Universidade do Minho
Escola de Economia e Gestão
Luís Carlos de Sousa Sá
Quality and Specialisation in Regulated
Markets with Semi-Altruistic Providers
Luís Carlos de Sousa Sá
Quality and Specialisation in R
egulated Mark ets wit h Semi-Altruis tic Pro vider s 16
Luís Carlos de Sousa Sá
Quality and Specialisation in Regulated
Markets with Semi-Altruistic Providers
Trabalho efetuado sob a orientação do
Professor Doutor Odd Rune Straume
Dissertação de Mestrado
Mestrado em Economia
Universidade do Minho
Escola de Economia e Gestão
DECLARAÇÃO
Nome:
Luís Carlos de Sousa Sá Número do Cartão de Cidadão: 13386813
Título da dissertação:
Quality and Specialisation in Regulated Markets with Semi-Altuistic Providers Orientador:
Professor Doutor Odd Rune Straume Ano de conclusão: 2016 Designação do Mestrado: Mestrado em Economia Endereço eletrónico: [email protected]
É AUTORIZADA A REPRODUÇÃO INTEGRAL DESTA DISSERTAÇÃO APENAS PARA EFEITOS DE INVESTIGAÇÃO, MEDIANTE DECLARAÇÃO ESCRITA DO INTERESSADO, QUE A TAL SE COMPROMETE.
Acknowledgements
First and foremost, a word of endless gratitude to my supervisor, Professor Odd Straume, for all the patience, guidance and support he has given me over the past four years in which we have been lecturer and student, supervisor and aspiring researcher, several times. He has my most profound admiration and respect. To Professor Paula Benesch for having introduced me to Health Economics and, above all, for the enthusiasm and immense belief in my doings. To Professor Rosa Branca Esteves for having taught me Game Theory, to Professor Francisco Carballo Cruz for having taught me how to think like an Economist, and to both for all the countless times they have helped me beyond their responsibility.
Be certain that the reach of my gratitude is not limited by words. For that reason, I ought to thank all the teachers and lecturers whom I have encountered throughout the years and whose name I have not written but surely not forgotten. From my school in my hometown Ponte da Barca to EEG. In some way, this dissertation and myself are the result of your hard work.
To my friends and family and, mostly, to Lu´ısa, for several months of hearing me talk about hospitals, quality, and equilibria. Thank you all for listening to me.
Writing this dissertation was a work of finding le mot juste—the right word. Also the right mathematics. If an inattentive reader captures the intuition of my results on a first reading, then my work was successful.
Quality and Specialisation in Regulated
Markets with Semi-Altruistic Providers
Abstract
In markets subject to price regulation, like health care, firms resort to variables such as quality and location to attract consumers. In a sequential game, semi-altruistic providers choose locations first and then quality levels strategically. The regulator devises a payment scheme to finance health care providers that consists, at least, of a prospective amount paid for each unit of the medical good supplied. When the regulator is able to commit to a prospective payment prior to any move by the providers and if altruism is sufficiently low, there is a trade-o↵ between quality supply and horizontal di↵erentiation, and the first-best outcome is generally not achieved. If altruism is above a certain threshold, there is oversupply of quality and underdi↵erentiation in equilibrium. When the regulator sets the prospective payment only after location choices are made, the first-best quality level is elicited by setting the appropriate prospective payment, and over-, but not maximum, di↵erentiation arises. In the absence of altruistic preferences, on the other hand, providers choose to maximally di↵erentiate. In a game with the same sequence of moves, it is also shown that, by letting the regulator place a restriction on the surpluses providers are allowed to retain prior to location decisions are made, the equilibrium level of horizontal di↵erentiation, though still inefficiently high, is reduced and welfare improved.
Qualidade e Especializa¸c˜ao em Mercados
Regulados com Empresas Semi-Altru´ıstas
Resumo
Em mercados sujeitos a regula¸c˜ao de pre¸cos, tal como nos mercados de cuidados de sa´ude, as empresas recorrem a vari´aveis como a qualidade e a localiza¸c˜ao para atrair consumidores. Num jogo sequencial, empresas semi-altru´ıstas definem local-iza¸c˜oes primeiro e depois n´ıveis de qualidade estrategicamente. O regulador desenha um esquema de reembolso para financiar os prestadores de cuidados de sa´ude (as empresas), que consiste, pelo menos, num pagamento prospetivo por unidade do bem m´edico fornecido. Quando o regulador estabelece um pagamento prospetivo antes de qualquer a¸c˜ao por parte das empresas e se o altru´ısmo for suficientemente baixo, aquele enfrenta um compromisso entre qualidade e diferencia¸c˜ao, n˜ao sendo o resultado socialmente ´otimo geralmente atingido. Se o altru´ısmo for suficiente-mente alto, excesso de qualidade e diferencia¸c˜ao insuficiente emergem em equil´ıbrio. Quando o regulador estabelece o pagamento prospetivo apenas depois de as empresas escolherem a sua localiza¸c˜ao, o n´ıvel socialmente ´otimo de qualidade ´e induzido pelo pagamento prospetivo apropriado e diferencia¸c˜ao excessiva, mas n˜ao m´axima, ocorre em equil´ıbrio. Na ausˆencia de preferˆencias altru´ıstas, contudo, as empresas optam por diferencia¸c˜ao m´axima. Num jogo idˆentico, ´e demonstrado que, ao permitir que o regulador imponha uma restri¸c˜ao aos excedentes que as empresas podem reter anteriormente `as suas escolhas de localiza¸c˜ao, o valor de equil´ıbrio de diferencia¸c˜ao horizontal, apesar de ainda ineficientemente alto, ´e reduzido e o bem-estar social incrementado.
Contents
1 Introduction 1
2 Literature Review 7
3 The Model 13
3.1 The Quality Subgame . . . 17
3.2 The Location Subgame . . . 21
3.3 Optimal Regulation with Full Commitment . . . 25
3.4 Optimal Regulation with Partial Commitment . . . 29
3.4.1 Optimal Prospective Payment . . . 30
3.4.2 The Location Subgame . . . 32
3.5 Optimal Regulation with Two Instruments . . . 36
4 Concluding Remarks 43
1
Introduction
In markets where firms are subject to price regulation, other variables are brought into play in order to attract consumers, whose choices are, then, mainly based on cri-teria such as quality and distance. This is a fairly accurate description of health care markets. In most OECD countries, out-of-pocket expenditures borne by patients are inexistent or amount only to a small share of the total cost of the service provided1—
making quality and distance much more relevant determinants of consumer choice—, and health care providers are compensated by a third payer (a public or private in-surer, the government or a regulator, for example) according to some reimbursement scheme. Markets for higher education in several European countries are also well described by this framework. Tuition fees play a relatively minor role in these coun-tries, and the quality and location of the institution are usually decisive factors of a potential student’s choice. Since funding is commonly based on student attendance, it is reasonable to consider that universities try to attract students through the qual-ity and the characteristics of the training they provide. While the former may be
1Automatic health coverage is provided to the entire population and financed from taxes in
13 OECD countries (Australia, Canada, Denmark, Finland, Iceland, Ireland, Italy, New Zealand, Norway, Portugal, Spain, Sweden and the United Kingdom); Ten countries rely on social health insurance, compulsory for all or almost all of the population and mainly financed through income-related social contributions (Austria, Belgium, France, Germany, Greece, Hungary, Japan, Korea, Luxembourg, and Poland); In the US, the Medicare and Medicaid social health insurance pro-grammes cover, respectively, individuals aged 65 and older who have worked and contributed to the system, and low income families.
related to faculty curricula and facilities, like laboratories and libraries, the latter is typically related to the type of courses o↵ered, or actual geographical location. For clarity of exposition, the analysis is focused on the case of health care markets. One should, nonetheless, keep in mind that it is not exclusively applicable to such markets.
In the absence of price as a strategic variable, firms normally face decisions along, at least, two dimensions. First, they must decide where to locate and, then, they ought to decide on the quality of the services provided. Since both decisions a↵ect not only costs, but also demand, and, in turn, the payments received from the third payer, firms are assumed to set quality (vertical di↵erentiation) and to choose location (horizontal di↵erentiation) strategically. Indeed, Tay (2003) finds empirical evidence that supports the hypothesis that distance and quality are important determinants of hospital choice. Quality is interpreted as a characteristic of health care that improves the e↵ectiveness of treatment in terms of its outcome or any dimension of patient well-being. It is a result of investment in medical equipment, diagnostic tests, and other amenities, as well as contracts with highly skilled physicians and nurses. Locations might be interpreted, at least, in three di↵erent ways. Firstly, the set of locations may be thought as the geographical space. In this case, a health care provider’s location is simply the place where its physical facilities were built. Secondly, locations might be understood as the product space. This is particularly the case of private practices that position themselves in the market through advertising in an e↵ort to di↵erentiate from the competitors. Thirdly, locations can be interpreted as the providers’ speciality mix or, in other words, the treatments and services they o↵er. Even if providers (are legally obliged to) supply health care from a wide range of medical specialities, they may, formally or informally, decide to concentrate e↵orts on improving their performance in a given field—i.e., they may specialise. Henceforth, the terms location and specialisation, travelling/transportation costs and mismatch costs, firms and providers, and consumers and patients are used interchangeably throughout the analysis.
The key contribution to the existing body of literature is to model providers as be-ing semi-altruistic. It is assumed that, in addition to profits, providers care, to some
extent, about consumers’ utility. This is a widely used assumption in the literature on health care markets, even though the case of location-quality competition is an exception. Health care providers—specially, hospitals—have complex internal struc-tures, in which a managerial hierarchy and a medical one coexist. In the literature on health care supply, physicians are often presented as being imperfect agents who trade-o↵ patient benefits against lower profits. More recently, the notion of mission oriented workers in the broader public sector has emerged in the literature, encom-passing, for instance, the cases of physicians who are determined to improve their patients’ condition, and educators who are committed to fostering their students’ knowledge. This may be due to the years of intense training they have submitted themselves to, to social welfare concerns, and even to moral stands. Correspond-ingly, management may have objectives of their own, like dimension, reputation or budget control. The balance of power within such firms is then di↵erent from that of firms operating in markets for other private goods, and it is, thus, reasonable to believe that their objectives, or, at least, some aspects of their decision-making, will be distinct. The ideia that hospitals di↵er from other firms dates back, at least, to Newhouse (1970). In the context of non-profit private firms, he proposed a util-ity maximising model in which a hospital’s objective is to maximise the utilutil-ity of its conflicting decision-makers (managers and medical sta↵) with respect to quality and quantity of care. A thorough theory of the internal organisation of hospitals was later developed by Harris (1977), according to whom hospitals are actually two separate firms—a medical sta↵ and an administration—, each of which having its own managers, objectives, strategies and constraints. The hospital is then the stage where the interaction (or confrontation) between these two sources of power takes place, and it is from such interaction that the decisions made by the hospital as a whole emerge.
Imperfectly competitive markets often fail to deliver the optimal levels of quality and specialisation, providing grounds for regulation. In addition to investigating the quality and location choices of semi-altruistic firms facing a fixed price, the welfare implications and the optimal payment scheme are tackled as well. When devising payment schemes, higher incentives are generally associated with higher risk for the
providers. E↵orts on activity and cost containment are fostered by prospective pay-ment schemes, which might, however, induce cream skimming and lower quality. Retrospective payments, on the other hand, encourage providers to supply suffi-cient quality, while dampening cost savings. In OECD countries, four main types of payment systems are used to remunerate hospitals: fee-for-service/payment for activ-ity, line-item budgeting, prospective global budgets, and Diagnosis-Related Groups (DRG) based payments. In a fee-for-service payment scheme, hospitals are remuner-ated retrospectively for every service and procedure performed during the treatment of a patient. A line-item budget corresponds to a total amount prospectively allo-cated to the hospital to finance its activities over the course of the year, with funds being assigned to specific cost items or services. A global budget is also a prospective amount hospitals are endowed with, although some degree of autonomy in deciding how funds are used is granted as well. The main di↵erence between these two bud-geting mechanisms is the discretion a hospital is given for allocating resources among its various activities. DRG-based payment, or payment per case, funds all the ser-vices and deliverables required during a hospital stay in one single payment. A DRG is a weight that specifies, according to a prearranged classification, the amount of resources necessary to treat a patient with a given diagnosis—typically covering both the clinical and non-clinical services provided, like accommodation and nursing—, and upon which the tari↵ paid to the hospital for every patient admitted in each group is determined. The payment a hospital receives is known prospectively and irrespective of the length of stay and the actual costs incurred. The main objective of the adoption of DRG-based payments is to give hospitals a financial incentive to reduce costs per patient and average length of stay, thus, improving technical efficiency. As long as the marginal revenue exceeds the marginal cost per patient, DRG-based payments foster activity by encouraging hospitals to treat more patients. Using data from the OECD Health System Characteristics Survey 2012, de La-gasnerie et al. (2015) report that countries with health insurance systems tend to use DRG-based payment systems as the main method to finance hospital inpatient activity: 14 out of 20 (70%) have opted for DRG-based payments to public hospitals, and 2 out of 20 use global budgets. Tax-funded national health systems, on the other
hand, generally finance hospitals using a prospective global budget: 9 out of 13 (70%) have chosen global budgeting as the main payment method to fund public hospitals, and 3 out of 13 use a DRG-based system2. It is this preeminence of the use of global
budgets and DRG-based payments that the model of Chapter 3 attempts to capture. The remaining of this dissertation is organised as follows. Chapter 2 reviews the existing body of literature on location-quality competition, semi-altruistic pref-erences, and payment schemes. Chapter 3 presents, solves and discusses a spatial model of health care providers location-quality competition and the optimal regula-tion of such markets under di↵erent assumpregula-tions and sequences of moves. At the end of Chapter 3, the possibility that the regulator places a restraint on the share of sur-pluses providers’ may retain is allowed for. This is also known as profit constraints. Chapter 4 o↵ers concluding remarks and some possible topics for future research.
2Countries such as Sweden, Norway and Portugal use a DRG-based classification for managing
2
Literature Review
Even though the link between quality and competition has received some attention in the literature—often with empirical studies yielding somewhat ambiguous results1—,
that has not been the case of location-quality competition.
Using a spatial duopoly model with exogenous locations and where firms choose quality and prices sequentially, Ma and Burgess (1993) conclude that firms under-invest in quality to dampen price competition, which gives grounds for policy in-tervention. Price regulation eliminates the distorting e↵ect of quality on prices and improves welfare by leading to higher quality levels without the corresponding price increase. In a similar model, Wolinsky (1997) finds that competition often results in an inefficient allocation. Moreover, even under complete information, price reg-ulation does not generally yield the first-best allocation. In both studies, however, location choices are not investigated.
Economides (1989) presents a duopoly model in which firms choose locations (va-riety) first and then quality levels and prices simultaneously, or quality and prices
1For example, Kessler and McClellan (2000) report that competition improved outcomes for
Medicare patients diagnosed with acute myocardial infarction (AMI) after 1990; Cooper et al. (2011) find that higher competition was associated with a faster decrease in 30-day AMI mortality after the formal introduction of patient choice in January 2006 in the English National Health Service (NHS); Gowrisankaran and Town (2003) present mixed evidence for the impact of increased competition on mortality rates; and the results of Shen (2003) show that increased competition adversely a↵ects short-term health outcomes after treatment for AMI, but do not a↵ect patient survival beyond one year after patients’ hospital admissions.
sequentially. Contrary to the principle of minimum di↵erentiation, the results show that, in both settings, firms prefer to maximally di↵erentiate their products in the product space. At the same time, quality and prices display minimum di↵erentiation. When the costs of quality improvements are sufficiently low, higher di↵erentiation increases quality, prices and profits, through a positive e↵ect on demand. Greater dif-ferentiation dampens competition, which allows firms to charge higher prices, whose negative impact on demand is dominated by the positive e↵ect of higher, relatively inexpensive, quality. Importantly, these results are achieved with quadratic quality costs that are independent of the level of output. Price regulation is beyond the scope of the article, though. Using the same specification for quality costs and incorpo-rating fixed prices but not considering optimal regulation, Calem and Rizzo (1995) develop a model of the hospital industry in which firms set locations and quality sequentially and bear a share of their patients’ travelling (mismatch) costs. In this sense, their model is symmetric to the one presented in Chapter 3, in which a share of the gross utility patients derive from health care consumption enters the firms’ objective function. They find that competition generally fails to deliver the socially optimum levels of quality and di↵erentiation and identify the distorting e↵ect of quality on specialisation: in equilibrium, higher prospective prices induce hospitals to di↵erentiate too much in order to dampen quality competition.
To the best of my knowledge, Brekke et al. (2006) were the first to address the issue of regulation when locations are endogenous. Assuming only the convexity of fixed quality costs but without specifying a functional form, in a model of spatial competition where firms choose their locations and the quality of the product, they derive the optimal price set by a welfarist regulator. If the regulator can commit to a price prior to the choice of locations, first-best outcomes are only achieved for a specific level of travelling costs—which may also be interpreted as the level of competition—, and the second-best price yields oversupply of quality and an insuffi-cient degree of horizontal di↵erentiation if transportation costs are suffiinsuffi-ciently high. When the regulator is not able to commit to a prospective price prior to the location choice, first-best quality is elicited by the optimal price, but horizontal di↵erentia-tion is maximum. Providers maximally di↵erentiate in order to induce the regulator
to set a higher prospective price that aims at fostering quality provision. Bardey et al. (2012) extend the framework of Brekke et al. (2006) by relaxing the assump-tion that quality costs are independent of the level of output. Including variable quality costs does not alter qualitatively the results. It opens, however, the possi-bility of using partial or total retrospective reimbursement of variable quality costs as an additional regulatory instrument. When a pure prospective payment leads to underprovision of quality and overdi↵erentiation, cost reimbursement under full commitment increments quality provision, while maintaining the price low enough to restraint di↵erentiation, if transportation costs are low. Furthermore, if the reg-ulator can only commit to the prospective payment prior to the choice of location, and transportation costs are sufficiently high or low, the resulting allocation can dominate the one with full commitment. Hence, the authors argue that, contrary to the trend in international experience, there are grounds for retrospective payments to health care providers. Although Bardey et al. (2012) do not consider any form of altruism on the part of providers, the literature on payment schemes that comprise both prospective and cost-based components—mixed payment schemes—is closely related with the literature on semi-altruistic health care providers.
The assumption that health care providers act, at least to some degree, in the in-terest of the patient—what is identified as health care providers’ altruism—is widely used and recognised in the literature. Ellis and McGuire (1986) developed a model of physician behaviour under the assumption that physicians maximise their payo↵s by balancing the hospital’s profits and the benefit to the patient. They show that, if physicians undervalue the latter relatively to the former, a prospective payment scheme may lead to undersupply of care, while a retrospective reimbursement sys-tem might result in oversupply of care. Under a prospective payment scheme, the socially optimum provision ensues only when the physician is a perfect agent for the patient—i.e., when the physician values the hospital’s profits and patient benefit equally. Ellis and McGuire (1986) then devise a mixed reimbursement system that combines a prospective payment with a partial cost-based payment and find that the level of cost sharing that elicits the socially optimum outcome is decreasing in the weight physicians place on patient benefit. Similarly, Chalkley and Malcomson
(1998) analyse the optimal payment scheme and conclude that the appropriate cost-sharing rate depends on the extent to which providers care about patient benefit. They find that, unless providers are fully-benevolent—the term the authors use for the case when providers equally value profits and patient benefit—, some degree of cost-sharing is generally optimal. Jack (2005) relaxes the assumption of complete information and lets altruism vary across physicians and to be private knowledge to them. He shows that, with asymmetric information, the optimal contract is typically a non-linear scheme that relates payment and costs. Exploring the issue of asym-metric information further, Chon´e and Ma (2011) let the weights attached to profits and patient benefit as well as the patient’s valuation of treatment (or severity) be the physician’s private information and unknown to the purchaser, the insurer. Again, the optimum mechanism depends on the physicians’ altruism and, furthermore, is shown to be insensitive to the patient’s valuation of care. Also tackling the issue of payment scheme design in health care markets with multitasking (multiproduct) providers, Eggleston (2005) considers that they care about patient benefits from treatment, as well as net revenues and disutility from e↵ort. Her results too suggest that, when some dimensions of quality are noncontractible and costly to provide, a mixed payment scheme should be favoured. Kaarbøe and Siciliani (2011) extend the work of Eggleston (2005) by exploring the complementarity and substitutabil-ity of the contractible and noncontractible dimensions of qualsubstitutabil-ity in a model with semi-altruistic providers.
This particular strand is part of a broader literature that allows for alternative objective functions of agents in the public sector, among which health care providers are preeminent. Akin to the assumption of altruism, agents are often modelled as sharing, to some extent, the objective function of the principal. Considering vocation-intensive sectors, such as nursing and teaching, Heyes (2005) models the vocation for performing a given task as a dichotomous random variable in a setting where indi-viduals in the profession they have the vocation for receive a higher payo↵. Brekke and Nyborg (2010) develop a model of work motivation in which the individual’s self-image—or the desire to be important to other people—enters his or her utility function. They demonstrate that the behaviour of workers paid according to their
marginal product is not influenced by the concern about others, whereas the e↵ort of those workers who cannot be accurately monitored—like nurses—and, for that reason, who are not paid according to their productivity, is increasing in the weight they attach to others’ welfare. Focusing on the higher education market, Del Rey (2001) formalises the idea that universities maximise a weighted sum of the increase in total productivity of its undergraduate students, defined as the number of students times the quality of teaching, and the total utility derived from research. Besley and Ghatak (2005) adopt the notion of mission-oriented agents, defining mission as the attributes of a project that make some agents value its success above any monetary income they receive in the process. Makris (2009) studies an agent-principal prob-lem in which a budget-constrained agency—that the author interprets as a group of citizens who have an expertise in the production of a non-marketable good—derives direct utility from providing the output it is mandated to. Tonin and Vlassopoulos (2009) present experimental evidence that action-based altruism, which is defined as the characteristic of a provider that derives direct utility from performing a task about which he or she cares about, accounts for an increase in e↵ort provision that is both statistically and economically significant.
Although location choices are not the subject of the study, a noteworthy result on the link between quality competition and altruism is due to Brekke et al. (2011). They demonstrate, in a Salop-type framework with semi-altruistic providers, that increased hospital competition will lead to lower quality if the hospitals are suffi-ciently altruistic. Empirical evidence that supports this theoretical prediction was recently reported by O'Kee↵e and Skellern (2015). Using data from the English Na-tional Health Service Sta↵ Survey collected before the introduction of patient choice to build an exogenous measure of providers’ altruism, they find that low-altruism hospitals improve quality supply in response to increased competition, while the response of high-altruism hospitals is greatly attenuated.
Finally, let the literature on profit constraints be briefly commented on. The anal-ysis of profit constraints if often done within the scope of non-profit firms. As far as the empirical evidence is concerned, Sloan (2000) o↵ers a review of the literature on the link between ownership and quality provision. Most studies find no significant
dif-ference between private non-profit and for-profit hospitals, but a few do report some evidence favourable to non-profits. Considering output instead of quality—although the framework could be extended to include preferences for quality—Lakdawalla and Philipson (2006) propose a neoclassical theory with an endogenous non-profit sector in which firms maximise utility over profits and output separately. The non-profit, thus, reflects a mix of altruism and profit motives, with output preferences repre-senting a desire to produce independent of monetary gain. As expected, the more the firm values output relative to profits, the higher is the output. Additionally, in their framework, non-profits are endowed with a competitive advantage vis-`a-vis their profit-maximising counterparts, because their preference for output allows them to absorb pricing below average cost. In a setting where the firm ownership status is chosen a priori, Glaeser and Shleifer (2001) investigate quality decisions by non-profit and for-profit firms, and show that the former supply a higher level of quality than the latter do. This is due to the non-profit status, which prevents owners from distributing profits, and, hence, reduces the incentives to shirk (costly) quality supply. Here, however, like in Brekke et al. (2012), the objective is to in-vestigate the e↵ects of profit constraints that are exogenously imposed to the firms, rather than explaining the existence of non-profits. Under price regulation, Brekke et al. (2012) show that, if firms are semi-altruistic and there is no cost containment and no disutility of providing quality, profit constraints lead to higher quality. In spite of considering exogenous locations, this set of assumptions is akin to that of the model presented in the next chapter, which makes that result one of particular interest.
3
The Model
Each of two identical single-product health care providers, indexed by i = 1, 2, choose sequentially a location xi on the line segment [0, 1] and a quality level qi q. Let
the provider whose location is xi be called provider i. In order to ensure that the
problem is well defined throughout the analysis and that a Nash Equilibrium exists, the location sets of providers 1 and 2 are, respectively, restricted to be [0, 1/2 x] and [1/2 + x, 1], where x is a small positive number. Providers are qualified to treat all patients in the market, regardless of their speciality mix (their location on the line)1. The lower bound q on health care quality represents the minimum treatment
quality providers are allowed to o↵er, with q < q being interpreted as malpractice. For simplicity, q is taken to be equal to zero. Providers are financed by a third-payer, the regulator, who o↵ers a prospective payment P for each unit of the good supplied and potentially a lump-sum transfer T . The marginal cost of production, c, is assumed to be constant and independent of both locations and quality levels2.
The cost of achieving quality qi is qi2/2, where is a postive parameter measuring
1Letting one of the providers be prepared to treat a given condition while the other is not
would, indeed, make competition significantly less meaningful, since two separate markets could be considered.
2The assumption that costs are separable in quality and output implies that quality has the
the relative importance of quality provision costs3.
A unit mass of consumers (patients) is uniformly distributed along the line, each of whom demanding one unit of the good. Patients perfectly observe the qual-ity level and the location of each provider, and higher qualqual-ity is always preferred4.
Transportation (mismatch) costs are quadratic in the distance travelled5, but no
out-of-pocket expenditures are borne by the patients. The common gross utility6
each patient derives from health care consumption, v, is large enough for the whole market to be covered for any value of qi. Under the speciality mix interpretation of
location choices, this is equivalent to stating that seeking treatment from a provider whose speciality mix is di↵erent from that the patient would ideally require is always preferred to not being treated at all for all levels of quality.
Providers are assumed to have altruistic preferences in the sense that they care about the utility their patients7 derive from being treated—or, to put it di↵erently,
the number of patients they treat—, and the utility patients derive from treatment quality. Providers are, however, insensitive to the travelling costs their patients incur by seeking medical treatment from a firm whose location is not ideal. In other words, it is assumed that providers care only about the clinically relevant component of patient utility. It might be argued that, under the speciality mix interpretation of location choices, mismatch costs borne by patients are clinically relevant as well, since they represent some (though, small) degree of treatment inadequacy. Even though patients would always prefer to be served by a provider whose location is closer to the ideal, it is not unreasonable to believe that what matters more for providers in terms of treatment e↵ectiveness is the overall quality of care they supply and the number
3This specification is common in the literature. See, for example, Economides (1989), Calem
and Rizzo (1995), and Bardey et al. (2012). Carey and Stefos (2011) present empirical evidence that the number of high technology services, the proportion of registered nurses in the nursing sta↵, and patient safety indicators, all used as measures of quality, a↵ect positively hospital costs.
4The underlying assumption here is that patients are sensitive to the level of quality provided.
Varkevisser et al. (2012) find empirical evidence that patients have a high propensity to choose hospitals with a good reputation.
5If transportation costs were linear in the distance, quality levels would not depend on locations.
6That is, homogeneity in the severity of patients’ conditions is being implicitly assumed.
7This is known as “warm-glow altruism”. A “warm-glow altruistic” provider is that who cares
of patients who benefit from this quality, contingent on being qualified to minister to those patients and independently of their conditions. Altruism is, then, modelled by including a share ↵2 [0, 1] of the aggregate gross utility of consumers attending provider i in its objective function8. Note that the possibility that providers are
perfect agents for their patients (↵ = 1) is allowed for, but the case in which they care more about patient benefit than profits (↵ > 1) is ruled out.
This particular formulation need not be interpreted as altruism stricto sensu, since it may reflect health care providers’ concerns about their practice rather than about patient well-being in itself. The return to several specialities requires an amount of training providers can only obtain by treating a sufficiently high number of patients. The same reasoning applies to the availability of high technology equipment and access to innovative procedures. Gaynor et al. (2005) report evidence supporting the practice makes perfect e↵ect, since they find quality enhancing scale economies that cause large hospitals to provide better quality of care and to improve outcomes. Modelling altruism in this manner allows this bias towards high volume and quality investment to be included in the providers’ objective function.
The key paramater is, thus, ↵, which measures the degree of altruism on the part of the firms. The pure profit-maximising case may be easily retrieved by setting ↵ equal to zero.
For the first part of the analysis, the following 3-stage game is considered: Stage 1: The regulator sets the prospective payment P ;
Stage 2: Providers simultaneously choose locations X ={x1, x2};
Stage 3: Providers simultaneously choose quality levels Q = {q1, q2}.
A game with this structure is called a game with full commitment, referring to the fact that the regulator is able to commit to a prospective payment prior to all providers’ moves are made.
8This formulation is consistent with, for example, Ellis and McGuire (1986), Jack (2005), Makris
(2009), Brekke and Nyborg (2010), Brekke et al. (2011), Kaarbøe and Siciliani (2011), Brekke et al. (2012), and Siciliani et al. (2013).
A patient located at z who purchases one unit of health care from provider i has a utility:
U (z, qi, xi) = v + qi t(z xi)2, (1)
where t is a transportation cost parameter.
Let the location of the patient who is indi↵erent between seeking treatment from either firm be denoted by z. It is given by:
z = 1
2(x1+ x2) +
q1 q2
2t , (2)
where = x2 x1 represents the degree of horizontal di↵erentiation between
providers. The minimum distance between providers is, thus, = 2x. With a uniform distribution of consumers, demand faced by providers 1 and 2 is D1 = z and
D2 = 1 z, respectively.
Finally, providers’ payo↵s are:
⌦i(Q, X, P ; ↵) = ⇡i(Q, X, P ) + ↵Bi(Q, X, P ), i = 1, 2, (3) with ⇡i(Q, X, P ) = (P c)Di q2 i 2 + T, (4) B1(Q, X, P ) = z Z 0 (v + q1)dz, (5) B2(Q, X, P ) = 1 Z z (v + q2)dz, (6)
where ⇡i(·) denotes profits, and Bi(·) is the aggregate gross benefit provider i’s
pa-tients derive from health care consumption.
To guarantee that the equilibria considered throughout the analysis exist and are well defined, it is assumed that ↵ < (4xt )/5. This is a refinement in the restriction on the providers’ location sets discussed above, and it naturally precludes any result in which there is minimum di↵erentiation ( = 0).
The two later stages of the game are considered first in order to investigate how quality and location choices are determined by the prospective payment. As usual, the game is solved by backwards induction for the pure-strategy Nash Equilibrium outcome.
3.1
The Quality Subgame
At stage 3 of the game, providers maximise a weighted sum of profits and aggregate patient benefit, subject to a limited liability constraint. Formally,
max qi ⌦i = (P c)Di 2q 2 i + T + ↵Bi s.t. ⇡ 0.
For given locations X = {x1, x2} and prospective payment P , the first-order
condition (FOC) for an interior solution is: (P c)@Di @qi qi+ ↵ @Bi @qi = 0, i = 1, 2. (7)
Each provider chooses quality supply by balancing profits and aggregate consumer benefit. Quality is optimal when the weighted sum of the marginal financial benefit from increased demand and the marginal non-financial benefit from aggregate patient utility equals the marginal cost of quality provision. Equation (7) shows how quality provision a↵ects providers’ payo↵s. The benefit from increasing quality is twofold. First, increasing quality supply increases demand, which, with constant marginal costs, yields higher profits. Second, providers care about the utility their patients derive from quality, but also about the number of patients who benefit from this quality. Hence, it is straightforward to see that increasing quality fosters aggregate consumer benefit, both through quality itself and through market share expansion. The negative e↵ect from increasing quality stems directly from quality provision costs.
demand and consumer benefit and inserting the results into equation (7) yields the optimal quality levels q⇤
i ⌘ q⇤i (X, P ) for providers 1 and 2, which are implicitly given
by: P c 2t q1+ ↵ ✓ v + q1 2t + z ◆ = 0, (8) P c 2t q2+ ↵ ✓ v + q2 2t + 1 z ◆ = 0. (9)
The first term in brackets corresponds to the demand e↵ect on aggregate patient benefit: all else equal, capturing an additional patient increases Bi by v + qi. The
remaning terms in brackets represent the quality e↵ect on aggregate patient benefit: all else constant, increasing quality augments the utility of all patients served by the provider, and Bi rises proportionally to Di.
The second-order condition (SOC) @2⌦
i/@qi2 < 0 is satisfied since it holds at any
equilibrium for ↵ < 2xt .
Let the subscript s identify symmetric equilibria9. It may be shown that, at a
symmetric equilibrium, the equilibrium quality level, q⇤
s ⌘ qs⇤( , P ), is:
qs⇤= P c + ↵(v + t )
2t ↵ . (10)
It follows immediately from (10) that the equilibrium quality level is higher than that that would emerge in the pure profit-maximising case (↵ = 0).
The equilibrium interior solution implies that the limited liability constraint is not binding—equivalently, that the Lagrange multiplier associated with the constraint is equal to zero. Thus, it remains to be ensured that playing the game according to q⇤
s( , P ) yields nonnegative profits for all locations such that x1+ x2 = 1. The
condition that has to hold is:
(P c)2+ (P c)⇥(2t ↵)2 2↵(v + t )⇤ ↵2(v + t )2+ T 0. (11) This is equivalent to requiring that P 2 [P , P ], where the bounds on P are the
9Taking into account the symmetric structure of the game, the analysis is limited to symmetric
roots of the second degree polynomial on left-hand side of (11). Note that, by o↵ering a lump-sum transfer to providers, the regulator leaves open the possibility of a prospective payment below marginal cost. If the providers are not o↵ered a lump-sum transfer, though, a prospective payment strictly above marginal cost is required for the condition to be satisfied.
Applying the Implicit Function Theorem to (8)-(9) and then imposing symme-try10 yields the following comparative statics results for the equilibrium quality:
@q⇤(X, P ) @↵ = P c + 2t (v + t ) (2t ↵)2 > 0, (12) @q⇤(X, P ) @P = 1 2t ↵ > 0, (13) @q⇤(X, P ) @t = [2 (P c + ↵v) + ↵2] (2t ↵)2 < 0, (14) @q⇤ 1(X, P ) @x1 = @q⇤2(X, P ) @x2 = t[2 (P c + ↵v) + ↵ 2] (2t ↵)2 + ↵t 2t 3↵ > 0, (15) @q⇤1(X, P ) @x2 = @q⇤2(X, P ) @x1 = t[2 (P c + ↵v) + ↵ 2] (2t ↵)2 + ↵t 2t 3↵. (16) As expected, quality increases with altruism. A marginal improvement in quality expands the share of the market served by the provider and augments the utility of each of its patients. Then, it follows that the higher the weight providers place on patient utility, the higher is the benefit for providers from a marginal increase in quality.
The result that quality provision increases with the prospective payment is also in-tuitive. Since marginal costs are constant and independent of quality, a higher price-cost margin produces greater gains from capturing a larger market share through quality increments.
The more costly it is for patients to travel —i.e., the greater the disutility from seeking treatment from a provider whose location is not ideal—, the lower are the gains in terms of demand from quality improvements since new patients are more
difficult to capture. Thus, a higher t dampens the incentives to supply higher quality. Consider, now, the link between own locations and the incentives for quality provision. The closer each provider locates to the middle of the line—and, thus, to the rival—, the stronger are the incentives to provide high quality levels, because demand is less captive due to the short distance between providers. This mechanism operates both through profits and patient benefit. In fact, it may be easily seen from (8) and (9) that locating towards the middle of the segment strengthens not only the demand e↵ect on aggregate patient benefit, but also the quality level e↵ect. All else equal, net revenues and the number of patients whose utility the provider cares about rise as a result of locating closer to the rival. This is represented by the first term in (15). Then, because Bi rises proportionally to Di, a marginal increase in
quality augments the utility of a higher number of patients, which yields stronger gains for providers in terms of aggregate patient benefit. This is represented by the second term in (15).
By the same token, the further the rival is located, the weaker are the incentives to provider high quality levels since demand is less responsive to quality adjustments. Unlike the case of own locations changes, however, a counteracting force arises. It may be easily checked from equations (8) and (9) that, when the rival moves away from the middle of the line, the quality level e↵ect on aggregate patient benefit is intensified. Due to the rival’s relocation towards the endpoint of the segment, the market share of the provider expands. A marginal improvement in quality will, then, enhance aggregate patient benefit proportionately more, providing incentives to in-crease quality supply instead. A sufficiently small ↵ ensures that (16) holds with a negative sign11. It is in this more interesting case that the subsequent sections will
focus on.
11Otherwise, the location subgame would yield a corner solution with minimum di↵erentiation,
because locating closer to the middle of the would simultaneously expand each provider’s market share, while still reducing the intensity of quality competition. In this case, it would be optimal for providers to locate as close to the rival as possible.
3.2
The Location Subgame
At stage 2 of the game, providers simultaneously choose their locations for a given prospective payment P . They maximise ⌦i(Q, X, P ) with respect to xi, anticipating
the quality pair Q ={q⇤
1(X, P ), q2⇤(X, P )} implicitly given by equations (8) and (9).
Using the Envelop Theorem, the FOC with respect to xi for an interior solution is:
(P c) ✓ @Di @xi + @Di @qj @q⇤ j @xi ◆ + ↵ ✓ @Bi @xi + @Bi @qj @q⇤ j @xi ◆ = 0, i6= j. (17) Equation (17) shows that, as is common in horizontal di↵erentiation models, two opposing e↵ects emerge: a market expansion e↵ect and a strategic e↵ect, each operating through profits and aggregate patient benefit. Consider, first, the for-mer. All else equal, there are incentives to locate closer to the rival—thus, reducing di↵erentiation—since it augments revenues and the number of patients ministered to. On the other hand, reducing the level of di↵erentiation triggers tougher quality competition, which drives providers to locate further apart. The market expansion e↵ect is represented by @Di/@xiand @Bi/@xi, while the strategic e↵ect is represented
by (@Di/@qj⇤)(@qj/@xi) and (@Bi/@qj⇤)(@qj/@xi).
At a symmetric equilibrium, equation (17), which implicitly defines the optimal level of di↵erentiation ⇤ s⌘ ⇤s(P ), simply reads: 1 2 1 2 2 (P c + ↵v) + ↵2 (2t ↵)2 ↵ 2t 3↵ = 0. (18)
The market expansion e↵ect and the strategic e↵ect are, respectively, given by the first and second terms in the left-hand side of equation (18). The second term in square brackets is the quality e↵ect on aggregate patient benefit discussed in the previous section. Reducing di↵erentiation triggers tougher quality competition, but the rival’s response is partially o↵set by the fact that its market share is reduced in the the first place. A lower market share implies that a marginal increase in quality has a smaller e↵ect on patient benefit, which dampens the quality response to reduced di↵erentiation.
Proposition 1 summarises the Nash Equilibrium outcome in the location-quality game.
Proposition 1. The unique equilibrium locations and quality levels at a symmetric equilibrium are: X = ⇢ 1 ⇤ s(P ) 2 , 1 + ⇤ s(P ) 2 and Q ={qs⇤(P ), qs⇤(P )} , where ⇤
s(P ) is implicitly given by equation (18), and qs⇤(P ) is given by equation (10)
evaluated at ⇤
s. Additionally, ⇤s(P )2]2t5↵, 1].
Proof. The existence and uniqueness of ⇤s(P ) are proved as follows. Equation (18) can be rewritten as:
[2 (P c + ↵v) + ↵2] (2t 3↵)
(2t ↵)2(2t 2↵) = 0. (19)
Denote the left-hand side of equation (19) by ( ), whose roots are the set of candidates for the optimal level of di↵erentiation, and note that = 5↵/2t ⌘ b is the minimum distance between providers. Then,
lim
! b+ ( ) =
2t 2(P c + ↵v) + ↵2t 60↵3
24↵2t ,
which is negative for a sufficiently low ↵. Additionally, lim
!1 ( ) =1.
Given that ( ) is continuous, a root must exist in the interval ] b ,1[. Uniqueness of ⇤ s(P ) follows from: @ ( ) @ = (2t 2↵)2(2t ↵)3+ 2t [2 (P c + ↵v) + ↵2] (2t 2↵)2(2t ↵)3 > 08 2] b , 1[,
where:
= 8(t )2 22↵t + 13↵2> 08 2] b , 1].
Hence, ⇤
s(P ) exists and is unique. It may be the case that the root is greater
than one, which yields a corner solution with maximum di↵erentiation: ⇤
s(P ) = 1
and X ={0, 1}.
Consider, now, the global maximality of ⇤
s(P ). Note that providers’ payo↵s may
be written as ⌦i = [P c + ↵(v + qi)] Di+ T ( /2)q2i. Thus, the SOC with respect
to x1 is: @2⌦ i @x2 1 = [P c + ↵(v + q1⇤)] ⇢ 1 2t 2 ✓ @q1⇤ @x1 @q⇤2 @x1 ◆ + q ⇤ 1 q⇤2 t 3 1 2t 2 @q2⇤ @x1 1 2t @2q⇤ 2 @x2 1 +↵@q1⇤ @x1 ✓ @D1 @x1 +@D1 @q2 @q⇤ 2 @x1 ◆ . Evaluated at ⇤ s(P ), this becomes: 1 2 2[P c + ↵(v + q⇤)] ⇢ 2 (P c + ↵v) + ↵2 (2t ↵)2 + 3↵ 2t 3↵ t @2q⇤ 2 @x2 1 . It can be shown that:
@2q⇤ 2 @x2 1 = 8(t ) 2[P c + ↵v 2t (t ↵)] (2t ↵)3 + 8t2 (t 3↵) 3(2t 3↵)2 + 2t(6t 2t + ↵) 3(2t ↵) . A sufficiently high v, thus, ensures that the SOC are satisfied at a symmetric equi-librium.
Implicitly di↵erentiating equation (19) yields the following comparative statics results for the equilibrium level of horizontal di↵erentiation:
d ⇤(P ) d↵ = 2⇤( v + ↵) ⇥A (2t 2↵)2(2t ↵)3+ 2t ⇥ , (20) d ⇤(P ) dP = 2 ⇤ (2t 2↵)2(2t ↵)3+ 2t ⇥ > 0, (21)
d ⇤(P ) dt = 2 ⇥ (2t 2↵)2(2t ↵)3+ 2t ⇥ < 0, (22) where: ⇥ = 2 (P c + ↵v) + ↵2 > 0, ⇤ = (2t ↵)(2t 2↵)(2t 3↵) > 08 2] b , 1], A = 2t (2t ↵) 2(2t 2↵)(2t 3↵).
An almost trivial, yet noteworthy, feature is that, at a symmetric equilibrium, the market expansion e↵ect is constant. This implies that location choices are only determined by how quality responses to changes in di↵erentiation are influenced by variations in the parameters. Therefore, the discussion shall focus on the behaviour of the strategic e↵ect.
It might be seen from equation (18) that an increase in altruism has an ambiguous impact on the size of the strategic e↵ect and, accordingly, on the equilibrium level of di↵erentiation. The intuition is as follows. When altruism is stronger, both the demand e↵ect on aggregate patient benefit and the quality e↵ect on aggregate patient benefit have a greater weight on quality choices. Locating closer to the rival fosters the former e↵ect, because demand becomes more responsive to quality changes and an additional patient is easier to capture through quality increments. Thus, the rival’s response will be more aggressive, driving providers to increase the level of di↵erentiation. On the other hand, locating closer to the rival dampens the latter e↵ect for the reason that a marginal increase in quality will improve the utility of a smaller number of patients. Consequently, the rival’s response is softened, and di↵erentiation is reduced. Since the relative importance both forces increases with ↵, it is unclear which one dominates.
A higher price-cost margin implies, all else equal, that the rival’s response to reduced di↵erentiation is more aggressive. In other words, because demand is more responsive due to the lower distance between providers, an increment in quality supply is more profitable when P is higher. Thus, an increase in the prospective
payment unambiguously strengthens the strategic e↵ect and drives providers to locate further apart.
When it is more costly for patients to travel, demand is less responsive to quality changes, which implies that the rival’s response is weaker. As can be seen from (18) the strategic e↵ect is decreasing in t so that the equilibrium distance is decreasing in the level of transportation costs.
Finally, consider the comparative statics for the equilibrium quality level in the interior solution of the two-stage game defined in Proposition 1:
dq⇤(P ) d↵ = (2t 2↵)2(2t ↵)2[P c + 2t (v + t )] (2t ↵) [(2t 2↵)2(2t ↵)3+ 2t ⇥ ] + (23) 2t ⇥ [t + v + (2t ↵)(P c + ↵v)] (2t ↵) [(2t 2↵)2(2t ↵)3+ 2t ⇥ ] > 0, dq⇤(P ) dP = (2t 2↵)2(2t ↵)3+ 2t ⇥ (2t ↵)[(2t 2↵)2(2t ↵)3+ 2t ⇥ ] > 0, (24) dq⇤(P ) dt = (2t 2↵)2(2t ↵)2⇥ (2t ↵)[(2t 2↵)2(2t ↵)3+ 2t ⇥ ] < 0, (25) where: := 4(t )2 12↵t + 7↵2> 08 2] b , 1].
For given locations, an increase in the prospective payment induces providers to improve quality supply (direct e↵ect). However, it also drives providers to increase the level of di↵erentiation in order to soften quality competition, which has a negative e↵ect on the equilibrium quality level (indirect e↵ect). Equation (24) shows that the direct e↵ect dominates, and this is the case of a change in alturism and in the level of transportation costs as well.
3.3
Optimal Regulation with Full Commitment
In this section, optimal regulation is analysed. As a benchmark with which to com-pare the Nash Equilibrium outcome summarised in Proposition 1, the first-bestout-come is defined as the one that maximises aggregate patient utility net of transporta-tion, output and quality provision costs12. The first-best outcome is, therefore, the
one that would result, if the regulator produced the medical good herself or himself, given locations and cost functions. Formally, welfare is specified as:
W (Q, X) = z Z 0 (v+q1 t[x1 z]2)dz+ 1 Z z (v+q2 t[z x2]2)dz 2 X i=1 ✓ cDi+ q2 i 2 ◆ . (26)
With the analysis confined to the case of symmetric equilibria, (26) simply reads: W (q, ) = v + q t[1 3 (1 )]
12 c q
2. (27)
The first-best levels of quality and di↵erentiation are13:
qF B = 1
2 , (28)
F B = 1
2. (29)
The first-best solution yields a quality level that equates the marginal benefit of quality improvements in terms of patient utility and the marginal cost of qual-ity provision. Furthermore, the first-best solution requires a pair of locations that minimises mismatch costs. With consumers uniformly distributed along the line seg-ment of length one, these are given by X = 14,34 , which is a standard result in the Hotelling framework.
For any symmetric equilibrium, at stage 1 of the game, the regulator sets the prospective payment P , anticipating the quality and location pairs Q ={q⇤
s(P ), q⇤s(P )}
12Note that this is equivalent to maximising the sum of consumer and producer surpluses net of
third-payer expenses and disregarding the altruistic component of the providers’ preferences.
and X =n1 ⇤s(P ) 2 , 1+ ⇤ s(P ) 2 o
. Thus, the regulator solves the following problem:
max P2[P ,P] W (P ) = v + q⇤s t[1 3 ⇤ s(1 ⇤s)] 12 c (q ⇤ s)2.
The FOC for an interior solution is14:
(1 2 qs⇤)dqs⇤ dP + t 4(1 2 ⇤ s) d ⇤ s dP = 0. (30)
The regulator sets the prospective payment by balancing the marginal social (net) benefit from increments in quality provision and the marginal social benefit from increased horizontal di↵erentiation.
Proposition 2 presents the welfare results for the symmetric Nash Equilibrium outcome in the three-stage game.
Proposition 2. For ↵ ↵ ⇡ 1/4, the first-best outcome is achieved for a unique value of the transportation cost parameter, bt = (2 + ↵ +p17↵2 20↵ + 4)/2 . If
t > bt, the second-best solution is characterised by underdi↵erentiation and oversupply of quality. If t < bt, the second-best solution is characterised by overdi↵erentiation and undersupply of quality.
For ↵ > ↵, the first-best is not attainable, and the second-best yields excess quality and insufficient di↵erentiation.
Proof. Consider the prospective payment bP that elicits the first-best level of di↵er-entiation. Setting = 1/2 in equation (19) yields:
b P = c ↵ ✓ v + ↵ 2 ◆ +(t 2↵)(t ↵) 2 4 (t 3↵) .
Before proceeding, note that, at P = bP , the exogenous parameter restriction be-comes ↵ < t /5.
The FOC is now given by:
(1 2 q⇤s)dqs⇤
dP = 0. (31)
Thus, bP elicits the first-best quality provision if and only if the FOC is satisfied and q⇤ s( bP ) = qF B.Using (10), qs⇤( bP ) = (t ↵)(t 2↵) 4 (t 3↵) + ↵ 2 .
Solving q⇤s( bP ) = qF B for t yields as real solutions t = 2+↵±p17↵2 20↵+4
2 for ↵ 2 h 0,10 4p2 17 i [h10+4p2 17 , 1 i
. It might be shown that t > 5↵ is satisfied only for t =
2+↵+p17↵2 20+4
2 ⌘ bt and ↵
10 4p2
17 ⌘ ↵. Equation (31) is naturally satisfied.
Consider, now, the case of t6= bt, in the light of the following result: @q⇤ s( bP ) @t = (t )2 6↵t + 7↵2 4(t 3↵)2 > 08(t ) > 5↵. If t > bt, qs⇤( bP ) > qF B, since @q⇤
s( bP )/@t > 0. With dqs⇤/dP > 0, the left-hand side
of (31) is negative, and the FOC is not satisfied. Therefore, bP cannot be optimal. Let the superscript SB denote variables at the second-best solution. Recall that d ⇤
s/dP > 0 and dqs⇤/dP > 0. It then follows from concavity of the
prob-lem that PSB < bP . Since ⇤
s is increasing in P , ⇤s(PSB) < 1/2, which
im-plies that (t/4)(1 2 ⇤
s) (d ⇤s/dP ) > 0. Satisfying the FOC then requires that
(1 2 qs⇤)(dqs⇤/dP ) < 0, which is true if and only if qs⇤(PSB) > qF B. The second-best
solution when t > bt is characterised by ⇤
s(PSB) < F B and q⇤s(PSB) > qF B.
The reverse reasoning holds for t < bt.
Finally, take in consideration the case of ↵ >↵. For such values of ↵, q⇤ s( bP ) >
qF B8t, and the left-hand side of (31) is negative so that the FOC is not satisfied.
Consequently, there is no P such that q⇤
s = qF B and ⇤s = F B. Again, concavity of
the problem implies that PSB < bP , and, by the same token as above, the second-best
solution is characterised by ⇤s(PSB) < F B and q⇤
s(PSB) > qF B.
Proposition 3 states that, when implementing the second-best solution, the reg-ulator faces a trade-o↵ between quality provision and horizontal di↵erentiation.
The intuition for the results when ↵ ↵ is as follows. When t > bt, the strate-gic e↵ect is relatively weak because the rival’s response to reduced di↵erentiation is milder, which drives providers to locate closer to the middle of the line. In other words, when it is sufficiently costly for consumers to travel, only a small degree of di↵erentiation is need to soften quality competition significantly. Since di↵erentia-tion is increasing in the prospective payment, achieving its first-best level requires a payment so high that excessive quality ensues. The regulator might then do bet-ter by trading o↵ adequate di↵erentiation against lower quality and, thus, setting a lower payment that yields both underdi↵erentiation and oversupply of quality. The symmetric argument holds for t < bt. These findings are indeed the ones presented by Brekke et al. (2006) and Bardey et al. (2012) for the pure profit-maximising case. This implies that, for sufficiently low levels of altruism, the presence thereof does not change the optimal regulatory scheme.
However, if altruism is sufficiently high (i.e., above the threshold ↵), providers are biased towards excessive quality supply so that the equilibrium level of quality is always above the first-best, regardless of how costly it is for patients to travel. In this case, welfare is maximised when the regulator forgoes optimal locations and sets a prospective payment that dampens quality provision.
3.4
Optimal Regulation with Partial
Commit-ment
Up until this point, it has been assumed that the regulator moves before the providers, or, in other words, that the regulator is able to commit to a prospective payment be-fore providers choose locations and quality levels. An alternative sequence of moves, arguably more realistic, is one in which the regulator sets the prospective payment only after providers choose their locations on the line. If these are interpreted as the geographical space, the application of such argument is straightforward. The physical location of health care facilities is usually a decision for the very long run, and, accordingly, it is likely that at least some aspects of a given regulatory scheme
are changed or updated at a posterior date. Even if locations on the line are un-derstood as the providers’ speciality mix or, in the case of private practices, how they decide to position themselves in the market through advertising, this timing is feasible. These decisions require plans that are generally designed for the medium or even the long run. In a thorough analysis of DRG-based hospital payment systems of twelve European countries, Busse et al. (2011) report that in ten of those countries15
the payment rate is updated annually. This strongly suggests that a revision in the prospective payment will occur while a provider’s plan to specialise in a set of med-ical fields or its marketing strategy are kept unchanged. It is, thus, quite plausible that a game in which the regulator moves after the providers’ first move is a more accurate description of how actual health care markets work. Formally, the game is now played as follows:
Stage 1: Providers simultaneously choose locations X ={x1, x2};
Stage 2: The regulator sets the prospective payment P ;
Stage 3: Providers simultaneously choose quality levels Q ={q1, q2}.
With this sequence of moves, the exogenous parameter restriction may now be relaxed to ↵ < (4xt )/3. Hence, the minimum distance between providers in a game with partial commitment is = 3↵/2t ⌘ e .
When the regulator is unable commit to the prospective payment prior to location choices, the whole quality subgame continues to be played exactly as is described in Section 3.1., with the equilibrium quality levels being given by (8) and (9).
3.4.1
Optimal Prospective Payment
With this sequence of moves, however, symmetry cannot be imposed ex ante in the welfare maximisation problem. At stage 2 of the game, for any locations X =
15Namely, England, Estonia, Finland, France, Germany, Ireland, the Netherlands, Poland, Spain
(Catalonia), and Sweden. In Portugal, the frequency of updates is irregular, and in Austria pay-ments rates are updated every four or five years or when necessary.
{x1, x2}, transportation costs are given. Therefore, maximising the welfare function
in (26) with respect to P is equivalent to solving the following problem:
max P2[P ,P] W (P, X) = 2 X i=1 n [v + q⇤i(P, X) c]Di(P, X) 2[q ⇤ i(P, X)]2 o .
The FOC for an interior solution is:
2 X i=1 ⇢ (v + qi⇤ c) ✓ @Di @qi @q⇤i @P + @Di @qj @q⇤ j @P ◆ + Di @qi⇤ @P q ⇤ i @qi⇤ @P = 0. (32) Using the fact that, at any equilibrium, @q⇤1/@P = @q⇤2/@P , the FOC simply reads:
1 (q1⇤+ q2⇤) = 0. (33) It may easily be seen from (33) that, at a symmetric equilibrium, the regulator is able elicit the first-best quality levels by setting the appropriate prospective payment.
The SOC for the regulator’s problem is always satisfied at any equilibrium since @2W (P, X)/@P2 = 2 (@q⇤
i/@P ) < 0.
Solving equation (33) for P yields the optimum prospective payment P⇤(X)⌘ P⇤
at any equilibrium16: P⇤(X) = c + (1 ↵)t ↵ ✓ v + 1 2 ◆ . (34)
A first noteworthy result that carries over from the pure profit-maximising case is that the optimal prospective payment is increasing in the level of horizontal dif-ferentiation. The greater the distance between providers, the less intense is quality competition, which implies that a higher prospective payment is required to induce the first-best quality supply. Additionally, the optimal prospective payment is be-low the one that ensues when ↵ = 0. Since the equilibrium quality provision is increasing the level of altruism, weaker stimuli are needed to induce first-best qual-ity. In fact, the optimal regulation does not require a payment above the marginal
cost of production. If altruism is sufficiently high so that there is overinvestment in quality, a prospective payment below marginal cost may indeed be necessary to discourage quality provision and achieve the first-best outcome. This result holds when providers receive a lump-sum transfer as well. Otherwise, condition (11) is only satisfied with a prospective payment strictly above marginal cost. Note that a regime in which a higher ↵ induces a lower P , being then compensated by a higher T , resembles the results presented in the literature reviewed in Chapter 2, according to which a quantity-based payment should be decreased as alturism increases and the fixed amount raised to maintain the same total payment per case (e.g. Ellis and McGuire (1986)).
Finally, consider the following useful result: @P⇤(X)
@x1
= @P⇤(X) @x2
= (1 ↵)t < 0. (35) Moving towards the middle of the line reduces the prospective payment. Lower di↵erentiation increases quality provision, which implies that a lower payment is required to induce the optimal quality level. This result conveys the additional e↵ect that this sequence of moves produces: when the regulator sets the prospective payment after location choices are made, providers must take into account the e↵ect thereof not only on the intensity of quality competition, but on the prospective payment as well.
3.4.2
The Location Subgame
At stage 1 of the game, providers choose locations, anticipating the quality pair Q ={q⇤
1, q⇤2} and the optimal prospective payment P⇤. Using the Envelop Theorem,
the FOC with respect to xi for the providers’ problem is:
(P⇤ c) ✓ @Di @xi + @Di @qj @qj⇤ @xi ◆ + ↵ ✓ @Bi @xi + @Bi @qj @q⇤j @xi ◆ + @P⇤ @xi Di+ (P⇤ c) @Di @qj @q⇤ j @P + ↵ @Bi @qj @q⇤ j @P = 0, i6= j. (36)
The first two terms on the left-hand side of equation (36) represent the market expansion e↵ect and the strategic e↵ect already present in the full commitment case. Recall that, while the former drives providers to reduce the level of horizontal dif-ferentiation in order to serve a larger share of the market, increasing revenues and aggregate patient utility, the latter induces providers to locate further apart to soften quality competition. The third term is the twofold payment e↵ect. As shown in (35), moving towards the middle of the line reduces the prospective payment set by the regulator, which generates two counteracting forces. On the one hand, there is a direct negative e↵ect, because revenues fall by (@P⇤/@x
i)Di. On the other hand,
there is an indirect positive e↵ect, because the rival’s quality supply will fall due to the lower payment, which causes demand to increase, improving both revenues and aggregate patient benefit.
Proposition 3 summarises the symmetric Nash Equilibrium outcome in the three-stage game with partial commitment. Let the superscript pc denote partial commit-ment.
Proposition 3. For sufficiently low levels of altruism, there is a unique equilibrium in the three-stage game that is characterised by first-best quality and over-, but not maximum, di↵erentiation. Equilibrium locations, prospective payment and quality levels are given by:
X = ⇢ 1 ⇤pc s 2 , 1 + ⇤pc s 2 , P⇤(X) = c + (1 ↵)t ⇤pcs ↵ ✓ v + 1 2 ◆ , and Q ={qF B, qF B}, where ⇤pc s = t + ↵ +p(t )2 4↵t + ↵2 2t 2 ⇤ F B , 1⇥, with (t )2 4↵t + ↵2 0.
Proof. First, note that, at a symmetric equilibrium, inserting (34) into (10) or, al-ternatively, imposing symmetry in equation (33), yields q⇤
s(P⇤) = qF B. Using (16),
(34), and (35), together with the definitions of D1and B1, the FOC (33) with respect
to x1 at a symmetric equilibrium may be written as:
✓
1 + ↵
2t 3↵
◆
1 = 0. (37)
It is straightforward to see that, if ↵ = 0, the FOC is satisfied for = 1 so that the equilibrium is characterised by maximum di↵erentiation. Solving (37) for gives the two candidate solutions:
= t + ↵± p
(t ) 4↵t + ↵2
2t 2] e , 1[.
Let the the smaller and the larger roots be denoted, respectively, by 1 and 2. It
may be shown that the SOC, evaluated at symmetric locations with = 1 and
= 2, yields: @2⌦ 1 @x2 1 = 1 = t " 4(1 ↵)(t )2 ↵ [(16 15↵)t 2↵(2 ↵)] [4(1 ↵)t ↵(8 7↵)]p(t )2 4↵t + ↵2 # 2(t 2↵ p(t )2 4↵t + ↵2) > 0, (38) and @2⌦ 1 @x2 1 = 2 = t " 4(1 ↵)(t )2 ↵ [(16 15↵)t 2↵(2 ↵)] +[4(1 ↵)t ↵(8 7↵)]p(t )2 4↵t + ↵2 # 2(t 2↵ +p(t )2 4↵t + ↵2) < 0. (39)
Conditions (38) and (39) hold provided that ↵ is sufficiently small. Thus, providers’ payo↵s are maximised at = (t + ↵ +p(t ) 4↵t + ↵2)/2t > 1/2.
The intuition for the results presented in Proposition 3 is better understood by first analysing how the strategic and the payment e↵ects interact. Recall, first, that each e↵ect has two opposing components. As far as the strategic e↵ect is concerned,
