(lecture notes in economics and mathematical systems 116) dr. kenichi miyazawa (auth.)-input-output...

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2/146

Lecture

Notes

in Economics and

Mathematical

Systems

Managing Editors: M. Beckmann and

H. P.

Kunzi

Mathematical Economics

116

K. Miyazawa

Input-Output Analysis and the

Structure of Income Distribution

Springer

Verlag

Berlin· Heidelberg·

New York 1976


3/146

Editorial Board

H. Albach'

A.

V.

Balakrishnan' M. Beckmann (Managing Editor)

P.

Dhrymes . J. Green . W. Hildenbrand . W. Krelle

H. P.

KUnzi (Managing Editor) . K. Ritter' R. Sato .

H.

Schelbert

P. Schonfeld

Managing Editors

Prof. Dr. M. Beckmann

Brown University

Providence, RI 02912/USA

Author

Dr . Kenichi Miyazawa

Hitotsubashi University

Kunitachi,

Tokyo, 1861Japan

Library

or

Congress Cataloging n Publication Data

Miyazawa, Ken' ichi , 1925-

Prof. Dr. H.

P.

KUnzi

Universitat ZUrich

8090 ZOrich/Schweiz

Input-output analysis and

the st ructure of

income

dis tr ibut ion.

(Mathematical economics) (Lecture

notes

in economics

and mathematical systems ; 116)

Bibliography: p.

Includes

index.

1.

Inter indust ry economics.

2.

Income

d is t r ibu t ion - ·

Mathematical

models. 3 . Japan--Economic

condi

i ons -

Mathematical models. I .

Ti t l e .

I I . Ser ies. I I I .

Se

r ies : Lecture

notes

in

economics

and mathematical

systems

;

116.

HB142.M59 339.2 76-000006

AMS Subject Classifications (1970): 90AlO, 90A15,

90A99

ISBN 978-3-540-07613-1 ISBN 978-3-642-48146-8 (eBook)

001 10.1007/978-3-642-48146-8

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or

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© by Springer-Verlag

Berlin'

Heidelberg 1976


4/146

FOREWORD

The purpose of this study is in keeping with the

shift

in concern over the eco

nomic problems

of

growth to

those of

income

distribution in recent years. Income

distribution problems may be

analyzed

by

not only the

traditional

procedures, but

also

by

some

extensions of the input-output technique

as

I shall demonstrate in

this

volume of the Lecture Notes. Some

fruitful

results are obtained by applying the

extended input-output technique to

income

analysis

as

well

as

to output analysis.

This

volume

consists

of three

parts.

These parts

may

be

viewed

along

two

veins,

with

some overlapping unavoidable: (1) Parts One and Two contain extensions of the

input-output analysis

and

(2) Parts One

and

Three contain studies of the

effects

of

the structure of

income distribution

on some other

economic relationships.

First, as

an

extension of the input-output analysis, we present a synthesis of

the Leontief

interindustry

matrix

multiplier and

the

Keynesian

income

multiplier

in

disaggregated form, and introduce a

new

concept

which may be called

the Interrela

tional Income Multiplier"

as

a matrix. It

is

designed

to

analyze the

interrelation

ships

among

various income-groups in the process of

income

formation through the

medium of industrial

production

activity.

Although

this

multi-sector

multiplier

follows

from Leontief's interindustry

matrix

multiplier, i t is

formulated

by

the

inclusion of the

income

generation process,

which

is omitted in the usual input

output

open

model,

and by

projecting the

multiplier

process

into

not only the output

determination

side,

but also

into

the income-determination side.

Secondly,

we

shall proceed

to

formulate a

method

of

partitioning off

the

origi

nal Leontief inverse in terms of the combined

effects

of "Internal"

and

"External"

matrix

multipliers

and their

induced

sub-multipliers. Because

the usual Leontief

inverse provides

us

with

knowledge

of only the ultimate

total

effects of interindus

try propagation and not the disjoined effects separable

into

the partial multipliers,

as

such, our

method may

well

be

applied to the various kinds of problems

that

re

quire

us

to trace

back

to

the

interactions

among

two

or

more

strategic

industry

groups.

Finally,

some

empirical applications of these

two models

are introduced, deal-


5/146

IV

ing with several cases from the Japanese economy and with

an

international comparison

of the interdependence between service and goods-producing sectors. The empirical

illustrations

also include the applications of

an

interregional version of the input

output

model

in the extended forms.

The other theme of

this

volume deals with the structure of income

distribution.

In

this context,

we

employ

two methods

of

entirely different

nature.

The first is an application of the above mentioned

interrelational

income multi

plier model, by which we clarify the effects of income-distribution-factors

on

the

income determination process.

In

the standard income analysis or in the standard

input-output open model, the

same

amount of autonomous expenditures cannot have vary

ing effects on the level of national

income

even

if

the expenditures consist of dif

ferent

commodity proportions.

The same criticism

holds for the Kalecki-Kaldor type

of models -incorporating income-distribution-factors as

far

as there are

no

changes in

the relative

income

shares.

But

in the real world this situation is not so.

It

will

be shown

that

in order to have the value of income vary in conjunction with the com

modity proportions of

demand,

i t

is

not

sufficient

to introduce the

structure

of

income

distribution by types of income-group alone, but we must introduce

at

the

same

time the

distribution-factors by

the types of

industrial

value-added for the

production structure also.

The second study is a differentials-analysis

especially

of

wages

and interests

as

rewards to the factors of production.

In contrast

to the above approach which

focuses upon the interindustry intermediate inputs

as

factors of production, we

concentrate our attention

directly

on income

distribution

among

the primary inputs

by the

size

of firms. The analysis is an integral part of the last Chapter's

in

vestigation of Japan's

dualistic

structure. The dualistic character of the Japanese

economy, reflected mainly in production techniques and financial arrangements, are

considered in relation to

distribution

and to economic growth.

While numerous individuals

have made

important suggestions

and criticisms,

I am

especially in debt to K. Ara, A.

S.

Bhalla. W.

H.

Branson,

S.

Masegi,

K.

Ohkawa.

M.

Shinohara.

Y.

Shionoya, and

T.

Watanabe. Special

gratitude

is extended to Ryuzo

Sato who read through the original manuscript with constructive criticisms

and

who


6/146

v

recommended this volume

for publication in

this

Series.

The

author

gratefully

acknowledges Gilbert

Suzawa

for correcting and improving the English content of the

original manuscript. Lastly, the author wishes to thank the various Journals,

as

noted in the footnotes to each chapter,

for

permission to reproduce the original

articles in various revised form.

Tokyo,

August 1975

Kenichi

Miyazawa


7/146

FOREWORD

INPUT-OUTPUT

ANALYSIS

AND THE STRUCTURE OF INCOME DISTRIBUTION

BY KENICHI MIYAZAWA

CONTENTS

PART ONE: INPUT-OUTPUT

AND

INCOME FORMATION

CHAPTER

1

INTERINDUSTRY ANALYSIS

AND

THE

STRUCTURE

OF INCOME

DISTRIBUTION

. . . . . .

.

I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

II. Interindustry Analysis and the Process of Distribution and Expenditure

of National Income.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1)

The

Leontief Multiplier,

Keynesian

Multiplier

and

Kalecki Multiplier

2) Gener&lization of the Input-Output Model

3) The

Coefficients of Inter-Income-Groups

III.

The

Relationship of Inter-Income-Groups

and

the Multi-Sector

Income

Multipl ier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1) The

Income

Multiplier

as

a Matrix

2) Accepted Multipliers

as

Special Cases

3)

Structure of the Propagation Process

IV. The Convergence Conditions of the Model

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14

1)

The

Properties of Leontief-type Matrices

2)

Convergence Conditions in the

Model

CHAPTER

2

INPUT-OUTPUT

ANALYSIS AND INTERRELATIONAL INCOME MULTIPLIER

AS A MATRIX

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

22

I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

II. Formulation of a Regional

Income

Multiplier in the

form

of a Matrix .. 23

III.

The

Interrelational

Income

Multiplier

among

Regions

. . . . . . . . . . . . . . . . . . 26

IV.

Composition

of Final Demand and the Regional Income-Distribution

. . . . .

29

V. Output Determination and Interregional Income Generation . . . . . . . . . . . . . 36


8/146

VIII

CHAPTER 3 FOREIGN

TRADE MULTIPLIER, INPUT-OUTPUT ANALYSIS

AND

THE

CONSUMPTION

FUNCTION

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

I. Introduction

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

43

II. The Foreign

Trade

Multiplier

and

the Circular Flow of Intermediate

Products

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

III. The Modified Multiplier and the Fundamental Equation for an Open

Economy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

IV. Interindustry Analysis and the Consumption Function . . . . . . . . . . . . . . . . . .

48

V.

Empirical Estimates

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

51

VI.

Formula

for the

Computation

of the Subjoined Inverse showing the Effect

of Endogenous Changes in Consumption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

1)

Derivation of the

Computation Formula

2) Propagation P ~ c e s s Combining Leontief's Multiplier and the

Keynesian Multiplier

PART TWO:

INTERNAL

AND

EXTERNAL

MULTIPLIERS

CHAPTER 4 INTERNAL AND EXTERNAL MATRIX MULTIPLIERS IN THE INPUT-OUTPUT MODEL . . . . 59

I. Introduction

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

59

II. Partitioned Matrix Multipliers

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

60

III.

Interregional Repercussion Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

IV. Some Extensions of the Model

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

71

1)

Extension

in the

Number

of Partitioned

Groups

2)

Inclusion of the Income Formation Process

CHAPTER

5 AN ANALYSIS OF

THE

INTERDEPENDENCE

BETWEEN SERVICE

AND GOODS-PRODUCING

SECTORS

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

I. Introduction

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

76

II. Income

and Employment

Analysis of Interdependency of

Two

Sectors

. . . . . 77

III. Input-Output Analysis of the Interdependency of

Two

Sectors . . . . . . . . . . 85

1)

Intersectoral Propagation Pattern


9/146

IX

2) Cost-Push Effects of Service-Prices

IV. International Comparison

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

93

PART

THREE: DUAL

ECONOMIC

STRUCTURE

CHAPTER 6

THE

DUAL STRUCTURE

OF THE JAPANESE

ECONOMY

AND

ITS GROWTH PATTERN

I. Introducti on . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .100

II.

Economic

Growth

and

Differentials

in Capital Intensity

by

Size of

Firm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

1) Schema of Capital Concentration

and Growth

of Enterprises

2) Differentials in Wages and Capital Intensity

3)

Permanence of the

Dual

Structure

III.

Differentials in Composition of

Funds and Interest

Rates . . . . . . . . . . . . .

111

1) Funds of Enterprises and Capital Accessibility

2) Differentials in Interest Rates on Borrowed

Funds

and Cost of

Funds

3)

Differentials in Interest Rates

and Unequal

Distribution of

Loans

IV. Structural Peculiarities of Capital Concentration . . . . . . . . . . . . . . . . . . . . 120

1) Factor Proportion

and

Differentials in Wages and Interest Rates

2) Structural Peculiarities of Capital Concentration in Japan

V. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

INDEX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133


10/146

PART ONE

INPUT-OUTPUT

AND

INCOME

FORMATION

CHAPTER 1

INTERINDUSTRY ANALYSIS

AND THE STRUCTURE OF

INCOME DISTRIBUTION*

I .

r n t n o d u c t i o ~

In

the standard interindustry analysis, consumption demand is treated

as

an

exogenous

variable,

so

that

the usual Leontief matrix multiplier analysis lacks the

multiplier process via the

consumption

function

that

one customarily finds in a

Keynesian

Model. In

order to treat consumption demand as an e ~ g e n o u o variable in

the Leontief system, the household sector

is

routinely transferred to the processing

sectors, and is regarded as an industry whose output is labor and whose inputs are

consumption goods. But the appropriate procedure in dealing with

consumption

is not

to regard i t as a fictitious production activity, but to introduce the Keynesian

consumptiqn_function on a disaggregated level.

To this

end, we

have

formulated a

matrix multiplier

which combines Leontief's propagation process with the

Keynesian

propagation process in the form of the Leontief inverse multiplied by a

~ u b j o ~ ~ e d

~ n v ~ e matnix. The subjoined inverse reflects the

effect

of endogenous

changes

in

consumption demand.

l

)

Nevertheless, this extension of the standard Leontief model

may

not adequately

deal with the

interrelation

between

the interindustry

and

consumption

structures.

The

reason for this

is'

that the

consumption

structure generally

depends

on the

structure of income-distribution. The income-distribution structure regulates the

consumption pattern in that the consumption pattern consists of the expenditure be-

havior of various income-groups.

* This is a revised and integrated version of two articles which are originally

published, under the same title, in M ~ o e c o ~ o ~ c a Vol.15 Fas. 2-3, Agosto

Dicembre

1963

(with collaboration of

Shingo

Masegi),

and

in the theoretical part

of "Input-Output Analysis and Interrelational Income Multiplier as a Matrix,"

H ~ o t h u b ~ h i J O U A ~ a t 06 E c o n o ~ ~ Vol.8, No.2, Feb. 1968.

1) K. Miyazawa [32J, especially Section IV and VI. See Chapter 3 in this

volume.


11/146

2

In

this

chapter,

we

shall

try

to incorporate the process of income distribution

and

expenditure into the input-output system. If we denote the

income

multiplier

manifesting the income-distribution-factors as the "Kalecki

multiplier ,2)

then our

task is one of

combining

the Leontief output multiplier and the Kalecki multiplier

into its disaggregated

and

generalized form.

I I .

In;tvUndu..6tJr.y

Anal.y.6,u, and .the PJc.Oc.e.6.6 06 V,u,:tJc..i..bution and

E x p e ~ e 06 National. Inc.ome

At the outset, in order to delineate the salient aspects our problem, we will

give a

brief

macro-numerical

example

of the model

to be

developed

later.

In the

standard input-output model, final demand

f

(= consumption C + investment I

=

10)

determines the level of output X

via

the input coefficient a = R/X = 3/4 (where R =

,

R

C

30

8

W 6

- - - - - - - - - -

p

4

total

X

40

f total intermediate inputs),

i.e.

,

I

I

I

I

I

I

I

I

I

2

I

\

X

40

1 - 1

X = f =

1 _ 3/4 . 10

= 40.

This is a

macrocosmic

expression of the Leontief output matrix multiplier.

But consumption C is originally induced by the income Y

( = wage W+

profit

p = 10). The consumption coeffi

cient is e =

e/y

= 8/10,

so

that the Keynesian income

multiplier equation is Y = 1 e I = 1 _18/10 . 2 = 10.

Thus by combining

the simple Keynesian income multiplier with the simple Leontief

output multiplier, we obtain the following output solution for an input-output model

with endogenous consumption demand:

3

)

1 f- 1 1 I .

X=r:a

= r : - a ~

( i)

2)

M.

Kalecki [25],

Chap.

5.

3)

This macro-multiplier (or its disaggregated form) is derived more convincingly by

tracing the propagation process from the initial injections. This

method

of

derivation is

utilized

in Chapter 3.


12/146

3

Of

course, the income-multiplier

-1_1---

holds only for a

particular

income-dis

- a

tribution

pattern. Let dl

=

w/y

and

d2

=

p/Y

denote the

relative

shares of

wages

and

profit,

respectively,

and let

a

l

= Gw/W and a

2

= Gp/P

the propensities

to

consume of

laborers and capitalists, respectively, then

we

have

the generalized Kalecki income

multiplier

= 1 _ a l d ~ + a

2

d

2

) , which

incorporates the income-distribution

factors.

4

)

The

input-output solution is then expressed in the form of (ii):

(i

i)

If

we let v = y/x =

1 -

a

denote the value-added

ratio, and

v

l

= w/X, v

2

= p/x

the value-added

ratios

of

wage-income and

profit-income,

respectively,

the output

solution takes the following

form:

(i

i i)

This equation

(iii)

is

the macro-counterpart of the matrix

multiplier

which

we

will

develop next.

2) GenVta..Uza;tion 06

the

Inpu.t-Ou.tpu.t

Model

The

value-added

sector

in the

interindustry model is

not only divided into

n

industry-groups along the column, but is also divided

into

r income-groups along the

row, as

our simple macro-numerical

example

illustrates.

Let

us

express the

income

of the kth group earned

from

the

jth

industry

as Y

k j

(j

= 1, . . .

n ;

k =

1, . . . r ) ;

1

4) If we let

al =

1 ,0

<

a

2

<

1, we have

Kalecki

's own

formulation

1

(1 _ dl)(l _ (

2

) as a special case of this expression. If we let a

l

= 1,

a

2

_-,--'-1-.----,------;-, I -

1/2,

then

we

have

Y =

T -

(ald

l

+ a

2

d

2

) -

1

1

- 1/2

1

(6/10

+

1/2.4/10) • 2

10 or

2 =

10.


13/146

4

this r x n income-formation matrix shows the most generalized pattern of income-dis-

tribution.

Corresponding to

this

income-distribution

pattern,

consumption

demand C

ik

is also defined as consumption for the ith

commodity

by the kth income-group (i = 1,

. . .

,

n

;

k

= 1,

. . . ,

r). The

coefficients

of our

model

are represented in Figure 1,

where:

Fi gure 1

A =

the n x n matrix

of input

coefficients

a . . =

x

. .

x.,

1.-J

1.-J

J

V =

the r x n matrix

of value-added

ratios v

k j

= Y

k

·Ix.,

J J

C =

the

n

x

r

matrix of consumption

coeffi cients

n

a ..

1.-J

x.

=

jth industry's

output,

J

Y

k

= kth income-group's income.

Let

x = a column vector of output,

fa = a

column

vector of consumption

demand,

i

, j

1, 2, . . . , n

k = 1 , 2 ,

. . .

,r

f

= a

column

vector of final demand other than consump-

(n > r)

ti on,

then, the input-output system

can be

conveniently expressed as

x = AX

+

fa

+

f·

(1

. 1 )

In

the standard input-output analysis

where fa'

as well as

f,

is treated as

an

exogenous variable, the following well

known

solution is obtained:

(1 .2)

But if we treat

the consumption

demand fa

as an endogenous set of variables and

re-

gard the household

sector as

a

distinct

decision-making unit instead of

as

a

ficti

tious production unit, the introduction of a disaggregated consumption function

is

necessary.

The

consumption function of our

model

can

be

written

as

follows:

f

=

Cv.x

=

~ k ) v k ) x - v

.x

l..

v -

l.. v k k .

a k=l k=l 1.-

J J

(1 .3)


14/146

5

(k) _ ( (k)

where

°

-

0lk' 02k' . . .

,

°nk)

, is

a column vector and v =

(v

kl

' v

k2

' . . .

,

v

kn

)

is a

row

vector.

If

we

add

nonhomogenous

terms, or

exogenous

elements to the

con

sumption function,

C

becomes the matrix of marginal

coefficients, and

in

this

case we

can include the

nonhomogenous

terms in f.

5

)

Substituting the

consumption

function

(1.3) into

(1.1),

we get

X = A X + C V X + f

Solving (1.4) for X,

we

obtain the following alternative expressions:

X = [I - A - CV]-l f

=B[I - CVBr I f

=B[I + CKVB]f

( i )

(i

i)

(i i i)

where, of course, B =

[I

-

A]-l, i .e.,

Leontief inverse matrix multiplier.

(1 .4)

(1. 5)

The

first

expression (i) in (1.5) gives

us

the ~ g e i n v ~

matnix

multi

p l i ~ showing the total effects of exogenous final demand

on

outputs via interindus-

try and

induced

consumption activities. The

existence of the inverse

[I - A - cv]-l

is generally verified. Expression (i) can be converted into the second expression

(ii),

namely,

the "original Leontief inverse"

B

postmultiplied

by

the inverse

The

conversion

is as

follows:

[

-1

I - A -

cv]

[{I -

CV(I

- A)-l}(I -

A)]-l

(I

-

A)-l[I

_

CVB]-l

=B[I -

CVBr

l

(1 .6)

We can

refer

to the inverse [I -

CVB]-l

as the hubjoined i n v ~ matnix. This in

verse

reflects

the effect of endogenous

changes

in

each

income-group's consumption

expenditure. Matrix mul tiplier equation (1.6) corresponds perfectly to the

macro-

5)

If

we

define

some

o(k),

which

is

the

capitalist

group's

coefficient,

as , our

model

formally contains the prob

lem

of

induced

investment.


15/146

6

multiplier

(iii)

in our previous example.

6

)

The

advantage of matrix multiplier formula (ii) in (1.5) is

that i t

distin-

guishes the inverse reflecting

endogenous consumption

activity from the inverse re

flecting production activity, in contrast to formula

(i)

which

does

not make

such

a

distinction. Moreover, if consumption coefficients and value-added ratios are not

as

stable

as

the input coefficients, i t is desirable to have the "subjoined inverse"

expressed in a form which can be easily computed and revised.

The

development of

such a practical computation formula is also useful

from

the standpoint of under

standing the theoretical aspects of inter-income

group

activity.

We

now

turn to

such

a task.

Let us write

k = l , . . . ,r

v = l ,

. . .

, r .

Then, as

we shall

show,

we can

prove

that:

B[I - CVBr

1

= B[I

+

CXVB].

(1.7)

The

third expression (iii) in (1.5) means that the n x n subjoined inverse

[I - cv.s]-l

can

be obtained, without inversing the matrix, by the means of using the

~ ~ e t t i o n l income m u t t p l ~ X whose order is r x r . A proof of the identity

between (ii)

and

(iii)

is as follows:

with the definition

then

X[I

- VBC] = I

'"

CX[I

- VBC]VB

=

CVB

CXVB[I

- CVB]

=CVB

I - CKVB[I - CVB] = I - CVB

I = [ I +

CXVB][I

- CVB]

: . [ I

-

CVBr

l

= I + CKVB,

where identity matrices I 's in the first and second equations have the order of

6)

If

we set r = 1 in (1.6), i.e.,

if

we

do

not

make

a distinction

among

the income

groups, the equation (1.6) coincides with the formula which

we have

derived else

where (see [32] p. 63 or (3.20) in

Chap.

3), and i t corresponds perfectly to the

macro-multiplier (i) in Section 1).


16/146

7

l

X 1 ' , and those in the third and subsequent equations

have

the order of n x n

respectively. In practical terms, since l in

most

cases is very much smaller than

n,

the

l

x

l

matrix K should be readily obtainable. Consequently, if we already have

the numerical table for B, we

can

renew the subjoined inverse

whenever i t

is neces

sary to do so.7)

3)

The

Coe6McLent6 06 IntVt- Inc.ome-GlWupb

W e may also work out the proof of formula (iii) in (1.5) by the method

which

traces the propagation process initiated by the original injections. This method

may, at the

same

time, reveal the

economic meaning

of matrices Land K.

Denoting

by

m

the numerical stage of the propagation process, we get

(m 1;

2)

Hence,

Thus,

x

=I X = Bf

+

BC( I L

m

-

2

)VBf.

m=l m

m=2

Hence,

if the term

I

L

m

-

2

(i.e.

I

Lm) is convergent,8)

m=2 m=O

X = B[I + C(I - L)-lVB]f

=B[I + CKVB]f.

The result again confirms (1.7).

(1.8)

(1. 9)

(1.10)

(loll)

The

matrix

L =VBC

may

be

interpreted as

an

array of coefficients which show

the interrelationship among income-groups in the process of propagation resulting

from each income-group's

consumption

expenditure pattern. In order to prove this

7) Our

model

can be

easily

extended to accomodate

an open

economy with foreign trade.

8)

The convergence conditions of our

model

will be examined in Section IV.


17/146

8

point,

we

take the vth

income group

as representative and trace its

consumption

ex

penditure

effect

on another kth income group's income.

-> ->

->

increase in output increase in income increase in consumption

of each industry of the vth

group

of the vth

group

->

increase in output of each

industry due to the additional

consumption

of the vth

group

increase in income of the kth

group

due

to additional

income

of the vth income group

Thus the element of

L,

i.e. Zkv' can be written as

(1.12)

That

is,

the coefficient Zkv shows how much income of the kth'income-group is gener-

ated by the expenditure

from

1 unit of additional income of the vth income-group.

Thus we

can

term L the "matrix of inter-income-group coefficients , and K "the

interrelational multiplier of

income

groups".

A proposition

arises

in connection with the matrix of inter-income-group co

efficients: the column sums of the matrix L equal the total consumption coefficients

of each income group, i.e.

i

L = i'VBC = i [ I - A]BC = i 'C

I

I

n n

where and i ; are

row-summation

vectors of order

n

and I respectively.

I I I .

The.

Re1.a.UonolUp

06

In.:teJt-Income.-GJtouP.6

and

the. Mutti-Se.etoJt Income. MuttiplieJt

(1.13)


18/146

9

1) The

Inc.ome MuLUpUeJl.

a6

a. Ma.tJvi.x

We

shall now project equilibrium output into equilibrium income.

As

before,

denoting by Y the

column

vector of r order

whose

elements are household

incomes by

income-groups, we get

Y = VX.

(1.14)

substituting

formula (iii) of (1.5) into this expression (1.14), the income equation

becomes

Y

=

VB[I + CXVB]f

[I

+

VECX]VEf

[I

+ LX]VBf,

in which r + LX = X because [r

-

L]K = I , so we obtain

Y = KVBf.

(1.15)

(1.16)

Justification for the existence of formula (1.16)

may

be attempted by tracing

the propagation process caused

by

the

initial

autonomous injection of f , or final

demand

excluding endogenous

consumption

expenditure.

Using

suffix

m

in parentheses

( ) to denote the numerical stage of propagation,

we

get

VX{l)

= VBf

VX (m) = VECY (m-

1 )

(1.17)

m-1

=

LY{m-1)

=

L Y{l) '

for m 2

This gives the expansion in powers as:

00 2 3

Y

=

m ~ r

(m)

=

Y

(1)

+

LY

(1)

+

L

Y

(1)

+

L

Y

(1)

+

2 3

=

[ r + L + L + L + . . . ]Y (1

) .

(1.18)

Hence, if the

term

L

m

is convergent, we obtain the following fundamental equation of

income

formation:

Y

=

[I

-

L]-l

VEf

=

KVBf·

(1.19)

We may des i gnate the r x

n

matri x

KVB

as the

m u L U - ~ e c . : t o J t

..[nc.ome

muLUpUeJl.

in

matrix form or simply ma.tJvi.x

muLUp£..{eJt

on ..[nc.ome

noJtma.t..[on.

This matrix has the


19/146

10

following composition: the interrelational income multiplier" K post-multiplied by

the coefficient matrix of induced income VB.

9

) Thus, equation (1.19) will give us

the direct and indirect induced incomes of each income-group attributable to the

initial autonomous demand.

10

)

This multi-sector

income

multiplier is a distinguishing feature of our model.

In

the conventional input-output analysis, where

consumption demand

is

entirely

exogenous, the outputs of various industries

have

different values depending

on

the

proportions of final

demand;

but as

far as

the value-added sector is concerned, in

come has the same value as final demand

and

does not depend on the proportions of

final demand. In contrast, as is evident in (1.19) of our model, incomes (both

total

income and group

incomes) have

different values depending on the proportions of final

demand,

and

this is due to the fact

that

our

model

takes explicitly into account the

structure of income

distribution.

2)

Ac.c.epted Mu1:UpUeM M Special.

e M U

This conclusion cannot

be

obtained

by

the introduction of

an

endogenous consump-

tion structure without some explicit consideration of the distribution-pattern.

The

reason for this is as follows.

(a) If we do not distinguish among the income-groups, i.e. if we let r =

1,

the matrix V becomes the row vector of

n

order and, correspondingly, matrix C be

comes

the

column

vector of n order. If we denote these vectors as

v '

and a,

respectively, and assume that all value-added in the national economy consists of

the income accruing to the household

sector,ll)

then

9)

An

alternative justification for formula (1.19) was suggested

by

W. H.

Branson

[4] at the Econometric Society Meetings,

Washington

D.C., 1967. Income gener

ated by exogenous expenditure is equal to VBf,

and

income generated through

endogenous

demand

as

a function of income is

equal

to VBCY, thus

income

Y is

10)

11)

given by Y = VBCY +

VBf

=

[I

- VBCr

1

VBf.

To

combine the income-effect in our model with the

relative price-effect,

we may

be

utilized R. Stone's "linear expenditure system". See [50], [52].

With

this

assumption,

v'

becomes

the vector of value-added

ratios for

the

whole

economy, and in an economy with

no

foreign trade and government activities, the

conversion

v'

= i [I -

A]

becomes possible. Then we get

v'B

= i [I

- A][I -

A]-l = i I =

i .

Of course, if the household sector accounts for only

one part

of the value-added

sectors in the national economy,

this

conclusion

must be

modified.


20/146

11

L

= VBC =

v'Ba

= i [ I

- A]Ba

= i 'a

=a

(=Keynesian

macro-propensity to

consume)

[

-1 1

K = I -

L]

= - -

1 -

0

(1. 20)

where

i

is a row-summation vector. So, the income multiplier equation (1.19) be-

comes

Y

= KVBf

= _ 1 _ v'Bf = _ 1 _

i f

=

_ 1 _

f

1 - 0

1 - 0 1 - 0

a

(1.21 )

where fa

is

a

scalar,

where fa

=

fl

+

f2

+

f3

+

. . .

+

fn'

and

the vector Y

becomes

a

scalar, too. This scalar multiplier coincides exactly with the

Keynesian

multiplier.

Thus

our conclusion

that

income

has different

values depending

on

the proportions of

exogenous demand

is not substantiated in the special Keynesian case.

12

)

(b)

Furthermore,

even

if we introduce income-distribution-factors in macro

economic form

as

in the Kalecki or Kaldor models, the above Keynesian result is not

improved. Denoting by

d the column vector of k order

whose

elements are

relative

shares of each income-group,13) we

may

rewrite the matrix Vas V =

dv',

and the

matrix L takes the following

form:

L = VBC = dv'BC =

di'C

= de',

where e ' = i 'C is the row vector of

k

order whose elements are the

total

propensities

to consume of

each

income-group.

Then,

we get L

m

=

(de,)m

=

d(e'd)m-l

e

,

=

dim-le',

where

i is a scalar

showing

the weighted average of propensities to consume of each

income-group. Thus, the

interrelational

income

multiplier in

this

case

is

K = [I - L]- l = I + I L

m

m=l

=

+ I

i

m

-

1

de' = I

+

_ 1 _

de'

,

m=l 1 - i

12)

The

output

multiplier

corresponding to

this

case will

be

X

=

B[I

+

_ 1 _

ai']f

1 - a

and this coincides with the

result

(3.20) in Chapter 3.

13)

Where,

of course, the

sum

of all elements of d is equal to

1,

i .e.,

2k

d + d +

. . .

+ d - 1 .

(1 .22)


21/146

12

and the fundamental equation takes the

form:

Y = KVBf

= [I

+

_ 1

_ de']

dv'Bf

=

[I

+

_1__ de']

di f

=

[d

+

~ d ] f

1-1.

1-1. 1-1.

= _l-df.

1 -

i

0

(1. 23)

In which case, the autonomous

demand

vector f becomes a scalar fO' and the equation

(1.23) coincides with the

Kalecki multiplier,

except when i t is expressed in some

generalized

form.

In order to convert the above equation into a scalar multiplier,

all that

is

required

is

to multiply both sides of the equation

by

summation

vector

i , i .e .

i Y =

i,_l_

df =

i d-

l

- f

=_1_

f •

l i 0 l i

O

l_ i

O

(1. 24)

If we assume the constancy of

relative

shares, the scalar 'i

always

takes a constant

value, and, after all, equation (1.24) ends

up

being formally equivalent to the

Keynesian multiplier (1.21).

(c) Again, if

we

regard consumption

demand

as

an exogenous

variable

as

is

customary

(f = f + f), the income multiplier equation becomes

()

Y

= VBf =v'Bf = i f = fO'

(1. 25)

and

income equals final demand irrespective of the proportions of final demand.

Thus, in order to conclude that the values of income

differ

depending

on

the

proportions of autonomous final

demand,

i t is necessary to introduce not only the

structure of consumption

demand,

but also the structure of

income

distribution.

3) Sbw.c.twr.e

06 .the PILOPll9ilioYl PILOC.e1>.6

If we

lump

together the above two

mechanisms

of output and income determination,

we

have the following system:

~ ] = [*J +

(1.

26)

Solving

this

system for

X

and

Y,

we

get


22/146

13

and

i t is expected that this solution

can

be converted to the

form:

(1.27)

where g

is

a

column

vector of exogenous income.

14

) The preceding separate solutions

(1.5) and (1.16) are equivalent to (1.27)

where

g is disregarded.

Now, let

us return to the output propagation equation (1.10). Equation (1.10)

can

be

interpreted

as the propagation process viewed

from

the income-formation side.

But the same propagation process can also be observed from the consumption side or

the production side as well. These three aspects of the propagation process are:

(a) the income-formation side

(VBC

=

L)

x

=

Bf + BC[I + VBC + (VBC)2 +

. . .

]VBf

= Bf

+

BC[I - L]-lVBf

(b) the consumption expenditure side

(CVB)

x

=

B[I + CVB + (CVB)2 + . . . ]f

= B[I + C(I -

L)-lVB]f

(c) the production side (BCV)

X =

[I

+

BCV

+ (BCv)2 + ..• ]Bf

[I

+

BC(I -

L)-lV]Bf.

(1

.28)

(1. 29)

(1. 30)

It is interesting to note that in all cases, we can obtain the computation for

mula

(1.7) by projecting the propagation process

into

the income-formation side L =

VBC.

On

the other hand, if

we

derive the

sum

of the geometrical progression

from

the

consumption side (CVB) or the production side (BCV), we

do

not obtain the computation

formula (1.7) directly, but instead obtain the equation (1.6) which is the product of

two inverse matrices. This means

that

the income-formation side

has

a homogeneous

14)

The proof of (1.27) is easily demonstrable by use of the following identity:

IPEI + CKVB] IBCK] [r - A I-C] = W ]

L KVB IK

L-V

0

iPr

I

The

expression (1.27) in

this

chapter

is

equivalent to the formula in

K.

~ i y a z a w a

[33], or (4.7) in Chap. 4, in which, if we let Bl = VB, B2 = BC and M = K, we get

(1. 27) .


23/146

14

character

which contrasts

strikingly with the

nonhomogeneous

character of both pro-

duction

and

consumption

activities.

One other

point

regarding the propagation process should be explained. Equa

tions

(1.28) (1.30)

assume

a propagation process in

which

the

entire

process

is

a

succession of

separate

two-step movements: in the

first,

the propagation

from

the

production

side is

represented entirely

by

the effect of matrix

B, and

in the next

step,

the propagation occurs on the income-formation and consumption expenditure

sides.

But

instead of

this

assumption,

we may

assume

that

propagation occurs simul

taneously in

all

three sides, i .e . , production, distribution and expenditure.

In

the

latter case, instead of equation (1.28) (1.30), the propagation equation may

be

rewritten

as

follows:

x = f +

(A

+ CV)f +

(A

+ Cv)2f +

(1.31 )

00

We write

A

+

CV = Q,

and,

if

we

assume

the term

L

to be

convergent, we

have

m=O

(l

.32)

which

coincides with

(i)

of

(1.5). By

formulae

(ii) and (iii)

of

(1.5), we

get

X =

B[I

-

CVB]-lf

=

B[I +

CKVBJf.

Thus, the two propagation cases, i.e. the case of (1.28) (1.30)

and

the case of

(1.31),

have

the same

sum,

but obviously the

~ n c t e d muttiplien

in the case of

(1.28) (1.30) has generally a larger value than the truncated

multiplier

in the

case of (1.31).

We turn

next

to

the analysis of the convergence conditions of these

two

cases.

IV.

The

Convengence CondLtiOn6 06

the

Model

So

far,

we have

assumed

the existence of a meaningful solution,

X

0,

for

our

fundamental equation, X = AX + CVX + f, (f

0),

i .e . , we

assumed

the existence of

1

1 -1

(I

- A - CV) =

(I

-

Q)

0

and

of K = I - L) .

In

order to

treat

these prob-

lems

and their relationships, we will first review the properties of the Leontief

type matrices as preparation for developing the convergence conditions of our model.


24/146

15

1) The PfWpeJLtie.-6 06 Leon.tie6-type

Ma:tJUce.-6

For

non-negative square matrices in general, the following properties are well

known:

[I] Let a

be

a n x n non-negative matrix. Then the conditions (1°) - (4°)

below are equivalent.

00

(1

0) Lam

converges

m=O

(2°)

All

characteristic roots of a are less than 1 in absolute value.

(3°) I - a is non-singular and

(I

-

a)-l

is non-negative

(4°) For any non-negative vector f , the equation (I -

a)x = f

has a

unique non-negative solution.

For Leontief-type matrices,. i.e. non-negative matrices with

no

column-sums

greater than 1,

Woodbury

gives the following

lemma:

15

)

[II]

Let

a

be

Leontief-type and I - a nonsingular. Then the equation

(I - a)x = f has a unique non-negative solution.

From

propositions

[I]

and

[II] we

obtain

00

Lemma

1.

Let

a

be Leontief-type. L

am

converges

if

and only

if I

-

a

is

m=O

nonsingular.

Now,

we may transform a into the form (1.33) below by some permutation matrix P

A2 •••••.••

A2k

(1 .33)

o

where A

l

, A

2

, . . . ,

Ak

are indecomposable square submatrices, and k > 2 or k

=

1

depending

on

whether or not

a

is decomposable.

Then we may improve upon another proposition of Woodbury's.16)

15)

M.

A.

Woodbury

[54],

p.

353,

Lemma

3.2.

16)

M.

A. Woodbury [54],

p. 357,

o n o L t ~ y 3.6,

where

the condition is stated

as

follows: at least one of the column sums

be

less than 1 for some column in

each block.

06 columlU>

of the matri

x"

.


25/146

16

[III]

Let a be Leontief-type. A necessary and sufficient condition that

any non-negative

v e t o ~

t,

the

equation

(I -

a)x =

t ha6

a non-negative ~ o ~ o

is

that at least

one of the column

sums be

less than 1 for some column in each

submatrices A

l

,

A

2

, . . . ,

Ak in (1.33).

Based on

[I],

another

form

of [III] is obtained by replacing the paragraph

italicized

in [III] with "all characteristic roots of a be less than 1" in absolute

value,

which

we call

[III'].

Solow's Theorem asserts that condition [III '] is a sufficient one.

17

)

W e

can

also

show

that

i t

is

necessary too.

18

)

A different form of [III] or

[III '] ,

more convenient for our purpose, is

Lemma 2. 19) Let a be Leontief-type.

00

(1°) If all column-sums are less than 1, Lam converges.

m=Q

00

(2°) If all column-sums are equal to 1,

L m

diverges.

m=Q

17)

R.

Solow

[47], p. 36, Theorem 1

and

p. 38, Corollary.

18) Proof of the necessity of condition

[III '] .

We show that

(i) implies (ii)

below.

(i) All characteristic roots of

a

are less than 1 in absolute value.

(ii) Each A

l

,

A

2

,

. . .

,

Ak has at least

one column-sum

less than 1.

Suppose that the condition (ii) does not hold. Then all column-sums of

some

A . are equal to 1. For m-dimensional vector j = (1, 1, . . .

, 1 ) ,

m being the

d ~ g r e e of A., jA. =

l . j ,

i. e. 1 is a characteri stic root of

A.. As

the

'I. 'I. ' .

characteristic

roots of A

l

, A

2

, . . . ,

Ak

are also

that

of

a,

1 is a characteris-

tic

root of

a,

unlike

(i).

19) Proof of Lemma 2.

(1°), (2°) and the necessity of condition (3°) are immediately evident from

[III '] and [I].

(As

to (1°) and (2°), see also R. Solow [47], p. 32, p. 37).

00

Sufficiency of condition (3°):

Suppose that

Lam diverges. Then,

by [III '] ,

m=Q

a

must be

decomposable

and, for

some A.

in (1.33),

all

column-sums are equal to

1. If

i = 1, let

Al

= A(l).

If i t -

1: A

l

. ,

. . .

,

A.

1 . are all zeromatrices

'I. '1.- ,'I.

(otherwise, at least one column-sum in the i-th block of columns of a

must be

greater than 1).

Hence

we can remove A in (1.23) to the upper

left

corner by

some

simultaneous permutation of rows and columns, without losing the character

of the

form

(1.33).

Then

we

let

A. =

A (1).

'I. n

In

either case,

a

is decomposable

into

the

form

(1.34).


26/146

17

(3°)

In

case some

column-sums

are equal to 1 and some

less

than 1,

00

L

am

converges

if

and

only

if

a

is

not decomposable into the

m

following

form (by

some simultaneous permutation of

rows

and

columns) :

[

~ l )

A(l2) ]

A(2) ,

where all column-sums

of A(l)

are equal to 1.

(1. 34)

The

assertions (1°) and (2°) in this proposition have nothing to do with the

decomposability of a. And as for (3°),

i t is

the

particular

and not the general

decomposability

which

matters.

To

be

precise, condition (3°) includes two cases: the case

where a is

indecomposable and the case

where a is

decomposable, but not into the form (1.34).

Now,

let

us

return to our model. We may assume that the matrices

A, V

and e

in the preceding sections have the following properties [pl] - [p4].

n

l

[pl]

L a .

+

L v

k

· = 1

i=l

1.-J

k=l

J

( j

1, 2,

. . .

,

n)

l '

n

[p2]

L v

k

·

> 0

or

La ..

< 1

( j

1, 2,

. . .

,

n)

k=l

J

i=l

1.-J

n

[p3]

LVkj

> 0 (k

1, 2,

00 . ,

1')

j=l

[p4] (k = 1, 2, . . . , 1'), where a

k

These assumptions are reasonable from an economic standpoint.

(A

generaliza-

tion of [pl]

is

to be examined later).

The existence of B = (I = A)-l

is

guaranteed

by

[p2J (See Lemma 2 (1°)). As

A, V

and

e

are respectively non-negative,

n

x

n, l

x

nand n

x

1',

matrices, VEe =

L

=

(Zk

) and A + ev = Q = (q .. ) are also respectively non-negative,

l

x

l

and n x

n,

v 1.-J

matrices, and the following

equalities

hold:

Lemma 3.

l

00) L Zkv

k=l

=a

V

(v

1,2,00 ' ,1 ' )


27/146

18

n

(2°)

I q . .

=

1

i=l

1-J

1 , 2 , . . . ,n ) .

From

the

equalities

and [p4]:

C o ~ o ~ y

Land

Q are Leontief-type.

2)

Convengenee C o n ~ O n 6 ~ n ~ h e Model

We can now consider the convergence properties of the propagation process in our

model.

' '

T h e o ~ e m

1.

The

convergency of

I

Lm

coincides with

that

of

I ~ .

m=O

m=O

= jn. Thus, R is also Leontief-type. Since I

+

R . . •

+

= I

+ C(I +

L

+

•••

m-l)

m m

+

L VB, the convergency of IE and that of are equivalent. Next,

from

Lemma

1,

Ign

and IQm converge

if

and only if

I

I - R

I

f 0

and I

I -

Q I

f 0,

respectively.

And, since I - Q = I - A - CV = (I - CVB) ( I - A), I I - Q I = I I - R I· I I - A

I ,

where

always I I - A I f O. Hence I I - Q I f 0 and I I - R I f 0 are equivalent.

This

means that I ~

converges

if

and only if

I ~

converges, and therefore

if

and only

if

IL

m

converges.

A simultaneous permutation of rows and columns in

A, reflecting

a change in the

order of industry groups, induces a permutation of the

columns

in V and that of the

rows in C. On the other hand, a permutation of

rows

in V, reflecting a change in

the order of income groups, induces a permutation of

columns

in

C,

and conversely.

For brevity, we call the former I-permutation and the latter II-permutation.

Then, as a convergence condition of

I ~ we

have:

n

T h e o ~ e m 2. Let a

k

= I a.

k

i=l

1-

(k

=

1,

. . . , X')

be the total propensities to consume of income groups.

(1

0) If all

a

l

, . . . ,

aX

are 1ess than 1, then I ~ converges.


28/146

19

(2°) If 01 = . . .

=

0 p

=

1, then LQ diverges.

(3°)

In

case some of ok (k =

1,

. . . ,

p)

are equal to 1

and

some less than

1,

LQ

converges

if

and only if A, V and C are not decomposable

by

any 11- and

I-permutations sumultaneously into the following respective forms:

A =

c =

(1.35 )

V

=

where 0 < h <

n,

0 < 8 < P , and all column-sums of C

l

are equal to 1.

n

Proof. From Lemma.

3 ,

(2°), Lq

. .

= 1

if

and only if

i= 1

1.J

(k = 1, . . . , p)

(1°)

ok < 1 (k = 1, . . . , p):

Suppose that

there exists a number j such

that

n

L

q

•. = 1. Then v

l

. =

.•.

= v . = 0 from (i). This contradicts [p2]. Hence,

i=l 1.J J PJ

n

L

q

.• < 1 for all j , and therefore LQ converges (Lemma. 2, 1°).

i=l 1.J

n

(i )

(2°) 01 = . . . = = 1: As equation

( i )

holds for all j, L

q

. . = 1 ( j =

1,

. . . , ~

" i=l 1.J

Thus,

LQ

diverges from

Lemma.

2 (2°).

(3°) A simultaneous permutation of rows and columns in

Q

induces a I-permutation of

A, V and C only, and conversely. A II-permutation leaves

Q

unchanged. Therefore

LQ

converges, from

Lemma.

2. (3°),

if

and only if Q is not decomposable by any I-permuta

tion of A, V and

C, into

the following

form:


29/146

20

where

all

column-sums

of

Q

1

are

equal

to 1,

0 <

h

<

n.

(i i)

We

shall prove that Q is decomposable into (ii)

if

and only if A, V and Care

decomposable into the respective forms represented in (1.35).

Let Q be

decomposed

into (ii). By a

suitable

II-permutation, without changing

Q,

we

may take

(O< t


30/146

21

The condition in (3°) of this theorem contains the following two cases:

(a) A is indecomposable, (b) A is decomposable, but not into (1.35) together with V

and C simultaneously.

As a consequence we have

Corollary

1. In the case where some c

l

' . . . , c

r

are equal to 1 and some less

than 1, L ~ comverges if A is indecomposable.

So we can treat Kalecki 's Model as a special case of o n o L t ~ y 1, where

n

= 1,

r

=

2.

According to

Theonem

1,

we

have

o n o L t ~ y 2. The convergence condition of L ~ given in Theonem 2 is also

that

of

LLm.

In

addition, from [I] we

have

o n o ~ y 3.

A necessary

and

sufficient condition that, for

any

f 0, the

equation

(I

-

Q)X

= f has a non-negative solution coincides with

the convergence condition given in

Theonem

2.

The zero parts of

A,

V and C in (1.35) may

be

as large, but

no

larger as V

2

and C

l

will vanish in

view

[p3] and [p4].

Moreover, we may adopt a

more

general assumption than [pl]:

n

r

[pl' ]

La . . + LV

k

· 1.

i=l

7-J k=l J

Then, the

possibility

of

L ~

diverging

becomes

more

limited: (1°) in

Theonem

2

remains

true,

but (2°) no longer holds.

As

for (3°), the given condition is a suf

ficient,

but not a necessary one.

20

)

20) One case represented

by

assumption [pl '] is an open economy with foreign trade.

As

shown in Section III, in the case of a

c i o ~ e d economy

with no foreign

trade,

the conclusion

that

income

has different

values depending on the proportion of

final demand can be derived only

by

introducing the

structure

of income distri

bution.

But

in the case of

an

open

economy

with foreign trade the

same

conclu

sion can

be

derived without introducing the distribution

structure,

because the

coefficients on

the production side enter

into

the income-formation process

through imports. For the

open

economy model, refer to Chapter 3.


31/146

CHAPTER

2

INPUT-OUTPUT ANALYSIS AND INTERRELATIONAL INCOME MULTIPLIER AS

A MATRIX*

I .

rntnoduction

As

an

extension of the input-output

analysis,

we have introduced in the previous

chapter a

new

concept

which might

be called the

interrelational income

multiplier"

in matrix

form.

It

was

designed to analyse the

interrelationships

among various

income-groups in the process of income formation, and in this respect i t

tells

us

how

much

of one group's

income is

generated

by

another group's expenditure from one

unit

of additional income via the

medium

of

industrial

production activity. Although this

multi-sector

multiplier

follows from Leontief's "interindustry matrix

multiplier ,

i t

is formulated

by

the inclusion of the income generation process,

which is

omitted in

the standard input-output open model, and

by

projecting the

multiplier

process into

the income-determination side

rather

than the output-determination side.

Our extended model contains a theoretical implication not found in the Keynesian

model

nor

in the standard Leontief input-output model. In the Keynesian income-

determination

multiplier model,

the same amount of autonomous expenditures cannot

have different effects on the level of national income, even

though

the expenditures

have different

commodity proportions. The same restriction holds for extended

models

incorporating income-distribution-factors, such as the Kalecki-Kaldor type, in so

far

as

there are

no

changes in the

relative

shares of

income

and

in the propensities to

consume of each income group. Similarly, in the Leontief input-output model, al

though

the outputs of industries vary depending on the proportions of autonomous

expenditures, the

total income is

independent of the composition of autonomous ex-

penditure. This result also holds in the case

where

household consumption expendi-

ture

is

treated as

an

endogenous

variable,

in

so

far as

we retain

the assumption

that

* This is a

slightly

revised version of the empirical application part of

article

which

was originally

published, under the same t i t le, in

H i t a ~ u b a h i

o ~ n a t 06

Eeanomic6, Vol. 8 No.2,

Feb.

1968.


32/146

23

the level of

income

and its use do not

depend on

the composition of production.

l

) In

the real world. however. autonomous expenditures of equal amount. but having differ

ent

commodity

compositions. appear to have different effects on income formation.

In

order to have an input-output model in which the value of income differs depending on

the proportions of

autonomous

demand.

i t

is necessary to introduce not only endoge

nous consumption in disaggregated form. but also the structure of

income

distribution

by

the type of income-group as

well as by

the tupe of industrial value-added. This

is exactly what our extended model accomplishes.

In

this

chapter

we

shall see whether or not our extended

model

is

consistent

with facts. We shall use the

model

to interpret input-output and

related

empirical

data.

I I .

Fotunula.tion

06

a Regional.

Inc-arne Au£.;t[p'ueJt in

;the

60tun 06

a MabUx

An

application of our model

is made

for interregional income-distribution.

Our

model is thus reformulated in a form

suitable

for this purpose. However. the gener-

al model itself is applicable to the study of class-distribution or size-distribu-

tion of

incomes

as well. The

ommision

of the

income

formation process in input-

output analysis

is

especially unwarranted in the interregional

interindustry

case.

because the location of production depends on

the location of consumption.

and

the

latter

cannot be determined separately from the calculation of

income

generated in

each region.

Let us divide k regions into n industry

sectors.

and express the coefficients

of the model as follows:

A =

[{jJ

... the nk x nk matrix of interregional input

coefficients.

~ ~ the amount of ith commodity produced in region r for

use

of 1

1.-J

unit

of output of the jth industry in region s

(i.

j

=

1. 2 •

. . . .

n;

r,s=1.2, . . . , k ) .

1)

This

is

true in the case of a closed economy with no foreign trade

and

government

activities. But in the case of

an open

economy with foreign trade and government

activities,

the same conclusion does not hold, because the composition of produc

tion plays a part in the

income

formation process through imports, subsidies and

taxes. .


33/146

24

v =

[ y ~ S J . . .

the k x nk matrix of value-added ratios of household sectors

J

in each region.

the income of a household in region r earned from 1

unit

of

J

production of the

jth

industry in region

S (j

= 1,2, . . . , n;

r , s = 1 , 2 ,

. . . , k ) .

the nk x k matrix

of coefficients

of regional consumption

expenditure.

rs

c

i

the consumption expenditure for the ith

commodity

produced in

region r

from

1

unit

of

income

earned in

s region's

household

sector

( i =

1,2, . . .

, n; r , s = 1,2 , . . . , k).2)

Using

the nk x nk matrix of the usual Leontief

interindustry

inverse:

B

= [I -

Ar

1

= [ b ~ ~ J

1.-J

(2.1)

the corresponding earnings by the household sectors in each region are easily deter-

mined as follows:

~

rs sr]

VB=

lv.b . . ,

• J J1.-

,s

(2.2)

which forms the k x nk matrix of coefficients showing induced income earned from

production activities among

industries

and regions. On the other hand, the induced

production

due

to endogenoU4 consumption per 1 unit of income in each

region's

house-

hold sector is given as the following nk x k matrix:

Be =

r b ~ ~ C x : s J

Lz:,r

J1.-

1.-

(2.3)

Joining these two expressions,

we

get the following square matrix:

(2.4)

2)

Instead of

this definition,

if we denote byeS the

total

propensity

to

consume in

region

s, and by h ~ s

the consumption-allocation

coefficients

for the ith

commodi-

1.-

ty produced in region r , then

we

get e h ~ s = cX:

s

1.- 1.-


34/146

25

since the multiplication of rectangular and square matrices V,

Band

C makes a

new

k x k square matrix whose order is

equal

to the number of regions. This square

matrix

L

may be interpreted

as

an array of coefficinets

which

shows the interrela

tionships among

incomes

of various regions through the process of propagation from

consumption

expenditure in

each

region. Its elements

Zrs

show

how

much income in

region r is generaged by the expenditure

from

1 unit of additional

income

in the

region

s.

Of course, the

income

propagation process does not terminate

after

the

one

round

of inducement indicated

by

the

coefficient

matrix

L,

because the next

round

of

earnings will be generated as a result of the production activity induced by the ex

penditure from the preceding

round

of additional

incomes.

This has come to

be

known

as the "successive income generating process". Ultimately, such a successive reper

cussion process naturally leads to the intersectoral income multiplier among regions

of the following type:

K =

[I -

Lr

1

=

[krsJ.

(2. 5 )

The matrix K will be called the interrelational regional income multiplier matrix".

This matrix

shows

the direct and

indirect

income-generaged per unit of

income

region

ally originated, where, of course, I is the identity matrix having the order of

k x k.

Denote by X a

column

vector of nk order whose elements are output of n indus

tries in

each

region, and

by

Y a column vector of k order whose elements are incomes

of the household sector in

k

regions. Let

f

stands for a

column

vector having

nk

order of final

demand

other than

endogenous

consumption expenditure, and

g

stands

for a column vector having k order of exogenous

income. Then we have

the following

system:

(2.6)

As

shown

in Chapter 1, solving this system for X and Y, we get

=

~ [ I

; ~ K V B J

I

iKJ

~ J

(2.7)


35/146

26

III. The

I n t ~ e l a t i o n a l Income M u t t i p t i ~ among

R e g ~ o n

An empirical application of our

model

is made for a three-region view of the

Japanese economy by utilizing the large 1960 interregional input-output table pub

lished in 1966 by MITI (the Ministry of International Trade

and

Industry). The table

required

more

than three years of preparation prior to its publication. For analyti

cal purposes, the original data tabulated in 9 blocks and 25 industrial sectors

was

aggregated into three regions and 25 sectors. All estimations of parameters and

other calculations for our

model were

done under the cooperation of the MITI-staff,

t

(lecture notes in economics and mathematical systems 116) dr. kenichi miyazawa (auth.)-input-output...

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