To Wait or to Pay for Medical Treatment? Restraining ex-post Moral Hazard in Health Insurance

Transcription

1 Jaob Univerity Bremen To Wait or to Pay for Medial Treatment? Retraining ex-pot Moral Hazard in Health Inurane Stefan Felder Prioriierung in der Medizin FOR 655 Nr. 11 / 008 Campu Ring Bremen Germany FOR 655 Working Paper erve to dieminate the reearh reult of work in progre prior to publiation to enourage aademi debate. Copyright remain with the author.

2 ie Reihe Prioriierung in der Medizin umfat Arbeit- und Forhungberihte der FG Forhergruppe FOR655 Prioriierung in der Medizin: eine theoretihe und empirihe Analye unter beonderer Berükihtigung der Geetzlihen Krankenveriherung (GKV). ie Berihte und weitere Informationen zu der Forhergruppe können abgerufen, werden unter: oder The erie Prioriierung in der Medizin onit of working paper and reearh report of the FG (euthe Forhunggemeinhaft, i.e., German Reearh Foundation) Reearh Group FOR655 Prioriierung in der Medizin: eine theoretihe und empirihe Analye unter beonderer Berükihtigung der Geetzlihen Krankenveriherung (GKV).(Prioritizing in Mediine: A Theoretial and Empirial Analyi in Conideration of the Publi Health Inurane Sytem) Report and further information an be found at or Impreum: Campu Ring Bremen Germany ISSN

3 To Wait or to Pay for Medial Treatment? Retraining ex-pot Moral Hazard in Health Inurane Stefan Felder Otto-von-Guerike Univerity Magdeburg We explore the hierarhy of two intrument, waiting time and oinurane for medial treatment, for optimally olving the tradeoff between the eonomi gain from rik haring and the loe from moral hazard. We how that the optimal waiting time i zero, given that the oinurane rate i optimally et. JEL Index: 8, H1, I18 Keyword: waiting time, oinurane, moral hazard, optimal inurane ontrat 1. Introdution Mot health inurane plan do not ue indemnity payment that depend on health tatu, but reimbure the ot of the are atually provided to onumer. The reaon i that the health tatu of a peron an only be imperfetly oberved. When inurane reimbure health are expene, it ubidize the prie of health are at the margin, giving rie to a dead-weight lo. An optimal inurane ontrat thu involve opayment by the onumer. Coinurane olve the tradeoff between rik haring and the inentive to onume inreaed medial are. Prof. r. Stefan Felder Intitute of Soial Mediine and Health Eonomi (ISMHE) Leipziger Str Magdeburg phone: tefan.felder@imhe.de FOR655 Nr. 11 / 008

4 To Wait or to Pay for Medial Treatment? Uer harge apply in oial health inurane ytem in many ountrie. Some ervie, uh a dental are, may be exluded from the oial inurane pakage, and dedutible and opayment rate are ued to retrain the demand for health ervie. Countrie with a national health ervie, however, do not generally employ oinurane to urb demand for health are. They intead ue waiting time for thi purpoe. There i a vat literature analying waiting time and waiting lit in the proviion of health are ervie (for an overview ee Culli, Jone and Propper, 000). Linday and Feigenbaum (1984) develop a model in whih waiting lit queue funtion a a rationing devie. Waiting time matter beaue the value of health are deay the longer treatment i potponed after the diagnoi. Iveron (1993) outline a upply ide model in whih waiting lit for inpatient are are the reult of a politial bargaining proe over reoure. Iveron (1997) and Olivella (00) explore the effet of a private etor on waiting in the national health are ytem. Another trand of literature explore the ope of uing waiting a a method of rationing health are in order to meet ditributional objetive. Hoel and Sæther (003) how that queuing in the publi health are ytem might entie high-inome peron to opt out and buy health are privately while till partiipating in the finaning of the publi ytem. If uffiient weight i given to equity, welfare an be improved by uing the elf-eletion mehanim. In the abene of ditributional motive, however, it i never optimal to have a poitive waiting time for health treatment. Smith (005) analyze uer harge for different treatment under an exogenou health are budget. For a given treatment, all onumer have the ame benefit, but, a the onumer wealth differ and marginal utility of wealth i dereaing, oinurane redue the demand for health ervie by diouraging poorer onumer from eeking treatment. In line with optimal taxation rule, the oinurane rate for a treatment i inverely related to the prie-elatiity of demand. Gravelle and Siiliani (006) extend the Smith model by introduing heterogeneity with repet to the onumer benefit from treatment and a publi health are ytem whih ue queuing to ration demand. Prioritization among treatment again follow an elatiity rule: the more elati demand i with repet to the waiting time, the longer the optimal wait i. Adding uer harge to the model, Gravelle and Siiliani demontrate that Ramey wait and prie apply imultaneouly. However, they do not hek whether the optimum require the ue of both intrument. In thi paper, abtrating from equity onern, we addre the omplementary appliation of waiting time and oinurane uing a onumer health model, whih inlude health and onumption a argument of the utility funtion and a linear oinurane heme deigned to retrit moral hazard. We add the invetment apet of health, determining the onumer inome in the proe, and waiting time, whih redue inome ine an untreated ill peron annot work. We how that the optimal oinurane rate i poitive (due to moral hazard) and maller than one (due to the onumer rik averion). For a given oinurane rate, the optimal waiting time may or may not be poitive. However, if the oinurane rate i optimally et, it i not optimal to have a poitive waiting time for treatment. FOR655 Nr. 11 / 008

5 Stefan Felder. The model We onider a one-period world in whih the repreentative onumer health tatu h in eah tate of the world depend on a ompoite of health ervie, denoted by m, and a random variable θ > 0 whih repreent exogenou hok to the onumer health tatu: h = m θ. (1) The health tatu determine the onumer inome in the labour market. We aume a prodution funtion with the following propertie: f = f ( h ); f ' > 0; f '' < 0; f '(0) =. () Waiting for treatment hinder the onumer from working and redue her inome by a fator τ, the waiting time, with 0 τ < 1. Sine θ i not obervable to the inurer, the amount the inurer pay to the onumer in eah tate intead depend on her purhae of health ervie m. Aume that the inurer harge a ontant oinurane rate λ, with 0 λ 1. Moreover, we retrit the onumer total expoure to rik by auming λ + τ 1. With an atuarially fair inurane premium, the onumer onumption flow in tate i retrited by the budget ontraint: ( 1 ) ( ) ( 1 ) = τ f h λm λ π m, (3) where unit are hoen in uh a way that all ommoditie ell at a prie of unity λ π m i the inurane premium with π a tate-peifi 1 and ( ) probabilitie ( π = 1). The inurane premium may alo repreent a lump-um tax to finane health inurane in a publi ytem where the tate i the only inurer. The onumer utility depend on onumption and health tatu h. For a given value of θ the onumer utility i given by u We aume ( θ ) u(, m θ ) =. (4) m mm m u, u > 0; u, u < 0 ; 0 m u = ; u ( 0, m θ ), u (,0) =. (5) The onumer fae a two-tep optimization problem. After realization of θ, he deide on a utility maximizing onumption and demand for health ervie m FOR655 Nr. 11 / 008 3

6 To Wait or to Pay for Medial Treatment? with a given waiting time τ and oinurane rate λ. Ex ante he olve for the optimal τ and λ whih maximize expeted utility, taking into aount her ex-pot deiion on and m. The firt-order ondition for the ex-pot maximum in tate i (, θ ) (, θ) '( θ)( 1 τ) λ (, θ) u m m u m f m u m + =. (6) emand for health are i optimal when the marginal utility of health are equal it marginal ot. The marginal benefit from health are arie either from the higher utility of health or from a higher inome, whih i tantamount to the onumption model and the invetment model of health are demand (Gromann, 197). The onumer net prie of health are demand i λ f ' h 1 τ > 0. ( )( ) From (6) and the budget ontraint (3) with a fixed premium, we find: ( f ( )) h λu λ '1 τ = < 0, (7) θ ( λ) u ( λ f ( τ) ) 1 ' 1 = > 0 and (8) θ ( f ( ))( ) λu λ '1 τ λ τ = < 0, (9) θ with ( λ ( τ) ) ( τ) = u hh + u f '1 + 1 f '' u < 0. The onumer will hooe more health ervie when the hok to the health tatu i large. emand for health and onumption will dereae and marginal utilitie of health and onumption will in turn inreae with a larger health hok. Similarly, we derive Slutky equation for the effet of a repetive hange in λ and τ on the demand for health ervie: ( λ ( τ) ) u u m f '1 = < 0, (10) ( λ ( τ) ) ' u f u f f ' 1 = < 0. (11) τ 4 FOR655 Nr. 11 / 008

7 Stefan Felder To ompare the relative ize of the ubtitution and inome effet, we alulate: ( )( ) ( ) λ ( τ) u 1 f ' u f ' 1 m f = τ. (1) The ubtitution effet of an inreae in oinurane i larger than that of an inreae in waiting time provided that f ' < 1 (whih follow from the firt-order m ondition (6) given that λ + τ 1 and u, u > 0 ). When the health tatu ha no effet on inome (i. e. f ' = 0 ), an inreae in waiting time indue an inome effet only. The inome effet of oinurane will be maller when it over a maller bae than the waiting time, i.e. m < f. A marginal inreae in the oinurane rate then involve a maller monetary rik for the onumer than a marginal inreae in the waiting time. 3. The optimal inurane ontrat The optimal ontrat (, ) λ τ maximizing the onumer expeted (indiret) utility reult from olving the program max EV ( λ, τ) = πu (, m θ) (13) λτ, ubjet to (3), 0 λ 1 and 0 τ < 1. For a marginal hange in λ, the envelope theorem yield: EV = + πu (, m θ) m πm ( 1 λ) π. (14) An inreae in the oinurane rate ha two oppoite effet on the onumer expeted utility. On the one hand, it lower expeted utility ine the onumer opayment inreae at the given health are demand m. On the other hand, it inreae expeted utility beaue the premium dereae (the lat two term in the braket). Abtrating from a hange in m (the lat term), expeted utility dereae with an inreae in the oinurane rate when the onumer i rik avere. With repet to waiting time, we find: EV τ = πu (, m θ) f ( m θ) ( 1 λ) π. (15) τ FOR655 Nr. 11 / 008 5

8 To Wait or to Pay for Medial Treatment? An inreae in τ dereae inome but lower the premium to the extent that the demand for health ervie dereae. Propoition 1 Full inurane ( λ = 0 ) and no inurane overage ( λ = 1) are both not optimal. Proof i) With 0 λ =, the optimal alloation i governed by '1 ( τ ) u u m f '1 ( τ ) λ = u f u m =, and (10) beome = < 0. Sine the onumer hooe the λ 0 ame onumption level aro the tate of the world, (14) redue to: EV = u π > 0. λ = 0 λ = 0 ii) 1 With λ = 1 and 0 τ < 1, (14) yield EV = π u π m m. The λ = 1 differene in the braket i poitive for mall value of m and negative for large value of m. Marginal utility of onumption i negatively orrelated to the value of m (ee (8) and (9)). Let m t be the maximal value for whih the u, m θ > u, m θ term in the braket i poitive. We then have ( ) ( t t t) for all m > mt and u (, m θ) < u ( t, mt θt) for all m mt EV π = 1, the right hand ide of thi inequality equal zero and thu <. Then, πu (, m θ) πm m < u ( t, mt θt) π πm m. From λ λ = 1 < 0. For an interior olution, λ ( 0,1), we need to hek the eond derivative. From (14) we obtain: ( λ ) A π u A u with ( ) A = πm m 1 λ π EV = +.(16) 1 The line of proof follow Breyer et al. (008). 6 FOR655 Nr. 11 / 008

9 Stefan Felder A m m π and from the budget ontraint (3) and We find = ( 1 λ) the firt-order ondition (6) (16) yield: ( ) ( ) u A = + u m. Inerting thee expreion into m EV u m = + + πu ( 1 π u λ) π u ( ). m Given u, u > 0, u < 0 λ < 0 and λ 0 m λ > 0 then i a uffiient but not neeary ondition for the eond-order ondition ( ( ) EV λ < 0 ) to hold. <, ( ) Auming that the eond-order ondition for an optimal interior olution of the oinurane rate i fulfilled, we an tate: Propoition For f m, the optimal waiting time i zero. Proof Inerting the ondition for an optimal λ ( EV λ = 0 ) from (14) into (15) yield: For EV = + + τ τ πu f m πm ( 1 λ) π. (17) f m, the differene in the braket i poitive (ee (1) for u > 0, f ' < 1, 0 λ > f '1 τ and < 0 ). It follow then that f m i uffiient for u <, ( ) EV τ < 0. If f m, oinurane not only i more effetive in retraining moral hazard than the waiting time. It alo involve a lower monetary rik, due to a maller inome effet. If the hok i very high, health are ot may exeed inome and the inome effet of a marginal inreae in the oinurane rate would be larger than that of a marginal inreae in the waiting time. Uing oinurane exluively may till be optimal. Only if the ditribution of θ were heavily tailed FOR655 Nr. 11 / 008 7

10 To Wait or to Pay for Medial Treatment? uh that exeive rik prevailed would an optimal heme ombine oinurane with a poitive waiting time. If the oinurane rate were not optimally hoen, a poitive waiting time might alo be appropriate. Tranforming (15) yield EV ( 1 λ) τ = πu f 1 + π. τ τ f τ m If the elatiity of health demand i omparably high, expeted utility varie poitively with τ. An inreae in waiting time will then uffiiently redue health demand o that the dereae in the inurane premium exeed the inome lo. 4. Conluion National health ervie ue queuing a a rationing devie like entrally planned eonomie do (Kornai, 1980). Soial health inurane ytem depend more on uer harge to ignal the arity of reoure to the onumer of health are. In thi paper, we demontrate that an optimized deign doe not generally ue a poitive waiting time for medial treatment. The reaon i that oinurane uffiiently olve the tradeoff between rik-haring and moral hazard, thereby rendering wait a redundant intrument. Still, a poitive waiting time might be optimal if the hoie of the oinurane rate i retrited or if health tatu i ubjet to an exeive rik. Thi paper retrit the analyi to a ontant oinurane rate in the tradition of Zekhauer (1970) and Feldtein and Friedman (1977), among other. With a non-linear heme, the optimal oinurane rate tend to be maller for high health are expene (Blomquit, 1997). Sine for high expene the optimal waiting time might be poitive, it would be intereting to analyze a non-linear oinurane heme ombined with waiting time, an iue we leave to future reearh. If an interior olution for waiting time exited, it ould be hown from (14) and (15) that π u μ τ τ π u f = λ π u μ m π m m with μτ and μ λ a the elatiitie of health are demand with repet to waiting time and the oinurane rate, repetively. Thi equation orrepond to Eq. (40) in Gravelle and Siiliani (006), who laimed Ramey prie and wait for the optimal prioritiation of demand for medial treatment. 8 FOR655 Nr. 11 / 008

11 Stefan Felder Aknowledgment Thi paper originated from my diuion of Siiliani and Gravelle (006) at the 7 th European Health Eonomi Workhop in Contane in May 006. I thank Peter Zweifel, Univerity of Zurih, and Matthia Kifmann, Univerity of Augburg, for helpful omment. Suggetion from three anonymou referee have improved the truture of the paper oniderably. 5. Referene Blomquit, Å, Optimal non-linear health inurane. Journal of Health Eonomi 16, Breyer, F., M. Kifmann and P. Zweifel, 008. Health Eonomi, Springer, Berlin (forthoming). Culli, J.R., P.R. Jone and C. Propper, 000. Waiting lit and medial treatment. In Culyer, A.J. and J.P. Newhoue (Ed.). Handbook of Health Eonomi, Elevier, Amterdam, Volume 1B, pp Feldtein, M. and B. Friedman, Tax ubidie, the rational demand for inurane and the health are rii. Journal of Publi Eonomi 7, Hoel, M. and E.M. Sæther, 003. Publi health are with waiting time: the role of upplementary inurane. Journal of Health Eonomi, Gravelle, H. and L. Siiliani, 006. Ramey wait and prie: prioritization when health are i rationed by waiting. Centre for Health Eonomi, Univerity of York, April 006, England. Groman, M., 197. On the onept of health apital and the demand for health. Journal of Politial Eonomy 80, Iveren, T., A theory of hopital waiting lit. Journal of Health Eonomi 1, Iveren, T., The effet of a private etor on the waiting time on a national health treatment. Journal of Health Eonomi 16, Kornai, J., Eonomi of hortage. North-Holland, Amterdam. Linday, C.M. and B. Feigenbaum, Rationing of Waiting Lit. Amerian Eonomi Review 74, Olivella, P., 00. Shifting publi-health etor waiting lit to the private etor. European Journal of Politial Eonomy 19, FOR655 Nr. 11 / 008 9

12 To Wait or to Pay for Medial Treatment? Smith, P.C., 005. Uer harge and priority etting in health are: balaning equity and effiieny. Journal of Health Eonomi 4, Zekhauer, R., Medial inurane: a ae tudy of the tradeoff between rik preading and appropriate inentive, Journal 10 FOR655 Nr. 11 / 008

13 Working Paper Serie FOR Hartmut Kliemt: Priority etting in the age of genomi, eember 007 (1). Marlie Ahlert: If not only number ount alloation of equal hane, eember 007 () 3. Stefan Felder: The variane of length of tay and the optimal RG outlier payment, eember 007 (3) 4. Jeannette Winkelhage, Adele iederih, Simone Heil, Petra Lietz, Felix Shmitz-Juten und Margrit Shreier: Qualitative Stakeholder-Interview: Entwiklung eine Interviewleitfaden zur Erfaung von Prioritäten in der medizinihen Verorgung, eember 007 (4) 5. Antje Kökeritz: A ooperative bargaining model for two group of patient, January 008 (1) 6. Marlie Ahlert and Hartmut Kliemt: Neeary and uffiient ondition to make the number ount, January 008 () 7. Stefan Felder and Andrea Werblow: o the age profile of health are expenditure really teepen over time? New evidene from Swi Canton, February 008 (3) 8. Marlie Ahlert, Wolfgang Granigg, Gertrud Greif-Higer, Hartmut Kliemt und Gerd Otto: Prioritätänderungen in der Allokation potmortaler Spender-Lebern Grundätzlihe und aktuelle Fragen, February 008 (4) 9. Marlie Ahlert, Stefan Felder and Bodo Vogt: How eonomit and phyiian trade off effiieny and equity in medially and neutrally framed alloation problem, February 008 (5) 10. Adele iederih and Hartmut Kliemt, Publi health are prioritie at the poll a note, Marh 008 (6) 11. Stefan Felder, To wait or to pay for medial treatment? Retraining ex-pot moral hazard in health inurane, April 008 (7)