A New Class Problem in the Calculus of Variations

ISSN1560-3547,RegularandChaoticDynamics,2013,Vol.18,No.6,pp.558–589.cPleiadesPublishing,Ltd.,2013.ANewClassofProblemsintheCalculusofVariationsIvarEkeland1*,YimingLong2**,andQinglongZhou2***1CEREMADEandInstitutdeFinanceUniversit´edeParis-IX,Dauphine,Paris,France2ChernInstituteofMathematicsandLPMCNankaiUniversity,Tianjin300071,ChinaReceivedJune19,2013;acceptedOctober11,2013Abstract—Thispaperinvestigatesaninfinite-horizonproblemsintheone-dimensionalcalculusofvariations,arisingfromtheRamseymodelofendogeneouseconomicgrowth.FollowingChichilnisky,weintroduceanadditionalterm,whichmodelsconcernforthewell-beingoffuturegenerations.Weshowthattherearenooptimalsolutions,butthatthereareequilibriumstrateges,i.e.Nashequilibriaoftheleader-followergamebetweensuccessivegenerations.Tosolvetheproblem,weapproximatetheChichilniskycriterionbyabiexponentialcriterion,wecharacterizeitsequilibriabyapairofcoupleddifferentialequationsofHJBtype,andwegotothelimit.WefindalltheequilibriumstrategiesfortheChichilniskycriterion.ThemathematicalanalysisisdifficultbecauseonehastosolveanimplicitdifferentialequationinthesenseofThom.OuranalysisextendsearlierworkbyEkelandandLazrak.Itisshownthatoptimalsolutionsaclassofproblemsraisingfromtimeinconsistencyproblemsintheframeworkoftheneoclassicalone-sectormodelofeconomicgrowth,andcontainsnewresultsinenvironmenteconomics.Withoutexogenouscommitmentmechanism,anotionoftheequilibriumstrategiesinsteadoftheoptimalstrategiesisintroduced.Wecharacterizedtheequilibriumstrategiesbyanintegro-differentialequationsystem.Fortwospecialcriteria,thebi-exponentialcriteriaandtheChichilniskycriteria,weestablishedtheexistenceoftheequilibriumstrategies.MSC2010numbers:49J40,91B02,49L99DOI:10.1134/S1560354713060026Keywords:minimizationproblem,sustainableeconomy,time-inconsistency,existenceDedicatedtoProfessorAlainChencineronhis70thbirthdayContents1INTRODUCTION5592THERAMSEYPROBLEM5613TIME-INCONSISTENCY5653.1EquilibriumStrategies5653.2TheHJBApproach5663.3TheEulerEquations5693.4TheControlTheoryApproach5704THEBIEXPONENTIALCASE5704.1TheEquations5704.2SolvingtheBoundary-valueProblem5744.3TheExistenceofEquilibriumStrategies579*E-mail:[e-mail]**E-mail:[e-mail]***E-mail:[e-mail]558ANEWCLASSOFPROBLEMSINTHECALCULUSOFVARIATIONS5595THECHICHILNISKYCRITERION581ACKNOWLEDGMENTS588REFERENCES5881.INTRODUCTIONIneconomictheory,andinoptimalcontrol,ithasbeencustomarytodiscountfuturegainsataconstantrateδ0.Ifanindividualwithutilityfunctionu(c)hasthechoicebetweenseveralstreamsofconsumptionc(t),0t,he/shewillchoosetheonewhichmaximizesthepresentvaluegivenby∞0u(c(t))e−δtdt.(1.1)Thatfuturegainsshouldbediscountediswellgroundedinfact.Ontheonehand,humansprefertoenjoygoodthingssoonerthanlater(andtosufferbadthingslaterthansooner),aseverychild-rearingparentknows.Ontheotherhand,itisalsoareflectionofourownmortality:10yearsfromnow,ImaysimplynolongerbearoundtoenjoywhateverIhavebeenpromised.Thesearetwogoodreasonswhypeoplearewillingtopayalittlebitextratohastenthedeliverydate,orwillrequirecompensationforpostponement,whichistheessenceofdiscounting.Ontheotherhand,thereisnoreasonwhythediscountrateshouldbeconstant,i.e.,whythediscountfactorshouldbeanexponentiale−δt.Thepracticeprobablyarisesfromthecompoundinterestformulalimε→0(1−εδ)t/ε=e−δt,whenaconstantinterestrateδisassumed,buteveninfinance,interestratesvarywiththehorizon:long-termratescanbewidelydifferentfromshort-termones.Asforeconomics,thereisbynowahugeamountofevidencethatindividualsusehigherdiscountratesforthenearfuturethanforthedistantfuture(see[18]forareviewupto2002).Thereisalsoanaggregationproblem:inasocietywhereindividualsuseconstant(butdifferent)discountrates,thecollectivediscountratemaybenon-constant(see[14]).Sothepresentvalueformula(1.1)shouldbereplacedbythemoregeneralone:∞0u(c(t))h(t)dt(1.2)wherehisadecreasingfunction,withh(0)=1.Butthenanewproblemarises,whichisnowwellrecognizedineconomictheory,buttoourknowledgehasnotyetreceivedtheattentionitdeservesincontroltheory.Itistheproblemoftime-inconsistency,whichrunsasfollows.Supposethedecision-makerhasthechoicebetweentwostreamsofconsumptionc1(t)andc2(t),startingattimeT0.Attimet=0,he/shefindsc1(t),yieldingthehighestpresentvalue:∞Tu(c1(t))h(t)dt∞Tu(c2(t))h(t)dt.(1.3)He/shethenchoosesc1(t).WhentimeTisreached,thepresentvaluesarenow:∞Tu(c1(t))h(t−T)dtand∞Tu(c2(t))h(t−T)dt.(1.4)Ifh(t)=e−δt,thentheorderingfoundattimet=0willpersistattimet=T.Indeed,∞Tu(c(t))e−δ(t−T)dt=eδT∞Tu(c(t))e−δtdt,sothatthetwotermsin(1.4)areproportionaltothetwotermsin(1.3).However,thisisapeculiarityoftheexponentialfunction,anditisnottobeexpectedwithmoregeneraldiscountREGULARANDCHAOTICDYNAMICSVol.18No.62013560EKELANDetal.rates.Thedecision-makerthenfacesabasicrationalityproblem:whatshouldhe/shedo?Tobemorespecific,assumethestatek(t)isrelatedtothecontrolc(t)bythedynamicsdkdt=f(k)−c(t),k(0)=k0,(1.5)c(t)0,k(t)0(1.6)andthedecision-makerisinterestedinmaximizing(1.2).Howshouldhe/shebehave?Intheexponentialcase,whenh(t)=e−δt,theansweristopicktheoptimalsolution:ifitisoptimalattim