When prices hardly matter: Incomplete insurance contracts and markets for repair goods

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1 When res hrdly er: Inolee nsurne onrs nd rkes for rer goods Mrn Nell Professor of Rsk nd Insurne Unversy of Hurg, Gerny hone: , fx: Andres Rher Asssn Professor of Insurne Illnos Se Unversy hone: , fx: Jörg Shller Asssn Professor of Orgnzon Theory WHU Oo Beshe Grdue Shool of Mngeen, Vllendr, Gerny hone: , fx: oerg.shller@whu.edu Ths drf: Jnury 3, 005

2 When res hrdly er: Inolee nsurne onrs nd rkes for rer goods Asr Ths er looks rkes hrerzed y he f h he dend sde s nsured. In hese rkes onsuer urhses good o oense onsequenes of unfvorle evens, suh s n den or n llness. Insurne oles n os lnes of nsurne se ndeny on he nsured s ul exenses,.e., he nsured would e rlly or oleely reursed when urhsng ern goods. In hs seng we dsuss he neron eween nsurne nd rer rkes y fousng, on he one hnd, uon he develoen of res nd he rke sruure n rkes wh nsured usoers, nd, on he oher hnd, he resulng klsh on ol nsurne onrng.. Inroduon Ths er s onerned wh rkes hrerzed y he f h he dend sde s nsured. In hese rkes, whh wll e referred o s rer rkes, onsuer urhses good or rer serve o oense onsequenes of ern unfvorle evens, suh s n den or n llness. Exles re segens of he rkes for r rer serves nd renl rs s well s he rkes for edl serves nd hreul rodus. The f h onsuers re nsured, would y self no use eono roles so long s nsurne ones re le o wre olee onrs ssgnng ndeny yens drely o ny ossle se of he world. Tylly, hough, he se of oenl ses of he world s rher olex lyng h wrng olee onrs would eher e ossle or

3 use dsroorone rnson oss. For exle, olee onr n uo nsurne would hve o resely defne he ndeny yle n se of ny ossle dge o he nvolved uos. As he ler s usully no rels oon, nsurne oles n os lnes of nsurne se ndeny on he nsured s ul exenses,.e., he nsured would e rlly or oleely reursed when urhsng ern goods. In erfe rer rkes he f h onsuers re nsured would hve no on he ul res, sne res orresond o rgnl os. However, s erl work suggess, nsurne desgn hs or uon rer rkes. D nde h rer rkes re ofen erfe nd, hus, res exeed rgnl oss. A srghorwrd ronle for hs s rke ower whh n resul fro rodu dfferenon. For he sngle onsuer, rnson oss nurred n he roess of onsung rer goods ofen dffer ross sulers, for nsne deendng on he loon of sulers relve o he onsuer. In he onex of r rer shos or renl rs, n llusron of hs n e seen n sl referenes. Anoher exle n e oserved n rkes for hreul rodus nd helh serves, where rke ower resuls fro onsuers desgned referenes for ern sulers. Gven suh referenes, s n orn sk o nlyze he lons of nsurne for onsuers dend desons n erfe rer rkes. An llusrve exle The Gern r renl rke I n ofen e oserved h n rer rkes re dsrnon eween nsured nd unnsured onsuers exs nd h res re sgnfnly hgher for nsured onsuers. As n exle for hs, onsder he Gern r renl rke. See, for exle, Anderln nd Fell (994, Segl (999, Mskn (00.

4 3 In hs rke or segen of nsured onsuers n e dened: The usness n den susue renl rs ouns for roughly 30 % of he enre rke. Consuers n hs segen eorrly susue vehle h ws dged n n den. They re eher oensed y her ollson loss nsurer or hey hve vld l for susue r gns he oher ry or, effevely, he oher ry s lly nsurer. 3 Therefore, hs segen onsss exlusvely of onsuers whose renl r exenses re overed y n nsurne ony, whle onsuers exenses n he renng shre of he rke re unnsured. In he 990s, dfferenes n res for susue nd non-susue renl rs n he Gern rke ould e esly nvesged, s rng nforon for hese segens were deerned nd ulshed on regulr ss. 4 The d re olleed for dfferen r lsses nd dfferen z ode res nd onss of nforon fro he os oulr rffs. The followng le lss verge res fro he yers 997 hrough 999 for he os frequenly rened r lss n 00 rndoly hosen z ode res. 3 4 See Bundesvernd der Auovereer e.v. [Assoon of Cr Renl Cones], Auovereung, Düsseldorf 998. Plese noe h uo lly nsurne (whou ny onsurne s ndory n Gerny. Therefore, n los ny se, hs lly l s overed hrough nsurne. The EuroxShwke GH ony regulrly ulshed survey onernng he res for renl rs n Gerny, whh dsngushed eween he den susue usness nd he so lled free usness nd reored he serely.

5 4 Tle : Averge res n he Gern renl r rke (r lss 5, 00 rndoly hosen z ode res Yer Dly Re Weekly Re Susue Crs Non-Susue Crs Dfferene Susue Crs Non-Susue Crs Dfferene DM ( DM ( % DM ( DM ( % DM ( DM (4. 5. % 44.8 DM ( DM ( % DM ( DM ( % 6.56 DM ( DM ( % Soure: Shwke-Bewerung GH & Co KG, ShwkeLISTE-Auoeressegel, Osnrük 997, 998, 999. Durng he sle erod, res n he susue r usness exeeded he res for nonsusue rs y %. More resely, hese nuers n e onsdered lower ounds for he ul re dfferenes, s he non-susue rffs were dused y ens of generl ddonl ollson overge oonen. 6 Surrsngly, only few heorel ers so fr hve del wh he nerdeendenes eween nsurne nd rer rkes. Freh nd Gnsurg (975, for exle, ddress he queson of how, n onools helh re rke, dfferen yes of nsurne enefs ffe re nd quny. They fnd, ong oher resuls, h n ny se oh reers wll nrese, wh res endng o nfny for he se of olee nsurne. However, sne, e.g., he rkes for edl serves or r rer serves ylly hve n olgools or os 5 6 The vlues gven n rkes re he erl sndrd devons. Ths exr re oonen ws dded, sne res n he Gern susue r rke generlly nlude lly s well s ollson nd orehensve overge, whle res for non-susue rs ofen only nlude lly nsurne nd ern ddonl rl overge, u he vlle d dd no nlude he ul rese rnge of nsurne overge. Therefore, for he non-susue r res s gven n he le, here s endeny of oversng he orre vlues.

6 5 sruure, he resuls of Freh nd Gnsurg (975 do no ure he suon n os of he rer rkes we re neresed n. Gynor e l. (000, nlyze he nerdeendene eween he degree of oeon n helh re rkes nd he exen of exess onsuon due o nsurne. Ther resuls nde h even n he resene of nsurne-ndued hnges n re elsy, onsuers enef fro nresed oeon n helh re rkes. The exsng reled erl lerure, whh lso for he os r ddresses he dend for helh re nd hreul rodus, s exensve. Mos of he fndngs re srghorwrd nd orresond o he heorel resuls enoned ove. For nsne, Newhouse e l. (993 found h ens wh full nsurne overge used sgnfnly ore helh re hn hose who hd o o-y drely. (The sudy lso showed h he dfferen nsurne lns he rng households hd een ssgned dd no sgnfnly ffe her helh suon. Hellersen (998 onenres on hysn s oson s n gen. Even hough her fndngs do no nde h n ndvdul en s nsurne overge ffes he resron erns of rulr hysn, she shows h he dsruon of yes of overge ong hysn s ens s orn for he lkelhood of resrng geners (s oosed o rde-ne drugs. In reen er, Pvnk (00 nlyzes how reduon of nsurne overge nfluenes hreul rodu res. Her resuls show h hese res derese onsderly s ens ou-of-oke exenses nrese. Severl sudes y Feldsen show h hysns n edl rkes rse her fees nd rove her rodus when nsurne overge eoes roder, nd even non-rof hosls resond o n nrese n nsurne y nresng he sohson nd he re of her serve (Feldsen 970, 97. More ornly nd roly soewh uzzlngly frs glne,

7 6 erl nlyss ndes h reduon of he ul dend of nsurne overge would ndue welfre gn,.e. ndvduls urhse oo uh nsurne (Feldsen 973, he ssue ws revsed y Feldn nd Dowd 99. Ths s surrsng, s one would exe h workng nsurne rkes rovde he ol oun of overge. Feldsen suggess h hs s due o rsoner s dle, s Peole send ore on helh euse hey re nsured nd uy ore nsurne euse of he hgh os of helh re. 7 One of he gols of hs er s o rovde heorel exlnon for hs fndng, whh onenres on he sruure of n nsurne rke, where no nforon syeres or rnson oss re resen nd overge s rovded urlly fr res. The reson why he neron eween nsurne nd rer rkes hs no ye een suded ore exensvely, resuly n e seen n he yl ereon of nsurne n he eonos lerure: Insurne onrs re usully nerreed s sef knd of fnnl onr, n whh he nsured n reurn for he reu qures l uon fuure seonngen yens. Mos resely, hs hs een sed y Arrow: nsurne s he exhnge of oney now for oney yle onngen on he ourrene of ern evens (Arrow, 965,. 45. Aordng o hs vew, nsurne onrs re onsdered olee n he sense h he oun of ndeny n e drely ed o he ourrene of ses of he world. However, s hs een sed ove, hs s no wh we oserve n orn lnes of nsurne, where he nsured, n se of loss, reeves overge sed uon hs or her ul rer exenses. Therefore, hese nsurne onrs re nolee, s he nsurer s yens re no unguously gven nd, n rulr, deend on he res for rer serves. 7 Feldsen (973,. 5.

8 7 In hs er, we dsuss he neron eween nsurne nd rer rkes y fousng, on he one hnd, uon he develoen of res nd he nuer of sulers n rkes wh nsured onsuers, nd, on he oher hnd, he resulng klsh on ol nsurne onrng. To kee hngs s sle s ossle, we ssue h no nforon syeres exs nd h nsurne s vlle urlly fr reus. Frons, however, exs n he rer rke. We onsder rer rke wh rodu dfferenon whh rovdes he sngle suler wh ern sl rke ower. The odel frework eloyed here s sed uon n roh nrodued y Slo (979. Bslly, he fous s on ndesrle onngenes n nsurne. We re neresed n he of nolee nsurne onrs on rer rkes. As he nroduon of nolee onrs ens susnl erfeness nd euse our nlyss s suosed o onenre on hs role, we wll sn fro oher erfeons n he nsurne rke. In onrs o he exsng lerure, we lso sudy new se of he role onernng he ol sruure of nsurne rkes. A reo-effen nsurne onr xzes he exeed uly of onsuers under furher onsrns. The n sk for he nsurer n he onsdered onex s o lne he rde off eween he rsk lloon nd he nsurne ndued re effe on he rer rke. Bu he lng effe of onsurne re on he rer rke re level deends on he rke shre of he offerng nsurne ony. In n os rke sngle nsurer s onr desgn only hs rgnl on he rer rke nd s re level. Consequenly, he equlru onsurne wll nrese n he rke shre of rulr nsurer or derese n he nuer of nsurne ones resevely. The render of he er s orgnzed s follows. Seon nrodues he odel. In Seon 3 we resen dfferen enhrks for he followng nlyss. Seon 4 dsusses he

9 8 of nolee nsurne onrs on he sruure of he rer rke, whle seon 5 ddresses effes n he nsurne rke. Seon 6 onludes.. The odel frework Our nlyss fouses on he ol nsurne desgn nd he nuer of frs n rer rkes wh nsured onsuers. We ssue h onsuers hve heerogeneous referenes. These referenes re nerreed s eng used y onsuers sl dsruon. We onsder n sulers, denoed,..., n h offer good resevely rer serve. Eh ony offers rer serve he re nd he sulers oee n res l Berrnd. The onsuers wh n nl welh of w 0 fe he rsk of loss wh roly π. In se of loss he sulers offer one rer un, whh fully resores he loss, u onsuers fe rnsoron os h nreses n he dsne x o he suler. The odel frework s sed uon he rulr y odel of Slo (979, where onsuers re unforly nd onnuously dsrued long rle wh ereer equl o / π. 8 Consuers hve uly funon U u( w x, where he uly s ddvely serle n he rer serve nd he rnsoron oss. w reresens he fnl welh of onsuers exludng ny rnsoron os. The onsuers re ssued o e rsk verse wh rese o he rer rsk nd rsk neurl onernng he rnsoron os. Therefore, u ( s we-dfferenle uly funon wh u ( > 0, u ( < 0. In he nsurne rke rsk-neurl nsurers, denoed,...,, sulneously offer onrs C ( α, wh onsurne re nd resulng ndeny of ( he fr reu α π (. We furher ssue h onsuers sufferng 8 Ths ssuon les h he ex os sze of he rer rke, fer he relzon of losses, s one.

10 9 fro loss lwys derve surlus fro onsung un of he rer good. Exly one un s urhsed. Through hese ssuons we sn fro he role of ex os orl hzrd (Puly 968, s he exen of urhsed rer serves s ndeenden of he oun of overge. Ths s lusle n suons where only one rer un s neessry nd overonsuon hs no vlue for onsuers. Assung h unnsured onsuers derve surlus fro urhsng he rer serve les h nsured onsuers wh onsurne onr srly refer o dend he serve n se of n den. C Sne n hs odel rke enry nd re re desons vrles, u sulers usness loons (or ore generl: rodu dfferenon re no, n ssuon s needed onernng he os-enry dsruon of sulers on he rle. We ssue xu rodu dfferenon,.e., for he sef odel onex of hs er, h sulers re equdsnly sred round he rle. Ths rese s founded on he resuls of D Asreon e l. (979, who hve shown h n he Hoellng (99 lner y frework sulers would hoose xu rodu dfferenon (onrsng Hoellng s orgnl resuls,.e. hey would loe her usnesses s fr fro eh oher s ossle. As he erssene of hs resul for he se of nsured onsuers s no ovous, we ddress hs ssue n he Aendx. The sequene of he onsdered ge eween nsurers, onsuers nd sulers s s follows: A sge, eh of he nsurne ones offers n nsurne onr C. Then sge, he oenl enrns n he rer rke sulneously hoose wheher or no o ener he rke. Referrng o he xu dfferenon resul fro he Aendx we resue h sulers h enered re equdsnly dsrued on he rle. As we nlyze he role of he nuer of sulers enerng he rke, we ssue h he oenl enrns fe fxed enry

11 0 oss of f > 0. Beuse of he free enry ssuon he equlru rof of enerng frs s zero. Fnlly, sge 3 he sulers h hve enered se her res, gven her loons. 3. Sol o As referene on for he followng nlyss, we ke look dfferen enhrk suons. Le us frs sr wh suons where olee nsurne onrs re fesle. These onrs nd he ssoed ndeny n e ondoned uon ny ossle se of nure. Under suh del rusnes he ol nsurne rrngeen s srghorwrd: sne nsurne ones n ne he (equlru rke re for rer un, he ndeny orresonds o hs re. Thus, he rer rke s neher ffeed y nsurne onrs nor y he sruure of he nsurne rke. Frs Bes When olee nsurne onrs re fesle, frs es rsk lloon n e rehed v full nsurne onr. However, one of he n resuls of he Slo odel s h n equlru oo ny sulers ener he rer rke. Thus, when he sruure of he rer rke s endogenous, overrdng he Slo oeon nd verlly negrng he rer rke leds o frs es suon. Sne onsuers re fully nsured under he frs es nsurne onr, res re rrelevn fro welfre erseve. The only reson for overrdng he oeon n he rer rke s o redue he nuer of oerng rer serve sulers. A onooly nsurer or olon of ll nsurne ones n eslsh rer serve nework wh frs es nuer of rer shos whh nze he su of sndng exenses, onsuers rnsoron os nd rer exenses.

12 n n nf n x dx n. ( 0 Therefore, he frs-es nuer of sulers FB n s n FB. ( f Seond Bes In seond es suon, olee nsurne onrs re sll fesle, u due o legl or oher resrons, nsurne ones re no le o overrde he oeon n he rer rke. As n he frs es suon, he rsk lloon s sll frs es. However, s onsequene of he Slo odel n equlru oo ny sulers ener he rke. Agn, sne nsurne onrs ondon uon he se of nure, he re effe s rrelevn. Only he nresed nuer of sulers leds o welfre loss ored o he frs es suon. Thrd Bes A furher welfre loss s nurred when nsurne onrs re nolee. The ol nolee nsurne onr rdes off he nsurne-ndued re effe on he rer rke nd rsk lloon. As we wll show n he followng seons, he sruure of he nsurne rke wll hve or on he rer rke s well s on sol welfre. 4. Effes n he rer rke Srng wh he re oeon sge 3, we ssue h n sulers hve enered he rke. In hs suon, onsuers dede ou deerns ouoes nd only hose who suffered loss urhse he rer un. We ssue h ll onsuers eed he se

13 nolee nsurne onr wh srly osve onsurne re ( > 0. 9 Beuse hey re loed syerlly, we onenre on syer equlru, where ll sulers hrge he se re. Eh fr hs only wo surroundng oeors. In order o derve sngle suler s dend funon, le us onsder suler. A onsuer loed eween suler nd one of s neghors (offerng rer un he re he dsne [ 0,] s ndfferen eween he wo oeors, f x fro suler x x (3 n holds (where denoes he rnsoron os er un of dsne eween he onsuer nd suler. (3 s To hghlgh he effes of nsured onsuers on he sruure of rer rkes, we rewre x x. (4 n The rnsoron os re ndes he sulers rke ower, s deernes o wh exen res of rer serves n exeed rgnl os. If usoer s nsured nd, hus, s elow one, he rke ower of rer frs s nresed. 9 Ovously, 0 n never e r of n equlru, euse n hs se: he onsuers dend s oleely re-nels, sulers n hrge n nfnely hgh re nd he nuer of enerng sules lso ends o nfny. Addonlly, he nsurne reu would exeed ny nl welh. Usng slr rguen, we n esly see h here s rl level of onsurne > 0, suh h he nsured s udge onsrn s ndng for <. Therefore, osve onsurne level,, suh h he nsured s ron onsrn nds for < lso exss wh.

14 3 The resulng dend funon of suler s gven y D (, x n (. (5 Eh fr xzes s rof funon x Π (, ( n ( f, (6 where denoes he er-un os of rovdng he rer good. The frs order ondon for rof xu n syer equlru wh s. (7 n The re level n he rer rke dereses n he nuer of enerng frs nd n he onsurne re. The nuer of enerng frs s herefore endogenously deerned y he followng zero rof onsrn Π f f 0. (8 n n n In he onex of free rke enry he nuer of frs n equlru s gven y n f. (9 Even whou nsurne, he nuer of sulers n rke equlru n s oo hgh, ored o he frs es ou (Slo 979, sne level n he rer rke s FB n > n holds. The equlru re

15 4. (0 In equons (9 nd (0 he se of unnsured onsuers refers o. Thus, nsurne leds o n nrese n he nuer of sulers s well s n he rke re. The nuon ehnd hese resuls s srghorwrd: The rke ower of frs s nresed y nsured onsuers. Ths rs new enrns, whh leds o derese n rofs. Sne rke enry uses ddonl sndng exenses, he zero rof ondon les h res hve o e hgher, f onsuers re nsured. 5. Effes n he nsurne rke Le us now onenre on he hrd es suon wh nolee nsurne onrs. Due o he olexy of he ses of nure, nsurers re unle o fully sefy he ehvor of usoers nd sulers n he se of loss. Consequenly, nsurne onrs n only e ondoned uon he onsuer s dend for he rer good. As srng on for our nlyss, we exlore he hrd es nsurne onr. In hrd es suon sol lnner offers nolee onrs wh onsurne re. Ths onsurne re rdes off he nsurnendued re effe nd rsk lloon. Evdenly, onooly nsurer offers he se onsurne re s he sol lnner. Thus, n he onsdered onex n nsurne onooly s never nferor o ny oher rke sruure. However, s we wll show n Prooson, he equlru onsurne re wll derese n he nuer of nsurne ones. Consequenly, he nsurne onooly s even srly sueror o ny oher rke sruure.

16 5 As long s he ron onsrn does no nd, he hrd es onsurne re under he onsdered rusnes s sefed y he followng exeed uly xzon role: x ( π u w π ( π u w 0 0 π (. 4 ( The frs order ondon for n neror soluon s gven y EU ( π π u ( w π u ( w l π n π ( 3 ( ( ( 8 ( ( or ( π ( π 8u ( w l ( ( π ( π ( ( 3 ( 3 u u ( wn ( w l (3 resevely, where w denoes he fnl welh of onsuers n he se of no loss nd n w l denoes he fnl welh n he loss se.

17 6 We know h > 0 nd herefore u u ( wn ( w l < holds. 0 The hrd es nsurne onr enls less hn full overge, n order o l he re effe on he rer rke. The sruure of hs resul s que slr o wh n e oserved n sndrd nsurne orl hzrd odels. Ol onrs derved fro hose odel freworks lso enl only rl overge, sne, lke n our odel, rde-off exss eween rsk lloon nd he vodne of neffenly hgh losses. However, whle n he orl hzrd onex hese neffenly hgh losses re due o redued refulness s onsequene of syer nforon, n our frework hey resul only fro he overge-ndued nrese n res. Sne onons of reers exs for whh he lef hnd sde n (3 s greer hn, we onlude h n neror soluon does no lwys exs. To ensure he exsene of n neror soluon, he onsuers uly n he se of loss ( u wl x ( us e deresng n. ( u( w x l u ( w π π ( 8 l < (4 Rerrngng hs yelds he ondon ( π ( π ( 3 > (5 8u ( wl The nsurne effe, ndeny less he nsurne reu, for onsn re level hs o exeed he re effe. In oher words, for gven loss roly π, roduon oss, 0 See foonoe 9. See, for exle, Shvell (979.

18 7 rnsorons oss nd uly funon u (, here wll lwys e rl onsurne re suh h he of rgnl nrese n overge s zero. Noe h key dfferene exss eween re nreses n rer rkes nd orl hzrd. In he ler role, n ol nsurne onr effenly solves he nenve role eween he wo onrng res nd does no hve ny on oher onrs. However, n he role suded here eh ndvdul nolee nsurne onr ffes he rke re for he rer serve nd herefore he ol onrng n oher nsurne relonshs, s he followng rooson llusres. Prooson The equlru onsurne re nreses srly n he rke shre of nsurne ones. Therefore, n nsurne onooly s srly sueror o ny oher rke sruure. Proof: see Aendx The ly o redue he re effe on rer rkes ndued y nsured onsuers delnes n he nuer of nsurers, s he fron of he rke ffeed y sngle nsurer s onsurne re vron dereses. Consder n os rke sruure. In hs suon, nsurne onrs offered y sngle nsurer hve neglgle on he re level on he rer rke. Therefore, n oeve nsurne rke wh role of exernles rses nd he syer Nsh equlru s no reo-ol. The dfferene eween he equlru onsurne re nd s he greer he hgher he nuer of nsurers. In hs sense, reduon of overge n oeve nsurne rke roves welfre. Ths rovdes heorel exlnon for Feldsen s erl resuls.

19 8 On he oher hnd, onools nsurer oleely kes he of he level of overge on he rer rke re level no oun nd, hus, offers onrs h enl onsurne of. Therefore, our odel rovdes n rguen for he oenl suerory of nsurne onooles n ern suons. Addonlly, nsurne rke regulon or ooeron eween nsurne ones n e oher enefl rohes o l overge. 6. Conluson In nuerous lnes of nsurne, suh s, for nsne, helh or uo nsurne, ndenes re sed on he ul exen of rer serves he nsured urhses. Insurne overge of hs knd, however, hs or uon he rer rkes, f hese rkes re no erfe: The re level for rer serves s well s he nuer of sulers nrese. The rsng re level gn ffes he ol nsurne onr desgn, sne even n erfe nsurne rkes wh olee nforon, n ol onr would ssgn shre of he loss o he nsured. I nno e exeed, hough, h nsurers n oeve rke offer he ol onr, s he re nrese ndued y nsurne overge would no our only for he sngle nsurer u ffes ll nsurers n he rke. Ths ens h n exernly exss. Therefore, nsurers wll offer onrs wh less onsurne nd hus ore overge hn solly desrle. Ths er s frs se owrd nlyzng he nerdeendenes eween nsurne nd rer rkes. Nurlly, we hd o leve orn ses for fuure reserh. Fro our on of vew, he followng roles ould e rher neresng os o e kled: In oher onexs uhors lso hve reenly rgued h nsurne onooles for ern res heve eer resuls hn oeve rkes. See, for nsne, he erl fndngs y Ungern-Sernerg (996 for he se hoeowner s nsurne nd he dsusson of nerdeenden seury roles y Kunreuher nd Hel (003. However, s noed y Bono nd Zwefel (00, onooly nsurers n orl hzrd onex y nde n exessve level of loss revenon. Therefore, oher effes n l he suerory of suh n nsurne rke sruure.

20 9 We ssue h he rodu se s oleely hoogeneous. Ths ens h no rodu s ror eer hn he oher. Ths ssuon sees deque e.g. for uo nsurne, sne onsuers referenes for rer serves re nly deerned y vlly nd onvenene. On he oher hnd, ens would ofen hve redeerned referenes for ern hreul rodus, s n rulr oyrgh-roeed rodus. I herefore sees fruul o lso look rer rkes wh heerogeneous rodu ses. In hs er, he ssuon hs een used h he nsured s lso he onsuer for he rer serve. Bu hs s no useful o hrerze lly nsurne where he v, who hs l gns he nsured, urhses rer serves. The v usully hs legl rgh o e fully oensed, suh h n lly nsurne he on rer rkes should e even ore sgnfn. When nsurers nno wre olee onrs nd, hus, he re level of rer serves rses, verl negron of nsurne nd rer rkes sees srghorwrd roh. 3 An nsurer ould self offer ern rer goods or ould o-oere wh suler of hese goods. Verl negron s, e.g., frly well-develoed n he Aern helh nsurne rke (Mnged Cre, whle n he Euroen helh seor s well s n uo nsurne n only e oserved n s nfny. For hs reson, he nroduon of verl negron sees o e n orn exenson of hs nlyss. 3 Verl negron n lso e owerful ool gns ex os orl hzrd.

21 0 Aendx Proof of Prooson We onsder n nsurne rke wh denl nsurers h oee sulneously n onrs. Frs we look he effes of sngle nsurer s vron of he onsurne re on he rer rke. f A onsuer loed eween sulers nd s ndfferen eween he wo oeors, x ( / n x f he onsuer s nsured y he nsurer nd x ( / n x oherwse. The fron of onsuers nsured y s, whle he fron of usoers no nsured y s. The resulng dend funon of fr s gven y D (, x n ( n (. (6 For syer equlru one ons n. (7

22 The zero rof onsrn les f n (8 nd. (9 Now we re le o deerne he ol onr for nsurer. I s gven y w u w u π π π π 4 x 0 0 (0

23 The frs order ondon for n neror soluon s gven y l n l w u w u w u 3 3 ( ( ( 8 ( π π π π ( As feure of he syer equlru wh, we derve l n l w u w u w u 3 3 ( ( ( 8 ( π π π π. ( The se ( orresonds o (3. Ovously, he rgh hnd sde of ( nreses n for gven level of onsurne. q.e.d.

24 3 Insurne nd Produ hoe In he followng we wll nlyze sulers dfferenon desons. We onsder, n he sr of Hoellng (99 nd D Asreon e l. (979, lner y odel. Consuers re unforly dsrued long he nervl of he lengh of. We wn o onenre on he of he nsurne frework on sulers rodu hoe. Therefore, n hs seon we sn fro ny rke enry desons of he sulers nd deled nlyss of he nsurne rke. Hene, s ssued h ll onsuers urhse n nsurne onr wh he se onsurne re. For he ske of sly, we onsder he lner y odel of Hoellng (99 wh only wo sulers ( n nd qudr rnsoron oss of er un of lengh. 4 The sulers ly wo sge ge n whh hey frs sulneously hoose her loons (, nd ferwrds her res. Fr s loed he on 0 nd fr (, where we se 0 nd, whou loss of generly, 0. The dend funons of he sulers re D D (, (, ( ( ( (, (3. (4 4 Ths ssuon s only used hroughou hs seon of he er euse, sne D Asreon e l. (979 hve shown, he ssuon of lner rnsoron oss whn he lner y odel under ern rusnes n led o he non-exsene of rke equlru.

25 4 Eh ony xzes he rof funon Π Π, (5 nd Π,. (6 We n solve he wo sge deson role y ens of kwrd nduon. A sge he sulers hoose he Nsh equlru res for gven loons, (, whh resul fro he nerseon of he followng reon funons 0 Π (7 nd 0 Π. (8 The syer equlru res re: 3 (9

26 5 nd 3 (30 resevely. The equlru re level n he rer rke nreses wh delnng onsurne re, nd he nsurne desgn hs desve on he re level n he rer rke. Heneforh, we del wh he ol rodu hoe of he sulers sge. For h reson sulers xze her rof funon for gven res Π, (3 nd Π,. (3 The ol rodu hoe of suler s gven y Π d d D D d d, (33 where we n dsngush he dend nd he sreg re effe of vron of he rodu hoe. Usng (3, (9 nd (30, he dend effe s D (34

27 6 Sne we del wh syer role, he nsurne desgn hs no on he dend effe for oh sulers. Fnlly, usng (3 nd (30, we hve o verfy he leverge of nsurne overge on he sreg effe D d d (35 Arenly, we on he se resul h he nsurne rrngeen hs no on he sreg effe, nd herefore on he sulers rodu hoe. Sne he rk-u ( s osve, d Π / d 0 holds, whh leds o he xl dfferenon resul of D Asreon e < l. (979. q.e.d.

28 7 Referenes Anderln, L. nd L. Fell, 994, Inolee Wren Conrs: Undesrle Ses of Nure, Qurerly Journl of Eonos 09: Arrow, K.J., 965, Ases of he Theory of Rsk Berng, Yrö Johnsson Leures, Helsnk. Bono, D. nd P. Zwefel, 00, Inforon ou Mulle Rsks: The Cse of Buldng nd Conen Insurne, Journl of Rsk nd Insurne 69: D Asreeon, C., J. Gszewz, nd J.-F. Thsse, 979, On Hoellng s Sly n Coeon, Eonoer 47: Feldn, R. nd B. Dowd, 99, A New Ese of he Welfre Loss of Exess Helh Insurne, Aern Eono Revew 8: Feldsen, M.S., 970, The Rsng Pre of Physns Serves, Revew of Eonos nd Sss 5: 33. Feldsen, M.S., 97, Hosl Coss Inflon: A Sudy n Nonrof Pre Dyns, Aern Eono Revew 6: Feldsen, M.S., 973, The Welfre Loss of Exess Helh Insurne, Journl of Poll Eonoy 8: Freh, H.E. nd P.E. Gnsurg, 975, Iosed Helh Insurne n onools rkes: heorel nlyss, Eono Inqury 3:

29 8 Gynor, M., D. Hs-Wlson nd W.B. Vog, 000, Are Invsle Hnds Good Hnds? Morl Hzrd, Coeon, nd he Seond-Bes n Helh Cre Mrkes, Journl of Poll Eonoy 08: Hellersen, J.K., 998, The Iorne of he Physn n he Gener versus Trde-Ne Deson, RAND Journl of Eonos 9: Hoellng, H., 99, Sly n Coeon, Eono Journl 39: Kunreuher, H. nd G.M. Hel, 003, Inerdeenden Seury, The Journl of Rsk nd Unerny 6: Mskn, E., 00, On ndesrle onngenes nd nolee onrs, Euroen Eono Revew 46: Newhouse, Joseh P. nd he Insurne Exeren Grou, 993,. Free for All? Lessons fro he RAND Helh Insurne Exeren, Hrvrd Unversy Press, Crdge. Puly, M.V., 968, The Eonos of Morl Hzrd: Coen, Aern Eono Revew 58: Pvnk, N., 00, Do hreul res resond o en ou-of-oke exenses? RAND Journl of Eonos 33: Slo, S., 979, Monools Coeon wh ousde goods, Bell Journl of Eonos 0: Segl, I., 999, Colexy nd Renegoon: A Foundon for Inolee Conrs, Revew of Eono Sudes 66: 57 8.,

30 9 Shvell, S., 979, On Morl Hzrd nd Insurne, Qurerly Journl of Eonos 93: Ungern-Sernerg, T. von, 996, The ls of oeon: Housng nsurne n Swzerlnd, Euroen Eono Revew 40:.