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Mi ti gating Non-co ntract abl e Ac ti ons by

Randomness

Roland Str ausz 3

Free University of Berlin

July 15, 1998

Abst ra ct

Thi s pap er studi es non-contractabi li tyofacon trac t de si gner's acti ons in an

age ncy mo d el withc ostl y monitorin g. It showsthat non-contractabi li tyma yl ead

to an expl i cit randomness, whi ch i s not optimal unde r full c ontractability. The

ran domnessmti iga tes n on -contractability. Its e ecti ve ness i ncreases wi tht he ex

p ost veri abi li tyo f the non-contractabl e vari abl e. Mi tigati on is p erfec t, i f the

non-c ontractabl eacti onisp erfec tlyexp ostveri abl e. Thepap er shows that

non-contractabi li tyi s l essseverethan somerecent literaturei ndi cates.

(JELc lassi cati on: D82)

Keywords: in comple te contractin g,r and om si gnals, sto chastic contracts,

non-contractabl emonitoring

3

FUBer lin,B oltzmannstr . 20,D-14195,Berl in,Ger many.Iw ouldlike to thankHe lmutBe ste r for

the inten se and clarifying discus sion s that have led to thispa per. I would fu rther more like to thank

DavidPe re z-Castrillofordrawingmyatte ntiontos omeinconsiste nciesinanearlie rdraft . Thank-you's

als ogoto SjaakH urkens ,theparticipantsof Quatschgrupp e2andthoseof thebrownbag seminarat

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1 Int roductio n

In rece nty ears the re has b ee n an in creased interest i n c ontracti ng mo d els, i n

whi cht he contract d esigner's commi tment i s l i mited. For two di erentr easons

the li terature views th ese mo d els of i ncompl ete contracti ng as a more reali stic

desc ri pti onofcon tracti ng environments. First, on part of the contrac t de signer

there may exist acti on s that are n on -observabl e. The non-observability l eads to

non-c ontractability and l i mits the c ontract desi gn er's( contractual ) commitment.

Examp le saremon itoringor,moregen erally,apri nci pal 's e ortinstraigh tforward

extensi ons of stand ard agency mo del s. 1

Se cond, the contrac t desi gner's l i mi ted

commitmentma y be due to an exogen ou s l imi tati on on enforceabl e contracts.

These may be the inabi l ity toexcl ude ex post ren egotiati on contractuall y or the

unavai labi l ityofrand omco ntracts.

This paper concerns itsel f with incompl ete contracting du e to

non-contractabi li ty. It showst hatno n -contractabilityma y b e miti gated byt he

in-tro d uction of a rando mn ess, whi ch i s not opti mal unde rful l contractabi li ty. The

i ntui tion b e hi nd this re sult mayb e expl ain ed asfo llows. In agency probl emsthe

pri nci palmotivates heragentb ya carrot ands ti ckapp roach, promi sin gan agent

a rewardfor ou tcomes,whi ch i nd icatethat theagentw asdi li gent, and pu nish ing

hi mforoutcomesthati nd icateshirki ng. Nowconsi derastrai ghtforwardext en sion

ofast andardagency prob lemi nwh ichthe prin ci palmaycho osesomehel pi ng

ef-fort. Th ehel pi nge ortwl til ypicallya ecttheprobabilitythatthe agentrec eives

thepromi sed carrot orstick. Si ncecarrots arecostly tothepri nci pal, whi le sticks

representcostsa vi ngs,apri nci palwil ltendtohavelittl einc entivestoprovi dehel p

whenshei s c ommi ttedtoa toughcarrotand sticksc hed ule . Thepri nci pali s abl e

tocircumventt h is probl em, when h er e ort is contractabl e. I n thi scase shecan

simp lyc ommi th erself c ontractuallyto somel evelof h elp ,p reventi nghersel f from

myopi call y mani pu lating the han d out of promi sed sticks and carrots duri ng the

game. Wi thnon-contractabi li tythi s i snotpossi bl e. Inth iscase onl yalax carrot

and sti ckapproach wi ll i ndu ceh er tocho osea posi tive level ofh elp . A lax carrot

and stick app roach has, however, the di sadvantage that i t decreases the agent's

i ncenti ves. An ex ante random c ontractma y al le viate thi s probl em. By mi xing

betweenato ughcarrotandstickapproach ,whi chin duce sn oh elp ,andal axcarrot

1

The non-obse rvabilitym aya lso c once rn s omeexan te non-ve ri able information on part of t he

contr ac tdes igner . Non-con tractabilityofthiski ndwillleadtheage nttoin ter pretthecontrac to e ras

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and stick approach,that i nduc es he lp, thep ri nc ipal i s abl e touse a tough carrot

and sti ckap proachthat stil l ind ucessomeh elp in expectati on.

R ecently,no n -contractabil i ty has expl ici tly been studi ed by Jost(1995, 1996)

and K hali l (1997). Wew i l lfol lowthe se pap e rs i nthei r assu mption that th e

non-contractabi li tyco n cerns the p ri nc ipal 'sm oni tori ng dec isi on. 2

Thi s permits us

to appl y ou r resul ts d irec tly t ot h ese mo del s and e nable s u s to sh owt hat

non-contractabi li tyism uchlse s severe than thi s recent literature i ndi cates. 3

More

preci sel y,w e wi ll showt hat the resu lts of Jost (1995, 1996) and Khal il (1997)

are more due to an undi scusse d exogenous l i mita ti on on en force ab le contracts,

the un availa bilityofra n dom contracts, thanto the ir exp li ci t assump ti on of

non-contractabi li ty. Thesep apers namely assume a moni torin g techn ol ogy,fo r whi ch

ran dom transfers area b le to miti gate the non-c ontractabilityp erfec tly. Th is

pa-p e rthereforewarns again st attribu ti ngtheresul tsofJost(1995,1996) andKh alil

(1997) sol elyto thei rassu mptionof non-c ontractability.

Al though the opti mality of randomness may not b e obvious at rst si ght, we

l ike to stress thatour form of randomi zati on isi n n orespect far-fe tche d. I n fact,

the type of randomiz ati on that we consi der i s standardl y app li ed i n th e full

con-tractabi li tymode.lItist h erefore rather natural toconsi der thi styp e of

random-nessal so in th ecaseof non-contractabi li ty.

Thus,weanal yzenon-c ontractabilityi nasi mp leagencymo delwi thc ostl y

mon-itorin g.Itisw ell known that for such mo del s opti mal contracts exh ib it rand om

monitorin g and de termi ni stictransfers,i f contracti ngis compl ete(e.g. Town send

(1979), Bord er and Sob el (1987),Mo okherjee and Png (1989)).W e show that if

one l eaves th ecompl ete contractin g setti ng and assumesthat mon itori ng i s

non-contractabl e, the opti mal contract wil l involve random transfers. We obtai n the

resul tthatrandomtran sfersareabletomi tigatethep ri nci pal 'snon-contractabi li ty

compl etely,if mo n itoringis expo st veri abl e. If the ex p ostveri abilityof m oni

-tori ng i s onl yp artial ,ran domtransfe rsarestil l stri ctlyoptimal, but al levi ate the

non-c ontractabilityonlyparti ally.

The rest ofthe paperi s organi zed asfollows. The next section i ntro d ucesthe

mo del , whic hisa na dapted versi on of the one use d in S trausz (1997). S ection

2

Onemays eemonitor ingbyth eprincipalas helpingtheage ntinmakinghisact ionve ri able.

3

Khalil st udie s an agenc y pr oblem with advers e se le ction r at her than moral haz ard. The

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3der ives the optimal contrac t und er compl ete c ontracti ng. The opti mal

con-tract is a sp eci al case of Mo okherjee and Png (1989). Sec tion 4 intro du ces

non-contractabi li tyo fm oni tori ng and shows how the optimal contract de p en ds on a

typeofr and omne ss,whi ch also arisesi ntheful l c ontractabilitysetti ng. Secti on5

analyze s th e optimal contract und er non-contractabl e moni torin g an d the

exoge-nousl i mitati onthatcontractsmaynotexhi bi trand omne ss. Thissettingissim ila r

to Jost (1995, 1996) and Khali l (1997) and enabl es u s tomea sure the value of

ran domization. Secti on6st u die scontinuou smoni torin gand shows that the

opti-mali tyofr and omtran sfersd o esnotde p en don th edi scretemo ni tori ngdec isi ono f

thepri nci pal . Fi nall y, Sec ti on 7concl ude s.

2 The model

Con sid erari skn eutral prin cip alwh oempl oys ariska verse agent. Theagent'sjob

ist oc ho osean un observabl e acti on. For si mplicityw e assume that the avai labl e

actionsareei thertowork,ortosh irk. Bycho osi ngtowork,th eagentinc ursacost

of e orte,w hilefor shi rki ngth ese costsar e z ero. The agent'su tilityissepar abl e

i nweal thand e ortandma ybe wri ttenas:

U(t ;e)=u(t)0e;

where we assume U(0;0)=0ist he agent's outsi de opti on. Furth ermore, the

agent's util ity is i ncreasi ng i n transfe r, i.e. u 0

(:) >0, and exh ib its riska versi on,

i.e. u 00

(t )<0.

Astheownerofthe rm,th eprinc ip alre cei vestheoutputrel ate dtotheagent's

action. Ifthe agentsh irks, th eoutpu ty

s

resu lts,wh il eworkin g l eadstoanoutpu t

y w , where y w > y s

> 0.Ou tput i s, however, unobse rvable an d, the refo re,

non-contractabl e.

Toprovi dei ncenti ves,thepri nci palmaymoni torth eagentatacostc2(0;y

s ).

Mon itori ngrevealst hea gent'sacti onwit hpro bability2(0;1]andd o esn otreveal

anyth ingwi thprobabi li ty1 0. 4

Ifa resu ltobtain s,i ti s veri able an d contracts

maybemadeconti ngentonit.W ewi ll thereforerepresent atransf erschedu letby

atr iple(t w ;t s ;t n ), wheret w ,t s

are the transfersfrom the p rinc ipal tothe agent,

when the evid ence shows that the agentw orked, or shi rked , re sp ecti vely. The

transfer t

n

i s made, when no evi de nce is availa b le. Thi s coul d ei th er be because

4

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monitorin gdi dnotrevealanythi ng,orb e causeth eprin cip aldi dnotmoni tor. Note

thatthe tri pl e(t

w ;t

s ;t

n

)ma yitsel f depend on otherv eri abl evari abl es.

Wefur the r assume that the agentisw eal th constrai ned . The agentca n not

b e forcedtopa y a positivea mount to the prin ci pal, i.e. transfers must b e

non-negative.

Toha veano n-t ri vialsettin g,weassu methatth epri nci palwantstoi nduc ethe

agenttow ork. Thi swillb ethe c asei fth edi erenc ey

w 0y

s

i s,i n compari son to

e,la rge en ough,i.e. y

w 0y

s

>>e .Itw i l lal sob eh el pfultod e ne anesucht hat

u 0 (u 0 1 (e=))(u 01 (e=)+ c=)=e=: 5

3 Contractable Mo nito ring

In this se ction we anal yze the mo del i n a compl etecontracti ng envi ronment and ,

thus, assu me that moni torin g is veri ab le and contractab le. Fol lowing standard

contract theory contractabi li ty i mpl ie s that the princ ip alma yco mmi t hersel f to

thosevari ab leswhic har econtractable . Thi scommitmentma yin volverand omn ess.

Therefore,when moni tori ngi scontractabl ethepri nci palc anco mmi ttoarand om

monitorin gstrategybyspeci fyi ngiti nthecontract. Wewi lldesc rib ethepri nci pal 's

monitorin g strategy by her probabil i tyo f monitorin g p. It then follows that a

generalcontract C consistsof acomb ination (t;p). 6

The p ri nc ipal 'spr obl em i s to o er a contract C =( t;p)t hat maximi zes her

payo ,whil e in duci ngtheagenttoacce ptthec ontractandw ork. Su ch acontract

C yie ld sth epri nci palthe util ity

V(C)= y w 0pt w 0(10p)t n 0pc: (1)

To i nd uce the agenttowork, thec ontractm ustmakethe agentweakly bettero

i fhecho osestowork. Gi ve nacontractC=(t;p)theagent'sutil i tyfromworking

is: U w (C)= pu(t w )+ ( 10p)u(t n )0e 5

Note that eisw ell-de ned. Due to ris k neu tralityl im

t!1 u 0 (t)=u(t)=0a nd, by assumption, lim t!0 u 0 (t) =u(t)= 1. Since u 0

(t) =u(t)iscon tinuous onthe int erval (0;1),th ere exists atl eas tone

 t 2(0;1) such that u(  t)= u 0 (  t)( 

t+c=). Thesolution isuni que , sinc e, fora givent, thederivative

w.r.t. t of the left hands ide is large r than the de rivative w.r.t. t of the rightha nd side . It follows

 e=u 01 (  t). 6

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whi le shirki ngyi eld shi m U s (C)= pu(t s )+( 10p)u(t n ):

Therefore,acontractC=(t;p)i nduc estheagenttoworkifU

w

(C) U

s

(C)whi ch

yi eld sth ei ncenti ve compati bi li tyco n diti on

p(u(t

w )0u(t

s

))e: (2)

Final ly,thecontractmustyie ldth eagentatlea sth isoutsi deopti on,i .e. U

w (C)

u(0)=0 .Ho wever,si ncecontracts mustb ei ncenti vecompatibl eand transfersare

non-n egati ve,i tfol lowsthatU

w

(C)U

s

(C)0. Hen ce,anyi nce ntivecompatible

contractC=(t;p)is a utomati cal lyind ivi dual ly rati onaland wema ydisregardi t.

The op tima l contrac twi ll bethesol utionto 7 max p;t V =y w 0pt w 0(10p)t n 0pc subject to(2):

Propositi on 1 Ifmo nitor ingisc ontracta bleande< e, theo ptimalcontractC 3 = (t 3 ;p 3 ) exh ibit st 3 s =t 3 n =0, p 3 =e=(u(t 3 w ))<1, and u(t 3 w )=u 0 (t 3 w )(t 3 w +c=).

Pro of: Sin ce the t ransfer t

s

do e s not ente r the pri nci pal's objecti ve fun ction

di rectly and a ect s onl y theincentive constrai nt (2),wema ysetthspi a ymentas

l owas p ossib le, i.e. t

s

=0. More over,t

n

e nterson ly negativel y i ntothe objective

func ti on. Hen ce, iti s alsoopti mal tose tt

n

=0. Fi nall y,thei ncenti ve constrai nt

wil l b e b in din g, i .e. p = e=(u(t

w

)). Wema y therefore rewrite the p ri nci pal 's

maximi zati onprobl em as:

max t w V(y w )= y w 0 t w +c= u(t w ) e; (3)

withtherequi re mentthatp=e=(u(t

w

))2[0;1].

The rst order c ond iti on yie lds

u(t 3 w )= u 0 (t 3 w )(t 3 w +c=):

Asexpl ai nedi nfo otnote5au ni quesol uti ont 3

w

existsan dwi llb esuchthatu(t 3

w )=



e=. Sin ce e< e,t he i ndu ced probabi li ty ofmoni torin gp=e=(u(t 3

w

))isst ri ctly

smal l erthanone. Last,n otethatthestati onarypointat t 3 w isin deed amaximum sin ce V 00 (t 3 w ) = 2u 0 (t 3 w )[u(t 3 w )0u 0 (t 3 w )(c=+t 3 w )]+u 00 (t 3 w )(t 3 w +c=) u(t 3 w ) 3 e = u 00 (t 3 w )(t 3 w +c=) u(t 3 w ) 3 e< 0: 7

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Q.E.D.

The intu iti on behi nd theopti mal contract iswel l kn ownand easil ye xpl ai ned .

To i ndu ce th eagenttow ork, theprin cip alo ersh im a reward i fthere i s posi tive

evi denc ethath eworked. Th eopti malsi zeof therewarddependsonthefol lowing

trade -o . On the one hand, a hi gh er reward lowersth e ne ed for moni tori ng and

thereforered uces thecostsofmoni tori ng. Ontheotherhan d itexp osestheagent

tomoreri sk,whi ch i scostly d uetoriskaversi on. 8

The contractual obligati on that th e pri nci pal mon itors with a c ertai n p

roba-bi li tyma y,a t rst si ght,seem probl ematic. Eve n if monitorin g i s veri abl e, how

can a c ou rt ve rify thatt h e prin cip al do es in deed u se the random strategy? The

stan dard justi c ati onist o argue that one maypur i fy themoni tori ng de cisi on by

condi tioni ng i ton some veri abl e,rand oms i gnal.

More sp ec i c ally, assume there exists a random and veri abl e si gnal s with

atoml ess d ensi ty f(s)and cu mul ativedtsi ri bution F(s ). The signal s is r eal i zed

after th eagent has taken his acti on, but before the pri nci pal de cid es whethe rto

monitor. 9

N ow de ne the signal s 3 such that F(s 3 ) = p 3 an d l et the contract

sp e cify that th e pri nci pal i s to moni tor for real i zati ons of s  s 3

and may not

monitorwhen s>s 3

. Thi ssp e ci cati onpuri estheprobabi li stic monitorin g such

thata c ourt can veri fyit i nastandard sense.

4 Non- contractable M onitoring

In thi s sec tion w elea vet he world of c ompl ete contractin g and assu me that, d ue

tonon-veri abilityornon-observability,monitorin g isn olongercontractabl e. We

li ket o stress that this is th e onl y assumption wec hange. More speci cal ly,w e

maintain the assumpti on that the exi stence and nature of monitorin g evi de nce

is still veri abl e. Henc e the pri nci pal may,b ym eansofaco ntract, commi t to

some tran sfer sched ul e t =( t

w ;t

s ;t

n

). Moreover, c ontractualco mmi tmentma y

invol verandomnessi nth esensedi scussede arl ie r. Th econtractmayha vetheform

8

For morede tails se eMo okherjeean dP ng(1989).

9

This re quir es that th eagent's dec isiontakes place b e fore thepr inc ipal's. Alter natively,one c ould

informt he principalabo ut ther ealization of s ,befor eshe takes he rac tion andinform theagentonly

aft erhehastake nhisac tion. Whatisne ede disthatth eagentdo esnotknowthere alizationatthet ime

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(t w (s);t s (s);t n

(s )),whereth esignalsiss peci edasb e fore. Iti s real i zedafte rthe

agentha s taken hi saction andbeforethe prin cip alcho oses tomon itor.

Taki ngaccountof thesignalse xpli ci tly,wemaysummari zetheg ameb e tween

pri nci paland agent asfoll ows:

 t=0: Thepri nci palo erssomecontract C=(t

w (s);t s (s);t n (s ))totheagent.

 t=1: Theagentac cepts orrejectsthe o er.

 t=2: The agent deci desto work or tosh irk. The pri nci pal do es notobserve

the agent's deci sion.

 t=3: Nature reveal sthe uni nformati ve signal s accordin g to th ecu mu lative

di stribu tion F(s ).

 t=4: Thep ri nci pal cho oses whether ornotto moni tor.

By assu mp ti on, an optimal contract i ndu ces the agentt owork. The opti mal

contractmustth ereforein duce th epri nci paltomoni tor,sin ceotherwise theagent

wil l shi rk. Gi ve nsome contractt(s),thereali zations, andthefactthattheagent

works, th epri nci pal hasanin centive tomonitorif and only if

t n (s)t w ( s)+(10)t n ( s )+c,t n (s)t w (s)0c=: (4)

We thereforeco n cl ude thattheopti malcontract must exhi bi tt

n (s)t

w

(s)0c=

forat l eastsomereal izationof s .

I n princ ip le the contract (t

w (s);t

s (s);t

n

(s ))ma y b e extremel y comp lex, as i t

may depend on s in any concei vable way.T he f ol lowin g l emmashows,ho wever,

thatthe opti malc ontrac t hasasi mpl estructure. 10

Lemma1 Wem ay assume th at th eoptima l contract t (s ) has the fo llow ingfor m

(t w (s );t s (s);t n (s))= 8 < : (t 1w ;t 1s ;t 1n ) if ss 3 (t 2w ;t 2s ;t 2n ) if s>s 3 with s 3

some element in the support S and transfers su ch that t

1s = t 2s =0 , t 1n t 1w +c=, andt 2n t 2w +c=.

Pro of: Assume wi thou t l oss of gene ral ityt hat some general contract t 3 (s )= (t 3 w (s);t 3 s (s);t 3 n

(s))i soptimalandi nduc esthepri nci paltocho osethed etermi ni stic

monitorin g strategy m(s): S !f Mon,NotMong in equi li bri um. We wil l show

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thatthere e xistsacontract ^ t=( ^ t w (s); ^ t s (s ); ^ t n

(s))whi chsa ti s esthecond iti onso f

thelemma an d yi el dsthepri nci pal just asmuch,whi l e i tyi el ds theagentw eakly

more.

De ne M as the se t of reali zations of s su ch that the prin ci pal mon itors. I t

foll ows M fs jt

n (s)t

w

(s )+ c=g.Lki ewise,d e ne N ast h esetof reali zations

of s such that the prin cip al do es not mon itor, i.e. N = SnM f s jt

n (s) 

t

w

(s )+c=g.

Conseque ntl y,workin g yiel ds theagent an expected util ityof

Z s2M fu(t w (s))+(1 0)u(t n (s ))0egdF(s )+ Z s2N ft n (s)0egdF(s)

whi le shirki ngyi eld s

Z s2M fu(t s (s))+ ( 10)u(t n (s))gdF(s )+ Z s2N t n (s)dF(s )

Si nce, by assumption, th e opti mal contract ind uces th e agenttow ork, i t must

sati sfy Z s2M [u(t w (s))0u(t s (s))]dF(s )e: Si nce t s

(s)i snot paidin equi li bri um and onl ya ec ts the incentive constraint,we

mayw.o.l.g. assumethat anopti mal c ontract sati s es t

s

(s )=0 foral l s.

Now, l et the fu nction (X)g ive the Le b esgue measure of a set X. Then

i n equi li bri um the p ri nci pal moni tors with probabi li ty (M)=(S) and d o es not

monitor wi th probabi li ty (N)=(S)=1 0 (M)=(S). De ne s 3 such that F(s 3 )= (M)= (S).No wde n efor ss 3 ^ t i (s)= Z s2M t i (s) f(s) R m 2M dF(m) ds;

wherei=w;n.Lki e wise,de nefors>s 3 ^ t i (s)= Z s2N t i (s) f(s) R m 2N dF(m) ds; withi=w;n.

I ti seasytoverifythati fth econtractti nduc estheagenttow ork,thecontract ^

t

wil lal soi nduc etheagenttow ork. Furt he rmore,und erthecontract ^

tthepri nci pal

monitors wi th the same probabi li ty (M)=(S). Final ly, note thatt he contract ^

t

yi eld sthe p rinc ip alt he same p ayo as the o ri gi nal contract t, whi le i t yiel ds the

agentweakl y more. Weco n clu de thatif t iso pti malthen ^

t mustalso beoptimal .

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The previous lemma givesal ready somei ndi cati onab out theuseful nessof the

ran dom si gn al s. It sh ows that the si gn al s en ab le s the prin cip al to use two

di e re nt determi nisti c contracts. On e c ontract that i ndu ces her to moni tor and

oneforwhich shewi ll notmonitor. Si nceeachcontracthastob ecomp ati blewith

the p ri nci pal 's moni tori ng i nte nti on, i t has to obey ce rtain restri ctions an d the

pri nci palisnotcomple tel yfreei ncho osi ngthem. Wewil lshowthatth epri nci pal ,

neverthel ess, gai ns,if sh euses thetwocontracts e e cti vel y.

More over,L emma1redu cesthecompl exi tyofth ep ri nci pal 's proble m

dramat-i cal l y. It reduc esth eprobl em to n di ngtwocontrac ts t

1 and t

2

and a probabi li ty

p whi chsolvesthefol lowin gmaxi mization prob lem

max p;t 1w ;t 1 n ;t 2 n ;t 2 w V =y w 0p(t 1w +(10)t 1n )0(10p)t 2n 0pc s.t. pu(t 1w )e (5) p(u(t 1w )+(10)u(t 1n ))+(10p)u(t 2n )e (6) t 1n t 1w +c= (7) t 2n t 2w +c=: (8)

In equali ty (5)represents the inc entive c ompatib il i ty. I nequal ity(6) e xpressesthe

age nt's ind ivi dual rational ity c on straint an d in equali tie s (7) and (8) are the

re-stri ctions that Le mma 1i mp oses on thecontracts t

1 and t

2

. Th e c ontrolvariabl e

p represents thech oi ceof thecut-o si gn al s 3

,i .e . F(s 3

)=p.

Propositi on 2 Ifm onito ringis non-contra ctablea nd e<e,the op timal contract

t(s)=(t w (s);t s (s );t n (s))is su ch th at t w (s )= 8 < : t w for ss 3 0 for s>s 3 t n (s)= 8 < : t w +c= for ss 3 0 for s>s 3 and t s (s)=0 fo r all s, where s 3 satis es F(s 3 )=e =(u(t w ))a nd t w is such that u(t w )=[t w +c=]u 0 (t w ).

Pro of: Fi rst, n ote th at du e to th e l imi ted l iabi l ity of the agent any in centive

compati bl econtractsati s estheind ivi dualrati onali tyconstrai nt(6). Notefurther

that, for any p 2 [0;1], the prin ci pal's objective functi on i s weakly dec re asi ng i n

t

2n

. A soluti on therefore e xhi bitst

2n

=0. Hence , constrai nt (8) is au tomati cal ly

sati s edandth evalu eoft

2w

isi rre levant. Wemaythu ssi mpl ifythemaximi zation

probl em to max p;t1w;t1 n V =y w 0p(t 1w +(10)t 1n +c) s.t. pu(t 1w )e t t +c=:

(11)

It is cl ear that p shoul d b eset as smal l as p ossib le, such that the i ncenti ve

com-patibi l itycondi tionb in ds, i .e. p=e=(u(t

1w )). Th is yi el ds max t1 w;t1n V =y w 0 t 1w +(10)t 1n +c u(t 1w ) e s.t. t 1n t 1w +c=:

Disregard in gtheconstrainton ewoul dsett

1w

asl argeasp ossi bl e,whi let

1n

assmal l

asp ossib le . I tfol lowsthat c on straint (7)b ind sattheop ti mum: t

1n =t 1w +c=. max t1w y w 0 t 1w +c= u(t 1w ) e (9)

The rstorder cond ition yiel ds

u(t w )=[t w +c=]u 0 (t w ): Q .E.D.

Wewil l d eferthed iscu ssi onof theoptimalcontrac ttothenext section,where

we wi llcontrasti ttotheop ti maldetermini sticcontract. Th ep roposi tion leadsto

two important insi ghts,whi chare expresse di nthefol lowi ngtwocoroll ari es.

Corol l ary 1 If the principal's action is non-co ntracta ble, th e optim al transfer

sch edu le t israndom.

R eferri ng to th e c omp lete contracti ng mo del , th erandomne ss of the transfers

i squi tenatu ral . Iti sactuall ythesamekin dofrandomnessthatthep ri nci pal uses

whensheis abl etoc ommi tcontractuall ytosomemoni tori ngprobabi li ty. Inboth

contracti ngenvi ron me ntsanun certai ntyexistsbefore theagentch o oseshi saction

and is re sol ved whe n the pri nci pal has to cho ose whe the r to mon itor. The on ly

di e re nce i s that with non-c ontrac tabi l ity the uncertainty con cerns the transfers

rathe rthan th emonitoring d eci sion.

Corol l ary 2 If mo nito ring is perfect (= 1), non-contra ctability o f m onito ring

does not a ect th e maximu m payo o f the principa l. If mo nitor ing is imperfect

( < 1), non- co ntractability of m onito ring red uces the m aximum payo of the

principal.

(12)

The i ntui ti onbehi ndCorol l ary2b e come scle arbyreferri ngtoL emma1. Th is

l emma shows th at the pri nci pal may use on e contrac t for th e case she moni tors

and anothe r forthe case she do es notmonitor. The contracts must, however, be

consi stent wi th the p rinc ip al 's mon itori ng i ntenti on. The contract t

1

, whi ch the

pri nci palintend stousewhe nshe mon itors,mustthe re forei nd ucehertomon itor.

It fol l ows thatthe payment t

1n

must b el arger than t

1w

,i .e. p ositi ve. Thi s i snot

costl y,i fmon itori ngisperfect,sin ceinth iscasethepaymentt

1n

isne vertriggered

when, in ac cord an ce wi th her i nte nti ons, the p ri nci pal mon itors. I f monitoring

i s i mp e rfec t, the posi tive t

1n

b ecomes costly, si nce i t is triggered with posi tive

probabi li ty. I tforcesthe prin cip al tole avea rent totheagent.

5 Non- co ntractable M onitoring a nd

Deter-ministic Transfers

In th is se cti on we maintain th e assumption that mon itoring i s non-c ontractabl e

and, in ad di ti on, assume that the p ri nci pal i s unabl e to use rand om transfers.

Thi sassu mp ti on en ab les u sto measure the useful nessof randomiz ati on .

Si nce th e age nt i s to work, moni torin g must o ccu r wi th p osi ti ve probabi li ty.

Ju stl ike i nthep re vioussecti onth is meansthattheopti mal contractmusti nd uce

the pri nci palto moni tor. As a c on sequenc e, th e op ti mal de termi ni sti c contract t

sati s es t n t w 0c=: (10)

The pri nci pal is i ndi e re nt abou t moni tori ng if (10) hol ds with equ al i ty. Si nce

the pri nci pal may in th is case also cho ose to moni tor wi th prob ab il ity one and

transfers are costl y to her, we may wi th ou t loss of gene ral ity assume that the

optimal contract exhi bi ts

t

n =t

w

+c=: (11)

The optimal c ontractwi ll the re forebethesol utiontothefoll owi ngmaxi mi

za-tionp robl em max p;tw;tn V =y w 0pt w 0(10p)t n 0pc

(13)

Propositi on 3 If monitoring is non- co ntracta ble a nd rand om tra nsfers are not

possible, a contract C may only specify the transfers t=(t

w ;t s ;t n ). The optima l contract C 33

exh ibit s u(t 33 w ) = e=, t 33 s = 0, t 33 n = t 33 w

+c=. The m onito ring

probabilit y in equ ilibrium isp 33

=1.

Pro of: Again i t is strai ghtforward tosee th at the soluti onrequi res t

s

=0. S

ub-stitutin g (11) i nto the pri nci pal 's objective fun cti on and acknowl edgi ng th at the

i ncenti ve comp ati bi li tyc on straintmustb ebi ndi ng, we must maximi ze

V(y w )=y w 0t w 0c=:

und er the restricti on that p = e=(u(t

w

)) mu st b esmal ler than on e. This yiel ds

thesol uti onu(t 33

w

)=e=.

Q .E.D.

Theuse fu ln essoftheran domsi gnalsi sbestunde rsto o dbyreferri ngtoL emma

1. Wi th ou trandomizationpossi bi li tie sthe pri nci palcoul donly useone de termi

n-i stn-i c contract, t , and thi s contract has to i nd uce a weak preferenc e to mon itor,

i .e. must sati sfy e quali ty (10). Lemma 1 shows that withthe random si gn al the

pri nci palis abl etouse two di sti nct de termi ni sti cc ontrac ts,t

1 and t

2

. These

con-tracts mu st,howe ver, sati sfycertai n condi tions. Contractt

1

has to i nduc e herto

monitor and must therefore sati sfy the same restric ti on as the contrac t t. It is

therefore of li ttl e hel p. Th e actual bene ci al e e ct of the random si gnal rel ates

to the avai l ab il i ty of the c ontrac t t

2

. Th is c ontrac t sh ou ld in duce no monitoring

and therefore re quires that the tran sfer t

2n

is not to o hi gh i n comparison to t

2w .

But thi s i s in ac cord an ce wi th th e pri nci pal's obje ctives. Si nce th e contract t

2

i nduc esh er nottomoni tor, sheis sure to paythetransfert

2n

. Consequ ently,she

woul d li ke to se ti t as low as p ossib le, i .e. t

2n

=0. In comp ari son totheop ti mal

determi nisti c t, th e pri nci pal may th erefore save on i mp le me ntation c osts, i f she

usesthe rand omsi gnals.

6 Continuo us M onitoring

On e mi ght su ggest that the ran dom transfers are op ti mal due to th e p ri nci pal 's

di scretechoice b etwe en mon itoringornotmoni torin gandthat th is featurewoul d

(14)

often l ead toequ il i briawhi ch on ly exi st in mixe dstrategi es and th at the se mi xed

strategyequ il ib ri aofte nd isappearwhen strategyspacesbecomeconti nuous. Th is

secti on showsthat such aconc lusi on wou ld b einc orrect.

To intro duc e a c onti nuous mon itori ng dec isi on of the p rinc ipal , sup p ose that

thep ri nci pal hastocho oseamoni torin ge ort,which determin esthee ec ti veness

ofmoni tori ng. Wemo del th is i deaby assumin gthat thepri nci palcho osesa

mon-i tormon-in gpreci sion p2[0;1]ata cost c(p). Thevariabl e prepresentstheprobabi li ty

that the agent'strue acti on i s reveal ed and impl i esthat n othi ng i s re veal ed with

probabi li ty (10p). Thi sformu lation general ize stheforme r mo del ,as theformer

obtai nswhe n weassu me cto h ave thequ asil in ear form

c(p)= 8 < : pc= if p 1 if p>:

In th is sec ti on we wil l stud y a d i erent versi on of th is gen eral mo del and assume

that c i s conve x, twi ce di erentiabl e,where th e rst deri vative exhi bi ts c 0 (0) =0 and li m p! 1 c 0

(p) = 1. We wi l l show that i f the cost functi on c satis e s these

condi tions, a de termi ni sti c contract t c an not be opti mal whe n mon itori ng i s not

contractabl e.

With non-c ontrac tabi l ityadetermini sticcontractdetermin es atransfersch

ed-ul et, wh ichi nd ucesthep ri nci paltocho oseamonitoringprobabi li typ2[0;1]that

min imi zes

p(t

h 0t

n

)+c(p):

Byassumption,theoptimalcontracti nd ucestheagenttowork. Thi simp li esthat

theop ti malcontract ind ucesa posi tive monitorin g probabi li ty. Therefore , were a

determi nisti ccontract ^

top ti mal ,i twoul d,du etoth eassumpti onli m

p! 1 c

0

(p)=1 ,

i nduc e astri ctp osi ti ve probabi li typ^<1,wh ichsatis es the rst order condi tion

c 0 (^p)= ^ t n 0 ^ t w : (12)

Thi simpl i es thatfora determi nisti ccontracti t nece ssari lyh ol dsthat ^

t

n >0.

I nste ad ofu sin g thedetermin istic contract ^

t, the princ ip al may usethe si gnal

s to ind uce a mi xture between a (de termi ni sti c) contract t

1

, wh ich i nd uces the

pri nci pal to monitor with p robabi l ity p

1

> p,^ an d a (determini stic ) c ontrac t t

2 ,

whi ch in duce sthe p ri nc ipal nottomoni tor, i .e . p

2

=0. By ch o osi ng th emixture

such that a compou nded moni tori ng prob ab il i ty of p^ resul ts, the princ ip al may

reduc e theexpected tran sfe r,asforthecontractt

2

sh e mayse tt

2n

(15)

pri nci paltoi ndu ceth emoni tori ng prob ab il ityp^bysomemi xtureth antoi nd ucei t

di rectly by on edetermini sticcontrac t. Incontrastto th ep re vious mo delrand om

transfers involve an i mple mentation cost to th e p ri nci pal . Thi s e ec t do es not

o ccu r,wh en,l ikeintheprevi ou ssections,thecost fun cti oni sl ine ar. I twi l lred uce

the e ectiven ess of usi ng ran dom contracts. However, si nc e it i s onl y a second

ord er e ect, whi le thepossi bi l ityof setti ng t

2n

=0 o ersa rst ord er gai n, some

mixi ngwi ll alwaysb eopti mal .

Propositi on 4 Theoptim al contract conditionsnon-trivially onth e signal sand

is thereforerandom.

Pro of: S uppose n ot, then some de termi ni sti c contract t = (t

w ;0;t

n

) i s op ti

-mal. Such a contract must i ndu ce the age nt to work and, due to the assumption

l im

p!1 c

0

(p)= 1, i nduc es the p ri nc ipal to mon itor wi th a prob ab il i typ 3 2(0;1) such that c 0 (p 3 )=t n 0t w

. Nowd e ne forsome 1 th eparameter ssuch that

F(s)= . Nowc on si de rthe fol l owi ng rand omc ontract

t w (s)= 8 < : t 3 w if s<s 0 if ss ;t n (s)= 8 < : t 3 w +c 0 (p 3 = ) if s<s 0 if ss

Note that the de te rmi ni sti c contract t 3

obtain s for =1. Furthermore, for any

< 1 the contract t(s) i ndu ces the pri nci pal to mon itor with probabil i ty p 3

=

when s < s whi ch o cc urs with probabil i ty . When s sthe prin cip al wi l l not

monitor. Fromtheage nt'sperspecti vemonitoringoc cursthereforewi thprobabi li ty

p 3

= =p 3

. Consequ ently,theran domcontractt (s ) in ducesth eagenttowork,if

this al soh ol ds forthe d etermi ni stic c ontrac tt 3

. Th e imp leme ntation costsof the

contractt (s) i s p 3 t 3 w +( 0p 3 )(t 3 w +c 0 (p 3 = ))+c(p 3 = ): (13)

Thed erivati ve ofexpressi on (13) is

(t 3 w +c 0 (p 3 = ))0( 0p 3 )c 00 (p 3 = )p 3 = 2 0c 0 (p 3 = )p 3 = 2 : Eval uati onat =1yi el ds t 3 w +c 0 (p 3 )0(10p 3 )c 00 (p 3 )p 3 0c 0 (p 3 )p 3 =t 3 w +(10p 3 )(c 0 (p 3 )0c 00 (p 3 )p 3 )>0:

Therefore the i mpl ementation cost at = 1 are stri ctly in creasing in . Th is

i mp li es that l owerin g i nc re ases th e p ri nci pal 's util ity. It thus fol lows th at a

determi nisti ccontractt 3

(16)

Al thou ghrandomtransfersareoptimal,th eywi l lb eun ab letomi ti gatethe

non-contractabi li typerfectl yfortworeasons. Fi rst,th econvexi tyofcmake sth emi xing

b e havi or of th eprin cip al costl y. Sec on d,the opti malmoni tori ngprobab il i tywith

ful l-c ontractabi l ity issmall er th an on e. Thisi mpl ies that the ex p ostveri abi li ty

of moni torin g i s not perfect. As in dic ated by Coroll ary 2 random transfers are

unabl eto miti gatethe non-contractabi li tyi nthi scase.

7 Conclusion

Thi s p aper stud ied the e ect of a n on -contractabil i ty of the contract desi gn er's

actions on op ti mal contracts. It showed that non-c ontrac tabi l ity may lead to a

ran domness,wh ichi snotopti malu nderful lcontractab il i ty. Foll owi ngrec entwork

onn on -contractab il i ty,we have chosentoshowthi sinasi mple age ncymo d elwith

costl y moni tori ng. It shoul d,however, b ecl earth at,asin di catedin thei ntro d

uc-tion,theresul tal soh ol dsi nothercontextsofnon-contractabi li ty. Inanysituation

i n whi ch a p ri nc ipal takes a non-contractabl e acti on expl i ci trandomness maybe

optimal. Thi swil lbethec aseasso onasthen on -contractableactiona ectsthe

un-derl yin g di stri buti on of the contractin g vari abl es. The non-contrac tabi li ty causes

a moral h azardprobl em on partof th eprin cip al . Asshown bythi spap er, in cl

ud-i ng exp li ci t randomness may al l evi ate this moral h az ard probl em. Exampl es are

promoti on alacti vi ti esbyth eprin ci pal,he lpi nge ort,orotherformsofpro d uctive

actions.

As shown the degree of miti gation dependson the ex post veri abil i ty of the

non-c ontrac tabl eaction. Agaugeofexpostveri abi l itymayb efoundi nthedegree

by which the n on -contractable acti on a ects the d istrib ution of the c ontractabl e

vari ab les. If th e action causes a comp le te shi ft in the u nderl yin g di stri buti on ,

suchthatveri abl eoutcome scanb eperfectlyattrib utedtosomenon-c ontractabl e

action, expl ici t ran domn ess miti gates the non-c ontractabi l ity p erfectl y and the

assumptionof non-c ontractabi l ity hasnoe ect.

References

[1] B order,K.andJ .Sob e l(1987),"S amuraiaccountant: atheoryofaud itin gand

p lun der",Reviewo fEconomic Stu dies54, 525-540.

(17)

prob-[3] J ost, P. (1991), "Mon itoring i n prin ci pal-agent re lationshi ps", Jou rna l of

In-stitut io nal a nd TheoreticalEconomics147,517-538

[4] J ost, P.(1996), "On the rol e of commi tme nt in a pri nci pal -agent re lationshi p

wi than i nformed prin ci pal ",Jo urnal of Economic Theo ry 68,510- 530

[5] K hali l , F. (1997), "Audi ting wi th ou t c ommi tment", RAND Jo urnal o f

Eco-nomics 28,629-640

[6] Mo okherjee, D. an d I . Png (1989) "Optimal Aud iti ng, I nsurance, and Redi

s-trib ution",Quarter ly Jo urnal of Econo mics104,399-415

[7] S trausz,R .(1997)"Del egati onofmoni torin gi naprin cip alage ntrel ati onshi p",

Reviewo f Economic Stud ies64, 337-357.

[8] Townsend ,R.(1979),"Op ti malcontractsandcomp e ti tive marketswi thcostly

References

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