THEORETICAL APPROACHES RELATED TO OPTIMAL CONTRACTS IN CASE OF MORAL HAZARD

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1 wwwararesscom/volumes/vol9issue3/ijrras_9_3_08df TEORETICA APPROACES REATED TO OPTIMA CONTRACTS IN CASE OF MORA AZARD Tudor Colomesch Deartmet of Comuter Scece, Stefa cel Mare Uversty of Suceava, Romaa E-mal: ABSTRACT Ths aer emhaszes a mortat ssue as cocers the theory of cotracts: the moral hazard The aer begs wth a descrto of the geeral model of moral hazard for dscrete cases, where the varat of two effort levels s solved The, solvg of the model s oted out, by the method of frst order aroxmato, cocludg wth a geeralzato of the moral hazard model for more effort levels Keywords: Moral azard Model, Asymmetrc Iformato, Prcal-Aget Model, Otmal Cotracts 1 TE INFORMATIONA ASYMMETRY GENERA REGARDS Whe a cotract s cocluded, the formatoal asymmetry s oted out whe oe of the artcats ow more or better formato comarg to hs or arters The asymmetry of formato kows three tyes of aearace: - the moral hazard (rsk) at whch the Prcal s ot able to kow the effort level carred out by the Aget For ths reaso, he wll clude some ctato such cotract; - the adverse selecto (couter-selecto) stuato where the Aget ows formato that the Prcal does ot kow before cocludg the cotract I such stuato, the Prcal wll roose to the Aget more cotracts, ad deedg the cotract chose, he wll otce the formato hdde by the Aget; - sgalg oe of the arts has a hdde art of formato, though such formato s sgaled by ts behavor to the other art, also The mortace of formatoal asymmetry roblem was better emhaszed 001, of Nobel Prze for Ecoomy, acheved by George Akerlof, Joh Stgltz ş Mchael Sece, for the studes carred out by these o markets of asymmetrc formato GENERA FORM OF TE MORA AZARD MODE FOR TE DISCRETE CASE The cocet of moral hazard aeared for the frst tme XVII th cetury, but for a log tme, t had a rofoud egatve result, thus volvg a false reteces behavor The frst aroach related to curret vew of the moral hazard belogs to Pauly (1968), whch roved that f the tye of surace affects the requremet, the total surace cotract s ot loger Pareto otmal Zeckhauser (1970) bult a frst model of moral hazard, aled to the satary system, followed by Sece ad Zeckhauser (1971), wth a geeral form of the moral hazard model The moral hazard occurs whe the Prcal caot otce the effort submtted by the Aget, or whe he receves some rvate formato after sgg the cotract, therefore, the artcats ow the same formato ust before cocludg the cotract, ad the formatoal asymmetry wll take lace durg ts rogress The stages of rug such cotract moral hazard codtos are the followg: the Prcal rooses a cotract to the Aget; the Aget accets (f he agrees) the cotract; the Aget accomlshes a effort (that caot be checked by the Prcal); the tye determes the state; the acto of the Aget s eded wth a result, for whch ths wll be rewarded The hyothess of the model assumes the stuato of the Prcal eutral to rsk ad of a Aget wth rsk averso We cosder that a cotract s cocluded betwee the Prcal ad the Aget, meetg the codtos ad rcles revously reseted The Aget oblges oeself to make a effort deoted by e, accordg to whch a result wll be acheved The set of otetal results (exressed moey) wll be rereseted by:, ad, 1, X x1, x,, x e P X x e sgfes the robablty of achevg the result x codtos of the effort e We wll assume that e 0, 1, ad e Ths model studes the behavor of the two arters towards the rsk, codtos of ucertaty, thus aalyzg accordace to ther utlty fuctos For such effort, the Aget wll be rewarded by the Prcal wth a wage W, ad ts utlty fucto wll deed creasgly by such wage, ad decreasgly by the effort carred out: 413

2 Colomesch Otmal Cotracts Case of Moral azard, U W e U W V e The form of the utlty fucto shows that t s adductve searable as related to the other two argumets W ad e, so ts atttude related to rsk wll be deedet wth the level of effort The wage wll be a fucto deedg of the result acheved: W W X The aget havg rsk averso has a fucto of utlty strctly cocave ad creasg as related to the wage, whch meas that: U 0, U 0 Also, a hgher effort of the Aget reresets for hm a decrease of utlty, therefore: V 0 The margal dsutlty of the effort s cosdered to be creasg, meag that V 0 The Prcal s the beefcary of the effort result secfc to the Aget, so that ts utlty fucto, deoted by B X W, wll deed creasgly by the level of result ad decreasgly by the wage ad to the Aget The Prcal s eutral to rsk, ad we otce that B 0 ad thus, B s costat I case of symmetrc formato, the model leads to towards a otmal wage, deedet of result, so that the Aget wll be charge of choosg the level of effort carred out the frst ste of the roblem, thus choosg the mmal effort I ths way, the utlty wll be maxmal So, the choosg of Aget ca be formally wrtte uder ths form: e arg max ˆ ˆ e U W x V e eˆ (1) The codto metoed above s kow uder the ame of ctato costrat or stmulato costrat It roves the resece of moral hazard stuato For the secod ste of the roblem, determg the otmal level of the effort carred out wth the cotract, the Aget decdes f he sgs or ot that cotract The accetato wll take lace f the utlty estmated wll be at least at the level of reserve utlty, meag: U : e U W x V e U Ths codto s kow uder the ame of artcato costrat or dvdual ratoalty costrat I the frst ste of the game, atcatg the acto of the Aget, the Prcal wll roose a cotract, by whch the roblem s soluto wll be rereseted Such otmzato roblem maxmzes the utlty, wth the two costrats, meag the artcatve ad ctato costrats: max e B x W x e, W x, e arg max ˆ ˆ eˆ e U W x V e e U W xv e U e 0; W x 0,, () 3 TE MORA AZARD MODE ON TWO EVES OF EFFORT We wll start ths secto by solvg the moral hazard model, dscrete varat, wth the stuato by whch the Aget ca choose betwee two levels of effort: a hgh level of effort, deoted by, ad a low level of effort, deoted by We wll have e e, e ad obvously, V e V e the X set of the ossble results that mght occur: X x x x 1 Also, we wll arrage creasgly We wll deote as forwards: e P X x / e 0, 1,, the robabltes of achevg the result x, 1, e P X x / e 0 for each the two varats of effort ad we wll assume that these robabltes are strctly ostve We wll also have: 414

3 Colomesch Otmal Cotracts Case of Moral azard The moral hazard occurs whe the Prcal asks to the Aget to carry out the hgh effort e O the cotrary case, solctg the low effort e, he wll ay to Aget the fx amout from the otmal cotract, stuato of symmetrc formato W U U V e We wll aalyze the case whe the Prcal solcts the carryg out of the hgh effort e But, the Aget wll choose to submt a effort e f oly the exected utlty s hgher tha the stuato of low effort I ths way, the ctato costrat s uder the form: U W x V e U W x V e 415 U W x V e V e (3) The otmal cotract wll be the soluto of the otmzato roblem () As the Prcal s rsk eutral, we wll acheve B costat ad therefore, Bx a x b, a 0, b The two costats that aear ca be elmated from the amed fucto I such case, the roblem () ca be rewrtte: max W x, W x 0,, x W x U W x V e U U W x V e V e The otmal soluto of ths roblem should verfy the Kuh-Tucker codtos agrage fucto assocated to the revous roblem s therefore: 1,,, W x x W x U W x V e U U W x V e V e 0, 1, ; 0; 0 W x Frst order codtos are rereseted by: From the frst order codto, we have: U W x U W x 0 : U W x 0,,,,, U W x Addg the relatos wrtte above, accordace to 1, ad takg to accout that wll acheve: (4), we Oe mght otce that 0, sce 0, 1, ad U 0 U W x stuato, Kuh-Tucker codto related to s met, ad the costrat of artcato wll be saturated We wll calculate forwards the secod order dfferetal quotet of agrage fucto, ad we get: W x U W x U W x I such

4 Colomesch Otmal Cotracts Case of Moral azard U W x,, W x 416 (5) Aalyzg (4) ad takg to accout that 0, 1, ad U 0, we ca see that 0, 1, As U 0, because the Aget has rsk averso, we wll deduct from (5) that W x 0, 1, Sce the other artal dervatves of secod order secfc to fucto ad related to wages have zero value, the essa matrx wll be egatvely defed, ad the statoary ot wll be rereseted by the local extreme ot The otmal codto of the cotract gve by (4) ca be rewrtte more smle, by dvdg the relato at 1 (we kow that 0, 1, ):, 1, UW x If we assume that 0, we wll see from the revous result that the wage s costat; ths asect cotradcts the ctato costrat (3), leadg to the stuato of symmetrc formato, whe a the Aget chooses the mmal effort level Thus, we eed 0, whch meas that the moral hazard brgs to the Prcal a strctly ostve cost, ad mlctly a roft lower tha the stuato of symmetrc formato cotract The multler s called shadow cost The Aget s wage wll deed for ths cotract uo the level of result ( 0 ), thus creasg as the decreases Ths reort, amed robablty rate, shows what ercetage the result x dcates the effort e (whe the reort decreases, the robablty that a result x to be acheved wth a effort e wll crease) The Prcal should offer a hgher wage, so that the Aget wll carry out a creased effort Cosderg the codtos of dfferece towards the rsk of the Prcal, the aymet deedg o result s carred out order to motvate the Aget For the Prcal, the advatage of establshg a wage related to result cossts fdg formato as related to the effort carred out by the Aget The ga deeds uo formato ad wll crease deedg o result, f a better result s assocated to a hgher effort Forwards, we wll determe a codto accordg to whch a hgher result wll volve a hgher wage From the revous relato, oe ca acheve the followg: 1 1 W x U,, We ca otce the followg: f the robablty rate decreases, 1 1 value, the ts verse wll crease, ad therefore the argumet of fucto U metoed above wll crease Sce ts argumet s creasg, we wll deduct that the wage W(x ) s creasg As cocluso, f the rate of robablty s decreasg, hgher wages wll corresod to hgher results 4 SOVING TE MORA AZARD MODE BY TE FIRST ORDER APPROXIMATION The moral hazard roblem () shows the coveet accordg to whch oly the ctato costrat s sgfcat Moreover, t s reseted uder the form of other roblem of otmzato I order to overass such weak ot, ths costrat was relaced wth the frst order codto: e U W x V e 0 As a matter of fact, oe should meto that ths frst order codto s oly ecessary order to maxmze (1), but s ot certaly eough (the roblem of otmum s ot certaly cocave)

5 Colomesch Otmal Cotracts Case of Moral azard Itroducg roblem () the above metoed codto, we wll acheve: max e x W x e, W x, e U W xv e U (6) e U W x V e 0 agrage fucto assocated to ths roblem s:, 1,,, e W x e x W x e U W x V e U e U W x V e Wrtg dow the frst order codto, as related to the wage, we acheve: W x 0,, e e U W x 1 euw x 0 : UW x e 0,, U W x e As the essa matrx a statoary ot s egatvely defed, the ecessary codto of frst order also becomes suffcet, so that such ot wll be extreme local If 0, the otmal cotract wll be dfferet as comarg to the symmetrc formato The results deed oly uo the reort e e If such reort s creasg as related to, we deduct from (7) that e (7), ad the wage wll deed uo results U W x s decreasg as related to Though U 0, sce the Aget has rsk averso, ad thereforeu wll be strctly decreasg ad we wll obta that wage W(x ) s creasg We reached the same cocluso as the stuato of two levels of effort, meag that a better result sgalzes that a hgher effort was carred out Wrtg the frst order codto as related to the effort of Aget e, for the roblem deoted by (6) (that s a ecessary codto), we wll have: e x W x e U W x V e 0 e 0 e U W x V e 0 e x ew x eu W x V e The revous equalty shows the way wch varato of Prcal s roft as related to the effort modfcatos s determed by the varato of wage, to modfcatos of effort ad ctato costrat 5 TE MORA AZARD MODE WIT MORE EVES OF EFFORT We wll forwards reset a geeralzato o a dscrete case of the moral hazard model wth two levels of effort, solved aragrah 3, to be aled the stuato whe a Aget ca choose betwee more levels of effort (or actos) a1, a,, am, m 3, the set of actos of the Aget, ad P X x a,, m, 1, the robablty by whch the result x s acheved, whe the Aget carres et s cosder 417

6 Colomesch Otmal Cotracts Case of Moral azard out the effort a As the revous cases, we assume that these robabltes are strctly ostve: 0,, m, 1, (therefore, some actos mght be excluded) The Prcal, ot beg able to otce the effort of the Aget, wll offer t a wage related to the acheved result, ths beg the oly elemet that ca be otced by the Prcal I order to smlfy the otatos, we deote by W W x, 1, the wage receved by the Aget, whe the result otced was x We wll cosder a Aget wth rsk averso, havg a utlty fucto strctly creasg ( U 0 ), strctly cocave ( U 0 ) ad addtvely searable, uder the form of U(W)-a, ad a Prcal eutral to rsk, havg the utlty fucto uder the form B(X-W)=X-W (we ca make ths assumto sce, beg rsk eutral, B costat t results Bt at b, a 0, b ; kowg that a ad b are costat, they ca be elmated from the roblems of otmzato) The Aget wll choose the otmal acto that wll esure hm a maxmal utlty, codtos of rewards offered by the Prcal: U W a, m max Therefore, he wll choose the acto a, wth 1, 418 (8) m, f: U W a k U W a k, k 1, m, k (9) Choosg the maxmum of the exected utlty from (8) wll lead us towards the satsfyg of those m-1 ctato costrats, from (9) Also, we should resect the costrat of artcato, accordg to whch he exects a utlty at least equal to the utlty of reserve, U : U W a U (10) The Prcal wll decde the level of wage offered to the Aget, followg the achevemet of the maxmum exected utlty: max x W W,, (11) The otmal cotract of moral hazard corresodg to the fx acto a, wth 1, m wll be rereseted by the otmal soluto of the roblem of otmzato, whch ams o maxmzg the fucto of Prcal s utlty (11), resectg the ctato costrats (9) ad the artcatve costrat (10): max x W W,, U W a k U W a k, k, m, k (1) U W a U agrage fucto assocated to the revous roblem s exressed uder the form:, 1,, k, 1,,, W k m k x W m U W a U W a U W a We U k k k k k deoted by, k k, m, k m, resectvely by, the Kuh-Tucker multlers attached to the costrats of ctato (9), resectvely to the artcatve costrat (10)

7 Colomesch Otmal Cotracts Case of Moral azard The codto of frst order otmal as related to wage W, 1, wll be thus wrtte: m 0,, UW UW W k k k k m 1 k U W 0 : U W 0 k 1, 1, U W (13) k To the saturated ctato costrats of roblem (1), strctly ostve multlers wll corresod to them, ad those actos wll brg the same utlty as the acto a The otmal cotract s gve by relato (13), where the fracto 1 s creasg as related to W, sce U s strctly UW decreasg The otmal wage ca be deducted from the revous relatosh uder the followg form: 1 1 W U,, m k k 1 k k Oe mght see that for such cotract, the wage s ot costat, as the stuato whe the Prcal ca otce the actos of the Aget, so that the two arters share ther rsk The varato of wage W wll deed uo the reort of robablty 6 CONCUSIONS I the stuato of model o two levels of efforts, the costrat of artcato of the Aget s saturated, ad the wage acheved wth the otmal cotract s o loger costat, cotradcto wth the symmetrcal formato; such thg shows that the moral hazard brgs to the Prcal a strctly ostve cost, ad mlctly a roft lower tha the cotract of symmetrcal formato The wage of the Aget deeds uo the level of result, aymet deedg uo the result beg doe oly order to motvate the Aget The advatage of establshg a wage related to result s rereseted for the Prcal by fdg out formato related to the effort carred out by the Aget The ga deeds uo formato, ad wll crease deedg uo result, f a better result s assocated to a hgher effort REFERENCES [1] Besako, D, Braeutgam R (005) Mcroecoomcs, d edto, Joh Wley&Sos, New York, ISBN [] Bolto, P, Dewatrot M (005) Cotract Theory, Massachusetts Isttute of Techology, ISBN [3] Crawford, T W, Kuerma A (006) Gamblg o umatara Iterveto: Moral azard, Rebello ad Iteral War, New York: Routledge, ISBN [4] Dembe, A E, Bode, I (000) Moral azard: A Questo of Moralty?, New Solutos, vol10, r 3, ag 57-79, ISSN [5] Gravelle,, Rees R (199) Mcroecoomcs, d edto, ogma Grou, odo, ISBN [6] olmstrom, B (1979) Moral azard ad Observablty, Belle Joural of Ecoomcs, vol 10, ag [7] affot, J J (1995) The Ecoomcs of Ucertaty ad Iformato, Massachusetts Isttute of Techology, ISBN [8] affot, J J, Martmort D (00) The Theory of Icetves: The Prcal-Aget Model, Prceto, ISBN [9] Macho-Stadler, I, Perez-Castllo J D (1997) A troducto to the ecoomcs of formato, Oxford Uversty Press, ISBN [10] Marescu, D, Mar D (009) Mcroecoome avasată Ecooma formaţe, ed ASE, Bucureşt, ISBN [11] Pauly, M V (1968) The Ecoomcs of Moral azard: Commet, Amerca Ecoomc Press, vol 58, r 3, ag , ISBN [1] Salae, B (005) The Ecoomcs of Cotracts: A Prmer d edto, The MIT Press Books [13] Sece, A M, Zeckhauser R (1971) Isurace, Iformato ad Idvdual Acto, Amerca Ecoomc Revew, vol 61, r, ag , ISSN [14] Zeckhauser, R (1970) Medcal Isurace: A Case Study of the Tradeoff betwee Rsk Sreadg ad the Arorate Icetves, Joural of Ecoomc Theory, r, ag 10-6, ISSN k k 419