Gatekeeping versus direct-access when patient information matters

HEALTH ECONOMICSHealth Econ. 19: 730–754 (2010)Published online 18 June 2009 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/hec.1506

GATEKEEPING VERSUS DIRECT-ACCESS WHEN PATIENTINFORMATION MATTERS

PAULA GONZALEZa,b,�

aDpto. Economıa, Metodos Cuantitativos e Historia Economica, Universidad Pablo de Olavide, Sevilla, SpainbFEDEA, Spain

SUMMARY

We develop a principal-agent model in which the health authority acts as a principal for both a patient and ageneral practitioner (GP). The goal of the paper is to weigh the merits of gatekeeping versus non-gatekeepingapproaches to health care when patient self-health information and patient pressure on GPs to provide referrals forspecialized care are considered. We find that, when GPs incentives matter, a non-gatekeeping system is preferableonly when (i) patient pressure to refer is sufficiently high and (ii) the quality of the patient’s self-health informationis neither highly inaccurate (in which case the patient’s self-referral will be very inefficient) nor highly accurate (inwhich case the GP’s agency problem will be very costly). Copyright r 2009 John Wiley & Sons, Ltd.

Received 30 January 2008; Revised 15 March 2009; Accepted 29 April 2009

JEL classification: D82; H51; I18; L51

KEY WORDS: general practice; moral hazard; incentives; patient self-health information; patient pressure; referrals

1. INTRODUCTION

This paper weighs the relative benefits of gatekeeping versus non-gatekeeping health-care systems,within the context of the current debate over the merits of giving general practitioners (henceforth, GPs)control over patient access to specialized care. In particular, it studies how patient self-healthinformation and patient pressure on GPs to provide them with referrals affect the efficacy of eachsystem, and thus contribute to our assessment of its desirability.

At present, two main types of health-care systems can be observed in most European countries. Incountries like Italy, the Netherlands, Norway, Spain, and the United Kingdom, a gatekeeping systemprevails and, hence, GPs control patient access to specialized care. There are other countries, likeBelgium, Finland, France, Germany and Sweden, where the gatekeeping role of the GP is very limited,as patients are free to choose between GPs and specialists.

Researchers have debated the pros and cons of these two systems for the past several years. On theone hand, it has often been argued that appointing GPs as gatekeepers to specialist care can help reduceoverall health-care costs.1 For many countries in Eastern Europe, where health care is now underreform (Hebing, 1997), this perceived financial advantage seems to be driving policymakers’ currentinterest in designing systems based on a gatekeeping strategy. Cost-containment concerns also seem to

*Correspondence to: Dpto. Economıa, Metodos Cuantitativos e Historia Economica, Universidad Pablo de Olavide, Carretera deUtrera, km 1. 41013 Sevilla, Spain. E-mail: [email protected]

1See, for instance, Delnoij et al. (2000), Franks et al. (1992), Martin et al. (1989), and Starfield (1994, 1996).

Copyright r 2009 John Wiley & Sons, Ltd.

have encouraged some direct-access countries to switch over to gatekeeping-like systems in recentyears.2

Nevertheless, there seems to be little evidence that gatekeeping actually serves to lower health-careexpenditures (see, e.g. Barros, 1998 and Kroneman et al., 2006). In fact, in countries such as the UnitedStates, where gatekeeping has been a central strategy in the cost-containment initiatives of managedcare organizations (Wolf and Gorman, 1996), some HMOs are now relaxing the restrictions on access tospecialized care (Ferris et al., 2001).

Closely related to the debate over the relative merits of these organizational models is that concerningthe role of GP incentives. It has been widely recognized in the literature that the regulation of the GPsand the payment structures they face have significant implications for costs in health-care systems (see,for instance, Scott, 2000). Because such incentives tend to significantly influence patient diagnosis andreferral – the two decision-making processes of greatest potential impact as far as overall costs areconcerned – they must be taken into account whenever gatekeeping and non-gatekeeping systems arecompared.3

To the best of our knowledge, Garcıa-Marinoso and Jelovac (2003) are the only to provide a uniformtheoretical framework for identifying the optimal GP payment system, which they then use to comparethe desirability of gatekeeping versus non-gatekeeping systems. They find that the optimal GP paymentscheme would require a combination of cost-sharing components: a cost sharing of the GP treatmentand a bonus for non-referral. This contract yields the right diagnosis and referral incentives to the GP.They conclude that, when GP incentives matter, gatekeeping must be viewed as superior to non-gatekeeping.

Despite the novelty of their analysis and its obvious relevance to this discussion, Garcıa-Marinosoand Jelovac focus only on GP incentives when defining the optimal system. In particular, they disregardthe role played by patient self-awareness and the effect of patient pressure on GPs for referrals ingatekeeping versus non-gatekeeping systems.

This paper aims to fill this gap by proposing a model that considers not only GP incentives, but alsopatient behavior as determinant of each system’s relative efficacy and desirability. To do so, we considerthree aspects of the patient’s role. First, we acknowledge that the patient’s self-health information willcertainly influence our view of which system is preferable, since it may be more efficient to allow apatient with an accurate understanding of her condition to freely choose her own medical provider thanit would be to require her to obtain a referral. Second, we take into account that when patients canfreely choose their medical provider, the design of a co-payment system to discipline patients who mightstrategically choose to visit a specialist or a GP should be addressed, since the extra cost to patients ofco-payments may reduce the attractiveness of a direct-access system. Third, we recognize that patientpressure on GPs to refer them for specialized treatment may ultimately undermine the efficiency of agatekeeping system. In this regard, it has been suggested that GPs are highly responsive to patientpressure as far as their referral behavior is concerned (see Virji and Britten, 1991; Armstrong et al., 1991and O’Donnell, 2000, among others).4 There is also some evidence that patient expectations may play a

2For instance, Belgium, France, and Germany recently showed initiatives to introduce a gatekeeping system. In Belgium, tostimulate the gatekeeping role of the GP patients pay lower co-payments if they register with one specific GP and accept that theirindividual ‘global medical file’ is kept by that GP (Schokkaert and Van de Voorde, 2005). Similarly, since July 2005 all thosebenefiting in France from health insurance coverage must choose their main physician (‘medecin traitant’). As a result, it will costthem more to consult a specialist directly, without being referred by their ‘medecin traitant’ (Sandier et al., 2004). In Germany,since 2004 sickness funds are obliged to offer the option to enrol in a ‘family physician care model’ with financial incentives forpatients who comply with the gatekeeping rules (Busse and Riesberg, 2004).

3There is empirical evidence that GP behavior is influenced by economic motives. See, for instance, Croxson et al. (2001) on thereferrals of GPs in the UK, and Iversen and Luras (2000) on the volume of services provided by GPs in Norway.

4Fleming (1992) in a European study of referrals, reported that pressure from patients about whether they should be referred‘influenced’ between 30 and 60 percent of referrals.

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Copyright r 2009 John Wiley & Sons, Ltd. Health Econ. 19: 730–754 (2010)

DOI: 10.1002/hec

role in many referral decisions and that GPs who feel pressured by their patients or who sense that thelatter may expect a referral may, in fact, be more likely to provide one (Webb and Lloyd, 1994).

In accordance with the approach that GP contracts should include appropriate diagnosis and referralincentives, we analyze how patient information and patient pressure on GPs to refer them to secondary caremay affect our choice of one system over the other. The model used here is a principal-agent model, in whichthe health authority acts as a principal for both a patient and a GP. Two types of informational asymmetriesarise in this context. First, the health authority faces a problem of moral hazard in its relationship with theGP, since neither the diagnosis nor its outcome is verifiable. Second, the patient has an informationaladvantage when selecting a physician, as her belief about her severity is private information.

We find that if the regulator wants to give GP incentives to making correct diagnoses and referrals,non-gatekeeping is optimal only when patient pressure on GPs is sufficiently high. This is becausegatekeeping systems may not be able to provide adequate GP incentives in such high-pressure contexts.In addition, we find that non-gatekeeping systems are optimal only when the patient’s informationregarding her own condition is neither too bad nor too good. If a patient’s signal is very uninformative,her self-referral will be very inefficient. When patient information is extremely accurate, however, non-gatekeeping is convenient for the patient and also saves some of the costs of specialized care.Nevertheless, this advantage is outweighed by the fact that in a non-gatekeeping system the accuracy ofthe patient’s self-diagnosis fosters primary care costs. Stated more clearly, since only patients who thinkthat their health condition is mild will visit the primary provider, this self-diagnosis is a source ofinformation that will dissuade the GPs from making a costly diagnosis. In this case, a health authoritythat wishes to profit from patient self-health information will find it more efficient to allow patients tofreely choose their medical providers, calling on the GP’s referral only if treatment fails.

Although primary care is recognized as the mainstay of many health-care systems in developedcountries, theoretical work by economists into general practice is still scarce. The study that comesclosest to our own – the aforementioned article by Garcıa-Marinoso and Jelovac (2003) – diverges fromours in its conclusion that gatekeeping systems are always better than non-gatekeeping ones when GPincentives matter. In contrast, we argue that the optimality of gatekeeping systems should be qualifiedwhen the role of patient information and patient pressure on GPs to refer are also taken into account.

Brekke et al. (2007) contributes to this discussion by analyzing the competition effects amongstsecondary care providers that arise when GPs are assigned a gatekeeping role. From this perspective,they also sustain that gatekeeping systems may not always be socially desirable.

Finally, the behavior of GPs has been explored from a number of other angles. Thus, Malcomson(2004) discusses which contractual agreements are most effective to induce GPs to exert effort indiagnosis, and finds that it is counterproductive to offer GPs incentive-based contracts in contextswhere patients are allowed to choose between a gatekeeper with an incentive contract and one without.Karlsson (2007), on the other hand, analyzes the desirability of competition among GPs as aninstrument for assuring the quality of primary care services. He finds that capitation systems havetrouble providing GPs with appropriate incentives, since the search activity of patients offsets the directeffects of a change in the capitation rate.

The rest of the paper is organized as follows. In the following section, we present our model. InSection 3, we analyze both patient and GP behavior. In Section 4, we derive the optimal patientco-payment levels and the optimal contractual payment scheme for GPs. In Section 5, we compare thetwo institutional frameworks discussed above. Finally, in Section 6 we conclude.

2. THE MODEL

Our basic set-up is in line with Garcıa-Marinoso and Jelovac (2003). There are three agents in oureconomy: a patient, a GP, and the regulator or health authority. In fact, there is implicitly a fourth

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agent: a provider of specialized medical care, but we will consider him as a passive agent, as the analysisof his behavior is out of the scope of this article.

We will now go through the three players of the model and outline their objectives, their choicevariables and their knowledge. Finally, we will set out the timing of the game. Appendix A provides asummary of all the relevant variables of the model.

2.1. The patient

The patient suffers from a certain illness. The severity of the illness is measured by a random variable s.We assume that s can only take two values: s and �s, which indicate whether the patient is either low orhigh severity. For the sake of simplicity, we assume that both types of illnesses are equally likely. Thepatient is perfectly aware that she is ill but does not know just how serious her illness is. Her symptoms,however, provide her with a private signal or belief about the severity of her health problemðsb 2 fsb; �sbgÞ. We assume that the probability that a patient receives a correct signal is b 2 ð12 ; 1Þ. Thevalue of b, therefore, measures the level of self-awareness of the patient as it captures the likelihood thatthe patient correctly assesses the severity of her condition. Formally:

Prð�sb j�sÞ ¼ PrðsbjsÞ ¼ b and Prð�sb jsÞ ¼ Prðsbj�sÞ ¼ 1� b

The patient, therefore, seeks health care from a medical provider. She will demand medical attentioneither from a GP or from a specialist. This decision depends on the existing institutional framework. Ingatekeeping, the patient has no choice and has to visit the GP. In non-gatekeeping, however, the patientcan choose to visit either the GP or the specialist.

We consider the patient to be endowed with a utility function that is separable in health and income.The patient’s health status and her available income are the same in all contingencies (minor or majorillness). Therefore, in this model, maximizing patient expected utility is equivalent to minimizing thevalue of her expected costs. These costs come from two main sources. First, from the health loss (l) thatthe patient suffers when she receives primary care and a referral is necessary. These losses can beunderstood as the cost of waiting for specialist treatment. Second, the patient may also incur amonetary cost. In non-gatekeeping, where patients can freely choose their medical provider, the healthauthority has to set certain co-payments to induce the patient to enter the health-care sector either onthe primary level, or directly on the secondary one. The set of co-payments is denoted byðpg; pgs; psÞ 2 R3

þ, where pg measures the monetary cost of visiting the GP, pgs represents the cost ofvisiting the specialist with a GP referral and, finally, ps is paid in case the patient decides to accessspecialized medical care directly.

As claimed in the Introduction, there exists empirical evidence suggesting that patient pressure onGPs for referrals may alter GP behavior. We model this pressure as the probability that the patientrejects GP treatment. In case of doing so, she will demand private specialist treatment at a cost f. Thisway, she avoids any potential health loss, although she bears the full cost of receiving specializedtreatment. In particular, we assume that there is a fraction r of patients that are obstinate, in the sensethat the GP cannot convince them that they have a minor illness when they believe they have a majorone. These patients always decide to reject GP treatment and pay for specialized services even if, as wewill see later, it would be worthwhile for them to follow the GP’s recommendation.5

5We have chosen this modelization with obstinate patients purely for the purposes of expositional clarity. The same qualitativeresults hold in a more complex model where all patients are fully rational but there is heterogeneity in their cost of waiting, in sucha way that those with high waiting costs find it optimal to reject the GP’s treatment recommendation. The details of thisalternative are available upon request.



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2.2. The general practitioner

We consider that the GP is able to cure a patient only if the severity of the health condition is low, whilethe specialist can heal both levels of severity. The GP, upon receiving a patient, is required to make adiagnosis in order to assess whether the patient requires specialized treatment or not. The diagnosisyields a signal about the severity of the patient’s condition ðsd 2 fsd; �sdgÞ that is correct with probabilityd 2 ð12 ; 1Þ. Formally:

Prð�sd j�sÞ ¼ PrðsdjsÞ ¼ d and Prð�sd jsÞ ¼ Prðsdj�sÞ ¼ 1� d

We consider that d4b, i.e. once the GP has made a diagnosis, his level of knowledge about the trueseverity of the illness exceeds that of the patient.6 In making a diagnosis, the GP incurs a disutility, thatwe denote by cd.

As part of the diagnosis, the physician also observes the patient’s belief regarding her own healthcondition. Combining the diagnosis with this piece of information, the GP should decide on treating thepatient or referring her to the specialist. If the GP prescribes a treatment that cures the patient, the gameends. Otherwise, the patient is referred to the specialist, bearing a health loss in those cases where theGP has not referred her directly.

As both the GP’s decision to diagnose a patient and the diagnosis are hard to verify, the incentivesincluded in the payment contract will crucially determine GP’ behavior.7 Our payment contract is basedon observable variables and consists of four non-negative components ðD;B;T;RÞ. First, the GP recei-ves a fixed payment D. Second, he receives a budget B to purchase a range of specialist services and tocover his prescribing. Third, he incurs a cost T when providing treatment to the patient. Finally, the GPpays R when the patient is referred to the specialist. This payment structure contains: (i) a capitationcomponent or payment per visit (D), (ii) the savings the GP makes on any cost-sharing scheme, either(B�T) if he does not refer the patient to specialized care or (B�R) if he does not prescribe treatment.8

We restrict our analysis to situations where the budget allocated to the GP is sufficiently large so as tocover any potential service the patient may require. Formally, this amounts to considering thatðB� T� RÞ � 0.9

2.3. The Health Authority

The third agent involved in the model is the health authority. The health authority pays the costs of thetreatment provided to the patient, and also the payments made to both the GP and the specialist.

We denote by cs the costs of the specialist services, which include not only the treatment costs, butalso the payments made to the specialist. As the costs of treatment by a specialist are generally higherthan the costs of treatment by a GP we normalize, without loss of generality, the latter to zero.

The health authority designs the GP contract and the patient level of co-payments so as to minimizethe sum of the financial costs both from primary and specialized health care (i.e. expected treatment

6For simplicity, we focus on the case in which both sd and sb are correlated with s, but patient and GP errors are conditionallyindependent.

7The remuneration methods for GPs differ across countries and experimentation with their contractual arrangements abounds. Ingeneral, the reforms depart from strict capitation or fee-for-service payments and introduce additional components aimed atcontaining costs and reducing referrals to hospital.

8This contract is in the same spirit of the fundholding scheme for GPs reintroduced in the UK in April 2005. In the UK, GPsfundholders are allocated a budget to provide primary health care and purchase some of the specialist services for which theyreferred patients. The ‘unspent’ share of their budget could be reinvested in their own practice (Dusheiko et al., 2007). Also, insome Italian ASLs (‘Aziende Sanitarie Locali’) GP contractual arrangements combine capitation with an additional payment thatrewards GPs with a proportion of the savings generated from meeting expenditure targets, including the cost of pharmaceuticals,laboratory tests, and therapeutic treatments prescribed by the GP (France et al., 2005).

9In the UK, budgets for GPs fundholders were intended to be sufficient to buy the bundle of services which the GP’s patients hadconsumed before the GP became a fundholder (Dusheiko et al., 2007).

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costs and payments to both the GP and the specialist) and the patient’s expected costs (which includesboth her expected health losses and her monetary expenses).10

We denote CGP for the expected financial costs associated with primary care, CSp for specializedtreatment and CPat for the patient expected costs. Formally, the problem faced by the health authoritycan be stated as follows:

minpg ;pgs ;psD;B;T;R

CPat þ CGP þ CSp

Our aim is to study whether it is socially useful to use patient self-awareness as a mechanism forprovider selection. Hence, the level of co-payments will be designed to ensure that patients use their self-health information and visit the specialist directly only if they believe a GP will not heal them. As we areinterested in providing the GP with right diagnosis and referral incentives, we mainly focus on contractsthat induce the GP to diagnose and follow the diagnosis, i.e. to treat the patient whenever the signalreceived from the diagnosis is sd and refer her if �sd.

2.4. Timing

The timing of the game consists of the following stages. First, the health authority sets the GP paymentcontract, which the GP can either accept or reject (in which case the game ends), and also sets thepatient level of co-payments. Second, the severity of the illness is realized, and the patient seeks healthcare from a medical provider. If she visits the specialist the game ends. If she visits the GP, then thedoctor makes a diagnosis, which provides him with a signal about the patient’s severity. In the thirdstage, after observing the signal, the GP decides whether to treat the patient himself or to refer her to thespecialist. If he decides to refer the patient, the game ends. In case he decides to treat her, she may acceptor reject this treatment. If she rejects it or, in case she accepts, if the patient recovers her health, thegame ends. Otherwise, the patient is referred to the specialist.

As usual, we solve the game by backward induction.

3. AGENT BEHAVIOR

In this section, we characterize the behavior of the patient and the GP in our model. First, we analyzehow the level of co-payments determines the decision of the patient to either visit the GP, or directlyrequest specialist medical treatment. This analysis only applies when considering systems where patientsare not obliged to compulsorily visit the GP. Second, we set out to derive the conditions that the GPpayment contract has to fulfill in order to ensure that he decides to costly diagnose the patient and,afterwards, follow the diagnosis. Figure 1 provides the extensive form of representation of the gameunder analysis.

3.1. Patient behavior

In our model, the patient can be either high severity or low severity, with an ex ante equal probability.However, once the patient observes her own symptoms and is aware of her personal circumstances, sheis able to update these probabilities. Then, the probabilities that the patient recognizes/misrecognizesthe severity of her illness, that are detailed in Appendix B, are

Prð�sj �sbÞ ¼ PrðsjsbÞ ¼ b ¼ 1� Prðsj �sbÞ ¼ 1� Prð�sjsbÞ ð1Þ

10Our objective function departs from a standard social cost function in that co-payments appear only as costs for patients but notas revenues for the health authority. We have chosen this modelization to avoid an indeterminacy in the characterization of theoptimal co-payments. For our qualitative results to remain valid with a standard social cost function, it suffices to introduce acost of raising public funds. Adding such cost, however, would complicate the comparisons without adding further insights.



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In non-gatekeeping, the patient has the choice between two alternatives: visit the specialist directly, orgo first to the GP. If the patient visits the specialist directly, her cost is given by the co-payment she hasto pay ps, but no health loss is borne. If the patient visits first the GP she always pays pg and, then, if sheis eventually referred to the specialist pgs. Moreover, she may also suffer from a health loss whenever shereceives treatment from the GP that does not heal her. Those patients who are obstinate always reject

Figure 1. GP and patient decision tree: GP payoffs (P) and patient costs (C)

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GP treatment if they believe to be in a severe condition. In this case, they do not incur either pgs or thehealth loss l, but they have to pay the private fee f.

With the help of Figure 1, patient costs in any circumstance can be easily computed.11 In comparingpatient expected costs when demanding first GP attention, or direct specialist care, we conclude thefollowing:

(i) If sb, a patient goes first to the GP whenever:

ps � pg þ bð1� dÞpgs þ ð1� bÞð pgs þ ð1� dÞlÞ

(ii) If �sb, a non-obstinate patient visits the specialist directly whenever:

ps � pg þ bð pgs þ ð1� dÞl Þ þ ð1� bÞð1� dÞpgs

(iii) If �sb, an obstinate patient visits the specialist directly whenever:

ps � pg þ bðdpgs þ ð1� dÞ f Þ þ ð1� bÞðð1� dÞpgs þ dfÞ

Taking into account that the co-payment levels have to provide appropriate incentives to any patient,we obtain the following lemma.12

Lemma 1. A patient visits the specialist directly when �sb and goes to the GP when sb if and only if:

� ps � pg � bð pgs þ ð1� dÞlÞ þ ð1� bÞð1� dÞpgs and� ps � pg � bð1� dÞpgs þ ð1� bÞðpgs þ ð1� dÞlÞ

This lemma shows that the higher the accuracy of the patient information is the milder bothrestrictions are. This is a natural result since what the health authority is trying to induce through theco-payments is, precisely, that the patient uses her self-health information when selecting the medicalprovider. The more accurate the understanding of her condition is, therefore, the smaller the expectedcosts of her self-referral.

3.2. GP behavior

In our model, the GP faces a population of patients that can be either high or low severity, with ex antethe same probability. In order to update these probabilities, the GP uses two pieces of information: thepatient’s beliefs and the signal received from the diagnosis.

Once this information has been acquired, the probabilities of correctly diagnosing a low severity, thatare detailed in Appendix B, are

Prðsjsd \ sbÞ ¼db

dbþ ð1� dÞð1� bÞ¼ 1� Prð�sjsd \ sbÞ

Prðsjsd \ �sbÞ ¼dð1� bÞ

dð1� bÞ þ ð1� dÞb¼ 1� Prð�sjsd \ �sbÞ

ð2Þ

11Throughout this sub-section it is considered that the GP behaves optimally, i.e. makes a diagnosis and follows it. The paymentsthat ensure this behaviour are computed in Sub-section 4.2.

12In order to avoid that no patient accepts GP treatment, the private alternative must be sufficiently costly. In particular, f mustexceed the patient’s expected costs associated with accepting GP treatment when the patient receives signal �sb (i.e. when thepatient is more interested in being referred to a specialist). Formally, this requires that f4ðbð1� dÞ=ðbð1� dÞþð1� bÞdÞÞðlþ pgsÞ.



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Analogously, the probabilities of wrongly diagnosing a low severity are:

Prðsj �sd\sbÞ ¼ð1� dÞb

ð1� dÞbþ dð1� bÞ¼ 1� Prð�sj �sd\sbÞ

Prðsj �sd\ �sbÞ ¼ð1� dÞð1� bÞ

ð1� dÞð1� bÞ þ db¼ 1� Prð�sj �sd\ �sbÞ

ð3Þ

Once the GP has diagnosed the true severity of the health condition, he then decides whether to treat thepatient or refer her to the specialist.13 If the GP refers the patient to the specialist, he receivesthe capitation payment D and keeps the unspent budget (B�R). If the GP recommends treatment, healways incurs the cost-sharing T on the treatment and, with a certain probability (i.e. if the patient has amajor illness and needs a referral) he also incurs R. With the help of Figure 1, the payments to the GP inany circumstance can be easily computed. In comparing the different payments that the GP receivesfrom prescribing either treatment or referral, we can conclude the following:

(i) If sd and sb, the GP treats the patient whenever T� ðdb=dbþ ð1� dÞð1� bÞÞR and refers herotherwise.

(ii) If sd and �sb, the GP treats the patient whenever T� ðdð1� bÞð1� rÞ=dð1� bÞ þ ð1� dÞbÞR� rðB� T� RÞ and refers her otherwise.

(iii) If �sd and sb, the GP refers the patient whenever T� ðð1� dÞb=ð1� dÞbþ dð1� bÞÞR and treatsher otherwise.

(iv) If �sd and �sb, the GP refers the patient whenever T� ðð1� dÞð1� bÞð1� rÞ=ð1� dÞð1� bÞ þ dbÞR� rðB� T� RÞ and treats her otherwise.

From these conditions, we see that both T and R have to be strictly positive, which means that theGP contract should include some cost sharing both on treatment and specialist services.

Observe also that the conditions that the GP payment scheme has to fulfill in order to effectively inducehim to follow the diagnosis are different for the two institutional frameworks. In non-gatekeeping, sinceonly patients who think that their health condition is mild visit the GP, the only relevant restrictions are (i)and (iii). In gatekeeping the four conditions matter, but it can be easily checked that the most demandingones that actually determine GP behavior are (ii) and (iii) This leads to the following lemma.

Lemma 2. The GP always follows the diagnosis if and only if:

� In non-gatekeeping:

� In gatekeeping:

13Throughout this sub-section, it is considered that patients behave optimally, i.e. in non-gatekeeping the patient demands primaryattention only if sb. The co-payments that ensure this behaviour are computed in Sub-section 4.1.

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In gatekeeping, the GP faces all kind of patients. In order to ensure that the GP always follows hisdiagnosis, we have to induce him to do so even in those cases in which this is contrary to the patient’sbeliefs. As a result, the higher the referral pressure (measured by r) the more difficult to induce the GP tostick to his diagnosis as he will be more tempted to over-refer patients. In non-gatekeeping, the GPalways receives patients who think they are in low severity. This implies that there is no pressure forreferral, which makes the restrictions less demanding.

In our model, the GP receives neither his signal nor the patient’s one until Stage 3 of the game.Before this stage, therefore, the GP has to decide whether to make a diagnosis or not, and whatto do in case he does not make it (either systematically treat or refer the patient). When the GPdecides to diagnose the patient, it could be the case that, afterwards, he might decide not tofollow the diagnosis. The conditions written in Lemma 2 ensure that the GP will stick to hisdiagnosis.

The derivation of the GP’s expected utility when the GP diagnoses the patient and followsthe diagnosis (U), for both gatekeeping and non-gatekeeping systems, is detailed in Appendix C.Once the GP’s expected utility has been computed, we can obtain the restrictions that determinewhen he decides to diagnose the patient. These restrictions come from ensuring that the abovestated utility is higher than both the utility the GP would obtain from systematically referring thepatient (UNgk

R ¼ UgkR ¼ Dþ ðB� RÞÞ or from systematically treating him, UNgk

T ¼ Dþ ðB� T� RÞ þRb in non-gatekeeping, and Ugk

T ¼ Dþ 12 ðB� TÞðbþ ð1� rÞð1� bÞÞ þ ðB� T� RÞð1� bþ ð1� rÞbÞ½ �

in gatekeeping.The following lemma summarizes the GP’s decision of making a diagnosis.Lemma 3. The GP decides to make a diagnosis if and only if:


� In gatekeeping:

The conditions to induce diagnosis are, as predictable, more demanding as the cost of the diagnosisincreases. As it is the case in Lemma 2, an increase in the accuracy of the diagnosis makes the conditionsless demanding.

Combining Lemmas 2 and 3, we obtain the following proposition that states how the referralpressure can be an unsolvable problem.

Proposition 1. Finding a contract ðD;B;T;RÞ that induces the GP to treat the patient when sd and torefer her if �sd:

� In non-gatekeeping it is always possible.� In gatekeeping it is possible provided r� r.

With r ¼ 1� ð1� dÞb=dð1� bÞ.



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Proof. See Appendix D. &

This proposition ensures that in non-gatekeeping, it is always possible to design a payment contractthat induces the GP to diagnose a patient and follow the diagnosis. In gatekeeping, however, this is notthe case, and the existence of such a contract depends on the intensity of patient pressure. If the referralpressure is sufficiently high, it is impossible for the health authority to find values of ðD;B;T;RÞ thatsimultaneously fulfill all the constraints. The reason for this is that patient pressure on GPs for referralsgenerates a problem of over-referral of patients with a mild diagnosis. The value of the cost sharing onthe specialist services that would induce the GP to treat a patient whenever sd is so high that it wouldeventually give incentives to the GP to treat also patients with a severe diagnosis.

Proposition 1 has shown an important implication of the presence of patient pressure for referrals.A gatekeeping system maybe unsustainable, since it may not be able to provide the GP with properincentives to diagnose the patient and follow the diagnosis, while this is not a problem in non-gatekeeping.

It can be checked that r is decreasing in b. This implies that as patient self-health informationbecomes more accurate, the maximum threshold of pressure compatible with gatekeeping decreases. Asthe patient has a more accurate understanding of her condition, the physician will be less willing toeffectively recommend treatment when this is contrary to the patient’s will.

4. THE HEALTH AUTHORITY’S PROBLEM

As stated in Section 2.3, the health authority aims at minimizing the sum of the financial costsassociated with primary and secondary care, and patient expected costs. CGP, CSp, and CPat are derivedformally in Appendix C.

The problem of the health authority can be analyzed in two steps. First, if the system is a non-gatekeeping one, the health authority has to design the set of co-payments that induces the patient tovisit a specialist directly if and only if he believes he is high severity. Secondly, the health authority hasto design the contract that provides the GP with incentives to make (and follow) a diagnosis.14

4.1. The optimal co-payment levels

The co-payment levels set by the health authority will be the ones that minimize CPat. The healthauthority has to take into account the constraints computed in Lemma 1, which ensure that the patientonly visits the specialist directly when �sb, as well as the fact that co-payments have to be non-negative.

The health authority’s optimization program is as follows:

minpg;pgs ;ps

CPat ¼ 12 ðps þ pg þ pgsðbð1� dÞ þ 1� bÞ þ lð1� bÞð1� dÞÞ

s:t

ps � pg � bðpgs þ ð1� dÞlÞ þ ð1� bÞð1� dÞpgs

ps � pg � bð1� dÞpgs þ ð1� bÞðpgs þ ð1� dÞlÞ

pg � 0; pgs � 0; ps � 0

8>><>>:

The following proposition characterizes the optimal level of co-payments.Proposition 2. If the health authority wants the patient to visit the specialist directly when �sb and to

go first to the GP if sb, the optimal level of co-payments ðp�g; p�gs; p

�s Þ is such that p�g ¼ p�gs ¼ 0 and

p�s ¼ ð1� bÞð1� dÞl.

14The problem can be solved in two steps since: (i) As long as the GP diagnoses the patient and follows the diagnosis CPat isindependent from the GP contract; (ii) CGP is not altered by the co-payment levels, provided they induce the patient to select themedical provider according to her belief about the severity; (iii) CSp depends only on the institutional framework.

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This proposition shows that setting only ps40 is enough to induce patients to follow their beliefwhen choosing their medical provider. This co-payment structure is in line with the very recent Belgianand French reforms, aimed at enhancing the gatekeeping role of GPs. In both countries, theco-payments are larger for those patients who go to the specialist directly, without a referral.15

It is worth noting that this simple structure for the optimal co-payments relies on the fact that inour model the patient is endowed with a linear utility in money, so income effects are absent andco-payments do not interfere with financial insurance issues.16

Finally, it is straightforward to see that the patient always benefits from a higher accuracy in theunderstanding of her condition. The reason is two-fold. First, the health losses she bears are lower, asher self-selection of medical provider is more likely to be correct. Second, the monetary expenses shefaces also diminish, as the co-payments needed to induce her an appropriate selection of medicalprovider are decreasing in the accuracy of her belief.

4.2. The optimal payment contract

The payments offered to the GP will be the ones that minimize CGP. The health authority has toconsider the fact that the GP’s expected utility (U) cannot be lower than his reservation utility(normalized to zero) ðPCÞ, and also that his liability constraints have to be fulfilled ðLLCÞ. We do theanalysis within this framework with limited liability constraints for the doctor, i.e. we impose that,under any circumstance, the doctor must receive a non-negative payment.

On top of this, we must include that the GP’s incentive compatibility constraints (IC) in the healthauthority’s optimization program. These are the restrictions that induce the GP to diagnose the patientand follow the diagnosis (defined in Lemmas 2 and 3).

The health authority’s optimization program is as follows:

minD;B;T;R

CGP

s:t:

U� 0 PC

D� cd LLC1

T� 0 LLC2

R� 0 LLC3

B� Tþ R LLC4

IC

8>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>:

ð5Þ

With CGP 2 fCNgkGP ;C

gkGPg and IC 2 ffICNgk

Pdi; ICNgk

Fdig2i¼1, fIC

gkPdi; ICgk

Fdig2i¼1g, depending on whether we are

in non-gatekeeping or gatekeeping.

15In these countries, co-payments also have a dissuasive purpose and, therefore, there is a positive (though small) level of co-payment for visiting the GP or the specialist with a GP’s referral. This is not necessary in our model as we do not deal withhealthy individuals who make unnecessary visits to the system.

16Garcıa-Marinoso (1999) provides a description of how the insurer can regulate access to specialist care by manipulating thepatients’ insurance contract in a model with income effects.



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From Proposition 1 we know that, in gatekeeping, designing a contract that induces the GP todiagnose the patient and to follow the diagnosis is only possible provided the referral pressure is not toohigh. Hence, hereinafter, we restrict our analysis to values of r such that r� r.

Let us define

er � d� bð1� bÞðd� bþ 2dbÞ

2 ð0; �rÞ

The following proposition characterizes the GP’s optimal payment contract.Proposition 3. If the health authority wants the GP to treat the patient when sd and to refer her if �sd

the optimal contract ðD;B;T;RÞ is as follows:


DNgk ¼ cd

BNgk ¼ TNgk þ RNgk

TNgk ¼cd

ð2d� 1Þð1� bÞ

RNgk ¼cd

ð2d� 1Þð1� bÞb

The health authority’s expected primary care costs are:

CNgkGP ¼

cd2ð2d� 1Þ

4dþb

1� b� 1

� �� In gatekeeping:

Dgk ¼ cd

Bgk ¼ Tgk þ Rgk

Tgk ¼2cdðð2d� 1Þ þ 2ð1� dÞGðd;b; rÞÞ

2d� 1

Rgk ¼4cdGðd;b; rÞ

ð2d� 1Þð1� ð1� bÞrÞ

The health authority’s expected primary care costs are

CgkGP ¼

cd2d� 1

½4dþ 2ðGðd;b; rÞ � 1Þ�

With Gðd;b; rÞ ¼1 if r� er

ð2d�1Þð1�ð1�bÞrÞðdð1�bÞþð1�dÞbÞ2 ð1�ð1�bÞrÞðd�ð1�dÞðdð1�bÞþð1�dÞbÞÞ�db½ �41 otherwise

8<:


The structure of the GP’s optimal payment contract is similar in gatekeeping and non-gatekeeping.First, in the worst possible contingency, the GP receives a capitation payment or a payment per visitthat covers the cost of making a diagnosis (DNgk ¼ Dgk ¼ cd). Second, to avoid systematic treatment ofthe patient, the contract includes some cost sharing of the GP’s treatment (both TNgk and Tgk are strictlypositive). Third, to avoid systematic referral, the contract also imposes some cost sharing of thespecialist costs (both RNgk and Rgk are strictly positive). Finally, the budget allocated to the physician(B) should be just enough to cover the expenses in which the GP may incur.17

17The qualitative insights of the optimal contract are similar to the ones in Garcıa-Marinoso and Jelovac (2003).

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We focus now on analyzing how the health authority’s expected primary costs are affected by ourrelevant variables (r and b). As far as patient pressure on GPs for referrals is concerned, we find thatsuch pressure raises the payments the health authority has to allocate to the physician in order to avoidan excessive number of referrals in a gatekeeping system. To be more precise, the savings the GP makesif he does not refer the patient to specialized care (Bgk � Tgk) are increasing in r. Nevertheless, we findthat, for values of pressure below er, the referral pressure does not affect the health authority’s expectedprimary costs. The reason is that the increase in the payments to the GP is compensated by the fact thatsuch payments are paid less often at equilibrium. For values beyond er, the increase in the paymentsneeded to provide the GP with proper incentives is so high that it always affects primary care costs. As aresult, these costs will be higher the larger the value of r.

Finally, the quality of the patient’s information unambiguously raises expected costs from primarycare. The reason, however, is of a different nature in non-gatekeeping and gatekeeping. Innon-gatekeeping, patient information generates a problem of ‘diagnosis substitution’. The factthat only those patients who think that their health condition is mild visit the GP fosters GP’sincentives to use the patient’s self-diagnosis as a substitute of his own diagnosis and systematicallytreat the patient. Hence, inducing the GP to make a diagnosis becomes extremely expensive. Ingatekeeping, patient information increases health costs through the referral pressure. For levels ofpressure above er, the more accurate the patient’s information is, the more difficult that the GP decidesto follow the diagnosis. This makes it more costly for the health authority to avoid an excessive numberof referrals.

4.3. Optimal contracts under uninformed strategies for GPs

So far we have followed the conventional wisdom in the literature that GP contracts should includeappropriate diagnosis and referral incentives. For this reason, we have computed patient co-paymentsand the GP’s contract for the scenario where the GP diagnoses the patient and follows the diagnosis. Inthis sub-section, we briefly discuss how the results would differ when the GP follows ‘uninformed’strategies, i.e. the GP does not condition his referral decision on the information gained during thediagnosis. In other words, the GP can either systematically treat or refer the patient to the specialistwhatever the diagnosis indicates.

Focusing, first, on the patient’s side of the problem, it is straightforward to compute the co-paymentsthat ensure that the patient only visits the specialist directly when �sb for the two uninformed strategies ofthe GP may follow. Consider first that the GP’s contract is such that he systematically refers all patientsto specialized treatment. In this case, patients face no potential health loss in visiting first the GP and,hence, no co-payments to discipline patients are needed (pRg ¼ pRgs ¼ pRs ¼ 0). When the paymentcontract induces the GP to systematically treat all patients co-payments do play a role. Following thesame line of reasoning as in Lemma 1 and Proposition 2, the new co-payments can be easily computedand are given by pTg ¼ pTgs ¼ 0 and pTs ¼ ð1� bÞl. One can see that co-payments share the same structureas when GP’s incentives matter. The only difference is that, now, the co-payment when visiting directlythe specialist is higher, since when the GP systematically treats all patients, it is more costly to providepatients with incentives to visit the specialist directly only when �sb.

When we concentrate on the GP’s side of the problem, it is straightforward to derive the optimalpayment contracts for the uninformed strategies we are considering. As in these cases, the GP ignoresthe diagnosis that is not worthwhile for the health authority to provide him with incentives to diagnose.The optimal contract, therefore, is a flat contract in which the GP: (i) receives no payment to cover thecost of making a diagnosis (D ¼ 0), (ii) receives a budget that covers his reservation utility, and (iii)faces no cost sharing either when treating or when referring the patient (T ¼ R ¼ 0).

Total expected costs for the health authority associated to each of these uninformed strategies can beeasily computed. If the health authority wants the GP to systematically treat all patients the costs are as



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follows:

CNgk T ¼ð1� bÞlþ 12 ð2� bÞcs

Cgk T ¼ 12 ðrfþ ð1� rÞlÞ þ 1

2 ð1� rÞcsð6Þ

On the contrary, if the health authority wants the GP to systematically refer all patients, the costs areCNgk R ¼ Cgk R ¼ cs.

5. ON THE CHOICE OF THE OPTIMAL SYSTEM

In this section, we provide a discussion on the global picture the health authority faces when choosingthe best organizational system to access health care. We mainly focus on the scenario where GPcontracts include appropriate diagnosis and referral incentives, although we also study how the resultsextend if the GP adopts an uninformed strategy. As it has become clear throughout the paper, the roleplayed by patient self-awareness, as well as their pressure on GPs for referrals, are the two key elementsthat will drive the health authority choice between gatekeeping and non-gatekeeping systems.

5.1. Partial comparisons

As a first step, we confront gatekeeping and non-gatekeeping focusing separately on each of thecomponents of the health authority’s expected costs: patient costs, primary care costs, and specialized costs.

First, focusing only on the patient’s side of the problem, we find the following.Proposition 4.When GP contracts include appropriate diagnosis and referral incentives, there exists a

threshold b�o1, such that:

� If b� b�, gatekeeping generates lower patient expected costs than non-gatekeeping.� If b 2 ðb�; 1Þ, non-gatekeeping generates lower patient expected costs than gatekeeping.


When we concentrate only on patient expected losses, non-gatekeeping may be the optimal system toaccess medical care. The reason is clear as when patients can freely choose their medical provider, thehealth authority relies on their information. As the quality of the patient’s belief increases, the self-selection becomes perfect and the costs associated with this system converge to zero. In gatekeeping, onthe contrary, as we force patients to disregard their own belief and always access primary care, we donot profit completely from their more accurate understanding.18

This reasoning extends directly to the case where the GP skips the diagnosis and adopts theuninformed strategy of systematically treating all patients (see Equation (6)).19

Considering only the GP’s side of the problem, we get the following.Proposition 5.When GP contracts include appropriate diagnosis and referral incentives, and focusing

only on primary care expected costs:

� If bo 1þ4d2þ4d non-gatekeeping is preferred to gatekeeping.

� If b� 1þ4d2þ4d there exists a threshold r�ðbÞ 2 ½~r; r� such that:

� If r� r�ðbÞ gatekeeping is preferred to non-gatekeeping.� If r4r�ðbÞ non-gatekeeping is preferred to gatekeeping.

18Note that in the limit case when b! 1 and, thus, d! 1, patient costs under the two systems converge to zero.19Recall that we have seen in Sub-section 4.3 that when the uninformed strategy is one of systematic referral of patientsgatekeeping and non-gatekeeping are completely equivalent in all dimensions.

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Proposition 5 illustrates the trade-off the health authority faces in terms of primary care costs. It canbe checked that, whenever a patient visits the GP, primary care costs are always smaller in gatekeepingsystems, provided the problem of pressure is not severe. However, these costs are incurred less often innon-gatekeeping, as not all the patients visit the GP. For this reason, we find that there exist values of bfor which non-gatekeeping is less costly, even in the absence of pressure.

As patient self-health information becomes more accurate, the GP’s incentives to skip the diagnosisincrease, which raises the costs of non-gatekeeping. When b is sufficiently high, then gatekeeping issuperior, unless there is a sufficiently important problem of pressure for referral, i.e. if r4r�ðbÞ.20

Note that neither patient self-health information nor referral pressure plays a role in the choicebetween gatekeeping and direct-access when the GP follows an uninformed strategy. This is due to thefact that both b and r affect the cost of providing adequate incentives to the GP and such incentives areabsent when dealing with uninformed strategies.

Finally, in terms of expected secondary costs, it is direct to see the following.Proposition 6. When GP contracts include appropriate diagnosis and referral incentives, expected

secondary costs are never lower in non-gatekeeping. Moreover:

� The higher is b, the closer the expected specialized costs in both systems.� The higher is r, the lower the expected specialized costs in gatekeeping relative to non-

gatekeeping.


Even if, in general, gatekeeping allows savings in specialist costs, the more accurate the patient’sunderstanding of her condition, the closer the costs in the two systems. The improvement in theaccuracy of self-referrals reduces the over-utilization of specialist treatment in non-gatekeeping. On thecontrary, a higher referral pressure on GPs implies more patients leaving the public sector, what reducesthe expected specialist costs in gatekeeping.21

It is straightforward to check that the same result holds when the GP adopts the uninformed strategyof systematically treating all patients.

5.2. Global comparison

This sub-section combines the previous results and provides an overall comparison of gatekeeping andnon-gatekeeping when both financial costs (GP and specialist costs) and patient costs aresimultaneously considered.

We start by considering the scenario where GP contracts include appropriate diagnosis and referralincentives. First, when the patient’s signal is extremely uninformative ðb! 1

2Þ both patient expectedhealth losses and co-payments are higher in non-gatekeeping and, hence, from the patient’s point ofview gatekeeping dominates. From the GP’s incentives point of view, however, non-gatekeepinggenerates lower costs (CNgk

Gp oCgkGp if b! 1

2Þ, but only because the GP is visited less often. Nevertheless,gatekeeping saves with respect to non-gatekeeping in terms of unnecessary visits to the specialist(Cgk

SpoCNgkSp if b! 1

2). Moreover, this difference in specialist costs will outweigh any saving that

20The reader should note that the range of parameters where gatekeeping dominates in terms of primary costs is actually rathernarrow. If we consider the lower bound on b computed in Proposition 5, together with the restriction that d4b, we find that theparameter space where gatekeeping dominates starts at d ¼ ð1þ

ffiffiffi5

pÞ=4, that is approximately 0.8.

21Note that this argument does not imply that it is optimal for the health authority to have a pure private health system. Resortingto private practice is costly for patients (f) and the health authority internalizes the cost of the private treatment through patientexpected cost.



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non-gatekeeping may yield in terms of primary care costs, provided specialized treatment is sufficientlymore costly than primary care. Therefore, in general, systems where patients freely choose their medicalprovider would not dominate.

Second, when patient information is extremely accurate ðb! 1Þ there are strong arguments in favorof non-gatekeeping. First, since patients make no mistakes when selecting their medical provider, atequilibrium they bear no health losses and the co-payments they pay become negligible. Second, there isnot an over-utilization of specialist services, as no low severe patient mis-interpret her symptoms. A highaccuracy of the patient’s information, however, has perverse effects on GP behavior and these are moresevere in non-gatekeeping. In particular, when b! 1, inducing the GP to diagnose the patient andfollow the diagnosis becomes prohibitively expensive when patients can freely choose their provider.This makes that, overall, gatekeeping dominates.

Finally, for intermediate parameter values, the optimal choice will depend on the relative strength oftwo opposite effects. On the one hand, non-gatekeeping, even if it allows to successfully use patientinformation, will generate a substitution of GP’s diagnosis by patient self-health information. On theother hand, gatekeeping suffers from the problem of patient pressure on GPs for referrals, that mayeven make a successful process of diagnosis and treatment/referral choice impossible.

We summarize the discussion above in the following corollary.Corollary 1. If the health authority wants the GP to diagnose the patient and follow the diagnosis,

and the patient to adequately select her medical provider, then:

� If r4r, gatekeeping is unsustainable. Non-gatekeeping dominates.� If r� r, then:� When b! 1

2 gatekeeping dominates.� For intermediate values of b, there exists a threshold in the level of patient pressure such that,

for values below it, gatekeeping dominates whereas, for values above it, the optimal system isa non-gatekeeping one.

� When b! 1 gatekeeping dominates.

When the health authority does not provide the GP with incentives to diagnose, the effect of theaccuracy of the patient’s self-health information on the choice of the best system to access secondarycare is much simpler and intuitive. In general, non-gatekeeping dominates provided the quality of thepatient’s information is sufficiently large, as the following corollary shows.

Corollary 2. If the health authority wants the GP to skip the diagnosis and either systematically treator refer the patient, and the patient to adequately select her medical provider, there exist two thresholdsb1 � 2l=ð2lþ csÞ and b2 � ðð1þ rÞðcs þ l Þ � rfÞ=ð2lþ csÞ such that:

� If b� maxfb1; b2g gatekeeping dominates. Moreover, the health authority wants the GP tosystematically refer (treat) the patient if cs � ð>Þðrfþ ð1� rÞlÞ=ð1þ rÞ.22

� If b� maxfb1;b2g non-gatekeeping dominates and the health authority wants the GP tosystematically treat the patient.

Two are the main insights that can be extracted from this corollary. First, consistently with thescenario where GP contracts include appropriate diagnosis and referral incentives, when patients arepoorly informed about their health status gatekeeping is superior. In this case, whether the healthauthority prefers the GP to systematically treat or refer the patient will depend on the relativemagnitude of the health loss the patient suffers when she receives primary care and a referral is

22Note that for cs � ðrfþ ð1� rÞlÞ=ð1þ rÞ, the dominance of gatekeeping is only weak, since for the uninformed strategy ofsystematic referral gatekeeping and non-gatekeeping are equivalent.

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necessary (l), vis-a-vis the costs of specialist services (cs). The higher (lower) the value of cs relative tothat of l, the more inclined will be the health authority towards a strategy of systematic treatment(referral) of patients.

Second, when patient self-health information is sufficiently accurate, the health authority should relyon this information and, therefore, should opt for a non-gatekeeping system coped with a systematictreatment on the part of the GP.

5.3. Discussion

At this point, it seems appropriate to briefly review the cases at hand. Our analysis suggests that, if thegoal is to provide GPs with incentives to diagnose patients and refer them only when necessary, a non-gatekeeping system is optimal only when (i) patient pressure on GPs to refer them for specialized care issufficiently high and (ii) the quality of the patient’s self-health information is neither very inaccurate (inwhich case the patient’s self-referral will be very inefficient) nor very accurate (in which case the GP willbe strongly tempted to skip the diagnosis and systematically treat). This result points to an apparentcontradiction: when patient self-health information is very accurate, it is not worth using. However,such a paradoxical recommendation can only be sustained when the analysis is restricted to thosesituations for which GP incentives matter, where it is impossible to design a contract that combines boththe patients’ prior knowledge and the physician’s eventual diagnosis. When we disregard incentives todiagnose and consider uninformed strategies in which the GP systematically treats or refers patients, amore accurate patient self-health information unambiguously favors non-gatekeeping.

In fact, if we look for the best system to access secondary care when patient knowledge is highlyaccurate, we draw the conclusion that it is impractical to provide the GP with incentives. The bestapproach in this case is to use the patient’s self-health information instead of (and not merely inaddition to) the GP’s diagnosis, for two reasons. First, when a patient knows a great deal about her ownstate of health, the added value of the physician’s diagnosis is small because the probability that thepatient will misinterpret her symptoms is very low. Second, providing the GP with incentives todiagnose is very costly. Consequently, when compared against any other alternative, a system in whichpatients are free to choose their medical provider and only receive the GP’s referral after a failedtreatment will always dominate. It allows the health authority to benefit completely from the patient’sknowledge, while eliminating the financial burden of giving incentives to GPs.

6. CONCLUDING REMARKS

We have developed a principal-agent model in which the health authority acts as a principal for both apatient and a GP, and have employed it to evaluate the appropriateness of using the latter as agatekeeper to secondary care. We have followed the conventional wisdom in the literature that GPcontracts should include appropriate diagnosis and referral incentives. In this setting, we have shownthat patient self-health information and referral pressure on GPs alter our outcome as to which systemoffers the best strategy for accessing specialist care. Specifically, we have demonstrated that these twofactors directly affect the expected health-care costs of patients and indirectly affect the diagnosis andreferral behavior of the GP, and thus have an impact on the expected primary and secondary costs ofcare.

In terms of policy recommendations, our analysis suggests that whenever GP incentives matter,which system we choose depends on the relative importance of two factors: patient pressure on GPs toprovide referrals for secondary care (in the gatekeeping system) and the use of patient self-healthinformation as a substitute for the GP’s diagnosis (in the non-gatekeeping one). In general, non-gatekeeping is optimal only if the pressure to refer is sufficiently high and if the patient’s information isneither very accurate nor very inaccurate. Two factors play a role here. On the one hand, if the patient’s



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signal is highly uninformative, her self-referral will be very inefficient. However, if her self-healthinformation is extremely accurate, the benefits of using a non-gatekeeping system will be outweighed bythe high expense of providing GPs with incentives to diagnose. In this case, a health authority thatwishes to put patient self-health information to efficient use may not find it worthwhile to encourage theGP to screen out the less serious cases. On the contrary, the authority may find it more efficient to allowpatients to freely choose their medical providers, calling on the GP’s referral only if treatment fails.

This insight opens up a potentially fruitful path of research centred on the potential substitutabilityand/or complementarity of the health information provided by patients and GPs, respectively, whichwould further our knowledge about the shape of optimal contractual agreements for GPs.

One potential criticism to our work is that the quality of the patient’s self-health information may bedifficult for health authorities to discern. However, one would expect such information to be more accurateamong patients whose condition is chronic or inherited, recurrent, or has easily recognizable symptoms thanamong those with other types of pathologies. And it is precisely for these types of illnesses with clearsymptoms where the comparison between gatekeeping and non-gatekeeping becomes relevant. For diseaseswith more diffuse symptoms, where patients are much more likely to choose the wrong specialist, gatekeepingwould clearly offer the additional advantage of favoring a correct match between patients and specialists.

For this article, we chose to focus on a single-physician framework, avoiding contexts in which GPscompete over patients. For such a setting, an alternative to patient pressure would be the patient’soption to obtain the desired referral from a competing GP. The above analysis ruled out this possibilityin order to keep our model analytically tractable. Nevertheless, all our qualitative insights regarding theeffects of patient pressure on GPs for referrals would remain valid in a model characterized by GPcompetition provided: (i) the patient bears a cost of switching to another GP (such as a searching cost)and (ii) the GP’s revenues diminish whenever he loses a patient.

There are other issues that have not been addressed in this work. First, we have not taken intoaccount how distributional concerns would affect the choice of the best system to access secondary care.Patient self-awareness is often highly related with income and, hence, a non-gatekeeping system mayfoster inequities as poorer individuals may not be able to assess so precisely their actual healthcondition. Consequently, systems with gatekeeping may provide better guidance for people with lowersocioeconomic position. Second, we have assumed that the probability that a patient exerts pressure onthe GP in order to obtain a referral is exogenous. Nevertheless, one can reasonably argue that the bettereducated may be better able to articulate their needs and insist on referral when they think it isnecessary, what again suggests the need to incorporate socioeconomic variables into future analyses onthe optimal organization of health care.

Finally, we would like to highlight that, although primary care is recognized as the basis of health-care systems in many developed countries, to date economists have done little theoretical research intogeneral practice. We hope that this study will open the door to further research into that area, and thatit will also stimulate the ongoing debate over the pros and cons of encouraging GPs to take on agatekeeping role. Both theoretically and empirically, further research is clearly needed before we canaccurately assess the relevance of the relationship between patient self-health information, patientpressure on GPs to refer and GP incentives.

ACKNOWLEDGEMENTS

This work was initiated while I was visiting CORE-UCL (Belgium), whose hospitality is gratefullyacknowledged. I am indebted to Maurice Marchand and Nicolas Porteiro for useful discussions andvaluable suggestions. I also would like to thank Louis Eeckhoudt, Karen Eggleston, Begona Garcıa-Marinoso, Javier Hualde, Izabela Jelovac, Robert Nuscheler, Motohiro Sato as well as seminarparticipants at XXVIII Simposio de Analisis Economico (Sevilla), 5th European Health Economics

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Workshop (York), Sixth Biennial Conference on the Industrial Organization of Health Care (Hyannis,Massachusetts), Universidad de Navarra, Universidad Carlos III de Madrid and University ofCopenhagen for their helpful comments. Comments and suggestions from the editor and two anonymousreferees have improved the paper considerably. I acknowledge financial support from Fundacion PedroBarrie de la Maza, Junta de Andalucıa (projects SEJ-01252 and SEJ-02936) and Spanish Ministry ofScience and Innovation (project ECO2008-04321/ECON). The usual disclaimers apply.

APPENDIX A: SUMMARY OF NOTATION

s 2 fs; �sg true severity of the illness.sb 2 fs; �sg patient belief.b 2 ð12 ; 1Þ accuracy of patient belief.l40 health loss if patient receives primary care and a referral is necessary.f40 cost of the private treatment alternative.r 2 ð0; 1Þ proportion of obstinate patients (rate of referral pressure).ð pg; pgs; psÞ Set of co-payments:

pg when visiting the GP.pgs when visiting the specialist with a GP’s referral.ps when visiting directly the specialist.

sd 2 fs; �sg GP’s diagnosis outcome.d 2 ðb; 1Þ Accuracy of GP’s diagnosis.cd40 Cost of GP’s diagnosis.ðD;B;T;RÞ GP’s contract:

D Capitation (or fee for service) payment.B Budget allocated to purchase services for the patient.T Cost borne when treating the patient.R Cost borne when referring the patient.

cs40 cost of the specialist services.CPat patient expected costs.CGP expected costs associated with primary care.CSp expected costs associated with specialist treatment.

APPENDIX B: GP AND PATIENT UPDATED PROBABILITIES

Let us consider three random variables s, sd and sb, such that s, sd, sb 2 f�s; sg.Both sd and sb are correlated with s However, we consider that patient and GP errors are not

correlated.In general, 8i; j 2 f�s; sg it is true that:

Prðs ¼ ijsb ¼ iÞ ¼Prðsb ¼ ijs ¼ iÞPrðs ¼ iÞ

Prðsb ¼ ijs ¼ iÞPrðs ¼ iÞ þ Prðsb ¼ ijs ¼ jÞPrðs ¼ jÞ

Then, (1) follows directly.It is also true that 8i; j 2 f�s; sg:

Prðs ¼ ijsd ¼ i\ sb ¼ jÞ ¼Prðs ¼ iÞPrðsd ¼ i\ sb ¼ jjs ¼ iÞ

Prðs ¼ iÞPrðsd ¼ i\ sb ¼ jjs ¼ iÞ þ Prðs ¼ jÞPrðsd ¼ i\ sb ¼ jjs ¼ jÞ



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Moreover,

Prðsd ¼ i\ sb ¼ jjs ¼ iÞ ¼ Prðsd ¼ ijs ¼ iÞPrðsb ¼ jjs ¼ iÞ

From here it is straightforward to derive the expressions in (2) and (3).

APPENDIX C: GP’S EXPECTED UTILITY, HEALTH AUTHORITY’S EXPECTEDFINANCIAL COSTS AND PATIENT EXPECTED COSTS

In gatekeeping: GP’s expected utility:

Ugk ¼Dþ PrðsÞ½Prðsd \ sbjsÞðB� TÞ þ Prðsd \ �sb jsÞð1� rÞðB� TÞ

þ ðPrð�sd\sbjsÞ þ Prð�sd\ �sb jsÞÞðB� RÞ� þ Prð�sÞ½ðPrð�sd\sbj�sÞ þ Prð�sd\ �sb j�sÞÞðB� RÞ

þ Prðsd \ sbj�sÞðB� T� RÞ þ Prðsd \ �sb j�sÞð1� rÞðB� T� RÞ� � cd

¼Dþ 12 ½ðB� RÞ þ ðB� TÞdðbþ ð1� rÞð1� bÞÞ þ ðB� T� RÞð1� dÞð1� bþ ð1� rÞbÞ� � cd

Health authority’s expected primary care costs:

CgkGP ¼ Ugk þ cd¼ Dþ 1

2 ½ðB� RÞ þ ðB� TÞdðbþ ð1� rÞð1� bÞÞ þ ðB� T� RÞð1� dÞð1� bþ ð1� rÞbÞ�

Health authority’s expected specialized care costs:

CgkSp ¼ ½Prð�sÞð1� rPrðsd \ �sb j�sÞÞ þ PrðsÞPrð�sd jsÞ�cs ¼

cs2ð2� d� rð1� dÞbÞ

Finally, patient expected costs:

CgkPat ¼PrðsÞPrð�sb\sdjsÞrfþ Prð�sÞ½Prð�sb\sdj�sÞðrfþ ð1� rÞlÞ þ Prðsb \ sdj�sÞl�

¼ 12 ½ð1� dÞðð1� bÞrlþ ð1� rÞlÞ þ rfðð1� bÞdþ ð1� dÞbÞ�

In non-gatekeeping: GP’s expected utility:

UNgk ¼Dþ PrðsjsbÞ½PrðsdjsÞðB� TÞ þ Prð�sd jsÞðB� RÞ�

þ Prð�sjsbÞ½Prð�sd j�sÞðB� RÞ þ Prðsdj�sÞðB� T� RÞ� � cd

¼Dþ ðB� T� RÞ þ T½dð1� bÞ þ ð1� dÞb� þ Rdb� cd

Health authority’s expected primary care costs:

CNgkGP ¼ PrðsbÞðUNgk þ cdÞ ¼ Dþ ðB� T� RÞ þ T½dð1� bÞ þ ð1� dÞb� þ Rdb

Health authority’s expected specialized care costs:

CNgkSp ¼ ½Prð�sÞ þ PrðsÞðPrð�sd\sbjsÞ þ Prð�sb jsÞÞ�cs ¼

cs2ð2� dbÞ

Finally, patient expected costs:

CNgkPat ¼PrðsÞ½Prðsb \ sdjsÞpg þ Prðsb \ �sd jsÞðpg þ pgsÞ þ Prð�sb jsÞps�

þ Prð�sÞ½Prðsb \ sdj�sÞðpg þ pgs þ lÞ þ Prðsb \ �sd j�sÞðpg þ pgsÞ

þ Prð�sb j�sÞps� ¼ 12 ðps þ pg þ pgsðbð1� dÞ þ 1� bÞ þ lð1� bÞð1� dÞÞ

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APPENDIX D

Proof of Proposition 1. In non-gatekeeping, it is easy to check that, for any value of b and d, thereexist values of R, T and B, such that ICNgk

Pd and ICNgkFd are fulfilled simultaneously.

In gatekeeping, however, ICgkFd1

and ICgkFd2

are mutually compatible if and only if:

ð1� dÞbð1� dÞbþ dð1� bÞ

R�dð1� bÞð1� rÞ

dð1� bÞ þ ð1� dÞbR� rðB� T� RÞ

Since B� Tþ R the best case for this inequality to be fulfilled is when B ¼ Tþ R: Taking this intoaccount, it is direct to check that the inequality can be true if and only if r� r, with�r ¼ 1� ð1� dÞb=dð1� bÞ. It can be shown, then, that for any r� r, there exist values of B, T andR, such that ICgk

Pdand ICgkFd are fulfilled simultaneously. This completes the proof. &

Proof of Proposition 2. The optimal level of co-payments is the solution to the program given by (4).The problem is one of linear programming. Hence, the solution lies on a vertex of the restricted domainof the program. It can be shown that the restrictions pg � 0, pgs � 0 and ps � pg � bð1� dÞpgs þð1� bÞðpgs þ ð1� dÞlÞ must be binding at the optimum. The solution, therefore, is given by p�g ¼ p�gs ¼ 0and p�s ¼ ð1� bÞð1� dÞl. This completes the proof. &

Proof of Proposition 3. We compute the optimal payment contract for non-gatekeeping andgatekeeping separately. To determine the relevant incentive constraints we use Lemmas 2, 3 and 4.

(a) In non-gatekeeping, the program the health authority faces is:

minD;B;T;R

12½Dþ ðB� T� RÞ þ T½dþ ð1� 2dÞb� þ Rdb�

s:t PC; fLLCig4i¼1; IC

NgkPd1

ICNgkPd2; ICNgk

Fd1; ICNgk

Fd2

First of all, it is straightforward to see that LLCi, i ¼ 1; . . . ; 4 imply the PC and that ICNgkFd are implied

by ICNgkPd . Therefore, the health authority chooses the cheapest contract compatible with the LLCi and

the ICNgkPD . It can be checked that LLC1 has to be binding at the optimum. The reasoning is the

following: the health authority’s costs are increasing in D and D does not appear in any of the incentive

constraints. As a result DNgk ¼ cd. An analogous reasoning allows us to ensure that LLC4 has to be also

binding and that, hence, BNgk ¼ TNgk þ RNgk.

It is easy to see that LLC3 binding cannot be a solution as ICNgkPd1

and ICNgkPd2

would be mutually

incompatible. Moreover, LLC2 and ICNgkPd1

binding cannot be a solution as ICNgkPd2

would not be fulfilled.

A similar argument rules out LLC2 and ICNgkPd2

binding as a potential solution.

The optimal solution of the problem, hence, has to be such that ICNgkPd1

and ICNgkPd2

are binding. Fromhere we obtain that:

TNgk ¼cd

ð2d� 1Þð1� bÞand RNgk ¼

cdð2d� 1Þð1� bÞb

The health authority’s expected primary care costs are:

CNgkGP ¼

cd2ð2d� 1Þ

4dþb

1� b� 1

� �� (b) In gatekeeping the problem the health authority faces is:

minD;B;T;R

Dþ 12½ðB� RÞ þ ðB� TÞdðbþ ð1� rÞð1� bÞÞ þ ðB� T� RÞð1� dÞð1� bþ ð1� rÞbÞ�

s:t: PC; fLLCig4i¼1; IC

gkPd1

ICgkPd2; ICgk

Fd1; ICgk

Fd2



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First of all, it is straightforward to see that LLCi, i ¼ 1; . . . ; 4 imply PC. Moreover, by a similarreasoning as in the non-gatekeeping case, Dgk ¼ cd at the optimum. Secondly, a simple, but tediousprocess allows to check that LLC4 has to be binding at the optimum. If B4Tþ R this wouldincrease the equilibrium values of T and R resulting, hence, in higher costs for the HA. Therefore,Bgk ¼ Tgk þ Rgk.

In order to proceed, let us define er � ðd� bÞ=ð1� bÞðd� bþ 2dbÞ 2 ð0; �rÞ. We solve the program bydistinguishing two cases:

� If r� er, then ICgkPd1

and ICgkPd2

imply both ICgkFd1

and ICgkFd2

. A completely analogous reasoningto the one followed for non-gatekeeping shows that the solution of the problem is such that both

ICgkPd1

and ICgkPd2

are binding. The optimal values, hence, are given by: Tgk ¼ 2cd2d�1,

Rgk ¼ 4cd=ð2d� 1Þð1� ð1� bÞrÞ.� If r 2 ðer; �r�, then it is not true that ICgk

Fd1and ICgk

Fd2are implied by ICgk

Pd1and ICgk

Pd2. First of all, it is

straightforward to see that neither T� 0 nor R� 0 can be binding at equilibrium. Therefore, theoptimal contract has to be on one of the vertexes determined by the set of IC constraints.

By pairwise crossing all the IC constraints we find:

� ICgkPd1

and ICgkPd2

binding violates ICgkFd2

.

� ICgkFd1

and ICgkFd2

binding violates both ICgkPd1

and ICgkPd2

.

� ICgkPd2

and ICgkFd1

binding violates ICgkPd1

.

� ICgkPd2

and ICgkFd2

binding violates ICgkPd1

.

� Finally, ICgkPd1

and ICgkFd1

binding, as well as ICgkPd1

and ICgkFd2

binding, are shown to be vertexes ofthe domain and, hence, potential solutions of the program.

It can be checked that the equilibrium values of T and R obtained from ICgkPd1

and ICgkFd2

binding aresmaller and, hence, this constitutes the optimal contract. Some algebraic manipulations yield:

Tgk ¼2cd þ4cdð1� dÞ2d� 1

ð2d� 1Þð1� ð1� bÞrÞðdð1� bÞ þ ð1� dÞbÞ2 ð1� ð1� bÞrÞðd� ð1� dÞðdð1� bÞ þ ð1� dÞbÞÞ � db½ �

� �

Rgk ¼4cd

ð2d� 1Þð1� ð1� bÞrÞð2d� 1Þð1� ð1� bÞrÞðdð1� bÞ þ ð1� dÞbÞ

2 ð1� ð1� bÞrÞðd� ð1� dÞðdð1� bÞ þ ð1� dÞbÞÞ � db½ �

� �Summarizing the results obtained in the two regions, we can write the GP’s optimal contract in agatekeeping system, and its associated costs, as stated in Proposition 3. &

Proof of Proposition 4. First, evaluating the equilibrium levels of CNgkPat and Cgk

Pat for one extreme of the

domain b ¼ 12, it can be checked that, for every r40, CNgk

Pat 4CgkPat. Conversely, when b! 1 and,

therefore, d! 1, it is easy to check that CgkPat ¼ CNgk

Pat . We now show that there exists a threshold b�o1

such that, if b 2 ðb�; 1Þ, then CNgkPat oCgk

Pat. First, after substituting the optimal co-payment levels, patients

expected costs under non-gatekeeping are CPatNgk ¼ 2lð1� dÞð1� bÞ=2. Under gatekeeping, those costs are

CgkPat ¼ ð1� dÞðð1� bÞrlþ ð1� rÞlÞ þ rfðð1� bÞdþ ð1� dÞbÞ½ �

In order to find a sufficient condition let us disregard the fraction of CgkPat that depends on the private fee f.

When doing that one can find that a sufficient condition for CNgkPat oCgk

Pat is that b41=ð2� rÞ. Therefore,

there exists a threshold b� � 1=ð2� rÞ such that, if b 2 ðb�; 1Þ, then CNgkPat oCgk

Pat. This completes the proof.

Proof of Proposition 5. By comparing CNgkGP and Cgk

GP, as defined in Proposition 3, we find that:

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DOI: 10.1002/hec

If boð1þ 4dÞ=ð2þ 4dÞ then CgkGP4CNgk

GP , for every value of r.If b� ð1þ 4dÞ=ð2þ 4dÞ then:

� If r� er, then it is straightforward that CgkGPoCNgk

GP .� If r4er, then Cgk

GP4CNgkGP , Gðd; b; rÞ4ð3� 2bÞ=4ð1� bÞ � d.

It can be checked that Gðd; b; rÞ is monotonically increasing in r. Therefore, if for r ¼ �r,Gðd;b; �rÞ4ð3� 2bÞ=4ð1� bÞ � d, then there exists a threshold r� 2 ½~r; rÞ such that if r� r� thenCgk

GPoCNgkGP . Conversely, if r4r�Cgk

GP4CNgkGP .

If for r ¼ �r, Gðd; b; �rÞ � ð3� 2bÞ=4ð1� bÞ � d, then for every value of r 2 ½0; �r� it holds thatCgk

GPoCNgkGP . Substituting the value of �r and checking the inequality, it can be shown that there exists a

value ~b 2 ðð1þ 4dÞ=ð2þ 4dÞ; 1Þ such that Gðd;b; �rÞ � ð3� 2bÞ=4ð1� bÞ � d if and only if b� ~b.This completes the proof. &

Proof of Proposition 6. Comparing CNgkSp and Cgk

Sp, as computed in Appendix C, we get that:

CNgkSp � Cgk

Sp ¼cs2½dð1� bÞ þ rð1� dÞb�40

Moreover, it is direct to check that @½CNgkSp � Cgk

Sp�=@bo0, while @½CNgkSp � Cgk

Sp�=@r40. This completesthe proof. &

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Gatekeeping versus direct-access when patient information matters

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