Bowles and Gintis ([1]). Models with heterogeneous labour have been dealt with by many, notably in Potron ([24]), Okishio ([22]), Morishima ([18]), Bowles.
Labour Values in Models with Heterogeneous Labour Takao Fujimoto∗
Abstract This note is to point out a flaw in the definition of necessary labour made in the literature for models with heterogeneous labour, and to present a correct definition. In so doing, our model can be so general that it may include joint production, alternative consumption baskets for the reproduction of each type of labour service, durable consumption goods, and it allows for direct inputs of various labour services in the consumption baskets of workers.
1
Introduction
This note is a sequel to the author’s memorandum Fujimoto ([9]), and is to correct an error made therein about my wrong understanding of the ‘inadequate’ concept of necessary labour defined in Bowles and Gintis ([1]). Models with heterogeneous labour have been dealt with by many, notably in Potron ([24]), Okishio ([22]), Morishima ([18]), Bowles and Gintis ([1], [2]), Steedman ([25]), Hollander ([13]), Krause ([14], [15], [16]), and Fujimori ([7]). A flaw in the definition of necessary labour of a particular type is shared among most of these authors: the exceptions are Okishio, Morishima, and Fujimori. That is, when the necessary labour to reproduce one unit of a type of labour is ∗
Faculty of Economics, Fukuoka University
1
defined, inputs required indirectly to sustain the lives of workers who render labour of other types are completely neglected: thus the amounts of necessary labour are grossly underestimated. This flaw becomes evident when a model allows for direct labour inputs in the consumption baskets of workers. In section 2 we explain the flaw by use of a simple Leontief model, and present our new definition of necessary labour based on Fujimoto and Opocher ([12]). Section 3 includes a method of reduction of heterogeneous labour to abstract labour for a general input-output model with joint production as well as durable consumption goods. Some numerical examples are presented in section 4. In the final section, some remarks are given.
2
Necessary Labour
Let us consider a Leontief-type simple input-output model without joint production and with heterogeneous labour. There are n kinds of commodities, and m types of labour. We adopt the symbols in Krause ([16]). They are: A : a given n × n matrix of material input coefficients, processes as columns, L : a given m × n matrix labour input coefficients, B : a given column n × m matrix of consumptions per unit of labour, In : the n × n identity matrix, Λ : the n × m matrix, L(I − A)−1 . In a way similar to that for models with homogeneous labour, an important role was played by the matrices, Λ ≡ L(I − A)−1 and H ≡ L(I − A)−1 B. Notable authors, however, have made one and the same incorrect interpretation of Λ or H . 2
Bowles and Gintis([1, p.186]) wrote: “Let the value of good i be given by λi = (λ1i , . . . , λmi ), a vector of the direct and indirect labour hours of each of our m types embodied in a unit of good i. If we let Λ = (λr i ), then we have Λ = ΛA + L, so Λ = L(I − A)−1 . We call λr i the ‘r-value of good i’.” Krause([16, p.174]) also stated: “the value/surplus-aspect can be analyzed in terms of the m × n-matrix H = L(I − A)−1 B as follows. The entry hij of H gives the amount of labour of type i required directly or indirectly to reproduce one unit labour-power of type j.” Potron ([24, p.70]) as translated by Mori([17, p.523]) put: “[(i, k)element of (I − A)−1 B is the production of [good i] [directly and indirectly] necessary for the subsistence of a consumer of type [k]; [(k, k)-element of L(I − A)−1 B] represents the labour that this subsistence demands [directly and indirectly] from the category of labourers belonging to type [k].” It is quite clear that each entry of the matrix Λ does not show the labour hours of a particular type required directly and indirectly to produce one unit of a particular commodity. This is simply because in order to produce a commodity, other types of labour may also be necessary, and thus to sustain those types of labour some more commodities, and so the labour type concerned, are required. We should either incorporate the necessary commodities required by other types of labour as material inputs in a way von Neumann adopted ([27]), or devise out a new method as is done in Fujimoto and Opocher ([12]). Let us explain the latter. We first define some new symbols: µ ¶ µ ¶ In 0 A B I≡ and A ≡ . L 0 0 Im In this manner, commodities and various labour types are treated in a symmetrical way: the matrix B is the material input coefficient matrix in the household activities to produce labour services. Two more symbols are defined with i > n. 3
[i]
[i]
[i]
[i]
[i]
[i]
[i]
[i]
Λ[ i ] ≡ (λ1 , λ2 , . . . , λi−1 , λi , λi+1 , · · · , λ[ni ] ), and [i]
[i]
Λ[ i ] ≡ (λ1 , λ2 , . . . , λi−1 , 1, λi+1 , · · · , λ[ni ] ).
The new symbol Λ[i] stands for the (n + m)-vector of values in terms of the labour type i . Then, this should satisfy [i]
Λ[ i ] · I = Λ[ i ] · A.
(1)
This equation simply says that the value of a commodity or a type of labour is the labour hours of type i required directly or indirectly to produce one unit of that commodity or that labour type. Eq.(1) can be transformed to [i]
[i]
[i]
Λ[ i ] · I = Λ[ i ] · A+ (0, 0, . . . , 0, 1 − λi , 0, · · · , 0), which is next rewritten as follows: [i]
[i]
v · Λ[ i ] · I = v · Λ[ i ] · A+ ei ,
(2)
where v is a positive scalar and ei is the row (n + m)-vector whose i-th entry is unity with the remaining elements are all zero. Note that v has to satisfy [i]
v · (1 − λi ) = 1, [i]
and can be positive only when 1 > λi v ≥ 1. From eq.(2), we have [i]
(3)
≥ 0, which in turn leads to
v · Λ[ i ] = ei · (I − A)−1 .
(4)
From eqs.(3) and (4), we can solve Λ[ i ] . All we have to assume is n+m such Productiveness Assumption: There exists an x ∈ R+ that x À ·Ax. 4
In more general models where joint production as well as durable consumption goods are allowed for, we had better introduce two symbols, B and A. Different from the above, these two matrices are now rectangular as in von Neumann models, and each element can be nonzero, implying the possible joint production in commodities and in labour types, the existence of durable consumption goods, and of alternative processes for the production of commodities and of labour services. In the general case, we first solve the following linear programming problem (DG): (DG) max qi subject to q 0 B ≤ q 0 A + b(i) and q ∈ Rn+m , + where b(i) is the i-th row of B. Next, the values can be calculated exactly as [i]
λi =
qi∗ − 1 [i] and λj = qj∗ /qi∗ for j = 1, . . . , n, j 6= i. qi∗
The reader is referred to Fujimoto ([9]) and Fujimoto and Opocher ([12]) for the detailed explanation on how to derive these formulas. The latter contains also new definitions of “skilled labour” as well as “bads”.
3
Reduction to Abstract Labour
Our method confirms Steedman’s assertion ([25], [26]) that it is not necessary to reduce various types of labour to a certain type, so long as values and exploitation of each type of labour are concerned. One may, however, wish to estimate the overall rate of exploitation without using the actual employment data of various groups of labour services. This can be done by the reduction to abstract labour or any one type of labour in a way similar to that in Potron ([24]) and Krause ([15], [16]).
5
Let us denote an m-vector of conversion rates among labour types by c ∈ S m−1 , where S m−1 is the (m − 1)-dimensional simplex. Next we calculate the combined labour input coefficient vector, ` ≡ c0 · L. Using this vector `, we then compute the labour values of commodities Λ. Finally by use of ` and Λ, we can deduce the vector of values of various types of labour in terms of combined labour, c∗ . This c∗ is now normalized so that it is included in S m−1 , denoted by c∗∗ . The map from S m−1 into itself defined by a correspondence from c to c∗∗ satisfies the conditions required in Kakutani fixed point theorem. Thus, there exists an c for which we can find a value vector under which the values of various labour types are proportional to c, which fact can be used to yield the equal rate of exploitation among labour groups provided that the wage rates and the prices of commodities are such that each type of labour can consume no less than the required inputs. By denoting this proportionality factor by α, we have the uniform rate of exploitation, e, as e = (1 − α)/α. (See Krause ([16]).)
4
Numerical Examples
Let us consider a simple numerical example.. In our economy without joint production, there are two commodities and two types of labour: n = 2 and m = 2. Various data are given as:
A≡
µ
0 0 0 0
¶
,L ≡
µ
0.1 0.3 0.4 0.2
¶
, and B ≡
µ
2 1 0 0
¶
.
We regard the second type of labour as the standard of value, i.e., i = 4, and first augment the input matrix A by incorporating the
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required inputs to pay the first type, thus creating µ ¶ µ ¶ 0.2 0.6 0.8 −0.6 + + A = ,I −A = , and 0 0 0 1 µ ¶ 1.25 0.75 + −1 (I − A ) = . 0 1.0 So, the labour values of two commodities become µ ¶ ¡ ¢ 1. 25 0.75 ¡ ¢ 0.4 0.2 = 0.5 0.5 . 0 1.0
Next, we compute these values in our method explained in section 2, that is, by using eqs.(3) and (4). 0 0 2 1 0 0 0 0 A≡ 0.1 0.3 0 0 . 0.4 0.2 0 0 e4 · (I − A)−1
1 0 ¡ ¢ 0 1 = 0 0 0 1 0 0 0 0 ¡ ¢ = 1.0 1.0 2.0 2.0 .
0 0 1 0
0 0 0 0 0 0 − 0 0.1 0.3 0.4 0.2 1
2 0 0 0
−1 1 0 0 0
Therefore, we get
Λ[4] =
¡
0.5 0.5 1 0.5
¢
¡
,
¢ confirming the two values of commodities as 0.5 0.5 . On the other hand, based upon Potron ([24]) and Bowles and Gintis ([1]), we have µ ¶ µµ ¶¶−1 µ ¶ 0.1 0.3 1 0 0.1 0.3 −1 = L(I − A) = 0.4 0.2 0 1 0.4 0.2 7
and L(I − A)
−1
B=
µ
0.1 0.3 0.4 0.2
¶µ
2 1 0 0
¶
=
µ
0.2 0.1 0.8 0.4
¶
.
From these numerical results, we know what Bowles and Gintis([1, p.186]) wrote is wrong, because their two values (0.4, 0.2) are smaller than the ‘true’ values (0.5, 0.5). To compensate their underestimation, Boles and Gintis needed to increase the necessary labour required by other types of labour actually employed. Thus, a severe criticism by Catephores ([6, p.278]) is pertinent. (The reply by Bowles and Gintis ([3]) does not answer this attack by Catephores.) On the other hand, what Potron ([24, p.70, in the footnote]) might thought and the explanation by Krause ([16, p.174]) are ambiguous when they ‘use’ the word “indirectly” simply because labour is heterogeneous: indirect requirements of labour are taken into account only through material inputs, and not through the use of other types of labour. The interpretation of Potron and Krause can be made valid only if labour is made homogeneous in some way or other.
5
Remarks
5.1. Bródy ([5, p.23]) named the matrix A, the complete matrix. Our definition of values based on the complete matrix is derived from that in Fujimoto and Fujita ([11]). 5.2. The method of handling heterogeneous labour by Okishio ([22], [23]), which is adopted also by Morishima ([18]), requires the life-long data on how various types of labour are created, and each type of labour should be formed in a unique way. After all the method can work only in models without joint production. 5.3. Fujimori’s approach ([7]) to the abstract labour is somewhat 8
similar to Okishio’s ([22]). He uses, however, the actual employment data of various labour types, thus the method can be affected by those data which are not technological nor biological. See also Hollander ([13]). 5.4. A view on exploitation among heterogeneous labour, which has been suggested in Bowles and Gintis ([1], [2], and [4]), is presented in Fujimoto ([10]). 5.5. The author will proceed to consider the Fundamental Marxian Theorem in disequilibrium, adopting the method of reduction to abstract labour explained in section 4 above. (See Fujimoto ([8] for the theorem in disequilibrium).)
References [1] Bowles S., Gintis H. (1977): ‘The Marxian theory of value and heterogeneous labour: a critique and reformulation’, Cambridge Journal of Economics, 1, pp.173-192. [2] Bowles S., Gintis H. (1978): ‘Professor Morishima on heterogeneous labour and Marxian value theory’, Cambridge Journal of Economics, 2, pp.311-314. [3] Bowles S., Gintis H. (1981a): ‘Labour heterogeneity and the labour theory of value: a reply’, Cambridge Journal of Economics, 5, pp.285-288. [4] Bowles S., Gintis H. (1981b): ‘Structure and practice in the labor theory of value’, Review of Radical Political Economy, 12, pp.1-26. [5] Bródy A. (1970): Proportions, Prices and Planning: A Mathematical Restatement of the Labor Theory of Value, Akadémiai Kiadó, Budapest. 9
[6] Catephores G. (1981): ‘On heterogeneous labour and the labour theory of value’, Cambridge Journal of Economics, 5, pp.273-280. [7] Fujimori Y. (1982): Modern Analysis of Value Theory, Springer-Verlag, Berlin. [8] Fujimoto T.(1978): ‘Exploitation, profits, and growth: a disequilibrium analysis’, Economic Studies Quarterly, 29, pp.268275. [9] Fujimoto T.(2008): ‘Labour values: an exposition’, Fukuoka University Review of Economics, 52, pp.383-393. [10] Fujimoto T.(2009): ‘The concept of exploitation in a general linear model with heterogeneous labour’, Investigación Económica (UNAM), 68, pp.51-82. [11] Fujimoto T., Fujita Y. (2008): ‘A refutation of commodity exploitation theorem’, Metroeconomica, 59, pp.530-540. [12] Fujimoto T., Opocher A. (2007): “Commodity content in a general input-output model”, to appear in Metroeconomica. [13] Hollander H. (1978): “A note on heterogeneous labour and exploitation”, Discussionsbeiträge zur Politischen Ökonomie, 14. [14] Krause, U. (1979): Geld und abstrakte Arbeit. Über die analytischen Grundlagen der politischen Ökonomie, Campus, Frankfurt am Main. [15] Krause, U. (1980): ‘Abstract labour in general joint systems’, Metroeconomica, 32, pp.115-135. [16] Krause, U. (1981): ‘Heterogeneous Labour and the Fundamental Marxian Theorem’, Review of Economic Studies, 48, pp.173-178. 10
[17] Mori K. (2008): ‘Maurice Potron’s linear economic model: a de facto proof of ‘fundamental Marxian theorem” ’, Metroeconomica, 59, pp.511-529. [18] Morishima M. (1973): Marx’s Economics: A Dual Theory of Value and Growth, Cambridge University Press, Cambridge. [19] Morishima M. (1974): ‘Marx in the light of modern economics’, Econometrica, 42, pp.611-632. [20] Morishima M. (1978): ‘S.Bowles and H.Gintis on the Marxian theory of value and heterogeneous labour’, Cambridge Journal of Economics, 2, pp.305-309. [21] Okishio N. (1963): “Mathematical note on Marxian theorems”, Weltwirtschaftlisches Archiv, 91, pp.287-299. [22] Okishio N. (1965): “The Fundamental Theory of Capitalist Economy”, (in Japanese), Soubunsha, Tokyo. [23] Okishio N. (1977): Marx’s Economics: Values and Prices, (in Japanese), Chikuma-Shobou, Tokyo. [24] Potron M. (1913): ‘Quelques propriétés des substitutions linéaires à coefficients = 0 et leur application aux problèmes de la production et des salaires’, Annales Scientifiques de l’É.N.S. 3e série, t.30, pp.53-76. [25] Steedman I. (1977): Marx after Sraffa, New Left Books, London. [26] Steedman I. (1980): ‘Heterogeneous labour and ¿classicalÀ theory’, Metroeconomica, 32, pp.39-50. [27] von Neumann J. (1945-46): “A model of economic equilibrium”, Review of Economic Studies, 13, pp.1-9.
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