Optimal Contracts in Principal-Agent Model with Hidden Information and Risk-Averse Manager, Study notes of Microeconomics

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Microeconomic

Theory

AndreuMas-Colell MichaelD. Whinston

and

Jeny R. Green

r

NewYork

Oxford

OXFORD UNIVERSITY PRESS

C H A P T E R

Adverse Selection,

Signaling,

T

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

, ^. f. 0

f o r e > 0

o " e l e , a ) \ : g

f o r e : 0.

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

10. A seeminglymore importantsourceof hiddeninformation

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

(14.C.2)

and

(14.C.3),

we seethat

u'(wfr

• g(efi,

0,))

: u'(wt

• g@T.,

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

( 1 4. C. 6 )

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

• g ( e f i ,? r ) ) = u ( w T. - g ( e T , $ ) ) ; t h a t i s ,

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.

(14.C.7)

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

• g(e,O,))= 71

\Managrr

Yetter

Off

{ ( w , e ) : r ( e ) - w = I I f }

w!

, I

• l l - \

u

' \ u )

Figure14.C.

The optimal

wage-effort pair for

state 0' lvhen statesare

observable.

Profitsof

'.

0

uwnerln

J

State 0,

I

(nl)

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

• l l - \

u

' \ u )

I n';{

n ; ]

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"

• g(e

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

(14.c.8)

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 -

• a. To seethis,notethat the

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

• (t - zi)ufll,whereu,: u(wr- gk,,O.S1fori: Z, 11.ff,L expected

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

• _ * , L À u f r + ( L _ ). ) u f l t r c

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

S. ;;:

1 s@r,g'))

if he insteadclaims

rhat it is state0r. Thus,he

will tell

the truth if

S ' :

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 '

Ï-

"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).

""'W.

aÂalyre

problem(14.C.8)

througha sequence

o[lemmas.Our arguments

for

I

th.r. results

makeextensive

useof graphical

analysisto build intuition.An analysis

oirnt,

problemusingKuhn-Tucker

conditionsis presented

in Appendix

B.

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 ) :

• 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

• g ( e , 0

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 ,

• g ( ê t ,0r ) )

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:

H]'

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

16. In modelswith more than two types,this proPertytakesthe form that only the incentive

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)

i*",

efi)

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 )

@i,ê')

êL ei

$,n, eï)

(wa,

etr)

(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

(rl,

ei)

(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

21. The model studiedin Section13.D with zi(f) = 0, correspondsto the limiting casewhere

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

::i"il;:T::::

'" (r4'c'r0)

where

wetake

r,= xi,

lvi= -i,'

û=0,

s(t,,01)=

-.u(xi,0;),

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,

.

(p,,T)'where

p,is

theanowed

retairprice

and{ isa transfe,

o"rJ:::;ïliJ;:ïff:tï

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

22. The regulator's objective function can be generarized

to

ionsumerand producersurplus,with greaterweighton .onrurn.rr.

",rilldependon 0,.

allow a weighted

averageof

In this case,

the functionz,(.)