Setting health plan premiums to ensure efficient quality in health care: minimum variance optimal risk adjustment

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1 Journal of Publc Economcs 84 (2002) locate/ econbase Settng health plan premums to ensure effcent qualty n health care: mnmum varance optmal rsk adustment Jacob Glazer *, Thomas G. McGure a, b a Tel Avv Unversty, Faculty of Management, Tel Avv, Israel b Harvard Medcal School, Boston, USA Abstract Rsk adustment refers to the practce of payng health plans a premum per person (or per famly) based on a formula usng rsk adusters, such as age or gender, and weghts on those adusters. One role of rsk adustment s to make sure plans have an ncentve to accept all potental enrollees. Another role, at least as mportant n our vew, s to lead health plans to choose the effcent level of qualty of care for the varous servces they offer. Most of the research and polcy lterature on rsk adustment focuses on the frst problem. Ths paper proposes a new way to calculate weghts n a rsk adustment formula that contends wth both problems. For a gven set of adusters, we dentfy the weghts that mnmze the varance n plan predctable health care costs that are not explaned by rsk adustment (addressng the access problem), subect to the payments satsfyng condtons for an optmal rsk aduster (makng sure plans provde the effcent qualty). We call the formula mnmum varance optmal rsk adustment (MVORA) Elsever Scence B.V. All rghts reserved. 1. Introducton Managed health care plans nsurance and servce organzatons responsble for provdng medcally necessary health care enroll about 75 percent of the U.S. populaton and large shares of European populatons. These health plans receve *Correspondng author. E-mal address: glazer@post.tau.ac.l (J. Glazer) / 02/ $ see front matter 2002 Elsever Scence B.V. All rghts reserved. PII: S (01)

2 154 J. Glazer, T.G. McGure / Journal of Publc Economcs 84 (2002) ndvdual or famly premums, pad by governments (e.g. Medcare or Medcad n the U.S.), or by a combnaton of employers and consumers n U.S. employmentbased health nsurance. One of the maor concerns wth the health nsurance/ managed care health care polcy s adverse selecton (Enthoven, 1993). Generally, plans may take actons to dscourage or encourage potental enrollees. For one thng, they may refuse some applcants, although overt actons to dscourage ndvduals are normally prohbted and may be readly montored. More troublesome s that plans may dstort the mx of the qualty of health care they offer to dscourage hgh-cost persons from onng the plan. As a number of papers have observed, decsons about what care s medcally necessary are fundamentally outsde the scope of drect regulaton (Mller and Luft, 1997; Newhouse, 1996). In ths paper, we consder how rsk adustment of premums pad to health plans can 1 address what we refer to as the ndvdual access problem and the qualty problem. Rsk adustment refers to the practce of payng health plans a premum for each person or famly based on a formula usng rsk aduster varables, such as age, gender, or ndcators of pror health care use, and weghts on those adusters. Rsk adustment researchers have sought to fnd the formula that does the best ob of fttng the dstrbuton of health care costs n a populaton. Rsk aduster varables are chosen that are lkely to be related to costs, and weghts on those varables are chosen by regresson technques to mnmze the sum of the squared resduals, a 2 practce we wll refer to here as conventonal rsk adustment. The ntutve appeal of regresson coeffcents as weghts on rsk adusters derves from a desre to address the ndvdual access problem. Matchng payments wth ndvdual costs as closely as possble may dscourage plans from denyng membershp to some and aggressvely recrutng others. When the problem beng addressed s plans dstorton of the qualty of servces 1 A comprehensve dscusson of health nsurance premums and rsk adustment would nclude analyss of the settng of the premums charged to enrollees. Except n some ndvdual nsurance markets, ndvduals receve some subsdy of ths premum. In the case of government programs n the U.S. and some European countres, ths subsdy s complete and the person or famly pays nothng. In the employment-related health nsurance context n the U.S., the employer pays a porton of the premum. Authors have consdered how to set the consumer contrbuton n order to gve consumers the ncentves to on the rght plans. Ths ssue can also be consdered wthn an adverse selecton framework and may nvolve rsk adustng consumer premum contrbutons. The most mmedate effect of the person-pad component of premums s to nfluence the consumer s choce of plan. In ths paper, as wll be made clear n the next secton, we wll analyze a stuaton n whch the consumer s contrbuton s zero. Ths s emprcally accurate for many people n the U.S., ncludng a large share of the employed populaton, as well as the entre populaton enrollng n managed health care n Medcare and Medcad. It s also common n Europe. 2 Age and gender are commonly used n regresson formulae, but alone explan only about 1% of the varance n actual costs. Research on mprovng the explanatory power of these regressons has focussed on usng sophstcated clncal algorthms to defne new varables based on the dagnoses from pror health care use. Researchers have been able to run regressons explanng up to 10% of the varance n actual costs, but rsk adustment systems presently n use explan much less. See van de Ven and Ells (2000) for a revew of the emprcal lterature. Keenan et al. (2001) descrbes rsk adustment practces by maor publc and prvate payers n the U.S.

3 J. Glazer, T.G. McGure / Journal of Publc Economcs 84 (2002) to affect the decsons of groups of potental applcants, however, the regressonbased approach lacks ntutve foundaton. Fgurng rsk adustment weghts to mtgate qualty dstortons requres a concepton of how health plans set the qualty of servces, and a concepton of what optmal qualty means n ths context. In ths paper we propose an approach to dervng weghts on rsk aduster varables that begns wth an explct statement of health plan behavor n a market wth heterogenety n the demand for health care servces, and an explct statement of the economc effcency problem. Wth two effcency problems n vew, ndvdual access and qualty dstortons, and one polcy nstrument, the formula for the premum pad to plans, t s natural to expect there to be some trade-off n meetng the two competng obectves. Although t has nspred most emprcal rsk adustment research, ndvdual access to health plans s not the maor socal effcency problem n the health plan market. In the U.S., Europe, Israel and Latn Amerca, governments and employers requre contractng plans to offer perodc open enrollments. Durng open enrollment perods (once a year for most U.S. employers, and every month n Medcare), plans must accept any applcant. It seems clear that open enrollment regulaton works well to ensure access. Smulaton research demonstrates that exstng rsk adustment systems leave some ndvduals as bg losers or wnners (Chapman, 1997; Shen and Ells, 2000), but there s no evdence that plans act to deny membershp to ndvduals. By way of contrast, qualty of care n managed care health plans s the maor polcy focus n the U.S. and elsewhere. We beleve that the qualty dstorton problem s more mportant and less amenable to other regulatory solutons than the ndvdual access problem, and s therefore the proper prmary target of rsk adustment polcy. Therefore, n the analyss below, when t comes to a tradeoff between polcy obectves, we put more weght on mantanng qualty than on ndvdual access. Plans ncentves to overprovde the qualty of some servces and underprovde the qualty of others derves from the dstrbuton of health care demands n the populaton. Wth data about the dstrbuton of those demands, the regulator can antcpate plans ncentves, and mpose a system of correctve taxes and subsdes at the person level, usng the many observable varables avalable for each ndvdual. Wthn ths framework we derve a smple rule that characterzes optmal rsk adustment: the relaton between the covarance between spendng on a servce and the rsk adusted premum and the covarance between spendng on that servce and the sum of spendng on all servces must be the same for all servces that the plan provdes. In general, there wll be many combnatons of weghts on rsk aduster varables that satsfy the rule for optmal rsk adustment. By choosng from among the set of optmal rsk adustment weghts the combnaton that leads to the best ft of premums to costs, we ntegrate our approach wth tradtonal regresson-based methods, n effect desgnng a rsk adustment formula that contends wth the two forms of adverse selecton problems dentfed above. We wll refer to our proposed method for rsk adustment as mnmum varance optmal rsk adustment (MVORA), optmal

4 156 J. Glazer, T.G. McGure / Journal of Publc Economcs 84 (2002) because t solves the qualty dstorton problem, and mnmum varance because t does so n a way that mnmzes the sum of the squares of the devatons between premums and costs at the ndvdual level. In lne wth the lterature on rsk adustment, we argue that ths statstcal crteron s motvated by a concern for ndvdual access. Our paper s related to several lnes of research n health and publc economcs. Many papers n health economcs are concerned wth effcency problems due to selecton n nsurance markets (see Cutler and Zeckhauser, 2000; Encnosa, 1999 and van de Ven and Ells, 2000 for revews). Researchers have long been aware that one sgnfcant adverse selecton-related problem assocated wth competton among health plans s servce competton to attract the good rsks/deter the bad (Newhouse, 1996). Lttle research has been done, however, on the mplcatons of ths concern for a rsk adustment formula. So far, rsk adustment of ths type has only been characterzed n smple cases. Glazer and McGure (2000) consder the two-servce case and one rsk aduster, and show that the best rsk adustment weghts are the soluton to a par of equatons. The weghts obtaned by ths procedure are generally dfferent from those obtaned from regresson coeffcents. From the perspectve of publc economcs, t s natural to vew rsk adustment as queston of optmal taxes and subsdes on the prces pad for health nsurance, a framework adopted n several papers (Glazer and McGure, 2000; Neudeck and Podczeck, 1996; Selden, 1999). By explct characterzaton of the dstortons emergng n markets for health plans, we are able to dentfy, Pgouvan fashon, the set of taxes and subsdes necessary to algn prvate ncentves to maxmze proft wth the socal obectve of producton of the effcent qualty of health care. In the context of rsk adustment, the taxes and subsdes to correct ncentves work n a unusual fashon. Normally, a tax or subsdy apples drectly to the actvty ntended to be affected: for example, a tax on the volume of polluton s ntended to reduce polluton. Obvously though, there are other ways to tax/ subsdze to ht a partcular polluton target. If the actual level of polluton were not verfable, for example, a regulator famlar wth the effect of polluton (on, say water qualty) could put a tax on ths effect n order to manpulate the frm nto choosng the target level of polluton. The dea n ths paper s the same. Qualty provded by health plans s usually not verfable and, hence, cannot be taxed or subsdzed drectly. However, the regulator can magnfy or dmnsh the revenue consequences to the health plan s qualty choces by choce of the rsk adustment formula. If older people value and on a plan n response to good care for cardac problems and the regulator s concerned that the qualty offered for these dseases 3 s too low, the regulator can nduce hgher qualty by subsdzng older people. 3 The health payment lterature has used ths nsght prevously, as n Rogerson (1994) where the optmal hosptal per dscharge payment was fgured n order to nduce the desred non-verfable qualty. In ths case, qualty s one-dmensonal, and the level of the payment was the only nstrument necessary. Another early paper along the same lnes s Ma (1994).

5 J. Glazer, T.G. McGure / Journal of Publc Economcs 84 (2002) A central nsght of ths paper s that there s an nstruments/targets feature of rsk adustment polcy. A plan sets qualty for many servces. The regulator has many varables n a rsk adustment formula. When the regulator has more nstruments than t needs, the qualty problem can be addressed. The fnal choce of rsk adustment formula can then be made n order to contend also wth ndvdual access. The paper s organzed as follows: Secton 2 presents our basc model of health plan behavor n an envronment wth no uncertanty and symmetrc nformaton. In Secton 3 we present the condtons for optmal rsk adustment and n Secton 4 we solve for the Mnmum Varance Optmal Rsk Adustment formula. An example of how to fgure MVORA s contaned n Secton 5. Secton 6 extends our analyss to the case of uncertanty and asymmetrc nformaton. 2. Health plan behavor Assume there are N ndvduals. Each one of them s about to choose a health plan. In ths secton, we wll analyze the behavor of one (representatve) health plan, takng the behavor of the others as gven. In Secton 3 we wll analyze ths behavor wthn a symmetrc equlbrum. The model presented n ths secton s based on that n Frank et al. (2000), hereafter referred to as FGM. The health plan s pad a premum (possbly rsk-adusted) for each ndvdual that enrolls. Indvduals dffer n ther need/demand for health care, and choose a plan whch maxmzes ther expected utlty. Health care s not a sngle commodty, but a set of servces maternty, mental health, emergency care, cardac care, and so on. A health plan chooses a ratonng or allocaton rule for each servce. The plan s choce of rules wll affect whch ndvduals fnd the plan attractve and wll therefore determne the plan s revenue and costs. We assume that the plan must accept every applcant, and we are nterested n characterzng the plan s ncentves to raton servces Utlty and plan choce The health plan offers S servces. Let ms denote the amount the plan wll spend on provdng servce s to ndvdual, f he ons the plan, and let: m 5 hm,m,...,m. The dollar value of the benefts ndvdual gets from a plan, 1 2 S u (m ), s composed of two parts, a valuaton of the servces an ndvdual gets from the plan, and a component of valuaton that s ndependent of servces. We assume these enter addtvely n utlty. Thus, where, u (m ) 5 v (m ) 1 m (1)

6 158 J. Glazer, T.G. McGure / Journal of Publc Economcs 84 (2002) v (m ) 5O v (m ) s s s s the servce-related part of the valuaton and s tself composed of the sum of the ndvdual s valuatons of all servces offered by the plan. v s(? ) s the ndvdual s valuaton of spendng on servce s, also measured n dollars, where v 9s. 0, v 99 s, 0. Assume for the moment that the ndvdual knows v (m ) wth certanty. (Ths assumpton wll be relaxed n Secton 6). The non-servce component s m, an ndvdual-specfc factor (e.g. dstance or convenence) affectng ndvdual s valuaton, known to person. From the pont of vew of the plan, m s unknown, but s drawn from a dstrbuton F (m ). We assume that the premum the plan receves has been predetermned and s not part of the strategy the plan uses to nfluence selecton. The plan wll be chosen by ndvdual f u. u, where u s the valuaton the ndvdual places on the next preferred plan. We analyze the behavor of a plan whch regards the behavor of all other plans as gven, so that u can be regarded as fxed. Gven m and u, ndvdual chooses the plan f: m. u 2 v (m ). For now, we assume that, for each, the plan has exactly the same nformaton as ndvdual regardng the ndvdual s servce-related valuaton of ts servces, v, and regardng the utlty from the next preferred plan, u. For each ndvdual, the plan does not know the true value of m but t knows the dstrbuton from whch t s drawn. Therefore, for a gven m and u, the probablty that ndvdual chooses 4 the plan, from the pont of vew of the plan, s: n (m ) F (u 2 v (m )) (2) 2.2. Managed care Managed care ratons the amount of health care a patent receves. Followng Keeler et al. (1998) and FGM, let qs be the servce-specfc shadow prce the plan sets determnng access to care for servce s. A patent wth a beneft functon for servce s of v (? ) wll receve a quantty of servces, m determned by: s v 9 (m ) 5 q s s s Let the amount of spendng determned by the equaton above be denoted by m s(q s). Note that Eq. (3) s smply a demand functon, relatng the quantty of servces to the (shadow) prce n a managed care plan. s (3) 4 An alternatve nterpretaton s that ndex descrbes a group of people wth the same v (m ) functon and n (m ) s then the share of ths group that ons the plan.

7 J. Glazer, T.G. McGure / Journal of Publc Economcs 84 (2002) Proft and proft maxmzaton Let q 5 hq,q,...,q be a vector of shadow prces the plan chooses and 1 2 S m (q) 5 hm (q ),m (q ),...,m (q ) be the vector of spendng ndvdual gets S S by onng the plan. Defne n (q) ; n (m (q)). Expected proft, p(q), to the plan wll depend on the ndvduals the plan expects to be members, the revenue the plan gets for enrollng these people, and the costs of each member. p(q) 5O n (q)[r 2O m (q )] (4) s s s where r s the rsk-adusted revenue the plan receves for ndvdual. The plan wll choose a vector of shadow prces to maxmze expected proft, Eq. (4). Defnng M5 os m s(q s) to be total spendng on a person, proft per person s p5 r2 M. Assumng that ndvduals share the same elastcty of demand for 5 any servce (allowng common elastctes to dffer across servces), the proft maxmzng condton for q s, s 5 1,2,...,S, becomes (see FGM): O nm s qs 5 ]]]]. (5) O F 9m sp 2.4. The effcency crteron The health plan allocates resources effcently when t acts so as to equalze the degree of ratonng across all S servce areas. Ths s (second-best) effcent n the sense that the value of a dollar of health care spendng s equalzed across all possble uses. Formally, ths requres qs 5 q, for all s, where q s some constant characterzng the overall degree of ratonng. When q 5 1, a frst-best effcency s acheved. When q. 1, second-best effcency s acheved but the overall budget for health s too small, and when q, 1, t s too bg. There s, however, no partcular reason to expect proft maxmzaton represented by Eq. (5) to lead to the same shadow prce for each servce s, unless the rsk adustment system s able to equalze the relatve ncentves to supply each servce. 3. Optmal rsk adustment In the analyss of plan behavor ust dscussed, the rsk adustment formula was taken as gven. We now consder how to structure the rsk adustment formula to 5 Note that the fact that all ndvduals share the same elastcty of demand for a certan servce does not mply that ther demand curves are dentcal.

8 160 J. Glazer, T.G. McGure / Journal of Publc Economcs 84 (2002) nduce the plan to provde servces at the (second-best) effcent qualty. A rsk adustment formula that acheves ths goal wll be referred to as optmal rsk adustment. The analyss carred out here s very much n the sprt of the standard prncpal-agent lterature. In that lterature, the prncpal constructs an ncentve scheme so that the agent, actng n ts own nterest, wll behave accordng to the prncpal s goals. In what follows the regulator (the prncpal) sets a rsk adustment formula (an ncentve scheme) to nduce the proft maxmzng plan (the agent) to provde the effcent qualty (the regulator s goal). We have n mnd a stuaton where there are several dentcal health plans. Thus, Eq. (5) above descrbes the proft maxmzng behavor of each of the health plans n equlbrum (takng the behavor of the other plans as gven). In a symmetrc equlbrum, each ndvdual has the same probablty of beng n the plan, so n5 n. Assume that F s unform and the same for all. Then, we can smplfy Eq. (5) to say that second-best effcency requres that for each s, Eq. (6) holds. O m s ]]]]]] 5 l, (6) O msr 2O msm for some l. 0. Eq. (6) can be rewrtten as cov(m s,r) 2 cov(m s,m) 1 ]]]]]]] 5 (M 2 r 1 ]) (7) m l where covarances are defned as wth s 1 cov(m,r) 5 ] O (m 2 m )(r 2 r ) and s N s s 1 cov(m,m) 5 ] O (m 2 m )(M 2 M ), s N s s O m s ; m s, M ; O M, and r ; O r ] ] ] N N N Eq. (7) embodes the man result of ths paper. To equalze ncentves to raton all servces, the covarance of the rsk aduster wth the use of every servce must track the covarance of total predcted costs wth the servce s use. Intutvely, the optmal rsk adustment formula must have the property that by spendng on a

9 J. Glazer, T.G. McGure / Journal of Publc Economcs 84 (2002) servce, the cost consequences to a plan (from servng the demands of new members) relates to the revenue consequences (from the premums they brng n) n the same way across all servces. When ths set of condtons holds, the plan has an ncentve to raton all servces wth the same strngency. It s worthwhle to contrast the condtons represented by Eq. (7) for the optmal way to set premums to the usual approach based on statstcal ft. Conventonal rsk adustment seeks to brng per person premums as close as possble to per person costs. The frst contrast to be made s that we have S condtons characterzng optmal rsk adustment rather than one crteron based on ft. If a plan chooses the qualty of S servces, t s necessary to address the condtons for effcency for each servce, so more than one crteron seems evdently necessary. Second, the statstcal crteron of ft of premums, r, to costs, M, s replaced by crtera represented by covarances that lead proft maxmzaton decsons of health plans to be dentcal wth condtons for effcency. When the extra revenue brought n by spendng on servce s (captured by the covarance between ms and r) s n the same relaton to the extra total costs ncurred by spendng on servce s (captured by the covarance between ms and M) for all servces, proft-max- mzaton s effcent. We are now ready to consder how to use a rsk adustment formula to set premums r n order to satsfy condtons for optmal rsk adustment. Suppose that a set of J rsk adusters (varables) s avalable to use n a rsk adustment formula. Call ths set X 5 (X 1,...,X J). The varable x represents the value of the aduster for person (x could be the age or gender or health status of ndvdual ). r s the rsk-adusted premum payment made on behalf of person. A rsk adusted premum can be wrtten as r 5 X b (8) where b s a vector of weghts on the rsk aduster varables. As can be seen from Eq. (8), for a gven set of rsk aduster varables, the ssue of desgnng a rsk adustment formula s a matter of settng the rght weghts b. Rather than usng a regresson, we fnd the b s as solutons to optmalty condtons. Usng Eq. (8), each of the S equatons n Eq. (7) can be wrtten as O bcov(m s,x ) 2 cov(m s,m) 1 ]]]]]]]] 5 M 2 r 1 ] (9) m l s Note that Eq. (9) s lnear n the coeffcents of the rsk adusters b. The RHS of Eq. (9) s the same for all S equatons. It s nterestng to note what Eq. (9) mples for the relatonshp between the

10 162 J. Glazer, T.G. McGure / Journal of Publc Economcs 84 (2002) covarance of the x s wth expected costs on a servce and the b s. If the cov(m s,x) 6 s small, n order to equalze ths to a target covarance, the b s must be larger. Solvng for M 2 r 1 1/l, we are left wth S 2 1 lnear restrctons on the b s. Each of these restrctons takes the form of Eq. (10). In addton, a gven budget b n the form of an average rsk adusted premum yelds another lnear restrcton. We defne optmal rsk adustment to be the weghts b 1,...,bJ that satsfy the set of S 2 1 equatons (10), and the budget constrant n Eq. (11). We label any set of o b s satsfyng these s condtons b. o o O b cov(m s,x ) 2 cov(m s,m) O b cov(m S,x ) 2 cov(m S,M) ]]]]]]]]] 5 ]]]]]]]]] m m (10) s S s 5 1,...,S 2 1 OO b x ;O b x 5 b (11) 1 o o ] N The system of Eqs. (10) and (11) wll have at least one soluton f a rank condton s satsfed, here, f J. S. We have assumed that the plan makes S ndependent ratonng decsons, and there are J rsk adusters, none of whch s a lnear combnaton of the others. It s unclear what should be sad about the magntude of S, the number of ratonng decsons made by a plan. On the one hand, a plan offers a very large number of dfferent servces. But on the other hand, from the standpont of what management may practcally pay attenton to, there may be many fewer decson varables. For purposes of management, servces mght be grouped n ust a few categores: ratonng may be done for example at the level of prmary care, pedatrcs, mental health, cardac care, and so on, leadng to perhaps groups. (There are 19 Maor Dagnostc Categores, MDCs, for nstance, that underle the DRG system). As for the number of rsk adusters, J, ths number could be very large. Consder age and gender alone. Whle only two varables there are many potental degrees of freedom n age and gender. Medcare explots these by creatng age-gender cells. In a populaton ranged n age from say 1 65, a payer mght feasbly create ten age 3 two gender cells for 20 mutually exclusve varables to use for rsk adustment. Addng zone of resdence and elgblty status (among workng people: employee, spouse, other dependent), geometrcally enlarges J. Furthermore, dagnoses from pror health care use have been grouped nto scores of categores n recently developed dagnoss-based classfcaton systems (Ells et al., 1996; Wener et al., 1996). It 6 Ths s a generalzaton of a fndng of Glazer and McGure (2000), where t was shown that wth one sgnal (rsk aduster varable), as the nformatveness of the sgnal deterorates, the weght on the sgnal ncreases n the optmal rsk adustment formula.

11 J. Glazer, T.G. McGure / Journal of Publc Economcs 84 (2002) thus seems qute safe to proceed on the assumpton that the number of rsk aduster varables, J, exceeds the number of decsons made by management, S. 4. Mnmum varance optmal rsk adustment (MVORA) For a gven per person budget b and a set X of rsk adusters, Mnmum Varance Optmal Rsk Adustment (MVORA) s a vector of rsk adustment weghts b * 5 ( b *,...,b *) that solves the followng (constraned mnmzaton) problem: 1 J 1 2 b,...,b ] 1 J N Mnmze O (M 2O b x ) (12) F cov(m s,x ) cov(m S,x ) cov(m s,m) cov(m S,M) s.t. O b ]]] 2]]] 5]]] 2]]] m m m m G s S s S s 5 1,...,S 2 1 (13) and O b x 5 b (14) The set of S 2 1 equatons n Eq. (13) s a rewrtng of the S 2 1 equatons n Eq. (10). MVORA contends wth both problems caused by adverse selecton. The constrants guarantee the rsk adustment formula s optmal, n the sense that no qualty dstorton wll take place. Mnmzng the sum of the square of the devatons addresses problems that may arse when payments devate from costs at the ndvdual level. The lterature on conventonal rsk adustment does not contan a formal statement of why conventonal rsk adustment s the soluton to an adverse selecton problem n the market for health plans. No one, so far as we know, has suppled the argument for why the economc loss assocated wth a devance between the rsk adusted payment and the predcted cost should go up accordng to the square of the dfference. In ths secton, we follow the exstng rsk adustment lterature and smply presume that a better statstcal ft of predcted to 2 actual costs, as measured by an R statstc, advances the cause of dealng wth the selecton problem related to ndvdual access. Before we go on to dscuss the soluton for MVORA, t may be worthwhle to present conventonal rsk adustment n the context of our model. Conventonal rsk adustment derves weghts from the soluton to an ordnary least squares

12 164 J. Glazer, T.G. McGure / Journal of Publc Economcs 84 (2002) regresson of M on X. Thus, conventonal rsk adustment s, n fact, the soluton to Eq. (12) wthout the constrants n Eq. (13) and s gven by: c 21 b 5 (X9X) X9M The conventonal rsk adustment formula s therefore, c r 5 X b c We can express the set of S lnear constrants n Eq. (13) n matrx form as Ab 5 C, where A s an S 3 J matrx wth elements cov(m s,x ) cov(m S,x ) ]]] 2 ]]] s 5 1,...,S 2 1 a 5 ms m s S 5x s5 S b s a J 3 1 vector of weghts on the aduster varables and C s an S 3 1 vector of constants wth element cov(m s,m) cov(m S,M) ]]] 2 ]]] s 5 1,...,S 2 1 Cs 5 ms ms 5 b s5 S Note that the soluton for MVORA descrbed above s the estmated coeffcents from a regresson of M on X (.e. as n conventonal rsk adustment), constraned by the S lnear restrctons n Eqs. (13) and (14) (Thel, 1961). Thus, we can wrte MVORA as c c b* 5 b 1 (X9X) A9[A(X9X) A9] (C 2 Ab ) (15) One can see that f the constrants n Eq. (13) are bndng, MVORA wll be dfferent than conventonal rsk adusters. One specal case n whch the constrants do not bnd s when conventonal rsk adustment s perfect n the sense of completely capturng the varance n health care costs. In ths case, o bx5 M, and condtons (13) are exactly satsfed. Except n ths extreme case, the constrants (13) wll be bndng and MVORA wll dffer from conventonal. We can also observe that addng a new rsk aduster varable x wll mprove the 7 A smple regresson of M on X gnores the dstrbutonal characterstcs of M. For many observatons, M 5 0 (no health care spendng). Furthermore, among the postve observatons, M s skewed to the rght. Most of the emprcal lterature studyng the effect of economc factors on health care use has employed a two-part model orgnally developed for the RAND Health Insurance Experment (Duan et al., 1983). In the frst part, a use no use equaton s estmated wth a logt or a probt regresson. In the second part, an OLS regresson s performed on a transformaton (square root, log) of the dependent varable for the observatons wth postve spendng. See Mannng (1998) and Mullahy (1998) for recent dscusson. The rsk adustment lterature reles prmarly on one-part lnear regressons. The man ratonale appears to be a desre for smplcty n the rsk adustment formula.

13 J. Glazer, T.G. McGure / Journal of Publc Economcs 84 (2002) regulator s ablty to mnmze Eq. (12). Addng a new rsk aduster wth at least some correlaton wth some element of cost gves the regulator another nstrument and must do at least as well n mnmzng Eq. (12). As the presentaton of MVORA makes clear, the rsk adustment formula we propose depends on plan proft maxmzaton. Strctly speakng, MVORA s the mnmum varance rsk adustment formula that sustans the effcent expendture decsons by plans as proft maxmzaton. The queston arses as to how to fgure MVORA when expendtures are smply data from some pattern of servce expendtures generated by some other process, for example a fee-for-servce system n whch there s no health plan pad by captaton, and n whch proft 8 maxmzaton by a health plan plays no role. Any change n the payment system, for example, ntroducton of a rsk-adusted captated system, wll change ncentves and change the pattern of servces. A MVORA system can be calculated wth frst and second moment nformaton from the data from the FFS system, but then ntroducton of the payment system would change the use patterns. It seems lkely that a recalculaton of the MVORA formula wth the shft n use patterns, and perhaps a further recalculaton, etc., would lead towards optmal servce patterns, but such a queston of the dynamc adustments to rsk adustment s not formally confronted here. We are makng only a weaker clam, that f the health plan s provdng the effcent level of servces whle proft maxmzng, MVORA wll keep t there. (The conventonal rsk adustment scheme would move the plan away from optmalty). 5. An example: conventonal rsk adustment and MVORA Lke conventonal rsk adustment, MVORA can be found by usng regresson technques n data. In ths secton we calculate conventonal rsk adustment and MVORA n a very smple emprcal example, to llustrate how MVORA works, and to show how t dffers from conventonal n the way premums are fgured. Suppose data on per person spendng comes from the followng smple envronment. There are two servces a and d. The populaton s dvded nto three age groups, age 1, age 2 and age 3. The proporton of the populaton n age, sa. Age s the only avalable rsk aduster. For each ndvdual, x 5 1 f the ndvdual s n age, and x 5 0 otherwse. If an ndvdual needs servce a, the dollar value of servces he wll receve s m, regardless of hs age group, and f an ndvdual needs servce d the dollar a value of servces he wll receve s m, regardless of hs age group. The proporton d of ndvduals who need servce a s the same n all three age groups and 8 Medcare s rsk adustment system to pay HMOs s, for example, calbrated on data from the FFS sector.

14 166 J. Glazer, T.G. McGure / Journal of Publc Economcs 84 (2002) normalzed to 1 and the proporton of ndvduals who need servce d n age s g, 5 1,2,3. In fgurng MVORA, the nformaton assumpton we make n ths example s the followng. There are two types of people, the healthy and the sck, dfferentated by ther use of servce d. The healthy do not need d at all, the sck always need d. We assume that people know ther type wth certanty. That s, healthy people can accurately forecast that they wll use m of servce a and none of servce d, and the sck type accurately forecast that they wll use ma and md of servce d. So, g s the share of the sck types n age group. a of servce a The age categores wll be correlated wth type so long as the g dffers for each age category. Age wll thus have value as a rsk adustor. Conventonal rsk adustment wll be unable to fully solve the qualty problem snce under conventonal rsk adustment prvate nformaton remans about type and a health plan would want to underprovde servce d and overprovde servce a to attempt to attract the healthy wthn each age category. MVORA fxes ths problem. The calculatons below are smplfed because the age categores are (0,1) and the antcpated use of md by the healthy type s zero. Ths allows us to avod explct summatons over types n the covarance terms below. c Let r denote the conventonally rsk adusted premum pad for an ndvdual n group, then: c a d r 5 m 1 g m (16) Conventonal rsk adustment smply pays for each ndvdual the average cost of an ndvdual n the age group the person belongs to. Movng to optmal rsk adustment, we need to defne some covarances for ths example. One should frst notce that snce all ndvduals n all age groups use the same level of servce a, and covsm a,xd5 0, for 5 1,2,3 (17) covsm,md5 0, (18) a where M denotes the total dollar value of servces the ndvdual receves. As for servce d, under our assumptons: where: covsm d,xd5 asmd2 mdd for 5 1,2,3 (19) m 5 g m d d (20) s the average dollar value of servces d used by an ndvdual n age, and

15 J. Glazer, T.G. McGure / Journal of Publc Economcs 84 (2002) m 5O a m d d 51 s the populaton average use of servce d. Furthermore, snce m s the same for all ndvduals, we get that: a (21) covsm,md5 varsm d (22) d d In order to calculate the optmally rsk adusted premum we assume that the plan breaks even namely, b 5 m 1 m (23) a d where b s the budget per person. Note that under the conventonally rsk adusted premum above the plan also breaks even, so the comparson between the two payments schemes wll be on an equal bass. Pluggng Eqs. (17) (23) nto Eqs. (10) and (11) we obtan that optmal rsk o o o adustment s a trple sb,b,b d that solves the followng two equatons: and Ob a(m 2 m ) 2Var m d5 0 (24) 51 o d d s d 3 o O ba5 ma1 m d (25) 51 The MVORA s the trple sb *,b * d that solves Mn O agm f s a1 md2 bd 1s1 2 gdsma2 b d g (26) b 1,b 2,b351 s.t. Eqs. (24) and (25). Table 1 presents a specal case of the example above. The frst column n Table 1 specfes the three age groups, the second specfes Table 1 An example of Conventonal and Mnmum Varance Optmal Rsk Adustment c Age a g ma md r r* group

16 168 J. Glazer, T.G. McGure / Journal of Publc Economcs 84 (2002) a, the proporton of each group n the populaton, the thrd column specfes g, sckness d probablty, the fourth column specfes m a, the dollar value of servce a, the ffth column specfes m d, the dollar value of servce d. The last two columns c compare r, the conventonally rsk adustment premum, wth r*, the MVORA premum, for each age group. As can be seen from the table, the conventonal and MVORA premums are qute dfferent. For the relatvely healthy age group 1 (the young ) MVORA pays much less than conventonal rsk adustment whereas for the other two age groups t pays much more. As s well-known, snce age s an mperfect ndcator of type, a conventonally rsk adusted premum wll leave a health plan wth some ncentves to attract the healthy types (those that would use only servce a) wthn each age group. The conventonal premums would therefore nduce a plan to oversupply treatment for servce a and undersupply treatment for servce d. MVORA premums counteract ths ncentve. By undersupplyng servce d, the plan would tend to dscourage relatvely more of the age 3 people. By payng more for the age 3 group, MVORA gves the plan a strong ncentve to keep them. Indeed, MVORA fgures the rsk-adusted weghts so that the ncentve to supply both servce a and d s ust balanced to effcently allocate a fxed budget. The marked dsparty between the conventonal rsk adustment and MVORA s a consequence of the extreme assumpton we make about prvate nformaton. If ndvduals could not perfectly forecast ther health care use, MVORA would be closer to conventonal. One can also return to Eq. (5) above to examne shadow prces f plans were pad by conventonal rsk adustment. In our extreme example, the stuaton would be dsastrous for servce d. Snce everyone who has postve spendng on servce d s a loser (negatve p), the denomnator of Eq. (5) s so small t s negatve! Essentally, a plan has an ncentve wth conventonal rsk adustment to reduce servce d to as low a level as possble. 6. Uncertanty and asymmetrc nformaton The analyss so far has assumed that both the plan and the ndvdual know wth certanty the spendng n dollars on each of the health care servces that wll be used by the ndvdual, m s. We shall now extend our model to the more realstc case where both the ndvdual and the plan only hold some belefs about the future health care needs of the ndvdual, and these belefs may not the same. We wll see that the two crucal elements n the analyss here are the plan s belefs about each ndvdual s expected use of each servce s, whch we denote by m s, and the plan s belefs about ndvdual s belefs of how much of each of the servces he s expected to use, denoted by m ˆ s. In Appendx A, we analyze plan s behavor n ths case and show that the proft maxmzng shadow prces are gven by:

17 J. Glazer, T.G. McGure / Journal of Publc Economcs 84 (2002) O nm s qs 5 ]]]]]]. (59) O F 9 m ˆ (r 2 M s ) A plan s expected proft for an ndvdual s the product of the probablty the person ons the plan tmes the proft or loss once he s a member. The probablty that he ons depends on what the person beleves he wll receve n terms of servces n the plan. The proft once he s enrolled depends on revenue and use n the plan. It s the plan s expectatons about these two elements that wll govern plan behavor n ratonng care. Note that wth respect to the decson to on a plan, the plan must form belefs about what the consumers expect. Ths s m. ˆ Once n the plan, and from the pont of vew of the plan, what consumers expect s no longer relevant what matters s what the plan expects a person to use. Ths s m. The placement of mˆ and m n Eq. (59) reflects these consderatons. See Appendx A for dervaton. Now MVORA can be redefned to be the soluton to the followng problem: 1 2 ] N b 1,...,bJ Mnmze O (M 2O b x ) (129) F cov(m ˆ,x ) cov(m ˆ,x ) cov(m ˆ,M ) cov(m ˆ,M s S s S ) s.t O b ]]] 2]]] 5]]] 2]]]] m m m m G s S s S s 5 1,...,S 2 1 (139) and O b x 5 b (149) The obectve functon n Eq. (129) s a modfcaton of Eq. (12) to take nto consderaton plan s belefs about ts expected loss (proft) on each ndvdual. Condton (129) s meant to capture the problem of ndvdual access. As such, t s plan s expectatons about what persons wll cost n the plan that govern plans decsons about access. We beleve t s reasonable to suppose that f M dffers from r, access problems may emerge. Therefore, mnmzaton of the sum of the squares of the devatons of r from plan s expectatons of costs, M, s a reasonable crteron. The constrant (139) comes from Eq. (59) n a way smlar to the way we have obtaned Eq. (13) from Eq. (5). The soluton for MVORA descrbed above s the estmated coeffcents from a regresson of M on X, constraned by the S lnear restrctons. Thus, we can wrte MVORA as b* 5 b 1 (X9X) A9[A(X9X) A9] (C 2 Ab ) (159) where, b 5 (X9X) 21 X9M and, A s an S 3 J matrx wth elements

18 170 J. Glazer, T.G. McGure / Journal of Publc Economcs 84 (2002) cov(m s,x ) cov(m S,x ) ]]] 2 ]]] s 5 1,...,S 2 1 a 5 ms m s S 5x s5 S b s a J 3 1 vector of weghts on the aduster varables and C s an S 3 1 vector of constants wth element cov(m s,m) cov(m S,M) ]]] 2 ]]] s 5 1,...,S 2 1 Cs 5 ms ms 5 b s5 S In the case of uncertanty and asymmetrc nformaton the conventonal rsk adusters wll be dfferent than MVORA for two reasons. Frst, even f the c constrants (139) were not bndng, the soluton to MVORA wll be b and not b. Second, f the constrants are bndng, the soluton wll be gven by the constrants and wll not come from an unconstraned regresson. 7. Dscusson The qualty of health care offered by competng managed health care plans s the foremost concern of health polcy n the U.S. and many other countres. Many elements of qualty cannot be controlled by regulaton, leavng open the opportunty for plans to manpulate the qualty they offer n an effort to acheve a proftable mx of enrollees. Another polcy concern s that plans may take actons to dscrmnate aganst partcular ndvduals. The method of rsk adustment proposed here contends wth both the qualty and the access problem n an emprcally mplementable way. Condtons for optmal rsk adustment can be derved from the means and covarances among rsk aduster varables and elements of health care spendng. A statstcal ft crteron can then readly be appled to select among the weghts satsfyng condtons for effcent qualty. Applcaton of our methodology requres that the dstrbuton of health care spendng n a populaton be avalable at the level of the servce, not ust n total. The concept of a servce represents the level of aggregaton at whch management of a health plan makes decsons about resource allocaton. In most health care clams data sets, nformaton on the dagnoss, locaton of servces (offce, hosptal, etc.), procedure conducted, and specalty or type of provder s typcally ncluded on the clam. Informaton s certanly avalable to classfy health care encounters nto the servces called for n our paper; the ssue s, however, how all ths data should be used to do so. Ths requres new work on classfcaton of health care use from the pont of vew of resource management. Theoretcal work can consder the mplcatons of usng too fne or too coarse methods of aggregaton for purposes of dervng condtons for optmal rsk adustment. Our method also calls for nformaton about expected health care costs,

19 J. Glazer, T.G. McGure / Journal of Publc Economcs 84 (2002) specfcally, plans expectatons about the consumers costs, and plans belefs about consumers expectatons. Whle ths may seem a dsmayng prospect, n ths respect we are no worse off than conventonal rsk adustment. Careful dscussons about the goals of conventonal rsk adustment have recognzed for some tme that the obectve of conventonal rsk adustment s to explan predctable costs, yet t s actual costs that we see n the regressons dervng recommended weghts. The same fall back of usng the dstrbuton of actual spendng as opposed to expected spendng s avalable to our method as well. Another way to look at the ssue s as a topc for research. Health plans make the decsons about resource allocaton and access on the bass of expectatons. Regulators attemptng to counter the adverse consequences of these decsons would beneft by knowng how plans make decsons and based on what nformaton. The salency of plans expectatons to the polcy ssues related to qualty and access makes ths a very mportant topc for emprcal research. Acknowledgements Ths paper was wrtten whle McGure was at Boston Unversty. Research support from grants R01 MH59254 from the Natonal Insttute of Mental Health and P01 HS10803 from the Agency for Health Research and Qualty are gratefully acknowledged. We are grateful to Margarta Alegra, Ana Balsa, Zhun Cao, Davd Cutler, Randy Ells, Rchard Frank, Alberto Holly, Mathas Kfmann, Kevn Lang, Joseph Newhouse, Mark Satterthwate, Manuel Tratenberg, and Alan Zaslavsky for comments on an earler draft. Remanng errors are the authors responsblty. Appendx A. Proft maxmzaton wth uncertanty and asymmetrc nformaton Let T denote the set of possble health states of each ndvdual and let t denote an element n T. Let vt5 hv t1(m t1),vt2sm t2d,...,v ts(m ts) denote the vector of S valuaton functons for the S servces, f an ndvdual s health state s realzed to be t. We assume that for each t and s, v ts(? ) satsfes the propertes dscussed earler. Let xˇ be some random varable, the value of whch depends on the state t, and let k be a dstrbuton functon defned over T. Let E k[x] ˇ denote the expected value of xˇ wth respect to the dstrbuton k. The order of moves s as follows: at the frst stage, the plan chooses ts level of shadow prces q 5 (q 1,q 2,...,q S). When choosng q, the plan s uncertan about each ndvdual s health state t, and t holds pror belefs about t, denoted by (the dstrbuton over T ) g. At the second stage, ndvduals observe the plan s level of shadow prces and decde whether or not to on. At ths stage, the ndvdual s

20 172 J. Glazer, T.G. McGure / Journal of Publc Economcs 84 (2002) uncertan about hs true health state t and he holds some pror belefs about t. Let f be a dstrbuton functon over T whch denotes the plan s belefs about ndvdual s belefs about hs health state t. Thus, both g and f are plan s belefs, the frst s the plan s belefs about ndvdual s health state and the second s ts belefs about what the ndvdual thnks hs health state to be. At the thrd stage, when servces are provded, the ndvdual s true health state s already known. Hence, for a gven shadow prce qs and a valuaton functon v ts, the plan s expendtures on ths ndvdual n servce s wll be mtssq sd, gven by: v 9ts(m ts(q s)) 5 q s. Let v t(q) 5 os v ts(m ts(q s)). Let u t denote the ndvdual s utlty f hs health state s t and he chooses the alternatve plan. The plan s assgned probablty that the ndvdual wll on s gven by (we drop momentarly the subscrpt from the analyss): n (q) F(E [u 2 v ˇ ]) f f t t The plan s expected proft on the ndvdual s S F GD p (q) 5 n (q) r 2 E O m ˇ (q ). fg f g ts s s Dfferentatng wth respect to q s9 yelds dp fg(q) ]] 5 F 9Ef vˇ 9ts9 mˇ 9ts9 r2 Eg O m ˇ ts(q s)] 2 ne[m f g ˇ ts9 9 ] dq s9 f gs F GD Usng v9ts 5 qs and m9ts 5 (esm ts)/qs for all t, where es s the elastcty of demand (common to all ndvduals) for servce s (see FGM, 1999), the rght-hand sde becomes S nm f s9 e F 9m ˆ (r 2 M ) 2]] s9 s9 q s9 D where mˆ s9 5 E f[m ˇ ts9] s the plan s belef about the ndvduals predcton of the health resources he wll consume, m 5 E g[m ˇ ts9] s the plan s predcton of ts expendture on the ndvdual, and M 5 os m s. Wth a populaton of N ndvduals, the proft-maxmzng qs wll therefore be 9 (59) from text. s 9 Inducng a plan to supply servces requres some proft. In the case of a perfect ft between a rsk-adusted payment r and M, expected proft on each person would be zero. In ths case, some fxed costs are necessary (as n any model of monopolstc competton).

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