A Diasertation Preaented t o the 7aoulty of the Oraduate Sohool

of Yale University

I n Caiididaoy f o r the Degree of Doctor of Philosophy

Doouiiiento de Trabajo In s t i t u to Torouat o D i Tella Centro de Investigaoiones

Zoon6uiloaa - ifoviembre 1965

Tho author waild l i k o t o oxpross hor thanks t o t h c ine t i tu -

t ions t h a t mado poseiblo hor etudios and roaoaroh a t Yalo Univorsity,

namoly, tho Coneojo Naoional do Invostigaoionos C i o n t ~ f i o o e y TÓoni-

G a s of Argontina, tho Org~in i ra t ion of Amorioan S ta tos and Yalo Uni-

vorsity. Elho is a leo indobtod t o Talo Univorsi ty f o r finanoing tho . *I

oomplter work a t tho Yalo Compitor Oontor and t o tho I n s t i t u t o ' P o r

m a t o D i Tol la of Argontina f o r providing tho f a o i l t t i o s and tho fi-

nan t i a l support during tho f i n a l thinklng and wr i t ing of t h i s d i s s e r

ta t ion.

Sho 1s pa r t iou la r ly g ra to fu l t o Profossors Horbort Soarf

and T j a l l i n g C. Kooprnans who a t two d i f fo ron t s tages of t h i s disaer-

t a t i o n moant so moh t o hor f o r t h o i r undorstanding, oommonts and

advioo.

Sho is nlso m y thankful t o Profossors Arthur Olrun, Jamoa

Tobin, Jamoe Friodman and John Foi f o r valunblo oomonts on o a r l i o r

draf t6 of tho prospootus and tho d issor to t ion .

Sho l a a l s o m o h indobtod t o Profossor J u l i o R. O. Olivora

of tho Univorsity of &onos Airos f o r h i s onocuragomont and suggos-

t ione on hor work.

Laat Irit not l o a s t , aho wishos t o oxproas hor doop in to l -

loo&al dobt t o har husband Rolf R. f i n t w l f o r hie po-nont o n o o u r

agmont, o o n s t ~ ~ o t i v o or i t io i sm a s woll a s f o r tho long and s t i n u l a t i n g

discusaione of hor work.

Tho toahnioal graphs a ra tho work of Mr* Martin Yrroa and

LL~. Donald Mo Nab wont through tho English toxt .

The object 0.2 the d i s s e r t a t i o n i s t o ~ r e s e n t a model of e- conornic f luo tua t ions víliioh recognizes t h e i a c t t h a t econoiiiio dynrti;iics i e not only oonoerned vati preferenoes and teohnology but a l s o with plana and expeotations.

It is e s s e n t i a l l y a genera l iea t ion of t h e Hiclcsirrn uodel of t h e t rade cycle don , - t h e following ~iaiil l ines .

1.- The prociuotion s i d e 0.2 the econony is 2orna l ly taken i n t o aocount under t h e f o i ~ i of a produotion funct ion honogeneoua oí? degree one and with constaat co&icients .

2.- A procese of i"ormation of c p e o t a t i o n s i e introduoed which beiongs t o t h e tx t r a2o la t ive type, ire. , a c e r t a i n f r e a t i o n of t h e l a t e s t ohange i n s a l e s i e sdded t o t h e l a t e s t observed sale. Sinoe the niodel w o r b i r i t h ooilstant prloea, i t is reasoneble t o con- s i d e r expectations on q m i t i t i e s der~wded.

3.- Investnent i i l f i s e d o a p i t a l i s not a n a l y t i o a l l y d.ivo+ ced f rom investrient i n invcntor ies , schieving i11 t h i s w w s q u a n t i t p adjustrnent ~~echanism at t h e lower turn ing point of t h e nonl inear oyole. Dernand f o r inventaries i s j u s t i f i e d under conditiona of ce- t a i n t y by analogy mith t h e d a a n d f o r money f o r t ransaot ions purpose.

The mathernaticai fonnulat ion ol" t h e nodel reduoes t o s ay* tem of seven d i f fe rence equations and t h r e e dynarjclo inequal i t iea . The non l inea r i t i e s of t h e nodel suggested the ~ c ~ l o r a t i o n by means of s i m l a t i o n . Por t h a t i>urpose, t h e p a r a i ~ e t e r values w e r e ohosen working bsckwards fror.1 t h e value of t h e i i ~ i i d roo t af t h e unoone- t ra ined model, so aa t o avoici an a r b i t r a r y eaplosion of t h e system. h t h e i r a o r e t h e initiai pos i t ion cf t h e s y s t e r ~ f o r t h e siiíniLation W a s

&ven by t h e e q u i l i b r i w eolut ion oí? t h e nodel both f o r t h e progroa- s i v e and t h e non-progressive econoay.

A r e s u l t obtaiiled with t h i a niore ooriplex ralodel i s t o give weight t o P r d . Hic!csi conjeoture about the a s l o s i v e na tu re of t h e unoonetrained cycle so t h a t t h e noi i l inear i t ies glay a c r u c i d r o l e f o r the I1orbital s t a b i l i t y " of the economy.

Another r e s u l t re levant t o eoonoüdo theory i s the f a o t t h a t the farnal in t roduct ion o l i i iventories pemfits a nonl inear and endo- genoua theory of the oyole whioh does not "fai.1" i n t h e non-progre* s ive eoonoqy.

Chapter one Briof desor ip t ion of Hioks' modo1 . . . . . . . . . . . . . . 8 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Disciussion of tho invostmont Ainotion adopted . 18

Chaptor two Themodel . . . . . . . . . . . . . . . . . . . . . . . . . . 21 Wumnary . . . . . . . . . . . . . . . . . . . . . . . . e . . 32 Equilibrium i n the S ta t ionary Eoonomy . . . . . . . . e . . . 33 Equilibrium i n thc Progrossivo Eoonorny . . . . . . e . . . . 34 Paramoters and tho timo poriod . . . . . . . . . . . . . . e 40

Chaptor threo Analyeis of tho modo1 . . . . . . . . . . . . . . . . . . . . 45

Chaptor four Bohaviour of tho modo1 e . . . . . . . . . . . . . . . . . . 53 Tho unoonstrainod oconany . . . . . . . . . . . . . . . . . . . 53 Tho nonlinoar ooonony Analpis of tho simulation . . 54 A typioa l o . . . . . . . . . . . . . . . . . . . . . . . . . . 58 Bonsi t ivi ty ana lys is and tho l i m i t oyolo . . . . . . . . . . 63 Tho pro$rossivo eoonany . . . . . . . . . . . . . . . . . . . 72

Pppondix Algorithm f o r tho dotominat ion of tho oharaotor i s t io roota

of tho unoonrrtrainod systom . . . . . . . . . . * . ~ . . . 74 Btum's thoorom and oasos of multiplo roots . . . . . . I e . 75 Mgorithm f o r tho caeos of simplo roots: roa1 roots . 77 Aigorithm f o r tho oasos of simplo rootsr oomplox roo t s . . . 83

"J.YO sc ienoe thr ivee i n t h e atmospñere of d i r e c t prac t ica1 &m end even prao t ioa l r e s u l t s a r e but t h e product of cttsinterested work at t h e problem f o r t h e problemla sake, Nobody who oraves I o r quiok and ehort anmers t o burning questions of the day zill oare t o entangle h inse l f i n d i f f i o u l t i e a \7hioh only pa t i en t l a b o r can o l e a r i n t h e oourse of many years".

J. Schumpeter, &onometrica 1933.

hypotliesis t h a t postulatea t h e axistenoe of endogenously generated

antidanped oycles prwonted by non l inea r i t i e s frorn prooeeding beyond

a c e r t a i n point.

Nwer the less t h r e e kinds of daf io ien t f ea tu res are

observed i n theee thoories.

1 ) Ii'rom tho formal point of vim? of model bui lding only a few

maoroeconomio va r i ab le s suoh as t o t a l output, aggregate oo*

suniption and invos t i~ent a r e t r e a t e d rnathomatioaily although

s i d e referenoos a r e nade i n a l i t o r a r y may t o o ther va r i ab le s

and even t o iiiore than one sector. Furthemore niost of those

I/ Hioks J.&# A Contribution t o t h e Thoory or" t h e Frnde Oxford 1950.

Ooodwin, It I "The N o ~ l i n e a r Aoce1er;ttor aad t he P e r s i s tenoo cf Business CyolosM, Eoonometrica, 1951.

Kaldor, N.; '?8-liodel of the Trade Cyclen. 2aonoi;do fournal, Maroh 1940.

cyolo uodels t h a t f l ou r i shod in t ho 50 's a r o t o bo considorod aa

do:.md dominated nodols s ince no e x p l i o i t formal raforenoo is ma

do t o tho production bounds o r t o tho naturo of tlio i n p l i c i t pro

duotive teohnology of t ho ooonomy.

2) The nonl inear aspoots of t h e rnodols a r o t roa tod only by gmphi*

al o r descr ip t ivo r.iothods s ince t h o i r mathomaticzl formulation

i s l imi tod t o systoms roduciblo t o a sooond ordor difforonco o-

quat ion vrith constant ooo2fioionto. I n othor nords tlxe mathemat - i o a i troatment r d o m t o unoonstrainod systons, L o . , i t fo-

l i s o s tho bohavior of t o t a l output i n b o t i - r o ~ tho oonstrainte. Y

3) i r i th rogard t o i i ivosti~ont, thoso nodola have oonoontrated on

f i x e d c a p i t a l 5-rith oooasional roforoncos t o invontoq- i n v e s t

nont. On t h e othor hand modela vrhioh oonsidor invontory f luotrr-

t i o (such as y), rn&o no r l e r o n c o t o i i x o d oai3itai.

3/Ba exception t o t h i s general stateuiont i s roprosentod by Goodv~inls model vrhoro ho a t tunp te a for in i l iac t ion of t h e no* l i n o a r i t i o s by graphioal i i i tograt ion of a Raloigh ttypo oquation ( roou l t of a Taylor ls x $ p r a z i i t i o n t o t h e mized difforonoe - dif- f o r o n t i a l equetion of h i s liiodoi). Navortheless by t h i a nethod, no general a a l y t i c so lu t ion i o obtained s inco t h e g r s p h i o d i n t e g r e t i o n correaponds t o one p a r t i c u l a r s e t of parmietor valuos. Mc Anulty J., lTainoa J., and S t r o t a R NGoodvinls nonl inear theory of tho Businosa cyclei an oloctroanalog solutionI1. &onouetrica, J y l y 1953, mplorod f u r t h o r t h e modo1 by us ing an elootro w a l o g comput or.

y ~ o v e l l U, "lknufacturorf s Invont or ies , Salos E x p e c t e t i o n ~ aild tilo acce iora t ion Principien. &bonomctrica, J u l y 1961.

I

As does ~io!c3/ , other authore distinguish even between two

idnde of inventoriesc i n working oapital, t e o h n i o ~ l l j neoessary

for produotive ?ur;loses, snd stoolce eoonomioally neoessary i f

sroduotian i u t o be oarried on "aomfortably". 3ut i-rhen i t

oomes t o the uoüel i t s e l f these im;>orteat variables are ignore6

by rpsking use nP an a r t i f i o i a l dePlaer the defini-kíon nof the

wriod. The 2eriod i s defined long enough so that the induoed

investrnent i n ~iorking oapital índuoed sn inoreaoe i n demand

juat offseta the deounulation i n stooke. 2//

Sinoe the t iue that EU&a pbl iehed hia book an the trade

oyole, there seems t o have been no f o r d atterngt t o generalize snd

ex%end bis uodel a l o n ~ the l inee su&@sted by Bhe forecoing

oomment s. 6 '

R. F'ishJ, reoognieed i t as a deiiiand dociinated model

when he e x p l i o i t l ~ introi-uoed an inequality i n a pro~rauuing f o r

mulation of the aio::oian uodel, representing bounds imnosed a t sny

period by the s i se o: the oagital stook. Nevertheless i t i s an

opt imi~at ion aodel miL as suoh it belonga t o the realm of normative

eoonomios where one han t o Beoide,W means of sone sooial C-eoision

on ends, on a general vreelPare funotion.

It i s the yurpose oil t h i s dissertat ion t o ,)reaent a des-

dHioks, J.R., op. oit . , pp. 47.

%&s, J.S., op. oit., pp. 53.

6/R. Trish, Parametilo 2 o l , ~ ~ , ~ ~ i n a of the IIioksian Model. Univeraity of Oslo, 1962.

o r i p t i v e model of eoononio f luotua t ione whioh i s essent ia l l j r a gene-

r a l i s a t i o n of the flio::sian nodel dong the l i n e s suggesto8 by the

th ree already mentioiled pointe.

Por t h a t pr.)ose l e t u s keep the univeree o: &isoourne

vfell i n mind. iie w i l l introduoe i n t o the Hlo::sian nodel oapite.1

stoolc, depreoiation, inveotuent i n inventaries, irivestment i n f ixed

o a p i t a l and e d e e a s ex-post endogenoue variables.

mirthermore, ríe w i l l Zormally oonsider the su-.]. l y oide of

the eoonoqy i n the Zorra of a produotion funotion exhfbi t ing oonatant

r e tu rns t o soale as vrell 88 f ixed ooefi 'ioients o9 pro&uotion sinoe

i t 1s the teo'hnolog impl i c i t i n the Hioksian theory.

a n a l l y , the ooneiCeration of the var iab le s a l e s r e .

~ r e a e n t i n g wgregatd Cemmd, suggests the introduotion of a non-

l i n e a z al looat ion ~ m l e ivhio:~ w i l l d i s t r i b u t e the f u l l y ouployed

resouroes avai lable aman; oonsuuers iind produoera. i'5 3oopnan $ pointed out; Ib3ow the l i u i t a t i o n (of the oe i l ing) i s brou;;ht t o

bear separately on oonauw-7tion and investuenf i s l e f t uns2eoi2ied

by ñioksb'.

The Biu i a t o inves t iga to the oonsequonoes of the generali-

eiation a s oonsioteiltly ss poesibie within the l i m i t oio the delined

universe.

h e t o the fwt t h a t no ana ly t ioa l solut ion oan be &ven

t o the r e su l t ing nosel -a e-ystem of seven 8ifferenoe e q u t i o n s

3/Koopmnns S, and Daueoh 4, Seleoted toe ic s i n e0~0mi0Ei involvina mathemationl-roaaoning. Cowlee F. Paper Bo 136.

mí tbee inequa l i t i e s - i n order t o enable the o l a s s i fy ing of al1

possible so lu t ions i n t h e ;>arameter spaoe, we s h a l l e x ~ l o r e i t

us ing si nnrlation.

S i d a t i o n i s here uncierstood t o be a t o o l f o r stuQring

tho behavior of ooiaplicate6. systema when they oannot be solvoc!. by

laown purely derluotive oethods. The experimentation u-rlll md:e

use of i t e r a t i v o mothodo iri~ioh aro desoribed i n the iolloiring pagee.

As Baunol pu ts i t "Pss t iou lar ly &n lar@ soa le 00m;mting

probleme involving equations o? high ooreer o r many var iab les o r

equations i t e r a t i v e tooUque6: oan be more e f f i o i e n t than m y d i r e o t

method of ao lu t ian even when d i roo t methods a r e available", S,

Undoubtedly, m e might objeot t o t h e f a c t thet tho

t e o h n o l o ~ , o r -for t h a t ua t to r - vrith the absenoe of a r i o e s as

e x p l i o i t variables.

Neoolansioal nodels of growth have alre- orossed the

bridge from t h e si6.e of f ixod ooefi'ioienta t o snooth i s o q u m t s

r e s ~ o n d i n g t o o h a n ~ e a i n i s o t o r prioes.

Nevertheless those neoolass ios l modela "i^ailx i n nnother

&:reotion. The aohievo6. i l e x i b i l i t y i n t h e sense of v a r i a b i l i t y

c , t i k l a lys i s of t h e concept o2 oii iulat io .~ ' , ' , aai Buskey, B., "A languago i o r si~.aulation", i n S s g o s i b ~ cn Si,.ilil+

m-..,

?>>2i;. ~odde&%~14.$~~~~-+:d-A~1i c a t i o n ~ 3 Q l l g $ B + y M Sbieiices. A. I i o ~ ~ a r t a l d P. Balderston, m i t o r s , Ohio, 1963. --

'Be~u:~,ol, Y., bbonoiLo Dyimuioa, N a Yorl:, '1959, p. 251.

of teohniques i s n o t ~on~ileuentod i n the r ea l i s t i o dimension of

var iabi l i ty of u t i l i z d i o n of the capi ta l etoclc. Along tl.iis

road the nao-~ieoolassicd modele of aotivi ty analysia and i ts

tierivations aeem t o take the leed, although a permanent oloavage

soems t o e s t hero betvreen the aot ivi ty a a l y s i s mo¿els whore

inequali t ies take ez.ilioit nooount of undemtil iaat ion of eoonomlo

rosouroes 8nd tho undei.ou:>l~ment theories. Tho f ormer attaoh,

through the duai analysis, a eero equilibrium sr ioc t o the unde*

u t i l i zed resouroe vrheroan underutilization uxiete i n the rea l l i f e

with market prioes diP2erent frorn ooro. The guestion i e whether

o r not this oleavam i s t o bo ex.plained by unoertdnty done.

In the next oimi,tar, a desoriation of Prof. Elo::sf model

w i l l be prenante6. ani. tbe uain points that determino& this

ganoralization wfl l be diaousoed,

In ohapter 2 vio ahsl l presont a c?osori:>tion of the d e l

and al1 i ts assum-tions. It v r i l l inolude tho stuciy of the

stat ianary equilibriun as ~ ~ e l l as the equilibrium i n the I1re&larly

progrcssive eoonowfl.

h oha;>tcr threo no mil1 b o g a the analysis o;" the wodel

;,,roi>aring tho way f or oq lo r ing the unoonstrained eoonow. The

dotai la of the algorithri f o r oaloulatinc: the maoi~iai root are l o f t

l o r the amondix t o tho thesis.

10 --/me necolassioal srgument wants t o show that any ten6onoy f o r the oapitcd stook t o outgrow the labor foroe oan be avoicled b~r uaing a toobique having a difforent oa-pita1 intensity.

11% ohaptor 4 tlio r o s u l t s of t h e sirnulation f o r tho vfholc

s y s t m írill be presontod i-dth tho e c p l i c i t oporation o1 t h e i i o h

l i n e a r oonstraints .

Fina;Lly tho api->o;ldix presents t h e d e t a i l s of t ho algorithm

dcvisod and adoptod f o r t h e systematio w o l u t i o i ~ of al1 roal, c o w

p lez uld n u l t i p l e roo t s of tho f i f t h dogree oharaotor i s t io polyn-

mial oquatio:~ v~hich -1-iithout ooonouiio oontont i n i t s o l f - reduoes t h e

model under oer ta i i i assur.~ptions.

C H A P T I R O M Y

Briof desoript ion of Bioks' medel - .-

Hiokel mcdel -a oimplif ioat ion @ his r iohe r theary- 1s

fundamentslly en o o o e l e r a t o r m l t i p l i e r m d e l oharaoter ised by an

inveetmont Ainotion whioh depende on t h e r a t o of ohange of wtwt and

s oonsumptiion funotion dopondlng on ou tp i t (inoome) of t h e provicurr

p a r i d , The ver iab los i n r e a l terme oonsidoraL a s flm i n period t

are: Y inoome ( o u t p t ) , C oonnumption demand, A autanancus invoatment

and 1 induoed inveetmont. There a r e trra wonomio parameters, tho

marginal propensity t o oonaime, o, and tho inveetmont ooeff ioiont v.

The pr inoipa l oauie of f l u o t u a t i o n s - l i e s i n tho e f feo t e f

ohan¿.oe of produotion on inveetment, i.o., iniruood investmont.

l'hus, wo have i n the s ta t i rmary casos

or, by using (1) and (2)

I n t e m e of deviat ions from t h s oquilibrium path, Ct , (3)

is roduoed t o tho following hamogoneme sooond order dlffaronoo equa t im

(4) Yt = 0 Ybl + v(ybl - yb2) 9 where y t - Y t - E t

P;I equation l i k e (4) iaay yield four t o t a l ly different patterns af

behavior ecoording t o the nature of the two eigenvalues i n turn

depending on the paremeter values. I/

They a re the oases of etesdy oonvergenoe, danped f l u o t u e

tions, exploeive Pluotuations and exploeion without fluotuatione or

steady divergenoe frocl equilibrium.

I n other worde, the nature of the n o t s , ri f o r I - 1, 2,

02 the ohareoteri~ilitio epuation oorresponding t o (4) j i.e., 2 r - (o + v) r + v P O, give four poeeible oesee among whioh the iiioat

2 rea l ia t io acoording t o the ende of the moda is w h a v >(1 +F) in suoh 8 oase the eoonowic eyetein i a arploeive without i"luotuation~,

but a ceiling, a "direot res t ra int upon u p a r d e s p w i o n i n the lorm

of eoarcity of erd*loyable resoumeaV5/prevente i t from peasing the

l imite of f u l l emplopent.

The ooefficientsaand v a re euoh that ea upward eh i l t from

the epuilibrium path w i l l oreate a moveinent tow- disequilibriun

oonPined by two barriem, the iiientioned oeil ing sad a f l o o r deteniti-

ned by the feot that gross inveatnent oannot be negative.

Eioks w~rke d t h a regularly progressive eoonony where

proüuotion ;jm et a oonetant r a t e @.ven erogenously by the r a t e of

g&h of autonanioua investaent.

'EiOkü J, o cit., -Ltho.iafiod bppan-.

d ~ i o b J , op. oit., psee 95.

4 eteady gmwth of t h i e lcind will induoe investment whioh

ni11 be a oonstant praportion af actual pmduotion. If the ra fe af

growth, g, of autonomoue investment is given, t o t a l output mst

g r m at the ame r a t e aad this l e neoessarg f o r progreesive epuili-

bdum.

Let g be the r a t e of gavwth of autonomous investment. Then

(3) beeomeea

It is a l se t rue that

Letting be the r a t e of powth of Yt we have that

hihitipiying (6) b l (l+h) and eubstreoting i t from (5) we have1

0 = 1 [ ( I+B) - ( l + h ) 1 from whioh it i e ahown that far a progreeeive equilibrium g n ~ e t

wusi A * '

The equilibrium gmwth path of the progreeaive wonomy i e

deduoed from the path of autonomous investmurt by appl- tbe

"eupemiultiplierll t o it. 1/

yThe eupemultiplier i e the siinple ~uul t ip l ier , t&ng eleo induoed i n v e s t m t in to aooount. ñioke, J., op. cit., pp. 103.

Solving f o r Y*, the equilibriuin path f o r output i e i i n a i l y obtained

Comente.

One week aageot of the Hioiraian model 1s the treatuient af

autonoimus investuerit as oposeed t o induoed investuent . By autonomous inves9uoPf Eioke meant "piblio invee tment ,

investment whioh ooours i n direot reeponee f o inventions end uuoh

of the lony range investment whioh i e only szpeoted t o pay f o r

i t s e l i over s 1011% periodW. d

Thie investuent playa a fundaraentel role i n the model,

te suoh en rrrctent thet the rembvai. o9 i ts trend w d d imply the

diaappearanoe ef the lower turning point oi? the oyole.

T h i s i e not t rue oi' the l i nea r theory (the only part wethe-

matioally formsli5ed) sinoe the reaiovai al autono~~ous iiiveetuent

misply turne the othemise nonhouogcuieow oharaoberistio squation

&ola, J., DD. oit., pp. 59.

i n t r a homogeneous one. Hence the sane pattern oL' oyolioi l behsvior

a re gossible.

Nwertheless when one begine t o explore the dizferent

phisea of the oyole with the nonlinear theory i n operation,Z/then

the model o w e t get &L' the f l o o r without en autononow investment

trend.

W s w i l l first show the inonotonic oonvergenoe t o the

a t a t i o n e q vaiue of the m l t i y l i e r during the seoond ghsse of the

trendiese oyole. I n suoh a phaae the value ef t h r ~ o e l e r a t o r

ohanges t o iero snd the inodel i s reduoed to,

Then the reduoed equation beoomes Yt - o Y + dt, and t l i t e aolution 1s

A where Ye - l_o i s the e~ui l ib r ium value of output (inoome) af

the f l oo r aud Yo l e the i n i t i a l value, Le. , the v d u e at the oeil ing

IiCombining the d i f f iou i t i es of diiferenoe equations with those of nonlAnear theory we get an &mal of a ferooioue ~haraoter . . .~ Ooodwin R. "A nonlinear theory vf the oyole." Beview of Ebonomioe aad Sts t i s t tos , i?ovember 1950.

Eenoe i t is seen that i n the seaond phaee of t h e nonlinear A path of output, t he syetefi~ converges %o-- . Por a reooverg t o be

l - r possible, the dii'ference (Y - yt ) should be posit ive. Let ua ase

t +l i f it i s t r u e i

Sinoe o < l and, s ince i t is reasonable t o assume that i n

t h i a phaae Y > A 1 (l-c) t he a p r e s l o n i s seen t o be negative. o

On the other hand, when outonouous investment i a subjeot

t o en upward t r e n a g, the reoovery beooriies poss ib le aa can be seen

in t h e Pollowing. The reduoed equs t i in becomes8

Let us o s l l r

Then (1) becomes

t Mviding through by (l+g) 8

Then, substituting in (2):

For a reoovery t o be posaible ( Y ~ + ~ - yt) should be poeitive, a s oan

If the r igh t hand s ide i e posltlve, then it impliee Qhat - '1

A s t inoreaeee, the f i r e t term l e l ike ly t o daabate. In

: t th ie aay the poeeibil i ty o f a reowery i e &m f r m the 1- turning

i poínt of output d e n autcmomous lnveatment keepe l t e upmd trend. Ne-

rertheleee, l e t us eee what thie impliee i n eocmamio termo. It m w e

that onoe the eocmamy h i t a the f loor of sero groee increetment the re-

oovery l e brought about by autonanouo inveritmcirit whoee trend prooeedr

8t a oairitant poeitive r a t e even during the alrsnp.

Thia impliee tha t the well known dual charaoter of investment

r damand generator a s well a s oapacity creator N i s ignored with re-

gard t o autonomous inveatment dome trend 1s Arrthermore unrelated t o

the produotive oapaakfy of the aoonomy When %he seoond property

cf inveetment - capacity creator - is not ignored, the exogenoua trend

woULd add t o exoeas capacity during the downswing, even when %he acce-

l e r a t c r 1s "shut offu, evidently an undesirable feature, a s excess ca-

paoity can never be absorbed

kí other words i n Prof. Hicka' model, autanomous investment

oreatea capacity only a t the poinf of reoovery. A t that point it i e

olearly needed, beoauae once excees capacity has been absorbed the le-

ve1 of output i s adjusted t o the oapital etock. Henoe output cannot

increase and the increment oannot be a llsignalll f o r induced investment

through a pcaitive acoelerator coefficient without an increaae l n the

capi ta l stock. The oapital stock cannot increaae without positive net

lnveatment which in turn repuires a change i n qutput. Hence a c i r c l e

i e cloaed.

When inventaries, fixed capital , and sales a re expl ic i t ly

introduced in to the mcdel - a s wi l l be dane in the next chapter - then

the r i g ld i t i ee of the downswlng tend t o diaappear and the lower tu-n-

ing point beccmes poasible without any trend in autanmous investmentd

This effect wi l l be a h m in chapter four.

A s autonomous investment i a remove6 f r m the model, it

o/ A s a matter of fac t in Hioka' madel even induced invest- itmt is unrelated t o the phyaical cap i ta l oapacity of the eocmomic myat en.

Ekoept in the rare case in whfoh the dspreoiaticm exceeds L

beomes pertinent t c diaouss why i t 1s oonsidered oorreat t o regard

a l 1 the investment ao t iv i ty of the eoonomy a s induoed by the past

oourse of business aotivity.

abany of the exogennue foroei impreoieely gathered under the

general name of autonanoue lnvestment (populatim ohanges, ohanges i n

wants) a re foroea whioh t h i s dieeertaticm asüumes t o be taken into a 2

oount by the lntrodwticm of the variable sa les representing demand.

It is assumed that the ourrent leve1 of eoonomio act iv l ty

and the oouree whioh has been fdlowed i n the reoent peat determine

the investmant behavior and deoieions of actual entrepreneura.

It seems a r t i f i c i a l t o distinguieh so sharply betwean the

ourrent and reoent past oourse of eocmcmio ac t iv i ty on the me hand,

and the so called "grawth trend over long periodo" cm the other; espe-

o i a l l y when the l a t t e r l e uurelated t o the exleting produotim bounds

oí the economy.

It is the "n& and the "yesterday" whioh can be expeoted t o

ahape the antioipations for ntanarrmlt - a oansequenoe of the essen-

t i a l l y omtinuous human existente of eomomic egents. Vihat might be

true 1s that the sane past and present receive different lnterpreta-

t i m a through different prooesaes of mental assimilatim. Bowever,

t h i s procese i s unknm in the present s t a t e of the eooial solenoes.

'Pa. other weak features of the Hiokeian model ñrought about

the ln t roduct im of tm additioplal and explioit oeillnge ln the form

of inequalities. The first 1s a oeil ing of oapital oapacity iinoe no

oapital reetr iot ion was oonsidered a t a l 1 ln the afore montlmed model.

Phyeioal capi ta l m e implioit ly aasumed t o bs permanently adjusted t o

output, period a f t e r period, during the upewing. Thie feature i a un-

r ea l i a t i o , quite apart fram the faot that the ta& of investment i s

preoisely the removal of bottleneoke. tilirthermore, i n Prof. Hioksv m 2

del nothlng a l t e r e i f the inveetment goods moe oreated - assuming

that they can be oreated by demand alane - are thr- away. This is

tn i e fo r h ie lnduoed inveetment ln the upwring and f o r autonomous in-

vestment i n a l 1 the phases of the oyole.

The aeoond additicmal oeil lng i s a o e i l i w of sa les or to-

tal d~mand. A oei l ing of d e s l e sa id t o d e t wha a t any time pe-

riod, t o t a l demand ex-ante tendi t o ezoeed the available t o t a l supply

oapoeed of t o t a l output plus a o o d a t e d lnventoriee.

It i s not poseible for demand t o increase ex-post a t a fa-

or r e t e than that a t whirh oapaoity output i s rieing. A s lnventcries

oannot be negative and - a s has been dieouesed abwe - a t any period

there i s an effective oonstraint of physioal oapital, omsumption

plus investment demand might t r y t o exoeed the available supply during

$he upsñlng, smetimee m11 before the w e r a l l oei l lng of f u l l employ

ment of labor i s reached.

A t t h i s point, i inoe no prioes enter exp l ic i t ly ln to the

model, a non-linear ru le oí' allocaticm or relaxation i s introduoed

which vil11 permlt the eoanomio eystem t o respond t o t h i s barrier.

Hicks l e f t the allooaticm of the outpuf of the oeiling bet-

ween oonsumptim and to t a l lnveetment unspeoified, a ocmsequenoe of

working everywhere with the variable output lnstead of sales.

The overall ceilFng of labor bringi about no need for exp l i

o i t relaxation rulea ln the caee of a me eeotor model.

Disouasion of tho invostmant funotion adoptad

Tho oonsumption finoticm 1s osson t i a l ly tho samo a# i n # i o h s t

model oxoopt for a d i s t r i t u t o d l a g of two pariods whioh has boon eoon

t o add a oortain smoothnoss t o tho dynamio systom.

A s f a r a s invostmont i e oonoornod, it oan bo sa id t h a t i n

t o m e of ompirioal var i f ioa t ions , thoro is sti l l no g ~ n o r a l l y aoooptable

thoory of invostmont . Attaoks woro nado on ompirioal &ounds agains t the

S/ aoooloration prinoiplo notably by 'Plnborgon, who oontrastod it with

what ho oal lod tho p r o f i t prinoiplo f inding tho l a t t o r a bot tor ox-

planation of invostmont f luc tua t ions . I n thoeo t o s t e Tinburgon ro joc ts

tho aoooloration prinoiplo on tho quostionabla pcunds t h a t tho ad-

d i t i o n of tho r a t o of incroaso of produotion doos not inoroaso fho s i s o

of tho oorrolat ion ooofi ioicnt - obtainod with p r o f i t s a s tho oxplana-

tory m r i a b l o - t o a s i ~ n i l i o a n t d e ~ o o .

A s Somor @ p o i n t ~ d art woll, both thoor?c>s aro not noooa-

s a r i l y mutually oxolusivo; thu p r o f i t prinoiplo r o a l l y inoludos tho

soooluration prinoiplo a s vtoll a s othor faotors. íiny closo aeaooiation

bstwoon p r o f i t s and invostmont might bo ovidoncu of tho oporation of

s/Tinborgon J. i t S t a t i s t i o a l ovidonoo on tho Aoooloration Principlol' Eoonaaioa, 1938, and

Tinborgon J. S t a t i s t i o a l t o s t i w of Wisinoss-Cyolo . thoorios. VOL. 1, (Conovo, Loaguo of Natione, 1939).

i %omors R. Govornmont Zkpondituro an& Buainoss Invostmont. 1 Chaptor 5.

the aoooloration grinoiple. This oan be soen both on e t a t i e t i o a l a s

well as on empirSoa1 pounds.

On e t a t i s t i o a l ,-rana lb/ t h e r o l e & other variables basidos

prof i ta is ~ r o a t l y undereatimatod, t o the extent that ool l inoar i ty

between tho addit ional variables and p r o i i t e should be oxseoted.

On thooret ioal ;rounds it oan bo s a i d thaf wlth a l i n e a r

hmogeneais produotion fimotion w-ith variablo o ~ & f i c t i u h ~ , i f prioee

of faotore aro g i v o q m&misation of prof i ta implios t h a t tho eoanomy

1s moving along the oxpanaion paCh, ( l m s of tangonoy botween t h e

isoqucuits and the prioo l ine) . I n t h i s oasoy moving a l o w the

oxpanaion path over timo noana an inoroaso i n tho o a p i t a l s toak

aeeooiatcd w i th an inoroase i n outputj tho fiinotional rc la t ionship

boing linoar.

If tho produotion funotion l e l i n o a r homojoncaie with

constant oouffioionte, thon no mattor how faoLor prioes vary,

invostment w l l l f o l l w Sdontioal pattoraa a s i n tho ?rovious oaeo. '.

Two addit ional and rooont s t a t i s t l c a l vorif ioat ions .soom

t o strengthen evon fur thor tha cause fo r tho aocoleration prinoiplor

llJ The first i e a papm by Waltors who oonoludoe t h a t one

importan+ roason f o r tho donnrard b ias of the principia l e the

oorrelat ion oxist ing batweon toohnioal proGTosa and o h w e e i n

outpit. Whon allawing f o r t h i s b ias the moaeurod valuo of the

aooolerator "cases qui te oloso t e oneta thoorotioal o x p ~ o t a f i o n s . ~ ~

-/Smxth, B, . , tf%pirioai ovidonoo on tho Aoooleration P r i n o i p l ~ ~ ~ , Tho Roviow of Cooncido S t u d m Juna 1964.-

uWalters A,, "The Aooalarstor and Toohniaal P r o , r o s ~ ~ ~ ,

Tho aooond ii a papor by Yisno 12/ who extondod t o the f io ld

of imoitmont the samo anslyiie urod i n tho t h o w of tho oonsumption

funo$ion, that $6 the pormenant and rola t ivo irmomo approsoh.

Ho fin& tha t the t rue rulat ion 1s bofaoon pormanont ohsngoi

i n outplt and invostmont and tha t tho roason fo r tho dmnward bias 1s

tho influenoo of tho tranaitory oompononta of tho ohango i n a t p i t e

Wlth th ia spproeoh Jiisner f inds tho aoooloration prinoiple rruporior

t o tho prof i t prinoiplo.

Whet wi11 bo proposod i n tho no* ohaptor a s a theory of

invostmont, is an oxtoneion of tho aoooloration pr indple . Tho main

foaturo i s that oxpeotationü aro explioit ly takon into oomidaration

i n oontrast to thc oommonly oonooivcd aooolurator, a so r t of toohnioal

rolafionahip basad on roaliications ami not on oxpootations.

%isner, R., "Investmont, faote and fanoy." Amorioan ~oonomio ñopiew, &y 1963.

C H A P T E R T W O

The Xodel

It is basically a oertainty rnodel. Neverthelees, use wi l l be

nade of severa1 ex-ante variables such ae anticipated sales, piauned

inventaries and planned oapacity, although without a rp l io i t l y facing

the problem of deoiaiaí making under unoertainfy. h othar worda, the

antioipationa of the nodel a re single-valued, whereas under unoertain-

t y they may be probability distr ibutions and plana may be stratsgiee.

The word an t ic ipa t ime i s used instead of arpeotatiaíe,

eince plana oan a l so be regarded aa expeotaticma i n a broad aenee.

dn t io ipa t imi w i l l r e l a t e t o the future oourse of variables

outside the direot oontrol of ecc~iomio decisian makere, whereas plana

r e l a t e t o the future oouree of aotion of the eomcunio *ente themselvea.

They a re not dec i s ims about future actima. They a re infarmed fore-

oasts aa t o what the beat course of aoticm wi l l turn out t o be g. This i a the eense 1x1 which the variable lloptimum i n v e n t ~ i e a ~ ~ w i l l be

used, aa part of the p l u s for inveatment i n inventoriea.

l o e t --ante variablea ñill play the r d e of intermediate

variables sinoe the analytical goal w i l l be t o establish relat icne o-

neofin# =-post variablei. They juet help t o determine the behaviaria-

t i 0 functione thraugh a prooeie of eliminatiai es w i l l be aeen.

Time is divided inlio dieorete hrmiqe~leour periodo. Eaoh time

u i i i i l i a n i p. mi& o d e a í , me ro ie oi m t i o i ~ a t i m s in eocpaaio behavior an - d the i r use in w m d o anslyiir, Urbana, 1964.

period mst be regarded aa an arbi t rary leag*h of time inhoee ohoioe l a

diotated by analytioal oanvenienoe. Nevertheleee i t i r neoeeeary *o

real izo the importanoe of the time period i n the preeent model einoe

it defines the geetation lag of ínvestment. Henoe the length of the pg

siod cannot be eaeily modified without introdwing axpl ic i t ly another

'parameter representing the geetakian lag.

In eaoh time period, firme have fo.make deoisime in regard

t o the magnitudea of f i v e variabler, narnely, produotim, malee, inven-

tory investment, inveetment in fixed capi ta l and capi ta l capacity. For

t h i e purpoee, they have t o take in to acocunt sane dynamic conatrainta

in the fomn of equal i t les cr inequal i t ies which limit the i r freedm of

aotion and determiiie the extent of the enforcement of deoiaione.

Dacisions rofer t o planned movss of the ourrent time pericd.

Betwean pericdas eoononiio agente implement the i r current deoieians t o

the aztent that j.s feas ible and adjust the i r aotual aatione *en deoi-

aione oannot be implemented.

Therefore, a t the beginning of period 6 , entrepreneurs make

t h e i r p o d w t i o n and inveetment plana for khat period.

For the prodwtion plan, it i e aeeumed that they entimate a

the anticipated malee fo r the period, St , by adjueting the ea le i of

the previoua period, S t-1' by ame fraction v of the inoreaee i n mlen

between the l a e t two perioda.

The adjwtment faotor i a oalled here the ~ h a i i a r i a t i o a002

l ea rn ing procesa when al loved t o vary through time as n m iir?orrmtion

i s gained. This oont ras t s witii t he oap i t a l coe i f io ien t , a tochnologi-

c a l parametor known i n the eoonomic l i t e r a t u r e as %he aooelerator".

The value of v i s a funotion of the degrees of cptiroism of

t h e entrepreneur end has no r e l a t i o n whatsoever t o t h e cap i t a l output

r a t i o whioh is exogenously given by tho p revs i l ing production taob-

nique.

On tho o the r hand, i f by "learningw, a r e l a t i v e l y permenent

change i n bohavior reaultingi"rom paet experionoe i e undorstood, t h i s

model has "once and f o r evor learning" sinoo v i s assumed t o be cona-

tant over time. A s long as tho pas t ohsnge i n s a l e s i s posi t ivo, in-

ves to r s roaot viith tho sano v.

This aooelorator i s the f o m e bohind invostinent i n phyai-

oa l oapi ta l , s inco it is tilo increaao i n demand r a t h a r than tho in-

oreaso i n output vvwoh i s u l t i u a t e l y responsiblo f o r t h e a t i ~ i u l e

t i o n of invostrnent.

It i s f u r t h e r aaaui.iod t h a t thero ex is ta an "optirnum" leve l of

invontorios, ? l i n e a r l y ro la ted t o an t ic ipa tod sa les , whero the coef- t' f i o i e n t r ind ica te s the rooiprooal cf t h e r a t e of turnover;

Thorefore, entropreneurs plan t o ad jus t invontorioe ex i s t ing

a t tha boginning of t h e period, V t c t h i s noptimumw l e v e l whioh i a a t'

simple avorago stook t o meet s a l e s during p e k o d t. The r a t o 02 adjust-

ment o r reaotion oooff iciont , f , which i s a i s o a peyohological c o e f f i r

oient , re~?loots the importwoe given by t h e deoiaion d c e r t o having

en optimai inventory, 2/

In general, firma may be a p e o t e d not t o adjuet i n s t a n t -

aneously t o t h e 'optiinun* l e v e l of inventories pa r t ly beoause they

aze not ~ e r t a i n .f t h e i r expeotatlont3. If < were t o un-

ohanged, ac tua l invent or ies , would tend t o appmaoh i t as t

tenda t o i n f i n i t y whenever t h e inventory plana are f u l l f i l l e d .

When f = O, entrepeneura plen t o d n t e i n the o ld l e v a

of inventar ies , V exioting a t the beginning of p e b o i t. When t ' f 1, they plan t o a t t a i n the optirnuui l eve l within the period.

For 8 &ven 9, the grea ter the differenoe between $ and V , the grea te r 1s t he r e t e per un i t of tiuie a t whioh they adjust t h e i r

demend.

The elements dieoussed permit eoonoaiic agente t o Pornulate

at the beginning of period t, t h e i r produotion plana, ~ f t , i'or t h e

period, designed t o iuieet ant ioipated ee le s and t h e pleaned inoreaee

i n inventodes .

One oan not ioe here the f l a i b i l i t y provided by the

a o o ~ a t i o n of inventories in a uodel with oonstant pdoes. The

eoonoiiiio syatem responda t o -ante d i equ i l ib r iums with an adjuetrnen*

2/liheee a r e the equations known i n t h e l i t e r a t u r e t h a t deals wlth inventory investnent suoh as Yetaler L. "The nature and s t a b i l i t y af inventory oyolesI1. Review oP &onom&os &.Statiatioe, 1 9 4 , en& Lovell &, "Butfer stocke, s a l e s expeotations and e t a b i l i t y l e d t i - eeotor ana lys is of t h e Inventory Cyolen. & o r r o n e t r i o ~ 1962.

LI regaras the yroduction plana of (4), the i'resdom of

ohoioe of entrepreneura l e reetr ioted by two dynemio oonetrainte.

The first 1s &ven by an equaiity ln t ema of the history of eoine

variable8 or i n i t i a i oondltiona f o r period t.

Thie auttisl inventory level m8y fk1 beiow the planned

level whenever t o t a l aotuai consurnption plus inveatnient demsnd

exoeeds the sales antioipated f o r the prevlous period. Comereeiy,

unplanned antual inventoriee iiuy be aoouimilated when aotual demend

f s l l a ehort of antioipated sales.

The seoond üynsnio oonstraint i e &ven by the produotion

funotion

Ralation (6) esys that if both the oapital stook of the

wonomy gaisting at the beginnilk. of period t, Kt, uid the labor

foroe, Lt, a r e euffioient, then entrepreneure -1 oerry UA* t h e i r

produotion plana i n fu l l , %.e., Xt - X: . Otherilse t h q -1

produce a8 moh s e e i ther oageoity o r labor allore.

The eyribols and b denote the labor and ospita3 ooeffioienta,

the l a t t e r of dirnensionality atookinput per unit of flaw output, that

i e , p i re time.

in other wolde, a oei l ing of oapital oapaoity or one cü

labor may be cncounteredy a faot that can be crxpreeod by the f o l -

lowing inequali t ieat

whioh inipliee a constsilt ooeflioients t eohnology.

Therefore, firma es t inate the planned oapaoity by multiplying

the capital output r a t i o by the planned praduotion. In this respeot,

Le. , i n regard t o investment plana, they iust i'irst s s t i ~ m t e K? t+l '

the oapaoity needed f o r oarrying out the produotion plan8 f o r the next

pedod, einoe a geststion l a g Zor inveetmant of one perlod 1s allowed.

T h i s ~ e s t a t i o n l a g i e defined aa being the period between the

investmont outlay i n real t e a and the nouent at whioh the n m oape

oi ty is ready t o be ut i l i sed.

(Equalfty since a c e a s oapaoity vvould not be planned i n 8

oerteinty model).

F Sinoo xP refem t o the next period, i t is aeaumed that

k t +l

j/It is aeeumed that a rizad proportion s r i e t s at any time between the eime of the population and that of the available labor Zoroe. Purtherniore the working day i e ins t i tu t ione l ly fi-ed and the d e l is oloeed t o foreign trario. In other words, there ootiea a time Whan the labor oeil ing i a an efzeotive constraint.

i t w i l l be estimsted i n the sane wa~r se current production plans.

The differenoe i e that the oorresponding psychologioal ooeffioients

a r e die t inct in principie beoause the e~rtrapolation owers two periode

i n s t a or" one. 9'

It is new poesible t o ooriipute the lwels of induoed groes

invee tnent # x: by adding t o the plenned inorease i n oapacity the

amcunts pertinent t o replaoementt

Dt depende on current output l w e i e vcith depreoiation being

a l inear funotion of uti l izat ion. d denotes the r a t e of dopreciaticn.

The ra to of groee investment, 1 at any time is subjeot t > to a uinimm, i.e., a f loor reflooting a jus t i f ied l w k of syuunetry

between the procesees of capital aocwuiation snd d e o ~ l a t i a n . Thia

faot eapl ioi t ly reoognizes that negative net inveatment is limited

inescapably t o deprociaticn ratee. ~%rtherroare, i t is in agreemant

with the observod Pacts i n the sense that i n actuaJ. eoonomio l i f e ,

there existe anoesa oapacity with i d i e plasta and maohines. It

recognizes f l e z i b i l i t y i n the var iab i l i ty of ut i l izat ion.

T h a t is, whenever the gmas induoement ta iavest i e negative,

i gross investment i a zero, or i n terma of inequalitiesr

autonoriious component represent ing eubsis tenco oonsuiiiption~

t vihere At - Po (l+g) and g representa the r e t e of growth of popula-

tion. Subsistence consm~ption g row with population.

This behavior on t h e pa r t af consumera does ~ i o t &e use of

eny ant io ipa tory data; eeoh plan represente a pure atrategy. I!

Total demsod, l e obtained by edding ooneumgtion and g.ross

dnvostment domands,

Vhen, at any &ven poriod, t o t a l demand tende -ante t o

ezceod the ourrent ava i l ab le supply ( inventoriee plus ourront output),

a o e i l i n g oi s a l e s 1s rcaohcd wiiioh oan be a p r e s s o d by the following

inequal i ty t

in o the r vuorcle, t he g a p l i c i t in t roduot icn oP invontorios as

an endogenous va r i ab le permite t h e f l a x i b i l i t y provhded by tho pos-

s i b l e diaorepanoy oi' actual ovor planned inventaries whenever s a l e s

expeotations a r e not f u l f i l l e d . h l o g o u s l y , t he re 1s another pas-

s i b l e souroe of inopual i ty of ac tua l mvos ovor plsnned noves rosul -

t i n g i'rord a possible tandenoy t o v i o l a t e the o a t l i n g of s a l e s on the

p a r t of d o o i ~ i o n &era.

Sonnaily, whonover tho oase i a auoh,prices wouid diatributo

tho available supply anong consumora and produoore. In thia reapoct

i t is assunod that tbo roaulting tondenoy f o r arioe inoreasea -1

affoot ci-d f o r al1 purpoaos i n a pmportionate way, i.e., thom

are no faroed savings i n th i s econouy. a/

One way o i introducing prices indirootly nithout a- to

the coriylaxity qf the nodel is by w i n g m i e s or" ralanatiaa or

ailooation.

It i a poesiblo t h w t o distinguish two oaswi One i a whan

the systcin is under the salo8 oeiling, Le. , whenc

in this oase, rolations (10) and (11) ara binding, tho

f o k r with positivo @rosa investnent.

The second poseibil i ty to be t&en in to aooount 1s when

the syetem tanda t o aood this oeiling, i.e., ii'hanr

Then (10) ~ n d (11) a r e transformad into8

i (10 '1 It - = (0 , $ ) Nt and

8/By forced savinga it i e underatood a procese suoh aa uonotarg inflation thmugh whioh inveetmeilt oould be i n o s o ~ t further at the coet af oum-ant ooneunptioa. See & Lovsll, 'Y Keynosien enslyeia of Boroed savini;~", Jn'c e r n a t i d Eoonowio Xevlar. Septarnber 1963.

Thia nonl inear a l loca t ion r e l a t i o n simply b i s t r i b u t e s s a l e s

i'n proportion t o t h e d~ina i ld~ of producers and ccnsurners, a c l u d i n g

aubsistence consumption s ince i t i s assumed that i t is geared t o PO-

pulat ion =ovrth. Whenever t h e system i s below this ce i l ing , Nt - 1 s ince the r a t i o of the r i g h t hand s i d e of (14) would be g rea te r than

S/ one.

It i s now possible t o eliminate t h e a -an te var iab les from

t h e above r e l a t i o n s and a p r e s a t h e i r r e l s t ionsh ip t o *post f h e d

investmant. By ueing (1) t o (41, (7) (8) we can writel

It can be cbeerved t h a t new gross investment i s needed t o

sh t i s fy r

1. the planned incrernent i n inventoriee) ( r S& - Vt), and - t 2. tho ant ic ipa ted increnent i n sale61 ( S+l S62) i/

Jurthemore, t h i s inves t~~ient funct ion peFmits t h e ohecldng

of the depx o l defioienoy i n t h e cap i t a l s tock at any period by

conparing the roquired capacity, b , vrith t h e h i e t o r i c a l c a p i t a l

stock, Kt, ax i s t ing a t tho period t. I n t h i s way, the

q/dn a l loca t ion r u l e I c r tho l a b o r fo rce and t h e cap i t a l oapacity o: t h e economy would only be neodeü i n e aisaggregated modeL

10 - iv and i t h e parameter8 r e s ü l t i n g fmm t n e gestationalag of investmont i a f ixed capi tal . v and f a r e usad t c estimate S wheroas ; and ? a r e used t o e s t i a a t o Sa

t' t +l

"Irrat ionali tyt t of odding t o azoess oapacity during a depressi.cn is

provented. Simtlarly, if tho ooonomy 1s growing, the o r a t i o n of

stroesa oapaoity i s prevented by alloning entreprenoum t o telíe arist ing

capacities in to aooount i n t h e i r plans.

Suuuuarining, the modo1 i s reducod t o tho follawing sysfem

of differenoe inoqua.tions, whioh oan be mritton i n a r o o u r s i ~ e WQ',

Le., one can syetomatioally go ffrom ono rolat ion t o the n a t , a

feature wbioh wll l be of holp during the airaulation.

i V* + xt (7) Nt - d n 1,

- 't i me= (O, 1 ) + a XtCl t + X t 2 1

i (9) It - (O, I t ) N*

[ - ~ / ( 1 - 6 ) ] t

Rbioh holds a6 long as tho syston i a ooneiatent.

T h i s case is s p u e t r i o t o oaso o.-, so tliat X can be takon

in a similar may.

Then, tho cquilibrium s ta r t ing valuee f o r d1 variabias in #

the progressive economy is found.

Parametera ai3u the time poriod

Pa regara tho ten paranatora of the model, (a, b, c, d, f ,

f, A , v, ; ami r ) , a t lwt i'our can be oonsidered aa Iiaving time

dimension. W Bs i a v r e i l l ~ o i m , the oapital output ra t io , b, can bo

oonsidered as pure timo, sinoe i t is d d i n e d as the m t i o of a stook

over a flow per unit of timo, both of which a re axpressed i n terme of

the samo conrnodity. An inoroaso i n the l M h of tho time poriod

implios, o e t e d a paribua, a reduotion in b.

The uffeot of tiixe on f , tho reaotion oodf ic ien t d

inventoriee, can be wpootod t o be the opposite. Bs the antreproneurla

decision poriod inoroasos he oan apoc t , oetor is paribua, t o a t t a i n

* ~ t i e noooseary t o noto that thoJo aro i n t h i s madel Zour further implioit paramotora vrhioh a ra relevant f o r the plenning and

8 deoision sota of e o o n o ~ o agenta. They a re the oonsuurption diatributed loge, the irwestment lag (botwoa tho ohange aaloa and the deoiaion

& fo invest) and the gostation lag f o r Pixed inveatmont. Bll af thom E &e taken t o be one period a s a first approximtion.

the optiinum leval of inventorles within the period, so thet f i n c r e e

ees. in general f can be ' 1. An f - 2 wouid m- tht inventoriee f a re planned t o be adjusted i n half a period.

The product llrf" remaine unaltered wlth reapect t o t i n e

sinoe r, the inveree oi? the r a t e oí? turnover cen be apec t ed t c

deorease as the length of the time period incresses.

The depreciation coefficient, d, i e .timedii.eneionleee eince

depreciation i e aseumed t o be a function of use, although it 1s

related t o depreoiation when oonaidered aa a lunction o+* time, through

the prevailing technology. Pssuiiie a servios l i f e o+" fwenty yeers f o r

plant and equipnent. T h i s imjjlies a depreoiation, due t o use,of 15$ if the aapi ta l cutput r a t i o 1s 3. How, l e t us shorten the period t o

sir msnths, whioh meU8 a depreciation as a function oz" time of 2.5%

which, taking in to aooount the inoreesed capi ta l outi~ut ra t io , leaves

the depreciation coa*fioient - a s a function of use - unchanged.

The behevicristic aooelerator, v, i e a pure nuri1ber eince i t

l e the r a t i o of two fnoreriiental flows. I t e value re f lec t s the I1degree

of optinisns o? the entrepreneurl'.

A s t o a and o, whero the sum denotes the overall marginal

propeaaity t o coneue, they a re a lso independent of the time unit,

althouyh implicitely they a re defined i'or a l a g of one anir *o periode,

respectively.

In regard t o the t i n e period of o u r modal, and i n generai

of 8ny other model, i t is r e lwan t t o dieouas eome points Which a re

often not f u l l y tsicen into aocount.

It hae been eaid regarding modele uring the aooeleration prí&

oiple that , einoe tho oapital output r a t i o has a time dimeneion, then

the bohavior of aggregate output depende m the ohoioe of the time unit

a s we oan eee by quoting lLodigllani:

"kw in the modele baeed m tho aooeloration prinoiplo, tho vg lue of the aooeleratLa ooeff ioimt 9ypiWlly turns out t b play a o r i t i oa l ro le i n determining whether the time path of eggregate output wi11 be damped or oxploeive. 'lt w d d appear therefore that tho e t ab i l i t y implioatione of a given uiodel oan be ohanged by the puroly f o rml tr iok of rhanging the ohoioe of the time unit. Onoe wo introüuoe our "epeed of ad- j ~ e t m e n t ~ ~ ooeffioienf, t h i e dieturbing poeeibil i ty tende t o dieoppijsr in that the e t ab i l i t y of the syetem ni11 generally depend on the produot of thin oooffioient and the mvent io- nal aooeleration ooeffioienl pnd' t h i e .product is independent of the ohoioe of '+he time u n i t l l . d

In the f i r e t plaoe, there i s eme oonfueion implied in th ie

etatement ae regarde the iden t i f ioa t im of the aooolorator coefficient

- eeontially timebimensionleea - wíth the oapital cooffioiont, a para-

meter refleoting pure time. This oonfuaion i s tho prodwt 04 foouaeing

attention exolusively cm the demand eide of deeieion makere inetead of

e p e o i f n g a omplete model.

Nevertheleis, apart from th i s , im can dlgreea for a while and

real ise what the atatement quoted abwe a o o ~ l i c r h e s .

The ohnge in the time unit of a modo1 hae a deoper omee-

puenoe than merely affeoting the numerioal value of a time dimeneianai

parameter, oonsepuenoe whioh might well be the tnt. resean fa? a l t e r i q

Madigliani 3'. 8 Comen* t o t tCa~oi*y , oapaoity u t i l i s a t i m and the aooeleration prinoiple8I. In =emr of C a ~ i i a l Porin&p. Stu- dies in Inoome --- andfllealth, Vol. 19, pp.457.

/

the degree of dampenlng or of explosivenese of a model.

It is t rue that - oeter ie peribui, - when the time unit beoomeq

ehorter, the numerioal value of the oapital output r a t i o inoreases sinoe

fo r a given oapital etook, the f im of output per uni t of time daoreamq.

The effeot of th ia i n a model auoh ae the Hiokaian model i s olearly

t o inoreaee the poss ibi i i ty of a omstrained explosion fo r a given valuq

of the eavlnge ooeffioient.

The r e a l queetion h m v e r i e whether or not the oeterie pari-

bus aeeunption 1s oorreot. Can me r ea l ly aeserf fhat ohengee in the

time un i t w í l l not have eubsoquent effeots on the modo1 through oorree-

pcnding ohangee i n the timing of deoieione of eoonomio e e n t s ?

The answer appears t o be that the ehortening of the period

would oblige eoonmio agents t o reaot more puiokly t o changea in eoono-

mio oonditions by aooalerating the timing of the i r deoieions. This aoca

le ra t ion would inorease the poeeibil i ty 09 errara whioh mQht therefore

beoome aooumulative fo r reasons quite different fram the numerioal

ohange ln a oapital coeffioient.

Even admitting that no errore oome from th i e eide or that de-

oieione are unaffeoted by the ohange i n the time unit, the quoted ata-

ment doee not take in to aooount tha following polnt.

It i e not euffioiont t o maneuver parametere so a s t o make

them lndependent of time beaause the effeot oannot be avoided - by

ohanging the time unit - of inoreaeing the nunber of complete oyo1.e

u I f m e Interprete hia o. as the oapital ooeffioient, whioh

wae never admitted a s suoh by the authar i n h i e bodc.

f o r a &ven ffme period 88 many t i m e 8 a8 tPe sliorter unit üiotatse.

I n 0th- worbe, the uathematioal aolutiaa o0 the -del

would be the aame sinoe poronetera ore &e (time - dimwionless)

.by uaing aome devioe, but the number of oyeles t o be heb would 0-8

n i t h ohanges i n the eoJe. TUS feature l a ee unrePliatio as that

whioh i e swght t o be evoided.

Thettmorol' t o be extra4t.d fron ill thia, 1s thpt the d&i-

n i t lon af the t im perlod tnuet oome Srom 'the theory itrrelf snd that

i t riust be justkfied by woaoiirio oota8iderationa. Iuter, the eagtdoai

egjl ioetlon 02 the modei tu;y ver t fy the va l id i ty & the oseumptionir.

C H A P T Y R T H R 2 Z

Analysis of the Nodo1

I n tho prosent sootian, tho oeonomio aystom dosoribod i n thc

proooding pages w i l l be subjoot t o sovoral assumptions. This w i l l

enablouo t o study tho oharaotoristio ~olynomial assooiated with the

difforonoc o~uatiryis syatem and m s t bo ooneiderod a s tho s ta r t ing

point f o r tho aforo montionod s imla t ion .

Tho stratogy implios a f i r s t s top assumption of all-arornd

equali t ies Cespito tha oxplioit oxistenoo of many inoquality

res t r io t ions i n tho modol undor oonsidoration.

It i s tnio that tho stuüy of t h i s oharaotoristio equation

M11 only dotormino tho behavior of tho ooonomy i n tho caso that no

inoqualitios ros t r io t tho plana and aotions of tho ooonmie agonte,

hero, whon tho ooonomy 1s to t a l l y liaoar.

Tho s t a b i l i t y of l inoar oooncmio eystoms, f u l l y dosoribod

by studying eharaotorietio polynomials of low dograo, is not inde-

pondat of tho i n i t i a l oonditiona, i n tho numbor indioatod by tha

nrdor of tho aeeooiatod difforonoo oquation. This implioa tha ao-

ooptanoo of tho statomont tha t tho magnitud0 and turning points of s

oyolo aro oanplotoly dotorminod by wonts whioh porhsps took plaoo

oany timo poriods boforo.

Non linoar thoory on tho othor band, auoh as that with whioh

tho Hiokaian modol 1s oonoornod,doos away with tho nooossity f o r

initbrrl oonditions.

i i

Tbo a m o happcns in aur p c s c n t caso. As soon as ono

inoquality c o i l l n g is h i t , thc oconofily "i'orgcts" in pare thc l a ~ t

i n i t i a l conditione. In mtthomatical torms t h ~ syntcm suddonly ohangoa

i n t o a naw charactoríetio cquation ~ t h onc incquality tranaformod

l n t o oqunli ty and thu magnitud0 of thu prcvima i n t u r i r ~ ~ a l oiLcnvrtluo

boing changod f n t o a, d i f fo run t l t i t ~ n t r~e.t. At thc ~amú t i m ~ anothor

oquality is rolaxod and tranefomnvd into an inaquality~ O t h a M s ~ tho

fixst inoquali ty tvc~zld tond f a br; violatodi

T b i ~ has tho of foo t o," damponing a syetom t h a t ofhomiso,

doponding on thc parmotor valuij8, could w o l l oxplodo, roaohing ratos

af g m f h qu i t c unwarrantod by any kiatorfoal axpori~ncü.

As 8oon as t h o systorn h i t 6 n coiliw thú scnlo or t h o s i g m .. .

of t h o o i ~ c n v o o t o r o a l t ~ r too. w5nt i s important ia tho m a l o af thc

voctor asoaoivtod rr i th tho mnxiriinl r o o t , l / e i n c ~ it is t k i s root tbv t

dominatos in tbc: 10% run. If the i n p n o t is r c l a t i vu ly smooth, so

that thoro i a a roduotion in t h ~ soalb but no chango ifi t h ~ s igns oP

tho oi~onvalua~, t h o systom wlll grow a ~ a i n and thus, v l r j l l go an

inda i fn i to ly . if thc impaot, on thc othor hnnd, cha%o~ tho si~n of

tho domiiant voctar a oumulativo dovwvi.ing micht s o t in .

I n s t a d oS studyiw m o o o ~ ~ i v t o h i ~ r n o t c r i s t i c polynomials

aocord iq to thc oye l icn l g b s o thu ooonorq i~ in, polyilomials whloh

cmld oasjly rciaoh vory hf gh do6;rcica, wo will r c so r t t o einnilatien

fox tho etucly of tbu oomplotc nonllnuar ay~tom.

1 i'%o l i r g c s t ruoi in modulus of tho dutorminantal oqueticn

/A - SI/- 0 ,

Bcvurtholoes, for s o v o r n l roasons, it is imi3ortant t o

viaualiz; how this rnorú compldt~ modo1 would bdhavi, if l c f t incon-

S t r a l n d .

.l. h r rcaeons of analogy as rogarcis thc Hioksian modo1 as w o l l as

o t h a r modols of tho t radu ayolc whioh oan bc ane i l y roducud t o a

s u a o i ~ C order dif foroncc oquation, íifhich is, by toclay, wc11 known

in ooononii~s.

2. To r o a l i c ~ tho Und of mnthomaticnl d i f f i o u l t i a s onc moountúra

as Boon as a rnodcl d ~ p n r t a fzom n r ~ s t r i c t c d numbur of variablus

or whon tho matrix of t h o e y a t ~ m dous not bolong to t hc oatugory

of alrcndy f h l l y o x p l o r ~ d systuns as v i i l l bc discuasbd latar.

3. To suz tho magnitudíis as i~iroll as tlic typcl of oizdnvrrluoa whioh

ara I i k ~ l y t o dominatu t h o p r u c ~ a t modo1 for r u a l i s t i c suts of

paramctcr vn luos .

4. To r t j a l i ~ u thc oxtont of t h o damponia, o f i o c t on tlic buhavior of

thc systi~ms by tho auocoasivo inuclual i t ice it goos on hittiw, so

tha t thc oconomy b~cornue orbitally s t a b l u in t h o l i m i t . z/

3n n c o o r d n n o ~ with th,, p r ~ c u d i n ~ d i s cus s i an un¿ in turma of

tht p r ~ v i m s chap t t r t t is aeaumbd t h a t

2 iOrbital st i lbi l i ty rnunns t b t thr, t i m o paths of t h c

variabloe cuilvorgo not tu a p u i n t but t o t h ~ path of s o m ~ poriodia rnotion, i a a . , in thú 10% run, tha gamu ayo10 i~ r o p o a t ~ d . Soo T. Wogtahi, " 0 r i thcj S t a b i l i t y of a Computitivo Xoonomyt', ~conomcfrioa, 1962, and h. Andronovt a i ~ d S. Chaikin, Tñioory of Oso311at30m. Moaoow, 1937 -

lTt = 1

(iqo ceiling of sa les y available sugply eatisf iss t o t a l denand)

* ( Production plana are almays euf orc ed)

To aiidplify the i io ta t ion let ue calli

Their, the horno;;.eneous ayatolli oi" equationa t o ~ t o r k With

The characteristic i i i a t rix o2 the syst eri, & t e r a:~proprlaf e

The resulting c iu rac toi.isi;ic polynokliial, P( z) , an dgebra lc :

equat ioi? ii? z , is oi' tbc ~ e v e a t h degree with no i l~do$ondei~t terr,~.

It has tmo zeso la tcnt r o o t s , lci10~10L&'e which perii~itsi us

t o I c m r t h o degree of tho polynol,iial to livc, by dividing P ( z ) by

the !uioLJiz linear $ac '~ora,

Yhe $01 ~ i i o i ~ i i a l t he11 li oconos i

Tbe caefi"icio~rts A (i = 1,. . ,5), are eharessed in terim i'

of the t eri paxar,iet era of the i~lodol,

They can be recuaed t o eight by i d ~ i n g the f o l l o ~ J S w plausi-

b l e a ~ s ~ i l g t i o ~ t

1. The r e o i y r o c a l oi" the rate of turnover, r, is o o n ~ i t a n t

throu~hcub~oripds t and t+l, ln our notation t lda üieaafl - t h a t r r.

2. Ths behavioristic accolerator, v, aid t h e seaotion coeffi-

c i e n t of inventosies, f, are p r o p o r t i o d l y relatad t a each

ot her i

v/f = rrhich io assuiied t o bo !mown "a p c i a r i t t by the

decision iA1a2~e2s. In t h i s way o n l y thxee o2 theae Tour p a r e

i-iet ers aro f nilependent.

Jt is n o t poas ib le to obtain ~ L I analytic s o l u t i o i ~ of the

abovo sgaten m I difiexencc equatione s ince no general aolution can

be &%ven to the assocfated obara~taris t ia poiyn~uial whioh a o e e d s

tbe T o u r t h de;ree, i~urtberi,iore ;lo o the r reduotion can be perzoriiied

in the praoeding SiSth degrce characteristic equation.

In t h i a r e ~ p e c t our task is c o ~ ~ p l i c a t e d due to the speciai

structurs of the ~ha~acteristic ,.iatri.x 02 our .,lodel. It doee not poa-

seas aiw of the l'good i"caturest' coialion t o othar econoi:dc ~iiodels such

3~ as the input-output iliodel ol LeonticG-, or the i i iu l t i -aectord u u l t i -

p l i e r ~ i o d e l 4' G o ~ d ~ ~ * ~ h i ~ ~ i a i i ~ ~ ~ ' ~ h f ohh w m ~ d allair us t o find a

relatively simple answor 2or the o r b i t a l s t i b i l i t y 6 / oi the eooiiorny

descl-ibed by our dodel , In thess cases, t h a Perron-Probalius t!heorerils

7 1 4 fornon-aegat ive , s q u a r e ~ d i i ~ d e c o ~ ~ o ~ a b l e i ~ i a t r i c e s a r o a p p l i c a b l e . It l o lrnown that such i,ia-trices have a, p o s i t i v e root not lesa in modulus

than any other root, vthic:i i i1 a d d i t i ~ n is siiiiple ai~d roal. Ii" the iiia-

tris: is decoinpo~able, t h e dciidnant roof is non-negative as w e l l a8

red .

I n other tirords, Lha behavior oi" econo~~liee described by such

natrices, a b ~ h a v i o r wiSch ia doi,,inated in tile l o n g run by t h e laten*

r o o t of t h e largest i ; l ~ d u l ~ ü , can be airiiply ascertainod by searching

f or the naxi~~ia l real r o o t s . She nu~ierical aialysi~ is thon ~ i i i ~ ~ ~ l i í ' i ~ d

ainae no coinplex 01- ~ : iu l t i -> l e r o o t e ara in the way,

'lHaiCcin8, D. arid Siiiioil, H. "iiotei Soi.10 Gondi tiona of hacr* econoaic Stabili ty. '' &onoi1mt rioa, 1949,

"leaishi, T. "01-1 the S t a b i l i ty o2 a CoiJ~i3,,etitivo Illcono~~~y, "

veraity Eress, 1364. A~~mxik.

881n8 i~ ~ N B i i i t h 8 OBBe 09 ~ t h e ~ ~ ~ W I B I ~ I B ? 8 ~ 0 b BB

those apjlicabl e to h o f i n i t e poai t ive iaatric es where a l 1 eiganvaluaa

are lnovfii t o be real,

Sinoe none of these propert iea apply t o the iiiatrij; w i t h

which wo are conaarnad, no o the r roati ia opcii a c e p t t b a t aolving

t he chasaoterietic polynori.ia1 20r se1 ec t ed paramet er valuea, by

iii-s o2 apecial al&oritlli:dc tecliniques and ail elec-broiiic coLnputer.

Several alg;.orith,,ia i o r t h e evaluatian of the ~ o o t s ol" poly-

noiniala oT degree groatar than f o u r exiet. The procedure t h a t has

been afioptod w i l l ae ar:~l&notl ii? the kp$endix.

e / lalow, iI. " O n t h o St ruo tu re of Linear Lodala, ", k o n o m e z - ca, 1956.

As has boori ;~rcviausly discusaod, t h c ro is no viay of

obtaiidii , &I m a l y t i c a i so-ution i'rola %$u i , iodol bacauso o: the

a i i s tcnoc af lnoyual i ty restiictionu and n o n l i n e a r i t i o e , Coila*

yuent ly tho ~ ~ i o d a l ~ ~ 1 1 1 bi: s ~ p l o r o d u a i l ~ ~ , the o i ~ i u l s t i o n tocluiiyuo.

Aa usoi; liar, the rrasd ci,,:ulatiori ,,~eai.is U E ~ F I Z ~ a coiiii3utor t o 5 r o r l ~ out

tbo cvolutlon 02 varia'oloa ov,r tiiiio vrithin t h e ;eaora l i'2aiL~ovrork

as ~ i v o n by ths nadol.

a s han boon sho-un i i i chagtor 3, evon na,loctin; the

n o i ~ l i n c a r i t i o s by A a k i ~ ~ ~ us e 02 t lio e p o c i a assuuy-tioi~a oi""cn 1'0ulb

i A 2 thc tool b@z,oi' ccoiio.dsts, thu degroe oi' tliu ~ I i a ~ a o ~ o r i a t i c poly-

no..lial is G U C ~ %ha$ no ,oiioral a l a l y t i c so lu t i on c m bo o'U.;;ained.

Zn t u s ~ l o s i i ~ ; ; . chsL>-ter tí0 val1 r o g o ~ t olí. t ho r c s u l t ~ o2

'aotn at-teiii2ts ; the s i i - u l a t i on eorreaL~ondin,- to t h o n o d i i i u a r econo~ily,

ami t h o roEu l t a of t110 algoXithil Jiscussod in t ho appondix corres2on-

cd;rs :*o Llio u c a n s i ; r a i a o d cgsterii irhFcl; d lobrad us to s e l o a t t h e par*

i ; i ~ t o r ~ ->ha% verc to be ussL,

Thc uncoi~oüra incd ec onoiLi3f ---.C. C.-....-.,-- -- -- *

1% í ~ a a saiL j.11 C ~ E L ~ ~ G F 3, 1ihi10 ~ L S C U G S F D ~ t h o ~ l c o i ~ s t r a i n o d

eccilo,,iy, t B a t it rr:auld be intoraoting t a soe tho L.m~-;riitudo u ~ d tyges

02 roots t k t '"Joro l i . rc ly t o c.oi.iinatc the ~ io i i o l f o r r o d ; i ~ t l c s o t ~ of

parmeter vaiues. Thia 5s i:,~?artailt becauso tbo maulta a2 t h i s mora

c o l r i ~ l a ,,.odel seei~i to givo i,lore w ~ i g h t t o Hicks' conjeo;Cure about t h e

s ~ p l o s i v o hypcthosis.-/ In Kiok ~ i model the paraustor spaoa can be

exhaustivoly dividad into Pour regions cosresponding to 2our t o t a i l y

d i f 2 oront typoe of dynaiic ii ollavi or, wong irhich t ho azpi osiva hy;~+

t h e s i s 1s on ly onc, I';io al,oTith,,i mhich solvod our í'ifth dogxoo ~ h &

rao te r i s t i r ; pol;rnoi.iialy20uarl tbo er.- lo si ve bjpathenis t o bo th r u i c

and not just a p o s s i b i l i t y . Thc ooi~iputcr vas capablo aP ooi.iirig up

kdth 1:arc: t h a i z onc thousand cnzios in a Ioi-f - . i i nu to~ , i"or s o t s of aar*

I iz d i i i o s t al1 caneg t h a iixa~iiitucio o* tbo ~j:odulua 02 the ~iw

:~i i ; i a l root waa graat o r t;iai~ ono, IJo possi'uii ity or" dar,ij?od 2luc tuationa

or stoacy Lonver,cnco 8oollicd i > l a u ~ i ' o l ~ * As rogarda t h o tyim o f cii~an-

vduos , t h c i , i a z i t i a l oncG lrrcre d i r a y a s iwplo ( a i t h o r roal or c o i n p l ~ )

and vd-ch a poaitive roal pai-t.

1Jo caaos o2 , i iu:'~iyl~ 1-mot8 viiore obsem~i l , n o i t l i o r roal ser

co1;1:3l~i;~ and al1 cascs oi ' co,,plgl;: r o o t s o t h e r tliail the l a r ~ e e t ono

vmre looe hliat~ one in i.iolulus,

b s t of tho analysis w í l l bc ~ o i s h o d to t h o ca8o of a "non

grogrossivo cconoi i~f ,~ ' t l i a t ia, t o tho case of a rat c ol" zro't-tth of tho

popuia t ion equal to aoxo. Aa is B~OTJII i n cbapter olla, it ie in t h e

dHiclrs, J.R., ap. cit., p. 92.

U 0 Q t ho ap.3 ondi;;.

stzt ioilary daso vhore L;zc l r ichian ir.odol does not psEiaaa a lotres

turning i;poin.t;. -do t i i l l 8 e ~ habr tha a p l i c i t i n t r o d u ~ t i o i i oi" iizve~triant

in invcn-torles adds tiio ruqu i rod f ' l a i b i l i t j r at t h i a yoint 02 rocovtiry,

T h i s is i . , iportant as o t h c n l i a o a ~10ngrowL1i;' 0001101;~ -ifoultl ; ~ o t havo

a~ry 2osa i ' h i l i t y 02 recovcriizg 2roi;l a daprcsaion w h a l c2 t t o sndog-

nouz forcea.

This is also illipo18tant i ' r o r~ t ho 2oinL 62 vio$-r of c c . o i l ~ ~ . l C

thoory sinco in t h i s T P ~ i'ra can hava an cndogonous non-liilaar t h o o r y

02 tLio cycle -urhicb doos not lYa i l ' l in tho ~ t ~ t i o n a r y cano,

In t h o r'irst l ) lnco, rro í-~ill s t ~ r t by i l l u s t r a t i i ~ g tho i;iodol

srith tilo study o2 a p a r t i c i i a ~ ' caue, r ~ p r ~ ~ ~ n t ~ d 111 i;ra$h n u i ~ b u r one,

nhicb can bc c*nsidui.oL aa a - typics l c:rcle, apgo.ril:iately a linit

cjrclu. 5'hia w i l l ilcrliit us -to soc t h o 11ray iu t-íhich t h o aconor;iy r c e

ponds to the dilx'creiit no:zlinoari ti es by ilit orpr o t i n ~ t l io ~zuribora ap-

pearin, at tho bot t o , ~ o2 t h o graph correoponding t o tbe r o s t r i c ' ~ i o n s ,

In the seconC place, a cal-12arative 6Lynanics stucq o: the

,dodel w i l l be yer20n~iod. S i n c e t h e i,iodel coi i ta ins nine parmleterg,

it is very díff icult t o givo a roproaentation af t h e pa rmete r apace

s h o w i n ~ rogLons which c ofrcaponii t o c l i f l e r sn t pat t erns o2 behavior.

lis sIial1 therei 'ore l i i , i i t oursolves to coilsidsring 8~1ial.l changes in

t h e va lues of t h e parardoters. The a e n s i t i v i t y oi' the econor.Qr t a these

changas \vil1 be d l s c u ~ ü e d in t e r i , ~ ~ o.; t he lol lowing i,lain characterie-

tica ay t h e cycl et

- average c!-uration

- lensth o2 t i i i i u r e q u i r o d Llor a b ~ e : , ~ ~r depress ion to trorlc iteel; out, asid

- res@ t i ons t o the ~ i i y o i e r t n o n l i n e ~ ~ i ties.

l inear thoory , Iii linear t hoo iy , li!.dt ~ y c l e s can o i ? l ~ ba had i 'or

'one particular s e t 02- i>aranoters, i,e,, t h o s c re~ultini; in a iilartitlal

r o o t 02 uni'i;arj. iliodulus, O d y - in t k a case da we hzve a i~ iono to~~ous

r e p o t i t i o i l 02 ccjrc2ical ga-Lhs; a s i i , d l dtipa7ture C ~ O L * t h a t uizitary

i , l u i i , i a l r o o t produces q u i t e a difÁ'e~eizt behavior, N i t h ilonlinear

theory it is p o s s i b l e t a have l i í i i i t ogoles fY*uiiioh are sson to be

811 ei i i lu la t io i~s fltart r v i th t l ~ e (uns tabla) equilibri~~~ sol*

with X c ff (ou tpu t r ~ l a n n -u l i ' i l lod) , vrith 1 li / b ( t h a systeili O O O

is j)i.oLiucing at Tul1 cagacity), aiid r r i t i i L 2 AA; or, in tami~s OS O O Y une~ilgloy~~ient ratos, sirioe tlm l a b o i ~ o u t s u t r a t i o is assu~icd t o be one,

x L o . X e r e U , tl-ie ra te o: unoiírployi+ient, is 1 0 j i .

o 1 -U

Purtharmoi-e, initially, the sgstem i a be1017 t h e ~ciling of

sal es, i. o , , N = 1 so tlnt no s e l ~ a t i o n r u l e 1s ncedeu in order t o O

a l loca te t h e t o t a l a u p i ~ l y a2 i aven to r i e s aild t o t a l 0 u t i 3 ~ t aiiiüng consu-

iliers and produc ora.

d. i i th d i i . i ' ~ ro i :~o equations it shoulri be na ta& that no ~ O P

Tect lii,~it cycloo are olitzined duo to ti;;io d i scon t i i iu i ty in rslatian to tho soi i - l ineari t ior ,

'de can rroii; dith k = 1 ainoe l abor c w be iaeasured i i i aiiy units. On t h c o t h e r hai~d, ~apacitg is also i,ieasured i i l output tsri,is, íct That is, capacity = _ - m - e 1i1 t h i a rray, capi ta i u& l abor aro l inde co~ik parsblc to t h e b íl oi:+variabl es 02 the ~ iode l . i7ev ortl-iel ea s in the case of i r , i i i~veei lor ias , tl-iay iicep the i r stoclc natur2-.

I n arder t o atuQ tbs o y c l i c d paths, a disturbarico is ii~ppe

-aed onz tbe equilibriu,,i path 02 the ecdnaiw i4 the f o m ol" a 1 ,i loaa

02 the capital ~tock.

Tho p a r t i c u l a r tgp af disturbance that has besn cbosm Ss

uni~lpor t ant i

1 a t because oP the non2i ;?oar i t ies mi" tbe modol. Tl~e i~~i t ia i equili- -- brin,i 2oa i t ion ,ilus the d i s t urbano* shoc!~ bccoine i r r o l evailt tm

eoon as the co i l inss are reachod. dhen thie happsue, new initial

cori&tions ara 68t -.,~hich are inilepenueat the poin-t 02 departi+

T8.

2 nd 'ihat is r s levmt i o r Lhe theory of econoi,fic Llu~tuativns ic tbo - l i d t cycleci th3 ..iodel i:~izht roach, T:my ara indopeadmt of t h e

i n i t i a l 00nGii i0 i1~ and aro t o be coasidered as an c; iBc!~si~n o?

ths coiirLK)n couicoi~t 02 a t a b i l i t y ia the B e i l B 0 oi' coilvergenoe t o

e i u i l i b riul.1 poiilt.

A sgst oiii bsn ol-bi'cal s t a b i l i t y o r ca l r ~ a c h a 1ii.d.t cycle ifhen,

a= ttiidie increascs, it coi?ver;;eB t o a ty;~e oi' oeoillatory pa th

tha t sougi-lly k c ~ ~ a repoatin, , i t s el2 over tiine.

111 ~ , i i s is the ;;:;.nf~ o: o s c i l l a t o i y i u t h that our ~ L ~ ~ i u l c B i o n e reaob .

a r o e 1 r r o It is a L>atb iihich is not iri-

i'luenccs. t;r -;ha initial diaturbanoe c r by tbe ~qcaili'uriuii depa*

t5ng ;,0:1:1 b a

\fe v f i l l ¿leacribe a coiiiplete cycle in e06noiilj.c teri.18 in ozder

t o i l l u s t r a t a liow tbe 1.iode1 rriar!é~+

Ths gxayb represonts the oscillatory patha obtained d t e r

peillod nixty, a aath wllich i a appr&ir.utaly rapeated t h e r o d t e r . The

grwupo 01 three nui,iber~ at the bottoiii 02 the ~ r a y h p e r d t a a t r a i ~ h t -

;FaiTrard reacliizg 02 t h e co:~straiíi ts the econoi;iy goea on hitting over

t i ~ ~ ú , They can be read accordiiig t o t h s Pollawing taiiloe, trhioh arw

~ r n a i s t c : i t w i t h t h e notation of chapter twar

Tabla 1

Table 2

Tho first t a b l o ad:.dt@ tlie p a e a i b i i i t i e a - accosdil l~ t o t h e

mi,ibars in tbs i ' irst coln;.a o;' the ~ r a g h - of a coi i inz o;' fui1 ei~gloy-

uent oi" l a b o r (ita 41, a c e i f ir& o2 ca;i.iacit:r (iP Z), tho Yul-'iilnont

oi" i i r \ r > C ~ c t i ~ i ~ 2lans l), oi- t l io c o b Y i i ~ a L i ~ i l OS t h ~ las% ~ W O possi-

b i l i t i e s (íio 3 ) .

The tliilui tabCe rci'ers t o invest..iant, The i ~ ~ a s i b i ; L t l o a aro

t l m t tjia sys te~ i rcaohoa t h e i i a o r aL" zero gro~fi l i a~ue tn i en t (iP 21, tha t

i ~ ~ d u c o i l i i ~ v e s t ~ . ~ e n t i s rec1;lzed (do I.) , or t b a t i.t c ~ m o t be 2ully re*

o/ lized (;To O, iv1ie.n t h o c o i l i t i , o; oales I s U t ) ,

Tha gsagh ~ ~ o s e a t u t b e ;aths of s i x endogenous varf abl ea i

~ o i ' i a ~ ~ . i ~ t i o i . i , alo os, ou-t;~ut, p i o d u e t i a ~ z >laxis , cagaGZtyl ani, Inveslto-

rioc. h o o a i: lv<:nt,ient in any ;?criad cm bo read by t h e v a r t i c a l di&

tmie botuoen sa les and o o n s v . ~ t i o n , s i m e by (10) S = Ct + Xt. t

She caee ei tu& s d here Y arreaponde t u the i"ol1owiilt; parmeter

valuea ;

k u n g abetrzct iona o2 shor t paalcs and troub-he %re aee here

a c o i a ~ l s t o ;r'ou+phaeiod cyr ; lo 20r outpi;t BE^ w e l l ae lor capacity

During t k o 2irz.t plknoa, tbo dorfnsi7in6, boain i l in~ st t - 60, víc hava a a l sw proceso of absorp t ion o1 a c o s a oapaoity 3uring whioh

grose irlvostinent i a zero wu t h u cag i ta l 8toc:r docui.iulatan st the pace

&c;tnta& b;f leprociation, Yurini, fb f s i i k a o , outtnit $lana are iiret by

aotual pmduction, p lana rrhich do i l o t a i r i i o i t h ~ r a t capacity or

Cull o n u l o y ~ ~ m n t , duo to the gloai~iy v i m e oT t h e deyreaa io i~~

321 e~ anu Coilc?i,.:;iti oa rol1 evi ~li~lilc3.r yatt ~rn8 during t h i s

ghase, and can absoma a procosa of slow a c ~ u i ~ u l a t i o n of inventariao

nrhich $vi11 bo mari t~ i,dco rocovery soarrible. Tbeae ii1veil"YrFea a r a

t h e psoGuct of emjll díscro,~cmcies between ou t i~u t a ~ d sa loe úur ing the

d@~~nmfin& 'de can s o 0 that OiICO ~ G O B B ~apa~ity han been 2ully absor-

bed (at t - 67) t he econo~.iy k i - ta tho i ' loor, a i t e r vUch rocoveq l a

pbsaibla b e a m o kheA5e -19 $o&itiv& &roa@ i l l vos ti.ieat whieh a t rf rcrt aiim

ply i la311t ains the e;rLs-ting caliitdi stoo:~.

Ijluation (10) of $he ¿.iodel su;(i~iaxiaeil on page X, t el18 u8

that this ~ i l l produce a chanze i n a a l e ~ , and by e%ua%ions (2) and (6)

ws ?aow t h a t the chango tal1 fl.tiiiulatu production plana m Well as the

induc~i.;ent tu invcs-t. In t h i a ay the ~apital stoulc is takQrr off tb8

f l o o r a& tbe prosper l ty 2base se ts in.

The reccvery is ilade i ~ o n s i b l a by t h e ondogonoue i'oruea 02

the QCOilOi.lJ ' s i i t l ~ o u t the h e l p oi" anjr aogeiloua treiid. L'i;.atly there

is just eiiougli invea t t iont -Lo lreop the exiati i ig c a p i t a l ntook at t he

?loor lovel , Jhat we havc to ~LBCUEEI ie the Iisecise 01%&1n of thoae

i;ooCLsr iit: han aoen that tlio;r cor~ie i ' r o ~ i acauíiulatod inventaries. Iri

th i6 i.rode1 i i l v~n t0 r i . e~ c o i ~ s i s t o;' coi~suiipt ion as w e l l ae iilveat,,ient

~ o o d s , How ~ h c n , shall we justi2y the fact that t h e cor roc t q u a t i t i e s

of I n ~ o a t , ~ i e n t gaods wll1 b e availa'lilu, t u , , iaintain t h ~ capital etoclr,

and Lo aad to t h t ca2acity du i ing the ups'uing?

Oizo anawer I .u ,~t b c t h a t ii' the case w r o that i i~vs i l tor i se

-vere j iost ly co:u;mscd oT consu..ii>.i;ion guods, it vmuld be ou t 0;' tbe

$ro¿uction or" thc per iod thnt ilivefitixi~t cvuld be sztisi'loil, The LE+

cu,i:iilation of i i lve i l tor iss tlut is ~ b ~ e i ~ e d as a riS e a-t thc be&-ii.ulin;

02 the upmvins, iu -Lo bo useil f c r ~oi:suii;i~t-ion In euch a iray that t h e

proLuctive capacity o2 the eCoiIO,,v Fc l i b e r d t e d to srociuc e iiivostment

goods that ldll add to cai~eci t j r i1.i the jr*cilloíuing peziod,

111 t h i a way, LIJC ::'irsi havn a loviex turning point f or salas

which b c g i i ~ to iacrcaso ( o u t 02 i r~ventor ies) ~ i v i n ~ tho p n i t i v o grasa

inveati lont noeded Por sil iiicrcauo in c apacity i n -tha Polloiiing p r i o c l .

'de havc - lfho i l vfe d l o w I"or t h e laga iilhersnt i n t h e mde l - a 10-uor kurning Soint ~ o r ou-tpuL as w a l l ao 20.r capaci'cy, irhioh tog*

t h e r i.Qtiatc tho n a t p b s s , the upswin~. The ceiiing 02 capacity

i~i h i t - a upocial c e i l i i ~ i r l~ich growe e l l&0g9a0~~ly ~ L t h the econoiiiic

~ y S t elil - alld output groWs at f u l l capscity util t h e gap 02 uilelbipl6y-

;,~ent o? l a b o r a l ~ ~ c s t di s x i ~ ~ ~ ears,

W b a t bappann i n tbs ,,iewt;lne i d t h c d o e c p s o t ~ t i o n a ami

plans? I-Y'hwi tbe econolri;r iiasnoa the ;Pir~t lower tumiiw point, that

of salos, ~yectatione bccorio te0 0 p t i 4 s f S ~ ; output denArad by yr*

duoun GGO& oapacity, aaii BU a concoquwioe so doea aotual i ~ r o d u ~ .

tion, u n t i l t h o tiae that entrcpreneurs "learn", ; iba i t l a tos late,

aria tho downsvfing has s e t ini

Since the eiitregroneur basco hfe a p o o t a t i o u Zor $zcsent

and A t u r e s d o e on h i ~ a c t u a l uparienoe bP tho 2aat, at tho be&*

nfng o: the u$swing, illloductiori IB nat enough t o oatiai3y dcL.iaild and

lnvantorf es docriiiuilato. T h i s iu so baoauae c a m o i t y i e tho red . .

bott2eiaock in tho uyeidn,-. klo 8 0 0 t h a t capar.ity Goes not s p a n d as

rapidly az the U i ~ o o t a - t i o i i s of s a l w diotata ea as t o bo aisle t o

r.iaintain t h o o i ~ t i ~ : u w 1 cvcl of i ; l v o n t o r i ~ ~ l ~ desirod sroduction - 1-rbat

we havc oetllod i i i ' odu~t i~ i l plms -. -;'ollo'i?a a 2ath that is clooer to

de- (sales) than that 02 actual proLuction. Thu upser turning

po in t s f o r P and K arr. a p l a i r i o d by v r b t h ~ p n a t e fho course of

sales. Tha Path of s d e a reachoe a u2ger turnfng point boforo P

aiid X , duo to tho c o n c ~ u a i c ua ~ C O B C oapacity mlieri output raaubost

the coiling 02 i'ull hii2loyiliont of labor. 170 m e that cap~tcity conti-

nucs to ,zotv boyond tho c a i l i n g 32 labor as a consecLuence úL" invest-

uent outlays during tho y r ~ v i o u s yori od,

'dlion a o o s s capncity appoars, the rsduc-Lion in invoet~zent

( s i : z c o b;,, (6) oiltropra~cura talco the a i ~ l t i n g ~ a s i t 8 3 st002: into a*

count in t h a i r d e c i s i o i ~ a ) a c a o d a ths increase in consru,i$tion in euoh

a way that t o t a l sales decliiie. TMs dooline i i 1 2 l u e n ~ ~ s - Tdtb a l a g

of ona period - the nst docis ion f o invest, and úonseyuently the path

capacity a l a o r o a c h ~ s aii u;j;Jor turiling poiiitm T h c r e d t o r thi8 path

roflcc.tn onco EIOXO the s l o w , ~ roccss o: a - ~ s o i p t i o n o2 G c e s s capacitjr

e-uring irliieh grms i i~vost: .~cnt rci:;iain~ at zoso.

- . k c t us Boe ~ ~ l l a t liali~ons to t h o path oi" t o t a l o u t ~ 3 u . t ~ ¡:e

rroc tliat Gurin;; t h o rccovery m d t h s upwiriiig this sath r'allm-JS t h e

c c i l i n g O~ capacity uitil t h c 20iiit vxhcro t he PuJ.1 or.zployi,~nt cei l laz

i a hit, Output reiains at this cc i l ing i 'or ~ l ca rca ly ano :)oriorl, b*

causo in t h s stationary caso it z ~ p r ~ ~ m t ~ an absoluto bar r l e r , Tll*

reaP$.,z output ini t ia tcs z slarr ds~riiai~ing i-iarlced by shert r s c i l l a t i o n s ,

in accmri ia~zc~ r i i t b thc i l c s i r cz to proauco mi' tthe antro;3ronwcr, ~ h o

sc0i.i to "lcarn" t h c r e d p o a a i b i i i t i o s oi' t h e econo"w rhr ing d o p e s s i o n

p e r i o ~ ~ .

i/c: o a n o-bxoivi: in -Ibis cdac that no relaxatioil r u l o is rie*

dad t o a l loca t o thc: o u t i i u t E: tlw cailiing bct:~con coasuiicrs and pr*

ducers b ~ c a u s i : no c u i l i n g nL- saic6: is h i t . Thia i.,ioana t h a t i i ivontarics

to irioclerate t h c i r dernand , or ca>ital gomda.

In t h i o sca t io i i w u will &-glore t he ou;isoquorzccs ol' e n d l

chm;os i i i t b e paraiictcr vnlucsi ai" the ciodol on t h c ~ d . n o h a r a c t o r i *

t ics of t h o limit cycles obtaincd.

'In t u s ?.ioCel, a o Ion, as i nven to r i o s aro l o a i t i v e , '10

can ilovor havc unsatio2bsd do,,ailiLI

Thcy ea11 r u i c ? r i:!ainZy ta t h o a t a t i o n a q hiconor.iy althou{a

t h e l a s t cask will be c;:plisitly c o i z c o ~ n o d k i t h tho a r ~ l o i - a t i o n o2

thc ooilscquoncoa a groi1-&1g 'irelid i i l t h o l a b o r ü e i l i n z .

ti0 r í i l l Bci coi~ce~.ncd i c t h tila co.,.gari son batírooil ¿.~-oupa o2

t i r o o r t h r e o gagha. Cliu i'irs-t g o u s oi' t h r o o graplia c a l be usad t o

t r a c e t h z a f f c c t s a chaqii?, ca2ital o u t i ~ u t rat io , according t o

tho ;ollolrLng tzbl e i

- II

Graph b a d i' O r v v f 1- b o t

2 1.00 0.4 0.05 o , l 0.3 0.25 0.5 1.5 0.3 1.4799

3 1.50 0.4 0.05 0.1 0.3 0.25 0-5 1 3 Qt3 1.9093

4 2.00 0.4 0.05 O,Z 0.2 0.25 0.5 1.5 0.3 4.0961

'$0 can cac t h z t des:jitc t h o aubatanoia; p o r c u n t a ~ o chango

in thu value of b , tho ~ o i , i ~ a ~ i ~ o i i ruvaaia tho ,Aiaii~tanancc 0.2 a cer-triin

a.t;rilcturd stability. 'Il-iia i l . ipl iue t b t altbough thc: ~ o o t 02 ~~iaxii<al

, ,ioddus ( ~ ' o x tho unco~loiraiirrod ~ j r s t r ? ~ ~ ) X S larg~r - otario ;>aribus - t h o largor thc valuc u; the cai2i"u ~ ~ ~ ' i ' i c i m t is, t h c d8iLlgening

c2:"i'cct oi t b c c o n s t r z i n t ~ is such that uveii tho d'f cc t or" a r o ~ t as

1ar;c as livo iadus a;:a;r a8 thc: 1 i . i - i c p l o ia aprjroacbad.

iIe can ohnorvo, thougAI that tho l a i g c y tho capLtaJ coeie

l i c i o i r t , t h o ai.ioothor tiie pi'ooo;o oi aosor2tion oí' esceBs caj.3acity.

Thc rcaüo;l is tha t in t h a p-aphs, K atailds L'OZ capaciiky and n o t rr'or

capital stoc::. Iioncc in the dovii~awin,, tho sai.ia dcpro~iation rahto

isipii03 a s:,dl e r ret iuctior~ in ~apacity whsn t h o c a p i f a l co&'9f cllont

is largor, T is pheno~~cnon in obsoxvcd in ilraphe 2, 3, 4, ao ~rel l a8

in j anid 6.

In al1 i'ive casoa t h o sama atructurd pat -~orn 0.2 tho typi-

cal caso 5s r d l o c t od. Yho tu rn ing golnts oi" aaloe loal t b e t u r n i n g

pointr o2 t hc othor variabloi. Tho ug&idrai, 3or oapacity an r i d l

as f o z output is procodoit by invontory docui i~datloi~ ~ i n c o it is the

chango in aaloa f aci l i tatod by tliis phanoi~onon n u c h uomtitutoa tbe

"~ igna l " f or cn-irupronuurs to i.1a2o t h o i r ilzvoetnei~t dooioions.

A coilsoquunco o2 tho cl ian~o in b 1s rieon to bo that tho la+

gor t lmt b , tho laygcr t h e capi ta l raquirornmta t o i i l c r o a ~ o prcduotion

are. i i i i t h i a rospüot an intoreating d f o c t can bu obecntcd. Poapite

the 2act that oneo t b o ceililzc aP s a l o e Se roacbod, t l io a i l oca t ion r u l o

ef oux tlodel i~ propost ional t o i l e ~ w d s of concu..igtion aad i:ivoatuonf,

tho a3i'oct is s o m ~ t o poi ' leot a aituation o1 i'orcoil saviii@ irliich 1s

utore acu ta m h m b is Zarger. Holv can this be orplained?

Talco t \ io ocono.,!icü, A 3 , t h a oiily d i f r'cronco b o t ~ o o i l thcd

buitlg t h t tho cccand has a h i ~ b o x capital cooí"~ioient . A ~ i v o n deoired

incrci.ioilt in psoductSon ruqui roe a h i zha r mount of invusti~ont ii1 thc

caso o2 c c o n o q 3 tliail iri t l i a t o: ooonozg A.

Suy-~ooo t h a t in both casos t h c i nc roasa in c a i ~ a c i t y rcquired

te LEC% thoac plan6 ~ c o o d a tho productiva capacity of tllc ocono,,;y plus

ti lo i i i von tos ics z í l i i ~ h havu accul.iulatod uy t o that i-ioiiont, Thon tho

cciliia;: o í aaluu 1s rcacliod a11 tho d f o o t o 2 thu o i ~ o w t i o i l oí' the al-

locat ion 1-u: o i o ;,iort: itu'lationaiy ( roc~uctf o n oi" roa3 conau,.ipti~n) i i 3

- r .

?lag u ~ A L u ; . ~ , u i 3 ; ; : IOI.~LL a ~ t ~ a i , t ii.~,i nliort J U C ~ I ~ U ~ ~ O ~ B .

tho oaao of ooonomy B , lirhich had tho highsr oapital ooaffiaient. b

incroasoa, tho poei t ion o: coneunora womena at t ha rooovary point*

ke far as tho cffoot on tba amplitud0 end poriod of the cyolo

fs conccrnod, wo sec t h a t t h o larger tho percontago ohango in tbo capi-

tal ood, ' ic iont tho longor tha amplitudo or" tho oyolo. T U As t o bo

a p o o t cd bocausc , of t ho ~ I b v i o r oapital doounulatf on p r o c o ~ s , whioh has

draady b aen diacuso od.

Wo oan d s o obsoivo that the larsor b is, t h o loas l i k o l y

1s tho syutom t o roach tho coiling of full ogplaymont oP labor. Tho

rcason i u that as tha oconorw booonioa moro o e p i t d insonsivo, tho coi-

ling of capacity bccomcs s uoro aFfcctivo oonetraint on prorluctfon,

and a downturn a c t a in b o f o r c tho full ornployinont point i a roaohod.

Tho influcnco o2 a chango in r - tho invor~o oZ tho ra to af

turnovor - on thu t i m o gath oC tho inodcl i~ vory cvidont a8 ano Wouid

mpcct, '4Jlion r inurcnsori n o h w o that f or tha sama reto 02 s d o e

a p c o t c d pcr poriod t h c ontsapranour iiiaintaina a bighor.optfmum l a v d

of invont o r i o e ovar t h o i n t orval coneidorod. Alturnativaly, givon tho

l cvc l of optimum invovltorioa, an lncroasod r ~ d l I dir,dnieh tho f l o w

of aaloa p o r pcriod. Thfe & f o o t io ru f lou tod by gragbs suoh as 7 and

8. In Graph O n r o l n t i v o l y h izhur lovcl of invontorios is i í iaintaind*

IIonco it bccomos o a s i c r 2o r tho oconaioy to osrpmdrwl.tbout roaching tho

c c i l i n g o1 aeblos t h an in caso nuiiibox 7. Thc rdo ro , at tho point of

rocovcry conauiiiiiption ie d f o c t o d to n loaaor dogroo and tho t r m ~ i t i o i i

phaao ba thcn moro P l a i ' X o ,

Anathor a3L'oot duo to tha inaroasu Ln x is that during tho

ups~*cina; t h o r o i s a highor rato oi" invostrnont, vhioh ooi.lris SiiimCdIat~ly

from oquation ( 6 ) . As t h c pode in t b ~ aocumuiation o f oap i t a l i s thon

highcr, tho dútr~nmin~ boaowos longor sinao tha absorptian procosa i s

sloirod dor-m,

Tho uppos tuxi~in, point for output oacura f or d i P f orunt ros-

sons i ~ i both cssoo. Grazh 7 bpra ly nooda t h o o a i l i n g af labor to turn

down, Thc 3 d c a trcnd alor7s doi;n aild output doolinos with lt, v ~ i t h o u t

wor roaching full omploynont, TIiis i s a o l a a r oaao of orriatoncu of

uppor tu rn ing p o i n t s . m o l u o i v a ~ y duc t o tho oporn t ion o2 tho nonlinoa-

r i t i o ~ o t h o r than t i ~ t 02 the l z h o r c a i l i n ~ . , fn N O E on ' ~ho otbuc

hruid, tbora in a si~lcllar c f í ' ú c t , but a l s o duo t o tha labor constrafnt.

íYith rcgarits to changos in f , v and G, v c arc bound t e

n o t i c o that thoy a r o t o bs t;t!con aa canriitiocod cbangoa aincu tbo value

o I f doponda an thor.1. Y?2 r u a t rorilorubar t h o aaauinption fimdo in chaptor

thrcor G'o supposod thcso t h a t $ho bchavior$stio aacolezator v and

t ho roaction i n v o n t o r y codf ic iont s' , arc in a rolct t ion vi" proporti* V v

nality to oach o thor* - f =T' no that sny c-a in v , f o r af-

f o c t s t b o valuo of 9 , in t h o sarila a r o c t i o n d prupoi-tion i n oaoo of

ohnngcn in m d f , aiid in tha opgooito dirac t ion in t h o onso of

chauigoa in v , Hence, in tho f f rst caso tbo df oct on tho syatom of

a chango o i t h e r in f or in i a r e i d o r c e d by a cbange in f , 1-iharoaa

if v changos, the total oi'teot 1s ~orno~hat ~ountcractod by a siiriulta*.

ncoua opposite chango i l a 7 As the intansify oP tho roaction ta a

chango in s a l o s incroaseu, t h a ontxcpronour~s raact ion t a tho adjust-

mont of invcntorics f or n a t period decroaaoa. Tha d f o a t un tbo u+

c o n e t d n c d oconomy is t o reduco tho m s i r i i a l root, aithough not to tho

pint of loaving thc csploaivc rogion, On tbo othor hand, thia daríp*

ning & oat i ~ i maintainod in tbo nonliaoar aoonorq* \¡o o m seo thio

L

Graph l T a b a d i' c r v v f J!& Root

9 1.00 0.4 0-03 0.1 0.4 0.5 0.5 1.5 0.3 1.4614

10 1.00 0.4 0.03 0.1 0.4 0.5 1.0 1.5 0.15 l .a .59

Tho amplitud0 of tho Pluctuatione in output, in capacity and

in genoral in t h c rcst of tho variables is mch sfid.lor as v incro*

soa, Thoso ~ m d l c r I 'luctuations occur vciry closo t o t h c f u l l omploy-

mmt coilin,. Tho s i ~ , ~ J t a n o o u s rotluctlan in t h o r a t c of adju~tmont

of inrrcntorics vihich accornpaios tha incronsc in v has tho d200t of

rcducin, tho o v c r f d ~ h ~ o n t o, s d o a mpoctat ions at tbo l o ~ i c r

tu rn ing point (ac tua l s d c a do no t w c u ü d mpootod salca sa r,~uch ae

in 9 , s i n c o invcntorios dccumulato t o a l o s s o x dogroc). i io soo that

thc uy2or t u rn ing point i 'o r n a i o a is much roduccd in graph No 10.

This ha^ tho conscqucnco 02 s l o r ~ i n g dovm thr: ra to 02 invostmmt and

i-dth it, t h o outputcsccca3-capacLty gap of tho syritcrii i n tho downsr-fing.

U 1 thoso d f c c t s scom t o suggost a largor aaneitivity of tho rr

systcrn t o chmgos in tho pnraniot or in P , r ~ h i c h i a uaod to ostimato

tho raquirod charigcs in capacity Tox tho coming poriod (soo cquatlon

Whon t h a chango in tho mto of invontory invcstiucint, f , i~ largo, tho bohavior of tho syatom booomo~ orratia, a inoo t h o d f o u t l a

roinforcod by s ohmgc i n f oi" t h c a m o proportion* l'urthamore, tho

p ~ o c oso of absorption o2 oaaoss oapaoity, bocomos i r r e d i a t i o a l l y long.

T h i s c m bo aoon in grnpbe No 11 arid 12, ro f loo t ing tho follo'c-iing vaiuoai

S i i d l a r conditionod d f m t r o s u l t a l'rorii a chango in 7 as c m

be a c w in casca No 13 and 14.

In both casco thc i , = i i - ~ d . (unconutraifiod) root i a la rgor

than onc. Ho7ürcvor, thc dai~ipcning cff cct of thu nonl inoaxi ty is m o h

acccntuztod in No 13 sincc thc p a t h of t h o vai.i:~b7oa oac i l l a toa in a

smoothor t-~zy around t h o cquiiibr:ul,i pctih. Tnc c ~ i l i n g of fui1 omploy-

m w t of l abor is ncvor rcnchcd a1d t h o ayo-iom iu eE083 t o P u l 1 oapcicity

oiost 02 tlic timo. Tho c o i l i i ~ g of c~zpaci ty i~ t h o o n l y of roa t ivo moving

conatraint i-fhich is l i k o l y to bc tho oniy ono r o ~ p o n s i S l a for tho

dovrntuxn of tho eyatom bcforo t h o oxploeian in any varlablo bocomos a

fC*t. Hm-rcvor, tba oconorny dooa n o t eoom t o t d c o advclntago of d l tho

capacit~r crontod, In r c d i t y ono obsorvos that c i t h o r oapacity cunnot

bo ful ly u8od bccauac of labor scarcitioa or d s o labor o m o f bo fu l ly

omployod b o c w ~ a oi" csnpacity roatriotions. Vo obeomc Ln thiri oaso

t h f both capacity mu labor ara not f ully usod bocauac cntrcpronoua

p r d o r that a l to rna t ive t o t h o nccurJl lat Ion o? iuivontorioa ainoo prw

duction c-ct bo mld. This phanomana dSeappcarB aEi t inorossos

( ~ r a p h lijO 14). Wo still soo that output dooo c o t tako aüvzntngo d

al1 tho capacity croatod in t h o upstdng but tha roason i a t o bo found

in thc oncountor of t h o 2~11 amploymwt coi l ing. An incrccsa in 7 i n c r ~ a a o s invcotsunt ; hcnc o tho systm doowwilat cn moro i nvontorioa at

thc point o0 rocovcry duo to t h o a a a d o m t i o n of capital sccuroulation

rit this point*

'Pko d x o c t 02 a chm,-o fn tho dogrociat ion rr- to, d, can bo

soon in gmpbs such as $ azd 15,

GraphTJO b a d I c r v - Y ef b Root R

C 1.00 0.4 0.03 0.1 0.4 0.25 0.5 1.5 0 - 3 194224

15 i,ai 0.4 0.05 0.1 0.4 0.25 0.5 1*5 0 - 3 1.4076

Tho main c f f oc t is t o nocolorato t ho rato 02 aboorption g f

u c o n s capncity, Honco thc pcrlod of thu cyclo doarcnsoa f r o m f i f t c o n

t o clcvw poriode, Tho ar,i?litudo, h o ~ ~ w o r , ramins apprazimataly tha

aamo m

Output haa t o r.idt a shortor ilumbor of porioder during t h u

dar-rnsrhng t o booomo adjustod to oapacity for thc rocovory. Conaquorr

t ly , f o r a &van i n t c r v d , tho oconomy romins ae an uvarago c l o s o r

to i"ul1 wqploymont, md t h u labor o o i l i n g i n o q u d i t y t u m ~ moro oft cn

into oqujlity,

A sccondarj cí'foct oP thc ohango in d is that moro invontw

d o s arc noodod An tho rccovory, so tbat thoy bocomo acarcor at th ia

point, tho l a rgcr d i a .

1 - Graph lJ'O b a d f c r v v f &X Root

5 1.00 0.3 0.03 0.1 0.3 0.25 1.00 2.00 0 * 2 146'174

ipinally, t bc modo1 d o ~ s not aoom to bo vory aonei t iva t o

changcs in t h c propcnsity to conBumo (Graph8 auch a B No 5 md No l b ) ,

cxcapt for lcss nuworouo ahor+rtm fluctuations in output during tho

dovixrsning w i t h an i n c r w o in tho proponsity t o conauriio. Tho largor

t h o propona i t j t o comur,~c, t h o h i ~ h a r i a t h o (unetabla) aquilibrtum

1 ovcl of output toTilard tthich tho do:mer-rfng convorgoe, and tho smnllcr

ia thc smplitudo ol thc cyclo of cnpacity. Al1 t h o oaacB obaomod

r o Z l a f t h o sama psttorn, ;dth changas o i t h o r i n a or in o .

Xn Grapbti No 17 a ~ i d No 18 v ~ o can ooo tha d f o c t of a chango

bn tho r a t o of population grovrth f r o m aoro t o S$

Tho daparting point ror tho grorídng oconomy i s d a o tha mui-

l i b r i ~ ~ i ~ a o l u t i o n for t hc modo1 dcsaribad in chaptor. trro to t M o h t h o

samo diaturbanoo \-$;LB irilpoaad ns in tho p rwioue aaaa.

Tho obvious C i r s t off oc t tro can obacmo i a that tho introdu*

t i o n 02 thc trcnd iiiakcs it unnoccriary to await t h o troctring out of al1

tho c n p i t d f ram thc pxeacding boom boTorc bcginaing t h a coming ono.

As g incroaaos t l lo syatcrn tonda t o arrivo moro oas i ly t o tho c c i l i n g

of l abor md to rcrnain on it lor igcr ~ i l i o o l i w i c c f U r # l it doos not r*

prceunt iu-i abeolut c barricr any inoro.

ilcprosaions a r o shor tor duo t o tho fnorcaao in autonomous

( subniatonco) conaumption, 8, nhich grovm at thc samo r a t o as g Thc

mi stcnao of tho trmd plzys ii,iport,lnt r o l c n t t ho point oP rocovcry

by making it r o l a t i v o l y Icss dcpondont upon the tcinporary documulation

of invontor ios , L c t ua l o o k at this morc cloaoly. Tho caao r o p r o s w

tod in Grnph No 17 is 8uch that tho l o v ~ o r tu rn ing point docs not invol-

vo a groat a t a n on tha part of invontoriu~ for t h o roaovory of tho

coonouy . On t ho o tho r h a d , in Graph 18, wo obsorvo t h a t , as tho oconorqy

booomofi accustonod to tho trand, (tho ilrst ayclo soon i o a t i l l ref-

foutod by tho i n i t i a i dogarting yosit iun) no docuinulation of invant*

r i o a i o Snvolvod for tho rocovcryi T u s i s i f o bo apuctod, s i n c o tho

t r o d on uutonomous (aubuf a t ano u) c onsuril~ition f ac i l i t s t o ~ tho rocovory

of tho pnth ~f a j l o s (dommd). Thia stsrbs a dof in i to t u m i n g point

Fn t o t a l output, rnadc poseibl o by unusod cspacity and unayploymont.

Y a cn u p e ~ t n t i o n s i i o ~ ? b o c o ~ o f ullfill od; production plnns

aro f u i l f i l l o d , and t u s cncouragaa a f nrii, rocovory in oapacity ~ U o h

ataAa a now acoumul&tlon patli.

Wc s w that t h o grovsing ooonomy r o v u d a s aortain optirLlsrn

fn tho Eioneo tbat capitai atart~ f t s aricu-wlation path b d o r o axoosa

capacity i s Sully absorbed. T l d s 1s so bwauso tha a p o c t o d i n o r o s ~ o

in saloa - vliich is carldca out - crñcoods t h o g o a e i b i l i t i o ~ of t b ~

wisting muaod capacity, Tha upst~ing gocs on until tho labor oc i l ing

is roachod. 'Ilio growlng acononu allorss output to croos d o n g t h i a

oci l ing ovor a groatsr numbor oi' poriods t i a in tho atationary aaao.

N c v c s t b d o s s , as tha output p o ~ r t h rato- U P a r both boforo and &Sor

t h o i r oncountcr of t h o r;uiliw, tho da-mawing i a inovitablo. Tho don-

~ J i n g fa slso frrfluancod by cuiothor f n c t a r i~i'hioh 1 4 an upjgr turning

point lor dcracuid ( s r i l ~ ) in tho prciviou~ porlod.

Thc peth of en lo : tums doid11 ~ U O Q U B U invostsonf $a rnuoh ra-

ducoil duo t o tho accumlation of capacity irhioh olrcoods tho f u i 1

ployncnt p o s a i b i l i t i c s of tho oooizoinyi

&J~o~r4tj&.l;i'&~ t b e d e t e r ~ . : & i i o o the ~baracteristic roots of thq UIC oiis-trained syo t ei:i. ---"-

The pxoccdure devised aiid sdopted i ' o r the systa,:utio w d u e

t io iz 02 real, coinpla: a i ~ d i . i ~ l t i i ~ l ~ roots i~ eynthetizeci i n the 901-

lowlng cbart i

One fiiterestil~;; ~ ' o a t u r e o*' th t i >ivcedure to be Coacribed is

that t b ~ sroblem of waluatiou o2 co iug la r o o t s can be tranefaiwed

into oilo of evaluatinz one dot.iilar~t real root. In t l ~ i e ww the y r e

bl o111 oi' yuessing at a starti:i, value ili a plan o2 tvfo di..iensians o a a

b e avoided.

A seoond f catara is that coiivergence Se accolorat oh and

ingrove6. on ivi th respect t o o t h e r nathods, in s u ~ h a way t h a t t h c

evaluation o2 r o o t s becoi,iea ,.lore I l a i b l e a ~ i the naed Lor atarting

near the real value of tha root i a obviated,

~ k l ~ ~ O r $ > 2 " i k ~ ~ ' ~ x t l ~ e iioterII.iiiatikn 02 t he ~haractoristic r o o t s of t h e u i l con~ t r a ined 8jr~tei.i. .-- --

The procodure d e v i ~ o d a i ~ d adogited I 'or the siy~tei;iatio walua-

t i o n oi' coioplf t~ auci L i u l t i s l o r o o t a is synthetized in the foZ-

One i i i t e r e s t i n ~ Yeanturc 02 t h u plwcedure t o be cloacribed is

t h a t the pro'slem oP waluatioii of coiuglw roots can be trai~si'omned

fnto o i ~ e of evaiuatii~g one do i . i i~a l t real r o o t . In this W a y the $ r e

blerii OA' gueesing at a startin; vaiue i ~ z a p l a n 02 two Oi-~eneiens ~ a n

A second T o a t u r e ia t b a t colzvergence i a aocderat od and

ir~lproved on u-fith respoct to other iilethoda, in auoh a way t h a t tbo

e v d u a t i o n oJ roota becmen iiiore i ' l a r i b l e as tha need 2or ota r t ing

near t k e red. value o2 t h e root is obviated.

~2urt.-iemlore tilo i.io*,hod incorgorat su a ayet e i a t i o way o2

de1iiliiiting the i n i t i a l bounds wi t h i n which a i l r o o t s iliuet f Le by

tddn, aGvantage at each s t c p 02 tha additional in20rílutzon @nBd+

The a lg~ r i t l i i ~ i Tras auhi - l i t t ad to an IBLi col-~puter (7040/7<354)

by ueiiic a s r o g r a i ~ s i ~ e c i a i l y writteii in d'ortran 1'1.

Thi8 sra,ral.i G O ~ ; , ~ J U ~ O B t be rive r o o t s f a 1/12 tli 02 o, eecond,

r o o t s correuposdii?,. t o oaoh ,iartfcular s e t o2 paral,ietar v a l u e ~ . T h i s

len,stli oJ" t i n e includes a l s o a t e a t w U ~ h v e r i P i e ~ tbo v a l i d i t y oaY

th0 r s o u l l s based on t i lo relati0113 aist iw betwesn t h e coelf i c ien t s

and t h e roots OS puly; ioidals ,

Stui:inls Thoorern and Caooc 02 2,iultiplp~e iloots .----m.--- -*.-..-.--. . . ,. -- ---- *.------* ..

GturlJi9 s tkcorsrii vas a,~sZiod in tlie I ' ir~t p laca as the

s tar t iz i~ poiilt . It is d e s i g o d t o i i~uicate t h e nu:iber oL' distinct

Ted. y ~ ~ t ~ WhiCh lie bct.lieon -two a y e ~ i i ' i c valuau oi' the c l l a r a c t e r i ~ t i c

var iab l e.

X r a c t a i>pl icat ion o* tho theorem o o u l t ~ ~ u i l t i p l a r o o t o but

onco, so that ii' there are ti.m ~ i , . i y l e roots and oize double root in a

glvein ia terval , oilly t h r o a roots are counted,

'20 Zind t i ie iii~.;bor 02 real m c t s , r7e avaluato t hose wbich

l L o betnoon winua anC ;ilue i i d i n i t y .

The application o;? t h a theorei-ri coizsis ts ceec i~t i iLly Ln

f indin; tho nigns OS the so c d l e d Sturmis Cunctionc; at t h o s e ~ t r o r i i e

values, Tbe Sirst Stun.:la Cu i~c t ion i e t h e o r i g i n a l polynoi~iial , the

second its Xrst derivative, the tliird i e dnua the re;-iainder obtained

2rom tiie i ' i r a t two Zun~ t i o i m ny a. $rocese o2 Ion& f i r l i ~ i o i l by tt$$lylnz L 2

the algorithi., oi' A c l i d e ami no ~ i í , * . ' ~ a . t o ~ ~ i r . i ~ at tha las* reiliainder

i ~ o t iden t i ca l ly nero,

Ii" thc last rai,iazn¿os 02 t h s d i v i ~ i i o i z i a 02 zoro del;ree

then al1 roots are ei, . lglo. Thocia o i l g l e roots c a l bo real or coi , ip lu ,

Ln tiro 2irst oaoo, they are to be evduatod by uoin, a co12aiilation of

Eevrto.zfcr ~.ietSod and ;lio ~iothoct oi' i"dm posizion to ba deacribed later,

11-i t h o eve:~.t t h a t th030 simple r o o t s are @ o i A : > X ~ iiwbers,

dizl'erent posaibill ti es ,:;uc t Le c;oiit erdplated ~vhic l i w i l l ba úisouss%d

in tho na-.t sect ioi iu. L' thc dosroe of tho last r u i ~ a d e r in t h e a;h

p l i c a t i o n 02 S t u r , ~ ' s tlicort3.1i Ui';Cors i'ro;:; &oro, then t h e ~ o will be

~iultislu roots.

Phis is so because t h o dc;roo oL' tth las* r ~ , , . ~ a i n d e r dif fermt

2ro:; zero sivea -Lhe nuil:lies oi ' l i i ~ o a r Zac'Lors co,.ulion t o n i i t l r the poly-

nolliial snd its r l o ? i v ~ t f vo,

;'Baso co~iii~on l inear i a c t o ~ s aro repcatod ilzctora 02 tbe orí-

~ i m l p o l y i ~ o ~ ~ i a l , L o t P(z) bo tho origiinal ;do;jrna.iial oL' o ~ d e a n ,

sed aa

where 4(a) i a a gcljli?o~iial oi' tbe (n - r) t h dci;rea.

';le have also that

. - -.- - --_C_-_

'/Bani.iel, J. ----m,-w ;ioonor,do Jxuaid c - Ch. 12

eo that 5 is seen t o be a r o o t oi" rnu l t ip l ic i ty (r- 1) 02 P 1 ( z ) . 1

2 o r oxampl e, B esrii~;;. in ~ ~ u n d t het o u r c 1mrac-t e r i n tic > o l y i i ~

iiliaL f a o2 t he f i i ' t h desrea, i2 t h o ro~taindar is a $ol;rlzoi.iial of the

*t$ird ilugroe, then it , m o t have at leaat one raat o: i i ~ u l t i d l ~ i t y ttro,

or at ~ i o s t , oile of ii!ul-tiglicity throo.

In th0 i'irst caao, i, e., it has oue aimpla root and one

o% t iult i~f ici ty t ~ v o , t iie siiilplc root i i s a double root 02 the or ig ina l

polynsi;.llal, whereas tlic r o o t 02 , i t u i t i ~ ~ L i c i t y t3Vo is a t ~ - i ? l e r o o t o2

t ho BW~ZQ polynoinial . I n t h e secona case, i: t h o three 200-ba of t he re i~a inder are

5 equal, i i u r , ii l i(z) - (z - B ) , thon z is a ~ ~ u a r r r u i ~ l e root o2 t h e 1 1

original >olynoiiiial, a root i ihich is cFl.gly ovaluated b; t d A i ~ s tho

cubic root oP t h c indc:~exidaiit terii, e i n ~ o , %he 2roauct o2 tho roats of

4 37 dividino t h e o r i g i i i a l polyno,..ilal P ( 2 ) b ( z 5 e 1i-

near q u a t i ~ n La ob ta iuod , YroiAl which tlie Í ' i i ' th root is evaluated.

811 t h e o t i ior ~ o s s i b i l l t i e a caii bo -i;a;ron $rito aucount i n a

oír-alar ivaal

So we soc t h t i;i tbo caac o;- i i iu l t i i~ la roota , a relativsly

str&giCorv~ard proceLurc c r ; m bo u t i l i z e d by t8kdng advutago of ~ii-1>10

theoreizG o2 th0 t h e o i y oi' ~ ~ l y r i o i . ; i a l ~ appl ied to our particular chara*

t a r i a tic cquation.

idi'; 'o~*itl~i1 -. 2ar tbe case 02 ci~,w:e rog t~xrs t t ) roota. -

In orLer to edaluata t h e real roote, a co .'Jiru:ion a f

iTewtonfs i,iethod ancL t h o r e ~ u l a L'alai i~othod 18 ut i l iaed . as4(

iieugton's i;iotbod i a baaed on ths a x g a ~ ~ a i o n 02 %he Zunction in

aoxiea about a n e i g l l b ~ u ~ i n ; p o i n t o2 the s o l u t i a n ; aerioo nhich are

ucua l ly Lirolren 03 i1it11 tila l inear tan\*

Tlio i;:otl.iod i~ n p a c i a l l y woll auit od t o tbo prob10~r.i oi" eva-

l ~ a t i i i ~ thoac r o o t o Socauso 02 t h e ~ i i , i p l o a igor i th i i i that is available

2or c a l c u l a t i u z b o t l i tlie valuo 02 a poiy~ioi~iial a:id of i t e firet dori-

vative (as - i l sL i a:> ~r:loriv~-tivea 02 Iligher arder if a yaator coiivorgeiloe

i~ a o u ~ l i t i i l aol,ie casos)

u te r t l io i n i t i a l ~ u o s a i n c : ~ o o a ~ , tlie secjuanco S 18 n detori.dnecl i'rolii t h o recurronce ro la t i a l r

D a a ~ ) i t c tlie Yact that the 1 oca1 quadratic Lonvoryonco of

iTmfto i~~s i,i~th02. can a l u o be e o t a - ~ l i s h e d Lor tlie caso of 2airs o2

coi:iplc;r c o n j u a t e rootu, tbe i ' o l l o w i n ~ po in t s ro,,iain unaizoneredi

1. Eíow to ~ u a r m t e o convorL;acice v ~ h ~ n the startin; valuos are

n o t ' lsv;i"ficio~~tl;; lo so" t o tbo ~301uti0ili

2. Hoy? to i ' i i~d a b o ~ Q T i r e t apprwii-ation in t h o tvto d i i~ona ion

al col,.pl a plano.

3. How to copo w i t h tho -iroblom 02 i i ~ u l t i g l o roots (already

disyoaed o2 in t h o groviouo s o c t i o n ) .

L.iong o t hers, Bairatol-rt B ,,iethod, 1i:re B o r n o u l l i ' 3, and tbs

Quotient Diii^erei~oe alGaritili,i) ~ a i i be epp l iod i;rith a vaqring dagreo

of succoss to t h e PLnding oL coiilplex r o o t a , but VIO are s t i l l left

with tl-ie t i f o o t h e r wiai~strereil p o i n t s .

Graeif cl s iilcthod', al so a a l l o d tlie root squaring i:iethod,

a ~ t o ~ n a t i o a i l y a i ~ m e r a the I a a t t w o pointa, a l t h o u ~ h at t h e cost o?

surpassing t h e size f l tho in t e rne1 iIiemory OT an IBIB ooiiiputox, &ter a

f ew i toxa t ione .

The algorit1ii.l dovisod in t h i ~ disaertstion to cvduats t h e

roal anL ahtiple rriots is a coi.lbination of Bmfton1o rnethod aiid the

fi i,iotho<i 02 i 'a l se p o s i t i o i l ,

Shia last i,ietliod aleo cal lad "ro,&a Talsi", coi i s i s ta of ap.

pra~iiiiatin;;' tiie graL)h o*' tbe po lyno ia id by a a t r a i ~ h t l i n e pasaing

th rough t h o points ( z n+i > ' 'n+i

) aild ( Z 3 P ( z ) ) I 'P~ 80iO i ) 1 i n n 4

Tho p o i n t 02 i i ~ t e r ~ o c t i o n 02 thie chord with tho horizoiztal - -

a x i a d¢;'inea the new approxi1:ation. dilo therofore nood to havo the

values i ' o r wh ich the function lza~ op i~os l t e ~cicns.

Binco the f u ~ c t i o n is cont inuous botween tho t v ~ o values,

t h a r o i.iuat be at l a a a t oiie root bettveoi? tbem,

T b i s t v+s t ago proceduro ha^ a yoint oi' c k g + o v c r nhich

can bo i rd ica ted as fo l l ownz

u t o r a s t a r t i n g i ~ ~ t c r v a l i a c b o s a ~ the algori t l -a b o ~ i n a by

vil1 ers, P. k a c t i c d &1aly8is, 1947 - --,----

aírt andlw i3erwtoni s imt bod so aa t o approxi~ia$e t b e curvs n o t by a t-

;en$ but by a cume o2 h i ~ b e r order. That i a , we t&e

or, negloctiog toriirci oí* highor orderi

8 atabi l i ty Jrcquirca Yox oonvariienoo in the oompltatian

t u s dorivativo bas boon oalmlatod, b o l - ~ l ~ t ~ thc last tam, glv ing

At Gvery a t o i ~ t h i s value waa aar4~guted f n order to uheok

wkathor or not convergonce could bs attainod. ,ihmmex Gonvargoaoo

1s not at-taiilod, th~ritwre~nula Z d s i iiisthod coziee in$o aot ion by

taidng thú a r i t b i e t i c weari 02 tbe last two 81s wldch oorraapona t ú

oppoai t e oi,;na 02 P ( z ) ,

Since i'ocr t h i a we noed bne plvduct y(an) P(z ) to be s+l

negativa {opposito sigis), obviounly the 1aethoL b u i h p ~ in ta a &fPicuY.ty

whon thu $olynoiaíal i e aP svm dv~ree, a i ~ c e i u B U C ~ a C & ~ Q f t &&y

w s l l bo that the l u n c t l o i ~ han valuos oi' eqEtqual ai6.n at tha er=tror,~es of

the choscn intorval. muiever t h i s $ G the a m e , thsro is no indioatfon

aa to tha l r c a t i o n ei" tlle r o o t aB the graph sbmv?ui

To cope with tbis 1m33ibility, t h o al,orit?i;.i n~stoi , iat ical ly

divides tlic i n t e rna1 i i i t o twa i ~ a r - t a alti uvaluates i t ~ ~ ; ~ S Q ; L ! B S , SO

t h a t Y i r ~ a l l y i n a T i n i t e nu~~!~cr 02 s t e s ~ , poi i l ta are obtainod at trhich

the Tunc t i o i l at taiiis o i~ l io s i te s i y a s .

Fhis ia s o bccausc Stunif s thoose;.i assuro:: - i:~ t h e case o?

the Zourth rlegrae - o? i'h c i s t oizce oi ' at least tiirro dis t i : lc t roa1

roots so that a 8nYl ' i~ia iz t ly a;.iall iiítemal caii;~ot coi~'taili t h e ~ i bo th .

Oiico t h c o-jpozi te a i ~ n s are Tound, t h o rmthod ;~rocoods ~ v i t h

either .Lb¢ iiewton o r +tIzc r e ~ ~ i a laloi 1,icthod accorci int t o ~-~l.iother or

n o t come2sonce 1s att&led, (as is a L p l a i n c d on pa;o 20).

AS 2ar as the c;l!oica of the s t a r t i n g interna1 i s concerned

t h e 207:.oi'ring bounds provida a c i r c l e lar@ o n o u ~ h 80 aa t o c o n t d n

al1 t h e ~ o o t 8 .

L o t 3 be a root 02 P ( B ) . The fol loí i in& general inequality i

s p r e a a e o t b t al1 real 02 co;ilcc: r o o t a are w i t b i i ~ a c i rc le having

(l+ins: / \ / ) as radiua, vhore A (i 1,. . . , 6 ) are the coei72icienta i '

In tho case ol" roai r o o t s tho circlo beooxoa a segiaent and

tke l o w e r &id u p - e r boulid i 'o r t h e rcote sro as i'ollowsr

Thosa aro g e i ~ e ~ d bousds wUch c a z bo inyroved on asi ebli-

t i o n a 3 incom:ation bccoiaoz avaflablo.

Ir' it is :cnavii~ t h a t L. O, then ono root h a s t o bo non-n+ 1T

gativa x ~ d iTe caui replace tllc l o f t s i r le or lov?ar bound o2 (1) with

zem. ( S i i ~ c e P ( O ) U A B ).

1;t' A > O a16- t h o p o l y i ~ o r ~ i a l is 02 an odil 9ogrea ?;re oan N

roplace in (1) the upper bouuid i i i t h aero. Wo i : l ig rnva~ei~ t is sosaible

when P ( z ) is o: evon dezreo i i i t h i e c w e .

In o the r viozdx, as izow i d o r r i i a t i o n i~ ~ a i n e d , t h e l any th of

t h o oogilent (1) can be ehortolieci with t h e cor ta in ty tbat the roots can

be Y o u ~ d on it.

g i n d l y , i s ' t h e I i l tcrest lierj in i'iriding tilo r o d r o o t of

largest i i i o d ~ l u ~ , (as in tlie oase o2 t h o po lynoa ia l o2 s i x t h degroe

P o r waluating tho coiriplaz root~), thon t h o s t z r t i n g i n t e r v d c a n t a l ~ e

advailta~o of tho i'olloiidii, propor tyt

It i a norv easy t o ii?lgrove the lovirer boui~il in t h i s case* It i/a

, b ec oues 1 A IT 1 ainco t h e product 09 al1 r o o t a OL' P ( z ) equaJ8 its

indepondont teru,

Alcorj.thm f or the case 02 eiu;jle rooteil ~ 0 f , l ? l ~ roots . - -

As has a l ready beon discusoad t h e natbewatical atfucture of

the rnodcl doce not perrriit tho applicat ion of tho Porror+i'robmius

theoren 2 o r nonncgat ivo j.iat r icos ,

T h i s iiiiplios tiiat t h e i joc t s ib i l i t y of t ha ~ y s t a11 boing doilii-

nated a o o n p l a i root c a ~ i ~ o t bc! di~cardcd. Tilord o r e a systei.iatic

pyoccduro t o avaluate it is noeded,

The c h r a o t o r i a t i c polynoiiiial ia o2 tho f i f t h dcpoe, tkon

aoiiio proi7ertFes can bo Ccclucod, A-t least ono r o o t :?ill be yoal

- a l t b o u ~ h not neccssarily tho do,.iinant OLIO - a inco al1 co;iplair r o o t e

carda in con juda to pairs.

Sincc one roo t ii:uot nccesaarily bo r u a l , wo can find it a d

roduco the ~ O ~ T C G 03 t h e polyiioi.dal bg ono. Lot us onur,ierato thon t h o

poss ib lc pattorna oi' co:,iple;r roots that i m ~ t be considerod.

Lot us cal1 a and a coriiplez r o o t and i t a coi, jugata ' 0% k 1;

poctivelg.

Thon t h o casas a m i

1. TOUT ~ ~ i w l e ~ m o t a z . ~ n d í2 zp , 1 1

with 1 sl 1 & 1 ap 1 2, Dvo real and tivo co:.ipl o= r oo t s ,

3. Pour cor@l& roots -r7uitl. oquai iioduli.

Thore i o ~ i o t b o r pos3i1dlc case,

111 oxder to co~ le idc r the 2ira.t caacl, t h o p01giloi.iial 0.L" i"oul"bh

d s ~ s c c iu transf onliod i i i t o one of s i x t h dogroo vrhioh ha^ a8 roots t h e

products ol' t h ~ r o o t s 02 t l ia Poriiior.

Tho idea o i t h o tila:lg;or;ilatiol~ is to i;~ti a 2 o l y ~ ~ l i ~ i ~ i a l t b t

is casicr t o aolvo, a ~ o i d i i l g t h o ~ I C C O S B ~ ~ Y of i;ia:iin, f u i t i d . guesses

in t ha t170 diaoaaional c o . . ~ l a planc*

Only the real x o o t ~ 02 tho g o l y i l ~ i ~ i i a l 02 a i x t h d o ~ r o o ayo

rolevar-it ior O U ~ Q U T ~ Q S C B , s i 3 c o as we ~ E l h Z t l l aaeg thoy aro t h o squaro

i f t b o : :odulus o2 thc co, . i l i la r o o t s 02 t b o g o l ~ r i ~ r . d a l 02 i 'ourth dogreo

; , 5 t h $v;Y~h me arc coilcú~i.rod,

I h o v ~ i n ~ onc ;~mdulus o; t h e >olyi1oiLlial of d o ~ r o o f ú u r i a

onnugb t o roduce t b e p r o b l o n aa can bc seon Tro;.i tho Pollot-riag. AseuiA~o

t h a t $ O , t h a t is, t h a t tha poly~ioiiiial has no cubic t o r . 1. This

assui.:ptior-i i~l ju~tif ieCt 3inc0 111 02dor t o dct c r i i~ i i :~ ti" t h o cquation

has a i l its iiloduli cqual wo Lavo t o chaizgo thc origLn o: tho g01ynoi~lia.l

4" If 2 1 - a,* - i b7-, :E E= 1, 2 and t b c i z conjugates

AL A- -L

aro t l ~ e f o u r (d i i ' i ' e ron t ) r o o t s , i i c !aor;~ that

Thus, iT 1 z1 ( i~ ~ i v e i i , r ~ o can i i n d 1 z21fro4i t h e r d a t i o n

Qn the othcr bancl, tlia f ollovi lng relat io iz holdat

Ronco

no that t h e probloril is aolvod,

l7hy t h o choioe 02 a i 'ui~ction af degrao s e ? Bcoaueo, t o

~ O ~ I Z i?ith, t l l~ s a o t ~ 02 the g o l y n o ~ i a d of d a í ; ' r ~ ~ f o u r i,lWt bo troatod

sy,:unetricaily s i i ~ c c t h ~ y aro iioi y o t i d o i i t i i i od. Tlic aiiLlglo5.t viay ie

t o i"oli&i suilia or product~ o* Croupo a2 r o o t ~ , Siilco our in toroer t 119s

in t h o moduli of tho roota , producta botv~oon pairs of r o o t z ,.iuat be

i'ori~ed, ailice a cor?ii~le;; iiuiiibor I .nalt ipl iod by i t s coiiju,ato gavos t h o

squarc 02 its L ~ ~ O U U ~ U B .

Tiius t h c Coa-ficionte tif t h o norii p01yi10i.dal - t h o ao-callod

elmontaqr ~yirir~iottric fuilctiona o2 i t a r o o t e - are a i a o oy.,i laotric func-

tiona vi' t h e roots 02 tjlo polynoldal of dog;.roo Tour,

&ch cordLicient 02 t h o poiyizoiidal. ai' a i x t h d o ~ r o o l a obta i -

nad f ron t h a t of degroe i'our by a i - i th i i~o t io oporationsr L a t pi, l o r

i = 1 . , n , bo tha coci ' f ic ionts oj: a po ly i~o~i i i a l 02 C;ot;xoo n in 3,

and l e t z be the roots* i

Tbun, irc I m o r i ~ that

I n the sa.r;lu rmy tho cooPPicioaha of t h o poly11o:-~ial of dcgxeo

aL: can bo a p r o n s c d as i i o l y r ~ o ~ : u l a l ~ o9 tho cocEicicnta of that o2

deGco Tour, sinct: by d d i n i t i o i l oach root o-ia t h ~ i'ondcr i a tho yrouuct

oZ al1 t h o 2 o s a i b l o pnira of roo t s o;;' tho la t tor .

L e t A , i O l,.. .- 4, stand f o r thu c o o f f i c í c n t s of t h c i

o r i g i n a l po1;ynumial and B. , i m 1,. , , , 6 , i'or thoso of thc now olic ?

?

'Ihcri t l io i'or*i.iiilao ddi i i i izg -tho ;B s ara i

Aa t o t h o r o o t ~ i o2 t h o 11mi >olynoiLSal, Qi ' ñ o havo that

Thcn thrcr? ~ i s e l c s c w > u ilral-~n i i l thc ~ o , ~ p l e x p lano rspro-

sentiil; tbc t rawa 'a i -od i30iL~;la;.iial I

Tbo largost c i r c l c has a radius oqual to 1z112, tho s r a l c s t - m d t h a cci~tral oiic

Tliu ir-tcrscctioli o;;' t h o tt'ro ctreilie c i l - c l os w i t h t30 posi t ivc

red ha:-a;:is givoa ~s tlic rcal and poaf t i v o coots, t h c oi:ly rolovant

oncs l o ? our yurpoaes.

Tho Your o t h e r z o a t ú , a l 1 c o i ~ p l a aro il? t h o i n - t~ r i~ iod ia to

c i n 1 E, s i n c o by czusu.,;>tioi~

I;rhat real ly , . ia t tora i:: tho f act t b a t tho rmt on tho l a rges t

c i r c l e iü rca3. poiitivc L ~ ~ ~ L C U il; GW, tlii3ii bo ovaluatod by usiiig

t h c algori thr ,~ of' thc prcc cdiilg S so~lion.

A, 'urt l- io~,>iaro, t l i i o ros1 antl p o s i t i v c roa* i~ t l l ~ 3c luar~ of

t b e ~.iodulus 01' thn co iiil c- r o o t a o; t h u l a rdes t i,~odulus of' tlic ;sol;rnw

s i a l oL - o u r t h dogr,c,

Ln otber ~ r o r t l s , t h o 1 a r ; o ~ t r o o t o2 t h c t r a ~ s f omod polync-

i9a l b 3 -Lhe fol1oirin;i ' t ~ a proporticsi

l. It is th r : s o o t o . ~ tlic lsyzosL iIiod~lusi 02 tthc polynoidal of

s L ~ ; ; t h d e ~ s ~ e , a 1:~ropcrty oaoy to voi-il'y s i n c ~ al1 ita dori-

va t ivos i.iu~t b e L:os i t ivo a t t h a t p o i n t ,

I'he roi. iaiizin~ ~ o o t s a r c u i i t h i i ~ t h o c i r c l e havini t l io o r i ~ i n

as G O ~ ~ ~ E X m d i i l tarsccti iz , t bc iar~oet r d r o o t ,

Tho s o ~ o i i d i 3 o a s i b l ~ pat,Lorn of cor,iplac r o o t s t o bo cons ihurod

i a tho case oP trro roa1 and t v ~ o coLiiplm r o o t s ,

ThiB Se t h e s i . : l ~ l o s t caso a i n ~ o t h e o i d g i n d poly i i~ iLia l can

thon bo roducod to a yuairztLc oguacion, Thia roduct ion i~ oarr icd

out by auiooss lvo d i v i o i o n o 0.2 P ( s ) by z - z ) m ( z - se), 1

w i t h z aizL z a a t h o roa1 r o o t s . 1 2

Thc tl-iixd a n ~ l 2 i ~ a l caso is -tlia ono al! ~01.1i3i71ti: r o o t s of

cqual i.ioduli, 1.t is thc oano I , I ~ ~ F G

T h i s case can bo i~ -ii;.oüui~tod ;;raghicaily i i ~ tlio c o i 1 ~ > l ~

?lano as a ~ c c t a n g l c vrhoea cxlroinoe a r o t h o roota thoiiisolvon. l e o a

bo G O C ~ Z in t h c ~raph 011 ijazc 87 anil *ron1 tho f 'omulao o: pago 8 6 , i11

th ia G a G c tho tr~ini 'or iL,ad iiol:ri~o;...la;l. 05 tho s i x t h do~roo hae ropoatod

r a o t s , so t1,at a a t r a i ¿ h t i ' o r , ~ a r d a p i ~ l i o a t i o n 02 i;owtonla i-~cthod is not

poaeiblo. l'llia i a ncciusaary oo aa t o bo ablo to rocogni3o t h i a caao

by G1ic Yomili t h c pol;iioi.iial 02 tho Your th d u ~ r o e .

Tho task cons i s t a then, of d e v i a i q a t a e t in t o m e of tho

ooefficiants of ~ ( a ) , whioh will i nd ioa t a Wh- tho m o t a form a raoG

L o t ? ( e ) bo tho fourth dogrco polynomial aiid l c t Ai , f o r

1. 1, ., 4 bo its a o o f f i c i ~ n t 8 .

Thon wo know,

Lot us narrumc for tho momont that al + a p O. lkom (1) t h t e

f i x t h o m o r o , cinoo tho raoduli a r o oqual,

On tho othor hand wo know tbat,

FLnally wo oonoludu thaf A 0 , In othcr worda, ff k = 3 3

- *1 O tho r o o t s foxm a rootnnglo. Convorsoly if A - O but A 4 0, 3

tho r o o t s oamot h a v ~ oqual moduli. Thon thu tost t o vorify if a

polynomial w i t h complox roota and A U O has al1 l t s rnoduli oqusl is 1

to soo mhothcr or not A 0. 3

Tima vro can ~ o o that m r provi~ional assumptios a + a = O, 1 2 is vory uacfi2~ t h o r o f o r o , imtond of fnvostigat5ng thu roota of our

oxiglnal polynomial, P( a) w i l l bo tranaformod into anothor polynooiial

Thu r u l n t i o n botivoon y and P, i s ~ l v c n by

4 2 G ( ~ ) - y + Ap y i B4 > a quadratio oquation in

If in ~ ( y ) , Uj O, wo Lrnow t ho t t h o m o b i l i oru di~tinot

and thc tranefomnntion of tho proviaus sootion oan bo applicd.

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