Advantageous Selection versus Adverse Selection in Life Insurance Market

Transcription

1 Covr Pag Advantagous Slcton vrsus Advrs Slcton n f Insuranc Markt Ghadr Mahdav [email protected] Post Doctoral Rsarch Assocat: Jaan Socty for th Promoton of Scnc JSPS, Graduat School of Economcs, Kyoto Unvrsty, Yoshda-onmach, Sakyo-ku, Kyoto , Jaan Contact Addrss: , amamatsubara cho, Nshnomya sh, , Jaan Emal: [email protected] Tl: , Fa:

2 Advantagous Slcton vrsus Advrs Slcton n f Insuranc Markt By Ghadr Mahdav* Abstract: Th convntonal thory of advrs slcton gnors th ffct of rcautonary fforts on th robablty of dath and also dosn t consdr th corrlaton btwn th atttud towards rsk and rsk osur. Th mlcaton of such gnoranc wll b th nsurrs nd u wth hgh-rsk ndvduals and th markt facs th nsuffcnt rovson of th olcs. owvr, ths thory s not suortd by most of th mrcal works. Th altrnatv advantagous slcton thory assums a ngatv corrlaton btwn rsk avrson and rsk osur and consdrs th ffct of rcautonary actvty on th dath rat. Undr ths assumtons, nsurrs nd u wth rlatvly low-rsk ndvduals, th markt offrs suffcnt of olcs and, th slcton ffct wll b rotous to nsurrs as mor rsk-avrs low-rsk ndvduals ar not only wllng to ay mor for rcautonary fforts but also ar mor nclnd to nsur. W show that undr crtan crcumstancs whn th ndvduals ar suffcntly rsk avrs, th robablty of dath s smallr than ts crtcal valu, and th rocssng cost s suffcntly larg th slcton ffct wll b advantagous to th markt. W also show that whn ndvduals ar not suffcntly rsk avrs and consuntly thr robablty of dath s not suffcntly small, th ncssary condton for havng advantagous slcton rgm s th rocssng cost to b smallr than ts crtcal valu. Kywords: Advrs Slcton, Advantagous Slcton, f Insuranc, rcautonary ffort. Gl Classfcaton: G22, D82, D41 1. Introducton Advrs slcton s orgnally dfnd n nsuranc thory Rothschld and Stgltz, 1976 to dscrb a stuaton whr th nformaton asymmtry btwn olcyholdrs and nsurrs lads th markt to a stuaton that th olcyholdrs clam losss that ar hghr than th avrag rat of loss of oulaton usd by th nsurrs to st thr rmums. Accordng to th convntonal thory of dmand for lf nsuranc undr asymmtrc nformaton S Donn, Dohrty and Fombaron, 2000, lf nsurrs consdr th rcvd mortalty rats of oulaton to st th rmum, whl th ndvduals can b dvdd nto two grous of rsk lvl, lt s say, low- and hgh-mortalty grous, and th nsuranc comans can't dstngush btwn thm but th ndvduals know what grou thy blong to. ow-rsk ndvduals ralz that thr mortalty rat s low and thy ar subsdzng hgh-rsk ndvduals so wll b rluctant to nsur, whl hgh-rsk ndvduals wll hav motvaton for urchasng mor nsuranc as thy ar ayng lss than thr ral rat and ar actually rcvng subsdy from low-rsk ndvduals. Consuntly, th avrag mortalty rats of urchasrs of lf nsuranc s hghr than th rcvd mortalty rats by nsuranc comans and thus th comans nd u wth olcyholdrs who ar of hghr than avrag rsk rats. *Post Doctoral Rsarch Assocat, Graduat School of Economcs, Kyoto Unvrsty. Emal: [email protected]. I would lk to rss my arcaton to Prof. M. Kjma for hs constructv commnts. I would also lk to thank Mr. S. Rnaz for hs usful suggstons and also rovdng th orgnal rogram that was usd for th grahcal manulaton. Th author gratfully acknowldgs a fllowsh from Jaan Socty for th Promoton of Scnc JSPS. Th usual dsclamr als. 1

3 Th tnt of advrs slcton s affctd by th rason th ndvdual urchass th roduct. If th olcy s comulsory or s offrd by th mloyr, th ffct of advrs slcton wll lkly b lss svr than th voluntary olcy. Th tnt of advrs slcton s also affctd by ag, s, ncom, walth, occuaton, currnt halth status and th sz of olcy ald for. It sms that th tnt of advrs slcton dclns ovr tm as ol can bttr guss thr halth stuaton for th nt yar than for many yars latr. So, th ty of olcy wll also hav sgnfcant ffct on th tnt of advrs slcton. For aml, w ct hghr lvl of advrs slcton n short-trm lf nsuranc olcy than whol-lf nsuranc. Th convntonal thory of advrs slcton contans th followng assumtons: 1 Th dffrnc n osur to rsk: Pol dffr n th lvl of ognously dtrmnd rsk osurs. For smlcty, w consdr that ol ar dvdd nto two grous of rsk lvls, hgh- and low-rsk grous. 2 Postv corrlaton btwn slf-rcvd rsk lvl and ral rsk lvl: Advrs slcton occurs whn th ndvduals blfs about thr mortalty and thr tru rats ar ostvly corrlatd. If not, thr wll not b a systmatc dffrnc btwn olcyholdrs and oulaton s mortalty rats and hnc no advrs slcton occurs. 3 No rlatonsh btwn th lvl of rsk avrson and rsknss: In othr words, thr s no way to clam whthr hgh-rsk ndvduals ar lss rsk avrs than low-rsk ndvduals and vc vrsa. 4 Customrs know mor about thr rsknss than th nsurrs and ffcntly us thr nformaton aganst th nsurrs. Th mlcaton of such assumtons wll b that nsurrs facng advrs slcton st th rmum hghr to b abl to afford th clams whch, rsults th t of good rsks from th markt and nsuffcnt rovson of th roduct. Dst ths straghtforward undrstandng from th convntonal thory of nsuranc dmand undr asymmtrc nformaton, ths thory s not suortd by most of th mrcal works. Thr ar many mrcal vdncs that aar to conflct wth th standard thory of advrs slcton n nsuranc markt. mnway 1990 fnds that at a hostal n Tas, th rcntag of nsurd ndvduals amongst hlmtd and unhlmtd motorcyclsts s 73% and 59%, rsctvly. also found that amongst drvrs, 40 rcnt of thos who wor thr sat blt bought nsuranc whl only 33 rcnt of thos not warng th blt urchasd th covrag. Both amls show that hgh-rsk ndvduals unhlmtd and not warng th blts urchas lss covrag. Cawly and Phlson 1999 usng U.S. Tachrs Insuranc and Annuty Data conclud that asymmtrc nformaton s not actually a barrr to trad n th lf nsuranc markt as thy couldn t fnd nough vdnc on stnc of advrs slcton n ths markt. Thy couldn t fnd any sgnfcant corrlaton btwn ndcator varabls for slf-rcvd rsk and th 2

4 uantty of trm nsuranc. Thy also couldn t fnd any sgnfcant ffct btwn actual rsk and th dmand for lf nsuranc. Surrsngly, thy could show ngatv covaranc btwn rsk osur and th dmand of lf nsuranc. Thy also found vdnc of bulk dscounts and ngatv rlatonsh btwn rc and uantty that ndcats th fact that low-rsk ndvduals urchas mor lf nsuranc. Othrws, th nsuranc comany wll not b abl to afford th lablts of hgh-rsk ndvduals wth lowr rmums. Thy concludd that ths can b du to ffctv undrwrtng olcy and th fact that nsurrs may know thr costs of roducton bttr than olcyholdrs and, th nsurrs rcvd rsk rats ar mor accurat than th rat rcvd by customrs. McCarthy and Mtchll 2003 found that th mortalty rat of UK and US mals and fmals urchasng trm- and whol-lf nsuranc s blow that of th unnsurds. For aml, thy found that mortalty rats for mal and fmal urchasrs of whol-lf nsuranc ar only 77.5 and 68.5 rcnt of th total oulaton mortalty rat for th UK, and 78.6 and 90.9, for th US, rsctvly. Sglman 2004 clams that th nformatonal asymmtrs ar n th favor of nsurrs not nsurds as nsurrs utlz varous stratgs of undrwrtng and rsk classfcaton that comnsat for or vn ovrcom nformatonal advantag of olcyholdrs. Morovr, th bhavoral or sychologcal factors hl to offst nsurds nformatonal advantags. For aml, whn thr s ngatv corrlaton btwn rsk avrson and rsk osur, th addtonal dmand of th hghr-rsk ndvduals wll cancl out. Mza and Wbb 2001 stat that n addton to rcautonary ffort that lans th ngatv corrlaton btwn nsuranc dmand and rsk lvl, htrognous otmsm also suorts ths ngatv corrlaton: gh rsks ar mor otmstc about th vnts to b mrobabl, so thy urchas lss nsuranc. Dachraou, Donn, Eckhoudt and Godfrod 2004 show that mor rsk-avrs agnts whos bhavor follow th md rsk avrson utlty functon may snd mor on slf-rotcton actvts whn th loss robablts ar blow 1/2. Julln, Salan and Salan 1999 gv th suffcnt condtons undr whch mor rsk-avrs agnts rt mor fforts to dcras th robablty of loss. Thy show that slf rotcton ncrass wth rsk avrson f and only f th ntal robablty of loss s low nough. Ths rsults rnforc th da of advantagous slcton n lf nsuranc markt, as th customrs mortalty rat s usually vry small. Ths mrcal vdncs that contradct th convntonal thory of dmand for nsuranc undr asymmtrc nformaton and advrs slcton thory lad us to vw th roblm from a nw rsctv. 3

5 To dscrb th contradcton btwn th convntonal thory and th mrcal rsults, w focus on th rcautonary ffort for avodng losss. Instad of th assumton that ol dffr n th lvl of ognously dtrmnd rsk osurs that dtrmns th nsuranc dmand, w concntrat on th assumton that hghly rsk-avodng ndvduals ar mor lkly both to try to rduc hazard by urchasng nsuranc and takng hyscal rcautonary fforts. In othr words, ol who buy mor nsuranc tnd to b mor safty conscous and thus ar mor nclnd to undrtak rcautonary fforts. Invrsly, lss rsk-avrs ndvduals ar lss lkly to buy nsuranc voluntarly, and thy ar th ons most lkly to lac thmslvs dlbratly n dangrous stuatons. Consuntly, n ths sttng, th slcton ffct wll b advantagous to th markt as nsurrs nd u wth a lot of cautous low-rsk ndvduals who ar lkly to ay for rcautonary fforts. In th nt scton w dvlo a modl to dscuss th ffct of rcautonary actvty on th lf nsuranc dmand and to fnd th condtons undr whch advantagous slcton occurs. In scton 3 w fnd th dmands for two grous of dffrnt rsk lvls and th otmal oolng rc. Scton 4 rsnts a numrcal aml to show why low-rsk ndvduals rfr to contnu urchasng at th markt vn though thy ar subsdzng th hgh rsks. Scton 5 dscusss th Drcton of th ffct of aramtrs on changng th rgm to advantagous slcton by grahcal manulaton. Scton 6 concluds. 2. Th Modl Suos all ndvduals hav th sam oortunty to lowr th robablty of dath by rvntv fforts. Each ndvdual facs th robablty of dath whr ndcats th rcautonary fforts and s assumd to affct th robablty of dath n th sam way for all th ndvduals. W assum < 0 whch mhaszs that rcautonary actvty mrovs th survval rat and has ngatv ffct on th robablty of dath. ttng th functon U. rrsnt utlty n th lf stat and V. dath stat, th ctd utlty of a olcyholdr s utlty for survvng mmbrs of th houshold n th EU,, W, Y = 1 [ U W Y ] V W 1. 1 Th varabl s th nsuranc unt rmum, W s th ndvdual s walth, Y s th ncom and, rfrs to th dmand for lf nsuranc whch s dfnd as th total covrag n th vnt of dath. Ths modl suggsts that th agnts nvst n both of rcautonary ffort for 4

6 rducng th robablty of dath and, lf nsuranc for handlng th rmanng rsk. Obvously, th amount of nsuranc dmand should b nonngatv, 0. W lan to amn th drcton of th ffct of dath rat uon th dmand for lf nsuranc to fnd th condtons undr whch th markt slcton s advantagous for nsurrs. In othr words, w want to show whthr advantagous slcton can occur n ths sttng. Whl th convntonal thory s basd on olcyholdr s ognous rsk osur, our thortcal stu s basd on th assumton that rcautonary fforts of olcyholdrs and ngatv corrlaton btwn rsk osur and rsk avrson dtrmn th lvl of lf nsuranc dmand. Th roblm can b statd as Ma EU,, W, Y = 1 [ U W Y ] V W 1. 2 Th frst ordr condton for mamzaton s d d U U V V. = d d 0, 3 whr U s th margnal utlty wth rsct to total asst. Th trms of V and d / d ar th margnal utlty of bust wth rsct to asst and th drvatv of lf nsuranc dmand wth rsct to th mortalty rat, rsctvly. Th scond ordr condton s 2 EU U... 1 [.. U θ U θ U θ U θ ] 2 V. V. φ V. φ [ V. φ V. φ ] 0 4 Whr d θ = 1, d φ = 1. d d Obvously, rsk avrson condtons U > 0, V > 0, U < 0, V < 0 ar not suffcnt to nsur th scond ordr condtons, but w assum th scond ordr condton s mt lttng th soluton b global mamum. From 3 th drvatv of lf nsuranc dmand wth rsct to th mortalty rat wll b found as d [ V. U.] 1 U. = 5 d { 1 U. 1 V.} 5

7 Our uros s to fnd th condtons, onc satsfd, th advrs slcton rgm changs to advantagous slcton. Th ncssary condton for advantagous slcton to occur s th lss rsky ndvduals urchas mor than hgh rsks. Thrfor, w should fnd th condtons that shft th sgn of 5 to ngatv. Insurrs do not rmt th customrs to urchas mor than thr ctd loss. In othr words, th customrs ar rmttd to urchas thr full-nsuranc amount of covrag or artal on. In a full-nsuranc condton whr total loss s comltly covrd by lf nsuranc comnsaton or n a artal covrag condton whr th covrag s lss than th ctd loss, th utlty from bust wll not cd that of th consumton [ V. U. 0]. Snc < 0 and 0 < 1, th followng suffcnt condtons togthr rsult n < advantagous slcton rgm d / d < 0. a U. > 1 6 U. V. b 1 U. 1 V. > 0 7 Condton 6 stats that rcautonary actvts should hav consdrabl ffct on ushng th robablty of dath down. Ths condton s satsfd whn th ndvduals valu th ffort so hghly that th ffct of rcautonary fforts on th robablty of loss cds ts crtcal valu. In othr words, th ndvduals should b suffcntly snstv to rcautonary ffort. If th ndvduals ar suffcntly rsk avrs that valu th fforts hghly, th ffct of rcautonary fforts on dcrasng th mortalty rat b consdrabl. Condton 7 s satsfd whn th robablty of loss s suffcntly small and th loadng factor s suffcntly larg. Th nsurrs can rcv th ovrall robablty of loss and dtrmn th rmums accordng to ths rcvd rsk rat and a loadng factor. Th uaton = 1 λ ndcats th rlaton btwn rc and th rcvd rsk lvl by nsurrs, whr λ ndcats th loadng factor. Th loadng factor s addd to th rmum to covr th rocssng cost, contngncs, and to guarant roft for nsurrs. Thrfor, whn rocssng cost s suffcntly larg, th rc wll bcom consdrably largr than. Ths condton togthr wth th condton of havng vry small guarants th lft hand sd of 7 to b ostv. Consuntly, whn ndvduals ar suffcntly snstv to rcautonary fforts and thr robablty of loss s suffcntly small whl th loadng factor s suffcntly larg, advantagous slcton wll b th stng rgm for th dmand undr asymmtrc nformaton d / d < 0. 6

8 Ths rsult s logcal as mor rsk-avrs low-rsk ndvduals can tolrat hghr ncras n rcs and dductbls ncurrd bcaus of any ncras n rocssng cost. Shortly sakng, whn ol ar suffcntly rsk avrs and thr corrsondng mortalty rats ar suffcntly small, th ncssary condton for havng advantagous slcton rgm s th rocssng cost to b largr than ts crtcal valu. If ndvduals ar suffcntly snstv to rcautonary fforts and loadng factors ar suffcntly larg, th mlct crtcal valu for robablty of loss that guarants th advantagous slcton rgm wll b found from 7 as C m U. = 8 U. 1 V. For valus lss than th crtcal valu < C m, th trm 7 wll bcom ostv and hnc th condton for advantagous slcton wll b nsurd. In an advantagous slcton condton, th ndvdual s fforts for avodng loss, whch dcrass th robablty of dath, has ostv corrlaton wth th dmand for lf nsuranc. To fnd th crtcal rocssng cost, w assum thr s no contngncs and roft. So, loadng factor wll b ual to th rocssng cost. Undr such assumton th mlct crtcal valu for rocssng cost wll b C C m 1 V. U. =. 9 1 U. V. If th ndvduals ar suffcntly snstv to rcautonary fforts that nualty 6 satsfs, thn th rocssng cost should b gratr than ts crtcal valu to rsult n advantagous slcton rgm. Rfrrng to 5, w can fnd anothr st of condtons that lad to th advantagous slcton: c U. < 1 10 U. V. d 1 U. 1 V. < 0 11 Whn rcautonary actvty s not suffcntly ffctv on dcrasng th robablty of loss, and th robablty of dath s largr than ts crtcal valu, th ncssary condton for havng advantagous slcton rgm s th loadng factor and th rocssng cost to b smallr than 7

9 8 thr crtcal valus. As a rsult, for lss rsk-avrs ndvduals, th markt s advantagous to th nsurr f robablty of loss s rlatvly larg whl th rocssng cost s rlatvly small. 3. Th Dmand vl In ths scton w dtrmn th dmand lvl for two grous of mor rsk-avrs low-rsk ndvduals and lss rsk-avrs hgh-rsk ndvduals to chck f advantagous slcton occurs n ths sttng S also Mahdav and Rnaz, W assum th utlty and bust functons ar of a class of CRRA. Th roblm wll b to mamz th ctd utlty functon, ,,, 1 1 W Y W Y W EU Ma α α α α = 12 subjct to th constrants that nonngatv amount of nsuranc s urchasd and, ovr-nsuranc s not rmttd n th markt ndcatng th nsurr dosn t rmt th customrs to urchas mor than thr ctd loss. Th frst ordr condton wll b as = W Y W α α, 13 that ylds th otmal dmand for lf nsuranc as K W K Y W = 1 *, 14 whr K α =. At th suly sd, th nsurr facs wth th rocssng cost of C for ach unt of covrag to offr th contracts to customrs. Actually, th nsurr cannot dstngush th rsk lvls of hs customrs but can rcv th ovrall condton of th whol oulaton. Undr rfct comtton 0 1] [1, = = C 15

10 Aftr nsrtng th obtand otmal dmand 13 for both grous of low-rsk ndvduals and hgh-rsk ndvduals, th nsurr s roblm n a rfct comtton condton wll b statd as [1 [1 C 1][ Y W K W ] K 1 16 C 1][ Y W K W ] K 1 = 0 4. Numrcal Eaml Th ky modfcaton to th convntonal advrs slcton modl s th ngatv corrlaton btwn ndvdual s atttud towards rsk and hs rsk osur. Morovr, mor rsk avrs ndvduals ar not only mor lkly to rduc rsks on thr own by takng good car of thr halth, but also thy ar mor avrs to fnancal rsks and hnc mor wllng to ay to lmnat such rsks through nsuranc. As a rsult, thy sm to b low-rsk longr-lvd ndvduals who would lk to urchas mor nsuranc. In th convntonal modl, th low-rsk ndvduals do not want to ool thmslvs wth th hgh-rsk grou snc vn though th rmum s actuarally far for all th nsurds as a whol; t s too hgh for thm. So, thr wll b a motvaton for thm to las. But f low rsks ar assumd to b mor rsk avrs than th hgh rsks, thy valu nsuranc contracts mor than thr hgh-rsk nghbors and wll hav motvaton to ool wth thm vn though th rats ar hghr than thr actuarally far rats s also Sglman,2004. Now assum th aramtrs for both grous of th ual sz ar as follows: W = W = 0, Y = Y = 100, = , α = 0.5, = 1, and = 0.001, α = 0.9, = 2 and C = 0.2. Th oolng ulbrum rc for ths numrcal aml and th otmal dmand for both grous ar obtand as * , * = 87.29, * = = Th answrs satsfy th constrants and logcal consdraton, as th dmand lvls ar lss than th full-nsuranc dmand. To amn whthr advantagous slcton may occur n th sttng, w should chck n trms of utlty whthr low-rsk ndvduals rfr to urchas th oolng nsuranc or would lk to go unnsurd. Th ndvduals ctd utlty n no nsuranc, full nsuranc at far rc, oolng ulbrum wth obtand otmal dmand cass and, th gans from nsuranc urchas for th ctd utlty 12 ar shown n th tabl. 9

11 Tabl 1-Utlty comarson ow-rsk Indvduals gh-rsk Indvduals Probablty of dath Incom Cost of Effort 2 1 Actuarally far rmum for ach unt Coffcnt of rlatv rsk avrson EU n no nsuranc cas EU n full nsuranc cas at far sarat rcs =0.001, Gan from nsuranc % EU n Poolng ulbrum *=0.0016, X*=58.24, X*= ow-rsk ndvduals ar mor rsk avrs and hav hghr coffcnt of rlatv rsk avrson whl th condton for hgh rsks s th oost. Aftr ntroducng nsuranc at far rmum, low-rsk nsurds obtan a gratr rcntag gan n utlty snc thy ay lowr rmum whl thy ar mor rsk avrs, valung nsuranc hghr and hnc, obtan mor satsfacton from any unt of nsuranc covrag. It s shown that low rsks ar mor satsfd wth oolng ulbrum rathr than lavng unnsurd. Evn though th rc s not far to thm and thy ar subsdzng th hgh-rsk olcyholdrs, stll thy rfr to urchas lf nsuranc olcy. That s bcaus thy ar suffcntly rsk avrs to ool wth thr hgh-rsk nghbors. In othr words, thy do not dro out of th markt as thy ar suffcntly rsk avrs that can tolrat hghr than far rcs. Ndlss to say, th hgh rsks rfr oolng ulbrum to no-nsuranc cas. gh rsks would also rfr th oolng ulbrum to full nsuranc at far sarat rcs f thr wrn t larg rocssng cost that maks oolng rc gratr than thr far sarat rc. But snc th oolng rc s gratr than thr far rats du to hgh rocssng cost, thy rfr th full nsuranc at far sarat rcs th bst. Shortly sakng, th ngatv corrlaton btwn rsk avrson and rsk osur maks th markt lausbl that low-rsk ndvduals do not t from th markt and consuntly th nsurr wll not fac th roblm of advrs slcton. Whn th low-rsk grou s suffcntly rsk avrs, thr s a tndncy for thm to rman at th markt whl som arts of hgh rsks dro out of t, lavng th good rsks to contnu urchasng th olcy. Morovr, th suffcntly rsk-avrs low-rsk ndvduals also undrtak rvntv fforts that dcras th mortalty rat furthr and mak th stuaton mor lausbl for nsurrs. 5. Th Effct of Paramtrs In ths scton w us grahcal manulatons of th numrcal aml to fnd th aramtrs that ar crucal to dtrmn th rgm. Th fgur 1 shows th otmal dmand for hgh- and low-rsk grous, and th corrsondng rc for 10,000 unts of nsuranc whn 10

12 rocssng cost changs from 0 to 1.5 whl othr aramtrs ar kt gvn as th numrcal aml. It s shown that th ga btwn and s gttng smallr whn th rocssng cost s ncrasng. At around C = 1 thy ntrsct and th advrs slcton rgm changs to th advantagous on as th otmal dmand of low-rsk ndvduals cds that of hgh rsks. Fgur 1- Dmands and rc whn C s ndognous, for th followng aramtrs: = 0.001, α = 0.9, = 2, W = W = 0, Y = Y = 100, = , α = 0.5, = 1. Th rgm changs to advantagous slcton at th crtcal valu of C = 1. In fgur 2 w just dcrasd th rsk avrson of hgh-rsk ndvduals from 0.5 to 0.1 n ordr to ncras th rlatv rsk avrson of low-rsk ndvduals. In dong so, th rgm changs to advantagous slcton for smallr rocssng cost C =

13 Fgur 2- Dmands and rc whn C s ndognous and α s dclnd, for aramtrs: = 0.001, α = 0.9, = 2, W = W = 0, Y = Y = 100, = , α = 0.1, = 1. Th rgm changs to advantagous slcton for a smallr crtcal valu of C = Ths two fgurs obvously show that ncrasng rocssng cost lads to advantagous rgm scally whn low rsks ar suffcntly rsk avrs comard to hgh-rsk ndvduals. Th ffct of rsk avrson s shown n fgur 3. W lottd th dmands and rc whn th low-rsk ndvduals rsk avrson s changng from 0 to 1 ndognously whl all othr factors ar fd as th orgnal numrcal aml. Evn though th otmal dmand of hgh-rsk ndvduals cds that of low rsks for all th rang, ts ga s gttng smallr whn low-rsk ndvduals bcom mor rsk avrs, ndcatng that rsk avrson has ostv ffct on swtchng th rgm to advantagous slcton. 12

14 Fgur 3- Dmands and rc whn α s ndognous for Paramtrs: = 0.001, = 2, C = 0.2, W = W = 0, Y = Y = 100, = , = 0.5, = 1. α Whn low rsks bcom rlatvly mor rsk avrs, th rgm tnds to chang to advantagous slcton. To amn th ffct of rsk avrson undr a hghr lvl of rocssng cost, w lottd th dmands and corrsondng rc whn th rocssng cost s 0.9. W obsrv that for th low-rsk ndvduals, whn rsk avrson s largr than , th rgm changs to advantagous slcton as th otmal dmand for low rsks cds that of hgh-rsk ndvduals. Ths condtons ar n corrsondnc wth th drvd condtons of th thortcal art 6 and 7 that suggstd undr hgh lvl of rsk avrson and low lvl of mortalty rat, a hghr lvl of rocssng cost s rurd to rsult n advantagous slcton. 13

15 Fgur 4- Dmands and rc whn α s ndognous and C s ncrasd, for aramtrs: = 0.001, = 2, C = 0.9, W = W = 0, Y = Y = 100, = , α = 0.5, = 1. Th rgm Changs to advantagous slcton for crtcal valu of α = In fgur 5, th robablty of dath s assumd to b a dcrasng affn functon of rsk avrson α assng through th onts α, = 0.9, and α, = 0.5, Obvously, two dmand curvs cross at dath ar dntcal at ths ont. Th othr ntrscton s at α = 0.5 as two grous rsk avrson and robablty of α = At ths two onts two grous urchas ually and th nsurrs rcvd mortalty rat wll b ual to th ral rat of th nsurds and, thr wll b no nformaton asymmtry. W can dscuss th slcton roblm n thr arttons: for th art 0.5 α < 1, th < hgh-rsk grou of dmands mor and thrfor th so-calld advrs slcton rgm rvals n th markt. For th art < α < 0. 5, vn though cds advrs-slcton rgm rvals as rvously calld low rsks wth th dmand lvl of, stll ar mor rsky at ths scton as thr rsk avrson s smallr than th rsk avrson for th grou whch was rvously calld hgh rsks wth th dmand lvl of. Th only scton whr 14

16 advantagous slcton can b obsrvd s th trm lft whr 0 α < as th < currntly lss rsky grou whos dmand s shown by urchass mor than th currntly mor rsky grou wth th dmand lvl of. Ths grah suorts th condtons 10 and 11 whch suggstd whn th rsk avrson lvl s rlatvly low and thrfor th robablty of loss s rlatvly larg and th rocssng cost s rlatvly small th slcton ffct wll b advantagous as obsrvd n th trm lft whr 0 α < < Fgur 5- Dmands, and rc whn α s ndognous and α assng through th onts 0.9,0.001 and s a dcrasng functon of 0.5, for aramtrs: = 2, C= 0.2, W = W = 0, Y = Y = 100, = , α = 0.5, = 1. W obsrv th advantagous slcton rgm only n trm lft whr rsk avrson s low, s rlatvly larg and C s rlatvly small. In fgur 6, th rocssng cost C s ncrasd from 0.2 to 0.9, th mortalty rat of hgh-rsk ndvduals s ncrasd from to to mak low rsks safr comaratvly, and th rlatv rsk avrson lvl of hgh rsks s dcrasd from 0.5 to 0.1 to ncras th rlatv rsk 15

17 avrson of low rsk ndvduals. Th rsult of such changs s satsfactory: th markt facs wth advantagous slcton for th rang 0.1 α < 1. < Fgur 6- Dmands, and rc whn α s ndognous and s a dcrasng functon of α assng through th onts 0.9,0.001 and 0.1,0.002 for aramtrs: = 2, C= 0.9, W = W = 0, Y = Y = 100, = 0.002, α = 0.1, = 1. W obsrv th advantagous slcton rgm for a wd rang of 0.1 < α < 1. Th rsult corrsonds wth th thory as th rlatv rsk avrson of low rsks and th rocssng costs ar ncrasd whl th rlatv rsknss of low rsks s dcrasd. Th rsult corrsonds wth th condtons 6 and 7 whch stat that, whn ndvduals ar suffcntly rsk avrs and thr robablty of loss s suffcntly small and th rocssng cost s suffcntly larg, advantagous slcton wll b th stng rgm for th dmand undr asymmtrc nformaton. 16

18 6. Conclusons Th classcal thory of dmand for nsuranc undr asymmtrc nformaton rsults n nsuffcnt rovson of olcs and advrs slcton. Ths conclusons sm to contradct most of th mrcal works n th fld. W try to rsolv ths contradcton by ntroducng th ffct of rcautonary actvts whch mrovs th survval rats, and assumng a ngatv corrlaton btwn rsk avrson and rsk osur. Undr ths two assumtons, th so-calld advrs slcton rgm can b substtutd wth a favorabl stuaton whch s calld advantagous slcton. In a numrcal sttng, w hav shown th cas whr low-rsk ndvduals rfrrd to contnu urchasng th olcy vn though thy wr ayng mor than thr far rc. W could also show that undr crtan crcumstancs whn th robablty of loss s smallr than ts crtcal valu, th olcy holdrs ar suffcntly rsk avrs and th rocssng cost s suffcntly larg, th slcton ffct wll b advantagous to th markt and th so-calld advrs slcton rgm rvals no longr. W could also show grahcally th cass that good rsks ar bttr off wth oolng ulbrum rathr than dro out of th nsuranc ool. As a rsult of ngatv corrlaton btwn rsk avrson and rsk osur, th low-rsk ndvduals rfr to urchas lf nsuranc olcy vn though th rc s not far to thm and thy ar actually subsdzng th hgh-rsk olcyholdrs. If th ndvduals ar not suffcntly rsk avrs and hav hghr robablty of dath, thn th ncssary condton for havng advantagous slcton rgm wll b facng a low lvl of rocssng cost smallr than ts crtcal valu. Eamnng th ffct of ncom and walth can tnd ths rsarch. To lmnat th ncom and walth ffct, w nd to dscuss how th dffrntals of ths factors chang th rgm from advrs slcton to advantagous slcton. 17

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