Jacobs University Breen The Variance of Length of Stay and the Optial DRG Outlier Payents Stefan Felder Priorisierung in der Medizin FOR 655 Nr. 3 / 7 Capus Ring 1 8759 Breen Gerany www.jacobs-university.de FOR 655 Working Paper serves to disseinate the research results of work in progress prior to publication to encourage acadeic debate. Copyright reains with the authors.
Die Reihe Priorisierung in der Medizin ufasst Arbeits- und Forschungsberichte der DFG Forschergruppe FOR655 Priorisierung in der Medizin: eine theoretische und epirische Analyse unter besonderer Berücksichtigung der Gesetzlichen Krankenversicherung (GKV). Die Berichte und weitere Inforationen zu der Forschergruppe können abgerufen, werden unter: http://www.for655.de oder http://www.priorisierung-in-der-edizin.de The series Priorisierung in der Medizin consists of working papers and research reports of the DFG (Deutsche Forschungsgeeinschaft, i.e., Geran Research Foundation) Research Group FOR655 Priorisierung in der Medizin: eine theoretische und epirische Analyse unter besonderer Berücksichtigung der Gesetzlichen Krankenversicherung (GKV).(Prioritizing in Medicine: A Theoretical and Epirical Analysis in Consideration of the Public Health Insurance Syste) Reports and further inforation can be found at http://www.for655.de or http://www.priorisierung-in-der-edizin.de Ipressu: Capus Ring 1 8759 Breen Gerany www.jacobs-university.de ISSN 1866-9 www.for655.de www.priorisierung-in-der-edizin.de
The Variance of Length of Stay and the Optial DRG Outlier Payents Stefan Felder Otto-von-Guericke University Magdeburg Prospective payent schees in health care often include supply-side insurance for cost outliers. In hospital reiburseent, prospective payents for patient discharges, based on their classification into diagnosis related group (DRGs), are copleented by outlier payents for long stay patients. The outlier schee fixes the length of stay (LOS) threshold, constraining the profit risk of the hospitals. In ost DRG systes, this threshold increases with the standard deviation of the LOS distribution. The present paper addresses the adequacy of this DRG outlier threshold rule for risk-averse hospitals with preferences depending on the expected value and the variance of profits. It first shows that the optial threshold solves the hospital s tradeoff between higher profit risk and lower preiu loading payents. It then deonstrates for norally distributed truncated LOS that the optial outlier threshold generally decreases with an increase in the standard deviation. The intuition for this result is that a higher variance increases the profit risk, which in turn leads hospitals to insure a larger part of the LOS distribution. JEL Index: G, I11 Keywords: Optial outlier DRG payents, supply-side insurance in health care, stop loss insurance Prof. Dr. Stefan Felder Institute of Social Medicine and Health Econoics (ISMHE) Leipziger Str. 44 D-391 Magdeburg phone: 391-67431 e-ail: stefan.felder@ishe.de FOR655 Nr. 3 / 7
The Variance of Lenth of Stay and the Optial DRG Outlier Payents 1. Introduction In the id-eighties, US Medicare introduced the prospective payent syste, under which hospital reiburseents for patient discharges are based on their classification into diagnosis related groups (DRGs). Prospective payents replaced the old cost-based reiburseent syste, which depended on a patient s length of stay (LOS). The change fro retro- to prospective payents transferred the loss risk fro the insurers to the providers and gave the latter an incentive to econoize patient treatent costs. Nevertheless, Medicare retained part of the forer syste by introducing outlier payents for long stays. Hospitals could charge the costs of treatent based on the actual LOS for patients staying longer than a stated LOS outlier threshold, while pure prospective payents applied to patients discharged within the LOS threshold. The new schee resebled an insurance contract with a deductible, as the hospital insures only the part of the LOS distribution beyond the threshold. In the last twenty years, ost industrialized countries followed US Medicare and introduced DRGs or siilar grouping systes for the reiburseent of inpatient services, copleented by outlier payents to share the risk of treatent costs between hospitals and insurers. Two different ethods are applied to define outliers. Soe countries calculate the outlier threshold by adding or 3 standard deviations to the ean of LOS (Gerany, Spain and the early US-Medicare syste). Other countries use a nonparaetric outlier ethod that is based on the inter-quartile range of LOS, which is ultiplied by a factor of 1.5 and added to the third quartile (England, Italy and Denark). France cobines the paraetric and the non-paraetric ethods to define the LOS outlier threshold. 1 The health econoics literature dealt with insurance aspects of DRG outlier payents early on when Medicare introduced its prospective reiburseent syste. Ellis and McGuire (1988) applied Arrow s principle, which states that full insurance after a deductible is the optial structure of insurance, to hospital reiburseent. They show that outlier payents should be based on the hospital s average loss per case rather than on individual case-level losses, since the hospital itself can pool the loss risk of individual cases to soe extent. Keeler et al. (1988), cobining an optial deductible with coinsurance on the arginal costs of expensive cases to reduce oral hazard, cae to the sae conclusion. Outlier payents serve as an insurance schee for hospitals against excessive losses and they itigate probles of access and underprovision of care for the patients in need of costly treatent. Keeler et al. also studied the optial policy for paying ore than one DRG when the outlier payents have to be ade case by case. Provided that the hospital s utility is quadratic, they show that the optial schee includes deductibles that are the sae for all DRGs if there are no coinsurance restrictions and there is a stop equal average loss policy for each DRG per outlier under a constant coinsurance rate. Since, with concave utility, the arginal value of oney is higher when losses are greater, the 1 See Schreyögg et al. (6) for details on the outlier threshold in European countries FOR655 Nr. 3 / 7
Stefan Felder optial outlier payent policy is to equalize the expected loss of each DRG by adjusting the deductible correspondingly. This paper s focus is on the relationship between the LOS standard deviation and the optial outlier threshold. Section presents the optial risk sharing between the hospital and the insurer. The existence of a positive threshold arises since we assue loading on the net insurance preiu and risk aversion on the part of the hospitals. We derive the optial outlier threshold for hospitals that exhibit ean-variance preferences over lotteries, and give coparative-static results with respect to the degree of risk aversion, the loading factor and the costs of stay per die. Section 3 deals with the adequacy of LOS outlier rules. We paraeterize LOS randoness, assuing that LOS is norally distributed and truncated fro below at zero. The optial outlier threshold is shown to increase with the LOS standard deviation provided that the latter increases the hospital s arginal profit risk. This condition cannot be proven to be fulfilled in general. However, the condition holds for the LOS distributions observed in the Geran hospital syste. Section 4 discusses the results and section 5 concludes that the optial outlier threshold decreases with an increase in the standard deviation of LOS.. Optial risk sharing between the hospital and the insurer To begin with, we set the costs per die of a hospital stay equal to one. The insurer is assued to pay the hospital depending on a patient s LOS t according to the following rule: if l F t < t if t, where is the outlier threshold ( > ), l( ) = tf ( t) dt and (1) f t is the density function of the LOS, with f () tdt= 1 and f ( t). With F() t f () t dt cuulative density function, F ( ) is the share of cases and l( ) t = as the F is the average LOS in the lower part of the distribution (LOS up to the threshold ). The reiburseent schee (1) includes an insurance contract covering the LOS beyond the outlier threshold. Assuing that the insurer loads the insurance preiu by the factor λ ( < λ < 1), with zero-profits, the insurer s per patient preiu aounts to ( 1 λ ) p = l + + h, () FOR655 Nr. 3 / 7 3
The Variance of Lenth of Stay and the Optial DRG Outlier Payents tf t dt where h( ) = and h( ) ( 1 F( ) ) is the average LOS in the higher part of the distribution, i.e. in the insured part of the LOS distribution. The hospital s expected profits per patient equal the difference between the expected reiburseent per patient and the per patient preiu (): μ π ( ) = l( ) + h( ) p( ) = λl( ). The variance of profits is zero in the insured part of the LOS distribution. Given (1), the variance of the expected profits per patient, thus, aounts to: σ π () ( ) (). (4) = l t f t dt = t l f t dt Preferences of the hospital are assued to follow the ean-variance criterion, i.e. hospitals axiize V ( μπ, σ π ). As Meyer (1987) and Sinn (1983) independently showed, this specification is a perfect substitute for the ore standard expected utility approach when one restricts attention to the linear distribution classes of rando variables. This restriction is not crucial in our odel as LOS is the only rando variable. For siplicity, we assue that the hospital s utility function takes the for r V ( μ ) π, σπ = μπ σπ ( ), (5) where r is a constant representing the hospital s degree of absolute risk aversion ( r > ). (3) Since σ is a onotonic transforation of σ, it is clear that (, ) the sae qualitative results. μ σ preferences would lead to 4 FOR655 Nr. 3 / 7
Stefan Felder Inserting (3) and (4) into (5), we can rewrite the hospital s utility as a function of the LOS threshold: r V ( ) = λl( ) ( t l( ) ) f () t dt. (6) Using the Leibniz rule and noting that h( ) = l( ) = f ( ) holds, we derive for a arginal increase in the threshold: V r = λf ( ) ( l( ) ) f ( ) + ( t l( ) )( f ( ) ) f () t dt r (7) = λf ( ) f ( ) ( l( ) ) tf () t dt l( ) f () t dt r ( l( ) ) = f ( ) λ l ( ) ( 1 F ( ) ). An increase in the threshold, on the one hand, reduces the preiu by f ( ) λ, which, in turn, increases utility. On the other hand, it increases the profit risk by f ( ) l( ) l 1 F, which lowers utility. Let ( ) ( l( ) ) k l 1 F (8) easure the change in the profit risk when the LOS threshold arginally increases, and rewrite (7) as: r V ( ) = f ( ) λ k( ). (9) This equation illustrates the two key factors governing the tradeoff for an increased threshold: the loading factor l, which deterines the benefits, and the additional profit risk k( ), which captures the costs. The optial threshold balances the two opposing effects, giving rise to: λ k( ) = (1) r in the optiu. FOR655 Nr. 3 / 7 5
The Variance of Lenth of Stay and the Optial DRG Outlier Payents k > since λ, r >. Utility is axiized provided the second-order condition V < holds. For =, Assuing that a solution exists, (1) requires that λ = ( r ) k( ) and thus ( ) λ ( ) that k( ) f,,,r V = f r k. It follows > in the utility axiu, given λ >. Proposition 1: The optial threshold i) decreases with an increase in the degree of risk aversion r and ii) increases with an increase in the loading factor λ. Proof: i) For infinitesially sall changes in and r, it holds around the axiu that: ( λ ) k d= r r dr, or d λ r = < dr k ii) As ( λ r) 1 λ k >., since dλ = dλ >, it follows that r d r dλ = k >. With higher loading, the optial LOS threshold increases. In other words, the hospital opts for a lower insurance coverage when the preiu becoes ore expensive. Furtherore, the threshold decreases when the degree of risk aversion increases. A riskneutral hospital ( r = ), by coparison, would not choose any insurance at all (i.e. = ). We have noralized the costs of a stay per die to one. If the costs per die are β, the effect of an increase in the threshold on expected utility becoes EU = f ( ) β( λ ( r ) βk( ) ) (see (8) and (9)). The optial threshold, thus, requires: λ k( ) =. (11) βr Hence, as with the degree of risk aversion, we can state Proposition : The optial threshold decreases with an increase in the costs per die β. Most DRG outlier threshold rules do not incorporate the costs of treatent. According to proposition, the optial LOS threshold should, ceteris paribus, decrease with the cost of treatent, reflecting the corresponding increase in the hospital s profit risk. 6 FOR655 Nr. 3 / 7
Stefan Felder 3. LOS standard deviation and the optial threshold As entioned in the introduction, DRG systes use properties of the observed LOS distribution to deterine the outlier threshold. A ore dispersed LOS distribution leads to a higher outlier threshold in both paraetric and non-paraetric calculations of the threshold. In the following, we concentrate on paraetric distributions and assue a norally distributed LOS according to: t = μ + σε, with E ( ε ) =, Var ( ε ) = 1, (1) where σ is the exogenous standard deviation. The density function of the noral distribution is () f t 1 = e σ π 1 = e σ π (( t μ) σ) ε. Since LOS is non-negative, we consider the noral distribution truncated below at point. The density function of the truncated distribution writes:, - t f () t = f () t, t, 1 F ( ) where F( ) = f ( t) dt and Using (1)-(14), we find: f t is defined as in (13). ( F ) ( a) σ 1 ε l ( ) = tf ( t) dt = ( μ + σε ) e dε π 1 ( F ) a σ ( ) σ ( ) 1 F ( ) (13) (14) a σ ( a) σ ( μ) σ ε ε e d e (15) σ 1 = μ ε σ π 1 a μ F F f f =. For the non-truncated distribution, the truncation is at. Since F f l = l = μf σ f. ( ) = ( ) =, it follows that FOR655 Nr. 3 / 7 7
The Variance of Lenth of Stay and the Optial DRG Outlier Payents With the truncated distribution, the optial LOS threshold (see (8) and (1)) changes slightly to: ( l ( )) λ k( ) l ( ) 1 F( ) + F( ) =. (16) r Proposition 3: The optial threshold decreases with an increase in the LOS standard deviation, provided the latter increases the arginal profit risk. d k( ) σ Proof: At the optial threshold, = holds. Since k dσ k ( ) >, d k it follows that sign = sign dσ. σ Fro (16), we derive the effect of a change in the standard deviation on the hospital s additional risk at the LOS threshold: ( ) k F F l = l( ) ( l( ) F( ) + F( ) ) σ σ σ σ (17) This equation reveals two effects of an increase in σ on the hospital s additional risk. First, a higher spread of the distribution changes the ass of LOS in the interval where the hospital bears the risk. Secondly, the increase in σ also changes the average LOS in the uninsured part of the distribution, which will lead to a change in the DRG payent. As we will see below, for the non-truncated distribution the two effects have opposing tendencies, as one would expect, while for the truncated distribution the second effect cannot be signed. For the three derivatives in (17), we first find fro (15): F F f f F + μ σ f f σ l l ( ) σ σ σ σ σ σ = Moreover, it holds that: F() t = f t σ () 1 F (18) t μ and (19) σ f () t () (( t μ ) σ ) 1 = f t. () σ σ Inserting these equations into (18) leads to: 8 FOR655 Nr. 3 / 7
Stefan Felder ( ) f ( ) σ + ( μ) + f σ + μl ( ) = σ σ 1 F ( ) l. (1) For the first effect of an increase in the LOS standard deviation, we find fro (19): ( ) F F μ μ = f ( ) f ( ). () σ σ σ σ Given > μ, this difference is negative, i.e. the ass of the distribution that falls into the uninsured part decreases when the standard deviation of the distribution increases. This, in turn, decreases the hospital s risk, which would, then, indicate that the LOS threshold should increase. The second effect, however, tends to be in the opposite direction. First, we observe l F F l,f,f 1, given > μ. + > as Hence, the sign of the second effect has the opposite sign of E σ. For the nontruncated distribution, we find l ( ) σ < fro (1). The saller average LOS in the uninsured part lowers the DRG-payent and increases the profit risk. This can best be seen when one considers the hospital s axial loss of treating a patient. It equals the difference between the threshold and the DRG payent: l ( ). This axial possible loss increases when l ( ) decreases. On the other hand, the axial possible gain of treating a patient increases when l ( ) oves toward zero. Altogether, this indicates that the hospital s profit risk increases. While this is true for the non-truncated distribution, we cannot be sure for the truncated distribution as the second effect cannot be signed here. It is forally not possible to sign the total effect of an increase in the LOS standard deviation on the hospital s arginal risk even when the distribution is non-truncated. Hence, we have to depend on siulations based on the factual distribution of LOS to evaluate the total effect. Table 1 presents characteristics of the top 3 Geran DRGs in 5, calculated fro a saple of roughly 7, hospital cases in the state of Saxony-Anhalt. The average LOS is 7.9 days and its average standard deviation 4.1 days. The next colun shows the actual threshold for the individual DRGs as published in 5. On average, the actual threshold is 15.86 days, which is close to two standard deviations above the ean. When we calculate the threshold according to Geran DRG outlier ethodology ( ( ( ( ( = exp( μ + σ ), where μ and σ are the ean and standard deviation of the log of LOS, respectively), the values in the fifth colun are obtained, the average being days. FOR655 Nr. 3 / 7 9
The Variance of Lenth of Stay and the Optial DRG Outlier Payents Table 1: The top 3 Geran DRGs (5) DRG Mean (μ) LOS LOS threshold The change of profit risk Std. dev. actual based on Based on % Based on (σ) ( % ) () k % σ k B7B 11.83 6.5 3 9..5 F6B 1.47 7.14 6 39.7.1 C8Z 3.1 1.88 5 6.79.37 G67C 4.19 3.19 8 13.88.1 F67B 6.13 3.48 1 19.33.1 I68B 9.3 5.1 3 6.8.3 O6C 3.93.1 7 1.31.5 F6C 1. 5.51 3.17.1 E71B 5.96 5.4 16.47.6 G6B.91.98 1 9.34.63 B69B 7.39 4. 15 1.3. E77C 8.98 4.98 17 7.34.1 G49Z 1.61.7 n.d 3 n.d.7 E77B 1.5 7.9 4 4.41.1 I44Z 14.9 3.99 5 -.4 -.1 G54Z 6.9 3.8 14 16.5.1 B8Z.96.59 6 8 1..51 F66B 5.83 4. 13.43. D3Z 6.6.75 11 13.13.5 E65B 8.91 5.5 19 7.3.1 F49C 1.83.53 n.d 3 n.d -.7 D63Z 4.83 3.16 9 14.66.9 G4Z 6.89 4.93 1 17.9.35 I48Z 14.6 4.3 5 3 -.7 -.18 I69Z 9.65 5.7 4 35.13. B76D 5.11 5.6 14 1.95.13 LZ 7. 4.71 13 18.71.4 F6D 8. 4.69 19 5.16.1 G48Z 9.81 5.96 5.38.11 G67B 6.5 4.48 1 19.77.11 Mean 7.9 4.19 15.86 19.97.41.9 Max 14.6 7.9 6 4 1..63 Min 1.61.53 5 3 -.7 -.1 σ Source: own calculation based on 68.977 DRG cases of AOK patients in Saxony-Anhalt, 5. 1 FOR655 Nr. 3 / 7
Stefan Felder We can evaluate the DRG threshold rule at the given thresholds and study whether the arginal risk increases with an increase in the standard deviation. The result of this is presented in two final coluns of Table 1, presenting the values of k / σ for the current and the calculated Geran thresholds. Notice that, except for two DRGs, the sign of the derivative is always positive. Thus, in alost all cases, the hospital arginal risk increases, so that a risk-averse hospital would like to extend insurance coverage, i.e. to lower the LOS threshold when the standard deviation increases. Figure 1: The effect of arginal profit risk fro an increase in σ for different μ and σ; truncated (tr) and non-truncated (non-tr) noral distribution 1.5 Effect on arginal risk 1..9.6.3. 3 5 7 9 11 13 15 threshold 17 19 1 3 5 μ= 3; σ=; tr μ= 6; σ=4 - tr μ=1; σ=6 - tr μ= 3; σ=; non-tr μ= 6; σ=4 - non-tr μ=1; σ=6 - non-tr As an alternative test of the Geran outlier payent syste, one can calculate the value of the derivative (17) for truncated and non-truncated LOS as a function of the threshold for given distribution paraeters. Figure 1 shows these derivatives for three stylized (μ, σ) pairs. 3 For the non-truncated distributions, the derivative is positive, declining to zero for large thresholds. For both the truncated and the non-truncated distributions, the 3 If we interpret the observed ean as the ean of the truncated distribution ( E ), the ean of the non-truncated (μ) can be derived using (15): μ = E σ f ( ) ( 1 F( ) ). If we eploy μ to calculate the derivative (17), the results regarding the sign of (17) reain unchanged. FOR655 Nr. 3 / 7 11
The Variance of Lenth of Stay and the Optial DRG Outlier Payents derivative is positive over the whole range of thresholds. When the threshold is close to the ean, an increase in the threshold raises the arginal profit risk. The axiu is attained within one to three days, depending on the LOS distribution characteristics. For larger thresholds, the effect, driven by the density function, quickly converges to zero. Again, we find a positive effect on the arginal profit risk, indicating that a larger LOS standard deviation should lead to a saller outlier threshold, contradicting the outlier threshold rule. 4. Discussion Since a hospital can influence a patient s LOS, the assuption of an exogenous LOS is soewhat unrealistic. Let us consider an endogenous LOS. The insurance coverage will then affect the hospital s choice of a patient s LOS. Beyond the threshold, reducing the Δ l lowers preiu loading payents and increases per patient patient s LOS by profit by Δ l( ) λ, while within the threshold a reduction of the LOS by Δ l( ) translates one to one into higher profits. In other words, insurance coverage dilutes the incentive to reduce the LOS. The optial policy then needs to address the tradeoff between risk spreading and appropriate incentives (Zeckauser, 1979), which can be solved by cobining a threshold with a coinsurance on the arginal costs of stay beyond the threshold. DRG systes reflect this, as outlier days are usually reibursed with a 4% rebate on the average cost per die. An endogenization of the LOS will not necessarily change the coparative statics of the optial threshold. Consider a siple case where efforts to reduce the patients LOS only shift the distribution to the left without changing its shape. In this case, the productivity of efforts is independent of the initial LOS. Abstracting fro the discontinuity at the threshold, efforts will have no effect on the LOS variance. Consequently, although the optial threshold will increase in order to give incentive for cost reduction for a larger range of the LOS distribution, the qualitative relationship between the standard deviation and the optial threshold will not change. A higher LOS variance will, ceteris paribus, be accopanied by a decrease in the optial threshold. The productivity of efforts to reduce the LOS ay, however, increase with the initial LOS, so that these efforts will have an effect on the LOS variance. Still, it appears that the following hierarchy of instruents would apply in this case. Moral hazard can be restrained using a non-linear coinsurance rate that increases with the observed LOS to give a stronger incentive to reduce the LOS when it is less expensive to do so. A higher LOS variance will have no effect on the efforts to reduce the LOS. The optial threshold will reflect these efforts but the coparative statics regarding the variance of LOS will not change its qualitative nature. Proving this conjecture would be difficult, given the discontinuity of the LOS distribution arising when efforts to reduce the LOS are introduced. 1 FOR655 Nr. 3 / 7
Stefan Felder Ellis and McGuire (1988) criticize the existing outlier payents based on individual cases and propose an insurance schee based on the average case. Risk pooling within the hospital will reduce the variance of the profit per patient and, thus, decrease insurance deand. This qualification, however, does not affect the optial threshold rule. A hospital which shoulders a higher risk due to a large case-ix index, ceteris paribus, will deand a lower threshold copared to a hospital with a lower case-ix load. The new generation of outlier payent systes in the USA is no longer based on the LOS, but on the patients costs of stay. 4 This reflects the epirical observation that, after controlling for DRG, the costs of stay are only weakly related to the LOS aong very long stay cases (see Keeler et al., 1988). Interestingly, the cost outlier schees do not define the thresholds as a function of the variance. Rather, a cost-to-charge factor deterines the threshold. In this case, a ean-preserving increase in the standard deviation will not affect the threshold. This rule is better than the forer one, which set the threshold two standard deviations above the ean. Australian outlier payents do not depend on paraetric distribution, being based on the arguent that the LOS is not norally distributed. 5 The threshold, called the high tri point, is often or 3 ties the average length of stay. Like the US Medicare cost outlier, this schee appears to doinate the original threshold rule, as it does not further aggravate the hospitals profit risk by increasing the threshold when the LOS standard deviation increases. Ma (1994) has analyzed payents systes designed to restore cost and quality incentives. When the provider can refuse expensive patients, a piecewise linear reiburseent rule arises. While prospective reiburseent applies to low cost treatent, cost reiburseent is the optial rule for expensive patients. This result points to another function of the outlier threshold, viz. the prevention of patient duping, which can occur when reiburseent is purely prospective. 5. Conclusion Prospective payent schees in health care often include supply-side insurance for cost outliers. In the early US Medicare and any current European DRG systes, the outlier schee fixes a length of stay (LOS) threshold, constraining the profit risk for the provider. This threshold increases with the standard deviation of the LOS distribution. The present paper addresses the adequacy of this outlier threshold rule for risk-averse hospitals with preferences depending on the expected value and the variance of profits. 4 See for instance, Departent of Health and Huan Services, Center for Medicare and Medicaid Services, 4 CFR Part 41, Federal Register, Vol. 68, No. 43, March 5, 3, p. 14-149. 5 For New South Wales, see http://www.health.nsw.gov.au/, for Victoria, see http://www.health.vic.gov.au/pfg/, and for South Australia, see http://www.health.sa.gov.au/. FOR655 Nr. 3 / 7 13
The Variance of Lenth of Stay and the Optial DRG Outlier Payents It first shows that the optial threshold solves a hospital s tradeoff between higher profit risk and lower preiu loading payents. A coparative static analysis reveals that the optial outlier threshold decreases with an increase in the hospital s degree of absolute risk aversion as well as with an increase in the per die cost of treatent. Higher treatent costs increase the arginal profit risk, iplying that a hospital will increase insurance coverage. With a given risk, a ore risk-averse hospital will also want to extend coverage, i.e. to lower the LOS outlier threshold. We then paraeterize the LOS distribution, assuing a truncated norally distributed LOS. An increase in the standard deviation has two effects. A larger spread first decreases the share of the distribution that belongs to the uninsured part of the distribution. Hence, this tends to decrease the hospital s profit risk. On the other hand, a larger spread decreases the average LOS of patients in the uninsured part, which lowers the DRG payent. Hence, with an unchanged threshold the hospital s risk increases. The su of the two opposite effects cannot be signed, so one has to depend on siulations to address the adequacy of the outlier threshold rule. The siulation results using Geran hospitals discharge data show that the hospitals arginal profit risk is larger for DRGs with a high standard deviation than for DRGs with a low standard deviation of LOS. We conclude that the optial threshold of a DRG should decrease with an increase in the LOS standard deviation. Author note I a grateful to Claudia Heinecke for technical assistance. The paper was presented at the annual conference of the health econoists group within the Verein für Socialpolitik in München, October 1-13, 7. I thank the referee, Matthias Staat, and the other participants for helpful coents. 6. References Ellis, R. P. and Th. G. McGuire (1988), Insurance principles and the design of prospective payent systes, Journal of Health Econoics 7, 15-37. Keeler, E. B., F. M. Carter and S. Trude (1988), Insurance aspects of DRG outlier payents, Journal of Health Econoics 7, 193-14. Ma, C.-T. A. (1994), Health care payent systes: cost and quality incentives, Journal of Econoics & Managdeent Strategy 3, 93-11. Meyer, J. (1987), Two-oent decision odels and expected utility, Aerican Econoic Review 77, 41-43. Schreyögg, J., T, Stargardt, O. Tieann, and R. Busse (6), Methods to deterine reiburseent rates for diagnosis related Groups (DRG): A coparison of nine European countries, Health Care Manageent Sciences 9, 15-3. 14 FOR655 Nr. 3 / 7
Stefan Felder Sinn, H. W. (1983), Econoic decision under uncertainty, Second English edition. Asterda et al.: North-Holland Publishing Copany. Zeckhauser, R. (197), Medical insurance: a case study of the tradeoff between risk spreading and appropriate incentives, Journal of Econoic Theory,, 1-6. FOR655 Nr. 3 / 7 15
Working Paper Series FOR 655 1. Hartut Kliet: Priority setting in the age of genoics, Deceber 7 (1). Marlies Ahlert: If not only nubers count allocation of equal chances, Deceber 7 () 3. Stefan Felder: The variance of length of stay and the optial DRG outlier payents, Deceber 7 (3) 4. Jeannette Winkelhage, Adele Diederich, Sione Heil, Petra Lietz, Felix Schitz-Justen, Margrit Schreier: Qualitative Stakeholder-Interviews: Entwicklung eines Interviewleitfadens zur Erfassung von Prioritäten in der edizinischen Versorgung, Deceber 7 (4)