Hospital Reimbursement in an Oligopolistic Industry

Transcription

1 Hospital Reimbursement in an Oligopolistic Industry Kurt R. Brekke Program for Health Economics in Bergen University of Bergen May 8, 2000 Abstract This paper analyses the design of hospital reimbursement in an industry with imperfect competition, scarcity of labour and socially costly government transfers. We find that hospital competition may increasebothquantityandqualityofhospitalcare,andifthisisthe case prospective payment has an unambigious positive effect on both quantity and quality. Hospitals may produce too little or too much health care in the Cournot-Nash equilibrium compared to first best depending on the intensity of competition, the labour scarcity and the shadow cost of public funds. While imperfect competition induces the hospitals to produce too little, scarcity of labour and costly government transfers have the opposite effect. Optimal policy in the case of overproduction of hospital care is to impose a tax (negative fixed price) on the hospitals. When the imperfect competition effect dominates, a subsidy (positive fixed price) should be imposed. Key words: Hospitals, reimbursement, oligopoly and welfare. JEL classification: D43, I11, L13. I thank Jan Erik Askildsen, Lars Sørgard, Andrew Jones, Trond Olsen, Fred Schroyen, Odd Rune Straume and seminar participants at the University of Bergen for valuable comments. A preliminary version of the paper was presented at the 22th annual national conference of economics in Bergen. The usual disclaimer applies. Department of Economics, University of Bergen, Fosswinckelsgate 6, N-5007 Bergen, Norway. Tel: Fax: kurt.brekke@econ.uib.no 1

2 1 Introduction Many countries are concerned about the cost of providing health care to their citizens. This concern has lead governments to carry through several reforms aiming at increasing the efficiency in providing health care. While competition between health providers has been prevalent in private health care systems, like the US, for many years, several countries have in the last decade introduced provider competition within the public health care system (see OECD, 1994). In UK, Sweden and New Zealand internal markets are implemented by separating the purchasing and provider functions. More recently some countries, like Norway, Sweden and Danmark, have delegated the choice of hospital to the patients. However, this effort to contain costs and increase the efficiency of providing health care may be counteracted by another important feature of the health sector, namely the scarcity of health personnel. Excess demand for physicians and nurses may indeed be a major source of cost expansion in several countries. The purpose of this paper is to analyse the design of hospital reimbursement when hospitals compete for patients, faces scarcity of health personell and receive government transfers that are socially costly. We offer three main findings. First, hospital competition may not only increase the quantity, but also the quality of health care provided. If this is the case prospective payment will have, contrary to what is often argued, a positive effect on both quantity and quality of hospital care. Second, while imperfect hospital competition induces the hospitals to produce too little health care compared to first best, scarcity of labour and socially costly government transfers increase the social cost of providing hospital care and may lead to overproduction. Tougher hospital competition may then not be socially desirable because it enforces the problem of cost expansion. Third, first best levels are attainable if the regulator can use a two-part tariff system (block grant and a fixed pricepercase). Ifthelabourscarcityeffect dominates the imperfect competition effect, optimal policy is to impose a negative fixed price (tax) on the hospitals. Otherwise, the hospitals should receive a positive fixed price (subsidy). Several studies of hospital markets address the question of whether competitive markets are characterised by higher prices (or costs) and/or higher quality. Dranove and White (1994), who give an extensive review of recent theoretical and empirical literature on hospital competition, argue that recent empirical studies have shown a positive relation between higher hospital prices and higher market concentration. Furthermore, they argue that hospital competition may also increase the quality of health care provided. These conclusions are to a large extent confirmed by Kessler and McClellan (1999) 2

3 who study the consequences of hospital competition for Medicare beneficiaries heart attack care from 1985 to They find that (after 1991) itwas approximately 8 percent more costly to be treated in the least competitive fourth of hospital markets, as compared to the most competitive fourth. And, the quality of care in competitive markets was higher as well. Patients in the least competitive fourth of hospital markets experienced approximately 1.5 percentage points higher mortality than those in the most competitive areas. These empirical findings are very much in line with the first result of this paper. Inthisrespectthemodelproposedheremayprovidesomerelevant theoretical intuition to these empirical findings. Several papers have analysed the design of hospital reimbursement using contract or incentive theory. Schleifer s (1985) model of yardstick competition is a significant contribution within this strand of literature. He shows that when the managerial effort is unobservable, the hospitals will produce efficiently if paid a prospective rate at the average of all other hospitals marginal costs. The reason is simply that managers will invest the socially optimal amount of effort if they keep the full reward of doing so. Newhouse (1996), who gives an extensive review of the literature so far, argues that under more general assumptions (heterogeneous patients, imperfect agency, uncertainty about the hospital costs) a pure prospective payment system will not be optimal. Instead a mixed reimbursement system will be optimal in such circumstances. While there is a relatively large amount of papers using contract or incentive theory to analyse the design of hospital reimbursement, the Industrial Organisation literature within this field is limited. A seminal paper is Pope (1989) who analyses nonprice (quality) competition in a prospective payment system. He argues that competition can play an important role both by increasing quality and reducing managerial slack, and that the qualityenhancing aspects of competition can be amplified through reimbursement of a proportion of incurred costs. A more recent contribution is Ellis (1998) who assumes that patients differ in severity of illness. He finds that while cost-based reimbursement results in overprovision of services (creaming) to all types of patients, prospectively paid providers cream low severity patients and skimp high severity ones. Our paper is closest to the work by Kesteloot and Voet (1998). They study the hospitals incentives to improve the quality considering both price and non-price (quality) competition. In their model quality improvements may increase or reduce costs, and create positive or negative spillover to rival hospitals. They find that quality improving effort rises with the rate of prospective payment, while the impact of retrospective reimbursement is ambiguous but likely to be negative for quality improvements that are highly 3

4 cost-reducing and create large positive spillover. In this paper we carry the work by Kesteloot and Voet (1998) further in several dimensions. First, we suggest that scarcity of health personell may affect the hospital cost function. Since labour is used to produce both output and quality, we assume the hospital cost function to be increasing and convex. In addition we argue that increased production may induce a negative spillover on rival hospitals due scarcity of labour. For example the hospitals may have to outbid each other in order to attract skilled physicians. Second, we model the regulator s problem and derive the firstbestlevels of output and quality such that we are able to compare the Cournot-Nash output and quality to first best. Finally, we make an explicit analysis of the optimal reimbursement system within this framework. Consistent with Kesteloot and Voet (1998), we find that prospective payment may increase quality as well as output. We argue that quality being proportionally related to output in the Cournot-Nash equilibrium is plausible, but depends on the specification of the cost and demand functions. In general this relation may be negative as well as non-linear implying that prospective payment does not have an unambiguously positive effect on quality as Kesteloot and Voet (1998) argue. We consider a special case where prospective payment gives the hospitals no incentive to improve on quality. Furthermore, we demonstrate that even if prospective payment induces the hospitals to provide more output and quality, higher levels may not be socially desirable. If the negative spillover effect due to labour scarcity dominates the imperfect competition effect, the Cournot-Nash equilibrium implies a provision of health care that is too costly from a social point of view. Optimal reimbursement policy is then to impose negative fixed prices (taxes) on the hospitals in order to reduce the production level in equilibrium. The paper is organised as follows. In section 2, we set out the analytical framework. In section 3, we derive the Cournot-Nash equilibrium and explore the characteristics of the equilibrium with respect to the hospital reimbursement system. In section 4, we analyse social welfare by deriving the first best levels of output and quality and compare the Cournot-Nash output and quality levels to first best. In section 5, we focus on optimal reimbursement policy by exploring the implementation of first best levels through the policy instruments. Finally, in section 6, the paper is concluded. 2 The model We consider a market that consists of two identical hospitals, denoted by i =1, 2, each producing a differentiated health service (service 1 and 2). 4

5 Hospital i faces the following inverse demand function p i =1 q i dq j + αx i i, j =1, 2; i 6= j (1) where 0 <d<1 and 0 < α < 1. p i is the patients marginal willingness to pay for health care produced by hospital i (the hospital price charged directly to the patient), q i is the quantity of hospital care (measured as the number of cases), and x i is the quality of the hospital treatment. The direct demand functions are given by q i = 1 d p i + dp j + α (x i dx j ) i, j =1, 2; i 6= j 1 d 2 The parameter d reflects that services from hospital 1 and 2 are perceived as imperfect substitutes by the patients, and thus represents the degree of horisontal differentiation. When d 1, the care produced by hospital 1 and 2 are close substitutes, while d 0 implies that the care produced by the hospitals are perceived as independent services by the patients. This parameter may reflect a set of hospital characteristics (e.g. location, reputation, etc.) that makes some patients prefer the care produced by hospital 1 and others prefer the care produced by hospital 2. The parameter α is the inverse price sensitivity of quality changes, and represents the degree of vertical differentiation. Higher levels of quality are perceived as better by all the patients. Following Kesteloot and Voet (1998) and several others we define quality widely including both care items that are crucial for patient outcome (e.g. highly skilled physicians, good medical equipment, efficient drugs) as well as care items that only improve on patient comfort (e.g. size of the rooms, quality of the meals, availabilities of TV). Let us now consider the supply side. The production of health care is typicallyverylabourintensive. Scarcityofhealthpersonnelmayaffect the hospital cost function. Since labour is used to produce both output and quality, we suggest the total cost of producing hospital care to be increasing and convex in both products. In addition, we find it plausible to assume that the costs of producing output and quality is not only influenced by a hospital s total amount of production, but also by how much that is already produced of either quality and output. The higher level of output produced, the higher marginal costs of producing one unit more of output compared to one unit more of quality (and vice versa). Furthermore, we focus on an additional effect of scarcity of labour. When hospital 1 increases its production, this induces a negative spillover on hospital 2 s costs. The intuition behind this effect is related to the fact that a hospital has to employ more physicians and nurses in order to increase 5

6 its output. When there is a given number of physicians and nurses in the labour market, an increase in output does not only increase its own cost but also the cost of the rival hospital due to higher wages and other employment costs. Hospitals might have to outbid each other to attract more physicians and nurses in order to increase their production. A cost function that encompasses these elements is the following. 1 C i = 1 x 2 2 i + qi 2 +(δxi + βq j ) q i i, j =1, 2; i 6= j (2) where 0 < δ < 1 and 0 < β < 1. 2 The parameter β is the negative spillover, representing the scarcity of labour. When β 1 the spillover is high, which is interpreted as a high scarcity of labour. The parameter δ (the cross partial derivative) values how the marginal cost of quality is affected by the initial level of output produced (and vice versa). When δ 1 the marginal cost of quality is equal to the marginal cost of output (if we ignore the spillover effect). In the other case when δ 0, the marginal cost of quality is unaffected by the level of output produced (and vice versa), which means that the costs of producing quality and output are additive separable. One important feature of the hospital sector is that the major part of hospital revenues does not come from patient out-of-pocket payments. Because of the high degree of health insurance, a significant share of hospital treatment costs are paid for either by the government or an insurance company. In the literature it is often distinguished between retrospective and prospective payment. Retrospective payment means broadly that hospitals receive a payment based on actual historical costs, e.g. the hospital submits its bill to the government or insurance company after medical care has been given. In a prospective system, payment rates are set prior to the period for which care is given and is a fixed price (or fee) per case often related to Diagnosis-Related Groups. In this paper we do not consider the trade-off between retrospective and prospective payment, but rather how to set the prospective payment. 3 The hospital reimbursement schedule is defined as T i = A + sq i i =1, 2 (3) 1 Since the major share of costs in a hospital consists of labour and material costs, fixed costs are ignored. 2 Increasing marginal costs and (strict) convexity implies that 0 < δ < 1 and β > 0. However, we find it plausible to assume 2 C i = 2 C i > 2 C i qi 2 x 2 q i i q j 1 > β. 3 Retrospective payment could well have been included in the analysis (by specifying the transfer as e.g. T i = A + sq i + γc i ). However, this makes the analysis more complicated and does not change the main results in this paper. 6

7 Here A represents a block grant (or lump-sum transfer) and s denotes the fixed price per case (or prospective rate). When considering an optimal reimbursement system, we allow the block grant and the fixed price to attain negative values as well as positive. To prevent a take-the-money-and-run strategy we assume that the hospitals have to commit to a positive production level to attain the block grant. In other words, if q i =0,thenA =0. Finally, the hospitals objective function needs to be specified. We assume that the hospitals maximise profit defined as π i = p i q i + T i C i i =1, 2 (4) Prima facie, profit maximising hospitals may not be a realistic assumption. Considering the ownership structure only a small fraction of the hospitals are private for-profit hospitals. The major part of hospital care is provided by either public or private non-profit hospitals. Furthermore, within a hospital there are different agents (physicians, nurses, administrators) that may pursue different goals. The hospital objective may then be a weighted sum of these goals, where the weights reflects the bargaining power of the different agents. In the literature, hospitals are sometimes assumed to be benevolent (or partially benevolent), which often means that the patients benefit of the hospital treatment is included in the objective function as well as hospital profit (see e.g. Chackley and Malcomsen, 1998). Other times the hospitals maximise a utility function. Glazer and McGuire (1994) suggest that the hospitals maximise utility as a weighted average of quality, size (quantity of services) and profit. However, this type of utility function does not change the main results in this paper. Thus for simplicity, but also consistent with severalpapers(bösanddefraja,1998, Danzon, 1982, Ma, 1994, Pope, 1989), we assume that the hospitals are profit maximisers. 4 We consider a two stage, quality-quantity game with complete information. The rules of the game are as follows. In stage 1, the regulator chooses the level of the fixed price per case s and the block grant A in order to maximise social welfare. In stage 2, hospital 1 and 2 observe the reimbursement schedule proposed by the regulator, and then simultaneously and independently choose the level of output q i and quality x i that maximises their profits. The game is solved by backwards induction. 4 See Dranove and White (1994) for a theoretical and empirical discussion of hospital objectives. 7

8 3 Hospital competition First we analyse the hospitals behaviour at stage 2, and characterise their choices of output and quality for any combination of the block grant and fixed price per case. Each hospital face the following problem max q i,x i 1 (1 q i dq j + αx i + s) q i + A x 2 2 i + qi 2 +(δxi + βq j ) q i s.t. q i 0 and x i x 0 where x represents the minimum level of quality required typically determined by government regulations. Quality levels below this level may be interpreted as malpratice. For simplicity we normalise x to zero. The levels of output and quality that maximise hospital i s profit aregivenbythe following first order conditions: 5 π i q i = 1 3q i (d + β) q j +(α δ) x i + s =0 i, j =1, 2; i 6= j (5) π i x i = (α δ)q i x i =0 i =1, 2 (6) From these first order condition, we may propose the following Proposition 1 x i > 0 and x i q i > 0 iff α > δ. Otherwise,x i = x =0. Proof. This follows straightforward from eq. (6). Thus it is only optimal for the hospitals to provide quality above the minimum required level (x i > x =0)if the patients value quality sufficiently much (α > δ). The critical level of the marginal willingness to pay for quality depends on the relation between output and quality in the cost function. If the costs of producing output and quality are separable (δ 0), α may be relatively low. As long as α > δ, the profit maximising level of quality is proportionally related to output. 6 The intuition behind this is that by raising quality, the hospitals can raise the price while keeping the same demand. This price increase can be passed on to all inframarginal patients, which generates 5 Second order derivatives are: 2 π i / q 2 i = 3, 2 π i / x 2 i = 1 and 2 π i / q i x i = 2 π i / x i q i =(α δ). Thus, the two first second order conditions are fulfilled. For the last condition to hold, we have to assume that 3 > (α δ) 2. 6 Note that this is plausible but not general. The relation between output and quality may be both negative and non-linear. This would be the case if we instead used a Cobb- Douglas cost function like C = q a x b. 8

9 an extra revenue. Quality will then be increased until the marginal revenue (αq i ) equals the marginal cost (x i + δq i ) of providing quality. In the other case, when α δ, the marginal cost of providing quality will always be higher than the marginal revenue, which means that it is never profitable for the hospital to increase the quality above the minimum level required. 7 It must be emphasised that the profit maximising level of output is not proportionally related to quality. This we see from eq. (6). Thus, the level of output that maximises a hospital s profit may indeed be positive (q i > 0), even if the profit maximising level of quality is to provide only the minimum level required (x i =0). In Kesteloot and Voet (1998), the profit maximising level of quality is related to output in a similar way as in our model. However, they assume that α δ > 0. The argument is that cost increasing quality improvements would only be implemented by rational hospital managers if the net effect on the hospital objective function is positive. We find it unreasonable to rule out the option of α δ for the following reasons: First, this restriction does not follow from the hospitals profit maximisation problem. Within this framework there exists a situation were profit is maximised and the hospitals produce a positive level of output but only the minimum required level of quality. Second, the proportional relation between output and quality is not general but depends on the specifications of the cost and demand functions. This is also surprising since we would expect a trade-off between providing output and quality. Thus, ruling out the possibility of α δ means that we restrict our analysis to an even more special case. Finally, as we shall see later on this assumption is crucial for the results regarding the effect of prospective payment on the levels of output and quality produced by the hospitals. The first order conditions above yield the following Cournot-Nash equilibrium outcome: q c = x c = 1+s 3+d + β (α δ) 2 (7) (α δ)(1+s) 3+d + β (α δ) 2 (8) Since the equilibrium outcome is symmetric, we have for simplicity dropped the hospital notation. Analysing the Cournot-Nash equilibrium yields the following proposition about existence: 7 If we allowed quality to attain negative values as well (e.g. interpreted as malpratice), then α < δ would imply that the profit maximising level of quality is negatively, proportionally related to output. 9

10 Proposition 2 q c > 0 iff s> 1. x c > 0 iff α > δ and s> 1. Otherwise q c =0and x c =0. Proof. The denominators of q c and x c are positive from the S.O.C. (α δ) 2 < 3 (see footnote 5). Then the result follows straightforwardly from eq. (7) and (8). The nonnegativity conditions for output and quality imply a lower bound on the fixed price per case. If a negative fixed price (tax), which exceeds this lower bound, is imposed on the hospitals, they will not produce a positive level of output (closing down). We will return to this problem when we solve the subgame perfect equilibrium and discuss optimal reimbursement in section 5. Furthermore, consistent with proposition 1, we observe that the hospitals will not produce quality above the minimum required level unless the patients value quality sufficiently high. The question whether a prospective payment system gives the hospitals incentive to provide an optimal level of quality, has been widely disputed (see Newhouse, 1996). It has been argued that prospective payment gives the hospitals poor incentives to provide quality, but good incentives with respect to output. As a first approach to this question, we analyse how a change in the fixed price per case affect the level of output and quality in the Cournot-Nash equilibrium. This is done by comparative statics, and leads to the following proposition: Proposition 3 qc s α > δ. Otherwise xc s =0. > 0 and β q c s < 0. x c s > 0 and β x c s < 0 iff Proof. By computation we get q c s = 1 3+d+β (α δ) 2 x c s = (α δ) 3+d+β (α δ) 2 β β q c s x c s = β (3+d+β (α δ) 2 ) 2 = β(α δ) (3+d+β (α δ) 2 ) 2 The denominators of qc xc and are positive from the S.O.C. (α s s δ)2 < 3 (see footnote 5). Then the proof follows straightforwardly. As expected we find that prospective payment gives the hospitals incentive to produce more health care in the Cournot-Nash equilibrium. An increase in the fixed price per case reduces (increases) the hospitals marginal costs (revenues), which in turn makes it profitable to increase the level of output. As long as quality is proportionally related to output, the quality of hospital care also increases with the rate of the prospective payment. 10

11 However, if the marginal cost of an increase in quality is not covered by the marginal increase in revenue (which is the case when α δ), prospective payment has an effect on output only, and the hospitals provide the minimum required level of quality. Furthermore, we find that the effect of prospective payment on hospital production decreases with the size of the negative spillover. If the scarcity of health personnel is high (β 1), increasing the fixed price per case has a lower effect on the output and quality of hospital care provided in equilibrium, than if the scarcity of labour is low. The intuition is that for a given level of output produced by hospital j, a higher spillover increases the marginal costs of output to hospital i. Ceteris paribus, this induces a lower profit maximising level of output. As long as quality is proportionally related to output this accounts for quality as well. 4 Welfare analysis In this section we consider the hospital industry from a social planner s point of view. We assume that the social planner is concerned about consumer surplus, producer surplus and any possible distortion in the economy generated by taxes. Thus we ignore any income distributional considerations, and focuses only on allocational efficiency. The social welfare function is defined as follows: W = U X i p iq i + X i π i (1 + λ) X i T i i =1, 2 (9) The parameter λ represents the shadow cost of public funds due to distortionary taxes, and by assumption λ > 0. The inverse demand functions definedineq. (1) come from the maximisation problem of a representative patient with the following (quasi-linear) utility function: U = X q i 1 hx i 2 i q2 i + d X i i Xj q iq j + α X x iq i (10) i where i, j =1, 2 and i 6= j. Byusingthedefinition of hospital profit (ineq. (4)), we may express the social welfare as follows: W = U + λ X i p iq i (1 + λ) X i C i λ X i π i i =1, 2 (11) The first term is the gross consumer surplus. The second term reflects the social benefit of patient copayments. Since copayments reduce the need for socially costly hospital reimbursement, this may be interpreted as a social benefit. The third term is the total social cost of producing hospital care. 11

12 Finally, the last term reflects the social cost of leaving profit in hospitals due distortions in the economy generated by taxes. The social planner s problem may be specified as follows: max W = U + λ X p iq i (1 + λ) X C i λ X π i i =1, 2 q 1,q 2,x 1,x 2,s,A i i i subject to π i 0 i =1, 2 As long as hospital reimbursement is socially costly, it is not optimal for the social planner to leave any profit in the hospitals. Since we have allowed for lump-sum transfers (the block grant A), we can assume the participation constraints to hold with equality, i.e. π i =0(i =1, 2). The regulator s optimisation problem is now unconstrained, and the social welfare is no longer a function of the policy instruments s and A. The solution is given by the following first order conditions: 8 W q i =1 q i dq j + αx i + λ (1 2q i 2dq j + αx i ) (12) (1 + λ)(q i + δx i +2βq j )=0 W x i =(1+λ)[(α δ) q i x i ]=0 (13) where i, j =1, 2 and i 6= j. Social welfare is maximised when marginal social benefit, which is the sum of the marginal utility and the marginal benefit of copayments, is equal to the marginal social cost of hospital production. Notice that these first order conditions simply are the Ramsey pricing equations (expressed in terms of output). They differs from the standard formula in two directions. First, we consider a negative spillover, which yields a higher marginal social cost than if there is no (or a positive) spillover. This implies a lower first best level of output. Second, we consider quality as well as several products. The first best level of quality is given by the second condition (eq. 13). Since there is no price on quality, it affects the optimal price (or output) of hospital care. Solving the first order conditions yield the following first best levels of 8 Second order derivatives are: 2 W = (2 + 3λ), 2 W qi 2 x 2 i (α δ)(1+λ). Thus, thetwofirst second order conditions are fulfilled. For the last condition, 2 W q 2 i 2 W x 2 i > = (1 + λ) and 2 W q i x i = ³ 2 W q ix i 2, to hold, we have to assume that 2+3λ > (α δ) 2 (1 + λ). 12

13 output and quality of hospital care. q w = x w = 1+λ 2+3λ + d (1 + 2λ)+2β (1 + λ) (α δ) 2 (1 + λ) (α δ)(1+λ) 2+3λ + d (1 + 2λ)+2β (1 + λ) (α δ) 2 (1 + λ) (14) (15) Note that the first best level of quality is proportional to the first best level of output in the same manner as in the Cournot-Nash equilibrium. Examination of the first best levels leads to the following proposition: Proposition 4 q w > 0. x w > 0 iff α > δ. Otherwise x w =0. Proof. The denominators of q w and x w are positive by the S.O.C. 2+3λ > (α δ) 2 (1 + λ) (see footnote 9). Then the proof follows straightforwardly from eq. (14) and (15). The social welfare is maximised for a positive level of hospital care produced. 9 However, this is not true with respect to the quality of care in all cases. If the marginal social cost of providing quality is larger than the marginal social benefit for all quality improvements above the minimum level (normalised to zero), then it is socially desirable to induce only this minimum level. Having derived the first best levels of hospital care, the next question is: Do the hospitals produce too little or too much hospital care? Comparing the Cournot-Nash equilibrium (without any fixed price per case) and the first best levels lead to the following proposition: Proposition 5 q w < (>) q c iff β < (>) β b 1 λd 1+λ. xw < (>) x c iff β < (>) β b 1 λd 1+λ and α > δ. Otherwisexw = x c =0. Proof. Let s =0. b β 1 λd 1+λ is found by setting qw = q c and solve for β. Aslongasα > δ, this applies to quality as well because x w =(α δ) q w and x c =(α δ) q c.otherwise,ifα δ, the nonnegativity constraint on (or minimum required level of) quality implies x w = x c =0. bβ is the degree of labour scarcity that implies the Cournot-Nash quality (in the case of α > δ) and output to be equal to the firstbestlevels. 9 For some health services the first best level could of course be zero. For exampel if the fixed cost of producing the service is very large, or if the marginal social benefit is smaller than the marginal social cost for all positive values of output. 13

14 This critical level depends on the competition intensity (d) and the size of the shadow cost of public funds (λ). As the hospital ³ competition intensifies, β the critical level of labour scarcity is reduced b = λ. The intuition d 1+λ is simply that harder competition induces the hospitals to produce more health care. This pushes the Cournot-Nash equilibrium up towards the first best levels. Then only a small labour scarcity effect induces the hospitals to produce too much output and quality. Second, if hospital reimbursement becomes³ socially more costly, this also reduces the critical degree of labour β scarcity b = 2+λ(1 d). The reason is that a higher shadow cost of public funds reduces the first best levels of output and quality relative to the λ (1+λ) 2 Cournot-Nash equilibrium. Then the degree of labour scarcity that induces the hospitals to produce too much, becomes lower. From the above proposition we may identify three different situations. First, if λ =0,thenβ b =1which implies q c <q w for all β. This means that if hospital reimbursement is not socially costly, the hospitals produce too little output and quality of health care in the Cournot-Nash equilibrium independent of the level of labour scarcity. Second, if λ 1 d,thenb β =0 which implies q c >q w for all β. This means that if the shadow cost of public funds is sufficiently large, the hospitals produce too much output and quality of health care in the Cournot-Nash equilibrium no matter how scarce the labour supply is. Notice that if hospital competition becomes softer (d 0), the critical shadow cost of public funds that induces overproduction becomes very high (and vice verca). Third, if 0 < λ < 1 d,then0 < β b < 1, which implies q c > (<) q w if β < (>) β. b In this case the hospitals may produce too much, too little or simply the first best level in the Cournot-Nash equilibrium dependent on the labour scarcity. Again the intensity of competition matters. If hospital competition is hard (d 0), the critical size of the shadow cost that induces overproduction becomes smaller. 5 Reimbursement policy Having stated that the hospitals may produce too little or too much health care in equilibrium, the following question arises: Is it possible for a regulator to implement the socially optimal levels of hospital output and quality in this type of industry? Assuming that the regulator may reimburse the hospitals by using a block grant (lump-sum) and a fixed price per case as specified in eq. (3), yields the following proposition. Proposition 6 There exists a fixed price per case s that solves q c = q w and x c = x w, and a block grant A that solves π (q c, x c,s )=π (q w, x w )=0. 14

15 Proof. By computation we find that q c = q w and x c = x w are solved for s 1 β λ (d + β) = 2+3λ + d (1 + 2λ)+ 2β (α δ) 2 (1 + λ) and π (q c, x c,s )=π (q w, x w )=0is solved for A (1 + λ)[1+λ (3 + 2d)+(1+λ)(2β α + δ)] = 2 2+3λ + d (1 + 2λ)+ 2β (α δ) 2 (1 + λ) 2 Thus, it is possible for a regulator to implement first best levels of output and quality in this hospital industry. The procedure is as follows: First, the regulator induces the efficient production level by setting the fixed price per case such that the Cournot-Nash output and quality are equal to the first best levels. Then, the regulator extracts the profit the hospitals earn at this production level, which is done by setting the lump-sum transfer (block grant) equal to this profit. The next question that raises is: How should the regulator set the fixed price and the block grant? Proposition 7 Optimal reimbursement policy implies s > 0 iff β < b β, s < 0 iff β > b β and s =0iff β = b β. Proof. We have that q w = q c and x w = x c are solved by s 1 β λ (d + β) = 2+3λ + d (1 + 2λ)+ 2β (α δ) 2 (1 + λ) (see proof of proposition 6). The denominator of s is positive from the S.O.C. 2+3λ > (α δ) 2 (1 + λ) (see footnote 9). Then the sign of s is determined by the numerator, which is positive (negative) when β < (>) b β 1 λd 1+λ. Optimal reimbursement policy for the regulator is to set a positive fixed price (subsidy) per case if the imperfect competition effect dominates. This is simply a restatement of the wellknown result that it may be socially desirable to subsidy an oligopoly (or a monopoly). However, a negative fixed price (tax) per case is optimal when the scarcity of labour or the shadow cost of public funds dominate the imperfect competition effect. A negative fixed price per case would turn the external cost of labour scarcity and/or the social cost of distortionary taxes into a private cost which the hospitals take into account in their maximisation problem. A severe labour scarcity increases the social cost of producing hospital care, but since this is an external cost, intensified competition leads to a too high production level of hospital care. 15

16 6 Concluding remarks This paper deals with how a regulator should design the reimbursement system when hospitals compete for patients and there is scarcity of health personnel. We consider a two-stage, complete information game where in stage 1 the government sets the level of the block grant and the fixed price in order to maximise social welfare. In stage 2, the hospitals independently and simultaneously choose output and quality of the health care in order to maximise profit. The main result of this paper, which has received little attention in the literature so far, is that there is a scope for negative fixed prices (taxation) within a prospective payment system. If there is sufficiently large scarcity of labour or hospital reimbursement is sufficiently costly, the hospitals provide too much output and quality of health care in the Cournot-Nash equilibrium. The intuition is that while the oligopoly effect induces the hospitals to produce too little output and quality compared to first best, the scarcity of labour and the shadow cost of public funds counteract this effect. If one or both of these two effects are sufficiently large, they dominate the oligopoly effect and thus push the Cournot-Nash equilibrium above the firstbestlev- els. In this case, we show that first best levels of output and quality are implemented by imposing a tax (negative fixedprice)onthehospitals. The second result, which contrary to the first has received a lot more attention in the literature, considers the question of whether prospective payment induces an optimal level of quality. Consistent with Kesteloot and Voet (1998), but contrary to several other papers (see Newhouse 1996), we find that output and quality of hospital care increases with the fixed price per case. However, we show that this is only the case when the costs of quality improvements can be covered through higher prices. If this is not the case, a higher prospective payment only induce the hospitals to produce more output, while keeping the quality at the minimum required level. First best levels of output and quality of hospital care are in fact attainable in this model. Prima facie, this result is surprising since the regulator only has two instruments (the fixedpricepercaseandtheblockgrant)to induce optimal levels of six variables (output, quality and profit for hospital 1 and 2). However, the result depends on two conditions: First, we only consider symmetric equilibrium levels of output and quality. Under more general assumptions like asymmetric demand or cost conditions the hospitals may produce different levels of output and quality in equilibrium. This would also be the case if hospital competition is a Stackelberg game. However, if we allow for idiosyncratic (hospital specific) reimbursement, prospective payment may implement first best levels even in an asymmetric equilibrium. Second, 16

17 and more crucial for the result, is the feature that quality is proportionally related to output (as long as the marginal willingness to pay for quality is sufficiently large). This follows from the specification of the demand and cost function. In general, the relation between output and quality may be negative as well as nonlinear. Then the first best levels of output and quality may not be possible to implement within a prospective payment system. However, these considerations are beyond the scope of this article and are suggested for further research. References [1] Bös, D., DeFraja, G Contracts for Health Services: Quality versus Excess Capacity. Discussion Paper No. A-578, University of Bonn. [2] Chalkley, M., Malcomson, J.M., Contracting for health services when patient demand does not reflect quality. Journal of Health Economics 17, [3] Danzon, P., Hospital profits : the effect of reimbursement policies. Journal of Health Economics 1, [4] Dranove, D., White, W.D., Recent theory and evidence on competition in hospital markets. Journal of Economics & Management Strategy 3, [5] Ellis, R.P., Creaming, skimping and dumping: provider competition on the intensive and extensive margins. Journal of Health Economics 17, [6] Glazer, J., McGuire, T.G., Payer competition and cost shifting in health care. Journal of Economics & Management Strategy 3, [7] Kessler, D.P., McClellan, M.B., Is hospital competition socially wasteful? NBER working paper [8] Kesteloot, K., Voet, N., Incentives for cooperation in quality improvement among hospitals - the impact of the reimbursement system. Journal of Health Economics 17, [9] Ma, C.A., Health care payment systems: cost and quality incentives. Journal of Economics & Management Strategy 3,

18 [10] Newhouse, J.P., Reimbursing health plans and health providers: Efficiency in production versus selection. Journal of Economic Literature 34, [11] OECD The reform of health care: A review of seventeen OECD countries. Health Policy Studies No. 5, Paris. [12] Pope, G., Hospital non-price competition and Medicare reimbursement policy. Journal of Health Economics 8, [13] Schleifer, A A Theory of Yardstick Competition. Rand Journal of Economics 16, 3: