Optimal Assignment of Durable Objects to Successive

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Optimal Assignment of Durable Objects to Successive Agents ? Francis Bloch † Nicolas Houy ‡ September 29, 2009 Abstract This paper analyzes the assignment of durable objects to successive generations of agents who live for two periods. The optimal assignment rule is stationary, favors old agents and is determined by a selectivity function which satisfies an iterative func- tional di?erential equation. More patient social planners are more selective, as are social planners facing distributions of types with higher probabilities for higher types. The paper also characterizes optimal assignment rules when monetary transfers are allowed and agents face a recovery cost, when agents' types are private information and when agents can invest to improve their type. JEL Classification Numbers: C78, D73, M51 Keywords: Dynamic Assignment, Durable Objects, Revenue Management, Dy- namic Mechanism Design, Overlapping Generations, Promotions and Intertemporal Assignments. ?We are grateful to Gabrielle Demange, Philippe Jehiel and seminar audiences at NUS (Singa- pore) and PSE (Paris) for their comments. Correspondence: and . †Department of Economics, Ecole Polytechnique, 91128 Palaiseau France, tel: , ‡Department of Economics, Ecole Polytechnique, 91128 Palaiseau France, tel: , nhouy@free.

  • agent

  • transfer rules

  • vickrey-clarke-groves mechanisms

  • optimal assignment

  • assignment policy

  • when monetary transfers

  • all agents


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OptimalAssignmentofDurableObjectstoSuccessive
Agents

FrancisBloch

NicolasHouy

September29,2009

Abstract
Thispaperanalyzestheassignmentofdurableobjectstosuccessivegenerationsof
agentswholivefortwoperiods.Theoptimalassignmentruleisstationary,favorsold
agentsandisdeterminedbyaselectivityfunctionwhichsatisfiesaniterativefunc-
tionaldi

erentialequation.Morepatientsocialplannersaremoreselective,asare
socialplannersfacingdistributionsoftypeswithhigherprobabilitiesforhighertypes.
Thepaperalsocharacterizesoptimalassignmentruleswhenmonetarytransfersare
allowedandagentsfacearecoverycost,whenagents’typesareprivateinformation
andwhenagentscaninvesttoimprovetheirtype.

JELClassificationNumbers:C78,D73,M51
Keywords:
DynamicAssignment,DurableObjects,RevenueManagement,Dy-
namicMechanismDesign,OverlappingGenerations,PromotionsandIntertemporal
Assignments.

WearegratefultoGabrielleDemange,PhilippeJehielandseminaraudiencesatNUS(Singa-
pore)andPSE(Paris)fortheircomments.Correspondence:
francis.bloch@polytechnique.edu
and
nhouy@free.fr
.

DepartmentofEconomics,EcolePolytechnique,91128PalaiseauFrance,tel:+331693330
45,francis.bloch@polytechnique.edu

DepartmentofEconomics,EcolePolytechnique,91128PalaiseauFrance,tel:+331693330
15,nhouy@free.fr

1

1Introduction
Thispaperconsidersdurableobjectswhicharesuccessivelyreassignedtoagents.
Theprimeexampleofobjectswhichareregularlyreassignedarepositionsinor-
ganizationsandbureaucracies,likeexecutiveo

cesordiplomaticpostings.Other
examplesincludeo

ces,roomsindormitories,computerservers,largescalescien-
tificequipmentliketelescopesandparticleaccelerators,hotelroomsandrentalcars.
Inalltheseexamples,agentsaregiventemporarypropertyrightsoverthedurable
objectforagivenperiod,andcannotbeforcedtoreturntheobject.Thesetempo-
rarypropertyrightscreateaconstraintontheoptimalassignmentpolicychosenby
abenevolentsocialplanner,andourobjectiveinthispaperistocharacterizethe
e

cientdynamicassignmentofadurableobjecttosuccessivegenerationsofplayers
takingintoaccounttheseindividualrationalityconstraints.
Weconsiderenvironmentswhereoverlappinggenerationsofagentsenterthemar-
ketdeterministically,andagentsareassignedtheobjectfortheirentirelifetime.
Agentsdi

erintheirvaluationsfortheobjects,ortheirqualificationsforthepo-
sitions.Agentscharacteristicsarethustwo-dimensionalandincludeatypewhich
determinesthevalueoftheassignment,andanagewhichdeterminesthelengthofthe
assignment.Ourobjectiveinthispaperistocharacterizeoptimalassignmentrules
inthistwo-dimensionalmodel,andconstructrevelationmechanismswhenagents’
typesareprivatelyknown.
Thebasictrade-o

betweenageandvalueisbestunderstoodinatwo-period
overlappinggenerationsmodel.Whenassigningthegoodtoanoldoryoungagent,
thesocialplannermakesthefollowingcomputation.Assigningthegoodtotheold
agentforoneperiodhasapositiveoptionvalue,asthegoodcanbereassignedatthe
endoftheperiod;assigningthegoodtotheyoungagentfortwoperiodsprevents
theplannerfromreassigningthegoodimmediately.Hence,iftheoldandgood
agentswereofthesametype,itwouldalwaysbeoptimaltoassignthegoodtothe
oldagent
.
1
Thislineofreasoningshowsthat,foranytype
θ
oftheyoungagent,
theplannerwillprefertogivetotheanyoldagentoftypegreaterorequalto
φ
(
θ
),
where
φ
(
θ
)
<
θ
.
Themaincontributionofthepaperistocharacterizetheselectivityfunction,
φ
(
θ
),whichisadoptedintheoptimalassignmentpolicy.Thisfunctionisdefinedby
aniterativefunctionaldi

erentialequation,whichissimilartotheequationsused
inphysicsandmathematicstostudydynamicalsystemswithstate-dependentde-
1
Theoptionvalueofgivingapositiontoanolderagentisawelldocumentedhistoricalfact.
Forexample,thehistoryofthepapacyrecordsanumberofelectionswherecardinalsvoluntarily
chosetheoldestcandidate.Oftentimes,these”transitionpopes”turnouttobethemostenergetic
ande

ectivepopesoftheirtimes.SeeCollins(2009)andhisaccountofthereignoftwofamous
”transitionpopes”,JohnXXII(1316-1334)andJohnXXIII(1958-1965).

2

lay(Eder1984).Whilewecannotprovideanaanalyticalsolutiontothefunctional
di

erentialequation,weproveexistenceanduniquenessoftheoptimalassignment
policyandshowthattheselectivityfunctionisincreasingandconvex.Weillustrate
theoptimalassignmentpoliciesdorunfiromandquadratictypedistributions,and
derivecomparativestaticspropertiesofthesolution.Undersomeregularitycon-
ditions,weshowthatwhenthediscountfactorincreases,orwhenthedistribution
oftypesisshiftedsothathighertypeshaveahigherprobability,thesocialplanner
becomesmoreselective,andassignstheobjecttotheyoungagentlessoften.
Inthesecondpartofthepaper,weconsiderdi

erentextensionsofthemodel.
First,weanalyzetheoptimalassignmentpolicywhenmonetarytransfersareallowed
andagentscanbecompensatedwhentheyreturntheobject.Ifthereisnorecovery
cost,theoptimalassignmentpolicyisidenticaltothefirst-bestpolicywithoutindi-
vidualrationalityconstraints:theobjectisassignedateveryperiodtotheagentwith
thehighestvalue.Ifanoldagentwhocurrentlyholdstheobjecthasasmallervalue
thantheyoungagent,theyoungagentcaneasilytransfermoneyinreturnforthe
objectandtheindividualrationalityconstraintceasestobebinding.Withpositive
recoverycosts,theoptimalassignmentstrategybecomesmorecomplex,andinvolves
twodi

erentselectivityfunctions,onewhichisusedatperiodswherenoagentholds
theobject,andonewhichisusedwhentheoldagentholdspropertyrightsoverthe
objectandneedstobecompensated.Wecharacterizetheoptimalassignmentpoli-
ciesassolutionstosystemsofdi

erentialfunctionalequations,bothwithfixedand
proportionalrecoverycosts.Wealsoillustratethesecomplexassignmentstrategies
fortheuniformandquadraticdistributions.
Inasecondextensionofthemodel,weanalyzedirectrevelationmechanisms
whenthetypesoftheagentsareprivatelyknown.Giventhetimestructureofthe
assignmentrule,wecanbuilddi

erentmodelsofrevelationmechanisms.Inthefirst
model,wesupposethatagentsareaskedtorevealtheirtypeswhentheyentersoci-
etyasyoungagents,whethertheobjectisreassignedornot.Inthesecondmodel,
weassumethatagentsareonlyaskedtorevealtheirtypes(andpayatransfer)at
periodswherethegoodisreassigned.Forbothmodels,weusestandardarguments
tocharacterizetransferrulesimplementingtheoptimalassignmentpolicy.Notsur-
prisingly,thesetransferrulesinvolveastepfunction,wheretransfersjumptoaflat
positivevaluewhentheagent’stypereachesthethresholdvalueforwhichthegood
isassignedtoher.
Finally,weinvestigatetheagents’incentivestoinvestinordertoimprovetheir
typesbetweenthetwoperiodsoftheirlives.Inthatmodel,ayoungagentwhodoes
notreceivethegoodinthefirstperiodhasanincentivetoinvestinhistypeboth
inordertoincreasetheprobabilityofreceivingtheobjectwhenold,andtoincrease

3

thevalueofthematch.Inthismodel,thesocialplanners’optimalassignmentpolicy
andtheagents’incentivestoinvestaredeterminedsimultaneously,asthesolutions
toasystemofdi

erentialequation.Again,whileweareunabletosolvethesystem
analytically,weprovideanumericalillustrationofthesolutionsfortheuniformand
quadraticdistributions.
Axiomaticcharacterizationsofassignmentrulesfordurablegoodshaverecently
beenstudiedbyKurino(2008)andBlochandCantala(2008).Kurino(2008)con-
sidersadynamicextensionofAbdulkadirogluandSo¨nmez(1999)’sstudyofhouse
allocationwithexistingtenants–thefirstexampleofanassignmentproblemwith
individualrationalityconstraints–,andanalyzeswhethertherulesproposedinthe
staticpaperstillsatisfye

ciencyandincentivecompatibilityinthedynamiccontext.
BlochandCantala(2008)consideramodelwhereagentsareassignedtodi

erent,
verticallyrelatedobjects,andcharacterizeMarkovianassignmentruleswhichsat-
isfymyopice

ciencyandfairness.Bycontrast,inthispaper,weconsiderasimpler
modelwhereagentsonlylivetwoperiodsandcanonlybeassignedonegood.Inthis
simplermodel,weareabletocharacterizedynamicallye

cientrules,andtodiscuss
theincentivepropertiesoftransferrules.
Thispaperisalsorelatedtotherapidlygrowingliteratureondynamicmecha-
nismdesign.ParkesandSingh(2003),AtheyandSegal(2007),Bergemannand
Valima¨ki(2006)andGershkovandMoldovanu(2008a,2008b,2008c)studydy-
namicassignmentproblems,whereagentsentersequentially,andparticipateina
Vickrey-Clarke-Grovesrevelationmechanismwhichdeterminestransfersandgood
allocations.TheyshowthatVickrey-Clarke-Grovesmechanismsandoptimalstop-
pingrulescanbecombinedtoobtaine

cientdynamicmechanisms.Inthesemod-
els,objectscanonlybeassignedonceatthetimeofentry.Someofthesestudies
(likeGershkovandMoldovanu(2008b,2008c))distinguishbetweenbenevolentand
revenue-maximizingplanners.Whenagents’typesareknown,theliteratureonyield
managementinmanagementscienceandoperationsresearch(seeTalluriandVan
Rysin(2004))providesanin-depthstudyofoptimalpricingstrategies.
Therestofthepaperisorganizedasfollows.WeintroducethemodelinSection
2.Weanalyzee

cientassignmentpoliciesinSection3.Section4isdevotedtothe
threeextensionsofthemodelandSection5concludes.

2TheModel
2.1Agents
Weconsidertheassignmentofasingledurableobjecttooverlappinggenerationsof
agents.Timeisdiscreteandrunsas
t
=1
,
2
,...

.Ateachperiod
t
,onenewagent

4

enterssociety.Agentslivefortwoperiods,sothatateachperiod,societyconsists
exactlyofoneyoungandoneoldagent.Allagentssharethesamediscountfactor
δ

(0
,
1).
Agentsarecharacterizedbytheirtype
θ
whichmeasurestheflowofutilitygen-
eratedbytheassignmentoftheobject.Wesupposethattypesaredrawnbefore
anagententerssocietyandlastfortheagents’entirelifetime.Typesofsuccessive
agentsaredrawnindependentlyfromthesamedistribution
F
,whichisassumedto
benon-atomic,havefullsupportonacompactinterval
Θ
=[
θ
,
θ
]
⊂￿
+
andadmit
acontinuousdensityfunction
f
.Weassumethattypesarealwayspositive,sothat
allassignmentscreatepositivesurplus.
2
Atanytime
t
,wedenoteby
θ
ty
and
θ
to
the
typesoftheyoungandoldagents.Themodelstartsatperiod0,wherethetypeof
theoldagent,
θ
0
o
isgivenandknowntoeveryone.
Theobjectwillbesuccessivelyassignedtoagentspresentinsociety.Wesuppose
thatagentscannotbeforcedtorelinquishtheobject.Inotherwords,
weassume
thatwhentheobjectisassignedtoayoungagent,theyoungagentkeepsitfortwo
periods.
Thisassumptionintroducesastrongasymmetrybetweenoldandyoung
agents.Itimpliesthat,atsomeperiods,theobjectwillnotbereassigned.Wecan
thusdistinguishbetweentwosetsofdates:aset
T
0
ofperiodsatwhichtheobject
is
notreassigned
becausetheyoungagentretainsitforanotherperiod,andasetof
dates
T
1
atwhichtheobjectisreassigned.Thisisillustratedinthefollowinggraph:
Inthisexample,therearethreedatesatwhichtheobjectisassigned,dates
t
=0
,
1
,
3.Atdate
t
=2,theobjectisnotassigned.Theobjectisassignedtothe
oldagentatperiods
t
=0
,
3andtotheyoungagentat
t
=1
.
Inthebaselinemodel,wesupposethattransferstoagentsarenotallowed,and
thattheplanneronlychoosestowhomtheobjectisassignedatanydate
t

T
1
.
Formally,theplanner’scontrolvariablearetheprobabilities
p
to
and
p
ty
withwhich
theobjectisassignedtotheoldandyoungagent.Theutilityflowforayoungand
oldagentatdate
t

T
1
aregivenby

U
ty
=
p
ty
θ
ty
,
U
to
=
p
to
θ
to

andatadate
t

T
0
by
2
Thisassumptionismadewithoutlossofgenerality.Ifsometypeswerenegative,wecouldas
wellconsideradistributionwherethesetypesformanatomatzero(whichwouldnota

ectour
analysis),asthegoodwillneverbeassignedtothem.

5

Figure1:
Overlappinggenerationsandassignments

y0=,UtooU
t
=
θ
t
.

2.2Socialplanner
Weassumethatthesocialplannerhasaninfinitehorizonandevaluatessequences
ofpayo

susingthesamediscountfactor
δ
astheagents.Ateachperiod
t

T
1
,the
socialplannerchoosesaprobabilitypair(
p
ty
,p
to
).Thesechoicesgenerateastochas-
ticprocessoverthesequencesofassignments(
θ
0
,....,
θ
t
,...
)andweassumethatthe
benevolent
socialplannerchoosestheprobabilitypairinordertomaximizethedis-
countedsumofutilitiesresultingfromtheassignment:
∞V
=
δ
t
E
[
θ
t
]
,
￿1=tWhilethesocialplannercaninprincipleconditionhischoice
{
(
p
ty
,p
to
)
,
}
atpe-
riod
t
ontheentirehistoryuptoperiod
t
,asimpleapplicationofwell-knownargu-
mentsshowsthatthereisnolossofgeneralityinrestrictingattentionto
Markovian
decisions
,wherethesocialplannerbaseshisdecisiononastatesummarizingthe

6

payo

-relevantpartofthehistory.
3
Inourcontext,thepayo

-relevantpartofthe
historyatanydate
t

T
1
isgivenbythetypesofagentspresentatperiod
t
,and
wedefineastateasavectoroftypesoftheyoungandoldagents,(
θ
y
,
θ
o
)in
Θ
×
Θ
.

3Optimalassignmentofobjects
3.1Characterizationoftheoptimalassignment
WenowcharacterizetheoptimalassignmentbyapplyingBellman’sprincipleofop-
timalitytotheMarkoviandecisionproblemofthebenevolentsocialplanner.Define
thevaluefunctionatastate(
θ
y
,
θ
o
)by

θθV
(
θ
y
,
θ
o
)=max
p
y
(
θ
y
(1+
δ
)+
δ
2
V
(
t
y
,t
o
)
f
(
t
y
)
f
(
t
o
)
dt
y
dt
o
)
￿￿p
y
,p
o
θθ
θ￿+
p
o
θ
o
+
δ
(1

p
y
)
V
(
t
y
,
θ
y
)
f
(
t
y
)
dt
y
.
(1)
θIntheexpressionabove,wedistinguishbetweenthetwochoicesofthesocial
planner:ifsheallocatesthegoodtotheyoungagent,theyoungagentkeepsitfor
twoperiods,andaftertwoperiods,thenewstatewillbecharacterizedbytwonew
typeswhichhavebeendrawnaccordingtothedistribution
F
.If,ontheotherhand,
thegoodisassignedtotheoldagent,hewillkeepitonlyforoneperiod,andin
yothenextperiod,thetypeoftheoldagentwillbeknown,andgivenby
θ
t
+1
=
θ
t
whilethetypeofthenewagentwillbedrawnaccordingto
F
.Fromexpression(1),
weimmediatelymakeoneobservation.Astypesareassumedtobepositive,the
planner’sobjectiveisincreasingin
p
o
for
p
y
fixed,soshewillalwaysassignthegood
tooneofthetwoagents.Wecanthusreplace
p
o
by1

p
y
.Giventhis,
θθV
(
θ
y
,
θ
o
)=max
p
y
(
θ
y
(1+
δ
)+
δ
2
V
(
t
y
,t
o
)
f
(
t
y
)
f
(
t
o
)
dt
y
dt
o
)
￿￿ypθθθ￿+(1

p
y
)(
θ
o
+
δ
V
(
t
y
,
θ
y
)
f
(
t
y
)
dt
y
)
.
θTheobjectiveislinearin
p
y
,sothatweobtainasimplecharacterizationoftheop-
timalpolicy:theplannershouldeitherassignthegoodtotheyoungortheoldagent
withprobability1,dependingonthesignof
θ
y
(1+
δ
)+
δ
2
θθθθ
V
(
t
y
,t
o
)
f
(
t
y
)
f
(
t
o
)
dt
y
dt
o

￿￿θ
o

δ
θ
V
(
t
y
,
θ
y
)
f
(
t
y
)
dt
y
.Next,definethemapping
φ
from
Θ
to
￿
by:
θ￿3
SeePutterman(1994)foranintroductiontotheliteratureonMarkoviandecisionproblemsand
theproofthatMarkoviandecisionrulesareoptimal.

7

￿
θ
￿
θ
￿
θ
φ
(
θ
)

θ
(1+
δ
)

δ
V
(
t,
θ
)
f
(
t
)
dt
+
δ
2
V
(
t,z
)
f
(
t
)
f
(
z
)
dtdz.
θθθWelabelthefunction
φ
the
selectivityfunction
associatedtotheoptimalplanner’s
decision.Thisterminologyiseasilyunderstood:thesocialplannerwillallocate
thegoodtotheoldagentifandonlyif
θ
0

max
{
θ
,
φ
(
θ
y
)
}
.Astheselectivity
functionfullycharacterizestheoptimalassignment,wenowstudythepropertiesof
thismapping.
Lemma1
Thefunction
φ
isstrictlyincreasing,andsatisfies
φ
(
θ
)
<
θ
forall
θ

[
θ
,
θ
)
,
φ
(
θ
)=
θ
.
Lemma1providesausefulcharacterizationoftheoptimalassignmentforanypair
oftypes(
θ
y
,
θ
o
).Itshowsthatoptimalassignmentsarecharacterizedby
threshold
rules
,determiningforany
θ
y
,theminimaltypeoftheoldagentwhoispreferred
to
θ
y
,andforany
θ
o
,theminimaltypeoftheyoungagentwhoispreferredto
θ
o
.
Ofcourse,theseoptimalthresholdrulesarenotindependent(oneistheinverseof
theother),andwefocusattentiononthethresholdrule
φ
(
θ
y
),whichdetermines
theminimaltypeoftheoldagentpreferredto
θ
y
.As
φ
(

)isstrictlyincreasingand
φ
(
θ
)
<
θ
forall
θ

[
θ
,
θ
),wenotethatthereexistsaunique
θ

suchthattheplanner
preferstoassignthegoodtotheoldagentforany
θ
y

θ

,andusestheselectivity
function
φ
(

)whenever
θ
y

θ

.ThesecondpartofLemma1capturesthe”option
value”ofassigningtheobjecttotheoldagent.Ifbothagentshavethesametype,
thesocialplanneralwayspreferstoassigntheobjecttotheoldagent,assheretains
theoptionvalueofassigningtheobjecttotheyoungagentnextperiod,butcan
alsodrawayoungagentwithhighertype.Thisoptionvalueexplainstheexistence
ofapositive”gap”,measuredby
θ

φ
(
θ
),betweentheminimaltypeoftheyoung
andoldagents.Finally,noticethatLemma1holdswithoutanyconditiononthe
di

erentiabilityoftheselectivityfunction
φ
.
Next,observethat,assumingthat
φ
isdi

erentiable,byasimpleapplicationof
theenvelopetheorem,wecompute:
θ∂
θ
V
(
t,
θ
)
f
(
t
)
dt

1
￿=
F
(
φ
(
θ
))
.
θ∂Usingthisobservation,wecaneasilycharacterizecontinuouslydi

erentiableselec-
tivityfunctionsassolutionstoafunctionaldi

erentialequation:

8

Theorem1
Thereexistsanoptimalassignmentpolicycharacterizedbyacontinu-
ouslydi

erentiableselectivityfunction
φ
(

)
whichistheuniquesolutionoftheiter-
ativefunctionaldi

erentialequation:
φ
￿
(
θ
)=1+
δ

δ
F
[
φ

1
(
θ
)](2)
withinitialcondition
φ
(
θ
)=
θ
.
Theorem1characterizestheoptimalpolicyasthesolutiontoaniterativefunc-
tionaldi

erentialequation.Theiterativefunctionaldi

erentialequation(2)belongs
toaclassofdi

erentialequationswhichhavebeenstudiedinphysicsandmathemat-
icstoanalyzedynamicalsystemswithstatedependentdelays(seeEder(1984),Si
andZhang(2004)).Existenceanduniquenessofsolutionstothisfunctionalequation
doesnotderivefromstandardtheoremsonordinarydi

erentialequations,butcan
neverthelessbeobtainedusingBanach’sfixedpointtheoreminfunctionalspaces.
Thisfunctionaldi

erentialequationdoesnottypicallyadmitclosedformsolutions.
4
Inspectionofequation2providesadditionalinformationontheoptimalselectivity
function
φ
:
Corollary1
Theoptimalselectivityfunction
φ
(

)
isstrictlyconcaveforany
θ

[
θ
,
θ
)
,and
φ
￿
(
θ
)=1
.
Corollary1andLemma1showthateventhoughequation(2)doesnotadmitan
analyticalsolution,theoptimalselectivityfunctionpossessesremarkableproperties.
Inaddition,equation(2)hasasimplerecursivestructure,whichstemsfromthefact
that,as
φ

1
(
θ
)
>
θ
,thevalue
φ
￿
(
θ
)onlydependsonthevalueofthefunction
φ
atpointslargerthan
θ
.Thisrecursivestructure,togetherwiththeinitialcondition
φ
(
θ
)=
θ
,allowsforane

cientalgorithmtocomputenumericalsolutionstoequa-
tion(2).Usingthisalgorithm,wecomputetheoptimalselectivityfunctionforthe
uniformandquadraticdistributionsover[0
,
1].
Thefirstgraphshows,forthreedi

erentvaluesof
δ
(
δ
=0
,
0
.
5and1),the
optimalselectivityfunction
φ
forthedistribution
F
(
θ
)=
θ
over[0
,
1].Thesecond
graphmapstheoptimalselectivityfunction
φ
forthesamevaluesof
δ
andthe
distribution
F
(
θ
)=
θ
2
over[0
,
1].
4
Ifthedistribution
F
isuniformon[0
,
1],thefunctionaldi

erentialequationbecomessimilar
toanequationstudiedbySiandZhang(2004)whoprovideoneanalyticsolutiontotheequation.
Unfortunately,thesolutiontheyproposedoesnotsatisfytheboundarycondition
φ
(1)=1.

9

1

.08

6.0)(θφ4.0

2.0

000.20.40.60.81
θFigure1:Optimalselectivityfunctionwithuniformdistribution
1

8.0

6.0)(θφ4.0

2.0

000.20.40.60.81
θFigure2:Optimalselectivityfunctionwithquadraticdistribution
Figures1and2clearlyillustratethepropertiesoftheselectivityfunctions
φ
,
whichareincreasing,concave,crossthe
x
-axisatapositivevalue
θ

,andsatisfy
φ
(
θ
)=
θ
and
φ
￿
(
θ
)=1foralldistributionsandallvaluesofthediscountfactor.The
figuresalsosuggesttwocomparativestaticspropertiesoftheselectivityfunction.

01