sele tion GuillaumeCarlier DamienGaumont May 25,2000 Abstra t 1
We study in entive- ompatible labor ontra ts in the ase where both
individualprodu tivityandsubje tivedis ountrateÆareunobservableby
the rm. We rst show that unidimensionalmanifolds of agents group on
the same ontra t. High , low Æ agents may hoose the same ontra t as
low,highÆ agents. Weshowexisten eand uniquenessofanoptimalwage
fun tion whi h is ontinuous and unbounded. This optimal wage fun tion
an bedeterminedbytheironing pro edure.
Keywords: AdverseSele tion,In entive-CompatibleContra ts,
Ironing Pro edure, Heterogeneous Time Preferen e.
J.E.L. Classi ation: C61, C62,D82, J31,
CEREMADE,CNRSUMR7534,UniversiteParisIXDauphine,Pla edeLattredeTassigny,
75016Paris,e-mail: arlier eremade.dauphine.fr
ERMES,CNRS,UPRES-A7017,UniversiteParisII(Pantheon-Assas),92rued'Assas,75270
ParisCedex06,e-mail:gaumontens.u-paris2.fr
1
We thank David Bell, Ivar Ekeland, Thierry Granger and Nizar Touzi for helpful
Inthe labormarket, adverse sele tion a ords therelationbetween
produ -tivity and wages, Yellen (1984). Adverse sele tion is indeed a reason for
rmsto set eÆ ien y wages. In thispart of the eÆ ien y wages literature,
workersonlydi erintheirindividualability,Aziariadis(1983), Mal omson
(1981),Stiglitz(1984),Weiss(1980). Inthatlineofresear h,labor ontra ts
allow rms to sele t most skilledagents by spe ifying wages that are non
de reasing with respe t to workers' produ tivity. Neverhtless, this theory
failstoexplaintwowell-knownempiri alobservations: xedwagesandwage
di erentials.
Fixed-wage theory points out that workers having the same wage do
not ne essarily have the same produ tivity, Aziariadis (1981), Grossman
(1981), Polemar hakis and Weiss (1978), Killingsworth (1988, p. 61-65)
and manyempiri alstudieshave testedthetheoryof xed-wage at the
mi- roe onomi level, Abowdand Ashenfelter (1981), Ham (1982). Moreover,
wage-di erentials literature emphasizes that wage di eren es still remain
partiallyunexplained. Empiri altestssolelybasedonworkers'produ tivity
fail to explain all the existing varian e of wages. Starting from the
semi-nalwork ofAbowdand Ashenfelter(1981), Krueggerand Summers(1988),
manyempiri alstudieshave shownthatheterogeneityofprodu tivityisfar
from apturing heterogeneityof wages amongagents.
In this paper, we try to explain why equally produ tive agents do not
ne essarily a ept identi al labor ontra ts. We argue here that a relevant
adverse sele tion parameter may be time preferen e. Indeed, sin e equally
produ tive individualdo not samely valuateleisure time, they do not
ne -essarilya ept thesame labor ontra t. Arguingthatheterogeneityoftime
preferen e aptures a trade-o between leisure and in ome, it may be a
reasonablewayto explainwage-di erentials.
We therefore onsidera model where bothprodu tivity and time
pref-eren e are unobservable to the prin ipal so that two-dimensional adverse
sele tion takes pla e. Our aim is to give a possible explanation for both
xed-wages and wage di erentials in a prin ipal agent setting. We
estab-lish that there may be perfe t substitutionbetween produ tivityand time
preferen e: in entive- ompatibility impliesthat very heterogeneous agents
re eive the same ontra t. More pre isely, very skilledbut little provident
workerson theonehandandless produ tive butvery providentworkers on
theotherhandmaya tuallygrouponthesame ontra t. Inotherwords,the
two parameters may ompensate ea h other. This substitution may
there-foreexplainbothwage di erentialsand xed-wages. Finally,thepossibility
ofbun hingo ersanotherpossibleexplanationfor xed-wages e e ts.
We rst show that the above mentioned substitution allows to redu e
spe i resultofourmodelisthatthereisno niteupperboundonoptimal
wages for agents whose both hara teristi s tend to the upper bounds on
produ tivity and subje tive dis ount rate. We believe that this somehow
surprisingresultisquite robustwhen theredu tion tri kwe useapplies.
Ourpaperisorganizedasfollows. Se tion2presentsthemodel. Se tion
3 hara terizesin entive- ompatiblelabor ontra tsinourtwo-dimensional
adversesele tionsetting. Se tion4solvestheemployer'sprogramand
har-a terizes the optimal labor ontra t. Finally, in Se tion 5, we dis uss our
resultsand some possibleextensions.
2 The model
We onsider a two-period model. In period 1, the rm produ es a single
good, denoted A. Produ tion plans are ompatible with the exogenous
demandforAand arephysi allypossible. The rmis ableto sellits entire
amount of output. Interest rate r lears the exogenous nan ial market.
Consumers work in period 1 and are retired in period 2. At ea h period
t, agents onsume a single good denoted C
i
;i = 1;2. We assume that
C
i
6=A;i=1;2and that pri esof A;C
1 ;C
2
areexogenous and equalto 1.
2.1 The agent
Ea h agent is denoted by his individual hara teristi s, i.e. a pair of
pa-rameters(;Æ) that he perfe tlyknows,where representshis produ tivty
andÆ hissubje tivedis ountrate. Duringtheworkingperiod,a (;Æ)-type
agent produ es l units of observable output at the real wage w, onsumes
C
1
and saves w C
1
. During the retirement period, ea h agent onsumes
C
2
equalto thedis ountedreturnof his rst periodsaving.
A (;Æ)-type agent is hara terized bya separableintertemporal utility
fun tion: U 0 (C 1 ;C 2 ;Æ;t)=v(C 1 )+Æv(C 2 )+t (1)
wherev isa on ave in reasingfun tion, trepresentsleisuretimeinperiod
1. If T denotes the total amount of time available in period 1, agent's
produ tion depends on his produ tivity, so that we have: l() = (T t)
whi hyields: t=T l=thatwerepla ein(1)toobtain: ~ U(C 1 ;C 2 ;;Æ;l)= v(C 1 )+Æv(C 2
)+T l=. Multiplyingthepreviousexpressionbyweobtain
thenew representation ofpreferen es:
U(C 1 ;C 2 ;l;;Æ)=v(C 1 )+Æv(C 2 ) l (2)
The agent maximizes his intertemporal utility (2), taking as xed by the
The rmsetsalabor ontra tinordertohirea ontinuumofheterogeneous
agentswhose(;Æ)areunobservable. Intheremainderofthepaper,the rm
knows thedistributionoftypeswhi his hara terized byadensityfun tion
f(:;:)thatisassumedtobeof lassC 1
,andstri tlypositiveonthere tangle
:= ; Æ;Æ (0;+1) 2 :
A labor ontra t is a dire t me hanism whi h both de nes the wage
fun tionw andtheindividualoutput fun tionl,withrespe ttotheagent's
type. Using Myerson's Revelation Prin iple we may fo us on
in entive- ompatible ontra ts.
Theproblemoftheprin ipal onsiststhenin ndingin entive- ompatible
ontra tsthat maximizeits pro t.
3 In entive- ompatible labor ontra ts
From nowonwe assumetheutilityfun tionto be exponential
v( )= e
(3)
sothattheAgent's programis:
(P ) V(;Æ;w;l) = 8 > > < > > : sup C 1 ;C 2 e C 1 Æe C 2 l s.t. 0C 1 w 0C 2 (1+r)(w C 1 )
Lemma 1 If the further assumptions are satis ed:
(H 1 ): Æ(1+r)<1, (H 2 ): v 0 2R satis es: 0> v0 +Æ> Æ(1+r),
then for all (;Æ)2, for all (w;l):
V(;Æ;w;l)v 0 )V(;Æ;w;l) = 2+r 1+r h [Æ(1+r)℄ 1 2+r e 1+r 2+r w i l (4)
Proof isgiven inAppendix1.
This resulthasthe followingimpli ations:
1. If Æ(1 +r) < 1 then all agents have a higher subje tive interest
rate than the nan ial market. Consequently, (H
1
) implies that all agents
onsumeinthe rstperiod.
2. (H
1
)and(H
2
)ensurethatallparti ipatingagentshaveahomogeneous
onsumptionbehavior. Thisex ludeszero-levelof onsumptionoverone of
thetwo periods. Fromourpointofview,ex ludingthoseextremebehaviors
onagentsthat do onsumeoverthetwo periods. A tuallytheothergroups
ofagents shouldbestudiedapart.
We intendtoshownowthatin entive ompatible ontra tsdependonly
on the redu ed s alar parameter = Æ 1
2+r
. This parameter is therefore
interpreted an exhaustive statisti whi h sums up all relevant information
fortheprin ipal. Su hanunidimensionalexhaustivestatisti aptures
sub-stitutionbetweenprodu tivityand dis ount rate.
Notethatunderassumptions(H
1
) and(H
2
)theindire tutilityfun tion
ofthe agents(iflarger thanthereservation levelv
0 ) isgiven by: V(;Æ;w;l) =W( ;w;l)= g(w) l (5) where =Æ 1 2+r and g(w)= ( 2+r 1+r )(1+r) 1 2+r exp( 1+r 2+r w): (6)
Letus now de nein entive- ompatibleand admissible ontra ts
De nition 1 1) A ontra t is a pair of fun tions (w;l):
:=[;℄[Æ;Æ℄!R 2
+
(;Æ)7!(w(;Æ);l(;Æ))
w is alled the wage partof the ontra t and l its produ tion or physi al
part.
2)A ontra t(w;l)isin entive- ompatibleifandonlyifforall(;Æ);( 0 ;Æ 0 )2 2 : V(;Æ;w(;Æ);l(;Æ))V(;Æ;w( 0 ;Æ 0 );l( 0 ;Æ 0 )): (7)
3)A ontra t (w;l)isadmissibleifandonlyifitisin entive- ompatible and
satis es the parti ipation onstraint:
V(;Æ;w(;Æ);l(;Æ))v
0
, for all (;Æ)2: (8)
Remark 1 Lemma 1 implies that if (w;l) is admissible,then
V(;Æ;w(;Æ);l(;Æ))=Æ 1
2+r
g(w(;Æ)) l(;Æ)
thatonlydependon . Letusde ne : [ ; ℄:=[:Æ 1 2+r ;:Æ 1 2+r ℄ (9)
and,forall 2[ ; ℄:
I :=f(;Æ)2:Æ 1 2+r = g (10)
thenadmissible ontra ts are hara terized by thefollowingresult:
Proposition 1 1)Let(w;l)besome ontra t andde netheasso iated
util-ity fun tion f W(;Æ):=Æ 1 2+r g(w(;Æ)) l(;Æ), for all (;Æ)2 (11)
then (w;l) is admissibleif and only if f W is a onvex fun tionof f W(;Æ)=W(Æ 1 2+r ) withW onvex (12)
andthe following onditions aresatis ed :
1. 8 2[ ; ℄;8(;Æ)2I ;g(w(;Æ))2W( ) 2. W( )v 0 :
2) Asa onsequen e, if(w;l) isan admissible ontra t then there exist two
non de reasing fun tions we and e
l from [ ; ℄ to R
+
su h that for almost
every (;Æ) 2: (w;l)(;Æ)=(w;e e l)(Æ 1 2+r ) (13) Proof:
1) Let(w;l) beadmissible,let be in( ; )and (;Æ) and( 0
;Æ 0
) bein I
in entive ompatibility onditionbetween(;Æ) and ( 0 ;Æ 0 ) yields: f W(;Æ)= f W( 0 ;Æ 0 ) f W istherefore onstant inI : f W(;Æ)=W(Æ 1 2+r ).
By in entive- ompatibility,wealso get:
W( )=sup
z2
g(w(z)) l(z)
W is then onvex sin e it is a supremum of aÆne fun tions. Writing
in entive- ompatibility using the de nition of W we get for all , 0 and z2I : W( 0 ) W( )( 0 )g(w(z))
onstraint of . Conversely assume thatthe ontra t (w;l) satis es 1. and
2. with W de ned aspreviously and onvex. By ondition1., we get that
(w;l) isin entive- ompatibleand also thatW isnon in reasingsin eg<0
sothat ondition2. ensuresthattheparti ipation onstraintis satis edfor
every type.
2) Let(w;l) be anadmissible ontra t and W bede nedasin1). Sin e W
is onvex, it isdi erentiablealmost everywhere:
W( )=fW 0 ( )g a.e. De nethen: B :=f(;Æ)2: W isnotdi erentiable at Æ 1 2+r g
one an easily he kthatB is Lebesguenegligible.
Letwe be su h thatg(w( ))e 2W( ) forevery and e
l by:
e
l( ):= g(w( ))e W( )
e
w is learly non de reasing (sin e g is in reasing and W is monotone)
and (13) issatis ed a.e. sin eB is negligible. It remainsto prove that e
l is
nonde reasing. Let and 0 bein[ ; ℄,sin eg(w( e 0 ))2W( 0 )wehave: e l( 0 ) e l( ) (g(w( e 0 )) g(w( ))):e
We an on ludethenthat e
l isnon de reasingsin ebothg and we are.
Remark 2 The ontra t(;Æ)7!(w;e e
l)(Æ 1
2+r
)isitselfadmissibleandyields
the same pro t to the prin ipal as (w;l). Thereis therefore no loss of
eÆ- ien yto fo us on ontra ts of .
4 Optimal labor ontra t and solution to the
prin- ipal's problem
4.1 The prin ipal's problem
UsingthepreviousSe tion, we areable to writetheprogram of the
prin i-pal in a fairly standard way. More pre isely we show that the initial
two-parameters-problem is equivalent to a unidimensional parameter adverse
(P p ) 8 > > < > > : sup (w;l)= Z [l(;Æ) w(;Æ)℄f(;Æ)ddÆ s.t.: (w;l) isan admissible ontra t
isequivalentto the following one:
( e P p ) 8 > > > > > > > < > > > > > > > : sup e (w)e := Z [g(w( ))( h( )e +H( )) w( )h( )℄d e s.t.: e
w isnon de reasing nonnegative (P1)
e l( )= g(w( ))e v 0 + Z g(w(s))dse 0 (P2)
wherehdenotesthedensityfun tionandH the umulativedistribution
fun -tion of (see Appendix2 for omputations andproperties of h and H.)
SeeproofinAppendix3.
Note thatthe expressionof e
(w)~ enables to point outthetwo opposite
e e tsof thewage onthepro t:
- a linearnegative e e t on ost: w( )h( )e
- a on ave positive e e t on indu ing higher types to produ e higher
quantitiesof output:
g(w( ))( h( )e +H( )):
4.2 Chara terization of the optimal wage fun tion and
iron-ing pro edure
Letus rst showan existen e and uniquenessresult.
Theorem 1 Theprin ipal'sprogram( e
P
p
)admitsauniquesolutiondenoted
w
.
Proof:
Firstnotethat theset of we satisfying (P1)and (P2) isnonempty and that
thevalue of ( e P p ), sup ( e P p ) is nite. Letw n
be some maximizingsequen e
of( e P p ) i.e. w n
satis es(P1) and(P2) forall n2N and:
lim n!+1 e (w n )=sup ( e P p ) (14)
Firstnotethat forall 2( ; ), the sequen ew
n
( )is boundedfor
other-wise,sin ew n isnon de reasing, e (w n
n
Helly's sele tion Theorem (see for instan e Natanson) implies that, up to
asubsequen e,w
n
onverges pointwiseto some nonde reasingnonnegative
fun tionw
. Obviously,w
satis es(P2)byLebesgue'sDominated
Conver-gen eTheorem. Finally,Fatou'sLemma yields:
e (w )limsup n e (w n )=sup ( e P p ) so that w is a solution of ( e P p
). Finally uniqueness follows from stri t
onvexityproperties of( e
P
p ).
To derive rst order onditionsand hara terize more pre iselythe
op-timal wage fun tion, we rst need a Kuhn-Tu ker type result that enables
usto integrate onstraint(P2) withinthe riterion:
Lemma 2 If w solves program ( e P p
) then there exists 0 su h that w
isthe solution of:
( e P p; ) 8 < : sup e (w)e := e (w)e + Z g(w( ))d e + g(w( ))e
s.t.: we is non de reasing nonnegative (P1)
is a Kuhn-Tu ker multiplier asso iated with onstraint (P2) of program
( e
P
p ) .
Sin etheLemmafollowsfromstandardseparationarguments,theproof
isleft to thereader.
The following proposition summarizes rst order ne essary onditions
and thusenablesto hara terize the optimalwage fun tionw : Proposition 3 1) w is the solution of ( e P p
) if and only if it satis es the
following onditions: (i) ( )0 (ii) ( ( ))dw =0 (iii) ( g 0 (w ( ))+( ))w ( )=0
with being de ned as in the previous Lemma, dw
denoting the variation
measure asso iated with the non de reasing fun tion w
, and being given
by: 8 2[ ; ℄:( ):= Z [g 0 (w (s))(sh(s)+H(s)+) h(s)℄ds:
2)Moreover, onditions (i), (ii)and(iii) implythatw
is ontinuouson
Letw
0
bethesolutionoftheproblemobtainedfrom( e
P
p;
)whenomitting
themonotoni ity onstraint; learlyw
0 an be omputed analyti ally: w 0 ( )= ( (g 0 ) 1 ( h( ) h( )+H( )+ ) if h( ) h( )+H( )+ (1+r) 1 2+r 0otherwise (15) Sin e g is on ave, w 0
is non de reasingon every sub-intervalon whi h
7! +
H( )+
h( )
is. The fa t that h is de reasing near and an easy
omputation showthen that w
0
is in reasing at the topof the distribution
(seeAppendix2.) Moreover, w
0
tendsto+1as tendsto sin eh( )=
0 + and (g 0 ) 1 (0 + )=+1.
Proposition 3 a tually enables to obtain w
from w
0
via the so- alled
ironing pro edure (see Mussa and Rosen). There exist indeed 2 kinds of
sub-intervalsof types .
1) Those on whi h there is stri t dis rimination dw
> 0, using the
ontinuity of the solution and ondition (ii), we obtain w = w 0 on su h intervals. 2) Those on whi h dw = 0 so that w
is onstant. Su h intervals are
alledbun hes. Onabun h,alltheagentsareo eredthesame ontra tforit
wouldbetoo ostlytodis riminateamongthem. Conditions(i),(ii)andthe
ontinuityofw
enableto hara terize theextremitiesofbun hes. Suppose
thatan intervalof types[
; ℄isa bun h withw ( )=w ( )=w.
Ontheone hand, bya ontinuityargument,we get:
w ( )=w ( )=w 0 ( )=w 0 ( )=w:
Ontheother hand,we have (
)=(
)=( ); and thelatteryields
Z [g 0 (w)(sh(s)+H(s)+) h(s)℄ds=0:
Those onditionsingeneral enableto fully hara terizebun hes.
Finally, the next proposition states that there is stri t dis rimination
amongagentswith typeatthetopofthedistribution: nobun hingo urs
for highest types. This is a rather standard and intuitive result sin e one
may naturally expe t the prin ipal to extra t more rent from high type
agents.
Proposition 4 Thereexists
0 2( ; )su h that w =w 0 on [ 0 ; ).
Let 1 2( ; )besu hthatw 0 isin reasingon[ 1 ; ). Firstitisimpossible to have w < w 0
on some interval of the form [ ; ) with >
1 , for
otherwise w
wouldbesuboptimalon [ ; ).
It isalso impossibleto have w >w 0 on aninterval [ ; )with > 1 ,for
thatwouldimply,forall s :
g 0 (w (s))(sh(s)+H(s)+) h(s)<0 and then: ( ) (s)<0
whi h implies, with (ii) that w
is onstant over [ ; ) ontradi ting w > w 0 . Sin ew 0 andw
are ontinuousthereexiststhen
0 2[ 1 ; )su hthat w 0 ( 0 ) = w ( 0
): It follows then from the optimality of w that w = w 0 on[ 0 ; )sin ew 0 isin reasing on[ 0 ; ).
Aswehave already remarked thatw
0
isunbounded we have:
Corollary 1 The optimalwage fun tionw
is unbounded:
w
( )!+1 as ! .
At rstsight,unboundednessoftheoptimalwagemaybethoughtasnon
intuitive. Thefa tthattheoptimalwagew
isunbounded anbeinterpreted
as follows: ideal -type agents are improbable for two reasons sin e both
highprodu tivityandhighsubje tivedis ountratehavetobe ombinedfor
thesame individual,sothat h( )=0. Evenif the rm hasa prior density
f(;Æ)>0ontheideal agent,itperfe tlyknowsthatinthe nitepopulation
itaimstohire,thereisno(;Æ)agentbutatbest( "
;Æ " Æ )with" and" Æ
positiveand unknown. The previouspropositiononlysays thatthe rm at
theoptimumisleadtoproposearbitrarilyhighwage- ontra tswhen("
;"
Æ )
be omesarbitrarilysmall. Indeed,when = onlythepositivee e tofan
in reaseoftheprodu tionsubsistssin eh( )=0. Itisthereforethedouble
s ar ity of the ideal -type that explains why it is optimalfor the rm to
set su h an unbounded wage fun tion. One an easily show that optimal
wagesremainboundedforinstan eifthe rmas ribesapositiveprobability
(Dira mass)to theideal (;Æ)-typeagent.
Fromanempiri alpointofview,thelowerboundonwageswhi hindu e
(high ,high Æ)-type individualsto a ept a labor ontra t spe ifyingvery
littleleisuretime,ifnotin nite, an bevery highinthelabormarket( hief
exe utive, traders,managers,footballplayers,topmodels...)
5 Dis ussion
Letusnowshowthatourbasi modelandourresultsmaybegivendi erent
sket h possibledevelopmentsof ouranalysis to othere onomi areas.
5.1 An alternative formulation
We aim to show that a very similar model may be interpreted in terms
of edu ation a quisition. Let us assume now that period 1 is devoted to
training whi h will be produ tive in period 2. In period 1, an individual
hooses how mu hto train,therestof thetimebeingdevotedtoleisure. In
period 2he hooses how mu hto work and onsumesthewholeofhis wage
in omew. In this asepreferen esare givenbythefollowingintertemporal
utilityfun tion: U 1 (C 2 ;Æ;t)=Æv(C 2 )+t (16)
where v is a on ave in reasing fun tion and t represents period 1 leisure
time. If T denotes the total amount of time available in period 1, agent's
produ tioninperiod2 depends onhis learningprodu tivityand training
time (T t), l() = (T t), whi h yields t=T l=. Repla ing the
previous in (16), using C
2
= w and multiplying by we obtain the new
representationofpreferen es:
U(w;;Æ;l)=Æv(w) l:
Using this representation, it is easy to show that our redu tion tri k an
be used again, de ning = Æ as a new exhaustive statisti .
In entive- ompatible ontra ts are therefore -dependent. This means that
substi-tution of parameters still o urs in that training model: agents with low
produ tivitymaygetsame wagesasmore produ tive agentsifthey are
suf- iently provident to spend enough training time in period 1. It is worth
notingthat in this ase there is no needfor a CARA spe i ationof v for
substitutionto o ur. One an alsoshow, assumingforinstan ev 0,that
the prin ipal's program whi h determinesthe optimal wage fun tion, is of
theform: 8 > > > > > > > < > > > > > > > : sup e (w)e := Z [v(w( ))( h( )e +H( )) w( )h( )℄d e s.t.: e
wis non de reasingnonnegative (P1)
e l( )= g(w( ))e v 0 + Z g(w(s))dse 0 (P2) (17)
Itis learthenthatthemainresultsofSe tion4stillhold: existen e,
unique-ness, ontinuity, ironing type hara terization. Moreover, if v satis es the
Inada type ondition v 0
(+1) = 0 then the statement of Proposition 4
Finally that model may be interpreted as an alternative approa h to
sig-nallingmodelsa la Spen e (1974). The main di eren ebetween those two
approa hes isthat, in our model,the level of training does not reveal
pro-du tivitysin eit isalso explainedbyheterogeneity of timepreferen e.
5.2 Wider approa hes
The design of optimal ontra ts with two-dimensional adverse sele tion is
not a widely settled area. As suggested in se tion 5.1, the problem we
adressed on erningthedesignoflabor ontra tsmaybegivenanalternative
interpretationwithoutanyparti ularutilityspe i ation.
We believethatthetri kusedinthepaper anbeappliedinseveral
rel-evant e onomi areas. E onomi ally,this redu tionof a multi-dimensional
s reening probleminto a single dimensional one is possible if there is
sub-stitution betweenthe di erent parameters, we give an example wheresu h
asubstitution o ursin thelabormarket. This kindofte hniquewasused
by Lewis and Sappington (1988) in the framework of regulation of a
mo-nopolist. Our resultshoweveremphasize thefa t that when thisredu tion
tri k anbeused,one mayexpe tunboundedsolutionssothatProposition
4 is robust. Moreover, introdu ing heterogeneity of time preferen e as an
adverse sele tionvariable an leadto further on lusions inawider lassof
Prin ipal-Agentproblems.
One anthink rstofbanktheoryandinparti ularoflending ontra ts.
Two majorproblems of redit banksare defaultpayments and anti ipated
reimbursement. Traditional expli ative individual variables for both
prob-lemsgenerally fo usonriskontheonehandandin omeontheotherhand.
Wethinkthereispotentials opeforalternativevariablesthat apturemore
orlessprovidentbehaviors. Asimilarapproa h ouldalsobefruitfullyused
inLifeInsuran e ontra ting. Inthat ase, timepreferen e heterogeneityis
interpretedasindividualwillingnessto bequestto relatives.
Finallyourapproa hmayalsobe onsistentintheframeworkoftaxation
theory. Inthatframeworktaxesare olle tedto nan efutureexpenditures.
A benevolent planner should then set a tax s heme knowing that the
dis- ountedvalueof future publi realizationsdi er amongagents.
Appendix 1: Proof of Lemma1.
Program of(;Æ)-agentsis:
(P ) 8 < : V(;Æ;w;l)=sup 1 e 1 Æe (1+r)(w 1 ) l s.t. 0 1 w
e 1 = 1 2+r [(1+r)w ln(Æ(1+r))℄: Sin e(P
)is on ave its solution,denoted 1 isgiven by: 1 = 8 < : 0if e 1 0 e 1 if0 e 1 w wifw e 1
Firstnotethat (H
1
) straightforwardlyimpliesthat e
1 >0:
Nowlet usassume:
V(;Æ;w;l)v 0 (18) where v 0 satis es (H 2
). Suppose then that for a (;Æ) agent 1 = w then(18) yields: e w Æ lv 0 )w e 1 = 1 2+r [w ln(Æ(1+r))℄ 1 2+r [w 0 ln(Æ(1+r))℄= 1 2+r ln Æ(1+r) ( v 0 +Æ) >0
the latter yields a ontradi tion with e
1
w so that the desired result is
proved omputing: V(;Æ;w;l)= 2+r 1+r [(Æ(1+r)) 1 2+r exp 1+r 2+r w℄ l:
Appendix 2: Level sets and distribution of .
Figure 4 6 - Æ Æ Figure 5 6 - Æ Æ
Let 2 L 1
([ ; ℄;R ) and let us ompute E( ( )), by the hange of
variables: (;Æ) 7!(; )=(;Æ 1 2+r ), we get: E( ( ))= Z Z Æ Æ (Æ 1 2+r )f(;Æ)dÆd = Z Z D ( )f(;( ) 2+r )) (2+r) 1+r 2+r dd
whereD isde nedby:
D:=f(; )2R 2
: 2[ ; ℄ 2[( );( )℄g
(:) and(:) are given by:
( )= ( if 1 Æ 1 2+r otherwise ( )= ( if 2 Æ 1 2+r otherwise , 1 :=Æ 1 2+r , 2 :=Æ 1 2+r
UsingFubini'sformulayieldsthen:
E( ( ))= Z ( )[ Z ( ) ( ) f(;( ) 2+r )) (2+r) 1+r 2+r d℄d :
Finallywe obtainh thedensityof :
h( )= Z ( ) ( ) f(;( ) 2+r )) (2+r) 1+r 2+r d: (19)
Note then that h is ontinuous sin e f(:;:), (:) and (:) are and h( ) =
h( ) = 0 for ( ) = and ( ) = : From (19) h is obviously positive
in [ ; ℄ sin e f > 0 and a straightforward omputation shows that h is
de reasingfor suÆ iently loseto .
Appendix 3: Proof of Proposition 2.
Proof:
First note that if (w;l) is a solution of (P
p
), and if W is de ned as in
Proposition1then itisne essary that:
W( )=v
0
(20)
for, otherwise ontra t (w;l+") would be admissible for small " > 0 and
in reasethe prin ipal'spro t. We may therefore with no loss of generality
add (20)asan additional onstraint in(P
p ).
Letthen (w;l) be admissibleand satisfy(20)and (w;e e
l) be asin
Propo-sition 1, 2). We already know that we and e
l are non de reasing. Re all
that:
W( )= g(w( ))e e
g(w( ))e =W ( ) a.e. (22)
togetherwith (20)weget:
e l( )= g(w( ))e v 0 + Z g(w(s))ds:e (23)
One an easily he kthat (w;l) is admissibleand satis es(20) if and only
if we is non negative non de reasing and e
l given by (23) is nonnegative,
sin e(21), (22),(23)and the onvexityofW implythat e
lisnonde reasing,
non negativity of e
l redu es to (P2). Finally (w;l) is admissible and
satis- es(20)ifandonlyifwesatis es(P1) and(P2)wherewe and e
l areobtained
from(w;l) by(21), (22) and(23). Now we have:
(w;l)= Z [l w℄f = Z [ e l w℄he (24)
Ontheother hand,using(23) and integrating byparts, we get:
Z e l( )h( )d = Z ( g(w( ))e v 0 )h( )d + Z [ Z g(w(s))ds℄h( )d e = Z g(w( ))h( )d e v 0 + Z g(w( ))H( )d :e (25)
From (24) and (25) we get (w;l) = e
(w)e v
0
so that Proposition 2 is
proved.
Appendix 4: Proof of Proposition 3.
We rst needthefollowingte hni alresult:
Lemma 3 De ne for all 2[ ; ℄:
( ):= Z [g 0 (w (s))(sh(s)+H(s)+) h(s)℄ds
let ">0, the following properties hold:
1) both integrals R " w d and R " ( ( ))dw exist and: Z " w d= Z " ( ( ))dw +(( ") ( ))w ( ")+( )w ( ) = Z " [g 0 (w (s))(sh(s)+H(s)+) h(s)℄w (s)ds
whered denotesthe Stiljes measureasso iated withthe absolutely
on-tinuous fun tion and is de ned as in Lemma 2.
2) lim "!0 + (( ") ( ))w ( ")=0
1) w
j[ ; "℄
isa boundedvariationfun tionand is absolutely ontinuous,
both integrals R " w d and R " ( ( ))dw
exist and the rst one
an be integrated by parts in Stieljes sense ( f. Natanson (1967)), whi h
yields: Z " w d= Z " ( ( ))dw +(( ") ( ))w ( ")+( )w ( )
We onlyhave to showthen:
Z " w d= Z " [g 0 (w (s))(sh(s)+H(s)+) h(s)℄w (s)ds
Ontheotherhand,de ning,8n; 8k=0;:::;n 1,u n k := + k n ( " ) we have: j Z " w d Z " g 0 (w (s))(sh(s)+H(s)+) h(s) w (s)dsj lim n!+1 j n 1 X k=0 Z u n k +1 u n k (w (u n k+1 ) w (s)) g 0 (w (s))(sh(s)+H(s)+) h(s) dsj C st lim n!+1 1 n (w ( ") w ( ))=0
thelastinequalityfollowsfromw
beingnon-de reasingandg 0 beingbounded. Thus1) is established. 2) Sin ew is nonde reasing: j(( ") ( ))w ( ")j Z " j(g 0 (w (s))j(sh(s)+H(s)+)+h(s)℄w (s)ds
holdsand thelatter integral tends to 0 as"!0 byLebesgue's Dominated
Convergen e Theorem,sin e hw 2L 1 and w g 0 (w ) isbounded.
We are now readyto prove Proposition3.
Proof:
Let w be a non negative non de reasing fun tion s.t. there exist two
on-stants A 0 and B su h that: w Aw
+B (whi h implieshw 2 L 1
+ ).
Thenforall t2(0;1) thefollowingholds:
1 t [ e (w +t(w w )) e (w )℄0: (26) First,wehave: lim t!0 + 1 t [g(w ( )+t(w( ) w ( ))) g(w ( ))℄=g 0 (w ( )(w( ) w ( )): (27)
f t (s)= 1 t [g(w (s)+t(w w )(s)) g(w (s))℄(sh(s)+H(s)+) we have: lim t!0 f t (s)=g 0 ((w (s))(w w )(s)(sh(s)+H(s)+) and jf t (s)j sup z2[w (s);w (s)+t(w w )(s)℄ g 0 (z)j(w w )(s)j(sh(s)+H(s)+) g 0 (A 1 w (s)+B 1 )(A 2 w (s)+B 2 ):(sh(s)+H(s)+) whereA 1 0,A 2 0,B 1 andB 2
are onstantsonlydependingonAandB.
Sin e w7!g 0 (A 1 w+B 1 )(A 2 w+B 2
) is bounded there isa onstant C su h
that: jf
t
(s)jC forall tand s. Thus, Lebesgue's DominatedConvergen e
Theoremyields: lim t!0 + f t (s)ds= Z [g 0 (w )(sh(s)+H(s)+)℄(w w )ds (28)
Then,passing tothe limitt!0 + in (26)yields: Z [g 0 (w )(sh(s)+H(s)+) h(s)℄(w w )ds + g 0 (w ( ))(w( ) w ( ))0 (29)
forallwnon negative, nonde reasingandsu hthatwAw
+B forsome
onstantsA0andB. Let bein( ; ),takingw(s):=w
(s)+1
s ,8s
in(29)exa tly yields: ( ) ( )0; 8 i.e (i).
Notethat( ( ))dw
isanonnegativemeasure,absolutely ontinuous
withrespe tto thenonnegative measure dw . Takingw=2w andthen w= w 2 in(29)yields: Z [g 0 (w )(sh(s)+H(s)+) h(s)℄w ds+ g 0 (w ( ))w ( )=0: (30) Sin e( ( ))dw is nonnegative,weget: lim "!0 + Z " ( ( ))dw = Z ( ( ))dw :
Onthe otherhand, we know fromthe previouslemma:
Z " [g 0 (w (s))(sh(s)+H(s)+) h(s)℄w (s)ds
= " ( ( ))dw +(( ") ( ))w ( ")+( )w ( ): (31)
(30) and part2) of Lemma3 yieldthen, passing tothe limitin(31):
Z ( ( ))dw =( g 0 (w ( ))+( ))w ( ): (32) Takingw=w ( )in(29) yields: w ( )( ) Z [g 0 (w )(sh(s)+H(s)+) h(s)℄w ds0: (33)
Passing to thelimitin(31) we get:
Z [g 0 (w )(sh(s)+H(s)+) h(s)℄w ds = Z ( ( ))dw ( )w ( ): (34)
(33)and (34) implythen:
Z ( ( ))dw 0 sothat( ( ))dw
=0i.e. (ii). Finally(ii)and (32)imply(iii).
Conversely, (i), (ii), (iii)are also suÆ ient onditions for optimalityby
onvexityproperties of( e
P
p; ).
2) Letusestablishthe ontinuityofw
: Supposethereexists in( ; )
su h that w ( + ) > w
( ), is an atom of the measure dw
and then
ondition(ii)implies: ( )=( )=inf
[ ; ℄
it iseasy to prove that is
rightand left di erentiableat every point of ( ; )then:
0 g ( )=g 0 (w ( ))( h( )+H( )+) h( ) 0 0 d ( )=g 0 (w ( + ))( h( )+H( )+) h( ) whi h implies w ( + ) w ( ) be ause g 0 is de reasing, ontradi ting w ( + )>w
( ). We an on ludethat w
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