Moral Hazard in Loss Reduction and the State Dependent Utility

Transcription

1 Moral Hazard in Loss Rduction and th Stat Dpndnt Utility Jimin Hong Businss School Soul National Univrsity Gwanak Ro, Gwanak Gu, Soul, 5-96, Kora S. Hun Sog Businss School Soul National Univrsity Gwanak Ro, Gwanak Gu, Soul, 5-96, Kora This Vrsion: 05.6 Authors ar gratful for th support from th Managmnt Rsarch Institut and th Institut of Financ and Banking of Soul National Univrsity.

2 Moral Hazard in Loss Rduction and th Stat Dpndnt Utility Abstract: W considr a stat dpndnt utility modl with binary stats whr moral hazard occurs in loss rduction. W find diffrnt rsults dpnding on th rlativ sizs of th marginal utilitis btwn th loss stat and th no loss stat. (i) If th marginal utilitis ar qual btwn th two stats, th optimal insuranc involvs full insuranc up to a limit and coinsuranc abov th limit, which corrsponds to th cas of th stat indpndnt utility. (ii) If th marginal utility in th loss stat is gratr than that in th no loss stat, thn th optimal insuranc includs full insuranc, and th moral hazard problm bcoms lss svr than undr th cas of th indpndnt utility. (iii) If th marginal utility in th loss stat is lss than that in th no loss stat, thn th optimal insuranc includs th dductibl up to a limit and coinsuranc abov that limit, and th moral hazard problm bcoms mor svr. W xtnd th modl into a two priod stting, and apply it to th cass of a dbt contract of a firm and a wag contract. Kywords: moral hazard, stat dpndnt utility, loss rduction ffort, dductibl, coinsuranc

3 Moral Hazard in Loss Rduction and th Stat Dpndnt Utility I. Introduction In th insuranc contxt, moral hazard ariss bcaus purchasing insuranc lowrs th incntivs to rduc risks, whn insurrs cannot obsrv th policyholdrs' fforts (actions). Moral hazard incurs costs by distorting th allocation of rsourcs. With no moral hazard, a policyholdr can achiv th utility at th (first-bst) fficint lvl by purchasing full insuranc. With moral hazard, howvr, th policyholdr purchass partial insuranc, failing to achiv th (first-bst) fficincy, as shown in Shavll (979), for xampl. Insuranc litratur oftn distinguishs btwn two typs of risk rduction: loss prvntion and loss rduction. Loss prvntion is to lowr th probability of loss occurrnc (frquncy), whil loss rduction is to lowr th loss siz (svrity) whn a loss occurs (s Ehrlich and Bckr (97) for th arlir conomic rsarch). Exampls ar fir alarms and mdical chck-ups for th formr, and car airbags and arthquak-rsistant construction dsign for th lattr. Moral hazard may occur in rgard both typs of risk rduction, although a majority of rsarch is concrnd with loss prvntion. It is known that moral hazard may lad to diffrnt insuranc contracts dpnding on th typs of risk rduction. Wintr (000) shows that th optimal insuranc is full covrag abov a dductibl in th cas of moral hazard in loss prvntion, whil it is full covrag up to a limit and partial insuranc abov th limit in th cas of moral hazard in loss rduction. On th othr hand, moral hazard, in most cass, is addrssd in montary trms undr th xpctd utility in litratur. This approach prsums that a loss is montary and th full compnsation for th loss can fully rcovr th lvl of utility. That is, a loss is a rplacabl good in trms of Cook and Graham (977). Whil this approach is convnint, thr has bn a concrn that thr may b th cass whr montary compnsation cannot fully substitut a loss. Cook and Graham (977) point out that th loss of an irrplacabl good may lowr th policyholdr s utility mor than th montary loss dos. For xampl, halth is an irrplacabl good, so full covrag of mdical xpnss is not a prfct substitut for good halth. To addrss this concrn, a so-calld stat dpndnt utility approach has bn utilizd in litratur. Undr th stat dpndnt utility, th loss occurrnc may also chang th utility function, so that th simpl montary compnsation may fail to rcovr th sam lvl of utility as th pr-loss on. Tchnically, th chang of utility function lads to th chang of marginal utility, which producs diffrnt outcoms from th standard stat indpndnt approach. Shioshansi (98), Schlsingr (984) and Huang and Tzng (006) analyz th optimal insuranc for th irrplacabl loss undr no moral hazard. Shioshansi (98) suggsts a Som confusion xists in trminology. For xampl, Ehrlich and Bckr (97) us slf-insuranc for loss rduction and slf-protction for loss prvntion. In th halth conomics contxt, Barigozzi (004) uss primary prvntion for loss prvntion and scondary prvntion for loss rduction.

4 simpl stat dpndnt utility approach to study insuranc for irrplacabl commodity. Schlsingr (984) dvlops on Shioshansi to allow marginal utilitis to vary on th incom lvl as wll as stats of natur. Both analyss of Shioshansi and Schlsingr ar basd on th assumption that th insuranc contract is of a coinsuranc form. On th othr hand, Huang and Tzng (006) study th optimal form of th insuranc contract and find that it includs a dductibl and coinsuranc abov th dductibl, which rcoups th rsults of Raviv (979). Not, howvr, that th rsults of Huang and Tzng ar basd on th stat dpndnt utility, whil thos of Raviv (979) ar basd on th positiv costs and a riskavrs insurr. Dionn (98) xamins th insuranc contract dsign undr stat dpndnt utility and moral hazard in loss prvntion. H finds that th optimal covrag is partial du moral hazard and th ffort lvl is affctd by th stat dpndnc of utility. H furthr argus that covrag is highr and ffort is lowr undr stat dpndnc than undr stat indpndnc if th marginal utility in th loss stat is gratr than that in th no loss stat. In th similar spirit of Dionn, w also invstigat th optimal insuranc dsign undr th stat dpndnt utility and moral hazard. Unlik Dionn, howvr, w study th moral hazard problm in loss rduction. Givn that our focus in on loss rduction, w ar concrnd with th cass of natural disastr outbraks such as typhoons, arthquaks, and hurricans whos occurrncs ar not influncd by human fforts. Popl can only act to rduc th damag. Rcntly, w hav obsrvd th hug damags from th tsunami in Japan (0) and th typhoon in th Philippins (03). W bliv that this study can shd light on th optimal insuranc contract against catastroph losss. Loss rduction and th stat dpndnt utility ar also important in th halth car and insuranc contxt. First, halth is oftn irrplacabl in that a full rimbursmnt of mdical xpnss may not fully rcovr th quality of lif prior to th halth loss. Thus, th stat dpndnt utility may b usful in undrstating halth insuranc. Scond, loss rduction (or scondary prvntion) is an important concrn for both tratmnt and prvntion of furthr dvlopmnt of a disas, onc th disas is dtctd (s Ellis and Manning, 007 and Barigozzi, 004). W find that th optimal insuranc form dpnds on th rlativ sizs of marginal utilitis in diffrnt stats. If th marginal utility in th loss stat is gratr than that in th no loss stat, thn th optimal insuranc form contains full covrag up to a limit and coinsuranc abov th limit. Full insuranc is possibl as wll whn stat dpndncy is sufficintly larg. On th othr hand, if th marginal utility in th loss stat is smallr than marginal utility in th no loss stat, thn th optimal insuranc includs that of th dductibl up to a limit and coinsuranc abov th limit. This scond cas is consistnt with Huang and Tzng in obtaining th rsult of Raviv, vn if w do not assum costs and a risk-avrs insurr. W also xamin th rlativ ffort lvl compard to th stat indpndnt utility cas. If w assum that th covrag lvl is qual btwn th two utilitis, thn th ffort lvl dpnds on th marginal utility of incom in th loss stat as wll. Whn th marginal utility of incom in th loss stat is qual to that of incom in th no loss stat, th optimal ffort lvl is also qual. Givn th indmnity lvl, if th marginal utility of incom in th loss stat is gratr than that of incom in th no loss stat, thn th rlativ ffort lvl is highr. Thus, w can stat that th moral hazard problm is naturally rlivd. In th rvrs cas, th ffort is lowr and moral hazard sms to b mor svr. W xtnd th modl into a two priod modl. Undr a stat dpndnt utility, in th

5 cas whr marginal utility of incom in th loss stat is highr than that of incom in th no loss stat, th loss rduction ffort at tim can b highr whn th loss occurs at tim compard to th ffort whn thr is no loss in th first priod. In th rvrs cas, th ffort in scond priod can b lowr. Byond ths rsults, w also find applications in th ral world, such as dbt contracts undr financial distrss situations, and wag contracts. If w assum that production cost function is stat dpndnt bcaus of rputation cost whn profit is low, thn th ffort to incras th oprating incom can b highr. Th rvrs cas is also possibl. W can find th optimal wag contract whn mploys hav a stat dpndnt utility. This papr procds as follows. Sction II dscribs th modl. Sction III prsnts th stat dpndnt utility cas undr no moral hazard. Sction IV analyzs th moral hazard problm with stat dpndnt utility and th optimal insuranc forms. It also compars th ffort lvl with that undr th stat indpndnt utility. Sction V xtnds th modl into a two priod contxt. Sction VI discusss th applications. Th last sction concluds. II. Modl dscription W considr a two stat modl. Uncrtainty is rprsntd by two stats rgarding loss occurrnc, dnotd by S : th no loss stat ( S 0) and th loss stat ( S ). Following th stat, th utility function is dnotd as u( W, S) whn W is incom. If loss occurs, th utility is lowr than th utility without th loss vn if th loss is rcovrd, so u( W, S 0) u( W, S ) for all W. Th utility function is twic diffrntiabl and strictly concav with rspct to an incom. That is, u '( W, S) 0 and u ''( W, S) 0 for all S. Th policyholdrs cost function for th ffort is c (). W suppos that c'( ) 0 and c''( ) 0. W also assum that all policyholdrs ar homognous. As a rsult, an advrs slction problm dos not xist in this modl. W assum that th probability of loss occurrnc is p and that p is not controlld by ffort. Th loss siz is x and x follows th distribution f ( x; ) on a support of [ xx, ] givn ffort. W impos furthr rstriction on th distribution givn, calld th Monotonic Liklihood Ratio Condition, MLRC (Milgrom, 98). This condition is that f ( x; H ) givn H L, is dcrasing in x. That is, as th ffort incrass, th liklihood f ( x; L ) of occurrnc of loss dcrass. If th ffort is incrasd, howvr, w suppos that th support of x dos not chang. Q is th insuranc prmium and I( x) is th indmnity. W also suppos that th insuranc markt is comptitiv, so th insurr s profit should b zro. III. No Moral Hazard Cas As a rfrnc, lt us not th first bst cass with stat indpndnt utility and stat dpndnt utility. W suppos that loss siz x has th dnsity function f ( x; ) givn. Th individual maximizs his utility. Thn th program is 3

6 Max [ p] u( W Q, S 0) p u( W Q x I( x), S ) f ( x; ) dx c( ) (), I ( x), Q s. t. Q p I( x) f ( x; ) dx () Th Lagrangian is, whr is th Lagrang multiplir L [ p] u( W Q, S 0) p u( W Q x I( x), S ) f ( x; ) dx c( ) [ Q p I( x) f ( x; ) dx] For notational simplicity, lt us writ u ( W) u( W, S 0) and u ( W) u( W, S ). 0 W also us th notation W0 W Q, W W Q x I( x). Thn th first ordr conditions for optimum with rspct to Q, I( x) and ar as follows. LQ [ p] u ' 0( W0 ) p u ' ( W ) f ( x; ) dx 0 (3) L pu ' ( W ) f ( x; ) pf ( x; ) 0, for ach x (4) I( x) (5) L p u ( W ) f ( x; ) dx c'( ) p I( x) f ( x; ) dx 0 L Q p I( x) f ( x; ) dx 0 (6) W hav followd th first bst solution, rarranging first ordr conditions. u ' ( W ) u ' ( W ) (7) 0 0 W know that in cas of stat indpndnt utility, full insuranc is optimal. That is, I( x) x, sinc th xprssion (7) changs into u ' 0( W0 ) u ' 0( W ) in th stat indpndnt cas. W also obtain th optimal amount of ffort by (5). Th ffort should satisfy quation (8). u0'( W) p xf ( x; ) dx c'( ) (8) Rcall that th LHS of (8) is transformd to u ' 0( W) p F ( x; ) dx bcaus xf ( ; ) x x dx = x F ( x; ) F ( x; ) dx F ( x; ) dx. As a rsult, (8) is positiv. Th x LHS of (8) implis th marginal bnfit of taking car to rduc th loss siz and consquntly to rduc th prmium. Th RHS rprsnts th marginal cost of taking such car. Howvr, if policyholdrs hav stat dpndnt utility, thn th optimal insuranc covrag and ffort lvl can b diffrnt from th covrag undr stat indpndnt utility dpnding on th rlativ siz of marginal utilitis of incom btwn th loss and th no loss stat. Now, lt us dfin that th stat dpndnt utility in th loss stat is as blow: 4

7 u ( W) u ( W) A( W), A( W) 0, W (9) 0 Thn, w hav th following rlation: Rlation. If u0'( W) u'( W), W, thn AW '( ) 0. That is, A( W) A, constant. Rlation. If u0'( W) u'( W), W, thn AW '( ) 0. Rlation 3. If u0'( W) u'( W), W, thn AW '( ) 0. A is a Ths rlations ar dpictd in figur,, and 3. According to Schlsingr, rlation can b xplaind by th conjctur that th loss maks popl mor apprciativ of additions to walth. Rlation 3 can b intrprtd in th opposit way. In addition, w dfin compnsation ( W) for th loss satisfying following xprssion. u '( W) u '( W ( W)), W (0) 0 This dfinition diffrs from that of Cook and Graham rgarding of compnsation. Cook and Graham dfin compnsation basd on utility, but w us marginal utility of incom. By th xprssion (0) and th Taylor xpansion, w obtain th following xprssion. A'( W) ( W) u ''( W) () By th concavity of th utility, w hav: A'( W) 0 ( W) 0 and A'( W) 0 ( W) 0 () Th xprssion () indicats that if th marginal utility in th loss stat is gratr than that in th no loss stat, individuals claim positiv compnsation vn if th walth lvl is th sam. On th contrary, if th marginal utility in th loss stat is lss than that in th no loss stat, ngativ compnsation is possibl. Lt us dfin AW '( ) as a stat dpndncy. Thn w hav AW '( ) ( W ) u ''( W) by (). Thus, w know that th compnsation incrass whn stat dpndncy incrass, or th curvatur of utility with loss dcrass. Manwhil, thr xists a rlationship btwn compnsation CW ( ) which is dfind by Cook and Graham and ( W). CW ( ) satisfis that u0( W) u( W C( W)), which movs th insurd to a utility lvl without accidnt stat. Using th Taylor xpansion, w know that: u0( W) u( W C( W)) u( W) C( W) u'( W) 5

8 Thus, w obtain C( W) u '( W) A( W) and A'( W ) u '( W ) A( W ) u ''( W ) A'( W ) A( W ) ARA ( W ) C'( W) (3) [ u'( W )] u'( W ) whr ARA ( ) W is th absolut risk avrsion in th loss stat. By (3), w obtain C'( W) 0 3 whn AW '( ) 0. In particular, as th absolut risk avrsion incrass, C'( W ) incrass as wll. On th othr hand, if AW '( ) 0 thn th sign of C'( W ) is ambiguous. Howvr, w can say C'( W) 0 whn AW '( ) or th absolut risk avrsion is sufficintly small. Now, w hav Lmma. Lmma. (No Moral Hazard Cas) Undr a stat dpndnt utility, w obtain following rsults. () Th optimal covrag is full insuranc whn u0'( W) u'( W), W. () Th optimal covrag is full insuranc if ovr-insuranc is not allowd and ovrinsuranc is optimal if possibl whn u0'( W) u'( W), W. (3) Th optimal covrag is partial insuranc whn u0'( W) u'( W), W. Proof. S th Appndix.// In th cas of u0'( W) u'( W), th utility aftr rciving full covrag is lowr than th utility without an accidnt. This is bcaus xcss paymnt for quivalnt utility in both stats may incras th prmium. Hnc, th marginal cost of insuranc is highr than th marginal bnfit of insuranc. As a rsult, full insuranc is optimal vn if th utility with an accidnt still lowr aftr paid bnfit. In th cas of u0'( W) u'( W), th optimal insuranc is ovr-insuranc which is dcidd at th lvl that maks marginal utilitis idntical in th loss stat and no loss stat, whil partial insuranc is optimal if u0'( W) u'( W). W also obtain Lmma in trms of an optimal ffort amount. Lmma. (No moral hazard cas) Undr a stat dpndnt utility, w hav following rsults. () Th optimal ffort is quivalnt to a stat indpndnt utility cas whn u0'( W) u'( W), W. () Th optimal ffort is gratr than that of a stat indpndnt utility cas whn 3 Cook and Graham indicat that if th irrplacabl goods ar classifid as normal goods, thn C'( W) 0. On th contrary, C'( W) 0 whn th goods ar considrd as infrior goods. Thy also argu that rplacabl goods can b viwd as a spcial cas of irrplacabl goods whn C'( W) 0. 6

9 u0'( W) u'( W), W. (3) Th optimal ffort is lss than that of a stat indpndnt utility cas whn u '( W) u '( W), W. 0 Proof. S th appndix.// Although utility is stat dpndnt, sinc th marginal utility of incom is idntical in both th loss stat and th no loss stat, th marginal bnfit of putting in an ffort to dcras th loss siz and prmium dos not chang. Thus, th optimal ffort lvl is unchangd. On th othr hand, if u0'( W) u'( W), thn th marginal bnfit of additional ffort incrass du to disutility. Consquntly, th policyholdr incrass his ffort until th marginal bnfit of taking car to lowr th loss siz is qual to th marginal cost of taking such car. Th u0'( W) u'( W) cas can b xplaind in opposit way. IV. Moral Hazard Cas In this chaptr, w dvlop th loss rduction modl with moral hazard basd on a stat dpndnt utility. Th insurd maximizs his utility, so th xpctd utility is as follows: Max [ p] u( W Q, S 0) p u( W Q x I( x), S ) f ( x; ) dx c( ) (4), I ( x), Q s. t. Q p I( x) f ( x; ) dx (5) arg max a [ p] u( W Q, S 0) p u( W Q x I( x), S ) f ( x; a) dx c( a) (6) Th scond condition is th incntiv compatibility constraint undr moral hazard. W suppos that th convntional first ordr approach is valid, thn th scond constraint (6) bcoms d p u( W Q x I( x), S ) f( x; ) dx c'( ) 0, whr f( x; ) dx f ( x; ) dx d (7) Lt us us th sam notation for abbrviation as in th prvious sction III. Lagrangian is xprssd as: Th L ( p) u ( W ) p u ( W ) f ( x; ) dx c( ) [ Q p I( x) f ( x; ) dx] 0 0 (8) [ p u ( W ) f ( x; ) dx c'( )] Th first ordr conditions ar L ( p) u ' ( W ) p u ' ( W ) f ( x; ) dx p u ' ( W ) f ( x; ) dx 0 Q 0 0 (9) L pu ' ( W ) f ( x; ) pf ( x; ) pu ' ( W ) f ( x; ) 0, for ach x (0) I ( x) 7

10 () L p I( x) f ( x; ) dx [ p u ( W ) f ( x; ) dx c''( )] 0 Rarranging ths conditions, w hav f( x; ) ( ) () u ' ( W ) u ' ( W ) f ( x; ) 0 0 From (), w obtain Lmma 3. Lmma 3. (Holmstrom) Undr a stat dpndnt utility, 0. Proof. S th Appndix.// Lmma 3 is consistnt with th rsult of Holmstrom (979). Th condition 0 mans that th insurr also dsigns th contract to induc mor ffort from th insurd undr a stat dpndnt utility. Bfor xamining th optimal covrag with moral hazard undr a stat dpndnt utility, w invstigat th covrag undr a stat indpndnt utility. Wintr (000) rmarks that th optimal covrag involvs full covrag for small losss and partial insuranc for high losss. W obtain th sam rsult as Wintr whn w suppos that u0'( W) u'( W), W. As is vidnt in th no moral hazard cas undr a stat dpndnt utility, th optimal covrag dpnds on not only th loss siz but also th marginal utility of incom. W considr thr cass following comparison with th rlativ siz of marginal utilitis, and find th optimal covrag in ach cas as wll. W obtain proposition. Proposition. (Moral hazard cas) Undr a stat dpndnt utility, th following rsults hold. () Th optimal covrag includs full insuranc up to a limit and coinsuranc abov th limit if u0'( W) u'( W), W. In particular, this is quivalnt to th optimal covrag lvl undr a stat indpndnt utility. () Th optimal covrag includs full insuranc up to a limit and coinsuranc abov th limit if u0'( W) u'( W), W. It also can b full insuranc for all loss whn a stat dpndncy is sufficintly larg. (3) Th optimal covrag includs a dductibl up to a limit and coinsuranc abov th limit if u0'( W) u'( W), W. It can also b full insuranc up to a limit and coinsuranc abov th limit whn a stat dpndncy is sufficintly larg. Proof. S th Appndix.// Rcall that th pnalty is givn only to th loss siz. From (), w know that insurrs pnaliz th insurd for th larg loss sinc th ffort affcts loss siz. On th othr hand, insurrs do not pnaliz and giv incntiv th insurd for th small loss to induc th ffort as much as possibl. As a rsult, up to a limit, th ovr-insuranc is optimal if it is allowd, and coinsuranc is optimal abov th limit for all thr cass. Assuming that th ovrinsuranc is not allowd, full insuranc is optimal up to a limit. In particular, if 8

11 u0'( W) u'( W), W thn, th covrag is idntical with th stat indpndnt utility cas. This is bcaus marginal utilitis ar qual for th sam lvl of incom, so th first ordr condition for th maximization is quivalnt to th cas with a stat indpndnt utility. W also can s that if u0'( W) u'( W), W, thn full insuranc is possibl whn stat dpndncy is sufficintly larg. Policyholdrs nd mor indmnity than th stat indpndnt cas, as th stat dpndncy is largr. On th contrary, if u0'( W) u'( W), W and stat dpndncy is sufficintly largr, thn policyholdrs may rquir a lowr lvl of indmnity than th cas with stat indpndnt utility. Thrfor, ngativ insuranc can b possibl up to a limit. Howvr, ngativ insuranc is not fasibl in rality, so th dductibl is optimal up to a limit. W can compar th rlativ ffort lvl btwn a stat dpndnt utility cas and a stat indpndnt utility cas as wll. This is Proposition. Proposition. (Moral Hazard Cas) Undr a stat dpndnt utility, if th covrag is qual to th covrag with a stat indpndnt utility for all covrag lvl, thn w hav followings. () Th optimal amount of an ffort is qual to an ffort undr a stat indpndnt utility if u0'( W) u'( W), W. () Th optimal amount of an ffort is mor than an ffort undr a stat indpndnt utility if u0'( W) u'( W), W. (3) Th optimal amount of an ffort is lss than an ffort undr a stat indpndnt utility if u '( W) u '( W), W. 0 Proof. S th Appndix.// In Proposition, () is intuitivly clar. Th loss dos not chang individual s attitud to th additional walth. Thrfor, th optimal ffort lvl may b th sam. Manwhil, if u0'( W) u'( W), thn individuals chrish mor th additional walth undr a stat dpndnt utility than undr a stat indpndnt utility. Thus, givn indmnity individuals mak mor ffort. Hnc, w can say that th moral hazard problm is lss svr. In th cas of u0'( W) u'( W), th ffort is lowr, so th moral hazard appars mor svr. As a rsult, th moral hazard problm is a still significant problm undr th stat dpndnt utility with loss rduction. Howvr, th concrn about th moral hazard problm might b xcssiv or should b highr dpnding on th marginal utility of incom. V. Extnsion Two Priod and Moral Hazard Cas In this sction, w xtnd th loss rduction modl from on priod to a two priod modl undr a stat dpndnt utility. Th stat rgarding loss occurrnc is dnotd by whr t is priod and. Th loss siz and ffort ar dnotd x t and t rspctivly. Following th loss siz and ffort lvl in ach priod, th indmnity is rprsntd as I( x ), I( x, x ). Th prmium at tim t is dnotd as Q t. W suppos that th stat variabls ar indpndnt ach othr and th ffort at tim dos not affct loss siz at tim. Consquntly, w can writ th joint dnsity of loss givn th ffort lvl as follows. S t, 9

12 f ( x, x ;, ) dx dx f ( x ; ) dx f ( x ; ) dx (3) W also suppos that th loss siz in ach priod has th sam distribution and that th distribution of loss siz givn still follows MLRC. Th ffort in priod dpnds on th loss siz at tim. That is, ( x ). W assum that both th insurr and th insurd commit to th long trm insuranc contract. W suppos th First ordr approach on ffort is also valid in a two priod modl as wll. Thus, th problm is, Max p u W Q S p u W Q x I x S f x dx c Q, Q, I ( x ), I ( x, x ),, ( ) (, 0) ( ( ), ) ( ; ) ( ) ( p)[( p) u( W Q ( x 0), S 0, S 0) p u( W Q ( x 0) x I ( x 0, x ), S 0, S ) f ( x ; ( x 0)) dx c( ( x 0))] p[( p) u( W Q ( x ), S, S 0) f ( x, ) dx p u( W Q ( x ) x I ( x, x ), S, S ) f ( x ; ) f ( x, ( x )) dx dx c( ( x )) f ( x ; ) dx ] (4) (5) s.. t Q p I( x ) f ( x; ) dx (9) Q ( x ) p I ( x, x ) f ( x ; ( x )) dx, whr x 0 (6) Q ( x 0) p I( x 0, x) f ( x; ( x 0)) dx (7) p u( W Q x ( I ), x S ) f( x; ) dx '( c ) p( p) u( W Q ( x ), S, S 0) f ( x, ) dx p[ p u( W Q ( x ) x I ( x, x ), S, S ) f ( x ; ( x )) dx f ( x ; ) dx c ( ( x )) f ( x ; ) dx ] 0 (8) p u( W Q ( x 0) x I ( x 0, x ), S 0, S ) f ( x ; ( x 0)) dx c'( ( x 0)) 0 p u( W Q ( x ) x I ( x, x ), S, S ) f ( x ; ( x )) dx c'( ( x )) 0 for ach x, x 0 (30) Conditions (8), (9), and (30) ar th incntiv compatibility constraints. From th xprssion (8), w know that th insurd should considr th ffct of th ffort at tim on th indmnity at tim. 0

13 Lik th prcding cas, w simply dnot utility function as ui ( W) u( W, S i) and u ( W) u( W, S i, S j),whr i, j 0 or. In addition, w writ walth of ach priod ij t as W0 W Q t, whr t,, W W Q x I( x ), W ( x 0) W Q ( x 0) x I ( x 0, x ), and W ( x ) W Q ( x ) x I ( x, x ) whr x 0. Lt us dfin th stat dpndnt utility for priod as blow. u ( W) u ( W) A( W) whr i 0, (3) i i0 This is similar to (9). By (3), w know that u ( W) u00( W) A( W). Using ths notations, w obtain Lmma 4 from abov problm. Lmma 4. (Lambrt) () Th optimal risk sharing rul is a. f ( x ; ) ( ) u '( W ) u0 '( W0 ) f ( x; ) b. f ( x ; ( x 0)) [ ( x 0) ] u '( W ) f ( x ; ( x 0)) u '( W ( x 0)) f ( x ; ) f ( x ; ( x )) '( ) ( ; ) ( ; ) ( ; ) '( ( )) c. [ ( x ) ] u0 W0 f x f x f x u W x () ( x) 0 for all x (3) 0 Proof. S th Appndix.// W know that th indmnity for th scond priod still dpnds on th loss siz for th first priod from 0. In addition, ( x) 0 implis that th highr loss siz in th first priod brings out th lowr indmnity of th scond priod for ach ffort. In th scond priod, if x 0 thn th insurd can fully hdg th risk. That is, u00 '( W0 ( x 0)) u0 '( W0 ( x )) f ( x ; ) Ex [ ]. On th contrary, E u x [ x 0 '( W ( x 0], so 0)) u '( W ( x )) f ( x; ) f ( x ; ) if x 0, thn th insurd may tak th risk as. This mans that if th ovrinsuranc is not allowd, th insurr dos not pnaliz th insurd by som lvl of loss f ( x; ) which occurs in th first priod. Howvr, th loss at tim xcds th critical valu, thn, th insurd is givn th pnalty in th scond priod. That is, th insurr can disprs th risk of th first priod into th scond priod. Now, w can compar th ffort lvl at tim dpnding on th stat dpndncy of

14 utility. W show that Proposition 3 is satisfid. Proposition 3. (Moral Hazard in Two Priod Cas) Undr a stat dpndnt utility, suppos that th covrag of th scond priod whn loss dos not occur in th first priod is qual to th covrag with a loss in th first priod. Givn covrag lvl, th following rsults hold. () Th optimal amount of ffort at tim with loss at tim is qual to an ffort with no loss at tim, if u0'( W) u'( W), W. () Th optimal amount of ffort at tim with loss at tim is mor than an ffort with no loss at tim, if u0'( W) u'( W), W. (3) Th optimal amount of ffort at tim with loss at tim is lss than an ffort with no loss at tim, if u0'( W) u'( W), W. Proof. S th Appndix.// Du to th stat dpndncy, th ffort lvl undr a stat dpndnt utility in th scond priod can b highr or lowr than that undr a stat indpndnt utility. In addition, th rsults of Proposition 3 show that undr a stat dpndnt utility, whn th accidnt occurs in th first priod, th moral hazard problm of th scond priod may b allviatd or dpnd, compard to th cas with no loss of th first priod dpnding on th marginal utilitis of incom btwn loss and no loss stats. Gnrally, it is known that th moral hazard problm is mitigatd in a long trm contract. Howvr, whn th loss is fortunatly low in th first priod, th stratgic bhavior whr th insurd slackns ffort in th scond priod still xists in a long trm contract. Proposition 3 shows that undr a stat dpndnt utility, th possibility of this typ of stratgic bhavior may b lowr in th cas of u0'( W) u'( W), W. In cas of u0'( W) u'( W), W, th possibility is highr as wll. VI. Applications Modls of loss rduction with moral hazard can xplain som cass in th ral markt world. First of all, as prviously statd, our modl can xplain th optimal insuranc form for catastrophs such as floods or arthquaks. Sinc this kind of insuranc is usually offrd by govrnmnt agncis such as FEMA (Fdral Emrgncy Managmnt Agncy) of th U.S., bcaus th loss siz is usually vry hug and it is not asy to stimat th probability of th natural disastr, our rsults may hold som policy implications. To b a littl mor spcific, if u0'( W) u'( W), W, th govrnmnt indmnity includs full insuranc to rcovr th insurd s utility. In addition, th ffort to rduc th loss can b highr than what popl may hav thought. That is, thr xists th possibility that th moral hazard problm can b ovrmphasizd. Th rvrs cas can also b tru. Scond, in halth insuranc, th x in this study can b considrd as th xpns for th halth loss xprssd as a montary trm. W can rgard that illnss as th accidnt as wll. If th halth xpns includs an xpns to rduc th risk of complications, thn th loss rduction ffort plays an ssntial rol in th insuranc paymnt. In addition, aftr th onst of illnss, quality of lif cannot b rcovrd. In th cas whr th utility dpnds on th loss occurrnc, th moral hazard problm may not b critical. Th ffort to lowr th

15 svrity of th potntial illnss may b highr, and mor indmnity compard to th stat indpndnt cas can nhanc fficincy whn u0'( W) u'( W), W. Not that th moral hazard with a loss rduction modl can b intrprtd as an x-post moral hazard problm in this cas. Part of th halth conomics litratur, such as Nyman (999a, 999b), Nwhous (006), Ellis and Manning (007), and Sog (0) suggsts that mor gnrous insuranc covrag can improv th wlfar as wll. Nyman indicats that th convntional approach ovrstats th wlfar loss du to th moral hazard. Nwhous argus that lowr cost sharing would rduc total cost, including futur cost, by spnding mor on halth srvics. Sog also claims that th optimal covrag should b highr whn tratmnt is not prvntativ. Although Ellis and Manning focus on th loss prvntiv ffort, th argumnt that mor covrag can b optimal is consistnt with our rsults. W can xplain othr xampls using our modl. Lt us discuss th optimal dbt contract. Thr is an ntrpris with two stats dnotd by S : output is lowr than dbt ( S ) and profit is highr than th dbt ( S 0). For simplicity, w assum that th probability with stat S is p and p. In a low profit stat, th profit givn ffort follows th distribution f( ; ), and th profit follows th distribution g( ; ) in a high profit stat. W can intrprt as oprating incom. Ths distributions follow MLRC. Th cost is dividd into two parts. Th cost function for ffort is dnotd by c. Th cost function v mans th production cost function, and v is convx. W suppos that v(, S 0) v(, S ) A( ). If th profit is low, thn th rputation cost is incrasd and th valu of th brand is dcrasd, so w can assum that A( ) 0. Th dbt principal is I and th intrst rat is r. W dnot th dbt rpaymnt schdul as B ( ) and B( ) is a non-dcrasing function. W first not that th production cost function is stat indpndnt. That is, v(, S 0) v(, S ). Th modl is as follows: [ ( ) ( )] ( ; ) ( ) [ ( ) ( )] ( ; ) ( ) Max p B v f d p B v g d c B( ), 0 s. t. p B( ) f ( ; ) d ( p) B( ) g( ; ) d ( r ) I 0 p B v f 0 d p B v g d c [ ( ) ( )] ( ; ) ( ) [ ( ) ( )] ( ; ) '( ) whr 0 B( ) min( v( ),( r ) I), ( r) I 0 v ( ) Solving this problm, w obtain Rsult. Rsult. Undr a stat dpndnt production function, if dbt intrst is qual, thn w hav following rsults. () Th ffort is qual to th ffort undr a stat indpndnt production cost function whn v'(, S 0) v'(, S ). () Th ffort is highr than th ffort undr a stat indpndnt production cost function whn v'(, S 0) v'(, S ). 3

16 (3) Th ffort is lowr than th ffort undr a stat indpndnt production cost function whn v'(, S 0) v'(, S ). Proof. S th Appndix.// Rsult is consistnt with Proposition 3. If th production cost function is stat dpndnt, thn th moral hazard problm of a borrowr may b lss svr than th problm undr a stat indpndnt cost function. This cas can b intrprtd that th xtra cost A( ), which is brand valu or rputation cost, dcrass as incrass. In th rvrs cas, th moral hazard problm is mor significant. Lastly, w invstigat th moral hazard problm in th wag contract whn th mploy has a stat dpndnt utility. W dnot th salary and th rsrvation utility as S( ) and u, rspctivly. Lik th prcding xampls, w assum that thr xist two stats. Stat 0 indicats that th profit is highr than th limit valu *, and stat prsnts th profit as bing lowr than *. In ach stat, th profit givn ffort has th dnsity functions g( ; ) and f( ; ). Both g( ; ) and f( ; ) satisfy MLRC. Th cost function for mploy ffort is c. W also suppos that u( S( ), S ) u( S( ), S 0) A( S( )) and AS ( ( )) 0. W can rgard this AS ( ( )) as a sntimnt valu or th rputation of an mploy. This mans that if prformanc is blow som critical valu, thn th mploy s utility is dcrasd additionally. Th modl is Max p * [ S ( )] f ( ; ) d ( p ) [ S ( )] g ( ; ) d S( ), 0 * 0 * s. t. p u( S( ), S ) f ( ; ) d ( p) u( S( ), S 0) g( ; ) d c( ) u * p u( S( ), S ) f ( ; ) d ( p) u( S( ), S 0) g ( ; ) d c'( ) 0 * * Th solution of this problm xhibits th following Rsult. Rsult shows that th moral hazard problm should b considrd cautiously as wll. Rsult. Undr a stat dpndnt utility, if th salary is givn thn w obtain following rsults. () Th ffort of an mploy is qual to th ffort undr a stat indpndnt utility whn u'( S( ), S ) u'( S( ), S 0). () Th ffort of an mploy is highr than th ffort undr a stat indpndnt utility whn u'( S( ), S ) u'( S( ), S 0). (3) Th ffort of an mploy is lowr than th ffort undr a stat indpndnt utility whn u'( S( ), S ) u'( S( ), S 0). Proof. S th Appndix.// VII. Conclusion This study considrs a stat dpndnt utility undr moral hazard, focusing on th loss 4

17 rduction ffort. W assum a two stat modl: loss occurrnc stat and no loss occurrnc stat. According to th marginal utility of incom on th stats, w analyz th optimal insuranc covrag. If th marginal utility of incom is qual btwn th two stats, th optimal indmnity and ffort lvl ar idntical with th stat indpndnt utility cas. Optimal insuranc involvs full insuranc up to a limit and coinsuranc abov th limit. If th marginal utility of incom in th loss stat is gratr than that of incom in no loss stat, thn th optimal insuranc includs full insuranc. On th contrary, if th rlation is th rvrs btwn th marginal utilitis, thn th optimal insuranc includs th dductibl up to a limit and coinsuranc abov that limit. In this cas, full insuranc also can b involvd. This papr also invstigats whthr th moral hazard is mor or lss svr undr a stat dpndnt utility as wll. As a rsult, if th indmnity lvl is qual to th indmnity undr a stat indpndnt utility, th ffort is highr than that of a stat indpndnt utility whn th marginal utility of incom in th loss stat is largr than that of incom in th no loss stat. That is, th moral hazard problm may b rlivd. In contrast, moral hazard can b mor svr whn th marginal utility of incom in th loss stat is lowr than that of incom in th no loss stat. W xtnd th stat dpndnt modl into a two priod modl. Th ffort lvl at tim whn th loss occurs in th first priod can b highr than th ffort whn th loss did not occur at tim, if th marginal utility of incom in th loss stat is gratr than that of incom in th no loss stat. This modl can b applid in a dbt contract and a wag contract modls. Rfrncs Barigozzi, F., 004, Rimbursing Prvntiv Car, Gnva Paprs on Risk and Insuranc Thory 9, Cook.P., Graham,D., 977, Th Dmand for Insuranc and Protction: Th Cas of Irrplacabl Commoditis, Quartrly Journal of Economics 9, Dionn, G., 98, Moral Hazard and Stat Dpndnt Utility Function, Journal of Risk and Insuranc 49, Ehrlich, I., and G.S.Bckr, 97, Markt Insuranc, Slf-Insuranc, and Slf-Protction, Journal of Political Economy 80, Ellis, R. P., and W. G. Manning, 007, Optimal Halth Insuranc for Prvntion and Tratmnt, Journal of Halth Economics 6, Hölmstrom, Bngt., 979, Moral hazard and obsrvability, Th Bll Journal of Economics : Huang, R.J. and Tzng, L.Y., 006, Th Dsign of an Optimal Insuranc Contract for Irrplacabl Commoditis, Gnva Risk and Insuranc Rviw 3, -. Lambrt, Richard A., 983, Long-trm Contracts and Moral Hazard, Th Bll Journal of Economics, Milgrom, P.A., 98, Good Nws and Bad Nws : Rprsntation Thorms and Applications, Bll Journal of Economics, Nwhous, J. P., 006, Rconsidring th Moral Hazard-Risk Avoidanc Tradoff, Journal of Halth Economics 5, Nyman, J. A., 999a, Th Valu of Halth Insuranc: Th Accss Motiv, Journal of Halth Economics 8,

18 Nyman, J. A., 999b, Th Economics of Moral Hazard Rvisitd, Journal of Halth Economics 8, Raviv, A., 979, Th Dsign of an Optimal Insuranc Policy, Amrican Economic Rviw 69, Schlsingr, H., 984, Optimal Insuranc for Irrplacabl Commoditis, Journal of Risk and Insuranc 5, Sog, S. Hun., 0, Moral Hazard and Halth Insuranc whn Tratmnt is Prvntiv, Journal of Risk and Insuranc 79.4, Shavll, S., 979, On Moral Hazard and Insuranc, Quartrly Journal of Economics 93, Shioshansi, F.P., 98, Insuranc for Irrplacabl Commoditis, Journal of Risk and Insuranc 49, Wintr, R.A., 000, Optimal Insuranc undr Moral Hazard, in: G. Dionn, d., Handbook of Insuranc, Boston: Kluwr Acadmic Publishrs, Appndix. Proof of Lmma. () This cas is quivalnt to th stat indpndnt cas, so is omittd. () By (7), w know that u0 '( W0 ) u '( W ). Thus, if u ' 0( W) u ' ( W), W, thn w can writu 0 '( W 0 ) u '( W 0 ( W 0 )) and ( W0 ) 0 by (0) and (). As a rsult, I( x) x ( W0 ). (3) This proof is similar to th (), so w would skip. //. Proof of Lmma. () In quation (8), u ' ( W ) u ' 0( W ) and indmnity is full insuranc, which is qual to th indmnity undr a stat dpndnt utility. Thus, th optimal ffort lvl is idntical with th ffort undr a stat indpndnt utility. () Rcall that th optimal indmnity is gratr than th indmnity undr stat indpndnt utility by lmma. (8) is changd as follows (A.) u '( W ( W )) p ( x ( W )) f ( x; ) dx u '( W ) p xf ( x; ) dx c'( ) Sinc th prmium is incrasd as I( x) incrass in (8), following xprssion is satisfid with rspct to * which is th optimal ffort lvl undr a stat indpndnt utility. u '( W ) p xf ( x; *) dx c'( *) As a rsult, th ffort lvl is incrasd. Furthrmor, if ovr insuranc is not allowd, thn 6

19 u '( W ) u '( W ) in LHS of (8) so th ffort has to b incrasd as wll. Thus, th ffort 0 lvl undr a stat dpndnt utility incrass whn u0 '( W0 ) u '( W ), W. (3) Th proof is similar to th proof of (), so is omittd.// Proof of Lmma 3. Cas ) u0'( W) u'( W), W At first, w suppos that 0. To satisfy quation (), I( x) x. Howvr, with th constraint that p u0( W) f( x; ) dx c'( ) 0, w know that th following condition is satisfid and this is a contradiction p u ( W ) f ( x; ) dx c'( ) pu ( W ) f ( x; ) dx c'( ) 0. (A.) On th othr hand, suppos 0. Lt us dfin that x { x f ( x; ) 0} and x { x f ( x; ) 0}. W also dfin I ( x ) Thus w hav f( x; ) ( ) u ' ( W ( I ( x))) u ' ( W ) f ( x; ) u ' ( W ( I x)) u ' 0( W ( I ( x))) u' 0( W ( I)) I ( x) I x (A.3) as th indmnity for x and I ( x) for x. In a similar way, w obtain, I ( x) I( x) x W obtain th following rlation as wll: I ( x ) f ( x ; ) dx [ I ( x ) x ] f ( x ; ) dx [ I ( x ) x ] f ( x ; ) dx [ I ( x ) x ] f ( x ; ) dx 0 x x (A.4) ( xf x ( x; ) dx x F ( x; ) F ( x; ) dx F ( x; ) dx 0) x Dominanc) This is a contradiction. As a rsult, 0 Cas ) u0'( W) u'( W), W (by First Ordr Stochastic Likwis Cas, by (3), u ' ( W ) u ' 0( W0 ) for x with f( x; ) 0, so w hav I( x) x,whr u' u' 0 whn I( x) x, 0. In this cas, by (0) and (), whr ( W0 ). W also know u ' ( W ) u ' 0( W0 ) for x with f( x; ) 0, so 7

20 I( x) x. If 0 thn, I( x) x so it is a contradiction that. 0 pu0( W) f( x; ) dx c'( ) 0. In addition, 0. This is also a contradiction, bcaus f( x; ) dx 0. Cas 3) u0'( W) u'( W), W By (8), u ' ( W ) u ' 0( W0 ) for x with f( x; ) 0,so w know I( x) x and by (0) and (), whr ( W0 ). In addition, it should b satisfid u ' ( W ) u ' 0( W0 ) for f ( ; ) 0 x, w hav I( x) x. At this tim, I( x) x to fulfill u ' ( W ) u ' 0( W0 ). In a way similar to Cas, 0 is a contradiction.// Proof of Proposition. W first prov that all thr cass involv full insuranc up to a limit and coinsuranc abov that limit. By (3) and th proof of Lmma, w obsrv that optimal insuranc includs full insuranc up to a limit. Now w prov that th coinsuranc abov th limit is includd in th optimum. Lt us assum that xl x x for x L such that f( xl, ) 0. Thn by MLRC, f ( x ; ) f ( x ; ) is satisfid, so w can driv th following rlation. u ' ( W Q x I( x )) u ' ( W Q x I( x )) u W Q x I x u W Q x I x ' ( ( )) ' ( ( )) x I( x ) x I( x ) I x x ( ) I( x ) x (A.5) As a rsult, coinsuranc can b possibl. () W know that u0( W) u( W) A, whr A is a constant. Thn, u0'( W) u'( W). W also hav, by (7), 0( ) ( ; ) '( ) 0 p u ( W ) f ( x; ) dx c'( ) p [ u ( W ) A] f ( x; ) dx c'( ) 0 p u W f x dx c bcaus Af ( x; ) dx 0. That is, two utilitis hav an idntical covrag and ffort lvl in optimum. () By proof of cas in Lmma, for f ( x, ) 0 w know that th L 8

21 u ' 0( W ) u '( W Q xl I( xl)) and I( xl) xl ( W Q). If th lvl of ( W Q) is sufficintly larg, thn full covrag is possibl. (3) In similar ways with (3), w s that I( xl) xl for f( xl, ) 0 by proof of Cas 3 in Lmma. Thus, if ( W Q) is sufficintly larg, thn optimal insuranc can b dductibl up to a limit and coinsuranc abov th limit. On th othr hand, ( W Q) is sufficintly small, thn optimal insuranc can b full insuranc as wll. // Proof of Proposition. () Th proof is similar to th proof of () of Proposition, so is omittd hr. () Undr a stat dpndnt utility, quation (7) is transformd as p [ u ( W ) A( W )] f ( x; ) dx c'( ) 0 0 0( ) ( ; ) '( ) ( ) ( ; ) (A.6) p u W f x dx c p A W f x dx For x whr f ( x ; ) 0, w can writ L L A ( W ) f xl ( x; ) dx x L A( W ) f ( x; ) dx A( W ) f( x; ) dx (A.7) W alrady know following rlation. x I( x ) x I( x ) for x x. Thus by (0), w hav xl xl A ( W ) f ( x ; ) dx A ( W Q x L I ( x L)) f ( x ; ) dx (A.8) x L L L xl A( W ) f ( x; ) dx A( W Q x I( x )) f ( x; ) dx x L A( W Q xl I( xl)) f( x; ) dx ( f ( x; ) dx 0) (A.9) Hnc, w obtain A ( W Q x I ( x )) f ( x ; ) dx 0 whn u0'( W) u'( W), W. As a rsult, th ffort should b incrasd if th indmnity is qual in a stat dpndnt cas and a stat indpndnt cas for all indmnity lvls. (3) Th proof is similar to th proof of (), so is omittd.// Proof of lmma 4. () From (4) to (30), w hav following first ordr conditions. LQ ( p) u 0 '( W0 ) pu '( W ) f ( x; ) dx p u '( W ) f ( x ; ) dx 0 (A.0) 9

22 LI( x ) Q( x 0) pu '( W ) f ( x; ) - pf x ( ; ) pu '( W ) f ( x ; ) 0 (A.) L ( p)[( p) u '( W ) p u '( W ( x 0)) f ( x ; ( x 0)) dx ] I ( x 0, x ) ( x 0)( p) ( x 0) p u0 '( W ( x 0) f ( x ; ( x 0)) dx 0 (A.) L ( p) pu '( W ( x 0)) f ( x ; ( x 0)) ( x 0)( p) pf ( x; ( x 0)) 0 ( x 0) pu '( W ( x 0)) f ( x ; ( x 0)) 0 (A.3) 0 LQ ( x) p[( p) u0 '( W0 ) f ( x, ) dx p u '( W ( x )) f ( x, x;, ( x )) dxdx ] ( x) p p( p) u0 '( W0 ) f ( x, ) dx L I ( x, x ) pp u '( W ( x )) f ( x ; ( x )) dx f ( x ; ) dx p ( x ) p u '( W ( x )) f ( x ; ( x )) dx f ( x ; ) dx (A.4) ppu '( W ( x )) f ( x; ( x )) f ( x; ) x ppf x x f x ppu '( W ( x )) f ( x ; ( x )) f ( x ; ) ( ) ( ; ( )) ( ; ) ( x ) ppu '( W ( x )) f ( x ; ( x )) f ( x ; ) 0 (A.5) Rarranging ths conditions, w hav followings u '( W ) 0 ( x 0) u '( W ) 00 0 ( x ) u '( W ) 0 0 W hav th following risk sharing ruls for priod. f ( x ; ) u W u W f x ( ) (A.6) ' ( ) ' 0( 0 ) ( ; ) W also hav th ruls for priod in cas of no accidnt in t = as blow: [ u '( W ( x 0)) u '( W )] f ( x ; ( x 0)) ( x 0) u '( W ( x 0) f ( x ; ( x 0)) f ( x ; ( x 0)) [ ( 0) ] u '( W ) f ( x ; ( x 0)) u '( W ( x 0)) x (A.7) Whn accidnts occur in t = : 0

23 [ u '( W ( x )) u '( W )] f ( x ; ) f ( x ; ) u '( W ( x )) f ( x ; ( x )) f ( x ; ) 0 0 ( x ) u '( W ( x )) f ( x ; ( x )) 0 u [ '( W ) 0 0 f ( x ; ) f ( x ; ( x )) ( x) ] f ( x; ) f ( x; ) f ( x; ) u W x '( ( )) (A.8) () Th proof () and (3) is similar to Lmma 3 and Lambrt, so it is omittd.// Proof of Proposition 3. W first prov (). () & (3) ar provd in a similar way with (). Undr a stat dpndnt utility, w can transform quations (9) and (30) as blow. [ 00( ( 0)) ( ( 0))] ( ; ( 0)) c x p u W x A W x f x x dx '( ( 0)) 0 (A.9) [ 00( ( )) ( ( ))] ( ; ( )) '( ( )) 0 (A.0) p u W x A W x f x x dx c x Th rmaindr of th proof is similar with th proof of proposition, so w omittd. rsult, th scond priod ffort is highr whn accidnt occurs if u' 0 u'. // As a Proof of Rsult. () W first prov rsult (). W hav th following first ordr condition. [ ( ) (, )] ( ; ) ( ) [ ( ) (, 0)] ( ; ) '( ) p B v S f 0 d p B v S g d c p [ B( ) v(, S 0) A( )] f ( ; ) ( ) [ ( ) (, 0)] ( ; ) 0 d p B v S g d c'( ) (A.) If v'(, S 0) v'(, S ), thn A'( ) 0 and th ffort lvl would b highr. * A( ) f ( ; ) d 0, so w obtain that () Th proof of rsult () and (3) ar similar to rsult (), so it is omittd. // Proof of Rsult. () W first prov rsult (). * p u( S( ), S ) f ( ; ) d ( p) u( S( ), S 0) g ( ; ) d c'( ) 0 *

24 * p [ u( S( ), S 0) A( S( ))] f ( ; ) d ( p) u( S( ), S 0) g ( ; ) d c'( ) (A.) 0 * If u'( S( ), S ) u'( S( ), S 0), thn A'( S( )) 0 and know that th ffort lvl would b highr. * A( S( )) f ( ; ) d 0, so w () Th proof of rsult () and (3) ar similar to rsult (), so is omittd. // Figur. Stat dpndnt utility function whn marginal utility with loss is idntical with marginal utility without loss u 0 A(W) u W Figur. Stat dpndnt utility function whn marginal utility with loss is gratr than marginal utility without loss u 0 A(w) u W Figur 3. Stat dpndnt utility function whn marginal utility with loss is lss than marginal utility without loss

25 u 0 A(w) u W 3