AN EQUILIBRIUM ANALYSIS OF THE INSURANCE MARKET WITH VERTICAL DIFFERENTIATION

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1 QUIIRIUM YI OF T IUR MRKT WIT VRTI IFFRTITIO Mahto Okua Faculty of conomcs, agasak Unvesty, 4-- Katafuch, agasak, , Japan [email protected] TRT ach nsuance poduct pe se s dentcal but the nsuance companes that sell ths dentcal nsuance poduct ae not necessaly dentcal. ecause the natue of nsuance s to cove damages fom accdents, consumes hope to get the nsuance moneys uckly fom the nsuance companes when accdents happen. In ths study, ths s ntepeted as aftecae and all nsuance companes ncopoate t nto the stateges. The nsuance companes n the maket ae assumed to compete not only on pce (nsuance ate) but also on ualty (level of aftecae). Thus, the nsuance maket s assumed to have vetcal dffeentaton. ITROUTIO onsumes have to decde fom whch nsuance company they puchase an nsuance poduct. These decsons can be affected by seveal factos. The most common facto s pce. If a decson on nsuance, n a compettve maket, can be explaned by the level of pce alone, all nsuance companes would set the same pce, as a esult of etand competton. oweve, pevous studes have confmed the exstence of pce dspeson n the nsuance maket. Fo example, Jung (978) has poved that thee s pce dspeson on dentcal automoble lablty nsuance n hcago, Illnos and thee ae statstcally sgnfcant dffeences when nsuance companes ae classfed by dstbuton system. Mathewson (983), ahlby and West (986), and osey and Yavas (995) have agued that pce dspeson stems fom the seach costs of consumes. ege et al. (989) pesented empcal evdence on swtchng costs and dffeental pces. chlesnge and chulenbeg (99, 993) explaned pce dspeson accodng to thee factos: poduct dffeentaton, seach cost, and swtchng cost smultaneously. In pevous studes, othe factos n the nsuance maket ae consdeed to explan pce dspeson. One of these s coveage. Undoubtedly, all consumes pefe moe coveage than less. oweve, othe thngs beng eual, t seems that coveage s not defntve to the exstence of pce dspeson. Ths can be explaned by the followng thee easons. Fst, n geneal, almost all of the nsuance companes offe dentcal nsuance poducts because the egulato wll not be allowed to offe any othe type of nsuance poduct. ven f thee wee

2 dffeent nsuance poducts, couts have feuently sad that they ae same (Mcowell, 989). econd, as Mcowell (989) has noted, even f thee wee total deegulaton of nsuance poducts, the smlaty of nsuance poducts would contnue because t s much moe expensve to custom-make o talo an nsuance poduct fo each consume than to sell a standad one. Thd, because nsuance poducts ae nvsble and defned po to puchase, t may be too dffcult fo common consumes to undestand the contents completely. Thus, many consumes may not be fully awae of the contents of nsuance poducts when they puchase them. nothe facto s the ualty level. chlesnge and chulenbug (99, 993) have ponted out that all nsuance companes ae not necessaly dentcal even f all the nsuance poducts ae dentcal because t s possble to set dffeent ualty levels among nsuance companes. Futhemoe, chlesnge and chulenbug (99, 993) have povded ualty fo nsuance companes eputaton, solvency chaactestcs, maketng methods, clam-handlng pocedues, and so on. In the eal wold, all the above atttudes ae compounded. oweve, to the extent that the scope of ths pape s estcted to the Japanese nsuance maket, t can be consdeed that clam-handlng pocedues ae the most meanngful facto. ecause the natue of nsuance s to cove the damage asng fom an accdent, consumes hope to get nsuance money pomptly fom the nsuance companes when an accdent happens. Ths can be ntepeted as aftecae. In contast, at least untl now, when Japanese consumes puchase an nsuance poduct, they may not egad othe atttudes, such as solvency, as mpotant because pehaps only a small numbe of nsuance companes have gone bankupt snce Wold Wa. In Japan, the maketng of non-lfe nsuance s mostly though agents and the maketng of lfe nsuance s though nsuance pacttones, except fo some foegn nsuance afflates. Then because both non-lfe and lfe nsuance polces ae sold though a sngle dstbuton system, maketng methods can be consdeed as dentcal. In ths pape, an attempt s made to analyze an nsuance maket whee nsuance companes not only offe nsuance poducts but also povde aftecae f In patcula, Japanese non-lfe nsuance poducts, such as fe, automoble and pesonal accdent, ae povded by dentcal clauses. The eason fo ths s to puchase ensuance fom ensuance companes and to puchase consuance wth othe nsuance companes. In the eal wold, the level of aftecae can be measued by the complant ato, whch s based on the numbe of complants eceved. ee oepnghaus (99), Wells and taffod (995) and oyt and Quey (999).

3 an accdent happens. The model nvolved has two mpotant featues. Fst, chlesnge and chulenbug (99), Gavell (994) and Tsutsu et al. (000) have analyzed a hozontal dffeentated nsuance maket that s assocated wth vaety. oweve, because consumes may pefe moe aftecae than less, ths pape ndcates the vetcal dffeentaton that s assocated wth ualty. econd, my model dffes fom geneal vetcal dffeentated models n the followng ponts. () The pce n an nsuance maket s epesented by the nsuance ate, whch euals the nsuance pemum dvded by the nsuance amount. 3 Thus, unlke othe makets, to set a pce n an nsuance maket, each nsuance company contols smultaneously the two vaables, nsuance pemum and nsuance amount. () In geneal, though an nsuance company can sell an nsuance poduct at dffeent pces to dffeent consumes, t cannot set at dffeent levels of ualty to dffeent consumes. Fo example, t s mpossble to pefom the dffeentaton that low-sk consumes eceve the nsuance money uckly and hgh-sk consumes eceve the nsuance money slowly. Futhemoe, such a dffeentaton s pohbted by Japanese nsuance law. In othe wods, egadless of the consume s sk type, all consumes who have the same nsuance company eceve same level of ualty f an accdent happens. (3) consume can eceve aftecae only f an accdent happens. Ths fact suggests that contay to pevous studes, two paametes, pecepton fo ualty and pobablty of an accdent, ae dectly elated to a consume s decson. Ths pape has two puposes: fst, to analyze how a vetcal dffeentaton n a compettve nsuance maket wll affect pce and secondly, to examne socal welfae whch s sum of consume welfae and poduce welfae as the egulato mposes mnmum ualty standads. T MO In ths secton, the basc model s defned. Thee ae two nsuance companes, and. uppose that two types of consume exst n an nsuance maket, whch nclude hgh-sk consumes wth accdent pobablty and low-sk consumes wth accdent pobablty, and assume that 0 < < <. Futhemoe, assume that both nsuance companes can 3 eeafte, n ths pape pce and nsuance ate ae used ntechangeably. 3

4 obseve the type of consume wthout cost. enote that s the numbe of hgh-sk consumes and stctly postve. et and s the numbe of low-sk consumes, whee both ae epesent the nsuance pemum and the nsuance amount, espectvely, of a consume of type puchased fom the nsuance company. oweve, each nsuance company cannot dstngush ualty n accodance wth each consume. Thus, nsuance company offes only one ualty, [ mn, max ], egadless of type. et mn [ 0, max ) ualty and (, ) max mn epesent the mnmum epesent the maxmum ualty. ssume that both nsuance companes have the same fom of the cost functon and the level of ualty has no nfluence on uanttes ( ) a whee a > 0. ome examples of such ualty featues ae the nvestment made educatng employees and payng out nsuance money moe apdly. It s assumed that both nsuance companes cannot avod payng ths nvestment cost, even f they sell nothng. The utlty of consumes s assumed to be sepaable n ncome and aftecae. Thus, the utlty functon s U u() v() () whee u () s the utlty that s elated to the ncome level and v ( ) s the utlty that s elated to the aftecae. ach consume has to choose the puchase of one unt fom the moe desable nsuance company. ow, applyng att (964), the consume s utlty, whch s elated to hs ncome, becomes IO IO ( ) IO Va () whee epesents the cetanty euvalent, epesents the expected payoff, and epesents the (vaable) payoff. ach vaable that s ndexed IO s 4

5 ncluded n the utlty u (). s the degee of absolute sk aveson and t s assumed to be constant egadless of the consume s type and wealth. Va ( ) denotes the vaance. IO onsde n uaton (). et W be the ntal wealth. enote as the value of an nsuable asset and, fo smplcty, thee ae only two states of the wold, total loss and no loss. If a consume of type puchases an nsuance poduct {, } δ fom nsuance company, ths consume s faced wth the stuaton whee hs payoff becomes W f an accdent happens and W f an accdent does not happen. Then IO can be wtten as IO ( ) W. (3) IO Usng uaton (3), ( ) Va n uaton () s gven by Va IO ( ) ( )( ). (4) ubsttutng uatons (3) and (4) n uaton () gves IO ( ) ( )( ) W. (5) et us next consde v (). Fst, assume that v ( ) s a lnea functon. Thus, R the cetanty euvalent,, whch s elated to the aftecae, s consstent R wth the expected utlty,. oth vaables that ae ndexed R ae ncluded n the utlty v (). Futhe suppose that all consumes have the own ualty valuaton, say θ, whch has unfom dstbuton on [ 0,]. 4 Gven these 4 ccodng to taffod and Wells (996) and oyt and Quey (999), the dffeent valuatons among consumes ae a functon of age, educaton level, and clam ecod. 5

6 assumptons, t can be wtten as R R θ. (6) Usng uatons (), (5), and (6), the maxmzaton poblem fo consume can be obtaned by Max W ( ) ( )( ) θ. (7) Ths pape sets out a two-stage game as follows: In the fst stage, both nsuance companes decde smultaneously. Thus, ths stage can be consdeed choce of ualty. Wthout loss of genealty, the ualty that s offeed by nsuance company s hghe than that submtted by nsuance company. That s, >. In the second stage, both nsuance companes decde the nsuance poduct δ smultaneously afte they obseve the ualtes. Thus, ths stage can be consdeed the choce of pce. et the magnal consume who does not dffeentate between two nsuance companes be denoted by θ. Fo smplcty, the esevaton pces fo consumes fo an nsuance poduct ae suffcently hgh to ensue that all consumes ae wllng to puchase. 5 Then ( ) ( )( )( ) ( ) θ (8) whee the fst tem shows the effect of the dffeence between the nsuance pemums, the second tem shows the effect of the dffeence between the nsuance amounts, and the thd tem shows the effect of uncetanty of consume afte puchasng an nsuance poduct. Futhe, all tems n the denomnato nclude ( ). Thus, all of the above effects decease wth nceasng the value 5 Moe pecsely, see ppendx. 6

7 of ( ). uaton (8) means that the consumes n the nteval [,θ ] nsuance company whle those n the nteval [,] 0 puchase fom θ puchase fom nsuance company. ecause θ s unfomly dstbuted on [ 0,], the aggegate demand fo the nsuance poduct n the type maket s θ and fo the nsuance poduct s θ. Two ponts should be noted. Fst, a dffeence n the accdent pobablty leads to not only a dffeence n demand fo aftecae but also a dffeence n the nsuance poduct. Thus, though consumes have the same dstbutonal fom egadless of the type, t does not necessaly mean that θ and θ ae dentcal. econd, n eulbum, 0 < θ < always holds. In othe wods, n eulbum, both nsuance companes have stctly postve demand. 6 ach nsuance company s assumed to be sk neutal and the expected poft functons take the fom ( Π ) ( ) θ ( ) a θ, (9) ( Π ) ( )( ) ( θ )( ) a θ. (0) YI OF T QUIIRIUM In ths secton, a subgame pefect ash eulbum s deved. Ths eulbum s obtaned by backwad nducton (elten, 975). t the begnnng of the eulbum analyss, t s shown that, gven each ualty, both nsuance companes set the pce smultaneously. ubsttutng uaton (8) n uatons (9) and (0), the expected poft functons fo the nsuance companes become: 6 oof of ths s defeed to the ppendx. 7

8 8 Π, a, () Π, a. () The fst-ode optmalty condtons ae 0 Π, (3) { } Π 0, (4) Π 0, (5) { } Π

9 ( ) ( )( )( ) 0. (6) y combnng these euatons, then 7 ( ), (7) 3, (8) ( ), (9) 3. (0) ach nsuance ate p can be denoted by euatons, each eulbum nsuance ate may be wtten as. Thus, usng the above p p ( ), () 3 ( ). () 3 The mplcatons of the above euatons ae as follows: Fst, fom uatons () and (), each nsuance ate ncludes not only the accdent pobablty but also the addtonal costs of dffeentatng ualty. In othe wods, both nsuance companes offe an actuaally unfa pce. econd, as compaed wth uatons 7 Thee ae thee solutons fo uatons (3) to (6). oweve, only uatons (7) to (0) satsfy the second-ode condtons fo the maxmzaton poblem. oof of ths s defeed to ppendx. 9

10 () and (), snce > then p p. Theefoe, fo all {, }, the pce > of nsuance poduct s hghe than the pce of nsuance poduct. Thd, uatons (8) and (0) show clealy that, n eulbum, both nsuance companes sell the full nsuance. hlch and ecke (97) and tgltz (977) have poved that wth pefect nfomaton, n eulbum, all nsuance companes sell the full nsuance. In ths model, t s clea that ths poposton s also satsfed even f poduct dffeentaton pevals. To consde the ualty choces n the fst stage of the game, substtutng uatons (7) to (0) n uatons () and (), each expected poft functon can be wtten as ( ) a Π, (3) 9 4 ( ) a Π (4) 9 whee. ee, denotes the expected numbe n case of an accdent. 4 uppose that < max and mn <. Then one asymmetc subgame 9a pefect eulbum exsts. ence, the nsuance companes set the ualtes as follows: 8 mn, (5). (6) ubsttutng uatons (5) and (6) n uatons () and (), they can be seen that 8 The followng solutons ae poved n ppendx. 0

11 p p ( mn ), (7) 3 ( mn ). (8) 3 uatons (7) and (8) can explan why the pce of the low-sk consume s sometmes hghe than that of the hgh-sk consume n the (Japanese) nsuance maket. In othe wods, even f a consume s low sk, he o she may face a hgh pce to puchase fom nsuance company that offes hgh ualty, and vce vesa. In ode to deal wth ths poblem, ccumstances ae examned unde whch the above stuaton occus. Mathematcally, t needs to confm the exogenous condton that satsfes can be obtaned that p p >. Thus, subtactng p fom p, t p p ( ) ( )( mn ). (9) 3 The fst tem shows the dffeence between the nsuance ates; the second tem s elated to the dffeence between the ualty levels. The fst tem s obvously stctly negatve. ence, t must be satsfed that the second tem s stctly postve wheneve p p >. mplfcaton yelds the condton > 0. Ths condton means that the dffeence between the two accdent pobabltes s elatvely small. oweve, the condton > 0 s meely the necessay condton but not the suffcent condton fo p p >. onseuently, even f ths condton s satsfed, t s possble that p p s not satsfed. Fom (9), > p p s > easly obtaned when ( mn ) s lage. Ths mples that the moe the ualty s dffeentated, the geate the pobablty that p p s satsfed. > OMRTIV TTI In ths secton, seveal comments ae made on the eulbum pces that ae

12 shown n uatons (7) and (8). y uatons (7) and (8), all eulbum pces ae affected by fve knds of exogenous vaable: ) accdent pobablty, ; ) value of an nsuable asset, ; 3) numbe of consumes, ; 4) cost functon fom, a ; and 5) mnmum ualty level, mn one exogenous vaable changes.. ow, consde the case whee only ) accdent pobablty ( ) atally dffeentatng uatons (7) and (8) wth espect to, p > 0, (30) p > 0, (3) p > 0 k fo k. (3) These devatves have a numbe of nteestng featues. Fst, fom uatons (30) and (3), all eulbum pces ae monotone nceasng and convex functons of the own accdent pobablty. Ths esult mples that each pce s a non-lnea (convex) functon of ts own accdent pobablty n spte of a pefectly nfomed nsuance maket. n ntutve explanaton fo ths s povded. n ncease n nceases both the expected nsuance money and the total aftecae cost. xpected nsuance money s a lnea functon of ts own accdent pobablty whose slope s eual to unty, whle the aftecae cost s a convex functon of ts own accdent pobablty because t leads to an ncease, not only n the expected numbe n case of an accdent, but also the eulbum ualty of nsuance company. econd, usng uatons (30) and (3), then p p > and

13 p p >. The change n the nsuance ates of nsuance company s lage than that of nsuance company because t only affects whle nsuance company always sets ts ualty mn. Thd, by uaton (3), all eulbum pces ae monotone nceasng functons of the othe accdent pobablty. Ths esult can be explaned as follows. n ncease n k mples that a consume of type k would be moe lkely to dese hgh ualty. Insuance company ases ts ualty. Ths ncease leads to an ncease n the eulbum pces of both types because of the addtonal ualty cost. Moeove, ts ncease also nceases the nsuance ates of nsuance company. oweve, ths featue stems not fom the exstence of the ualty vaable, but fom the mpossblty of ualty dffeentaton between consume types. If t wee possble to set the ualty n accodance wth consume types, an ncease n k would only affect the pce and ualty of type k. ) value of an nsuable asset ( ) ffeentatng uatons (7) and (8) wth espect to, p < 0, (33) p > 0. (34) uatons (33) and (34) ndcate that all eulbum pces ae monotone deceasng and convex functons of. The ualty cost pe nsuance poduct s constant egadless of the nsuance amount. ccodngly, the moe the nsuance amount nceases, the moe the ualty cost pe nsuance amount deceases. Thus, an ncease n the value of an nsuable asset bngs about a decease n all eulbum pces because, n eulbum,. In othe wods, thee ae economes of scale wth espect to ualty cost. It can be also ntepeted as band gadng n the nsuance nsttuton. 3

14 3) numbe of consumes ( ) The followng popety s eadly vefed: p > 0. (35) The ntuton undelyng ths devatve (35) s eadly shown. n ncease n the numbe of consumes nceases the facton of the consumes who wll be pessng a clam. Then, nsuance company nceases ts ualty and the nsuance ates of both nsuance companes also ncease. 4) cost functon fom ( a ) Fom uatons (7) and (8), the followng devatve can be deved: p < 0. (36) a n ncease n a mples that the cost functon becomes steepe. It leads to a decease n the ualty of nsuance company and both nsuance ates also decease. 5) mnmum ualty level ( mn ) uaton (7) and (8) s patally dffeentated wth espect to mn : p < 0. (37) mn Thus, an ncease n the mnmum ualty level deceases all eulbum pces because t lmts the ange n whch the nsuance companes can dffeentate ualty and then they face an ntensfed pce competton. T FFT OF MIIMUM QUITY TR In ths pape thus fa, the stuaton whee mnmum ualty level s an 4

15 exogenous vaable has been consdeed. oweve, n eal lfe, the egulato often sets the mnmum ualty level endogenously. Fo example, consde the Japanese lfe nsuance contact. In pncple, t bnds the nsuance company to pay nsuance money wthn fve days afte the pocedual condtons have been completed. These contacts ae uled by the egulato and so all nsuance contacts ae almost dentcal. amely, the tme of payment, whch s contaned n the clams handlng pocedues, s the egulato s endogenous vaable. To cte anothe example, snce 996, a lfe nsuance company would be pemtted to ente the non-lfe nsuance maket n Japan, and vce vesa. oweve, the egulato euested lfe nsuance companes that ae gong to ente the non-lfe nsuance maket to employ the own assessos. Ths euement means that a lfe nsuance company must epesent a cetan ualty level when enteng n the non-lfe nsuance maket. In ths secton, the effect of the mnmum ualty standads on socal welfae s analyzed. eveal theoetcal models of mnmum ualty standads have been developed, ncludng eland (979), Ronnen (99) and Vallett (000). common theme of these eseaches s that mnmum ualty standads can educe an exteme ualty dffeentaton and ase ualty. Imposng mnmum ualty standads necessaly mpoves consume welfae. oweve, at the same tme, poduce welfae deceases because not only the pce deceases but also the ualty cost nceases. Thus, t s not clea whethe the mposton of mnmum ualty standads s desable o not. Fst, consde consume welfae. y uatons (8) and (0), all consumes puchase full nsuance. Thus, a consume of type deves the followng utlty fom buyng the nsuance poduct δ. U ( ) θ u W. (38) ubsttutng uatons (7) to (0) n uaton (8), t can be wtten as θ fo {, }. (39) 3 et W be consume welfae. Usng uatons (38) and (39), then 5

16 6 { } { } d u W d u W W, 3 / 0 3 / mn θ θ θ θ. (40) In contast, let W be poduce welfae. It s eual to { } 3 a mn W { } 3 a. (4) Futhe, denote that the socal welfae s W. The fom of W s W W W. (4) ffeentatng uaton (4) wth espect to mn { } { } [ ] W u W u W u W u d dw mn mn a. (43) whee mn u u. ow, accodng to eland (979), to nvestgate whethe the mposton of mnmum ualty standads s desable o not, t needs to confm whethe uaton (43) at 0 mn s postve o not. In ths case, a W u a W u d dw mn mn a W u a W u. (44)

17 The sgn of uaton (44) s ambguous. onsde the stuaton whee uaton (44) s postve. When thee s geate magnal utlty (that s, u () s lage), uaton (44) s postve. Thus, fo the nsuance maket wth small ntal wealth (W s small), a slght slope fo the cost functon ( a s small), and a hgh value of an nsuable asset ( s lage), to mpose mnmum ualty standads may be socally desable. In contast, the effect of accdent pobablty ( ) and the numbe of consumes ( ) have an ambguous effect. These esults can be descbed n detal. Fst, a decease n a leads to an ncease n the ualty of nsuance company and so both nsuance ates se. Then, some consumes, whose θ s elatvely low, have a decease n the utltes but othe consumes have an ncease n the utltes. Futhe, both nsuance companes ncease the expected pofts. oweve, f a becomes smalle than a cetan level, the ualty dvesty becomes too lage and many consumes may have the utltes deceased. Thus, mnmum ualty standads may be socally desable when a s small. econd, consde the case whee W s small and/o s lage. y assumng that each consume s sk avese, even f mnmum ualty standads cause nsuance ates to fall a lttle, the utlty nceases dastcally. Thus, n the stuaton whee W s small and/o s lage, t s advantageous to mpose mnmum ualty standads. Thd, consde why the effects of and these s descbed as follows. n ncease n and/o ae ambguous. The eason fo ases a consume s magnal utlty u ( ) and so the fst tem n uaton (44) nceases. oweve, at the same tme, the second tem n uaton (44) deceases because these nceases lead to an ncease n. ccodngly, whethe o not mnmum ualty standads may be desable s unclea. Fnally, the esults pesented n ths secton pont out that mnmum ualty standads may be desable when each consume s not wealthy. Many fms have moe wealth than ndvduals. Thus, geneally speakng, mnmum ualty standads may be desable fo ndvduals but not fo fms. Ths ntepetaton concdes wth the fact that the Japanese egulato mposes moe sevee egulatons upon the nsuance maket fo ndvduals than fo fms. 7

18 OUIO RMRK ccodng to chlesnge and chulenbug (993), each nsuance poduct pe se s dentcal but the nsuance companes that sell ths dentcal nsuance poduct ae not necessaly dentcal. In ths study, ths pont can be ntepeted as aftecae and each nsuance company ncopoates t nto ts own stategy. The Japanese nsuance maket s nvestgated whee competton s not only on pce (nsuance ate) but also on ualty (level of aftecae). Thus, ths pape consdes the nsuance maket wth vetcal dffeentaton. In ths dscusson, some lght s shed on seveal eal lfe uestons. Fo example, n the Japanese nsuance maket, thee ae two types of nsuance company. One s domestc, the othe s foegn. It s well known that domestc nsuance companes set elatvely hghe pces than foegn nsuance companes. Many foegn nsuance companes compete aggessvely on pces. In contast, domestc nsuance companes focus on sevce competton nstead of pce competton. amely, domestc nsuance companes tend to offe hgh pces and hgh ualty, whle foegn nsuance companes tend to offe low pces and low ualty. Ths fact concdes wth the esults of ths pape. 9 In anothe example, as descbed above, mnmum ualty standads may be desable fo ndvduals but not fo fms. ence, to mpose a dffeent type of egulatons between ndvduals and fms may be atonal. uch nsstence can be seen n a consdeable numbe of woks n the nsuance feld. oweve, the esults n ths pape ae dffeent fom othe eseaches. atculaly, othes emphasze the weakness of ndvduals n tems of nfomaton gap, baganng powe, and level of knowledge. In ths pape, whethe mnmum ualty standads ae desable o not s decded by the level of wealth. Thus, even f thee wee the consumes who have pefect knowledge and stong baganng powe, the egulato should mpose some egulatons fo ndvduals. oweve, the above analyss s ncomplete on seveal ponts. The followng two ponts ae patculaly nteestng. Fst, n ths model, both nsuance companes and consumes have pefect nfomaton. Thus, all nsuance companes do not need to deal wth advese selecton. oweve, n the eal wold, because they cannot vefy each consume s type, nsuance companes may have to decde pce and ualty due to tell the tue types. Futhemoe, assumng pefect nfomaton s also elated to the consume sde. Recently, pce nfomaton has become cleae because mao consumes can easly compae pces usng the Intenet. oweve, some 9 Thee s good evdence that domestc non-lfe nsuance companes have fa moe sevce centes to povde clam-handlng pocedues than do foegn companes. 8

19 uncetanty emans. That pce cannot be decded as long as the polcy tem contnues because consumes get polcy dvdends at least once a yea. The level of ualty n the tmng of contacts does not necessaly concde wth that n the tmng of accdents because the polcy tem s nomally long. econd, ths pape explctly analyzes the nsuance maket wth vetcal dffeentaton whee aftecae s the only ualty vaable. oweve, thee ae, n fact, othe factos that can be consdeed as ualty vaables. Fo example, the sevce that s offeed ex ante o ntem the polcy tem seems to be a dsegaded facto. 0 Fnancal obustness (degee of solvency) wll become mpotant. Futhe, hozontal dffeentaton may need to be ncopoated nto the model. These ponts ae stll open uestons. Much addtonal wok s eued to elax the above estctons and they ae left to possble futhe eseach. oweve, seveal esults n ths pape nclude mpotant poltcal mplcatons fo the nsuance maket. 0 Fo example, osby and tephens (987) have confmed that consumes who had been ecevng n hgh-level sevces befoe they puchase ae wllng to pay hghe nsuance pemums than consumes who had not ecevng. Ths has been efeed to n pevous studes, fo example, chlesnge and chulenbug (99), tchett (994) and oyt and Quey (999). 9

20 IX If a consume of type does not puchase an nsuance poduct, the consume's cetanty euvalent, 0, s ( ). () 0 W Thus, the consume's puchasng (o patcpatng) condton should be wtten n ths fom: IO 0 () o fnally ( )( ) θ 0. (3) To nduce the condton whee all consumes puchase an nsuance poduct, one can confne attenton to the consume whose θ s zeo. ubsttutng θ 0 nto uaton (3) and solvng that fo : ( ) ( )( ). (4) ence, all consumes puchase an nsuance poduct f and only f uaton (4) holds. IX uppose that θ. Then, the demand fo nsuance company n the type maket s zeo. oweve, now nsuance company has an ncentve to alte ts ~ ~, ~ δ, whee ~ ( ) ε (ε s a own nsuance poduct to { } vey small postve numbe) and ~, because t always gets addtonal 0

21 expected poft. Thus, θ s mpossble and, hence, < θ. mlaly, t s easy to pove 0 > θ. IX In ths appendx, the smultaneous euatons, shown by uatons (3) to (6), ae solved. Fst usng uatons (3) and (5), the followng euatons ae deved. 3, () 3. () ubsttutng uatons () and () n uaton (4), t can be expessed as 0 3. (3) mla to above, substtutng uatons () and () n uaton (6), then 0 3. (4) In ode to hold uaton (3), ethe euaton must satsfy the followng: 0, (5) 0. (6)

22 In ode to hold uaton (4), ethe euaton must satsfy the followng: 0, (7) ( ) ( )( )( ) 0. (8) ccodngly, the combnatons to be satsfed wth both uatons (3) and (4) ae fou. oweve, the combnaton (6) and (8) s neve satsfed because the sum of uatons (6) and (8) s 3( ) 0 appendx s to check the emanng thee cases. ase : combnaton (5) and (7) It s easy to obtan that. Thus, the emande of ths. (9) Thus, both nsuance companes offe full nsuance. ase : combnatons (6) and (7) Fom uaton (7), t s gven by. (0) ubsttutng uaton (0) n uaton (6), t becomes ( ) ( ). () ecause ( ), ( ) s an mpope soluton.

23 3 ase 3: combnaton (5) and (8) Fom uaton (5), t s gven by. () ubsttutng uaton () n uaton (8), t becomes 4. (3) lthough thee ae thee knds of solutons n these smultaneous euatons, ase s the only soluton fo the maxmzaton poblem. The othes ae meely saddle-pont solutons. To pove that, fst defne the second-ode condtons fo the maxmzaton poblem fo nsuance company as follows: 0 < Π, 0 < Π, 0 > (4) whee Π Π Π. ecause nsuance company has the same fom, ts condtons ae omtted. To confm the second-ode condton, the followng euatons ae deved. Π, (5) { } Π, (6)

24 ( Π ) { }. (7) uatons (5) and (6) ae always stctly negatve. oweve, the sgn of s unclea. et us fst consde ase. The followng euaton can be wtten as 3 ( ) ( ) > 0. (8) Fom uaton (8), ase s always satsfed wth the second-ode condtons fo the maxmzaton poblem. In contast, consde ase. The followng euaton can be wtten as ( ) < 0. (9) Fom uaton (9), ase s neve satsfed wth the second-ode condtons fo the maxmzaton poblem. mlaly, t can be confmed that ase 3 s also neve satsfed. ence, ase s the only soluton fo the maxmzaton poblem. Fnally, substtutng uaton (9) nto uatons () and (), all eulbum nsuance pemums (7) and (9) ae deved. IX uppose that both nsuance companes choose the same ualty level ˆ. Then n the second stage they must offe an actuaally fa ate. Thus, the expected pofts ae ( ( ˆ )) Π ( ˆ ) aˆ Π. () Fo any ualty level ˆ > mn, ethe nsuance company has an ncentve to lowe ts own ualty because t leads not only to an ncease n evenue but also to a decease n ualty cost. 4

25 Fo ualty level ˆ mn, the expected pofts ae ( Π ( )) Π ( ) mn mn offes anothe ualty level a mn ~ mn. ow, consde that nsuance company ε, whee ε s a vey small postve numbe. Then, the expected poft of nsuance company s gven by ~ ( ( ~ 4 )) ( ~ a Π mn ). () 9 In ode to pove the nonexstence of a symmetc eulbum, t must be shown that Π ( ) s lage than Π. ubtactng Π fom ( ) ~ t can be seen that ( ( )) Π ( ) mn mn ( ~ ) 8 9a( ~ ) Π ~, Π ~ mn mn { mn } 8. (3) Thus, f uaton (3) s stctly postve, then 8 9 ~ a mn >. (4) ( ) 0 In addton, uaton (4) becomes 8 ε < mn. (5) 9a The neualty uaton (5) s satsfed f mn <. ence, n the case of ˆ mn, ethe nsuance company has an ncentve to change own ualty level. 3 3 In contast, f mn, thee s no ncentve to change the ualty level fom ˆ mn. Thus, t has an unnteestng symmetc eulbum {, mn mn }. 5

26 Fom the above descpton, attenton can be confned to the asymmetc case, that s >. It s easy to calculate that ( Π ) 9 a < 0 mn, (6) ( Π ) 4 4 a 0 9 9a. (7) RFR ege,..,.r. Klendofe, and. Kuneuthe, 989, ynamc Model of the Tansmsson of ce Infomaton n uto Insuance Makets, Jounal of Rsk and Insuance, 56, osby,.. and. tephens, 987, ffects of Relatonshp Maketng on atsfacton, Retenton, and ces n the fe Insuance Industy, Jounal of Maketng Reseach, 4, ahlby,.g. and.. West, 986, ce speson n an utomoble Insuance Maket, Jounal of oltcal conomy, 94, oepnghaus,.i., 99, n nalyss of omplant ata n the utomoble Insuance Industy, Jounal of Rsk and Insuance, 58, 0-7. hlch, I. and G.. ecke, 97, Maket Insuance, elf Insuance and elf-otecton, Jounal of oltcal conomy, 80, Gavell,., 994, Remuneatng Infomaton ovdes: ommssons Vesus Fees n fe Insuance, Jounal of Rsk and Insuance, 6, oyt, R.. and T.J. Quey, 999, Qualty and onsume atsfacton: What Factos Really Matte n Insuance? Jounal of Intenatonal Insuance,, Jung,.F., 978, utomoble Insuance Rates n hcago, Illnos, Jounal of Rsk and Insuance, 45, eland,.., 979, Quacks, emons, and censng: Theoy of Mnmum Qualty tandads, Jounal of oltcal conomy, 87, Mathewson, F.G., 983, Infomaton, each, and ce Vaablty of Indvdual fe Insuance ontacts, Jounal of Industal conomcs, 3, Mcowell,., 989, eegulaton and ompetton n the Insuance Industy (ew Yok: Quoum ooks). att, J.W., 964, Rsk veson n the mall and n the age, conometca, 3,

27 tchett, T.., 994, otental ce and oduct Qualty hanges fo Insuance elveed by anks, n: Gustavson,.G. and.. angton (eds.), Insuance Rsk Management, and ublc olcy ssays n Memoy of Robet I. Meh (oston: Kluwe cademc ess). osey,.. and. Yavas, 995, each Model of Maketng ystems n opety-ablty Insuance, Jounal of Rsk and Insuance, 6, Ronnen, U., 99, Mnmum Qualty tandads, Fxed osts, and ompetton, Rand Jounal of conomcs,, taffod, M.R. and.. Wells, 996, The ffect of emogaphc Vaables on eceved lams evce Qualty, Jounal of Insuance Issues, 9, chlesnge,. and J.-M.G. chulenbug, 99, each costs, wtchng osts and oduct eteogenety n an Insuance Maket, Jounal of Rsk and Insuance, 58, chlesnge,. and J.-M.G. chulenbug, 993, onsume Infomaton and ecsons to wtch Insues, Jounal of Rsk and Insuance, 60, elten, R., 975, Reexamnaton of the efectness oncept fo ulbum onts n xtensve Games, Intenatonal Jounal of Game Theoy, 4, tgltz, J.., 977, Monopoly, on-nea cng and Impefect Infomaton: The Insuance Maket, Revew of conomc tudes, 44, Tsutsu, Y., M. ekguch, and T. hano, 000, The emum-vdend ompetton n the ewa Japanese fe Insuance Industy: Game-Theoetc Intepetaton, Japanese conomc Revew, 5, Vallett, T.M., 000, Mnmum Qualty tandads unde ounot ompetton, Jounal of Regulatoy conomcs, 8, Wells,.. and M.R. taffod, 995, evce Qualty n the Insuance Industy: onsume eceptons Vesus Regulatoy eceptons, Jounal of Insuance Regulaton, 3,