What should (public) health insurance cover?

Transcription

1 Journal of Health Economcs 26 (27) What should (publc) health nsurance cover? Mchael Hoel Department of Economcs, Unversty of Oslo, P.O. Box 195 Blndern, N-317 Oslo, Norway Receved 29 Aprl 25; receved n revsed form 7 August 26; accepted 9 August 26 Avalable onlne 15 September 26 Abstract In any system of health nsurance, a decson must be made about what treatments the nsurance should cover. One way to make ths decson s to rank treatments by ther ratos of health benefts to treatment costs. If treatments that are not offered by the health nsurance can be purchased out of pocket, the socally optmal rankng of treatments to be ncluded n the health nsurance s dfferent from ths standard cost-effectveness rule. It s no longer necessarly true that treatments should be ranked hgher the lower are treatment costs (for gven health benefts). Moreover, the larger are the costs per treatment for a gven beneft cost rato, the hgher prorty should the treatment be gven. If the health budget n a publc health system does not exceed the socally optmal sze, treatments wth suffcently low costs should not be performed by the publc health system f treatment may be purchased prvately out of pocket. 26 Elsever B.V. All rghts reserved. JEL classfcaton: H42; H51; I1; I18 Keywords: Health nsurance; Prortzaton; Cost-effectveness 1. Introducton In any health nsurance system, publc or prvate, one must make a decson about what treatments the health nsurance should cover. Health economsts have often argued that costeffectveness analyss should play an mportant role n choosng what should be offered by health Useful comments have been gven by partcpants at a semnar at the Ragnar Frsch Centre for Economc Research,n partcular by Kjell Arne Brekke, and by Per-Olov Johansson, Albert Ma and two anonymous referees. I gratefully acknowledge fnancal support from the Research Councl of Norway through HERO Health Economc Research Programme at the Unversty of Oslo. Tel.: ; fax: E-mal address: mhoel@econ.uo.no /$ see front matter 26 Elsever B.V. All rghts reserved. do:1.116/j.jhealeco

2 252 M. Hoel / Journal of Health Economcs 26 (27) nsurance. Cost-effectveness s n ths context usually defned as the mnmum cost for a gven health beneft, or equvalently, maxmal health benefts for gven expendtures on health care. 1 There s a large lterature that s crtcal to ths type of analyss. One lne of crtcsm s that costeffectveness analyss requres an aggregate measure of health benefts. Whether ths measure s qualty adjusted lfe years (QALYs) or some other measure, one needs severe restrctons on a general preference orderng over lfe years and health qualty of each lfe year to be able to represent preferences by any smple aggregate measure. 2 A second lne of crtcsm has been that whatever aggregate health beneft measure one uses to represent preferences at the ndvdual level, one mght queston the ethcal or welfare theoretcal bass for aggregatng health benefts across ndvduals. 3 The present paper gnores the above-mentoned problems wth cost-effectveness analyses of prortzaton ssues. The focus s nstead on a dfferent mportant ssue: at least for publc health nsurance, most of the lterature that dscusses how a health budget should be allocated across potental medcal nterventons explctly or mplctly assumes that the health nterventons that are not funded by the publc budget are not carred out. However, both under publc and prvate health nsurance t s often possble to purchase treatment out of pocket f treatment s not covered by the health nsurance. Examples of treatments that typcally may be purchased out of pocket are surgcal sterlzaton, asssted fertlzaton, cataract surgery, dental care, prescrpton medcne. Comparng dfferent nsurance arrangements one wll fnd that they dffer wth respect to what s covered and what s not. When treatments of the type above are not covered by the health nsurance, they are nevertheless avalable for those who want to fnance the treatment out of pocket. The paper dscusses the use of cost-effectveness analyses for prortzng a health budget for a publc health system or a prvate nsurance company when an out of pocket opton exsts. It s shown that when there s an out of pocket opton, a smple cost-effectveness crteron of maxmzng the sum of some aggregate measure of health benefts for a gven budget s not necessarly the best way to allocate the health budget. In partcular, such standard cost-effectveness analyss does not maxmze the sum of utlty levels of the members of the health nsurance. The reason for ths s that the beneft of ncludng a partcular treatment n the nsurance program can no longer be measured smply by the gross health mprovement ths treatment gves: some of the health care would otherwse have been performed n any case, so the net health ncrease s lower than the gross ncrease. On the other hand, by ncludng a treatment n the health nsurance, there are reduced personal costs of treatment fnanced out of pocket. Ths cost savng should be ncluded n the beneft sde of ncludng the treatment n the health nsurance. In order to add the benefts of mproved health wth the personal cost savng one s thus forced to make a monetary valuaton of the net ncrease n health benefts. The paper shows that maxmzng the sum of utlty levels of the members of the health nsurance (gven the budget) gves a dfferent outcome than smply maxmzng gross or net health benefts for the gven health budget. A comparson s also 1 For a further dscusson of analyses based on the cost-effectveness see e.g. Wensten and Stason (1977), Johannesson and Wensten (1993), Garber and Phelps (1997) and Garber (2). 2 See e.g. Broome (1993), Mehrez and Gafn (1989), Culyer and Wagstaff (1993), Blechrodt and Quggn (1999) and Gafn et al. (1993). 3 Crtcsm of ths type of aggregaton has been gven by e.g. Harrs (1987), Wagstaff (1991), Nord (1994), Olsen (1997) and Dolan (1998), whle e.g. Blechrodt (1997), Blechrodt et al. (24) and Østerdal (25) have provded axomatc analyses showng how one can aggregate ndvdual QALYs to reach a socal objectve functon wth a sound welfare theoretcal bass.

3 M. Hoel / Journal of Health Economcs 26 (27) gven between the rankng that maxmzes the sum of utlty levels and the standard cost-effectve rankng of dfferent treatments. Whle most of the content of the present paper apples both to publc and prvate health nsurance, t s perhaps most relevant for publc health nsurance. For publc health nsurance, the members of the health nsurance wll typcally be exogenous, and often equal to the total populaton. It then makes good sense to allocate an exogenously gven health budget to maxmze the sum of the members utlty levels (although ths s not the only concevable socal objectve). As a normatve recommendaton, ths also makes sense f the health nsurance s prvate. However, n the latter case health nsurance wll often be voluntary and there wll typcally be a choce between nsurance companes. The present analyss has nothng to say about what s optmal from the pont of vew of an nsurance company, or how the allocaton of persons across nsurance companes may depend on what treatments the nsurance companes offer. Fnally, we dsregard the ssue of a possble mx of prvate and publc nsurance, as dscussed by e.g. Blomqvst and Johansson (1997), Allesandro (1999) and Hansen and Kedng (22). The rest of the paper s organzed as follows. In Secton 2 the basc model s ntroduced, and the socally optmal rankng of dfferent treatments s defned as the rankng that maxmzes the sum of utlty levels of the members. The standard cost-effectve rankng (.e. rankng accordng to ratos between health benefts and costs) s compared wth the socally optmal rankng. When there s no out of pocket opton these rankng are dentcal. Wth an out of pocket alternatve, however, the rankngs dffer. An mportant result n ths secton s that t s not obvous how the costs of treatments should affect the rankng. It s shown that for dentcal health benefts, a treatment wth hgher costs may be ranked hgher than a treatment wth lower costs. Moreover, the larger are treatment costs for a gven beneft cost rato, the hgher prorty should the treatment be gven. Whle the health budget s assumed exogenous n Secton 2, the socally optmal budget s derved n Secton 3 for the case of publc health nsurance. It s shown that f the budget s not hgher the optmal level, treatments wth suffcently low costs should not be covered by the health nsurance, no matter how large the health benefts are. Secton 4 dscusses some extensons, and Secton 5 concludes. 2. Prortzng for an exogenously gven health budget The utlty level of a healthy person wth net ncome y s gven by an ncreasng and strctly concave utlty functon u(y). If ths person gets an llness j the person s utlty s reduced from u(y)tov U j (y) f no treatment s gven. However, f treatment s gven for ths llness (wthout any payment) the utlty level wll nstead be v T j (y), whch by assumpton s hgher than vu j (y). It s also reasonable to assume that v T j (y) u(y). Snce the occurrence of an llness wll be assumed exogenous, we can wthout loss of generalty assume that treatment gves full recovery n the sense that v T j (y) = u(y).4 Fnally, we assume that the utlty loss n the absence of treatment (= v T j (y) vu j (y) = u(y) vu j (y)) s ndependent of ncome, and we denote ths utlty loss by h j. Utlty as healthy or treated after an llness s thus u(y), whle utlty wth an llness j that s untreated s u(y) h j. The assumpton that the utlty loss due to an llness s ndependent of ncome makes the analyss smpler, and also makes t possble to aggregate health benefts of 4 If v T j (y) <u(y) there would be an unavodable utlty loss of u(y) vt j (y) n the case of llness j, ndependent of whether or not treatment was gven for ths llness, and an addtonal utlty loss of v T j (y) vu j (y) f treatment was not gven.

4 254 M. Hoel / Journal of Health Economcs 26 (27) treatments by aggregatng the h j -terms over persons wthout havng to consder whch persons one s aggregatng over. Notce also that the assumpton of utlty losses beng ndependent of ncome s consstent wth the wllngness to pay for a treatment beng ncreasng n ncome. 5 However, the assumpton of utlty losses beng ndependent of ncome s not consstent wth the emprcal fndng of e.g. Vscus and Evans (199) that the margnal utlty of ncome s hgher the better health one has. If the margnal utlty of ncome s rsng wth health, our varable h j must be ncreasng n y nstead of ndependent of y as we have assumed. We return to ths ssue n Secton 4. There are n mutually exclusve potental llnesses, and each person gets llness j wth an exogenous probablty π j. The cost of treatng llness j s c j. As explaned above, the health beneft of treatng llness j s h j. Standard cost-effectveness analyses recommend that treatments are ranked accordng to the ratos h /c, and that the health nsurance (publc or prvate) should cover all treatments for whch ths rato s above some threshold determned by the sze of the health budget. In the absence of the possblty of fnancng treatment out of pocket, such a decson rule wll also maxmze the sum of all persons utlty levels. To see ths, let F(y) be the ncome dstrbuton functon,.e. F(y) tells us what share of the populaton has net ncome below or equal to y. Wthout loss of generalty we assume that the lowest and hghest ncome s and 1, respectvely, mplyng F(1) = 1. To smplfy the analyss we also assume F() =, but none of the results depend on ths assumpton. The densty functon correspondng to F s f(y)=f (y). At the level of the socety, the probabltes π are shares of persons wth each of the n llnesses. We defne a vector of polcy varables δ 1,..., δ n, where δ j = 1 f treatment for llness j s covered by the health nsurance and δ j = otherwse. 6 In the absence of the possblty of fnancng treatment out of pocket, llnesses that are not covered by the health nsurance are not treated. In ths case the sum of utltes of all persons, henceforth called socal welfare, s therefore gven by 7 : W = δ π u(y) f (y)dy + (1 δ )π [u(y) h ]f (y)dy (1) In Appendx A t s shown that maxmzaton of W subject to an exogenous budget constrant mples that all treatments satsfyng h j c j >λ (2) should be covered by the health nsurance, where λ s the shadow prce of the budget constrant, and thus s lower the hgher s the budget. 8 5 The wllngness to pay for treatment of llness j, denoted WTP j s defned by u(y WTP j )=u(y) h j, and s ncreasng n y due to the concavty of u. 6 It s assumed throughout that there are no co-payments for treatments covered by the health nsurance. Introducng co-payments would not change our results as long as the co-payments are so small that they do not deter anyone from choosng treatments that are offered by the health nsurance. 7 Wth ths setup we have mplctly assumed that π = 1. If there s a state of no llness, ths state s ncluded n the lst of n llnesses wth h = for ths state. 8 Due to the nteger problem, there wll typcally be one margnal treatment, gven by equalty nstead of the nequalty n (2). Ths treatment should be partally ncluded,.e. some but not all persons should be offered treatment. The proporton of persons treated s determned so that the budget s exactly used up. Ths nteger problem s from now on gnored.

5 M. Hoel / Journal of Health Economcs 26 (27) We now ntroduce the possblty of fnancng treatment out of pocket. Assume that f a partcular treatment j s not covered by the health nsurance, each person has the opton to buy treatment out of pocket at the same prce as the cost would have been for the publc health system or the prvate nsurance company. If treatment for llness j s not covered by the health nsurance a person thus has the choce between buyng treatment, gvng hm/her the utlty level u(y c j ), or gong untreated, gvng hm/her the utlty level u(y) h j. The person wll choose the alternatve that gves the hghest utlty level: f u(y c j )<u(y) h j for a person who gets llness j, ths person wll prefer to be untreated than to pay for treatment out of pocket. If u(y c j )>u(y) h j for a person who gets llness j, ths person wll prefer to pay for treatment out of pocket rather than go untreated. If u(y c j )<u(y) h j for all y, then no one wll buy treatment that s not offered by the health nsurance. If u(y c j )>u(y) h j for all y, everyone wll buy treatment f ths treatment s not covered by the health nsurance. The most nterestng case s the case where there exsts a crtcal value ξ(h j, c j ) (, 1) defned by u(ξ(h, c) c) = u(ξ(h, c)) h (3) In ths case the persons who have ncomes below ξ(h j, c j ) wll choose to go untreated f treatment s not covered by the health nsurance, whle persons wth ncomes above ξ(h j, c j ) wll buy treatment out of pocket. 9 When there s an out of pocket alternatve socal welfare W s no longer gven by (1), but nstead by the followng expresson: W = δ π u(y) f (y)dy + ξ(h,c ) (1 δ )π [u(y) h ] f (y)dy + (1 δ )π ξ(h,c ) u(y c ) f (y)dy (4) The frst term n ths expresson gves the welfare level n the states of llness for whch the health nsurance covers treatment. The second and thrd term gve the welfare level n the states for whch treatment s not covered by the health nsurance. The second term s the welfare of those who choose to be untreated, and the thrd term s the welfare of those who choose treatment and pay for ths out of pocket. In Appendx A t s shown that maxmzaton of ths expresson subject to the budget constrant mples that a treatment j should be covered by the health nsurance f and only f R(h j,c j ) >λ where λ as before s the shadow prce of the budget constrant, and the rankng functon R(h, c) s defned by { R(h, c) = 1 } 1 hf(ξ(h, c)) + [u(y) u(y c)]f (y)dy (6) c ξ(h,c) The followng lemma s proved n Appendx A (where subscrpts denote partal dervatves): (5) 9 More precsely, ξ(h, c) s defned as follows: () ξ(h, c) = for h u() u( c) (and thus u(y c) u(y) h for all y [, 1] snce u(y) u(y c) s declnng n y due to the concavty of u), () ξ(h, c) = 1 for h u(1) u(1 c) (and thus u(y c) u(y) h for all y [, 1]), and () by (3) for u(1) u(1 c)<h < u() u( c).

6 256 M. Hoel / Journal of Health Economcs 26 (27) Lemma 1. The functon R(h, c) defned by (6) and (3) has the followng propertes: R(h, c)<h/c for h > u(1) u(1 c) and R(h, c)=h/c for h u(1) u(1 c) R h (h, c)>for h < u() u( c) and R h (h, c)=for h u() u( c) R c (h, c)>for h > u() u( c) and R c (h, c)<for h < u(1) u(1 c) v R(αh, αc)>r(h, c) for α >1and h > u(1) u(1 c) v lm R(h, c) = c u (y)f (y)dy Several nterestng results follow from ths lemma. The frst result concerns the mportance of the avalablty of the out of pocket opton: t has so far mplctly been assumed that all treatments can be fnanced out of pocket f not covered by the health nsurance. In practce, there may for varous reasons be some treatments that wll not be offered at all f they are not covered by the health nsurance. One reason for ths could be fxed costs (not formally ncluded n our model). If the number of persons wshng to purchase treatment out of pocket s too small, such treatment may not be offered. A second reason could be that n a country wth compulsory publc health nsurance, the government could forbd varous types of prvate treatment. 1 A thrd reason s related to the fact that for many llnesses there may be several alternatve treatments. Consder an llness wth two treatments A and B, wth health benefts and costs hgher for A than for B. If one of the treatments for sure s gong to be offered by the health nsurance, the relevant ssue s whch of the treatments the health nsurance should offer. In our analyss, the relevant health beneft s the dfference between A and B and the relevant cost s the cost dfference between A and B. If B but not A s offered by the health nsurance, several persons mght wsh to pay the cost dfference between A and B and thus get treatment A nstead of B. However, both n publc and prvate health nsurance ths s often not permtted: f one wants to have A nstead of B one has to pay the whole cost of A, and not only the cost dfference between A and B. Gven ths, A wll often not be a relevant alternatve to purchase out of pocket. For a treatment for whch no out of pocket alternatve exsts, we can smply set ξ = 1 nstead of beng determned by (3). It s straghtforward to verfy that all of the analyss above remans vald, so that R(h j, c j )=h j /c j nstead of (6) for a treatment that s not avalable f not covered by the health nsurance. From part of Lemma 1 we know that for llnesses for whch some persons would fnance treatment out of pocket f not covered by the health nsurance, the rankng functon s lower when out of pocket fnanced treatment s avalable than when t s not. The followng proposton therefore follows: Proposton 1. If there are treatments that are not avalable f not covered by the health nsurance, such treatments should be gven hgher prorty as a canddate for ncluson n the publc health program than treatments that have the same rato h j /c j of health benefts to costs but are avalable even f not covered by the health nsurance, provded some persons would choose such treatment. Loosely speakng, ths proposton says that treatments wthout an out of pocket opton should be gven hgher prorty than treatments for whch an out of pocket opton exsts. The ntuton s that when an out of pocket opton exsts, some persons wll n any case get treatment. For these persons the beneft of offerng treatment through the health nsurance s only the personal cost 1 In e.g. Norway there s a legal regulaton prohbtng new npatent facltes (some beds were accepted before the law came nto practce n 1986). Treatments requrng npatent facltes can thus only be purchased prvately by gong abroad, mplyng a cost consderably hgher than the cost of the domestc publc treatment.

7 M. Hoel / Journal of Health Economcs 26 (27) savngs by not havng to pay for the treatment. These personal costs are lower than the utlty loss of not havng a treatment at all, so treatments wth an out of pocket opton should therefore be valued lower than treatments wthout. Part of Lemma 1 says that the rankng functon s ncreasng n health benefts for any llness for whch some persons would choose to be untreated f the nsurance program dd not cover the treatment. We therefore have the followng proposton: Proposton 2. Consder llnesses for whch some persons would choose to be untreated f the nsurance program dd not cover the treatment. For such llnesses one should gve hgher prorty to a treatment j the hgher s the health beneft h j of the treatment for a gven cost c j. Ths result s very smlar to what follows from standard cost-effectveness analyss. A more nterestng result follows from part of Lemma 1. Ths part of Lemma 1 s llustrated n Fg. 1. The fully drawn curve s R(h, c), and the dashed hyperbole s h/c. For suffcently small treatment costs,.e. values of c that are so low that h > u(y) u(y c) holds for all ncomes (c<c * n Fg. 1), everyone wll choose treatment even f t s not covered by the health nsurance. Includng ths treatment n the nsurance program s thus smply an nsurance aganst monetary costs as long everyone n any case chooses treatment. Ths nsurance beneft s ncreasng more than proportonally wth costs due to rsk averson,.e. the concavty of the utlty functon u. Ifon the other hand treatment costs are suffcently hgh,.e. c s so hgh that that h < u(y) u(y c) holds for all ncomes (c>c ** n Fg. 1), no one wll choose treatment f t s not offered by the health nsurance. In ths case R(h j, c j )=h j /c j,.e. the rankng functon s lower the hgher s the treatment cost. The exact propertes of the functon R between these two cost lmts s ambguous, and depends on both the utlty functon u and the dstrbuton functon F. Although t s assumed sngle-peaked n Fg. 1, ths need not be the case. From Fg. 1 we have the followng proposton: Proposton 3. The rankng of treatments dfferng n costs but havng the same health benefts s ambguous. In partcular, there may exst three treatments wth dentcal health benefts but dfferent costs c 1 < c 2 < c 3 where treatment 2 should be ranked hgher than both 1 and 3. Fg. 1. R(h, c) and h/c for a gven value of h.

8 258 M. Hoel / Journal of Health Economcs 26 (27) Part v of Lemma 1 gves us the followng proposton: Proposton 4. Consder llnesses for whch some persons would choose to be untreated f the nsurance program dd not cover the treatment. For such llnesses one should gve hgher prorty to a treatment j the hgher s the cost c j of the treatment for a gven rato h j /c j of benefts to costs. Proposton 4 ponts at an mportant dfference between the present case and the case wth no out of pocket alternatve. One can no longer smply rank treatments by ther beneft cost ratos, the cost per treatment s also an mportant factor to take nto consderaton. The reason for ths result s the concavty of the utlty functon: the hgher s the cost of a treatment, the fewer wll choose to fnance treatment out of pocket, even f health benefts ncrease proportonally wth treatment costs. Ths means that the term h/c n (6), whch s larger than the second term, gets ncreased weght (hgher F)asc and h are ncreased proportonally. 3. The optmal sze of the health budget So far t has been assumed that the budget s exogenously gven. For the case of publc health nsurance fnanced through taxes, t s of nterest to ask what sze of the budget maxmzes the socal welfare level W gven by (4). To answer ths queston, one must make an assumpton about how taxes are dstrbuted among dfferent persons. We shall assume that an ncrease n the health budget reduces all net ncomes by the same amount. Ths can be justfed by assumng that the ntal tax system s optmally desgned, where ths optmzaton has taken nto consderaton possble dstrbutonal preferences. Startng at such an optmum, t makes no dfference whch element of the tax system one changes n order to fnance a small ncrease n the requred tax revenue. Snce B s the budget per capta, we thus assume that an ncrease n B reduces all net ncomes by the same amount,.e. du/db = u (y)( 1). Usng ths and maxmzng socal welfare W wth respect to B gves (see Appendx A for detals) λ = u (y)f (y)dy + (1 δ )π ξ(h,c ) [u (y c ) u (y)]f (y)dy (7) The second term n ths expresson would vansh f t was not possble to purchase treatment that was not covered by the health nsurance (.e. f ξ = 1). In ths case (7) n combnaton wth (2) would have a straghtforward nterpretaton: treatments for the dfferent llnesses should be ncluded n the publc health system f and only f the health gan (n utlty unts) per Euro exceeds the populaton average of the margnal utlty of ncome (.e. the nverse of the margnal wllngness to pay for a health mprovement). When treatments that are not covered by the health nsurance can be fnanced out of pocket, the second term n (7) s postve due to the concavty of u. The shadow prce of the budget s thus hgher for ths case than for the case where there s no out of pocket alternatve. Moreover, we know from part of Lemma 1 that the rankng functon R s not hgher wth than wthout an out of pocket opton. It therefore follows that fewer treatments pass the crteron for ncluson n the publc health program n the present case than n the case when there s no out of pocket opton. In other words: Proposton 5. The optmal budget of a publc health system s smaller when treatment also s avalable fnanced prvately out of pocket than t s when there s no such opton.

9 M. Hoel / Journal of Health Economcs 26 (27) From part v of Lemma 1 we know that the frst term n (7) s equal to lm R(h, c), mplyng c that lm R(h, c) <λ. Snce the shadow prce λ of the budget constrant s larger the smaller s the c budget, the followng proposton follows from Fg. 1: Proposton 6. If the health budget s equal to or smaller than the socally optmal sze, treatments that may be fnanced out of pocket should not be ncluded n the publc health program f the costs of these treatments are suffcently low, no matter how hgh the rato of health benefts to treatment costs are. 4. Some generalzatons In the formal analyss above we have used several smplfyng assumptons. However, several of the results are lkely to hold also under more general assumptons. In ths secton we shall consder three possble generalzatons. In the analyss, t was assumed that the utlty loss from an untreated llness s the same for everyone. However, as argued n Secton 2, t mght be more realstc to assume that the utlty loss of an untreated llness h j s ncreasng wth ncome, and s thus dfferent for dfferent persons. Even among persons wth the same ncome, the utlty loss of an untreated llness wll typcally vary among persons. The way the model s set up, t seems natural to thnk of h j as a measure of the severty of llness j (f untreated). An example of such an nterpretaton s physcal llnesses that may be assocated wth dfferent levels of pan for dfferent persons. In other cases t may be just as natural to thnk of h j as a parameter descrbng a person s preferences. An example could be a couple who can only have chldren through asssted fertlzaton. The term h j s n ths case a varable reflectng how much worse of the couple feels wthout chldren than wth,.e. a varable measurng the strength of the preferences for havng chldren. Whatever the background for the dfferences n the h j varables among persons, ths dstrbuton wll affect who wll choose out of pocket treatment and who wll not. Ths choce wll thus no longer only depend on ncome as assumed n the prevous analyss. However, there s good reason to beleve that the results above wll hold also when there s heterogenety n the populaton n the h j varables n addton to ncome. An mportant factor behnd several of the results was that the benefts of ncludng a treatment n the health nsurance to those who would get ths treatment n any case s lower than the benefts as measured by the utlty gans h j. Also for ths case we therefore get a rankng functon resemblng (6), wth the terms n the ntegral beng lower than the term h j. In an earler verson of the present paper t s demonstrated that all of our results are vald wth heterogenety of preferences/llness severty for the specal case of ncomes beng the same for all. See also Hoel (26), where some of the results are derved for the case of heterogenety both n ncome and preferences. In the formal analyss t has been assumed that the llnesses are mutually exclusve. In realty, there s also the possblty of co-morbdtes,.e. two llnesses occurrng smultaneously. Such co-morbdtes may cause problems even for standard cost-effectveness analyses, as the utlty loss from two llnesses 1 and 2 need not be equal to h 1 + h 2. If e.g. h 1+2 > h 1 + h 2 (n obvous notaton) t may be the case that llnesses 1 and 2 are rejected based on the rankng crteron (2), whle nevertheless h 1+2 /(c 1 + c 2 )>λ. For the case wth an out of pocket alternatve, co-morbdtes may create problems for the rankng even f h 1+2 = h 1 + h 2 : from Proposton 4 t follows that we cannot rule out the possblty of R(h 1 + h 2, c 1 + c 2 )>λ even f R(h 1, c 1 )<λ and R(h 2, c 2 )<λ. In practse, t may be dffcult to nclude treatment of 1 and 2 when they occur smultaneously, but

10 26 M. Hoel / Journal of Health Economcs 26 (27) not nclude them when they occur ndependently. One possble practcal soluton could then be to let the decson of whether or not to nclude treatments n the nsurance program be affected by how frequently the llnesses occur jontly as opposed to ndvdually. In the formal analyss, t was assumed that the cost per treatment was ndependent of whether t was fnanced by the health nsurance or out of pocket. Ths need not be the case. If there s a publc health system and treatments out of pocket are suppled by prvate supplementary health provders, the reason for cost dfferences could be dfferences n effcency between the publc and the prvate sector. Proft margns could also dffer dependng on whether a treatment was purchased by an nsurance company or prvately out of pocket. If cost dfferences between treatments covered by the nsurance and purchased out of pocket vary across treatments, ths may obvously affect the optmal rankng of treatments. However, f there are dfferences n costs but the proporton between out of pocket costs and nsurance covered costs are dentcal across treatments, our analyss s easy to generalze. In ths case we let the cost of the nsurance company for treatment j be c j (as before) and the out of pocket treatment cost be kc j, where k s some postve number (smaller or larger than 1). In our analyss the terms ξ c and y c n (3) and (4) would be replaced by ξ kc and y kc. Gong through the proofs n Appendx A wth ths modfcaton t can be shown that all of our results reman vald (wth y c n the rankng functon replaced by y kc). 5. Concludng remarks The precedng analyss has shown that the exstence of an out of pocket opton has mportant consequences for the rankng of treatments n a cost-effectveness analyss of what health nsurance should cover. The conclusons are summarzed n Propostons 1 6. To be able to make an exact rankng over alternatves, one would n addton to the health benefts that enter a standard costeffectveness analyss also have to compute the second term of the rankng functon gven by (6). However, even wthout such an exact calculaton of ths term, our qualtatve results may be of some use n how to rank dfferent treatments. In partcular, for treatments wth roughly the same beneft cost ratos (h/c) that are all close to the threshold level λ, our analyss suggests that the health nsurance should cover the treatments wth large costs per treatment but not those wth relatvely small costs per treatment. An example of treatment wth low costs could be surgcal sterlzaton (at least for men). Ths s a once n a lfetme treatment and the cost s only a small fracton of 1% of the lfe tme ncome of most people. The present analyss therefore gves a good justfcaton for not ncludng ths treatment among the treatments covered by the health nsurance. Prescrpton medcnes for chronc dseases are on the other hand often qute costly (over a lfe tme), and there are thus good reasons for coverng such expenses by the health nsurance, even n cases where the health beneft to cost rato s lower than for some treatments that are not covered by the nsurance. Appendx A. Dervaton of results A.1. The optmal rankng Wth the notaton ntroduced n Secton 2, the budget constrant (per capta) s δ π c B (8)

11 M. Hoel / Journal of Health Economcs 26 (27) where B s exogenous. The Lagrangan correspondng to maxmzng (4) subject to (8) s L = δ π u(y) f (y)dy + ξ(h,c ) (1 δ )π [u(y) h ]f (y)dy + ( 1 (1 δ )π u(y c )f (y)dy + λ B ) δ π c ξ(h,c ) (9) A treatment j should be ncluded n the publc program f L s ncreasng n δ j but not ncluded f L s declnng n δ j. It follows from (9) that { } L 1 = π j h j F(ξ(h j,c j )) + [u(y) u(y c j )]f (y)dy λc j δ j ξ(h j,c j ) = π j c j [R(h j,c j ) λ] (1) where R s defned by (6). Ths expresson s postve f and only f (5) holds. For the case wth no out of pocket opton, we have ξ = 1, and the dervatve n (1) s postve f and only f (2) holds. A.2. The optmal budget The optmal budget s found by maxmzng W wth respect to B gven that the ncrease n B reduces all net ncomes by the same amount,.e. du/db = u (y)( 1). Usng ths and settng the dervatve of (9) wth respect to B equal to zero gves Eq. (7). Proof of Lemma h > u(1) u(1 c) mples that ξ < 1 (cf. (3) and footnote 9), and thus F(ξ) < 1. Snce u(y) u(y c)<h for those who choose to pay for treatment out of pocket, the frst part of Lemma 1. follows. h u(1) u(1 c) mples that ξ = 1, cf. footnote 9. Ths proves the second part of Lemma h < u() u( c) mples that ξ > and thus F(ξ) >. From (6) t follows that R h = F c +{h [u(ξ) u(ξ c)]}f (ξ)ξ h(h, c) (11) where the term n curly brackets s zero from the defnton of ξ. We therefore have R h > for F >.Ifh > u() u( c)wehaveξ = 1 and thus F(ξ)=,andR s ndependent of h. 1. h > u() u( c) mples that ξ = and thus F(ξ) =. From (6) t follows that R(h, c) = u(y) u(y c) c f (y)dy (12) The fracton n the ntegral s ncreasng n c due to the concavty of u, mplyng R c >. h < u(1) u(1 c) mples that ξ = 1 (cf. footnote 9), and thus F(ξ) = 1. In ths case t follows from (6) that R(h, c)=h/c, mplyng R c <.

12 262 M. Hoel / Journal of Health Economcs 26 (27) v Let h = βc. Keepng β constant and ncreasng c gves { } dr(βc, c) h u(ξ) u(ξ c) dξ(βc, c) = f (ξ) dc c c dc d u(y) u(y c) + f (y)dy (13) ξ dc c The term n curly brackets s zero. Moreover, when the ntal values of h and c are such that h > u(1) u(1 c), we have ξ < 1, and the dervatve n the ntegral s postve due to the concavty of u. A proportonal ncrease of h and c thus ncreases R. 1.v When c s suffcently small, everyone wll choose treatment even f t s not covered by the health nsurance, mplyng that ξ = for c suffcently small. 1.v then mmedately follows from L Hoptal s Rule. References Allesandro, P., The optmal socal health nsurance wth supplementary prvate nsurance. Journal of Health Economcs 18, Blechrodt, H., Health utlty ndces and equty consderatons. Journal of Health Economcs 16, Blechrodt, H., Quggn, J., Lfe-cycle preferences over consumpton and health: when s cost-effectveness analyss equvalent to cost beneft analyss? Journal of Health Economcs 18, Blechrodt, H., Decdue, E., Quggn, J., 24. Equty weghts n the allocaton of health care: the rank-dependent QALY model. Journal of Health Economcs 23, Blomqvst, A., Johansson, P.-O., Economc effcency and mxed publc/prvate nsurance. Journal of Publc Economcs 66, Broome, J., Qalys. Journal of Publc Economcs 5, Culyer, A.J., Wagstaff, A., QALYs versus HYEs. Journal of Health Economcs 12, Dolan, P., The measurement of ndvdual utlty and socal welfare. Journal of Health Economcs 17, Garber, A.M., 2. Advances n cost-effectveness analyss of health nterventons. In: Culyer, A.J., Newhouse, J.P. (Eds.), Handbook of Health Economcs, vol. 1A. Elsever, Amsterdam, pp (Chapter 4). Garber, A.M., Phelps, C.E., Economc foundaton of cost-effectveness analyss. Journal of Health Economcs 16, Gafn, A., Brch, S., Mehrez, A., Economcs, health and health economcs: HYEs versus QALYs. Journal of Health Economcs 12, Hansen, B.O., Kedng, H., 22. Alternatve health nsurance schemes: a welfare comparson. Journal of Health Economcs 21, Harrs, J., QALYfyng the value of lfe. Journal of Medcal Ethcs 13, Hoel, M., 26. Cost-effectveness analyss n the health sector when there s a prvate alternatve to publc treatment. CESfo Economc Studes 52, Johannesson, M., Wensten, M.C., On the decson rules of cost-effectveness analyss. Journal of Health Economcs 12, Mehrez, A., Gafn, A., Qualty-adjusted lfe years, utlty theory, and healthy-years equvalents. Medcal Decson Makng 9, Nord, E., The QALY a measure of socal value rather than ndvdual utlty? Health Economcs 3, Olsen, J.A., Theores of justce and ther mplcatons for prorty settng n health care. Journal of Health Economcs 16, Østerdal, L.P., 25. Axoms for health care resource allocaton. Journal of Health Economcs 24, Vscus, W.K., Evans, W.N., 199. Utlty functons that depend on health status: estmates and economc mplcatons. Amercan Economc Revew 8, Wagstaff, A., QALYs and the equty-effcency trade-off. Journal of Health Economcs 1, Wensten, M.C., Stason, W.B., Foundatons of cost-effectveness analyss for health analyss and medcal practces. New England Journal of Medcne 296,