COMPARING ALTERNATIVE REIMBURSEMENT METHODS IN A MODEL OF PUBLIC HEALTH INSURANCE

Transcription

1 XIV CONFERENZA IL FUTURO DEI SISTEMI DI WELFARE NAZIONALI TRA INTEGRAZIONE EUROPEA E DECENTRAMENTO REGIONALE coordinamento, competizione, mobiità Pavia, Università, 4-5 ottobre 2002 COMPARING ALTERNATIVE REIMBURSEMENT METHODS IN A MODEL OF PUBLIC HEALTH INSURANCE FRANCESCA BARIGOZZI pubbicazione internet reaizzata con contributo dea società itaiana di economia pubbica dipartimento di economia pubbica e territoriae università di Pavia

2 Comparing Aternative Reimbursement Metods in a Mode of Pubic Heat Insurance Francesca Barigozzi y First version: October 1998 Tis version: June 2000 Abstract I compare in-kind reimbursement and reimbursement insurance. I expicity consider outpatient and inpatient care in a mode were iness as a negative impact on abor productivity. Consumers are eterogeneous wit respect to intensity of preferences for treatment wic is teir private information. Ten te socia panner as a coice of two kinds of reimbursement structure: pooing (uniform) and sef-seecting aocations. Anayzing pooing aocations I sow tat reimbursement insurance weaky dominates in-kind reimbursement. Wie considering sef-seecting aocations I sow tat te two reimbursement metods are, from a socia wefare point of view, equivaent. Keywords: eat insurance, in-kind transfers, reimbursement insurance, adverse seection. I tank Aberto Bennardo, Giacomo Cazoari, GianLuca Fiorentini, Umberto Gamarini, Aessandro Lizzeri, Eric Main, Aessandro Pavan, François Saanié and especiay Hemut Cremer for epfu comments. Paper presented at te Association of Soutern European Economic Teorists (ASSET), Boogna 1998, and at te CNR worksop: Equity, eciency and te pubic provision of private goods, Boogna, January y GREMAQ, University of Tououse and University of Boogna. E-mai: barigozz@spbo.unibo.it

3 1. Introduction Risk averse consumers demand eat insurance. Tey insure against te nancia risk associated wit buying medica care. I wi study aternative eat insurance reimbursement metods to nd out wic kind of payment is sociay preferabe. In an idea word, te optima insurance contract from te socia panner s point of view woud pay ump-sum transfers contingent on te eat status. If iness occurs consumers woud receive a cas payment reated to te severity of disease, so tat consumer s sovereignty woud be competey preserved. In reaity we normay do not observe tis type of reimbursement. Instead of a cas payment, we generay observe eiter in-kind reimbursement or reimbursement insurance (ater on reimbursement on treatment cost). Generay, wen reimbursement is in-kind, consumers are payed directy in medica services. Payment is contingent on disease as it woud be for cas reimbursement, but, in te case of in-kind, consumers are not free to coose te quantity of treatment tey prefer. On te contrary, wen reimbursement is on treatment cost, insurance payment depends on consumers expenditures upon eat care. In tis case instruments as coinsurance and deductibe are used to imit overconsumption. Te representation of tese reimbursement pans tat I coose in te mode is not abe to capture a teir compex features but provides a treatabe framework. To be as simpe as possibe I assume tat wen reimbursement is in-kind (IK), access to care is free and consumers receive a quantity of treatment determined by te insurer. Imposing a ceiing on treatment avaiabe to consumers, insurance is abe to prevent ig demand for care. Tis impies tat in-kind reimbursement aows cost-containment. At te same time an evident disadvantage of IK reimbursement is te cost on socia wefare due to te imposition of a consumption constraint to te insured peope. Concerning pysician s fee, an important consequence of free access to care is tat, wit IK reimbursement, eat care providers are payed directy by te insurer. Considering pubic eat insurance systems wic use to reimburse in-kind, we generay refer to Nationa Heat Service type organizations. Great Britain, Germany, Itay 1 and, ony for inpatient care, aso France 2 are an exampe. 1 In Itay outpatient care reimbursement is rater compicate, but we can say tat at east GP services are provided in-kind. 2 In some cases, in France, tird party payer principe may take pace aso for outpatient care. However it occurs ony for cronic or very serious diseases or, more often, as te consequence of 971

4 Conversey, wen reimbursement is on treatment cost (TC) I assume tat consumers are free to coose te quantity of treatment tey desire. A consequence is tat, not internaizing te entire eat care cost, tey demand an excessive quantity of it (overconsumption). Tis is an ex-post mora-azard probem. Concerning again pysician s fee, wit treatment cost reimbursement eat care providers generay are payed by consumers. Te atter, after te insurance caim, receive a partia reimbursement from te insurer. As an exampe tis reimbursement is used, ony for outpatient care, in France. Figure 1 summarizes te trade-o, caracterizing te two reimbursement metods, between consumers freedom in coosing treatment quantity and consumers incentive to not overconsume. Te importance of a uniform consumption constraint (in a sense wic wi be cari ed ater) directy depends on te eve of eterogeneity caracterizing te popuation. Tis suggests tat it coud exist a tresod vaue in consumers eterogeneity suc tat wen eterogeneity is not too ig in-kind reimbursement is better, wie, wen eterogeneity is sucienty ig, treatment cost reimbursement is preferabe. To my knowedge tis institutiona comparison between aternative reimbursement metods is sti an unexpored issue. Moreover tis work provides a treatabe framework for systems wic use a mix of in-kind and treatment cost reimbursement: outpatient care are reimbursed on treatment cost and inpatient care are reimbursed in-kind. Tis is ust te reimbursement pan used in France. In fact, in te case of outpatient care consumers sare a part of treatment costs, but tey maintain an important eve of freedom in coosing treatment quantity. On te oter side, for inpatient care, access to care is free and treatment quantity is normay decided by te pubic insurance. compementary private insurance purcase. Te Frenc system for outpatient care eave compete freedom to consumers: tey coose te provider (bot generaist and speciaist) and te number of examinations. Moreover consumers directy pay for te services and te treatment prescribed. Later tey ask for reimbursement to te Socia Insurance Administration and tey are paid back approximatey from te 60 to te 80% of teir expenses. Concerning Frenc consumers freedom, te oce based doctors convention introduced a vountary sceme in 1987 wic o ered te possibiity to doctors of becoming médecin référants. Patients wo oin tis sceme ave a mora commitment not to visit a speciaist directy. Te aim of tis sceme was essentiay to ceck te ecacy of a possibe cost containment measure. But most doctors were reuctant (up to te end of 1997, ony te 12,5 per cent of tem ad oined te sceme) because of te fear tat tey may be more controed by te eat insurance system. In fact in tis sceme tey are obiged to keep detaied patient records. 972

5 Freedom in coosing quantity Reimbursement insurance In-kind reimbursement Cost-containment Figure 1.1: trade-o between consumers freedom and cost-containment. Concerning te reated iterature, rst, I ave to mention te modes on moraazard in eat insurance, one of te semina papers being Zeckauser (1970). Te way I treat treatment cost reimbursement represents a particuar case of te more genera reimbursement scedue of is mode. Second, concerning in-kind reimbursement, I reate to te iterature on in-kind transfers and optima taxation (among oters Cremer and Gavari (1997)). In tat iterature te sef-seecting property of in-kind transfers in second-best economies as been anayzed. Tird, more generay I refer to te iterature on income taxation wit uncertainty in wic taxation is used to insure consumers against various types of wage and eat risks (as an exampe, Varian (1980) and Cremer and Gavari (1995)). In te next pages I compare IK and TC reimbursement in a mode of pubic eat insurance 4. I assume tat consumers are eterogeneous wit respect bot to teir state of eat and to teir preferences for treatment consumption. Te pubic insurer pays te roe of te socia panner and e is fuy informed on 4 A pubic eat insurance as been anayzed for te rst time in Bonqvist and Horn (1984). Te autors sow tat, if individuas di er in teir earning abiity and aso in te probabiity of faing i, ten a pubic eat insurance is an ecient too to redistribute wefare wen income taxation is inear. Togeter wit teir focus on pubic eat insurance, Bonqvist and Horn (1984) presents anoter simiarity wit respect to tis paper: in bot te modes consumers utiities are statedependent. 97

6 consumers state of eat. In te rst part of te work I constrain te insurance pan to be uniform in te sense tat tastes eterogeneity is not taken into account. Anayzing pooing aocations I nd tat TC dominates IK reimbursement. Tis means tat, contrary to intuition, tere is no trade-o between TC and IK depending on consumers eterogeneity. In te second part, I consider sef-seecting aocations, i.e. aocations were consumers can coose insurance pans wic takes into account teir preference for treatment. In tis case I sow tat te two reimbursement metods are, from a socia wefare point of view, equivaent. 2. Te mode Let us consider a representative consumer and tree possibe states of eat. Consumer can be eaty, not seriousy i and seriousy i. Wen not seriousy i, consumer needs outpatient care, wie wen seriousy i, consumer needs inpatient care. More precisey, wit probabiity p 1 ; consumer is in good eat and as a fu earning abiity, is margina abor productivity is w 1 (abiity is normaized to equa te wage rate). Wit probabiity p 2 ; consumer is i and, as a consequence, e partiay ooses is earning abiity; is margina abor productivity fas to w 2 <w 1 : Finay, wit probabiity p =1 p 1 p 2 ; consumer is seriousy i and ooses a is earning abiity (e is ospitaized); in tis case margina abor productivity fa to w =0: 5 Consumer s preferences are state-dependent and twice separabe: U i (C; X; L) =u (C i) v (L i ) H i + µ i Á i (X i ) i =1; 2; indicates eat status as stated above, C is an aggregated consumption good taken as numeraire, X is eat care consumption and L is abor suppy. H i is a xed, state dependent, utiity oss wic can be partiay recovered troug eat care consumption. Te term µ i Á i (X i ) indicates utiity from eat care consumption. In particuar Á i (X i ) is eat improvement from treatment consumption, wie te parameter µ i (0 <µ i µ i ; i =2; ; = ; ) represents intensity of preferences for treatment, i.e. eterogeneity in consumers tastes. Wit probabiity ¹ consumer as ow preference for eat care consumption (e is ow-type), wie, wit probabiity ¹ =1 ¹ e as ig preference for eat care consumption (e is ig-type), and tis for bot states of iness (bidimensiona eterogeneity). 5 As I wi sow in section 5.1, iness severity pays an important roe aso wit respect to consumers tastes on consumption eves. 974

7 Standard ypotesis on utiity functions od: u 0 (C i ) > 0; u 00 (C i ) < 0; v 0 (L i ) > 0; v 00 (L i ) > 0: H 1 = 0 < H 2 < H. Te function Á (X) is suc tat: Á 1 (X) =0 8X; Á i (0) = 0; Á 0 i (X) > 0; Á00 i (X) < 0; i =2; : Moreover, H i >µ i Á i (X i ) ; 8i =2; ; 8 = ; and 8X; suc tat consumer s utiity is aways greater wen in good eat tan wen i. Te socia panner wi be concerned wit making comparisons of utiity eves across consumers types. Tus, I assume fu comparabiity of consumers utiities. Te timing of te mode is as foows: at t 1 (interim) consumer earns is type and at t 2 (ex-post) te eat-risk is reaized and consumer earns is state of eat too. As it is sown in gure 2, te socia panner decides interim, wie consumer decides ex-post. t 1 (types) t 2 (state of nature) t Socia Panner Consumer Figure 2: timing. In tis mode I focus on te reationsip between consumer and pubic insurance, te eat care provider is not expicity considered. Te situation described ere ts bot te case of a pubic provider (verticay integrated wit te pubic insurer) and of a private one in a competitive market. In bot cases, assuming a inear tecnoogy, te eat care unitary cost is constant. Tis aows us to say tat consumer and te pubic insurer face te same treatment price (q) : Moreover, I assume tat te provider beaves as a perfect agent for is patient. Concerning te informationa structure of pubic insurance, in te mode consumer as potentiay two private informations: is eat status (captured by te margina abor productivity w i ) and is type µ i (ig/ow taste for treatment). I assume tat consumers eat status is observabe. Tis means tat, concerning tis aspect of te examination (as opposed to treatment purcase), cousion between patient and pysician is impossibe: pysician acts as a perfect agent for insurance. As a consequence reimbursement can be contingent to te eat status. Conversey, I assume tat, in eac state of eat, preference for treatment 975

8 is not observabe and pubic insurance can reimburse consumers according to a pooing aocation or a sef-seecting one. Te structure of te work is as foow. In te rst part I wi sow te rstbest, ten I wi compare te aternative insurance pans wen te Government impements pooing aocations. In particuar te considered pans are: cas, in-kind, treatment costs reimbursement and, nay, a mix of te two previous metods. In te second part I wi anayze te same reimbursement metods wen te socia panner impements sef-seecting aocations. To summarize, te considered cases are:. First-best Pooing aocations: - rst-best - uniform pans: - cas reimbursement - in-kind reimbursement - treatment cost reimbursement - mix of reimbursement types Sef-seecting aocations: - cas reimbursement - in-kind reimbursement - treatment cost reimbursement - mix of reimbursement types I assume tat, exibiting an iness certi cation provided by a pysician, patient is entited to receive reimbursement. First-best is represented by a contract contingent bot to te eat status and to preference for treatment, tat is a pan caracterized by ve non-uniform monetary transfers (P; R 2;R 2;R ;R ): Consumption in te tree states of eat is: C 1 = w 1 L 1 P C 2 = w 2 L 2 + R 2 qx 2 C 2 = w 2 L 2 + R 2 qx 2 C = R qx C = R qx were P is premium payed by eaty consumer, R 2 ( = ; ) is net from premium reimbursement for outpatient care and R ( = ; ) is reimbursement 976

9 for inpatient care. Wit cas reimbursement consumer decides to purcase te quantity of treatment e prefers. Note tat in state of nature cas transfer R must be enoug to et consumer purcase bot eat care and aggregated consumption. Te socia panner maximizes te utiitarian 6 socia wefare function SW = ¹ EU(µ )+¹ EU(µ ),wereeu(µ ) is ow-type consumer s expected utiity and EU(µ ) is ig-type consumer s expected utiity. Expected utiity of ow-type and ig-type individuas are respectivey mutipied for te proportion of owtype and ig-type consumers in te popuation 7 : ow and ig type consumers ave te same weigt for te socia panner. Note tat, wen eaty, te two consumer types are identica. Te socia panner soves: 8 Max p 1 [u(w 1 L 1 P ) v(l 1 )] + P;R i ;L i ;X i X +p 2 ¹ >< u w2 L 2 + R 2 qx 2 H2 + µ 2Á 2 X 2 v(l 2) + =; X +p ¹ u R qx H + µ Á X =; >: s:t: : p 1 P = p 2 (¹ R2 + ¹ R2)+p (¹ R + ¹ R ) (.1) Two remarks can be usefu. First, premium is fair. Second, because of te way te eterogeneity parameter µ i enters te utiity functions, socia wefare is increasing wit respect to eterogeneity. Tis is equivaent to say tat igtype consumers ave te igest weigt in tis economy. As a consequence an utiitarian socia wefare function redistributes from ow to ig-type individuas 8. 6 Concerning te coice of te socia wefare function it is interesting to say tat te maximin principe of Raws is ess appicabe to cases wic deas wit eat and te aocation of eat care. In fact wie te need for Raws primary goods (e.g. food and coting) are more or ess te same for a, tere is a muc more unequa distribution of te need for eat care re ecting te natura ottery. Tere are consequenty muc wider variations in te resources required to meet suc unequa distribution of needs. In particuar te crucia issue for te maximin criterion is te severity of te worst o. As ong as it is feasibe to improve te eat of tis individua, resources woud be directed to im irrespective of te forgone improvement for te oters. For an interesting discussion on tis subect see J.A.Osen: Teory of ustice and teir impication for priority setting in eat care, Journa of Heat Economics 16, 1997, Considering a arge number of representative consumers, ¹ is equivaent, ex-post, to te proportion of te -type. 8 For tis reason, in te socia wefare function, giving a iger (tan ¹ ) weigt to ow-type 977

10 >From FOCs we nd te fu-insurance resut 9 : Moreover it is: C 1 = C 2 = C 2 = C = C (= C) (.2) L 1 : w 1u 0 (C) =v 0 (L 1 ) (.) L 2 : w 2u 0 (C) =v 0 (L 2 ) (.4) X 2 : µ 2Á 0 2 X 2 = qu 0 (C) ; = ; : (.5) X : µ Á 0 X = qu 0 (C) ; = ; : (.6) As we expected, in every state of eat abor suppy and treatment quantity are determined suc tat margina bene t equas margina cost, as a consequence MRS L2 ;X 2 = q. Moreover in state of eat 2 it is: L 2 = L 2 and X 2 >X 2 ; w 2 and in state of eat it is X >X : Concerning te monetary transfers, not surprisingy one nds: Ri >Ri;i=2; : Note tat te coice of X i and L i can be decentraized because consumers face prices w i and q. As a consequence te socia panner can obtain rst-best o ering te rst-best contract and etting consumers coose (ex-post) abor suppy and treatment quantity. Here I brie y introduce te structure of in-kind and treatment cost reimbursement wit fu information on consumers preferences. Te two reimbursement pans wi be treated in detai in te case of asymmetric information in section 4.2 and 4., 7. and 7.4. In-kind reimbursement Reca tat, wen reimbursement is in-kind, access to care is free and consumers receive te package of care X ¹ i wic is determined by insurance. I assume tat te transfer X ¹ i as to be entirey consumed: no intermediate eves of consumption are possibe. Tis interpretation of in-kind reimbursement, wic represents a good approximation of reaity, wi become important anayzing sefseecting aocations (section 7.). Individuas consumption in te tree states of eat is: consumers woud be more equitabe. In fact it woud aow to redress te reative importance of te two consumers types. I eave it for future researc. 9 Here utiities are state-dependent and separabe. Ten, for any given income eve, i eat does not ater te margina utiity of income. As a consequence fu insurance is optima. Moreover te fu insurance condition concerns ony aggregate consumption. 978

11 C 1 = w 1 L 1 P IK C 2 = w 2 L 2 + R IK 2 ; X 2 = X ¹ 2 C = RIK ; X = X ¹ were = ;. Note tat seriousy i consumers are obiged to consume accordingtotetransferr IK. Tis means tat in-kind reimbursement imposes a doube constraint on seriousy i consumers: aggregate consumption and treatment quantity. In section 4.2 it wi be cear tat tis doube constraint concerns aso not seriousy i consumers. Reimbursement on treatment cost Wit reimbursement insurance te socia panner uses a cost-saring parameter i 1(i =2; ; = ; ) 10 to reduce eat care overconsumption. As it was said before, te mora-azard probem due to te subsidization of eat care corresponds to te main disadvantage of reimbursement on treatment cost. As in te case of cas reimbursement, consumers coose teir preferred treatment quantity, te di erence is tat ere te socia panner modi es treatment consumption prices. Consumption in te tree states of eat is: C 1 = w 1 L 1 P TC C 2 = w 2 L 2 + R TC 2 2qX 2 C = R TC qx It is evident tat under perfect information in-kind and treatment cost reimbursement are bot equivaent to rst-best. In fact, wit bot reimbursements metods te socia panner can use four additiona instruments (respectivey ¹X i wit in-kind and i wit treatment cost reimbursement, i =2; ; = ; ) suc tat it can do at east as we: Actuay wen monetary transfers contingent to preference for treatment are avaiabe, tese additiona instruments are useess. Concerning treatment cost reimbursement, obviousy wit fu information te socia panner sets i =1suc tat prices are not distorted. Tis resut is stated in te foowing proposition: Proposition 1 Under perfect information, cas, in-kind and treatment cost reimbursement are equivaent. Tis resut is in ine wit Arrow s intuition. In fact, in is semina paper (Arrow (196), page 962), e says tat, in an ypoteticay perfect market, te 10 Wit fu information considering eiter i or simpy i is equivaent: 979

12 existing di erent metods of treatment costs coverage soud be equivaent. In gure i consumers rst-best aocation is sown. As te reader can see, te sope of ow-type utiity function is iger tan ig-type one, tis appens because dc i dx i = µ Á 0 i(x i ) i u 0 (C i ) : C i U i U i C i = C i X i X i X i Figure : i consumers rst-best aocation. Wen µ is not observabe; rst-best aocations cannot be impemented. In fact, in state of eat 2, rst-best payment impies: C2 = C2 ; X2 >X2 and L 2 = L And in state of eat : C = C ; X >X: Tis means tat ow-type consumers woud mimic ig-type ones. 4. Uniform pans Deaing wit ow-types incentive constraints te pubic insurance as a coice of two kinds of reimbursement structure. Tose in wic te insurer is unabe to distinguis (ex-post or ex-ante) among individuas: tis corresponds to a pooing aocation. And tose in wic te ig-type and te ow-type can (ex-post) 11 In particuar it is: R IK i = R IK i = R i qx i = R i q ¹ X i : 980

13 be identi ed as a resut of te action undertaken by te di erent groups: tis corresponds to a sef-seecting aocation. Te reimbursement pans tat we observe in te European pubic eat insurance systems are essentiay uniform: pubic insurance does not o er contracts wic di er according to teir recipients taste. In fact te same pan is proposed to consumers wit te same iness witout caring about teir (di erent) preferences for treatment. A possibe reason for impementing pooing aocations is ust te presence of an idden incentive constraint as te one treated in tis work: facing o te ack of information on consumer s type, in te rea word pubic insurance cooses to provide uniform reimbursements. Maybe tis coice is due to te presence of poitica constraints and/or administrative costs. Anyway, to investigate on tis interesting issue is not in te aim of tis work Cas reimbursement Tis pan is de ned by tree monetary transfers: (P C ;R C 2 ;R C ): I consumers receive reimbursement and coose in te market te preferred treatment quantity. Consumptionintetreestatesofeatis: C 1 = w 1 L 1 P C C 2 = w 2 L 2 + R C 2 qx 2 C = R C qx Note tat te ony di erence wit respect to rst-best is tat ere we add te uniformity constraint R i = R i = R C i : Recaing tat consumer maximizes ex-post, wit cas reimbursement consumers programs are as foow. Good eat: max L 1 u(w 1 L 1 P C ) v(l 1 ) Ten in state of eat 1 abor suppy is de ned according to te foowing equation: L 1 : w 1 u 0 (C 1 )=v 0 (L 1 ) (4.1) Outpatient care: max u w 2 L L 2 + R2 C qx 2 H2 + µ 2Á 2 X 2 v(l 2) 2 ;X 2 Ten in state of eat 2 abor suppy and purcased treatment quantity are respectivey de ned according to equations: L 2 : w 2 u 0 (C 2)=v 0 (L 2); = ; ; (4.2) 981

14 Inpatient care: >: X 2 : µ 2 Á0 2 X 2 = qu 0 C2 ; = ; : (4.) =; max X u R C qx H + µ Á X Ten in state of eat purcased treatment quantity is cosen according to equation: X : µ Á 0 X = qu 0 C ; = ; : (4.4) Te socia panner soves te foowing program: 8 Max p 1 u(w1 L P C ;R2 C 1 P C ) v(l 1) + ;RC X +p 2 ¹ u w2 L >< 2 + R2 C qx 2 H2 + µ 2Á 2 X 2 v(l 2 ) + =; X +p ¹ u R C qx H + µ Á X s:t: : p 1 P C = p 2 R C 2 + p R C were abor suppies L i and treatment quantities X i (4.1), (4.2), (4.) and (4.4). >From FOCs we nd: verify consumer s FOCs u 0 (C 1 )=E [u 0 (C 2 )] = E [u 0 (C )] 12 (4.5) As equation (4.5) sows, ow and ig-type consumers coose di erent aggregated consumptions. Fu insurance is preserved ony in average. Tis distortion from te fu insurance represents te cost te uniformity constraint imposes. Totay di erentiating equations (4.2) and (4.) one nds tat dl 2 > 0 and dx 2 dx 2 > 0. Ten, not surprisingy, it is: X2 >X2 dµ and L 2 >L 2: As a consequence 2 we are not abe to say neiter if C2 is iger or ower tan C 2 nor wic not seriousy i consumers type is caracterized by te iger utiity eve 1. In te same way, totay di erentiating equation (4.4), one nds tat dx > 0 so tat dµ 12 E [u 0 (C i )] = ¹ u 0 Ci + ¹ u 0 Ci : 1 In fact it is: U2 U2 = u C2 u C 2 + µ 2 Á 2 X 2 µ 2 Á 2 X 2 + v L 2 v L 2 7 0; were U2 and U2 respectivey are -type and -type not seriousy i consumer s utiity and X2 >X2;L 2 >L 2;C2 7 C2. 982

15 X >X: Ten C >C : But, again, we cannot a priori say wic seriousy i consumers type is caracterized by te iger utiity eve In-kind reimbursement Tis pan is caracterized by tree monetary transfers 15 and by two packages of care: (P IK ;R2 IK ;R IK ; X ¹ 2 ; X ¹ ): Individuas consumption in te tree states of eat is: C 1 = w 1 L 1 P IK C 2 = w 2 L 2 + R2 IK; X 2 = X ¹ 2 C = R IK ; X = X ¹ Wit respect to rst-best I added bot te uniformity constraint R i = Ri = Ri IK and te consumption constraint X ¹ i = X ¹ i : Tat is one more constraint wit respect to (uniform) cas reimbursement. Tis aows us to say tat, wit in-kind reimbursement, consumers cannot be better o. In session 5 it wi be cear tat consumers are aways worse o (proposition 2). Heaty consumers program is te same I sowed in te previous case and equation (4.1), were P IK substitute P; sti ods. Outpatient care: max u w 2 L 2 + R2 IK H2 + µ L 2 2Á ¹X2 2 v(l2 ) Ten, in state of eat 2, abor suppy is: L 2 : w 2 u 0 (C 2 )=v 0 (L 2 ) (4.6) Note tat ere, because of te separabiity of preferences, bot types of not seriousy i consumers ave te same abor suppy L 2. As a consequence C 2 = C2 = C 2 : Tis means tat imposing te constraint on treatment quantity, in-kind reimbursement yieds to te same aggregate consumption for bot consumers types. Te same ods for inpatient care. In fact consumers are constrained to R IK and ¹X and teir utiity is: u(r IK ) H + µ Á ( X ¹ ): In oter words in-kind reimbursement imposes te foowing two constraints to i consumers: Ci = Ci = C i and X ¹ i = X ¹ i = X ¹ i : 14 In fact it is: U U = u C u C + µ Á X µ Á X 7 0; were U and U respectivey are -type and -type seriousy i consumer s utiity and C <C;X >X. 15 In fact, being te eat status observabe, te socia panner is aways abe to use te monetary transfers R i : Note tat te transfer R2 IK can be, and presumaby is, negative (te socia panner can coect resources aso from not seriousy i consumer). 98

16 Wit in-kind reimbursement, in bot states of iness it is: U i Ui = Á ¹Xi i µ i µ i > 0 i =2; were Ui and Ui respectivey are ig-type and ow-type utiity. Tis inequaity means tat ig-type utiity is aways arger tan ow-type utiity: wit in-kind reimbursement ow-type consumers are aways worse o 16. Te di erence between te two utiity eves is proportiona to eterogeneity. Te pubic insurance program is: 8 Max p 1 u(w1 L P IK ;Ri IK 1 P IK ) v(l 1) + ; ¹Xi >< +p 2 u(w 2 L 2 + RIK 2 ) H 2 + e i µ 2 Á ¹X2 2 v(l 2 ) + +p u R IK H + e i µ Á ¹X >: s:t: : were e µ i = P =; p 1 P IK = p 2 R IK 2 + q ¹ X 2 + p R IK + q ¹ X ¹ µ i : In fact, to impement te pooing aocation, te Government maximizes te utiity of a representative consumer: te mean µ-type consumer. Not surprisingy, from FOCs wit respect to P IK ;R IK 2 and R IK we nd te fu-insurance condition: C 1 = C 2 = C = C ¹ (4.7) Moreover treatment packages are determined according to: ¹X 2 : e µ2 Á 0 2 ¹X2 = qu 0 ( ¹ C) (4.8) ¹X : e µ Á 0 ¹X = qu 0 ( ¹ C) (4.9) Obviousy neiter type of i and seriousy i consumers receive te optima quantity of treatment (determined respectivey by equation (.5) for outpatient care and by equation (.6) for inpatient care) because ¹ X2 and ¹ X are determined according to te mean-µ-type consumer. It is evident tat, wen tere is no eterogeneity µ 2 = µ 2;µ = µ ; we are back to rst-best. 16 Wie wit cas reimbursement it was U i U i

17 4.. Reimbursement on treatment cost As I said before i represents te coinsurance parameter, it is di erent for outpatient and inpatient care. Te uniform pan is caracterized by tree monetary transfers 17 and by te two coinsurance parameters: P TC ;R TC 2 ;R TC ; 2 ;.Individuas consumption in te tree states of eat is: C 1 = w 1 L 1 P TC C 2 = w 2 L 2 + R TC 2 2 qx 2 C = R TC qx Wit respect to rst-best te uniformity constraints on te monetary transfers (Ri = Ri = Ri TC ) and on te coinsurance parameters ( i = i) as been added. Consumers programs are te foowing. Heaty consumers decision is te same I sowed in te previous cases and equation (4.1) sti ods. Outpatient care: max u w 2 L L 2 + R2 TC 2 qx 2 H2 + µ 2Á 2 X 2 v(l 2) 2 ;X 2 As a consequence abor suppy is determined, again, according to equation (4.2), moreover treatment quantity is: X 2 : µ 2Á 0 2 X 2 = 2 qu 0 C2 ; = ; (4.10) Inpatient care: max X As a consequence treatment quantity is: u R TC qx H + µ Á X X : µ Á0 X = qu 0 C ; = ; (4.11) 17 As in te case of in-kind reimbursement, te transfer R TC 2 can be negative. 985

18 Te pubic insurance program is: 18 8 Max p 1 u(w1 L P TC ;Ri TC 1 P TC ) v(l 1 ) + ; i X +p 2 ¹ u w2 L 2 + R2 TC 2 qx 2 H2 + µ 2Á 2 X 2 v(l 2 ) + >< >: s:t: : =; X +p =; ¹ u R TC qx p 1 P TC = p 2 (1 2 ) q P +p (1 ) q P =; =; H + µ Á X ¹ X 2 + p 2 R TC 2 + ¹ X + p R TC >From FOCs wit respect to P TC ;R TC 2 and R TC one nds te foowing equation: E [u 0 (C i )] = u (C 1 ) 1+(1 i )qe i =2; TC i It is interesting to remark tat equation (4.12) woud be equivaent to (4.5) for i =1;i=2; : However it is easy to sow tat, because of consumers eterogeneity, tis wi never be te case at te optima treatment cost reimbursement poicy 19.Ten i ;i=2; are aways di erent from 1 for positive eve of eterogeneity. Obviousy wen tere is no eterogeneity it is optima to impose i = 1; i =2; and rst-best is obtained. >From FOCs wit respect to te coinsurance parameters one nds: (1 i ) E + E (X i ) u 0 (C 1 )=E[X i u 0 (C i )] i Te interpretation of equation (4.1) is as foows: te eft and side represents consumers margina cost and te rigt and side consumers margina bene t 18 As it is normay te case in eat insurance modes, I do not impose any constraint on te coinsurance parameters i ;i=2;, and I wi verify ex-post if tey are ess or iger tan unity. 19 In fact, from equation (4.14) beow, wic describes te optima coinsurance parameter, we know tat i =1impies u 0 (C 1 )= E [X iu 0 (C i )], i =2; ; and from equations (4.12) tat i =1impies u 0 (C 1 )= E (X i ) E [u 0 (C i )] ;i =2;. Tis means tat it must be E [u 0 (C i )] E (X i )=E[X i u 0 (C i )] ; wic is impossibe because C i depends aso on X i. 986

19 from a negative variation of i (a fa in treatment price). Wen i decreases, consumers out of pocket expenses decrease as we, wie insurance reimbursement expenses increases. As a consequence insurance premium must increase as we. Margina cost is measured by margina variation of insurance premium (in bracket) mutipied for margina utiity of consumption in state good eat. In fact premium is paid by eaty consumers. In te rigt and side te positive income e ect from a negative variation of i is measured by te product of treatment quantity and consumption margina utiity in te iness status. Mean vaues appear because a uniform pan is impemented. To nd te optima coinsurance parameters, equation (4.1) can be rewritten as: i =1 u0 (C 1 )E(X i ) E [X i u 0 (C i )] u 0 (C 1 i Te coinsurance parameters are positivey correated to treatment demand mean derivatives wit respect to i ; tese terms are a measure of mora azard. i is reated to price easticity of demand for treatment so tat equation (4.14) reminds us te inverse easticity rue in Ramsey taxation: te commodity wose demand is more ineastic is subsidized more. 20 Verifying ex-post if it is optima to impose a subsidy ( i < 1) or a tax ( i > 1) ; we nd 21 tat a sucient condition to subsidize treatment is: u 0 (C 1 ) <¹ u 0 Ci + ¹ u 0 Ci i =2; (4.15) wie a necessary condition to tax treatment is te opposite of (4.15). Te rigt and side of te previous inequaity is average margina utiity of consumption in te iness status. Margina utiity being decreasing, te interpretation of (4.15) is te foowing: it is optima to subsidize treatment if eaty individuas consumption is arger tan a particuar mean of te i individuas one. Tis means tat te socia panner may impose a tax on treatment Mix of reimbursement types Consider now a reimbursement wic pays on treatment cost for outpatient care and wic pays in-kind for inpatient care. Te mix of reimbursement types (MT) 20 As it wi be cari ed in section 5.1, easticity is iger for outpatient care. Ten we expect tat 2 > : 21 >From (4.14) te foowing yieds: i < 1, cov(x i ;u 0 (C i ))+E (X i )[E (u 0 (C i )) u 0 (C i )] > 0; were cov(x i ;u 0 (C i )) is positive: 987

20 is caracterized by te foowing instruments: (P MT ;R2 MT ;R MT ; 2 ; X ¹ ): Consumption eves are: C 1 = w 1 L 1 P MT C 2 = w 2L 2 + RMT 2 2 qx 2 C = R MT ; X = X ¹ As before, eaty consumers coose teir abor suppy according to FOC (4.1). Not seriousy i consumers coose simutaneousy teir abor suppy and treatment quantity suc tat tey respectivey verify FOCs (4.2) and (4.10). As wit in-kind reimbursement, seriousy i consumers utiity is u(r MT )+µ Á ¹X : Te pubic insurance program is: 8 >< >: Max P MT ;R MT X +p 2 p 1 u(w1 L i ; 2 ; X ¹ 1 P MT ) v(l 1) + ¹ u w2 L 2 + R2 MT 2 qx 2 =; +p u R MT H + e i µ Á ¹X H2 + µ 2Á 2 X 2 v(l 2 ) + s:t: : p 1 P MT = p 2 (1 2 ) q P =; ¹ X 2 + p 2 R MT 2 + p R MT + q ¹ X >From FOCs wit respect to P MT and R MT one nds tat: C 1 = C = R MT : As we expected, fu-insurance concerns ony eaty and seriousy i consumers aggregated consumption. Moreover FOCs wit respect to R MT 2 ; 2 and X ¹ determine respectivey equations (4.12), (4.14) and (4.9). 5. Comparing te aternative uniform reimbursement pans >From proposition 1 te foowing coroary ods. Coroary 1 In te case of pooing aocations if consumers are omogeneous, in-kind, treatment cost and a mix of types reimbursement are identica and equivaent to uniform cas payment. In fact wen consumers are omogeneous te uniformity constraint as no consequence on socia wefare and we are back to rst-best. Reintroducing eterogeneity, te foowing resut ods In te foowing te ranking among reimbursement scemes wi be stated wit te symbos º and ¼ respectivey for weak dominance and equivaence. 988

21 Proposition 2 In te case of pooing aocations a rst reimbursement metods ranking is te foowing: treatment cost reimbursement º cas reimbursement º in-kind reimbursement. Proof. (i) Uniform cas weaky dominates uniform in-kind reimbursement. In fact, recaing te discussion in session 4.2, cas reimbursement is caracterized ony by te uniformity constraint wie in-kind reimbursement as one more constraint on treatment consumption. Once te monetary transfers R 2 and R ave been xed it is aways better to et consumers coose treatment, being consumption prices not distorted. (ii) Uniform treatment cost weaky dominates uniform cas reimbursement. In fact cas reimbursement is caracterized by tree monetary transfers (P C ;R2 C ;R C ) wie treatment cost is caracterized by tree monetary transfers and by two tax/subsidies on treatment price 2 (P TC ;R2 TC ;R TC ; 2 ; ), i.e. treatment cost as two more instruments. Moreover, for 2 = =1; treatment cost is equivaent to cas reimbursement. As a consequence treatment cost is at east as we as cas reimbursement. Remark tat te parameters 2 and tat we ave introduced as te source of mora-azard in treatment cost reimbursement, actuay do not represent a cost. In fact te distortion tey impose on treatment price as a positive e ect on socia wefare. Te reason is tat 2 and are used to smoot consumption between di erent consumers types in te same eat status, suc tat TC optima aocation can approac fu insurance. In oter words 2 and aow to indirecty and partiay avoid te consequences of te uniformity constraint 24. Comparing uniform mix of types wit uniform in-kind and uniform treatment cost reimbursement one nds: Proposition In te case of pooing aocations a second reimbursement metods ranking is te foowing: treatment cost reimbursement º amixoftypes º in-kind reimbursement. Proof. A mix of types pays on treatment cost for outpatient care and in-kind for inpatient care. From proposition 2 we saw tat treatment cost dominates 2 See te discussion at te end of section In fact, oosey speaking, if we consider for exampe inpatient care, (< 1) ets C = C C = q X X be ower wit respect to C (cas) =q X X : (Te same argument ods for C 2 ) Note tat for =0it woud be C = C, but in tis case moraazard woud be too costy. Tis is te standard trade-o between mora-azard and optima risk spreading. 989

22 in-kind. Ten, considering MT wit respect to TC, seriousy i consumers are worse o, wie not seriousy i consumers are indi erent. As a consequence TC weaky dominates MT. On te oter side, considering MT wit respect to IK, not seriousy i consumers are better o, wie seriousy i consumers are indi erent. As a consequence MT weaky dominates IK. One can concude tat treatment cost weaky dominates a mix of types wic weaky dominates in-kind. To nd a more genera resut, tat is a ranking of cas and MT, I wi introduce in te next section an assumption on te structure of consumers eterogeneity Unidimensiona eterogeneity Regarding eterogeneity, empirica evidence sows tat, in te case of serious iness, te price easticity of demand for treatments is sma 25. A reasonabe interpretation is tat patients, for suc an iness, ave te sentiment tat tere is ony one appropriate treatment. Moreover, tis aows us to say tat, in te case of inpatient care, eterogeneity is sma and, as a consequences, a uniform consumption constraint wi ave a ow impact on socia wefare. Given tese considerations I set µ 2 µ 2 µ µ 0; so tat eterogeneity is ower in te case of serious iness. Te particuar case of unidimensiona eterogeneity (µ 2 µ 2 >µ µ =0)is interesting because it can represent a good approximation of reaity. Considering te uniform mix of reimbursement types in te particuar case wit no eterogeneity on serious iness te foowing remark ods: Remark 1 In te case of pooing aocations if seriousy i consumers are omogeneous, a mix of types is equivaent to treatment cost reimbursement. In fact, if seriousy i consumers are omogeneous, from coroary 1 we know tat a te reimbursement metods are equivaent to cas. Tis impies tat, concerning inpatient care, treatment cost and a mix of types are equivaent. Wie concerning outpatient care, treatment cost and a mix of types are te same by de nition. As a resut it is possibe to de ne a compete ranking of te four reimbursement metods wen seriousy i consumers are omogeneous: 25 Resuts from te RAND Heat Insurance Experiment sow tat eat care price easticities beong to te range [-0.1, -0.2]. In particuar, concerning serious iness treatment consumption, resuts sow tat tere are no signi cant di erences among te coinsurance pans in te use of inpatient care services. (Manning and oters (1987), page 258) 990

23 Proposition 4 In te case of pooing aocations if seriousy i consumers are omogeneous (unidimensiona eterogeneity), te compete reimbursement metods ranking is as foows: treatment cost reimbursement mix of types º cas reimbursement º in-kind reimbursement Bidimensiona eterogeneity Wit bidimensiona eterogeneity (µ 2 µ 2 µ µ 0) itexistsatrade-o between cas reimbursement and a mix of reimbursement types suc tat it can be eiter -cas dominates a mix of types- or te opposite. In fact, from proposition 2, cas weaky dominates in-kind, as a consequence seriousy i consumers are better o wit cas reimbursement. At te same time treatment cost weaky dominates cas, as a consequence not seriousy i consumers are better o wit te mix of reimbursement types. Indeed, concerning tis probem, one can make te foowing remark: te socia wefare function used in te mode suggests tat uniform mix of types weaky dominates uniform cas reimbursement. In fact, rst of a, in reaity inpatient care are ess frequent tan outpatient ones: p <p 2. Second, I assumed tat eterogeneity is ower in te case of serious iness: A simpe way to represent tis situation it is to normaize µ 2 = µ =1suc tat µ i 1 measures eterogeneity. Ten it is µ 2 µ 1. In tis way, te eterogeneity structure gives even more weigt to not seriousy i consumers. As a consequence, a mix of types, giving more utiity to not seriousy i consumers, soud reac te iger eve of socia wefare. After te previous considerations we expect tat, wen eterogeneity is bidimensiona, te compete reimbursement metods ranking is as foows: treatment cost reimbursement º a mix of types º cas reimbursement º in-kind reimbursement. Notice tat tis is not a genera resut because it depends on a more assumption (p <p 2 ) and on a speci c normaization (µ 2 = µ =1): As we said in te introduction, it seems natura to expect tat te trade-o between in-kind and treatment cost reimbursement is a ected by te degree of eterogeneity (see gure 1). Proposition 2 sows tat tis is not te case. Te reason is tat treatment cost makes use of two more instruments and imposes no constraints on consumption. Tis resut is stricty reated to a crucia assumption of te mode: te eat status is observabe. Tis assumption impies tat te socia panner can aways use monetary transfers contingent on te eat status. 991

24 6. Sef-seecting aocations Deaing wit sef-seecting aocations, te previous ranking of reimbursement metods may be substantiay a ected. Wit tis respect, a rst important fact is tat now te constraint on treatment quantity imposed by in-kind reimbursement becomes a usefu instrument. Directy providing (indivisibe) in-kind eat services, te socia panner can observe te treatment consumed by i individuas, as we sa see. As bot te eat status (captured by consumers margina abor productivity w i ) and te pre-tax revenue (w i L i ) are observabe, abor suppy is known and is aways part of te contracts proposed by te socia panner to consumers. 26 Consider now treatment quantity. Except wen reimbursement is in-kind, treatment quantity is not observabe by te socia panner and mimicking on eat care consumption arises. 27 Obviousy, if treatment consumption X i is not observabe, ten consumer s aggregate consumption (C i = w i L i +R TC i ( i )qx i ) is not observabe too. Wit in-kind reimbursement, on te contrary, treatment quantity is observabe. Tis foows from te interpretation of in-kind transfer X ¹ i as an indivisibe package of care (see section ). Te socia panner s programs addressed in tis section are standard cases of mecanism design under adverse seection. Looking for te optima mecanism of eac reimbursement sceme, I wi ten empoy te we known Reveation Principe 28. Hence, I wi study direct mecanisms in wic consumers (trutfuy) announce teir type µ and te insurer o ers an aocation wic speci es a te reevant variabes in te contractua reationsip wit consumers. Notice tat for eac reimbursement metod we sa ook for te socia panner s optima aocations attainabe witin eac reimbursement sceme. Tis means tat, as we sa make cearer, te avaiabe reimbursement pans wi 26 See te concusions for an extension of te mode, in ine wit te Optima Taxation iterature (Stigitz (1987), among oters), wit asymmetric information (aso) on te eat status. 27 In te case of cas reimbursement it is evident tat insurance as no way to contro consumers treatment purcase. On te contrary, in te case of treatment cost reimbursement, consumers present te pysician s invoice and ten receive reimbursement from te insurance. As a consequence, in tis case we can say tat treatment quantity is ex-post observabe. However, in te rea word tis information is generay not used by te insurance (wic, in fact, impements inear commodity tax on treatment, represented in tis mode by te parameter i ). For tis reason in te mode treatment quantity is not observabe wit treatment cost reimbursement as we. I wi come back to tese considerations ater (see note (1)). 28 Myerson (1979), among oters. 992

25 not necessariy aow to obtain te second-best optimum. Te reason is obviousy tat some of tem are instrument-constrained. To ave consumers trutfuy report teir type, te socia panner as to maximize is obective function under (aso) te incentive compatibiity constraints. As it as been sown at te end of section, te ow-type consumers are te mimickers. Standard mecanism design tecniques wit discrete types (see Fundenberg and Tiroe (1991), pages ) sow tat it is optima to make te mimikers incentive compatibiity constraints binding tus impying tat a te oter constraints are satis ed. 29 As a consequence, to recover te separating aocations I wi add two incentive constraints to te socia panner s program: one for te ow-type not seriousy i consumers and anoter for te ow-type seriousy i consumers Cas reimbursement Separating cas reimbursement is caracterized by four monetary transfers P; R2;R and by consumers abor suppies (L i ); = ; ; i =1; 2. In particuar insurance o ers te foowing contracts: (P; L 1 ) for eaty consumers, te coupe of contracts L 2;R2 and L 2 ;R2 respectivey for ow and ig-type not seriousy i consumers and te uniform transfer R for bot seriousy i consumers types as, in tis case, te ony variabe te socia panner can contro is te monetary transfers and ten no separation can be obtained in te seriousy i state. Notice tat tis means te socia panner cannot discriminate between te two seriousy i consumers type and is obiged to o er a pooing contract. Tus, ony ow-type not seriousy i incentive constraint appears in te insurance program. Wit respect to treatment, not seriousy and seriousy i consumers wi respectivey coose treatment quantity according to equations (6.1) and (6.2): X 2 : µ 2Á 0 2 (X 2 ) qu 0 w 2 L 2 + R 2 qx 2 =0 (6.1) X : µ Á0 (X ) qu 0 R qx =0 (6.2) wie te mimicker wi coose te preferred quantity according to te foowing equation: X 2 : µ 2 Á0 2 (X 2 ) qu 0 w 2 L 2 + R 2 qx 2 =0 (6.) 29 A forma proof of tis resut is standard and ten omitted. A compete proof is avaiabe from te autor. 99

26 Te socia panner program ten is: 8 Max p 1 [u(w 1 L 1 P ) v(l 1 )] + P;R 2 ;R X;L i +p 2 ¹ u w2 L 2 + R 2 qx 2 µ 2;L 2;R 2 H2 + µ 2Á 2 X 2 =; >< v L X 2 + p ¹ u R qx H + µ Á X >: =; s:t: : p 1 P = p 2 ¹ R 2 + ¹ R 2 + p R ( ) µ 2;L 2;R 2 + u w 2 L 2 + R2 qx 2 µ 2 ;L 2;R2 + µ 2 Á 2 X 2 µ 2 ;L 2;R 2 v(l 2 ) u w 2 L 2 + R2 qx 2 µ 2 ;L 2;R2 + µ 2 Á 2 X 2 µ 2 ;L 2;R2 v(l 2 ) ( ) were 6= 0 and 0 are respectivey te budget constraint Lagrange mutipier and te incentive constraint Kun Tucker mutipier. >From FOCs wit respect to P and L 1 one respectivey nds: u 0 (C 1 ) =0 (6.4) w 1 u 0 (C 1 ) v 0 (L 1 )=0 (6.5) wic impy >0. Moreover, as we expected, eaty consumer s aocation is suc tat margina bene t equas margina cost of abor suppy. >From FOC wit respect to R 2 one nds: suc tat, using equation (6.4), C 2 >C 1. >From FOC wit respect to L 2 it foows: p 2 ¹ + u 0 C2 =0 (6.6) p 2 ¹ w 2 u 0 C 2 v L 2 =0 (6.7) and tere is no distortion for te ow-type not seriousy i consumer. >From FOC wit respect to R 2 one nds: u 0 C 2 u 0 C2 =0 (6.8) p 2 ¹ were C 2 is te mimicker s aggregate consumption. From equations (6.4) and (6.8) it foows tat C 2 <C

27 >From FOC wit respect to L 2 one nds: w 2 u 0 C2 v L 2 w 2 u =0 (6.9) 0 C2 v L 2 p 2 ¹ Substituting (6.8) in (6.9) and rearranging one nds: p 2 ¹ v 0 L 2 w2 =0 (6.10) p 2 ¹ Suc tat it must be p 2 ¹ >0. Moreover, using (6.6) and (6.7), (6.10) sows tat L 2 <L 2: Concerning ig-type distortion, for te (6.) te mimicker wi coose X2 <X2 ; ten C2 >C2 : As a consequence, (6.9) does not contradict tat p 2 ¹ >0ony if w 2 u 0 C2 v L 2 <w2 u 0 C2 v L 2 < 0, and ten w 2 u 0 C2 <v L 2 : Tis means tat ig-type not seriousy i consumer is forced to suppy too muc abor and to under-consume (wit respect to aggregate consumption). Finay, from te FOC wit respect to R one nds te same resut obtained for uniform cas reimbursement: E [u 0 (C )] = u 0 (C 1 ) Hig-type consumers, aving a iger preference for treatment, coose an iger treatment quantity wit respect to ow-type consumers. Seriousy i consumers pooing aocation in te case of cas reimbursement is represented in gure 4. C U U C C X X X Figure 4: seriousy i consumers aocation wit cas reimbursement. 995

28 Note tat te two point in gure 4 are incentive compatibe (no type prefers te consumption bunde of te oter type) but, obviousy, no one of te two incentive constraints is binding (reca tat in te seriousy i state no separation can be obtained because a unique instrument can be used, te monetary transfer). Tis anticipates tat cas wi be easiy dominated by oter reimbursement scemes. Te foowing proposition summarizes te previous resuts concerning te separating cas aocation: Proposition 5 Te optima cas sef-seecting aocation is suc tat: contracts L 2 ;R 2 and L 2 ;R 2 verify L 2 <L 2 and C2 >C2 : Tere is no distortion for ow-type consumer. On te contrary ig type consumer is forced to suppy too muc abor and to consume too ess aggregate consumption ( w 2 u 0 (C2 ) <v 0 (L 2)). Bot seriousy i types receive a monetary transfer R and coose treatment and aggregate consumption suc tat X <X and C >C : 6.2. In-kind reimbursement In-kind reimbursement is caracterized by ve monetary transfers (P; R2;R ); by consumers abor suppies (L i ) and by te transfers X ¹ i ;i=2; ; = ;. AsIanticipated in section, it is reasonabe to assume tat te in-kind transfer X ¹ i is an indivisibe package of care 0 suc tat te socia panner can observe treatment consumption. As a consequence, wit in-kind reimbursement, aggregate consumption, treatment quantity and abor suppy are a observabe. Te contracts proposed in te tree states ten are (C 1 ;L 1 ) ; (C2;X 2;L 2); (C2 ;X2 ;L 2); (C;X ) and (C ;X ). It is interesting to notice tat in-kind represents te unconstrained direct mecanism in te sense tat, given te agent s type announcement, a te reevant variabes are cosen by te socia panner. As a consequence we can anticipate tat te in-kind optima aocation corresponds to te aocation wic weaky dominates te oters. 0 Letting consumers coose te preferred treatment quantity under te constraints X i ¹X i ; te in-kind reimbursement sef-seecting aocation woud be very simiar to te cas reimbursement one. Te ony di erence woud be tat wit IK C i is observabe. 996