Optimal Health Insurance for Multiple Goods and Time Periods

Transcription

1 04 R.P. Ells, S. Jang, and W.G. Mannng Optmal Health Insurane for Multple Goods and Tme Perods Randall P. Ells a,, Sheny Jang b, Wllard G. Mannng a Department of Eonoms, Boston Unversty, 70 Bay State Road, Boston M 05, US. <[email protected]> b Renmn Unversty of Chna, Bejng, P.R. Chna. <[email protected]> Harrs Shool of Publ Poly Studes, The Unversty of Chago, Chago IL US. <[email protected]> knowledgements: We have benefted from omments durng presentatons at Boston Unversty, Harvard Unversty, Renmn Unversty, Pekng Unversty, The Unversty of Chago, the E annual meetng, and the meran Soety of Health Eonomsts onferene. We also thank Davd Bradford, Kate Bundorf, lbert Ma, Tom MGure, Davd Meltzer, Joe Newhouse, Kosal Smon, Frank Sloan, Wenja Zhu, and two anonymous referees for ther useful omments. The opnons expressed are those of the authors, and not those of Boston Unversty, Renmn Unversty, or The Unversty of Chago.

2 bstrat We examne the effeny-based arguments for seond-best optmal health nsurane wth multple treatment goods and multple tme perods. Correlated shoks aross health are goods and over tme nterat wth omplementarty and substtutablty to affet optmal ost sharng. Health are goods that are substtutes or have postvely orrelated demand shoks should have lower optmal patent ost sharng. Postve seral orrelatons of demand shoks and unompensated losses that are postvely orrelated wth overed health serves also redue optmal ost sharng. Our results ratonalze overng pharmaeutals and outpatent spendng more fully than s mpled by stat, one good, or one perod models.. Introduton entral theme n health eonoms s the behavor of patents and provders n the presene of partal health nsurane that overs some but not all of the ost of health are. The eonom motvaton for suh overage s that rsk-averse ndvduals an redue ther fnanal rsk by poolng rsks wth others through nsurane that effetvely shfts funds from healthy to sk states of the world to redue the fnanal rsk from unertan health. Partal, rather than omplete, nsurane an be desrable to offset the adverse effets of moral hazard when the margnal payment pad by the onsumer for a serve s less than the soal ost of produng that serve. Gven that patents respond to lower out-of-poket pres of health are, health nsurane nreases the amount of the are that they purhase, generatng a deadweght loss from the nreased use. There s a great deal of empral support for the law of demand applyng to health are n the lterature, omng from observatonal studes, natural experments, and randomzed experments (Newhouse et al., 98; Mannng et al., 987; Zwefel and Mannng, 000; Baker and Fnkelsten, 0). We fous here on the normatve ssues of what the seond-best onsurane poles should be over multple treatment goods and multple perods. Ths top has taken on new mportane n the reent meran debate on health are reform, where demand-sde ost sharng has reemerged as a entral ost ontanment strategy. Cost sharng plays a entral role n the health nsurane exhanges, whh promote a seres of plan ters (platnum, gold, slver, bronze) that dffer prmarly n ost sharng. Ths ssue s also entral n Value-Based Insurane, whh promotes hgher overage for more effetve serves (Fendrk et al, 00; Choudhry et al, 00). Key ssues n ths poly dsusson are what essental benefts should be overed, and

3 how to justfy hgher or lower onsumer ost sharng. In an earler paper, we examned optmal nsurane where the tradeoff s between one treatment good and one preventve serve (Ells and Mannng, 007); n ths paper, we fous on the new ssues that arse from more than one dmenson of unertanty omng from multple treatment goods and multple tme perods. lthough we are fully aware of the new nsghts for nsurane from behavoral eonoms (e.g., Kunreuther, Pauly and MMorrow, 0), we nonetheless adopt a neolassal approah and derve results based on a model of expeted utlty maxmzaton n whh health are demand urves reflet true margnal benefts, onsumers are rsk averse, and only demand sde ost sharng s avalable to nfluene osts. Even f onsumers make the wrong hoes beause of mspereptons, nerta, or other behavoral frtons, t an stll be of nterest to model optmal nsurane more formally as a gold standard wth whh to ontrast mperfet hoes. Even under alternatve assumptons, there wll always be a tradeoff between deadweght losses from moral hazard and welfare gans from the rsk proteton offered by nomplete nsurane. Our work s not the frst to arefully onsder multple health treatment goods (Goldman and Phlpson, 007), but to our knowledge t s the frst to examne the ssues that arse from multple, orrelated soures of unertanty, whether the orrelatons arse due to orrelated demands aross dfferent health are goods at a pont n tme, orrelated demand shoks over tme, or the orrelated losses that arse from unompensated losses (e.g. job loss) that aompany the treatment ost shoks that an be overed by the nsurane plan. We derve four new fndngs about seond-best optmal nsurane:. Holdng varanes onstant, postve ontemporaneous orrelatons among goods justfy lower onsurane rates, whle negatve orrelatons justfy hgher onsurane rates. Ths fndng modfes prevous fndngs on the role of omplements vs. substtutes.. Postve seral orrelaton over multple perods (suh as wth hron are or presrptons flled) justfes lower onsurane rates. Stat models of only one perod mss the mpled aumulaton of rsk over tme. 3. Health-related losses that are not fully ompensated by nsurane (suh as lost nome or tme osts of treatment) justfy lower onsurane rates. 4. Savngs behavor modfes optmal onsurane rates and and ts effet depends on the nature of seral orrelaton. In the followng setons of the paper, we provde a systemat revew of the exstng lterature, hghlghtng that our model generalzes two lmtng ases that are already n the 3

4 lterature. fter dervng omparatve stats for our model, we numerally evaluate the relevane of our approah usng empral estmates from a set of large employment based nsurane plans. s we proeed, we show how more tradtonal rules for onsurane rates hange as a result of our extensons beyond the one good, one perod model.. Lterature Revew Muh of the eonom lterature on optmal health nsurane fouses on the fundamental tradeoff of rsk spreadng and approprate nentves (Cutler and Zekhauser, 000, p. 576). Spefally, t examnes ether the dead weght losses from moral hazard, the tradeoff between moral hazard and the gans from nsurng aganst fnanal rsk, or the dfferental overage of multple goods wth varyng degrees of rsk. Muh of ths work employs a one-perod, one health are good model wth unertanty about health states or unertanty about levels of health are expendture [rrow, 963; Besley, 988; Cutler and Zekhauser, 000; Dardanon and Wagstaff, 990; Pauly, 968; Spene and Zekhauser, 97; Zekhauser, 970]. Papers that derve the optmal nsurane ontrats usng ths framework have employed varants of the tradeoff between the rsk premum (as refleted by the rrow-pratt approxmaton) and the deadweght loss from moral hazard (as refleted n Harberger s loss or related measures) or the ompensatng varaton (Marqus and Mannng, 999). lmost all of ths lterature has been based on ether a one-perod model or a two-perod model where the onsumer selets the onsurane before knowng her realzed state of health, but hooses her health are expendtures ondtonal on the state of the world that ours. The ommon result n ths lterature s that one should selet overage n a plan wth a onstant onsurane rate suh that the margnal gans from rsk reduton from a hange n the onsurane rate just equals the margnal osts of nreasng moral hazard. The onsensus of the empral stran of ths lterature s that optmal levels of ost sharng usually nvolve nether full nsurane (zero out-of-poket ost), nor no nsurane. Dependng on the formal model approah and the data employed, optmal onsurane rates range from the perent range (Feldsten See Feldsten (973), Feldsten and Freedman (977), Buhanan and Keeler (99), Mannng and Marqus (996), Newhouse et al. (993), Feldman and Mannng (997) for other theoretal and emprally-based examnatons of optmal nsurane. Blomqvst (997) extends the theory to nonlnear nsurane shedules. 4

5 and Fredman, 977; Mannng and Marqus, 996) down to values that are n the md 0 perent range or lower, possbly wth a dedutble and/or stop-loss (Blomqvst, 997; Buhanan et al., 99; Feldman and Dowd, 99; Feldman and Mannng, 997; Newhouse et al., 993). In ths paper, we reonsder the optmal nsurane for health are n markets wth one nonhealth good and two or more health are goods. We do ths n the ontext of both one and two perod models. We are aware of only three papers that have onsdered multple health goods. Besley (988) provdes a mult-good extenson to ths lterature n whh demands for health are goods are stohast, but does not model ether omplementarty or substtutablty between goods or onsder orrelated shoks n demand. Goldman and Phlpson (007) model two goods n one perod to show how omplementarty and substtutablty of health are serves affets optmal ost sharng n an expeted utlty format. However,they do not model multple dmensons of unertanty nor do they dstngush the separate roles of orrelated rsks and omplementarty vs. substtutes n ther expeted utlty framework. Ells and Mannng (007) model the ase of one treatment good and one preventon good to hghlght how optmal nsurane rules dffer for the two types of goods, but they nlude only one health demand shok and do not model multple treatment goods, orrelatons over tme, or savngs behavor. To keep the model tratble, we take a step bakwards from Ells and Mannng and do not examne the preventve effets of health are spendng. In dong so we also dfferentate our work from Hall and Jones (007) n whh health are affets mortalty and lfe expeteny, but not utlty, dretly. Usng the same utlty struture as Ells and Mannng, we examne how optmal ost sharng s affeted by the orrelaton struture of random shoks affetng demand for health are both aross goods and over tme. We also examne how savngs and unompensated health losses affet the optmal nsurane alulatons. The mportant nsght here s that, all other thngs beng equal, health are goods that are postvely orrelated should be more generously nsured than those that are negatvely orrelated or unorrelated. Ths holds both for ontemporaneously orrelated health are treatment goods and shoks that are serally orrelated over tme. The bas log s that f the demand for two goods or over two perods s unertan, then the ex ante ombned varane s larger f they are 5

6 postvely orrelated than f there s no or negatve orrelaton. Rsk averse ndvduals wll prefer more generous nsurane (lower onsurane rates) to redue ther fnanal rsk when the health shoks affetng health goods are postvely orrelated than f demand shoks are ndependent or negatvely orrelated. We frst study the optmal demand-sde nsurane overage for health are treatment when there are two health are treatment goods. fter developng a general analytal model as our base ase, we do omparatve stats on a seres of speal ases. key attraton of our spefaton s that we not only solve for the optmal ost sharng rate as a losed-form soluton to rereate the results from the bas model but also address a rher lass of eonom behavor. These nlude new results nvolvng unompensated health are losses, orrelated health are shoks, and ross pre elasttes of demand wth multple goods. Our seond set of analytal results onsders a two-perod model n whh health are shoks persst over tme due to hron ondtons. In a mult-perod ontext, f a onsumer s savngs reat to healthare shoks, then both the ost of rsk and the optmal ost sharng rates hange. Postvely serally orrelated shoks (e.g. hron llness) mply that healthare should be more generously overed (lower ost sharng) than when shoks are ndependent or negatvely orrelated aross perods. Our next set of analytal results onsders the two good, two perod model, wth both ontemporaneous and seral orrelatons, whh gves nsghts nto optmal overage for aute versus hron ondtons. Optmal opayments wth two perods are hgher unless shoks are perfetly orrelated over tme. The onludng seton of the paper dsusses a few empral results that have a bearng on our analytal fndngs. We brefly dsuss empral estmates of the varane of three broad sets of serves and the magntudes of ontemporaneous and ntertemporal orrelatons that shed lght on the empral relevane of our fndngs. 3. Model Followng rrow (963), Pauly (968), and Zekhauser (970), we fous only on the demand-sde whle examnng optmal ost sharng. We onsder rsk averson, moral hazard, s shown below, the detals are more omplated beause optmal onsurane also depends on the varablty n demand, own and ross pre effets, orrelated error shoks, and (n the ases of multple perods) the nterest and dsount rates. 6

7 and unompensated losses wthout attemptng to model other onerns that nfluene optmal nsurane overage suh as the followng: orretng for externaltes, suh as those that an our wth ommunable dseases (Hofmann, 007); altrusm or publ good arguments for nsurane overage (Coate, 995; Rask and Rask, 000 ); effets of nsurane on tehnologal hange (Goddeers, 984); dstrbutonal onerns nludng goals lke the elmnaton of poverty or ahevng soal soldarty (ndruls, 998; tm, 999; Maarse and Paulus, 003; Sn et al., 003), orretons of nformatonal problems (.e. onsumers make the wrong desons) (Doherty and Thstle, 996; Choudhry et al, 00); or nsurane that fosters more omplete oordnaton among health are provders (Duggan, 004; Lhtenberg, 00; and Newhouse, 006). Wthout denyng the relevane of these other fators affetng optmal nsurane, we set them asde and reexamne the effeny-based arguments for nsurane wth multple goods and perods and derve new results whh refne our understandng of the value of generous nsurane overage from the onsumer s pont of vew. We assume the ndvdual s utlty funton nvolves one non-health onsumpton good, Y, and two health treatment goods, X X, X, frst wthn one perod and then over two perods. 3 Pres of the three goods are P,, Y P and P, and the onsumer s nome n eah perod s I. In the underlyng behavoral model, there s a health produton funton that transforms health are X nto health status. Followng muh of the lterature (for example, Cutler and Zekhauser, 000), we examne only health nsurane plans wth a onstant onsurane rate 0 for both health treatment goods. There are no new nsghts ganed from ntrodung loadng osts (as ths s well overed n rrow, 963, and Ells and Mannng, 007); therefore, we smplfy our model and assume loadng osts to be zero throughout ths paper. Insurane premums,, are ompettvely determned and depend on the onsurane rates and the demand struture, but do not vary aross ndvduals. The nsurane poly s thus a pure onsurane plan wth no dedutble, stop-loss, or lmt on the maxmum expendture or level(s) of overed serves. We assume the followng sequene of steps n our full model: 3 In an earler verson of the paper, we used more general notaton that allowed for an arbtrary number of health are goods, but we found that all of the relevant ntuton s obtaned usng only two health are goods. 7

8 . The nsurer hooses the premum and onsurane rates, treatment for goods X and X, respetvely. for health are. Nature dedes the onsumer s state of llness as a vetor of random health shoks,, that affet the demand for the vetor of health are goods, X. 3. The onsumer hooses quanttes X and Y to maxmze perod utlty. 4. If a two-perod model, repeat steps and 3 for Perod. Condtonal on the avalable tehnology, the demand for medal are serves has been shown by many empral studes to be hghly nome nelast, wth.., (Chernew and Newhouse, 0). 4 For smplty, we assume that X s perfetly nome nelast: X I 0. Whle ths nome elastty assumpton s strong, t buys us a great deal of smplty that enables many losed form solutons for ases wth multple health treatment goods. Whle we make ths strong assumpton about nome elasttes, we make weaker assumptons about other parameters below. We avod onerns about orner solutons by further assumng that nome s always suffent to pay for at least some of all other goods Y after payng for X. Utlty n every perod s separable n health status and the utlty of onsumpton. orollary of ths s that health are shoks do not have any effet on the margnal utlty of nome, other than through ther effet on medal expendtures. Health shoks affet the margnal utlty of nome through spendng on medal are, but do not dretly affet the margnal utlty of nome. Ths s a ommon theoretal assumpton, used n many empral studes (e.g., rrow, 963; Zekhauser, 970; Mannng and Marqus, 996). 4. Sngle perod model wth two health goods Demand urves for eah of two goods are assumed to be lnear. 4 ross ountres or over tme, the nome elastty s greater than one (Newhouse, 993; Hall and Jones, 0). Most lkely beause the avalable tehnology grows wth nome, but for an effeny analyss of optmal ost sharng, the relevant elastty s the short run nome elastty, whh s very low. Holdng soal, demograph, envronmental and supply sde attrbutes onstant, low and hgh nome ndvduals demand very smlar levels of health are even aross dramatally dfferent nome levels n generous health plans. By assumng zero nome effet, we also separate the moral hazard loss from the pre effet and the nome transfer effet. Nyman (999) ponts out the nome transfer effet on medal are onsumpton s elmnated when nome transfers are elmnated, and ths would only our f the probablty of llness were, whh s the ase we onsder as the soure of our ex post effeny problem. 8

9 () X - BP / PY GP / PY X - B P / P G P / P Y For smplty, we normalze the margnal osts of all goods to be one, and express all pres n terms of the share of ths margnal ost pad by the onsumer. Hene, PY and P, and the ost share s the onsumer pre of the th health are good X. Note the presene of the (symmetr) ross pre oeffent, G, whh wll play a key role later. s n Ells and Mannng (007), we also norporate the dea that llness may nvolve other unnsured (unompensated) losses. These losses an be of two types: unompensated out-of-poket osts (whh we assume are proportonal to overed osts), and L, the unompensated health shok L. 5 losses, The margnal beneft funton for eah medal serve Y X s assumed to be a lnear demand urve wth a onstant slope B on the pre regardless of the realzaton of the random health shok. Stohast health treatment demand s ntrodued by lettng, where ~ F, wth E 0. ssumptons about the varane/ovarane vetor of are made below but throughout we assume that the dstrbuton does not depend on the out-of-poket pre or nome. Ths orresponds to the horzontal nterept of the demand urves havng a mean of when the full onsumer pre Ls zero. Usng ths onventon, a sngle onsumer s demand urve for eah medal serve X an be wrtten as () X B L L + G + j j j 5 rrow (963) was the frst to rase the ssue of unompensated losses n thnkng about health nsurane. Doherty and Shlesnger (983) show results smlar to our fndngs below, whle Gravelle and Rees (004,Chapter 9) reah smlar onlusons. We nlude these losses beause earler results are based largely on fndngs that assume rsks are ndependent of the health rsk. Seton 4.3 dsusses the senstvty of the usual seond best results to ths ssue. 9

10 In order to ntrodue rsk averson, we apply a monotonally nreasng onave funton V wth the property of onstant absolute rsk averson to the ndret utlty funton onsstent wth the demand funton n Equaton (). Usng ths notaton, we wrte the one perod ndret utlty funton wth two treatment goods as B( L) B( L) J ( L) (3) V(, I C, ) V L ( ) L ( ) ( L) G( L)( L) where J I, That ths utlty funton orresponds to the above demand funtons (nludng ther symmetr ross pre terms G ) an be verfed by applyng Roy s dentty. Usng the lnear demand equaton for X, the nsurer s break-even ondton for the nsurane premum s (4) ( ) B ( L) G ( L ) ( ) B ( L ) G ( L). In a two-perod model, we assume that the same premum s harged n both perods. Whle we have used the somewhat restrtve assumptons of lnear demand, addtve errors, and zero nome effets, ths spefaton has two attratve features. The error terms only nterat wth ost sharng ( ) n a smple multplatve form. Ths faltates ntrodung multple goods and multple perods. The lnear spefaton also allows us to onsder ross pre elasttes n a natural way. We now turn to the soal planner s problem of hoosng the optmal onsurane rates when there are two health are treatment goods ( X and X ) and a omposte all-other-goods ommodty, Y. We develop the model usng a general spefaton and then derve varous ases of nterest as speal ases. The optmal ost sharng rates for health are treatment wll maxmze the expetaton of Equaton (3). Takng ts partal dervatve wth respet to and settng equal to zero yelds an equaton that haraterzes the soal optmum. Sne ths expresson wll not n general have a smple losed form soluton, we take a Taylor seres approxmaton of the partal dervatvev I, evaluated around the nonstohast arguments of the utlty funton. Ths soluton an be wrtten as 0

11 (5) VI J K VII J K L L E V 0 E B( L) G( L) where J I, K ( L) ( L) B( L) / B( L) / G( L )( L ), ( ) B( L) G( L) ( ) B( L) G( L), B B BL G G GL Defne the absolute rsk averson parameter R V E, E, E, and use E E II V I, let 0. We show n the appendx that ths result an be rearranged to obtan frst order ondtons haraterzng the optmal onsurane rates and as follows. (6) R L L 0 B B BL G G G L B( L) G (7) R L L 0 B B BL G G G L B( L) G We onsder a varety of ways of nterpretng these two equatons below. In broad terms, the frst lne of eah equaton gves the margnal ost of nreasng the ost-sharng rate, whle the seond lne, whh nvolves R gves the margnal beneft n terms of redued ost of rsk. 4. One health are good, base asel L B G 0 The Base Case of a sngle health are good n a sngle perod wth no unompensated osts orresponds to L L B G 0. Ths yelds the well-known result from the lterature haraterzng the seond-best optmal nsurane ontrat for onstant onsurane rate plans when there s a smple tradeoff between moral hazard and the ost of rsk:

12 B R 0 (8) Where B 0 and the frst term (the margnal osts due to moral hazard) s nreasng n the pre response B. The gans from rsk poolng are nreasng n the varane of health are demand. Solvng for the optmal onsurane rate yelds Equaton (9), where the optmal base ase onsurane rate,, s nreasng n the pre response, B, and dereasng n the underlyng varane n demand,. (9) B BaseCase B R Intutvely, there s a tradeoff n hangng to derease rsk proteton as the patent pays more of the pre, and to redue moral hazard. The optmal onsurane rate s the rato of the loss from moral hazard to the net loss from hangng rsk proteton plus moral hazard. If demand s perfetly nelast B 0, and there s any rsk at all ( 0) then the optmal onsurane rate s zero. If there s no varane or the onsumer s rsk neutral( R 0), then the optmal onsurane rate should be. For all nterestng ases the optmal onsurane rate les BaseCase between 0 and : ddng unompensated health-related losses n the one good ase L 0 Inorporatng unompensated health losses so that L 0 also affets optmal ost sharng for overed treatment goods, nreasng the desred overage (dereasng the ost sharng rate ) as onsdered n the lterature on nomplete nsurane markets from Doherty and Shlesnger (983), Gravelle and Rees (004), and Ells and Mannng (007). The expresson for the optmal ost sharng rate for health good X beomes (0) R L BaseCase B R. The optmal onsurane rate s dereasng n the sze of the unompensated loss L beause all of the terms after the mnus sgn are postve. Wth unompensated losses, t s also possble for to be negatve or to reah a orner soluton where 0 for ether large L or

13 small B. Equaton (0) provdes an effeny-based ratonale for why full nsurane an be seond-best optmal: the absene of omplete nsurane markets to fully transfer nome nto hgh-ost health states of the world means that onsurane rates are set at or loser to zero than they would have been f the alternatve nsurane markets were omplete and onsumers were able to nsure aganst all health are losses. There are many health serves and ondtons whh have substantal unompensated health are related losses. Ths s partularly true n developng ountres where dsablty and unemployment nsurane are rare and produtvty losses from ll health are often large. 6 Thus, nomplete nsurane markets provde a ratonale for more generous nsurane overage of health are treatment, even when welfare losses due to moral hazard and nsurane loadng may be large. 4.3 Multple health are treatment goods 0, 0, B 0, L L 0 n mportant motvaton for modelng two rather than one health are treatment goods s to be able to examne the role of ross pre elasttes and orrelated health shoks. If L L 0, then (6) and (7) above an be solved for the optmal ost share for health good and smplfed to: G R BR BR Base Case () wth a smlar expresson for the seond health are good. Beause all of the terms multplyng and G are non-negatve, the negatve sgn means that the frst order effet of nreasng ether term s to redue the optmal ost share we have G B R 0, redung renfores the fat that relatve to the base ase. In ths ase, G 0. Hene, when two health serves beome stronger gross substtutes n the sense that G X s nreased, then both serves should have lower ost sharng rates. Ths fndng replates the fndng of Goldman and Phlpson (007) that as goods beome stronger omplements ( G 0) they 6 Wagstaff (007) provdes doumentaton of the large magntudes of nome losses from llness n Vetnam. 3

14 should have hgher ost sharng. We now refne ths onluson n lght of allowng for seond order hanges that affet demand for these two health goods. The mplatons of the ovarane (or orrelaton) between shoks affetng health goods on optmal nsurane an be omplex. s a result of the nd order effet of on 4, we annot unambguously sgn for all possble values of G and. We an, however evaluate ths dervatve for varous speal ases. In the ppendx we show that and G an be unambguously sgned to be negatve when ether of the followng holds true: ) both health are goods have the same demand urve slope and varane (B B, ) or ) as the lmtng ase where and G both approah zero. The ambguous ases, as dsussed below, our when the two parameters dffer n sgn. We use a graphal approah to ontrast optmal nsurane as G and hange by examnng how they hange relatve to the Base Case when G and are both equal to zero. In eah of Fgures through 3, the equlbrum expresson () and ts ounterpart for are shown as sold lnes, and are used to examne the net effet of hangng (Fgure ) and hangng G (Fgures ), and hangng both (Fgure 3). The Base Case llustrates the ommonly ted result for optmal onsurane that 0 when health are shoks for unorrelated goods are nether substtutes nor omplements (.e., G =0). Fgure llustrates how the soluton lnes pvot as nreases, whh fores both optmal onsurane rates to derease, as the ovarane term nreases from zero to a postve level. If goods and are gross substtutesg 0, and there s no ovarane ( 0 ), then as n Fgure, there s an unambguous derease n the optmal onsurane rates for both goods. Fgure 3 llustrates the orrespondng ase where the two goods are substtutes and postvely orrelated, that s, G 0, and 0. The substtuton effet and a postve orrelaton renfore eah other to further ut the optmal ost sharng. (The ounterpart fgures for omplements are shown n Fgure (a)-(b).) If both G and are of the same sgn t s possble that the partal dervatves of ether parameter on an be ambguous beause of seond order effets, whh

15 means the seond order dervatves an be postve or negatve. s shown n Fgure (a)-(), both equlbrum lnes are rotatng and shftng beause of the log llustrated n the Fgures (non-zero ovarane) and (substtutes). It s possble that the seond order effet of the two lnes movng wll leave one hgher whle the other s made lower. Our model s parameterzed n terms of varane and ovarane terms. However, sne n the preedng exposton we were holdng the varane terms onstant, ths mples that an nrease n the ovarane s dretly nterpretable as an nrease n the ontemporaneous orrelaton. Restatng our man result n terms of orrelatons: for demand strutures where goods are ether substtutes or nether substtutes nor omplements, nreasng orrelaton always warrants more generous nsurane overage. If two goods are suffently strong omplements, ( G 0), then nreasng orrelaton does not neessarly warrant more generous nsurane overage and t s possble that optmal nsurane rates an nrease, so that base. Conversely, for suffently negatvely orrelated shoks, optmal overage an also derease when goods are strong substtutes. It s straghtforward to show that other elements of the onventonal results for nsurane are also valdated n ths model, even f the overall level depends on omplementarty n demand or ovarane nformaton. It s shown n the appendx that < 0, < 0, > 0, and < 0. B B We an now add that the nrease n the ovarane, or more spefally the orrelaton wll lead to the reduton of at least one optmal onsurane rate, possbly both. It s that ambguty about whh s the affeted health are good, that leads us to put the results n parentheses n Table 5. Multple perod model Our framework an also address the ase of multple perods wth orrelated health are demands. In the ppendx, we derve optmal savngs rules for models wth T > perods, whle here we fous on the ase where there are only two perods (ndexed by and ). In ths seton we fous on the ase where there s only one good n eah perod, before turnng to two goods n two perods n the next seton. We assume that the ost share s the same n both perods, and 5

16 hene n ths seton use. We also assume there are no unompensated losses, so that L L 0, and we fous on the ase where demand s the same n eah perod exept for the health shoks, X - B+. ddtonally, we fous on the ase where the parameters and pre struture are onstant over tme: I I I,, and. To allow for the possblty that the health are demands n the two perods are orrelated, we assume that the, where. Ths assumpton sgnfantly seond perod health shok s dfferentates our paper from Goller (003), whh examnes the demand for nsurane n a lfeyle model wth no seral orrelaton n the nsurable rsk. In a dynam model, we need to ntrodue savngs, whh we assume to be optmally hosen. In our two-perod model, net savng s deded n perod after s known and all nome and savngs are spent n perod (.e., there are no bequests). Optmal savngs wll depend on all of the parameters of the model, but of speal nterest s that savngs wll depend on the ost sharng rate and the frst perod health shok q, S (, ) wth S / 0. In the ppendx, we show that the objetve funton to be maxmzed through the hoe of an be wrtten as follows: () V J K S(, ) EV E, E V J K ( r) S(, ) where J I B K B Exept for the savngs funton and the ntroduton of dsountng,, ths formulaton s very smlar n struture to what was used above for the ase wth one perod wth multple states of the world. The soluton for the optmal hoe of s derved n the appendx. We make the followng three further assumptons n dervng our soluton: 6

17 Savngs s optmal so that for all q, V... ( r) E V... s the nterest rate. I where r The utlty funton an be approxmated usng a seond order approxmaton wth onstant absolute rsk averson. Consumers an earn a return on savngs ( r) that s the nverse of ther dsount rate, so that ( r). In the ase of the quadrat utlty funton that we have used n our analyss, the optmal I savngs s approxmated by S (, ) S s (shown n appendx), where the expeted (ex ( r) ante) savngs are S and the optmal (ex post) savngs are redued by the R ( r) ( r) proporton s ( r) multpled by the out-of-poket health payments n tme perod. The term s s the margnal propensty to save. In partular, f ( r) (as assumed), then S(, ) s redued to a smple funtonal form S 7. Under these assumptons, when r n the optmal ost share s B B (3). ( ) BR s BR ( r)( ) Ths result s very smlar to the Base Case equaton (9) for the ase of a sngle perod and one health are good wthout any unompensated losses L 0. Equaton (3) dffers from the Base Case by the addton of a new savngs-related term n the denomnator, ( ) ( r )( ). 7 In the more general ase of any n perods, where the autoorrelaton terms are allowed to have an arbtrary pattern rather than a frst order autoorrelaton, we show n the appendx (44) that the optmal savng funton stll has a losed form. 7

18 s a funton of the orrelaton oeffent between perod and perod health shoks,,the nterest rate, r, and the onsumer dsount rate,. Note that s non-dereasng n, r, and, and that s dereasng n. s before, we nterpret the optmal ost sharng result for a varety of speal ases. Frst, onsder the ase where the margnal propensty to save s zero, s 0, so that savngs does not respond to health shoks. In ths ase, and the one perod model results reman orret even wth two perods. The onsumer must absorb all health shoks fully n the frst perod, so there s no dfferene between the stat and dynam hoes of. However when s 0, optmal opayments are hgher unless shoks are perfetly orrelated. Savng s a sort of selfnsurane that smooths rsk over the both perods and affets optmal nsurane by redung the need for generous nsurane when shoks are not perfetly orrelated n the presene of seond best nsurane markets. Seond, onsder the ase where the perod and perod shoks are perfetly orrelated, so that. One agan and the one perod model results hold. lthough savngs s possble, there are none f the onsumer knows exatly what the shok wll be n perod. There s no dversfaton aross perods n the burden of health shoks. In ths lmtng ase, nsurane should be the same as wth no savngs. Thrd, onsder the ase where health shoks are unorrelated over tme, so that 0. The dsount rate s a number lose to one, and t s onvenent to onsder the ase where and r 0 so that there s no nterest or dsountng. The plausble result n ths two perod model s that s wll be lose to one half, and half of the burden of a health shok s born n perod and half s deferred to perod. Sne the ost of rsk goes up wth the square of the devaton from ertanty, the savngs redues rskness n the frst perod to one-quarter of the one perod value and, hene, the ost of frst perod rsk (proportonal to the varane) s redued to one quarter of the one perod model results. Sne ths burden s shared between two perods, the net reduton n rsk s by one half of the varane. The reason that only delnes to 0.75 s that n a two perod model there s no opportunty to redue the burden of shoks n the seond perod. So whle savngs an redue the burden of frst perod shoks to one quarter of ther unertanty ost, savngs annot redue the burden of seond perod shoks. 8

19 Ths result wth no dsountng or nterest rates losely approxmates the result wth dsountng, sne the two terms ( r)( ) wll be approxmately 4 f onsumers use the market nterest rate for dsountng. 6. Two-perods wth two-goods So far, we have addressed the ases of two treatment goods and two perods separately. It s also mportant to onsder the ase of multple treatment goods wth more than one perod so that we an address ssues suh as the dfferental overage of aute and hron health are. We fous on the ase where there are two goods and two perods. The onsumpton of two goods n two perods s ndexed by,,, 9 X X X X wth subsrpts ndatng goods, and supersrpts tme perods. Followng the notaton n the onegood multple perod ase, we assume,,, I I Iand. In the X X appendx we allow for L L L 0, but here n the man text we fous on the ase of no other unompensated osts, so that L L 0. To make the results more onrete, we onsder the ase where X s a medal treatment good for an aute ondton and X as a medal treatment good for a hron ondton. Then, the aute good shok n the frst perod s not orrelated wth ts own future amount, whle the shoks affetng the hron ondton good two n both perods are postvely orrelated, where, 0<. We also assume that E( ) E( ) E( ) E( ) 0 and VR The objetve funton and frst order ondtons haraterzng the optmum are shown n the appendx. The man new result of nterest relates to how onsumers optmally hange savngs n response to aute rather than hron ondtons. If dsountng exatly equals +r, so that ( r). Then, we show n the appendx that the optmal savng funton s approxmated by, r r (4) S (,,, )

20 From ths expresson one an see that the margnal savngs out of the unexpeted health spendng from the two types of shoks dffer, wth s r and s. The expeted (ex r ante) savngs S are the same wth the two-perod one-good ase. The margnal propensty to saves s lower than s beause good s hron are and patents who reeve a large shok for hron are antpate a orrelated shok the followng perod and, hene, do not adjust ther savngs as muh as for an aute are shok. Results for the general ase are presented n the appendx. For the speal ase where demand shoks have zero ovarane n eah perod and the goods are nether omplements nor substtutes, so that 0, G 0, and L=0, the two-good, two-perod ase redues to the onegood two-perod ase. The optmal ost sharng s approxmated by (5) (6) B BR B B R r r The optmal ost sharng rates are dental to equaton (3) wth 0 for aute are. Thus, the orrelaton of spendng for hron are n two perods does not affet the optmal ost sharng of the aute ase. Thus, our fndngs ndate that hron are should have better overage than aute, all other thngs equal, gven that hron are s postvely serally orrelated, whle aute are s unorrelated over tme. In the appendx we also examne the ase where the two goods are substtutes or omplements and where the two goods are postvely or negatvely orrelated ontemporaneously, wth derved expressons (56) and (57). Whle savngs and postvely serally orrelated errors makes optmal ost sharng more senstve to these two parameters, and onsumer dsountng makes them less senstve, the arguments that substtutes and postvely ontemopraneously orrelated shoks deserve better nsurane overage than mpled by the one perod model ontnue to hold wth two perods. For the speal ase where 0and G 0, expressons 0

21 (5) and (6) reveal that optmal ost sharng satsfes We have extended the theoretal lterature on effeny-based models of optmal nsurane to address ssues that arse from orrelated soures of unertanty, whether the soure of the orrelaton s due to orrelated demands aross dfferent health are goods at a pont n tme, orrelated demands over tme, or the orrelated losses that arse from unompensated losses that aompany the losses overed by the nsurane plan. By usng a quadrat ndret utlty funton and, hene a lnear demand spefaton wth zero nome effets on the demand for health are treatment, we have been able to derve losed form expressons haraterzng optmal ost sharng for health are treatment when there are two health are goods and two tme perods. Table summarzes the omparatve stats fndngs n ths paper for the varous parameters onsdered for health are treatment goods and multple tme perods. In some of the ases that we have onsdered, we an only sgn the effets of a parameter on optmal ost sharng rates for ertan parameter values. These ases have the omparatve stat results n parentheses. The frst four rows of Table reaffrm onventonal results found n the prevous lterature, whle the bottom nne rows reflet new or at least newly nuaned results that extend the prevous lterature. It s well establshed that optmal ost sharng on health are treatment should be hgher as demand beomes more elast, onsumers beome less rsk averse, or the varane of spendng dereases. These frst four fndngs are onsstent wth the fndngs from Besley (988) and others. Our theoretal fndngs are also onsstent wth those of Goldman and Phlpson (007) on omplements and substtutes that all other thngs equal, ost sharng should be hgher for omplements than substtutes. Our man new fndng are: () postvely orrelated losses aross health are goods or over tme should almost always lead to more generous overage (lower ost-sharng) than unorrelated or negatvely orrelated losses; and () r < 0 and <0. lthough we expet these frst order effets to domnate, beause the real nterest rate r and have a omplex effet on savngs and onsumpton desons, we ould not unambguously sgn the effet of these varables on optmal nsurane n the general ase. 7. Theoretal Summary

22 optmal savngs behavor shfts optmal nsurane overage dependng on the nature of the shoks and orrelaton of eah good over tme. Chron ondtons, whh by ther nature dsplay postve seral orrelaton, deserve greater overage than aute ondtons that are not serally orrelated sne hron ondtons mpose more rsk and savngs annot make up for ths addtonal rsk as easly as when shoks are unorrelated. 8. Empral Relevane In ths seton, we brefly examne the empral magntudes of the two nnovatons of our model: the role of ontemporaneous orrelatons aross multple goods and of seral orrelaton over tme. We use data from the Thomson-Reuters (now Truven nalyts) MarketSan database from the perod on a populaton of non-elderly (age <65) enrollees n employment-based ommeral plans. We have restrted the sample to FFS, HMO, PPO and POS plans whh overed outpatent pharmay serves n addton to outpatent physan and npatent serves. We nluded only those ndvduals who were ontnuously enrolled for the full fve year perod. Ths yelds a sample of,335,448 ndvduals. Besdes the large sample sze, these data have two major advantages. The frst s that the enrollees are followed for several years, allowng us to study orrelatons by type of health are over tme. Seond, all of the enrollees had pharmay overage, allowng us to ontrast pharmay expendture patterns wth those of both npatent and outpatent are. Table summarzes key means, standard devatons and orrelatons from our fve year sample, deomposed nto three broad types of serves npatent (falty, not npatent physan) serves, outpatent serves, and pharmay serves. 8 The frst two olumns n the top half of Table reaffrm that npatent spendng, whle not the largest expeted ost, s by far the most rsky n a one year framework. Table also shows that the seral orrelaton oeffents for spendng for eah of the three serves dffer meanngfully, wth pharmaeutal spendng havng the hghest seral orrelatons and npatent spendng the lowest. The seral orrelatons also reveal that followng a health shok spendng returns to normal levels muh more slowly than a 8 Some researhers may expet a hgher proporton of spendng to be on npatent are. The MEDSTT ommeral lams do not nlude Medad or Medare enrollees, who have hgher hosptalzaton rates. MEDSTT lams haves 4% of all overed harges for npatent are n 004. Our samplng frame of usng only people wth fve onseutve years of nsurane overage has somewhat lower proporton of perent of spendng n npatent serves.

23 smple autoregressve, R(), pattern would suggest. Chron ondtons obvously explan ths pattern. The larger orrelatons for pharmay versus outpatent are and for outpatent versus npatent are suggest a larger orreton for pharmay than outpatent from the usual results for a one perod model. The fnal olumn summarzes the mplatons of usng multple years of spendng to alulate fnanal rsk by presentng the standard devaton of fve year sums of spendng rather than one year spendng. Whereas npatent spendng has nearly four tmes as muh varaton as pharmay usng a one year horzon, t s less than twe as varable f a fve year horzon s used. Our theoretal model shows that ontemporaneous orrelatons between multple health are goods an also be mportant. Table 3 presents two orrelaton matres for spendng on npatent, outpatent, pharmay and total spendng. The top half s for one year (004) whle the bottom half orresponds to orrelatons of fve year sums of eah of the three omponents and total spendng. The top orrelaton matrx shows that one-year npatent spendng s the most losely orrelated wth total spendng and that orrelatons among other serves are relatvely modest the onventonal vew. Takng nto aount the autoorrelaton effet, we generate the bottom half of the table. It reveals that the pattern for npatent spendng s weaker usng fve year total spendng. Outpatent spendng beomes the ategory most losely orrelated wth fve year total spendng, and pharmay spendng has a sgnfantly hgher orrelaton wth total spendng when ompared to the one-perod ase. Consderng the ontemporaneous orrelaton and the autoorrelaton, our mult-perod model suggests better overage for outpatent and pharmay spendng than suggested by onventonal rules for nsurane overage. Needless to say, the MarketSan data do not have nformaton on the range of unompensated losses. Thus, we are unable to omment on the magntude of the orreton for orrelated unompensated losses. Moreover, n the absene of estmates of the underlyng demand elasttes for these three serves, or, even more hallengng, estmates of the degree of omplementarty among them, t s dffult to determne how dfferent the onsurane rates under our rules would be from those based on Besley s formulaton or older approahes. But the dreton s lear. Wth ther hgher postve orrelaton over tme, pharmaeutals should have a lower onsurane rate than would our under the tradtonal rules for one-perod models. Spendng on npatent serves, whh s less serally orrelated than outpatent and pharmaeutals, would hange the least. 3

24 9. Dsusson The new results that we fnd most nterestng are those that () fous on the roles of unompensated losses dfferentally over health are goods and tme perods, and () those that address the role of orrelatons aross goods and tme. s n our earler work (Ells and Mannng (007), as well as Gravelle and Rees (004) and Doherty and Shlesnger (983)), unompensated health losses that are related to nsured serves should nfluene the level of ost sharng for orrelated health are goods. These losses provde a ratonale for both redung outof-poket osts for those goods whh tend to have larger unompensated losses, suh as tme lost due to hosptalzaton and reovery, tme lost by gong for a physan vst, or opayments. The ntuton s lear. If onsumers fae unertan nome losses whh are orrelated wth health are spendng shoks on ertan treatment goods, then overnsurng those treatment goods s a seond best response to redue ths ombned rsk from the ompensated and unompensated elements. In the tradeoff of moral hazard aganst rsk proteton, the key term n unertanty n the demand for health goods n the sngle perod, two-good model s L L where the θ s are unertan ex ante. The rsk premum depends on the varane of ths whole expresson, whh n turn depends on the sze of the unompensated losses (the L s) ompared to the out-of-poket opayments (the s). Unompensated losses nrease both the benefts from rsk reduton (the varane term from above) and the osts of nsurane (the demand / moral hazard term.). Our fndng that optmal treatment ost sharng should be lower for postvely orrelated treatment goods and goods wth postve ross-pre effets augments the fndngs of Besley (988) who fouses on multple goods wth unorrelated demand shoks. It makes strong ntutve sense that postvely orrelated varables ause greater varane whh needs to be partally offset by lower ost sharng. 9 The empral sgnfane of these results s dffult to assess, sne relatvely lttle researh has foused on estmatng ovarane and substtuton parameters. They may nonetheless provde gudane on overage for goods suh as brand name drugs, spealty uratve goods, or the overage of serous hron llnesses, whh may be strong 9 Besley (988, ) ndates that ross-elasttes and ovaranes jontly affet whh good s more generously overed. But ovaranes are the produt of rho and the two standard devatons. Thus hs result s atually applable to orrelaton ρ, beause he only uses whether ovarane s zero or not, whh s equvalent to ρ = 0. 4

25 substtutes or omplements. Ths framework also provdes a ratonale for more generously overng serves that are often jontly provded, suh as spealty serves and the lab tests or drugs that spealsts presrbe, sne suh goods wll be postvely ontemporaneously orrelated. Fnally, our mult-perod model shows the key role that savngs desons and orrelated errors play n settng optmal ost sharng. We are not aware of any papers n the health eonoms lterature that have emphaszed ths top, although there s a szeable lterature on how large unovered health losses an lead to dssavng and bankrupty. Whle there s a lterature on how onsumers respond to health spendng shoks, the mplatons for optmal health nsurane desgn deserves reexamnaton. Expensve, hron ondtons, whh exhbt strong postve seral orrelatons over tme, provde an eonom ratonale for more generous nsurane overage beause onsumers annot use ntertemporal savngs to redue the burdens of suh spendng. Thus, n a world where hron and aute are look to be otherwse equvalent n terms of pre responses and varablty, the stronger orrelaton n health are spendng for the hronally ll would lead to better overage than the standard one perod model would suggest. Gven that so muh of pharmaeutal use exhbts the same property, ths fndng also supports better pharmaeutal overage. It s worth hghlghtng the lmtatons of our study. s s ommon n ths lterature, we develop all of our models usng a lnear demand struture, wth addtve errors that have onstant varane, and our demand for health are treatment s perfetly nome nelast. 0 By makng these smplfatons, we assume away nome effets and orner solutons, whh are partularly relevant n equty dsussons of optmal health nsurane. In our model, nreasng nsurane overage does not subsdze low nome people but rather subsdzes those wth poor health. Obvously f poor health and low nome are orrelated, then transfers are mpled. We aknowledge that we use strong assumptons, but beleve our results make ntutve sense and are more general than many other models that use only onsumer surplus, or assume only two health states or one health are good. 0 The empral lterature fnds that demand s nome nelast overall, espeally n the absene of adverse seleton on nsurane overage (Mannng et al, 987; Newhouse et al, 993). But demand for spef health treatment serves may be more hghly nome elast and yeld dfferent results. 5

26 We have repeatedly used a seond order approxmaton of the utlty funton, whh s onsstent wth approxmatng the utlty funton wth a onstant absolute rsk averson funton. We are not espeally troubled by ths assumpton beause our results should hold as an approxmaton for any arbtrary funton, as long as the absolute rsk averson parameter does not vary too muh aross states of the world. Our unompensated loss funton and optmal savngs funton were also approxmated usng lnear funtons. gan, our results should hold as an approxmaton for more general nonlnear funtons. The other restrtve assumpton that we have made for tratablty s sake s that the varane n health are expendture s a onstant, ondtonal on the health state. Spefally we have assumed that the varane and the other hgher order moments n healthare treatment do not depend on the level of ost sharng, that s / 0, or other observable fators n the demand funton. n extenson of the urrent work would allow for the ommon observaton that the varablty n health are expendtures s an nreasng funton of the mean or expeted value of expendtures gven the ovarates n the model. Our models hghlght the empral sgnfane of demand and ost parameters. Of all of our parameters, the ost sharng demand responsveness of varous health are treatment goods has been studed the most. The RND Health Insurane Experment and other studes have establshed that spendng on npatent are s less responsve to ost sharng than spendng on outpatent are, whh s less responsve than spendng on pharmaeutals. The varanes and means of spendng on dfferent types of treatment goods are also well understood. Inpatent spendng s muh more varable than outpatent spendng and drug spendng, justfyng greater overage for npatent are than other health serves. These onlusons follow from the prevous lterature as well as our framework. When the varane beomes an nreasng funton of the mean, we pk up an extra term n the ost-of-rsk part of the frst order ondton that dd not exst f demand for health are treatment X( x) had onstant varane, ondtonal on health status. s we nrease the onsurane rate, we have the usual nrease n the term related to the varane of out-of-poket expenses. Wth nreasng onsurane also dereasng the mean, t also dereases the varane. Thus, we have a partally offsettng term to nlude n the ost of rsk. The magntude of ths reduton depends on how pre responsve demand s. s long as the demand for health are treatment s nelast wth respet to ost sharng, the qualtatve pattern desrbed earler n ths seton prevals. See Feldman and Mannng (997) for suh an extenson to the bas model that we onsdered brefly n Ells and Mannng (007) that allowed for a onstant oeffent of varaton for health are expendtures nstead of a onstant varane assumpton. 6

27 Less well studed are the mplatons of ross pre effets, ontemporaneous orrelatons, and seral orrelatons over tme of spef treatment goods. Drug spendng and outpatent spendng have hgher ontemporaneous orrelatons wth other types of spendng than npatent are, suggestng they may deserve greater nsurane overage than would otherwse be the ase. Cross elasttes of demand for drug and outpatent are are hgher than for npatent are, justfyng greater overage. Some evdene on ths s provded n Meyerhoefer and Zuvekas (006) who demonstrate that ross pre elasttes between non-mental health drugs and spendng on treatment for physal health are moderately large and statstally sgnfant. Relatvely lttle work has explored the ntertemporal orrelatons of spef treatment serves. We do know that spendng on some drugs and outpatent are s muh more hghly orrelated over tme than spendng on npatent are. Ells, Jang and Kuo (03) provde reent evdene on ths ssue n examnng more than 30 dfferent medal serves defned by type of serve, provder spealty and plae of serve. Our evdene from the Thomson-Reuters MarketSan data also suggests that these orrelatons are szeable and mportant for both outpatent and for pharmay, but stronger for pharmay than outpatent. Perhaps the area most n need of empral work s to doument the magntude of unompensated health losses that are orrelated wth health are spendng. Sgnfant unompensated osts provde a ratonale for zero or even negatve ost sharng on treatment goods. It would be nterestng to know how muh the onventonal model results must be adjusted gven ther magntudes. 7

28 REFERENCES tm, C., 999. Soal movements and health nsurane: a rtal evaluaton of voluntary, nonproft nsurane shemes wth ase studes from Ghana and Cameroon. Soal Sene & Medne; 48; ndruls, D. P., 998. ess to are s the enterpee n the elmnaton of sooeonom dspartes n health. nnals of Internal Medne 9; rrow, K. J., 963. Unertanty and the welfare eonoms of medal are. meran Eonom Revew 53; Baker, K., Fnkelsten,., 0. The Effets of Medad Coverage Learnng from the Oregon Experment. New England Journal of Medne 365; Blomqvst,., 997. Optmal non-lnear health nsurane. Journal of Health Eonoms 6; Besley, T., 988. Optmal rembursement health nsurane and the theory of Ramsey taxaton. Journal of Health Eonoms 7; Buhanan, J. L., Keeler, E.B., Rolph, J.E.,Holmer M.R., 99. Smulatng health expendtures under alternatve nsurane plans. Management Sene 37; Choudhry, N.K., Rosenthal, M.B., Mlsten,., 00. ssessng The Evdene For Value-Based Insurane Desgn.Health ffars 9; Chernew, M.E., Newhouse, J.P. 0. Health Care Spendng Growth. In: Pauly MV, MGure TG, Barros PP. (Eds),Handbook of Health Eonoms, vol.. North-Holland: msterdam,. p. -6. Coate, S., 995. ltrusm, the Samartan's dlemma, and government transfer poly. The meran Eonom Revew 85; Cutler, D.M., Zekhauser, R.J The anatomy of health nsurane. In: Culyer, Newhouse JP (Eds), Handbook of Health Eonoms, vol.. North Holland: msterdam; p Dardanon, V., Wagstaff,., 990. Unertanty and the demand for medal are. Journal of Health Eonoms 9; Doherty, N.. and Shlesnger Optmal nsurane n nomplete markets. Journal of Poltal Eonomy, 9(6): Doherty, N.., Thstle, P.D., 996. dverse seleton wth endogenous nformaton n nsurane markets. Journal of Publ Eonoms 63; Duggan, M., 004. Does ontratng out nrease the effeny of government programs? Evdene from Medad HMOs. Quarterly Journal of Eonoms,

29 Ells, R.P., Jang, S, Kuo, T., 03. Does serve-level spendng show evdene of seleton aross health plan types? ppled Eonoms 45; Ells R.P., Mannng W.G Optmal health nsurane for preventon and treatment. Journal of Health Eonoms 6; Feldsten, M., 973. The welfare loss of exess health nsurane. Journal of Poltal Eonomy, Unversty of Chago Press, vol. 8(), pages 5-80, Part I, M. Feldsten, M, Fredman B., 977. Tax subsdes, the ratonal demand for health nsurane, and the health are rss. Journal of Publ Eonoms 7; Feldman, R, Dowd, B., 99. new estmate of the welfare loss of exess health nsurane. The meran Eonom Revew 8; Feldman, R., Mannng, W.G., 997. Une Formule Smple du Taux Optmale pour Une Pole D- ssurane Malade (Englsh Translaton: smple formula for the optmal onsurane rate n a health nsurane poly). In: Jaobzone S(Ed),Eonome de la Sante: Trajetores du Futur, InseeMethodes, vol Frenh Natonal Mnstry of Health. p Fendrk.M., Smth D.G., Chernew M.E., Shah S.N., 00. beneft-based opay for presrpton drugs: patent ontrbuton based on total benefts, not drug aquston ost. meran Journal of Managed Care 7; 86 Goddeers, J.H., 984, Medal nsurane, tehnologal hange, and welfare, Eonom Enqury : Goldman, D., Phlpson, T.J Integrated nsurane desgn n the presene of multple medal tehnologes. The meran Eonom Revew 97; Goller, C., 003. To nsure or not to nsure?: n nsurane puzzle. The Geneva Papers on Rsks and Insurane Theory, 8: 5-4. Gravelle, H. and Rees, Ray Mroeonoms, Thrd Edton. Prente Hall: Harlow, England. Hall, R.E., Jones, C.I., 007. The value of lfe and the rse n health spendng. Quarterly Journal of Eonoms ; Hofmann,., 007. Internalzng externaltes of loss preventon through nsurane monopoly: an analyss of nterdependent rsks. The Geneva Rsk and Insurane Revew 3; 9-. Lhtenberg, F., 00. The effets of Medare on health are utlzaton and outomes. Fronters n Health Poly Researh, Vol. 5, ed. By lan Garber (MIT Press) Kunreuther, H.C., Pauly, M.V., MMorrow, S. 0. Insurane and behavoral eonoms. Cambrdge: Cambrdge Unversty Press. Maarse, H, Paulus,., 003. Has soldarty survved? omparatve analyss of the effets of soal health nsurane reform n four European ountres. Journal of Health Polts, Poly and Law 8;

30 Mannng, W.G., Newhouse, J.P., Duan, N. et al., 987. Health nsurane and the demand for medal are: Evdene from a randomzed experment. meran Eonom Revew 77; Mannng, W.G., Marqus, S.M., 996. Health nsurane: the trade-off between rsk poolng and moral hazard. Journal of Health Eonoms 5; Marqus, S.M., Mannng, W.G., 999. Lfetme osts and ompensaton for njures. Inqury 36; Meyerhoefer, C., Zuvekas, S., 006, Coverage for mental health treatment: Do the gaps stll persst? Journal of Mental Health Poly and Eonoms, Sep. 9(3): Newhouse, J.P., Mannng, W.G., Morrs, C.N., et al., 98, Some nterm results from a ontrolled tral of ost sharng n health nsurane. New England Journal of Medne; 305; Newhouse J.P. and the Insurane Experment Group, 993. Free-for-all: Health nsurane, medal Costs, and health outomes: the results of the health nsurane experment. Cambrdge: Harvard Unversty Press. Newhouse, J.P., 006. Reonsderng the moral hazard-rsk avodane tradeoff. Journal of Health Eonoms 5; Nyman, J.., 999. The eonoms of moral hazard revsted. Journal of Health Eonoms 8; Pauly, M.V., 968. The eonoms of moral hazard: Comment. meran Eonom Revew 58; Rask, K.N., Rask, K.J., 000. Publ nsurane substtutng for prvate nsurane: new evdene regardng publ hosptals, unompensated are funds, and Medad. Journal of Health Eonoms 9; -3. Spene, M, Zekhauser, R., 97. Insurane, nformaton, and ndvdual aton. meran Eonom Revew 6; Sn, D.D., Svenson, L.W., Cowe, R.L., Man, P.S., 003. Can unversal aess to health are elmnate health nequtes between hldren of poor and nonpoor famles? ase study of hldhood asthma n lberta. Chest 4; Wagstaff,., 007. Health nsurane for the poor: ntal mpats of Vetnam's health are fund for the poor. Poly Researh Workng Paper Seres 434, The World Bank Zekhauser, R., 970. Medal nsurane: a ase study of the tradeoff between rsk spreadng and approprate nentves. Journal of Eonom Theory ; 0 6. Zwefel, P.J., Mannng, W.G Consumer nentves n health are. In: Culyer, Newhouse JP (Eds), Handbook of Health Eonoms, vol.. North Holland: msterdam p

31 Table. Comparatve Stats on Optmal Consurane Rate Effet of on Own demand slope B + Other good demand slope Bj - Rsk averson parameter Varane of spendng on own Unompensated losses affetng nome Unompensated losses affetng utlty dretly Varane of spendng on other good j Covarane of spendng on and j Correlaton between spendng on goods and j Cross pre term for other good R - - L - L 0 j - j (-) j (-) G j (-) Real nterest rate r (-) Consumer dsount fator (-) Correlaton wth next perod error (-) Note: results n parentheses only hold for spef values of key parameters. 3

32 Table Means, standard devatons, fve year autoorrelaton rates of npatent, outpatent and pharmay spendng Usng unadjusted spendng Standard utoorrelaton wth spendng n year Mean devaton Standard devaton of 5 year sum 004 npatent spendng $ 87 $ 8, $ 6, outpatent spendng $,50 $ 7, $ 7, pharmay spendng $ 977 $, $ 7, total spendng $ 3,999 $, $ 3,875 Note: Results are based on healthare serves for MEDSTT Marketsan data for ndvduals aged < 65, who were ontnuously enrolled durng N=,335,448. 3

33 Table 3 Correlatons aross spendng on healthare serves 004 unadjusted spendng Inpatent Outpatent Pharmay Total Inpatent Outpatent Pharmay Total (symmetr) Fve year total spendng Inpatent Outpatent Pharmay Total Inpatent Outpatent Pharmay Total (symmetr).000 Note: Results are based on healthare serves for MEDSTT Marketsan data for ndvduals aged < 65 n 004, who were ontnuously enrolled durng N=,48,3. 33

34 Fgure Optmal onsurane rates for G 0 as Two Health Care Goods, One Perod: Inreasng orrelaton Lne () base base Lne () (, ). Lne () and Lne () orrespond to the solutons of Equaton, when there s no ovarane between the two goods and they are nether substtutes nor omplements. WthG 0, eah optmal onsurane rate equals the Base Case value and the optmal onsurane rates do not depend on eah other. Holdng the varanes terms and onstant whle nreasng the ovarane mples nreasng the orrelaton,. Both Lne () and Lne () rotate nward away from the Base base Case. Hene, as. Case shown s for G 0, but the pvots happen n the same dreton for arbtrary G R as 34

35 Fgure Optmal onsurane rates when ( G from0) & 0 Two Health Care Goods, One Perod: Substtutes base (, ). base s goods beome stronger substtutes, nterepts derease whle slopes nrease, suh that nterseton ponts are below base ase. Thus, substtutes unambguously lead to lower than the Base Case. Smlar results an be shown for general, sne only affets the slopes but not the nterepts. 35

36 Fgure 3 Optmal onsurane rates when G 0& 0 Two Health Care Goods, One Perod: Substtutes base (, ). base Startng from Fgure 3 and rasng the orrelaton from zero holdng, onstant to preserve the Base-Case omparson. Here we have an unambguous ase where the effet of G and renfore eah other, pushng both toward zero. The dash-dot-dash lnes have ( G 0 & 0) whle the sold lne s the base ase ( G 0) and the dashed lne orresponds to ( G 0 & 0). In ths ase, the dashdot-dash nterseton s southwest of the dashed nterseton, whh means the two effets renfore eah other. 36

37 ppendx Ths appendx derves seleted analytal results n the man paper. For onvenene, Table - presents our notaton. Equaton numbers shown wthout an suffx orrespond to numberng n the man text. Table - Notaton X X = quantty of health are treatment of serve Y = quantty of all other onsumpton goods I = onsumers total nome J = dsposable nome after premums = premum pad by onsumer PP=, demand pres of Y X and Y = ost sharng rate on treatment X (share pad by onsumer) = random shok affetng health and demand for X = mean of health are spendng on B = slope of demand urve when wrtten n the form X B G j = ross pre effet of j on = dsount rate used by onsumer X X and also on r = nterest rate reeved on savngs by onsumer S = savngs n perod = varane of and also varane of health are spendng j = ovarane of and j V = the onsumer s utlty funton V R = absolute rsk averson onstant II I L = per unt unompensated osts of treatment V L( ) = unompensated health losses that redue effetve nome from random shoks X X j X 37

38 -. Optmal ost sharng rates for health are treatment ssume there s no preventve good, and that treatment goods X and X have lnear demand urves of the form () X - BP / PY GP / PY X - B P / P G P / P Y These demands are onsstent wth a rsk neutral ndret utlty funton of () V I BP BP P P PP S G PY PY PY PY PY PY Y Usng the normalzatons PY, P L, P L,, lettng J I, S applyng the onave transformatonv, ths yelds an ndret utlty funton that an be wrtten as B( L) B( L) S S J ( L) (3) V V L( ) L( ) ( L) G( L)( L) Takng partal dervatves wth respet to yelds S V S I J K ( L) ( L ) X V 0 E B( L) G( X L ) X where (4) J I, K ( L ) ( L ) B ( L ) B ( L ) G ( L )( L ), ( ) B( L) G( L) ( ) B( L) G( L), BB BL G G GL Ths model s onvex quadrat n health are goods and we apply a rsk adverse (onave) funton to spendng on all other goods. Hene objetve funton s strtly quasonave, and the onstrants are all lnear or strtly onave. unque maxmum s assured. 38

39 Takng a frst order approxmaton of S VI around J-K, we an wrte (5) S V J K V J K L L S S I II V E X B ( L) G( L) S VI J K B( L) G( L) S E VII J KL L B( L) G( L) S V, usng E ( ) 0, the frst Defnng R II,, S E E, E VI order ondtons for the maxmum, one dvded through by the (nonstohast) V S ( J K) an be approxmated as follows. I X B ( L) G( L) (6) 0 E R ( L) ( L) B( L) G( L) B( L) G( L) R ( L) ( L). Fnally replang and rearrangng slghtly, we get expressons that are lnear n and (7) (8) B B BL G G G L B( L) G R L L 0 B B B L G G G L B( LX ) G R L L 0 39

40 In the man text we examne a number of speal ases. -. One good, no unompensated losses B G L L B B B R 0 (9) 0 (0) B B R -. One good, unompensated losses. B G L, but L 0, FOC (7) smplfes to 0 () B B B R L 0 () B R L B R --3 Two goods no unompensated losses, general demand struture If LL 0, then equatons (7) and (8) an be solved for as (3) ( G B )( R G ) ( G B)( R B ) ( R B)( R B) ( R G) It s straghtforward to show from (3) that (4) <0, <0, >0, <0, B B The expressons for and G annot be sgned for all possble values of G and, however ther sgns are unambguous for ertan speal ases. We onsder two speal ases. If --3- Symmetr demand slopes and varanes B B and then by fatorng the denomnator and smplfyng, t s straghtforward to show from (3) that = B R G R B G 40

41 For ases where 0, t s straghtforward to verfy that 0 and G 0 for all and G Lmtng ases where G and approah zero These partal dervatves an also be sgned for the lmtng ase where G and both approah zero. In ths lmtng ase the two partal dervatves beome (5) R B 0 ( )( ) R 0, 0 B G R B (6) R 0 G ( R B )( R B ) 0, G 0 Ths means that for suffently small G and, as the ovarane of the errors between two serves nreases (beomes more postve), then the optmal onsurane rate dereases. The more general relatonshp for varous ombnatons of andg shows that these dervatves annot be sgned unambguously, as dsussed n the man text. 4

42 -3. Two perod model It s well known that there are lose parallels between models wth multple states of the world and models wth multple perods. One key dfferene s that there s the possblty of orrelated outomes n dfferent perods, ether beause health shoks are serally orrelated or beause the two perods are lnked by savngs. We examne here a two-perod model wth one health are treatment good n eah perod, allowng both savngs and orrelated errors. The demand struture and premums are assumed to be the same n both perods. We use the dret utlty funton whh s the dual to the ndret utlty funton used thus far. We also fous on the ase where II Iand L L L, hene. We also assume onstant varanes over tme and ( r). (7) V X, Y, L ( ) Max EV E, E V X, Y, L ( ) X, Y, S where S ( X) V X, Y, V ( Y ) B S ( X ) V X, Y, V ( Y ) B onsumers dsount fator B ( L) K K ( L) B( L) s long as nome s suffent to always buy the optmal amount of X, then the same amount of X wll be purhased as n the one perod ase. (8) Hene we an use: X B( L) X B( L) Savngs wll equlbrate the expeted margnal utlty of nome n perod wth the margnal utlty n perod. Hene we have 4

43 (9) V ( ) ( r) E V I I Let the optmal savngs funton be S(, X) (derved below) and wrte the problem usng ndret utlty as (0) B( L) V J K S(, ) ( L) L ( ) EV E, E V J K ( r) S(, ) ( L) L ( ) where J J J I onsumers dsount fator B ( L) K ( L) Exept for the savngs funton, ths formulaton s very smlar n struture to that used for multple states of the world. Dfferentatng wth regard to yelds () VI J K S(, ) ( L) S(, ) B ( L) EV E, VI J K ( r) S(, ) ( L) E S(, ) ( r) B( L) S(, ) By the envelope theorem, the terms nvolvng n the above expresson wll anel out due to the assumpton of optmal savngs. Hene we an rewrte ths as: () VI J K S(, ) ( L) B( L) EV E V I J K ( r) S(, ) ( L) E B ( L) Exept for the fat that two dfferent perod utlty funtons are used, and the appearane of the savngs funton as an argument of the V ths s dental to the earler spefaton. Takng a I 43

44 frst order Taylor seres approxmaton of the smlar expansons (3) V I and V I funtons, as before, we obtan the R S(, ) ( L) B( L) 0 E E R ( r) S(, ) ( L) B( L) (4) VI J K VII J K S(, ) ( L) B( L) EV E V I J K VII J K( r) S(, ) ( L) E B ( L) ( ) B BLR ( L) S(, ) B( L) E R ( rs ) (, ) B ( L) E, S(, ) ( r) If onsumers use the same dsount rate as that s mpled by ther real nterest rate on savngs, then ( r) and the top expresson n brakets wll be zero, and we an further smplfy (5) R E, S(, ) 0 ( ) B BL R ( L) Usng B BL B. (6) 0 ( ) BB R ( L) R E S (, ), Ths expresson annot be smplfed further wthout explt savngs funton form S(, X ). s s showed below, optmal savngs rule an be approxmated by the followng lnear funton 44

45 (7) S (, ) S s ( L) ( r) ( r) Where S, s, R ( r) ( r) Ths mples that there s an average savngs level S but that savngs s redued by proporton s for all losses (ompensated or unompensated). Sne S wll be unorrelated wth, t wll drop out one expetatons are taken and we an wrte (8) 0 ( ) B R L s Rearrangng yelds the followng ondton for optmal ost sharng wth multple perods, (9) B R L s B R s If we plug n funton form of s, then r (30) ( ) BR L ( r )( ). ( ) BR ( r )( ) Now we turn to dervng optmal savng funton. s our model set up, savng s determned after health shok s revealed n the st perod. ssumng ndvduals are not budget onstraned, the optmal amount of X wll be purhased n both perods and wll not be affeted by the optmal savng deson, however savng does affet the amount of Y. Due to ths feature, we solve our optmal savng funton after takng optmal hoes of X as gven. Ths approah brngs us the same soluton for the optmal savng funton as used n the hoe of X, Y and S smultaneously n the frst perod. 45

46 (3) S Max EV V J K S( L) L ( ) E ( ) V J K r S( L) L ( ) where J I onsumers dsount fator B ( L) K ( L) B( L) The optmal savng funton S satsfes (3) VI J K S L r E VI J K r S L ( ) ( ) ( ) ( ) 0 Takng a frst order Taylor seres approxmaton of V I funton at J K, (33) VI J K S L V J K r E VI J K ( ) II ( ) ( rs ) ( L) VII JK VI J K S ( L) VII J K ( r) VI J K ( r) SVII J K ( )( ) ( ) (34) L r VII J K E s s assumed, E ( ), we obtan VI J K S ( L) VII J K ( r) VI J K ( r) SVII J K (35) ( r)( L) V J K (36) S ( L) R ( r) ( r) SR ( L)( r) R II Rearrangng the above equaton, we solve the optmal savng funton as followng ( r) ( r) (37) S ( L), whh mples that R ( r) ( r) ( r) ( r) (38) S, s R ( r) ( r) 46

47 Gven our assumpton ( r), S an be smplfed further as (39) S ( L) and S 0, s r r -4 Optmal savng for model wth multple perods T Now we allow more general ases. We assume that E ( t ) t, for any t t, wthout makng any restrtons on the stohast proess for t. It an readly be shown that (40) S (4) S (4) ( r) for T=3. ( r) ( r) r ( r) ( ) 3 3 S ( r) ( r) ( r) for T=4. ( r) ( r) ( r) for T=5 ( r) ( r) ( r) ( r) By nduton, t s easy to see that the optmal savng result for anyt s (43) S T t ( r) ( T t ) t T t t ( r). We do not try to derve optmal ost sharng rules for multple perods, other than to note that these more general savngs rules mply that rsk spreadng of health shoks over multple perods wll depend n a natural way on nterest and dsount rates. -5 Optmal savng for model wth two perods and two goods The onsumpton of two goods n two perods s ndexed by subsrpts for goods and supersrpts for perods,,, X X X X. Followng the notaton for the multple perod ase, we assume,,; I I I, L L L, L L L and. X X 47 X X X X Consder Xas an aute medal treatment good whh s not orrelated wth ts own future and X as a hron medal treatment good that s postvely orrelated:, 0<.

48 ssumng that E( ) E( ) E( ) E( ) 0 we have VR s s assumed that nome s suffent to support the optmal amounts of medal serve, the demands of two treatment goods n perod are (44) X B( L) G ( L ) X B ( L ) G ( L) Smlarly, the demands n perod are (45) X B( L) G ( L ) X B ( L ) G ( L) If we let the optmal savngs funton be S (,,, ), then we an wrte the objetve funton usng the expeted ndret utlty as V J K S ( L) ( L) L ( ) L ( ) EV E, V J K ( r) S ( L) ( L) E,, L ( ) L ( ) where (46) J J J I onsumers dsount fator B( L) B( L) K ( L) ( L) GXY ( L)( L) B ( L) G( L) B ( L ) G ( L ) The optmal savng funton S satsfes 48

49 VI J K S ( L) ( L) (47) ( re ) V ( ) ( ) ( I JK rs L L) 0,, Takng a frst order Taylor seres approxmaton of V I funton at J K, VI J K S ( LX ) ( L) VII J K (48) ( re ) VI JK ( rs ) ( L) ( L) VII JK,, (49) VI J K S ( L) ( L) VII J K ( r) VI J K ( r) SVII J K ( L )( r) V J K E ( ) ( L )( r) V J K E ( ) II II,,,, Havng assumed, 0<, E ( ), we obtan,, (50) VI J K S ( L) ( L) VII J K ( r) VI J K ( r) SVII J K ( L) ( rv ) II JK (5) ( ) ( ) S L L R ( r) ( r) S R ( L) ( r) R Rearrangng the above equaton, we solve the optmal savng funton as follows: S (,,, ) Ss( L) s ( L ), where the savngs and overall savngs rates ( r) are S R ( r) s ( r) and s (. The expeted (ex ante) r) ( r), savngs S are same wth the two-perod one-good ase. The margnal propensty to save s s lower than s beause good s hron are and patents redue the unertanty of expeted spendng of good by treatng t better n perod. In partular, f ( r), s redued to a smple funtonal form S ( ) ( ). L r r L Dfferentatng wth regard to yelds S (,,, ) 49

50 S(,,, ) VI EV B( L) B( L) (5) E, S (,,, ) ( r) E V I,, B( L) B( L) S (,,, ) VI EV B( L) B( L) (53) E, S (,,, ) ( r) E V I,, B( L) B( L) S (,,, ) By the envelope theorem, the terms nvolvng n the above expresson wll anel out due to the assumpton of optmal savngs. The optmal ost sharng rates and must satsfy (54) G s( ( ) B ( ) R L ) R ( L) s0 If we smplfy the problem by assumng that there are no unompensated ost related to health are (by assumng L L 0 ), the optmal ondtons beome (56) G ( ) B ( ) R L sr ( L) s 0 (55) ( )( B G ) ( )( G R ) R s ( )( B R ) R s ( )( B R ) R s (57) ( )( B G ) ( )( G R ) R s ( )( B R ) R ( ) ( )( B R ) R s ( ) s 50

51 Parallelng the analyss n seton..3 above for two goods n one perod, we an sgn the effets of G and on and only for speal ases, suh as when they are both n the neghborhood of zero. graphal analyss an be used to show results smlar to those n Table. Savngs tends for both aute and hron ondtons rases optmal ost sharng, but has a larger effet on aute ondtons whh are serally unorrelated. 5

52 Fgure (a) Optmal onsurane rates when ( G from0) & 0 Two Health Care Goods, One Perod: Complements (, ) base. base s two goods beome stronger omplements, nterepts nrease whle slopes derease, so that base equlbrum lnes shft outward suh that new nterseton pont les between and. Thus, omplements unambguously lead to hgher than Base Case when there s no orrelaton. 5

53 Fgure (b) Optmal onsurane rates when G 0& 0 Two Health Care Goods, One Perod: Complements (, ) base. St base Startng from Fgure and dereasng the orrelaton from zero holdng, onstant to preserve the Base-Case omparson. Here we have an unambguous ase where the effet of G and renfore eah other, pushng both away from zero. The dash-dot-dash lnes have ( G 0 & 0) whle the sold lne s the base ase ( G 0) and the dashed lne orresponds to ( G 0 & 0). In ths ase, the dashdot-dash nterseton s northeast of the dashed nterseton, whh means the two effets renfore eah other. Note that the two slopes of the dash-dot-dash lnes must be both postve/zero/negatve at the same tme. 53

54 Fgure (a) G R : 0, G from0 base base The sgn of when 0 and G from0 depends on whether G remans lower G than/exeeds R (.e. dependng on whether the sgn of slope swthes when G goes up from zero) as well as the relatve movement of the two lnes determnng whether resultng and move n the same dreton. The fgure 3(a)-3(C) orrespond to the ases when G R, G R, and G R. Note: these fgures do not exhaust all the possbltes of hanges n and. 54

55 Fgure (b) G R : 0, G from0 base base 55

56 Fgure () G R : 0, G from0 base base