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On technical change, differential rates of profit and exploitation-香川大学学術情報リポジトリ

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Kagawa Unzversity Eronomμ Revzew Vol 64 No 4, February1992,205-218

On T

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Ranade** and A

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The dynamics of capitalistic development had been one of the central themes of the classical economic theory. With able minds pondering over the question in more recent times it seems surprising that one still has to go to Marx for a comprehensive ana!ysis on the subject. On second thougths, it is probab!y not surprising given the depth with Marx theorizes The role of the innovators and technical change in the process of capitalism, however, is something that goes deeper than what Marx went into We will try to establish this in the following pages We will be building a case for the untenebility of a classical assumption of the uniformity of the rate of profit across the industries We also resurrect the Marxian concept of exploitation (see Morishima [4]for an able exposition) as a valid analytical tool This runs counter to the view taken in a formidable critique of the rate of exploitation by Roemer in [9], [10], among other places

*The work on this paper was done while Ravindra R Ranade was a visitor at the University of Alicante during December1988へJanuary1989The financial aid and the friendly hospitality is grateful1y acknowledged by him

**University of Kagawa, Takamatsu, 760, J.APAN

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-206- Kagawa Unzver.s!ty Econom!c Rev!ew 898

2. We adopt the standard constant returns to scale Leontief input-output model for the formal and the indestructible com model (the adjective being applicable to the word model and not to corn) for an informal elucidation The matrix A = [aij]nxn represents the Leontief technology with the assumption that A is not decomposible and it is productive making the maximun uniform rate of profit meaningful A vector L=[Ij]lxn consists of man-hours that go along with the n columns of A respectively into the n industries to produce the unit levels of outputs.. L is assumed to be positive Labour is taken as a numeraire e.. the money wage rate W = 1 Each of the n industries charges its own non-negative rate of profit industry j charging at the rate rj R is the diagonal matrix with0, rz,..., rn on its diagonal.As the profits are charged as mark ups on the amounts of capital employed in the industies the prices must satisfy the equation : p=(pA十L)(I+R) ..(1) or more explicitly : p=L(I+ R)[I -A(I + R)]一1 引 一 (2) In the hypothetical case of industries operating at zero profit levels the prices are equal to the labour values and the corresponding equations are λ=(AA+L) 一(司 or explicitly : A=L(I-A)一l ーリ

ω

)

It is elementary that p dominates A for all possible values of R and further, p is a monotonically increasing function of the diagonal elements of R To switch to the one-sector com story, A units of corn and L units of labour produce one unit of corn with the capitalists charging at the rate R so that the price of corn is L [{1/(1 + R)} -A]-l・Itis easy to see that the price would rise as a result of a rise in any of the terms L, A and R The maximum rate of profit possible is clearly Rmax=(1/ A)ーl

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899 On Techincal Change Differential Rates of Profit and Exploitation -207ー In the n-sector case Rmax will denote the diagonal matrix of the the maximum possible uniform rate of profit given by the difference between the reciprocal of the dominant characteristic root of A and unity.. Next, let the wage bundle recieved by the unit of labour be denoted by the vector b Itis assumed to be semi-positive As the money wage rate is unity, pb is also unity. The real wage rate, then, given by W /pb is also1. IfN denotes the total employment, the total wage basket is Nb and the necessary laboUI for the basket is (L(I -At1Nb) orNAb. The total labour, on the other hand, is same as total wages since we have taken W as 1 These total wages are then spent on buying the basket Nb at prices p.. Thus, total laboUI is Npb The surplus labour, then, is N(pb-Ab) The Marxian rate of exploitation e is defined as a ratio of surplus labour to necessary labour : (see Morishima [4] and Okishio [7] for a detailed exposition and Roemer [8], [9], [10] for a criticism of the rare of exploitation)・

e=(pb-Ab)/Ab=(pb/Ab)-1 (5)

To move back to the corn story, the wage-profit curve is given by the rule w=[ {1/(1

+

R)} -Al/Lb and it is not difficult to see that the curve is sloped negatively and is convex to the origin“ The rate e as a function of R

is given by R/ { 1 -A(1十R)}“Inthe Marxian conception of the capitalistic development characterised by capital using and labour saving (CULS) technical change, e.i . an increase in A and a decrease in L, the wage-profit CUIve shifts.. This results in a decrease in the maximum rate of profit and if the change is cost reducing as well then in an increase in the maximum wage rate. The well known Marxian dictum of falling rate of profit can now easily be seen as below. After a rearrangement, R= {(l-A)e} / {1

+

Ae} and if e is kept constant, it is clear that R must fall as a result of an increase inA The contraposition of this dictum, as pointed in Fujimoto [2], is : given that the rate of profit does not change, it follows

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-208- Kagawa Unwer'sity Eιonomic Review 900 that the rate e goes up as a result of the CULS change.

In the n-sector model the technical change is defined to be of the CULS type if A' is semi-positive and L' is semi-negative (whereinafter the prime would indicate the time derivative of the variable). Further, a change is called cost reducing if (pA' + L') is semi-negative Now, differentiating the equations (2) and (4) respectively, after some manipulation (see appendix for the simple differentiation of the inverse of a matrix), we get

p' =(pA' + L')(I十R)[I-A(I + R)]→+(pA+ L)R'[I-AO+R)]-l れ(6)

λ'= (λA'十L')(I-A)一1 リ…(7) Now, as the vector (pA' + L') is semi-negative and the matrix [I-A

(I+R)]一1is positive, if the rates of profit do not change then it is clear from

(6) that p' is negative Further, as p dominates A we know from the cost reducing assumption that the vector (AA' + L') is negative too and from (7) it follows that A' is negative. On these and related results (under the assumption of the uniformity of rate of profit) see Fujimoto [1], Nakatani [2], Okishio [7] and Roemer [8] among other contributions. Next, differenti -ating the equation (5),

e'=[{p'b十pb')λb}一{pb(A'b+Ab')}](λ

bt

2

Now, the price changes, as a result of technical change, in their turn may influence the working class to change the composition of the vector b..As far as the rate of exploitation is concerned, what matters, however, is the number of bundles b, more or less than,1 that a worker can buy as a result of the change in price vector p given that the money wage rate does not change. The problem of exploitation must not depend on the subjective changes in the tastes of the working class. Thus, we may take b' to be equal to kb with k being the constant of proportionality.The expression above, then, becomes

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901 On Techincal Change Differential Rates01Profit and Exploitation -209-or alternatively :

e' =[(p'bAb)一(pbλ'b)](λb)→ 山 内 山 内 い(8)

Itmay be noted here that since both the vectors of prices and values fall it is not c1ear as to what is happening to the rate of exploitation. Fujimoto and Ranade [3] show that the one sector result of an increase in e after the cost reducingCULS technical change is not generalizable to the n-sector case and they provide sufficient conditions for the result to go through under the assumption of the uniform rate of profit Most of Marx's conception of capitalistic development was based on the tenet of the rate of exploitation remaining unchanged.. This made sense in an era of the working c1ass having been forced to subsist at the minimum possible level of exixtance.. In modern capitalism such a c1assical assumption is no more valid. There is no mechanism in a modern economy which keeps the unobservable rate e in check It seems much more natural to treat e as an index of the worsening of the situation for working c1ass from its ideal utopia. In other words, it is just an index to see how much do the prices differ from the values because of the phenomenon of profits Intelligent union leaders may regard this as an important index of distribution between 'capitaJ'and labour..Its possible use in a debate on appropriate technology can not be underestimated. This also has some bearing on our presumption of the composition of the wage bundle b not changing after a change in prices In absence of the data regarding the subjective elements involved in demand behaviour of working c1ass the trade unionists would have to, naturally, continue with the consumtion vector b if the rate of exploitation is to be used as an argument for or against a new technology 3 As seen earlier, the cost reducing technical change ofCULS type dips

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-210ー Kagawa Unzverszty Economzc Revzew 902 the prices all around if the old rates of profits prevail across the industries This results in the drop of the value of the capital used in the industries十 The constancy in the rate of profit, then, means a drop in the total profit levels of the capitalists.. 1t is quite inconcievable that a technical change could get adopted if it causes a dent in the profits. There is a major difference between owners of industries accepting a fall in profits due to conditions of market demand or losing out to the working class in sharing of the surplus on one hand and consciously adopting a profit reducing and self-denying technical change on the other..The main motive for the innovators, as so well emphasised by Schumpeter [11], is to reap profits by adopting new techniques. After all, if the profits are going to fall because of the constancy of rates of profits after the change then there is nothing to prevent the capitalists to shift back to the pre-change technique which guarantees them a certain level of profit instead of continuing self-deniaL 1t is to be expected that the innovators reap additional mark ups after a new technique is introduced until the new technology loses its glitter in the competition. This means the prices fall after a while. During the time the proverbial hay is made, even if the sun shines only for a little while“There is, however, nothing to push the prices down to the levels at which the new profits are less than which were obtainainable before the technique was introduced The industry would have a potent threat in the game of the choice of techniques, between old and new, to maintain its pre-change level of profits.. This would obviously involve differential rates of profit between industries.. A higher rate would accompany the cheapening of capital for maintaining profit levels.. We will put this now formally. We saw before :

p' =(pA' + L')(I+ R)[I -A(1 + R)]-l +(pA+ L)R'[I-A(I+R)]-l

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903 On Techincal Change Differential Rates of Profit and Exploitation -211-prices are reduced and the matrix of the rates of profit adjusts immidiately after the change. That is to say, at the old prices (p' =0) (pA+L)R'=一(pA'+L)(I+R) .(9) We will consider the case of technical change taking place only in the first m industries throughout these pages.. In such a case, as the valuation of capital in the last n-m industries does not change, the rates of profit do not change either. The burden of adjustment (or does one say the pleasure of it) is borne entirely by the rates of profit in the first m industries There the reduction in costs in each of the industries is matched by a propor tionate increase in the rate of profit to leave the prices constanL Intuition demands that such an increase would result in windfall gains for the innovating industries..It can be seen as follows. The profit vector isπ

= (pA + L) R and after differentiation,

π'= (pA'+L')R+p'AR+(pA+L) R' 川 引 .(10)

Using (9) and (10), we get π, = (pA'+L')R一(pA'+L')(I十R)=ー(pA'十L')

which is semi-positive. Thus the obvious :

Proposition 1 A cost reducing technical change of CULS type is more profitable for the capitalists at the pre-change prices

The increments in the rates of profit are given by the equation (9) As the innovation gets older in time, the new entrants push the rates of profit down due to competition and the prices adjust in response.. The advantages of the change start being shared with the consumers.. The prices, however,

have a lower limit as a result of the availability of the pre-change techniques. The farthest they go is to allow no additional profits from the pre-change levels. Formally, from (6) after a rearrangement, we get

(pA + L)R' =p'[I -A(I+R)]一(pA'+ L')(I + R) and substituting into(10) and equating π, to 0 we get

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-212ー Kagaωa Umverszty Economzc Review or p'[AR + 1 -A -AR] =(pA' + L')

or p'ニ(pA'+L')(I-At1

904

(11) Since we have assumed A to be non-decomposible it is clear that all the prices must necessarily fall since the vector (pA' + L') is semi-negative. Also we know from (7) that the vector入must fall as a result of the technical change..Itis, then, a non-trivial question as to what is happening to the rate e. Noting the numerator of e' in (8) and substituting for p' and A' from (7) and (11), [(pA'+L')(I-At1bL(I-At1b]一[L(I+ R){I -A(I+ R)} -lb

(AA'十L')(I-At1b].. Now, subtracting and adding the term [(AA'十L')

(I-At1b L (I-At1b] to above and manipulating the terms, it is not very hard to see that the numerator of e' becomes an expression given by the sum of [(pA' -AA')(I -At1b L(I -At1b] and the negative of [(λA'+L')

(I -At1b L{(I + R)(I -A(I + R)t1ー (I-At1}b]..As vector p dominates λ, pA' becomes greater than λA' and first term can be seen to be positive Next,

we know that (AA'十L') is semi-negative and R is non-negative which

makes the second term also positive This means that the numerator of e' is positive. Thus a dictum in the classical Marxian tradition

Proposition 2 A cost reducing technical change of the CULS type increases the rate of exploitation if the rate of profit in various industries adjust to maintain the profits at the original levels. This result seems to depend on the coalitional behavioUI on the part of the capitalists against the laboUI. It assumes that all the industries earn their pre-change profits and not just those which undergo the change.. It is probably not very surprising that the rate of exploitation rises when the capitalists make sure that none of them get hurt as a result of the change. The case of the innovating industries maintaining their profits at the cost of the lesser mortals is not so straightforward.. In what follows we take the first m idustries protecting their profts and if this sounds like a coalition

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905 On Techincal Change Differential Rates of Profit and Exploitation -213-between them, then one may take m to be unity and the problem of technical change in more than one industry can be then treated as a sequential application of the following analysis

Let J denote a n X n matrix formed by augmenting the m X m unit matrix

by zeroes.. The first m industries keeping their profits implies a multiplicat -ion of the equation(10)by J and equating it to0 Therefore,

π'J =O=(pA' + L')RJ +p' ARJ +(pA + L) R'J

or (pA+L)R'J=一(pA'+L')RJ-p' ARJ ,(1~

Also from (6), after a rearrangement, it was already seen that

(pA+L)R' =p'[I- A(I+R)]一(pA'+ L')(I+ R) “(13)

Since there is no reason for the last n-m industries to demand and ensure higher rates of profit due to the competition, the most they can do is to look at the innovators with envy and charge their mark ups at the old rates This implies that the left hand sides of(12) and (13) are the same and so the right hand sides can be equated., Therefore, after a rearrange

-ment of the terms,

p'=(pA'十L'){I+ R(I-J)}[I -A {I十R(I-J)}

1

一l , ,,(1~ Thus, ensuring the profit levels to be maintained for the innovators does result in the prices to fall all around They fall, however, to the levels different from what is given by (11)The fall in the costs and the valuation of capital is compensated in the first m industries by a rise in the rates of --i profit to leave the profits unchanged and the last m industries suffer,

ending up with reduced profits., Just as in the earlier case, it is not trivial

to see what is happening to the ratee..The numerator of e' from the equation (8), after the substitution for p, A, p' and λ, in it from equations (2), (4), (14) and (7) respectivel仇 yields [(pA' + L')(I + R(I-.J)){ 1 -A(I + R

(I_J)}-l b L(I-A)ーも]-[L(I十R){I-A(I+R)}-lb (AA'+L') (I-A)一lb]. By subtracting and adding the term [(AA' + L')(

I+

R){I -A(

I+

R)} -lb L(I -At1b] to

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-214 Kagawa UmVer'Slty Economic ReVleW 906

above, the numerator of e' can be seen to be a sum of the two expressions given by the first being [(pA' + L')(1 + R(I-J)){1-A(

I+

R(I-J)} -1 b L(I-At1b]

一[(λA' + L')(I+ R){1 -A(1 + R)}ーlbL(I-A)一1b] and the other being

一[L(1+ R){1 -A(

I+

R)} -lb (λA' + L')(I -At1b] +[(λA' +L')(I+R){1-A(1+R)}-lb L(1-At1b]

Noting that the vector一(AA'+ L') dominates一(pA'-L') which in turn is positive and further, that the vector(I+ R){ 1 -A(I+ R)} -1 b dominates the vector (1 + R(1一日{I-A(1 + R(I -.J)} ~1 b which is again positive, it is easy to see that the first term in the sum of tne numerator of e' is positive The second term, unfortunately, is not conclusively so. It is, however, a pointer to sufficient conditions for the sign of e' to be positive

The algebraic manipulation of the above technical conditon does not seem to yield any general conclusion about the effect of technical change on the rate of exploitation 1t is, however, not very difficu1tto suggest plausible economic scenarios where the rate e is indeed pushed upwards.. As an example we sketch the fol1owing two sector possibility. Let the first good be an overwhelmingly wage good not much entering into the input requirements and the sector that produces it does not charge a very high mark up whereas the other good is exactly the opposite to the extent that the two fol1owing inequalities are satisfied.

(b

.

j

b2

Y

~註主 品[(rれlρ2){(

1十Iれω2)ν

/

(

1

+rlω1)}伶(a12/a1剖2刈1

(

仇νr乃2/βrれ叫)1三孟主 [{札l一

(

1十ωI乃2)河ad/{札l一

(

1+r口ω1)泊all}

]

The scenario is not unlike a third world rural agricultural capitalist already using a highly mechanised technology (possibly handed down by an external agency as a symbol of development) and unable to charge high mark up whereas the urban capitalist using again a very highly mechanised technique (wel1entrenched into an international state of the art technolog

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-F 内 υ , , a q L On Techincal Change Differential Rates of Profit and Exploitation 907 ical circuit) and in a position to be able to charge a very high mark up. In such a state of affairs, a further cost reducing mechanisation of agriculture at the cost of labour results in a higher exploitation.Itis easy to multiply theexampl~s of economies which satisfy the above conditions and result in exploitativetechnical change The appendixcarnes two numerical the

examples indicating the possibility of an increase as well as a decrease in the rate e. The basic truth, however, is :

Propos.ition 3 A cost reducing CULS technical change in the case of the concemed capitalist charging a higher rate of profit to maintain his The Marxian dictum previous profits may OI may not increase exploitation of the technological advances being exploitative is crucially dependent on some arithmatical accident of the production structure An introduction of a new cost reducing technique, thus, ensures higher exploitation of questJon the leaving mnovator the for profit of rate The existance of two alternative techniques, then, raises the undecided technique respectively given by [(p+p')(A+A') +(L+L')] and [(p+p')A+L] respectively The difference in costs is given The first part is negative because of the assumption by (pA'+L')+p'A'

of cost reduction and second is negative because the vector p' is negative and the matrix A' is semi-positive The older technique is thus seen to be costlier at the new prices and any reswitching is ruled out.

The analysis so far brings out some central issues of the process of 4 the m fundamental IS innovators of role The development econom1c

process. They push through new techniques to earn higher profits and even after competition ensures that the gains of the change are passed on to the consumers, they continue to eam higher rates of profit than others because and the older The costs of the new pnces are of reswitches new evaluated at the problem

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-216- Kagawa U nzver.sity Economic Review 908 of the threat of availability of the old technique.. This rationale fOr the phenomenon of differential rates of profit is apart from, for example, the rationale of monopolistic competition as emphasized by Okishio [6] among others. 1n any case, even if one had started with the assumption of uniform rate of profit, it is inevitable that one would have had to contend with differential rates of profit while analysing technical change.. The process of capitalist development can not be understood otherwise 1n commenting on the antagonistic connection between the wages and profits, one must talk about some averaging of the different rates of profit.1t is quite clear that this indexing must involve the amounts of capital in various industries which, in turn, would involve the prices. A much more direct question of interest would, natural1y, be of the degree to which the price vector differs from the value vector, thanks to the phenomenon of profits. The rate of exploitation provides an ideal index of this degree of difference.. Most of the non-trivial critiques, including Roemer [8], [9], of the rate e are based on the argument of it not being a good enough proxy for various basic economic characteristics that the Marxian literature seeks to study..Most of these critiques, however, talk of profitsat a uniform rate.Once one takes into account the existance of differential rates of profit and the possibility of changes in them over time, one loses a hitherto pivotal variable the uniform rate of profit r, for the analysis of the process of accumulation The critique of the rate e has to be reconsidered in this light..Accumulation involves change in almost al1 of the economic variables like technology, values, prices, profits (even the vector of goods constituting wages which has been assumed as given in this paper because of a need to simplify) and everything else including the rate e.. Unlike the others, however, the variable e, being an index, is a scalar giving a simple and valid analytical tool to study accumulation Itwould admirably serve the purpose of

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909 On Techincal Change Differential Rates of Profit and Exploitation -217-classifying alternative paths of accumulation (exploitative or otherwise) and the central issues in economic theory like choice of technique can be adequately dealt with basing the arguments on the cIass interests

Appendi.x

Let a square matrix and its inverse be denoted by D and D-1 respectivel -y As DD-1 =1, differentiating both sides yields D'D-1

+

D[D-T =0 and after a rearrangement, [D-T = -D-1D'D-1

As for the numerical examples required, consider the foIIowing 2-sector case.. all=0..5, aI2=0..5, a2l=OJ, a22=02 Labour vector L=[O.,l 0..9].. The technical change takes place only in the first sector and the changes are a'll=O..0,1 a'12=0, a'21=0..0,1 a'22=0 and L'=一[0..08,0].. The value vector before and after the change are about [0,49, 143] and [0,.36, 1.34] respectively, The maximum uniform rate of profit possible before the

change is 0..61 and the technical change is cost reducing for the rate being less than 0.44 approximately. In the situation of rl =0..l1 and r2=0.l,. the price vector before and after the change is [0..69, 175] and [0,56, 166]

respectively.Ifwe take the vector b as [,10, 0.1]then the rate of exploitation goes up from 0.366 to 0.391 as a result of the change,. In

another scenario of the rates being rlニ0.01 and r2=0 02, the price vector

before and after the change is about [0..5,1148] and [0,.37, 1.39] respectivel

-y For the same vector b above the rate of exploitation can be seen to go down from 0.039 to 0..030 as a result of the change, It may be worth

pointing out that the figures in the example are rounded upto two decimal points for the presentation

References

CI

J

Fujimoto T“,Note on Tecnical Change and the wage-Profit Curve", Economic

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-218ー Kagawa Universzty EconomzじRevzew 910

C2J Fujimoto T“,Falling Rate of Profit in the Grundrisse", Kagawa Univer.sity Economic Review, 53, 1980, 109-115

[3J Fujimoto T. and Ranade R R ,"On how Exploitation Becomes Difficult in Two Sector Models", Kagawa Univer.sity E正onomicReview, 6,11989, 45-60

C4J Morishima M, Marx'.sEconomic,s.Cambridge University Press, U K 1973 [5J NakataniT, "Profit, Wage, and Technical ChangeConsidering the Case of

Durable Equipment", (in Japanese), Economic Review (Iwanami), 29, 1978,

7277

[6J Okishio N,“Monopoly and the Rates of Profit", Kobe Umversity Economic Review, ,11955, 71-88

C 7JOkishioN, The Fundamental Theory 0/Capitali.st Economy, (in Japanese),

Sobunsha, Japan, 1965

[8J Roemer J ,“Technical Change and the Tendancy 01 the Rate 01 Profit to Fall", Journal oj Economic Theor.,ァ 16, 1977, 403-424

[9J Roemer J, A General Theory 0/Exploitat!on and Clas.s, Harvard University Press, U S A, 1982

c

1

OJ Roemer J ,“Should Marxists be Interested in Exploitation?", Phoilosophy and Public Ajjairs, 14, 1985, 30-65

C 11 J Schumpeter J, Capitalism, Sociali.sm and Demo正racy,George Allen and

Unwin, U K, 1976 (5th edition)

ー !

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