the implications of technical change in a marxian framework (dietzenbacher)

8/11/2019 The Implications of Technical Change in a Marxian Framework (Dietzenbacher)

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Vol.

5

(1989), No. 1,pp. 35 -46 Joumoi

of

EcOttOn^ S

ZaUtiMlh fOr t

by Springer-Verlag 1989

The Implications of Technical Change in a Marxian

Framework

By

Erik Dietzenbacfaer Groningen, The Netherlands*

(Received June 8,1988; revised version received April 20,1989)

1. Introduction

The effects of technical change on the price structure are well

documented for a Sraffa-Leontief model with a con stant profit

rate. See, for instance , H erre ro, Jimenez-R aneda, and Villar (1980),

Fujimoto, Herrero, and Villar (1983) and Dietzenbacher (1988).

Within a Marxian framework on the other hand, it is precisely the

change in the rate of profit that has been extensively discussed.

The question whether technical change causes the rate of profit to

fall, as posited by M arx, or not, h as led to a vivid controv ersy. Fo r

a recent con tribu tion see the d eb ate o n Sha ikh (1978), with

comments by Arm strong and Glyn (1980), Bleany (1980), Naka tan i

(1980) and Steedman (1980), and his own reply (Shaikh, 1980). See

further e. g. Roemer (1977, 1979), Bowles (1981) and Salvado ri

(1981). A central role in the discussion on the falling rate of profit

has been played by the fam ous Okishio theorem (1961). It states

that a technical change which reduces the cost of production

(measured in current prices) implies a rise in the rate of profit. In a

Marxian framework, little attention has been paid so far to the

* An earlier version of this paper was presented at the Third Annual

^ongress of the Europiean Economic Association, held in Bologna,


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36 E. Dietzenbacher:

price effects of technical change, whereas they occur simulta-

neously with a change in the profit rate.

In the present paper we focus on price changes in case of a

rising, or falling, rate of profit. As such, we do not go into the

debate on the falling rate of profit. We first prove in section 2 that

for an innovation that leads to, for instance, a rise in the profit

rate, the price in the innovating sector falls and, moreover, falls

relatively the most. As a consequence, the percentual increase

(resp. decrease) in the profits (per unit of output) is the least (resp.

the most) for the innovating sector. Second, in section 3 we show

that the criterion of cost-reduction is not only sufficient for the

rate of profit to rise, but also necessary. Hence the converse of the

well known Okishio theorem is also valid. This result implies that,

given the mode under consideration, Marx's falling rate of profit

can only be brought about by a technical change that extends the

cost of production. Third, in section 4, we present similar results

for three extensions of the basic model.

Consider a pure circulating capital model where A is the nxn

input matrix, I is the 1 x n row vector of direct labour inputs, b is

the n X1 column vector which gives the subsistence wage bundle,

jiis the equilibrium rate of profit and p is the 1 x n row vector of

production prices in equilibrium. TTie wage rate is taken as unity.

The equilibrium is specified by the following equations:

p = (l-l-;r)(pA-t-l), (1)

l = pb. (2)

Let the augmented input matrix be defined as M = A-l-bl, then (t)

can be written as

(3)

We assume that

A, b and I are

semi-positive^

and

that

M is

inde-

composable

and

prod uctive . These conditions imply that

the

^

For vectors and matrices we adopt the following notations and

expressions. x>0, non-negative, means JCf >0 for ali

i;

x >0,semi-positive,

means x>0 and x^O; x>-0, positive, means x,>0 for aili

For a non-negative, indecomposable matrix M that is productive,

there exists an output vector x>0 such that x>Mx Ax+blx. This

means that the technology is capable of prodi^ng a surplus over the

requirements for subsistence. These three conditions are sufficient to guar-


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The Implications of Technical Change in a Marxian Framework 37

equations (2) and (3) have a unique positive solution, i.e. n>0

and

f>0.

2. The Effects of a Change nthe Profit Rate

Suppose that a technical cha nge app ears in sector (or process)

i.

This affects A', the j-th column of the input matrix A, and/or /,.

The augm ented inp ut m atrix M is therefore also changed in its j-th

column M'. As a consequence the rate of profit and the prices

change. We assume that the subsistence wage bundle remains the

same. Using bars over the symbols to denote the new technology,

the equilibrium after the technical change is specified by the

following equations:

4

I = p 6 . (5)

It is assumed that also A and I are semi-positive and that M is

indecomposable and productive. Consider the relative changes in

the production prices, that isPj/pj. The following theorem asserts

that the relative price changes are bounded by the relative change

of the price in sector

i,

in which the techn ical chang e h as taken

place.

Theorem 1:

\f n>n: pj/pj

>

ip,/p^

(1H- * ) / ( l

+n > p^/pi

for all

j

(,

\^ n,

= 1

-I-

x ) piVPfor

j

^ s

MJ

=

M^.

F or

j

= rthere are three possibilities, (i)

//

> 1+

n

pM' ,

thus p>(l-i-;ir) plVLJThe Subinvariance theorem* implies that the

Frobenius root of M, that is 1/(1

-hx ,

is smaller than _l/ (;i -f

n ,

thus

n>ji,

which is a con tradiction, (ii)

/>;

< (1-I-;r)pM ' implies

P< (l +

;r p M

and

7i,

= (1H-;r)pM ' must hold. Thus

P= ( l + :^ )pM. Q .E.D .


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F or

a

rising rate

of

profit for ins tan ce , this the ore m states that

the largest relative price decrease willbeobserved insector i.If

pr od uc t / is the only pro du ct th at is requ ired for the subsistence of

the wo rkers ( i . e . A ,> 0 a nd

bj = O

for

all

j i),

then

Pi = Pi

and

Pj>Pj,

as

follows from p b

=

p b

=

1 . H ow ever

if

there

is a

prod-

uct j other tha n j , tha t isrequi red forsubs istence (i. e. bj > 0 for

s o m e

/ V I , it

follows that

the

pr ice

in

sector

/

falls,Pin: Sj/sj> (. /s (1-f )/(l { n)> s,/Sifor allj#i,

if nn, therateof

profit rises. The corollary asserts that for the innovated process the

(per unit) profit incre ases relatively the least or dec reases relatively

the most.

At

first sight, this result seems

to be

surprising,

a

few

remarks however are

in

place.

First,

the

statem ent hold s also with respect

to

to ta l profits,

provided that the output isno t affected by th e tech nica l change.

Total profit in sec tor i increases pe rce ntu ally th e least or decreases

percentually them ost. It is obviou s however, that the absolute

increase (decrease)

of

the total profit doe s

not

need

to be the

smallest (largest)

in

sector

i

Second, consumers may be expected

to react on the price changes. Substitution will leadto a different

final demand vector and consequently to

a

differen t ou tp ut vector.


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Therefore it is very well possible that the re is a decrease in the (pe r

unit) profit of sector

i

and an increase in its tota l profit du e to a

rise of its output. Third, it is not possible to state whether technical

change induces a rise or a fall in the (per unit) profit of sector

i

It

can only be asserted that there is at least one sector for which the

(per unit) profit increases, as follows from

s s)

b = x / ( l

-*-

^ )

As an illustration of this latter remark, we present two

numerical examples of technical change. The first is labour-saving

and yields an increase in the (per unit) profit of each sector. The

second

is

capital-saving a nd leads to a decrease in the (per unit)

prof-

it of the innovating sector and to increases for the other sectors:

[0.35 0.05

A= 0.15 0.45 0.05 |, b = | 1/3 | , l=(0.15, 0.15, 0.15), thus

0.15 0.15

ro.40 0.1

= 0.20 o.f

[0.20 0.2

a o 0.301

M

= j

0.20 0.50 0.10 .

[0 .20 0.20 0.40 J

Note that all rowsums are equal to 0.8, hence the Frobenius

root of M, that is

1/ 14-;r),

yields 0.8 and n= 0.25. All column-

sums are also equal to 0.8, so that p = (1 ,1 ,1 ) and s =

7rp/(l -f ;r) = 0.2 p = (0.2, 0.2, 0.2).

When the direct labour input in sector

1

reduces to 0.03, the

augmented input matrix becomes

0.36 0.10 0.301

0.16 0.50 0.10 .

0.16 0.20 0.40j

The rowsums equal 0.76, hence

1/ 14-^)

= 0.76 and

;T=

0.316.

P= (0.857, 1.118, 1.025) and s =;*p /( l- f ^) = 0.24 p = (0.206,

0-268, 0.246).

Starting from the original situation again, consider a capital-

saving technical change where each element in the first column of

A decreases with 0.10. Then

= [0.30 0.10 0.30

^A

0.10 0.50 0.10


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3 . Cost-ReductioD

In the discussion on the falling rate of profit, an im po rtan t role

has been played byO kishio s theorem (1961). Sup pose that capi-

talists co ntem plate the ado ptio n of a technica l innov ation in sector

i

It

is

assu m ed, often implicitly, that

the

m arkets

are

fully compet-

itive

so

tha t

the

decision

of no

individual firm

has a

measurable

impact on,

for

ins tance,

the

relative p rices. Supp ose that each firm

decides

to

in t roduce

the new

technology

if it is

cost-reducing,

w here the pro du ction costs are evaluated

at

cu rren t p rices . Clearly,

their short-run profit rate rises, yielding super profits

and a dis-

equil ibr ium. However , af ter the prices hav e readjusted , so as to

equil ibrate

the

rate

of

profit again,

the new

rate

of

profit will

be

higher than the oldrate. S o, rough ly spea king, O kishio s theorem

states that

the

rate

of

profit will rise

if

technical change

is

intro-

duced

in

sector

/

when

it is

co st-reducing

at

current prices.

The

/th

e lement

of the

vector

pM

gives

the

production costs

(evaluatedat prices p) in s ec tor / an d is equa l to pM . Th e criterion

of cost-reduction means that the following ineq uali ty ho lds:

Ipjdj, l


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heorem 2:

Cost-reduction

o n> yt,

cost-extension

n

implies cost-reduction.

From tjieorem

1

follows_that

pj/Pj>Pt/Pi

for all

j^i.

T h en ; p M '

i M i i / P p J i

i / d

z J I / W

j j j j

= p M '. ' Q. E. D .

To our knowledge, the converse of the Okishio theorem has

never been stated explicitly before. This is rather surprising

because in proving the first assertion it is not necessary to use the

price inferences from theorem 1. The one-to-one correspondence

above is altematively obtained by proving (i) cost-reduction

implies

7i>7t,

(ii) cost-extension implies

7i ;r) ,_theorem 1

ipj/Pj>pi/Pi

for


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4.

Extensions of tbe Basic M odel

We conclude this paper by presenting the results for three

extensions of the basic model as given by (2) and (3). ' First, the

subsistence wage bundle is allowed to change, second, non-depre-

ciating fixed capital is taken into account and, third, heteroge-

neous labour is taken into consideration.

Clearly, the assumption that the real wage bundle remains

fixed is unlikely to hold in real life. Also, it is well known that the

rate of profit m ay fall d ue to a cost-reduc ing tech nical change if

the wage bundle is allowed to vary. Take, for example, an inno-

vation that reduces the costs of production (evaluated in current

prices) but at the same time leads to a higher demand for labour.

As all firms w ould ad op t this new tech nolo gy, it is conc eivable that

the real wage rises. It may even rise so much that the equilibrium

rate of profit falls.

Let the equilibrium for the new technology be specified by the

following equations, instead of by the equations (4) and (5):

p = ( l - l - ;*)pM and I = p 6 ,

where b denotes the new wage bundle. Cost-reduction is defined

again by equation (6). It should be noted that in the present case

equation (6) is not equivalent to equation (7). The assertions with

respect to the relative price changes are slightly weakened in

comparison with those of theorem 1.

Theorem 3:

If

n>it: pj/pj

>

[(1+n /il + n ]

m in

{p,//j,; 1}

for all

j =i,

if

7i


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The Im plications

of

Technical C hange

in

a Marxian Framework43

Theorem

4:

If p6>pb: cost-reduction n,

cost-equality

=>

n

nn,

cost-equality o n

= M

cost-extension

n^,

cost-equality

=>n>n

cost-extension


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of the new direct labo ur inp ut of sectori as induced by the change

in the wage bundle. For pB denotes the value of the new wage

bundle in terms of the old wage rate. The value of the new wage

bundle in terms of the new wage rate equals one of course, as the

wage rate is taken as the numeraire.

Knowledge with regards to pBpb, enables us furthermore

to sh arpen the results in theorem 3. Fo r instance, pB>pb and

pB = pb= l imply pB>pB. Now consider the case

7i>jt

and suppose that

pi'^p,.

Theorem 3 the n yields

pj/pj>

1

--

n)/il + n)>\

for all

j

# / or, equivalently, p

B

> pB, which is a

contradiction. Therefore, in case pB>pb and ^ > ; ; it follows that

piipi/pi)il jT)/i\ n)

for all y # i . The expres-

sions of theorem 3 can th us be further refined by use of the

following assertions.

Corollary 3:

If n>n: pB > pb=> pi< Pi,

pB < p b Pi,

if

M=Tt: ph> phopi>Pi,

pB < pb /,

pb ,>^pE'>(H-7r)pM'.

The following result asserts that th eorem 2 rem ains valid while

theorem 1 is only weakened slightly.

Theorem 6:


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As a final extension of the basic model, heterogeneous labour

is taken into consideration . We adop t the form ulation of Bowles

and Gintis (1977) and Krause (1981). Assume there are mtypes of

labour and let the direct input coefficients be given by the

mxn

matrix L. Each type of worker has its own subsistence wage

bundle, given by the co lumns o fth e

nxm

matrix B, which is taken

constant. The different wage rates are given by the 1 x m row

vectorw = pB. Replacing (1) and (2), equilibrium is specified by

the following equations;

wL), (8)

w

= p B . (9)

These equations can be rewritten as (3) with M = A

-I-

BL. It is

apparent that theorems

1

and 2 remain valid. In addition, the

following, similar expressions hold for the wage rates under the

assumption that each type of worker uses at least one product

(other than i) for subsistence.'

Corollary 4:

The following expressions ho ld for all /c=

l . . .

m:

If n>

Tt:

w^/Wk> pj/pi,

ii ii = IpjBjt = Eipj/pj)PjBjk> ipi/pi) ^PjBjt

j

~ Pi^Pi) n't. Strict inequality follows from the assumption, which

states that for all

fc

here is a j # iwith

B,*

>0

Q.

E. D.

Note that the prices

Pj

and the wage rates w^ as determined by

(8) and (9) are un ique up to ascalar In the basic model (1) and (2)

the wage rate was set at unity. As a consequence

pi


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46 E. Dietzenbacher: Technical Change in a Marxian Framework

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the implications of technical change in a marxian framework (dietzenbacher)

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