HAZARD MORAL MODELS WITH THREE STATES OF NATURE

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1 Quantitativ thods Inquirs HAZARD ORA ODES ITH THREE STATES OF NATRE Danila ARINESC PhD Associat Prossor nivrsity o Economic Studis ucharst Romania danila.marinscu@csi.as.ro Dumitru IRON PhD Prossor nivrsity o Economic Studis ucharst Romania dumitru.miron@as.ro Dumitru ARIN PhD Prossor nivrsity o Economic Studis ucharst Romania dumitru.marin@csi.as.ro Abstract: Th papr analyss a moral hazard modl with thr stats o natur. Th modl is solvd using as variabls th inormational rnts and ort lvls. Finally w dtrmin th aturs o th optimal contracts in asymmtric inormation. Ky words: moral hazard; asymmtric inormation; inormational rnts; optimal contract Dspit 30 yars o studis in conomics o inormation th cts o asymmtric inormation on dirnt markts ar ar to b complt known. In act this asymmtric inormation constituts th cntral point in conomics o inormation and corrsponds to th situation whr a contractual partnr has mor inormation or bttr inormation that th othr partnr about th transaction charactristics. Th conomics o inormation concntrats on studying th incntivs to gt som potntial gains rom having privat inormation in a transaction. Th incntivs ar prsnt in almost all conomic activitis: thr ar incntivs to work with high productivity to produc good quality products incntivs to study incntivs to invst or to sav mony. A dirnt part o conomics o inormation corrsponds to moral hazard modls. This typ o modls analyss th conomic agnts bhavior whn acting on dirnt markts: labor markt inancial markts insuranc markts agricultur contracting tc. Th macroconomic litratur about th problm o icint wags corrlatd with th Agnt s ort startd with th paprs o Solow and Salop 1979 Shapiro and Stiglitz 1984 and 3

2 Quantitativ thods Inquirs was latr dvlopd by Carmichal 1985 cod and alcomson 1987 Saint-Paul 1996 Krishnan 007. Holmstrom and Tirol 1994 dvlopd a crdit rationing thory basd on moral hazard modls. Dav and Kastnr 009 analyzd th cts o pur xant moral hazard on halth insuranc markt and Duhnam 003 proposd a moral hazard modl or th lasing markt. Rcnt rsarch shows that th modls bcam mor and mor complx most o thm bing mixd modls with moral hazard signaling or scrning. Fudnbrg and Tirol 1990 proposd a mixd modl whr th Agnt s actual choic rgarding th ort is an ndognous advrs slction variabl at th rngotiation stag and this aspct gnrats inicincy. This problm was partially solvd by atthws 1995 and a Pag prsntd a mixd modl with moral hazard and advrs slction and Jullin and Salani 007 xtndd th moral hazard modl or th situation whr th Agnt s risk avrsion constituts his privat inormation such that th modl prsnts also an advrs slction problm. Such approach was also usd by Richlin and Siconoli 004; thy gnralizd th pur advrs slction modl o Rotschild and Stiglitz including som moral hazard variabls. ylovanov and Schmitz 008 studid a two priod moral hazard modl whr th Agnts ar risk nutral with limitd liability and thr idntical activitis. Introduction Th most usd modls o hazard moral ar th modls whr th Principal dosn t hav dirct control on th Agnt s ort. Thr ar also som modls o hazard moral not so usd in th litratur th Agnt s bhavior constituts hiddn inormation ithr bcaus this bhavior is not obsrvabl or vn it is obsrvabl th Principal can not know xactly which is th bst Agnt s dcision rgarding th lvl o ort. [] In th latr situation th scond typ o moral hazard onc th contract is signd th Agnt gts inormation about th stats o natur and knows which is th bst choic rgarding th ort h xrts. This inormation is not obsrvabl or vriiabl by th Principal. From this point o viw thr ar two typs o hazard moral modls: - th modls with an x-ant participation constraint. In this typ o modls th Agnt has a givn xpctd utility whn signing th contract and i h accpts th Principal s or h can not brach th contract in th utur. - th modls with x-post participation constraints th numbr o constraints is qual to th numbr o unknown or unprdictd situations such that th Agnt gts an xpctd utility which is always qual or gratr than his rsrvation utility or such unprdictd situation. will analyz a modl rom th irst typ prsntd abov this modl is not so otn discussd in th litratur with thr stats o natur. Th structur o th papr is as ollows. Sction 1 prsnts th modl. In Sction w transorm th modl using a wll known concpt in conomics o inormation litratur inormational rnts. Sction 3 studis th optimal contract in th situation o asymmtric inormation and in th last part Sction 4 w prsnt th aturs o th optimal contract and som concluding rmarks. 4

3 Quantitativ thods Inquirs 1. Th modl suppos that atr signing th contract th Agnt obsrvs knows th markt conditions i ths conditions ar good or bad. dnot by th paramtr that charactrizs th markt conditions with th businss whil indicats a avorabl situation or corrsponds to a mdium situation a mdium stat o natur. A high lvl o this variabl bad situation o unavorabl situation on th markt implis som dcisions and rgarding th ort with a highr cost than th othr ons. It is obvious thn that. also suppos that th Agnt will xrt a total lvl o ort dnotd by E but this ort lvl costs mor whn th markt conditions ar bad. considr that E whr th Agnt s dcision rgarding th ort lvl is costly but dosn t. Th Agnt will choos th costly ort with rspct to th inormation h gts rom. Th Agnt atr signing th contract obsrvs th tru valu o th variabl or. Th Principal obsrvs th total dcision E; bcaus h cannot distinguish btwn th markt conditions th Principal dosn t know th ort lvl xrtd by th Agnt. This mans that th latr could xrts a high lvl o ort or a mdium or low lvl o ort. Th Principal acs six typs o incntivs constraints 3 pairs o constraints som o thm bing local constraints 4 upward and downward incntiv constraints and th othr two constraints bing global incntiv constraints on upward constraint and on downward constraint.[ 7] Th irst typ o constraints shows that th Agnt dos not prtnd that th markt conditions ar or or whn th tru conditions ar or or or or or. Th scond typ o constraints shows that th Agnt dos not announc that th markt conditions ar or or whn th tru conditions corrspond to th othr typs. Subjct to ths constraints th Principal will or a mnu o contracts whr and rprsnt th costly ort and th Agnt s. wag or ach stat o natur avorabl mdium or unavorabl with considr that th rspctiv probabilitis o th thr natur stats ar and strictly positiv with 1. I th Principal is risk nutral and th Agnt is risk advrs than - using th usual notations - th mathmatical modl P or driving th optimal contract in th situation o asymmtric inormation is: ax s.t. 1 u 3 5

4 Quantitativ thods Inquirs 6 4 P Rmarks Th objctiv unction maximizs th Principal s xpctd nt proit. Th xprssion rprsnts th dirnc btwn th total rvnu th quivalnt o th total ort E and th wag rcivd by th Agnt paid by th Principal i th stat o natur is charactrizd by th paramtr. Th constraints givn by 3 5 and 6 7 ar local constraints upward and downward constraints and th constraints 4 and 8 ar global constraints on upward constraint and on downward constraint. Th utility unction charactrizs th Agnt s risk avrsion and has th ollowing proprtis: 0 and 0 strictly incrasing and strictly concav. Th unction rprsnts th cost unction o th ort th ort disutility and has th ollowing proprtis: 0 and 0 strictly incrasing and strictly convx. For xampl th trm rprsnts th cost o ort whn th total ort is E and th stat o natur is dscribd by th paramtr s valu. Th transormd modl using th variabls: inormational rnts and costly ort lvls t b th Agnt s utility lvls obtaind in ach stat o natur. Thror w can xprss ths inormational rnts as: also considr th unction : R R with or simplicity and without any loss o gnrality th sprad o uncrtainty on th markt conditions. Now th constraints 3-8 bcom:

5 Quantitativ thods Inquirs Ths nw constraints ar asy to driv. For xampl th constraint 10 is a transormation o th rlation 4 as w can s blow: Or th constraint 14 th global downward constraint is a transormation o 8 as w can s blow: must not or th ollowing propositions that th unction has th proprtis: i 0 ii 0. Ths aturs ar asy drivd using th ort cost unction. Proposition 1. I th st o asibl solutions o th program P is nonmpty thn th ollowing inqualitis ar satisid: i ; ii. Proo i us th local upward and downward constraints. Summing up th rlations 9 and 1 w gt: or: From th proprtis o th unction it ollows that: or Nxt rom th constraints 11 and 13 by summing up w gt:

6 Quantitativ thods Inquirs or: sing th monotonicity o th unction or th latr inquality yilds to:. Th condition rprsnts th implmntability condition or monotonicity constraint or th scond bst contracts in th situation o asymmtric inormation. ii Now using th constraint rom 3 and th implmntability condition w obtain: 0 Thn and so. From 5 w gt: It is obvious now that 0. To conclud w can stat that. Th optimal contract in th situation o asymmtric inormation Coming back to our sttings rom Sction 1 w ar now intrstd in solving th incntiv problm P. To simpliy th analysis and ind th rlvant binding constraints w procd as ollows. First w ignor or th momnt th local and global downward incntiv constraints givn by 6 7 si 8. It is almost obvious that th most icint typs would want to li upward and claim that thy ar lss icint. Scond in th inal stp w chck x post that th incntiv constraints ar indd not binding nonrlvant and ar satisid by th optimal solution. nd irst to proo th ollowing proposition. Proposition. Th global upward constraint 4 is implid by th two local upward constraints 3 and 5 whn th monotonicity constraint holds. Proo To show this rsult w us th constraints 9 and 11 which ar quivalnt with th constraints 3 and 5 and wr obtaind using th chang o variabls. Suppos that th ollowing inqualitis and ar satisid. 8

7 Quantitativ thods Inquirs 9 Summing up th abov two rlations w gt: or. It asy to show that. This is bcaus or or. This last xprssion corrsponds xactly to th implmntability condition assumd to b tru ith this simpliication o th Principal s program th only rmaining rlvant constraints ar 3 and 5. Th corrsponding Kuhn-Tuckr multiplirs or th constraints 3 and 5 ar dnotd by and. Thror th agrang unction it is writtn as: ; u Th irst ordr th optimality conditions Kuhn-Tuckr conditions ar: Proposition 3. Th participation constraint and th local upward constraints 3 and 5 ar binding at th optimum. Proo Adding up th rlations 18-0 :

8 Quantitativ thods Inquirs. From this it rsults that 0 and so th x-ant participation constraint is binding. Thror w gt: u Nxt w analyz th optimal valu o th two variabls and w considr th ollowing cass: Cas This is not an intrsting situation sinc it corrsponds to th cas o symmtric inormation. Cas. 0 and 0 Th irst ordr conditions rom and 0 yild to th inquality: and so. rsult In th sam way w also obtain can writ thn: rom Proposition 1. Cas 3. 0 and 0 or 1 and or sing th sam rlations as abov w obtain: but this contradicts th bing a contradiction o th rsult rom Proposition 1. Cas 4. Th irst thr cass ar not possibl solutions. Thror th only possibl cas corrsponds to 0 and 0. Th immdiat consqunc is that th local upward incntiv constraints ar binding. Anothr consqunc ollows: it is impossibl that th global upward incntiv constraint to hold with quality to b binding. or using th implmntability condition and th prvious rsults w can stat that th downward incntiv constraints hold strictly. proo this statmnt in th nxt proposition. Proposition 4. I th multiplirs and ar strictly positiv thn th ollowing ar tru: i ii 10

9 Quantitativ thods Inquirs iii Proo i I 0 and 0 thn th corrsponding constraints ar binding: and sing th irst quality w obtain: quivalntly. This is tru du to th implmntability condition. or Rmark: or than it was shown th constraint holds strictly maning that th Agnt is not intrstd to claim to announc th bst stat o natur whn th tru stat is th mdium on. Indd i thn rom 3 or rom th quivalnt rlation 9 w gt. From 16 and using 19 it rsults: and so: or Thror w hav. ut which is a contradiction to. Th conclusion is immdiat. and this implis that ii Th binding constraint 11 yilds to: that Th lattr inquality is tru bcaus w know rom th implmntability condition. hav alrady provd that th rlations i and ii ar satisid. Summing up th trms rom th two sids w gt: whr corrsponding to th implmntability condition or. 11

10 Quantitativ thods Inquirs 4. Conclusions drivd in th last sction th optimal solution o th Principal s problm. can now summariz th charactristics o this optimal solution in th ollowing thorm: Thorm. Th main aturs o th optimal contract in th situation o asymmtric inormation ar: A. Th Agnt s xpctd utility is xactly th outsid opportunity lvl o utility u th rsrvation utility lvl.. I th markt conditions corrsponds to th most avorabl situation th contract is Parto-optimal i... In this cas thr is no distortion with rspct to th irst bst solution. C. For th othr two markt conditions stats o natur th contract is no longr Partooptimal. In this cas th ollowing rlations ar satisid: and. Indd using th rlations 16 and 19 and th abov rsult 0 w gt: or. On th othr hand using 17 0 and 0 w gt: or. D. I th stat o natur is stats and th Agnt gts positiv inormational rnts with rspct to th. Th Agnt gts also a positiv inormational rnt in th stat rspct to th last avorabl stat o natur. with REFFERENCES 1. akr.p. Incntiv Contracts and Prormanc asurmnt. Journal o Political Economy pp olton P. Dwatripont. Contract Thory.IT Prss 004 pp ; 3. Fudnbrg D. Tirol J. oral Hazard and Rngotiation in Agncy Contracts. Economtrica pp

11 Quantitativ thods Inquirs 4. rossman S.J. Hart D.O. An Analysis o th Principal-Agnt Problm Economtrica pp Holmstrom. oral Hazard and Obsrvability. ll Journal o Economics pp Jullin Salani. Salani F. Scrning Risk-avrs Agnts undr oral Hazard: Singl Crossing and th CARA Cas; Economic Thory pp aont J.J. artimort D. 00 Th Thory o Incntivs. Th Principal-Agnt odl. Princton nivrsity Prss 00 pp arinscu D. arin D. Pooling Equilibria and Shut Down o ast Eicint Typ Policy on Crdit arkts Economic Computation and Economic Cybrntics Studis and Rsarch No. ASE Publishing Hous ucharst 003 pp Acknowldgmnt This work was coinancd rom th Europan Social Fund through Sctoral Oprational Programm Human Rsourcs Dvlopmnt projct numbr POSDR/1.5/59184 Prormanc and xcllnc in postdoctoral rsarch in Romanian conomics scinc domain. 13