Production Duality and the von Neumann Theory of Growth and Interest (Mathematical Systems in Economics)

MATr|EMATICAL SYSTE INECONOMICSt1 S .N . A F R I A T ft \ ir..JA,ate++oo 'G. BAMBERG Augsburg Editedby til. elcnnonru Herausgegeben von Karlsruhe ,,

l,{il

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( . . 1 , , .

G.HAMMER Augsburg R.HENN Karlsruhe R. KAERKES Aachen

K. NEUMANN Karlsruhe H. NOLTEMEIER Grittingen

o. oPrTz lnnsbruck

B. RAUHUT Aachen J. ROSENMULLER Karlsruhe R . W .S H E P H A R D Berkeley

Production Duality and the ' . vonNeumann Theory 0f Growth andfnterest S.N.Afriat

VERLAG A1{T(IIT HAII{' MEISENHHM Ail GLAI{

MATHEMATICAL SYSTE r1{EC0N0M|CS11 Editod by Herausgegeben von

S .N . A F R I A T Weterloo G.8AM8ERG Augsburg W.EICHHORN Karlsruhe

G.HAMMER Augsburg R .H E N N Karlsruhe R,KAERKES Aachen

K. NEUMANN Karlsruhe H. NOLTEMEIER Gdttingen O. OPITZ Innsbruck

B. RAUHUT Aachen J. RGENMOLLER Karlsruhe R.W. SHEPHARD Berkeley

Production Duality andthe vonNeumann Theory of Growth andfnteres S.N.Afriat FacultV of Mathematics Universityof Waterloo W a t e r l o o ,O n t a r i o , C a n a d a

VERLAG ANT(|T{ HAII{. MEISEI{HEIM AiI GLAI{

- Meisenheim am Glan @ 1974 Verlag Anton Hain KG Meisenheim am Glan Herstellung: Veilag rtnton Hain KG Printed in GermanY ISBN 3445{l I l3-3

PREFACE Von 1gs'nqnnre econoolc nodel, 1938, has had a steady attentron, ln

trangratl0n

part

ln 1g45.

of rrhlch ls

the orlglnal lntereat

hae frequentry

the origlnal partlcular

nodel. forn

(f956).

eepeclarly

can be traced

relatlon

a rDore abetract

flrst

eepeclally synnetrlcal

of such productlon

ln regard to featurea connectlon

deflned

wlll

economlc 6ense ' ls

relatl'one,

that

slnce the productlon

reratlon

thle

dual

result

plane rhere

no ahort

cute for

maxlmum growths

aeverar

greater

wllr

be treated

relatlon

of producte

nathenatlcal

growth

acroaa each elngle

froo

18 the product

an

of productlon the theoren

of the duale. ln the von perlod,

An apprlcetlon

of

theorentt,

ln

together

than 18 poaalble perlod.

slth

acroaa a elngle

are taten

a

of identical

senee froo

type of t'turnplke

perloda

for

which derlve

power of the dual.

leadg to a nodlfled

long-run

by Davld Gale

acroas any extended perrod

le the correepondlnt

of

be ln the

A baaic operatlon,

Neunann econouy la a poner of the relatloo lte

The

theory.

a further

the fornatlon

the dual of a product

whlch

paper.

glve a posltlon

of duality,

and 1t galne a further

(Sec. 1l)

will

relatlone

wlth

for

or elaboratrone

consideratlon

type but ln the dual apace of prr-cea. lnnedlate

here,

whlch appears in the noder,

von Neurnannfs growth and lntereet

The theory

llteraEure,

back to that

Ilere the lntereat

paper of

eince the repubrlcatlon

gone towards varlatlons

The study of euch relatlons

revlewlng

in hls

in the blbltography

of production

glven

and wae flrst

nlth

There r.e now a subetantlal

represented

stluulatlon

dealt

there

that are

by coapouodlng

ThereareotherthcoreogaboutthepreeervatlonofProPertles

theorem shoss a duallty

tyPe of

Another

of the duel.

property

a correspondlng

I'lth

of a relatlon

! proPerty

connectlnS

propertiear

between

or duallty

or equival'cocee of propertles'

ln producta,

betweencostanrdreturnfunctlona'wlthanexchangeofrolebetleen

of

vork

related

Thla type had lta

and 1ts dual'

a relatlon

(1967)

Rockafellar

froo

suggestlon

orlglnal

the notatlon

, and aleo

heueedwhlchexpreaaesananalogywlthcooJugacylnalinearspace ueed here (Sec' 10)'

has euggeeted that

(12-18)

sectlons

Later

results

and certaln

topology

theory,

extenaLona

baeed on hla

and flnite

frorn three

algebra'

arguoent'

Hle @rn topologlcal

II'W' Kuhn tellg

ln

full

me ln thie

lnportance

out to von Neunann the

from convexlty

derlve

theoreu, le preeented

lnteraectlon

necessary nodlflcatlon. he pointed

the nethode

Hle

feeture'

ln the duallty

then are revlesed

of

ln vhlch

posl'tl-ons,

alternative

brlnge

fashlon

and ln a typlcal

of a productlon

on the baels

theory which von Neunann'elaborated relatlon

the growth and lntereet

deal wlth

wlth

a certaln

connectton

that

of the regularl'ty

conditlonsvhlchappearlnthelatertheoryofDavldGale(1956)but are mleeing ln hle diecueelon lt

(c.f.

bears in von Neunannta actual

there

ls

another Polnt

Together which

1g a rate

wlth

Kuhn (1971)).

The oatter

argument le elaborated

here'

a8 but

beeldea'

the oaxlnr'u

at whlch

at leagt

grortth

rate

of the von Neumann theoryt

aone goode can Srorr'

lt

18 natural

any poealbly

to conelder can gror thle

particular

naxlnum,

of grovth

conslderatlon group of goode.

goode almrltaneouely and aleo

rate

at nhlch

ratGa laaocatGd

The upper llnlt

ig here coneldered the reclprocala

(Sectlon

the four 13).

of all

four

of the correapondlng

support

fron

revlsed

at the Unlversity

the Netlonal

ln the Sunner, 1973.

Unlverslty

at thls

flrst

at Chapel Hlllr

of Canada.

Ednin Burmelster

It

I wlsh to acknowledge wtth eemLnar, especlally

of Karleruhe, contJ-nulng earlLer

to Llonel

the von Neunann nunbere

for

of lntereet. palrs

ln a project

part

wlth

support

of Karleruhe

held

thanke the beneflt W. Eichhorn

dlscusslons

at Berkeley.

W. McKenzle of the Universlty of the Unl-verslty

partlal

was preeented at the

of the manuscript,

of Callfornia

wlth

(CS 21.95), and then

Theory at the Unlveratty

Shephard of the Unlveralty

all

prepared ln 1968 at

of Waterloo tn 1973, wlth

when he read and crltl-clzed

lndebted

for

of the von Neumann model

Sclence Foundation

Reeearch Councll

Synposlun on Productlon

of dlecueslona

the rate

factore

(Sec. 16).

of North Carollna

of the Natlonal

for

property

of a nanuscrlpt

Thls work le part the Unlverelty

wlth

any

numbere thue obtaLned are equal ln certain

Then a dletlnctlve

le the equallty

slth

of grosth

together

the dual systen which have ltrterpretatlon Generally

goods ln the economy

all

l{. }lcKenzle (1967) nakee an approach to

Llonel

together.

ln hle

lesser

of the

at Waterl-oo

and nlth

R. W.

I an slnllarly

of Rocheeter,

of Pennsylvanla.

and

ApaperofR.T.Rockafellarhasacloseconnectlonwlththe nork here:

Monotone Processes of Convex And Concave TyperMernolre

of the Amerlcan Mathernatlcal soclety No. 77, L967. T(x)

stands for

the set Rx here.

to the dual hbre. duallty

relatlons

Hls deftnltlon

Evldently

In hls notatlon

hle polar

le to dlrectly

whlch appear here ln sectlon

correeponds

make true

10 as theorema.

the

TASLEOF COMEN.TS

Introductlon 1.

Productlon relatlone

2.

Cleeelcel properttes

3.

Input aod output llnlte

4,

DoDlnatlon

5.

Claaelcel clorure

6,

Propcrtlce

7,

Dual quantlty

8.

Syoctrlcal

9.

Support functlooe

of conpormda and prlcc

rclatlona

duellty

10.

Co.t rnd rGveouc duellty

11.

Cotpound durllty

L2,

Grorth od

13.

Roclproclty

14.

von Ncuoeon rcdcl

15.

Topologlcal uthod

16.

Algcbralcel uthod

L7,

Gap-cycloe aod coqouod fectore

18.

Coopouadgrorth arrd lntcrcrt

ElblLogrephy

lotcrcet

factora

thtoror

Introductlon The econouLc groltth the theory

for

lnltlatlon

deflned

le there

relq,tlon

goode ln any productlon

slnllar

can aa well

functlon,

Sone developments

defined

relation

duallty features.

where these are dlfferent of productlon

concept

of productlon

Ln theory

context

of

analyalet 1g

and lts

lnvolved

dual,

dual of a product of dual Bymetrles

betrreen the

and proPertLee

of one are

There le a parallel

1s the product for

of the duals. lnstance

wlth

relatlons,

has expreaelon

set of the dual;

or the growth

the

There are a varlety

the coet functlon

Lnput 8et of a correspondence

the

sre sddltlonal

honogeneous productlon

claeslcal

besldee,

le a dual

asgoclatlon

of closed convex cone8, but there

Thue wlth

w111

For the claeelcal

apace.

18 a symetrlcal

wttlch

relatlonl

the von Neumann nodel,

Eepeclally

ln those of the other.

on an output

Eut relatloas

a8 the case where there

then arleea

ln the dual prlce

relatlon

theory

of

are

and number of goode.

The flret

there.

homogeneoue caae, there productlon

and outPut

Thua the lnput

and propertles

thus from the lnlttal

agaln have appllcatlon

reflected

and output8

lnpute

good.

Just one outPut

proceed

betneen poselble

be consldered.

the production

Ttre

relation.

perlod.

constructlon

an

of a type of productlon

ln terme of the aame klnd

deecrlbed wlth

of von Neuusnn (1959) glvee

nodel

as a return factor

for

on 8n functlon

a dual palr

glvlng

thls

are reclprocela,

of reletlon8

expreealon

for

the theoreu

of von Neunannwhere the reclprocsl grosth factor for the dual

of growth factore

Beslde the upper llnlt quantlty hle

factor.

becones the interegt

relatlon

vectora

nodel,

von Neurnrnnte theorem ghol'e ig attalned

, Vhich

and thue a naxlnum,

and equal

for

factor

correaponding for

al-nllarly

ln resPect

upper llnlt

prlcee

positlve

rrtth

to PosLtlve

here at the eane tloe

as von Neulannte

are not valld.

A theoreD of

For the more ldentifl-

by Gale 0956 ) theee reclprocal

treared

catlons

theoren.

Eouever,

another

theoren

and

le proved agaln

th18 conslderetlon

lnvolves

relatLong

the

quantltlesr

the dual relatlon.

McKenzle CI967 ) whlch

general

growth

the naxluuo

can also be coneldered

there

the dual relatlon,

of

Ln

lntereat

to the nlnlurm

whlch here appears as the reclprocal

factor,

eemlpoeltlve

for

be proved

nlll

under the weaker ae8ltrPtion8. The von Neuoenn conditlon cycllcal

caae of nore extended sene conclualona repreaenta

together

lt

le that

lt

ls

rrlth

a further

and thue do at leaet

bctter

lqo8elble

by tsklng to do better.

whlch

concluelon

theorem.

18 10 any caae poaelble

acroaa each perlod

power, and poealbly

type of condltlon,

type of "turnpikett

a nodlfled

acroaa N perlods rate

but

a * b > 0 w111 be exhlblted

aa a

glvca

the

ehlch

In grorth

to Srort at the naxinn as rcll

I ttahortrcut".

ea the Nth The concluelon

3

Given any two relatlone flrst

are the eaoe as the lnput

product frdn

A, B where the output goods of the aecond,

AB where z la AB-produclble

eone y whlch

N perlods

ln

goods of

le A-produclble

from x if fron

the von Neuoann nodel

x.

lt

the

they have a

ls B-produclble

Thua the relatlon

ie derlved

acroaa

a8 the Nth powe,

N

R" of the glven flxed Varloue propertles

relatlon

of a productlon

whlch are preeerved for The relatlon palr

R ln

products,

relatlon

(arb)

producti.on perlod.

whlch wllr

tn partlcular

for

be coneldered

powers.

le derlved

from a

thue,

: < R y = ( v g ) x > a t , b t > Y ,

18, R - AB nhere xAt = x > at,

The t here le interpreted from the lnput

x

produces

of the matrices.

the row-orders

tBy = bt I y .

as an actlvlty

and nhich

co@on columr-order nodel

any alngle

the von Neunann nodel

hatricea

of non-negatlve

(1)

that

R for

of the matricea

vector,

the output llhlle

the fornula

that

R ls

dlfferent,

y.

Ite

order

ln the context

are the eame and equal

nunber n of goods ln the economy, the foroula when theee are generally

which ts produced

(l)

aay n and n.

etlu It

le

the

of the to the

makee sense followe

from

1

cloecd:

(1f)

conlcal:

(111)

sddltlve! (rl)

(1),

51nce wlth

*Y

t,

",

xRY 'o xlRYl

(l

+

xRY

3 0)

(x*r')

xRy' xtRyt +

1e equlvalent

R(yly'). theEe ProPertle6

to convellty,

convex cone in 0E t

R le a cloeed

aaaert

*

t,

trRyr,

(1)

On.

hae the further

lt

Alao

propertlee (1v)

dlaPoeal

free of

xRY:

of outPut:

propertlee

The claaelcal be cloeed property

convex and wlth (11)

glvee

free

xtRY

Yt +

for

a productlon

relatlon

lt

that

Thus R t 0n

returns.

by (1) le claeetccl

ale

the honogenelty

Ihen

dlepoeal.

S . 0n x Onodetermlned

ta the relatton

xRYt'

the caae of conetant

(a,b)

fron

deternlned

: : xRY

xt

lnPut:

"

nm

hornogeneoue; and eo aleo

by

PSq = Pa > qb.

(2)

glven by (1) and (2) are dletlngulahed

The conatructlons polytope

and polYhedral.

For any R.

nn t

whatever,

the dual

S * R*, where S c 0rrx

0r,

pSq = xRy =+ px I 9Y.

(3) In ldentlcal

ldentlcal

0D

by

le deflned

The dual

as

fashlon,

of any relatlon wlth

any s has a dual

but lrLth exchange of rolee, is

homogeneoue.

claeelcal

the dual of lte

dual

if

and only

lf

s*.

Alao any relatLon lt

le ltself

l"s

claselcal

homogeneous. For example, R and S glven by (1) and (2) are such thet t

t

R = S for

*

and S = R", as can be verifled

llnear

palr

a, b by (1),

the polytope

wlth

( 4 )

theory

R glven from a

relatlon

the condltion

a * b > 0

Then Gale (1956 ) introduced homogeneous relatlon

whlch together

of a general

conslderatlon

R wlth

condltlons

o(R):

O R Y= + y = 6

can deflne

these conditions

regularlty

R.

for

applled by (2). relatlons

correspond to conditlons

Evidently

then, elther

> 0

one of these condltions

to R glven by (1) 1s equlvalent

to the other

a general theorem, that

Thls lllustrates

applled

for

to S glven

any dual palr

of

R,S o(R) 0, t > 0 and u > 0 such that

equlvalently,

that

A consequence of thie

one is regular

von Neumann consldered existence

to R glven by

Ae applled

(q) t > 0 -+ at > 0 (B) P > O:Pb respectlvely.

claeelcal

x R y , y > 0 f o r s o m ex , y

B(R):

(1),

uslng duallty

lnequalities.

von Neuman considered matrix

dlrectly

pat/pbt

pa > pbb, of the frmction (p > 0, t ) 0),

6

(4) er8ureE

rtrlch

I'ttth

and flnlte.

to u posltlve

(B) corrcapood

(c) 8nd

condltlone

The regularlty

la nell-deflned.

F''nartP:rR:rP, xzo]

le that

the.conclualon

p-', o- are both deflned,

and

and flnlte,

Posltlve

of

equivaleot

to S' another

ln reaPect

correepondlngly

and E deternl.ned

Introduclnt

Io-1.

i - auPtP: xR:rP,x > ol, 1a the eddltlonal

there

McKenzle (1967),

cen generally

that

o-- l

rtreni

lenot

Productloo A etock

vactora. 0 belng

y)

coluul

vectora

p ln the dual

Tten the exchange value the non-negatlve

dual,

all

f.

But

d i-

1, i 6-

by a vector

x ln the

of order

n.

Ttre prlcee

epace O' of non-negatLve

of quantltlee

x at prlcee

1

are rot

p la px e 0'

ilrders.

actlvlty

cooverta

a atock

of aone n goode y e nn, eo lt

of solc

of productlon

are deecrlbed

m goods x e nn

le descrlbed

e nn t 0m, x beLng the Lnput and y the output.

poeelbtlittee

lta

o fe 6 d:

the reclProcal relatlott.

of sone n goode le repreeented

A productlon

(x,

and S lc

of [,

hmogeneoue

claeelcal

rclatlone

by 6 vector

i.nto a etock

regular

be eald of the relatt'on

apace nn of non-negatlve represented

"otteepondlngly

dnetead.

are eetabllahed,

1.

deteroined

i

ae by Gale (1956),

va1ld

way by

la another

obtatned

However, nhen a general

R 1r coneldered,

rrlatloo

i' - i,

wlth

and slnllarly

to S.

lD rcapect

concluelon

by a palr The the

by a productlon

reletion

7

R c 0n r 0D, where the condltlon wlth

lnput

x and output

recorded

for

relatlon

introduced

growth.

If

lnput

there

to the standard

y, or that

and output

a feaslble

y is produclble

fron

ls a slngle

operatlon x.

rf

the goodg

are the seme then n = m. such 1s the type of

by von Neunann in the treatment output

of econonlc

good then m = 1 and R corresponde

concept of a productlon

For any productlon xR c 0n glven

xRy stgnlfles

functlon.

R c Qn x em and x e On, the aet

relation

by x n = [ y : x R y

deflnee the output possibllity

set of R for

any y e 0n the set Ry c fln given

for

an lnput

x.

Slurllarly

by

R y = [ x : x R y ] ls the lnput posslblllty

set of R for

an output y.

Glven two productlon relatlons A.0r lnvolvlng

r * s + t goode partlt,loned

class of s goods belng both output product

x ns, B.

in three

ea * Ot

classes,

for A and input

the nlddle

for

B, a thlrd

AB c Qr t 0E is determlned where

relation

xABz = (vy)xAyBz Thus the condltlon

that

z be AB-producible

a y wtrlch 1s A-producible A productlon x6Rys(s = L,2, ls

relatlon

R . Qm r Qn ls contlnuous y" o y (s + -)

to say R is a closed subset of em t en.

that

all

its

lnput

sets Ry . 0t

(x e nn) are closed.

x ia that

there

existg

from x and from whlch z ts B-produclble.

and x" - *,

...)

fron

lnplies xRy, whlch

Continulty

(y g Xm) snd output

Also R is convex if

lf

for

R inplles

sets xR c em

8

xRy' x|Ryl-+

ie a convex subset of 0n x 0n.

whlch ls to say it

A productlon lnput

and output

input

lts

all

that

luplles

(xl + xltr')R(yr + yll')(l,tr'

Convexity

R

for

sets are convex'

R has the Property

relatlon

- 1),

o,tr + lt

2

of free

for

disposal

lf xr :

and for

output

xRY '+

xtRY ,

Y'+

xRY' ,

l-f xRY ?

and thus for

both lf x R y Z y ' Z

x ' ] Equlvalent

st-atements of lnput

o + x r R y r .

and output

are

free disposal

x t a x + 1 f t c 1 r ! ' , Y aY"+

RY c RY' ,

and agaLn x e R y r x t l x + x t e R Y , y e x R , y : Y t + Y r e x R , provldedy']o.AccordenoteetheProductof R wlth

>

Thus the characterlstic

of a productlon

reLation

whlch

haefreedlsposalforlnputandoutputlsthatltsinputandoutput

9

seta ere orthogeneousr convexr

2.

they

ln the oppo'lte

are called

Claeelcal

orthoconvex

convex and wlth

productlon

free

output

poeelblllty

further

poeslble

dlepoeal

tor

as'uttrptlon for

at leaet

none decreased, by a strr.ct

output

claeelcal

output

they are

respectlvely.

a production

productLon

becones the catagraph, Thus nlth

the correspondlng

a claselcal

productlon f(x)

and A

R le that

> y ,

lnputs.

property

when there

corresponds

which

functlon

or the reglon

lnput

can always be lncreaeed

functlon

aa a coucave eenl-lncreaaLng

relatlon

of all

thls

rta

and orthoconcave.

(vyt)xtRyt

increaee

to be contlnuou',

and lnput.

aome output

R wrth

of a claaelcal

functlon.

ie deflned

aeta are cloaed orthoconvex

and whlch meana that

concept

also

and orthoconcave

relatlon

xr > xRy .+

output,

rf

oropertlee

A clagelcaL

elngre

aenaea.

of

below the graph,

productlon

le a

to the

deternlneg lnpute.

and

uaxlmuo

rn fact

R

of euch a R c 0n x 0

relatlon

functlon

- nex[y

le concsve, and semi-lncreasing,

: xRy] e e (x e nn; that

is

x r > x - f ( x r ) t f ( x ) . A production

relatlon

showe constant

returns

to scale

lf

xRy+,.F 1o whlch case 1t 1s also output'

so lt

1s llnearly

corresponds

honogeneous,

said

to be homogeneous.

to a productlon

functlon,

rf

there

is Just

then that

one

functlon

t0

A product{on

R le addltlve

relatlon

(r

= f(x)I

f(xl)

: 0)

lf

(x * x')R(Y * Y') '

xRY, xtRYr +

Glventhataproductlonreletlonlshomogeneoue,ltlgconvexlfarrd onlylfltleaddltlve.Dlecussionhereglllbenalnlyaboutclaael.cal relatlong'

homogeneoua production

3.

Llmlt

LnPut and output

For any lnPut

(Y t * { t * t Y +

e l n f A =

and then there

le deternlned

where

A le deternlned

set A t 0n, a eet inf

Y t I ( v z )z e L ' x < z S !

the 8et

nlnA-AnlnfA. If lnf

A c A' 80 that

A tg closed then lnf A la called

a lower

' and any Polnt

llmlt

Ihe orthogeneous cloaure

lnf

nln A'

orthogeneoua,

set wtrlch contalns If

A ls closed

contalna

Any polnt

of of A'

of mln A a mlnluuo'

of A is

i - t * : x t : x , x ' e A l Ttrls ls

A'

A and ls

contalned

A, and A le orthogeneoua lf

. ln cvery

and only

lf

orthogeneoue A'

['

then

i - (rrn l)' ThuelfAlsaclassicallnputset'thatlsclosedorthogeneouElnthe lnput

space 0n, then 6 -

Thua a cloeed

orthogeneous

eet A ls

(nln A)< determined

Juet by lta

DlntDs'

ll

Further,

for any set B, 6 = f

Also mln A, whlch

. o m l n A c

is now glven

B c

A

by

x e mln A . 0 .

to

equlvalent

appear that

wlll

free

R wlth

relation

For a homogeneoua productlon for

x'

yR ls bounded.

xR ie bounded if

Bo that

*,

y:

bounded there exists y such that x e X+

xls

and lf

lf

also R la contLnuous,

then another

equlvalent

1e o R y +

Y = 0 ,

whlch le to say Ry does not contaln conditlon

ls preserved

hol'ds for

Another condition ls preserved

for

products.

for

on a production

thls

it

holds

ls

for

Y = 0, so that

relatlon

A and B oABy impLles

whlch obvlously

the feaslble

output

for

all

condltion

Y 2 0 for

For R whlch is honogeneous and has free

obvlously

laet

Thls

1s

Ry 1s non-enpty

relatlon.

y > 0.

AB.

products

Thls can be called

all

For lf

x = 0, and then lnplies

then oA: 0.

Then qY" 1s unbounded, and hence qyt * -

subsequence yj

of y".

The corresponding

-1

-1

b y h o n o g e n e r t (yq, y l ) ' * l R y l ( q y l ) - ' ( " '

pomt ! of the rl(cri)-l, Yl euch that

a e u b s e q u e n c ey ' o f for It

yil(qy')

of R, that

by contlnu1ty

of x" le

Aleo *lRVl,

ao that'

-)'

-1 .

N o wq ( v i ( q v ; )

1,2,...).

auch that qi - 1, 8o that i > 0, and - 1^

xi'

the correspondlng subsequencex'of follows,

some

I ls compact' Hence there exLsts

and the 8et of y such that qy' a llnlt

Teke

(s + o) for

subsequence xl

- O (s + -).

bounded, so ttrat xj(Afi)-l

stlll

Y" such that

sequences *e,

- y (e - -)' xi(wi)-l

strfl

CIR!, which,

Then aLeo, * O'

since ! > 0, completee

the proof.

7.

Dual productlon For an actlvlEy

lnput

and output

are px, qy e relatlon

and prlce wlth

prlces

relatlons

input

x e Qn and outPut y e 0n rrhen the

are p e 0r, and g e 0m, the coet and the return

Q so the proflt

ls

qy = px e t fl.

R . on t f,]mhas associated

Rtt c Qn x Qr, nhlch

holds

with

between input

lt

Any productlon

a dual prlce

and output

prlcee

relatlon at which

1'

l1

posltlve

profit

ls

lnposelble

(1) In correapondlng dual productlon

productlon

with

pR*q = xRy o

px

any prlce

realtlon

faehlon,

I

eubJect to R.

Thus

qy S c 0o r 0, deternlnes

S* c 0n r 0m, where

relatlon

xS*y=psq+pxl9y Thue S* deacrlbee non-posltlve Now let

productlon

all

at all

lnput

actlvltlee

and output

T denote any relatlon

prlces

for whlch proflt 1n the relatlon

0n r nn or O

ln elther

D

x fl .

Theoren7.1 T^ . T,'+ U

l

T*: U

Tt

Thle 1e lmedlate

I

fron

the definltlon

Theoren 7.2 TcT** By deflnitton

of T* xTy a pT*q ..+.

px I

9I

equlvalently xTy .+. But by deflnltlon

pT*q.+

px l, 9I

of T** xT**y

. g x r . e . D > O

pEq .

.-. Aleo

pEq.-.

p:

q

- x : y = D P x : P y : q y Thus D* - E, and alnllarly Sinllarly

lt

PDttq. E* - D .

can be seen that

for

any u :

0, the relatlons

glven by x G y = x 2 y y , forn

p H q = u p : q

a dual palr.

Theorem 8.3 For any a e 0m, b e en, the relatl.one R. ScQ

-u

xfl

qlvenbv fr

xRy = (vt

> at,

e Qr)x

p S q = p a > q b ,

forn

nn r 0n,

bt

p e Q m ,

:

q € f l n

a dual pair. Thue, wlth

p : 0,

q

> 0 : 0 and r

pR*q=xRy:px:qy . - x i a r , b t z y + p x ] g I qat

< - p a > q a = p S q . Thue R* = S.

It

reral-na

to shorr' that

S* = R.

y

G, H

20

non-negatlve'

varlablea

all

Thue' wlth

xSiy:paiqb+pxlQY. That 1s, xS*I

and onlY lf

lf

holds

px < QY

pa I qb, has no solutLon

(P, q) I 0.

for

But

(pq)f"l:o (pq)[*l'o |.-bJ (nl)

hae no solutlon

1 0 lf

and only

f *'l

l-'j hae a eolutlon

t I 0, that

The relatlons

Corollary.

[-Y]

f ''l t

-

|.-uJ

1s, lf

and only

1f xRy, ae requ{red'

R, S given by PSq = Pa > q

xRy = x > aY' forn

lf

a dual palr. Wlth reference

to the theoremr let a -

be the natrLces

a, b partltloned

^n ,, b, e Q" (i = f,...,

ao K ls

the relation

(ar, br),

(ar ...

I - 1,.'.'

clasaical

b -

ln thel'r

(bl ... br) columrs a, e nn,

l),

andlet

K -

[(at, br) : I - 1,..., r]

descrlbed r.

by a flnlte

,

set of lnPut-outPuts

Clearly the dual of K ls S'

K c R and K t R, nevertheless, R ls the classical

"r),

K* = R*, so K** - R'

homogeneous closure

homogeneous relation

of K, that

contalnlng

K'

is,

T t r u aw h l l e It

aPpears that

the enalleat

2l

Theoren 8.4 A neceesary and sufflcient 8y@etrlcally

dual ls

The consldered contlnuoua neceeelty deflnltlon

R c R**.

be classlcal

condltlon

of these propertlee of a dual.

Thue, conslder

It

ln

renalne

is

(tv)

that

the arguuent

Rt* - R.

Slnce R le orthogeneoue,

fron

totether

theee propertles

the

they are

relatlon

But for

The

theee propertles.

R wlth

a prlce

lf

be (1)

lt

proved dlrectly

a productl.on relatlon

to shors that

to be

orthogeneoua.

w111 be proved that

hae to be ahorm that

Thus lt

a relatlon

honogeneoug.

on the relatlon

is readlly

Now lt

le no dlfference

lnetead).

lt

for

(11) hornogeneoue (111) addltlve

eufflclent. (There

that

condltlon

ie coneldered

any relatlon, inply

I

c iIT.

by (1v),

I ] 0 . + . x R y < - r * R l y and elnce R le a closed convex cone, by (1),

(11) and (ili),

ao ia

R > . Conalder x:

o, y ] 0 such that x[y. beceuee R > le a closed convex cone, there

rhen friy, erlsts

and hertce,

(p, q) r 0 such

that (a)

x'R I Y' -t

px' 3 9y'

(b) px 0. (ll),

the monotonlclty

Now (a) wlthp, as requlred.

A]0

(b) Y>0 0 ioplies

23

A relatlon

T can be called I - o,

oTy Thenthe theorem shows rhat lf

for

if

it

hae the propertles

(vx, y)xTy, y > o

any dual palr

one 1e regular

if

and only

the other ls,

Corollary. if

regular

xR ls bounded for

and only lf

all

* I 0, equl-valent1y for

Sq ls non-enpty for all

q:

O, equlvalently

sone x > 0 for

some

q > 0. The regularlty can be lllustrated derlved

condltlons,

appearlng

1n Theorern 8.5 and Corollary,

by the von Neumann type of production paLr (a, b) and lte

from a matrlx

The condltlon

for

R to be regular

relatlon

R

dual S, ae glven by Theoren 8.3.

are

(1) ORy-y-9 (11) xRy, y > 0 for In terns

of

(1)t

some x, y.

(a, b) they are equlvalently ar - 0 +

br - 0, for all

c E 0n

(11)' bt > 0 for aorle t. Slnce they are equlvalent

to

(c)"

a_, 2 0 for r)

(B)"

br. e 0 for every good l. (r

every activity

r,

Conslder the condltlons ( o )

p > 0 = + p a > 0 ,

( B )

t > 0 : > b r > 0 .

Aleo (o) implles (1)' , and (B)' inplles

(1), while

is equivalent to (11)',

(B) is equlvalenr to (il).

so rhat

(o)

24

| doee not But (1)

nortevcr) (ll)

for

coadltloaa

neccltary

(c),

lnp1y

'

r.)

fron

(1)'

Thue,

eny case wtren Ln

requlree

that

Partlcular

0 -

every r euch that

the .th

if

thern ia unaltered

= 0.

.r)

br)

- 0 for

(a, b),

for

holde

condltlon

of thle

(o) ln

valldatlng

defect'

rrhlch renoves thle

holde,

and (B) as

(c)

of R'

regularlty

le e aenae for

there

ln

Eakes e defect

and thle

r

all

then the relation

froo

le deleted

"o1,-,

R rrhlch derlvee a, b for tt)

a whlch resulte'

In the natrlx

t 0

forallr,eo(cr)''laeatlafled.Itfolloweth8t(1)l.upllee(c)' provlded

(".) tr)) = o for

each actlvlty lnput

wlthout

r,

unaltered

the 1let

of baelc

whenever null actlvlties,

The provlao

is

condltlon

a * b > 0, slnce

lt

and lt

ls

+ br)

ae lrell ln

Theoren 8.6

'' o,.;ni anv For il:;;"."='".T.,, (1)

QRY"+Y'0

(11)

xRY, Y > 0 for

some x'

Y

that

ell

are

the von Neumann

> 0, lnplylng

a . > 0 or b . 2 0, aa requlred. t) t)

the condltlons

actlvltlee.

are added or removed

actlvltlee

glvee .r)

0) whtch le

anong the baslc

ln any caae autouatlc

reuoved.

(0'

actlvlty

lncluded

ia not

or output,

But R renalne fron

le the null

that

that

25

and

( a ) t > o : a t > 0 ( B ) p > 0 - p b > 0 are such that (o) : also

(1) -+

(1),

(B) .o

(o) provided ("r)

Now conalder

the regularlty

(1i) br))

; > 0 for

a1l r.

conditlons

(1)* osq:q=6 ( 1 1 ) * p s q , 9 > 0 f o r s o m ep , q S = R*, where, by theoren g.3,

on the dual relatlon

p S q = p a : q b It

ls obvlous that

equlvarent Obvlously

to also

(1)* ls equlvalent

ro (g),

ao then (1)* 1s

(11), whlch provides an illustration (11;'t Le equlvalent pa > qb for

and hence, by a general

of theoren g.5.

to,

some (pg) r 0,

theoren an lnequalltles,

to the denlal

that

(at,-b)0, equlvalently,

slnce,

a b > 0, to the denlal

that

a t = 0 , b t > 0 f o r s o n e a 2 0 , equivalently a t = 0 ? b t - 0 f o r a l l t , whlch ls

(1) r.

Thus (11)* is equlvalenr theorem 8.5 again.

ro (1)r,

and hence to (1),

whlch lrru'trates

26

Thefollowlnghasappllcatlonlngrowthandlntereettheory 1ater. Theoren 8.7 Ee,nareclaseicalhomogeneousproductLonreletlonewlth one of A n 8r,

duale u, V then elther

The the converae Bt of B le glven x(A n B')Y t*

by xEty

ln cotmon then there

exlsta

(1)

:rAy -+ px I 9Y

(1i)

YBx

Then input

-o

= yBr'

Ttrus

(p'

they have no Lnterlor

q) tOsuch that

for A wlth

(i)

lnpltee

N o w( i ) a n d ( 1 1 ) w l t h p l 0 ,

and for

p:0,

Thus

Also l n t ( A n B r ) t 0 + U n V r = 0 . (x, y) e int(A

n Br) then,

since (x, y) e lnt

PUq: and slnce

Px > qY

(y, x) e B' qvP+qv:px

Thua pUq :

qlp'

as required.

Thus

i n t ( A n B ' ) = 0 < + U n V ; * 0 , and ldentlcally

wich duals lnterchanged.

B wlth

q:0Ehowthat

i n t ( A n B t ) - 0 ' + U n V t t 0 .

For if

lf

qY I Px

free dlsposal

(11) lnpllesql0. pUq, qVp.

1f and onlv

xAY' YBx.

slnce A, Bt are closed convex cone8, lf point

le null

Lnterlor.

has non-enpty

the other

u n vr

A,

27

9.

Support functlons Glven an lnput

the lower 1ftolt

pos6lb11lty

of input

(1)

aet A c en and lnput

prlcee

cost ls

[p, A] - lnf [px : x e A] = n a x l t : x e A

As a functlon

of p e er,

(2)

[p, A] ls

[]p, Al - ltp,

llnearly (l

Al

+ p x > t ] . honogeneous,

: 0) ,

and superaddltlve. ( 3 )

[ p + q , A ] l [ p , A ] + [ q , A ] ,

whlch ehowe lt ( 4 )

to be concave.

Also

A c B + [ p , A ] : [ p , B ] Slnce

( 5 )

x e A : p x l [ p , A ] ,

the set (6)

a -

t" : px:

[p, A], p:

Ol

ls such that

(7)

a.e

Hence, by (a) and (7)

(8) But alao, (9)

[p, A] I tp, ,ij for

p : 0, by (1) and (5)

tp, Al = inf [px : x e A] = inf[px I

It

,

follons,

(10)

from (8) and (9),

: qx

]

[q, A], q I Ol

[P' A] thar

[p, A] = tp, nl

(p : Ol

p e 0rr

2E

Thus A = i that,

I t

w111 nosr be ehown

then A - i'

orthoconvex

A be closed orthoconvex'

has to be shorm now that

It

orthoconvex.

closed

A 1s closed

if

conversely, Thus let

it

A ls

tnplfes

A ls cloaed orthoconvex'

(6) that

fron

apPears lrmedlately

It

Slnce Ln any case A c i'

l' there

exl8tg

Thue p > 0'

Now

Then, elnce A ls cloeed convex'

Suppose * . I. p*0andCsuchthat

( a ) x ' e A : P x ' l C ( b ) P x < C Now slnce A 1s orthogeneous, that

ls

x e A r * t : * - x t e A , froro (a) that

follows

and non-empty,

lt

(a) shows that

[p, A] I c'

p > 0'

Hence, by (10)'

* . A o p x l [ p , A ] + p x > C

;

Thus (b) shows that

x e A, as required'

Itlsnowfoberemarkedthat,foranyA,Atsldentlcalwlth theclosedorEhoconvexclosureofA,thstiethelntersectlonofall closed orthoconvex

sets containlng

A, and contalned

containing

A, thls

belng cloeed orthoconvex'

ln every closed orthoconvex

aet contalnlng

A.ItJuSthastobeseenthatlfBlscloeedorthoconvexarrdB>A then B,

A.

FlrsE

it

has to be remarked that,

Br A+ i'

for

i,.

any sets A' B

29

(4) and (5).

Thls appeare fron B - B, go B: It x e I

t t

A, lnplles

has already

But lf

B le closed orrhoconvex,

A, ae requlred.

been eeen that

if

A ie closed

then there exl.sts p I 0 such that

follows

dlrectly

that

lf

for

[p,B] for all

p I0

. 0, so that

eA,

If0

pa > 0'

Then the set

p > 0'

Other$tlse

takex=0'

a minimum on lt'

is non-empcy comPact' so px attains

for all

[x : px : pa' x e A] But this

is

also a minimum of Px on A' EvidentlY,

for

anY ser A and P I 0' [p, A] = [P, inf Al

,

and hence

a = (inr n) Further,

if

[p, A] has a solutlon

x then x e min A'

Theorern 9 .3 If

solution

A c Qn ls

for all

and min A 1s bounded

closed

then

[p,

A]

has a

P e Qn.

For then rnin A is

comPact and

inf[px

: x e A] = inflPx

: x e min A]

=min[Px:xeminAl = m i n [ P x : x e A ] . Given an outPut posslbillty

set B c on and output prlces

o' e Qn , E h e u p P e r 1 l r n i t o f r e v e n u e f r o m o u t p u t

ls denoted

[ B , q ] = s u p l q y : Y e b ] = m i n [ t : y e B : q y : t ]

The cheory of revenue functions functions,

10.

nostly

derlves

from that

for

Cost

wlth obvlous adaptatlons.

Cost and revenue duallt

T h e o r e r n1 0 . 1 Ig!,4je91

palr

R, S lp, Ryl:

Ips, y]

Since pSo, necessarlly [pS, y] = 0, I 0, so in case Ry so that

[p, Ry] = -,

the lnequality

llow suppose Ry t 0.

is verlfl.ed.

Slnce R, S are duals,

xRy, pSq -

for all

px I yg

Hence psq .>

lnf [px : xRy] I yq

and hence 1nf[px : xRy] I

srplyq

: psql ,

ae requlred. Theoren 10.2 E

n, S are a polyhedral

dual palr

lp, Ryl and provlded

Ry la non-enpty,

[ps, y]

there

then ,

ex16t x,

xRy, pSq, [p, Ry] = px - qy Wlth such R, S xRY - (v3)1 > 8t'

bt : Y

p S q = p a : q b , all

elements belng teken non-negatlve.

Then

q such that [pS, y]

x,

q

5Z

[P, RY] = tnf[Px:

x > at' bt:

Y]

: bt 2 yl

= inflpat Also

[pS, Y] = suP[qY : Pa ] qbl The linear

Program max[qy:palqb]

is

since q = 0 is a feasible

feasible,

and only

bounded if

if

solutlon'

Therefore

it

ls

the dual problem, mln[pat : bt ] Yl

'

1s feasible,

which is when Ry is non-empty'

have optimal

solutions,

and the oPtinal

Then both problems

values,

which are [pS, y]

and [p, Ryl, are equal. Corollary.

Ry is non-ernpty for

bounded for

all

and only

lf

pS is

P : 0.

For, given anY P, [pS, y] pS ls bounded.

y I 0 if

all

is bounded for

Thls ls a speclal

any 1f 1f and only if

case of theorem 8'5'

Theorem10.3 For dual S, R 1f etther

of

then they both do and the optimal If

[p, Ry] has a solution,

[p, Ry], values

[PS, yl have optirnal solutlons are equal'

let

[p, Ry] = px, xRy . Then xtRy : equivalently, set [(xf,

y)

pxt I Px ,

the closed convex cone R > ls dlsjolnt : px'

. Px].

fron

the convex

Thence thele exl-sts u' v such that

55

but

(il)

9 = trv. lnplles

(1)

xrR : yr -

ux' I vy'

(11)

pxf < px :

uxt < vy .

lnplles

p i

Then (i), that

,

px - trux, I r 0, and that

lu,

wich rhe roontoniclty of R becauee 1t has a dual,

p, Q ] 0 and rhen thar psqt o

pSq.

Flnally,

wlth

xRy,

px > gty ,

whlch, wlth px = qy, psq, shows thar px = [pS, y]. argument, lf

px - qy, where

[pS, y] has a solutlon

By slnilar

then so does [p, Ry], and again the

values are equal.

11.

Conpound duallcy

Theorem 11. I If

R0, Sg and Rt , Sr are dual palrs (1)

min maxIqx : pSOg, xRt]l x q

(11)

r n a xm l n I q x : p S o 9 , x R r y ] = [ p S o S r , y l q x

provided in each equallty lf

then

=p, RoRr]l

that both sldes have solutlons,

and only if

elther

side does.

The left

of (i)

is nin[ [pSs, x] x =

nln[[p, x

=

mlnInlnIpz x z

=

rnlnIpz: xr2

=

ninIpz z

thle belng

: xRlyl

Rox] : xRlyl

(by Theoren 10.3)

: zR6x] : xRiyl

zRgxRly]

: zR6R1y]-

[p, RoRry] ,

T h e o r e m1 1 . 2 and conpact

For closed convex A . fJn not containlng the orlgln

x e A r P e B s u c h

B c Q , c o n t a i n l

tiratfi'0a!4 ( x e A , p € B )

p i l p i 5 p x Deflne B* c 0n bv

x e B * = p e B - p x < 1 so B* ls closed and convex. posltlve

ls bounded since B contains

AlSo lt

element' and contains

a nelghbourhood of 0 since B ls bounded'

Slnce also B 1s closed and convex, given f, such that x e B* wh1le !i Slnce A is

= 1, for sosre i closed

e B*,

r t i.

wlthout

whlle

loss in generallty

Othersrlse there 1s no essentlal

x e B*: But then,

since i

there

i*

follows

]

Thls completes the proof. is

that

n a x m l n [ p x : x e A , P e B ] p x m i n u r a x [ p x : x e A r p e B ] x p both exist

and are equal.

from B* for

I

that

P e B : P ; : 1

conclusion

> 0 such

I = 1'

I such that

exists

x e A +

e B*, it

i

A be replaced by Al

further

An equlvalent

exist

0, there

lf

al1

f, e B'

that

can be taken that

lt

lx < 1,

< for

fx

Atr is dlsjoint

alteratlon

and convex,

Stnce A and B* are closed

so also Fi = f.

follows

lt

and does not contaln

that AI cuta B*, say ln a Point i, all

a

, f, e B, and

35

If

Corollary.

Rg, 56 and Rl, Sl are dual palrs,

pso ls bounded and contalns contain

both

0,

exist

and are

and the

nin

max mln[qx q x

: pS6q, xRly]

rnin maxIqx x q

: pS6q,

equal

and posiElve.

there

ls

no ambiguity

A classlcal is

non-empty

it

is

if

a regular

and only dual

charac[erlzaElon a1l

bounded,

qolqq.l

relation

equivalently,

denoting

both

the

max

Rryl

homogeneous productlon

and does not

regular

ln

xRly]

max by l p S o'

S are

element and Rly does not contain

then

Accordingly, min

a positlve

and p, y such that

of

if

palr.

1ts

that

one which

regular

0 for dual

Then,

a regular

and again, is

contain

S is

palr

Ehe sets regular

and classical

Sectlon

thac

the

Sq are

and the

regular

Then,

regular.

is Ry,

R is

y > 0.

all

as seen in

dual

is

relatlon

that

8,

another

dual

xR,

case R,

pS are

non-empty. of

its

Ry

theoren

In

sets

all

by

1f

A

dua1,

homogeneous.

Theorem 11.3 If

R9,

56 and R;.

St are

regular

dual

palrs

and p > 0,

y > 0

then IpSoSr , y], have

solutlons This

and are

follows

equal

IpSo, Rry],

Ip,

RORiy]

and posltive.

by combinatlon

of

the

foregoing

proposltlons.

8.5

T h e o r e m1 1 . 4 I f R s , R t a r e r e g u l a r c l a s s i c a l h o m o g e n e o u sa' n d h a v e d u a l s and has dual SgSl'

Sg, St, then R6R1is also,

is nornal.

IE is already seen Ehat a product of nonnal relations t 0, Y ) 0' Thus R = RoRtand S6Si are both normal and, for P IP, RoRtYl

IpSoSt , y], both have solutions let

dual be S.

its

Slnce RgRl is normal,

and are equal and positive.

slnce [p, RoRtyl

Then [pS, Y] also has a solution,

does, and has the same value,

It

by theorem 10.3.

follows

thatr

a 1 l p > 0 , IpS, y] = [pSoSr,Y] for

y > 0.

all

theorem 9.1,

implies'

Since pS, pSgSl are orthoconcave this

chat pS = pSsSl

for all

p I 0.

Hence S = SOS1.

Theorem 11.5 product AB' (AB)* r

For anY A' B with Thus, by definltion

A*B*.

of A*, B*,

xAyBz, pA*qB*r :> px I iy so that,

of AB, A*B't,

by definitlon

xABz, pA*B*r : and hence, by deflnitlon

of

pA*B*r

Px Z tz

,

(AB)*, .:>. .=rz

xAi.z => px > rz . p(AB)*r ,

,

by

for

37

The followlng

ls an aLternatlve

If

A, B are regular

It

has t.o be shown that,

(2).

of theoresr 11.4:

homogeneoue then (AjB;:t - 4*g*

classlcal

for any given p, q,

xAyBz say (l):,

proof

px > rz

-.

(vq)pA*qB*r ,

(2) implles there exisrs y such that

The contrary ot

IpA*, y] < [y, B*r] Wlth A, B regular

classlcal

homogeneous,thls

Ip, Ay] . by theorem 10.3, whlch 1nplles

contrary

and thls followlng

to

(1).

wlth

,

there exlst

xAyBz,

px < rz

A*B* )

(AB)* ,

x,

z such that

,

Thua

the prevlous

glves

IyB, r]

is equlvalenE to

proposltlon

gives

the requlred

result.

The

lllustratlon.

For example, Iet

R - AI where xAt

= x > at'

tBy 3 bt

> Y I

then R* ! A*B* r.J,here p A * u = p a : u , u B * q = u > g b , 5o that pR*q 0, and if P = s u p [ p ( x ) : x > 0 ]

then

O . p . - , and

iR$

for

some i

> o

42

fhls

the theoren of Gale (1956) '

ls

To show i

. -,

so that

-r . + 0 since p > 0 and p"; = 1' *loi-t

follows

now by continuity

and contradlcts

that

st11L pr + -'

and P, * P.

such that * i.

pi

= r

there

- 1- n x i '

whlch by regularity

rt

lnpllee

0'

i'

(r - L,2r"

, so that

i

')

euch that px, -

I'

polnt

x'

the x. have a linlt

L - i ^ - -

t L ^

iRIi

> 0 such that

i

exiats

Then, as beforer

-

L - " a

a

1 { n { }

exl-sts a subeequence

- 0, and there

But the corredpondlng

But xlRxlor,

Slnce xioi

> O.

a sequencex.

Ttrus, there exlsts

*i

thac G'i,

remalns to show that

It

*.kro,

i

'

rhen pi - 1r eo

For the corresponding subsequence o| of the 0r'

> O.

i

* i'

and a subsequence xf such that x'

i,

(r + -)

px = I are cornpact' ao the x. have a llnlt

But the x > 0 such that point

xrRxrga, 9, * -

= l, ] o, Pxr

*,

Take any p > 0'

f, = -'

that

and g, such that

r = Lr2r..'

xr'

Ihen there exists

posslble

suppose if

subsequence p' - 6 , ao that

and hence by contlnulty

x'pr

+ [f,'

inip'

Theoren 13.2 E for all

R, S are a dual palr

wlth

grovth

functions

p, x p x > 0 - o ( P ) P ( x ) I 1 Since xRxp(x),

pSo(p)p 1t follors px :

o(p)pxp (x)

for all p, x. Corollary (1). Corollary (1i).

=o 66 . t. Fi ' o ;P : l ,

that

o p : 1 .

'

p(x),

o(p)

then

+J

Forallp. for

p there existsx>

some F r O. p . i,

for all

Bur now i* ir

follows

0such thaExRxp. A1so6= o 0 - = >b t > 0

( t ) '

P > 0 , t > 0

Also consi-der the relations

P a t * P b t> 0 '

:

R t Qn " Qn' S t Q'

Qr',where

xRY - (vt e Qm) x > at, bt : Y p S q . p a l q b Thesearebothclassicalhomogeneous.Intheorem8.3theyareshown to form a dtral Palr, s = R *

R = S * .

For the condltlons M ( R ) . o R Y - Y = 0 W(R) = (vc, Y) xRY, Y > 0 defined on R, and similarly Yl(R) <w(R)

0l

of von Netnann lnvolve

The reeults recovered

p-".tp[p:xRxP,x>

wlth

together

theorems of

sectlon

further

6 ana 6 '

theorens

Then von Neumannrg topolo8lcal

basls.

lnvolvlng

13, whlch have convexlty

nolr be

Theee wlll p "na

and llnear

uelng

the

eeparatlon

ae

i,

have a review,

argubent wl1l

inwhlchtherelevanceoftheregularltycondltlonewlllbedealt with.

Flnally,

a flnite

algebralcal

lndependent

of both sectlon

the reeulta

agaln,

rnethod w111 be used, which Ls

13 and von Neumannrs approach but ylelds

and aleo leads to further

theory

on compound Srowth

and lnterest. ThevonNeumannnodelwillnowlllustratethetheoremaofthe prevlous

sectlon

vhlch

are appllcable

to any claeeical

homogeneous

47

relatlon. apeclflc

But the epeclal feeturee

nethods shonn later

explolt

the more

of the nodel.

Glven R, S derlved

fron

(a, b) eubJect to

14.1 sh@re theae are each regular

claealcal

(o) and (B),

theoren

hooogeneous relatlona.

llence by theorem 13.1 (Gale), O . p - .- r 0 < o - .- ; alao p- and d, rtrLch are deflned hlve

aa upper ll-Elts,

are attalned

and

naxina. Thue theoremg 13.1 and l4.l

glve:

Theoren 14.2 (a) end (B) lnplv 0 < ; < - r 0 < o - < -

end

6-naxtp:xRxp,x>0] i-naxlo:pSop,p:O]. Let C . [ p : x R x p , x > 0 ] a n d

t t - [ p : b t > 8 t p r r > O ] .

fheoren 14.3. G c llr and (a) 1npl1ee G - H Suppoae g e Gr that

is xRxp, x > 0 for

x > at, bt > xp , for

eome x,

t.

Ttren, provided

p > 0,

bt > xP > o ao that

t > o.

Arao

x > 0

aone x, equlvalently

b t > x P : a t p r so that bt I atP. Thus, for

g > 0, P e G iraPlles t 2 0 ,

b t > a t P '

( * )

Stnce0 eHnGit

eH'

that lsp

f o r s o m et '

Now suppose 0 e H, that x = at,

condltlon

(*)

is

(o) gives x > 0'

for

i s c o n c l u d e d t h a tG c H '

some t'

Then' taking

Thus

x > a t r b t : x P r x 2 0 , shons that

P e G, and hence H c G'

is equlvalent

Sl-nce in any caae G c H' thls

to G = H.

Theoren 14.4 (s) and (B) lnPtY 0 . p- . -,

0 < E < -

and

6 - r " * [ p : b t ] a t P , t > o l o-- nax [o : pa: This follows

opb, P > 0]

by comblnation of theorem 14'2 and 14'3'

Theoren 14.5 (y) funplles that,

for

all

x and P,

( 1 )

x R x P , x > 0 r g > 0

(il)

pSoq,P>0,o>0

and

lnply

(111) px > 0

49

of R, (1) 1s equlvalent

From the defi.nitlon (1)' for (1i)

some t,

x>

at, bt]xpr

and thls

x)

t > 0.

lnplies

Eo

0,pt 0 A1so, from the deflnitlon

of S,

is ( 1 1 ) '

and thls

p a > o p b , p > 0 , o t 0 ,

lrnpiles pat>opbt>opatp >0 pat = 0 is inpossible,

From thlsr

also pat = 0, contradlcting

for with

o > 0 lt

would lnply

(y) since t > 0, p > 0.

and slnce x > at frorn (1)'

lt

Thus pat > 0,

folLows that

p x a p a t > 0 , showlng (li1). Theoren 14.5 (c),

(B)

?nd

(.r) lnply

(1)

66-r

( 1 1 )

; = ; ,

and also

Fron (c) posltlve

; - ;

and (B) lt

has been concluded thet

f,, 6 are flnlte

and

and also xnxd,

x ) 0,

pSEp, p 2 0, for

some p, x.

that

px > 0.

concluslons

By theoreu Flnally,

follow.

4.4,

from th18 nlth

(v)

frorn theorero 1.3.3, Corollarlee,

lt

la concluded the requlred

50

nlght gror

to actually

be unable

aluuJ,taneouely

that,

though some gooda

the rnaximrm growth p-, all

etteLn

et any leeser

rate,

horever

for

polytope

theorem 1s proved again

Thla

6 - p f"

of the conclualon

The luportance

close

can

to f,.

correspondences

ln

algebra.

theorem 16.2 by flnlte Theorern 14.7 (c),

( 1 )

of p, t and u such that

('1) ftnply the existence

(B),

b t > a t u , t 2 0 ,

(11) pb a t l , t 2 0

deflnlng

a grohtth factor,

and any v such that

pb < vpa, p > 0, an interegt

factor,

lt

Hls

appears that

v D a t > p b t > p a t t r a> 0 .

part,

u as ln and whlch

the concluelon

of

(1v).

5l

Then pat = 0 ls

finposslble,

a contradl-ctlon

of

every l-nterest and (tt)

(y).

factor

1np11e8 also pbt - 0, glvlng

slnce thla

Thus pat > 0, and hence v > I ls

at leaet

every grorrth factor.

u appears as both a grorrth and an interest

follows

that

the nlnlnuq

lr 1s slmultaneously of lnterest

Fron thle

Whlle theoreur 13.1, of Gale (f9s6),

general{zes

a further

po - f.

relatlon

p- -i

result duallty,

lt

does thle

the more generally

15.

present

duallty,

valld

p-ri -1,

relatfone

factors

theoren

and

13.3

wtrich produces the

the addltlonel

ln McKenzie (1961),

ln the duallty

If

extenda an aapect of the

wlth

aleo glves baeie for

It

factor.

nust be unLque.

aapect dependlng on duallty

already

but

le unlnvolved

lt

(1)

But in

the maxlmum of growth

factors.

von Neumann cheory whlch

Accordtngly,

klnd

of

not dependlng on

framework wtrich here provldea io - l.

TooologlcaL oethod The folloving

lntereectlon

(1938) and used to obtaln

for

hls

economlc nodel.

V are compact convex and E, F S c U x V are cloeed and

I!U,

the seta xE, Fy are non-emDty convex for

xEy, xFy for Corollary

results

(von Neunann)

Theoren 15.1

such that

theorem rraa proved by von Netrnan

all

x, I !!gg

eoqe x, y.

(Kakutanl).

and euch that

If

U is compact convex and E c U x U ie cloaed

xE is non-enpty

convex for

a1l. x then xEr &_.g

x.

For take V - U and xFy = 1 - t. Kakutanl

(19+1),

after

giving

a sLmpler proof

of the corol,lary,

52

deduced the theorem frorn lt.

Ihus,

given such E, F let

W- U x V

anddeflnel6l.lxWby (x, y)T(x', Then T le closed

= r'FY ^ xEYt .

I')

and (x, y)T - Fy x xE (x, y)T(x,

ie convex and hence, by the corollary, eome xr Y.

xPy, xEy for

The proofa polnt

y) and equlvalently

theoren

of von Neumann and Kakutanl

depend on the

both

fLxed

of Brouwer (19)9).

Aa an appllcation

of von Neunsnn's

theoreD Kakutanl

deduced

the foLlowing. Theoren 15.2. If valued

U, V are compact convex and f(x, on U x V euch that

functlon

le a continuous

ly :f (x' Y) I tl

then

max min f(x,

-

y)

nln nax f(x, yeV xeU

xeU yeV Cloaed E, F c U x V auch that

xFY = f(x,

by von Neunannrg theoremr rnaxf (xr, mln ytEV IteU

But in

any caee

Y) '

nax f(xt, XreU

xEy, xFy for

y')

Y)

xE' Fy are convex are deflned

xEy = f (x, y) - roln f(x, yt ev

Ilence,

real

the setg

lx : f(x, y) : tl, are conver

y)

(x, y) -: -1 f

by

Yt)

Y) .

eome x, gax x'eU

y.

Then

mln f (xr, YteV

Yt).

53

mln max f (x, yeV xeU

y) > max ml.n f (x, xeU yeV

y)

.

For let g(x) = min f (x, y), yev

- rnax f (x, y) xeU

h(y)

,

80 that e(x)If(x,y)jt'(v) and hence < urln h(y) yev

nax g(x) xeU Thus the theorem ls proved. functlon

It

ls valld

le in the cloeure of the real nunbere.

Another statenrent of the concluslon a eaddle polnt,

that

iB there exlet f (x',

for all

x',

the functlon

x, y such that

f (x, y) : f (x, yr),

can have at most one eaddle-value.

be two saddle polnts,

For let

that is

g(x, yo) l8(xo,

t;,

g(x, yr) : g(xt, y) , for

all

x, y.

Then in particular,

g(xo, yo) 5 S(xr, yr) j and elmilarly

has

y) 1s a saddle-value.

A functlon yi)

y) I

le that

y'. .

Then u = g(x,

(xr,

also when the range of the

wtth 0, I

g(xr, yo) I

interchangerl, so rhat

gix,r,yc) = g(xt, yt)

.

a(xo, y0),

(x0, yO),

54

order

Thua, wlth

1r, Qn.

of

by row vectors

end sun to 1: end let

ere non-negatlve

m where elements

be sinllarly

in 0r, described

(n-l)-eftop1ex

Let U be the

J e Qo, I e 0n havlng

elemente

V all

l, U = [ p : p e 0 r r , p I - l ] ' v - t t : t e f , l m , J t - 1 ] Any p e 0o where p > 0 hae a normaltzatlon where the ray through p cuts u.

the polnt t(Jt)

-l-

(pr)-l

P e U, whlch ls

slnllar1y

any t > 0 glves

The sets U, V are compact convex.

e V.

Conslder - pbt/Pat

g(p, t) It

deflned

ls well

not both pat ! 0 pbt = 0, and thls

provlded

asaured by the condltlon

(P > 0, t > 0)

(v).

It

le hoaogeneoua of degree zero and

thue he6 the Bam€ valuea when p, t are replaced ln U, V. lt

True,

lt

ls apeclfled

ls

by Juet ite

by their

normallzatlon

valuee on U, V'

Aleo

18 contlnuoua. For00 It

followe

that,

for

all

u and t ' the set

[p: ls

convex.

Aleo,

lf

and only

lf

for

g(P,t) > u]

glven p and t,

bt > atu

thls

conditlon

holde

for

all

p

55

Stnilarly

the setg : t(p,

lt are convex. all

t lf

Also,

and only

for

t) :

ul

given y, p the condltton

g(p,

t)

:

u holds

for

lf pb . Upa

Ttle condltlons that

f(p,

thls

concluelon

t)

of theorem 15.2 are satlafled,

has a saddle polnt ls equlvalent

(p, t),

with

the concluglon

and eaddle value u.

to the exletence

A1eo,

of p, t and p where

p e U r t e V a n d 0 < ! . such thet b t > B t u , Thls laet le that

concluelon

p b < u p a

correeponda to that

the condltl-ons

(c),

the denlal'of

the denlal

of

(c)

and lmedlately

here becomee ldentlcal The result

But it

le equivalent

(B) le equlvalent

theee trro apeclal

nith

that

of the foregolng

to a saddle velue

ro a saddle velue verlflable

lnpllee (1)

le well

deflned. ( 1 1 )

t ) 0

p b < u p a ,

p : 0

and

u - O.

Putting the content

in theoren 14.7. dlecuaalon

1.8 stated

(p > 0, t > 0)

and has a saddle polnt. b t > a t u ,

here,

u r o, Eod

caseg aslde,

the functton g(p, t):pbt/pat

A dlfference

can be eeen dlrectly

Theoren 15.3 (y)

14.7.

(B) are not in the hypotheale

0 < u < - le not In the concluslon. thet

ln theoren

Also

aa fo11orr8.

t6

are the condirlons

a 6add1e-point

for

(p'

t)'

u'

4.saM

von Neunsnnpolnted out that (.r) assuresthe functlon glven bv(r)lsrrelldeflned,gndalsothatcondittons(11)arethecondltlona forasaddle.polntandseddle-value.Itcanbenotedthatthe ldentlflcatlon

(il)

of u in

a saddle value lmredlately

with

the uniqueness of such ! '

establlahee

Theorem 15.2, which Kakutani obcalned as a consequence of

to establlsh

applled

thle

Instead

route.

dlrectly,

of a saddle-polnt

the existence

he proposed to uae his (for

oethod

by the followlng

of Ehe functlon, follow

But von Neumann dld not

(tl).

to

end hence of a solution

has here been

theorem 15'2,

theorem,

von Neunannrs intersectl-on

lntersectlon

theorem

whlch a rnodlflcation

is

necessery) . ConelderE,FcUxVwhere pEt = bt

' Pbt - Patl

> at^

Pbt '

pFt = Pb 5 vpa, o, u belng

vPat

'

Thue ln pEt'

deternlned.

finpllcttly

'

wlth

a(i

ros I of a, flret o{, and then,

t 3 t(l

t^

for all

i

becauee P > 0, neceasarllY - a{ttr

for

r - nh[b11

t/a(lr

blrt

sone 1'

Thla glvee : .1t

r r 0]

,

dependlng'"' r";:'= 1", ;.;'"" =

[ p : b (r 1, t

> 3 / - t (1

I o

- 0 ] , p - r.

denotlng

5'.7

that

16, Et ls

the face of the slnplex

such that br, t t a/{ tl. (1 (1 for

some 1 lt

Thus Et is convex, and slnce b,. (i

ls non-empty (There ls difficulty

ln whlch case tr = -). slmilar

Grantrng al.so that

discussl.on holds

conditlons

U on which pt = 0 for

bre present

conclusLon that

(lt

for

does not)

an appllcatlon

E n F * 0r that

since pbt - patl

l,

ls applicable

Thus pat > 0.

Then fron

only lf

and that

.I .

a

of theorem 15.1, wlth

v it

ls

so rhen also pbt - 0, glvlng

(Thls argunent

t- - -a(, r- t-

F and the seta pF, the

ls pEt, pFt for

For such p, t and the correspondlng

1

here when at = 0r

E is crosed,

for

all

I < -r

the

some p, t. lmpossible

a contradlctlon

or sirnllarly

pat - o,

that

of

(V)

v < -).

patlBpbr=vpat lt

l - v.

follor^rs that

Such p, t therefore

glve the requlred

concluelon. Modlflcatlons where there

can uake good the defecte

le vlolatlon

(c) or

of

(B),

that

in thla

and then renolnrng

are satisfled,

Wlth (o) (a1so requlred

deflned,

convex and non-empty.

closed.

For,

pat > 0.

In thls

for

r ( -,

Also lt

reoarked,

wlth

case PEt oPb, Pat

where

PatP' Pbto

HereElsaabeforebuttherelsanodlflcatlonlnF.Nor(c)and

Then theorern 15.1 can be applled

sete pF.

9 ,

o

pFt for

pEt, < -

E and the sete Et'

(B) and ('r) give the same condltlona

and, ldentlcally,

that

for

condltlons

give the requlred

(r)

some P' t.

pbt - PatP, Pat ' thls

oPbt

lnPlles pat > 0, Pbt > 0.

Also pbt - PatP vhlch,

vlth

Pbt > 0, lnPll'es p o - 1 .

Thue, with

u where

for

F and the

to glve the conclualon

For such p, t and the correepondlng

t

Wlth (v),

aa before;

'

oPbtP ,

59

-I P

p, t give the requlred the addltlonal 0 < !r < -,

16.

!

-

o

,

conclueion.

hypotheeis

(o),

Thls provee theoren 15.3, wlth

(B) and the addlrional

conclualon

as requlred.

Algebraical

method

Conslder a palr wlth

t

elenents

of matrlces a, b e en, rhat

in the non-negative

number 0; and condltlons

( c ) t > 0 - a t > O ( B ) p > 0 - p b > O (v) t > 0, p > 0-

par*pbr

> 0,

where t e 0m, p e 0r, i equlvalently p > 0 -

(c)'

pa > O

( g ) " t > o - b t > o (V)' a*b

> 0

Let f - e u p l t : b r > a t l , r > O ] , J - l n f l v : p b < v p a , p > O l , and i = s u p l l : b t > a t l , t r 0 ] , v - i n f [ v : p b I v p a , p > 0 ] , eo in any caee ^ . lr Obvlouely (B)'

+

i

' 0

(o)t

:

i

. -

u :

v

is of order D x tr,

Thus ( c r ) -

v < v < -

( B ) - s . i 5 I Also,

I and P' v

anY t'

for

b t > 4 t t r , t t 0 pb-. Pa, P>0 implles 0 < patl Thus pat = 0 inplles (y)

irnplies

< vPst' :; Pbt

also Pbt - O' glvlng

Tttle thoets thet

u'

pat > 0, end then f:

a contradictlon

(.r)-^:i, provlded

i

. -.

But thle

The follotring

18 true

ln any caee lf

hae been Proved'

Theorem 16.1 For anv a, b

i S ^, u : i , ( o ) - - v t -

(B)==-i'0, and ( v ) o I S r , so ( o ) , ( B ) , ( Y )-

o ' i :

Nolr the followlng

wlll

Theoren 16.2 For any a, b

^ :

t :

be shosn

i ' -

'

; - -'

of

(Y)'

eo

6l

i : I ,

i : ,

( a ) - r - 1 , ( g ) : i = u It

glves an algebralcal

result

proof

ln theorem 14.6.

Lenuna.

For_-ry.

for

polytope

The followlng

correspondences of the

1s requlred

a, b and Lr bt > atu for some t > 0 or pb . upa for

(1) elther

some p 2 0

and not both. (11) either

bt > atu for sone t > 0

or pb < lrpa for

some

p > 0 and not both. Thls

follows

from the theorem that,

ct > 0 forgomet >0orpc for

(1) take c - b - lar Flrrt

lt

(c) lnpllee,].

for

u for sone lr.

Thus

replace c by - c'.

(il)

w111 be shonn the (o) lrpllee "o,].

c, elther

0andnotboth.

and then for

-,

any matrlx

i:

r.

By rheorern 16.1,

By deflnltlon

of i,

for

any ltuch U. p b : 1 , ' p a ' p t 0 , for

eome p and ut where

u , u ' 3 , l. But (o) lnplles follorre

pa t 0, slnce p > 0.

Wlt.h thls,

and y > ut lt

that p b < y p a r p > 0

But thl6,

by the Lema,

lnplles

that

b t > e t u r t ) 0 le lmpoeelble tt

for

ls Bhoun that,

all

t.

for all

By deflnltlon u, u t i -

of f thls ,:

i,

lnplJ,es tL : I.

end Ehis

Thus

.

lmp1les v:

:

A.

will

trow lt

be shown that

l-n anv case v i

there ls nothlng more to Prove' So suppose f let

u > f.

Then, from the deflnltlon

of I,

< -' for

all

I

lf

and t lt

ls

inposslble

that b t r a t u r t 2 0 . Hence, by the Lemta, pb < ;rpa, p > 0 for

some P.

But Ehen J.nnedlatelY p b < u p a r p > 0

for

by the definition

some p, and this

of i

lrnplles u l;'

Thus

p t f > r l ; , and hence ^ : t. Theproofoftherestofthetheoremcorresponds,wlthobvlous modlficatlon. Corollary.

For any a, b

i : ^ vrr V",g9j]g9i:I, V


0.

has a solutlon

But, nlth

of pb < ppa for some p .6, bt > atp has no solutlon have such a solutlon, Thus the denlal Corollary. t,

of (1)

of (1),

by the Lema,

6 t 0, thls

then le also a solutlon

and, agaln by the Lerma, this

t > 0.

But wtth p.;,

by deflnltlon

of f,, so there ls a contradicElon.

For any a, b the conditlon

(1) lnplles

p and p such that

and also that

b t > a t u ,

t z 0 ,

p b : ! p a ,

p z 0 ,

such u is unlque,

fuoplles

Ehls [lrsE

The proof of (il)

is false.

pa < 6pb

lnplies

and (c)

lnplles

1s slnilar. that

there

exlst

64

;r > 0 .

u < - and (B) lmplies

wlth

the

y where

and taklng

to theorem 16.2,

corollary

by conbinatlon

follows

this

the uniqueness,

But for

I = u = i , slnce f = i, ls

Thus, wlth

-l',

i

: v p , o a] p r b , defttted

deflne

2, ...

btN > at1l,

I * - s u p [ l : b t 1 , =^ t 2 , . . . ,

= inf[v

of von Neumann'

and compound factors

For any a, b e Qn and N = l,

i*

and

14.

shown at the end of Sectlon

Gap-cycles

such v is unique,

by the argumelt

ls best establlshed

thus glven,

already

17.

That (V) lnpUes

sub1ect to (f).

tl

] 0] ,

pN-la I PNb, pH > 0l

...,

the case N = l

as before,

glves

I, - I , i, = ; Stnce bt > atl

glves also btr > atr2,...,

lt

follows

bttrN-l r at trN,

that

t'

N r =T '

N i

and siurilarly - - N u N I u Since, from theoren 15.2, corollary, Theorem 17.1 For anv a, b and N, N u N < v

: N < A

i l n N

i:

I

' the follorrlng

ls obtalned'

65

(Vn) wlll

A condltlon ldentlcal

with

be shonn whlch ln case N - I becomes (y).

the von Neumann condltlon

15.2, corollary,

lnplles

Neunann theory,

(v*),

and (B) lnplles

I,

f = i,

together

= in,

this wlth

Wtrlle (V), by theoreur

belng a maLn reeult the regularlty

of the von (c)

condltlons

and hence, because of theorem 17.1, alao

I - J , and at the saxoe ttne : : N ^N=^ ' Thls last

reeult,

productlon

growth,

le re-establlshed

theory

theorero" - that

type of "turnplke

for nore rapld

perlod

- N uN=u

when put in a growth and lnteresc

expreas.ea a modlfled no short-cuts

-

or greater ae a block

proflt,

context,

there

are

wtren the

of N origlnal

perlods. Ttre von Neunann condltlon (v)

for

sir

> 0 or Ora , 0,

r every good la elther

For an ordered palr 8ap If

lt

ls a + b , 0, ls restated

to

every good I and baelc actlvlty

actlvlty

that

> 0 a. +b. 1r 1r

and le equlvalent (y)'

(y),

ls nelther

second, that

r.

ln every baelc

an Lnput or all output.

of activltlee

an output

That ls,

for

(r,

the flrst

e),

a good 1 deflnea

nor an input

for

a

the

ls

orr'o'"i"-o Then a gap-cvcle

ls deflned

by a cycllc

a gap between every succeselve pair.

aequence of actlvltlee

wlth

66

It

a gap-cycle

provldes

goods i,

J,

...,

(q, r),

that

ls

the succesgor of q is

that

k whlch aPPear as gaPs between (r,

b1.-o'

O n O- 0 ,

that

...,

rr 8'

for aII le,

remalng unaltered

(H)

( a ) ls

aleo for

The part

The condltlon Ls self-dual'

k'

between goods and activltles

condltlon

by whlch lt

roles' (G) ls

for

condition

the regularity

of the conditlon

condltlons

(G) whlch

or of goods, where there

of actlvttles'

interaedlate

..',

a > 0 a n d b > 0

eufftclent

ldentlcal

to the condltlon

a > 0 or b t 0, stronger

and the etlll

0

when theee exchange their

An obvioue eufflclent

""

* an. > o

...

q and 1, J,

hae I syurctry

lt

t)t

"t.'

la equivalent

* oJ" *

(s'

J J

J O

(G) brr *.1"

s)'

"1"=o 3.. - 0

b.^ = 0,

The abaence of gap-cycles

f agein'

sequence of

cycllc

ls a further

there

if

sequence of

q represent a cyclic

understood

belrg

lt

actlvl-tl-esr

...r

..tr

s'

rr

Then let

wlth

the von Neunann condltlon

betlteen

(c)

applles

le a slngle (Y),

and (8).

to cycllc

sequences

element ls

eo (G) te a condltlon

(H) end (Y) '

(H)e(c)-(y) Let

('y*) be the part

of the condltton

(G) whlch appllee

to

67

cyc1lc

sequences of N elenento,

ao (G) ls

the conJunctlon

of

(y*)

for al1 N,

(c) Assuming (a) and (8), ( 1 ) for

some l,

v.

(11) for some tI

I* I i* .

suppos" i,

. I*

Then

O I i H < v < t r . I n I Now tr < f*

from (1) tnplles

bt1 > at2, btz I ar3, ..., btN I ati l,

, ...,

c* e Qm. Also i,

< v impliea

t1 2 0,

68

vPna I Pl b, Pla i

(lil)

tt

> 0-

btN > 0D

at1 tr> 0+

att > 0'+

and I > 0 frorn (i)'

(o)'

Fron (11)t wlth

P,, e Qo'

...r

for somePl,

P N - 1 8I P w b ' P N > 0 '

Pzb, ""

tN = 0

and then aN = 0 -

> O+

.tN

baN-l

= 0 +

> 0

t"-,

and generallY .1_1z 0. (1 '

ai - 0: It

N)

2, ...'

that

follows

Sln11arly

tt,

fron

(111), wtth

(v) Alao fron

pI,

an = o

...,

(1v)

...r

(B) '

P o l> 0

(11) and (111) ' (vl)

vpNatl I prbtt

I Prat2 I Pzbt2

:

...

. . . i n * u t * : p w " t rr : o , ao (v11)

vPnatl

i

But wlth

v < I and P*at1 > 0,

But thl8

lupllea

every

(vltr)

plbtl

But from (1v), tl, r,

6, ...,

t.

- Pratz

Also fron

br, gtvLng a contradictlon

(vlt)

term ln

"1" of

ls

zero.

only lf

Pnatl'0'

Hence P*btn - Pnatt

- P2bt2

(v) pr t P2t ..., k.

...'

(vl)

ls poaslble

tN each have eomeelement Posltlve'

t2,...,

poaltLve, say 1-, J,

1

Itatl

r 0' say

PN have eomeeleuente

fhen (v111) lnplles

- o J " - . . . - b k a- . a r - o , (1*).

Ihus,wlth

the aesunptlon

of

(c)

and (B)'

59

(rn) lnnllee the orlglnal

suppoeltlon v-n . i* 1" inpoesl-ble, eo

the theorem le proved. Glven (c) a n d ( B ) ,

Corollary.

(Y ) tnpllee N -

-l'I u M - '

for

H : tl, eo (G) inplles

all

The argurent

for

thle

-M

'

thls

''M

for

all

M, and (f)

lnplles

ls glven ln the dlscueslon

before

; = I

.

the

theorem.

18.

Conpound growth

and lnteregt

Coneider an econouy ln eucceesive perloda the actlvlty

ln each perlod

of sone n goods, and lt (1)

lnput-output

(2\

the output

1s descrlbed

of productlon,

by an lnput

where

and an output

is aeguned that poselbllltles

of one perlods

are the same ln all ls

the resource

for

perlode the lnput

of

the next (3)

there 18 free dleposal of both ourpur and lnput.

BV (1), poseLbllltles (x, y),

some relatlon

R c 0n t 0n gives the lnput-output

acrosa any perlod,

6(', y')

and by (2) for

to be succeeslve lt

ls requtred that y i

the condltlon

for any (x1, yl),

of actlvLtles

through N auccesslve perlods

(4)

xrRyr i

The resultant

x2Ry2:

...\RyN

...,

(5,

*t.

Thue

yn) to be a poselble chain ia

xrRyr I xzRyz i

of such a chain 1e the input-output

acroas a apan of N perlods.

lnput-outputg

These reeultants

(xt,

...

: **nr"

yN) lt

of poesible

determines

chalns of

70

length N describe the lnput-output It

derlves

as the Nth.orpound

perlod,

deflned

rdcursively

perlods. unlt

relatlon

Possiblllty

\

across N R across a

of the relstlon

by Rt = R wlth N-1

(5)

*\y

= (vx', y')xR^'^ Y'1x'RY (3) le

The assumptlon of free-dlsposal x'

(6) Wlth this,

: 1 xRY : Y'

(5) ls equlvalent

(7)

to = (vz)xn*-t

*\v

whlch establlshes

simply

\

for

property

the slnple

for

Free-disposal

x'

Also lf

'r xRNy I yt

that

follows

lf

R is

Maxtnuro Sroltth by repetltion alngle

-

N

d*:6",

planned

for

can be no better,

plan coverlng

6 deter:nlned fron -

regular

the entlre

*tRNyt

the same

.

or convex, so 1s RN. lmply the same for

l-e homogeneous and has bounded output rt

lnplles

RN ,

R does not generally

for

R obvlouely

relatlon

R le homogeneousr addltlve'

continulty

'N-IRY

z*-t)xRztR "'

the compound relation

(e)

znY '

as the Nan Potrer RN of R, glven by

xRNy= (v21,...,

(8)

x'RY' .

then lt

crasslcal

perlod.

R, and correspondlngly

ls not

RN, but

if

R

does by theoren 5.1.

homogeneous then so ls RN.

one perlod and night

It

and extended

over N

conpare unfavourably Thus wlth [N

with

a

the growth factor

from RN, ln any case

p r o v l d e d R l s h o r o o g e n e o u s ,a n d t h e r e c a n b e e n q u i r y a b o u t

the poesibllfty

p" , !N.

taken as the accounttng

Wlth a succession period,

ae well

of N baslc

perlods

as the production

being

perlodr

the

7l

interest fron

i*

factor

derives

the dual R* of R,

fron

the d,ral RN* of RN, Just as i

Wlth R regular

rh dual of Ehe N-" power ls

classical

derives

homogeneous, the

rh the N-" power of rhe dual,

RN* = R*N , by'theoren

11.3.

grohrth factor lnterest

Ttrus wlth

S = R* as the dual of R, wlth

o-, SN 1" ldentical

factor

ir,

wlth

the dual of RN.

being the reclprocal

N*

R.' , becones ldentlcal

naximum

The N-perlod

of the growth factor

\.rlth the reclprocal

of

of the growth factor

o"

N

of SA.

But, from hourogenelty, 6u i

6N , and hence, fron

the reclprocals,

- - N v N : v Theoren 18. I If a unlt

R is a regular

perlod

wlth

are the grovth

classlcal

honogeneous productlon

growth and intereet

and lnterest

p-, i

factore

over N perlods

factors

relatlon

and if

6",

for

i,

then

i * 5 i N : 0 " : o * . Thls ls

shown ln che foregoing

theorem 13.3, corollary Thls wlll

dlscueelon,

together

(1), which ahowethet;:

now be applled

to a productlon

Ne,mann fotm R = AB where A, B are deternined

wlth

o . the von

wlth

relatlon

from matrlces

a, b b,

x A t = x : a t , t B x : b t > x , and for whlch the conditions and the last followlng

sectlon,

applles

(o),

(B) whlch appear ln Sectlon

are the regularlty

ln any relatlon

For anv relatlon

conditione.

Flrat,

A, B and ln particular

R deflne

u(R)-Bup[u:xRxp,xz0]

8, the

to these.

12

Thue, wlth

R a productiotr

f.

relatlon,

and S = R*,

p ( R ) 'd = u ( S ) ,i = d - l

Theoren 18.2 If

A le hooogeneous and B regular

then

t [u : xA3xp' x ) 0, U > 0] For lf

[u : tBAtu, t > 0'u > 0]

B ls regular, t B x , x 2 Q : t ) 0 ,

wtth u > 0, and by horoogenelty of A,

so that,

xAtBxu, x ) 0 :

tBxuAtu,

t 2 0,

and hence (vt)

(vx)xABxu, x 2 0'+

tBAtu, t 2 0,

aa requlred. (1).

Corollary

A ls honogeneous and B regular

If

then

u(AB) : u(BA) (il).

Corollary

A, B are honogeneous and B regular

If

u((AB)N) i

then

u((na)N)

For (AB)N - CB,

(nl;N = ss

where c - l(sl)N-l

s

A, B are hooogeneous then so la C.

Also lf Corollary

(111).

E

n, I are homogeneoue and regular

then

u((AB)N) - u((BA)N). Now wtth

the von Neunann relatlon

R - AB detetmlned

from matrlces

I-t

a, b the dual is S = A*B* lthere pSq = pa > qb, and chls shows thac the lnterest treated (c),

factor

ln the prevlous section.

i*

colncldes

wlth

the in

A1so, subJect to the conditlons

(B) so that A, B are regular. -

N

N

DN=u(R^')= u((BA)") But BA is

for whlch

the relatlon

sBAt=sb>ta, and thls i-N

shorvsthat,

treated

subJect to (q) and (g),6*

ln rhe laet

aectlon.

colncldes rrlth the

Theorem 17.2 and lts

corollary

thus glve the following. Theoren 18.3. For the von Neumann productlon matrlces a

and b sublect

the absense of gap-cvcles

to the regularltv lmp1les that

/v perlods is Just the Nth power of that and slurilarly

wlth

R deternlned

relation

the rninlsrun lnterest

condltlons

(o) and (B),

the naxlnum grosth factor factor.

from

factor

acroae a elngle

acroas perlod,

74

BIBLIOGMPHY

Afrlat,

s.N.

Reclproclty

Cremonese, L967. L-64.

Edlzlonl

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