The vertical differentiation model in the insurance market

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1 The vertal dfferentaton model n the nsurane market Mahto Okura * bstrat Ths note exlores the vertal dfferentaton model n the nsurane market. The man results are as follows. Frst, the eulbrum re dfferental s not a lnear funton of the hghest ualty valuaton (adent robablty) and the mum and mum ualty dfferentals. Seond, a hgh ualty nsurane frm does not always reeve greater eulbrum exeted roft, even f ts average ost s the same as that of a low-ualty nsurane frm. Fnally, a hange n the hghest ualty valuaton has an ambguous effet on the eulbrum exeted roft dfferental. Keywords: Vertal dfferentaton model, Insurane JEL lassfaton: D, G, L. Introduton It s well known that vertal dfferentaton s a useful strategy for addressng fere re ometton (e.g., Shaked and Sutton, 8; Shy, 5). For ths reason, vertal dfferentaton ommonly exsts not only n markets for tangble goods but also n markets for ntangble goods, suh as fnanal serves. For examle, Shlesnger and Shulenburg (, ) argue that the level of lams handlng roedures s a ualty fator n the nsurane market. In fat, followng an adent, all onsumers wsh to reeve knd and frendly adve and ayment of ther nsurane as * Mahto Okura s an assoate rofessor, faulty of eonoms, Nagasak Unversty, Jaan. Emal: okura@nagasak-u.a.j

2 soon as ossble. Thus, even f the roduts er se do not have ualty fators, onsumers an stll ereve ualty through the lams handlng roedure offered by eah frm. From ths vewont, nsurane frms omete not only on re (the nsurane rate) but also on ualty (the level of lams handlng roedure). There are at least two good reasons for exlorng the vertal dfferentaton model n the nsurane market. The frst s that very few studes nludng vertal dfferentaton models fous on the nsurane market, though some of the extant lterature does onern horzontal dfferentaton (e.g., Shlesnger and Shulenburg, ; Hofmann and Nell, 8). The seond reason s that onsumers an enjoy the serves from lams handlng roedures only f an adent takes lae. Ths suggests that the robablty of an adent an hel exlan the ualty valuaton arameter of a vertal dfferentaton model n the nsurane market. It s then easy to dedue that onsumers wth a hgh adent robablty wsh to urhase nsurane roduts from a hgh ualty nsurane frm and ve versa. From ths ersetve, even f the average ost (amount of nsurane er lam) s the same aross nsurane frms, the exeted average ost n a hgh ualty nsurane frm s hgher than n a low ualty nsurane frm. In other words, the ualty valuaton arameter affets not only onsumer demand for nsurane but also the exeted average ost of the nsurane frms themselves.. The model Suose there are two rsk-neutral nsurane frms ( and ) n the market. Wth loss of generalty, we assume that nsurane frm s a low ualty nsurane frm and nsurane frm s a hgh ualty nsurane frm. Here, [, ] denotes the ualty (level of lams handlng roedures) offered by the nsurane frm {, }, where [ ) and,, reresent the mum and mum levels of ualty, resetvely. Let be Shlesnger and Shulenburg () s the only known vertal dfferentaton model onernng nsurane. However, they do not onsder the desons of nsurane frms on re and ualty. Furthermore, they mltly assume that all onsumers have the same ualty valuaton arameters.

3 the adent robablty of onsumers, assumed to le on the unform dstrbuton [ ],, where ( ], s the hghest adent robablty of onsumers. Let denote the re (nsurane rate) offered by nsurane frm. The utlty of eah onsumer s assumed searable n re and ualty, as n Mussa and Rosen (8). The utlty funton of onsumer j urhasng nsurane from frm s then j j u. lso, assume that eah onsumer has to urhase one nsurane rodut from a more desrable nsurane frm. Let be the margnal onsumer who does not dfferentate between the two nsurane frms. Usng the onsumer utlty funton, the margnal onsumer s. () The exeted roft funtons of the two nsurane frms, denoted by, are sds, () sds, () where, reresents the average ost unrelated to the level of ualty. Ths note sets out the followng two-stage vertal dfferentaton model. In the frst stage, both nsurane frms smultaneously hoose ther ualty ( ). fter they observe eah ualty, they smultaneously hoose ther re ( ). The subgame erfet eulbrum s a sutable eulbrum onet for ths model and s derved though bakward nduton. From Es. (), () and (), the frst-order ondtons n the seond stage are, (). (5)

4 From Es. () and (5), the eulbrum res are derved as follows: {, () {. () To onsder the frst stage of the model, by substtutng Es. () and () nto Es. () and (), the exeted roft funtons of the frms an be exressed as 5 8, (8) 5 8. () From Es. (8) and (), the frst-order ondtons n the frst stage are { } 5 8 <, () { } 5 8 >. () From Es. () and (), and are the eulbrum ualtes.. Imlatons In order to llustrate roerly the mlatons of our model, we frst rovde the results of the seal vertal dfferentaton model where the ualty valuaton arameter s unrelated to the exeted average ost, that s, the roft funtons are and There are two solutons for Es. () and (5). However, only ths soluton satsfes the seond-order ondtons.

5 ( )( )( ), where ( ) ( ). In the same manner, we an derve the followng eulbrum values: ( ) ( ), ( ) ( ),, ( ) ( ), ( ) ( ). Usng Es. () and (), the eulbrum re dfferental n our model s, ( ) ( ) ( ). () In ontrast, the eulbrum re dfferental n the seal vertal dfferentaton model s ( ). () From Es. () and (), we know that an nrease n and ( ) exands the re dfferental n both models. However, unlke E. (), E. () s not a lnear funton of and ( ), beause the hange n and ( ) through the hange n the margnal onsumer. hanges the exeted average ost Further, from Es. (8) and (), the eulbrum exeted roft dfferental n our model s 8 5 ( ) ( ) ( ). () In ontrast, the eulbrum roft dfferental n the seal vertal dfferentaton model s ( ). (5) When usng E. (), whh nsurane frm reeves more eulbrum exeted roft s ambguous, even f the average ost s the same n our model; n ontrast, usng E. (5), the hgh ualty nsurane frm always reeves more roft n the seal vertal dfferentaton model. The reason s that the hgh ualty nsurane frm has to aet onsumers wth a hgh adent robablty, and so ts exeted average ost beomes hgher. When s hgher and ( ) s lower, average exeted ost s hgher than roft through hgh ualty sulyng, and a low ualty advantage aears n the nsurane market. lso, an nrease n ( ) exands the 5

6 eulbrum (exeted) roft dfferental n both models. In ontrast, a hange n has an ambguous effet on the eulbrum exeted roft dfferental n our model, whle always exands the eulbrum roft dfferental n the seal vertal dfferentaton model, beause an nrease n nreases not only rofts through dfferentaton but also losses through the nrease n exeted average ost.. Conluson Ths note exlored the vertal dfferentaton model n the nsurane market. The man results are as follows. Frst, the eulbrum re dfferental s not a lnear funton of the hghest ualty valuaton (adent robablty) and the mum and mum ualty dfferentals. Seond, a hgh ualty nsurane frm does not always reeve greater eulbrum exeted roft, even f ts average ost s the same as that of a low ualty nsurane frm. Fnally, a hange n the hghest ualty valuaton has an ambguous effet on the eulbrum exeted roft dfferental. knowledgements The author gratefully aknowledges the helful omments of artants n the Non-lfe Insurane Study Grou sonsored by the Non-lfe Insurane Insttute of Jaan. The author would lke to aknowledge the fnanal suort by the Mnstry of Eduaton, Sene, Sorts and Culture, Grand-n-d for Young Sentsts (),. Referenes Hofmann,., Nell, M., 8. The Imat of Intermedary Remuneraton n Dfferentated Insurane Markets. Workng Paers on Rsk and Insurane, Hamburg Unversty. htt:// Mussa, M., Rosen S., 8. Monooly and rodut ualty. Journal of Eonom Theory 8,.

7 Shlesnger, H., Shulenburg, M.G.V.D.,. Searh osts, swthng osts and rodut heterogenety n an nsurane market. Journal of Rsk and Insurane 58,. Shlesnger, H., Shulenburg, M.G.V.D.,. Consumer nformaton and desons to swth nsurers. Journal of Rsk and Insurane, 5 5. Shaked,., Sutton, J., 8. Relaxng re ometton through rodut dfferentaton. Revew of Eonom Studes,. Shy, O., 5. Industral Organzaton: Theory and latons. (MIT Press, Cambrdge, M).