Nº 504
ISSN 0104-8910
On choice of technique in the Robinson-Solow-Srinivasan model
On Choie of Tehnique in the Robinson-Solow-Srinivasan Model
M.AliKhan
y
andTapanMitra
z
July2003
Abstrat: We report results on the optimal \hoie of tehnique" in a model originally formulated
byRobinson,SolowandSrinivasan(heneforth, theRSSmodel)andfurther disussedbyOkishioand
Stiglitz. Byviewing thisvintage-apitalmodelwithoutdisountingasaspei instaneofthegeneral
theoryofintertemporalresourealloationassoiatedwithBrok,GaleandMKenzie,weresolve
long-standingonjeturesintheformoftheoremsontheexisteneandpriesupportofoptimalpaths,andof
onditionssuÆientfortheoptimalityofapoliyrstidentiedbyStiglitz. Wedisposeoftheneessity
oftheseonditionsinsurprisinglysimpleexamplesofeonomiesinwhih(i)anoptimalpathisperiodi,
(ii)apathfollowingStiglitz'poliy isbad,and(iii)thereisoptimalinvestmentindierentvintagesat
dierenttimes. (129words)
JournalofEonomiLiteratureClassiationNumbers: D90,C62,O21.
Key Words: Choie of tehnique, overtaking riterion, golden-rule stok, golden-rule pries,
value-loss, yling, average turnpike property, prie-support property, optimal program, Stiglitz program,
-program,long-run,transitiondynamis.
RunningTitle: TheRSSModel
Theworkreportedhereispartofaprojetwithalonggestationperiod: itwasinitiatedduringMitra'svisittothe
DepartmentofEonomisattheUniversityofIllinoisin1986,reeivedinvaluableimpetusfromProfessorRobertSolow's
presentationattheSrinivasanConfereneheldatYaleinMarh1998,wasontinuedwhenKhanvisitedtheDepartment
ofEonomisatCornellinNovember1998,Otober2000andinJuly2003,andtheEPGE,Funda~aoGetilioVargasin
January 2001and Deember2002. Theauthorsaregratefulto allofthese institutionsfortheirhospitalityandto the
CenterforaLivableFutureatJohnsHopkinsforresearhsupport.KhanalsothanksProfessorsAbhijitBanerjee,Jimmy
Chan,AvinashDixitandDebrajRayforstimulatingonversation,andChrisMetalfforhisarefulreading.
y
DepartmentofEonomis,TheJohnsHopkinsUniversity,Baltimore,MD21218.
z
1 Introdution 2
2 The Model and its Redued Form 7
2.1 Tehnology . . . 7
2.2 Programs . . . 7
2.3 Preferenes . . . 8
2.4 ConversiontotheReduedForm . . . 9
3 The Existeneand Uniqueness ofa Stationary OptimalStok 10
4 The Existeneofan OptimalProgram 11
5 Choie of Tehniques in the Long-Run 15
6 An Exampleofan OptimalPeriodi Program 16
7 Choie of Tehniques with a Linear Feliity Funtion: AnExample 17
8 Choie of Tehniques in Transition: A LinearFeliity Funtion 18
9 Choie of Tehniques with a Non-Linear FeliityFuntion: An Example 21
10The Prie-Support and Other Propertiesof an OptimalProgram 22
11Choie of Tehniques in Transition: A SuÆientCondition 24
12Conluding Remarks 26
Inthelatesixtiesandearlyseventies,underthegeneralheadingof\tehnialhoieunderfull
employ-ment in asoialist eonomy," Robinson (1960, 1969), Okishio (1966) and Stiglitz (1968, 1970, 1973)
studied the problem of optimal eonomi growth in a model of an eonomy originally formulated by
Robinson(1960,pp. 38-56),Solow(1962b)andSrinivasan(1962a)(heneforth, theRSSmodel).
1 The
workgeneratedontroversy. Stiglitzargued,withjustiation,thattheRobinson-Okishioassumptionof
axedlaboralloationbetweentheonsumptionandinvestmentsetorshadnoplaeinanexerisethat
soughtto determinetheoptimal growthpath, and therebyanoptimumtime-path ofthealloationof
labor. 2
Heidentiedadevelopmentpoliy,heneforthStiglitz'poliy,
3
underwhihthereisinvestment
only in the typeof mahine that minimizes eetivelabor osts and simultaneouslymaximizes the
steadystateonsumption,andautilizationofonlythosetypesofmahineswhoseoutputpermanratios
arehigher thanthe eetivelabor ost ofproduing: Stiglitz observedthat the \numberof workers
workingintheonsumption-goodssetorinreasesmonotonially(apital`widening'oursinasmooth
way),outputofonsumptiongoodsneednotbemonotoniallyinreasing",
4
andpresribedforthe
eon-omyatanypointin timeanoptimalhoieoftehniques,bothtouseandtoprodue,andtherebythe
(instantaneous)optimallevelsoftehnologialobsolesene{presriptionswhihareallindependentof
thefeliity funtion. RobinsonommentedonStiglitz'solutionbyritiizing hisassumptionof axed
positivedisountrate,ontinuoustimeandthelinearityassumptioninthespeiationoftheplanner's
feliityfuntion.
5
Robinson'sobjetionswereexpliitlyaknowledgedbyStiglitz,
6
andasarstapproximatestep,
7
heextendedhisearlieranalysistotheaseofaminimumonsumptiononstraintinasettingwith
on-tinuoustimeandapositiverateofdisount. However,heemphasizedthattheimportantmodiations
onernedtransition,ratherthanlong-run,dynamis.
8
Evenifthere isaminimumonsumption onstraintandanitegestation period, thepath
of development will, after an initial\adjustment" period, look exatlyas I havedesribed
it. [Unlike℄ long-runneolassialmodels withmalleable apital[where℄ theoptimal poliy
1
InKhan(2000,p.3),themodelisreferredtoastheSolow-Srinivasanmodel;alsoseeSolow(1962,onludingthree
paragraphs)andKhan(2000,Footnote12)forthewayitisseeninearlierwork.
2
See(1968,Paragraph1;1970,p. 421). Stiglitz(1970,p. 421)writes\Theremaybesomespeialsituations...where
theemploymentalloationisthe sameforallsteady-state paths,buteventhen,ingoingfromonesteady-statepathto
another, oneannot infer thatthe employmentalloation isunhanged { and it isthisdynamiproblem that we are
disussing." 3
ThisisformalizedinDenition8below. Asweshallsee inthe sequel,Stiglitz'poliyanbeusefully omparedto
Faustmann'ssolutiontotheforestryproblem,asformalizedinMitra-Wan(1986).
4
SeeStiglitz(1973,pp.143-144).Inthedisussionofhispoliy,Stiglitzalsodrewattentiontopreliminaryinvestigations
ofBruno(1967).
5
WerestateRobinson'sritiismsinourownterminology;shephrasesthemintermsofa\disountratehosenone
andforall,...negligiblegestationperiods,...[and℄easingtoonsumeandlivingonairduringtherstphaseoftheplan."
6
SeeStiglitz(1973,p.144)andalsoCass-Stiglitz(1969,p. 614).
7
ThusCass-Stiglitz(1969,Footnote19)sawtheinstantaneousutilityfuntionU(C)= 1forC<
CandU(C)=C
forC
C;
Caminimumonsumptionlevel,as\oneapproximationtothegeneralinstantaneousutilityfuntionsatisfying
U 0
(C)>0withlim
C=0 U
0
(C)=1andU
00
(C)0:" 8
Stiglitz's(1970)responseisimportantforthereord; thisessayanalsobeseen asafurtherinvestigationintothe
its long-run equilibrium value there is always a period of zero onsumption, after whih
onsumptionjumpstoitslong-runequilibriumvalue,whereasinourex-postxedoeÆients
modelonsumptioninreasessteadily toitslong-runvalue.
Given the primary interest in the undisounted ase, Stiglitz interpreted the undisounted ase as a
situation when the disount rate is\negligible"; he developed the intuitive ideasin disrete-time and
thenhose totranslatethemto theontinuous-timeframework.
9
InhisreentrevisitofSrinivasan(1962a),Solow(2000,p. 7)asksforasolutiontothe\Ramsey
problemforthismodel."SineStiglitzhadalreadyprovidedasolutionwitha\linearutilityandpositive
timepreferene",theopenquestions onernarigoroustreatmentoftheundisountedaseandof the
disountedasewitha\stritlyonavesoialutilityfuntionforurrentperapitaonsumption."Like
Robinson, Solow also mentions that an \adoption of this [linear utility℄ riterion an indeed lead to
unjustiablenegletofearlyonsumption,"andifonewastoshare\Ramsey'sbeliefthattheonly
ethi-allydefensiblesoialrateoftimeprefereneiszero,asuÆientlysharply-onaveutilityfuntionwould
enforealoserapproahtointergenerationalequality."Inshort,Solow'squestionremainsunanswered,
andthegeneralizationofStiglitz'sworkinthediretionsitpromptsremainsyet tobeaomplished.
10
Inthispaper,weaddressthisgeneralquestion,andmostimportantlyfromamethodologialpoint
ofview,dosointhesettingofthemoderntheoryofoptimalintertemporalalloationinitiatedoriginally
byRamseyandvonNeumannandbroughttoompletionatthehandsofBrok,GaleandMKenzie.
11
Sine thistheory was beingdeveloped at thesametime asthe\apitalontroversy"betweenthetwo
Cambridges, 12
it has not been brought to bear on the fundamental issues. Suh an appliation has
the advantage of illustrating the power and exibility of the modern theory: it is ideally suited to
dealwiththis problem, andthe generalresultsof this theoryanbe readilyapplied to thispartiular
ontext, using extremely elementary methods. As suh it is perhapsoverdue. A seondarybenetof
thisappliation onernsthetheoryitself; itoersinsightsintoitssopeandsuggestsdiretions along
whih it may ndfruitful extension.
13
It alsopointslearlyto issues towhih thegeneraltheory has
(andbyitsverynature,willhave)verylittletooer,therebyindiatingthatevenaftermuhtheoretial
progresshas been made, someof the questions that were askedfty yearsago aboutthe appropriate
hoieoftehniqueremainhardunansweredproblemsthat needtobeapproahedasebyase.
Interms of the speis of this paper, we truly treatthe Ramsey problem; that is, onsider a
9
SeeFootnotes1and3onpage608andthedisussionofthe\orret"priingsystemonpage606inStiglitz(1968).
10
Forsomepartialattemptsatsolution,andforanumerialexample,seeStiglitz(1973);alsoseeStiglitz(1968,Footnote
2,p.608)andCass-Stiglitz(1969).However,inKhan(2000,p.15),thesituationisexpressedas\Thelooseendremains
loose." 11
TherelevantpapersareGale(1967),MKenzie(1968)andBrok(1970).Inthesequel,whenwerefertothe\general
theoryofoptimalintertemporalalloation",weshallbehavingthesepapersinmind.
12
Itisofoursenotourintentiontorevisitthisdebatehere{theinterestedreadermaywanttoseeBirner(2002)and
hisreferenes.
13
Thus,adistintionhastobedrawnbetweenapplyingatheoremfromapplyingitsmethodsofproof.Asthereaderwill
seeinthesequel,thehypotheses ofBrok'sexistenetheoremandofMKenzie'sprie-supportpropertyarenotliterally
of strutural stability of the model at the zero disountrate, an assumption at best roundaboutand
at worst dubious. We are by this time veryfamiliar with the overtakingriterion of Atsumi (1965)
andvonWeiszaker(1965),andunder this riterion,anoptimalpathin theundisountedaseanbe
shown to exist and its properties an be rigorously studied. Our treatment of time is disrete: the
generaltheoryofintertemporalalloationisdevelopedinthesimplest andmostelegantwayin suha
setting(seeMKenzie (1986)foramasterlypresentation),andweanworkwithareduedformofthe
RSSmodel
14
in whih the tehnologialpossibilitiesare given by atransition possibility set, and the
objetivefuntionbya(redued-form)utilityfuntiondenedonthisset(thatis,denedonbeginning
and end of period apital stok vetors). We appeal to the methods of Brok (1970) and MKenzie
(1968)toshowtheexisteneof anoptimalprogram,and furthermore,to establishthat, startingfrom
anarbitraryinitialstok,itonvergesasymptotiallytoasubsetofthetransitionset,theso-alledvon
Neumannfaet, onsisting of allplans whihhave \zerovalue-loss" atthe golden-rulesupport pries.
Intheaseofastritlyonavefeliityfuntion,thevonNeumannfaetshrinks toapoint,andsowe
haveasymptoti onvergene to the golden-rulestok. These resultsfurnish aomplete resolution of
the problem of the long-runhoie of tehniqueand thereby illustrate the power and elegane of the
generaltheory.
It is important to appreiate the methodologial signiane of this reformulationof the RSS
model. Thestandardtreatmentis in terms ofPontryagin'spriniple,
15
and in the appliationof this
priniple, as in the work of Stiglitz (1968), Sen (1968) and others, oneappeals to the transversality
onditions in the study of the dierential equations pertaining to the state and auxiliary variables
obtained by substituting the values of the ontrols that maximize the instantaneous Hamiltonian.
16
Thustherelevaneoftherestpointsisestablishedonlytowardstheendoftheanalysis.Here,webegin
with therest points, thegolden-rulestok and thegolden-rulepries, and usethe value-loss funtion
andtheso-alledaverageturnpikepropertyofgoodprogramsto yieldtheoptimalprogram.
17
Stiglitz
investigatestheonvergene(turnpikeproperty)ofapaththatfollowshis(derived)poliypresriptions
asto thehoie oftehniques(referredto earlierand formalizedasaStiglitz'programin Denition 8
below), 18
whileweneedtoinvestigatewhether theoptimalpathanditsturnpikepropertyissustained
by these presriptions. To antiipate, this annot be established, in general, for either a linear or a
stritly onave feliity funtion, but only in speial (identied) ases of either formulation. Thus,
14
Wehoose to workwith the versionpresented inStiglitz (1968) ratherthan that in Solow (1962b)or Srinivasan
(1962a). AllofthesevariantsanbeviewedasspeialasesofthemodelsonsideredinBruno(1967)orthemoregeneral
treatmentinKoopmans(1971)andKoopmans-Hansen(1972). Weleavetheanalysisofthesepapersasataskforfuture
researh;alsoseethethirdparagraphoftheonludingSetion12below.
15
ThusDixit(1990,p.5)writes,\NowadaysHamiltoniansandphasediagramsareeverydaystuforthetypial
seond-yeargraduatestudent",andquotesFrankHahn'sreferenetothe\unseemlyhastetogetdowntotheHamiltonian."
16
IntheontextofStiglitz(1968),seehisFootnote2bothonpage605andonpage607.
17
Asthereaderwillseebelow,weappealtoMKenzie's(1983,1986)prie-supportpropertyonlytoestablishournal
resultpertainingtotransitiondynamisintheaseofastritlyonavefeliityfuntion(Theorem7below).
18
AsalludedtoinFootnote3,weseetheStiglitzpoliyastheanalogueoftheFaustmannsolutionintheeonomisof
forestry,anditwouldbeinterestingtopursuethe analytisofthisanalogy;seethefourth paragraphoftheonluding
straightforwardin another. 19
Intermsofsubstantiveresults,ourmethodsyield surprisesevenfor theaseofalinearfeliity
funtion. Wepresentsimpleexamplesof an eonomyonsisting of onlyasingle typeof mahine and
therebyposingnoissueastothehoieoftehniqueinprodution,
20
inwhihonsumptionandapital
stokexhibitaperiod-twoylealonganoptimalprogramoraperiod-fourylealongaStiglitzprogram
whih is thereby shown to be bad.
21
The rst shows the optimality of periodi over-building and
over-onsuming relative to the golden-rule levels, and the seond, the non-optimality of a no exess
apaity poliy { phenomena that Stiglitz (1968) apparently does not enounter in his
ontinuous-time formulation of the model.
22
In the ase of a general onavefeliity funtion, we establish the
non-optimality of a Stiglitz poliy { an aÆrmative answer to the question as to whether there is a
ompelling reason to ever produe a mahine (other than the golden-rule mahine) whih we know
would eventuallybedepreiated tozero? Wepresent aonreteexample ofan eonomy onsistingof
twotypes of mahines in whih this phenomenon anour. Stiglitz (1973, pp. 146-148) had earlier
presentedan exampleof afour mahine eonomywhere suh aphenomenon anour,but it iswith
disountingandaminimumonsumptiononstraint.
23
Alloftheseexhibitinadramatiwayboththestrengthandtheweaknessofthegeneraltheoryof
intertemporalalloationalludedtoearlier,andtherebyrevealexatlywhythehoieofanappropriate
tehnique is suh adiÆult and multi-faetted problem. Sine we have a omplete resolution of the
problemofthelong-runhoieoftehnique,thenaturalquestionarisesastothehoieoftehniquein
transitiontothesteadystate,theissueoftransitiondynamis: adeterminationofthetypeandamounts
ofmahinesthatareproduedandusedintheshortrun. Unfortunately,itisonthishardproblemthat
thegeneraltheoryhaslittletooer,anditisfairtosaythattheliteraturehasnoonreteresultsofany
generalityon thisissue.
24
However,weanoera(novel)set ofsuÆient onditionsunder whih the
Stiglitz'poliyisindeedoptimalforaneonomywithageneralonavefeliityfuntion,andafortiori,
foraneonomywithalinearutilityfuntionand aminimumonsumptiononstraint.
ThesesuÆientonditions,inpointingto aninterestingdistintion betweenhoieoftehnique
thatisappropriateintheshort-runfromthatwhihisappropriateinthelong-run,alsoonnetstothe
19
Stiglitz(1968,Footnote2,p. 608)reognizesthattheresultsforthelinearutilityfuntionmightnotarryoverto
thegeneralonavease,and inasubsequentanalysisoftheproblem,onewithaminimumonsumptiononstraint,he
referstothediÆultyofshowingthatthereisonlyonetypeofmahinethatmaximizesp(x;t);seeStiglitz(1973;p.145).
HealsorefersinthisonnetiontoBliss(1968)andCass-Stiglitz(1969). Forthisseptiismonerningthe linearase,
alsoseeSolow(2000)inadditiontoRobinson(1969).
20
Thequestionofthehoieoftehniqueintermsofuseofourseremains: shouldallofthestoksofthemahinebe
utilizeduntilprodutionisundertaken?
21
Asiswell-knowntostudentsofthe generaltheoryofintertemporalresourealloationassoiatedwithBrok,Gale
andMKenzie,thisisatehnialterm;seeDenition6below.
22
Part ofthereason forthisundoubtedlyliesinthe proedureof applyingresults forthe disounted asebysetting
thedisountratetobezero(negligible). However,itisworthre-emphasizinginthisonnetionthatBrok's1970results
werenotavailabletoStiglitzin1968.
23
Inthiswork, Stiglitzisprimarilyinterestedinwhat heallsthe phenomena of\reurrene"{ a situationwhena
mahineoneinservieisputoutofservietobebroughtbakintoservieagain.
24
literatureofthesixtiesonplanninginIndia(andelsewhere). Thisliterature omesaslosetostating
theproblemaspreiselyasanbeexpetedinthepre-Pontryaginperiod.
[T℄he whole point of the exerise is to get alternative time-paths of onsumption, out of
whih hoie anbe made aording to irumstanes. ... So, the hoie, in this system,
shouldnotbeputsomuhasonebetweenafasterandaslowergrowthrate,butasahoie
betweenonstantonsumption,oritsonstantrateofgrowth,oritsonstantrateofgrowth
ofrateof growth,andsoon. Itshould benotedthatfrom thelong-runpointofview,eah
ase is better than theearlier one, though the position is the reverse in the short period.
Whihtypeofpoliywehoosewilldependverymuhonthetimeneessaryforthelong-run
eets toompensatetheshort-runresults.
26
However, this literature and earlier work notwithstanding, it will generally be reognized today that
whether weareinterestedin this issuefrom aplanningperspetiveorfrom themodernperspetiveof
aompetitiverepresentativeagent,theproblemof anappropriatehoieof tehniqueshould reallybe
viewed aspartof thegeneral theoryof eonomi growth. A subsidiary motivation of this paperis to
failitatethisre-orientation.
27
We onludethis introdution with ashemati outline of the paper. Insetion 2wepresent
themodelandshowhowitanbeonvertedtoitsGale-MKenziereduedform. InSetion3,undera
standinghypothesison thenite set of parametersthat denethe RSSmodel, weshowthe existene
and uniqueness of the golden-rulestok, and its values to beindependent of the feliity funtion; we
alsoidentifyasetofgolden-rulepries. InSetion4,weshowtheexisteneofaprogramthatisoptimal
startingfrom anygiveninitial stokofmahines. InSetion5,weonsider thequestionof theorret
hoie oftehniquefor thelong-run,andthroughtheidentiationofthevonNeumannfaet,present
resultsforbothlinearandstritly onavefeliityfuntions. Thismaterialappliesthestandardtheory
totheRSSmodelandsuggeststheexampleofylialoptimalpaths(presentedinSetion6)andthat
ofbad Stiglitz programs(presentedinSetion 7) inasimpleone-mahinesetting withalinearfeliity
funtion. These examplesalsosuggestasuÆient parameterizationunder whih aStiglitz programis
optimalanduniquelyoptimalintheaseofalinearfeliityfuntion. WepresenttheseresultsinSetion
8. InSetion 9,weturn toprograms, labeled-programs, in whihthere is nospeiationasto use
but onlyto theonstrutionof themahine of type: Wepresentan examplethat showsin asimple
two-mahine setting with anon-linear feliity funtion that an optimal program is not a-program.
This, along with Setions 6and 7,is onerned with transition dynamis. InSetions 10 and 11, we
onsiderthesettingofageneralfeliityfuntion,andafteradevelopmentoftheprie-supportandother
25
InadditiontoRaj-Sen(1961)andSen(1960)inpartiular,alsoseeDobb(1956,1960,1961,1967),Halevi (1987),
Mirrlees(1962),Naqvi(1963),Solow(1962a);andforanopen-eonomyperspetive,Bardhan(1971).
26
SeeRaj-Sen(1961,pp. 48,51andonludingsetion). WeremindthereaderthatanearlierversionoftheRaj-Sen
paper(withadierenttitle)waspublishedinArthanitiin1959,andthatNaqvi(1963)isafollow-uptotheRaj-Senpaper
ashisleadingfootnotelearlyindiates.
27
Foranearlyemphasisonthis,seeMirrlees(1962)andSrinivasan(1962b);Okishio(1987)representsthealternative
onludingSetion 12liststhesalientresultsandidentiesproblems thatremain open. Thetehnial
andomputationaldetailsofsomeoftheproofsareolletedinanAppendix.
2 The Model and its Redued Form
Webeginwithsomepreliminary notation. LetIN(IN
+
)betheset ofnon-negative(positive) integers,
IR(IR
+
) theset of real (non-negative) numbers. We shall work in nite-dimensionalEulidean spae
IR n
withnon-negativeorthantIR
n +
=fx2IR
n
:x
i
0; i=1;;ng:For anyx;y inIR
n
; lettheinner
produtxy=
P n i=
x i
y i
;andx>>y; x>y; xy havetheirusualmeaning. Lete(i); i=1;;n;be
thei
th
unitvetorinIR
n
;andebeanelementofIR
n +
allofwhoseoordinatesareunity. Foranyx2IR
n ;
letkxkdenote the Eulidean normof x: Theempty set isdenoted by;and set-theoretisubtration
betweenAandB byA=B:
2.1 Tehnology
OurhoieofIR
n
is ditatedbytheonsiderationofaneonomyapableofproduinganitenumber
n of alternative types of mahines. For every i = 1;;n; one unit of mahine of type i requires
a i
>0units oflaborto onstrutit, andtogether with oneunit oflabor,eah unit of it anprodue
b i
>0 units of asingle onsumption good. Thus, the prodution possibilities of theeonomy anbe
represented by an (labor) input-oeÆients vetor, a = (a
1
;;a
n
) >> 0 and an output-oeÆients
vetor,b =(b
1 ;:::;b
n
)>>0: Without lossof generality
28
weshall assumethat thetypesof mahines
arenumberedsuhthatb
1
b
2
b
n :
Weshall assumethat allmahinesdepreiateat arated2(0;1): Thus theeetivelaborost
of produing a unit of output on a mahine of type i is given by (1+da
i )=b
i
: the diret labor ost
of produing unit output, and the indiret ost of replaing the depreiation of the mahine in this
prodution. 29
We shall work with the reiproal of the eetivelaborost, the eetive output that
takesthedepreiation into aount,and denote
30
itby
i
for themahine of typei: Throughout this
paper,weshallassumethatthereisauniquemahinetypeatwhihthiseetivelaborost(1+da
i )=b
i
isminimized,oratwhihtheeetiveoutputpermanb
i
=(1+da i
)ismaximized. Thus,weshallassume:
Thereexists2f1;;ngsuhthatforalli=1;;n; i6=;
>
i
: (1)
2.2 Programs
Foreahdatet2IN;letx(t)=(x
1
(t);;x
n
(t))0denotetheamountsofthentypesofmahinesthat
areavailablein time-period t,andletz(t+1)=(z
1
(t+1);;z
n
(t+1))0bethegrossinvestments
28
NotethatStiglitz(1968)assumesthatb
i >b
j
impliesthata
i >a
j
;whereasthisisanaturalhypothesis,wemakeno
suhassumption.
29
SeeStiglitz(1968,pp. 608-609)ona\labourtheoryofvalue"interpretation.
30
Asweshallseebelow,iisthevalueofthesteady-stateonsumptionpermanifonlymahinesoftypeiareusedand
of net investment and of depreiation. Let y(t) = (y 1
(t);:::;y n
(t)) be the amounts of the n typesof
mahines used for prodution of the onsumption good, by(t); during period (t+1):
31
Let the total
labor fore of the eonomy be stationary and positive. We shall normalize it to be unity. Clearly,
grossinvestment,z(t+1)representingtheprodutionofnewmahinesofthevarioustypes,willrequire
az(t+1)unitsoflaborinperiodt:Also,y(t)representingtheuseofavailablemahinesformanufature
of the onsumption good, will require ey(t) units of labor in period t: Thus, the availability of labor
onstrainsemploymentin theonsumptionandinvestmentsetors byaz(t+1)+ey(t)1:Notethat
boththeowofonsumptionandofinvestment(new mahines)arein gestationduringtheperiodand
availableat theend ofit. Wenowgiveaformalsummary ofthistehnologialstruture.
Denition1 A programfrom x
o
in IR
n +
isasequene
32
fx(t);y(t)g with (x(t);y(t))2IR
n +
IR
n +
suh
that x(0)=x
o
; andfor allt2IN;
x(t+1)(1 d)x(t); 0y(t)x(t); a(x(t+1) (1 d)x(t))+ey(t)1:
Aprogramfx(t);y(t)gis simplyaprogramfrom x(0):
Denition2 Assoiatedwith any programfx(t);y(t)g is agrossinvestment sequenefz(t+1)gwith
z(t+1)2IR
n +
; andaonsumptionsequene fby(t)gsuhthat for allt2IN;
z(t+1)=x(t+1) (1 d)x(t):
Denition3 Aprogramfx(t);y(t)gisalledstationaryifforallt2IN;(x(t);y(t))=(x(t+1);y(t+1)):
2.3 Preferenes
Thepreferenesoftheplannerarerepresentedbyafeliityfuntion,w:IR
+
!IR;whihis assumed
tobeontinuous,stritlyinreasingandonave,anddierentiablewhenstritlyonave. Wesuppose,
asintheliteraturetakingitsleadfromRamsey(1928),thatfuturewelfarelevelsaretreatedlikeurrent
onesintheplanner'sobjetivefuntion. Thepreiseriterionofoptimalitythatweworkwithisdueto
Atsumi(1965)andvonWeiszaker(1965).
Denition4 A programfx
(t);y
(t)gfromx
o
isalledoptimal if
liminf
T!1
T X
t=1
[w(by(t)) w(by
(t))℄0
for every program fx(t);y(t)g fromx
o
: It isalled astationary optimal programif itis stationary and
optimal.
31
Thereadermayhoosetothinkoftheonsumptioninperiodtasthesalar(t+1);with
i
reservedforb
i
=(1+da
i );
weavoidthisnotationinthetexttopreventanyambiguity.
32
Notefx(t);y(t)gisanabbreviationoffx(t);y(t)g
t2IN
fx(t);y(t)g; x(0)=x o
; anumber">0and atime period t
"
suhthat
P T t=1
[w(by(t)) w(by
(t))℄>
"forallT t
"
: Thus anoptimal program is onein omparison to whih no other programfrom the
sameinitial stokiseventuallysigniantlybetter,foranygivenlevelof signiane.
2.4 Conversionto the Redued Form
Following MKenzie (1968), weonvert the abovemodel into its \redued form", and as emphasized
intheintrodution,therebyexploitasfaraspossibletheresultsofthegeneraltheoryofintertemporal
alloationforourpartiularase.
Dene the transition possibility set asaolletionof pairs (x;x
0
);suh that it is possibleto
obtaintheamountsofthe ntypesofmahinesx
0
inthe nextperiod (tomorrow)fromthe amountsof
thentypesofmahinesxavailable intheurrentperiod(today). Formally,
=f(x;x
0
)2IR
n +
IR
n +
:x
0
(1 d)x0anda(x
0
(1 d)x)1g:
For any (x;x
0
) 2 ; one an onsider the amounts y of the n types of mahines available for the
produtionoftheonsumptiongood. Formally,wehaveaorrespondene: !IR
n +
givenby
(x;x
0
)=fy2IR
n +
:0yx andey1 a(x
0
(1 d)x)g:
Forany(x;x
0
)2;weshalldenotethenumberofmahinesthatareproduedintheperiod(x
0
(1 d)x)
byz: Note that z 0:Finally, thereduedform utility funtion, u: !IR
+
; is dened on suh
that
u(x;x
0
)=maxfw(by):y2(x;x
0 )g:
Weleaveittothereaderto hekforherselfthatourassumptionsonwimplythattheredued
formutilityfuntion,u;isuppersemiontinuous
33
andonavefuntionon;andthatitisinreasing
initsrstargumentanddereasingin itsseondargument.
Giventhedesriptionofthetransitionpossibilityset;andofthereduedformutilityfuntion,
u;itislearthatforanyprogramfx(t);y(t)gfromx
o
; (x(t);x(t+1))2andy(t)2(x(t);x(t+1))for
allt2IN:Also,foranyoptimalprogramfx
(t);y
(t)gfromx
o
; w(by
(t))=u(x
(t);x
(t+1))forallt2
IN; andforeveryprogramfx(t);y(t)gfromx
o ;
liminf
T!1
T X
t=0
[u(x(t);x(t+1)) u(x
(t);x
(t+1))℄0:
Insummary,thebasidataofthemodeldenotedbythetriple(w;(a
i ;b
i )
n i=1
;d)summarizingthe
feliityfuntionw,thetehnology(a
i ;b
i )
n i=1
;andthedepreiationrated;isonvertedtothepair(u;)
summarizingtheredued-formutilityfuntionuandthetransition possibilityset:
33
Itisnowwellunderstoodthatontinuityofwdoesnotneessarilyimplytheontinuityofu;seeDutta-Mitra(1989)
A stationary optimal program is of speial signiane, and in this setion we take the rst step in
establishing the existene of suh aprogram. We show the existene and uniqueness of astationary
optimalstok(synonymously,golden-rulestok),andsimultaneously,providea\priesupport"property
ofsuhastok. Thisonstitutesastationaryprie-supportpropertyforthestationaryoptimalprogram.
Weinvoketehniquesfrom thegeneraltheory ofoptimalintertemporal alloation,but ratherthan an
appeal to theKuhn-Tukertheorem, weexploit theonrete strutureof theRSSmodelto providea
purely onstrutive proof of our laims. This has the additional advantage that we an identify the
shadowpriesin termsofthebasidata ofthemodel.
Webeginwithadenition.
Denition5 A stationary optimal stok (synonymously, a golden-rule stok) is x^ 2 IR
n +
suh that
(^x ;x)^ isasolutionto theproblem: maximizeu(x;x
0
)subjetto(i)x
0
x;(ii)(x;x
0
)2:
If we limit ourselves to a stationary program in whih only a mahine of type i is used and
produed,theonstraintoflaborallowsustomaintainthestok
34
(1=(1+da
i
)andobtainastationary
onsumption stream in the amount b
i
=(1+da
i
) =
i
: Sine we have assumed (in (1) above) that a
mahine of type is the onethat uniquely minimizes eetivelaborosts, wesee that it is also the
typethatuniquelymaximizestheonsumptionperunit oflabor.
35
Denotey^=(1=(1+da
))e();and
note that if we arein suh astationary state, b^y =(b
=(1+da
))and w
0
(by)^ the marginalutilityof
outputprodued. Furthermore,sinethelaborostofamahine oftypeiisa
i
;and aunit oflaboris
worth((1+da
i )=b
i )
1
units of output, amahine is wortha
i
(b
i
=(1+da
i
)in termsofoutput, and
w 0
(by)(a^ i
(b
i
=(1+da
i
)))intermsofutils. Weanthenidentifyastationarypriesystem(^qintermsof
theonsumptiongoodandp^intermsofutils)
36
forthevarioustypesofmahinesasq^
i =(a i b i =(1+da i ))
andp^
i
=w
0 (b^y)^q
i
foreahi=1;;n:
Wean nowpresentasimplebutimportantresult.
Lemma1 w(b^y)w(by)+px^
0 ^
pxfor any (x;x
0
)2; andforany y2(x;x
0 ):
Proof: Nowforany(x;x
0
)2andy2(x;x
0
);wehave
37
by^ by q(x^
0
x) =
by q(x^
0 x) = by q(x 0
(1 d)x)+dqx
=
(1 ey az)+
ey+
az by qz+dqx
=
(1 ey az)+
n X i=1 ( b i )y i + n X i=1 ( i )a i z i
+dqx (2)
34
Thelabor requirementsofthe onsumptionsetor inthe amount(1=(1+da
i
) plusthoseofthe investmentsetor
arisingfromreplaementfordepreiationintheamountda
i
=(1+da
i
)adduptothetotallaboravailable.
35
AsalludedtoinFootnote 30above.
36
Whenthefeliityfuntionislinear,themagnitudesofp^andq^areidential,thoughtheirunitsremaindierent. Note
alsotheidentitiesq
i =a i i and i +dq i =b i foralli: 37
=
(1 ey az)+
X
i=1 (
i )y
i +
X
i=1 (
i )a
i z
i
+dq(x y) (3)
Sine (x;x
0
)2 ; z 0: Sine y 2 (x;x
0
); x y and 1 ey az 0: We an now appeal to our
standinghypothesisasdesribedin(1)to assertthat
by^ by q(x^
0
x)0 =) by by^qx^ qx:^ (4)
Givenourhypothesesonthefeliityfuntion w;weobtainasaonsequeneof(4),
w(by) w(b^y)w
0
(by)(by^ b^y)w
0
(b^y)(^qx qx^
0
)=(^px px^
0 )
Asimpletranspositionoftermsompletestheproof.
Weannowstatetheprinipal resultofthis setion.
38
Theorem1 Thereexistsauniquestationary optimal stokx^=(1=(1+da
))e():
Proof: Let y^= ^x =(1=(1+da
))e(); and hek that (^x ;x)^ 2 ; and y^2 (^x;x):^ Next,appeal to
Lemma1toassertthat(^x;x )^ isasolutiontotheproblemspeiedin Denition5,andhenethatx^is
agolden-rulestok.
We an also show that it is a unique solution to this problem. Suppose to the ontrary that
(~x ;x~ 0
)isanothersolutionwithaorrespondingy~2(~x;x~
0
)andz~
0
=~x
0
(1 d)~x :Sinew()isstritly
inreasing,b~y=b^y=
:Onsubstituting ~x;y~andz~forx;y andzin (3)above,weobtainthefatthat
the right hand side of (3)equals zero, whih implies that eahof its four terms is zero. This implies
that y~
i
=0=z~
i
for i6=; that x~
i
=y~
i
forall i;and that y~
+a
~ z
=1:Couplingtherst assertion
with theequality by~=
; weobtainthat y~
= 1=(1+da
); and henefrom the third assertionthat
~
x=1=(1+da
)e():Fromthelastassertionweanthenobtainthat ~z
=d=(1+da
)andhenethat
~ x 0
=z~ (1 d)~x=(d=(1+da
)+(1 d)=(1+da
))e()=(1=(1+da
))e():Thedemonstrationis
omplete.
4 The Existene of an Optimal Program
In this setion, we provethe existene of an optimal program from an arbitrarily given initial stok.
WefollowthemethodsofBrok(1970)whihin turnbuildonthoseofGale(1967){thismethodology
relies ontheoneptof agoodprogramand theexploitstheassumption ofauniquegolden-rulestok
todeduetheaverageturnpikepropertyofsuhaprogram. Wefollowthesameoneptualbenhmarks
in the ontext ofthe RSSmodel andpresentaunied treatment both to highlightertain stepsthat
areruialforsubsequentargumentationandtoavoidpossiblyonfusingross-referening.
39
38
Weremindthereaderofourstandinghypothesisasexpressedin(1).
39
Note that weannot diretlyapply the relevant theorems inGale, Brok (1970) or MKenzie(1968, 1987) sine
the assumptionsofthesetheoremsarenot diretlysatised;instead, theonretestrutureof theRSS modelallowsa
Denition6 A program fx(t);y(t)g is alled good if there exists G 2 IR suh that T t=0
(w(by(t))
w(b^y))Gfor allT 2IN: A programisalled badif lim
T!1 P
T t=0
(w(by(t)) w(b^y))= 1:
Proposition 1 There existsagoodprogramfromany arbitraryinitial stokx
o
2IR
n + :
Proof: Foreaht2IN; letz(t+1)=d^x e():Deney(0)=0;andy(t+1)=(1 d)y(t)+d^x e()for
t 2IN: Then, y(t) ismonotonially non-dereasing,andonvergesto x^ast !1: Givenan arbitrary
initialstok,x
o
;denex(0)=x
o
,andforeaht2IN; x(t+1)=(1 d)x(t)+z(t+1):Then,itiseasy
to hekthat fx(t);y(t)g is aprogram from x
o
: Given the denition of the sequene fy(t)g, we have
(by(t) bx)^ =(1 d)
t
(by(1) b^x) fort2;andby(t)db^xfort2IN
+
:Thus, wehavefort2IN
+ ;
[w(b^x) w(by(t))℄w
0
(by(t))(b^x by(t))w
0
(db^x )(b^x by(1))(1 d)
t 1
:
Thus,thesequenefw(b^x ) w(by(t))g issummable,andso fx(t);y(t))g isagood programfrom x
o :
Proposition 2 For anyprogramfx(t);y(t)g;thereexistsm(x(0))2IR
+
suhthatx(t)m(x(0))efor
any t2IN:
Proof:Theaset=0isatriviality. Fort2IN
+
; (x(t 1);x(t))2impliesax(t)1 + (1 d)ax(t 1)
P
t 1
=0
(1 d)
+(1 d)
t
ax(0):Sine0<d<1;weobtainax(t)(1=d)+ax(0):Leta
j =min 1in a i : Sinea i
>0foralli=1;2;n;weobtainx
i
(t)(1=a
j
)((1=d)+ax(0))m(x(0))foralli=1;;n;
andompletetheproof.
Proposition 3 Foranyprogramfx(t);y(t)g;thereexistsM(x(0))2IR
+
suhthatforanyt
1
2IN;and
any integert
2 t 1 ; P t2 t=t1
(w(by(t)) w(b^y))M(x(0)):
Proof: From Lemma 1, for any t
2 t 1 ; P t2 t=t1
w(by(t)) w(b^y) p(x(t^
1
) x(t
2
+1)) px(t^
1 ) m(x(0)) P n j=1 ^ p j
:LetM(x(0))=m(x(0))w
0 (b
=1+da
) P n j=1 a i b i
=(1+da
i
)toomplete theproof.
Proposition 4 Anyprogramthat isnotgoodisbad.
Proof: For any program fx(t);y(t)g that is not good, and for any N 2 IR; there exists T
N suh that P TN =0
(w(by()) w(b^y)) N M(x(0)); M(x(0)) the real numberwhose existene is asserted
in Proposition 3. By hoosing t
1
= T
N
+1 and t
2
= t > T
N
+1 in Proposition 3, we obtain that
P t =T
N +1
(w(by()) w(b^y))M(x(0)) forallt>T
N
+1:Onaddingthesetwoexpressions,weobtain
that P
t =0
(w(by()) w(b^y))N forallt>T
N
;andompletetheproof.
Proposition 5 Anyoptimal programisgood.
Proof: Let fx(t);y(t)g be an optimalprogram, and suppose it is not good. ByProposition1, there
exists a good program fx
0
(t);y
0
(t)g starting from x(0): Hene there exists G 2 IR suh that for all
T 2IN
+ ; P T t=0 (w(by 0
(t)) w(by))^ G:Pikany">0;and appeal toProposition 4to guaranteethe
existeneof t
"
suhthat
P T t=0
(w(by(t)) w(b^y))<G "forallT t
"
:Puttingthesetwoexpressions
together, we obtainthat
P T t=0
(w(by 0
(t)) w(by(t))) >" forall T t
"
; and heneaontraditionto
T!1
=(^x;y);^ wherex (T )=(1=T)
P
T 1
t=0
x(t) andy(T) =(1=T)
P
T 1
t=0
y(t)for allT 2IN
+ :
Proposition 6 If the golden-rule stokx^ is unique, every good program exhibits the average turnpike
property.
Proof:Foranyprogramfx(t);y(t)g;Proposition2,andthefatthaty(t)x(t)forallt2IN;guarantee
that thesequenefx(T);y(T )ghasat leastoneonvergentsubsequene. Let(x
1 ;y
1
)bethelimitof
onesuhsubsequene. Weleaveittothereadertohekthat(x
1
;x
1
)2andthaty
1
2(x
1 ;x
1 ):
Sinefx(t);y(t)g isagoodprogram,thereexistsG2IRsuhthatforallT 2IN
+ ;
G=T (1=T)
T 1
X
t=0
(w(by(t)) w(by))^ w(by(T)) w(b^y);
wheretheseondassertionisaonsequeneoftheonavityofw():Ontakinglimitsas T !1along
thehosensubsequene,andonappealingtotheontinuityofw;weobtainw(by
1
)w(b^y):Henex
1
isagolden-rulestok. Sinethegoldenrulestokisunique,x
1
=x ;^ andsinewisstritlyinreasing,
y 1
=y:^ Sineweworkedwithanarbitraryonvergentsubsequene,theargumentisomplete.
For anyy2(x;x
0
)andany(x;x
0
)2;let
Æ (^x;p)^
(x;x
0
)=w(by)^ w(by) p(x^
0
x)=p(x^ x
0
) (w(by) w(b^y)): (5)
Wheneverthere isno possibilityof onfusion, weshall abbreviateÆ
(^x;^p)
(x(t);x(t+1))by Æ(t) forany
program fx(t);y(t)g: We shall refer to fÆ(t)g as thevalue-loss sequene assoiated with the program
fx(t);y(t)g:
Proposition 7 Thevalue-loss sequenefÆ(t)g
t2IN
ofany program fx(t);y(t)g isnon-negative, and
T X
t=0
(w(by(t)) w(by))^ =p(x(0)^ x(T +1))
T X
t=0
Æ(t) for allT 2IN:
Proof: Foreaht2IN;letÆ(t)=p(x(t)^ x(t+1)) (w(by(t)) w(by)):^ Sinefx(t);y(t)gisaprogram,
weanappeal toLemma 1toassertthat Æ(t)0forallt2IN:On summingovert;andrearranging,
weompletetheproofoftheassertion.
Wenowdenetheaggregatevalue-lossassoiatedwithanyprogramstartingfromx
o as
(x o
)=inff
1 X
t=0
Æ(t):fx(t);y(t)g isaprogramfromx
o g:
Ournexttworesultsassertthatthisinmumisanitenumberandthatitanbeattained.
Proposition 8 Thevalue-loss sequenefÆ(t)g
t2IN
of anyprogramfx(t);y(t)g issummableifandonly
if itisgood. Henelim
t!1
allt2IN;
T X
t=0
Æ(t) = p(x(0)^ x(T +1))
T X
t=0
(w(by(t)) w(by))^
p(x(0)^ x(T +1)) Gp(x(0))^ G:
Sine P
1 t=0
Æ(t)isanitenumber,ertainlylim
t!1
Æ(t)=0:Ontheotherhand,therstequalityand
Proposition2allowsus toassertthat aprogramwithasummablevalue-losssequeneisgood.
Proposition 9 There exists a program fx
0
(t);y
0
(t)g from an arbitrary initial stok x
o
suh that its
assoiatedvalue-losssequenefÆ
0 (t)g satises P 1 t=0 Æ 0
(t)=(x
o
) where0(x
o
)<1:
Proof: Proposition1guaranteesthatthereexistsagoodprogramfx(t);y(t)g fromx
o
;andhenefrom
Proposition7,weobtainthat(x
o
)<1:SinefÆ(t)gisanon-negativesequene,ertainly0(x
o ):
Forany2IN
+
;thereexistsaprogramfx
(t);y
(t)gfromx
o
suhthatitsassoiatedvalue-loss
sequenefÆ (t)g satises 1 X t=0 Æ
(t)(x
o
)+(1=): (6)
From Proposition 2,the sequenefx
(t)g
1 =1
isbounded independentlyof (and oft but not ofx
o ).
Siney
(t)x
(t);thesameistrueofthesequenefy
m (t)g
1 m=1
:Weannowappealtoadiagonalization
argument (see Rudin (1967; Theorem 7.23) to extrat a subsequene indexed by
k
; k 2 IN
+
; that
onvergesforall t2IN: Letfx
0
(t);y
0
(t)g bethelimit ofthissubsequene. LetÆ
0
(t)=p(x^
0 (t) x 0 (t+ 1)) (w(by 0
(t)) w(b^y))forallt2IN:Sinew()isaontinuousfuntion,Æ
k
(t) !Æ
0
(t)forallt2IN:
Itiseasytohekthatfx
0
(t);y
0
(t)gisaprogramfromx:SinefÆ
0
(t)gisitsassoiatedvalue-loss
sequene,ertainly P 1 t=0 Æ 0
(t)(x
o
):Ifwenowonsider thesetofnaturalnumbersINasameasure
spaeequippedwith aountingmeasure,weanappealto Fatou'slemma(see Rudin(1967;Theorem
11.31)via(6)to assert
(x o
) lim
k !1 1 X t=0 Æ k (t) 1 X t=0 liminf k !1 Æ k (t)= 1 X t=0 Æ 0 (t):
Thisompletestheproofofourlaim.
40
Proposition 10 Aprogramfx(t);y(t)gwhoseassoiatedvalue-losssequenefÆ(t)gsatises
P 1 t=0
Æ(t)=
(x(0)) isoptimal ifitexhibits the average turnpikeproperty.
Proof: WeknowfromProposition8thattheprogramfx(t);y(t)g isgood. Nowsupposethatitisnot
optimal. Thenthereexistsaprogramfx
0
(t);y
0
(t)g; x
0
(0)=x(0);anumber">0andatimeperiod t
" suhthat P T t=1 [w(by 0
(t)) w(by(t))℄>"forallT t
"
:Sinefx(t);y(t)g isgood,fx
0
(t);y
0
(t)gisgood,
andbyProposition6bothprogramssatisfytheaverageturnpikeproperty. LetfÆ
0
(t)gbethevalue-loss
40
T t " ; "< T X t=0 (w(by 0
(t)) w(by(t)))=p(x(T^ +1) x
0
(T+1))+
T X t=0 Æ(t) T X t=0 Æ 0 (t):
Fromtheminimalityof
P 1 t=0
Æ(t);thereexistst
0 "
suh thatforallT t
0 "
; ("=2)<p(^x (T +1) x
0
(T+
1))=p(^x(T +1) x^+x^ x
0
(T+1)):OntakinglimitswithrespettoT;weobtainaontradition.
Theorem2 Forany arbitrary initial stok, x
o
2IR
n +
; thereexistsan optimal programfrom x
o
: If the
initial stok x
o
equals x^ = y^ = (1=(1+da
))e(); then the stationary program f^x;^yg is an optimal
program fromx
o :
Proof: Considertheprogram whose existeneis assertedin Proposition9. From Proposition 8, itis
ertainlyagoodprogram,and henefrom Proposition6satisestheaverageturnpikeproperty. Allof
thehypothesesofProposition10arefullled,andweanappealtoittoompletetheproofoftherst
laim. Fortheseond laim,note thattheaggregatevalue-lossofthe stationaryprogramis(trivially)
zero and that it trivially satises the average turnpike property. Again, an appeal to Proposition 10
ompletestheargument.
5 Choie of Tehniques in the Long-Run
Wearenowinapositiontodesribewhattheeonomylookslikeinthelong-run. Towardsthisend,we
beginwithaharaterizationofthevonNeumannfaetasdesribedin MKenzie(1968,1986). Itisof
interestthat underourstandinghypothesis asdesribedin(1), thisredues toaline intheEulidean
spaeofdimension2n:
Lemma2 The von Neumann faet f(x;x
0
)2 : Æ
(^p;^x)
(x;x
0
)=0g is a subset of f(x;x
0
)2 : x
0 i
=
x i
=0;i6=; x
0 =(1=a )+ x g;
=1 d (1=a
);withequalityifthefeliityfuntion wislinear.
If thefeliity funtion isstritlyonave,the faetisthe singletonf(^x;x)g:^
Proof: Pik(~x ;x~
0
)2andy~2(~x;x~
0
)suhthatÆ
(^x;^p) (~x ;x~
0
)=0:From(5)weobtainw(b~y) w(b^y)+
^ p(~x
0 ~
x )=0:Onappealing totheonavityofw();thisredues to
w(b~y) w(b^y)w
0
(b^y)(b~y by)^ =)b^y by~ q(~x
0 ~
x)0: (7)
Thisombinedwith(4)and(3)yields
(1 ey az)+
n X i=1 ( i )y i + n X i=1 ( i )a i z i
+dq(x y)=0:
Thisimpliesthatz~
i
=0=y~
i
=x~
i
=x~
0 i
foralli6=: Furthermore,thaty~
=x~
andthat ~ y a ~ z
=1=)x~
+a
(~x
0
(1 d)~x )=1=)~x
(7)andtherebyontradit(4). Thusb~y=by^=
:Onappealingtotheomputationsabove,weobtain
thaty~
=1=(1+da
)=~x
;andhenethatx~
0 =(1=a )+ ~ x
=1=(1+da
):
For the reverse impliation in the linear ase, pik (x;x
0
) 2 suh that x
0 = (1=a )+ x ; x 0 i =x i
=0; i6=; andy
=x
:On substitutingthese valuesin theleft handside of(3),weseethat
itisequalto zero. ButthatispreiselyÆ
(^x;^p)
(x;x) inthelinearase.
Weannowpresent
Theorem3 Anyoptimal programfx(t);y(t)g onvergestothe vonNeumannfaet, andthus
lim t!1
x i
(t) =lim
t!1 y
i
(t) = lim
t!1 z
i
(t) = 0for all i 6=: If the feliity funtion w() is stritly
onave, lim
t!1
x(t)=lim
t!1
y(t)=(1=1+da
)e() andlim
t!1
z(t)=(da
=1+da
)e():
Proof: Suppose that there exists " > 0 suh that for all k 2 IN
+
; there exists t(k) k suh that
P i6=
kx(t(k))k>":Weanthenassertthat forthevalue-losssequenefÆ(t(k))g
k 2IN +
; thereexistsÆ
o
andk
o
2IN
+
suhthat forallkk
o
; Æ(t(k))Æ
o
:Iftheassertionisvalid, weobtainaontradition
to Proposition 8and omplete the proof of therst laim. Thus, suppose that the assertion isfalse.
Thenweanmanufatureasequeneofintegersfk
i g
i2IN
+
suhthatlim
i!1 Æ(t(k
i
))=0:Nowonsider
thesequenef(x(t(k
i
));x(t(k
i
)+1))g
i2IN
+
andappeal toProposition2toguaranteetheexisteneofa
subsequenethatonvergestoapoint(~x;~x
0
):Sineislosed,andw()isontinuous,Æ
(^p;^x) (~x ;x~
0
)=0:
Wenowappeal toLemma2toobtainaontraditiontoourinitialhypothesis.
Fortheaseofastritlyonavefeliityfuntion,repeattheargumentabovebutwithkx(t(k))
^
xk+kx(t(k+1)) xk^ >": Inthis ase,(~x ;x~
0
)=(^x ;x);^ andweagainappeal toLemma 2toobtaina
ontraditiontoourinitialhypothesis.
6 An Example of an Optimal Periodi Program
Inthissetion, wepresentanexampleofasimpleeonomywithalinearfeliityfuntion in whih the
optimal path yles around the golden rule stok. The eonomy has available to it only one typeof
mahinewhose(labor)inputandoutputoeÆients(a
1 ;b
1
)aregivenby(2=3;1);thedepreiationrate
dby1=2;thefeliityfuntion byw(by)=y;andtheinitialstokofthemahinex
o
=1=2:Theredued
formoftheeonomyis givenby:
= f(x;x
0
)2IR
2 +
:(1=2)x+(3=2)x
0
(1=2)xg;
(x;x
0
) = fy2IR
+
:yxand y1+(1=3)(x 2x
0
)g=fy2IR
+
:ymin[1+(1=3)(x 2x
0
);x℄g;
u(x;x
0
) = maxfw(by):y2(x;x
0
); (x;x
0
)2g=min [1+(1=3)(x 2x
0 );x℄:
Considertheprogramfx(t);y(t)ggivenbyx(t)=y(t)=3=4;withagrossinvestmentofz(t+1)=
x(t+1) (1 d)x(t)=3=4 (1=2)(3=4)=3=8;forallt2N:Welaimthatthisisastationaryoptimal
program from x(0) = 3=4: Towards this end, we show that (3=4;3=4) is the unique solution to the
First observethat u(3=4;3=4) = min[1 (1=3)(3=4);3=4℄ = 3=4; and that u(x;x) = min[1
(1=3)x (2=3)(x
0
x);x℄: Nowif0x<3=4; u(x;x
0
)<(3=4)=u(3=4;3=4):Thus supposex >3=4:
Inthis ase,x
0
x implies1 (1=3)x (2=3)(x
0
x) 1 (1=3)x<3=4;and henethat u(x;x
0 )<
(3=4)=u(3=4;3=4):Theargumentisomplete.
Next,onsider aprogramsuhthaty(t)=x(t)forallt2IN; x(t)=1=2forallevent2IN;and
x(t) =1forall oddt2IN:It easy tohekthat this isaprogram that startsfrom 1=2andosillates
around3=4:All thatweneedtoshowisthat itisanoptimalprogramstartingfrom1=2:Towardsthis
end, wenote that p^=q^=1=2;and that thisprogram makesazerovalue-lossin eahperiodat these
pries:
Æ(t)=
(1=2)+(1=2) (1=2)(1=2) 3=4=0 fort=0;2;
1+(1=2)(1=2) (1=2) 3=4=0 fort=1;3;
Sine it also satises the average turnpike property, an appeal to Proposition 10 then ompletes the
argument.
7 Choie of Tehniques with a Linear Feliity Funtion: An Example
Inthissetion,weturntothequestionofwhihmahinesareoptimallyusedandprodued{thehoie
oftehniques{whenthefeliityfuntionislinear. Inpartiular,weexaminetheoptimalityofapoliy
presribedinStiglitz(1968). Towardsthisend,letD=fi2f1;;n):b
i
=b
=(1+da
)gbethe
set ofmahine-typeswhoseoutput perunit laborratiosare notlessthantheeetiveoutput perunit
laborratioof mahinesof type:
41
Weshall refertosuhtypesasdesirableand to thosenotin D as
undesirable. UnderaStiglitzpoliy,laborisalloatedineahtimeperiodtoasetofavailabledesirable
mahines withahighertypeofmahinehavingapriorityoveralowerone,
42
andanyremaininglabor
alloatedtowardsproduingonlyonetypeofmahine,that delineatedby. Moreformally,
Denition8 Aprogramfx(t);y(t)g withan assoiatedgrossinvestmentsequenefz(t+1)gissaidto
beaStiglitz program iffor any t2INthe followingpoliy presriptionsarefollowed.
(i) If D=;orx
i
(t)=0for alli2D;lety(t)=0andz(t+1)=e();
(ii) If 0
P i2D
x i
(t) 1; let y
i
(t)= x
i
(t) for all i 2 D; y
i
(t)= 0 for all i 62 D; and z(t+1) =
((1 P
i2D x
i (t))=a
)e();
(iii) If
P i2D
x i
(t)>1andx
1
(t)1;lety(t)=e(1)andz(t+1)=0;
(iv) If
P i2D
x i
(t)>1andx
1
(t)<1;lety
i
(t)=x
i
(t)for alli=1;;i
o
1; y
io
(t)=1
P i
o 1 i=1
x i
(t)
andz(t+1)=0;wherei
o
2D suhthat
P
io 1
i=1 x
i
(t)<1and
P io i=1
x i
(t)>1:
Wean nowaskwhethertheset ofoptimalprogramsisidentialto thesetofStiglitzprograms,
and it is perhaps surprising that this is deisively not the ase. We present an example of a simple
41
SeeFootnote 30andtheassoiatedtext.
42
isbad,leavealoneoptimal. Weturntothedetails.
Theeonomyhasavailableto itonlyonetypeofmahine whose(labor)inputand output
oef-ients(a
1 ;b
1
)aregivenby(2=5;1);thedepreiationratedby1=2;thefeliityfuntion byw(by)=y;
andtheinitialstokofthemahinex
o
=1=2:Thereduedformoftheeonomyis givenby:
= f(x;x
0
)2IR
2 +
:(1=2)x+(5=2)x
0
(1=2)xg;
(x;x
0
) = fy2IR
+
:yxand y1+(1=5)(x 2x
0
)g=fy2IR
+
:ymin[1+(1=5)(x 2x
0
);x℄g;
u(x;x
0
) = maxfw(by):y2(x;x
0
); (x;x
0
)2g=min [1+(1=5)(x 2x
0 );x℄:
Considertheprogramfx(t);y(t)ggivenbyx(t)=y(t)=5=6;withagrossinvestmentofz(t+1)=
x(t+1) (1 d)x(t)=5=6 (1=2)(5=6)=5=12;forallt2N:Welaimthatthisisastationaryoptimal
program from x(0) = 5=6: Towards this end, we show that (5=6;5=6) is the unique solution to the
problemdelineated inDenition5,andhenethat3=4istheuniquegolden-rulestok.
First observe that u(5=6;5=6) = min[1 (1=5)(5=6);5=6℄ = 5=6; and that u(x;x
0
) = min[1
(1=5)x (2=5)(x
0
x);x℄: Nowif0x<5=6; u(x;x
0
)<(5=6)=u(5=6;5=6):Thus supposex >5=6:
Inthis ase,x
0
x implies1 (1=5)x (2=5)(x
0
x) 1 (1=5)x<5=6;and henethat u(x;x
0 )<
(5=6)=u(5=6;5=6):Theargumentisomplete.
Next, onsider a program suh that for all t 2 IN; x(4t) = 1 = y(4t); x(4t+1) = 1=2 =
y(4t+1); x(4t+2)=3=2;y(4t+2)=1; x(4t+3)=3=4=y(4t+3): It easy to hek thatthis is a
programthatstartsfrom 1andreturnsto itafter fourperiods. Itisalsoeasyto seethatitisaunique
Stiglitzprogramstartingfrom1. IntermsofDenition 8,D=f1g;andin threeofthefourperiodsof
the4-period yle,ondition(ii)applies andusage andprodutionlevelsareuniquelyset to maintain
fullemploymentandnoexessapaity. Inotherwords,in theseperiods, allofthedesirablemahines
areutilized,andalloftheremaininglabor(noneinoneofthe3periods)isalloatedtotheonstrution
ofnewmahines. Intheoneremainingperiod,4t+2;thereisfullemploymentbutalsoexessapaity.
Now suppose the Stiglitz program dened aboveis a good program. It is easy to hek that
u(1;1=2) = 1; u(1=2;3=2) = 1=2; u(3=2;3=4) = 1; and u(3=4;1) = 3=4: Hene for all n 2 IN
+ ; P
4n t=0
[u(x(t);x(t+1)) (5=6)= (1=12)℄;aontradition. FromProposition5,weanthenonlude
thattheStiglitz programisnotoptimal.
SinethisisauniqueStiglitzprogramstartingfromx(0)=1;theoptimalprogramfromx(0)=1;
whih existsbyvirtueofTheorem 2,isnotaStiglitz program.
8 Choie of Tehniques in Transition: A Linear Feliity Funtion
Inthelightof theexamplepresentedin Setion 7,weanask foronditionsontheRSSmodelunder
whih thesetofStiglitz programsoinidewiththesetofoptimalprograms. Weturntothisquestion.
Thepointto be notedabouteah of thetwoexamples presentedabove isthe partiular value
of the parameter 1 d (1=a
1
): We have already referredto this parameter in Setion 5as
1
suÆientondition forthe optimalhoie oftehniquethat wepresentin this setion then 43
requires
that
1:
Theorem4 Withalinearfeliityfuntionw;andwith1>
1;anyStiglitzprogramisanoptimal
program. 44
Thistheoremisaonsequeneofthefollowinglemma.
Lemma3 With alinear feliity funtion w; and with 1>
1;the aggregate value-losses of any
Stiglitz programstartingfromx(0)equal(x(0)):
TheproofofLemma 3reliesruially onthesouresofvalue-lossalreadyidentiedin theproof
ofLemma1. On rewriting(2), weobtain
Æ(t)=by^ by(t) q(x(t^ +1) x(t))
=
(1 ey(t) az(t+1))+
n X
i=1 (
b
i )y
i
(t)+
n X
i=1 (
i )a
i z
i
(t+1)+dqx(t)
= (t)+
X
i2D (
b
i )y
i
(t)+
X
i62D (
b
i )y
i
(t)+
n X
i=1 (
i )a
i z
i
(t+1)+dqx(t) (8)
where (t) =
(1 ey(t) az(t+1)) is the value-loss from unemployment.
45
This is a ve-fold
deomposition 46
of the value-loss at any time-period: the other four termsonern value losses from
inorretusageandinorretinvestment. Theproofannowbeexeutedbytheomparison,periodby
period, of themagnitudes ofthe value-lossesofthe Stiglitz programand thoseof any otherandidate
program. WerelegatethedetailstotheAppendix, andturntoa
Proof ofTheorem 4: Letfx
s
(t);y
s
(t)g;withanassoiatedvalue-losssequenefÆ
s
(t)g;beaStiglitz
program. Sineonlyonetypeofmahine isonstrutedunder aStiglitz'poliy, andsine
1;
weanassertthat theStiglitzprogramisagoodprogram,andtoProposition6toassertthattherefore
itexhibitstheaverageturnpikeproperty. Inordertoompletetheproof,weneedtoestablishthat the
hypothesesof Proposition 10are satised,whih is tosay thatfÆ
s
(t)g satises
P 1 t=0
Æ s
(t)=(x(0)):
AnappealtoLemma 3thenompletestheproof.
Nextweaskwhether,undertheonditionsidentiedinTheorem4,aStiglitzprogramisuniquely
optimal. Towardsthisend, weanpresent
Theorem5 Withalinearfeliity funtion w;andwith 1<
<1;anyoptimal programfx(t);y(t)g
isaStiglitz program.
43
NotethatbydefaultD=f1g=fgineahoftheone-mahineexamplesonsideredabove.
44
Notethat
<1:Ifnot,thend< (1=a
);aontradition.
45
Introduedonlyforthetypographialreasonofreduingthelengthoftheexpressionbelow.
46
For a general disussion of the importane of deomposition priniples aross timein problems of intertemporal
unlikeTheorem4,Theorem5doesnotoverthease
= 1:Indeed,Theorem5isfalseforthisase.
IntheexamplepresentedinSetion 6,there isan optimalprogramwhihis notaStiglitz program.
47
Thus,weneedtoruleouttheasewhentheoptimalprogramstaysinthevonNeumannfaetbutdoes
not onvergeto the golden-rule values. Towards this end wean present aresult whih strengthens
theonlusionsofTheorem3intheaseofalinearfeliityfuntion andalsoshowsthem toholdfora
Stiglitzprogram.
Proposition 11 With a linear feliity funtion w; and with 1<
< 1;for a program fx(t);y(t)g
that is either an optimal or a Stiglitz program, lim
t!1 y
i
(t) = lim
t!1 z
i
(t) = 0 for all i 6= ; and
lim t!1
y
(t)=lim
t!1 x
(t)=^x=1=(1+da
); lim
t!1 z
(t)=d=(1+da
):
WerelegatetheomputationaldetailstotheAppendix, andturntoasharpeningofLemma3.
Lemma4 In a setting with a linear feliity funtion w; and 1
< 1; let fÆ(t)g be the
value-losssequeneofaprogramthatisnotaStiglitzprogramandfÆ
s
(t)gthe value-losssequeneof aStiglitz
programstartingfromthesameinitialstok. Thenthereexists">0suhthat
P 1 t=0
Æ(t) P
1 t=0
Æ s
(t)>
":
Weindiate in theAppendix howthe proofofLemma 4is astraightforwardmodiationof the
om-putationspresentedin theproofofLemma3. Weannowpresent
Proofof Theorem5: Supposeto theontrarythatthereexistsanoptimalprogramfx(t);y(t)gwith
anassoiatedvalue-loss sequenefÆ(t)g that isnotaStiglitz'program. Letfx
s
(t);y
s
(t)g beaStiglitz
programstartingfrom x(0)andwithanassoiatedvalue-losssequenefÆ
s
(t)g:An appealto Lemma4
andtoProposition7yieldsforallT 2IN
+ ;
T X
t=0
(by(t) by
s
(t)) = p(x^
s
(T+1) x(T+1))+
T X
t=0 Æ
s (t)
T X
t=0 Æ(t)
< p(x^
s
(T+1) x(T+1)) ":
Weannowassertthatliminf
T!1
P T t=0
(by(t) by
s
(t))>0:Ifnot,thenweobtainlimsup
T!1
^ p(x
s
(T+
1) x(T +1))>":NowanappealtoProposition11leadsustoonludethatp"^ <0;aontradition.
Thisveriesthetruthoftheinitial laim,and ompletestheargumentthat anyoptimalprogram isa
Stiglitzprogram.
47
Adetailedveriationofthislaimwouldleadusoutsidethesopeofanalreadylongpaper;seeKhan-Mitra(2002)
InSetions5and6,weonsideredoptimalprogramsin theontextofboththelongandtheshortrun
whenthefeliityfuntion is linear;hereweseekto understand in whatwaythisaseis dierentfrom
theonepertainingto astritlyonavefeliityfuntion. Essentially,thesituation isthis. Whenfuture
utilities arenot disounted, utilities are summedarosstime, and in the linearasethis translatesto
summingonsumptionlevelsarosstime. Thus,there isnopartiulardisadvantagein postponingpart
oftoday'sonsumption toafutureperiod(orvie versa)solongasthesumoftheonsumption levels
inthetwoperiodsremainsthesameandonsumptioninallotherperiodsareunaeted. Inthestritly
onavease,in onsideringsuhswaps,onehastotakeintoaounttheoriginallevelsofonsumption
inthetwoperiods,beauseaeterisparibustransferofaunitofonsumptionfromaloweronsumption
levelperiod tothehigheronewilldenitelyreduesoialwelfare.
The senario in whih there an be a dierene in the hoie of tehniques is one where the
short-run onsumption requirements are quite dierent from the long-run onsumption requirements
on anoptimal program. As we haveseen in Setion 5, the uniquegolden-ruletype of mahine, , is
thebestmahine touseformeetingthelong-runonsumptionrequirements,regardlessofwhether the
soialwelfare funtion islinearorstritly onave. In theshort-run,however,theimportantquestion
iswhihmahinebuilttoday willprovidethemostonsumptiontomorrow,givenanavailable amount
oflaborfornewmahineprodutiontoday,andwithouttakingintoaountthefat thatthemahine
depreiates. Thisislearlyqualitativelydierentfromthelong-run(golden-rule)problemandpointsto
b i
=a i
ratherthantob
i
=(1+da i
):Whentheorderingsofthesetwomagnitudesdier,onemahineisbest
fortheshort-runproblemandanothermahineisbestforthelong-runproblem. Thisseemstosuggest
that,in transition,onemayproduethemahinethatisbest fortheshort-run,but asymptotiallyrun
itdownviadepreiation,sothatin thelong-run,onlythegolden-rulemahineisproduedandused.
Whilethebasiintuitionislear,itrequiresonsiderablemoreworktoshowthatsuhsenarios
anindeed our. First,wepresentthefollowing
Denition9 Aprogramfx(t);y(t)g withan assoiatedgrossinvestmentsequenefz(t+1)gissaidto
bea-programif for anyt2IN; z
i
(t+1)=0for alli6=:
By appealing to the example presented in Setion 7, we an see that a -program is not in general
an optimalprogram. Ourquestion annow be phrased asto whether every optimalprogram is a
-program. Towardsthis end,wepresentan exampleofasimpleeonomyin whih there aretwotypes
ofmahines(n=2)whoseinputoeÆientsvetorisgivenbya=(2;3); theoutputoeÆientsvetor
byb=(4;5)andthedepreiationrate,d;by0.45(m(1 d)=0:55):Notethat
b 1
a 1
=2>(5=3)=
b 2
a 2
and b
1
(1+da
1 )
2:1052<2:1276
b 2
(1+da
2 )
;
denedasfollows:
w(y)=
y 2 fory2
1000(y 2) for0y<2
Theinitialstokof mahinesisspeiedas x
o
=(0:5;0):
We know from the analysis of Setion 5that in the long-run only mahines of type 2will be
produed and used along an optimal program. We are interested in demonstrating that mahines
of type 1 will nevertheless be produed in some time period along an optimal program from x: Our
methodofdemonstratingthisistosuppose,ontheontrary,thatmahinesaretype1arenotprodued
inperiod1alonganoptimalprogram. Weshowthataonsequeneofthisis thatanoptimalprogram
will suer alarge disutility (negative utility, large in absolute value) in either the rst or the seond
periodofonsumption,whihresultsinalargedisutilityeveninthelong-run. Weonstrutaprogram
fromx
o
whihproduesmahinesoftype1initially,andreahesthegolden-rulestokinanitenumber
(speially,eight)ofperiods;ithasnon-negativeutilityinallperiods,and,ofourse,(positive)
golden-ruleutilityfrom periodnine onwards. This showsthatourhypothesisthatmahinesoftype1arenot
produed on the optimal program in period 1must be false, and ompletes the demonstration. We
relegatetheomputationaldetailsto theAppendix.
Inonlusion to this setion, note that the feliity w used in the aboveexample is non-linear,
but notstritlyonave. Wean hek that allthealulationsshown in theAppendix below remain
validwithastritlyonavewdenedasfollows:
w(y)=
(53=47)2(y 2)=(y 1) fory2
1000(y 2) 0:5(y 2)
2
for0y<2
10 The Prie-Support and Other Properties of an Optimal Program
So farwehaveworkedonly withthe golden-rulepriesystem, and in this setion weuse MKenzie's
priesupport propertypresentedas Theorem 6 below to make avarietyof observationsregarding an
optimalprogram: there is apositiveinvestmentof mahines of type foraninnity oftime periods,
mahines valuablein the presentwerevaluable in thepast,there is investmentin mahinesof type
onlyiftheyarevaluable,thatsuhmahinesarealwaysvaluable,anequationforthepriesofprodued
mahines, thepriesystemis boundedand nally, apropertyof aprie-system usingthemahinesof
type asanumeraire.
49
Notethatnoneoftheseresultsrelyonstritonavityofthefeliityfuntion;
they exploit the average-turnpike property of good programs, and as suh on the uniqueness of the
golden-rule stok. Thus the standing hypothesis presented as (1) above ontinues to be the driving
fore.
We begin the formalities with a prie-support property for the RSS model. Sine we do not
exlude the situation where the eonomy has no stok of mahines, x
o
= 0; the result is a diret
onsequeneofmethodsavailablein MKenzie(1986; ProofofLemma 1),rather thanaorollary. We
48
Notiethesharpnon-linearityinthewelfarefuntionneartheonsumptionlevelof2. Higheronsumptionlevelsare
rewardedatamarginalrateof1,whileloweronsumptionlevelsarepenalizedatthemarginalrateof1000.
49
fullled inourontext,andallowhis prooftowork.
Theorem6 Letfx(t);y(t)g be an optimal programstarting from an arbitrary initial stok, x
o
2IR
n + :
Then thereexistsasequenefp(t)g
1 t=0
; p(t)2IR
n +
;suhthat for all(x;x
0
)2andy2(x;x
0 );
w(by(t))+p(t+1)x(t+1) p(t)x(t)w(by)+p(t+1)x
0
p(t)x:
Nextweturntothederivationof propertiesthatanbeusedtoshowthat anoptimalprogram
isa-programbut arealsoofindependentinterest.
Proposition 12 Thereexistsasequeneft
i g
i2IN+
suhthat z
(t
i
)>0for alli2IN
+ :
Proof: Supposethis notto bethease. Then thereexists T 2INsuh thatfor allt T; z
(t)=0:
Sineallmahinesdepreiateattherated2(0;1),thisimpliesthatx
(t)!0ast!1;andtherefore
that the time-average of x
(t); x
(t) ! 0 ast ! 1: An appeal to Propositions 5 and 6furnishes a
ontraditionandompletestheargument.
Proposition 13 Forany t2IN; andanyi=1;;n; p
i
(t+1)>0=)p
i (t)>0:
Proof: For any time-period t and any mahine of type i; let x = x(t)+e
i
"; z = z(t+1); x
0 =
(1 d)x+z; y = y(t) where " > 0: Then, (x;x
0
) 2 , and y 2 (x;x
0
): Then from Theorem 6,
p i
(t+1)(1 d)" p
i
(t)" 0 =) (1 d)p
i
(t+1) p
i
(t): In partiular, if for some t 2 IN; and
i=(1;;n); p
i
(t+1)>0;thenwemusthavep
i (t)>0:
Proposition 14 Forany t2IN; z
(t+1)>0impliesp
(t)>0:
Proof: Suppose that for some time-period t; z
(t+1) > 0 and p
(t) = 0: Then Proposition 13
implies that p
(t+1) =0: Pik " suh that 0<" <a
z
(t+1); and dene x = x(t)+"e(); x
0 =
x(t+1)+((1 d)x
x
(t+1))e();andy=y(t)+"e():Theney=ey(t)+"anda(x
0
(1 d)x)<
a(x(t+1) (1 d)x(t)) a
z
(t+1) <az(t+1) ": Thus (x;x
0
)2; and y 2 (x;x
0
);and from
Theorem 6, w(b(y(t))+p(t+1)x(t+1) p(t)x(t) w(by(t)+b
")+p(t+1)x
0
p(t)x: This yields
w(by(t))w(by(t)+b
");aontraditiontothefat thatwisstritlyinreasing.
Proposition 15 Forany t2IN; p
(t)>0:
Proof: Supposethatthereexists t2INsuhthatp
(t)=0:FromProposition12,there existsa
time-periodt
i
>tsuhthatz
(t
i
)>0:FromProposition14,thisimpliesthatp
(t
i
1)>0:Ift i
=t+1;we
obtainaontradition. Ift
i
>t+1;wemakeasmany(nite)appealstoProposition13asisneessary
toobtainaontradition.
Proposition 16 Forany t2IN; andanyi21;;n; x
i
(t)>y(t)=)p
i
(t+1)(1 d)=p
