A Geometri Investigation
M. Ali Khan y
and Tapan Mitra z
August 17,2002
Abstrat
Weharaterizeoptimalpoliyinatwo-setorgrowthmodelwithxedoeÆientsandwithno
disounting. Themodelis aspeialization toasingletypeofmahineofageneralvintageapital
modeloriginallyformulatedbyRobinson,SolowandSrinivasan,and itssimpliity isnotmirrored
initsrihdynamis,andwhihseemtohavebeenmissedinearlier work. Ourresultsareobtained
byviewingthemodelasaspeiinstaneofthegeneraltheoryofresourealloationasinitiated
originally byRamseyandvonNeumannand broughtto ompletionbyMKenzie. Inadditionto
the more reent literature on haoti dynamis, we relate our results to the older literature on
optimalgrowthwithonestatevariable: speially,totheone-setorsettingofRamsey,Cassand
Koopmans, as well as to the two-setor setting of Srinivasanand Uzawa. The analysis is purely
geometri,andfromamethodologialpointofview,ourworkanbeseenasanargument,atleast
inpart,fortherehabilitationofgeometrimethodsasanengineofanalysis.
1 Introdution
Inthelate sixties,under thegeneralheadingof \tehnialhoieunder full employmentin asoialist
eonomy," Robinson [33, pp. 38-56℄, [34℄, Okishio[29℄ and Stiglitz [43℄, [44℄, studied the problem of
optimaleonomigrowthin amodelofaneonomyoriginallyformulatedbyRobinson[33℄,Solow[39℄
andSrinivasan[41℄(heneforth RSSmodel). 1
Theworkgeneratedontroversy. Stiglitz[43,Paragraph
1℄, [44, p. 421℄argued that the Robinson-Okishioassumption of axed labor alloationbetween the
onsumptionandinvestmentssetors hadnoplaein anexerisethatsoughttodeterminetheoptimal
Theprojet of whihthis paper isa part hashad a long gestation period { it began inIllinois in1986, reeived
invaluableimpetusfromBobSolow'spresentationattheSrinivasanConfereneheldatYaleinMarh1998,andresearh
support whileKhan was at EPGE, Funda~ao Getulio Vargas. It is a pleasure to aknowledge the enouragement of
ProfessorsSen,SolowandSrinivasan. Thispaperrepresentsworkinprogressandisnottobequotedwithouttheauthors'
permission {it isintendedfor RonJones' seventieth birthday {to the authors, ateaher, friendand a geometer par
exellene.
y
DepartmentofEonomis,TheJohnsHopkinsUniversity,Baltimore,MD21218.
z
DepartmentofEonomis,CornellUniversity,Ithaa,NewYork14607.
1
In[11 ,p.3℄,themodelisreferredtoastheSolow-Srinivasanmodel;alsosee[39 ,onludingthreeparagraphs℄and[11,
growthpath, and thereby an optimumtime-path of thealloation of labor. On her part, Robinson
ritiized theassumption of apositivedisountrate, ontinuoustime and thelinearityassumption in
thespeiationoftheplanner'sfeliityfuntion. 3
Stiglitz's[44℄responseis importantforthereord.
Evenifthere isaminimum onsumptiononstraintandanitegestation period, thepath
of development will, after an initial \adjustment" period, look exatlyas I havedesribed
it. [Unlike℄ long-runneolassialmodels withmalleable apital[where℄ theoptimal poliy
is alwaysof theso-alled bang-bang variety { ifthe initial apital laborratio is less than
its long-run equilibrium value there is always a period of zero onsumption, after whih
onsumptionjumpstoitslong-runequilibriumvalue,whereasinourex-postxedoeÆients
modelonsumptioninreasessteadily toitslong-runvalue.
InhisreentrevisitofSrinivasan[41℄,Solow[40℄asksforasolutiontothe\Ramseyproblemforthis
model."SineStiglitzhadaalreadyprovidedasolutionwith\linearutilityandpositivetimepreferene",
theopenquestiononerns\astritlyonavesoialutilityfuntionforurrentperapitaonsumption."
Indeed,Solowalsomentionsthat\adoptionof thisriterionanindeedleadtounjustiable negletof
earlyonsumption,"andifoneweretoshare \Ramsey'sbeliefthat theonlyethiallydefensiblesoial
rate of time preferene is zero, a suÆiently sharply-onave utility funtion would enfore a loser
approahtointergenerationalinequality."Inshort,thegeneralizationofStiglitz'sworkinthisdiretion
ofanon-linearfeliityfuntionremainsyet tobeaomplished. 4
In this paper, we do not report any results regarding this open problem, 5
but rather reonsider
Stiglitz'analysiswithalinearfeliity funtion,butwithoutdisountingandindisrete time.
Further-more, we totally abstainfrom the questionof the hoie of tehniqueand onsider an eonomywith
onlyone type ofmahine. Indeed,withallthese simpliations {themodel's parametersare redued
totworealpostivenumbers{itisnaturaltowonderwhetherthereisverymuhlefttodetermine;the
problemseemsatrstglanetohavebeensimpliedintoatriviality. Inthelightofallthis,theresults
goagainstStiglitz'intuitionsin awaythat isnothingshortofdramati. Weidentifyparametervalues
under whih the optimumprogramsonsist of two-period yles aroundauniquegolden-rulestokof
mahines, and the amplitude of eah yle is dierent for dierent values of the initial apital stok
lyinginanidentiableinterval. Outsidethisinterval,thereisonvergeneinnitetimetooneofthese
yles,butonethatisnotneessarilyofamonotonitype. Movingtootherparametervalues,weshow
onvergenetothegolden-rulestok,againnotneessarilyofamonotonitype,andonethattakesnite
orinnitetimedependingontheinitialstokofapitaltheeonomyisendowedwith. Thus,inseveral
instanes, Ramseyoptimalityrequiresover-buildingandunder-depreiating relativeto thegolden-rule
stok. Inpartiular,thedynamisystemfailsinapartiularlysharpwaythereentriterionof\history
independene"proposed in[26℄.
Thoughtheresultsappearsurprisingintermsofthevolatilityoftheoptimalprogramswhenviewed
againstthebakdropofearlierwork,theyseemtameinthelightofmorereentworkonhaoti
dynam-is. Ourmodel, despiteits simpliity, isa bona detwo-setormodel, and reentwork of Nishimura,
SorgerandYano presentsrobustmethodswhihallowonetoonstruttwo-setormodels ofthe
stan-dard Srinivasan-Uzawa type (see [42℄, [49℄) whose optimal programs exhibit haos under its various
2
Stiglitz[44,p.421℄writes\Theremaybesomespeialsituations...wheretheemploymentalloationisthesamefor
allsteady-statepaths,buteventhen,ingoingfromonesteady-statepathtoanother,oneannotinferthattheemployment
alloationisunhanged{anditisthisdynamiproblemthatwearedisussing."
3
WerestateRobinson'sritiismsinourownterminology;shephrasesthemintermsofa\disountratehosenone
andforall,...negligiblegestationperiods,...[and℄easingtoonsumeandlivingonairduringtherstphaseoftheplan."
4
Forsomepartialattemptsatsolution,see[45℄. In[11,p. 15℄,thisisexpressedas\Thelooseendremainsloose."
5
bealternativelyviewedasshowinghowoptimalprogramsloosetheirvolatilitywhenthereisdisounting
andoneofthesetors,theinvestmentsetor,usesonlyonefator, namelylabor. However,itis worth
observingthat in either ase,thefull impliationsofallof thisworkhaveyetto bedrawnoutfor the
eldwhihservedastheirprimarymotivation{developmenteonomis. 6
Whereas theresultspresentedin thispaperonstituteitsprimarymotivation,aseondaryonern
is to report what we see as a rehabilitation of geometri methods as an engine of analysis. rather
thansimplyas aninstrumentof illustration. Ourmethods allowus to gobeyondastatementof mere
onvergene,andharaterizethestrutureoftheoptimalpoliyinsomedetail,inludingitsdynamis
intheshort-run. Inpartiular,weanomputethenumberofperiodsittakesforanoptimalpoliyto
onvergeto thegolden-rulestok, and what it impliesfor the therange in whihthe initial stoksof
apitalmustlie. Thegeometrialtehniques thatweelaboraterestontwobasimathematialresults:
theKuhn-Tukertheorem ofoptimization theory, and theBrok theoremon theexistene of optimal
programs obtainedasthose minimizing value-loss at suitably dened pries; see [48℄, [3℄ respetively.
By showing how basi and beautiful ideas in the general theory of intertemporal resoure alloation
are amenable to geometri manipulation, albeit in the spei ontext of oursimple model, wehope
to make them aessible to a wider audiene. 7
In this movebeyond alulusto areourse to global
methods, weareverymuh inthetradition ofthelassialtheory ofinternationaltrade. 8
Wearealso
in tunewithasimilarmovein reentinvestigationsin eonomidynamis, asforexamplein thework
ofNishimura-Yano [27℄.
The paper proeeds asfollows. In Setion 2,wedevelopthe model and plae it in theontext of
thetwo-setor neolassialsetting originally assoiated withSrinivasan and Uzawa. 9
Setion 3and 4
lay outthebasigeometrial apparatus. Despite ourbest eorts,this materialmayperhapsproveto
bediÆultand tediousreading asaresult ofthe modern reader'srustyreall of Eulideangeometry.
To ease this, we have summarized at the end eah setion all that we hopeto takefrom it, and the
readerwilling toaeptthese assertionsanproeeddiretly to theharaterizationand disussionof
optimalpoliiesin Setions 5,6and7. Setion 8onsiders extensionsandommentsonissueshaving
todowithomprativedynamis. Setion9onludesthepaperwithanoverviewandalistingofsome
openquestions.
2 The Model and its Anteedents
Given itsfamiliarity,weintrodue theRSSmodel in termsof thestandardnotationof thetwo-setor
settingespeiallyfamiliartostudentsof(old) tradeandgrowththeory. 10
Letthere betwosetors
pro-6
Wemakeabeginningin[13 ℄.
7
Fora quiklistof the oneptsexposited inthispaper,the readeranseethe lastbutoneparagraph ofthe next
setion.Giventheauthors'emphasisonreursivemethods,noneoftheseoneptsmakeittothetextbookofStokey-Luas
[46℄,forexample.
8
SuhatraditionofexpositionandinvestigationofoursebeginswithMarshall's\PureTheoryofForeignTrade"and
ontinueswithSamuelson,Meade,Jones,Johnsonandothers. Intheontextofgrowththeory,seeKoopmans[14℄.Fora
suintstatementontheadvantagesofthegeometrimethod,seeforexample[17,p.5℄and[10,pp.9-10℄. Fordiering
reationstotheuseofgeometry,seeKurz's[15℄andShakle's[37℄reviewsofMeade'swork.
9
AlsoseeShell[38℄andHaque[7℄intheontextofoptimumgrowth.Theuseofthetwo-setormodelinthetheoryof
internationaltradehasalongtradition;see[9℄,[10℄andtheirreferenes.
10
InadditiontoSrinivasan[42℄,Uzawa[49℄,Shell[38℄andHaque[7℄intheaseofoptimumgrowthwithdisounting;
alsosee[25℄forreferenestomodernwork. Asiswellknown,themodelhasbeenusedandexposited byUzawa,Solow,
andthroughtheuseoftwofatorsapitalandlabor. Foreahdate t; t=0;1;;letthetehnologies
bestationaryand givenby
C(t+1)=F
C (K
C (t);L
C
(t))=minfK
C (t);L
C (t)g;
Z(t+1)=F
Z (K
Z (t);L
Z
(t))=(1=a)L
Z
(t)wherea>0;
and thealloationoffators (K
C (t);L
C (t);K
Z (t);L
Z
(t))is non-negative. Let K(t)>0and L(t)>0
denotetheamountsofapitalandlaborthatareavailableinperiodt:Thenwehavehavethefollowing
(materialbalane)onstraintsonlaborandapitaluseineahperiod:
L
C (t)+L
Z
(t)1andK
C (t)+K
Z
(t)K(t):
Sinetheinvestmentgoodsetordoesnotuseapitalat all,weareworkingunder theextremeaseof
the assumptionthat the onsumption good setoris more apital-intensivethan theinvestment good
setor. 11
Inthistwo-setorsetting,Z(t+1) onstitutesthegrossinvestmentduring period(t+1)and
undertheassumptionthatapital depreiatesattherated2(0;1);weobtainforeahtime-period
Z(t+1)=K(t+1) K(t)+dK(t):
It follows that\investmentis irreversible"; by theveryspeiationof themodel, gross-investmentis
restritedto benon-negativeateah date. Underthe assumptionof apositivedisountfator Æ<1;
andastationaryfeliityfuntionU();leadingtothemaximizationof P
1
t=0 Æ
t
U(C(t))subjettoagiven
initial stokof laborand apital(K(0);L(0))0; we obtainaspeial aseof thetwo-setor optimal
growthmodel exposited,forexample,inMitra[25℄.
We nowfurther speializethis modelby assumingthat thedisount fatorÆis unity, 12
thefeliity
funtion U() is linearand normalized sothat U(C(t)) =C(t); and that laboris stationary and
nor-malized to unity. 13
To reet the latter assumption, we shall use lower-ase letters (t) and z(t) for
theonsumptionandinvestmentgood,and x(t)forthestokofapital. Sinetheapital-laborin the
onsumptiongoodsetorisunity,laboruseinthatsetorequalstheamountofapitalbeingusedwhih
inturnequalstheamountsofoutputandutilityproduedin eahperiod. Weshalldenotetheommon
value of allthese four quantities by y(t); but taking itin the rstinstane to referto the amountof
mahinesbeingusedintheonsumptiongoodssetor.
Insupposingthatfuturewelfarelevelsaretreatedlikeurrentonesintheplanner'sobjetivefuntion,
weofoursetakeourleadfromRamsey[31℄, butratherthantheassumptionofa\blisspoint"orthat
of\apitalsaturation", asin [31℄,[36℄and[35℄, wefollowtheliteratureandworkwiththe\overtaking
riterionof optimality" assoiatedwith is due toAtsumi [2℄ andvonWeiszaker[51℄. Thus, given an
initial apital stokx
0
0; we work with programs startingfrom x
0
: These are simply sequenes of
apitalstoks(non-negativenumbers)fx(t)g 1
t=0
suhthatx(0)=x
0
andwhihsatisfythetehnologial
oftheuseofthismodelinHeksher-Ohlin-Samuelsontradetheory,asexpositedforexamplein[9℄and[10℄.
11
Weinvitethe interestedreadertoredothe geometrianalysisof[8℄ underthe fatorintensityassumptionsof this
paper.
12
Fromnowon,Æwillhaveatotallydierentmeaninginthesequel.
13
Thelatterassumptionseemstobeinkeepingwiththetoneofthetime;see[1,Setion1.2℄fortheexpositionofwhat
istermedthereastheCass-Koopmans-Ramseymodel.SuhanassumptionisalsomadebySrinivasan[41℄andbyStiglitz
andmaterialbalaneonstraintslaidoutabove. Aprogramfx (t)g
t=0
startingfromx(0)issaidtobe
optimalifforanyotherprogramfx(tg 1
t=0
startingfromx(0)
liminf
T!1 T
X
t=1
((t)
(t))0;
where f(t)g 1
t=1 and f
(t)g 1
t=1
represent the onsumption sequenes assoiated with eah of the two
programsunder onsideration. Note thatthe liminf operationanalternativelystatedto saythat for
theprogramfx(t)g 1
t=0
; givenany">0;thereexists atimeperiodt
"
suhthat
T
X
t=1
((t)
(t))"forallT t
" :
Thusanoptimalprogramisonein omparisonto whih nootherprogramfrom thesameinitial stok
iseventuallysigniantlybetter,foranygivenlevelofsigniane. Aprogram 14
fx(tg 1
t=0
issaidtobe
stationaryifitisonstantovertime,i.e.,x(t+1)=x(t)forallt=0;1;:Aprogramissaidto bea
stationary optimal programifitisstationaryandoptimal.
Sofar,inemphasizingthemaximizationofaplanner'spreferenes,wehavenotfoussedsuÆiently
on the assumption of a xed oeÆients tehnology in our model. It is this assumption that looks
towards the multi-setoralsetting and thereby takes its lead from Von Neumann [50℄. The work of
Gale[5℄, Brok[3℄andMKenzie [18℄ anbe seenasstemming from vonNeumann'spaper,and nds
itsulminationin ageneralsoalled \reduedform"model 15
summarizedbytwobasiparameters: a
period-to-periodtehnologysetina(produt)spaeofapitalstoksinitialandterminaltotheperiod,
and a planner's utility funtion dened on this set. 16
Suh a formulation is exible enough to yield
asspeialasesthevarietyofgrowthmodelsstudied inthe literatureand therebyqualify asageneral
theoryofintertemporalresourealloation. 17
Itisthroughthevoabularyofthistheorythatweanalyze
thesimple two-setor model studied here. 18
In additionto Brok'stheorem, suh aformulationleads
us to disuss in the sequel the golden-ruleapital stok and its assoiated golden-rule prie system,
value-lossperperiodanditsaggregateovertime,thevonNeumannfaetanditsprivilegedsubset,good
programs,andtheaverageturnpikeproperty. IntermsofaomparisonwiththemethodsofPontryagin,
as used in our ontext in [43℄, perhapsthe ruial dierene is the possibility of a omplete analysis
basedonthegolden-rulepriesystemfromtheverystart,ratherapriesystemthatorrespondstothe
optimalprogram,andisthentypiallyshowntoonvergetoit. 19
14
Ifwedonotspeifythestartingpointofaprogram,thereadershouldtakeittomeanthatitstartsfromitsvalueat
timezero.
15
In[23,p. 389℄,MKenziedatesthis\reduedform"modelto1964,anddesribestheworkofGale,MFaddenand
himselfasa\fusionoftheRamseyandvonNeumannmodels. Malinvaud's1953paperisalsoanimportantstepinthis
evolution.
16
Inhiswork,MKenziedoesnotgiveanamefortheanalogueinageneralsettingofthesetthatwedenoteby. In
[25℄,itisreferredtoasthetransitionpossibilityset.Theevolutionofthissetisofsomeinterest:in[18℄,itinludesvalues
ofutilities,while[19℄retainsthenotationalonventionof[4℄wherepositivenumbersstandforoutputsandnegativeones
forinputs.
17
Forstatementsofthetheoryanditssope,see[21 ℄,[22 ℄,[23℄and[25℄.
18
Asweshallindiateinthesequel,therelevanttheoremsofthistheorydonotdiretlyapply,butthemethodsleading
totheproofsofthesetheoremsextendtooursimplesettinginastraightforwardway.
19
Thisisofoursenottosaythatapriesystemassoiatedwithanyoptimalprogramhasnoroletoplayinthetheory
studyinthesubsequentsetionsissummarizedbytwoparameters: positiverealnumbersaanddwith
d<1:
3 The Basis of the Geometry
WebeginwithFigure1inwhihthe45 o
-degreelineservesasanimportantbenhmark. Theamountof
apitalstok(numberofmahines), 20
x(t);availabletodayismeasuredontheX-axisandthatavailable
nextperiod,saytomorrow,x(t+1);ontheY-axis. Figure1highlightsthefatthatweworkindisrete
time; when the partiular time period is not to be emphasized, we shall use the symbols (x;x 0
) for
(x(t);x(t+1)):
Next,wefousonthelineODlyingstritlywithintheoneformedbytheX-axisandthe45 o
-degree
line. The slopeof OD measurestherate ofdepreiation: 21
aunit of today'sapital stokdepreiates
to(1 d) unitstomorrow. Theline representsaonstraintembodyingapreisetime-invariantformof
depreiation, onethat doesnotdistinguishonthis sorebetweenmahinesprodued today andthose
produedinthepast,aswellasthefatthatapitalannotbedisposedothroughamarketorinany
otherway. Thusit alreadyalertsthereadertothefat that weworkwithirreversibleinvestmentand
thatthereisaboundto disinvestment. Anypoint,sayS
d
;onODrepresentsasituationinvolvingzero
netinvestment,andwithnegativegrossinvestmentbeingequaltothedepreiationontheapitalstok.
ThelineMLisdrawnparalleltoODandrepresentstheonstraintoftheavailabilityoflaborineah
period. 22
Thelength 23
OM representsthemaximalamountofapitalstokavailabletomorrowifthere
isnoavailableapitalstoktodayandallofthelaborforeisemployedintheinvestmentsetor. Sine
aunitsoflaborarerequiredtoprodueoneunitoftheapitalstok(mahine),thislengthequals(1=a):
Sinethelines MLand OD areparallel,thedistane 24
betweenthemis aonstantequaltoOM, the
onstantmaximalamountofnetinvestmentthatanbeundertakenatanyperiod. Itsonstanyreets
boththestationarityoflaborandthefat thatmahinesarenotneededtomanufaturemahines.
Thefatthatonemahineisusedalongwithaunitoflabortoprodueoneunitoftheonsumption
good, the tehnologial speiation of the onsumption setor, is not expliitly graphed in Figure
1. 25
Given this, the tehnology is represented by the \open" parallelogram enlosed by the lines
OM; MLandOD. 26
Weshallrefertoelementsofthisperiod-to-periodprodutionsetasone-period
plans. 27
Note thatunlike[4℄,theseare notsimplyprodutionplansbut alsoontainaspeiationof
onsumption levels that theypermit. This will beome leareras we proeed. Forthe moment, two
featuresofoughttobenotedinourontext: (i)theabseneoffreedisposal,alreadymentionedabove,
20
Weshallusethesetwointerpretivephrasesinterhangeably.
21
Theequationofthislineissimplyx 0
=(1 d)x:
22
Theequationofthislineissimplyx 0
=(1 d)x+1=a:
23
Thereaderiswarnedtobealerttoidentialsymbolismbeingusedtodesignatelengthsandlines.
24
Notethatwedepartfromtheonventionalnotionof(Eulidean)distanebetweentwosetswhihinvolvesthelength
oftheprojetionofapointonelinetotheother.
25
In[20,p. 846℄,Mkenzie writes\Whenourinterestisanasymptotipropertyofthepathofapitalstoks,thereis
noneedtoshowhowutilitydependsonprodutionandonsumptionduringtheperiod...thesignianthoiefromthe
viewpointoftheintertemporalmaximizationproblemisthehoieofterminalstoksgiveninitialstoks. Thisxesthe
ontributionoftheperiodtotheoptimalprogram."However,asweshallseebelow,inourspeiontext,thediagram
anbeusefulevenforthedelineationofonsumptionlevelsandofdynamisintheshort-run.
26
Formally,=f(x;x 0
)2IR
+ IR
+ :x
0
(1 d)x0anda(x 0
(1 d)x)1g.
27
OM;nextperiod. 28
Beforeweturntothedepitionofpreferenesinthe(x;x 0
)-planerepresentedbyinitialandterminal
apitalstoks,weonsiderthelinesegmentMV:Thisprovestobea(most)valuableonstrutforthe
geometridevelopmentof themodel; itdelineatesseveralimportantaspets ofthemodeland weturn
totherstofthem. Note,tobeginwith,thattheslopeofthelineMV isgivenby
MV 0
V 0
V
=(1 d) 1=a; to bedesignatedby: (1)
SinetheoordinatesofthepointsM and V areknowntobe(0;1=a)and(1;1 d) respetively,itis
now asimplematter to getthe equationof theline MV: However,itis moreuseful rstto reet on
whatsuhanequationequates. 29
Towardsthisend,note thatunliketheinvestmentsetorwhih does
notuse apitaland isdependentsolely on labor,the onsumptionsetorusesbothlaborandapital,
andistherebysubjettotwoonstraints{oflaboraswellasthatofapital. Sinetheonlyprodution
tehniqueavailable to this setor entails a unit apital-laborratio, we an use thesame symboly to
representtheamountoftheonsumptiongoodbeingproduedinagivenperiod,aswellastheamounts
ofapitaland laborbeingused. (Sinethemarginalutilityoftheonsumptiongoodisalsounity,y is
simultaneouslyaproxyforthenumberofutils thattheplannerobtains,butthisanbebrakettedfor
the moment.) Thus, any one-periodplan (x;x 0
) onstrains y by the apitalavailability x and by the
residuallabor available one thelaborrequirementsof theinvestmentsetorhavebeenmet. Thus, in
symbols,
yx(the apitalonstraint); y1 a(x 0
(1 d)x)(thelaboronstraint): (2)
The line MV then represents asituation where the apital and labor amounts available to the
on-sumptionsetorareidentialandthereforeidentiallyequalto theamountofonsumptiongoodbeing
produed. Assuh,MV isthelousofinitialandnalterminalstoksforwhihthereisfullemployment
andnoexessapaityofapital. Itanbereferredtoasthefullemployment,noexess-apaityline.
Atapointof\below"thelineMV,thepointsS
3 andS
d
forexample,allmahinesarebeingutilized
butthereissurpluslabor. Similarly,atapointof\above"MV,pointsS
1 andS
`
forexample,there
isfullemploymentbut idlemahinesandexessapaity.
Sine this is a ruial onsideration of the model, a further elaboration of this fat is perhaps
warranted. Note that MV divides into two parts: the triangle MOV and the \open" remainder
LMVD:Consideranyone-periodplaninthetriangleMOV;saySrepresentedbythegenerioordinates
(x;x 0
):AttheinitialapitalstokxorrespondingtoS;anetinvestmentofSS
d
leadstofullemployment
oflabor. Onappealingtopropertiesofsimilartriangles, 30
weobtain
SS
d
MO =
x 0
(1 d)x
1=a =
VS
d
OV
=1 x: (3)
Sinetheapital-laborratiointheonsumptionsetorisunity,xalsorepresentstheemploymentinthis
setor,andhene(3) furnishestheharaterization(equation)of thelineMV as thefull-employment
linewhenitisrewrittenas
1 a(x 0
(1 d)x)=x=)x 0
=(1 d 1=a)x+(1=a)=x+(1=a): (4)
28
Thisdoesnot ofourseimplya\free lunh" {laborwhihiskeptinthe bakground,isbeingusedtoproduethe
output. MKenzie'sintuitionofthissetin[23,Figures7and8℄,forexample,bearsomparisonwithFigure1.
29
TheequationofthelineextendingthesegmentMV isgivenbyx 0
=x+1=a;see(4)below.
30
AtthepointS
2
;forexample,theamountofterminalapitalstokrequiredtogeneratefullemployment
is given by the ordinate of the point S; and sine net investmentat S
2
is only S
2 S
d
; theordinate of
S
2
falls short by theamountSS
2
. ThissegmentSS
2
thenisameasure ofsurpluslabor. Similarly,at
theone-periodplan, S
3
;thereissurpluslaborin theamountS 0
1 S
3
:Thus thetriangleMOV represents
planswithnoexessapaityofapitalbut surplus,unemployedlabor.
When we turn to a one-period plan, say S
1
in LMVD; the situation is reversed. At the initial
apital stokx orrespondingto S
1
; wehavealready seenthat anet investmentof S
d
S supports full
employmentof labor. But at S
1
; there is more investment than this, and onsequently lesslabor to
manalltheavailable stokof apital. Atthis stage,note analternativeinterpretation ofthelinesOD
andML;theyareiso-employmentlinespertainingtotheonsumptionsetor. MLrepresentsthelabor
onstraintin(2)abovewithy equatedtozero,andOD withitequatedtounity. Thusthelinepassing
through S and parallel to ML and OD represents theemployment in the onsumption setorat the
initialapital stokx:Its interesetionwith thelineMV furnishes uswith theapitalstokthat this
reduedamountoflaboranuse,andatS
1
;itisgivenbytheabsissaofthepointS 0
1
:Thisanbeseen
againbyonsideringsimilartriangles.
S
` S
1
S
` S
= MS
0
1
MS =
S 0
1 S
00
1
x 0
S
= apaityutilizationrate,tobedesignatedby: (5)
ThusattheplanS
1
;thereisfullemploymentoflabor,but idleapaityoftheamountx S 0
1 S
00
1 :Only
partof thestokof available apital, asmeasured byS 0
1 S
00
1
is used. Similarly,at theone-period plan,
S
`
;theapaityutilizationrateiszerowhileatS itisunitywithalltheavailableapitalstokbeing
used. Wethus havetheharaterization(equation) oftheline MV asthenoexess apaityline. In
summary,LMVD representsplanswithexessapaityofapitalbut fullemploymentoflabor.
It isworthnotingat thisstagethatat agiveninitial apitalstokx;asinFigure 1,themovement
from S
` to S
d
traesoutaprodutionpossibility surfaein thespae ofonsumption and investment
goods in Figure 2. It is useful to see how the arguments furnished above in the ontext of Figure
1 translate to Figure 2. If all labor is employed in the investment setor, the eonomy is at S
` in
Figure2, withompletespeialization, fullemploymentandzeroutilization rate:As wemovedown
the prodution possibility surfae, dereasedinvestment and onsequent releaseof labor leadsto the
produtionoftheonsumptiongood,andtoaninreasein:BeyondS,thereisnofurtherinreaseinthe
onsumptionisblokedbytheapitalonstraint,andtheonlyeetisaninreaseinunemployment. 31
Weannowundersorethefatthatapartiularone-periodprodutionplaninFigure2represents
morethansimplyinitialandterminalapitalstoks. Giventheparametersweworkwith,foranyplan
inthetriangleMOV;itsabsissa(rstoordinate),infurnishingthefullyutilizedstokofapitalbeing
usedin theonsumptionsetor,alsofurnishes theamountoflaborthat isbeingusedthere, aswellas
theamountof theonsumption goodandtheutilitythat itprovides. Ontheotherhand,foranyplan
in LMVD;asdisussedabove,onean determinetheapitalstokthat isbeingused,and fromit all
oftheotheramounts. 32
All these onsiderationsare useful when we turn to the depition ofthe planner's preferenes. In
Figure 3, onsider the point I
1
on the MV line. By speiation and onstrution, the amounts of
labor and apital being used in the onsumption good setor are all idential to the amount of the
31
NotehowRybzynski'stheorempertainingtoanequilibriumwithoutspeializationholdsinoursetting. Asxinreases,
theapital intensiveonsumptiongoodinreasesandthe investmentgood(withzeroapitalintensity)dereases. This
onlusionisreversedwithaninreaseoflabor.Fordetails,seeforexample[8,9℄,andalso[10 ℄.
32
1
Thisimpliesadereaseintheterminalapitalstokx 0
andaonsequentinreaseinthelaboravailable
totheonsumptiongoodsetor. Butthis setorisonstrainedbyapitalandhenenoinreasein the
onsumptiongood,andtherebyinutility,ispossible. Intermsofourpreviousdisussion,laborisbeing
madeinreasinglyavailableinazonein whihthereisalreadysurpluslabor. Ontheotherhand,move
outwardsfrom I
1
inadiretion parallelto MLor OD:This moverepresentsaninreasein theinitial
apitalstokxandaonsequentinreasein theapitalavailable totheonsumptiongoodsetor. But
no labor is being released and this setor now faes a labor onstraint. Hene, again no inrease in
theonsumption good, andthereby in utility, is possible. Intermsof ourpreviousdisussion, apital
is beingmade inreasinglyavailable in azone in whih there is alreadyidle apaity. Thusweobtain
anindierene urveI
1 I
1 I
1
of theLeontief typewithakink atits intersetionwith MV: Iftheinitial
apitalstokisinreasedandterminalapitalstokdereased,relativetoI
1
;asatI
2 orI
3
forexample,
aninreasedamountofbothapitalandlaborismadeavailabletotheonsumptionsetor,resultingin
ahigherleveloftheonsumptiongood,andtherebyofutility. ThelineMV thustakesonanewidentity
havingsolelytodowithpreferenes;itpegsamapofkinkedindiereneurvesin theinitial, terminal
apitalplane. Theplanner'spreferenesareompleteinthesensethatwenowhaveanindierenemap
overalloftheperiod-to-periodprodutionsetwithOMLmarkingthezero-utilityindiereneurve.
Weshalldenotetheutilityfuntion 33
orrespondingtothisindierenemapis by(x;x 0
) !u(x;x 0
):
Thebasisofourgeometrialapparatusarenowalllaidout: the45 o
-line,thelinesMLandOD
on-stituting;thepointV;andthelineMV delineatingbothaprivilegedsubsetofandtheindierene
maporrespondingto u. 34
4 Determination of the Benhmarks
Therst unknownto bedetermined isa standardbenhmark in thetheory of intertemporalresoure
alloation{thelevelofthe apitalstok thatallows amaximalsustainableutilitylevel,the so-alled
golden-rulestokx:^ Inthewordsof[23,p. 389℄,\atapointofmaximalsustainableutility,theterminal
apaitalstoksmustbeaslargeastheinitial stoks,andtheutilitymustbeas largeaspossiblegiven
thisondition." 35
It iseasily seenin Figure4to betheuniqueone-periodplan Gobtainedbytheintersetionof the
45 o
-degreeline withMV:Butthegeometry atuallyenablesusto omputeit onewereall theslope
ofthelineMV identiedin (2)above.
MG 0
G 0
G =
(1=a) ^x
^ x
= =) 1=a
^ x
=1 =d+1=a=)x^= 1
1+ad
: (6)
The golden-rule stok is solution to a maximization problem in whih the objetive funtion u
maximized overtheonstraintset and theadditional sustainability onstraintthat terminal stoks
arenotlessthat theinitialone. 36
As suh,there isashadowpriep^assoiatedwiththesustainability
33
Analytially, the utilityfuntion is given by u(x;x 0
) = maxfy 2 IR+ :0 xandy 1 a(x 0
(1 d)x)g =
min[1 a(x 0
(1 d)x);x℄:
34
Theperiod-to-periodprodutionsethasbeenspeiedinFootnote 26andtheutilityfuntionuinFootnote 33.
35
MKenzie[23,p. 389,paragraph 3℄also refersto thisas a\von Neumann point". Fora formaldenition inthe
ontextofthismodel,see[12 ℄,andmoregenerally[5℄,[3℄.
36
Analytially,wemaximizeu(x;x 0
)subjettox 0
xforall(x;x 0
onstraint. Onappealing toUzawa'sversionoftheKuhn-Tukertheorem[48℄, weanwrite
u(x;x 0
)+p(x^ 0
x)u(^x;x)^ forall(x;x 0
)2: (7)
WeannowfollowRadner[30℄ anddene thevalue-lossÆ
(^p;^x) (x;x
0
) atthegolden-rulepriesystemp^
assoiatedwiththeone-periodplan(x;x 0
)byre-writingtheaboveas 38
Æ
(^p ;^x) (x;x
0
)=u(^x ;x)^ u(x;x 0
) p(x^ 0
x)forall(x;x 0
)2: (8)
Allthis isastandardrehearsalofakeyoneptinthegeneraltheoryofintertemporalresoure
alloa-tion. 39
Whatis possiblynewisthat wehaveallthemahinery weneedto furnish aleargeometrial
representationofthisidea.
Webeginwiththedeterminationofthegolden-rulepriesystemp:^ Towardsthisend,onsiderFigure
4,andnote thatthezeronetinvestmentone-periodplanV;givenby(1;(1 d));anbesubstitutedin
(7)toyield
1 p(1^ d 1)x^=)p^ 1 x^
d
=)p^ V
0
V
PV
: (9)
Bythesametoken,themaximalnetinvestment,zeroonsumptionone-periodplanM;givenby(0;1=a);
anbesubstitutedin(4)to yield
0+p(1=a^ 0)x^=)p^ ^ x
(1=a)
=)p^ OG
00
MO
: (10)
Now,onusingthevalueofx^in(3), aneasyomputationyieldsthat
1 x^
d = 1 1 1+ad d = ad
d(1+ad) = 1 1+ad 1=a = ^ x (1=a) :
Henetheweakinequalitiesareallequalitiesinthefollowingexpression
V 0
V
PV =
1 ^x
d p^ ^ x (1=a) = OG 00 MO : (11)
Butthen in termsofthegeometrial development,wehaveshown thatthe angle\OMG 00
equalsthe
angle\V 0
PV:SinethetriangleMOG 00
isongruenttothetriangleOM 0 G 00 ; \OM 0 G 00 =\OMG 00 =\V 0
PV =\PV 0
G;
whih implies that the line M 0
O is parallel to the PV 0
: We have thus shown that the golden-rule
prie-systemisgivenbytheslopeofeither ofthelinesM 0
O andPV 0
: 40
37
Notethatalloftheonditions fortheinvokingofthistheoremaresatised: uisaonavefuntion,isaonvex
set,andthepositivityofaleadstoSlater'sformoftheKuhn-Tukerqualiationbeingsatised. AlsoseeGale[6℄fora
generalizationrelevanttoontextofgrowththeory.
38
NotethatwearedeningafuntionÆ
(p;^^x)
: !IR+:Wedepartfromtheliteratureinretainingthesubsript(^p;x)^
inÆ
(p ;^^x) (x;x
0
)eventhoughthegolden-rulepriesystemwillremainxedinthesequel.
39
Assurveyedforexamplein[21℄.OnemayquotehereGale's[6,p.22℄statementthattheneessityandsuÆienyof
(7)forthex^tobethegolden-rulestok\providesthesinglemostimportanttoolinmoderneonomianalysisbothfrom
thetheoretialandomputationalpointofview".
40
one-periodprodutionplansforwhih Æ
(^p;^x) (x;x
0
)=0wherefrom(5),
Æ
(^p ;^x) (x;x
0
) = u(^x ;^x) u(x;x 0
) p(x^ 0
x)
= [u(^x ;x)^ x p(x^ 0
x)℄+[x u(x;x 0
)℄ (12)
= shortfallfrom golden-ruleutilitylevel+idleapaity
TherstpointtobenotiedinthisonnetionisthattheslopeofthelineMV whihwasidentiedto
be in (1)turns outto beharaterized alsoby thegolden-rulepriesystemin apartiularly simple
form. WorkingwithFigure4,weobtain
= MG
0
G 0
G =
MO G
0
O
OG 00
= 1
G 0
O
MO
OG 00
MO =
1 p^
^ p
; (13)
wherethelastequalityfollowsfromtheaboveindentiationofthegolden-rulepriesystem. Onethe
slopeofthelineMV isdetermined,weneedonlyitsintereptforafullidentiation. SineMV passes
throughthegolden-rulestokpointx^onthe45 o
-degreeline, weobtainitsequationas
x 0
=
1 p^
^ p
x+C=)^x= ^ x
^ p
+C=)px^ 0
+(1 p)x^ =x^=u(^x;x):^ (14)
Wehavealreadyseentheline MV servingthree distint roles: asthefull employment line, asthe no
exess apaity line and and as benhmark line for delineating the planner's preferenes. The latter
yielded, in partiular,the onlusionthat theutility level,u(x;x 0
);of anyone-period produtionplan
(x;x 0
)onMV isx:Onsubstituting(14)in(12),weobtain
^ px
0
+x px^ =u(^x ;x)^ =)u(x;^ x )^ u(x;x 0
) p(x^ 0
x)=0=Æ
(^p;^x) (x;x
0
); (15)
and thereby disovera fourth identity of the line MV in this geometri development: it is the zero
value-loss linewith value-loss governedbythe golden-rulepriesystem. It is worthpointing outhere
thatalltheone-periodprodutionplansonthezerovalue-losslineonstitutewhat isreferredtoasthe
vonNeumannfaet. 41
But now, we are in a position, through Figure 5, to determine the value-loss of any one-period
prodution plan, whih is to say, of any point (x;x 0
) 2: As (12)makeslear, this value-loss is not
determinedsolelybythediereneininterepts(say,onthe45 o
-degreeline)ofMV andalineparallel
toitandpassingthroughtheone-periodprodutionplan. Foronething,thiswouldleadustoonlude
that there is value-gain asthe line MV moves outwards. Lines parallel to MV are indeed
iso-value-losslines, but theydepit the value-loss after taking exess apaity into aount. Of ourse, for all
one-periodprodutionplans, sayS
1
,in thesurpluslabortriangleMOV; thereisnoexessapaityof
apitalandheneitsutilityisfurnishedbyitsrstoordinate,leadingtotheseondtermin(12)being
zero. Hene its value-loss onsists onlyof its shortfall from golden-ruleutility level, therst term in
(12). Thisisgivenbythedierene betweenx^andtheabsissaofthepointof intersetionS of aline
M 0
V 0
paralleltoMV andpassingthroughS
1
: Toseethis,lettheoordinatesofS begivenby(x ;x 0
);
41
SeeMKenzie[21℄,[22℄,[23℄. In[23,p. 391℄, MKenziewrites\Thevon-Neumannfaetplaysaruialroleinthe
multi-setorRamseymodelone the assumptionofstritonavity oftheredued utilityfuntionisdropped."Asthe
andhenetheequationofthelineM V isgivenby
x 0
=
1 p^
^ p
x+C=)x 0
=
1 p^
^ p x+ x 0 + 1 p^
^ p x : (16)
If we now denote the absissa of S as x o
; and realling that S is the intersetion of M 0 V 0 with the 45 o
-line,weobtaintheshortfallfromthegolden-ruleutilitylevelthatweseek:
x o ^ p = x 0 + 1 p^
^ p
x
=)u(^x ;x)^ x o
= u(^x;x)^ p^x 0
(1 p)^x
= u(^x;x)^ x p(^x 0
x )=Æ
(^p;^x) (x ;x
0
): (17)
Inthisdemonstration,wehavealsoshownthatanyone-periodplanonM 0
V 0
hasthesamevalueloss.
Next,weturntoone-periodplansinthe\open"parallelogramLMVD:Inthisase,value-lossstems
from bothexessapaityand from thenegativeshortfallfrom the golden-ruleutilitylevel. We have
alreadyseenthat this shortfallisthe sameforallplans onthe lineS
2 S
5
parallel to MV;and isgiven
by thedierene betweenx^and the absissaof the pointof intersetion S
3 of S
2 S
5
and the45 o
-line.
Inorder to show that S
2 S
5
is aniso-value-lossline, allthat remains isfor us to showthat the exess
apaityassoiatedwith anyone-periodprodutionplanonit, sayS
2 ;S 3 ;S 4 ;S 5
;is idential. Butthis
iseasy from ourproedurefor omputingexessapaity: allofthetriangles withvertiesS
2 ;S 4 and S 5
exhibitedinFigure 5areongruent,andhenetheirbasesareequal.
Next, we haveto showthat the valuelosses inrease asiso-valuelosslines move\away" from the
zero-valuelossline MV in either diretion. This islear whenwelimitourselvesto the full apaity,
surpluslabor triangleMOV: ThediÆultyonerningone-periodplansin thefullemployment,exess
apaity area LMVD lies in the fat that asMV movesoutwards, both the negative shortfall from
golden-ruleutility aswellastheexessapaityinrease. However,thelatterinreasesmorethan the
former. Toseethis,onsidertheparallellinesM 0 V 0 andM 00 V 00
inFigure6. Theinreaseintheshortfall
amountsto x
1 x
2
; whereastheinreasein theexessapaityistheamountW
1 W
2
: Toseethat W
1 W
2
is always greater than x
1 x
2
; draw a line M 0
F parallel to the 45 o
-line, and simply observe that the
diereneintheabsissaeofthepointsF andM 0
(whihisx
1 x
2
sinetriangleswithvertiesF andF 0
areongruent)isgreaterthan W
1 W
2
: Andthisis alwaysso byvirtueofthefat that theslopeof the
45 o
-lineissteeperthantheslopeofOD;whihisanotherwayofsayingthat therateofdepreiationd
isalwayslessthanunity.
We now have a basigeometri tool, iso-value-loss lines, to haraterize optimal poliies, and we
turntoprovidingsuh aharaterization.
5 Optimal Poliies ( = 1): Equilibrium Cyles
Return to Figure 1, and note an inidental aspet that wehave ignored so far in our disussion: the
line MV is orthogonal to the 45 o
-line. This orthogonality is simply a onsequene of how the two
parameters, the input-output ratio a and the depreiation rated relate to eah other. In partiular,
from(1),and thefat thatthe\OMV is45 o
;
=1=a (1 d)= MV
0
V 0
V
=1=)a= 1
2 d
: (18)
42
ofparametervaluesofthemodel,albeitsingular,underwhihweworkinthissetion. 43
ReturningtoFigure4,wenotethattheorthogonalityofMV andthe45 o
-degreelineimpliesthatthe
trianglesGPV andGPP 0
areongruenttoeahother,andthereforethesegmentPV equalsPP 0
;whih
impliesthatPP 0
QV isasquarewiththegolden-rulestokpointGasitsenter. Withthis,thegeometry
bringstolightinterestinginvariantspertainingtotheasethatweareonsidering. Irrespetiveofhow
a and d hange, but provided that they ontinue to relate to eah other asgraphed in Figure 6, the
goldenrulestokx^isgivenby1 (d=2);theabsissaofQis(1 d);andnally, thegolden ruleprie
system,p;^ asgivenbytheratio,V 0
V to PV;isalwaysonehalf andtherebyindependentofaandd: 44
WenowturntotheharaterizationofoptimalpoliiesbyonsideringthesquarePP 0
QV inFigure
8. Forthis,weneedtoreallBrok'sTheorem[3℄whihguaranteesthatanyprogram(trajetory)that
minimizesthe aggregateofthe sequeneall (non-negative) value-lossesoverthelongrunis indeed an
optimaltrajetory,providedthegolden-rulestokisuniqueandsomeadditional(onvexity)assumptions
onerningandu(;)aresatised.Weaskthereadertoaeptthatthistheoremappliesinourase. 45
Butthis makesitevidentthat theoptimalprogramstartingfrom any initialstokin theintervalQV
ylesaroundthegolden-rulestok. ItbearsemphasisthatthisonsequeneofBrok'stheoremfollows
onlyfromtheharaterizationofMV asthezero-value-lossline(thevon-Neumannfaet);thefatthat
lossesinreaseas wemoveawayfromMV in eitherdiretion isonlyusedwhen wewanttoassertthat
theidentied program is uniquelyoptimal. Inanyase, it isa resultthat hasasurprise to it, ruling
outas it does, other plausible intuitions aboutthe optimalpoliy. Thus, it rules asnon-optimal, say
startingfrom x(0);theentirelyfeasibletrajetoryof maintainingurrentonsumptionlevelsin exess
ofwhatoptimalityrequiresandinvestinglesssoastohavethegolden-rulestokofmahinestomorrow.
An important aspetof these optimalprograms deservestobehighlighted. Thegeometry, in
par-tiular thefatthat PP 0
QV isasquarewith Gasits enter, makesitevidentthattheaverageof the
optimal programstarting from anyinitial stok in the intervalQV is preisely thegolden-rule stok.
ThisispreiselyBrok'saverageturnpikepropertywhihassertsthat theaverageoftheapitalstoks
ofall\goodprograms"andnotonlyoptimalonesonvergetothegolden-rulestok. 46
Thisispreisely
theasewhenthevonNeumannfaetisnotstable. 47
When we turn to initial apital stoks that lie outside the interval QV; we retain this aspet of
optimalityunder whih developmentplanningentailsover-buildingandthenover-depreiatingrelative
tothe golden-rulestokx:^ Theexeptionsto thisare situationswhen thetheinitialapitalstokis x^
itself, orx(0) 00
in the intervalbetween 1 and 2, orother identiable and omputably haraterizable
initial stoks in the intervals[2 n 1
;2 n
℄; where n = 2;3;: We leave it to the readerto use Figure
4to showfor herselfthat for all initialapital stoksless thanthat orrespondingto the pointQ; or
forthose in ℄1;2℄; theoptimalpoliy onvergesto thetwo-period equilibrium ylein twoperiods. In
general,for aninitial apital stokin [2 n 1
;2 n
℄;it takesn+1periods to onvergeto theequilibrium
yle,wheren=1;2;:Ingeneral,optimalpoliy isto followthepathharted bythetwolinesMV
43
Notealsothefuntionrelatingatodisaninnitelydierentiableonvexfuntion.
44
Ofourse,weouldhaveequallywellexpressedallthesevaluesintermsoftheinput-outputratioa;weleavethisto
theinterestedreader. Onethegeometryallowsustoseetheseinvariants,itisasimplemattertoalsoobtainthemfrom
theformulaepresentedin(1),(3)and(8).
45
Oneoftheassumptionsin[3℄isthatbeompatwhihislearlynotfullledinourase. ItisnotadiÆultmatter
toshowthatBrok'smethodsextendtoyieldatheoremthatdoesapplytooursituation;fordetails,see[12℄.
46
See[5℄foradenitionof\goodprograms",and[3℄fordetailspertainingtothelaim. Inpartiular,notetheimportane
oftheassumptionthatthegolden-rulestokisunique.
47
ItisalsotheasewhenInada's1964assumptionthat\allpathsthatremainonthefaetforeveronvergeuniformly
x(t+1)=max[ x(t)+1=a;(1 d)x(t)℄ forallt=0;1;: (19)
Wearenowinapositiontomaketwoobservations. Therstrelatestowhat istermedas\history
independent"dynamialsystemsin [26℄. Theoptimalpoliyfuntion underlying theoptimal program
leadsto adynamialsystemwhih isanythingbut historyindependent. The speedof onvergeneto
the equilibrium, but perhaps more importantly, the partiular equilibrium and its amplitude, are all
dependenton theinitialstokof mahinestheeonomyisendowedwith. Seond,note that thesetof
equilibriato whih theoptimalprogramsonvergeareofunountable ardinality;and whereasitdoes
notmakesense to saythat this ardinalityis inversely relatedto therate ofdepreiation, we ansay
thatthelengthoftheintervalQV serving asaproxyforthesetinreasesaswedereasebothdanda
inawaythatis onsistentwiththerelationshippituredin Figure7.
Weonludethissetionwiththenaturalquestionastohowtheoptimalpoliiesharaterizedhere
hangewhen theparametervaluesaanddarenotfuntionallyrelatedasshown inFigure 6. Weturn
tothisin thenextsetion.
6 Optimal Poliies ( < 1) : Convergene in Finite Time
Areaderfamiliarwithonlythebasisofthetheoryofdiereneequationswillappreiatethefatthat
the qualitativeproperties of the optimal program analyzed in the previous setion hinge on the unit
root of the underlying harateristiequation orrespondingto the line MV { the onlyompliation
relates to the fat that there is a kink in the dierene equation. 48
In this setion we grapple with
the diÆulties arisingfrom thefat that the absolute value of this root is greater than one. 49
What
isinterestingis thatdespitethis, perhapsbeauseof it,ratherthan adivergenefrom thegoldenrule
stok,weobtainpartiularlyfastonvergeneto itgolden-rulestok. However,thegeometrianalysis
beomessomewhatmoreintriate.
Thebasiintuitionproeedsasfollows: Oneanfallbakonthepreviousaseto supposethat the
optimalpoliyistrakedbythelousMVD:Thisiseasilyseentobefalsebyonsideringaone-period
produtionplan suhasS: Theoptimal poliy ylesaroundthe golden rule. Not onlyis theaverage
turnpike prpoertynegated but also that program is bad. Thus the optimalpoliy is to jump o the
pathtrakedbyMVD:Thequestioniswhenand towhere. Theanswertotheseondpartisobvious
{sinethevaluelosses arezeroatthegolden rulestok,and itissustainablethroughtime,itis lear
that optimal program must end there. And one the solution is phrased this way, rst part of the
questionalsoadmitstheonlypossibleanswer: namely,\assoonaspossible."Theonlyreasonwhythis
annot be aomplished in one period is beause of the irreversibility of apital { the eonomy may
besoapital-rihthat ittakestime todispose ofitsapital. Wherethis intuitionfails isin situations
where theinitial stokofapitalis lessthanthegolden-rulestok {theapital-pooreonomy. Inthis
ase,there is alwaysover-buildingofapital followedbysubsequentdepreiationto thedesired stok.
Asweshallseebelow,thisis relatedto thedesirabilityofalwaysmaintainingfullemployment. Inany
ase, ourmethods enable us not only to show onvergene in nite time but to be able to ompute
thenumberof timeperiodsrequiredto attainonvergene,andto beablesayhowtheyarerelatedto
48
In[22,p. 714℄,MKenziewrites\Thebehaviorofpathson[thevonNeumannfaet℄F maybestudiedbymeansof
diereneequations"andreferstohis1963paperwherethiswas\doneexpliitlyforthegeneralizedLeontiefmodel.
49
thelousMVD;andthisiseasytoanalyze.
The rst point to be established is that optimality requires, for any time period t; t = 0;1;;
andforallinitial apitalstoksx
(t)falling intheinterval[0;x℄;^ theperiod'sterminalapitalstokis
hosentobeonthenovalue-losslineMV.Thisis simplytosaythatforallsuhinitial apitalstoks,
full employment and full apaity be maintained the next period. If this was not so, and there was
anoptimalprogramfx
(t)g 1
t=0
suhthatforsometimeperiod
t; x
(
t+1) wasstritly belowMV;the
readeranhekforherself in theontext ofFigure 8,orof Figure6, that there existsan alternative
programwhih diersfromtheoptimalonlyattime
t+1;andinhavingastokhigherthanx
(
t+1)
then. Butsuhaprogramhasalowervalueloss,and thereforeontraditstheoptimalityofthegiven
optimalprogram. Note thatthis argumenthasnothingtodowiththeslopeof theMV line, andthat
iswhyweanworkwithFigure 6orFigure 8.
WeannowgainmoreintuitionintothebasisofourlaimbyonsideringFigure9. Fullemployment
guaranteesthatforallstoksintheinterval[0;x^noone-periodprodutionplanishoseninthesurplus
laborzone. Theiso-value-losslineguaranteesthatnohoiebemadeothesegmentMGorV
1
D:Thus
allthatremainsistheoptimalityofthesegmentGV
1
;sotospeak.
Next,webeginwithanoverviewoftheoptimalprogram,andonewehavealaim,weanfurnish
its substantiation. Towardsthis end, webegin with anidentiation of benhmarks forvalues of the
initial apital stok. First, in Figure 8, read o some invariantslabelled
1 ;
2 ;;
n
in theinterval
[0;x℄;^ wherenissuhthat
n 1
>0and
n
<0:Sinethezero-value-losslineintersetswiththeY-axis
(at M),and OD hasapositiveslope, suh anis well-dened. 50
Asanbeseenfrom Figure 8,these
benhmarksorrespond,indeedaredeterminedby, 51
apitalstokvalues1;2;4;;2 n
; n=1;2;:
Next we turn to series of values of apital stoks to the right and left of the unit apital stok
orresponding to thekink V at the terminationof thezero-value-loss line. Theones on the rightare
given by 52
r
n
; n = 1;2;; and the ones on the left, by `
n
; n = 1;2;;n; where n is suh that
`
n 1
>0and `
n
<0:Again,sinethezero-value-losslineintersetswiththe Y-axis(at M),and OD
hasapositiveslope,suhanis well-dened. 53
Asanbeseenfrom Figure8,allofthese benhmarks
orrespond,indeed aredeterminedby, 54
Wehavenowallthatis neessaryto showthat theoptimalpoliyisto followthepathharted by
thethreelinesMV;GV
1
andOD;whihistosay,thepathtrakedbythefollowingdiereneequation:
x(t+1)=max[ x(t)+1=a; x ;^ (1 d)x(t)℄ forallt=0;1;: (20)
Weshalldeveloptheargumentinaseriesofsteps. First,weshowthattheproposedpoliyisoptimalfor
apartiularvalueofaninitialapitalstoklyingintheinterval[^x;1℄:Thisistheruxoftheargument,
andthe nextstepis touse itto show that theproposed poliyis optimalfor any initialapital stok
lyingintheinterval[^x;1℄:Suessivestepsaretoworkthroughsuessiveintervals,alternatingbetween
leftandright,untiltheintervalsontheleftareexhausted, andthefousshiftsonlytointervalsonthe
right. 55
50
Thisissimplytosaythatanitenumberofsuhintervals[
n ;
n+1
℄; n=1;;overtheinterval[0;x℄:^
51
Thespeiformulaeareasfollows:
n
=(1=)(2 n 1
(1=a)); n=1;2;;n:
52
Thesupersriptrin r
n
standsforright,andthesupersript`in `
n forleft.
53
AversionofFootnote 50isalsorelevanthere.
54
Thespeiformulaeareasfollows: r
n
=x=(1^ d) n
;n=0;1;2;;and `
n
=(1=)((1=a) r
n
)=(1=)((1=a)^x=(1
d) n
); n=0;1;2;;n:
55
[1;x℄;^ [1; r
1
℄;[2;1℄; [ r
1 ;2℄; ℄;[
` 1 ;1℄; [2; r 2 ℄; [0; r 1 ℄;[ r 2
;4℄;[2 n ; r n ℄; [ r n ;2 n+1
thatthenextperiod'shoiebethegolden-ruleapitalstok,whihistosay,theone-periodprodution
plan J in Figure 9. Towardsthis end, onsider thevertialline going throughthepoints P andV to
whihtheplannerislimitedinhishoieofnextperiod'sapitalstok. Allone-periodprodutionplans
onthislinehigherthanJ havegreatervaluelossesandarethereforeruledout.
Wenowmagnify thepartofFigure9aroundGasFigure10,andonstrutthreeprodutionplans
J
1 ;J
2 andJ
0
:J
1
isaone-periodplansuhthatmaintenaneoffullemploymentnextperiodleadstoan
unhangedinitialstokofapital. Thus,J
1
isapointonthelinePV suh thatityieldsaunitapital
stoknextperiod. J 0
is aone-periodprodution plan suh that zeroemploymentin theonsumption
setoryields the samevalue-loss as the plan J. Thus, it is the intersetion of the line OD with the
aline parallel to MV and going throughthe point J: Finally, J
2
is aone-period plan withan initial
stokofapital equalto unityandsuh thatmaintenaneoffull employmentnextperiodleadsto the
initialstokofapitalastheplanJ 0
:Thus,J
2
isapointonPV suhthatityieldsaapitalstoknext
periodidentialtotheinitialapitalstokofJ 0
:ItislearthattheordinatesofJ;J
1 andJ
2
arestritly
dereasing.
Wenowadoptaonventionthatshortendesriptions:whenwesaythataplannerhoosesapartiular
pointin;weare(loosely)referringtoaone-periodplanwhoseinitialstokistheabsissaofthepoint
and whih the planner takesasgiven, and whose terminal stok is theordinate of the pointthat she
hooses.
Allone-periodprodutionplansonPV equaltoorlowerthanJ
2
havealowervalue-lossthisperiod
thanJ;butahighervaluelossnextperiod,andthereforeannotbepartofanoptimalprogram. Hene,
allthatremainsforonsiderationistheintervalofone-periodprodutionplans stritlybetweenJ and
J
2 :
Suppose the planner hose J
1
: Sine she an also hoose it next period, and sine this was her
optimal hoie this period, stationarity of ourmodel ditates that it is also her optimal hoie next
period. Butthisleadsto aprogramthat keepsaumulatingvaluelossesad innitumandis noteven
agoodprogram,leavealoneanoptimalone.
Thussupposethattheplanner'soptimalhoieisaone-periodprodutionplanbetweenJ
1 andJ
2 :
Fullemploymentimpliesthatnextperiod'sinitial stokishigherthanunity, whih istosaythat next
period's planis limitedtoavertialline totherightofPV: Butanypointhosenonsuh aline gives
ahigher value-loss than before, and these losses will keepon aumulating unless the planner puts a
stop by hoosing a plan on the extension of GJ: But she ould have done this in an earlier period,
ontraditingtheoptimalityofthehoieofaplanbetweenJ
1 andJ
2 :
Thusallthat remainsis toonsiderthehoie ofaprodutionplan betweenJ andJ
1
{thedarker
trajetory in Figure 9. Towards this end, werst observethat the dening harateristiof the ase
that we analyzeis that the no-value-loss line MV is steeper than the 45 o
-line, and this fat has not
beenexpliitlyusedsofar. WedosonowinFigure11;thisalsoallowsustoappreiate whatisspeial
abouttheaseoftheprevioussetion. WhatistruehereisthatGJ isstritlygreaterthanJJ
1 :Tosee
this, onstrutthelineP 0
S orthogonaltothe45 o
-line, reallhowJ
1
wasderived{from J to P toP
2
toP 0
2
:SineMV issteeperthanP 0
S; PP
2
islessthanPS;thisompletestheproofofthelaim.
Theabovefat allowsus toassertthat theline parallelto MV from anypointin thesegmentJZ
intersetsthesegmentGJ:
Consideraplan,sayZ ;hosenbetweenJ andJ
1
: Bythehoie ofZ ;asopposed toJ; theplanner
savesvaluenotexeedingWJ:Givenfeasibilityandthemaintenaneoffull-employment,thehoieof
Z takestheeonomytoaapitalstokgivenbythevertiallineP
1 P
0
1
throughZ
0 ;Z
1 andZ
2
;whihis
1 0 1
ButanypointhigherthanZ
0 onP
1 P
0
1
hasevengreatervalue-lossthanatZ
0 :
ThussupposetheplannerhoosesapointinthesegmentZ
0 Z
1
:Undersuhahoie,theleast
value-lossintheperiodwouldbeatZ
1
;butagain,sineW
1 Z
0
equalsWJ;wouldstillbegreaterthaninitially
hoosing J:
Thusallthat remainsis theonsiderationofahoieonP
1 P
0
1
belowZ
1 :ButZ
1
isplayingtherole
forZ thatJ
1
playedforJ;andsobyanragumentidentialtowhat wehaveseenbefore,thevalue-loss
atZ
0 isless.
Again,asintheaseof= 1;weover-buildand under-depreiate,butinawaythatthe
disrep-anies go to zeroin a nite numberof timeperiods, this numberdepending on themagnitude of the
stokof mahinesthat weinitiallystart with. Butthere areexeptionsto this statement wherethere
ismonotonionvergene.
Thegolden-rulestokx^andthegolden-rulepriesystemarenolongerinvariants. Byinspetionof
Figure8,weansee thattheyareinverselyrelatedtotherateofdepreiationdanddiretlyrelatedto
theinput-outputratioa:Andinthese parametrihangeswearenolongerontheknife-edgepitured
inFigure 6buthave\roomtomanoeuvre".
The dynamial systemis \history independent" in terms of [26℄. The equilibrium is unique, and
optimal programsirrespetiveof theinitial apital stok theybegin with, onvergeto it. To be sure,
the speedof onvergene, and the qualitative feature of the optimal program (amplitude, monotoni
versusnon-monotonionvergeneet.),doesdependontheinitialapitalstok,butthesefeaturesare
notgivenprominenein thedenition of\historyindependent" systemsthat weareworkingwith.
7 Optimal Poliies ( > 1): Monotoni Convergene
Inthissetion, weturnto theasegrapplewiththediÆulties arisingfrom thefat thattheabsolute
valueof therootisgreaterthanone. 56
Three sub-ases,noneof themanalytiallydiÆultbut perhapsofsomeinterestfrom aneonomi
pointofview: 1 < <0; =0; 0<<1 d:Essentiallyamodiationofguressentearlier. But
aslightlydisorientationwhenMV hasapositiveslope.
8 Possible Extensions
9 Conlusion
SummarizeandgobaktoStiglitz'quoteintheintrodution. Nolivingonair.Bang-bang. monotoniity.
Fixingtherateofprottobezero. relateto therateofdisount. laborsurplus. laboralloation.
Openproblems.
56
[1℄ Aghion,P.andP.Howitt (1998)EndogenousGrowthTheory,Cambridge: MITPress.
[2℄ Atsumi, H. (1965) \Neolassial Growth and the EÆient Program of Capital Aumulation",
Reviewof Eonomi Studies,Vol.32,pp.127-136.
[3℄ Brok,W.A.(1970)\OnExisteneofWeaklyMaximalProgrammesinaMulti-SetorEonomy",
Reviewof Eonomi Studies,Vol.37,pp.275-280.
[4℄ Debreu,G.(1961)Theoryof Value,NewYork: JohnWiley.
[5℄ Gale, D. (1967) \On Optimal Development in a Multi-Setor Eonomy", Review of Eonomi
Studies,Vol.34, pp.1-18.
[6℄ Gale, D. (1967)\A Geometrial Duality Theorem with EonomiAppliations", Review of
Eo-nomiStudies,Vol.34,pp.19-24.
[7℄ Haque,W.(1970)\SeptialNotesonUzawa's"OptimalGrowthinaTwo-SetorModelofCapital
Aumulation", and a Preise Charaterization of the Optimal Path", The Review of Eonomi
Studies,Vol.37, pp.377-394.
[8℄ Jones, R.W. (1965)\Dualityin InternationalTrade: aGeometrial Note",Canadian Journalof
Eonomis andPolitial Siene,Vol.31,pp.390-393.
[9℄ Jones,R. W. (1965)\The StrutureofSimple GeneralEquilibrium Models", Journalof Politial
Eonomy,Vol.73,pp.557-572.
[10℄ Johnson,H.G.(1971)TheTwo-SetorModelofGeneralEquilibrium: TheYrjoJahnssonLetures,
Chiago: Aldine.
[11℄ Khan,M.Ali(1999)\SrinivasanonChoieofTehniqueAgain",Chapter2inRanis-Raut(1999).
[12℄ Khan,M.AliandT.Mitra(2002),\OptimalGrowthunderIrreversibleInvestment: AnExample,"
mimeo.
[13℄ Khan, M. Ali and T. Mitra (2002), \On Choie of Tehnique in the Solow-Srinivasan Model,"
mimeo.
[14℄ Koopmans,T.C.(1964)\EonomiGrowthataMaximalRate",QuarterlyJournalofEonomis,
Vol.XVIII,pp.355-394.
[15℄ Kurz,M. (1970)\Reviewof \The GrowingEonomy" byJ. E. Meade",The Journalof Finane
Vol.25,pp.190-3.
[16℄ Majumdar,M.,T.MitraandK.Nishimura(2000)OptimizationandChaos,Berlin: Springer-Verlag.
[17℄ Marshall, A. (1879) \The Pure Theory Foreign Trade", reprinted in 1930 by London Shool of
Eonomis,London.
[18℄ MKenzie, L. W. (1968) \Aumulation Programs of Maximum Utility and the von Neumann
Faet", in J. N. Wolfe(ed.) Value, Capital andGrowth, Edinburgh: Edinburgh University Press,
pp.353-383.
[19℄ MKenzie,L.W. (1971)\CapitalAumulation Optimalin theFinal State", Zeitshrift f ur
43,pp.841-865.
[21℄ MKenzie,L. W. (1986) \OptimalEonomiGrowth, TurnpikeTheorems and Comparative
Dy-namis",inK. J.Arrowand M.Intrilligator(eds.)Handbook of Mathematial Eonomis, Vol.3,
NewYork: North-HollandPublishingCompany,pp.1281-1355.
[22℄ MKenzie,L.W. (1987)\TurnpikeTheory",in J.Eatwell,M. MilgateandP. K.Newman (eds.)
TheNewPalgrave,Vol.4,London: MaMillan,pp.712-720.
[23℄ MKenzie,L.W.(1999)\Equilibrium,TradeandCapitalAumulation",TheJapaneseEonomi
Review,Vol50,pp.369-397.
[24℄ Mirrlees,J.A.andN. H.Stern(eds.)(1973)Models of Eonomi Growth,NewYork: John-Wiley
andSons.
[25℄ Mitra,T.(2000)\IntrodutiontoDynamiOptimizationTheory,"pp.31-108in[16℄.
[26℄ Mitra, T. and K. Nishimura (2001) \Disounting and Long-Run Behavior: Global Bifuration
AnalysisofaFamily ofDynamialSystems",Journalof Eonomi Theory,Vol.96, pp256-293.
[27℄ Nishimura,K.andM.Yano (2000)\Non-linearDynamisandChaosin OptimalGrowth: A
Con-strutiveExposition,"Chapter8in [16℄.
[28℄ Nishimura, K., G. Sorger and M. Yano (2000) \Ergodi Chaos in Optimal Growth Models with
LowDisountRates,"Chapter9in [16℄.
[29℄ Okishio,N.(1966)\TehnialChoieunderFullEmploymentinaSoialistiEonomy",Eonomi
Journal,Vol.76,pp.585-592.
[30℄ Radner,R.(1961)\PathsofEonomiGrowththatareoptimalonlywithrespettoFinalStates",
Reviewof Eonomi Studies,Vol28,pp.98-104.
[31℄ Ramsey,F.(1928)\AMathematialTheoryofSavings",EonomiJournal, Vol.38, pp.543-559.
[32℄ Ranis, G. and L. K. Raut (eds.) (2000) Trade, Growth and Development: Essays in Honor of
Professor T.N. Srinivasan,Amsterdam: NorthHolland.
[33℄ Robinson,J.(1960)Exerises in Eonomi Analysis,London: MaMillan.
[34℄ Robinson, J. (1969)\A Model for Aumulationproposed by J. E. Stiglitz", Eonomi Journal,
Vol.79,pp.412-413.
[35℄ Samuelson,P. A.(1965)\A CatenaryTurnpikeTheorem involvingConsumption and theGolden
Rule",Amerian Eonomi Review,Vol.55,486-496.
[36℄ Samuelson, P. A. and R. M. Solow(1956) \A Complete Capital Model involvingHeterogeneous
CapitalGoods",QuarterlyJournalof EonomisVol.LXX,pp.537-562.
[37℄ Shakle,G. L.S. (1956)\Reviewof\AGeometryof InternationalTrade"byJ.E. Meade",
Eo-nomiaVol.90,pp.179-180.
[38℄ Shell, K. (1967) \OptimalPrograms of Capital Aumulation for an Eonomyin whih there is
ExogenousTehnialChange",Essay1inK.Shell(ed.)EssaysontheTheoryofOptimalEonomi
Eonomi Studies,Vol.29, pp.207-218.
[40℄ Solow,R.M.(1999)\Srinivasanonhoieoftehnique",Chapter 1in[32℄.
[41℄ Srinivasan,T.N.(1962)\InvestmentCriteriaandChoieofTehniques ofProdution",Yale
Eo-nomiEssays,Vol.1,pp.58-115.
[42℄ Srinivasan,T.N.(1964)\OptimalSavingsin aTwo-SetorModelofGrowth",Eonometria,Vol.
32,pp.358-373.
[43℄ Stiglitz,J.E.(1968)\ANoteonTehnialChoieunderFullEmploymentinaSoialistEonomy",
Eonomi Journal,Vol.78, pp.603-609.
[44℄ Stiglitz, J. E. (1970) \Replyto Mrs.Robinson on the Choie of Tehnique", Eonomi Journal,
Vol.80,pp.420-422.
[45℄ Stiglitz,J.E.(1973)\TheBadlyBehavedEonomywiththeWell-BehavedProdutionFuntion",
Chapter6in [24℄.
[46℄ Stokey, N.L.andR. E.Luas, Jr.(1989)ReursiveMethodsin Eonomi Dynamis,Cambridge:
HarvardUniversityPress.
[47℄ Stiglitz,J.E.(1973)\ReurreneofTehniques inaDynamiEonomy",Chapter7in [24℄.
[48℄ Uzawa,H.(1958)\TheKuhn-TukertheoreminConaveProgramming",pp.32-37inArrow,K.J.,
Hurwiz,L.,Uzawa,H. (eds.)StudiesinLinear andNon-linear Programming,Stanford: Stanford
UniversityPress.
[49℄ Uzawa, H. (1964) Optimal Growthin a Two-Setor Model of Capital Aumulation", Review of
Eonomi Studies,Vol.31, pp.1-24.
[50℄ vonNeumann,J.(1935-36)\
Uberein
OkonomishesGleihungs-SystemundeineVerallgemeinerung
desBrouwershenFixpunktsatzes",inK.Menger(ed.)ErgebnisseeinesMathematishen
Kolloqui-ums, No. 8. Translated as \A Model of General Eonomi Equilibrium", Review of Eonomi
Studies,Vol.XIII,pp.1-9.
[51℄ vonWeizsaker,C.C.(1965)\ExisteneofOptimalProgramsofAumulationforanInniteTime
