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Transcript of 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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38 E. Dietzenbacher:
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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The Implications of Technical Change in a Marxian Framework 39
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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40 E. Dietzenbacher:
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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The Implications of Technical Change in a Marxian Framework 41
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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42 E. Dietzenbacher:
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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44 E. Dietzenbacher:
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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The Implications of Technical Change in a Marxian Framework 45
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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