UNIVERSITY OF OSLO HEALTH ECONOMICS RESEARCH PROGRAMME

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1 UNIVERSITY OF OSLO HEALTH ECONOMICS RESEARCH PROGRAMME Allocatng health care resources when people are rsk averse wth respect to lfe tme Mchael Hoel Department of Economcs & The Ragnar Frsch Centre for Economc Research Workng Paper 2001: 10

2 Allocatng health care resources when people are rsk averse wth respect to lfe tme by Mchael Hoel* Health Economcs Research programme at the Unversty of Oslo HERO 2001 JEL classfcaton: D61, D81, H43, H51, I18 Key words: Health management, rsk averson, QALY, HYE * Department of Economcs, Unversty of Oslo, P.O. Box 1095, N-0317 Oslo, Norway. Emal: mchael.hoel@econ.uo.no 2001 HERO and the author Reproducton s permtted when the source s reffered to. Health Economcs Research programme at the Unversty of Oslo Fnancal support from The Research Councl of Norway s acknowledged. ISSN , ISBN

3 2 Abstract The crteron of cost-effectveness n health management may be gven a welfaretheoretcal ustfcaton f people are rsk neutral wth respect to lfe years. Wth rsk averson, the optmal allocaton of health expendtures change: Compared to the costeffectve allocaton, more resources should be allocated to health cases for whch the expected outcomes even after treatment are worse than average. The consequences of medcal nterventons are usually not known wth certanty. Gven ths type of uncertanty, smple applcaton of cost-effectveness analyss would recommend maxmzaton of expected health benefts gven the health budget. We show that when people are rsk averse wth respect to the number of lfe years they lve, the uncertanty assocated wth dfferent types of nterventons should play a role on allocatng the health budget.

4 3 1. Introducton To prortze among dfferent types of health expendtures, economsts often argue that cost-effectveness analyss should play an mportant role. Cost-effectveness s defned as the mnmum cost for a gven health beneft, or equvalently, maxmal health benefts for gven expendtures on health care. To be able to use such an analyss, one needs some measure of health benefts. Whle some analyses smply used the sum of lfe years saved due to a medcal nterventon, s more common so also take mprovements n health status nto account. A frequently used measure combnng the number of lves lved and the health qualty of these years s qualty adusted lfe years, or QALYs. The use of QALYs as a welfare measure, and as an approprate varable n cost-effectveness analyses, has been extensvely dscussed n the lterature. 1 A number of authors have crtczed the smple use of mnmum cost per QALY as a crteron for allocatng the health budget. A man crtcsm has been that the summaton of QALYs across ndvduals lacks a good ethcal or welfare theoretcal bass, see e.g. Harrs (1987), Wagstaff (1991), Nord (1994), Olsen (1997) and Dolan (1998). The crteron of cost-effectveness may, however, be gven a welfaretheoretcal ustfcaton under certan crcumstances: Imagne that a person must choose all health expendtures behnd the Rawlsan vel of gnorance,.e. before he/she knows hs/her health state. If ths person has preferences satsfyng the axoms of expected utlty theory, a cost-effectve allocaton of health expendtures wll be optmal, provded the person s rsk neutral wth respect to hs or her number of lfe years. However, rsk neutralty wth respect to lfe years s not a partcularly realstc assumpton (see e.g. the dscusson gven by Blechrodt (1995)). In ths paper we therefore consder how rsk averson wth respect to lfe years affects the optmal allocaton of health expendtures. 1 See e.g. Wensten and Stason (1977), Plskn et al. (1980), Mehrez and Gafn (1989), Broome (1993), Johannesson and Wensten (1993), Gabler and Phelps (1997), Blechrodt and Quggn (1999), and Nord (1999).

5 4 The consequences of medcal nteractons are usually not known wth certanty. Gven ths type of uncertanty, cost-effectveness analyss s typcally formulated as a recommendaton to maxmze expected health benefts gven the health budget. We show that when people are rsk averse wth respect to the number of lfe years they lve, the uncertanty assocated wth dfferent types of nterventons should play a role on allocatng the health budget. The rest of the paper s organzed as follows. The man assumpton regardng types of health states and preferences over these s gven n Secton 2. We avod the use of the QALY concept, as t can be argued that QALYs can only be a representaton of ndvdual lfe cycle preferences f people are rsk neutral wth respect to lfe years (see Plskn et al. (1980), Blechrodt (1995) and Blechrodt et al. (1997)). Instead, we use the concept of healthy year equvalents, or HYEs (see e.g Mehrez and Gafn (1989), Culyer and Wagstaff (1993), Gafn et al. (1993), and Blechrodt (1995) for a dscusson of ths concept) 2. In Secton 3 we derve the allocaton of health expendtures that would be chosen by a person decdng behnd the Rawlsan vel of gnorance. The specal cases of rsk neutralty and no uncertanty regardng health outcomes for a gven health state are dscussed n Sectons 4 and 5, before the propertes of the general case are dscussed n Secton 6. A bref summary of the man conclusons s gven n Secton Health states. A health profle s characterzed by a partcular lfetme and a partcular tme profle of varous health attrbutes durng ths lfetme. We assume that at the ex ante stage when people do not know ther health profle, they have an dentcal preference orderng over health profles. Moreover, we assume that for every health profle we can defne a healthy year equvalent (HYE),.e. a specfc number of years n perfect health that 2 If we are only concerned wth the number of lfe years, and not the health qualty of these years, both HYEs and QALYs are dentcal to the number of years lved. Moreover, f preferences have the property that they can

6 5 the decson-maker regards as equvalent to the health profle consdered. In ths way, each health profles may be summarzed by a sngle number measurng health standard by HYEs. We assume that there are m dfferent health states at the ex ante stage, wth health state havng probablty p of occurrng. Let h denote the number of HYEs n health state. Obvously h wll depend on what health care one s gven. Moreover, for a gven level of health care n a state, the number of HYEs s generally random. The number of HYEs n health state s thus gven by a functon h (c ;θ), where the uncertanty s represented by the parameter θ, wth each realzaton havng a known probablty q θ. In partcular, the HYEs n the absence of any treatment n health state s h (0;θ). By assumpton, the h functons are ncreasng n ther frst arguments. Moreover, we shall smplfy our analyss by assumng that all h functons are dfferentable and concave n ther frst argument. More precsely, usng h and h to denote frst and second order dervatves wth respect to c, we assume that h 0 and that h <0 for all h >0. In realty, as health expendture ncreases, there wll typcally be stages where one moves from one type of treatment to another. Therefore, the functon may be dscontnuous, and certanly non-dfferentable, at some ponts. However, for the general deas presented n ths paper ths s of mnor mportance. We therefore stck to our analytcally smple h functons. The decson-maker s assumed to have a von Neuman-Morgenstern utlty functon over HYEs, denoted by U(h (c ; θ)). Ths functon s assumed to be strctly ncreasng and concave. The case of rsk neutralty n lfe years s a lmtng case. For the case of rsk averson n lfe years U s strctly concave. be represented by the number of QALYs, then ths number s equal to the number of HYEs (see Blechrodt (1995).

7 6 3. The optmal allocaton of the health budget At the level of the socety, the probabltes p are shares of persons n each of the m health states. From the notaton of the prevous sectons we can wrte the government s budget constrant (per capta) as p c C (1) where C s an exogenously gven health budget. The decson-maker must choose all health expendtures c behnd a vel of gnorance, and does ths so that hs or her expected utlty s maxmzed 3. In other words, the followng maxmzaton problem s solved ( ) θ p θ θ subect to (1) (2) Maxmze q U h ( c ; ), whch gves the optmalty condton { U ( h ( c θ )) h '( c ; θ )} = λ ; E =1,,m (3) or, equvalently ( h ( c θ )) Eh '( c ; θ ) + cov{ U ( h ( c ; θ )), h ( c θ )} = λ E U ; =1,,m (4) ; Before dscussng the general case, we shall brefly consder two specal cases n the next two sectons. 3 A smlar approach s used by Pratt and Zeckhauser (1996) for a related problem.

8 7 4. The specal case of rsk neutralty If we have rsk neutralty wth respect to lfe years, U s constant, so that (3) may be rewrtten as Eh ' λ θ = U ' ( c ; ) (5) It follows from (5) that under rsk neutralty the optmal allocaton s characterzed by the expected margnal health benefts (measured by HYEs) of addtonal health expendtures beng the same for all types of health expendtures. Ths s the same allocaton as one would get from maxmzng the sum of expected HYEs for a gven budget for the sum of drect health expendtures. In the lterature, ths allocaton s often referred to as the cost-effectve allocaton, see e.g. Wensten and Stason (1977) for a further dscusson. 5. The specal case of a health state defnng a non-random health outcome As mentoned n the Introducton, t seems plausble that people are rsk averse wth respect to lfetme, and thus wth respect to HYEs. Let us frst gnore uncertanty regardng health outcomes for a gven health state. Formally, let h depend only on c, and not on θ. If ths s the case, we can rewrte (3) as ( h ( c )) h '( c = λ U ) =1,,m (6) It s clear from (6) (and the concavty of the functons U and h ) that the health budget should be allocated so that the margnal health benefts (measured by HYEs) of addtonal health expendtures should be hgher n states where the equlbrum number of HYEs s low than when ths number s hgh. In other words, rsk averson mples that health expendtures drected towards more serous health problems (measured by HYEs) should be gven a hgher prorty than they would n the smple case of cost effectveness.

9 8 6. The general case of uncertan health consequences of health care We want to nvestgate the consequences of uncertanty of a specfc health defect. To do ths, let the functons h (c ;θ) be gven for all. We compare the case of uncertanty n the relatonshp between c and h wth the case of certanty where the certan relatonshp between health expendture and HYEs s equal to the expected value of the functon h (c ;θ), denoted Eh (c ;θ). (The latter case corresponds to the degenerate case n whch the functon values h (c ;θ) are ndependent of the value of θ). Snce we are changng the h -functon for only one, t s reasonable to expect λ to have (approxmately) the same value for the two cases compared. Assume frst that the margnal health beneft of health expendtures n health state s non-random,.e. that h (c ;θ) s ndependent of θ. Ths means that although health outcomes may be uncertan, the dfferences n health outcomes due to dfferent levels of health care are certan. In ths case the covarance term n (4) s zero. The l.h.s. of (4) s larger or smaller wth uncertanty than wthout, dependng on whether E(U ) s larger or smaller under uncertanty than under certanty. Ths n turn depends on the sgn of U. If U s postve (as t s e.g. under constant relatve rsk averson), then E(U ) s larger under uncertanty than under certanty. From the second order condton of the optmzaton problem, t therefore follows that health expendture drected towards health defect should be hgher under uncertanty than under certanty. The opposte wll be true f U s negatve. To see the mportance of the covarance term n (4), let us now assume that U =0. From the result above we know that f the covarance term n (4) s zero, then the health expendture drected towards health defect should not be affected by the presence of uncertanty. Consder next the case n whch { ( h ( c ; θ ), h '( c ; θ )} 0 cov U > (7)

10 9 For any level of health expendture, U s lower for good values of θ than for bad values of θ (snce U <0). The assumpton (7) therefore means that the margnal beneft of health care (measured n HYEs) s lower for good health outcomes than for low health outcomes. In other words, health care reduces the uncertanty of the health outcome. When U =0 and (7) holds, the l.h.s. of (4) s hgher under uncertanty than under certanty. From the second order condton of the optmzaton problem, t therefore follows that health expendtures drected towards health defect should be hgher under uncertanty than under certanty. Consder the opposte case from (7),.e. { ( h ( c ; θ ), h '( c ; θ )} 0 cov U < (8) In ths case health care ncreases the uncertanty of the health outcome. We then get the opposte concluson from above: Health expendtures drected towards health defect should be lower under uncertanty than under certanty when U =0 and (9) holds. 7. A comparson wth a smple rule of cost-effectveness We have shown that a smple type of cost-effectveness s optmal f the decsonmaker s rsk neutral wth respect to lfe years. Wth rsk averson, the optmal allocaton of health expendtures changed. The analyss ndcates how the optmal allocaton devates from the cost-effectve allocaton. Loosely speakng, we have shown the followng: More resources should be allocated to health cases for whch the expected outcomes even after treatment are worse than average.

11 10 If the utlty functon has the property that U >0 (mpled by e.g. constant rsk averson), more resources should be allocated to cases for whch the health outcome s more uncertan than average, unless the treatment ncreases ths uncertanty. Even f medcal treatment for a partcular health defect ncreases the uncertanty of the health outcome, t s not obvous that less resources should be allocated to such a health defect than to a health defect wth a less uncertan development, snce we cannot rule out the possblty that U <0.

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