Equilibrium in competitive insurance markets with ex ante adverse selection and ex post moral hazard

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1 Journal of Pulc Economcs 84 (2002) locate/ econase Equlrum n compettve nsurance markets wth ex ante adverse selecton and ex post moral hazard Wllam Jack* Department of Economcs, Georgetown Unversty, Room ICC 580, Washngton, DC 20057, USA Receved 4 May 2000; receved n revsed form 10 August 2000; accepted 12 Novemer 2000 Astract Exstence of pure strategy equlra s studed n health nsurance markets that exht oth ex ante adverse selecton of the Rothschld Stgltz Wlson type, and ex post hdden nformaton moral hazard. It s found that ex post moral hazard has two offsettng effects on the exstence of equlrum, and that n general t s dffcult to say whether an equlrum s more or less lkely to exst. Numercal smulatons, and an analytcal example, confrm that moral hazard may ncrease the lkelhood of equlrum. These results are nterpreted as evdence that Health Mantenance Organzaton (HMO) markets could e less lkely to exht stale equlra than are fee-for-servce nsurance markets Elsever Scence B.V. All rghts reserved. 1. Introducton In mportant papers, Rothschld and Stgltz (1976) and Wlson (1977) (henceforth RSW) dentfed a fundamental weakness of nsurance markets. In the presence of asymmetrc nformaton aout ex ante rsk attrutes, a compettve equlrum may fal to exst. Some authors have responded to the prolem of non-exstence y modfyng the structure of the orgnal RSW model to estalsh exstence of equlra. For example, Dasgupta and Maskn (1986a,) show that f mxed strateges are permtted an equlrum always exsts. Alternatvely, Wlson (1977) extended hs asc model to nclude frms explct dynamc reactons to *Tel.: ; fax: E-mal address: wgj@georgetown.edu (W. Jack) / 02/ $ see front matter 2002 Elsever Scence B.V. All rghts reserved. PII: S (01)

2 252 W. Jack / Journal of Pulc Economcs 84 (2002) new contract offers, and found that pure strategy equlra generally exst. These papers ncreased the lkelhood of an equlrum exstng (to unty) y, respectvely, wdenng the strategy space and narrowng the set of feasle equlrumreakng contracts. In ths paper, I examne the exstence of pure strategy compettve equlrum n nsurance markets that exht oth adverse selecton and, n addton, ex post moral hazard. In ths set-up, nsurers can oserve nether the ex ante rsk characterstcs of purchasers of nsurance (as n RSW), nor the state of the world 1 that occurs ex post. Ths nformaton structure s meant to provde a more realstc descrpton of orthodox fee-for-servce medcal nsurance n whch nsurance companes are unale to montor accurately the true health status (.e., medcal care needs) of clamants. I wll argue elow that the standard RSW model wthout ex post moral hazard s more descrptve of pre-pad nsurance polces offered y US-style Health Mantenance Organzatons (HMOs), and that a comparson of the two models allows an analyss of the relatve market performance of the two organzatonal forms. The ntuton for the non-exstence result n the standard model gven y Rothschld and Stgltz s that f an equlrum exsts, t must e a separatng equlrum n whch ndvduals wth dfferent rsks purchase dfferent nsurance contracts. (No poolng equlrum exsts.) Ths separaton mposes a cost on low rsks, who are only partally nsured due to the self selecton constrant. They are prepared to pay to reduce ther rsk exposure, and f there are enough of them an nsurer could offer such a contract, and earn suffcent profts from the low rsks to offset any loss ncurred on hgh rsks who also opt for the new contract. That s, a poolng contract mght exst to reak the potental separatng equlrum. When the model s extended to nclude ex post moral hazard, oth hgh and low rsk ndvduals typcally purchase less than full nsurance, even n the asence of adverse selecton. Ths s ecause the unoservalty of the state of the world precludes optmal redstruton across such states, makng full nsurance too costly for oth rsk types. The costs to low rsks of separaton may thus e relatvely less, and a poolng equlrum that reaks the potental separatng equlrum less lkely. Moral hazard could thus ncrease the range of parameters over whch an equlrum exsts. The man contruton of ths paper s to confrm ths ntuton. Smple condtons under whch fee-for-servce markets are more lkely to support an 1 What I refer to here as ex post moral hazard has een laelled hdden nformaton moral hazard y, e.g., Hart and Holmstrom (1987). I do not consder what mght e termed ex nterm, or hdden acton moral hazard, wherey ndvduals can take unoservale actons that affect the proalty dstruton over states of nature. Also, I do not consder the ntermedate case of costly state verfcaton (e.g., Townsend, 1979) n whch a prncple (the nsurer) chooses an optmal audtng polcy. Instead, the two cases consdered here (wth and wthout moral hazard) can e consdered to represent nfntely costly and costless verfcaton, respectvely.

3 W. Jack / Journal of Pulc Economcs 84 (2002) equlrum than HMO markets are not avalale. However, wth the help of an analytcal example and numercal smulatons I am ale to show that for some parametersatons of preferences, an mproved nformatonal envronment n whch ex post moral hazard s asent (.e., n whch the state of the world s oservale) s detrmental to the attanment of equlrum. To understand these results, t s useful to renterpret the orgnal RSW ntuton regardng the condtons under whch an equlrum wll not exst. It s not so much the cost (n terms of expected utlty) of separaton that makes a poolng contract oth proftale to nsurers and attractve to low rsk ndvduals. Instead t s more useful to recognse that n a separatng equlrum, low rsk ndvduals are constraned to purchase a su-optmal quantty of nsurance at actuarally far rates. The relevant descrpton of the mpact of ncomplete nsurance s n terms of ts effect on the wllngness of low rsk ndvduals to purchase unconstraned quanttes of nsurance at actuarally unfar rates. The less far the rate.e., the hgher the prce at whch low rsk ndvduals would e wllng to purchase unconstraned quanttes of nsurance the less lkely s an equlrum to exst. Ex post moral hazard has two effects on the wllngness of low rsk ndvduals to purchase nsurance at actuarally unfar rates. Frst, moral hazard reduces the demand for nsurance at all prces, so the hghest prce at whch otherwse constraned low rsk ndvduals would e wllng to purchase nsurance falls. I refer to ths as the nternal ncentve effect as t s not a drect consequence of the 2 presence of hgh rsk ndvduals n the market. The second effect s that, ecause hgh rsk ndvduals also prefer to purchase ncomplete nsurance, the constraned level of nsurance offered to low rsk ndvduals at a separatng equlrum s reduced. Ths external ncentve effect ncreases the prce at whch low rsk ndvduals are wllng to purchase unconstraned quanttes, essentally ecause 3 the value of ther outsde opton s reduced. These two offsettng effects make t dffcult to analytcally sgn the change n the crtcal nsurance prce. Secton 5.1 elow presents an example n whch there s no external ncentve effect, so that the net mpact of moral hazard s to unamguously ncrease the lkelhood of equlrum. More generally, the numercal smulatons provde examples of when each effect domnates. Of course, when an equlrum does exst n oth types of market (.e., wth and wthout ex post moral hazard), the HMO allocatons wll Pareto domnate the fee-for-servce ones. Thus, condtonal on an equlrum exstng, more nformaton (the removal of ex post moral hazard) s etter, as would e expected. However, n those cases when mproved nformaton leads to non-exstence of equlrum, t s arguale that more nformaton s actually worse, as t causes the market to reak down relatvely more often. 2 It s, of course, just a consequence of moral hazard, ut overuse of ths term seems mprudent. 3 Ths effect derves from the mpact of moral hazard on the adverse selecton ncentve constrant. Agan, solely for reasons of clarty, avodance of these terms seems advsale.

4 254 W. Jack / Journal of Pulc Economcs 84 (2002) There s a danger n comparng two nsttutonal arrangements wth dfferent equlrum exstence propertes. Ideally we should search for a more approprate equlrum concept, perhaps wthn the context of a dynamc model, that assures exstence for all parameterzatons. Although speculatve at ths stage, one mght expect that n cases when a standard RSW equlrum exsts, the equlrum n the more general model would concde wth t, ut that when an RSW equlrum does not exst the equlrum of the more general model could exht cyclng or some other form of non-statonarty. If there are suffcent welfare costs assocated wth the second type of equlrum (for example, due to a lack of lfe-tme nsurance), a more defntve comparson of nsttutonal arrangements would e possle. The only other paper to have examned the ssue of equlrum exstence under dfferent nsttutonal structures that I am aware of s Chernew and Frck (1999). They allow nsurance companes to offer contracts wth oth a cost-sharng (.e., fee for servce) component and a non-fnancal cost of otanng care, dentfed as the degree of managedness. Managedness reduces a consumer s utlty ecause t constrans hs ex post choces. The exstence of ths second contractual dmenson expands oth the degree to whch low rsks can dfferentate themselves from hgh rsks, and the set of contracts that can reak any potental separatng equlrum. In ths sense, the authors fndng that the effects of managed care on equlrum exstence are amguous are not too surprsng. In the current paper however, the essental dfference etween the two nsttutons s an mprovement n the nformaton structure, whch nonetheless can have detrmental effects on equlrum exstence. Adverse selecton n health nsurance markets has recently receved growng attenton n the emprcal lterature. Cutler and Zeckhauser (1997) and Cutler and Reer (1998) oth fnd evdence that when employers changed the mplct susdes to, and hence relatve prces of, alternatve nsurance packages, consumers responded n the way predcted y the standard theory. In a study of a natural experment nvolvng states n the US, Buchmueller and DNardo (1999) found lttle evdence of an adverse selecton spral, akn to Akerlof s (1970) no-trade equlrum, ut dd fnd that the data were consstent wth the RSW model. None of these studes however have attempted to tackle the ssue of exstence. The next secton presents a formal model of health and nsurance choce under full and symmetrc nformaton aout ex ante rsk types and ex post health status. Secton 3 ntroduces adverse selecton, whle retanng the assumpton of full and symmetrc nformaton ex post, and examnes the equlrum propertes of the nsurance market. The model s essentally the same as that of the orgnal RSW framework, and represents the nformatonal structure faced y HMOs. Secton 4 models fee-for-servce nsurance y ncludng ex post moral hazard through the assumpton that health status s not oservale y the nsurer. It examnes the effects of the addtonal nformaton constrant on the nature and exstence of equlrum. Secton 5 presents an analytcal example of a stuaton n whch moral

5 W. Jack / Journal of Pulc Economcs 84 (2002) hazard must ncrease the chance of an equlrum, and reports the results of numercal smulatons. The fnal secton concludes. All proofs are relegated to Appendx A. 2. Insurance choce wth full and symmetrc nformaton In ths secton I present a smple model of health nsurance under full and symmetrc nformaton aout ex ante rsk types and ex post states of nature. Followng the analyss of Jack and Shener (1997), an ndvdual s utlty u(c,h) s assumed to e a well-ehaved functon of health h (not to e confused wth health care servces) and other consumpton c. Exogenous ncome s denoted y w. The shock aganst whch ndvduals wsh to nsure s the prce of health, u, and ndvduals purchase nsurance to maxmse ther expected utlty over states of the 4 world. Consstent wth the emprcal lterature, I wll assume that the elastcty of demand for health s less than unty, so that scker people spend more on health. Jack and Shener assumed a contnuous dstruton of u n ther model of nsurance choce wthout ex ante adverse selecton. Here, as I wll later want to examne the nteracton of ex ante adverse selecton and ex post moral hazard, I smplfy the ex post structure of the model and assume that u can take on one of just two values, u n the good state of the world, and u.u n the ad state. As g g n RSW, ndvduals dffer ex ante accordng to ther rsk of eng n the good or h ad state: hgh rsk ndvduals have a proalty p of eng n the ad state (and h (1 2 p ) of eng n the good state), whle the correspondng parameter for low l h rsk ndvduals s p, p. Low rsk ndvduals make up a proporton l of the populaton. When ndvduals rsk types are oservale y nsurers, nsurance contracts can dffer accordng to rsk. In ths case, assumng a compettve nsurance market and zero profts, nsurance contracts are offered ndependently to each type. Also, as the ex post state of the world (.e., health status) s oservale, contracts can e made state-contngent, and optmally are charactersed y lump-sum transfers etween states of the world. Thus, for ndvduals of type the contract s g descred y a par of transfers ht,t j, wth the components ndexed to the states of the world. The transfers are nterpreted as net transfers from ndvduals to the 4 Ths nterpretaton can e derved from a smple health producton functon model. Assume that u represents generc health status, and health h, s produced under constant returns to scale from nputs z, wth h 5 f(z,u ) 5 z/u. (Introducng decreasng returns does not alter the ntuton of the paper.) Thus u determnes the productvty of health nputs. If nput prces are p, then the mnmum cost of attanng health h n state u s c(h,u ) 5 puh. Ifp s normalzed to unty, u can e nterpreted as the prce of health.

6 256 W. Jack / Journal of Pulc Economcs 84 (2002) nsurance company, so t s natural to expect that t. 0. t. The optmal contract for each rsk type solves the followng optmsaton prolem: maxe [v(u,w 2 t ) s.t. E p 5 0 ht,t g j where v(u,w 2 t) s the ndrect utlty functon and p s the proft an nsurer earns from type ndvduals. More explctly, the optmal contract for type ndvduals satsfes g g max p v(u,w 2 t ) 1 (1 2 p )v(u,w 2 t ) ht,t g j g s.t. pt 1 (1 2 p )t 5 0 The soluton to ths prolem s charactersed y full nsurance, n whch the margnal utlty of ncome s equated across states of the world. Ths can e seen y smply susttutng the constrant equaton nto the ojectve functon, whch yelds 2 2 g g v (u,w 2 t ) 5 v (u,w 2 t ) where the suscrpt on the utlty functon denotes partal dfferentaton wth respect to the second argument. In standard models of ncome (as opposed to prce) rsk (as n RSW) optmal nsurance s charactersed y equalty of ncome, and hence of the margnal utlty of ncome, across states of the world, and the optmal contract s located on the certanty lne. Here, the optmal contract s located on the full nsurance lne, whose equaton s denoted m5 f(m g), where ms5 w 2 ts s net ncome n state of the world s, and whch s shown n Fg. 1. The contracts for oth hgh and low rsk ndvduals are shown, denoted H and L, respectvely. Note that the axes of the fgure represent nomnal ncome n each 0 state of the world, and so the ntal endowment s on the certanty lne at pont E. However, the equal margnal utltes or full nsurance lne along whch the optmal contracts le s to the north-west. g 0 3. Adverse selecton In ths secton, I assume that nsurers cannot dstngush etween hgh and low rsk ndvduals, ut that f and when an llness or accdent occurs, they can costlessly oserve the nature and extent of medcal need. In terms of the stylzed model, nsurers can oserve the ex post prce of health, u. Ths assumpton s meant to characterse n a smple way the nformatonal structure faced y HMO nsurance companes. Because HMO staff nclude physcans, who are assumed to e at least as good at dagnosng health condtons as ndvduals, t seems reasonale to ascre to them the alty to oserve u. On the other hand, n

7 W. Jack / Journal of Pulc Economcs 84 (2002) Fg. 1. Equlrum contracts wth full and complete nformaton. assumng that the HMO cannot oserve rsk characterstcs, I endow ndvduals wth more knowledge of lkely health-related outcomes regardng ther lfe-styles 5 and predsposton towards dsease contracton and accdents. If nsurers cannot dstngush etween dfferent rsk types ex ante, then the optmal contracts charactersed aove wll not e sustanale n equlrum. If the two contracts H and L are offered, oth hgh and low rsk ndvduals wll choose 0 0 the contract desgned for low rsks, L, and profts wll e negatve. The 0 equlrum contracts now solve the followng jont optmsaton prolem for 5 l,h: max E [v(u,w 2 t ) ht,t g j s.t. p 5 0 j and E [v(u,w 2 t ) $ E [v(u,w 2 t ) Profts contnue to e constraned to e zero for each rsk type separately (.e., there s no cross susdsaton). The second constrant requres that ndvduals 5 Of course, t s reasonale to assume that HMOs can engage n some screenng of rsks at the tme of enrollment. However, such screenng s also carred out y fee-for-servce nsurers, and as I wsh to dfferentate etween the types of nsurance accordng to ex post oservalty of states of the world, the assumpton of ex ante adverse selecton appears reasonale.

8 258 W. Jack / Journal of Pulc Economcs 84 (2002) self-select and choose contracts approprate to ther types. As RSW show, the mpact of these constrants s that equlrum contracts must dffer etween ndvduals (.e., there s no poolng equlrum), and that the equlrum nsurance coverage of low rsk ndvduals s ncomplete, at pont L n Fg Whle the contract they purchase transfers ncome etween the two states at the actuarally far prce, they are constraned from purchasng ther desred quantty of nsurance at that prce. Hgh rsk ndvduals contnue to e fully nsured at pont H. 0 The par of contracts hh,l j represents the only possle equlrum. But t 0 1 may not e sustanale. If there are suffcently many low rsk ndvduals so that the average market odds lne EA s close enough to the zero proft lne for low rsks, pl 5 0, then the potental separatng equlrum wll not e sustanale. The exstence of a contract such as g n Fg. 2 wll reak the equlrum, snce oth hgh and low rsks prefer t, and t earns postve total profts. In general there exsts some cut-off proporton of low rsks n the populaton, l *, such that f the actual proporton of low rsks l s greater than l *, an equlrum wll fal to exst. l* s that proporton of low rsks that results n the average market odds lne EA eng tangent to the low rsk ndfference curve through L 1. The smaller s l* the less lkely s an equlrum to exst. Thus I dentfy l* wth the chance or lkelhood of a pure strategy equlrum exstng. It s clear from Fg. 2 that, ceters parus, the further s L1 from the full nsurance pont L 0, the smaller s l*. In partcular, for a gven shape of the low Fg. 2. Equlrum n the RSW model of adverse selecton.

9 W. Jack / Journal of Pulc Economcs 84 (2002) rsk ndfference curves the greater the gap etween full nsurance and the actual nsurance that low rsks can otan, the less lkely s an equlrum to occur. Low rsk ndvduals would e wllng to purchase nsurance at a prce hgher than the l l actuarally far rate of p /12 p, as long as they faced an unconstraned quantty choce. The hghest prce at whch they would e wllng to purchase nsurance s l h p*/12 p* where p* 5 l*p 1 (1 2 l*)p. Snce hgh rsk ndvduals wll also wsh to purchase nsurance at ths prce, there needs to e suffcently many low rsk ndvduals n the populaton (.e., l must e greater than l*) to nduce nsurers to offer contracts at such prces. Note however that l* also depends on the shape of the low rsk ndfference curve n Fg. 2. Heurstcally, the hgher the curvature n the relevant range, the lower wll l* e, and the less lkely wll an equlrum e to exst. Equvalently, the hgher the curvature, say due to greater rsk averson, the hgher wll e the prce aove the actuarally far rate at whch a low rsk ndvdual wll e wllng to purchase an unconstraned quantty of nsurance. Agan, f there are suffcently many low rsk ndvduals to susdse the hgh rsks who also purchase nsurance at the ntermedate prce, then nsurers wll e wllng to supply nsurance at these rates and make postve profts. 4. Adverse selecton and moral hazard In ths secton I ntroduce ex post moral hazard nto the model. I retan the assumpton of no ex nterm or hdden acton moral hazard, oth ecause ths may e of secondary mportance n the health nsurance context due to the often sustantal non-pecunary costs assocated wth ll health, and ecause some choces that affect health status rsk are nowadays contractle (e.g., smokng). However, ex post moral hazard s thought to play an mportant role n determnng ehavour n health care and health nsurance markets (see, e.g., Feldsten and Fredman, 1977; Pauly, 1986; Mannng et al., 1987). In partcular, the nformaton structure, n whch nsurers cannot oserve the state of the world, s meant to characterse fee-for-servce nsurance. To analyse the equlrum contracts that are offered n ths settng, t s necessary frst to descre the nsurance choces of ndvduals n the asence of adverse selecton, and then to analyse market equlrum Incentve compatle contracts The asc characterstc of feasle contracts under condtons of ex post moral hazard s that they must ase net transfers not on the unoservale state of the world, ut on health expendtures. Ths charactersaton s consstent wth the oservaton that most fee-for-servce nsurance plans ase enefts on ncurred costs (n a possly hghly non-lnear fashon). When the possle states of the

10 260 W. Jack / Journal of Pulc Economcs 84 (2002) world form a contnuum, t s natural to wrte the net transfers defned y the nsurance contract for ndvduals of rsk type as a functon t (q) where q 5uh s the expendture on health y an ndvdual n state of the world u. For example, an nsurance contract for rsk type that stpulates a premum P and a constant consurance rate k (eng the proporton of expendtures pad y the nsured) 6 would e represented y the net transfer functon t (q) 5 P 2 (1 2 k )q. In the present case wth just two possle states of the world, a generc contract stpulates a net transfer and level of health care expendture n each state of the world. That s, the contract for rsk type s a par of allocatons h(q,t ),(q,t )j, g g s s s s 7 where q 5 u h s the amount of health expendture n state s, and t s the monetary transfer made to the nsurance company y the ndvdual n state s. For the purposes of descrng feasle contracts under moral hazard, let us drop the rsk superscrpt momentarly. For any par of transfers ht g,tj, the health expendture allocatons hq g,qj must satsfy the followng three condtons. 1. q must e feasle, gven t. That s, w 2 t 2 q $ 0. When an ndvdual n s s s s state s9 chooses contract ( q s,t s) her consumpton of non-health goods s c 5 w 2 t 2 q and her consumpton of health s h 5 q /u. s s s s9 2. q and qg must e ncentve compatle. That s, wrtng n(q s9,t s9,u s) 5 u(w 2 t 2 q,q /u ), t s necessary that s9 s9 s9 s n(q,t,u ) $ n(q,t,u ) s s s s9 s9 s for s9 ± s. 3. qs maxmses n( q,t s,u s) suject to the feaslty and ncentve compatlty condtons. The last condton reflects an assumpton that due to competton, for a gven par of transfers and hence a gven proft, nsurers wll offer ndvduals the most attractve consumpton undles suject to them eng feasle and ncentve compatle. Ignorng the ncentve compatlty constrant, the maxmal utlty attanale n each state gven the transfer ts s v(u s,w 2 t s) 5 u(w 2 ts2 q * s,q* s /u s), where v(.,.) s the ndrect utlty functon, q * s (t s) 5ush*(u s,w 2 t s), and h*(.,.) s the Marshallan demand functon for health. When there s no confuson, I shall just wrte q * s. Smlarly, c * s ; c*(u s,w 2 t s) 5 w 2 ts2 q s* (t s) s the effcent demand for the non-health consumpton good. In fact, t turns out to e more 6 Ths mples a net out-of-pocket expense of P 1 k q. 7 We are mplctly nvokng the revelaton prncple (Myerson, 1979). A typcal fee-for-servce contract s an ndrect mechansm, that stpulates a monetary transfer t as a functon of endogenous expendtures q. Such a polcy can always e mplemented va a drect mechansm that assgns a transfer and requred expendture par ased on a report of the parameter u, that s (t(u ),q(u )). For an exposton of oth the nary and contnuum models n the somorphc case of non-lnear monopoly prcng, see Trole (1988) Chapter 3.

11 W. Jack / Journal of Pulc Economcs 84 (2002) convenent to descre the ncentve compatlty constrant n terms of demand for the consumpton good, c. For a gven transfer par (t g,t ), consder the equaton v(u,w 2 t ) 5 u(c,[w 2 t 2 c/u ). g g g For all transfer pars (t g,t ) such that t g$ t, ths equaton has two solutons c (t,t ) and c (t,t ), wth c # c 1 g 2 g 1 2. Consderng only contracts that ncrease nomnal ncome n the ad state relatve to the good state, the followng partal charactersaton lemma otans: Lemma 1. Assume c s normal and that the prce elastcty of demand for health s less than one. When t g$ t, the contract par h( q g*,t g),( q *,t )j s ncentve compatle f and only f c*(u,w 2 t ) # c (t,t ). 1 g Ths lemma s llustrated n Fg. 3. When the transfer par satsfes the condtons Fg. 3. For ncome transfer (t g,t ), the effcent consumpton undles are ncentve compatle.

12 262 W. Jack / Journal of Pulc Economcs 84 (2002) of the lemma, the ndvdual s expected utlty across states of the world s V(t,t ) 5 pv(u,w 2 t ) 1 (1 2 p)v(u,w 2 t ). g g g When the transfer par s such that c*(u,w 2 t ). c 1(t g,t ) and the effcent health expendture allocatons are offered, ndvduals n the good state of the world wll prefer to choose the (q*,t ) contract (as wll those n the ad state). To ensure that those n the good state choose the contract desgned for them, swtchng to the ad state allocaton must e made less desrale. Ths s acheved y dstortng the consumpton undle n the ad state away from (c*(u,w 2 t ),h*(u,w 2 t )), as llustrated n Fg. 4. In partcular, y settng c equal to ether c (t,t )orc (t,t ), and q 5 w 2 t 2 1 g 2 g c, the ncentve for ndvduals n the good state to swtch can e removed. The partcular choce of c s determned y whch of the two allocatons (c,h ) ; 1 1 (c,[w 2 t 2 c /u ) and (c,h ) ; (c,[w 2 t 2 c /u ) s oth feasle and yelds Fg. 4. Wth larger ncome transfers (t,t ), the effcent consumpton undles ecome ncentve g ncompatle.

13 W. Jack / Journal of Pulc Economcs 84 (2002) the consumer hgher utlty n the ad state. It s expected that n most cases of nterest, c 5 c 1(t g,t ) wll e optmal. Suffcent condtons for ths are gven n the followng lemma: 8 Lemma 2. Suppose that for all m. 0, for all u [ (u g,u ), and for c [ (0,m), FS D Sc, DG. 0. d m 2 c u m2 c dc u h u Then for all transfer pars (t g,t ) such that t g. t and (q* g,q * ) s not ncentve compatle, u(c,h ). u(c,h ), and the health expendture allocaton that provdes maxmal utlty n each state and s ncentve compatle for ndvduals n the good state s q 5 q *(t ) and q 5 [w 2 t 2 c (t,t ). g g g 1 g Thus under the condtons of Lemma 2, an ncentve compatle nsurance 9 contract leads to excessve health expendtures y those n the ad state. Note that to show that the allocaton (q g,q ) s ncentve compatle for ndvduals n the ad state, t s necessary to show that those n the ad state wll not wsh to choose the contract (t g,q ). That s, t s stll requred that u(c 1,h 1). u(c * g,[w 2 tg2 c * g/ u ). Lemma 3. Suppose preferences over consumpton and health are quas-lnear, u(c,h) 5 c 1 z(h), and suppose the coeffcent of relatve rsk averson assocated wth the gross surplus functon z(.) s always greater than unty, that s, 2 hz 0(h)/ z 9(h). 1 for all h. Then gven a transfer par (t g,t ) such that t g. t, the expendture allocaton q 5 q *(t ) and q 5 [w 2 t 2 c g g g 1 s ncentve compatle for ndvduals n oth states. n As an example, suppose the gross surplus functon s gven y z(h) 52a/h for some postve constants a, n. Then n u(c,h) 5 c 2 a/h 99 and 2 hz /z 95(n 1 1). 1, so the condtons of Lemma 2 are satsfed. 8 Note that dstortng the consumpton undle of ndvduals n the good state, gven ther transfers, wll only ncrease ther ncentve to swtch, ecause such dstorton makes them worse off. Thus such a dstorton s not warranted. 9 In a contnuous state space model, a lnear contract (premum plus consurance rate) wll lead to excessve health expendtures n all states. But under a fully optmal non-lnear contract, the health expendture of the ndvdual n the est health state wll e undstorted. Ths s just the famlar no dstorton at the top result from optmal tax theory (Mrrlees, 1971).

14 264 W. Jack / Journal of Pulc Economcs 84 (2002) In general, for any transfer par (t,t ) wth assocated ( q (t,t ),q (t,t )) g g g g satsfyng condtons 1 3, expected utlty s V(t,t ) 5 pu([w 2 t 2 q,q /u ) 1 (1 2 p)u([w 2 t 2 q,q /u ) g g g g g Now suppose there s a contract consstng of equal transfers, t 0, w, n each state 10 of the world (t 0 could e less than zero). In a trval sense the par h( q * g (t 0),t 0),( q * (t 0),t 0)j s effcent and ncentve compatle. The ndvdual s welfare wth ths allocaton s V(t 0,t 0) ;V 0, and as the elastcty of demand for health s less than one, c*(u,w 2 t ), c*(u,w 2 t ) 5 c (t,t ). Now consder 0 g reducng the sze of the transfer to the ndvdual n the good state of the world (.e., ncreasng t g), and ncreasng the transfer to her n the ad state (.e., reducng t ), so as to leave expected utlty unchanged at V. At each successve 0 transfer par, the health expendture allocatons n each state must adjust so as to contnue to satsfy condtons 1 3 aove. That s, they must reman oth feasle and ncentve compatle, whle affordng the consumer maxmal utlty n each state suject to these constrants. For small changes n (t g,t ) n the drecton ndcated, the effcent health expendture allocatons ( q *(t ),q*(t )) satsfy the g g requred condtons. However, as the process contnues, at some pont the ncentve constrant nds, and the effcent allocaton wll cease to e ncentve compatle. More formally, let t 5 t (t ;V ) solve the equaton V(t,t ) 5V. Then g 0 g 0 the followng lemma otans. Lemma 4. Assume c s normal and t, w. There exsts t ˆ 0 g(t 0). t0 such that for all t [ (t,t ˆ g 0 g(t 0)) the effcent allocatons are ncentve compatle,.e., q (t,t (t ;V )) 5 q *(t ) and q (t,t (t ;V )) 5 q *(t ), whle for all t. t ˆ g g g 0 g g g g 0 g g(t 0), the effcent allocatons are not ncentve compatle. Now consder the mappng 2 g :(2`,w) R 1 ˆ ˆ t (w 2 t (t )),(w 2 t (t )) 0 g The mage of ths map n R defnes the oundary of two regons, and the equaton of ths oundary s denoted y m5 t(m g). It s straghtforward to confrm that t9(m g). 0. t s called the transton lne: to the rght of t the effcent allocatons are ncentve compatle, and to the left they are not. As well as characterzng transfers under whch effcent allocatons are mplementale, the transton lne tells us how moral hazard effectvely alters ndvduals preferences over mplementale nsurance polces. To each value of t 0 there corresponds a curve of constant expected utlty that crosses the transton 10 Clearly these transfers yeld postve (negatve) profts whenever t. (, )0. At ths stage, I am only nterested n descrng ndvduals preferences over net ncome n the two states.

15 W. Jack / Journal of Pulc Economcs 84 (2002) lne. To the rght of the transton lne the ndfference curves concde wth those that arse when there s no moral hazard. However, they are not dfferentale along the transton lne, and have dscontnuously greater slopes (n asolute value) than the smooth ndfference curves that exst n the asence of moral hazard. In partcular, n the asence of moral hazard, the ndfference curve s smooth at all ponts, ncludng ht ˆ (t ),t ˆ g 0 (t 0)j, and has slope there gven y dm /dmgu V5V o. Ths s also the value of the rght-hand dervatve of the ndffer- ence curve wth moral hazard at ht ˆ (t ),t ˆ (t )j, snce for m. w 2 t ˆ g 0 0 g g(t 0) the ndfference curves wth moral hazard are dentcal to those wthout. However, the left-hand dervatve of the ndfference curve wth moral hazard s (algeracally) less than dm /dmgu V5V, snce a margnal ncrease n t g (whch lowers m g) must e o offset y a larger reducton n t (ncrease n m ) than n the asence of moral 11 hazard Equlrum The poston of the transton lne relatve to the full nsurance lne, f(m ), g determnes whether ex post moral hazard has any effect on the potental equlrum contracts. If the transton lne s to the north-west of the full nsurance lne, then the ex post ncentve constrants only nd at very hgh transfer levels. These transfers are greater than those that would e chosen wthout moral hazard, so the potental separatng equlrum s not altered, as shown n Fg. 5. Smlarly, the lkelhood of an equlrum exstng s unchanged, as the crtcal value l* s 12 unaffected. However, when the transton lne les to the south-east of the full nsurance lne, then oth the potental equlrum contracts and the lkelhood of an equlrum exstng n general change. In ths case, at a potental separatng equlrum oth hgh and low rsk ndvduals otan ncomplete nsurance, at ponts H and L, respectvely. The contract for hgh rsks, H, s charactersed y the fact that t provdes maxmal expected utlty suject to earnng zero profts. As drawn n Fg. 6, ths occurs at the ntersecton of the transton lne and the p 5 0 h locus, due to the non-dfferentale knk n ndfference curves at that pont. It s plausle that H actually les further to the left of the transton curve (ut stll to 2 11 In models of hdden acton (or ex nterm) moral hazard (e.g., Arnott and Stgltz, 1988), non-convextes can easly arse. However n the current model of ex post moral hazard the ndfference curves reman convex. 12 Recall that n the asence of ex post moral hazard, all low-rsk ndfference curves cross the full nsurance lne wth a slope of 2 (1 2 p l)/p l. Thus, n the asence of moral hazard, the pont of tangency etween the market odds lne defned n terms of the crtcal proporton of low rsks, l*, and the relevant low rsk ndfference curve, s elow the full nsurance lne. As the potental equlrum contracts are unaffected y the exstence of moral hazard when the transton lne s aove the full nsurance lne, nether s ths tangency pont, and the crtcal proporton l* of low rsks s unchanged.

16 266 W. Jack / Journal of Pulc Economcs 84 (2002) Fg. 5. The poston of the transton lne relatve to the full nsurance lne determnes whether moral hazard wll affect the exstence of equlrum. Fg. 6. Impact of ex post moral hazard on potental equlrum when the transton lne les elow the full nsurance lne.

17 W. Jack / Journal of Pulc Economcs 84 (2002) the rght of the full nsurance lne), ut for llustratve purposes t s useful to assume that the knk n the ndfference curves s large enough to avod ths 13 posslty. As hgh rsk ndvduals have less than complete coverage, the ncompleteness of the coverage of low rsk ndvduals s further ncreased relatve to the case wthout moral hazard. The ntuton for ths s that, ecause the coverage of low rsks must e suffcently unattractve to hgh rsks to stop the latter from choosng t, the level of that coverage must fall. However, the cost of adverse selecton to low rsk ndvduals, n terms of ther reducton n welfare, may well e sustantally reduced when there s moral hazard, snce the choce of nsurance wthout adverse selecton s tself ncomplete. For example, the nsurance level chosen y low rsks n the presence of ex post moral hazard, ut wthout adverse selecton, s at pont L9 n Fg. 6. It s clear from the dagram that the change due to moral hazard, f any, n the crtcal value l* aove whch an equlrum wll fal to exst, cannot e unamguously sgned. On the one hand, ecause of the lower level of nsurance at the potental separatng equlrum, low rsk ndvduals wll e wllng to purchase an unconstraned quantty of nsurance at hgher prces than prevously, whch reduces l* and the lkelhood of an equlrum. Ths s the external ncentve effect mentoned n the ntroducton. But on the other hand, ecause of the knk n ther ndfference curves, they wll e unwllng to purchase nsurance at such a hgh prce as n the case of no moral hazard. That s, transferrng ncome etween states of the world at actuarally unfar rates s less attractve to low rsk ndvduals than efore, snce they wll e constraned to consume ncome n the ad state neffcently. Ths nternal ncentve effect ncreases l* and the lkelhood of an equlrum. 5. Examples In ths secton, we frst defne a set of preferences under whch moral hazard must ncrease the lkelhood of equlrum exstence. We then report results from numercal smulatons for a roader range of preferences An analytcal example I construct an analytcal example n whch the ntroducton of moral hazard must ncrease the chance of an equlrum exstng. Only the ntuton ehnd ths 13 Indeed, as one ojectve s to fnd stuatons n whch an equlrum wth moral hazard s more lkely to exst, t s suffcent to have an upper ound on the extent to whch the low rsk contract, L 2 moves down the p 5 0 lne. H provdes such an upper ound. l 2

18 268 W. Jack / Journal of Pulc Economcs 84 (2002) example s provded here, all detals eng relegated to Appendx B. Suppose at low ncome levels preferences are Leontef, ut that at hgher ncomes they are quas-lnear. Such ordnal preferences mply that at low ncomes the Engel curve for health s lnear, and that for hgher ncomes t s flat, gvng health the propertes of a necessty good. Ths general relatonshp s not nconsstent wth much emprcal evdence on the demand for health (e.g., Mannng et al. (1987)). In ths case, for low ncomes there are no dstortonary effects of prce susdes (.e., expendture-ased nsurance), so transfers n the ad state must e very large relatve to those n the good state n order to e ncentve ncompatle. The transton lne thus les aove the full nsurance lne. On the other hand, n the quas-lnear range at hgher ncomes, the dstortonary costs are postve, and net transfers from the good to the ad state ecome ncentve ncompatle earler. The transton lne therefore les elow the full nsurance lne n ths regon. At an ntermedate pont, the two lnes ntersect. Now assume that endowments are such that a fully nsured hgh rsk ndvdual consumes at the ntersecton. Ths stuaton s shown n Fg. 7. In ths case, the ntroducton of moral hazard has no effect on the potental equlrum contract offered to hgh rsks, as optmal nsurance for them contnues to e complete. Thus Fg. 7. An example n whch there s no external ncentve effect, and moral hazard can only mprove the chances of equlrum exstence.

19 W. Jack / Journal of Pulc Economcs 84 (2002) there s no external ncentve effect on low rsks, and the only mpact of moral hazard s the nternal ncentve effect whch mples that the hghest prce at whch otherwse constraned low rsk ndvduals are wllng to purchase an unconstraned quantty of nsurance falls, and the lkelhood of an equlrum exstng ncreases Numercal smulatons The analytcal example of the prevous su-secton used a very specfc utlty functon. In ths secton, I present the results of numercal smulatons usng a smpler representaton of preferences, n part to show that the specal analytcal case s not exceptonal. In partcular, I now assume that preferences are quaslnear over all ncome ranges. Thus for all c and h, the utlty ndex s of the form n c 2 a/h, where a and n are postve constants, so that the elastcty of demand for health s equal to 1/(n 1 1) [ (0,1). Indvduals are assumed to have dentcal atttudes towards rsk, represented y a constant coeffcent of rsk averson j, yeldng a complete descrpton of preferences over health and consumpton across states of the world of 1 n (12j ) u(c,h) 5 (c 2 a/h ). 1 2 j Usng ths representaton of preferences, I examne the condtons under whch the exstence of moral hazard has an effect on equlrum nsurance contracts, the extent of the changes n coverage that t nduces, and the effect of moral hazard on the lkelhood of an equlrum exstng. As shown n Appendx B for quas-lnear preferences, the full nsurance lne s gven y m 5 f(m ) 5 m 1 f g g 0 where f0 5fz(h(u g)) 2 egg2fz(h(u )) 2 eg5 sg2 s, es s unconstraned health expendture n state s, and ss s the surplus from health consumpton n state s. Smlarly, the transton lne s defned smply as m 5 t(m ) 5 m 1 t g g 0 where

20 270 W. Jack / Journal of Pulc Economcs 84 (2002) u 0 f g gg FSu D G 0 g t 5 z(h(u )) 2 e 2 z h(u ) 2 e, f. Thus, over all ncome ranges, the full nsurance lne and the transton lne are parallel, and the latter les always to the south of the former. In ths case then, moral hazard always affects potental equlrum contracts. Smlarly, t s easy to 14 check that t 0. 0, so the transton lne les aove the equal ncomes lne. Even wth these relatvely smple functonal forms, t s not possle to arrve at smple condtons under whch the external or nternal ncentve effect domnates, and thus under whch the lkelhood of an equlrum wll decrease or ncrease. Tale 1 thus reports some llustratve results from numercal smulatons usng the CRRA/ quas-lnear specfcaton of utlty aove. Entres n the tale represent the percentage ncrease n the crtcal value l* under condtons of ex post moral 15 hazard. For these smulatons, t was assumed that health expendtures n the good state of the world amounted to aout 2% of ncome (u g 5 0.1, m 5 10), and that the low rsk ndvduals proalty of eng n the ad state s 0.1. Tale 1 Percentage ncrease n the lkelhood of equlrum n the presence of ex post moral hazard j u h h p h p h 14 When n [ (21,0) and the elastcty of demand s greater than unty, t, Blank cells represent parameter values for whch the smulatons dd not converge.

21 W. Jack / Journal of Pulc Economcs 84 (2002) Smulatons are performed for two values of the elastcty of demand for health, h(51/(n 1 1)), consstent wth emprcal estmates of aout 0.2. The prce of health n the ad state vares etween 2, 10, and 18 tmes the prce n the good state, and health expendtures are correspondngly 4, 13, and 20% of ncome n the ad state when n 5 3. The results from Tale 1 confrm that the mpact of moral hazard on the lkelhood of a pure strategy equlrum exstng can e postve or negatve. In general t s not to e expected that monotonc relatonshps should necessarly exst etween the varous parameters of the model and the sze of the mpact of moral hazard on the lkelhood of exstence of a pure strategy equlrum. However, some general propertes of the model can e estalshed from the numercal smulatons. Frst, for many specfcatons, the mpact of moral hazard seems to e rather small, alterng the chance of an equlrum exstng y only a few percentage ponts. On the other hand, there are some parametersatons that yeld large percentage changes n the crtcal value of l when moral hazard s ntroduced, n oth the postve and negatve drectons. The results suggest that the mpact of moral hazard on equlrum exstence s lkely to e postve when the ad state s not too ad (.e., when u s relatvely low), at least for those cases n whch ndvduals are not too rsk averse. Tale 1 also suggests that ex post moral hazard appears more lkely to ncrease the chance of equlrum exstence when the elastcty of demand for the good whose prce s uncertan (.e., health) s lower. Thus, not only are the dstortonary costs due to ncomplete nsurance smaller when demand elastctes are smaller, t seems that the market s also relatvely less lkely to fal, n the RSW sense, n such stuatons. In addton, the mpact s lkely to e postve when the two rsk types have sgnfcantly dfferent proalty dstrutons over the two states (.e., when p s h suffcently greater than p ). Ths second property s llustrated further n Tale 2, l whch presents percentage changes n the crtcal value of l as the rato of the proalty of eng n the ad state for good and ad rsks vares. In partcular, p l Tale 2 Effect of the dvergence of rsks on the mpact of moral hazard on the lkelhood of equlrum pl p h % Change n l*

22 272 W. Jack / Journal of Pulc Economcs 84 (2002) vares from 0.02 to 0.4, whle ph s set equal to (1 2 p l). The smulatons are reported for h5 0.1, ug5 0.1, u5 1.8, and j Concludng remarks Ths paper has egun an nvestgaton of the exstence of pure strategy equlra n nsurance markets under condtons of ex ante adverse selecton and ex post hdden nformaton moral hazard. Snce the work of Rothschld and Stgltz, and Wlson, t has een well known that the frst type of nformaton asymmetry can lead to adly functonng markets, n the sense that a pure strategy equlrum may fal to exst. The ncluson of the second nformaton asymmetry s partcularly motvated y the common concern amongst health economsts and polcy makers that the exstence of fee-for-servce health nsurance leads to overspendng on health care. The model shows that under plausle assumptons, the ntroducton of ths further nformaton asymmetry can have the surprsng effect of ncreasng the lkelhood that the nsurance market wll work well, n the sense that an equlrum wll exst. Ths result s oth theoretcally nterestng and of relevance to the polcy deate. The dea that more nformaton s not always etter s not new, partcularly when the addtonal nformaton s held asymmetrcally amongst agents. In fact, the orgnal RSW model s a case n pont: when nether ndvduals nor nsurance companes know rsk attrutes, an equlrum wll always exst, ut when ndvduals know ther own rsk types, the market can reak down. However, ths paper has shown that, for some parametersatons at least, makng some asymmetrcally held nformaton pulc (.e., removng ex post moral hazard y makng the state of the world oservale) may well make matters worse. On the polcy sde, the result suggests that US-style Health Mantenance Organzatons could have amguous effects on the health nsurance market. If an HMO market exhted an equlrum, the asence of (or at least reducton n) ex post moral hazard due to the amalgamaton of nsurance and medcal servces may well ncrease aggregate welfare (assumng that there are no other negatve effects of HMOs, e.g., related to the qualty of servces). However, f the growth of HMOs led to poorly functonng nsurance markets wth many people unnsured, then the welfare mplcatons could e less engn. As mentoned n Secton 1, comparson of two nsttutonal arrangements wth dfferent equlrum exstence propertes s suject to questons of nterpretaton. A more complete model would deally admt the posslty of a mxed equlrum, n whch oth HMO and fee-for-servce nsurance contracts coexst. Whle there s a wdespread elef that rsk selecton n the US may well e affected 16 Of course, there s no reason that p 1 p should e constraned to equal unty. Ths s just a l h convenent parametersaton.

23 W. Jack / Journal of Pulc Economcs 84 (2002) through choce of organzatonal form (.e., wth lower rsk ndvduals optng for HMO coverage), such an nterpretaton reles on qualty dfferences etween the two forms. In ths paper I have assumed the only dfference etween them to e one of nformaton, so a more elaorate extenson to a model admttng mxed equlra awats future research. Acknowledgements I thank Mark Armstrong, Jeff Borland, James Dow, Perre-Yves Geoffard, Smon Grant, Jeff Klne, Ben Polak, and Jean-Charles Rochet, as well as partcpants n the 1999 ISPE and two anonymous referees for helpful comments, whle retanng responslty for remanng errors. Appendx A Proof of Lemma 1. ( ) Suppose c*(u,w 2 t ) # c 1(t g,t ). It s frst necessary to check that t s unproftale for ndvduals n the good state to choose the expendture-transfer par desgned for those n the ad state. But, f c * ; c*(u,w 2 t ) # c (t,t ), then 1 g n(q *,t,u g) 5 u(c *,[w 2 t2 c * /u g) # u(c,[w 2 t 2 c 1 1/u g) 5 n(q *,t,u ). g g g To check that ndvduals n the ad state do not choose the (q* g,t g) contract, let B(s,s9) e the udget set of an ndvdual n state s who chooses a contract wth transfer t s9. I show that n(q *,t,u ) $ n(q g*,t g,u ). Snce t g$ t, B(,g) 7 B(,), and snce q* maxmses utlty suject to the constrant B(,), the nequalty must hold. ( ) Proceed n a contrapostve fashon. Suppose that c * ; c*(u,w 2 t ). c * 1(t g,t ). Then y the elastcty assumpton, c, c*(u g,w 2 t ) # c 2, so c * [ (c,c ). But then 1 2 n(q *,t,u g) 5 u(c *,[w 2 t2 c * /u g). u(c,[w 2 t 2 c 1 1/u g) 5 n(q *,t,u ) g g g so the effcent allocaton s not ncentve compatle for ndvduals n the good state. h

24 274 W. Jack / Journal of Pulc Economcs 84 (2002) Proof of Lemma 2. Suppose (t g,t ) s such that ( q * g,q * ) s not ncentve compatle. Then, wrtng h9 5 [w 2 t 2 c /u, g u(c,h 9) 5 u(c,h 9) (1) wth c, c 1 2. The condton of the lemma says that as one moves down the udget lne n the good state (.e., as c ncreases), the change n utlty from a proportonal ncrease n h, holdng c fxed, ncreases. Snce for each 5 1,2, h /h9 5u g/u, gven (1), ths mples that u(c,h ). u(c,h ) as requred. Thus (q,q ) 5 (q*(t ),[w 2 t 2 c g g g 1) s the allocaton that s ncentve compatle for ndvduals n the good state and that provdes maxmal utlty for ndvduals n oth states. h Proof of Lemma 3. Frst show that the condton of Lemma 2 s satsfed. Indeed, d m 2 c u m2 c d m 2 c m2 c FS D Sc, DG 5 FS Dz 9S DG dc u h u dc u u z 9((m 2 c)/u ) m 2 c z 0((m 2 c)/u ) 52 2S D u u u 1 52 fhz 0(h) 1 z 9(h) u g. 0. Thus, the allocaton s ncentve compatle for ndvduals n the good state. Next, the allocaton s ncentve compatle for ndvduals n the ad state f and only f n(q,t,u ) 5 c11 z(h 1). c* g 1 z s(mg2 c * g )/ud5 n(q g,t g,u ). Recallng the fact that c* * g 1 z s(mg2 c g )/ugd5 c1 1 z (m 2 c 1)/u g, so that c* g 2 c 1 5 z s(m 2 c 1)/ugd2 z s(mg2 c g* )/u gd, ncentve compatlty for those n the ad state s equvalent to z s(m 2 c )/u 2 z (m 2 c *)/u. z (m 2 c 1 d s g g d s 1)/ugd2 z s(mg2 c g* )/ugd or that z(x) 2 z( y). z(ax) 2 z(ay) for x. y and a. 1. But ths condton s satsfed f and only f the functon hz(h) s decreasng n h, whch s equvalent to the condton of the corollary. h Proof of Lemma 4. By the elastcty assumpton, at the ntal transfer par (t g,t ) 5 (t 0,t 0), c*(u,w 2 t 0), c*(u g,w 2 t 0) 5 c 1(t 0,t 0), so q g(t g,t ) 5 q g* (t 0) and q (t g,t ) 5 q * (t 0). Normalty of consumpton mples that c*(u,w 2 t )/ t, 0, and c*(u,w 2 t ) ` as t 2`. Smlarly, c / t, 0 and c 1 g 1 / t. 0 jontly mply that c 0ast 2`. By contnuty, there exsts a t ˆ (t ). t, and an 1 g 0 0 s d