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Publicly Available Published by De Gruyter September 3, 2020

Survival of altruistic gatekeepers: Kickbacks in medical markets

  • Erwin Amann and Stefan Felder ORCID logo EMAIL logo
From the journal German Economic Review


Patients often rely on the advice of their general practitioner (GP) to decide which treatment best fits their needs. Hospitals, in turn, might influence GPs’ referral decision through kickbacks. We present a model with a monopolistic hospital and competitive GPs who vary in the degree of altruism towards their heterogeneous patients and show that an equilibrium without crowding out exists that separates GPs into referrers and care providers. Naïve patients visit purely selfish (referring) GPs, while rational patients sort themselves between the two groups of GPs. Finally, we investigate the scope for regulation, including an optimal coinsurance rate.

JEL Classification: D47; D82; I11; I18; L50

1 Introduction

In many countries, patients have no direct access to hospital treatment or medical specialists but need a referral from their general practitioner (GP) for specialized care. At the same time, price competition between health care providers and their access to health care markets are often heavily regulated. As Pauly (1979) noticed early on, administered prices that are above marginal cost incentivize profit-maximizing providers to pay kickbacks to physicians in return for patient referrals. Patients, however, might also benefit from kickbacks that give them access to specialized care. In fact, although mostly hidden, kickback payments are quite common.[1] In contrast, professional ethical codes and most countries’ legislations prohibit physicians from accepting kickbacks for referrals.[2] Kickbacks are seen as a gateway used by profit-maximizing physicians to crowd out altruistically motivated colleagues.

This paper analyzes the referral behavior of competitive GPs with different degrees of altruism towards patients’ benefits. Patients are heterogeneous as they differ with respect to the utility that they obtain from specialized care, which they need if they are severely ill. Our approach considers price competition in primary care and administered prices in specialized care, explicitly differentiating between the two sectors, as it is typical in many health care systems. Moreover, we assume that patients have coinsurance, which could lead to an overutilization of medical services. Health services, diagnosis and treatment, are credence goods. Patients do not know whether they are severely ill or are only suffering from a minor illness. A severe illness requires the patient’s referral to a specialized care provider. A minor illness can be treated in all settings, but at a higher cost if treatment is not provided by a GP. We assume that there is only one specialized care provider, a hospital, whose price is administered. The hospital pays kickbacks to GPs who are willing to refer patients with minor illnesses. The GP diagnoses the severity of the patient’s illness and makes the referral decision, whereas the patient cannot verify the diagnosis.

In this setting, we investigate whether kickbacks drive out altruistically motivated GPs in a competitive environment. We differentiate between patients with naïve expectations and patients with rational expectations and analyze the corresponding equilibria. Whereas naïve patients expect that their GP will act in their best interest, rational patients anticipate that their GP might unnecessarily refer them to the hospital. Moreover, we analyze the scope of a regulator who can administer the hospital price or set the coinsurance rate, and also investigate the effect of a kickback ban.

The following, Section 2, relates our paper to the literature. Section 3 presents the model and characterizes the sequence of decisions that leads to a subgame perfect Nash equilibrium. Section 4 analyzes the GPs’ referral decision. Section 5 examines how rational and naïve patients choose a GP. Section 6 derives the competitive referral and treatment prices in the outpatient sector and the sub-game equilibrium when only rational patients exist and when both rational and naïve patients are present. Section 7 determines the monopolistic hospital’s profit-maximizing kickback and derives the equilibria. We show that an equilibrium exists that separates GPs into referrers and care providers. Whereas selfish GPs belong to the referral group, the care-providing group comprises all non-selfish GPs. No crowding-out occurs. Rational patients sort themselves to referrers and care providers, depending on the specific utility they experience from specialized care. Naïve patients all visit the referrers. Section 8 addresses a world without purely selfish GPs. It appears that also the most selfish GPs treat some of their patients. Section 9, investigating the scope for regulation, shows that the hospital price has no impact on the referral decisions of GPs. A regulator can influence the allocation by setting the coinsurance rate or by banning kickbacks. With naïve patients present in the market, regulation cannot lead to a first-best allocation. A second-best equilibrium, however, is feasible, in which the optimal coinsurance rate leads rational patients to choose the first best. Section 10 concludes.

2 Related literature

There is a broad literature on gatekeeping in health care. Brekke et al. (2007) investigate the informational role of GP gatekeepers in secondary health care markets in which two hospitals compete in quality and specialization. They complement the multi-task agency literature on the economics of general practice, e. g., by Garcia Mariñoso and Jelovac (2003) and Malcomson (2004). Optimal payment systems are derived that induce GPs to conduct diagnoses and incentivize efficient referral and treatment decisions. Allard et al. (2011) study how referral to secondary care is affected by incentive contracts for GPs. The role of price competition is not captured in this literature. Instead, price regulation is present, as in Brekke et al. (2007) or the principal-agent structure between a public insurer and a GP under asymmetric information is highlighted. Godager et al. (2015), while also assuming administered prices, introduce Hotelling competition in overall care quality, captured by GPs’ degree of altruism. In the equilibrium, both GPs choose the same degree of altruism. In this setting, they study the effect of competition intensification in the physician market on GP’s referrals of patients to specialty care and derive two opposing results. Intensified competition increases GPs’ altruism, which, in turn, leads to more referral. If more competition increases GPs’ net revenues, on the other hand, they have a stronger incentive to provide treatment and thus refer less often. Altruism in Godager et al. (2015) is in the GPs’ choice set and observable by the patients. Our approach differs as we study potential crowding out of intrinsically motivated GPs in a perfectly competitive market, where altruism is not directly observable but only indirectly via specialization. In contrast to Godager et al. (2015), the variation in the degree of altruism is exogenous and patients differ with respect to their preference for primary and secondary care.

Health care services often are credence goods as patients are usually unsure about the specific treatment they need and, in some cases, even about the sort of treatment they receive. The credence goods literature generally shows that, provided liability (an expert cannot provide the cheap treatment if the expensive treatment is appropriate) and verifiability (an expert cannot charge for an expensive treatment if the cheap treatment is provided) apply, competitive markets solve the fraudulent expert problem at no cost (Emons 1997a,b, Dulleck and Kerschbamer 2006). Bester and Dahm (2018) sustain observability and verifiability of treatment but assume that the expert’s diagnosis effort is not observable and its outcome noisy. They show that under these conditions, the first best is still contractible if diagnosis and treatment are separable. When verifiability does not apply and liability is only limited, Bester and Ouyang (2018) prove that the optimal contract induces inefficient undertreatment. In health care, the outpatient market is typically competitive, whereas markets for inpatient and for specialized care are not. GPs act as gatekeepers and provide treatment for minor ill patients. In fact, hospitals often have regional market power due to high setup costs and patients’ preferences for nearby inpatient care. Without market power, competition would drive inpatient prices down to opportunity costs (Dulleck and Kerschbamer 2006) and kickbacks could not persist. For the competitive outpatient sector, we consider that GPs are not exclusively driven by pecuniary incentives, but that they might also take into account patients’ utility, an aspect that is usually neglected in the credence goods literature.

Starting with Arrow (1963), assuming that physicians are altruistic towards their patients has been standard in the theoretical health-economics literature (see Ellis and McGuire 1986, for a seminal paper). Kolstad (2013) analyzes the effects of performance assessment on physician behavior and finds that physicians are willing to forgo profits for better quality provision. The experimental literature, focusing on physicians’ reactions to different payment systems, confirms the existence of altruism (Hennig-Schmidt et al. 2011; Brosig-Koch et al. 2016). Godager and Wiesen (2013), also in a lab experiment, find that physicians positively weigh their patients’ health benefits in their own preferences, but that they vary substantially in their degree of altruism. Jack (2005) considers the heterogeneity of physicians with respect to their intrinsic benefits from high-quality health care, in which altruism only accounts for the quality provided and not patients’ utility. He shows that under certain conditions, a menu of prices can differentiate between heterogeneous physicians and provide incentives for efficient care. Choné and Ma (2011) allow for heterogeneity, considering not only the degree of physicians’ altruism but also the severity of patients’ health problem. Patients’ utility and physicians’ consideration of the health problem are assumed to be positively correlated, albeit not perfectly in accordance. The payment scheme, then, can only take the physicians’ degree of altruism into account and not perfectly match patients’ benefit.

In Makris and Siciliani (2013), patients are heterogeneous with respect to their negatively correlated benefits and cost of treatment. Providers differ in efficiency and have opportunities to refuse patients. Altruistic providers experience an additional utility according to patients’ benefits. The degree of altruism is fixed and common knowledge. With moderate levels of altruism, the first best can be achieved. That is, inefficient providers do not mimic efficient ones if the purchaser maximizes benefits minus cost. With strongly altruistic agents, by comparison, efficient providers need to treat more than the efficient number of patients in order to sufficiently separate from inefficient providers. In our model, physicians do not differ in their degree of efficiency, avoiding the pooling effect in Makris and Siciliani (2013). In contrast, we derive the result, that in perfectly competitive markets, the degree of altruism need not have an impact on physicians’ decision in equilibrium, as long as a minimum altruism with respect to the patient’s benefit exists. Liu et al. (2018) analyze a principle agent setting with a hospital and two types of physicians with one basic technology and time inputs each. The patients’ benefit from treatment is assumed to be proportional to the output. The degree to which the experts’ altruism may differ is unknown to the principal but common knowledge for the two experts, who decide how to split the revenues. If the experts are intrinsically motivated, a first best subgame-perfect outcome exists, in which the expert with lower primary care cost and lower productivity acts as the gatekeeper and the specialist as the secondary care provider.

Kickbacks have hardly been addressed in the literature so far. Owen (1977) discusses the role of kickbacks that providers of title-insurance services pay to real estate brokers to steer homebuyers. Pauly (1979) shows that kickbacks might induce GPs to refer their patients rather than perform lower quality procedures themselves, and thus benefit their patients. Inderst and Ottaviani (2012) analyze Hotelling competition between two sellers through kickbacks to intermediaries who advise consumers. They show that competitive kickbacks need not be inefficient, as they allow the more efficient firm to gain a higher market share. Then, caps on kickbacks can be inefficient. The same applies if the provider is forced to disclose kickbacks. Our framework differs in three ways. First, we assume that the hospital is monopolistic. Second, GPs face price competition. Third, GPs’ altruism is proportionally to consumer utility and not only expressive of the honest advice they give to the patients about their illness. In this setting, kickbacks can also be welfare improving against a situation in which kickbacks are banned. Kickbacks are forbidden in many countries. Nevertheless, although mostly hidden, kickback payments are quite common. Liu et al. (2018) show in a principle-agent model with primary health care physicians as gatekeeper and a more efficient specialist, constituting an almost perfect symbiosis between hospital and GP, that an efficient outcome can be achieved if experts are altruistic and the degree of altruism is common knowledge to the experts. In our model, there is no perfect relationship, but the opportunity to pay (potentially hidden) kickbacks exists. We show that kickbacks, even then, need not crowd out altruistic GPs in a competitive environment, as long as patients are aware of GPs’ incentives. Competitive markets are rarely subject to health economic research. However, competitive markets with incompletely insured patients can serve as an incentive device for self-selection, avoiding the problem in Choné and Ma (2011), in which only the physicians’ altruistic motive can be taken into consideration by an incentive mechanism and not the patient’s valuation for health care. We show that with rational patients, the first best can be achieved by optimally setting the coinsurance rate.

3 The model

We assume a competitive primary care market and a monopolized secondary care market. The monopolistic hospital chooses the optimal kickback it pays to GPs for their patient referrals. There are two types of patients, minor ill and major ill, whose treatment costs differ and depend on the sector of provision. The treatment of a minor ill patient is by an amount Δ c _ more costly in the hospital. Patients, seeking treatment for their illnesses, do not know whether they are severely ill or only suffering from a minor medical problem. Verifiability is also not satisfied, i. e., patients can verify neither the diagnosis nor the treatment they receive. However, we shall assume liability of the providers in the sense that undertreatment can be detected by the patient and is verifiable so that legal rules will hold an expert liable for the provision of inappropriate, poor-quality care.

When their health is restored through treatment, patients achieve monetized gross utility r if treated by a GP, and r + s with s s _ < 0 < s ¯ and s ¯ > Δ c _, if treated in hospital. s might be called patients’ extra utility of hospital care, which is not related to the severity of illness, but patient-specific, reflecting, e. g., differences in opportunity cost of time and preferences in general. Patients reveal their s to the GP whom they perceive as their agent. GPs cannot treat severely ill patients, but always have to refer them to the hospital. From an efficiency point of view, minor ill patients with s < Δ c _ should be treated by their GP, whereas patients with s Δ c _ should receive inpatient care. Patients are drawn independently from the cumulative distribution function G s with a differentiable strictly positive density g s . The latter and the population share of minor ill patients μ are assumed to be common knowledge.

Patients are partly covered by health insurance and pay a coinsurance rate equal to λ of the price of health services. As they have no direct access to inpatient care, patients visit their GP who acts as a gatekeeper. They have no specific information on the characteristics of the GPs, except for their treatment and referral prices. GPs diagnose patients and decide whether to treat them or to refer them to inpatient care. GPs receive a kickback payment κ 0 from the hospital for referring a patient with a minor illness. We assume that GPs are heterogeneous with respect to the degree that they internalize their patients’ net utility from treatment. A GP of type α internalizes a share α of monetized patient net utility. GPs are also imperfect agents in the sense that they would misreport the diagnosis to a patient with a minor illness whom they want to refer to inpatient care.[3] The hospital price p H refers to the price for the treatment of the severe illness.

The sequential structure of the model is as follows:

  1. The hospital price p H and the coinsurance rate λ are given or administered by the regulator.

  2. The hospital sets kickback κ.

  3. Competitive outpatient prices p r and p t are chosen.

  4. Patients decide which GP to visit.

  5. GPs make the referral decisions.

  6. Treatments are performed and payoffs are realized.

The game is solved by backward induction leading to a subgame perfect Nash equilibrium.

In most countries, prices in medical markets are administered. Under these circumstances, patients are neutral with regard to physician’s characteristics and will not reveal their preferences about medical services by their selection of the type of physician. Still, with administered prices in place, GPs might compete by differentiating their services. In the following, we assume that price differentiation reflects differences in the marginal cost of specific services.[4]

4 GPs’ referral decision

For GPs, the referral price p r , the kickback κ they receive for referring minor ill patients, the treatment price p t , the treatment cost of a minor case c _ and the cost of diagnosis d are given. Their utility consists of the profit they earn by treating or referring patients and patient utility that they internalize in their decisions. A utility-maximizing GP of type α determines the pivotal patient s ˜ α among the minor ill patients for whom he is indifferent between treatment and referral:

(1) max s α u G P α = d + 1 μ p r + μ G s α p t c _ + 1 G s α κ + p r + α μ s _ s α g s r λ p t d s + s α s ¯ g s r + s λ p r + p H d s + α 1 μ s _ s ¯ g s r + s λ p r + p H d s ,

in which G s α is the share of patients with a minor problem that GP α will treat. The FOC for an inner solution of the utility maximization ( s _ < s ˜ α < s ¯) can be written as

(2) p t c _ κ + p r + α λ p H + p r p t s ˜ α = 0 .

Thus, GPs take into account the profit and a share α of patients’ utility from treating as compared to referring their patients. From this, we derive the threshold function for type α GPs, separating their patients into those who they treat ( s s ˜ α ) and those who they refer ( s > s ˜ α ):

(3) s ˜ α = λ p H + p r p t κ + p r p t c _ α .

The critical value s ˜ α is strictly increasing in α if and only if κ + p r p t c _ > 0, i. e., if referring a patient is more profitable than treating him. Fig. 1 illustrates this increasing incentive in α.

Whereas GPs with a very low α refer all their patients, GPs with a degree of altruism above α ˆ will treat at least some of their minor ill patients. The more altruistically motivated a GP is, the lower the share of patients that he refers will be: for α > α ˆ, we have s ˜ α > 0 and s ˜ α < 0. If s ˜ α intersects with the horizontal line at Δ c _, some of the strongly altruistic GPs are treating minor ill patients who, from an efficiency perspective, should receive inpatient care. At the same time, foremost selfishly motivated GPs refer minor ill patients whom they should treat. For a perfect altruist, (3) becomes s ˜ = λ p H p r p t 1 λ κ c _, which simplifies to s ˜ = p H κ c _ if patients pay the full price of medical services ( λ = 1). In that case, treatment and referral prices have no effect on the referral decision as the perfect altruist equally weighs his profit and the patient’s utility.

Figure 1 
The threshold function 




\tilde{s}\left(\alpha \right).
Figure 1

The threshold function s ˜ α .

5 Patients’ choice

Patients know their additional utility or disutility from specialized care s. Rational patients ( r a) take into account that their GP might refer them even if they only have a minor medical problem and no advantage from hospital treatment. An equilibrium might then exist that separates GPs into ‘referrers’ and ‘care providers’ at the prevailing prices.[5] Referrers (R) are GPs who do not treat patients: ( p r R > 0, no treatment). Care providers (C) post the price pair ( p r C , p t C ). Patients’ utility, depending on their s and whether they visit a type R or a type C GP, then, is

(4) u r a R , C , s = u r a R s = r + s λ p H + p r R u r a C s = μ r λ p t C + 1 μ r + s λ p r C + p H .

The higher s, the more attractive it is to visit a referrer. For the patient who is indifferent between the two types of GPs, u r a R s = u r a C s , we obtain

(5) s ˆ r a = λ p H p t C + p r R 1 μ p r C μ .

s s _ , s ˆ r a will visit a care provider, whereas s s ˆ r a , s ¯ will visit a referrer. (5) is the base for evaluating changes in the rational patients’ respective demand for health services of the hospital and the two types of GPs.[6],[7] A marginal increase in p H increases the patients’ price by λ, which, in turn, decreases demand for inpatient care. If p r R increases, demand for referrers will fall, whereas demand for care providers will increase. If p r C or p t C increases, demand for care providers will fall and vice versa for the demand for referrers. The share of patients who visit a care-providing GP increases in the coinsurance rate λ, provided that the term in the parenthesis of (5) is positive, which will be the case if the hospital price is sufficiently high. With full insurance, prices would have no influence on patients’ choice of GP. Rational patients with s 0 would visit a care provider, while patients s > 0 would consult a referrer, independently of the prevailing prices.

Naïve patients ( n a), given the prevailing prices and their utility of treatment, will maximize the following utility function:

(6) u n a s = r μ λ p t + 1 μ s λ p r + p H .

Unlike with rational patients, naïve patients’ utility from specialized care s has no influence on their GP choice.

6 Competition

We assume perfect price competition in the outpatient care market. This requires that patients face a positive price for medical services ( λ > 0). A GP’s profit, depending on which group he belongs to, is

(7) π G P R , C = π G P R = p r R + μ κ d π G P C = μ p t C c _ + 1 μ p r C d .

Then, the following zero-profit condition holds for referrers R and care providers C, respectively:

(8) p r R = d μ κ ,
(9) μ p t C + 1 μ p r C = μ c _ + d .

Proposition 1 (Rational patients).

A separating equilibrium of the subgame exists, in which GPs of type α = 0 join the referral group R and GPs of type α > 0 belong to the care provider group C. Depending on s, patients decide which group of GPs they will visit.


  1. Inserting the zero-profit conditions (8) and (9) into (5) yields

    (10) s ˆ r a = λ p H c _ κ .

    Patients above this threshold prefer to visit referrers; patients below prefer care providers. The characteristic of the marginal patient depends on κ, without observing κ directly, but only via competitive prices.

  2. A referral price lower than d μ κ is not feasible for referrers, as this would drive them out of the market. Assume that the referral price of care providers is marginally lowered to p r C = p r R ε, with ε 0. From (9), the treatment price is p t C = c _ + d + 1 μ κ + ε 1 μ / μ, in which we insert the marginally lower referral price from (8). These prices are sufficient to give referrers no incentive to mimic care-providing GPs.

  3. The patients’ decision which type of GP they visit is independent of ε. The zero-profit conditions (8) and (9) imply (10): s ˆ r a = λ p H c _ κ . The care providers’ referral decision follows s ˜ α = λ p H c _ κ + ε 1 / α λ s ˆ r a for ε 0. Thus, care providers will not refer any minor ill patient who is consulting them. But then, they have no incentive to reduce the price below p r R . Thus, ε = 0 holds in the equilibrium.[8]

  4. If equilibrium prices are announced, p r C = p r R = d μ κ, p t C = c _ + d + 1 μ κ, patients sort themselves according to (10).

  5. Inserting equilibrium prices in the GPs’ decision (3) gives the critical patient who is still treated: s ˜ = λ p H c _ κ . This is independent of α as the numerator in (3) vanishes under competitive prices. With competitive prices, we thus have s ˜ = s ˆ r a . If patients sort themselves, only referring GPs α = 0 receive patients above the threshold. Any GP with α > 0 treats all his patients.



  1. The equilibrium is separating, both from the GPs’ perspective, differentiating between α = 0 (referrers) and α > 0 (care providers), and the perspective of patients with s s ˆ r a visiting a care provider and s > s ˆ r a consulting a referrer. Inserting equilibrium prices in GPs’ referral decisions (3), gives κ + p r R = d + 1 μ κ = p t C c _. The numerator in (3) vanishes, and all GPs with α > 0 will refer all patients above the threshold. These patients visit a referrer anyway, so in equilibrium, no kickbacks are received by care providers.

  2. The equilibrium is pooling for all care providers, i. e., GPs of type α > 0. They incur zero profits for all patients and will act according to patients’ needs, i. e., they will treat patients with s s ˆ r a = s ˜, and refer those with s > s ˆ r a = s ˜. The increasing function in Fig. 1 is a horizontal line at the level s ˆ r a = s ˜ = λ p H c _ κ .

  3. If price competition in outpatient care is excluded by administered prices, separation via price differentiation is not feasible unless there is competition and differentiation in services. We assume that the costs of services are included in the relative price between care provision and the referral activity. This will not change the results in qualitative terms.

  4. If also service competition were excluded, administered prices could be set equal to or by a constant above marginal cost. This would not change GPs’ referral decisions, which are determined by patient utilities in this case.

  5. Without GPs of type α = 0 in the market, all GPs, independent of α, will provide care to all patients below the threshold s ˜ and refer all patients above s ˜, accepting kickbacks. Separation occurs not by the patients’ choice of GPs, but within the individual practice. GPs will treat and refer and make the referral decision in the best interest of patients, given that prices are competitive.[9] Note also that GPs cannot differentiate between naïve and rational patients. GPs’ choice only depends on patients’ s, which is observable to GPs, even though it is not correlated with patients’ illness.

Consider then the case when both rational and naïve patients are present in the market.

Proposition 2 (Rational and naïve patients).

In the competitive equilibrium of the sub-game, all GPs are present in the market. All naïve patients visit referrers, i. e., s ˆ n a = s _ and, thus, are referred, while rational patients separate around s ˆ r a = λ p H c _ κ .


In the equilibrium, referrers set p t R = 0 and p r R = d μ κ, while care providers post p r C = d μ κ and p t C = c _ + d + 1 μ κ, and patients choose their type of GP as proposed. We have to show that defection is not profitable for any GP. First, no GP can attract an additional rational patient without incurring a loss, as rational patients are treated at zero profit in the equilibrium (Proposition 1). Since patients maximize (6), a defecting care provider D could attract naïve patients only if μ p t D + 1 μ p r D μ p t R + 1 μ p r R = 1 μ d μ κ . The profit of the defector, attracting naïve patients and treating some of them is:

(11) π G P D p t D , p r D = μ G s ˜ α p t D c _ + μ 1 G s ˜ α κ + p r D + 1 μ p r D d = d + μ G s ˜ α p t D + 1 μ p r D μ G s ˜ α c _ + μ 1 G s ˜ α κ + p r D d + 1 μ d μ κ μ G s ˜ α c _ + μ 1 G s ˜ α κ + p r D

The last summand decreases in G s ˜ α and increases in p r D , profits converge to π G P D = 0 for G s ˜ α = 0 and the treatment price maximized at p r D = p r R = d μ κ. Thus, a defecting care provider attracting naïve patients cannot avoid losses, and therefore s ˆ n a = s _.

Rational patients, on the other hand, taking into account that referrers do not treat, decide whether they visit a care provider instead. Competitive prices for care providers are then p r C = d μ κ and p t C = c _ + d + 1 μ κ (see Proposition 1). At these prices, according to (5), the indifferent rational patient is s ˆ r a = λ p H c _ κ .  □

In contrast to the GPs’ choice for rational patients, care providers cannot be competitive with naïve patients, as they would incur a deficit. Thus, if α is low, GPs with positive α will still refer all patients (see the lower left part of s ˜ α in Fig. 1). The disutility of referring patients is lower than the loss from treating them.

Self-selection leads to a separating equilibrium for rational patients and a corner solution for naïve patients. Referrers are purely selfish, while care providers are GPs with varying degrees of altruism towards their patients.

7 Hospital sets the kickback

The monopolistic hospital H does not incur a diagnostic cost, as it is assumed that it knows a patient’s diagnosis from the referring GP. The treatment cost of major ill patient is c ¯, whereas treating a minor ill patient in hospital costs c _ + Δ c _. We assume that the hospital treatment price is administered such that patients visit their GP[10] and that p H c ¯ > c _ + Δ c _, which ensures that the hospital treats major ill patients and has an incentive to pay kickbacks to physicians in return for referrals of minor ill patients.

The hospital’s profit maximization problem with regard to the kickback can be written as follows:

(12) max κ π H = 1 μ p H c ¯ + μ p H κ c _ Δ c _ 1 G s ˆ r a κ ,

in which s ˆ r a κ = λ p H c _ κ is the marginal rational patient who visits a referrer (see (10)). The FOC for the profit-maximizing kickback κ and the corresponding marginal patient s ˆ r a = s ˆ r a κ reads

(13) d π H d κ = μ 1 G s ˆ r a + μ p H κ c _ Δ c _ g s ˆ r a λ = 0 ,


(14) λ g s ˆ r a p H κ c _ Δ c _ = 1 G s ˆ r a ,

in which the LHS indicates the profit increase at the intensive margin, whereas the RHS equals the additional kickback payments for infra-marginal referrals in response to a marginal increase in the kickback.

The optimal kickback amounts to

(15) κ = p H c _ Δ c _ 1 G s ˆ r a λ g s ˆ r a ,

and from (10), we obtain for the marginal patient

(16) s ˆ r a = λ Δ c _ + 1 G s ˆ r a g s ˆ r a .

Since the effective price patients pay for hospital treatment is equal to the marginal patient’s extra utility of inpatient care s ˆ r a , we can define the minor ill patient’s price elasticity for hospital treatment as follows:

(17) ϕ s ˆ r a = d ln 1 G s ˆ r a d ln s ˆ r a = G s ˆ r a / 1 G s ˆ r a 1 / s ˆ r a = s ˆ r a · g s ˆ r a 1 G s ˆ r a ,

in which g ( s ˆ r a ) / 1 G s ˆ r a is the hazard rate, i. e., the probability of the marginal minor ill patient being referred to the hospital, conditioned on those minor ill patients who have a higher value of s, and thus are receiving inpatient care.[11] Using (17), we obtain for the optimal kickback from (15)

(18) κ = p H c _ Δ c _ ϕ s ˆ r a ϕ s ˆ r a 1 ,

and for the marginal patient from (16)

(19) s ˆ r a = λ Δ c _ ϕ s ˆ r a ϕ s ˆ r a 1 .

Thus, κ is increasing and s ˆ r a is decreasing in the elasticity.[12] We assume that the SOC for a profit maximum is satisfied:

(20) d 2 π H d κ 2 = 2 g s ˆ r a 1 G s ˆ r a · g s ˆ r a g s ˆ r a < 0 .

Proposition 3 (Rational patients).

i) A competitive inefficient equilibrium applies if the coinsurance rate λ is close to zero or close to one. In the former case, too many, in the latter, too few patients are referred, and ii) equilibrium demand at the competitive patients’ price for inpatient care s ˆ r a is elastic ( ϕ s ˆ r a > 1).


  1. (19) reveals that for λ = 0, we have too many referrals ( s ˆ r a = 0 < Δ c _), while for λ = 1, too few referrals occur ( s ˆ r a > Δ c _).

  2. The hospital incurs no losses when treating a minor ill patient ( p H c _ Δ c _ κ 0) only if ϕ s ˆ r a ϕ s ˆ r a 1 1, and thus only if ϕ s ˆ r a 1. It pays kickbacks if κ = p H c _ Δ c _ ϕ s ˆ r a ϕ s ˆ r a 1 > 0. This gives ϕ s ˆ r a ϕ s ˆ r a 1 < p H c _ Δ c _ or, equivalently, ϕ s ˆ r a > p H c _ p H c _ Δ c _ > 1.


For illustration, consider the case of a uniform distribution of s, for which 1 G s ˆ r a = s ¯ s ˆ r a / s ¯ s _ holds. This provides us with s ˆ r a = s ¯ s ¯ s _ 1 G s ˆ r a , i. e., a linear negatively sloped inverse demand curve for inpatient care. The density is given by g s ˆ r a = 1 / s ¯ s _ and the elasticity is ϕ s ˆ r a = s ˆ r a / s ¯ s ˆ r a , which decreases from infinity at s ˆ r a = s ¯ to zero at s ˆ r a = 0. From (18) and (19), we obtain for the equilibrium s ˆ r a = λ Δ c _ + s ¯ / 2, which satisfies 0 < s ˆ r a < s ¯ since Δ c _ < s ¯, and κ = p H c _ λ Δ c _ + s ¯ / 2 λ > 0, which is ensured by the non-negative profit requirement and Δ c _ < s ¯. Finally, ϕ s ˆ r a = s ¯ + λ Δ c _ / s ¯ λ Δ c _ > 1 as λ > 0.

We again extend the model to include naïve patients with a share of β present in the market. According to Proposition 2, all naïve patients visit a referrer. Then, the hospital’s profit maximization problem becomes

(21) max κ π H = 1 μ p H c ¯ + μ p H κ c _ Δ c _ β + 1 β 1 G s ˆ r a κ ,

in which s ˆ r a κ = λ p H c _ κ is the rational patient who is indifferent between consulting a referring or a care providing GP.

From the FOC β + 1 β 1 G s ˆ r a = 1 β p H κ c _ Δ c _ g s ˆ r a λ and (10), the equations for the optimal kickback and the marginal patient become

(22) κ = p H c _ Δ c _ ϕ s ˆ r a ϕ s ˆ r a 1 1 + β / 1 β λ Δ c _ · g s ˆ r a and
(23) s ˆ r a = λ Δ c _ ϕ s ˆ r a ϕ s ˆ r a 1 1 + β / 1 β λ Δ c _ · g s ˆ r a .

The presence of naïve patients decreases kickbacks and increases the patients’ price for inpatient care. An increase in the share of naïve patients leads to more referrals, which, in turn, induces the hospital to lower kickbacks. This results in an increase in the referral price and in a decrease in the treatment price. Care providers become more attractive to rational patients. The effect on the hospital’s market share is ambiguous in this case: although the number of naïve patients, which will all be referred to inpatient care, will increase, the share of rational patients who will visit a referrer will decrease.

It can easily be shown that Proposition 3 also extends to the model that includes naïve patients. Two additional insights, however, arise. Firstly, for all λ > 0, the equilibrium cannot be efficient since naïve patients are always referred as long as they exist at all ( β > 0). Secondly, the presence of naïve patients decreases the kickback. This results in an increase in the referral price and to a decrease in the treatment price. Care providers become more attractive to rational patients. The effect on the hospital’s market share is ambiguous in this case: although the number of naïve patients, which will all be referred to inpatient care, will increase, the share of rational patients who will visit a referrer will decrease.

8 A world without purely selfish GPs

Assume that the altruistic parameter is bounded away from zero: α α _ > 0. In such a world, nothing changes if all patients are rational. With naïve patients in the market only and if the minimum altruistic GP has α _ > α ˆ, crowding-out occurs to the lowest α. However, the equilibrium still comes with some care provision. Again, providing care implies a loss, which is, however, covered by profits earned from referrals. Thus, GPs who treat some of their patients drive out GPs with a higher α. Assuming that prices are non-negative, competitive prices are p t R = 0 and

(24) p r R = d + μ G s ˜ α _ c _ μ 1 G s ˜ α _ κ 1 μ G s ˜ α _ .

Solving the hospital’s maximization problem, applying (12) with G s ˜ α _ in which s ˜ α _ satisfies (3) and prices are competitive, we find for the pivotal patient:

(25) s ˜ α _ = λ Δ c _ ϕ s ˜ α _ ϕ s ˜ α _ 1 1 + λ α _ 1 d + μ c _ 1 μ p H + Δ c _ λ Δ c _ 1 μ G s ˜ α _ .

GPs again refer patients with s s ˜ α _ . This can lead to inefficiently high or inefficiently low referrals, depending on α _ and the distribution of G s .

Note that for λ = α _ = 1 (25) is equal to (19). Thus, the equilibrium with naïve patients coincides with the rational equilibrium if GPs are perfect altruists and patients pay the full price of health care services. Since perfectly altruistic GPs face the same incentives to refer patients as in the situation with heterogeneous GPs, the hospital chooses the same level of kickback. At the same time, too few referrals would be observed in this particular equilibrium.

Consider the case of a uniform distribution of s. Given G s ˜ α _ = s ˜ α _ s _ / s ¯ s _ and ϕ s ˜ α _ = s ˜ α _ / s ¯ s ˜ α _ , we obtain from (25)

(26) s ˜ α _ = 1 2 λ Δ c _ + s ¯ + λ α _ 1 d + μ c _ 1 μ p H + Δ c _ 1 μ s ˜ α _ s _ / s ¯ s _ .

For λ = α _ = 1, s ˜ α _ = Δ c _ + s ¯ / 2 > Δ c _. For other parameter values, a binomial equation of the form

(27) 2 μ s ˜ α _ 2 s ˜ α _ 2 s ¯ 1 μ s _ + s ¯ + λ Δ c _ + s ¯ s _ λ α _ 1 d + μ c _ 1 μ p H + Δ c _ + s ¯ + λ Δ c _ s ¯ 1 μ s _ = 0

arises. Figure 2 illustrates the solution of (27) for specific parameter values.[13] The threshold is increasing in α _ and in the coinsurance rate. GPs treat more patients the larger their degree of altruism is. Stronger altruism also leads to more treatment in the outpatient sector if patients pay a higher share of the health services price. Efficiency ( s ˜ α _ = Δ c _ = 2) would be achieved for λ = 1 if minimum altruism among GPs were α _ 0.64. For λ < 0.8, by comparison, an efficient solution does not exist for any value of α _, given the assumed parameter values.

Figure 2 
The threshold function with naïve patients as a function of the type of surviving GPs 


_\underline{\alpha }.
Figure 2

The threshold function with naïve patients as a function of the type of surviving GPs α _.

Consider, finally, the case in which both rational and naïve patients are present, and α α _. Then, GPs of type α _ will crowd out all GPs α > α _ for naïve patients by setting p t R = 0 and asking a competitive referring price according to (24). Naïve patients visit a referrer and are referred according to (3) if

(28) s n a s ˜ n a = λ p H + p r R κ + p r R + c _ α _ .

Less selfish GPs α > α _, on the other hand, can undercut p r R by asking p r C = d μ κ and p t C = c _ + d + 1 μ κ. These are the competitive prices for care providers according to Proposition 1 (rational patients).

Rational patients will consult a care provider and are treated if s r a < s ˆ r a = λ p H c _ κ and referred otherwise. Then, s ˜ n a < s ˆ r a holds since λ α _ 1 × p r R + κ + c _ < 0. As s ˜ r a s ˆ r a , independent of the optimal kickback κ , the equilibrium is always inefficient.

9 Scope for regulation

A regulatory body has several options to change the allocation as it can administer the hospital price and set the coinsurance rate. For the outpatient sector, in which numerous self-employed physicians compete for patients, it does not appear to be feasible to administer GP prices. However, regulation can ban kickbacks, as is common in most countries.

Lemma 1 (Rational patients).

The patients’ price for inpatient care is increasing in the coinsurance rate ( d s ˆ r a / d λ > 0), while unaffected by the hospital price ( d s ˆ r a / d p H = 0).


  1. From (15) d κ / d p H = 1 for given s ˆ r a . As the hospital price and kickbacks move in parallel, the rational patients’ GP choice is unaffected (see (10)). Thus, d s ˆ r a / d p H = 0.

  2. From (16), we derive

    (29) d s ˆ r a d λ = Δ c _ 1 g s ˆ r a 2 g s ˆ r a 2 d s ˆ r a d λ + g s ˆ r a d s ˆ r a d λ 1 G s ˆ r a = Δ c _ d s ˆ r a d λ 1 + g s ˆ r a 1 G s ˆ r a g s ˆ r a 2 = Δ c _ 2 + g s ˆ r a 1 G s ˆ r a g s ˆ r a 2 .

    The denominator is positive according to the SOC (20), which implies d s ˆ r a / d λ > 0.


The profit-maximizing kickback increases with the hospital price; in fact, it cancels out any change in the hospital price ( d κ / d p H = 1). This implies that the rational patients’ GP choice is unaffected (see (10)), which, in turn, explains why s ˆ r a is independent of a change in the hospital price, and is also true if the hospital were free to set its price. In that case, we have a full equilibrium.[14] Adding naïve patients does not change the result. Naïve patients will still all visit a referrer, as the care providers under competition will not be able to sufficiently lower the treatment price. Administering the hospital price, thus, has no consequences for the allocation, except for distributional effects between patients and the providers of health care.

In a competitive equilibrium with a kickback ban, s ˆ r a = λ p H c _ applies for rational patients. In this case, efficiency for rational patients can be achieved by setting the hospital price at p H = c _ + Δ c _ / λ. This price will not ensure that all naïve patients will be treated in the appropriate sector, except in the case in which all GPs are perfect altruists. Then, we have s ˆ r a = s ˆ n a = Δ c _.

If the coinsurance rate increases, at given prices and kickbacks, patients’ price of hospital care increases (see (10)) and all GPs will treat more patients as they take into account patients’ disutility from higher prices (see (3)). As a reaction, the hospital tends to increase the kickback (see (18)).[15] This will result in a decrease in the competitive referral price and to an increase in the competitive treatment price (see (8) and (9)). However, these second round effects are not sufficiently strong to offset the increase in patients’ price of inpatient care.

Proposition 4 (Rational patients).

The optimal coinsurance rate is intermediate, i. e., 0 < λ r a < 1. If kickbacks are allowed, the optimal coinsurance rate is increasing in the price elasticity of demand.


Efficiency requires s ˆ r a = Δ c _. Introducing this expression into (19) and solving for λ directly leads to the optimal coinsurance rate λ r a = ϕ Δ c _ 1 ϕ Δ c _ . As ϕ Δ c _ > 1, it follows that 0 < λ r a < 1. Furthermore, d λ r a / d ϕ Δ c _ > 0. With a kickback ban, (10) satisfies s ˆ r a = λ p H c _ , which leads to λ r a κ = 0 = Δ c _ / p H c _ in the efficient equilibrium. Since r > p H c ¯ > c _ + Δ c _, 0 < λ r a κ = 0 < 1 holds.  □

With a uniform distribution of s, the optimal coinsurance rate is λ r a = 2 Δ c _ s ¯ / Δ c _ and ϕ Δ c _ = Δ c _ / s ¯ Δ c _ . The optimal kickback satisfies κ = p H c _ Δ c _ / 2 Δ c _ s ¯ , which is positive if s ¯ < Δ c _ 2 Δ c _ / p H c _ . If, on the other hand, s ¯ > Δ c _ 2 Δ c _ / p H c _ , no kickbacks are paid and the first best is the same as with a kickback ban, i. e., λ = Δ c _ / p H c _ .

Proposition 4 also applies for the case in which all GPs are perfect altruists. The optimal coinsurance rate when kickbacks are forbidden also works for naïve patients, if only perfect altruistic GPs are present in the market. With exclusively selfish GPs in the market, on the other hand, efficiency cannot be achieved by optimally setting the coinsurance rate. A second-best policy might be feasible. If Δ c _ < 0 1 s g s d s, a kickback ban would improve the allocation, whereas for Δ c _ 0 1 s g s d s no regulation would be needed. The same reasoning applies to the equilibrium with naïve patients, where only selfish GPs survive in the market. If the altruistic parameter is bounded from below to be positive, the optimal kickback is positive. Forbidding kickbacks, in this case, would generally not imply a welfare improvement. Instead, the regulator can set the optimal coinsurance rate according to (25). For uniformly distributed s, starting from (27), it can be shown that d λ β / d α _ s ˜ α _ = Δ c _ < 0, i. e., as more altruistically motivated GPs tend to treat a higher share of minor ill patients, the coinsurance rate must decrease to achieve the optimal referral rate. As can be gleaned from Fig. 2, health services might need to be taxed, i. e., λ β > 1, if GPs’ degree of altruism is low.

Proposition 5 (Rational and naïve patients).

The optimal coinsurance rate is second best only. It is smaller than the one in the rational equilibrium and it decreases in the naïve patients’ population share.


From (23), we find for s ˆ r a = Δ c _: λ β = ϕ Δ c _ ϕ Δ c _ 1 β / 1 β Δ c _ · g s ˆ r a . This coinsurance rate implies an efficient allocation of rational patients. First best cannot be achieved as all naïve patients are referred to the hospital ( s ˆ n a = s _). Furthermore, as β > 0, we obtain λ β < ϕ Δ c _ 1 ϕ Δ c _ = λ r a Finally, d λ β / d β = 1 β 2 · Δ c _ · g s ˆ r a 1 < 0.[16]  □

We know that care providers become more attractive for rational patients if the population share of naïve patients β or the coinsurance rate λ increases, due to a change of prices in favor of care providers. Thus, if β increases, the regulator will react with a decrease in λ to offset the change in outpatient prices.

If every GP shows at least a minimal degree of altruism, the level of coinsurance has an effect on the referral decision of the most selfish GPs. They will treat more patients if the regulator increases the coinsurance rate. Still as s ˆ r a s ˆ n a , the regulator cannot achieve a first-best allocation. A kickback ban would foster outpatient treatment, but again, would not lead to efficiency.

10 Conclusion

We investigated the market outcome of kickbacks paid by a monopolistic hospital to competitive GPs in return for patient referrals. Kickbacks can incentivize defrauding behavior by physicians who refer their patients to the hospital, which, in turn, might lead to a crowding-out of altruistically motivated GPs.

We proved the existence of an equilibrium that separates GPs into referrers, who refer all patients to the hospital, irrespective of whether they are severely ill or only suffering from a minor illness, and care providers, who only refer major ill patients but treat minor ill ones. This result holds true for any administered hospital price, as long as this price is above marginal cost. It would even apply for different hospital prices, depending on the type of treatment. However, two hospital prices would imply verifiability of treatment, which would fundamentally alter the model and GPs’ incentives.

Rational patients select which group of GPs (referrers or care providers) to visit, depending on their anticipated net utility from treatment. Naïve patients will all opt for the referrers. GPs join their respective group, based on the degree of their altruistic motivation towards patients. Purely selfish GPs will join the referrer group, whereas GPs with some degree of altruism will all belong to the group of care providers. This equilibrium is inefficient as all naïve patients choose the referring GPs. A policy that sets the coinsurance optimally achieves a second best, in which rational patients make an efficient choice. It may seem unrealistic that some GPs operate as pure referrers. This however is due to the assumption that purely selfish GPs exist. An interesting result arises if all GPs show a positive degree of altruism. In this case, again, all naïve patients end up being treated by the least altruistic GPs. However, these GPs do not refer all of their patients to the hospital but treat those whose disadvantage from inpatient treatment exceeds a threshold. A regulator can influence this threshold by setting the optimal coinsurance rate, but cannot optimally steer both the rational and the naïve patients, having only one instrument available. If the population share of naïve patients increases, he will lower the coinsurance rate to offset the change in outpatient prices.

The assumption of a competitive outpatient market might be questionable. Patients do not necessarily choose the GP who offers primary care at the lowest price, but also have demands for the quality of service, like treatment time or waiting period. If we assume that these services are costly and that GPs are efficient in their provision, results will qualitatively not change.

In our setting, GPs face not capacity constraints. If such constraints apply, as in Emons (2013), crowding out would be further limited. Prices would then be determined by the marginal referring GP and less altruistically motivated referrers would have positive profits in the equilibrium. In contrast to Godager et al. (2015), a higher altruism will not lead to more referrals. This is because patients in our setting differ with respect to their preference for treatment and referral. Figure 1 suggests that more altruistic GPs even refer less patients, because minor ill patients receive less expensive treatment in the outpatient sector. This effect, however, cancels out as soon as prices are competitive. Then, altruism is the driving force for GPs’ referral decision, independent of their specific degree of altruism.

We assumed that patients can verify neither the diagnosis nor the treatment. Instead, one might consider that patients are able to verify the kind of treatment they receive, but not their diagnosed health status before treatment. This excludes the possibility of overcharging; i. e., a patient who has a minor illness receives the appropriate inexpensive treatment but pays an excessive price. The hospital’s price for a minor treatment will then be bounded from above by the outpatient price. Partial verifiability, however, gives rise to overtreatment, as the hospital might have an incentive to employ the expensive treatment, which still allows it to charge the monopoly price (Dulleck and Kerschbamer 2006). From a societal perspective, overtreatment should be prevented, as it leads to an overuse of resources. Prohibiting kickbacks can be beneficial in this environment. However, introducing competition in the inpatient market will lead to marginal cost prices and to the disappearance of kickbacks. With full verifiability, i. e., if the diagnosis were also verifiable, GPs would need to react to the preferences of their patients, both rational and naïve ones. The corresponding equilibrium would be the pure rational one of Proposition 1.

An equilibrium, in which heterogeneous GPs split into two separate groups according to their functional roles, by either treating their patients or referring them to hospitals with a diagnosis, is similar to some contractual arrangements in certain health care systems. A market in which providers are allowed to engage in selective contracting and clients can enroll in specific health insurance plans, a self-sorting of providers and clients to different contracts might occur. As third-party stakeholders, insurers can design the reimbursement scheme for the providers and set appropriate user prices. Whether competition between insurers allows for prices that deviate from marginal cost is another story, which we leave for future research.


We presented different versions of this article at the annual conference of the German Association in Basel, March 2017, the biennial iHEA conference in Boston, July 2017, the annual conference of the German Economic Association in Vienna, Sept. 2017, the annual meeting of the Health Economists Group within the German Heath Economic Association in Linz, Oct. 2017, and the biennial conference of the European Health Economic Association in Maastricht, July 2018. We thank all participants as well as Beat Hintermann, Matthias Minke, Robert Nuscheler and the GER’s referee for helpful comments. The usual caveat applies.


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Published Online: 2020-09-03
Published in Print: 2021-02-02

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