Discussion Papers in Economics

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1 Dscusson Papers n Economcs No. No. 2000/ /15 Dynamcs Ramsey of Output Wats: Growth, Allocatng Consumpton Publc Health and Physcal Servce Captal n Resources Two-Sector when Models there of s Ratonng Endogenous by Growth Watng by By Hugh Gravelle and Lug Sclan Department of Economcs and Related Studes Unversty of York Heslngton York, YO10 5DD

2 Ramsey wats: allocatng publc health servce resources when there s ratonng by watng Hugh Gravelle Lug Sclan y 5 June 2007 Abstract The optmal allocaton of a publc health care budget across treatments must take account of the way n whch care s ratoned wthn treatments snce ths wll a ect ther margnal value. We nvestgate the optmal allocaton rules for health care systems where user charges are xed and care s ratoned by watng. The optmal watng tme s hgher for treatments wth demands more elastc to watng tme, hgher costs, lower charges, smaller margnal welfare loss from watng by treated patents, and smaller margnal welfare losses from underconsumpton of care. The results hold for a wde range of welfarst and non-welfarst objectve functons and for systems n whch there s also a prvate health care sector. They mply that allocaton rules based purely on cost e ectveness ratos are suboptmal because they assume that there s no ratonng wthn treatments. Keywords: Watng tmes; Prortsaton; Ratonng; Cost e ectveness ratos. JEL numbers: H21, H42, I11, I18 Natonal Prmary Care Research and Development Centre, Centre for Health Economcs, Unversty of York. Emal: hg8@york.ac.uk. NPCRDC receves core fundng from the Department of Health. The vews expressed are those of the authors and not necessarly those of the DH. y Department of Economcs and Related Studes, and Centre for Health Economcs, Unversty of York, Heslngton, York YO10 5DD, UK; C.E.P.R., Goswell Street, London EC1V 7DB, UK. Emal: ls24@york.ac.uk. 1

3 1 Introducton We nvestgate how a xed health care budget should be allocated across treatments when patent charges are xed and care s ratoned by watng. 1 Snce an ncrease n the supply of a treatment reduces the tme patents wat for t, optmal allocaton of the budget across treatments s equvalent to determnng the optmal watng tmes for the d erent treatments. Watng tmes for electve surgery are used as a ratonng mechansm n many countres wth tax or publc health nsurance nance. Examples nclude Australa, Canada, Denmark, Fnland, the Netherlands, Span and the Unted Kngdom. Average watng tmes for procedures, such as hp and knee replacement, cataract surgery, and varcose vens, of sx months are not uncommon (Sclan and Hurst, 2004). There s consderable varaton n watng tmes across treatment. Some examples from England n 2003/4 are 125 days for cataracts; 65 days for mouth or throat procedures category 2; 77 days for percutaneous translumnal coronary angoplasty; and 223 days for prmary hp replacement. The lterature on the optmal allocaton of the health care budget across treatments consders the mx of treatments whch wll maxmse total health gan (Garber, 2000). It suggests that resources should be allocated to d erent treatments accordng to ther cost e ectveness ratos (CERs): the rato of cost to health bene t. The lterature assumes that there s an exogenously determned number of patents for each treatment, all of whom have the same health gan from the treatment. Treatments are ranked by ther CERs and treatments wth the lowest CER are funded untl the budget s exhausted. The CER allocaton rule mples that there s no wat for treatment for funded treatments and an n nte wat for the unfunded. The key assumpton underlyng the CER allocaton rule s that margnal health bene t from a treatment s constant. But t s d cult to reconcle ths assumpton wth two types of evdence. Frst, sub-group analyses n trals show that bene t typcally vares across sub-groups recevng a treatment (Douglas, Buxton and O Bren, 2003). Second, f all patents had the same gans from treatment, demand curves for health care would be horzontal up to the number of patents. But utlsaton studes show that demand curves are negatvely sloped wth respect to tme and money prces (Martn and Smth, 1999; Martn et al., 2007; Newhouse et al, 1993; Rngel et al, 2002). Gven the d erences n health bene t for patents recevng a treatment, the assumpton of constant margnal bene t s equvalent to mplctly as- 1 Gravelle and Sclan (2007) consder the optmal choce of patent charges and watng tmes. They show that under plausble assumptons about patent preferences the optmal watng tme s zero f the welfare functon s utltaran. 2

4 sumng that patents are drawn at random from those who could bene t. But there are almost no examples of random ratonng n publc health care systems. Publcly funded health care s generally ratoned by money or watng tme prces. Such ratonng mples that treated patents have a hgher perceved bene t than those not treated and that the health gan of the margnal patent falls as supply s ncreased and the money or watng tme prce s reduced. Thus CER based rules for allocatng budgets across treatments rest on an ncorrect spec caton of the way the supply of a gven treatment s ratoned. Even f one accepts the value judgements underlyng them, CER based allocaton rules requre mod caton because the ratonng mechansm wthn treatments a ects the margnal value of treatments. Only one paper on prorty settng across treatments has taken account of the way n whch patents are ratoned wthn treatments. 2 Smth (2005) consders a health care system whch s nanced by a mxture of an exogenous budget and user charge revenue. The volumes of d erent treatments are determned by the patent demands at the user charges set by the health planner. It s assumed that all patents have same health bene t from a gven treatment but, because patents have d erent ncomes and utlty of ncome s concave, ncreases n the user charge reduce demand by dscouragng poorer potental patents. The optmal prces whch maxmse total health gan are determned by the exogenous CERs and by the money prce elastcty of demands. Treatments wth a hgh CER are not provded, those wth a low CER are provded at no charge, and those wth ntermedate CERs are provded at a below cost prce. In contrast to the usual Ramsey publc sector prcng rules (Ramsey, 1927), charges are hgher (and the volume of treatment smaller) the more elastc s demand wth respect to the charge. The result s smlar to that derved by Besley (1988) for optmal health nsurance contracts. In the usual Ramsey case prces are set to rase a gven sum n revenue and so must exceed margnal cost. In the case of health care, ether because of rsk averson (Besley, 1988) or because of an extra welfarst 2 Hoel (2007) shows that the smple CER based allocaton rules requre mod caton when some patents wll buy the treatment n the prvate sector f t s not provded n the publc sector. Consequently not all treatments provded n the publc sector have lower CERs than those not provded. For example, a treatment for whch there s a prvate sector alternatve may not be provded n the publc sector even though t has a lower CER than a treatment whch s provded but for whch there s no prvate sector alternatve. The ntuton s that when a prvate sector alternatve s avalable the gan from publc provson s the cost savng to the patents who would have bought t n the prvate sector. Ths cost savng s less than the utlty gan from treatment compared wth no treatment. Thus the average welfare gan from a treatment s lower f t has a prvate sector alternatve. Hoel (2007) does however assume that f a treatment s provded t s provded to all patents. Hence he does not need to consder how care s actually ratoned wthn treatments. 3

5 desre to ncrease consumpton of health care (Smth, 2005), the am s to spend a xed sum and so prces are less than margnal cost. But, whether the constrant s to rase a gven revenue or to spend a gven budget, there s less welfare loss n departng from margnal cost prces the less elastc s demand. In the standard Ramsey revenue rasng case, prces are rased further above margnal cost for goods wth less elastc demands. In the budget spendng case of health care, prces are lowered further below margnal cost the less elastc s demand. Hence prces (or consurance rates) are hgher for goods wth more elastc demands. In ths paper we examne allocaton across treatments for a heath care system where, n contrast to Smth (2005), watng tme rather than money prce ratons demand. We also allow for the possblty that patent health bene t from a treatment vares across patents and we consder a more general welfare functon whch ncludes health gan maxmsaton as a specal case. Unlke a money prce, a watng tme prce mposes a pure deadweght loss: t s a cost to patents whch s not o set by any gan to the provders. Longer watng tmes reduce the value of treatment because of lost expected bene t, temporary dscomfort and pan, and, for some pathologes, the hgher rsk of permanent reductons n health (see Hoel and Saether (2003) for some detaled examples of the cost of watng). The change n watng tme costs ncurred by patents when supply of a treatment s reduced s determned by the ncrease n the wat and the reducton n the number of patents who wat. The reducton n the number watng s equal to the reducton n the supply of treatment. A gven reducton n the supply of treatment wll generate a smaller ncrease n the watng tme the more elastc the demand wth respect to watng tme. Thus the more elastc the demand wth respect to watng tme, the smaller the ncrease n watng tme costs when supply s reduced and the watng tme ncreased. Hence, when there s ratonng by watng wthn treatments, the optmal allocaton of resources across treatments should result n longer watng tmes for treatments where demand s more elastc wth respect to watng tme. Watng tmes should also be longer for treatments where treated patents su er less from ncreased watng tmes. The above argument holds whether or not the welfare functon s paternalstc. If the welfare functon does not respect ndvdual choces between treatment and no treatment there s a further factor n uencng the watng tme: the welfare loss assocated wth nd erent patents dsplaced as watng tme ncreases. Wth a paternalstc welfare functon patents may place too low a value on treatment relatve to no treatment, so that there s a welfare loss assocated wth the margnal patent s decson not to be treated. We show that watng tmes should be lower, ceters parbus, for treatments 4

6 where there s a larger welfare loss for the margnal patent. Secton 2 sets out the model. Secton 3 derves and dscusses the man results for allocaton rules between treatments. Secton 4 extends the analyss by allowng patents to opt for a prvate sector and shows that the exstance of a prvate sector makes no essental d erence to the results. Secton 5 concludes. 2 Model spec caton 2.1 Preferences, demand, and watng tmes All health care s produced n the publc health care system. Demand for treatment s ratoned by watng. To reduce notatonal clutter we assume there are no user charges. We show n secton 3.3 that allowng for postve xed user charges makes no essental d erence to the results. Indvduals have the same preferences but d er ex ante n havng d erent ncomes y 2 [y mn ; y max ] and ex post n havng d erent health. Allowng for d erence n preferences merely complcates analyss and does not alter the results. Wth probablty ndvduals are ll wth condton and wll bene t from treatment. No ndvdual has more than one condton. If ll wth condton and not treated an ndvdual has utlty v NT (y), vy NT > 0. 3 Treated patents have a wat of w before recevng one unt of treatment whch produces a bene t (health gan) b 2 [b mn ; b max ]. Utlty f treated s v T (y; b; w ), whch s ncreasng n ncome and health gan and decreasng n the wat: vy T > 0, vb T > 0, vt w < 0. 4 Health bene t and ncome for patents wth condton s dstrbuted wth densty f (b; y) and dstrbuton functon F (b; y).the margnal dstrbuton functon for ncome s F y (y). The total populaton s normalsed to 1. The planner knows the dstrbuton functons but cannot prortse ndvdual patents on the bass of ther health gan or ther ncome. 3 The formulaton allows for the possblty that recevng treatment can ncrease ncome. Let y be ncome f treated, assume condton reduces ncome f not treated by L (y), and denote utlty f not treated as ^v NT (y L (y)). Then we can wrte v NT (y) = ^v NT (y L (y)). 4 We assume that watng tmes vary across treatments but are constant wthn treatments. Gravelle and Sclan (2006) analyse the e ect of watng-tme prortsaton wthn a treatment when bene t s partally observable through a contnous varable (lke age). They nd that prortsaton wthn treatments can ncrease welfare but has ambguous e ects on the margnal value of treatment. Thus ts e ect on allocatons across treatments s also ambguous. 5

7 The key assumpton s that ncreases n watng tme reduce the utlty from treatment compared wth the no treatment alternatve. The most salent form of ratonng by watng tme s ratonng by watng lst for electve care. Indvduals bear a cost n gettng on the watng lst for treatment. In systems wth gatekeepng general practtoners, patents rst have to consult ther general practtoner to get a referral and then ncur further costs n attendng hosptal outpatent department to be seen by a specalst who wll then place them on a watng lst. The longer the tme potental patents know they wll have to wat on the lst, the less the dscounted value of the treatment and the less lkely are they to be wllng to ncur the ntal costs of jonng the lst (Lndsay and Fegenbaum, 1984; Martn and Smth, 1999; Farnworth, 2003). In some systems there s ratonng by watng n lne (queues). Watng for treatment has an opportunty cost of forgone work or lesure tme, as well as possble e ects on the health gan. Ratonng by watng lne can be used for mnor alments n hosptal accdent and emergency rooms and for general practtoner consultatons. Our spec caton encompasses both ratonng by watng lst and by watng lne and does not restrct the way n whch longer wats reduce the utlty of treatment relatve to no treatment (Hoel and Saether, 2003). 5 The emprcal evdence shows that ncreases n watng tme reduce demand for health care (Gravelle, Smth and Xaver, 2003; Martn and Smth, 1999; Martn et al, 2007). An ndvdual wth llness demands treatment f and only f v T (y; b; w ) v NT (y) 0 () b ^b (w ; y) (1) where ^b (w ; y) s the threshold bene t level such that all those wth a smaller bene t do not seek treatment. The threshold s ncreasng n the watng tme snce ^b w (w ; y) = v T w=v T b > 0. The e ect of ncome on the bene t threshold s ^by (w ; y) = vt y vy NT vb T (2) whch may negatve or postve dependng on the e ect of llness and treatment on the margnal utlty of ncome. 6 If ncome s addtvely separable 5 Our spec caton ncludes the orgnal Lndsay and Fegenbaum (1984) formulaton of the model of ratonng by watng f we wrte v T = v (y) +be rw a and v NT = v (y), where a s the cost of gettng on the lst. 6 Emprcal studes suggest that, controllng for a wde range of morbdty and other soco-economc characterstcs, ncome has lttle mpact on the number of GP vsts but s postvely assocated wth specalst vsts and hosptal stays (van Doorslaer et al, 2004; Morrs et al, 2005). 6

8 from health and watng tme, the ncome sub-utlty functon s una ected by treatment, ll health has the same e ect on the ncome of the treated and untreated, then (2) s zero and those on hgher ncomes demand care as much as ndvduals wth low ncome. Demand for treatment s D (w ) = f (b; y)dbdy (3) ^b (w ;y) and D w (w ) = ^bw (w ; y)f (^b (w ; y); y)dy < 0 (4) where D s the per capta demand for treatment from ndvduals wth llness. The supply of treatment s z. The watng tme for treatment s determned by the market clearng condton D (w ) z 0; w 0; w [ D (w ) z ] = 0 (5) The equlbrum watng tme, when postve, s determned by (5) as w = w (z ; ); w z = 1=( D w ) < 0 (6) Resource allocaton decsons result n three categores of treatment. Some treatments are not provded by the health servce, whch s equvalent to settng a su cently hgh watng tme (w w o, D (w o ) = 0) to drve demand to zero. Treatments whch are provded are ether emergences, where the level of supply s such that there s no watng tme ( D (0) = z > 0), or electves, where there s a postve wat (w > 0, D (w ) = z > 0). 2.2 Welfare Welfare for an ndvdual wth ncome y s s(y; w) = X " ^b (w ;y) # s NT (y)f (b; y)db + s T (y; b; w )f (b; y)db ^b (w ;y) where s NT ; s T are welfare f the ndvdual has condton and s not treated or treated and w s the vector of watng tmes. The socal welfare functon s S(w) = s(y; w)df y (y) (8) (7) 7

9 The welfare formulaton s qute general and s compatble wth utltaransm, more complcated Bergsonan ndvdualstc welfare functons, and wth extra welfarst (Culyer, 1991) value judgements, ncludng smple health gan maxmsaton as n the allocaton lterature. The only restrcton we mpose s that welfare for a treated ndvdual s ncreasng n health gan and decreasng n the watng tme: s T b > 0, st w < 0. The welfare maxmsng bene t threshold b S for consumpton of health care by ndvduals wth ncome y s de ned by s T (y; b; w ) s NT (y) 0 () b b S (w ; y) (9) Indvduals may choose to consume too much (b S > ^b ) or too lttle (b S < ^b ) care. It s not possble to control use drectly, so that t s determned by ^b (w ; y), not by b S (w ; y). But ^b (w ; y) and b S (w ; y) vary wth watng tme, so that the welfare maxmsng allocaton may be n uenced by the e ect of watng tme on the dscrepancy between welfare maxmsng and actual bene t thresholds. In the utltaran welfare functon s T = v T, s NT = v NT, so that ndvduals decsons on health care consumpton are respected n the sense that ^b (w ; y) = b S (w ; y). Even f t was possble to drectly alter the bene t thresholds (and hence demand) at gven watng tmes, there would be no welfare gan from dong so. Polcy n health care markets often re ects paternalstc value judgements whch mply that ndvduals consume too lttle or too much health care (Musgrave, 1959; Sandmo, 1983). A common example s the belef that the use of health care should depend only on need, de ned as capacty to bene t from health care b, and not on characterstcs such as ncome. We can capture ths noton, whch s closely related to horzontal equty, wth a welfare functon n whch s NT (y) = v NT (y o ); s T (y; b; w ) = v T (y o ; b; w ) (10) The welfare maxmsng treatment threshold de ned by (9) s b S (w ; y) = ^b (w ; y o ) whch s the same for ndvduals rrespectve of ther ncome level (and any other characterstcs deemed rrelevant n assessng need). y o s the reference ncome requred to generate the welfare maxmsng treatment threshold gven the watng tme. Indvdual decsons on consumng health care wll not be welfare maxmsng except for those wth y = y o. If ^b y s negatve, expected use wll be too hgh for those wth y > y o and too low for those wth y < y o. The lterature on CER allocaton rules embodes the extra welfarst value judgement that the am of the health care system s to maxmse the total 8

10 health gan of the populaton (Garber 2000; Smth 2005). We could capture ths value judgement by wrtng welfare as dscounted health gan s T (y; b; w ) = (w )b; s NT (y) = 0 (11) where b s the health gan from health care wth no watng, and (w ) s a dscount factor satsfyng (w ) 2 (0; 1]; (0) = 1; r (w ) w (w )= (w ) > 0 (12) wth r beng the proportonate rate of decrease of the dscount factor. 3 Prorty settng between treatments 3.1 Optmal watng tmes Decson makers n the publc health care servce allocate resources amongst d erent treatments subject to an exogenously determned health servce budget constrant: M P c D (w ) 0 (13) where c s the constant average and margnal cost of treatment, and M the xed health servce budget. Health servce decson makers allocate treatment (z ), or equvalently choose watng tmes, to maxmse the socal welfare functon. The cost of meetng all demand generated by zero watng tmes exceeds the budget: P c D (0) > M, so that the budget constrant bnds. The Lagrangean s L = s(y; w)df y (y) + M P c D (w ) and the optmal allocaton sats = ^bw (w ; y)f (^b(w ; y); y)dy s NT s T (14) + ^b s T w(y; b; w )f (b; y)dbdy c D w 0; w 0 (15) wth complementary slackness, for all. 7 7 Although there always exsts a vector of watng tmes such that the constrant s sats ed as a strct nequalty, su cency of the rst order condtons requres restrctons on the second dervatves of the socal welfare functon, utlty functons and the dstrbuton functons. 9

11 Increasng w has two e ects on welfare. Frst, ncreasng the watng tme reduces welfare from treatment. The margnal welfare cost of watng per treated patent s = s T w(y; b ; w )f dbdy=d > 0 (16) ^b Second, there s a welfare loss arsng from the patents who decde not to seek treatment when the watng tme ncreases. The margnal threshold cost per treated patent s = (s T s NT )^b w (w ; y)f (^b (w ; y); y)dy=d (17) s T (y ; ^b (w ; y); w ) s NT (y) s the socal welfare reducton when the margnal patent wth ncome y fals to seek treatment. When the watng tme ncreases, the bene t threshold at whch patents seek treatment s ncreased (^b w > 0) and s T s NT s forgone. The margnal patent has zero personal bene t from treatment (v T (y; ^b (w ; y); w ) = v NT (y)) so that, f the welfare functon respects ndvdual relatve valuatons of treatment and no treatment, = 0. If all ndvduals place too low a value on treatment relatve to no treatment then s T (y; ^b (w ; y); w ) > s NT (y) at all ncome levels and > 0. For example, suppose that we care about need and the socal welfare functon s gven by (10) and that the reference ncome s y o = y max. Then all ndvduals ought to consume the same expected amount of care as the rchest ndvdual. If vy T > vy NT, so that the probablty of consumpton care when ll ncreases wth ncome, then all ndvduals consume too lttle care and > 0. Wth a smaller reference ncome some ndvduals have too hgh an expected consumpton and some too low, so that t s possble that < 0. We can use (16) and (17) to wrte (15) as ( + ) c " w w 1 0; w 0 (18) where " w = D w w =D < 0 s the elastcty of demand for treatment wth respect to the watng tme. Proposton 1 The optmal allocaton has three types of treatment: () treatment s not provded: w w o, where D (w o ) = 0, and lm w!w o ( + ) c " w w 1 0 (19) 10

12 () treatment s an emergency treatment: w = 0, and lm w!0 ( + ) c " w w 1 0 (20) () treatment s an electve treatment: w 2 (0; w o ) and w = c + " w (21) The optmal watng tme for electve care s hgher the greater the cost of treatment and lower the greater the margnal cost of watng ( ) and the margnal threshold welfare cost ( ) from patents dsplaced by hgher watng tmes. Watng tme should be hgher f demand s more elastc wth respect to watng tme. A gven reducton n provson wll generate a smaller ncrease n watng tme the more elastc the demand. Watng tme s a deadweght loss: none of addtonal cost to consumers of havng to wat longer accrues as a gan to anyone else (by contrast user charges pad by patents accrue to funders). Hence we should be more wllng to reduce supply for treatments wth more elastc demands. 3.2 Allocaton n terms of cost e ectveness ratos Suppose that welfare from treatment s addtvely separable n ncome and multplcatvely separable n the bene t and watng tme as: s T (y; b; w ) = ~s T (y) + (w)b (22) The margnal cost of watng per treated patent s = w (w )bf dbdy=d = r (w) (w ) ^b bf dbdy=d ^b = r (w) (w )B (w ) (23) where r (w ) = w (w )= (w ) s the proportonate rate of declne n the dscount factor and B (w ) s the undscounted health gan per treated patent. Note that B depends on the wat because w a ects the number of patents beng treated through ^b. The margnal threshold welfare cost of dsplaced patents per treated patent s = h ~s T (y) + (w)^b s NT (y) ^bw f (^b (y; w ))dy=d (24) 11

13 When ~s T (y) = s NT (y), the margnal threshold cost s postve because the welfare functon takes account only of dscounted health gans from treatment and gnores the factors whch n uence patent decsons on whether to seek treatment. An ncrease n expendture on treatment of 1 ncreases supply of the treatment by 1=c. A unt ncrease n supply reduces the watng tme by 1= D w. A unt reducton n watng tme reduces watng tme costs at the rate D = r B D and welfare arsng from addtonal patents treated ncreases at the rate D. Thus, when the welfare functon has (22), the margnal socal value n terms of health gan from an addtonal 1 spent on treatment s 1 1 [ c D D + D ] = w D D w B + r c c (25) Note that the margnal value of expendture depends on the bene t-cost rato B =c whch s the nverse of the CER. We can restate the condtons descrbng an optmal allocaton of the budget across treatments n terms of the margnal value of expendture on treatment (25) and the shadow value of the budget constrant (): Proposton 2 If welfare from treatment s addtve separable n ncome and multplcatvely separable n watng tme and bene t then at the optmal allocaton () f treatment s not provded then lm w!w o D D w B + r c c (26) where w o s the watng tme at whch demand s zero (D (0) = 0) () f treatment s provded wth zero wat then D B lm + r (27) w!0 D w c c () f treatment s provded wth a postve wat then D B + r = (28) D w c c A pure CER rule would provde treatment f and only f B (0)=c exceeded the shadow margnal value of the health sector budget. Although the optmal allocaton depends on the cost-e ectveness ratos for d erent 12

14 treatments, the CER based rule of allocatng resources n reverse order of cost e ectveness s not optmal. Snce patents are not drawn at random from the pool of potental patents the undscounted health bene t per treated patent (B (w )) declnes wth the number of patents treated. Thus the CER for a treatment s determned by the allocaton decson rather than determnng t. Suppose that all patents would have the same bene t b from a treatment, so that undscounted health gan per treated patent B = b s constant and does not depend on the amount of treatment. Suppose that v T = u T (y) + (w )b a where a s an access cost of gettng treatment. If there exsts an ncome ^y 2 (y mn y max ) such that v T (^y ; b ; w ) = v NT (^y ) and f vy T vy NT s postve, then demand for treatment s decreasng n the watng tme. Then even when the constant b replaces B (w ) n Proposton 2, the decson to provde treatments s not based solely on ther bene t cost ratos. Although a hgher bene t cost rato makes t more lkely that a treatment wll be provded t s also necessary to take account of the responsveness of demand to watng tme, the margnal threshold welfare loss and the rate at whch dscounted health bene ts declne wth the watng tme. A hgher dscount rate r makes t more lkely that a treatment wll be suppled snce the cost of smaller supply (mplyng a longer wat) are greater. A hgher threshold cost for patents dsplaced by a smaller supply wll also ncrease supply. Fnally, the more responsve demand s to watng tme the smaller wll be the supply and the hgher the watng tme. 3.3 User charges The assumpton that user charges were zero was made to smplfy the exposton and notaton. Allowng for postve xed user charges makes very lttle d erence. Thus suppose that treatment carres a charge of p < c. Then the threshold bene t level s ^b (y; p ; w ) determned by v T (y p ; b; w ) = v NT (y) and the demand functons also depend on the charge D (w ; p ). After nsertng ^b (y; p ; w ) and D (w ; p ) n the welfare functon (8) and the budget constrant (13), the resultng rst order condtons d er only n havng c p replace c. The only change to Proposton 1 requred s that producton cost c s replaced wth the net nancal cost c p to the publc sector the net nancal cost to the publc sector. Wth the socal welfare functon satsfyng (22), the value of an addtonal 1 spent on treatment (25) becomes D B 1 + r (29) D w c c 1 13

15 where = p =c s the copayment rate: the proporton of the unt cost of treatment recovered by a charge to the patent. Proposton 2 now holds wth (29) replacng (25). A treatment s more lkely to be provded, or have a larger supply, f t has a hgher exogenous copayment rate. 4 Publc sector allocaton n the presence of a prvate sector In many countres wth publc health care systems patents also have the opton of buyng treatment from prvate sector provders. We now show that the exstance of the prvate sector a ects the form of the demand functons for publc care but does not alter our man conclusons about the factors determnng the optmal allocaton of resources across treatments n the publc sector. Indvduals wth condton can obtan prvate sector treatment wth no wat at prce of m or publc sector treatment at no charge but after a wat of w. We assume that utlty s separable n ncome and health, and concave n ncome, to smplfy the dervaton of the results, though they do not depend on the assumpton. Publc treatment yelds utlty v GT = v T (y; b; w ) = (y) + u (b; w ) a. Utlty from prvate treatment s v P T = v T (y m ; b; 0) = (y m )+u (b; 0) a. We assume that margnal utlty from health gan s hgher f the wat s smaller. Utlty from no treatment s v NT (y) = (y). Patents prefer publc to no treatment f v GT > v NT. The bene t threshold above whch patents prefer publc treatment to no treatment s ^b (w ), GN de ned by u (b; w ) a = 0. Patents prefer publc to prvate treatment f v GT (y; b; w ) > v P T (y m ; b; 0). b GP (y; w ; m ) s the bene t threshold below whch patents prefer publc treatment to prvate treatment. It s decreasng n watng tme and ncome. Patents prefer prvate to no treatment f v T (y m ; b; 0) > v NT (y). The bene t threshold above whch patents prefer prvate treatment to no treatment s b P N (y; m ). The threshold s decreasng n ncome. Fgure 1 shows the choces of d erent types of patents. The locus where patents are nd erent between prvate and publc treatment (b GP ) and the locus where they are nd erent between prvate and no treatment (b P N ) are downward slopng n (b; y) space. The locus where they are nd erent between publc treatment and no treatment (^b GN ) s horzontal snce t depends only on the publc sector watng tme. Patents who are nd erent between 14

16 publc and prvate treatment and nd erent between publc and no treatment, must also be nd erent between prvate and no treatment. Hence b GP and b P N GN ntersect on ^b. It can also be shown that the locus b GP cuts the locus b P N from above. Patents wth hgh bene t and low ncome demand publc treatment. Patents wth hgh bene t and hgh ncome demand prvate treatment. Patents wth low bene t demand no treatment. The demand for publc treatment s D(w ) = y max ^bg (y;w ) y mn ^bgn (w ) f (b; y)dbdy (30) where ^b G (y; w ) = max[mn[b GP ; b max GN ]; ^b ]. Increases n watng tme reduce demand for publc treatment: D w (w ) = y max y max f (^b GN ; GN (w ) dy+ f (^b G ; y mn y G (w dy < 0 (31) When watng tme ncreases some publc sector patents decde not to be treated ( rst term n (31)) and some patents opt for the prvate sector (second term n (31)). In terms of Fgure 1, an ncrease n w shfts the locus ^b GN upward and so patents along the locus who have ncomes y 2 [y mn ; y GN ] swtch out of the publc sector and are not treated. The ncrease n w also shfts the locus b GP downward. Hence patents along the locus wth y 2 [y GP ; y GN ] decde to swtch from the publc nto the prvate sector. Total welfare s the sum of the utlty of publc patents, prvate patents and patents wth no treatment: S = S GT + S P T + S NT where S GT = X S P T = X S NT = X y max ^bg (y;w ) y mn y max y mn ^bgn (w ) b max ^bp y max ^bn y mn s GT (y; b; w )f (b; y)dbdy (32) s P T (y m ; b; 0)f (b; y)dbdy (33) s NT (y)f (b; y)dbdy (34) b mn 15

17 where ^b P (y; w ) = max[mn[b GP ; b max ]; b P N ] and ^b N (y; w ) = mn[^b GN ; b P N ]. An ncrease n watng tme has no e ect on the nframargnal untreated patents or prvate patents. It reduces the utlty of the publc patents drectly and also changes the thresholds at whch people choose publc treatment rather than prvate treatment or no treatment. Although the margnal patents are nd erent, there wll be a welfare gan or loss assocated wth the change n the number of patents treated n the publc sector unless the socal welfare functon respects the patent decsons. The margnal socal value of an ncrease n watng tme for treatment s, va the Lagrangean S P c D, y GN y mn h s NT (y) s GT + + y GN y GP h y max ^bg y mn ^bgn s GT GN (y; ^b ; w ) ^bgn w f (^b GN ; y)dy (y; b GP ; w ) s P T GP (y m ; ^b ; 0) ^bgp w f (^b GP ; y)dy GT sw (y; b; w )f (b; y)dbdy c D w (35) The rst term s the e ect on welfare va the dsplacement of patents who are forced out of the publc sector and who do not get treated. The second s the e ect va the dsplacement of patents from the publc nto to prvate sector. Dvdng (35) through by the number of treated patents ( D ) we can wrte the rst order condtons on watng tmes n an analogous manner to (18): + GN + GP c " w w 1 0; w 0 (36) where GN ; GP the publc sector nto no treatment or nto the prvate sector. GP are the margnal threshold costs for patents pushed out of lkely to be zero even wth an extra welfarst welfare functon snce the margnal ndvdual s merely shftng between consumpton of prvate and publc health care. Thus the form of the rules for allocatng a gven publc health care sector across treatments are una ected by the exstance of a prvate sector alternatve to publc care. 16

18 5 Conclusons We have nvestgated the optmal allocaton of a xed health care budget across treatments when there s ratonng by watng and user charges are exogenous. Our man ndng s that the optmal watng tme s hgher for treatments wth demands whch are more elastc to watng tme, hgher costs, lower charges, and smaller margnal dsutlty from watng. In addton, f the welfare functon does not respect patent choces between treatment and no treatment, the watng tme should be lower for treatments where underconsumpton of health care has a greater welfare cost. The general message s that optmal allocaton across sectors must take account of way care s ratoned wthn sectors. Hence, allocaton rules based purely on cost e ectveness ratos are suboptmal because they assume that there s no ratonng wthn treatments. References Besley, T.J Optmal rembursement health nsurance and the theory of Ramsey taxaton, Journal of Health Economcs, 7, van Doorslaer, E., Koolman, X. and Jones, A.M Explanng ncome-related nequaltes n doctor utlsaton n Europe. Health Economcs, 13, Culyer A.J The normatve economcs of health care nance and provson, n A. McGure, P. Fenn & O. Mayhew (eds.), Provdng Health Care: The Economcs of Alternatve Systems of Fnance and Delvery, Oxford Economc Press. Douglas, C., Buxton, M.J., O Bren, B.J, 2003, "Strat ed cost-e ectveness analyss: a framework for establshng e cent lmted use crtera", Health Economcs, 12, 5, Garber, A.M Advances n cost-e ectveness analyss, n A. J. Culyer and J. P. Newhouse (eds), Handbook on Health Economcs, Amsterdam: Elsever Farnworth, M.G., 2003, A game theoretc model of the relatonshp between prces and watng tmes, Journal of Health Economcs, 22(1), Gravelle, H., P.C. Smth and Xaver, A., 2003, Performance sgnals n the publc sector: the case of health care, Oxford Economc Papers, 55, Gravelle, H. and Sclan, L Is watng tme prortsaton welfare mprovng?, Department of Economcs Dscusson Paper 06/13. to appear n Health Economcs. 17

19 Gravelle, H. and Sclan, L Optmal wats and charges n health nsurance, February. Department of Economcs Dscusson Paper 07/02. Hoel, M., Saether, E.M., 2003, Publc health care wth watng tme: the role of supplementary prvate health care, Journal of Health Economcs, 22, Hoel, M., 2007, What should (publc) health nsurance cover?, Journal of Health Economcs, 26(2), Lndsay, C.M., and Fegenbaum,B., 1984, Ratonng by watng lsts, Amercan Economc Revew, 74(3), Martn, S., and Smth, P.C., 1999, Ratonng by watng lsts: an emprcal nvestgaton, Journal of Publc Economcs, 71, Martn, S., Rce, N., Jacobs, R., Smth, P.C The market for electve surgery: Jont estmaton of supply and demand, Journal of Health Economcs, 26, Morrs, S., Sutton, M., Gravelle, H Inequty and nequalty n the use of health care n England: an emprcal nvestgaton. Socal Scence and Medcne, 60, Musgrave, R The theory of publc nance, McGraw-Hll, New York. Newhouse, J. P., and the Insurance Experment Group Free For All? Lessons from the Health Insurance Experment, Harvard Unversty Press, Cambrdge. Pauly, M., and Blavn, F., 2007, "Value based cost sharng meets the theory of moral hazard: medcal e ectveness n nsurance bene t desgn", NBER Workng Paper Ramsey, F., 1927, "A Contrbuton to the Theory of Taxaton", Economc Journal, 37,145, Rngel, J.S., Hosek, S.D., Vollard, B.A., and Mahnovsk, S The Elastcty of Demand for Health Care: A Revew of the Lterature and Its Applcaton to the Mltary Health System. Rand Organsaton. Sandmo, A., 1983, "Ex post welfare economcs and the theory of mert goods", Economca, 50, Sclan, L., and Hurst, J., 2004, Explanng watng tmes varatons for electve surgery across OECD countres, OECD Economc Studes, 38(1), Smth, P.C. 2005, User charges and prorty settng n health care: balancng equty and e cency, Journal of Health Economcs, 24,

20 beneft b max GP y Publc treatment b ( y; w, m ) GP Prvate treatment ˆ GN b ( w ) No treatment b PN ( ybm,, ) b mn y mn GN y y max ncome Fgure 1. Patent characterstcs and choce amongst publc treatment, prvate treatment, and no treatment for condton. 19