Existence, optimality and dynamics of equilibria with endogenous time preference

ContentslistsavailableatScienceDirect

Journal

of

Mathematical

Economics

j o u r n al ho m e p age :w w w . e l s e v i e r . c o m / l o c a t e / j m a t e c o

Existence,

optimality

and

dynamics

of

equilibria

with

endogenous

time

preference

Selman

Erol

a

_,

_Cuong

_Le

_Van

b,c,d

_,

_Cagri

_Saglam

a,∗

a_{DepartmentofEconomics,BilkentUniversity,Turkey} b_{CNRS,ParisSchoolofEconomics,France}

c_{ExeterUniversityDepartmentofEconomics,UnitedKingdom} d_{DEPOCEN,VietNam}

a

r

t

i

c

l

e

i

n

f

o

Articlehistory:

Received25July2009 Receivedinrevisedform 20December2010 Accepted26December2010 Available online 19 February 2011 JELclassification:

C61 D90 O41 Keywords:

Endogenoustimepreference Optimalgrowth

Competitiveequilibrium Multiplesteady-states

a

b

s

t

r

a

c

t

Thispaperstudiesthedynamicimplicationsoftheendogenousrateoftimepreferencedependingon thestockofcapital,inaone-sectorgrowthmodel.Theplanner’sproblemispresentedandtheoptimal pathsarecharacterized.Weprovethatthereexistsacriticalvalueofinitialstock,inthevicinityofwhich, smalldifferencesleadtopermanentdifferencesintheoptimalpath.Indeed,weshowthatadevelopment trapcanariseevenunderastrictlyconvextechnology.Incontrastwiththeearlycontributionsthat considerrecursivepreferences,thecriticalstockisnotanunstablesteadystatesothatifaneconomy startsatthisstock,anindeterminacywillemerge.Wealsoshowthatevenunderaconvex–concave technology,theoptimalpathcanexhibitglobalconvergencetoauniquestationarypoint.Themultipliers systemassociatedwithanoptimalpathisproventobethesupportingpricesystemofacompetitive equilibriumunderexternalityanddetailedresultsconcerningthepropertiesofoptimal(equilibrium) pathsareprovided.Weshowthatthemodelexhibitsgloballymonotonecapitalsequencesyieldinga richersetofpotentialdynamicsthantheclassicmodelwithexogenousdiscounting.

© 2011 Elsevier B.V. All rights reserved.

1. Introduction

Theoptimalcapitalaccumulationmodelshavebeenatthecore

of thetheoryof economicgrowth anddynamics. Based onthe

dynamicconsumptionandsavingdecisionsoftheeconomicagents

driven by intertemporal utilitytrade-offs between current and

futureconsumption,thekeycomponentsofthesemodelsturnout

tobetherateoftimepreferenceandthetechnology.The

classi-caloptimalgrowthmodelsandmuchofthesubsequentliterature

ongrowthfocusontheconvexstructuresofthetechnologyand

preferencesthatguarantee themonotonicalconvergenceof the

sequenceofoptimalstockstowardsauniquesteadystate.Sucha

structure,however,implythatthemodelcannotbeusedto

under-standthedevelopmentpatternsthat differconsiderablyamong

countriesinthelongrun(seeQuah,1996;Barro,1997;Barroand

Sala-i-Martin,1991).

∗ Correspondingauthor.Tel.:+903122901598;fax:+903122665140. E-mailaddresses:selman@bilkent.edu.tr(S.Erol),levan@univ-paris1.fr (C.LeVan),csaglam@bilkent.edu.tr(C.Saglam).

Toaccountforthesenon-convergentgrowthpaths,avarietyof

one-sectoroptimalgrowthmodelsthatincorporatesomedegree

of marketimperfectionsbased ontechnologicalexternaleffects

and increasingreturnshavebeenpresented.Withinamodel of

capital accumulation withconvex–concave technology, Dechert

andNishimura(1983),MitraandRay(1984)havecharacterized

optimal paths and prove the existence of threshold dynamics

that generate developmentor poverty traps(see Azariadis and

Stachurski,2005,forarecentsurvey).Inthesemodels,economies

withlowinitialcapitalstocksorincomesconvergetoasteadystate

withlowpercapitaincome,whileeconomieswithhighinitial

cap-italstocksconvergetoasteadystatewithhighpercapitaincome.

Indeed,theintroductionofincreasingreturnsalsomakesitpossible

forprivatereturnstotheaccumulationofcapitalstocktobe

com-plementarywiththeaggregatestockleadingtoindeterminaciesor

continuumofequilibria(e.g.,BenhabibandPerli,1994;Benhabib

andFarmer,1996;seeNishimuraandVenditti,2006forextensive bibliography).

Ageneraltendencyinthesestudieswithmultiplesteadystates,

indeterminacyorcontinuumofequilibriaisthattheyaremostly

devotedtotheanalysisofthetechnologycomponentleavingthe

timepreferenceessentiallyunalteredwithanexogenouslyfixed

0304-4068/$–seefrontmatter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.jmateco.2010.12.006

geometricdiscounting.Inthispaper,yettheothermechanism,the

endogeneityoftimepreferencewillbeputforwardtoexplain

the-oreticallywhythedifferencesinpercapitaoutputlevelsamong

countriespersistinthelongrun.

Consideringthepreferencesoftheagentstoberecursive,the

earlycontributionsonthetheoryofendogenoustimepreference

postulate that anagent’s discountratedepends onthelevel of

present and future consumption (e.g.,Uzawa, 1968; Lucasand

Stokey,1984;Epstein,1987;Obstfeld,1990).Thesemodelsnearly

exclusivelyassumeanincreasingmarginalimpatiencesothatthe

agentsgetmoreimpatientastheygrowricher.Sucha

specifica-tionensuresstableoptimalcapitalsequencesthatconvergetoa

uniquesteadystateindependentoftheinitialconditions.However,

recentworkboththeoretically(e.g.,BeckerandMulligan,1997;

Stern,2006)andempirically(e.g.,Lawrance,1991;Samwick,1998; Fredericketal.,2002)proposethattheagentsgetmorepatient

astheygrowricherandimposethat thediscount ratedepends

closelyuponthestockofwealth(seeHamadaandTakeda,2009,

forarecentsurvey).Thisisinparalleltotheideathatthestockof

wealthisakeytoreachbetterhealthservicesandbetterinsurance

marketssothatthemorewealthiertheagentgets,themorepatient

itbecomes.Byconsideringdiscountfactorasafunctionof

con-sumption,Mantel(1998)studiestheimpactofdecreasingrateof

timepreferenceonoptimalgrowthpathofaneconomywitha

pri-maryfocusonthemonotonicitypropertiesofoptimalconsumption

andinvestment.Das(2003)derivesasetofsufficientconditions

forstabilityanduniquenessinthesamecontextinacontinuous

timeoptimalgrowthmodel.Stern(2006),alongthelinesofBecker

andMulligan(1997),letindividualsspendresourcestoincrease

theappreciationofthefuture.Heprovidesnumericalexamplesof

multiplesteadystatesandaconditionallysustainedgrowthpath.

In this paperweadapt theclassicoptimalgrowthmodel to

includeanendogenousrateoftimepreferencedependingonthe

stockofcapitalandanalyzetheimplicationsontheequilibrium

dynamics.Todoso,theplanner’sproblemisfirstpresentedandthe

optimalpathsarecharacterized.Weshowthatadevelopmenttrap

canariseevenunderastrictlyconvextechnology.Inotherwords,

weprovethatthereexistsacriticalvalueofinitialstock,inthe

vicinityofwhich,smalldifferencesleadtopermanentdifferences

intheoptimalpath.IncontrastwiththeearlycontributionsbyKurz

(1968),whoassumescapitaldependentpreferences,byBealsand Koopmans(1969),Iwai(1972),whoassumerecursivepreferences,

andmorerecently,byStern(2006),whoassumesendogenoustime

preferencedependingonfutureorientedresources,wenotedthat

thecriticalstockisdefinitelynotanunstablesteadystatesothat

ifaneconomystartsatthisstock,anindeterminacywillemerge.

Ontheotherhand,underasimilarconditiontothatofDechertand

Nishimura(1983),wealsoshowthatevenunderaconvex–concave

technology,theoptimalpathcanexhibitglobalconvergencetoa

uniquestationarypoint.Later,themultiplierssystemassociated

withanoptimalpathis proventobethesupportingprice

sys-temofacompetitiveequilibriumunderexternalityanddetailed

resultsconcerningthepropertiesofoptimal(equilibrium)paths

areprovided.Weshowthatthemodelexhibitsgloballymonotone

capitalsequencesyieldingarichersetofpotentialdynamicsthan

theclassicmodelwithexogenousdiscounting.

Thekeyreasonofourresultsisthatweconsiderthattherateof

timepreferencedecreaseswiththestockofwealthincontrastwith

exogenouslyfixeddiscountfactorassumedinstandardmodels.In

particular,thediscountfactorattributedtotheutilityof

consump-tionatperiodtincreaseswiththelevelofcapitalstockavailable

forproductionatperiodt.Accordingly,thelowerthestockof

cap-ital,thehigherthesacrificeofpostponingpresentconsumptionin

exchangeforfutureconsumption.InlinewithBeckerandMulligan

(1997),Stern(2006),animmediateimplicationofthisassumption

isthatastheagentgrowswealthier,thelevelofinvestmentin

capi-talthatwillbeappreciatedinthefutureincreasesandthereforethe

agentwillbecomemorepatient.Indeed,recentempiricalstudies

usingexperimentaldataalsosupportthevalidityofthishypothesis

ofwealthcausespatience(e.g.,Harrisonetal.,2002;Ikedaetal.,

2005).

Incorporatingsuchanhypothesis,animportantaspectofour

modelisthatwhileallowingvariationintherateoftime

prefer-ence,atthesametimeitmaintainstimeconsistency.Aremarkable

featureofouranalysisisthatourresultsdonotrelyon

particu-larparameterizationoftheexogenousfunctionsinvolvedinthe

model,rather,itprovidesamoreflexibleframeworkinregardsto

thediscountingoftime,keepsthemodelanalyticallytractableand

usesonlygeneralandplausiblequalitativeproperties.

Therestofthepaperisorganizedasfollows.Section2describes

themodelandprovidesthedynamicpropertiesofoptimalpaths.

Section3presentstheexistenceofacompetitiveequilibriumwith

externalityandstudiestheequilibriumdynamics.Finally,Section

4concludes.

2. Model

Themodeldiffersfromtheclassicoptimalgrowthmodelby

theassumptions ondiscounting. Ratherthanassumingthatthe

levelofdiscountonfutureutilityisanexogenousparameter,we

assumethatitisendogenousdependingonthepathofcapitalstock.

Formally,themodelisstatedasfollows:

max{c_t,x_t+1}∞_t=0 ∞



t=0



_t



s=1 ˇ(xs)



u(ct), subjectto

∀

t,ct+x_t+1≤f(xt),

∀

t,ct≥0,xt≥0, x0≥0,given,

Wemakethefollowingassumptionsregardingthepropertiesof

thediscount,utilityandtheproductionfunctions.

Assumption1. ˇ:R+→R++iscontinuous,differentiable,

stric-tly increasing and satisfies supx>0ˇ(x)=ˇm<1, supx>0ˇ

(x)<+∞.

Assumption2. u:R+→R+ is continuous,twicecontinuously

differentiableandsatisfiesu(0)=0.Moreover,itisstrictly

increas-ing,strictlyconcaveandu(0)=+∞(Inadacondition).

Assumption3. f :R+→R+ is continuous, twicecontinuously

differentiableandsatisfiesf(0)=0.Moreover,itisstrictlyincreasing

andlimx→+∞f(x)<1.

Foranyinitialconditionx0≥0,whenx=(x0,x1,x2, ...)issuch

that0≤xt+1≤f(xt)forallt,wesayitisfeasiblefromx0andtheclass

ofallfeasibleaccumulationpathsisdenotedby(x0).Itmaybe

easilyverifiedthatifx0<x0,then(x0)⊂(x0).Aconsumption

sequencec= (c0,c1,...) isfeasiblefromx0≥0whenthereexists

x∈(x0)with0≤ct≤f(xt)−xt+1.

Notingthattheconstraintswillbebindingattheoptimumas

theutilityandthediscountfunctionsarestrictlyincreasing,we

introducethefunctionUdefinedonthesetoffeasiblesequencesas

U(x)= ∞



t=0



_t



s=1 ˇ(xs)



u(f(xt)−xt+1). (1)

ThepreliminaryresultsaresummarizedinthefollowingLemma

whichhasastandardproofusingTychonovtheorem(seeLeVan

Lemma1. Let ¯x bethegreatestpointx≥0suchthatf(x)=x.Then,

(a)Forany x∈(x0), we have xt≤A(x0) for all t, where A(x0)=

max



x0, ¯x



.

(b)(x0)iscompactintheproducttopologydefinedonthespaceof

sequencesx.

(c)Uiswelldefinedanditiscontinuousover(x0)withrespectto

theproducttopology.

Itisnowclearthattheinitialoptimalgrowthmodelisequivalent

to:

max{U(x)|x∈(x0)}.

2.1. Existenceofoptimalpaths

Theexistence ofan optimalpath followsfromthefact that

(x0)iscompactfortheproducttopologydefinedonthespace

ofsequencesxandUiscontinuousforthisproducttopology.The

positivityofoptimalconsumptionandcapitalstockpathsfollow

fromtheInadacondition.

Proposition1.

(i)Thereexistsanoptimalpathx.Theassociatedoptimal

consump-tionpath,cisgivenby

ct=f(xt)−xt+1,

∀

t.

(ii) Ifx0>0,everysolution(x,c)totheoptimalgrowthmodelsatisfies

ct>0,xt>0,

∀

t. (2)

Proof. Itiseasy. 

2.2. Valuefunction,Bellmanequation,optimalpolicy

Inordertocharacterizethebehavioroftheoptimalpaths,we

willproceedbydefiningthevaluefunctionandanalyzingthe

prop-ertiesoftheoptimalpolicycorrespondence.ThevaluefunctionV

isdefinedby:

∀

x0≥0,V(x0)=max{xt+1}∞t=0



_∞



t=0



_t



s=1 ˇ(xs)



u(f(xt)−x_t+1)|

∀

t,0≤x_t+1≤f(xt),x0≥0,given



. (3)

Theboundsondiscountingtogetherwiththeexistenceof

maxi-mumsustainablecapitalstockguaranteeafinitevaluefunction.

UndertheAssumption(1)andtheAssumption(2),onecan

imme-diately show that the value function is non-negative, strictly

increasingandcontinuous(seeStokeyetal.,1989).Giventhese,

Bellmanequationfollows.

Proposition2.

(i) V(0)=0,andV(x0)>0ifx0>0.Ifxisanoptimalpath,then

V(x0)= ∞



t=0



_t



s=1 ˇ(xs)



u(f(xt)−x_t+1).

(ii)Visstrictlyincreasing.

(iii)Viscontinuous.

Proposition3.

(i)VsatisfiesthefollowingBellmanequation:

∀

x0≥0,V(x0)=max



u(f(x0)−x)+ˇ(x)V(x)|0≤x≤f(x0)



. (4)

(ii) Asequencex_∈(x0)isanoptimalsolutionifandonlyifitsatisfies:

∀

t,V(xt)=u(f(xt)−xt+1)+ˇ(xt+1)V (xt+1). (5)

Proof. SeeLeVanandDana(2003). 

Theoptimalpolicycorrespondence,:R+→R+isdefinedas

follows: (x0)=argmax



u(f(x0)−y)+ˇ(y)V(y)|y∈[0,f(x0)]



.

It is important to note that although the utility function is

strictlyconcave,thesolutionmaynotbeuniqueasthe

multiplica-tionofdiscountfunctiondestroystheconcavestructureneededfor

uniqueness.Thenon-emptinessandtheclosednessoftheoptimal

correspondenceanditsequivalencewiththeoptimalpathfollow

easilyfromthecontinuityofthevaluefunctionbyastandard

appli-cationofthetheoremofthemaximum.

Proposition4.

(i)(0)={0}.

(ii)Ifx0>0andx1∈(x0),then0<x1<f(x0).

(iii)isclosedandhenceuppersemi-continuous.

(iv)Asequencex∈(x0)isoptimalifandonlyifxt+1∈(xt),

∀

t.

(v)The optimalcorrespondence isincreasingsothat if x0<x₀,

x1∈(x0)andx1∈(x0)thenx1<x1.

Proof. (ii)Followseasilyfrom(2).(iii)FollowsfromtheTheorem

ofMaximum.(iv)Followsfrom(5).(v)SeeDechertandNishimura

(1983),orAmiretal.(1991). 

Theincreasingnessofiscrucialfortheconvergenceofoptimal

paths,hencefortheanalysisofthelong-rundynamics.In Stern

(2006),assuminganendogenous timepreference dependingon

thefuture-orientedresources,theincreasingnessofhasbeen

provenbyusingthestrictconcavityofˇ.Sucharestrictiononthe

curvatureofthediscountfunctionisnotnecessaryinoursetup.

Moreover,wehavealsoproventhattheoptimalcorrespondence,

isnotonlyclosedbutalsouppersemi-continuous.

Withthepositivityoftheoptimalconsumptionandthestockof

capital,Eulerequationeasilyfollows.

Proposition5. Whenx0>0,anysolutionx∈(x0)satisfiestheEuler

equation:

∀

t,u_(f(xt)−x

t+1)=ˇ(xt+1)u(f(xt+1)−xt+2)f(xt+1)

+ˇ_(x

t+1)V (xt+1) (6)

Proof. Recallthat0<xt+1<f(xt),

∀

t.Takeanyn,andconsiderthe

pathofcapitalxy_{definedasfollows:x}y

t =xt,

∀

t /=n+1andxy_n+1=

y∈Y,whereY=

f−1_(xn+2),f(xn)

_{.Notethatitisawelldefined}

openinterval includingxn+1 becausext+1<f(xt)and xt+2<f(xt+1).

Bychoosingy∈Y,weguaranteethaty<f(xt)andxt+2<f(y)sothat

xy_∈(x

0).FromtheoptimalityofxwehaveU(x)≥U(xy),

∀

y∈Y.

Let(y)=U(xy_{).Then(y)ismaximizedatzeroinY,which}

sug-geststhat_{(0)=0.Withsomealgebra,thisendstheproof. }

Inastandardoptimalgrowthmodelwithgeometricdiscounting

andtheusualconcavityassumptionsonpreferencesand

theproperties oftheoptimalpath iseasilyfoundby usingthe

firstorderconditionstogetherwithenvelopetheoremby

differ-entiatingthevaluefunction.However,inourmodel,theobjective

functionincludesmultiplicationofadiscountfunction.This

gen-erallydestroystheusualconcavityargumentwhichisusedinthe

proofofthedifferentiabilityofvaluefunctionandtheuniquenessof

theoptimalpaths(seeBenvenisteandScheinkman,1979;Araujo,

1991).

Tothisend,thenextpropositionshowsthatthevaluefunction

isdifferentiablealmosteverywhereandthereexistsaunique

opti-malpathfromalmosteverywherewithoutanyassumptiononthe

curvatureofˇ.

Proposition6.

(i)LeftderivativeofVexistsateveryx0>0andV−(x0)=u(f(x0)−

(x0))f(x0),where␪(x0)=min(x0).

(ii)RightderivativeofVexistsateveryx0>0andV+(x0)=u(f(x0)−

(x0))f(x0),where(x0)=max(x0).

Proof. SeeAskriandLeVan(1998);LeVanandDana(2003)forthe

proofsbasedontheClarkegeneralizedgradientsandthestandard

optimalgrowthtools,respectively. 

Nowwe are abletoseethe relationbetween

differentiabil-ityofthevaluefunctionandtheuniquenessoftheoptimalpath.

Noticethat,givenx0,ifoptimalx1isuniquesothat(x0)is

single-valuedand␪(x0)=(x0)=x1,fromtheaboveproposition,wesee

thatV

−(x0)=V₊(x0),henceVisdifferentiableatx0,andviceversa.

Thisanalysiscanbegeneralizedtoanyperiodtotherthanzero.

Theseallowustoclaimthattheoptimalcorrespondenceissingle

valuedanddifferentiablealmosteverywhere.

Proposition7.

(i)Ifxisanoptimalpathfromx0,thenVisdifferentiableatanyxt,

t≥1.Ifxisanoptimalpathfromx0,thereexistsauniqueoptimal

pathfromxtforanyt≥1.

(ii)Visdifferentiableatx0>0ifandonlyifthereexistsaunique

optimalpathfromx0.

(iii) Visdifferentiablealmosteverywhere,i.e.,the optimal pathis

uniqueforalmosteveryx0>0.

(iv)isdifferentiablealmosteverywhere.

Proof. SeeLeVanandDana(2003). 

Wehaveprovedthattheoptimalcorrespondenceissingle

valuedanddifferentiablealmosteverywhere.Inadditiontothis,

wehavealsoshownthatthereexistsauniqueoptimalpathfrom

almostanyinitialcapitalstock.Theseresultswillprovetobecrucial

inanalyzingthedynamicpropertiesoftheoptimalpaths.

2.3. Dynamicpropertiesoftheoptimalpaths

Apointxs_{isanoptimalsteadystateifx}s_{= (x}s_),sothatthe

stationary sequence xs= (xs_,xs_,...,xs_{,...) solves the problem:}

max{U (xs_{) :x}s_{∈ (x}s_)}.Ifxs_{isdifferentfromzero,thenthe}

asso-ciatedoptimalsteadystateconsumptionmustbestrictlypositive

fromInadacondition.Hence,fromEulerequation(6),thissteady

statewillsolve:

u_{(f(x)−x)=ˇ(x)u}_(f(x)−x)f_(x)+ˇ_{(x)V (x). (7)}

ByProposition3,weknowthatthestationaryplaneveryperiod

equaltoxs_{isoptimalfromx}s_{ifandonlyifitsatisfies}

V

xs

_=u(f(xs_)−xs_)+ˇ(xs_)V

_xs

_{. (8)}

Followingfrom(7)and(8),thissteadystatewillsatisfy:

u(f(x)−x)=[1−ˇ(x)][1_ˇ −ˇ(x)f(x)]

(x) u

_{(f(x)−x). (9)}

Wewillnowprovethattheendogenousrateoftimepreference

preservesthemonotonicityoftheoptimalpathsandprovidethe

conditionunderwhichtheconvergencetoanoptimalsteadystate

isguaranteed.

Proposition8. Theoptimalpathxfromx0ismonotonic.

Proof. Sinceisincreasing,ifx0>x1,wehavex1>x2.By

induc-tion,xt>xt+1,

∀

t.Ifx1>x0,usingthesameargumentyieldsxt+1>xt,

∀

t.Nowifx1=x0,thenx0∈(x0).Recallthattheoptimalpathis

uniqueaftert=1.Sincex0∈(x0),xt=x0,

∀

t. 

Itisimportanttonotethatthemonotonicityoftheoptimalpaths

hasbeenprovedwithoutanyassumptiononthecurvatureof

nei-thertheproductionnorthediscountfunction.Itisalready well

knownthatinmulti-sectornonclassicaloptimalgrowthmodels,

onecaneasilyrefertolatticeprogrammingandTopkistheoremin

ordertoprovethattheoptimalpathsaremonotoniciftheplanner’s

criterionfunctionissupermodular(seeAmiretal.,1991).However,

followingthisapproachinamodelwithtimepreferencedepending

onthefuture-orientedresources,Stern(2006)assumesastrictly

concavediscountfunction.

Asamonotonerealvaluedsequencewilleitherdivergeto

infin-ityorconvergetosomerealnumber,thefactthatoptimalcapital

sequencesare monotone proves tobe crucial in analyzing the

dynamicpropertiesandthelong-runbehaviorofourmodel.

Proposition9. Thereexistsanε>0suchthatifsupx>0f(x)<((1−

ε)/ˇm),anyoptimalpathconvergestozero.

Proof. Lettheoptimalpathxconvergetox>0.Then,by

consid-eringtheEulerequationast→∞,weobtain

u(f(x)₋x) u(f(x)−x) = [1₋ˇ(x)][1−ˇ(x)f_(x)] ˇ_(x) ≥ 1_{− ˇ}m ˇ sup



1−ˇmsup x>0f _(x)



, where ˇ

sup≡supx>0ˇ(x). Note that (u/u)=(((u)2−

u.u_)/(u₎2_{)>0,i.e.,u/uisincreasing.LetB=argmax}

x>0(f(x)−x).

Inaccordancewiththese,wehave:

u(f(B)−B) u_{(f(B)−B) ≥} u(f(x)−x) u_{(f(x)−x) ≥} 1−ˇm ˇ sup



1−ˇmsup x>0 f_(x)



.

Defining ε=(ˇ_sup /(1−ˇm))((u(f(B)−B))/(u(f(B)−B))),

supx>0f(x)≥(1−ε/ˇm) when the optimal path converges to

x>0.Thus,supx>0f(x)<(1−ε/ˇm)impliesthattheoptimalpath

convergestozero. 

Inaclassicoptimalgrowthmodel,iff(0)≤1/ˇ,thenthe

opti-malpathconvergestozero.Asweconsidertheendogenousnature

oftimepreference andallowfornonconvexities inthe

produc-tiontechnology, we have slightlydeviated from this condition.

However,itmustbenotedthattheintervalforsupx>0f(x)that

guaranteestheconvergencetozerowillalwaysbeasubsetof(0,

1/ˇm)andthiscannotbeimprovedinanessentialway.

Wewillnow presenttheconditionunderwhich the

conver-gencetoanoptimalsteadystateis guaranteedand analyzethe

behaviorof theoptimalpathswhen we would have uniqueor

multipleoptimalsteadystates.

Proposition10. Assumex0>0.Letinfx>0ˇ(x)=ˇ.Iff(0)>1/ˇ,

thentheoptimalpathconvergestoanoptimalsteadystatexs_>0.

Proof. Sinceˇ(·)isincreasing,ˇ=ˇ(0).Notethat[fˇ](·)isa

pathanditconvergestozero.Since[fˇ](·)iscontinuous,there

existsNsuchthatn>Nimpliesf(xn)ˇ(xn)>1.

IfxisanoptimalpaththenitsatisfiestheEulerequation.Hence,

foreveryt,inparticular,fort≥N,wehave:

u_(f(xt)−x t+1) =u(f(xt+1)−xt+2)ˇ(xt+1)f(xt+1) +ˇ_(x t+1)V (xt+1) >u_{(f(xt+1)−xt+2)ˇ(xt+1)f}_(xt+1) >u_(f(x t+1)−xt+2), implyingthat f(xt)−xt+1<f(xt+1)−xt+2.

Then,weobtainthat:

0<f(xN)−xN+1<lim

y→0[f(y)−y] =0,

leadingtoacontradiction.

Sincethesequencexismonotonicandbounded,itconvergesto

somexs_>0.Nowx

t→xs,xt+1∈(xt),xt+1→xs.Hence,bytheupper

semi-continuityofthecorrespondence,wegetxs_∈(xs_{). }

Withconvextechnologyand exogenouslyfixedtime

prefer-encerateinastandardoptimalgrowthmodel,iff(0)>1/ˇthen

thereexistsaunique(non-zero)steadystatetowhichalloptimal

pathsconvergeindependentlyoftheirinitialstate.Inourmodel,

duetotheexistenceofthemaximumsustainablelevelofcapital

stock,whenf_{(0)>1/ˇ,theoptimalpathconverges}

monotoni-callytowardsanoptimalsteadystate.Thisconditioncanberecast

asF(0)>r+ıbydefiningtherateofinterestrby1/(1+r)=ˇand

thefunctionfbyf(k)=F(k)+(1₋ı)kwhereFistheproduction

func-tionand ıistherateof depreciationofcapital. Accordingly,in

ourmodel,whenthecostofinvestmentislow,theoptimalpath

convergestoanoptimalsteadystate.

Wewillnowconsidertheoptimalpathdynamicsinthelong

runandshowthatourmodelcansupportuniqueoptimalsteady

statewithglobalconvergenceandmultiplicityofoptimalsteady

stateswithlocalconvergence.Indeed,wewillshowthatourmodel

exhibitsglobalconvergenceevenunderconvex–concave

technol-ogyandmultiplicityofoptimalsteadystatesevenunderconvex

technology.

Case1(Globalconvergence). Assumex0>0.Letinfx>0ˇ(x)=ˇand

f_{(0)>1/ˇ.Considerthecasewherethereexistsauniquesolution}

xs_{to(9).ByProposition10,weknowthattheoptimalpaths}

can-notconvergeto0.Then,anyoptimalpathconvergestotheunique

optimalsteadystatexs_{,irrespectiveoftheirinitialstate.}

Whenthereexistsauniquesolutionto(9),itisclearfromabove

thatitwillbeanoptimalsteadystate.However,whenthereexist

multiplesolutionstothesteady-stateequation(9),thestationary

sequencesassociatedwiththesesteadystatesmayormaynotbe

optimal.Anaturalquestionisthenwhetherthesesolutionswill

constitutea long-runequilibrium intheoptimalgrowthmodel

andifso,willthatenableustoshowwhetheraneconomywith

wealthdependentendogenousdiscountingcanexhibitthreshold

dynamicsevenunderconvextechnologyornot.Devotedtothis

end,first,wewillconsiderthecasewherewehaveexactlytwo

optimalsteadystates,forthesakeofsimplicity.Weshallprovethat

thiswouldimplytheexistenceofacriticalstockandlocal

conver-gence.Second,wewillshowthatourmodelcanindeedsupport

twooptimalsteadystatesinanexample.

Case2(Localconvergence). Assumex0>0.Letinfx>0ˇ(x)=ˇand

f_{(0)>1/ˇ.Considerthecasewherewehaveexactlytwooptimal}

steadystates,xl<xh,forthesakeofsimplicity.Supposethatxhis

unstablefromtheright.Giventheexistenceofamaximum

sustain-ablecapitalstock,asanyoptimalpathfromx0>xhhastoconverge

toanoptimalsteadystate,therewillbeanothersteadystatelarger

thanxh,acontradiction.Hence,xhisstablefromtheright.Itisalso

impossibletohavexlunstablefromtheleft,sincetheoptimalpaths

cannotconvergeto0.Thesealreadyimplytheexistenceofacritical

stockandtheemergenceofthresholddynamics.

Whenever both of theoptimal steady states turn out tobe

saddle-pointstable,itisclearthattherewillexistacriticalstock

ofcapitalbelowwhichtheoptimalpathwillconvergetoxland

abovewhichtheoptimalpathwillconvergetoxh.However,xhcan

bestablefromtherightbutunstablefromtheleft,andsimilarlyxl

canbestablefromtheleftbutunstablefromtheright.Thesecases

canoccuronlywhentheoptimalpolicyistangenttoy=xlineatxh,

andxl,respectively.Itisimportanttonotethatevenunderthese

casesthresholddynamicsemerge.Thefollowingproposition

pro-videstheformalanalysisofthelocalconvergencewhenwehave

twooptimalsteadystates.

Proposition11. Assumex0>0.Letinfx>0ˇ(x)=ˇandf(0)>1/ˇ.

Supposethereareexactlytwooptimalsteadystates.Letthehighand

thelowoptimalsteadystatesbe,respectivelyxhandxl.Thenthere

existsxc∈[xl,xh]suchthatanyoptimalpathxstartingfromx0,

con-vergestoxlifx0<xc,andconvergestoxhifx0>xc.

Proof. Take anyinitialcapital levelsy<z, and let y,z beany

two correspondingoptimalpaths.Since theoptimal

correspon-dence is increasing,weiteratively obtainyt<zt for allt. Hence

limt→∞yt≤limt→∞zt.Thus,ifzconvergestoxl,sodoesy.Ify

con-verges toxh, sodoesz. Let X1={x|there existsan optimalpath

fromxthat converges toxl}, x∗1=supX1, and X2={x|there exists

anoptimalpathfromxthatconvergestoxh},x∗2=infX2.Fromthe

above argument, x∗

1≤x2∗. Suppose otherwise, then there exists

small

1

,

2

≥0suchthatx∗

1−

1

∈X1,x2∗+

2

∈X2,andx1∗−

1

>

x∗

2+

2

.However,thiscontradictswiththeabove,bysettingx∗2+

2

=y<z=x∗

1−

1

.Nowwewanttoshowthatx∗1=x2∗.Suppose

not,thenthereexistsx∗

3suchthatx1∗<x3∗<x∗2.Butthenan

opti-mal path from x∗

3 can neither converge toxl norto xh by the

definitionsofx∗

1 andx∗2.Therefore,x∗1=x2∗.Definexc=x1∗=x∗2.If

x0<xc=x₁∗,takesomesmall



₁≥0withx0<x∗₁−



₁∈X1.Bythe

aboveargument,anyoptimalpathfromx0 convergestoxl.

Simi-larly,ifx0>xc=x∗₂,takesomesmall



₂ ≥0withx0>x2∗+



2∈X2,

which wouldshowthatanyoptimalpathfromx0 convergesto

xh.Noticethatxl∈X1andxh∈X2.Therefore,xl≤supX1=x∗₁=xc=

x∗

2=infX2≤xh,i.e.,xc∈[xl,xh]. 

Remark1. Asithasbecomeclearbynow,ifisnottangentto

y=xlineateitherxlorxh,botharelocallystableandxc∈(xl,xh).

Notethatifisnottangenttoy=xlineateitherx_lorx_h,itmustbe

thecasethatjumpsovery=xlineatxc.Inthiscase,xcisnotan

unstableoptimalsteadystatebutagenuinecriticalpointleading

tothethresholddynamics.

Ifistangenttoy=xlineatxl,itcanonlybefromabovethe

y=xlinebecausexlisstablefromtheleft.Inthiscasewegetxc=xl.

Similarly,ifistangenttoy=xlineatxh,itcanonlybefrombelow

they=xlineandwegetxc=xh.

Inordertoprovideabetterexpositionofouranalysis,following

Stern(2006),wewillspecifyfunctionalformsinanexampleand

showthatourmodelcanindeedsupporttwooptimalsteadystates

Fig.1.Optimalpolicyafter300iterationsontheinitialzerovaluefunction.

Fig.2. Lowsteadystate=0.0012isoptimal.

Example1. Supposethat

u(c) =₁c1− ₋ , f(x) =Ax˛_+(1−ı)x, ˇ(x) =−e−(x+)ε , where 0<{A,},0<{˛, ,ε}<1,0<<eε ,ande−ε<<

1.Checkthatf,u,andˇsatisfytheassumptionsets.Weemploy

thefollowingsetoffairlystandardcoefficients:

A=0.5, ˛=0.3, ı=0.15, =0.5, =0.97, =2.5,

=1, ε=0.9.

ItturnsoutthatthemaximumsustainablecapitalstockisA (x0)=

max



x0, ¯x



,where ¯xis5.58431,andthereexistthreesolutionsto

(9).Theprecisevaluesare,xl=0.0012,xm=0.5865,andxh=3.8105.

Inordertodeterminewhichoftheseareactuallytheoptimalsteady

states,weanalyzetheoptimalpolicybymakinguseofBellman’s

operator.Fig.1showstheoptimalpolicyforiterationsofthe

Bell-manoperatoronthezerofunctionandindicatesthatxlandxhare

stableoptimalsteady states.Figs. 2–4present thedetailed

pic-turesoftheoptimalpolicyintheneighborhoodofthexl,xmand

xh,respectively.

Atfirstsight,onemightthinkthatthemiddlesteadystatexmis

anunstableoptimalsteadystateasitissurroundedbytwostable

optimalsteadystates.IncontrastwithStern(2006),eventhough

xmisasolutionofthestationaryEulerequation(9),Fig.1strongly

indicatesthatitisnotanoptimalsteadystate.Indeed,ifitwere,

Fig.3.Middlesteadystate(xm=0.586505)isnotoptimal!

Fig.4.Highsteadystate(xh=3.81057)isoptimal.

wouldhavetocrossy=xlineatxm.Moreover,weseethatthereis

genuinecriticalpointatxc=0.6548.Itisclearlynotanoptimalor

nonoptimalsteadystatebecausethenitwouldhavetosatisfy(9).

Asxcisnotanoptimalsteadystate,cannotcrossy=xlineat

xc,i.e.,xc∈/(xc).Ontheotherhand,Fig.1showsthatthegraphof

jumpsovery=xlineatxc.Asisuppersemicontinuous,itmustbe

thecasethatthereexistx

1∈(0,xc)∩(xc)andx1∈(xc,∞)∩(xc).

Therefore,xcisacriticalpointwhichisnotanunstablesteadystate.

Foranyinitialcapitalstocklevellowerthanxc,thesystemwill

faceadevelopmenttrap,enforcingconvergencetoaverylow

cap-itallevelxl,evenunderastrictlyconcaveproductionfunction.On

theotherhand,foranyinitialcapitallevelhigherthanxc,the

opti-malpathwillconvergetoxh.However,ifaneconomystartsatxc,

anindeterminacywillemerge:thesystemcanoptimallyfollowx

1,

andatthesametimeitcanoptimallyfallintoadevelopmenttrap

followingx

1.

Theexistenceofcriticalvalueisactuallyrecognizedsincethe

papersbyClark(1971);MajumdarandMitra(1982);Dechertand

Nishimura(1983)indiscretetimeandSkiba(1978)andAskenazy and Le Van (1999) in continuous time horizon. These studies

are mostly devoted to the analysis of the technology

compo-nent leaving thetime preference essentially unaltered withan

exogenouslyfixedgeometricdiscounting.Theyassumeaspecific

convex–concave technology under which the low steady state

turnsouttobeunstableandhighsteadystateturnsouttobe

sta-blesothatanoptimalpathconvergeseithertozeroortothehigh

steadystate.However,weshowthatevenunderstrictlyconcave

productionfunction,theeconomycanexhibita“trap”sothata

smalldifferenceswillleadtopermanentdifferencesintheoptimal path.

In this respect,ouranalysisis in linewiththeearly

contri-butionsthat emphasizetheimportanceof thesubjective utility

functionandreleasetheoptimalgrowththeoryfromthealmost

takenforgrantedRamsey’sadditiveutilityfunction(e.g.,Bealsand

Koopmans,1969;Iwai,1972).Baseduponthestationaryordinal

utility function that implies a broad class of preference

struc-turesthatarerecursiveandcontainthetime-additivepreference

asa specialcase, theyshowtheexistenceofa criticalstock so

thattheinitialconditionscanaffectthelong-run optimalpath.

Adoptingastrictlyconcavediscountfunctiondependingonthe

future-orientedresources,Stern(2006)analyzesaseriesof

numer-icalexamplesthatalsoexhibitmultiplicityofsteadystateswith

merelocalconvergence.Besidesthese,Kurz(1968)introduces

cap-italasanargumentintheRamseytypeutilityfunctionandshows

alsothattheremayexistmultipleturnpikes.

Inallof thesestudiesmentioned above,thecritical stockof

capitalbelow which theoptimalprogram leadsonlytoforever

diminishingcapitalandgraduallyconvergingtodevelopmenttrap

isanunstableoptimalsteadystate.Ifitsohappensthatthe

ini-tiallevelofcapitalstockequalsthecriticalstock,theoptimalpath

willremainonthatgeneralizedturnpikeforever.However,inour

modelthecriticalstockisdefinitelynotanunstablesteadystateso

thatifaneconomystartsatthisstock,anindeterminacyemerges.

Comparedtotheoptimalgrowthmodelswithexogenoustime

pref-erence,this introducesafundamental differenceintheoptimal

pathdynamics.

3. Competitiveequilibriumwithexternality

Fromnowon,wewillassumethattheproductionfunctionfis

strictlyconcave.Wefirstdefinetheconceptsofequilibriumwith

externalityand competitiveequilibrium. Suppose weare given

asequenceofcapital ˜x =(x0, ˜x1,..., ˜xt,...)∈ [0,A(x0)]∞andthe

associatedsequenceofdiscountfactors ˜ˇ=( ˜ˇ1,..., ˜ˇt,...)where

˜

ˇt=ˇ (˜xt) ,

∀

t≥1. Giventhis fixedsequence ˜ˇ∈

ˇ,ˇm



∞,

con-siderthefollowingproblem:

maxx ∞



t=0



_t



s=1 ˜ ˇs



u (f(xt)−x_t+1) subjectto

∀

t,0≤x_t+1≤f(xt), x0>0,given.

Let the solutionto theabove problem bex=(x0, x1, ...,xt,

...).Itdependson ˜ˇ,hence ˜x.Wewritex= ( ˜ˇ)= (ˇ(˜x)),and

hencex=(˜x).Anequilibriumwithexternalityassociatedwith

x0isasequenceofcapitalstockx∗=(x0,x₁∗,...,x∗t,...)suchthat

x∗=˚(x∗_{).Alistofsequences(x}∗_,c∗_,p∗_,q∗_{)isacompetitive}

equilib-riumwithexternalityofthiseconomyifthefollowingaresatisfied:

(a) c∗∈ ∞

+,x∗∈ ∞+,p∗∈( 1+\{0}),q∗∈R++,

(b)c∗solvestheconsumer’sproblem:

maxc ∞



t=0



_t



s=1 ˇ(x∗ s)



u(ct) subjectto ∞



t=0 ∗ p tct≤q ∗_x 0+∗

where∗_{isthemaximumprofitofthefirm,}

(c)x∗solvesthefirm’sproblem:

∗_=maxx ∞



t=0 p∗ t(f(xt)−xt+1)−q∗x0 subjectto

∀

t≥0,0≤x_t+1≤f(xt), x0>0,given,

(d)themarketsclearateveryperiod:

∀

t≥0,c∗

t +x∗_t+1=f(xt∗),x∗0=x0.

Thepurposeistoprovetheexistenceofacompetitive

equi-libriumofthisintertemporaleconomy.Inordertodoso,wewill

firstprovethatthereexistsafixedpointof_˚,namelyan

equi-librium.Later,underanadditionalassumption,wewillshow

thatthisequilibriumisindeedacompetitiveequilibriumofthe

intertemporaleconomywithendogenoustimepreference.

Thevaluefunctionassociatedwiththeproblem(PE)takesthe

followingform: V(x0,ˇ1,ˇ2,ˇ3,...)=maxx ∞



t=0



_t



s=1 ˇs



u(f(xt)−xt+1) subjectto 0≤x_t+1≤f(xt), x0>0,given.

VsatisfiestheBellmanequation:

V(x,ˇ_t+1,ˇ_t+2,..)=maxy ∈[0, f (x)]

u(f(x)−y)+ˇ_t+1V(y,ˇ_t+2,ˇ_t+3,..)



.

Proposition12. ThefunctionViscontinuouswiththetopologyin

Rforx0andtheproducttopologyforˇ∈[ˇ,ˇm]∞.Moreover,itis

strictlyconcaveinx0. Proof. Let U(x0,ˇ,x)= ∞



t=0



_t



s=1 ˇs



u (f(xt)−x_t+1)

wherex∈(x0).OnecanshowthatUiscontinuousin(x0,ˇ,x)for

thetopologyinRforx0andtheproducttopologyfor(ˇ,x).Itiswell

knownthat(·)isacontinuouscorrespondencefromR+intothe

spaceofsequencesendowedwiththeproducttopology(see,for

instance,LeVanandDana,2003).Since

V(x0,ˇ)=max



U(x0,ˇ,x)|x∈(x0)



,

fromthemaximumtheorem,Viscontinuous.Thestrictconcavity

ofVwithrespecttox0followsfromthestrictconcavityoffandu.



Lemma2. Themapˇ iscontinuouswithrespecttotheproduct topology.

Proof. Itisstandard. 

Proposition13. ˚iscontinuouswithrespecttotheproduct topol-ogy.

Proof. Fromthemaximumtheorem,thesolutionx= (ˇ)isupper

semi-continuouswithrespecttoˇ.Asthesolutionto(PE)isunique,

itisacontinuousfunctionwithrespecttoˇ.Since ˜ˇ=ˇ(˜x),themap

Toattainanequilibrium,theinitialsequenceofdiscountinghas

tobeconsistentwiththelevelthatis assumedwhentheagent

makeshersingledecisions.Thissuggeststhatthefixedpointsof˚

arecandidatesforcompetitiveequilibria.

Proposition14. ˚hasafixedpoint.

Proof. ˚isacontinuousmappingfrom[0,A(x0)]∞into[0,A(x0)]∞.

Sincethedomainiscompactfortheproducttopologyandalsoa

convexset,Schaudertheoremconcludesthatthereexistsx∗_∈[0,

A(x0)]∞whichsatisfiesx∗=(x∗). 

Proposition15. If(x0,x∗₁,x∗₂...)isanequilibriumwithexternality

associatedwithx0,then(x1∗,x2∗,x3∗...)isanequilibriumwith

exter-nalityassociatedwithx∗

1.

Proof. LetV(x,ˇt+1,ˇt+2,..)bethevaluefunctionatperiodt+1.

Foranytwehave

V(x∗

t,ˇt+1,ˇt+2,..) =maxy∈[0,f(x)]

u(f(x)−y)+ˇ_t+1V(y,ˇ_t+2,ˇ_t+3,..)



=u(f(x∗t)−x∗_t+1)+ˇt+1V(x∗t+1,ˇt+2,ˇt+3,..)

Particularlyfort≥2,weseethat(x∗₁,x∗

2,...)isanequilibrium

withexternalityassociatedwithx∗

1. 

Proposition16. Anyequilibriumwithexternalityx∗associatedwith

someinitialcapitalstocksatisfiestheEulerequation:

u_(f(x∗

t)−x∗_t+1)=u(f(x∗_t+1)−x∗_t+2)ˇ(x∗_t+1)f(x_t+1∗ ). (10)

Proof. Itiseasy. _

Fortherest,weassumethatthefunction[ˇf](·)isdecreasing

and[ˇf](0)>1.Notethatlimx→∞[ˇf](x)<1.

Proposition17. Theequilibriumwithexternalityassociatedwithx0

isunique.

Proof. Taketwoequilibriawithexternality(x0,x1,x2,...)and

(x0,x1,x2,...)associated withx0.Supposethatx1>x1.Let the

associatedconsumptionpathsbe(c₀,c

1,...)and(c0,c1,...).

First,wewillprovebyinductionthatx

t>xt,forallt≥1and

c

t<ct for all t. Trivially, c0 =f(x0)−x1<f(x0)−x1=c0. Then

by (10), u_(c

1)[ˇf](x1)=u(c0)>u(c0)=u(c1)[ˇf](x1) which

impliesu_(c

1)>u(c1)as[ˇf]isadecreasingfunction.Hence,c1<

c

1.

Now suppose x

t>xt and ct<ct for some t. f(xt)−

x

t+1=ct<ct =f(xt)−xt+1, so xt+1 >xt+1. Again, by (10),

u_(c

t+1)[ˇf](xt+1)=u(ct)>u(ct)=u(c_t+1)[ˇf](x_t+1) which

impliesu_(c

t+1)>u(ct+1 )as[ˇf]isadecreasingfunction.Hence,

x

t+1>xt+1andct+1 <ct+1.

Nowconsiderthemaximizationproblemgiventhediscounting

sequenceˇ(x_).Thepathx

isfeasiblefromx0,andyieldsstrictly

higherutilitythanxgiventhediscountingsequenceˇ(x_)because

c

t<ctforallt.However,thisisacontradictionwithxbeingafixed

pointwhich wouldimplythat x itselfistheuniquemaximizer

giventhediscountingsequenceˇ(x_).Thereforex

1=x1.Letx1=

x

1=x1.

Notice that Proposition 15 implies that (x1,x2,x3,...) and

(x1,x2,x3,...)aretwoequilibriawithexternalityassociatedwith

x1.Thenapplyingthesamearguments,wegetx₂=x₂.Finally,by

inductivelyapplyingProposition15andtheabovearguments,we

getx

t=xt forallt. 

Corollary1. If[ˇf](x0)=1forsomex0,thentheuniquefixedpoint

associatedwithx0isthestationarysequence(x0,x0,x0,...).

Proof. Letˇ=ˇ(x0)andconsiderthebasicneoclassicalmodelwith

fixeddiscountfactorˇ.Astheinitialcapitalisalreadyx0whichis

steadystatecapitallevelintheneoclassicalgrowthmodel,(x0,x0,

x0,...)willsolvethesocialplannerproblem.Thismeansthat(x0,

x0,x0, ...)istheuniquemaximizergiventhediscountsequence

(ˇ,ˇ, ...).Thus,itisafixedpointassociatedwithx0,andbythe

abovepropositionitistheuniquefixedpointassociatedwithx0.



Proposition18. Any equilibriumwithexternality monotonically

convergestoapointxs_{satisfying[ˇf}_](xs_{)=1.Moreover,ifthereare}

multiplesuchpoints,itmonotonically convergestotheonethat is

closesttotheinitiallevelofcapital.

Proof. Onecanshowthatthesetofsolutionsto[ˇf](x)=1

con-stituesacompactinterval,sayX=[xs_min,xs

max].Notethatitiswell

possibletohavexs

min=x s

maxorxmins <x s

max.Lettheinitialcapital

levelbex0andtheassociatedequilibriumwithexternalitybex∗.

ByCorollary1,ifx0∈Xwehavex∗=(x0,x0,...)whichconverges

totheclosestsolutionx0to[ˇf](x)=1.

Considerx0<xsmin.Supposex∗doesnot–monotonically

con-verge–toxs

min.Thenx∗asasequenceeitherpassesfromtheregion

(0,xs

min]totheregion(x

s

min,∞),orstrictlydecreasesintheregion

[0,xs

min],atleastonceatsomepointintime.Formally,thereexists

someteitherwithx∗

t ≤xsmin<x∗t+1orwithx∗t+1<x∗t ≤xsmin.

Supposethatforsomet,xt≤x_mins <x_t+1.As1=[ˇf](x_mins )≥

[ˇf](x_t+1∗ ),(10)impliesu_(c∗

t)≤u(c_t+1∗ ).Thenbyconcavityofuwe

getf(x∗_t)−x∗

t+1≥f(x∗t+1)−x∗t+2,whichyieldsx∗t+1<xt+2∗ asx∗t <

x∗

t+1.Hence, x∗t+1<x∗t+2 and xsmin<x∗t+2.Inductively we obtain

xs

min<x∗t+1<xt+2∗ <....Duetothemaximumsustainablecapital

stock,x∗cannotdiverge,soitconvergesmonotonicallytoalevelx

thatisstrictlyhigherthanx∗

t+1.Takingthelimitof(10)forthefixed

pointx∗astgoestoinfinity,weseethatthelimitxofx∗hasto

satisfy[ˇf](x)=1.Thusxs

min<xt+1∗ <x≤xsmaxandhencex_t+1∗ ∈X.

ThenCorollary1impliesthattheequilibriumwithexternality

asso-ciatedwithx∗

t+1is(x∗t+1,xt+1∗ ,...),however,Proposition15implies

thatitis(x∗_t+1,x∗

t+2,...).Recallthatthefixedpointfromx∗t+1 is

unique,sowegetacontradictionasx∗

t+1<x∗t+2.

Now suppose that for some t, x∗

t+1<x∗t ≤xsmin. As 1= [ˇf](x_mins )≤[ˇf](x∗_t+1),(10)impliesu_(c∗ t)≥u(c∗_t+1).Thusweget f(x∗_t)−x∗ t+1≤f(x∗t+1)−xt+2∗ ,whichyieldsx∗t+1>x∗t+2asx∗t >xt+1∗ . Hence,x∗

t+2<x∗t+1<xsmin.Inductivelyweobtain...<xt+1∗ <x∗t ≤

xs

min.Takingthelimitof(10),weseethattheonlypossibilityisx∗

convergestozero.Ifx∗convergestozero,thereexistssomelargeT

suchthat[ˇf](xt∗)>1forallt>T,because[ˇf](0)>1.Thenu(ct∗)>

u_(c∗

t+1)fort>Tby(10).Hence,f(xt∗)>f(x∗t)−x∗_t+1=ct∗>c∗T>0

forallt>T.However,f(x∗_t)convergestozeroalongwithx∗bythe

continuityoff.Contradiction.

Thereforewehaveshownthatthefixedpointassociatedwith

x0<xsminmonotonicallyconvergestotheclosestsolutionx

s minof

theequation[ˇf](x)=1.

Thecaseofx0>xsmaxisanalogoustox0<xs_min. 

Justtorecall,thefixedpointx∗of˚,namelytheequilibrium

withexternalityassociatedwithx0,solvestheproblem

maxx ∞



t=0



_t



s=1 ˇ(x∗ s)



u(f(xt)−xt+1) subjectto 0≤x_t+1≤f(xt), x0>0,given.

Wewillnowprovethatsuchafixedpointisindeeda

competi-tiveequilibrium.

Theorem 1. Assume that f_{(0)ˇ>1. Define q}∗_=1,p∗ t =

(



t_s=1ˇ(x∗_s))u_(f(x∗

t)−x_t+1∗ ), c∗t =f(x∗t)−x∗_t+1,

∀

t. Then, (c∗, x∗,

Proof. First, we prove that p∗ is in 1 +\



0



. Clearly p∗

t >

0. We need to prove that

∞



t=0 p∗ t = ∞



t=0 ( t



s=1 ˇ(x∗ s))u(f(x∗t)−x∗_t+1)

is bounded. We have shown that x∗ convergesto some xs_∈X.

[ˇf](0)>1suggestsxs

min>0,hencethesequenceu(f(x∗t)−x∗t+1)

isbounded.Thus,thesum

∞



t=0

p∗

t existsinR+.

Since x∗ ∈[0, A(x0)]∞ we get x∗∈ ∞+. By the constraints of

theproblem(PE),f(A(x0))≥f(x∗t)≥x∗t+1≥0.Hence,f(A(x0))≥c∗t =

f(x_t∗)₋x∗

t+1≥0,i.e.,c∗∈ ∞+.

Theproofsthatx∗isasolutiontotheproblem(PP)andc∗solves

theconsumer’sproblem(CP)arestandard(see,e.g.,LeVanand

Dana,2003).

Finally,onecaneasilynotethatthemarketclearingcondition,

c∗

t +x∗_t+1=f(xt∗)issatisfied. 

ByProposition18,theoptimalpathsconvergetoapointx∗with

f_(x∗_)ˇ(x∗_{)=1. (11)}

Clearly,suchapointisuniqueif[ˇf](· )isstrictlydecreasing.

Forsimplicity,letusassumeaCobb–Douglasproduction

tech-nology,f(x)=Ax˛,whereA>0denotesthetotalfactorproductivity.

Theremayexistuniqueorcontinuumofsteadystatesinthismodel

withendogenoustimepreferencedependingonthecharacteristics

ofthediscountfunction.

Example2. Letˇ(x)=−e−(x+)ε_{,where0<,0<ε<1,0<<}

(1−˛)eε

,and(e−ε/1−˛)<<1.Notethatˇsatisfiesallofour

assumptions.Wewillnowshowthatˇ(x)f_{(x)isstrictlydecreasing.}

Wehave

ˇ(x)f_(x)

₌



_−e−(x+)ε



_A˛x˛−1





=A˛x˛−2



_e−(x+)ε

_{1−˛+εx(x+)}ε−1

_−(1−˛)



_.

Then,forx>0,

ˇ(x)f_(x)

_{<0ifandonlyif}

h(x):=exp



−(x+)ε



1−˛+εx(x+)ε−1



−(1−˛)<0.

Itisclearthathisdifferentiable,henceobtainsitsmaximumatthe

boundariesorcriticalpoints.Attheboundaries,

h(0)=exp

−ε

_{[1−˛] −(1−˛)} =(1−˛)

exp

−ε

_−



_{=−(1−˛)ˇ(0)<0,and} lim x→∞h(x)=x→∞lim(1−˛)e −(x+)ε_{+ lim} x→∞e −(x+)ε_εx(x+)ε−1 −(1₋˛)=−(1−˛)<0.

Soatanycriticalpointx,wehaveh(x)=0.Accordingly,

e−(x+)ε_ε(x+)ε−1



₋

_{1−˛+εx(x+)}ε−1

_{+1−(1−ε)} x

x+



=0.

Hence,(1_{− ˛}+εx(x+)ε−1_{)=1−(1−ε)(x/x+).Thenwehave}

h(x)=e−(x+)ε



_1−(1−ε) x

x+



−(1−˛)

atanycriticalpointx.Itisclearthattheexpressione−(y+)ε_[1−

(1−ε)(y/y+)] is maximized at y=0. Therefore, we obtain

thath(x)≤e−ε

−(1₋˛).Recallthat(e−ε/1−˛)<,hence

e−ε

−(1−˛)<0,implyingthatatanycriticalpointx,h(x)<0.

Thus, h is a negativevalued function, implyingthat ˇ(x)f_{(x) is}

strictlydecreasing.

Anydiscountfunctionˇunderwhichˇ(x)f_{(x)turnsouttobea}

strictlydecreasingfunctionimpliesauniquesteadystateandglobal

convergence.Theuniquesteadystateisbothlocallyandglobally

determinate.Moreover,themodelmayevenpossessacontinuum

ofsteadystates,anditdependsontheinitialconditionastowhich

oneisrealizedinthelongrun.Hence,theeconomyexhibitsno

tendencytowardglobalconvergence;animportantdeparturewith

respecttotheneoclassicalgrowthmodel.

Example3. Considerthefollowingdiscountfunctionthatconsists

ofthreeparts: ˇ(x)=

⎧

⎪

⎪

⎨

⎪

⎪

⎩

mx+n, x<a; 1 f(x) = 1 A˛x1−˛, a≤x≤b; −_x, b<x.

where A>0, 1>˛>0, (A˛/2−˛)1/1−˛_{>b>a>0, m=1−˛/A˛a}−˛_,

n=(1/A)a1−˛,=(2−˛/A˛)b1−˛_{,and=(1−˛/A˛)b}2−˛_.Letscheck

theassumptions.ˇisalreadydifferentiableintheregions(0,a),(a,

b),(b,∞).Somesimplealgebraforcheckingtheleftandrightlimits

ofˇandˇ_{atthepointsaandbshowsthatthisspecificationofm,}

n,andmakesˇisdifferentiableataandb.Thereforeˇis

differ-entiable,alsoimplyingcontinuity.ˇ(0)=n>0,henceˇrangesinto

R++.Also,asm>0,A>0,˛∈(0,1),and>0,ˇisstrictlyincreasing.

Asˇisstrictlyincreasing,thesupremumofwhichis,isstrictly

lessthanoneby(A˛/2−˛)1/1−˛_{>b.Moreover,thesupremumof}

thederivativeofˇisalsoboundedwhichiseasytoseebysome

algebra.

Noticethatthroughout thewholeinterval[a,b],ˇ(x)f_(x)=1,

thereforewehaveacontinuumofsteadystates[a,b].Aswehave

proventhatanyequilibriumwithexternalitywithinitialcapital

x0∈[a,b]willbeconstantovertime,theinitiallevelofcapitalstock

lowerthanawillleadtoanincreasinglymonotonicconvergence

towardsawhereastheinitialtheinitiallevelofcapitalstockhigher

thanbwillresultindecreasinglymonotonicconvergencetowards

b.

Remark 2. Existenceofthecriticalvaluexc andcontinuumof

equilibriawithstrictlyconcaveproductionfunctionarepeculiarto

ourmodelofendogenoustimepreference.

4. Conclusion

Inthispaper,wepresentthedynamicimplicationsof

endoge-noustimepreferencedependingonthestockofwealthinaone

sectorgrowthmodel.Weprovewithoutanyassumptiononthe

curvaturesoftheproductionandthediscountfunctionsthat

opti-malpolicyissinglevaluedanddifferentiablealmosteverywhere

in theplanner’sproblemand theoptimalpathsaremonotonic.

We consider the optimal path dynamics in the long run and

showthatourmodelcanexhibitglobalconvergenceevenunder

aconvex–concavetechnologyandmultiplicityofoptimalsteady

states even under a convex technology. Indeed, we show that

there existsa critical stockof capital belowwhich theoptimal

programleadsonlytoforeverdiminishingcapitalandgradually

convergingtodevelopmenttrapandabovewhich theeconomy

convergestothehighsteadystate.Wealsoshowthatifitso

hap-pensthattheinitiallevelofcapitalstockequalsthecriticalstock,

whichisnotanoptimalornonoptimalsteadystate,an

indetermi-nacyemerges.Moreover,themultiplierssystemassociatedwith

anoptimalpathisproven tobethesupportingpricesystemof

concerningthepropertiesofoptimal(equilibrium)pathsare pro-vided.

Extensionstoourmodelareobviouslypossibleandincludethe

considerationsofuncertainty,heterogenousagentsandstrategic

interactions.Theseareinourresearchagenda.

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