On a policy of transferring public patients to private practice

HEALTH ECONOMICS

Health Econ. 14: 513–527 (2005)

Published online 20 October 2004 in Wiley InterScience (www.interscience.wiley.com). DOI:10.1002/hec.946

PUBLIC^PRIVATEMIX

On a policy of transferring public patients to private practice

Paula Gonz!aalez*Universidad Pablo de Olavide and centrA, Sevilla, Spain

Summary

We consider an economy where public hospitals are capacity-constrained, and we analyse the willingness of healthauthorities to reach agreements with private hospitals to have some of their patients treated there. When physiciansare dual suppliers, we show that a problem of cream-skimming arises and reduces the incentives of the healthauthority to undertake such a policy. We argue that the more dispersed are the severities of the patients, the greaterthe reduction in the incentives will be. We also show that, despite the patient selection problem, when the policy isimplemented it is often the case that health authorities decide a more intensive transfer of patients to privatepractice. Copyright # 2004 John Wiley & Sons, Ltd.

JEL classification: I11; I18; L51; H51

Keywords public–private health services; physician’s incentives; patient selection; waiting-lists

IntroductionWorldwide public health services are plagued byexcess demand and lengthy waiting-lists. Thisunsatisfactory situation has persisted from thevery inception of most public health systems and,far from improving, it seems to get moresystematic over the years. In contrast, the privatesector is usually characterized by relatively shortor even zero waits and idle capacity. As aconsequence of this, several national healthauthorities have turned to private hospitals andclinics for assistance in reducing their waiting-lists.

The purpose of this paper is to study theincentives of health authorities to divert patientsfrom public sector waiting-lists to the privatesector. In contrast with other analyses, we restrictattention to a particular regulatory regime inwhich health authorities fully finance somepatients’ access to the private sector. There areseveral examples of the application of such kind of

policies in the real world. At the Spanish regionallevel, for instance, the Valencian Region has beenundertaking a policy of transferring patients toprivate hospitals over the last years. As aconsequence, more than 100 000 social securitypatients were treated at private hospitals from July1996 to June 2000, and 4.26 million Euros werespent in the year 2000 to defray the debt to privateclinics that have participated in the ‘Impact Plan’for reducing surgical waiting-lists. In Galicia 12%of the operations financed by the public sector areperformed at private hospitals.a

The model in this article aims at pointing out thebenefits and drawbacks that health authorities facewhen implementing this sort of policy. Finding thecorrect balance between cost-containment andimprovements in the provision of health careservices will be a major task for public decision-makers.

We study the effects that exploitation of publicwaiting-lists by third parties may have over

Copyright # 2004 John Wiley & Sons, Ltd.Received 6 February 2003

Accepted 13 April 2004

*Correspondence to: Dpto. Econom!ııa y Empresa, Universidad Pablo de Olavide, Carretera de Utrera, km 1. 41013 Sevilla, Spain.E-mail: [email protected]

governments’ costs when a policy of transferringpatients to private clinics is undertaken. Inparticular, we concentrate on doctors’ incentivesto influence and manage waiting-lists forelective surgery to their own private benefit. Thisproblem worsens if, as is common in countrieswith public health services and waiting-lists,doctors who work for government hospitals alsoengage in private practice. In the UK, for instance,most private medical services are provided byphysicians whose main commitment is to theirpublic sector duties.b Furthermore, there is usuallya significant difference between the forms ofpayment to the doctors in the public and privatesectors. In the public sector, it is common forphysicians to receive a fixed salary, with limitedfinancial incentives related to workload. In theprivate sector, on the contrary, the use of tailoredincentives seems to be more widespread. Inparticular, it is common that physicians arepartners or shareholders in private hospitals, withtheir remuneration being linked to the perfor-mance of the hospital (some form of profit-sharing).

These two features, doctors acting in bothprivate and public sectors and different remunera-tion schemes in both sectors, create a basicproblem. In a simple model in which the policy-maker reaches agreements with private hospitalsto have some patients on the public waiting-liststreated there, we show that patient selection(cream-skimming) by the physicians occurs, i.e.physicians have incentives to strategically divertthe easiest cases to their private practice.c

We show that the presence of cream-skimmingreduces the incentives of the health authority toundertake such a policy in order to shorten publicwaiting-lists. As the physician only transfers theless severe cases to private practice, the averageseverity of those patients who remain in the publicsector is increased, along with the costs that thehealth authority faces.

There are two main variables that determine theresponse of the health authority. First, the relativedispersion of the patients’ severities. The higherthis dispersion, the more the physician earns fromselecting patients and, at the same time, the greaterthe impact on the costs borne by the healthauthority. Secondly, the patients’ health lossesassociated with waiting for treatment. The largerthese losses are, the more willing is the healthauthority to resort to private clinics, even in thepresence of patient selection.

It is also shown that when the policy oftransferring patients is implemented, the healthauthority’s reaction to the problem of patientselection is, in many instances, a more intensivetransfer of patients to private hospitals, i.e. fewerpatients are kept on the public sector waiting-lists.The intuition behind this result is the following: Asthe patients who remain in the public sector are themost severe ones, the marginal cost of providingtreatment in this sector increases. Hence, morepatients are diverted to the private sector, despitethis fostering the capacity of the physician to selectpatients.

Two main assumptions are made in thisanalysis. First, the public sector is capacity-constrained. Hence, there is a waiting time beforetreatment is performed. In the private sector, onthe contrary, waiting times are assumed to be zero.This assumption is typical in the literature (see, forinstance [3]). The private sector does indeed reportalmost zero waiting times in most westerneconomies. For instance, Bosanquet [4] states that‘at present, there is under-occupation in privatehospitals (in the UK), with occupancy rates at50% or less’.

This paper is related to studies dealing with theinteraction between public and private healthsectors and waiting-lists. Iversen [5] and Olivella[3] have investigated the optimal decision of thehealth authority in terms of waiting times, focus-ing on the effects of the presence of a privatealternative. In our model, however, waiting-listsare taken as given and we concentrate on thewillingness of governments to turn to the privatesector in order to alleviate such lists. We model thephysician’s behaviour, whereas Iversen and Oli-vella deal with patients’ decisions as individualschoosing between waiting for free treatment in thepublic sector and paying for immediate treatmentin the private sector. As such, Iversen and Olivellaclearly rule out the possibility of cream-skimmingon the part of the doctors.

In [Barros PP, Olivella P. Waiting lists andpatient selection. J Econ Manage Strat, forth-coming] the doctor’s strategic behaviour plays animportant role. However, the different systems ofremunerating the physicians in either sector (whichis the crucial variable in our model) is notconsidered in their work. Patient-selection arisesmainly from a combination of the rationing policyundertaken by the health authority, the criterionof the private physician regarding which severitieshe is willing to treat and the decision of the

Copyright # 2004 John Wiley & Sons, Ltd. Health Econ. 14: 513–527 (2005)

P. Gonza¤ lez514

patients to leave the queue in the public sector andresort to having to pay for private treatment. Inour model, patients are fully insured irrespective ofthe sector where they are treated and no rationingpolicy is considered. This gives the full power todecide to the physician and, thus, always generatesa situation of ‘full cream-skimming’.d

There are some works where possible contribu-tions from the private sector to reduce publicwaiting-lists are analysed. Cullis and Jones [6] andHoel and Sæther [7] argue that a subsidy to privatetreatments may be an optimal policy instrument toreduce the public sector waiting time. In contrastwith these analyses, we restrict attention to aparticular regulatory regime in which healthauthorities purchase private services, by fullyfinancing some patients’ access to the privatesector.

The rest of the paper is organized as follows:First the model is presented. Next, we compute theoptimal policy in the benchmark scenario. After-wards, we study the behaviour of the physicianconcerning the selection of patients and theresponse of the health authority in a generalcontext. Then, we provide a complete character-ization of the results for particular functionalforms. Finally we conclude. All of the proofs are inthe appendix.

TheModel

There are three players in this economy: acontinuum of patients, a physician and the healthauthority. We model their relationship as a simplegame in which the health authority maintainssome patients waiting for treatment in the publicsector, and reaches agreements with privatehospitals to have the remaining patients treatedthere.

Let us now detail the decision variables, para-meters, and utility functions that are relevant inthe model.

There is a continuum of individuals in need ofhealth care, all of whom require elective treatment.We consider that the size of this population ofpotential patients is exogenously given and it isnormalized to 1. These patients are homogeneous,except for their degree of severity, which ismeasured by the random variable s. This variableis distributed according to a continuous andpositive density function dFðsÞ defined on ½

%s; %ss �.

Let Ds ¼ %ss�%s be the difference between the

extremes of the domain for s, and #ss the averageseverity of the population of patients.

We assume that the public sector is capacityconstrained. We do not model this assumption as amaximum threshold on the treatments that can beprovided by the public hospital. We consider,instead, that the public sector suffers fromcongestion and, hence, there is a waiting timebefore treatment is performed. Let x denote thenumber (or percentage) of patients treated in thepublic sector. We may interpret this variable in themodel as a proxy for the length of the publicwaiting-list. The remaining 1� x patients will betreated in the private sector.e As the private sectoris assumed to be operating under capacity, thereare no waiting-lists for private sector treatment.Moreover, since we are dealing with public healthsystems, we assume that patients are fully insuredirrespective of the sector where they are eventuallytreated.

A patient is assumed to obtain a benefit fromtreatment defined by Q. When the patient istreated in the public sector, this benefit is under-mined due to the existence of waiting-lists, becausethe patient suffers the costs associated with waitingfor treatment. We consider this loss to be higherthe fewer patients that are diverted to the pri-vate sector (i.e. the higher the congestion in thepublic system). We define d > 0 as the patient’smarginal loss associated with waiting for treat-ment. The patient’s benefit from being treated inthe public sector is, therefore, given byQpb ¼ Q� dx. In contrast, since in the privatesector patients suffer no loss due to waiting,Qpv ¼ Q. Under this formulation, Qpb5Qpv andlimx!0 Qpb ¼ Qpv. Social expected health benefitsderived from treatment are, hence, given byxQpb þ ð1� xÞQpv ¼ Q� dx2.

Let kpb and kpv denote the unit cost oftreatment (disregarding physician’s compensa-tions) in the public and private sector respectively.We take the cost of the public treatment to belinear to simplify exposition. The qualitativeresults of the model, however, can be easilyextended to other more general cost structures.With regard to the unit cost in the private sector,we consider that it is not too restrictive to assumethat is linear if the private sector operates undercapacity, as is assumed throughout all ouranalysis. Still, as is the case for the public unitcost, the analysis is robust to other cost specifi-cations. For the sake of clarity, we define

Transfer of Public Patients to Private Practice 515


Dk ¼ kpb � kpv as the difference between the unitcosts of treatment in the two sectors.

To be consistent with real-life observations, weconsider that the health authority makes aconstant payment for each operation performedin the private sector ðwÞ. Although we take thevalue of w as constant, one may think that thispayment is computed according to the averageseverity of the whole population of patients. Thisspecification, however, does not alter the qualita-tive results of the paper.

We model the physicians’ behaviour as beingthat of a single representative agent. As we arguedin the Introduction, the fact that the same doctormay work in both private and public practice is acommon feature in Europe. We model this byassuming that the doctor who undertakes theoperations in the public sector (in the morning,say) also works for a private hospital (in theafternoon). In defining the utility of the physician,therefore, we must not only take his revenues andcosts in the public sector into account, but those inhis private practice as well.

The physician incurs costs in providing treat-ment to public and private patients. These costscapture the physician’s disutility from treatingpatients, and depend not only on the number ofoperations performed in each sector (x and 1� x,respectively), but also on the average severity ofthe patients (#sspb and #sspvÞ. As such, the physician’scosts in each sector are given by:

Cpb ¼ Cpbðx; #sspbÞ

Cpv ¼ Cpvðð1� xÞ; #sspvÞ:

The functions Cið�Þ, with i ¼ pb;pv are increasingand convex in both the amount and the severity ofthe patients treated. Moreover, denoting x ¼ xpb

and 1� x ¼ xpv, we assume that @2Ci=@xi@#ssi > 0and Cið0; #ssiÞ ¼ @Ci=@xijxi¼0 ¼ 0; 8i ¼ pb; pv. Theconvexity of Cið�Þ reflects the fact that the higherthe amount and the greater the severity of patientstreated by the physician, the greater his marginaldisutility. Moreover, this formulation also cap-tures the idea that physicians’ working capacity inboth sectors is limited.

As the physician performs two independent(although related) duties from which he receivesdifferent payments, we consider a constructionwith separable cost functions as more appropriate.We should also take into account that the timedevoted by the physician to each activity mayaffect his costs. It may be reasonable to think, for

instance, that performing the same task in ashorter time might be more costly. For this reason,although working times are not explicitly mod-elled, they are implicitly captured by the fact thatthe physician’s costs functions are different acrosssectors.

This modellization we have proposed is onlyappropriate if the working time of the physician inthe public sector is fixed and exogenously given.This assumption can be justified as it reflects theactual situation in most European mixed healthcare systems, where full time physicians’ workinghours in the public sector range between 35 and40 h per week.f

Concerning physician’s revenues, and accordingto empirical evidence, we consider that theydiffer across sectors. First, in many publicsystems, physicians receive a salary conditionalupon them providing a (loosely defined) amount oflabour. The physician commits to perform xoperations and receives, in exchange, a paymentthat covers the costs which he has incurred.We assume, hence, that the physician receives inthe public hospital a transfer T that includes afixed amount of money ðRÞ plus a full reimburse-ment of the costs, i.e. T ¼ RþCpbðx; #sspbÞ.It is assumed that the health authority cannotobserve the severity of each patient. Hence, thepayment received by the physician ðTÞ is a functionof the physician’s total costs (see [8] for a similarset-up).

Secondly, as is often assumed in the literature(see, for instance [9,10] in the US context), weconsider that the physician’s private income isdirectly related to hospital profits. In particular,we define physician’s private net revenue as aconstant share of the private hospital profits ðpÞ.

Physician’s total utility is given by the sum of hisnet revenues in the public and in the privatehospital. Hence, we can define the utility functionof the physician as follows:

Us ¼T�Cpbðx; #sspbÞ þ p½ðw� kpvÞð1� xÞ�

�Cpvðð1� xÞ; #sspvÞ: ð1Þ

The health authority’s expected surplus derivedfrom the care provided is given by the differencebetween the patient’s expected health benefits fromtreatment and the total costs that the provision ofservices generates. Hence, the health authority’sobjective function is given by:

H ¼ Q� dx2 � ½Tþ kpbxþ wð1� xÞ�: ð2Þ

P. Gonza¤ lez516


The timing of the game is as follows: At a stageprior to the starting-point of our model, the healthauthority and the private hospital bargain over thevalue of the private fee w that the private hospitalwill receive per operation performed.g At the firststage, the health authority decides on two vari-ables: the number of operations that are divertedto private clinics ð1� xÞ, and the payment thephysician will receive from his public activity (Rplus a reimbursement of the costs). At the secondstage, the physician selects the patients he wants totreat in the public sector. The remaining patientswill be transferred to the private hospital. Finally,the whole population of patients receives treat-ment and the payoffs are realized.

We confront two different frameworks. In thefirst one, it is assumed that the physician can notselect patients on the basis of their severities. Stage2 is, therefore, not active in this initial set-up. Inthe second scenario we consider that, since theactual severities of the patients can only be knownby the physician, the health authority cannotmonitor the physician in his selection of patients.

We start by analysing the optimal policy in thefirst setting.

Benchmark scenario

In this section, we assume that the health authoritycan prevent the physician from selecting thepatients he wants to treat in each sector. Thiscan be understood as a situation in which thephysician can not manipulate the waiting-list and,hence, patients are uniformly distributed betweenthe public and the private sectors.h We can thusensure that the average severity of the operations isthe same in both sectors, i.e. #sspb ¼ #sspv ¼ #ss.

In order to guarantee the existence of an interiorsolution in this framework, we make the followingassumption.

Assumption 1. kpb5w5@Cpbðx;#ssÞ@x jx¼1

þ 2dþ kpb:

Under this assumption, the fee paid to theprivate hospital has to be bounded between twovalues. With this we are only requiring that: Onthe one hand, the private fee cannot be too low. Ifthis was the case, the health authority couldpurchase all the health services from the privatesector, i.e. it would be trivially optimal to send allpatients to the private sector. On the other hand,we also require that the private fee is not so high

that the policy of transferring patients is notundertaken, even in this framework where manip-ulation is not possible.

Since stage 2 is not active under the benchmarkscenario, we move to the health authority optimi-zation problem. The health authority maximizesits objective function, subject to the constraint thatthe payment made to the physician allows him toattain an exogenously fixed level of utility ð

%UÞ.

This level can be understood as the physician’sutility reservation. The optimization problem thatthe health authority faces is as follows:

maxx;T H ¼ Q� dx2 � ½Tþ kpbxþ wð1� xÞ�

s:t: T �%UþCpbðx; #sspbÞ:

The following lemma characterizes the optimalsharing of patients between public and privatepractice and the associated payment to thephysician.i

Lemma 1. In the benchmark scenario, theoptimal number of patients treated in the publicsector ðxnÞ and the payment the physician receivesðTnÞ are such that:

Cpbx ðxn; #ssÞ þ kpb þ 2dxn ¼ w;

with Tn ¼%UþCpbðxn; #ssÞ:

With the above lemma we have computed theoptimal policy in the first best scenario. This willbe our reference case for comparison with theresults in the next section.

As one may expect, the health authority decidesthat the public sector performs the number ofoperations that equates the marginal costs oftreating patients in both sectors. It is obvious thatthe optimal level of patients treated in the publicsector is increasing in the private fee w. Moreover,as the fee per operation paid to the private hospitalis fixed, the optimal number of patients treated inthe public sector is decreasing in the averageseverity of the population ð#ssÞ. Finally, the largerthe marginal loss associated with waiting fortreatment ðdÞ the more patients will be transferred.

It is worth mentioning that if we allowed thehealth authority to strategically select whichpatients to treat, the outcome would be that onlythe most severe cases were transferred to theprivate sector. This is given by the fact that thepayment to the private sector is a fixed fee peroperation, while the public sector costs areincreasing in the average severity of the patients.j



We proceed now to study the effects of dealingwith a physician who can choose strategicallywhich patients are to be treated in each sector.This will allow us to analyse the consequences ofthis potential strategic behaviour on the will-ingness of the health authority to undertake thepolicy.

Patient selection

Our concern in this section is to analyse whetherthe results differ when the physician has the abilityto select patients on the basis of their severity fortreatment in one or other sector.

The actual severity of a patient is often difficultto ascertain. In any case, it is obvious that doctorsand hospitals have better information than healthauthorities on any patient’s severity (see, forinstance, [11,12]). We consider, then, that thehealth authority is not able to monitor the severityof each patient treated. Although the overallaverage severity can be inferred ex-post (throughthe observability of the physician’s costs), thehealth authority cannot prove whether the physi-cian actually modified the pool of patients thatwere to be treated, i.e. it can not monitor ex-antethe severities of the patients that are chosen toreceive treatment.k As the patients will no longerbe randomly distributed between the public andprivate sectors, we cannot ensure, in general, thatthe average severity of the patients treated in bothsystems is the same.

To characterize the solution in this framework,we proceed to solve the game by backwardsinduction. At the second stage, the physiciandecides which severities he wants to treat in eithersector, subject to the restriction that x operationshave to be performed in his public practice.Therefore, he does not have complete freedom inthe choice of #sspb, as there may be values that arenot compatible with the sharing of patients setby the health authority. The physician will choosethe value of #sspb (and therefore also of #sspv) in orderto maximize his total revenue. Since he is a dualsupplier, he will consider the effects of his strategicbehaviour concerning his two sources of income.

In the following proposition we characterize thephysician’s behaviour concerning the selection ofpatients.

Proposition 1. For a given sharing of patientsbetween the two sectors (x and 1� x), the

physician will transfer the least severe cases tothe private sector. Formally:

A patient with severity s will be treated in thepublic practice if and only if:

s 2 ðs0ðxÞ; %ssÞ; with s0ðxÞ ¼ F�1ð1� xÞ:

This proposition shows that the physician wants totreat only the less severe cases in his privatepractice, leaving the more severe patients for thepublic sector. This behaviour, known in theliterature as ‘cream-skimming’, is caused primarilyby the difference between the physician’s remu-nerations from the two systems. In the publicsector, the physician has his costs fully reimbursed,independently of the amount and severity of thecases treated, whereas, in the private sector, hereceives a share of the hospital profits. As aconsequence, he has incentives to divert the ‘easier’cases to his private practice as this will have adirect impact on his revenues. Note that whattriggers physician behaviour is the fact that thepayment structure in the private sector provideshim with incentives for cost-containment (in orderto maximize hospital profits), and this is not thecase in the public sector.

As a consequence of Proposition 1, it is easy tosee that the average severity in the public sectorð#sspb ¼ #sspbðxÞÞ depends negatively on the amount ofpatients that are kept on the waiting-list, i.e.@#sspbðxÞ=@x50. The more patients that are keptwaiting, the smaller the physician’s capacity toselect patients strategically and, hence, the lowerthe expected level of severity that the healthauthority faces.

We shall now study how this problem of cream-skimming affects the decision of the healthauthority on when to undertake a policy ofdiverting patients to private practice, and on theamount of patients that should be transferred tothe private sector.

In the first stage, when the cost of implementingthe policy is considered, the health authorityshould take into account the fact that the mostsevere cases will be treated in the public sector.

The maximization problem of the healthauthority in this scenario is as follows:

maxx;T Q� dx2 � ½Tþ kpbxþ wð1� xÞ�

s:t:T �

%UþCpbðx; #sspbÞ

#sspb ¼ #sspbðxÞ:

8<:

P. Gonza¤ lez518


The first-order necessary condition for an interiorcandidate to a solution ðxmÞ is such that:

� 2dx� kpb þ w�@Cpbðx; #sspbðxÞÞ

@x jxm

�@Cpbðx; #sspbðxÞÞ

@#sspb@#sspbðxÞ@x jxm

¼ 0:

The decision of the health authority, whenchoosing the fraction of patients that will bediverted to the private sector, is driven by twomain effects: On the one hand, there is a directeffect of changes in the number of patients kept inthe public sector over the health authority’s costs.On the other hand, there is an indirect effect,caused by the problem of patient selection. Thehealth authority takes into account that thephysician’s capacity to behave strategically linksthe proportion of patients he has to treat in hispublic sector with the final distribution of theseverities across sectors.

These effects have an immediate and importantconsequence, presented in the following proposi-tion.

Proposition 2. The presence of patient selectionmay induce the health authority to decide not toimplement the policy of transferring patients toprivate practice.

The presence of patient selection makes thestrategy of reducing waiting-lists by transferringpatients to the private practice, less attractive forthe health authority. By diverting some patients,the health authority incurs increased marginalcosts for those who remain in the public sectorsince, by the selection of the physician, they are themost severe cases.

The next step is to assess the impact ofpatient selection when the policy is actuallyundertaken. That is, assuming that the solutionto cream-skimming is interior (i.e. xm51), will thehealth authority decide to maintain longeror shorter waiting-lists than in the benchmarkscenario?

In order to analyse this issue, the curvature ofthe health authority’s objective function is crucial.The added complexity of the problem means that,in general, we cannot ensure the objective functionto be concave in all its domain.l Assuming that thefirst order condition is sufficient (i.e. the programis generically concave), it is straightforward toshow the following:

Proposition 3. The distortion caused by patientselection on the proportion of patients diverted toprivate practice cannot be unambiguously ranked.

In particular, the number of patients kept in thepublic sector waiting-lists will be smaller underpatient selection (i.e. xn > xmÞ if and only if:

@Cpbðx; #sspbðxÞÞ@x jxn

�@Cpbðx; #ssÞ

@x jxn

> �@Cpbðx; #sspbðxÞÞ

@#sspb@#sspbðxÞ@x jxn

:

The left-hand side of the inequality aboveð@Cpbðx; #sspbðxÞÞ=@xjxn � @Cpbðx; #ssÞ=@xjxn > 0Þ mea-sures the increase in the marginal cost of treatmentin the public sector due to cream-skimming. Thiseffect induces the health authority to increase thefraction of patients transferred to the (cheaper)private sector. However, an indirect (strategic)effect emerges, that pushes in the opposite direc-tion. This is reflected by the right hand side of theinequality ðð@Cpbðx; #sspbðxÞÞ=@#sspbÞð@#sspbðxÞ=@xjxnÞ50Þ.This term captures how a decrease in the numberof patients transferred to the private sector reducespublic sector marginal costs, by indirectly affectingthe average severity in the public system. Asalready mentioned, by increasing the number ofpatients that remain in the public sector, the healthauthority limits the physician’s strategic capacityto select patients.

At this level of generality, the relative impor-tance of the two effects described above cannot beassessed and, therefore, no clear-cut predictionabout the optimal transfer of patients to privatepractice can be made.

Next, we consider a specific physician’s costfunction, as well as a particular distribution for theseverities. This will allow us to fully characterizethe effects of the cream-skimming phenomenon.Moreover, it will show how the relative dispersionof the patients’ severities, as well as the healthlosses associated with waiting for treatment, arethe two key elements that determine the responseof the health authority and its willingness toundertake the policy.

Patient selection in the uniform-quadratic case

Throughout this section, we assume that thephysician’s cost of treatment in each sector is



given by a standard quadratic function:

Cpb ¼ 12 ðx#ss

pbÞ2 and Cpv ¼ 12 ðð1� xÞ#sspvÞ2

On top of this we consider that the severities aredistributed uniformly in the interval ½

%s; %ss�. These

two assumptions allow us to obtain explicitsolutions to the problem with patient selection,and to derive meaningful comparisons to thebenchmark scenario.

Before proceeding to analyse the optimalresponse from the health authority, we need tostudy the curvature of its objective function. Forthe sake of notational clarity, let d ¼ Ds=#ss denotethe relative dispersion of the patients’ severities.The following lemma characterizes the curvatureas a function of x.

Lemma 2. The curvature of the health author-ity’s objective function, under patient selectionðHmÞ, is as follows:

1. If d � 38 Ds

2, or if d538 Ds

2 and d � eddð dDs2Þ,

then Hm is always concave.

2. If d538 Ds

2 and d > edd dDs2� �

, then:

(a) For any x 2 ½0; %xxðd; dDs2Þ�, Hm is concave

at x.(b) For any x 2 ð %xxðd; d

Ds2Þ; 1�, Hm is convexat x.

With edd dDs2� �

�4�2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3�8ðd=Ds2Þ

p1þ8ðd=Ds2Þ , and %xx d; d

Ds2� �

¼

1d þ

12

� ��

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1d þ

12

� �2 13 �

43

dDs2

q.

Lemma 2 allows us to study the curvature of thehealth authority’s objective function in terms ofthe relative dispersion of the severities and thepatients’ health losses associated with waiting fortreatment. If patients’ health losses are very high,the function is still concave in all the domain and,therefore, the problem has a unique candidate foran optimum. When the value of such losses issmall, then the relative dispersion of the severitiesturns out to be crucial. In this case, if the severitiesare sufficiently dispersed, the objective functionhas a convex section that is bigger the higher thedispersion. As a consequence of this feature, wemay have two candidates to optimum: an interiorone and the boundary solution (no transfer ofpatients to private practice).

The following proposition presents the solutionto the health authority’s maximization problem.

Proposition 4. In the presence of patient selec-tion, the health authority decides to undertake the

policy of transferring patients to private practice ifthe value of the private fee is below a giventhreshold. The higher the relative dispersion of thepatients’ severities and the lower the health lossesassociated with waiting for treatment, the moredemanding this condition is. Formally:

1. xm51 if w52dþ kpb þ #ss2G d; dDs2

� �.

2. xm ¼ 1, otherwise.where

G d; dDs2

� �¼

� d2 if d > 3

8Ds2; or if d53

8Ds2 and d � eddð d

Ds2Þ

gðd; dDs2Þ otherwise:

(

is a continuous function, gðd; dDs2Þ is such that

@gðd; d=Ds2Þ=@d50 and limd!1gðd; d=Ds2Þ ¼ 12.

As was already advanced in the previoussection, this proposition shows that the presenceof ‘cream-skimming’ can lead to a situation inwhich the health authority is no longer willing totransfer patients to the private sector.

The first insight that emerges from the proposi-tion is that the larger the health losses associatedwith waiting for treatment, the more likely that thehealth authority decides to transfer patients toprivate practice. The waiting cost is a great sourceof inefficiency for the public sector even in thepresence of cream-skimming. In addition to this,the threshold of the private fee at which the healthauthority is willing to carry out the policy isdecreasing in the relative dispersion of the patients’severities. The reason is that the strategic beha-viour of the physician (in transferring the mildercases to private practice) is fostered by the widerrange of severities, since his gains in divertingpatients are higher. As a result, when d is low thehealth authority does not suffer a substantialproblem of cream-skimming (since patients havesimilar levels of severity). In this case, the policy oftransferring patients to private practice is under-taken for a wide range of values of w. Inparticular, when d ! 0, the condition for under-taking the policy converges to the one required inAssumption 1. However, as the relative dispersionof severities increases, the condition necessary forthe health authority to reach agreements withprivate hospitals becomes more demanding. Sincethe treatment of some patients in the private sectorincreases the cost per operation in the publicsector, the health authority decides to retain the

P. Gonza¤ lez520


size of the waiting-list, unless the private fee issufficiently low.

The particular functional forms we have as-sumed allows us to compare the interior solutionin this setting with the optimal one in the bench-mark case. The result is the following:

Proposition 5. When the health authority iswilling to undertake the policy, the existence ofpatient selection implies that:

(i) If the relative dispersion of the severities issufficiently low, the health authority keepsfewer patients in the public sector, providedthat the private fee is below a given value.Otherwise, more patients are kept waiting.

(ii) If the relative dispersion of the severitiesexceeds a critical value, the health authorityalways keeps fewer patients in the publicsector.

Formally, there exists a threshold d0ðd=Ds2Þ > 0such that:

1. xm5xn if d � d0 dDs2� �

, or if d5d0 dDs2� �

and

w52dþ kpbþ #ss2F d; dDs2

� �.

2. xm > xn if d5d0 dDs2� �

and w > 2dþ kpbþ#ss2F d; d

Ds2� �

.

With F d; dDs2

� �¼ 1

4d 3dþ 6�ð� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

d2 þ 4dþ 36p

Þ

� dDs2 2d

2�.

This proposition shows the outcome of theunresolved conflict of effects presented in Proposi-tion 3. We see how, when the policy of transferringpatients is undertaken, the consequences of cream-skimming on the number of patients transferredare driven by the relative dispersion of theseverities.

When the relative dispersion is sufficiently high,the negative effect of treating more severe patientsin the public sector always dominates. Althoughthe capacity of the physician to select patientsdecreases with the amount of public operations hehas to perform, fewer patients are kept waiting.For this range of parameters, therefore, the healthauthority’s reaction to the problem of patientselection is a more intensive transfer of patientsfrom the waiting-list to private hospitals.

In contrast, when the relative dispersion is low,the final result is determined by the value of theprivate fee. A high level of w implies a relativelylow transfer of patients in the benchmark scenario;

we show that, in the present case, the presence ofcream-skimming leads to an even smaller transfer.In this region, therefore, cream-skimming leads toa smaller utilization of the private sector by thehealth authority and, thus, longer waiting-lists.Conversely, while the first best outcome was totransfer a high proportion of the patients (low w)to private practice, the distortion created bycream-skimming implies transferring even more(see Figure 1).

Figure 2 compares the objective functions of thehealth authority and the optimal sharing ofpatients under the alternative scenarios we havestudied: Hn denotes the objective function in thebenchmark case, whereas Hm stands for the one

Figure 1. Sharing of patients with and without patient selection

H ( )*H x

( )mH x

*xmxx

1

Figure 2. Health Authority’s objective functions when

d > d0ðd=Ds2Þ



under patient selection. This illustration is madefor the case in which the policy is implemented, thevalue of d is not very high, and the relativedispersion of the severities is sufficiently largeðd > d0ðd=Ds2Þ).

So far, we have taken the negotiations betweenthe health authority and the private hospital,about the payment for those patients transferred,as given in our model. It is crucial, however, toknow whether the equilibrium values we havecomputed leave room for such a bargainingprocess. That is, if there are values of w that makethe health authority willing to undertake the policy(i.e. w52dþ kpb þ #ss2Gðd; d=Ds2Þ) and, at the sametime, are profitable to the private hospitalððw� kpvÞð1� xÞ � 1

2ðð1� xÞ#sspvÞ2 � 0Þ. The resultis presented in the following remark:

Remark 1. In equilibrium, for each value of d,there exists a threshold Dkmin50 such that ifDk > Dkmin, the bargaining set is not empty (forevery value of d), i.e. the maximum fee the healthauthority is willing to pay exceeds the minimumthat the private hospital requires to accept publicpatients.

This remark shows that in all cases in which theprivate sector is more efficient than the publicsector and, even in some cases in which it is not(when Dkmin5Dk50), there are values of w thatare compatible to both the private hospital and thehealth authority, independently of the relativedispersion of the severities. Moreover, the regionwhere agreement is possible is larger, the higher isthe patients’ loss associated with waiting fortreatment ðdÞ. Figure 3 illustrates the bargaining

set between the health authority and the privatehospital for Dk � Dkmin.

Concluding remarks

This paper has analysed the incentives of healthauthorities to divert patients from the public to theprivate sector in a framework in which the publicsector is capacity-constrained, while the privatesector is not.

We have found that, due to the differentstructure of the physician’s remuneration in eachsystem, a problem of cream-skimming arises, i.e.physicians prefer to treat the less severe cases intheir private practices. This problem has beenshown to make the health authority more reluctantto implement a policy of transferring patients toprivate practice.

We have characterized the range of parametersthat make the health authority find it moreprofitable to keep longer waiting-lists instead ofmoving patients to private practice. There are twocrucial variables, apart from the private fee, thatinfluence the trade-off the health authority faces:On the one hand, the relative dispersion of thepatients’ severities. The higher the dispersion, themore the physician earns from selecting patientsand, at the same time, the greater the impact onthe costs borne by the health authority. On theother hand, the patients’ health losses associatedwith waiting for treatment. The larger these losses,the more willing the health authority to resort toprivate practice.

When the health authority decides to undertakethe policy, the amount of patients that are finallytransferred to the private sector cannot bedetermined unambiguously. We have found,however, that for a wide range of parametervalues, fewer patients are kept waiting. Thenegative effect of cream-skimming on the marginalcost of providing treatment in the public sectordominates (as the most severe patients remain init). Hence, more patients are transferred, despitethis increasing the ability of the physician to selectpatients.

In terms of policy recommendations, our resultsuggests that when designing policies to reducewaiting-lists by transferring patients from thepublic to the private sector, policy-makers shouldconsider the incentive effects of the differentreimbursement systems of the two sectors. More-Figure 3. Bargaining set for Dk � Dkmin

P. Gonza¤ lez522


over, the decision to undertake the policy or notmust be influenced not only by the fee peroperation agreed with the private sector, but alsoby the type of illness. In particular, the dispersionof illness severity is shown to be very important.The wider the range of severities, the more seriousthe problem of patient selection becomes and,therefore, the less beneficial the policy to thehealth authority.

The spirit of this work is eminently positive.Given the institutional framework and the con-tractual arrangements in existence, we haveidentified a potential problem of patient selection.A more normative analysis, in which possiblesolutions to the problem can be addressed, is leftfor further research. In spite of this, our articleprovides some indications on how to address theissue. One may think of two possible types ofinstruments. The first concerns physician reimbur-sement. Different contractual arrangements wouldalter physicians’ incentives and, hence, theydeserve a careful analysis.m A second option is toconsider measures that limit the physician’sstrategic capacity. Possible measures include theintroduction of exogenous rules for allocatingpatients between the two sectors and legal exclu-sions that prevent physicians from providingprivate treatment to patients that were on theirwaiting-lists.n

One potential criticism of our analysis is that wedo not consider how a physician’s professionalethics may alleviate cream-skimming. However,this criticism does not apply to our model since thecosts associated with waiting for treatment in thepublic sector are equal among agents and, hence,an altruistic motive would not alter physician’sincentives to perform cream-skimming. If thedirect cost of waiting were increasing in theseriousness of the patients’ condition, it is truethat doctors’ altruism may partially mitigate theirincentives to select the less severe patients. In thissense, our analysis could be considered as abenchmark for the worst-case scenario.

There are other issues that have not beenaddressed in this work. First, we have notconsidered any difference between the quality ofthe services provided by the public and privatesectors. This makes our analysis appropriate fortreatments when the patient’s condition is not life-threatening in the absence of treatment, as thesemedical conditions usually require facilities thatboth the public and the private sectors possess.Moreover, these non-urgent treatments are pre-

cisely the ones included in most plans for divertingpatients. Secondly, the analysis would require afirst step to study the negotiations betweenthe health authority and the private hospitalsconcerning the payment for those operationstransferred. Endogeneizing this process would bean interesting exercise that would provide thehealth authority with a new regulatory tool totackle the cream-skimming problem. However, aslong as the payment to the private sector is a fixedfee per operation, our results would continue tohold.

Certainly, more theoretical and empirical workin this line of research is needed. We believe,however, that this work can inform the debates onhealth policy and contribute to the development ofa better policy making process.

Appendix

Proof of Proposition 1

The optimization program the physician faces is asfollows:

max#sspb

Us ¼T�Cpbðx; #sspbÞ þ p½ðw� kpvÞð1� xÞ�

�Cpvðð1� xÞ; #sspvÞ;

with @#sspv=@#sspb50. Since T ¼ RþCpbðx; #sspbÞ, wecan rewrite this problem as:

max#sspb

Us ¼Rþ p½ðw� kpvÞð1� xÞ�

�Cpvðð1� xÞ; #sspvÞ:

From this expression, it is straightforward to seethat the physician’s utility is higher the lower is#sspv (and, therefore, the higher is #sspb). This impliesthat the physician chooses to transfer to theprivate sector those patients on the left tail of thedistribution of severities. Formally, there exists as0 2 ð

%s; %ssÞ such that:

If s 2 ð%s; s0Þ the patient is transferred to the

private sector.If s 2 ðs0; %ssÞ the patient remains in the public

sector.In order to compute the value of s0, we have to

take into account that a fraction x of patients haveto be treated in the public sector. Thus:Z %ss

s0ðxÞdFðsÞ ds ¼ x ) s0ðxÞ ¼ F�1ð1� xÞ:



Therefore, a patient with severity s will be treatedin the public sector if and only if:

s 2 ðs0ðxÞ; %ssÞ; with s0ðxÞ ¼ F�1ð1� xÞ

This completes the proof. &


In the benchmark case, Assumption 1 (rewritten asw� 2d� kpb � @Cpbðx; #ssÞ=@xjx¼150Þ was enoughto ensure that xn51. However, in the problemwith patient selection, the f.o.c. evaluated at x ¼ 1is given by:

w� 2d� kpb �@Cpbðx; #ssÞ

@x jx¼1

�@Cpbðx; #sspbðxÞÞ

@#sspb@#sspbðxÞ@x jx¼1

:

This expression is larger than w� 2d� kpb

� @Cpb

ðx;#ssÞ=@xjx¼1and, therefore, Assumption 1 is not

enough to ensure an interior solution for theproblem with patient selection.


Proof of Lemma 2

We study the curvature of the health authority’sobjective function in the restricted domain givenby the behaviour of the physician at stage 2. FromProposition 1 we have #sspb ¼ %ss� xDs=2. However,it is known that T ¼ Rþ 1

2½x#sspb�2. By substituting

these two constraints into the problem, we cancharacterize the curvature of the objective functionas a function of x. The optimization program atthe first stage is as follows:

maxx;R Hm ¼ Q� dx2

� Rþ 12 x %ss� xDs

2

� �� 2þkpbxþ wð1� xÞ

h is:t: R �

%U:

The f.o.c. is given by:

� x %ss�xDs2

� �� 2þx2Ds2

%ss�xDs2

� ��

þ w� kpb � 2dx ¼ 0:

Computing the second order condition andrearranging terms yields:

@2Hm

@2x¼ �%ss2 þ 3xDs%ss�

3

2Ds2x2 � 2d:

This second order derivative is increasing in x. Wecompute its roots and find:

x1 ¼%ss

Dsþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi%ss

Ds

� �21

3�

4

3

dDs2

s> 1 and

x2 ¼%ss

Ds�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi%ss

Ds

� �21

3�

4

3

dDs2

s:

Simple manipulations allow us to re-write x2 as

x2 ¼ 1d þ 1

2

� ��

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1d þ

12

� �2 13 � 4

3dDs2

q� %xx d; d

Ds2� �

:

It can be verified that @ %xxðd; d=Ds2Þ[email protected] here, since x � 1, it can be shown that:

(i) If %xxðd; d=Ds2Þ � 1, @2Hm=@2x50 8x 2 ½0; 1�,i.e. the objective function is always concave.

(ii) If %xxðd; d=Ds2Þ51, @2Hm=@2x50 for everyx � %xxðd; d=Ds2Þ. Thus, we can ensure thatfor x 2 ½0; %xxðd; d=Ds2Þ� the objective function isconcave, and for x 2 ½ %xxðd; d=Ds2Þ; 1� it isconvex.

In re-writing the conditions for %xxðd; d=Ds2Þ interms of the relative dispersion of the severities, wefind that:

%xxðd; d=Ds2Þ� 1, d �3

8Ds2; or if d5

3

8Ds2 and

� edd dDs2

� ��

4� 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3� 8d=Ds2

p1þ 8d=Ds2

%xxðd; d=Ds2Þ51 otherwise:



From the f.o.c. computed in Lemma 2 it isstraightforward to verify that x ¼ 0 can be nevera solution, since:

@Hm

@x jx¼0¼ w� kpb > 0:

This, together with the other conditions found inLemma 2, provides a complete characterization of

P. Gonza¤ lez524


the optimization problem:

1. If d � 38Ds

2, or if d538Ds

2 and d � eddðd=Ds2Þ,there exists a unique candidate to optimum(x). This solution will be interior, i.e. x 2 ð0; 1Þ,if and only if:

@Hm

@x jx¼1¼ w� kpb � 2d� #ss #ss�

Ds2

� �

50 , w� kpb � 2d5#ss2 1�d

2

� �:

2. If d538 Ds

2 and d > eddðd=Ds2Þ, there exists, atmost, a single interior candidate to optimum(x), such that x 2 ð0; %xxðd; d=Ds2Þ�. The bound-ary solution x ¼ 1 is also a potential candi-date to optimum, provided that:

@Hm

@x jx¼1> 0 , w� kpb � 2d > #ss2 1�

d

2

� �:

This is a necessary, but not sufficient condition, forx ¼ 1 to be a solution. Hence, to choose theoptimal level of x in this region we need tocompare the value function for both candidates.We check the conditions under which it is better tooperate a fraction a of the patients, rather thanperform all the operations. We find that:

If w� kpb � 2d5#ss2g d;dDs2

� �;

with g d;dDs2

� �

¼ maxa2ð0;1Þ

1

2ð1þ aÞ � a2d

ð1� aÞ4

dþ 1

� ��

�dDs2

2ð1� xÞd29an 2 ð0; 1Þ

such that Hmðx ¼ anÞ > Hmðx ¼ 1Þ:

We can see that gðd; d=Ds2Þ is such that@gðd; d=Ds2Þ=@d50, limd!1gðd; d=Ds2Þ ¼ 1

2.If w� kpb � 2d exceeds the above threshold,

then we can ensure that the boundary solution isoptimal, since for this parameter configuration@Hm=@xjx¼1 > 0. Therefore, the solution to theprincipal’s problem is:

(i) xm51 if w52dþ kpb þ #ss2G d; dDs2

� �,

where

G d;dDs2

� �¼

1�d

2if d >

3

8Ds2; or if d5

3

8Ds2 and d� edd d

Ds2

� �gðd; d=Ds2Þ otherwise

8><>:is a continuous function, gðd; d=Ds2Þ is such that@gðd; d=Ds2Þ@d50 and limd!1gðd; d=Ds2Þ ¼ 1

2.(ii) xm ¼ 1, otherwise.

Finally, from @Hm=@xjx¼1 it is also straightfor-ward to see that, in both regions, the larger is d themore likely it is that the interior solution is optimal.



From the previous proposition we know that xm isinterior if:

w52dþ kpb þ #ss2G d;dDs2

� �:

The f.o.c. of the health authority’s problem in thepresence of patient selection was given by:

@Hm

@x¼ � x %ss�

xDs2

� �2

þx2 Ds2

%ss�xDs2

� �þ w� kpb � 2dx ¼ 0:

Considering that %ss ¼ #ssþ 12Ds, we can rewrite this

in terms of #ss as:

� x #ssþ ð1� xÞDs2

� �2

þx2Ds2

#ssþ ð1� xÞDs2

� �þ w� kpb � 2dx ¼ 0:

The f.o.c. of the health authority’s problem undernon-manipulability was given by:

@H

@x¼ �x#ss2 þ w� kpb � 2dx ¼ 0:

Performing some algebraic manipulations, we findthat:

@Hm

@x>@H

@x, xDs#ss �ðxÞ2

d

2þ

3

2x 1þ

d

2

� �� 1þ

d

4

� �> 0:



From the inequality above we find that:

@Hm

@x jx¼xn

> 0 , w� kpb

> #ss21

4d3dþ 6�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffid2 þ 4dþ 36

p �� :

Rewriting it, we get:

w� kpb � 2d > #ss2

"1

4dð3dþ 6�


pÞ:

In order to complete the characterization of thesolution xm; we need to check when the inequalityabove is compatible with the condition guarantee-ing that xm is interior. On combining the abovecondition with that of Proposition 4, we find thatthere exists a threshold d0ðd=Ds2Þ > 0 such that:

(1) If d5d0ðd=Ds2Þ, then:If w52dþ kpb þ #ss2Fðd; d=Ds2Þ, then: xm5xn.If w > 2dþ kpb þ #ss2Fðd; d=Ds2Þ, then: xm > xn.With Fðd; d=Ds2Þ ¼ ½ð3dþ 6�


pÞ=

4d� d2d2=Ds2�.(2) When d � d0ðd=Ds2Þ then: xm5xn.This completes the proof. &

Acknowledgements

This is a substantially rewritten version of a paperentitled ‘Policy Implications of Transferring Patients toPrivate Practice’. Earlier versions of this paper werewritten at Universidad de Alicante (Spain) and atCORE-Universit!ee catholique de Louvain (Belgium). Iam truly grateful to Carmen Herrero, Javier L !oopez-Cu *nnat and Nicol!aas Porteiro for their invaluable help andsupport. I also would like to thank Pedro P. Barros, TorIversen, Izabela Jelovac, In!ees Macho-Stadler, XavierMart!ıınez-Giralt, Pau Olivella and Andrew Street forvaluable discussions and helpful suggestions. Sugges-tions from two anonymous referees have improved thepaper considerably. Financial support from the General-itat Valenciana GV01-371 and from the InstitutoValenciano de Investigaciones Econ !oomicas (IVIE) isgratefully acknowledged. The usual disclaimers apply.

Notes

a. This information has been obtained from the journal‘Informaci !oon’ (4th January 2001) and the Journal‘Faro de Vigo’ (22nd September2002).

b. A report by the Competition Commission [1],estimated that 61% of NHS physicians in the UKhave significant private practices. In addition to this,and according to Yates [2], an NHS specialist

undertakes, on average, two private operations aweek. In the Southern European countries, thisphenomenon seems to be even more common.

c. Cream-skimming may also appear in other frame-works. For instance, the editorial of The Economist(1998) addresses the criticism directed towardsHealth Maintenance Organizations in the US forexcluding people with high expected costs frompurchasing health insurance.

d. According to Barros and Olivella, ‘full cream-skimming’ is a situation in which all the mildestpatients end up being treated in the private sector.

e. In this work it is assumed implicitly that medicalguidelines have set minimum thresholds of acutenessfor a patient to be admitted for treatment. Thedecision of the health authority is to determine theamount of patients, from those who will receivetreatment, that should be transferred to privateclinics.

f. As pointed out by an anonymous referee, if workingtime in the two sectors adjusted perfectly to thenumber of patients treated (instead of being exogen-ous), it would be more reasonable to consider thatthe total cost of the physician depends only on thetotal number of patients treated and on the averageseverity.

g. We show later that, in equilibrium, the bargaining setis not empty, i.e. the maximum fee the healthauthority is willing to pay exceeds the minimum theprivate hospital will accept for attending to publicpatients.

h. We are ruling out the possibility that the healthauthority strategically decides to select wherepatients are treated, as a way to decrease the costsborne. We will discuss this issue briefly at the end ofthe section.

i. The proof of this lemma is straightforward andtherefore we omit it.

j. We thank an anonymous referee for pointing out thepossibility of patient selection by the health authority.

k. As a consequence of this, payments to publicphysicians are a function of total costs and numberof patients treated.

l. Using the fact that in the absence of patient selectionthe problem is always concave, it can be shown thatthe concavity is preserved in two cases. First, whenthe health loss associated with waiting for treatmentðdÞ is sufficiently large. Secondly, if the dependenceof #sspbðxÞ on x is relatively weak. This latter conditioncan be seen as a restriction on the distribution of theseverities. In particular, we will show in the followingsection that, for an uniform distribution of severities,it holds if the relative dispersion of the severitiesðDs=#ssÞ is below a certain threshold.

m. Ellis [8], Ellis and McGuire [9,10], Blomqvist [13],Dranove [14], Ma [15], Rickman and McGuire[16] and Selden [17] have written extensively on thisissue.

P. Gonza¤ lez526


n. In this regard, some health authorities have imposedrestrictions on the private earnings of the publiclyemployed physicians to limit their strategic capacity.Gonz!aalez analyses in detail this issue [18].

References1. Competition Commission. Private Medical Services:

A Report on the Agreements and Practices Relatingto Charges for the Supply of Private MedicalServices by NHS consultants. London: HMSO,1994.

2. Yates J. Private Eye, Heart and Hip: SurgicalConsultants, The National Health Service and PrivateMedicine. London: Churchill Livingstone, 1995.

3. Olivella P. Shifting public-health-sector waitinglists to the private sector. Eur J Polit Econ 2003;19: 103–132.

4. Bosanquet N. A Successful National Health Service.Adam Smith Institute: London, 1999.

5. Iversen T. The effect of a private sector on thewaiting time of a national health service. J HealthEcon 1997; 16: 381–396.

6. Cullis JG, Jones PR. National health service waitinglists: a discussion of competing explanations and apolicy proposal. J Health Econ 1985; 4: 119–135.

7. Hoel M, Sæther EM. Public health care with waitingtime: the role of supplementary private health care.J Health Econ 2003; 22–4: 599–616.

8. Ellis RP. Creaming, skimping and dumping: provi-der competition on the intensive and extensivemargins. J Health Econ 1998; 17: 537–555.

9. Ellis RP, McGuire TG. Provider behavior underprospective reimbursement: cost sharing and supply.J Health Econ 1986; 5: 129–152.

10. Ellis RP, McGuire TG. Optimal paymentsystems for health services. J Health Econ 1990; 9:375–396.

11. Boadway R, Marchand M, Sato M. An optimalcontract approach to hospital financing. J HealthEcon 2004; 23–1: 85–110.

12. Propper C. Agency and incentives in the NHSinternal market. Soc Sci Med 1995; 40: 1683–1690.

13. Blomqvist A. The doctor as a double agent:information asymmetry, health insurance, andmedical care. J Health Econ 1991; 10: 411–432.

14. Dranove D. Rate-setting by diagnosis relatedgroups and hospital specialization. Rand J Econ1987; 18–3: 417–427.

15. Ma C-TA. Health care payment systems: costs andquality incentives. J Econ Manage Strat 1994; 3–1:93–112.

16. Rickman N, McGuire A. Regulating providers’reimbursement in a mixed market for health care.Scott J Polit Econ 1999; 46–1: 53–71.

17. Selden TP. A model of capitation. J Health Econ1990; 9: 397–410.

18. Gonzalez P. Should physicians’ dual practice belimited? An incentive approach. Health Econ 2004;13: 505–524.



On a policy of transferring public patients to private practice

Documents

Transcript of On a policy of transferring public patients to private practice