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The optimal contracts in a principal-agent model with hidden information and a risk-averse manager. The manager's effort level and utility depend on a state of nature that is only observed by the manager. The document derives the conditions for optimal contracts, including the manager's marginal utility of income being equalized across states and the optimal level of effort in state 0 satisfying the condition of equating the marginal benefit of effort with its marginal disutility cost.
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Typology: Study notes
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C H A P T E R
Adverse Selection,
Signaling,
and Screening
Introduction
One of the impiicit assumptions
ol the fundamentai welfaretheorems
is that the
characteristics of all commodities
areobservable to all marketparticipants.
Without
thiscondition,
distinctmarkets cannot existfor goods havingdiffering characteristics,
and so the complete marketsassumption cannothold.In reality,however,
this kind
of informationis often
asymmetrically
held by market participants. Considerthe
foliowingthreeexamples:
(i) When a firm hiresa worker,the firm may know lessthan the worker
does
about the worker's innateabilitv.
(ii) Whenan automobile insuranc..orpunyinsures
an individuai, theindividual
mayknow morethanthecompany
aboutherinherent drivingskill
and hence
about her probabilityof havingan accident.
(iii) In the
used-car market,theseiler of a car may havemuchbetterinformation
abouther car'squalitythan
a prospective buyerdoes.
A number
of questionsimmediately arise about thesesettings
of asynmeîic
information: How do we characterize rnu.k., equilibria in thepresence of asymmetric
information?
What are the properties
of theseequilibria? Are therepossibilities
for
welfare-improving
market intervention?
In this chapter, we study thesequestions,
whichhavebeenamongthe
most activeareas of researchin microeconomic
theory
duringthe last
twentyyears.
We begin, in Section
13.8,by introducing asymmetric information inro a simple
competitive marketmodel.We seethat in the presence
of asymmetric information,
marketequilibriaoften
fail to be Paretooptimal.The tendency
for inefficiency in
thesesettings can be strikinglyexacerbated
by the phenomenon known as ad.uerse
selection. Adverse selection ariseswhenan informedindividual's
trading decisions
dependon her privately held informationin a manner that adverselyaffects
uninformed marketparticipants.
in the used-car market,for example,
an individual
is morelikelyto decide
to sellhercar whensheknowsthat it is not verygood.When
adverse selection is present, uninformed traderswill be wary of any informedtrader
who wishes
to tradewith them,and
their willingness to pay for the productoffered
436
.?{F
C H A P T E R 1 4 i T H E
p R t N C t p A L - A c E N T
p R O B L E M
then the managergets exactly
the same expected utility unde r w(rc)as under w(n,y)
for any levelof efforthe
chooses.Thus,the manager's effortchoice will be unchanged,
and he
will still accept the contract. However, becausethe managerfacesless risk,the expected
wage
payments are lower and the owneris betteroff (this
againfollowsfrom Jensen's
inequality:
for all z, u(Elw(n, y)lnl) > Elo(w(n, y))ln), and so w(z)
< Efw(n,y)
lzl).
This point can be pushed further. Note that we can always write
f (n,
yle)
:
fr(nle)fr(yln,
e).
rl fz}lz,e) doesnot depend
on e, then the
f2(.) termsin condition(14.B.11) againcancel
out and theoptimalcompensation
package
doesnot depend on y. Thiscondition
oi 1rçy1n, e
is equivalent to the statistical
concept that z is a suficient statisticfor
y with respect to e. The
converse is also true:As long as n is nota sufticient
statisticfor y, then wages
should be made
to depend on y, at leastto somedegree.
SeeHoimstrom(1979) for furtherdetails.
A numberof extensions
of this basic analysishavebeenstudied
in the literature.
For
example, Hoimsrrom(r9gz),
Nalebuffand Stiglitz(19g3),
and Green and Stokey
examine casesin whichmanymanagers
are beinghired and consider
the use
of relativeperformance
evaluation in suchsettings;
Bernheim and Whinston(19g6),
on the other hand,extendthe modelin the other direction,
examiningsettings
in
which a single
agent is hired simultaneously
by severalprincipals;Dye (19g6)
considers cases in whicheffortmay
be observed throughcostlymonitoring;
Rogerson
(1985a),
Allen (1985), and Fudenberg,
Holmstrom,and Milgrom (199b)examine
situations in which the agency
relationship is repeated
over many periods, with a
particularfocus
on the extentto which long-termcontracts
are more effective
at
resolving agency problems than is a sequence
of short-term contracts of the type we
analyzed
in this section. (Thisiist of extensions
is hardlyexhaustive.)
Many oithese
analyses focus on the casein which effort is single-dimensional;
Holmstrom and
Milgrom (1991) discusssome interesting
aspectsof the more realisticcase
of
multidimensional
effort.
Holmstrom and Miigrom (1987)have pursuedanotherinteresting
extension.
Bothered by the simplicity of real-world compensation schemes relativeto theoptimal
contracts derivedin models like theonewehavestudied
here,theyinvestigate
a iodel
in which profits
accrueincrementaily
over time and the manager
is able to adjust
his effort
during the courseof the projectin response
to eariy profit realizations.
They identifyconditions
underwhichthe ownercan
restricthimselfwithout lossto
the useof compensation
schemes that arelinearfunctionsof the
project'stotal profit.
The optimality of linear compensation
schemes arisesbecause of the need to offer
incentives
that are
"robust"
in the sense
that they continueto provide
incentiv.es
regardless of how earlyprofit reaiizations
turn out. Their analysis illustrates a more
generalidea,
namely,that complicating
the nature of the incentiveproblem
can
actuallylead to simplerformsfor optimalcontracts.
For illustrations
of this point,
seeExercises
14.8.5and 14.8.6.
The exercises
at the end of the chapter exploresomeof these
extensions.
l4.c HiddenInformation
(andMonopolistic
Screening)
In this section,we shift our focus to
a setting in which the postcontractual
informational
asymmetry takesthe form of hiddeninformation.
S E C T I O N H l o o E N
T N F O R M A T I O N ( A N O
M O N O P O L I S T I C S C R E E N I N G )
Once
again,an owner wishesto hire a managerto run a one-timeproject.
Now,
however,
the manager's
effortlevel,denotedby e, is lully observable.
What is
not
observable
afterthecontractis signed
is therandomrealizationoi the manager's
disutility
from effort.For example,
the managermaycometo find himself
wellsuited
to thetasks
requiredat thefrrm,in whichcase
highefforthasa relatively
low disutility
associated
with it, or the oppositemay be true.However,
only the manager
comes
to knowwhichcase
obtains.ro
Beforeproceeding,
we note that the techniques
we develophere
can also be
applied
to modelsof monopolistic
screeningwhere,in a settingcharacterized
by
p'rrrortrnrtual
informational asymmetries,
a singleuninformedindividual offers
a
m.nu of contractsin order to distinguish,
or screen,
informedagentswho have
differing
informationat the time ol contracting
(see Section13.Dfor an analysis
of
;H,#iïilî1,îili#,i?,1.T11,ru'.,:ï::,'n:"':1'-':l::,i'.1:il!,',:,:.'.:"
:1':
:j
To formulateour hidden
informationprincipal-agent
model,we suppose
that
effort
can be measured
by a one-dimensional
variablee e [0, .o). Gross profits
(excluding
any wagepayments
to the manager)are a simpledeterministic
function
of effort,
z(e),with z(0):0,n'(e) > 0, and
n"(e)< 0 for all e'
The manager
is an expected
utility maximizer
whoseBernoulliutility function
overwages
and effort,u(w,e,g),
dependson a stateof nature'0that is realized
after
the
contractis signedand that only the manager
observes.
We assumethat 0 e R,
and
we focuson a special
form of u(w,e,0)that is widely
usedin the literature:Ir
u(w,e,0) = u(w-
g(e, 0))'
The
functiong(e,g) measures
the disutilityof effortin monetaryunits.We assume
that
g(0,0) : 0 for all
g and,lettingsubscripts
denote
partialderivatives,that
9 " " ( e , 0 )
g e @ , 0 ) < 0
f o r e > 0
f o r e : 0
for all e
for all e
g " ( ' , ù { '?
L : U
Thus,the manageris averse
to increasesin effort,and this aversionis larger
the greater
the current ievel of effort. In addition, higher valuesof 0 are more productive
states
in the sensethat both the manager's total disutility irom effort, g(e,0), and his
marginal disutiiity from effort at any current effort level,g"(e,0), are lower when 0
between
managers
and owners
is that
the managerof a firm oftencomesto know more
aboutthe potentialprofitability
of various
actionsthan does the owner. In Section 14.D,
we discussone hybrid hidden action-hidden
information
modelthat captures
thisalternativesort of informational
asymmetry;
its formalanalysis
reduces
to that oi the modelstudiedhere.
ll. Exercise 14.C.
asksyou to consider
an alternativeform for the manager's
utility function.
S E C T I O N 1 4. C : H I O D E N I N F O R M A T I O I I ( A N D M O N O P O L I S T T C S C R E Ê N t N c ) 4 9 1
Theseconditionsindicatehow the two objectives of insuringthe manager
and
makingeffort
sensitiveto the stateare handled. First, rearrangingand combining
conditions
and
we seethat
u'(wfr
0,))
: u'(wt
0")), ( 1 4. C. 6 )
so the manager's
marginalutility of incomeis equalizedacrossstates.This is the
usualconditionfor a risk-neutral
party optimallyinsuringa risk-averse individual.
C o n d i t i o n
i m p l i e st h a t w f - g ( e f r ,? o ) : w T - g @ T , 0 1 ) ,w h i c h i n t u r n
i m p l i e s
t h a t u ( w f
t h e m a n a g e r ' s
u t i l i t y i s
equalized
acrossstates.Given the reservationutility constraintin (14.C.1),the
manager
thereforehasutility levelu in eachstate.
Now considerthe optimaleffort levelsin the two states.Since
g"(0,0): 0 and
ft'(O)
0, conditions(la.C.a)and (14.C.5)must hold with equalityand ef > 0 for
iF*,, i :1,2.
Combining condition (14.C.2)with (14.C.4),and condition (14.C.3)with
(14.C.5),we seethat the optimal level of effort in state9i, ef, satisfies
n'(ef)
g"(ef ,0)
lor i : L, H.
This conditionsaysthat the optimal
levelof effortin state0, equatesthe marginal
beneûtof effort in termsof increased
profit with its marginaldisutilitycost.
The pair (rI, eI) is illustratedin Figure 14.C.1(note that the wageis depicted
on thevertical
axisand theeffortlevelon thehorizontalaxis).As shown,themanager
is betteroff as we move to the northwest(higherwagesand lesseffort),and the
owneris betteroff as
we movetoward the southeast.Becausethe managerreceives
utility levelù in state9,, the owner seeks
to find the most profitablepoint on the
manager'sstate9, indifferencecurvewith utility levelri. This is a point ol tangency
between
the manager'sindifferencecurveand oneof the owner'sisoprofitcurves.At
this point, the marginalbenefitto additionaleffort in termsof increased
profit is
exactlyequaito the marginalcostborneby the manager.
The owner's
profitlevelin state0, is flf
n(ei)
u-'(u)
g(ef
,0'). As shown
in Figure 14.C.1,this profit is exactlyequalto the distancefrom the origin to the
point at which the owner'sisoprofitcurvethroughpoint (rI, eI) hits the vertical
{(w,
e):u(w
\Managrr
Yetter
Off
{ ( w , e ) : r ( e ) - w = I I f }
w!
, I
u
' \ u )
Figure14.C.
The optimal
wage-effort pair for
state 0' lvhen statesare
observable.
Profitsof
'.
0
uwnerln
J
State 0,
I
P R O B L E M
4 9 2 c H A p r E R 1 4 : T H E
p R t N c t p A L - A c E N T
u ( w - g ( e , 0 ) ) = ù
u ( w - g l e , 0 r ) ) = t
z ( e ) - w = l l l
z ( e ) - w = f l f
u
' \ u )
t
axis
[sincen(0)
0, if the wagepayment
at this point on the verticalaxis
is ]a < 0,
the owner's profit at (wf, ef) is exactiy-f,].
From condition (I4.C.7),we see that g"r(e,0)< 0, rc,,(e)
< 0, and g,"(e,0)> 0
impiy rhatefi
ef. Figure 14.C. depicrsthe optimalconrract,
l@fr,eîr),(wi,ei)].
These observations are summanzed in proposition 14.c.1.
P r o p o s i t i o n 1 4. C ' 1 : I n t h e p r i n c i p a l - a g e n t m o d e lw i t ha n o b s e r v a b l e
s t a t ev a r i a b l e 0 ,
theoptimal contractinvolves an effortlevele,ïin stateg,
suchthatæ,(ef
g
"(eî
, 0i)
and fully insuresthe manager,
settinghis wage in each
state
g,
at the ièvet razî
s u c ht h a t v ( w I -
O @ 1 , 0 , 1 1
s.
Thus,with a strictlyrisk-averse
manager, the first-best contractis characterized
by two basicfeatures: first,the ownerfully insures the manager againstrisk; second,
he requires the manager to work to the point at whichthe
marginalbenefitof effort
exactly equalsits marginaicost.Because
the marginalcostof effortis lower in state
0uthan in stategr, the contractcallsfor more
effortin statego.
The State 0 is ObseruedOnly by the Manager
As in Section14.8,the desireboth to insurethe risk-averse
managerand to eiicit
the proper levelsof effort comeinto conflictwhen informational
asymmetriesare
present. Suppose, for exampie, that the owner offersa risk-averse
managerthe
contractdepictedin Figure 14.C.
and relieson the managerto revealthe state
voluntarily.
If so,the ownerwill run into problems.
As is evident in the figure, in state
0o, the managerpreferspoint (wf, ef) to point (wfi,e[). consequently,
in state0,
he will /ie to the owner,claimingthat it is actuallystate0r. As is alsoevidentin the
figure,this misrepresentation
lowersthe owner's profit.
Giventhisproblem, whatis theoptimalcontract for theownerto offer? To answer
this question, it is necessary to start by identifying the set of possible contracts that
the owner can offer.One can imaginemany different
forms that a contractcould
conceivably take.For example, the ownermight offera compensation
functionw(z)
that pays the manageras a functionof realizedprofit and that leavesthe effort
wl.
wi
Figure 14.C.
The optimal
contn.
with full obsqvebll,
of 0.
4 9 4 c H A p r E R 1 4 : T H E
p R t N c t p A L - A c E N T
p R o B L E M
by the manager and yieldsexactlythe sameoutcome
as the initial contract. In fact,
a similar argumentcan be constructed
lor any initial contract(seeChapt
er 23),
and so the owner can restrict
his attention without loss to truthful revelation
mechanisms.
I o
To simplify
the characterization
of the optimal contract,we restrict
attention
from this point on to a specific
and extreme caseof managerial risk aversion:
infnite
risk aversion. In particular,
we take the expected utility of the managerto equalthe
manager's lowestutility levelacross
the two states. Thus,for the manager
to accept
the owner'scontract,it must be that the manager
receives a utility of at leastu in
eachstate' As above,efficient
risk sharingrequires
that an infinitely
risk-averse
managerhavea utility levelequalto u in eachstate.If, for example,
his utility is ,
in onestate and u' > û in the other,thenthe owner's expectedwagepayment
is larger
than necessary for givingthe manager
an expected utility of ù.
Giventhis assumption
aboutmanagerial risk preferences, the revelationprincipie
allowsus to write the owner,s
problemas foliows:
Max )"ln(er)
wul + (1 - À)ln(e")_ wt)
w H , e H
O , w L , e L > Q
reseruation utility
(or indiuidualr
arionality)
constraint
(iii)w"
n,
0n) 2 w,
s(e r, 0,)
'ër"i::i:r-i"rirp
atibitit
v
(iv) w. - g(eu 0r) > w, - g(er, g))
or self-s.election)
constramts.
The pairs (wo,eu) and (w",er) that the contractspecifies
are now the wage
and
effortleveis that resultfrom differe
nl announcemefis or the stateby the manager;that
is, the outcomeif the manager
announces that the stateis g, is (*,,e,). constraints
(i) and (ii) make
up the reseruation
utility (or indiuidual rationality)constrainr
for the
infinitelyrisk-averse
manager; if he is to accept the contract,he must be guaranteed
a utility of at least
n in eachstate.Hence,we must have
u(w,_ g(ei,g,))> u for
i : L , r l o r , e q u i v a l e n t l! , w i - g k i , 0 ) > n - ' ( u ) f o r i : z , Ë 1 .c o n s t r a i n t s ( i i i ) a n d
(iv) are the
incentiue compatibility(or truth-telling
or self-selection) constrajnrs for the
manager in states 0u andgr, respectively. consider,for example, constraint(iii).
The
s. t. ( i ) , r
g @ 1 , 0 ) > , - t ( t ) ]
(ii) w, - g(eu,g")
u-rfu))
14' one restrictionthat we haveimposedherefor expositional
purposes is to lirnit the outcomes
specified following the manager's
announcement to being nonstochastic (in fact, much of the
literature doesso as well). Randomization can sometimes be desirable in thesesettings because it
can aid in satisfying
the incentive compatibility constraints that we introducein problem (14.C.g).
SeeMaskin and Riley (198aa)
for an example.
15' This can be thought of as the limiting casein which,
starting from the concaveutilrty
function u(x), we take the concavetransformationur(u): -u(x)o
îor p <0 as the manager,s
Bernoulliutility functionand let p -
manager'sexpected utility over the
random outcomegiving (wn - g@a,0n))with probabilityÀ and,
(w,I
S@r,d.)) with probability
(1 - l') is then EU = -Uufr
u t i l i t y i s c o r r e c t l y o r d e r e d
b y ( n g y r r o : l h u f i + ( 1 - S , 1 u g 1 t r o. N o w a s
i
Min
{un,
ur} (seeExercise
3.C.6).Hence, â contractgivesthe manageran expected
utility greater
t h a n h i s ( c e r t a i n )
r e s e r v a t i o n u t i l i t y i f a n d o n l y i i M i n
{ u ( w ,
_
S @ r , 0 ) ) , u ( w r _
g ( e t , 0 r ) t } > ù.
s E C T t o N
1 4. C : H T O O E N T N F O R M A T T O N
( A N D M O N O P O L t s T I c S c R E E N I N G ) 4 9 5
f"j, *anager's
utility in state
gr
is u(wr
g(es,oH) if he tells
the truth, but it is
1 s@r,g'))
if he insteadclaims
rhat it is state0r. Thus,he
will tell
the truth if
o k , , 0 r ) 2 w r -
g ( e u g " ). C o n s t r a i n t
( i v ) f o l l o w ss i m i l a r l y '
"n
NJù inui ,tt. first-best
(full observability)contract
depictedin Figure14.C.
does
i not sarisfy
the
constraints
of problem(14.C.8)
because
it violatesconstraint
(iii).
aÂalyre
problem(14.C.8)
througha sequence
o[lemmas.Our arguments
for
th.r. results
makeextensive
useof graphical
analysisto build intuition.An analysis
oirnt,
problemusingKuhn-Tucker
conditionsis presented
in Appendix
L e m i l a
1 4. C. 1 :W e c a n i g n o r e c o n s t r a i n t
( i i ). T h a t i s , a. c o n t r a c t i s a s o l u t i o n t o
u " " ' o r o 6 ' " r
( 1 4. C. 8 )
i f a n d o n l y i f i t i s t h e s o l u t i o n
t o t h e p r o b l e md e r i v e df r o m ( 1 4. C ' 8 )
d r o p P i n g
c o n s t r a i n t
( i i ).
F.Frtr*
''' -' '
Proof,
Whenever both
constraints
w 1 1
g ( e n , 0 r )> w t -
g ( e u 0 r ) 2 w t
alsosatisfied.
This impliesthat the set
from (14.C.8)
by droppingconstraint
contracts
in Problem
(1a.C.8)'r
fl' ula (iii)
"are
tAtn.jd'it
mustbe that
g(eu0r) >- u-
1(u),
and so constraint
(ii) is
ol feasible
contractsin the problem
derived
(ii) is exactlythe sameas the set of feasible
i'i'
Lemma14.C.
is illustrated
in Figure 14.C.3.
By constraint
(i), (wr,e.) must lie
in the shaded
region
oi the figure.But by constraint
(iii), (w", e") must lie on or
;i
abovethe state 0r, indifference
curve through point (wr,
e1)' As can be seen,
,r, thisimplies that the manager's
state
g" utility
is at leastù, the
utility
he
getsat point
(*, e)
(u-
i(t),
i'
Therefore,
from this point on we can
ignoreconstraint
(ii).
t l. r r "
1 4. c. 2 :A n o p t i m a t
c o n t r a c t
i n p r o b l e m
( 1 4. c. 8 )m u s t
h a v ew 2 -
g @ L , 0 1 ) :
v t u t.
Proof:Suppose
not, that is,
that thereis an optimalsolutionl(*",er),(w",et)]
in
whichw. -
g(et,0r)> r-
1(ù).
Now, consider
an alteration
to the
owner'S
contract
u ( w
ù )
= û
u ( w
g ( e , 0 n ) )
= u n > û
u
' ( u )
Figure 14.C.
Constraint
(ii) in
problem(1a.C.8)
is
satisfied
by any
contractsatisfying
constraints
(i) and (iii).
u ( w - s ( e , 0 ) ) = a
( " r , e r )
u(w- 9(e,0r.))= ri
u ( w
g ( e , 0 " ) )
= u ( r ô ,
r ( e ) - w = f 1 , ,
(wi,ei)
( f r t , ê t )
u
' ( u )
é L Y L
and,asis
evidentin Figure14.C.5,thetruth-telling
constraints
arestillsatisfied.
Thus,
a contractwith ê1> el cannotbe optimal.
Now consider
part (ii). Given any wage-effort
pair (frr,ê") with êr (
ef, such
asthat shownin Figurel4.C.6,the
owner'sproblemis to find
thelocation
for (wr' e")
in the shadedregionthat maximizes
his profit in state
gH.The solutionoccursat a
point of tangencybetweenthe manager's
state9s indifference
curvethroughpoint
(fr,, êr) and an isoprofitcurvelor the
owner.This tangency
occursat point
(frs, eT,)
in the figure,and necessarily
involveseffort levelef because
all points of tangency
between
the manager'sstate
g, indifference
curvesand the owner'sisoprofit
curves
occur at effort level ef [they are characterized
by condition(14'C'7)for
j:
Note that this point of tangency
occursstrictly
to the right of effort
ievelé" because
ê r < e T < e f r..
A secondary
point emerging
from the proof of Lemma 14'C.
is that only the
truth-tellingconstraintfor state
g" is bindingin the
optimalcontract.
This property
is commonto many
of the other
applications
in the literature.l
Lemma 14.C.
In any optimal
contract,eL<eti
that is, the effort level in state
g. is necessarily strictlybelow the level
that would arise in state
01 if 0 were
observable.
Proof: Again,this point can
be seengraphically.
Suppose
we start with (w7,et):
(r'T.,ef),asin Figure 14.C.7.By Lemma14.C.3,
thisdetermines
thestateott outcome,
denotedby (vi", ef) in the
figure.Note that
by thedefinition
o[(wf, ef),
the isoprofit
curvethroughthis point is tangent
to the manager's
state0" indifference
curve.
Recallthat the absolute
distance
between
the origin and the point whereeach
state's
isoprofitcurvehits the verticalaxis
represents
the profit the ownerearnsin
that state.The owner'soverallexpected
profit with this contractoffer
is therefore
constraintsbetween
adjacenttypesbind,and theydo so
only in one
direction'
(SeeExercisel4'C'l')
u ( w
s ( e , 0 ) ) :a
u ( w - g ( e , 0 n ) ) = u n > ù
Isoprofit
Curves
Figure14.C.
(left)
An optimal contract
has e, :
g;.
Figure14.C.
(right)
The best contract with
e - : P t
(mo' ei)
6
State
6
S t a t e
l g , I
oH
i
prontl
Profit
I
4 9 8 c H A p r E R T H E P R I N C I P A L. A G E N T P R O B L E M
u ( w - g ( e ,? n D = u n > u
Isoprofit
Curves
} J
B '
u ( w - g ( e , 0 r ) ) = r i
State 0,,
indiffercnce
Curves
u -
' ( t )
ProfitLoss
in State 0r{-
u ( w - g ( e , 0 ) ) = u
1 w i.
e i )
êL ei
$,n, eï)
(il,",ef
)
u - ' ( t )
êL ei
( b )
Figure14.c.8 (a) The changein profits in state
0, from lowering e, siightly below ef. (b) The changein profitsin llllt
from lowering e, slightly below ef and optimally adjusting w".
equal to the average of thesetwo profit levels(with weightsequai to the relative
probabilities of the two states).
We now arguethat a change in the state0, outcomethat lowersthis state'seffort
levelto one slightlybeiow ef necessarilyraisesthe owner's expected
profit. To see
this,startby movingthe state0, outcometo a slightlylowerpoint,(frt,êr), on the
manager's state9r indifferencecurve.Thischange is illustratedin Figure14.C.8,along
with the owner'sisoprofit curve through this new point. As is evidentin Figure
14.C.8(a), this changelowersthe profit that the ownerearnsin state0r. However, it
also relaxes the incentiveconstraint on the state
gn
outcomeand, by doing so, it
allowsthe ownerto offera lowerwagein that state.Figure14.C.8(b) showsthe new
state0r, outcome,say (rô", efi), and the new (higher-proût)isoprofit curve through
this point.
Overall,this change resuitsin a lowerprofit for the ownerin stateAranda higher
profit for the owner in state0". Note, however,that because we startedat a point
of tangencyat (wf, ef), the profit lossin state0. is smallrelative to the gain in state
0". Indeed,if we wereto look at the derivative of the owner's
profit in state0r with
respect to an infnitesimal changein that state'soutcome,we would find that it is
zero.In contrast,the derivativeof profit in state0, with respect to this infinitesimal
changewould be strictly positive. The zero derivativein state 9, is an envelope
theoremresult:because we startedout at the first-best levelof effort in state0., a
smallchangein (wr, e") that keeps themanager'sstate0r utility at ù hasno first-order
effecton the owner'sprofit in that state;
but because
it relaxes the state0" incentive
constraint,for a small-enough changethe owner'sexpectedprofit is increased.r
How far shouldthe ownergo in lowering e'!In answeringthisquestion,the owner
must weighthe marginallossin profit in state0, against
the marginalgain in state
(fruê)
C H A P T E R 1 4 : T H E P R t N C t P A L - A G E N T P R O B L E M
n'n(e)> n'r(e)> 0 for all e ) 0, the analysisof this modelfollowsexactlyalongthe
linesof the analysiswe havejust conducted(seeExercise 14.C.5).
As in the caseof hiddenaction models,a number of extensions of this basic
hiddeninformationmodei have beenexpioredin the literature.Someof the most
generaltreatmentsappearin the context of the
"mechanism
design" literature
associated with socialchoicetheory.A discussion of thesemodelscan be found in
Chapter23.
The Monopolistic Screening Model
In Section13.D,we studieda modelof competitiue
screening
in which firms try to designtheir
employmentcontractsin a manner that distinguishesamong workers who, at the time of
contracting, havedifferentunobservableproductivitylevels(i.e.,thereis precontractual asym-
metricinformation).The techniques that we havedevelopedin our studyof the principal-agent
model with hidden informationenableus to formuiateand solve a model of monopolistic
sueeningin which, in contrastwith the anaiysisin Section13.D, only a singlefirm offers
employmentcontracts(actually, this might more properlybe calleda monopsonisric screening
model because the singiefrrm is on the demandsideof the market).
To seethis,supposethat,as in the modelin Section 13.D,thereare two possibletypesof
workerswho differin theirproductivity. A workerof type0 hasutility u(w,tl0):,
SQ,0)
when he receives a wage of w and lacestask levei r. His reservation utility level is u. The
productivitiesof the two typesof workersare 0, and 0r, with 0" > 0L> 0. The fractionof
workersof type 0, is Àe(0,1).We assume that the firm's profits,which are not publicly
observable,are given by the function nr(t) îor a type 0, worker and by zr(r) for a type 0,
worker,and that n'n(t)> nL)> 0 for all r > 0 [e.g.,as in Exercise13.D.1,we could have
n,(t) : 0,(l -
pt) for p > 0f."
The firm's problem is to offer a set of contractsthat maximizes its profits given worker
self-selectionamong,and behaviorwithin, its offeredcontracts.Once again, the revelation
principlecan be invokedto greatlysimplifythe firm's problem.Here the firm can restrictits
attentionto offeringa menuof wage-taskpairs[(wo, tu),(wu rr)] to solve
Max ),lnr(tr) - wxf + (1 - À)lnr(t") - w")
w r , t t à
0 , w z , t s l O
(i4.c.10)
s. t. ( i ) , ,
g ( t r ,? r ) > u
. ( i i ) w n -
Q ( t n , 0 u )
,
( i i i ) w " - g ( t
a , 0r )
- ,
g ( t
L , 0H )
( i v ) w , - g ( t u 0 r ). 2 w o
g ( t o , 0 " ).
This problem has exactly the same structure as (14.C.8)but with the principal's (here the
firm's) profit being a function of the state.As noted above,the analysisof this problem follows
exactly the same lines as our analysisof problem (14.C.8).
This class of models has seen wide application in the literature (although often with a
continuum of types assumed).Maskin and Riley (1984b),for example,apply this modelto the
study of monopolistic price discrimination.In their model, a consumer of type 0 has utility
u(x,0) - I when he consumesx units of a monopolist'sgood and makes a total payment of
?"to the monopolist, and cân earn a reservationutility level of u(0,0) : 0 by not purchasing
from the monopolist. The monopolist has a constant unit cost of production equal to c > 0
r - 0.
o n o H I o D E N I N F o R M A T I o T { :
H Y E R I D
M o D E L S
S 0 1
and seeks to offera menuoi (x,,
d) pairsto maximize its profit.The monopolist,s
problem
Baronand Myerson's
(1982)
analysis of optimalregulation
of a monopolist
with unknown
costsprovidesanotherexample.
There,a regulated
firm faces market demandfunctionx(p)
and hasunobservable
unit costsof 0. The
regulator, who seeks
to design a regulatory
poricy that maximizes
consumer
surplus, faces the monopolist
with a ch,
.
the firm' The regulated
firm is able
to shut down if it cannotearnprofits
of at leastzerofrom
any of the regulator's
offerings'
The regulator's
problemth.n .o...spondsto (r4.c.10)with
ti :
Pi,
wi =
T, û :
0, g(:ti, 0,) = -(p,_
0,)r(p,),and
r,(r,) = g
"1r; ar.r,
,noj.lrttttttt
l4'c'7 to l4'c'9 ask
fou to'stuay some.-"*piË!
"r monoporistic
screening
) HiddenActionsand HiddenInformation:
Hybrid Models
tt
Althoughthe hiddenaction-hidden
information
dichotomization
serves as a usefur
startingpoint for understanding
principal-agent
models,
-uny real-world
situations
(and someof the literatureas weil)
invorve erements of both probiems.
To consider
an example
of sucha moder,
suppose that we augment
the simple
hidden information
model considered
in Section14.c in the followingmanner:
iet the levelof effort
e now be unobservable,
and tet profits
be a stochastic
function
of effort,
described by conditionaldensityfunction
f(nle).In essence, what we
now have is a hidden action model,
but one in which the owner arsodoes
not
know something
about the disutilityof the manager
(whichis captured
in the state
variable
0).
Formal
anarysis of this moderis beyondthe scope
of this chaprer,
but the basic
thrust of the reveration
principreextends
to the anarysis
or ,h.r. typesof hybrid
problems'In particurar:as Myerson (i9g2) shows,ih.
o*rr., can now restrict
attentionto contracts
of the followingform:
(i) After
thestateg
is reaiized, themanager
announces
whichstate
hasoccurred.
(ii) The contractspecifies,
for eachpossibre
announcement
d e €l, the
effortlever
e(Ô ttrat
the manager
shourdtake
and a compensation
scheme w@l$.
(iii) In everystateg,
the manager
is willing to be both truthfurin
stage(i) and
obedient
followingstage
(ii)
[i.e.,he
findsit optimalto choose
effortievele(g)
i n s t a t e
0 1.
This contractcan
be thoughtof as
a revelation
game, but one in whichthe
outcome
of the manager's
announcement
about the staù is a hidden
action-style
contract,
that is, a compensation
scheme and a
"recommended
action."The requirement
of
"obedience"
amountsto an incentive
constraint
that is like that
in the hiddenaction
to
ionsumerand producersurplus,with greaterweighton .onrurn.rr.
",rilldependon 0,.
allow a weighted
averageof
In this case,
the functionz,(.)