On Privacy, Adverse Selection and Genetic Databases

Transcription

1 On Prvacy, Adverse Selecton and Genetc Databases Completed Research Word Count: 4991 Track: Socety, Polcy, & Regulaton Abstract Stored genetc samples wll provde more nformaton n the future on medcal rsks due to better genetc technology, than was magned when the samples were orgnally collected. Ths poses a threat to ndvdual prvacy due to secondary use of the collected genetc nformaton. In ths paper we study the mpact of genetc testng on health nsurance markets. Consumers who have taken a genetc test are assumed to be nformed about ther rsk of a genetc dsease. The nformed low rsk consumers have an ncentve to reveal genetc nformaton to the nsurer so as not to be clubbed wth the hgh rsk types. We characterze the exstence and nature of nsurance contracts when ndvduals can reveal genetc nformaton to nsurers but where revelaton of genetc nformaton s assocated wth a loss of prvacy. We then examne the welfare mplcatons of dfferent polcy proposals regardng genetc testng, wth the decson of the consumer to take a genetc test and to reveal genetc nformaton, beng endogenous.

2 1. Introducton Government and prvate medcal nsttutons are ncreasngly usng electronc databases to manage ndvdual medcal records (ehrman 000). An example of such databases s the vast amounts of data created by the uman Genome Project. Genetc nformaton stored n these databases can be used to estmate the lkelhood that an ndvdual wll suffer from a varety of condtons n the future (Roche et al 1996). A slew of new genetc methods, tests and knowledge promse to revolutonze medcne now that scentsts have cracked the human genome map. But those same advances could land health-care provders and nsurance companes n a quagmre of stcky ssues, ncludng conflcts wth patent prvacy (Wred.com 000). The ethcal, legal and socal mplcaton (ESI), of mappng and sequencng the human genome, has dentfed prvacy as one of the hgh-prorty research areas (Mesln et al 1997). Genetc tests attempt to detect the presence of genes that are assocated wth dsease or predspose those who nhert the genes to dsease (Mesln et al 1997). Genetc testng exacerbates the adverse selecton exstng n nsurance markets f the results of the test are not made accessble to the nsurers. owever the nsurers request for a level playng feld contrasts wth efforts of consumer groups to ncrease the prvacy protecton of genetc nformaton (Subramanam et al 1999). Snce the DNA molecule s stable once removed from a person s body and stored, t can become the source of an ncreasng amount of nformaton as more s learned about how to read the genetc code (Troy 1997). In other words stored genetc samples wll provde more genetc nformaton than was magned when the samples were orgnally collected. Threats to ndvdual prvacy may therefore be presented due to secondary use of the collected nformaton (Roche et al 1996). Because of the nfluence that genetc knowledge can have on an ndvdual s rght to prvacy, the Genetc Prvacy Act (GPA) suggests that the acquston, use, and dsposton of genetc knowledge s best placed n the hands of the ndvdual. 1

3 In ths paper we study the mpact of genetc testng on health nsurance markets. Consumers who have taken a genetc test are assumed to be nformed about ther rsk of a genetc dsease. The nformed low rsk consumers have an ncentve to reveal genetc nformaton to the nsurer so as not to be clubbed wth the hgh rsk types. We characterze the exstence and nature of nsurance contracts when ndvduals can reveal genetc nformaton to nsurers but where revelaton of genetc nformaton s assocated wth a loss of prvacy. We then examne the welfare mplcatons of dfferent polcy proposals regardng genetc testng, wth the decson of the consumer to take a genetc test and to reveal genetc nformaton, beng endogenous. The rest of the paper s organzed as follows: Secton covers the background lterature on prvacy and genetc testng. Secton 3 covers the assumptons, model framework and man results. Implcatons of the results obtaned are summarzed, and lmtatons and drectons for future research are dscussed n Secton 4.. Background Prvacy s defned as the moral clam of the ndvduals to be left alone and to control the flow of nformaton about themselves(coase 1960; Westn 1967). Revealng genetc nformaton to nsurers consttutes a loss of prvacy because of the nherent loss n control over the use of ths nformaton after t s released. The loss of prvacy could be the fear of secondary use of the genetc nformaton revealed or the fear of dscrmnaton at a later date due to some nformaton present n the genetc nformaton whch the consumer s not aware of currently. It s well known that nsurance markets are characterzed by adverse selecton (see Rothschld and Stgltz 1976) - asymmetry n nformaton on rsk types between the nformed or partally nformed consumers and unnformed nsurers has an adverse mpact on the profts of nsurers. Genetc testng ncreases the asymmetry n nformaton f the results are avalable only to consumers, thus makng the adverse selecton more acute. There s a lot of debate on how and whether the results of the genetc tests should be used (see Tabarrok 1994). There are four broad

4 polcy proposals regardng the use of genetc tests for nsurance purposes. Under the frst proposal, nsurers do not nqure, and are not permtted to nqure whether applcants have been tested. In the second proposal, genetc nformaton s revealed to the nsurer only wth the consent of the consumer. The thrd polcy alternatve s to permt nsurers to nqure whether applcants have been tested and to get genetc nformaton/test results wth the consent of the consumer. The fourth polcy alternatve s to requre or permt nsurers to requre that applcants be tested, and allow nsurers to use the test results-.e. mandatory release of genetc nformaton for nsurance purposes. (Doherty and Thstle 1996). rshlefer (1971) dstngushes between prvate nformaton (avalable only to a sngle ndvdual) and publc nformaton (avalable to everyone). Our work s closer to the prvate nformaton stream of lterature Cremer and Khall (199), Crocker and Snow (199) and Doherty and Thstle (1996). Agents receve a prvate sgnal n our model and they choose whether to make the sgnal publc or not 1. Crocker and Snow (199) have shown that f the nsurers know whether the consumers are nformed or unnformed about ther rsk type, then the prvate value of nformaton on rsk type for the consumer s negatve. Doherty and Thstle (1996) have looked at the case where nsurers cannot dstngush between nformed and unnformed agents and the case where nformaton on rsk type can be concealed or revealed to the nsurers. In ther settng unnformed can take a test (both costless and costly) to get nformed about ther rsk type and can reveal nformaton to nsurers are zero cost. The polcy holders knowledge of ther rsk type s therefore endogenous. They characterze the nature of equlbrum n all these cases and fnd prmarly that wth costly nformaton some unnformed may choose to reman unnformed. They also examne the welfare mplcatons of the endogenous nformaton model for publc polcy regardng genetc testng. In our settng, revelaton of nformaton has postve costs assocated wth t n contrast to Doherty and Thstle (1996). We characterze ths loss to be the loss of prvacy the fear of 1 We also consder a purely publc sgnal n the last secton. 3

5 dscrmnaton because of beng classfed as a hgh-rsk type n a later perod, due to better genetc tests. All genetc testng polcy proposals except the frst one where nsurers are nether permtted to nqure nor use genetc nformaton are affected f the loss of prvacy s taken nto account. As far as the mpact on consent law (Proposal ), we need not consder the prvacy concerns of the hgh rsk type snce presumably only the low rsk types are affected. owever n the case of mandatory testng and revelaton of nformaton, the loss n prvacy of the hgh rsk type s also taken nto account. 3. Model The nsurance market settng s compettve wth rsk neutral nsurers. There s a contnuum of rsk-averse consumers, dentcal n all respects except for accdent probabltes. They possess dentcal von Neumann-Morgenstern state ndependent utlty functons U (.), whch are twce contnuously dfferentable and strctly concave. Ther wealth s W state 1 (no loss) and W D n the loss state, where W > D. π ; = are the probabltes of loss of the hgh and low, rsk consumers. We also assume that π >, 0 π, π < 1, and that these probabltes are π < out of the agent s control, so that no moral hazard problem arses. We assume that each agent knows her rsk type. Informed consumers n our settng are consumers who have taken a genetc test n a prevous perod for some non-nsurance related reason e.g. job screenng. The proporton of low rsk (hgh rsk) consumers n the economy s θ ( θ ). If agents do not purchase nsurance they obtan expected utlty, V 0, π ) for =, ; where ( V ( 0, π ) = π U ( W D) + (1 π ) U ( W ); =., The nsurance contract company specfes the premum P and the net ndemnty I 3 pad ncase of a loss. The use of consurance rates and/or deductbles s not permtted. We use the notaton Moral hazard problem arses due to the fact that nsurers cannot observe the consumers actons how much care she takes to avod a loss. By assumng that consumers cannot affect the probablty of an accdent by ther actons we assume away the moral hazard. 3 Net Indemnty s the payment made to the consumer n case of loss mnus the premum 4

6 δ = P, I } to denote the sngle perod nsurance contract offered to type =,. The { expected utlty of a consumer wth, the probablty of loss under the nsurance polcy s V( δ, π ) = πu( W D+ I ) + (1 π ) U( W P); =, π δ If consumer releases genetc nformaton to the nsurer then the consumer faces a loss due to loss of prvacy. Ths loss beng a non-monetary loss s separable from the utlty from the nsurance contract. The net utlty s V ( δ, π ) Intally let us assume that consumers cannot reveal nformaton to the nsurers,.e. = 0. Consumers choose the polcy that maxmzes expected utlty. Competton drves down nsurance premums to an actuarally far rate. In the frst best case (no asymmetry n nformaton on rsk types), both rsk types get complete nsurance. The contract n the frst best case s δ = { π D, (1 π ) D}; =, owever under asymmetrc nformaton (Consumers have a pror knowledge of rsk types whle nsurers don t), the frst best contract can t be acheved (Rothschld and Stgltz 1976). The second best contract satsfes the constrants V ( δ, π ) V ( δ, π ); = (, ), j, j V ( δ, π ) V (0, π ); =, The separatng contracts are { δ, δ } (see Fg. 1). The hgh rsk consumer get complete nsurance whle the low rsk agent get less than full nsurance. 5

7 We now assume that a proporton λ U 0 λ 1of the consumers does not have a pror < U knowledge of ts rsk type. Out of these unnformed consumers a proporton p (p ) are hgh rsk (low rsk). The proporton of nformed hgh rsk type (low rsk types) s λ (λ ), 0 λ, λ < 1 4. θ = λ + p λ θ = λ + p λ U U < All these proportons are assumed to be common knowledge. The pror probablty of loss for the unnformed s π U = =, p π. The tmng s as follows, the nsurers frst choose the set of contracts to offer based on ther belefs on the actons by the unnformed (take genetc test or not) and the nformed low rsk (reveal nformaton or not). The unnformed then decde whether to take a genetc test or not, and the nformed consumer decde whether to reveal nformaton to the nsurer or not. The choce at each node affects the payoffs of all agents. P W P U V 0 45 P V V U δ U δ δ " δ U δ " δ E W 1 Fg. 1. Insurance Contracts 4 Strct nequaltes are assumed here for smplcty. Thus there are always nformed hgh and low rsk consumers. 6

8 Insurance contracts for the dfferent rsk types under dfferent nformaton structures are shown n Fg. 1. W and W are the wealth n the no loss and loss state respectvely. The 45 1 lne s the complete nsurance lne equal wealth n both states. EP, EP U and EP are the far odds lne (zero proft for nsurer) for the hgh rsk, unnformed and the low rsk consumer. V, VU, and V are representatve ndfference curves for the hgh rsk, unnformed and the low rsk agents 5. Under full nformaton the contracts offered are ( δ, δ, ). Under asymmetrc U δ 0 nformaton, wth no unnformed the contracts are ( δ, δ ), whle wth unnformed consumers " " U, δ the contracts are ( δ, δ ). Wth asymmetrc nformaton, only the hgh rsk consumers get full nsurance. It s easy to see that the followng constrants should be satsfed. " " " V( δ, π ) > V( δ, π ) ; V( δ, π ) > V ( δ, π ) U " " " V( δ, π ) = V( δ, π ) ; V( δ, π ) > V ( δ, π ) U U U U U U " " V( δ, π ) = V( δ, π ); V( δ, π ) > V ( δ, π ) U V( δ, π ) = V( δ, π ); V( δ, π ) > V ( δ, π ) We consder two cases, one where the nsurers cannot observe whether the consumers are nformed about ther rsk types or not and the second where the nsurers can see the consumers nformaton status. Doherty and Thstle (1996) gve the example of a specfc dagnostc test e.g. an IV test where the consumers can get to know for sure whether they have tested postve or not. If the nsurers always get to know f the consumer has taken a test, then they know for sure that the consumer knows hs rsk type. In contrast to ths specfc test, physcal exams gve no such specfc result and so nsurers never get to observe the nformaton status of the consumers. Crocker and Snow (199) have looked at the observable nformaton status case whle Doherty 5 The slope of the ndfference curves of low rsk, unnformed and hgh rsk consumers at the full nsurance (1 π) (1 πu ) (1 π) lne are > > π π π U 7

9 and Thstle (1996) have looked at the unobservable nformaton status case. Both of these papers fal to address the ssue of prvacy of genetc nformaton. In our model we model the loss of prvacy due to revelaton of nformaton to nsurers and ths dfferentates our work form thers so from now on > 0. In secton 3.1 we consder the case where all consumers are nformed. In secton 3. we look at the case where the nformaton status s not drectly observable but there s an ncentve for consumers to reveal verfable nformaton (negatve test result or genetc nformaton that reveals that the consumer s a low rsk consumer) and testng s costly for the unnformed. The observable nformaton status case wth costly revelaton and costly testng s consdered n secton 3.3. The mandatory testng and revelaton of genetc nformaton s consdered n secton Unreported postves, verfable negatves and costly revelaton. We frst consder the case where all consumers are nformed about ther rsk type.e. λ = 0. Proposton 1 Assumng all consumers are nformed about ther rsk types, nsurers cannot observe rsk type and consumers can report verfable nformaton to the nsurers at non zero cost, the equlbrum contract are ( δ, δ ) and ( δ, δ). The equlbrum ( δ, δ ) only exsts f < V( δ, π) V( δ, π) and Pareto domnates (, ) f < V( δ, π) V( δ, π). Proof. See Appendx δ " " Denotng V( δ, π ) V ( δ, π ) = and V( δ, π ) V ( δ, π ) =. and can δ U be consdered to be dfferent levels of dsutltes due to loss of prvacy wth " <. The contract " ( δ, δ ) exsts only when <.e. for small dsutltes due to loss of prvacy. For " hgh losses due to revelaton of nformaton,, the second best contract ( δ, δ) s the only equlbrum 6. It should be noted that the second best contract s always an opton to the 6 The exstence of ths equlbrum requres the Rothschld-Stgltz condton of there beng a suffcent number of low rsk consumers. We assume for smplcty that ths condton s always satsfed where relevant. 8

10 nsurer but the frst best contract can only exst for small value of losses due to revelaton. If < then the contract ( δ, δ ) Pareto domnates ( δ, δ ). The followng conclusons can be drawn from proposton 1 " a) For 0 <, ( δ, δ ) s the Pareto optmal equlbrum " b) For <, both ( δ, δ ) and ( δ, δ are possble equlbrum. ( δ, δ) " c) For, s the only possble equlbrum. ) Now we consder the case λ U > 0. If we assume that testng s costless, then testng s a domnant strategy for the unnformed. All unnformed consumers would take the test and become nformed. Thus wth costless testng, there s no mpact on these results f we assume that there are some consumers who are unnformed. 3. Unreported postves, verfable negatves, costly revelaton, unnformed consumers and unobservable nformaton status. Unnformed consumers can take a costly test to know ther rsk type. Also by consent law they can choose who gets to see the results of the test. It s clear that only the low rsk agents have an ncentve to release nformaton to the nsurers. Proposton. Assume unnformed consumers can observe ther rsk type at cost c, nformed and unnformed consumers cannot be dstngushed, rsk type s not drectly observed by nsurers and consumers can report verfable nformaton to nsurers at non-zero cost. Denotng c p by c, four contracts are feasble: ) ( δ, δ ) f c < V (δ, π ) V( δ, π ) ) ( δ, δ ) f < V( δ, π ) V( δ, π ) c " " " ) ( δ, δ, δ ) f V( δ, π ) V ( δ, π ) c < < V( δ, π ) V( δ, π ) U U U " " " " v) ( δ, δ, δ ) f > V( δ, π ) V( δ, π ) c. U U 9

11 Proof. See Appendx " " " DenoteV( δ, π ) V ( δ, π ) =,V( δ, π ) V ( δ, π ) =, V( δ, π ) V( δ, π ) = and V( δ, π ) V ( δ, π ) =. Assumng, t s easy to see that 0 < 1 < < 3 < 4 < where ; = 1..4 are dfferent levels of aggregate dsutltes 1 due to loss of prvacy and due to the cost of testng. Also representng contracts ( δ, δ) δ, " 3 " " 4 ( δ, δ ) δ, ( δ, δ, δ ) δ and ( δ, δ, δ ) δ. Fg. shows the regon where these contracts would exst. U U 3 1 U U 4 3 > c = δ, δ 4 δ δ, δ, δ δ, δ, δ c = δδδδ,,, 1 4 δ, δ 1 4 δ, δ, δ 1 + c = 1 δ, δ 1 + c = 1 4 δ, δ, δ + c = Fg. Exstence of dfferent Insurance Contracts by Regon. The dsutlty due to loss of prvacy s represented on the X-axs whle the scaled cost of testng s plotted on the Y-axs 7. For small values of aggregate dsutltes (dsutlty due to loss of prvacy + dsutlty due to testng), none of the unnformed stays unnformed. The contracts 1 offered are δ and δ. For moderate values of aggregate dsutlty, there are unnformed who prefer to reman unnformed and based on specfc values three or four contracts could exst. For 7 The appearance of Fg. vares based on the relatve magntudes of aggregate dsutlty but the basc ntuton remans the same. 10

12 hgh values of aggregate utltes both low rsk nformed prefer not to reveal nformaton and unnformed prefer to reman unnformed or unnformed become nformed whle nformed low rsk don t reveal nformaton or unnformed stay unnformed and nformed low rsk reveal nformaton. Wherever δ ( δ ) and δ ( δ ) are offered smultaneously and are the only contracts, δ ( δ ) Pareto domnates δ ( δ ). 3.3 Unreported postves, verfable negatves, costly revelaton, unnformed consumers and observable nformaton status. Everythng s same as n secton 3. except that now the nformaton status can be observed by the nsurers.e. the nsurers know whch consumers are nformed and whch consumers are unnformed about ther rsk type. Proposton 3. Assume unnformed consumers can observe ther rsk type at cost c, nformed and unnformed consumers can be dstngushed, rsk type s not drectly observed by nsurers and consumers can report verfable nformaton to nsurers at non-zero cost. Denotng c c p =, four contracts are feasble: ) ( δ, δ ) f c < V ( δ π ) V( δ, π ), ) ( δ, δ ) f < V( δ, π ) V(, π c δ ) ) ( δ, δu, δ ) f < V( δ, π) V( δ U, π) v) ( δ, δu, δ ) Proof: See Appendx Denotng V( δ, π ) ( δ, and, same as before and assumng that V ) U, π = <, we have 0 < < < <. Also denotng ( δ, δ, δ ) δ, U 6 1 ( δ, δ, δ ) δ and δ and δ are same as n secton 3., the regons of exstence of the U 11

13 5 6 varous contracts are shown n Fg. 3. Where δ and δ are offered smultaneously, and are the 5 6 only contracts offered, the contracts δ Pareto domnates δ. c = δ, δ 6 δ δ, δ, δ 1 6 δ, δ, δ c = δ, δ, δ, δ 1 6 δ, δ 1 6 δ, δ, δ + c = Fg. 3 Exstence of dfferent Insurance Contracts by Regon. 3.4 Mandatory testng and release of nformaton for nsurance The prvacy loss of the hgh rsk (low rsk) consumer due to release of nformaton s ( ) where >. All consumers have to reveal genetc nformaton to the nsurers to get nsured. Ths mples that all unnformed have to get nformed n order to get nsurance. Proposton 4. Assume unnformed consumers can observe ther rsk type at cost c, rsk type s not drectly observed by nsurers and hgh rsk (low rsk) consumers can report verfable nformaton to nsurers at non-zero cost ( ) where > and reportng nformaton s mandatory for nsurance, then denotng V = V ( δ, π) and V = V (0, π ) =,, U ) If V < V and V < V nobody buys nsurance. )If V < V ; V > V and pv + pv c< V only nformed low rsk buy nsurance and the U only contract offered s δ. 1

14 ) If V < V ; V > V and pv + pv c> V U, then unnformed become nformed and the δ low rsk buy the only contract offered. v) If V > V and V > V and pv + pv c< V, then unnformed reman unnformed U and reman unnsured, both hgh rsk and low rsk reveal nformaton and are offered the contracts ( δ, δ ). v) If If V > V and V > V and, pv + pv c> V then unnformed become nformed, U both hgh rsk and low rsk reveal nformaton and are offered the contracts ( δ, δ ) Proof: Straghtforward comparson of the utltes of each type of agent wth or wthout nsurance wth the clause that nsurance requres mandatory testng leads to above results. 4. Dscusson Nature of nsurance contracts under each of the polcy proposals has been summarzed n the propostons. Under the frst proposal where nsurers are nether permtted to ask whether the consumer has undergone any test nor ask for test results, the contracts offered are - full nsurance for the hgh rsk and ncomplete nsurance for the unnformed and the low rsk. These contracts rely on pure self-selecton a la Rothschld and Stgltz. If release of nformaton s allowed by consent law and nsurers cannot observe the nformaton status (Proposton ), then for the low aggregate dsutltes due to testng and loss of prvacy, the low rsk consumers can also get complete nsurance. Snce the unnformed cannot be dstngushed from the hgh rsk, the nsurer has to dstort the contract for the unnformed so that the hgh rsk does not prefer ths contract to her own. For hgher values of aggregate dsutltes, the nsurer s better off offerng the same contract that t offers under the frst proposal. Thus from a welfare perspectve, Proposal s better than Proposal 1. The results of Proposton 3 are relevant to Proposal 3, where the nsurer always knows the nformaton status of the consumers. The results suggest that the unnformed can also get full nsurance f they choose to become nformed. The key dfference between 13

15 proposals and 3 s that the nsurers can dentfy the nformed hgh rsk from the unnformed. Snce ths nformaton s conveyed to the nsurers wthout any nherent loss to the consumers of any type, from a welfare perspectve Proposal 3 domnates Proposal. Ths result should be kept n mnd by polcy makers when they choose from the dfferent proposals on genetc testng. Mandatory testng and mandatory revelaton of nformaton does the worst n terms of overall socal welfare snce, t mposes addtonal costs on the hgh rsk and unnformed agents wthout provdng any benefts to the hgh rsk and provdng benefts to the unnformed only n some cases. Based on the relatve magntudes of the loss n prvacy and the cost of testng, there could be 1-4 equlbrums n a gven regon as s shown n Fg. and Fg. 3. It s possble that some of these Nash equlbrum can be elmnated by the ntutve crteron. All the equlbrum calculated are separatng equlbrum. Poolng and partal poolng equlbrum can be ruled out snce they won t exst n the compettve nsurance settng descrbed here. If all nsurers offer a partal poolng or a poolng contract then there s an ncentve for an nsurer to devate and offer separatng contracts. References Coase, R.., The problem of socal cost, Journal of aw and Economcs, 3,Oct 1960, pp Cremer, J., Khall, F. Gatherng Informaton before Sgnng a Contract, Amercan Economc Revew, June 199, 8(3), pp Crocker, K.J., Snow, A., The socal values of hdden nformaton n adverse selecton economes, Journal of Publc Economcs, 199, 48, pp Doherty, N.A., Thstle, P.D., Adverse selecton wth endogenous nformaton n nsurance markets, Journal of Publc Economcs, 1996, 63, pp Edwn S. Flores Troy, "The Genetc Prvacy Act: An Analyss of Prvacy and Research Concerns", Journal of aw, Medcne & Ethcs, (4), pp

16 rshlefer, J., The prvate and socal value of Informaton and the reward to ncentve actvty, Amercan Economc Revew, 61, pp ehrman, S., Keepng your Genes Prvate, Gene etter, March 000, Mesln, E.M., Thomson, E.J., Boyer, J.T., The ethcal, legal, and socal mplcatons research program at the Natonal uman Genome Research Insttute, Kennedy Insttute of Ethcs Journal, 1997, 7(3), pp Rothschld, M., and Stgltz, J., Equlbrum n Compettve Insurance Markets: An Essay on the Economcs of Imperfect Informaton, Quarterly Journal of Economcs, 1976, pp Roche, P., Glantz,.., Annas, G.J., The Genetc Prvacy Act: A proposal for Natonal egstlaton, Jurmetrcs, Fall 1996, pp Subramanam, K., emare, J., ershey, J.C., Pauly, M.V., Armstrong, K., Asch, D.A., Estmatng Adverse Selecton Costs form Genetc Testng for Breast and Ovaran Cancer: The Case of fe Insurance, The Journal of Rsk and Insurance, 1999, 66(4), pp Tabarrok, A., Genetc testng: an economc and contractaran analyss, Journal of ealth Economcs, 1994, 13, pp Westn, A.F. Prvacy and Freedom, Atheneum, New York, Proof of Proposton 1. get the contract APPENDIX We verfy that t s an equlbrum for the low rsk consumers to reveal nformaton and δ, and then show that not revealng nformaton can also be an equlbrum. If nsurers expect the low rsk to reveal nformaton, then they offer the contracts ( δ, δ ). The value of revealng nformaton to the low rsk s 15

17 u = [ V( δ, π ) ] V( δ, π ) 8 If < V( δ, π ) V( δ, π ) then u > 0. Snce there s value to revealng nformaton all low rsk consumers would reveal nformaton. If > V( δ, π ) V( δ, π ), then value to revealng nformaton s negatve (u < 0). All low rsk consumers would choose the contract. owever Rothshld and Stgltz (1976) have shown that a poolng contract cannot be equlbrum. So (, δ δ ) s an equlbrum f < V( δ, π ) V( δ, π ). If the nsurers expect the low rsk not to reveal nformaton, then they offer the contracts ( δ, δ ). The value of revealng nformaton s u = [ V( δ, π ) ] V( δ, π ) = Snce value of nformaton s negatve, low rsk do not reveal nformaton. The hgh rsk types are ndfferent between and, so ( δ, δ ) s also an equlbrum. δ δ δ The nsurers and the hgh rsk types are ndfferent between the contracts ( δ, δ ) and ( δ, δ ). The low rsk types prefer ( δ, δ ) f < V( δ, π) V( δ, π ). Thus the equlbrum ( δ, δ ) Pareto domnates the equlbrum ( δ, δ ) f < V( δ, π ) V( δ, π) Proof of Proposton. There are three players n the game the unnformed, the nformed low rsk type and the nsurer. The strategy space for the unnformed s to get nformed or to reman unnformed 9. The strategy space for the nformed low rsk s to reveal nformaton or not to reveal nformaton. The strategy space for the nsurer s to desgn the set of contracts. We adopt the followng approach to calculate the equlbrums - the nsurer pcks the best contracts gven one of the four possble 8 We assume for smplcty throughout that the low rsk prefer the contract meant for the hgh rsk to remanng unnsured.e. V( δ, π ) > V (0, π ) 9 The nformed hgh rsk consumer always end up wth the contract and are not part of the game. δ 16

18 combnatons of strateges of the other two players. If none of the other two players has an ncentve to devate gven the contract offered by the nsurer and the strategy of the other player, then the contract s an equlbrum. We calculate all the equlbrums n four steps; each step corresponds to the nsurers belef on the four possble strategy combnatons of the other two players. Step 1: Assume that the nsurers expect the nformed low rsk type to reveal nformaton and the unnformed type to become nformed, then t would offer the contracts ( δ, δ ). The value of revealng nformaton for the nformed low rsk types s u = [ V( δ, π ) ] V( δ, π ) If < V( δ, π ) V( δ, π ), u > 0 and so all nformed low rsk types would reveal nformaton. The value of nformaton for the unnformed s I = { p V( δ, π ) + p [ V( δ, π ) ] c} V( δ, π ) U I = p [ V ( δ, π ) V ( δ, π ) ] c If c < V( δ, π) V( δ, π) c where c =, then I > 0, all unnformed would become p nformed. Thus gven that the nsurers expect the nformed low rsk to reveal nformaton and the unnformed to become nformed, the nformed low rsk have an ncentve to reveal nformaton and the unnformed have an ncentve to get nformed. So ( δ, δ ) s an equlbrum. If > V( δ, π ) V( δ, π ) c, then all unnformed would reman unnformed and ( δ, δ ) cannot be an equlbrum. Step : Assume now that the nsurers expect the nformed low rsk not to reveal nformaton and the unnformed to get nformed. In ths case, the nsurers offer the contracts ( δ, δ ). The value of revealng nformaton for the nformed low rsk s u = [ V( δ, π ) ] V( δ, π ) = 17

19 Snce the value of revealng nformaton s negatve, the nformed low rsk wll not reveal nformaton. The value of gettng nformed for the unnformed s I = { p V( δ, π ) + p V( δ, π ) c} V( δ, π ) U I = p [ V( δ, π ) V( δ, π )] c If c < V( δ, π ) V( δ, π ) then the value of nformaton I s postve, unnformed wll become nformed and ( δ, δ ) wll be an equlbrum. ( δ, δ ) cannot be an equlbrum f c > V( δ, π ) V( δ, π ). Step 3: Assume now that the nsurers expect the nformed low rsk type to reveal nformaton and " the unnformed to reman unnformed. In ths case the, nsurers offer the contracts ( δ, δ, δ ). The value of revealng nformaton for the nformed low rsk types s u = [ V( δ, π ) ] V( δ, π ) " U U " If < V( δ, π ) V( δ, π ), then u > 0, and all nformed low rsk reveal nformaton. The U value of gettng nformed for the unnformed s " I = { p V( δ, π ) + p [ V( δ, π ) ] c} V( δ, π ) U U " I = p [ V( δ, π ) V( δ, π ) ] c U " If < V( δ, π ) V( δ, π ) c, then the value of gettng nformaton s postve, they choose U to become nformed and so the contract " ( δ, δ, δ ) U cannot be an equlbrum. If " > V( δ, π ) V( δ, π ) c, then unnformed contnue to reman unnformed. Thus for the U " range V( δ, π) V( δu, π) " c < < V( δ, π) V( δu, π), the set of contracts " ( δ, δu, δ) s also an equlbrum. 18

20 Step 4: Assume now that the nsurers expect the nformed low rsk type not to reveal nformaton and the unnformed to reman unnformed. In ths case the nsurers offer the contracts " " ( δ, δ, δ ). The value of revealng nformaton for the nformed low rsk type s U " " u = [ V( δ, π ) ] V( δ, π ) = Snce the value of revealng nformaton s negatve, the nformed low rsk wll not reveal nformaton. The value of gettng nformed for the unnformed s " " I = { p V( δ, π ) + p [ V( δ, π ) ] c} V( δ, π ) U U " " I = p [ V( δ, π ) V( δ, π ) ] c U " " If < V( δ, π ) V( δ, π ) c, then the value of gettng nformed s postve and so U " " unnformed wll become nformed. Thus ( δ, δ, δ ) cannot be an equlbrum. owever f " " > V( δ, π ) V( δ, π ) c, then the value of nformaton s negatve and so unnformed wll U U " " choose to reman unnformed and the contracts ( δ, δ δ ) wll be an equlbrum. U, Proof of Proposton 3. As before there are three players n the game the unnformed, the nformed low rsk type and the nsurer. We calculate all the equlbrums n four steps; each step corresponds to the nsurers best response based on belefs on the four possble strategy combnatons of the other two players. Step 1 and Step are dentcal to the correspondng steps n Proposton. ( δ, δ ) s an equlbrum f < V( δ, π ) V( δ, π ) c and ( δ, δ ) wll be an equlbrum f c < V( δ, π ) V( δ, π ). Step 3: Assume now that the nsurers expect the nformed low rsk type to reveal nformaton and the unnformed to reman unnformed. Note that now snce the nsurers can dentty the nformed 19

21 from the unnformed, the, nsurers offer the contracts ( δ, δ, δ ). The value of revealng nformaton for the nformed low rsk types s u = [ V( δ, π ) ] V( δ, π ) U U If < V( δ, π ) V( δ, π ), then u > 0, and all nformed low rsk reveal nformaton. The U value of gettng nformed for the unnformed s I = { p V( δ, π ) + p [ V( δ, π ) ] c} V( δ, π ) U U I = [ p V( δ, π ) + p V( δ, π ) V( δ, π )] ( +c) U The term n the square bracket s negatve so I < 0 and unnformed contnue to reman unnformed. Thus ( δ, δ, δ ) s an equlbrum for < V( δ, π ) V( δ, π ). U ( δ, δ, δ ). The value of revealng nformaton for the nformed low rsk type s u = [ V( δ, π ) ] V( δ, π ) = I = { p V( δ, π ) + p [ V( δ, π ) ] c} V( δ, π ) I = [ V( δ, π ) V( δ, π )] ( + c) < 0 menu of contract ( δ, δ, δ ) wll be an equlbrum. U Step 4: Assume now that the nsurers expect the nformed low rsk type not to reveal nformaton and the unnformed to reman unnformed. In ths case the nsurers offer the contracts U Snce the value of revealng nformaton s negatve, the nformed low rsk wll not reveal nformaton. The value of gettng nformed for the unnformed s U U U U U The value of nformaton s negatve, unnformed wll choose to reman unnformed and the U 0