 1, to maximize rofits given by (15) or (16). The Socs are always satisfie an the Focs are given by: 1 ( ) Π Π kt t + t λ kγ + ϑ + λ = = < 0 x x kt (45) if >, or 1 ( + )( t + ) Π Π λ t γϑ γϑ = = < 0 x x 4kt 3 (46) if 0 <. It follows that hositals always maximally ifferentiate, regarless the rice chosen by the regulator or the number of atients that ecie to the visit a GP. Q.E.D. Proof of Proosition 7 Let us suose first that γ > t. Substituting the equilibrium values exresse in (4) into (3), we obtain the exression of welfare as a function (strictly concave) of rice, maximize (imlying otimal quality rovision) for = =, when in the final stage of 4a the game octors receive a ositive salary. If instea we substitute the equilibrium values exresse in (5) into (3), we obtain the exression of welfare as a function (strictly concave) of rice, maximize (imlying otimal quality rovision) for ( ϑ ) ( γϑ ) = = + +, when in the final stage of the game octors receive salary 4at 1 / t equal to zero. Given that γ > t, is not feasible, since ( ) equilibrium rice. If ( γ ) ( ϑ ) 0 a 4k t 1 16k < < + +, ( ) > an is the caniate Π < 0 an the regulator has to choose the rice that at least ensure non negative hosital rofits (not otimal quality rovision): ka ( a γ ) aϑ a k ( t ( a γ )( ka kγ ϑ )) = > 0. 35

36 If 0 < γ t, is not feasible, since < an < a < ( + ϑ )( t + γϑ ) kt, ( γ ) ( ϑ ) 0 1 /16 is the caniate equilibrium rice. If ( ) < < + +, ( ) 0 a 4k t 1 16k Π < 0 an the regulator has to choose the rice that at least ensure non negative hosital rofits (not otimal quality rovision): ( ) quality rovision choosing rices ( ( γ ) ( ϑ) ) a 4k t k or if 0 γ t = 64 kt a / t + γϑ > 0. The regulator can ensure otimal or resectively if t < an ( )( ) γ > an a 1 + ϑ t + γϑ /16kt.Q.E.D. References Armstrong, M. "Cometition in Two-Sie Markets." 005. Armstrong, M. an Wright, J. "Two-Sie Markets, Cometitive Bottlenecks an Exclusive Contracts." 004. Barey, D. an Rochet, J-C. "Cometition among Health Plans: A Two-Sie Market Aroach " 006. Barros, P.P. an Martinez-Giralt, X. "Public an Private Provision of Health Care." Journal of Economics an Management Strategy, 00, 11(1), Brekke, K.; Nuscheler, R. an Straume, O.R. "Gatekeeing in Health Care." Journal of Health Economics, 007, 6, "Quality an Location Choices uner Price Regulation." Journal of Economics an Management Strategy, 006, 15(1), Caillau, B. an Jullien, B. "Chicken an Egg: Cometition among Intermeiation Service Proviers." Ran Journal of Economics, 003, 34(),

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