Heterogeneous Labour and the Fundamental Marxian Theorem

The Review of Economic Studies, Ltd.

Heterogeneous Labour and the Fundamental Marxian TheoremAuthor(s): Ulrich KrauseSource: The Review of Economic Studies, Vol. 48, No. 1 (Jan., 1981), pp. 173-178Published by: Oxford University PressStable URL: http://www.jstor.org/stable/2297130 .

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Review of Economic Studies (1981) XLVIII, 173-178 0034-6527/81/00140173$02.00

? 1981 The Society for Economic Analysis Limited

Heterogeneous Labour and the

Fundamental Marxian Theorem ULRICH KRAUSE University of Bremen

1. INTRODUCTION

As was pointed out by Morishima at the end of his famous book on Marx's Economics, the heterogeneity of labour is a "serious dilemma" for Marx's theory of exploitation, especially for the so called Fundamental Marxian Theorem. The problem, which should be separated from the peculiarities of joint production, consists in finding an appropriate reduction of the various types of labour to one common unit. Recently Bowles and Gintis have undertaken an interesting reformulation of the Marxian theory of value with respect to heterogeneous labour by avoiding any reduction at all (Bowles and Gintis 1977, 1978. Cf. also Steedman 1977, Holliinder 1978). But for the purpose of the Fundamental Marxian Theorem this only shifts the problem from the heterogeneity of labour to the heterogeneity of the respective rates of exploitation.

The aim of the present paper is to extend the Fundamental Marxian Theorem (without joint production) from homogeneous to heterogeneous labour (Section 3). This extension is based on a new concept called the standard reduction of labour, because of its dual relationship to Sraffa's standard commodity (Section 2). For some arguments a certain type of matrices, called Sraffa matrices, is needed, which generalize the quasi- irreducible matrices as introduced by Bowles and Gintis (Appendix). Standard reduction implies always a uniform rate of exploitation, whereas the same is true for the reduction by wage rates only in special cases (for example: uniform consumption; labour inputs only; prices proportional to labour values. For the first two cases cf. Morishima 1978). Hence, in general standard reduction and reduction by wage rates are different.

2. THE STANDARD REDUCTION OF LABOUR

Analysing the production of n commodities by means of these commodities and by m types of labour, we use the following notation (thereby disregarding the peculiarities of joint production):

A the n x n-matrix of material-input coefficients L the m x n-matrix of labour-input coefficients B the n x m-matrix of consumptions per unit labour x the n x 1-vector of gross outputs (the state of the system) p the 1 x n-vector of prices w the 1 x m-vector of wage rates

Obviously all these entities should be non-negative (?0) and A productive, i.e. p(A) < 1, where p( * ) denotes the dominant eigenvalue of a matrix which is square and non- negative. Assuming an equalized rate of profit r and wages paid in advance, the price system can be written as p = (1 + r)(pA + wL). Assuming furthermore, that workers do not save, i.e. w = pB, the price system becomes

p = (1 +r)pM, whereM=A +BL. (1) 173



174 REVIEW OF ECONOMIC STUDIES

Whereas the price/profit-aspect therefore can be analysed in terms of the n x n-matrix M, the value/surplus-aspect can be analyzed in terms of the m x m-matrix H = L(I - A)-'B as follows. The entry hi1 of H gives the amount of labour of type i required directly or indirectly to reproduce one unit labour-power of type j. Let y = Lx be the total amount of labour of various types in the state x. By a reduction of labour we mean a 1 x m-vector a which is semi-positive (20).

Using a reduction a, the total value added in state x is given by ay and the total reproduction value of the labour-power is given by aHy. We therefore define with respect to y and a the rate of surplus value e(a, y) as

ay -a(Hy (2) e(a, y)= aHy -al2

aHly

This rate is well-defined, if value added and reproduction value are not both equal to zero. (It turns out, that the rates of surplus value considered by Bowles and Gintis (1977, 1978) and Morishima (1978) are special cases of the above e(a, y) for suitable a and y:) Now, in the following we shall consider a type of reduction which plays a crucial role in analysing the heterogeneity of labour, similar to the role played by Sraffa's standard commodity in analysing the heterogeneity of capital. A reduction is called a standard reduction, if for some scalar e

a=(1+e)aH, whereH=L(I-A)F1B. (3)

From (2) and (3) it follows, that in the case of a standard reduction e = m (a, y) for all y such that value added and reproduction value are not both equal to zero. Hence the scalar e in (3) is a uniform rate of surplus value, called the standard rate corresponding to the standard reduction. By the Perron-Frobenius Theorem the equations (1) and (3) possess solutions p -0, r > -1 and a ? 0, e > -1, such that the rates r and e are given by the dominant eigenvalues p (M) and p (H), that is

1 - p(M) e=1 - p(H) 4 p (M) pe= (H) (4)

Therefore, if r and e are given by (4), the interplay between the rate of profit and the rate of surplus wanted by the Fundamental Marxian Theorem turns out to be a question of the relationship between the dominant eigenvalues of the matrices M and H.

3. A FUNDAMENTAL MARXIAN THEOREM WITH HETEROGENEOUS LABOUR

The crucial step in proving this theorem is the following inequality for the dominant eigenvalue of the sum of two matrices.

Lemma. LetS, Tbe two non-negative square matrices and s, t two real numbers such that s p (S + T)-t and p (S) < s, t. Then

t * p ((tI -S)-1 T) --p (S + T)-,e-s * p ((sI - S)-1 T).

Proof. For p(S) < h there holds the following identity

(hI-(S + T)) = (hI-S)(I-(hI-S)f1 T).

(1) Let p(S + T) < h'< h. Hence h'I - (S + T) is invertible. Because of p(S) < h', from the above identity it follows, that I-(h'I-S)-'T is invertible too. Therefore p ((h'I - S)f1 T) < 1, and because of h'< h hp ((hI - S)f1 T) '- h'p ((h'I - S)-1 T) -h'. By taking the infimum over all h', it follows that

hp((hI-S)f1T)_-'p(S+T) for p(S+T)<h.



KRAUSE FUNDAMENTAL MARXIAN THEOREM 175

Now, to prove the first inequality of the lemma, let p (S + T) _ t and p (S) < t. If p (S + T) < t, then the inequality just proved gives what is wanted by putting h = t. If p (S + T) = t, then

hp((hI-S)f1T) _-p(S+ T) for all h such that t<h.

By taking the supremum over all h, it follows because of p (S) < t, that t p ((tI - S)-1 T) _ p(S + T).

(2) To prove the second inequality of the lemma, let p (S) <s p (S + T). Hence p(S) < h for h = p(S + T), and hI - (S + T) is not invertible. From the identity at the beginning, it follows, that I - (hI - S)-1 T is not invertible. Hence 1 ? p ((hI - S)-1 T). Because of s _ h, from this it follows, that

h _ h p ((hI-S)-1 T) _ s p ((sI-S)-1 T),

which proves the second inequality of the lemma. ||

Theorem. Let the rate of profit r and the standard rate e be given by the dominant eigenvalues of M and H (as in (4)).

(i) If r O, then e r. (ii) If r O, then r e.

(iii) r > O if and only if e > 0.

Proof. Let us first see, that p(H) =p((I-A)f1BL). If U= (I-A)F1B, then (I-A)f1BL= UL and H=L(I-A)f1B=LU. Now, it holds generally, that p(LU)= p (UL) for two matrices U and L for which the products LU and UL are defined. Next, we apply the lemma to S = A and T = BL. Putting s = 1 and taking p (A) < 1 into account, we get the following implication:

If p(M) _ 1, thenp(M) _ p((I-A)-1BL)=p(H).

Putting t = 1, we get as a second implication:

Ifp(M)?_ 1, thenp(H)=p((I-A)-1BL) _ p(M).

Now, by assumption we have

1+r' 1+e

Therefore, from the first implication we get (i), and from the second one (ii). Putting (i) and (ii) together, we get (iii). 11

If the matrices M and H are irreducible, then by the Perron-Frobenius Theorem the assumption made in the above theorem is satisfied, that is, r and e are given by the respective dominant eigenvalues. More generally, using the concept of a Sraffa matrix as introduced in the Appendix, the following conclusion can be drawn from the theorem.

Corollary. LetMandHbe Sraffa matrices. Then there exista unique solution r> -1, p > 0 resp. e > -1, a > 0 of the price system resp. value system (up to a scalarforp and a) and the statements (i), (ii), (iii) of the above theorem are valid for r and e.

Proof. The existence of unique solutions mentioned in the corollary follows from Theorem 2 of the Appendix. The uniquely determined rates r and e then must be given by the respective dominant eigenvalues. Hence the corollary is implied by the theorem.

Remark 1. Statement (iii) of the theorem is an extension of the usual Fundamental Marxian Theorem from homogeneous to heterogeneous labour. In the case of




homogeneous labour, i.e. m = 1, L becomes an 1 x n-vector 1 and B becomes an n x 1- vector d. Hence the matrix H reduces to the real number h = Ad, where A = 1(I -A)-1 denotes the vector of homogeneous labour values. Standard reduction and standard rate in this case are uniquely determined as ac= (up to a scalar) and e = (1- Ad)/Ad. Therefore (iii) states that r > 0 if and only if Ad < 1. The statements (i) and (ii) provide additional information to the Fundamental Marxian Theorem as stated by (iii), this even in the case of homogeneous labour.

Remark 2. Statement (iii) may be false, if the assumptions made in the theorem or in the corollary are not satisfied. Even in the case of homogeneous labour then the Fundamental Marxian Theorem may break down as the following simple example shows. Let

m=1 n=2 and A=[o ?1 B=d=[j, L=l=[1,1].

From this

M= [o 2j H= h = 1

and therefore M and H are both Sraffa matrices. The unique solutions mentioned in the corollary are e =0, a = 1 resp. r =0, p = (1, 2) (up to a scalar for a and p) and according to the corollary (iii) is valid for e = 0, r = 0. But the price system has also the solution f = 1, p = (0, 1), and for the rates e, F statement (iii) is false. Considering these rates e, F the assumptions of the corollary are not satisfied, because not F > 0 and the assumption of the theorem is not satisfied, because F is not given by the dominant eigen value p (M) = 1.

Remark 3. The characteristic feature of the Fundamental Marxian Theorem as stated above is that it connects the rate of profit (which is uniform) with a uniform rate of surplus value. Because e is uniform, e does not depend on the state x of the system, and therefore (iii) holds independently of the state x of the system. Especially it is not required that the system is in a self-reproducing state x, that is Mx x. (This requirement is the central topic in the discussion between Bowles, Gintis and Morishima, cf. Bowles, Gintis 1978 and Morishima 1978.) It seems to be the essential point of any Fundamental Marxian Theorem that it gives a relationship between r and the reproduction in terms of labour values, but not reproduction in physical terms. That these two types of reproduction differ can be seen from the following example (of course, reproduction in physical terms implies reproduction in terms of labour values):

m = n = 2, A= ] B=[? ] L= [2 ]

From this

M [= 23] H=[8 24] and O<p(M), p(H)<1. 10 8i 8

Because of p(M) <1 there exists a self-reproducing state, but the state x = (2, 1) for example is not self-reproducing. Nevertheless, the rate of surplus value e (a, Lx) is strictly positive for all reductions a. In this example the Fundamental Marxian Theorem as stated above is true, also in the absence of physical reproduction. This matters, because it may be advantageous for a system not to reproduce itself physically, as is shown in international trade theory.



KRAUSE FUNDAMENTAL MARXIAN THEOREM 177

APPENDIX

Every semipositive square matrix T can be transformed by a simultaneous permutation of columns and rows in the canonical form of Gantmacher, consisting of block submatrices Tij, with 1 ' i, js, such that

(a) Ti = Ti is square and irreducible for all i and Tij = 0 for i < j. (b) for some g with 1-g-' s: If i-' g, then Tij = 0 for all 1-j-i - 1. If i > g, then

Tij 1 0 for some 1 _- j- i - 1. Using the formula p(T) = maxi p(Ti) for the dominant eigenvalue p(), a theorem

proved by Gantmacher can be formulated as follows (Gantmacher 1959, Chapter XIII, Section 4).

Theorem 1. Tx = Ax has a solution A _ O, x > Oif and only if the canonicalform of T possesses the following property:

p(Ti)= p(T2)=** =p(Tg)> p(Tg+) for 1 is-gi (G)

Remark 4. A in the theorem is uniquely determined as A = p(T). Furthermore, it follows from the theorem, that, if the canonical form of T possesses (G) and if yT =,y with A ' 0, y 0 O, then ,u = p (T).

Definition. A semipositive square matrix S is called a Sraffa matrix, if the canonical form of the transpose S' possesses the Gantmacher property (G) together with g = 1.

The term "Sraffa matrix" is employed, because this type of structure seems to be considered by Sraffa as the only case of reducibility which is economically meaningful. (Cf. the example of beans in Sraffa 1960.) The importance of Sraffa matrices lies in the following fact.

Theorem 2. yS = Ay has a unique solution A -0, y > 0 (up to a scalar) if and only if S is a Sraffa matrix.

Proof. Putting T for the canonical form of S', the assertion of the theorem becomes: Tx = Ax has a unique solution A 0 O, x > 0 (up to a scalar) if and only if T possesses

property (G) together with g = 1. Taking an appropriate partition x = (xi, . .. , xs), the equation Tx = Ax can be written as

Tjxj1=Axi for 1 ' j-' g and

Tjxj+ i=jTjixi=Axj forg+1 _ j _ s.

(1) Assume first, T possesses (G) and g = 1. By Theorem 1 there exist A 0 O, x > 0 such that Tx = Ax. Let :0, z >0 be another solution, that is Tz = Az. From the irreducibility of T1 it follows that A = A = p(T1) and z1 = c x1 for some scalar c. Assume that for some k ' 1 zi = c x Xi is already proven for all 1? i - k. Because of g = 1, this implies Ac * Xk+1 -Tk+lc * Xk+1 = AZk+1 -Tk+lZk+l. Because of (G) together with g = 1 A >p(Tk+1), and therefore AI- Tk+j is invertible. Hence Zk+1 = C * Xk+, and by induction z = c * x.

(2) Assume next, Tx = Ax has a unique solution A 0, x >0 (up to a scalar). By theorem 1 T possesses (G). Define for 1 _ i _ g zi = i * xi. Because of (G) A > p (Tg+k) for 1 - k, and therefore, the definition

Zg+k = (AI - Tg+k)f1 Eg=k1 Tg+k,iZi for 1 _ k ' s - g

is meaningful, defining recursively Zg+l, Zg+2, ... , Zs. If Z (z1, ... , Zg, ... ., Zs) then z >0 and Tz = Az. Uniqueness then implies z = c x for some scalar c, especially cxi = zi = i * xi for 1-i-g. Hence c = i for 1 ' i-g, which implies g = 1. 11




Remark 5. Every quasi-irreducible matrix (cf. Bowles and Gintis 1977) is a Sraff a matrix, but not vice versa.

Remark 6. The notion of a Sraffa matrix can be used, to characterize irreducible matrices as follows: S is irreducible if and only if S and S' are Sraff a matrices. The concept of a Sraffa matrix is, roughly speaking, the "dual half" of the stronger concept of an irreducible matrix.

First version received June 1979; final version accepted June 1980 (Eds.).

Helpful criticisms from two referees and the editor of this journal on an earlier draft are gratefully acknowledged.

REFERENCES BOWLES, S. and GINTIS, H. (1977), "The Marxian Theory of Value and Heterogeneous Labour: A Critique

and Reformulation", Cambridge Journal of Economics, 1, 173-192. BOWLES, S. and GINTIS, H. (1978), "Professor Morishima on Heterogeneous Labour and Marxian Value

Theory", Cambridge Journal of Economics, 2, 311-314. GANTMACHER, F. R. (1959) Applications of the Theory of Matrices (New York: Interscience). HOLLANDER, H. (1978), "A note on heterogeneous labour and exploitation" (Diskussionsbeitrage zur

Politischen Okonomie, Nr. 14). MORISHIMA, M. (1973) Marx's Economics (Cambridge: CUP). MORISHIMA, M. (1978), "S. Bowles and H. Gintis on the Marxian Theory of Value and Heterogeneous

Labour", Cambridge Journal of Economics, 2, 305-309. SRAFFA, P. (1960) Production of Commodities by Means of Commodities (Cambridge: CUP). STEEDMAN, I. (1977). Marx after Sraffa (London: NLB).



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Heterogeneous Labour and the Fundamental Marxian Theorem