CENTRE FOR HEALTH ECONOMICS Disouning for Heal Effes in Cos Benefi and Cos Effeiveness Analysis Hug Gravelle and Dave Smi CHE Tenial Paper Series 20
CENTRE FOR HEALTH ECONOMICS TECHNICAL PAPER SERIES Te Cenre for Heal Eonomis as a well esablised Disussion Paper series wi was originally oneived as a means of irulaing ideas for disussion and debae amongs a wide readersip a inluded eal eonomiss as well as ose working wiin e NHS and parmaeuial indusry. Te inroduion of a Tenial Paper Series offers a furer means by wi e Cenre s resear an be disseminaed. Te Tenial Paper Series publises papers a are likely o be of speifi ineres o a relaively speialis audiene, for eample papers dealing wi omple issues a assume ig levels of prior knowledge, or ose a make eensive use of sopisiaed maemaial or saisial eniques. Te onen of e paper and is forma are enirely e responsibiliy of e auor, and papers publised in e Tenial Paper series are no subje o peer-review or ediorial onrol, unlike ose produed in e Disussion Paper series. Offers of furer papers, and requess for informaion sould be direed o Franes Sarp in e Publiaions Offie, Cenre for Heal Eonomis, Universiy of York. Oober 2000 H. Gravelle and D. Smi 2
Disouning for Heal Effes in Cos Benefi and Cos Effeiveness Analysis Hug Gravelle * Dave Smi ** Absra: Wen eal effes an be valued in moneary erms, as in CBA, ey sould be disouned a e same rae as oss. If eal effes are measured in quaniies (eg QALYs), as in CEA, and e value of eal effes is inreasing over ime, en disouning e volume of eal effes a a lower rae an oss is a valid meod of aking aoun of e inrease in e fuure value of eal effes. We presen individualisi and welfare models o argue a e rae of grow of e value of eal effes g v is posiive. Te welfare model suggess a g v is a weiged average of e rae of grow of e value of e dire effe of eal on uiliy, e grow rae of inome, and e grow rae of inome imes e elasiiy of e marginal uiliy of inome. We also sow a e Keeler-Crein parado, ofen used as an argumen agains disouning eal effes a a lower rae an oss, as no relevane for e oie of disoun rae in CEA. Keywords: disouning, eonomi evaluaion, value of eal JEL Code: I8, H43 * Naional Primary Care Resear and Developmen Cenre, Cenre for Heal Eonomis, Universiy of York, Heslingon, York YO0 5DD; email: g8@york.a.uk. We are graeful for ommens from pariipans in e CHE eonomi evaluaion seminar and e Alan Williams Blak Bull seminar. Suppor from e European Commission under onra F4P-CT96-0056 and from e Deparmen of Heal o e Naional Primary Care Resear and Developmen is aknowledged. Te views epressed are no neessarily ose of e funders. ** Cenre for Heal Eonomis; email ds2@york.a.uk 3
Inroduion Tere is an ongoing meodologial debae abou e appropriae way o ake aoun of fuure eal effes in evaluaions. Te majoriy view, mos reenly and ompreensively epounded in Lipsomb, Weinsein and Torrene (995), is a benefis and oss sould be disouned a e same rae. Te view dominaes e reommendaions on disouning by governmen agenies, regulaory bodies, learned journals and leading eal eonomis es (Smi and Gravelle, 2000). A smaller body of lieraure favours a lower rae for eal effes an for oss. Te mos influenial eample is Parsonage and Neuburger (992), wrien by wo UK governmen eonomiss and laer refleed in e UK Deparmen of Heal reommendaions for evaluaion of eal affeing inervenions (Deparmen of Heal, 996). We sugges in is paper a a leas some of e differenes beween e wo sools of oug arise from differen implii assumpions abou e deision one. We sow in seion 2 a os benefi analysis (CBA) of inervenions affeing eal requires proedures wi direly or indirely are equivalen o disouning e value of fuure eal effes a e same rae as oss (Cropper and Porney, 990; Cropper and Sussman, 990; Jones-Lee and Loomes, 995). In os effeiveness analysis (CEA), were eal effes are measured in volume raer an value erms, one valid meod of allowing for grow in e value of fuure eal effes is o disoun e volume of fuure eal effes a r = r - g v, were r is e disoun rae applied o oss and g v e rae of grow of e value of a uni of eal. An equivalen proedure in CEA is o adjus e volume of eal effes by g v and o disoun a e same rae as oss. Tus, providing e one is orrely speified and aoun properly aken of e anging value of eal, e wo views an be reoniled. Unforunaely, e preponderane of offiial and semi-offiial reommendaions for CEA is o use e same disoun rae for oss and benefis and no o adjus e volume of eal effes o allow for e grow in eir value (Smi and Gravelle, 2000). One barrier o reoniliaion of e wo views on disouning is e parado se ou in Keeler and Crein (983). Tey sow a under CEA erain ypes of worwile projes will be indefiniely posponed unless e same disoun rae is used for oss and eal effes. Previous responses o e Keeler-Crein parado ave argued a su projes are very peuliar and never our in praie or a e parado does no arise if onsrains on funding in any period are reognised (Parsonage and Neuburger, 992; van Hou, 998). Bu e disouning proedure used in assessing projes sould give e orre answer irrespeive of e proje. Te Keeler-Crein parado poins o a logial problem wi using differen disoun raes for oss and eal effes in CEA and anno be dismissed on empirial or praial grounds. We demonsrae in seion 3 a e Keeler-Crein parado reveals a fundamenal diffiuly wi CEA, oug no wi CBA. I does no arise beause of e use of differen disoun raes for oss and eal effes. Te parado is simply irrelevan o e oie of disoun rae for eal effes.
Te ruial issue is weer e value of eal effes is onsan over ime. A number of auors (Parsonage and Neuburger, 992; Visusi, 995; van Hou, 998; Brouwer, van Hou and Ruen, 2000) ave suggesed a e value of eal grows over ime and a as a onsequene e disoun rae on eal effes sould be less an e disoun rae on oss. Lipsomb, Weinsein and Torrene (995) are e mos influenial proponens of e majoriy view a oss and eal effes sould be disouned a e same rae in CEA. Tey reognise e possibiliy a e value of eal may be inreasing over ime and is impliaions for disouning eal effes. Bu ey onlude a e ase for su global adjusmens in CEA ondued from a soieal perspeive as ye o be fully made, in our judgemen. (Lipsomb, Weinsein and Torrene, 995, page 234.) Teir reluane o aep e impliaions of a posiive grow rae in e value of eal may in par be due o e absene in e lieraure o dae of argumens based on eplii models wi onvenional assumpions. Aordingly, we se ou in seion 4 wo simple models o underpin e more informal argumens wi sugges a e value of eal grows over ime. Te firs is based on a beavioural model of individual oie of eal affeing aiviies. Te seond uses e soial welfare framework familiar from disussions of e oie of e soial disoun rae (Layard and Glaiser, 994, Inroduion). Te framework as been used o argue a e disoun rae o be applied o fuure inome anges sould be r = ρ + gε, were ρ is e rae of disoun applied o fuure uiliy, g is e grow rae in inome and ε is e elasiiy of e marginal uiliy of inome. We eend e soial welfare framework o inorporae eal wi is valued in is own rig and wi may affe inome. We sow a e rae of grow of e value of eal (g v ) is a weiged average of e rae of grow of e dire uiliy effe of eal (k), e rae of grow of inome g, and e rae of grow of inome imes e elasiiy of marginal uiliy of inome (gε). Te weigs depend on e een o wi e inome loss from ill eal is borne by e individual or is overed by insurane. If eal as no effe on inome and e uiliy effe of eal is onsan over ime, e value of fuure eal effes in erms of fuure inome grows a e rae a wi a e marginal uiliy of inome falls over ime: g v = gε. As a onsequene e only reason for disouning fuure eal effes would be a fuure uiliy is inrinsially less valuable. Te disoun rae on eal effes (r ) would be e rae a wi fuure uiliy is disouned: r = r gv = ρ < r. In anoer speial ase, wen eal affes inome bu ad no dire uiliy effe, g v is equal o e grow rae in inome g and e disoun rae on eal effes is r = ρ + (ε ) g < r. Typially i is suggesed (Arrow, 995) a ρ is abou %, ε around 2 and g is 2% o 2.5%, yielding disoun raes on oss of 5% o 6%. Te disoun rae on eal effes in e wo speial ases is abou % wen eal as no effe on inome and abou 3% o 3.5% wen i only affes inome. Allowing for su differenes beween disoun raes on eal and oss an ave marked impliaions for rankings of inervenions (Parsonage and Neuburger, 992) and i is imporan a e orre disouning proedures be used. Bu e overwelming majoriy of publised CEAs of eal are inervenions use e same 2
disoun rae for eal and oss (Smi and Gravelle, 2000). We ope a e argumens in e urren paper will elp o onvine evaluaors of e need o ake aoun of e impliaions of e grow in e value of eal for eir oie of disoun rae. 2 Disouning for deision making: wo equivalen proedures Deisions abou inervenions wi onsequenes for ime sreams of oss (reduions in inome) and eal requires a se of judgemens abou e relaive values of eal and inome a differen daes. Consider a wo period eample of an inervenion wi anges presen and fuure oss by 0 and and e quaniies of presen and fuure eal by 0 and. We an summarise value judgemens or soial preferenes over inome and eal sreams in a soial welfare funion W y,, y, ) were y, are inome and ( 0 0 eal in period (Jones-Lee and Loomes, 995). Te welfare funion embodies judgemens wi deermine e rae a wi we are willing o sarifie one good (eal or inome a some dae) for anoer. Te marginal soial valuaion of eal in period in erms of period inome is e rae a wi we are willing o give up period inome in eange for period eal. I is e (negaive of) e soial marginal rae subsiuion beween inome and eal in period : W v () W were W y = W / is e marginal soial welfare from an inrease in eal in period and W is e marginal soial welfare from inome in period. Similarly for e marginal y value of fuure inome in erms of urren inome Wy + r W y 0 and e marginal value of fuure eal in erms of urren eal W + r W 0 (2) (3) Te value judgemens embodied in W define e soial rae of disoun on inome or oss r in erms of e willingness o sarifie urren inome for fuure inome and e soial rae of disoun on eal r in erms of e willingness o sarifie urren eal for fuure eal. In general, e marginal soial welfare from anges in eal or inome in any period depend on bo inome and eal in a period and possibly on inome and eal in oer periods. Unil we speify bo e form of e welfare funion and e levels of eal and inome we do no know weer r is greaer or less an r. Figure is illusraes ese definiions. Aloug ere are four marginal valuaions in e wo _ period ase ( v v,( + r ),( + r ) ey are no independen: one ree of em are 0, ) speified e oer is deermined. Consiseny requires a e marginal value of one good (eal or inome) in erms of anoer is e same waever e roue by wi ey are ompared. 3
2. Cos benefi analysis To deide weer an inervenion is worwile e marginal soial valuaions are used o onver all e onsequenes ino equivalen amouns of a ommon uni of aoun (inome or eal a some dae). Convenionally e uni of aoun is inome in e presen period., 0 and mus be onvered ino equivalen anges in 0 wi are summed o give e presen value of e inervenion. Su os benefi analysis (CBA) is no e prevalen form of evaluaion in eal eonomis beause of e diffiuly in valuing eal effes. However i is insruive o sar wi an ouline of disouning in CBA beause i as an eplii welfare eorei foundaion. Te presen value of e inervenion an be derived in wo equivalen ways. Te dire proedure values eal effes in ea period in erms of inome of a period and en disouns e fuure value a e rae of ineres on inome r. Te presen value of e inervenion under e dire proedure is v + v0 0 0 r ( + ) ( + r ) (4) Te indire meod of alulaing e presen value of e inervenion differs from e dire proedure in is reamen of. I onvers e ange in fuure eal ino an equivalen ange in urren eal and en applies e value of urren eal in erms of urren inome. Te presen value of e inervenion wi e indire proedure is v 0 + v0 0 0 r ( + ) ( + r ) (5) Sine e wo proedures are equivalen (4) and (5) mus be equal, so a v v0 = or + r + r + r + r v = v 0 (6) Te disoun raes on eal and oss are e same (r = r ) only if e value of eal in a period in erms of inome in a period is e same in bo periods (v = v 0 ). If e value plaed on eal grows over ime (v 0 < v ) en ere mus be a lower disoun rae on eal effes an on inome or oss (r < r ) and vie versa. Defining g v = (v - v 0 )/v 0 as e grow rae of e value of eal, (6) an be rearranged o ge r v0 = ( + r ) = v v 0 v r + r = + 0 r gv ( + g ) ( ) + g (7) v v Some of e disagreemens over disouning of eal effes may arise from a failure o spell ou 4
weer one is referring o disouning of e value of eal effes (v ) in a fuure period in erms of e inome of a period or o e quaniy of fuure eal effes ( ) wa is being assumed abou e rae of grow of e value of eal effes (g v ). Wen e value of eal effes is disouned e rae of disoun for inome r sould be used. If e volume of eal effes is disouned e rae of disoun for eal effes r is orre. Te disoun rae on e quaniy of eal effes is less an e disoun rae on oss (r < r ) if e grow rae in e value of eal is posiive (g v > 0). Te wo proedures require ealy e same informaion and judgemens abou e marginal valuaions of fuure os and eal effes in erms of presen inome. Te firs proedure, valuing ea in a period in erms of e inome of a period and en applying e disoun rae appropriae for inomes is peraps more inuiive. I is also in line wi e reommendaions of Feldsein (972). Feldsein suggesed a wen an inervenion as ompliaed onsequenes beause of is knok on effes on fuure invesmen, all e effes of an inervenion be epressed in erms of onsumpion anges wi are en disouned a e rae of disoun appropriae for onsumpion. 2.2 Disouning in CEA In os effeiveness analysis (CEA) e invesigaor is limied o quanifying e eal effes and does no plae a moneary value on em. Te aim is derive an inremenal os effeiveness raio (ICER) for e inervenion, defined as e disouned presen value of inremenal oss divided by e disouned sum of inremenal eal effes. Wen projes are muually elusive and quesions of e sale or divisibiliy of projes an be ignored, inervenions wi lower ICERs are preferred o ose wi iger ICERs. Inervenions sould be underaken wen eir ICER is less an some riial value λ: ( + r ) o ( + r ) + 0 + 0 λ (8) Te ruial issue, over wi mos of e debae in e eal eonomis lieraure on disouning of eal effes as foused, is e disoun raer o o be applied o eal effes in CEA o alulae e ICER. Te CEA rierion is used wen ere is insuffiien informaion on e value of eal effes o ondu a CBA. I seems reasonable o require a e CEA rierion would yield e same deisions as CBA if ere was informaion on e value of eal effes. 2 CBA would aep projes wose disouned presen value, given by (4) or (5), is posiive. Rearranging e ICER deision rule (8) e proje is aeped under CEA if λ + λ o 0 0 0 (9) ( + r ) ( + r ) 5
Using (6) and (4) or (5), e ICER rierion is equivalen o e CBA deision rule if and only if λ = v 0 (0) o v0 r = ( + r ) = r r gv v () Hene in os effeiveness analysis eal effes sould be disouned a e rae r o = r r - g v. Te same disoun rae sould be applied o eal effes in CEA as in e indire proedure under CBA. 3 We argue in seion 4 a e value of fuure eal in erms of fuure inome grows over ime (g v > 0), so a fuure eal effes sould be disouned a a lower rae an oss if no adjusmen is made o e volume of eal effes o refle eir growing value over ime. Te alernaive, dire, way o ake aoun of e anging value of fuure eal effes in CEA is o adjus e quaniy of effes. Te real quaniy of fuure eal effes an be defined as ˆ = θ, were θ is an adjusmen faor o allow for e ange in e value of fuure effes. Te CEA rule wi e same disoun rae applied o oss and o e real quaniy of eal effes is o aep e proje if: λ θ + λ 0 0 0 (2) ( + r ) ( + r ) wi is equivalen o e CBA rule if λ = v 0 (3) θ = ( + g ) (4) v Te impliaions of grow in e value of eal for CEA are reognised in e lieraure (Lipsomb, Weinsein and Torrane, 996; Parsonage and Neuburger, 992; Visusi, 995; van Hou, 998) bu ave made no impa on CEA praie (Smi and Gravelle, 2000). Visusi (995) and Parsonage and Neuburger (992) sugges adjusing e disoun rae o allow for e grow in e value of eal effes. Lipsomb, Weinsein and Torrane (995) favour dire adjusmen of e volume of eal effes. Tere are no logial grounds for preferring one approa o e oer. Te dire adjusmen as e advanage of dealing wi issue of e grow in e value of eal epliily and separaing i from e issue of e rae of disoun o be applied in CEA. If e value of eal is growing over ime some meod of allowing for i in CEA mus be found. I is simply inorre o use e same disoun rae for eal and os effes if e value of eal is growing. Unforunaely mos of e offiial reommendaions do no ake aoun of e possibiliy a g v is posiive and sugges a e same disoun rae be used for oss and eal effes (Smi and Gravelle, 2000). 6
2.4 Iner and inra-generaional disouning In disussion of weer eal effes ourring a dae + sould be given e same weig as eal effes ourring a dae, i is imporan o be lear abou weer one is omparing e effes on individuals wo will be aged a years a bo daes (inergeneraional effes) or individuals wo will be aged a a dae and a+ a dae + (inrageneraional effes). Te value v of e eal effes of an inervenion may depend on e age of e individuals affeed as well as e dae a wi ey our. A number of auors ave suggesed a meod disouning fuure eal effes wi disinguises iming and generaional aspes (Lipsomb, 989; Cropper and Sussman, 990). Te effes on individuals aged a a dae are disouned bak o e bir dae of e oor a dae -a and en e disouned value a dae -a are disouned bak o e deision daa 0. If e same disoun rae is applied a bo sages e proedure is equivalen o sandard approaes. Te proedure allows e possibiliy of using differen disoun raes in e wo sages. For eample, we mig be willing o respe individuals ineremporal preferenes as regards anges in eir eal or inome and use disoun raes derived from sudies of eir beaviour o disoun inome and os anges affeing em. Bu we may feel a ey undervalue e welfare of fuure generaions in eir ineremporal deisions (Sen, 967) and wis o use differen ineres raes wen disouning eir presen values (a oor bir daes) bak o e presen dae. Te wo sage proedure provides a nea meod of reoniling respe for individual preferenes over deisions wi affe em direly wi a soial onern for iner-generaional equiy. Reogniion of e disinion beween iner and inra generaional disouning does no aler e onlusions abou e relaionsip beween e disoun rae for eal effes and oss. Fuure eal effes a dae aruing o individuals aged a sould be aken ino aoun by valuing em in erms of e inome of aged a individuals a dae and en disouned bak o dae -a a e rae used for inome of aged a individuals a dae. Te disoun rae applied o e weal of e oor born a dae a- an en be applied o alulae e presen value of e eal effes. 3 Keeler-Crein parado Keeler and Crein (983) make a mu ied argumen for disouning eal effes and oss a e same rae (r o = r ) in CEA. Tey onsider e following iming problem. A single period proje an be underaken one only. Te oss and eal effes are e same waever e period in wi e proje is underaken. Te deision problem is o oose now e period in wi e proje will underaken. If e proje is underaken in period e disouned presen value of e os is /(+r ) and e disouned eal effe is /(+r o ). Te ICER for e proje underaken a dae is o + r = (5) ( + r ) ( + r ) + r 7
wi dereases wi if r o < r. If r o < r e ICER indiaes a e proje beomes more worwile e longer i is delayed. Wi an infinie ime orizon e CEA rierion will lead o e proje being deferred indefiniely even if i as a very favourable os effeiveness raio ( / ) if underaken in e presen period. Keeler and Crein (983) argue a, beause e deision maker is paralysed if r o < r in su projes, i is orre o se r o = r in CEA. Wi r o = r e deision maker is indifferen as o e iming of e proje using e ICER rierion and would be willing o pik a sar dae a random. We disagree: e reason wy Keeler-Crein projes presen diffiulies under e CEA rierion is a e CEA deision rule is inerenly inomplee and anno ope wi issues of e iming of deisions. Te soluion, suggesed in Keeler and Crein (983), of using e same disoun rae on eal and os effes in CEA fails o address e underlying problem, wi is in e CEA rule, no e rae of disoun. Keeler-Crein projes do no presen a problem wen e CBA deision rule is used and ave no impliaions for e oie of disoun rae o be used in CEA. Suppose a e Keeler-Crein proje is wor doing in period 0 under e CBA rule: v 0 - > 0. I is beer under e CBA rierion o defer e proje from period o period + if v+ v + > ( + r ) ( + r ) (6) wi is equivalen, sine v = v 0 (+g v ), o ( ) ( v ) D r v 0 + gv r g > 0 (7) Te beaviour of D depends on e grow rae in e value of eal effes. Tere are ree possible ranges of g v wi differen impliaions for ow D varies over ime and erefore for e opimal iming of e Keeler-Crein proje: (i) 0 gv g$. For small enoug grow raes D is negaive for all : e proje sould be done immediaely in period 0. 4 For eample, if g v is zero D = r - v 0 r < 0 or, equivalenly, wi v 0 = v = v, v ) /( + r ) < v ( (ii) $g < gv < r. For inermediae grow raes D is posiive for small and en beomes negaive: e presen value of e proje a firs inreases wi and en dereases. Hene i is opimal o delay e proje bu no indefiniely. For eample if = 00, = 70, v = 0 2, r = 0.06, g v = 0.025, en e proje sould be delayed unil = 8. 8
(iii) r gv. Wen e grow rae of e value of eal eeeds e disoun rae on oss D is posiive for all : e presen value of e proje inreases e longer i is delayed. In ase (iii) were r gv e deision maker would be paralysed under e CBA deision rule sine e presen value of e proje inreases e laer i is underaken. Case (iii) seems igly implausible sine i implies a one sould be willing o sarifie an arbirarily large amoun of urren inome o aieve a perpeual inrease in eal from dae in e fuure, no maer ow disan and no maer ow small e inrease in eal. 5 Te CEA rule wi e Keeler-Crein reommendaion a e deision maker use a disoun rae on eal of r o = r leads o e deision maker being indifferen as o e sar dae for Keeler-Crein projes. Only by ane will se en oose e orre sar dae. If e deision maker onfroned wi Keeler-Crein projes uses a CEA rule bu wi r o = r = r - g v se will be indifferen as o e sar dae wen g v = 0 sine e disouned os effeiveness raio will be onsan wi respe o e sar dae. Te orre deision is o underake e proje immediaely. Wen g v > 0 se will be led o defer e deision indefiniely wi is inorre eep in e igly implausible ase in wi r gv. Hene e CEA rule will lead o deisions wi are sub opimal in e sense of no maimising e disouned presen value of e proje. Tus CEA leads o inorre deisions wi Keeler-Crein projes, irrespeive of e oie of disoun rae. Te Keeler-Crein parado poins o a diffiuly wi CEA for erain raer unusual projes bu is irrelevan for e debae abou e appropriae rae of disoun for eal effes and os effes. 6 We onlude a, as far as e oie of disoun raes for eal in CEA is onerned, is parado is deeased. 4 Is e value of eal onsan over ime? Te grow rae in e value of eal effes g v is ruial for e oie of disoun rae. We ouline wo models, one individualisi and one soieal, o argue a e value of eal grows over ime. 4. Beavioural model Individuals an aler eir eal, or eir probabiliy disribuions over eal saes, roug onsumpion of eal are, eir lifesyles (Burgess and Propper, 998) and oupaional oies (Visusi and Moore, 989; Moore and Visusi, 990). We derive an epression for e value of eal v in erms of e individual s preferenes, eal enology and marke pries and en onsider ow i anges over ime. 4.. Value of eal Sine e lieraure as been summarised before (Joansson, 995) we an be brief. Consider a very simple eample of an individual wi inome y wo an buy eal are wi improves eal (, s) ( > 0). Te same onlusions old for any aiviy wi 9
affes eal and wi may direly affe uiliy. Te parameer s is a measure of e sae of medial enology. Uiliy is u( y p, (, s), ), were p is e prie of eal are. Te firs order ondiion for opimal onsumpion of eal are is u + u = pu. Dividing roug by e marginal uiliy of inome gives e marginal willingness o pay for eal as p my v = (8) (, s) (, s) Sine a uni inrease in inreases eal by (, s), a uni inrease in eal permis a reduion in onsumpion of by / wils keeping eal onsan. An addiional amoun p/ is freed o spend on oer goods. Tere is also e effe of e reduion in on uiliy. m = u / u is e marginal willingness o pay for an addiional uni of ignoring is effe y y on eal. If onsuming eal are direly redues uiliy en m y < 0 and e valuaion of eal in erms of urren inome is inreased. Te prie of e ommodiy p is observable bu, even if e eal are good ad no dire effe on uiliy ( m = 0), e marginal effe of on eal mus be known o alulae v. If y e eal are good also direly affes uiliy, e poenially observable p/ will under or over sae e marginal value of e eal ange. Pessimisi onlusions abou using marke pries o reveal e value individuals plae on eir eal are unanged even wen we ake aoun of insurane markes were ey an rade inome wen well agains inome wen sik or apial markes were ey an rade inome a one dae for inome a anoer (Gravelle, 2000). 4..2 Grow in value of eal Aemps o value eal by revealed or saed preferene eniques yield a wide range of esimaes. Su diffiulies in measuring e value of eal are one jusifiaion for os effeiveness analysis. However, wils CEA does no require a eal be valued, i does require an esimae of e grow in e value eal. Wide variaions in e esimaes of e value of eal render esimaes of is grow rae even more problemai. Consider e epression (8) for e valuaion of eal by an individual wo onsumes eal are. 7 Suppose a eal are as no dire effe on uiliy (m y = 0 ) so a v = p / (, s). Even in is simple ase v will ange over ime for ree reasons: anges in e prie of, anges in amoun of onsumed, and sifs in e eal produion funion due o enial progress in eal are. Te opimal amoun of are onsumed will vary over ime wi e produiviy of are, is prie and inome. Te grow rae in e value of eal, wen m = 0 an be wrien as 8 y y 0
g v = g e e g e g e g e g p + + p p y s s (9) s s were g p is e rae of grow in eal are prie, g e rae of grow of inome, g s is e rae of enial progress wi sifs e eal produion funion, e is e elasiiy of e marginal produiviy of eal are wi respe o, s e is e elasiiy of e marginal produiviy of eal are wi respe o e enology sif faor s, and e, e, e are elasiiies of e onsumpion of eal are wi respe o is prie, inome and enology. Te rae of grow in e value of eal depends on ree faors: e grow rae in e prie of are, e rae of grow of eal are onsumpion and is effe on e marginal produiviy of are and e rae of enial progress and is effe on e marginal produiviy of are. Te grow rae of e prie of are is plausibly posiive: g p > 0. Te sign of e seond erm in (9) is ambiguous. Aloug e marginal produiviy of eal are is diminising ( < 0), e erms in e brake in e seond erm are plausibly of differen signs: onsumpion of are delines wi is prie, and inreases wi inome. Te effe of enial progress on e demand for are is ambiguous sine e inrease in e marginal produiviy of are ends o inrease demand and e inrease in eal o redue i. Te evidene suggess a e onsumpion of eal are inreases over ime (Blomqvis and Carer, 997), so a e seond faor (inreases in eal are reduing is marginal produiviy) ends o inrease e value of v = p/ (,s). Beause enial progress inreases (,s), e ird faor ends o redue g v wi ould be erefore be negaive or posiive. We need assumpions abou e magniudes of e erms in (9) as well as eir signs. A posiive inome elasiiy of demand for eal or eal improving goods is neier suffiien nor neessary for e value of eal o inrease over ime. However, if only inome anges over ime (9) redues o p y s g v = e e g > 0 (20) y so a e grow rae in e value of eal depends on e rae of grow of inome (g > 0), e inome elasiiy of demand for eal are wi, from e inreasing sares of inome spen on eal as inome inreases is greaer an uniy (e y >), and e elasiiy of e marginal produiviy of eal e < 0. Esimaing e grow rae in e value of eal is learly diffiul, even in e simple ase we eamined ere, bu we believe a i is likely o be posiive. 4.2 Welfare model We an rea a similar onlusion using an enirely differen approa in wi we speify a soial welfare funion and use i derive e value of anges in fuure eal in erms of
fuure inome and ene o derive e grow rae of e value of eal. Te approa is insruive beause i is an eension of a well known framework for disussion of e rae of soial ime preferene for onsumpion or inome wen only inome eners e soial welfare funion and ere is no unerainy (Layard and Glaiser, 994). 9 Suppose a all individuals live for one period, are idenial eep for e period in wi ey live and ere are an equal number of individuals in ea period. Te resuls are no maerially differen bu are more ompliaed o derive wen individuals live for more an one period (Gravelle, 2000). Te soial welfare funion an be wrien in per apia erms as = 0 EU W = β (2) were e epeed uiliy of e represenaive individual is U = π u y ) + ( π ) u ( y ) (22) f( d d Te pure disoun faor on uiliy β = /( + ρ) allows for e possibiliy a a ange in e uiliy of a generaion ouns for less solely beause i arises a a laer dae. Sae f is e ealy or fi sae in a e individual is beer off in sae f oer ings (inome) being equal: u f (y) > u d (y). We an inerpre π as e probabiliy of e eal sae in a world were eal ouomes are independen or as e proporion of e populaion wo are ealy. Endowed inome wen diseased is y and wen ealy is y + l were l is e effe of ill eal on inome. We assume a endowed inome in bo saes grows a e rae g. To mainain omparabiliy wi e lieraure on e disoun rae under erainy, assume a e uiliy funions ave onsan elasiiy of marginal uiliy ε ε y f yd u = + K; ud = φ (23) ε ε were e elasiiy of e marginal uiliy of inome is ε and if ε > uiliy is bounded above. Wen φ = e marginal uiliy of inome is no direly dependen on e sae: u ( y) = u ( y), bu ere is a dire uiliy loss of K from being in e unealy sae. Wi f d 0 < φ < marginal uiliy is smaller wen ill an wen ealy and e ase in wi e sae d is dea, raer an diseased ould be allowed for wi φ = 0. 0 We use e welfare funion o derive e disoun rae on inome (oss), e value of eal in erms of inome and e disoun rae on eal. We firs illusrae e proedure for a simple ase in wi ere is insurane and e marginal uiliy of inome is no sae dependen, so a e opimal insurane seme is full over agains inome losses from ill eal. A planner wo as aess o an auarially fair insurane seme in ea period ooses y f, y d o maimises W subje o e insurane pool breaking even in ea period. Te Lagrangean for e problem is L = W + σ π ( y + l y ) + ( π )( y y )] (24) [ f d 2
Te firs order ondiions are β π u ( y ) σ π = 0; β ( π ) u ( y ) σ ( π ) = 0 (25) f f d d Given e assumpion a e marginal uiliy of inome is sae independen, e opimal insurane seme gives e insured an inome in ea sae equal o epeed inome: * * * y = y = y = y +πl (26) f d Te soial value of addiional inome y is σ marginal uiliy in period. = β ) ε E u = β ( * i y were ui E is epeed Te marginal soial value of an addiional uni of period + inome in erms of period inome (e inome disoun faor) is, using e envelope eorem on e Lagrangean (24) + r W / y W / y σ β Eu β( y + * ε + + i+ + = = = = β( ε σ β ( ) + g * Eui y ) ) ε (27) were g is e grow rae in inome. Using r ln( /(+ r )) and remembering a β = /(+ρ), e disoun rae on inome is r ρ + gε (28) We ave e sandard resul (Layard and Glaiser, 994) a e planner sould disoun fuure inome relaive o urren inome beause i is less valuable. Firs, fuure uiliy is valued less igly per se an urren uiliy (ρ). Seond, e inrease in fuure uiliy as inome is ransferred from e urren o e fuure period is smaller an e reduion in urren uiliy beause inome grows over ime (g) and e marginal uiliy of falls as inome inreases (ε). Te same resul an be derived for e ase in wi marginal uiliy is sae dependen. I also olds if we drop e assumpion of opimal insurane and reinerpre L as a soial welfare funion. In is ase we also inerpre π y + l y ) + ( π )( y y ) as e ( f d epeed surplus from a possibly subopimal insurane seme, and assume a a erain as e same soial value werever i arues: σ = β E u. 2 Using e more general inerpreaion of (24) as a soial welfare funion, e period value of an inrease in eal (an inrease in e probabiliy of e ealy sae) in erms of inome in period is ε ε L / π β ( u f ud ) + σi K ( y f yd )( ε) v = = + + I (29) L / y σ Eu Eu were I is e gross amoun of ompensaion e individual is paid if unealy (e amoun of over agains ill eal). An inrease is eal is valuable for wo reasons: Firs, i raises uiliy direly (e firs erms in e las epression in (29)). Te value of e dire inrease in uiliy in erms of urren inome depends on e size of e uiliy gain and on e marginal uiliy of inome. A iger levels of inome marginal uiliy is smaller and ene e moneary value of a given inrease in eal is greaer. i i i 3
Seond, ere is an inrease in epeed inome beause ealy individuals are more produive (e seond and ird erms in e las epression in (29)). Te value of e inrease in epeed inome depends on e insurane arrangemens. If ere is full over insurane ( I = l ) so a individual ges e same inome weer ealy or ill all e produiviy gain arues enirely via e las erm and is equal o e inrease in epeed inome. If insurane is inomplee ( I < l ) some of e gain arues o e individuals beause ey ave a greaer probabiliy of being in e ealy sae were ey ave a iger inome. Using (29) we ge d ln v gv = = ( b )[ a k + ( a) g( ε ) + gε] + b g (30) d ε ε were a = K /[ K + ( y f yd )( ε)], b = I / v and k is e rae of grow of K e dire uiliy gain from beer eal. Te marginal value of eal (29) a any dae depends on four faors: e dire impa of eal on uiliy, e inrease in uiliy from aving a iger inome wen ealy, e marginal epeed uiliy of inome, wi onvers e firs wo uiliy effes ino inome erms, and e epeed reduion in e os of insurane. Te grow rae in e value of eal is e weiged average of e rae of grow of ese faors were e weigs in general vary over ime. Wen ere is no produiviy gain from beer eal (so a l and ene I are zero) and eal merely as a dire effe on uiliy en g v = k + gε. Te dire effe of beer eal on uiliy may vary over ime beause of e effe of anges in publi goods or e environmen on uiliy. Te value of eal in erms of inome grows more rapidly e larger e grow rae of inome g and e rae a wi marginal uiliy falls wi inome ε: bo inrease e willingness o give up inome in eange for eal beause reduions in inome ave a smaller uiliy onsequene. If ere is a produiviy gain from beer eal bu no dire effe on uiliy ( K = 0 ) en g v = g. In e simple ase in wi ere is full over insurane e inrease in epeed inome arues enirely o e insurane pool and e grow rae of epeed inome is g sine we assume a e endowed inome of e sik grows a e same rae as e endowed inome of e ealy. If ere is inomplee insurane e differene beween uiliy from inome wen ealy and inome wen sik dereases so a epeed uiliy gain from beer eal falls. Wi inome in e unealy sae proporional o inome in e ealy sae e epeed uiliy gain falls a e rae (-ε)g. However, epeed marginal uiliy of inome falls even faser a e rae gε, so a e value of addiional uiliy in erms of inome grows a e rae g. Hene, waever e een of insurane and e saring of e inreases in epeed inome beween e individual and e insurane pool, g v = g if ere is no dire effe of eal on uiliy. Table summarises e impliaions of alernaive assumpions for e grow in e value of eal and e rae of disoun on eal effes. I makes lear a, in addiion o 4
assumpions abou soial welfare (embodied in ρ and ε), enology and resoures (embodied in g), e appropriae rae of disoun on eal effes in CEA also depends on e impa of ill eal on individual inome and e een of insurane. If uiliy from inome is bounded (ε > ), as is usually assumed, en, in e limi as inome beomes large, e effe of eal on inome beomes unimporan ompared o e dire effe of eal on uiliy, g v ends o k + gε and e disoun rae on eal ends o ρ k. Te urren Englis Deparmen of Heal reommendaion (Deparmen of Heal, 996), based on Parsonage and Neuburger (992), is a eal effes be disouned a.5% and oss a 6%. A number of auors (for eample, Arrow, 995) ave suggesed a ρ is around % and a ε is abou 2. Our resuls sugges a e reommendaion on r are broadly orre, bu ey suppor e pariular reommendaion on r only if less obvious furer assumpions are made, for eample a e value of e dire effe of eal on uiliy is onsan and large relaive o e value of e effe of eal on inome. 5 Conlusions Our onlusions an be summarised us if i is believed a value of fuure eal effes in erms of fuure inome grows over ime, is mus be allowed for in CEA and CBA, eier direly by adjusing e eal effes or indirely by adjusing e disoun rae on eal effes adjusing e eal effe as e meris of being eplii and separaing ou e issues of e value of eal effes in moneary erms and e disoun rae o be applied o fuure inome for os benefi analysis all eal effes sould be valued in e inome of e period in wi ey our and en disouned bak o a presen value using e rae of disoun appropriae for oss for os effeiveness analysis, were eal effes anno be valued in inome of e period, e nominal quaniy of eal effes sould be adjused o a real quaniy o refle e grow in e value of fuure eal effes and e same disoun rae be applied o oss and real eal effes evaluaions sould be eplii abou e approa aken o disouning eal effes and e reasons underlying i, su as assumpions abou e grow in e value of eal effes Te suggesion a e disoun rae for eal effes sould be less an e disoun rae for oss beause e value of eal grows over ime is no new. However, e fa a e orre disouning proedure for CEA was reognised bu en dismissed in e aper on disouning by Lipsomb, Weinsein and Torrene in e influenial ompendium of bes praie in eal eonomi evaluaions ommissioned by e US Publi Heal Servie (Gold e. al., 996), e failure o use e orre proedure revealed in publised sudies, and e inorre proedures reommended in many offiial guidelines (Smi and Gravelle, 5
2000), sugges a e ase needed o be made more firmly. We ope a by using simple eplii models of ineremporal deision making, we ave srengened e ase for aking aoun of e grow in e value of eal in eonomi evaluaions. Evaluaions based on CEA rieria require esimaes of e grow in e value of eal and CBA is impossible wiou esimaes of e value of eal. Aenion sould now be urned o e fundamenal issue for deision making in eal are: e value of eal. 6
Noes Noe a ineres raes are dimensionless, so a i makes sense o ompare e magniudes of r and r. v as e dimension of inome per uni of eal, so a v and v 0 an also be ompared. 2 Te welfare foundaions of CEA are disussed in Garber (2000) were i is argued a under erain irumsanes CEA an lead o welfare maimising deisions. See also Pelps and Muslin (99). A onrary view as been epressed by Donaldson (998) wo argues a CEA and CBA are aemps o answer differen quesions, raer an aemps o answer e same quesion wi differen amouns of informaion. He suggess a CBA is onerned wi alloaive effiieny and CEA wi enial effiieny. 3 In a onsiseny argumen frequenly ied in e debae, Weinsein and Sasson (977) assume a g v = 0 and en sow, by omparing wo projes direly agains ea oer and indirely via a sequene of equivalen proje, a if e wo omparisons are o yield e same resul en r o mus equal r. Epression () eplains wy eir argumen is logially orre wen g v = 0 bu as no relevane o e ase were g v > 0. 4 Te riial value a wi e CBA rule is indifferen beween saring in period 0 and in period is ( ) g$ = r v / v (, r ). 0 0 0 5 Wi r gv, r is zero or negaive and e presen value of e fuure perpeual inrease in eal is infinie 6 Te diffiuly wi e CEA rule is analogous o e problem wi omparing e inernal rae of reurn wi some arge rae of reurn as a means of aking invesmen deisions. Te inernal rae of reurn rule an lead o orre deisions (ie ose wi maimise ne presen value) only in a resried lass of projes (Hirsleifer, 970, 5-56). 7 In wa follows we invesigae e ange over ime in e value of a given eal effe in erms of inome of e period in wi e effe ours. We are onsidering e value of a ange ourring o individuals wo are e same age a differen daes, no e same individual a differen ages. 8 Differeniae ln v = ln p ln (, s ) wi respe o. 9 van Hou (998) also uses a welfare framework o derive an epression for e disoun rae on eal bu adops a soial welfare funion wi is non linear in e eal variable, so a i anno be inerpreed as per apia epeed uiliy, and does no allow for inreases in per apia inome arising from an inrease in eal. 0 Wen φ < we assume a inome is always greaer an e level required o ensure a uiliy wen ealy is greaer an uiliy wen diseased a e same inome level. Essenially e same resuls an be derived in more omple seings in wi being ealy as a dire effe on marginal uiliy so a e opimal insurane seme does no equalise inome aross saes (Gravelle, 2000). 2 To derive e resuls in e simple form below we also assume a e possibly subopimal insurane seme mainains a onsan raio of individual inome in e ealy and unealy saes and a iniially e probabiliy of e ealy sae is e same in all periods. 7
Referenes Arrow, K. J. (995). Inergeneraional equiy and e rae of disoun in long erm soial mimeo, paper for Inernaional Eonomis Assoiaion, World Congress, Deember. Blomqvis, A.G. and Carer, R.A.L. (997). Is eal are really a luury good? Journal of Heal Eonomis 6, 207-230. Brouwer, W., van Hou, B. and Ruen, F. (2000). A fair approa o disouning fuure effes: aking a soieal perspeive, Journal of Heal Servies Resear and Poliy, 5, 4-8. Burgess, S.M. and Propper, C. (998). Early eal relaed beaviours and eir impa on laer life anes: evidene from e US, Heal Eonomis, 7, 38-400 Cropper, M.L., and P.R. Porney. (990). Disouning and e evaluaion of lifesaving, Journal of Risk and Unerainy 3:369-79. Cropper, M.L. and Sussman, F.G. (990). Valuing fuure risks o life, Journal of Environmenal Eonomis and Managemen, 9, 60-74. Deparmen of Heal. (996). Poliy Appraisal and Heal: A Guide from e Deparmen of Heal. London. GO7/038 390, February. 996 Donaldson, C. (998). Te (near) equivalene of os-effeiveness and os-benefi Parmooeonomis 3, 389-396. Feldsein, M.S. (972). Te inadequay of weiged disoun raes, In R. Layard (ed.), Cos Benefi Analysis, 40-55. Penguin Books, London. Garber, A.M. (2000). Advanes in os-effeiveness analysis of eal inervenions, in Newouse, J.P. and Culyer, A.J. (eds.), Handbook of Heal Eonomis, Nor Holland, in press; also as Working Paper 798, June 999, Naional Bureau of Eonomi Resear. Gold, M.R., Siegel, J.E., Russell L.B., and Weinsein, M.C. (eds.) (996). Cos- Effeiveness in Heal and MediineOford, Oford Universiy Press. Gravelle, H. (2000). Valuing and disouning fuure eal anges, Deparmen of Eonomis, Disussion Paper No. /2000, July, Universiy of York. Hirsleifer, J. (970). Invesmen, Ineres and Capial, Prenie Hall, New Jersey. Hou, van B. (998). Disouning oss and effes differenly: a reonsideraion, Heal Eonomis, 7, 58-594. 8
Joansson, P.O. (995). Evaluaing Heal Risks: An Eonomi Approa, Cambridge Universiy Press. Jones-Lee, M. W. and Loomes, G. (995). Disouning and safey, Oford Eonomi Papers, 47, 50-52. Keeler, E. B. and Crein, S. (983). Disouning of life-saving and oer nonmoneary effes, Managemen Siene, 29, 300-306. Layard, R. and Glaiser, S. (994). Cos-Benefi Analysis, Seond Ediion, Cambridge Universiy Press. Lipsomb, J. (989). Time preferene for eal in os-effeiveness analysis, Medial Care 27:S233-S253. Lipsomb, J., Weinsein, M.C. and Torrane, G.W. (996). Time preferene, in Cos- Effeiveness in Heal and Mediine, Gold, M.R., Siegel, J.E., Russell L.B., and Weinsein, M.C. (eds.), Oford, Oford Universiy Press. Parsonage, M., and H. Neuburger. (992). Disouning and eal benefis, Heal Eonomis :7-76. Pelps, C.E. and Muslin, A. I. (99). On e (near) equivalene of os-effeiveness Inernaional Journal of Tenology Assessmen in Heal Care, 7, 2.-2. Sen, A.K. (967). Isolaion, assurane and e soial rae of disoun'', Quarerly Journal of Eonomis, 8, 2-24. Smi, D. H. and Gravelle, H. (2000). Te praie of disouning in eonomi evaluaions Cenre for Heal Eonomis, Tenial Paper No. 9, Mar, Universiy of York. Visusi, W.K. (995). Disouning eal effes for medial deisions, in F.A. Sloan (ed.) Valuing Heal Care: Coss, Benefis, and Effeiveness of Parmaeuials and Medial Tenologies, New York: Cambridge Universiy Press. Weinsein, M.C., and W.B. Sason. (977). Foundaions of os-effeiveness analysis for eal and medial praies, New England Journal of Mediine 296:76-2. 9
W W v y W W y y0 + r W W 0 + r 0 W W 0 y 0 v 0 0 Figure. Valuing anges in fuure eal in erms of urren os (or inome) direly: v /(+ r) ; and indirely: [ /(+ r )] v0 20
Table. Rae of grow of value of eal (g v ) and disoun rae on eal effes (r ). Heal affe inome? No Dire effe of eal on uiliy? Yes No g v = 0 r = ρ + gε = r g v = k + gε r ρ k = Yes g v = g r = ρ + g( ε ) < r gv = ( b )[ ak + ( a) g( ε) + gε] + g v k + gε if ε > b g r = ρ + g( ε)[( b )( a) + b ] ( b ) ak r ρ k if ε > g: rae of grow of inome; ε: elasiiy of marginal uiliy of inome; ρ: pure uiliy disoun rae; k: rae of grow of dire effe of eal on uiliy; r : disoun rae on inome (oss); b = I / v, ε ε a = K /[ K + ( y y )( ε)]. In all four ases r = r g v and r = ρ+ gε. f d 2