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 eecti 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
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 dierentr 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 eortinstraigh 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 ngeortwl til ypicallyaecttheprobabilitythatthe 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 eort 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 riable information on part of t he
contr ac tdes igner . Non-con tractabilityofthiski ndwillleadtheage nttoin ter pretthecontrac toe ras
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 riable.
3
Khalil st udie s an agenc y pr oblem with advers e se le ction r at her than moral haz ard. The
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 eorte,w hilefor shi rki ngth ese costsar e z ero. The agent'su tilityissepar abl e
i nweal thand eortandma 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.
Astheowneroftherm,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
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 edierenc 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 ene 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 oer 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
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 aect 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
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 justic 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 ic 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
(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 paloerssomecontract C=(t
w (s);t s (s);t n (s ))totheagent.
t=1: Theagentac cepts orrejectsthe oer.
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
thatthere e xistsacontract ^ t=( ^ t w (s); ^ t s (s ); ^ t n
(s))whi chsa ti sesthecond iti onso f
thelemma an d yi el dsthepri nci pal just asmuch,whi l e i tyi el ds theagentw eakly
more.
Dene 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 ene 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 yaec ts the incentive constraint,we
mayw.o.l.g. assumethat anopti mal c ontract sati ses 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). Dene s 3 such that F(s 3 )= (M)= (S).No wden 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 ayoas 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 .
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 ee 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 satises 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 sestheind 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 sedandth 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=:
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 aect 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.
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 ses 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
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 ee 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
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 geort,which determin estheeec 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 ierent 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 satise 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 therst 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
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 eectiven ess of usi ng ran dom contracts. However, si nc e it i s onl y a second
ord er eect, whi le thepossi bi l ityof setti ng t
2n
=0 oersa 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 ene 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
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 eect 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 -contractableactionaectsthe
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 ngeort,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. Agaugeofexpostveriabi l itymayb efoundi nthedegree
by which the n on -contractable acti on aects 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 ,
suchthatveriabl 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 hasnoeect.
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