How to regulate heterogeneous hospitals

This paper presents two alternative payment systems to reduce hospital ine¢ ciency. In both systems, one part of the payment is xed ex ante and allows for observable patient and hospital heterogeneity. The rst system is a mixed payment that retrospectively reimburses unobservable hospital heterogeneity specied by hospital xed e¤ects, but does not reimburse costs due to transitory moral hazard. The second system sets a prospective payment for all the non-observable characteristics, without reimbursing cost deviations due to either transitory moral hazard or hospital specic e¤ects. The advantage of the rst payment system is that it creates incentives to reduce transitory moral hazard while guaranteeing high quality of hospital services. Econometric estimates are performed on a sample of 7,314 stays for acute myocardial infarction observed in 36 French public hospitals We are grateful for helpful comments from Werner Antweiler of the Faculty of Commerce at the University of British Columbia, Alberto Holly of the University of Lausanne and Michel Mougeot of the University of Besançon. We also thank the participants of the NBER Summer Institute Workshop (Boston) as well as participants of the Crest-LEI and Delta seminars in Paris and participants of the twelfth European Workshop on Econometrics and Health Economics (Menorca) for useful comments. We are also wish to thank to two anonymous referees whose comments helped to improve the paper. This study was funded in part by grants from the DREES (Direction de la Recherche, des Etudes, de lEvaluation et des Statistiques) of the French Ministry of Labor and Solidarity. Any errors are our responsibility. yCorresponding author: Brigitte Dormont Thema UPX, Bâtiment G, 200 avenue de la république 92001 Nanterre cedex, France. Tel: 00 33 1 40 97 78 36. Fax: 00 33 1 40 97 59 73 . E-mail: dormont@u-paris10.fr. zThema-CNRS, University of Paris10-Nanterre, France and the Institut dEconomie et de Management de la Santé (IEMS), Lausanne, Switzerland. xDelta-CNRS, Ecole nationale Supérieure, Paris, France and the Institut dEconomie et de Management de la Santé (IEMS), Lausanne, Switzerland.


Introduction
This paper proposes a payment system that creates incentives to increase hospital e¢ ciency when hospitals are heterogeneous, without reducing quality of care.
In most Western European countries, a global budget system was introduced for cost containment purposes in the late 70's or early 80's. In these countries, most hospitals are public or publicly …nanced. The global budget system is still widely used in many European countries. It is also applied in Switzerland and in the USA for the hospitals managed by the Department of Veterans A¤airs.
This method of payment consists of an annual budget …xed in advance which does not vary with the volume of services delivered. It has a number of drawbacks: underservice, risk selection or ine¢ ciency. The type of drawbacks depends on whether the budget constraint is hard or soft. At present, there is growing pressure to reform hospital reimbursement systems through the introduction of a Prospective Payment System per DRG. In France, a gradual introduction of a PPS is planned for [2004][2005]. This study explores di¤erent optimal reimbursement systems for heterogeneous hospitals as an alternative to the current global budget regime in France.
However, the implications of this paper are not restricted to the French case and go beyond the scope of hospital payment systems. Our approach could be applied to other areas of health care …nancing. We show how to identify a certain component of moral hazard, one that can have a sizeable impact on cost variability. Our identi…cation method can be applied in situations where there is controversy about the sources of cost variability and the respective roles of ine¢ ciency versus legitimate permanent heterogeneity.
An example in the USA is the controversy regarding the use of Adjusted Average Per Capita Cost (AAPCC) to calculate Medicare Managed Care Reimbursement. AAPCC is based on a blend of risk adjusted rates and of average expenditures computed at the local level. Wennberg et al. (2002) observe that Medicare spending based on AAPCC varies widely between regions. For instance, the di¤erence in lifetime Medicare spending between a typical sixty-…ve-year-old in Miami and one in Minneapolis is more than $50,000.
The variations persist even after di¤erences in health are corrected for. The controversy is about the reasons for the observed di¤erence: is it due to moral hazard or to other di¤erences that are not captured by risk adjusted rates ? Using our method, it would be possible to isolate one component of moral hazard and e¢ ciency could be improved by an appropriate method of payment, while still reimbursing a part of unexplained cost heterogeneity between regions.
The theoretical foundations of a fully prospective payment system per stay have been de…ned by the yardstick competition model of Shleifer (1985). However, this model is based on rather unrealistic assumptions: homogeneity of hospitals, homogeneity of patients for the same pathology, …xed quality of care. Many studies have pointed to possible negative e¤ects of careless implementation of a PPS, namely patient selection and lower care quality (Newhouse (1996)).
In order to avoid these drawbacks, many authors have advocated a mixed payment system, combining a lump sum and the actual cost. However, such a system is rather di¢ cult to put into practice: its speci…cation can depend on unobservable variables or functions. This leads to questions that we take up in the case of France. How can we identify the costs corresponding to e¢ cient activity ? To what extent should patient and hospital heterogeneity be allowed for in a payment system? Drawing on Shleifer's theory of yardstick competition, we develop an econometric model where hospital variability is explained by patient and hospital characteristics. From the regulator's perspective, some of these characteristics are observable and some are not. We propose two alternative payment systems in order to reduce hospital ine¢ ciency.
We use a three dimensional nested database of 7,314 stays for acute myocardial in-farction observed in 36 French public hospitals over the period 1994 to 1997. Information is recorded at three levels: stays are grouped within hospitals and hospitals are observed over several years. The structure of our panel data allows us to identify one component of unexplained cost variability: short term moral hazard.
This article is organized as follows. In section 2, we describe the data. In section 3, which is devoted to the theoretical background, we propose an extension of Shleifer's basic model and de…ne an optimal payment rule. The speci…cation of the cost function is given in section 4, which shows how we identify some components of unexplained cost variability and de…nes our two payment methods. Our results are presented in section 5, together with the methods and speci…cation tests. In section 6, we simulate the implementation of our two payment methods and evaluate the potential budget savings. Section 7 concludes.

Description of the data
We have at our disposal a sample of 7,314 stays for acute myocardial infarction (AMI) observed in 36 French public hospitals from 1994 to 1997. In France, public 1 hospitals account the large majority of total admissions (2/3 of admissions for AMI). Our sample was extracted from the PMSI 2 cost database. Classi…cation of stays by Diagnosis Related Group (DRG) is performed on the basis of diagnoses and procedures implemented during the stay. In order to obtain a high degree of homogeneity in pathologies, we selected patients who where at least 40 years old with acute myocardial infraction (AMI) as the main diagnosis and grouped in the same DRG: uncomplicated AMI (DRG 179).
For each stay, we have information about the cost of the stay, secondary diagnoses, procedures implemented, mode of entry into the hospital (coming from home or transferred from another hospital), mode of discharge (return home or transfer), length of stay, age and gender of the inpatient.
The database gives access to rich, detailed information about stays. However, we cannot follow the same inpatient through successive hospital stays. There is no information about the patient's quality of life after the stay, about readmission just after the observed stay, about infections contracted during the stay. In addition, we have no information about the quality of services provided in terms of comfort or alleviation of pain. Participation in the cost database program is voluntary for hospitals and the number of participating hospitals is limited. They consent to give detailed information about their costs, which means that they must have accounting systems that enable them to provide such information. 3 Our panel data exhibit a rather complex structure. Information is recorded at three levels. The panel is unbalanced in several dimensions: not only does the number of stays recorded vary across hospitals for a given year but also the length of the observation period varies across hospitals.

Patients and hospitals
Together with drug therapy (aspirin, beta blockers, etc.), uncomplicated AMI patients (DRG 179) can receive various treatments such as thrombolytic drugs, cardiac catheterization (hereafter denoted as CATH) and percutaneous transluminal coronary angioplasty (PTCA). Catheterization is a specialized procedure used to view the blood ‡ow to the heart in order to improve the diagnosis. Angioplasty (PTCA) appeared more recently than bypass surgery. It is an alternative, less invasive procedure for improving blood ‡ow in a blocked artery.
In France, the use of an innovative procedure such as catheterization or angioplasty does not lead to classi…cation of a stay into a speci…c DRG. 4 These innovative procedures are most often performed within DRG 179: 76.1 % of CATHs and 82.8 % of PTCAs. Since they do not lead to classi…cation in a speci…c DRG, these costly procedures would not lead to a speci…c payment under a prospective payment system. A payment system which does not take these procedures into account would therefore penalise the innovative hospitals which use them and give hospitals incentives to select patients. proportion of hospitals never perform catheterization or angioplasty. These procedures require speci…c skills and high-tech facilities. For a given year, a hospital is considered to be innovative (INNOV) if it has performed catheterization for at least 2 % of the stays or at least one angioplasty. A hospital can be non-innovative one year and perform high-tech procedures the year after. On average over the four years, 60% of hospitals are classi…ed as innovative and these hospitals account for 71.5 % of the recorded stays.
To complete our database, we have also recorded information about hospital type from the SAE survey. 6 There are three types of hospitals: a CHR (Centre Hospitalier Regional ) is a public teaching hospital with research activities; PRIV stands for a private not-for-pro…t hospital (these hospitals have only recently been subject the global budget system PRIVs are innovative hospitals. Table 4 shows correlation coe¢ cients between hospital type, innovative hospitals and averaged indicators computed at the hospital-year level (95 observations). CHRs are innovative and have a low rate of discharge through transfer to another hospital. Private not for pro…t hospitals (PRIVs) are characterized by a high rate of use of innovative procedures and a high rate of admissions through transfers. Other public hospitals are rather non innovative. Patient ‡ows towards innovative hospitals appear clearly in (i) the positive correlation coe¢ cients we …nd between admission rates through transfers and CATH or PTCA rates; (ii) the negative correlation coe¢ cients we …nd between discharge rates through transfers and CATH rates. Table 5 gives average costs. Average cost per stay is equal to 4,198 e with a standard error of 2,863 e. On average, a stay is more costly when an innovative procedure has been implemented. As concerns hospital characteristics, stays are more expensive in teaching and in private not-for-pro…t hospitals. Stays are also costlier in innovative hospitals.

Historical context
In France, public hospital budgets have been based on a global budget system for more than ten years, including the years 1994-1997 that we study. A complete information system which classi…es inpatient stays by DRG has been set up, but a PPS has not been implemented. No reform of …nancing was undertaken from 1994 to 1997 (a gradual introduction of a PPS is planned for [2004][2005]. Budgets have no direct link to the actual 7 The SAE survey provides other indicators on hospitals, such as the number of beds, the occupation rate of beds, the diversi…cation of activities within hospitals. However, the high number of missing observations makes a complete descriptive analysis impossible. On the basis of a restricted number of observations, we …nd that CHRs are large hospitals with highly diverse activities. On the other hand, private not-for-pro…t hospitals (PRIVs) concentrate on a small number of activities. production of hospitals. Hospitals are managed by salaried administrators and do not keep the gains resulting from cost reduction e¤orts. In practice, the actual budget depends on the outcome of negotiations between the regulator and the hospital manager. In addition, hospitals are subject to a more or less soft budget constraint. This regulation leads to inequity and ine¢ ciency in the allocation of ressources (Mougeot (1999)).

ici 3 Theoretical background
The models used to study hospital payment systems are devoted to the general problem of local monopoly regulation. They consider the theoretical framework of an agency relationship between the regulator and the hospital, where the regulator has poor information about the cost reduction e¤ort provided by the hospital manager (moral hazard). For a particular disease, one assumes that the cost of one stay in a hospital h is given by: where a h and e h are private information of a hospital. a h is a technology parameter which represents the hospital's cost characteristics. It is a decreasing function of hospital productivity. e h represents the manager's e¤ort to reduce cost. The higher the e¤ort provided, the lower the moral hazard. A hospital exerting e¤ort level e h incurs a disutility denoted by '(e h ): '(:) is a continous function with ' 0 (:) > 0 and ' 00 (:) < 0: The services provided by hospital h generate a surplus S h > 0. In return, the regulator compensates the hospital through a monetary transfer P h . Hospitals are supposed to keep the rent earned through cost-reducing e¤orts and to face a hard budget constraint. Thus, each hospital h chooses its level of e¤ort in order to maximise its utility given by : Each hospital is supposed to be a local monopoly. One assumes that there is no collusion between hospitals. The regulator has to de…ne the levels of transfers which maximise social welfare subject to the constraint that hospitals must not be in state of bankruptcy ( takes distortions from taxation into account): 1 The yardstick competition model A prospective payment system (PPS) leads hospitals to exert the …rst-best level of e¤ort and to have a balanced budget (with no rent and no de…cit). A PPS is a …xed price contract. Since the payment is a lump-sum de…ned irrespective of actual cost, it gives the hospital a perfect incentive for cost reduction (' 0 (e ) = 1). At this stage, the problem is solved in part only. Indeed, a h is a private information of the hospital: the level of the lump-sum …xed by the regulator can lead the hospital to bankrupty or generate rents.
Thus, the problem of the regulator is to …nd the level of payments which is equal to the cost arising when the hospital is e¢ cient.
The yardstick competition model (Shleifer (1985)) solves the problem of informational asymetries by assuming that the technology parameters are all identical between hospitals: a h = a 8h. In this case, di¤erences in costs are only caused by moral hazard: The yardstick competition scheme consists in o¤ering to each hospital a rule of payment de…ned on the basis of the average costs observed for all other hospitals than h at the end of the year. The payment rule is: H is the number of regulated hospitals.
Here, C h is de…ned so as not to be in ‡uenced by C h : the resulting payment is equivalent to a …xed price contract. Since the payment rule is announced at the beginning of the year, the average C h is ex post equal to the cost corresponding to the …rst-best level of e¤ort: Transfers P h are such that each hospital breaks even: Expression (3) shows that P h is a lump-sum equal to the level of cost corresponding to an e¢ cient activity. In other words, P h is equal to the level of costs of a hospital when there is no moral hazard. Given our notations, the additional costs induced by moral hazard is equal to (e e h ). The payment rule leads ex post to: e h = e ; 8h: There is no longer moral hazard and the hospitals receive a payment (3) equal to the sum of the minimum level of costs (a e ) and of the disutility of the optimal level of e¤ort '(e ).
This ideal representation sets up the theoretical foundations of a fully prospective payment system. This model is based on rather unrealistic assumptions: homogeneity of hospitals, homogeneity of patients for the same pathology, …xed quality of care.
Many studies have underscored the great diversity in the conditions of care delivery for hospitals (teaching status, share of low income patients, local wage level, etc.). For instance, Pope (1990) shows that input prices can di¤er according to location, and that a hospital can be characterized by speci…c quality of services or severity of illness of admitted patients. These studies point out the risks of a fully prospective payment system: patient selection and lower care quality.
In order to avoid these drawbacks, many authors have tried to improve the basic model by removing hypotheses such as patient and hospital homogeneity. (Keeler (1990), Pope (1990), Ma (1994Ma ( , 1998, Ellis (1998), La¤ont and Tirole (1993)). It is also possible to consider extensions which introduce endogenous levels of number and quality of treatments (Ma (1994), Ellis (1998), Chalkley and Malcomson (2000)). Using various theoretical frameworks and hypotheses, some authors show that the social welfare can be improved by a mixed payment system combining a lump-sum and a reimbursement of the actual cost of treatment. However, the implementation of a mixed payment system is not straightforward: the proportions of the lump-sum and the actual cost are de…ned very di¤erently, depending on the theoretical model used, its main hypotheses and its parameterisation. Moreover, its de…nition often relies on unobservable variables or functions (such as the e¤ort disutility function, in La¤ont and Tirole's model).
In this paper, we consider an extension of the basic Shleifer's model, where the regulator is supposed to use the information available about observable sources of hospital cost heterogeneity.

Extension of the basic model
Consider C i;h; t the cost of stay i in hospital h during year t. We now suppose that the sources of hospital cost variability are partially observable. The regulator is able to observe the share e C iht of the costs which is linked to observable patient and hospital characteristics.
One has: (4) X 0 i;h;t represents individual patient characteristics such as age-gender cross e¤ects, admission and discharge modes, length of stay. W 0 h;t are observable hospital characteristics which can vary over time: the hospital's ability to perform innovative procedures, the implementation rates of high-tech procedures, the rates of admission or discharge through transfer.
Q 0 h are observable hospital characteristics which do not vary over time, such as the type: teaching, private not for pro…t or other public hospital.
In expression (4) the observed cost has two components: The second one, denoted ht ; is equal to the cost heterogeneity considered by the basic Shleifer's model (see expression (1)), where e ht is not observed by the regulator: ht = a e ht : Given these notations, the additional costs induced by moral hazard are, like in the basic model, equal to (e e ht ). Consider : where H t is the number of hospitals observed in year t.
The payment rule is now de…ned by: Here, h is de…ned 8 so as not to be in ‡uenced by ht . Assuming that the explanatory variables of e C iht are exogenous, i.e. that the hospital cannot manipulate their level in reaction to the proposed payment, the result of the payment is a …xed price contract. As explained before, the average h corresponds ex post to a situation with no moral hazard: The payment rule leads ex post to: e ht = e 8 h; t: There is no longer moral hazard. On the basis of rule (7), each hospital receives a payment corresponding to the minimum level of costs, for a given activity.
Each hospital breaks even with transfers P iht equal to: 4 Econometric speci…cation of the cost function When h are assumed to be random, the disturbance h + " h;t + u i;h;t has a , Our information is recorded at three levels (stays-hospitals-years), including the individual level of hospital stay. 9 The transition to the econometric speci…cation makes it necessary to take into account disturbances which are linked to patients' and hospitals' unobserved heterogeneity, omited variables and measurement errors.
Theoretical model (4) thus becomes: 8 Here h is de…ned as an average computed over several years. This is to consider a de…nition in accordance with the computation done in our empirical approach, where h is de…ned over 4 years. Notice that this doesn't change any prediction of the e¤ect of the payment rule, as soon as it is announced that the computation of h is updated every year. 9 Therefore, our approach is di¤erent of papers which evaluate e¢ ciency using data relative to average costs per hospital. A synthetic survey of this literature can be found in Linna (1998).
In contrast to theoretical model (4), we consider in econometric speci…cation (10) a hospital speci…c e¤ect h ; a hospital-year speci…c e¤ect " ht and a random error term at the patient level u i;h;t . The structure of our data results in a nested structure of the disturbance, which means that each each successive component of the error term is imbedded within the preceding component. The random error term u i;h;t is assumed to be iid (0; u 2 ): It takes unobservable patient heterogeneity into account. " h;t is a disturbance supposed to be iid (0; 2 " ) and uncorrelated with u i;h;t : We provide below detailed interpretations of h and " ht . Notice right here that the unobserved level of e¤ort e ht ; which appears in theoretical model (5), is a component of the term h + " h;t : As stated in the data section, the costs we observe result from an activity …nanced on the basis of a global budget system. Cost variability is therefore in ‡uenced by several factors: patient characteristics, hospital characteristics and ine¢ ciency.
Why should ine¢ ciency in ‡uence our "real" data ? Because of the way the global budget is implemented in France: as stated above in section 2:3, budgets have no direct link to the actual production of hospitals and the budget constraint is rather soft. In fact, ine¢ ciency is more or less possible, depending on the generosity obtained by the hospital manager from the regulator when bargaining for the budget.
Given patient characteristics, cost variability can stem from hospital characteristics such as hospital type (CHR, PRIV, PUB) and size, diversi…cation of activities, quality of services provided (performance of innovative procedures, comfort, alleviation of pain), skill level of nurses and doctors, quality of hospital management. Some of these factors are observable, some of them cannot be observed.
In this paper, we assume that the regulator has the same position as the econometrician.
More exactly, we assume that the regulator has an access to our database (the PMSI database) to set the payments. Therefore, the sharing out of variables between observable and unobservable components is the same for the regulator and the econometrician. The observable characteristics are the variables X 0 i;h;t for the patients and the variables W 0 h;t et Q 0 h for the hospitals: In (10), c t is a …xed temporal e¤ect, which can be linked to technological progress, the pace of price growth and the general trend of hospital budgets.
Given the observable characteristics, cost variability depends, in speci…cation (10), on the term:

Interpretation of hospital speci…c e¤ects h
The hospital speci…c e¤ects h can be assumed to be random or …xed. These e¤ects allow us to specify the time-constant unobservable hospital heterogeneity. In our theoretical framework, we have considered that the regulator has poor information about the cost reduction e¤ort provided by the hospital manager (moral hazard) but has information about observable sources of hospital cost heterogeneity e C iht : However, the cost variability can also be in ‡uenced by unobserved hospital characteristics explaining its e¢ ciency (adverse selection). Our theoretical framework considers only moral hazard. It does not address the issue of designing an optimal contract to deal with adverse selection. 10 Nevertheless, the costs we observe are in ‡uenced by unobserved hospital characteristics. Therefore we include adverse selection parameters into our econometric speci…cation.
Then, h can be seen as the result of three components : The components of h are interpreted in the following way: as h is an adverse selection parameter. The hospital's activity is more or less costly, depending on its infrastructure or on the existence of economies of scale or of scope. 11 mh h represents long term moral 1 0 On this issue, see for instance La¤ont and Tirole (1993 Within the implementation of a Prospective Payment System, we think that the regulator would be well advised to classify a priori these incidents as moral hazard, in order to give hospitals incentives to declare them, when the extra costs they induce are justi…able and exceptional. But such a regulation is not applied to the hospitals of our sample. In our data, the variability of " tr h;t can therefore be a¤ected by transitory unobserved shocks, the importance of which has to be evaluated empirically, in order to identify the share of " h;t due to short term moral hazard. Given the facts that (i) measurement errors are likely to be negligible; (ii) " tr h;t is mainly in ‡uenced by transitory shocks which are probably scarce, we think that the in ‡uence of " tr h;t on the variability of " h;t is negligible. An econometric test based on the stochastic cost frontier (SCF) approach gives an empirical support to this conjecture (see section 5.3).
Given this result, we can consider that the variability of " tr h;t is negligible and interpret the perturbation " h;t as an indicator of transitory moral hazard.

De…nition of two methods of payment
Econometric speci…cation (10) can be written as follows: where e C iht is the observable hospital heterogeneity. Consider the hospital-year means de…ned by C :;h;t = 1 where N ht is the number of stays recorded in hospital h in year t. Computing means at the hospital-year level eliminates the perturbation u i;h;t linked to the sample distribution of stays (u :; h;t P ! 0 when N ht is large 12 ). Therefore, one has: In our theoretical model, we have C i;h;t = e C iht + ht and the optimal payment is de…ned by (7): P iht = e C iht + '(e ) + h . In order to put this payment into practice, the regulator has to establish a link between the theoretical concept ht and the perturbations of the econometric speci…cation h + " h;t : In other words, he has to establish a link between the additional costs induced by moral hazard ht h = e e ht and h + " h;t : The arguments presented above, together with our SCF analysis, allow us to consider that " h;t can be interpreted as transitory moral hazard (" mh h;t ' " h;t ). The main di¢ culty concerns the hospital e¤ect: is h a legitimate hospital heterogeneity (which would be part of e C iht if it were observable) ? Or is h long term moral hazard, which must be crushed by an appropriate method of payment ? We have seen above that h can be seen as the result of three components ( h = as h + mh h + q h ) and that the moral hazard entails only one of these components ( e ht = mh h + " h;t ): Given the fact that the components of h cannot be identi…ed separately, the regulator is reduced to considering two extreme cases, whether h is supposed to be legitimate heterogeneity ( mh h = 0 and e ht = " h;t ) or to be entirely due to moral hazard ( h = mh h and e ht = h + " h;t ).

Taking or not unobservable hospital heterogeneity into account
In our theoretical model, the unobserved cost heterogeneity is equal to (5): ht = a e ht : As stated above, the regulator can consider two cases as regards the components of moral 1 2 On average, N h;t is equal to 77, with a minimum equal to 19 and a maximum equal to 250.

a) First method of payment
It relies on the assumption that hospital e¤ects h are linked to a legitimate heterogeneity. Given our notations, this comes down to suppose e ht = " ht . Thus: 1 ht = a + " ht : The rule of payment is given by: with 1 h de…ned by (6).
Assuming that hospitals keep the rent earned from more e¢ cient operations, they will exert the optimal cost-reducing e¤ort e . Ex post, the following equality will thus be veri…ed : 1 h = a e : With payment rule (12), the regulator takes the observable characteristics X 0 i;h;t , W 0 h;t and Q 0 h into account. In addition, the payment P 1 allows for permanent unobserved hospital heterogeneity h , assuming that it is due to adverse selection or to the care quality.
Nevertheless, this payment method still gives incentives to hospitals : cost deviations attributable to transitory moral hazard " ht are not reimbursed.

b) Second method of payment
The second method of payment is de…ned on the assumption that hospital e¤ects h are entirely due to moral hazard. In this case, e ht = h + " ht and: 2 ht = a + h + " ht (13) This second payment rule takes observable patient and hospital characteristics into account, but "crushes" unobserved heterogeneity h + " h;t . Implementing payment rule (14) comes down to interpreting all unobserved hospital heterogeneity as resulting from moral hazard.

Evaluating the ex post payments
We have seen in section 3.2 that payment rule (7) leads each hospital to provide the …rst best cost reduction e¤ort e . Therefore, each hospital receives ex post a payment corresponding to the minimum level of costs, for a given activity.
To evaluate the payments which can arise from the implementation of such a payment rule and the corresponding potential budget savings, we must evaluate the level of costs linked to an e¢ cient activity. Given our theoretical model the costs associated to an e¢ cient activity are equal to the payments arising ex post when rule (7) is implemented.
The estimation of cost function (10) allows us to evaluate the costs associated to an e¢ cient activity. However, the de…nition depends on the assumption relative to the components of moral hazard.

a) First method of payment
h being considered as a legitimate heterogeneity, one can estimate the ex post payments where we are using consistent estimates of the parameters and disturbances of model (10).
Under assumption (11) where we are using consistent estimates of the parameters and disturbances of model (10).
Here, we assume again that the most e¢ cient hospital-year observation corresponds to an e¢ cient activity (in other words, that the most e¢ cient hospital-year provides e ). But now we suppose that all the unobserved heterogeneity is resulting from moral hazard (13).

Estimation and results
We have chosen a linear speci…cation for the cost function: the dependent variable is C i;h;t and not Log(C i;h;t ). It is well known that health care expenditures generally have a very asymmetric distribution. In our case, however, the distribution is truncated on the right because of the selection of stays grouped in DRG 179 (uncomplicated AMI). More costly stays are grouped in other DRGs: complicated AMI or AMI treated by bypass surgery.
The tests we have carried out on the distribution of C i;h;t have led us to the conclusion that it is closer to a normal than to a lognormal distribution. More exactly, normality tests led to reject the null hypothesis for both C and Log(C): When we drop the 1% highest observed costs, the skewness is closer to the normal for C (S = 0:509) than for Log(C) (S = 1:117): Taking Log displaces the distribution to the left, leading to a negative value of the skewness. In addition, one of the results presented above provides an ex post justi…cation of our speci…cation choice: we …nd that the estimates of " ht do not increase on the raw scale as average hospital costs increase. 14

Estimation methods and speci…cation tests
In model (10)  A speci…cation test 15 led us to reject this hypothesis. Therefore, we specify h as a …xed 1 4 We thank one referee for this remark. 1 5 See Dormont and Milcent (2004). This test is not quite straightforward because our panel data is unbalanced in several dimensions: not only does the number of stays recorded vary across hospitals for a given year but also the length of the observation period varies across hospitals. Therefore our threecomponent error model (when h is random) is di¤erent from the unbalanced nested error component model considered by Baltagi, Song and Jung (2001) and we have to use the maximum likelihood estimator (MLE) de…ned by Antweiler (2001). To test for the independence of h , we have used an extension of the speci…cation test proposed by Mundlak (1978) for the standard error component model. Writting the correlation between h and the explanatory variables as follows: h = X 0 :;h;: 1 + W 0 h;: 2 + h , where h is iid (0; 2 ) and assumed to be uncorrelated with " h;t nor with u i;h;t , the independence test of h is equivalent to the restriction test for H0 : 1 = 2 = 0 in the model (estimated by MLE): In this case, the model includes hospital dummies and it is not possible to identify parameters which re ‡ect the in ‡uence of time-invariant variables Q 0 h : Speci…cation (10) becomes: This model is a standard error component model, with a disturbance equal to " h;t +u i;h;t . In this case, feasible generalized least squares (FGLS) lead to a consistent and asymptotically e¢ cient estimate if " h;t is not correlated with the explanatory variables.
Two speci…cations were estimated, related to di¤erent lists of explanatory variables Hausman's tests allowed us to validate the hypothesis that e¤ects " h;t are not correlated with the explanatory variables. Notice that the usual statistic of the Hausman test do not allow to consider variables W 0 ht . So, there is no di¤erence between the tests on models A and B. This test (denoted Hausman test 1) is equivalent to a test for no correlation between X 0 i;h;t and " h;t . It led not to reject the null hypothesis (table 7).
To test for the exogeneity of W 0 ht we used intrumental variables to build another Hausman's speci…cation test (denoted Hausman test 2). Here, we compared the estimator known to be consistent under the null and alternative hypotheses (the error component two-stage least square estimator, EC2SLS (Baltagi, 1981)) with an estimator which is consistent and e¢ cient under the null hypothesis (the feasible generalized least squares estimator, FGLS).
Instruments are the secondary diagnoses of the patient. A Sargan test has been imple-mented in order to check the validity of the instruments used for this Hausman's test. In addition, we examined whether this test could be subject to the weak instrument problem (Staiger and Stock, 1997). For this purpose, we tested for the signi…cance of the instruments in several equations, where each instrumented variable is explained by the instruments and the exogenous regressors. Hausman tests 2 led not to reject the null hypothesis for model (A) and (B) and the Sargan tests did validate the exogeneity of the intruments (table 7).
In addition, we found a large signi…cance of the partial correlation between instruments and endogenous explanatory variables, with high statistics 16 and levels of signi…cance lower than 10 3 : All these tests validate the hypothesis that W 0 ht are not correlated with " ht nor u iht : Given the fact that both e¤ects have components related to moral hazard, we had to examine whether " ht could be correlated with the hospital e¤ects h : For that purpose, we implemented a third Hausman test, comparing the FGLS applied to C i;h;t = X 0 i;h;t t + a + c t + h + (" h;t + u i;h;t ); where h is supposed to be …xed and " h;t is supposed to be random and not correlated to h nor the X 0 i;h;t , to the OLS applied to C i;h;t = X 0 i;h;t t + a + c t + h + " h;t + (u i;h;t ); where h and " ht are supposed to be …xed. The test relies on the fact that the OLS applied to the second model are consistent even when the " h;t are correlated to the h : This test led us not to reject the null hypothesis, with a Wald statistic equal to 23.2 and a p-value close to 1 (the corresponding 2 has a degree of freedom equal to 64).
All these tests provide evidence that we cannot reject the hypotheses that the variables X 0 i;h;t and W 0 ht are exogenous. Model (17) can be consistently estimated by the FGLS.

Results
The estimated coe¢ cients of the individual characteristics X 0 i;h;t are reported in table 6.
The in ‡uence of individual stay characteristics are in accordance with the results generally obtained when studying costs of stays for acute myocardial infarction. The most costly stays are observed for men and cost is a decreasing function of age. One additional day induces, ceteris paribus, an average additional cost of about 330-400 Euros. In addition, the estimation of an incomplete speci…cation using only individual patient characteristics giving an ex post justi…cation to our linear speci…cation.

SCF analysis
This analysis was implemented to give empirical support to the assumption that " ht is entirely due to transitory moral hasard. The basic SCF approach relies on the canonical "half normal" model (Greene, 2004), which uses a parametric speci…cation in order to identify the ine¢ ciency component. The disturbance is split into two components: a normal one, related to statistical noises and a half normal component, related to ine¢ ciency. In our case, we have: " h;t = " mh h;t + " tr h;t ; where " mh h;t is the transitory moral hazard and where " tr h;t is linked to measurement errors and transitory shocks. The SCF speci…cation relies on the following assumptions: " tr h;t N (0; 2 " tr ) and " mh h;t = j ht j ; with ht N (0; 2 ): The asymmetry parameter, = " tr gives an evaluation of the magnitude of the ine¢ciency component. In section 4.2, we explained that 2 " tr is likely to be negligible. If this conjecture is right, one should …nd 2 " tr ! 0 and ! 1: In this case, " h;t ' " mh h;t : Consider model (17). It can be written as follows: In a …rst regression, we estimate (19), where the v h;t are speci…ed as …xed e¤ects and where u i;h;t is supposed to be iid (0; 2 u ). Given our assumptions and the fact that N ht is large enough, the v h;t can be consistently estimated by OLS. This …rst step makes it possible to eliminate the patient dimension from the data variability and to get observations at the hospital-year level.
In the second step, we use the …rst-step estimatesv h;t and consider the SCF speci…cation, assuming (18) to estimate (20) by the maximum likelihood estimator. This allows us to identify the components of " ht : Notice that the constant a has been deleted from the …rst step regression (19). In second step speci…cation (20), the constant is taken by the hospital …xed e¤ects into account (there is no reference hospital). To avoid multicolinearity, we deleted one year dummy.
This treatment of the constant is adopted in order to avoid any a priori constraint upon the distribution of hospital e¤ects h : Our speci…cation di¤ers from the basic versions of panel data formulation of the SCF approach. In these versions, the ine¢ ciency is supposed to be time-invariant and re ‡ected by the individual e¤ect (here, h ). We think that this formulation is not appropriate in our case. As stated repeatedly above, h is not only a¤ected by moral hazard, but also by heterogeneity and care quality. These two factors can be symmetrically distributed. Our three dimensional database allows us to consider a less constraining hypothesis.
The estimation of (20) by the maximum likelihood estimator (using the 95 observations of the …rst-step estimatesv h;t ) led to: b " tr = 0:00025 and b = 605:7816: Thus b ! 1 ( = 2:394 10 3 ): This result gives an empirical support to our conjecture that the variability of " ht is entirely attributable to the transitory moral hazard. It was reasonnable to expect that measurements errors had a negligible importance at the hospital-year level.
But we didn't know the share of the variability of " ht due to transitory shocks. This parametric SCF analysis show that it is likely to be negligible too.

Simulation of two methods of payment
Our econometric estimates encourage the implementation of a prospective payment system.
Indeed, our results have revealed that the transitory moral hazard is far from negligible. As we have seen in section 4.3, the payment rule adopted by the regulator depends on whether h is supposed to be legitimate heterogeneity or to be entirely due to moral hazard. In the …rst case, it is de…ned by (12), in the second by (14). Under the assumptions of the theoretical model, these payment rules should give to each hospital an incentive to provide the …rst best cost reduction e¤ort, leading ex post to payments We can simulate the implementation of the two payment rules on our data. The …rst method of payment exerts a softer constraint on hospitals than the second method of payment. Indeed, payment P 2 ignores all unobserved heterogeneity ( h + " h;t ). With payment P 1 the regulator takes the time-invariant unobservable heterogeneity ( h ) into account, whether it is due to ine¢ cient management or to particularly good care quality. Table 9 gives the potential budget savings which can be expected from the implementation of such payment rules. 19 They are computed by measuring the di¤erence between total costs C iht and total ex post paymentsP 1 i;h;t orP 2 i;h;t . We can observe that the bracket de…ned by P 1 and P 2 is quite wide: the payment rule P 1 leads to potential savings of about 20 %; the payment rule P 2 leads to potential savings of between 51 % and 56 %, depending on the model considered (B or A).
P 1 is indeed the least constraining payment system. Yet, it still leads to substantial potential savings (20 %) because (i) it provides incentives to reduce the costs due to transitory moral hazard " h;t , (ii) the variability of costs due to transitory moral hazard is sizeable. We thus recommend this method of payment. It avoids using the hospital with the poorest care quality as a benchmark for cost. It takes permanent unobservable differences of quality between hospitals into account. This strategy is advisable, given that quality is a variable that cannot be veri…ed by the regulator.
The next step is to determine which model should be used to establish payments.
-In our estimations and simulations, we have taken the length of stays into account.
Nevertheless, the type of payment system that we suggest implementing should not be retrospective in the sense that it should be calculated by stay and not by day.
Therefore, we propose reimbursing on the basis of the estimated coe¢ cient of the length of stay in the cost function multiplied by a suitable indicator of the length of stay (an average indicator taking di¤erences in patient and hospital characteristics into account).
-The main di¤erence between the models A and B is that model B integrates charac-teristics such as the frequency of innovative procedures. The reason for integrating procedure rates into the payment system is to avoid patient selection and skimping on treatment. On the other hand, there is a risk of creating incentives for excessive use of procedures (McClellan, 1997). We notice that all variables W 0 ht are not signi…cant when estimating …xed e¤ects model (table 6) and that potential budget savings do not di¤er when implemented on the basis of model A or B. It is interesting to notice that taking into account heterogeneity through the hospital …xed e¤ects lead to a non signi…cant in ‡uence of variables such as T I and T X (rate of admissions or discharges through tranfers) as well as the frequency of innovative procedures. Indeed, these variables can be manipulated by the hospitals in the short run. It should be more di¢ cult for the hospitals to manipulate their own value of h , which derive from the estimation process.
Table 10 records correlation coe¢ cients between costs and payments. A high correlation rate means that the incentives for selecting patients are limited. We observe that substantial budget savings displayed in table 9 are compatible with high correlation coe¢ cients, especially in the between dimension, which is based on the yearly mean by hospital.
6.2 Share of retrospective payment in the …rst method of payment Payment method P 2 can be seen as a prospective payment, relaxed by the kind of risk adjustment resulting from the fact that we take observable patient heterogeneity into account. On the other hand, the …rst method of payment is partly retrospective because it reimburses costs di¤erences due to the hospital e¤ects h : More exactly, one can distinguish the following prospective and retrospective components of the …rst method of payment: We have obtained (for model A):^ = 46; 2 %; with a standard error equal to 12,9 %.
We have to underline that this sample mean (^ ) provides an evaluation which is not a rule of payment. It results from an ex post computation, which allows us to know the weight of retrospective payment induced by the implementation of payment rule P 1 :

Conclusion
Hospital heterogeneity is a major issue in de…ning an optimal reimbursement system.
In this paper, we have considered an extension of the basic yardstick competition model, allowing for the existence of observable sources of heterogeneity. We have applied an econometric approach to the identi…cation and evaluation of observable and unobservable sources of cost heterogeneity. The use of a three dimensional nested database makes it possible to identify transitory moral hazard, and to estimate its e¤ect on hospital cost variability.
In our speci…cation, observable hospital characteristics and hospital speci…c e¤ects enable us to take hospital heterogeneity into account. We obtain two alternative payment systems. The …rst takes all unobservable hospital heterogeneity into account, provided that it is time invariant, whereas the second ignores unobservable heterogeneity. Simulations show that substantial budget savings -at least 20 % -can be expected from the implementation of such payment rules.
The …rst method of payment seems advisable to us: it has the great advantage of reimbursing high quality care. It leads to substantial potential savings, because it provides incentives to reduce costs linked to transitory moral hazard, whose in ‡uence on cost variability is far from negligible. Thus, our study shows that: (i) one component of moral hazard can be easily identi…ed with three-level panel data: transitory moral hazard (ii) this component of moral hazard is sizeable. Therefore, substantial budget savings can be obtained from the implementation of a payment rule which eliminates only this component.
Moreover, this payment system is easy to implement, provided the regulator has information about costs of hospital stays. One drawback is that it would give higher reimbursements to hospitals which are costlier because of permanently ine¢ cient management. The choice between the two methods of payment depends on the weights assigned to e¢ ciency and care quality in the social objective function used by the regulator.
Our payment rules could be extended to other areas of health care …nancing. Considering again the example of AAPCC to calculate Medicare Managed Care reimbursements, our method would make it possible -not to identify the sources of geographical cost heterogeneity -but to identify the transitory moral hazard in the local-year dimension. If this component of moral hazard has a sizeable in ‡uence on cost variability, the savings derived from eliminating it can be substantial.
In order to induce e¤ective budget savings, the implementation of our payment rules requires the following: hospitals would have to earn the rents arising from improved e¢ciency and they would have to face a hard budget constraint. What can be infered from our simulations is limited by the fact that they are carried out under the hypothesis that behaviors are supposed to remain unchanged except as concerns moral hazard. In other words, hospitals are supposed not to adopt strategic behaviors in reaction to a reform of the payment system. Moreover, our evaluations of the budget savings assume a constant level of activity. Our …nding of a potential saving of 20 % means that greater e¢ ciency could have induced a saving of 20 % to …nance the hospital activity observed during the period 1994-1997. One important di¤erence between the PPS and the global budget is that the level of activity is in principle not capped within the PPS. An increase in activity could make hospital expenditures rise, even if hospitals progressed in e¢ ciency.