versão impressa ISSN 0185-1667
Inv. Econ vol.68 no.267 México jan./mar. 2009
The concept of exploitation in a general linear model with heterogeneous labour
El concepto de explotación en un modelo lineal general con trabajo heterogéneo
Fukuoka University <email@example.com>
Received October 2007
Accepted August 2008
En este trabajo consideramos el concepto de explotación y el llamado teorema fundamental marxiano en un modelo lineal general en el que se toman en cuenta la producción conjunta, el trabajo heterogéneo y el consumo de bienes durables. Una de las características de nuestro enfoque es que nos ocupamos de los bienes y de diferentes tipos de trabajo de una manera totalmente simétrica en lo que concierne a los datos tecnológicos. En un comienzo no hay capitalistas, sino un espectro de trabajadores que ofrecen diferentes tipos de trabajo así como otros servicios. Algunos trabajadores pueden ahorrar o tener propiedades que rindan ingresos diferentes de los salarios. Cuando los precios y los salarios están dados, podemos determinar qué tipos de trabajo son explotados y cuáles explotan a otros. Todas las proposiciones principales son validas independientemente de si nuestro modelo de economía está en una suerte de equilibrio.
Palabras clave: explotación, teorema fundamental marxiano, trabajo heterogéneo, modelo de von NeumannMorishima.
Clasificación JEL: B24, B51, D46
In this paper we consider the concept of exploitation and the socalled Fundamental Marxian Theorem in a general linear model in which proper joint production, heterogeneous labour, and durable consumption goods are all allowed for. One of the features of our approach is that we deal with goods and various types of labour in a completely symmetrical way so long as technological data are concerned. There are no capitalists in the beginning, but a spectrum of workers who supply various types of labour as well as other services. Some workers may make savings, or have properties which yield incomes other than wages. When prices and wage rates are given, we can determine which types of labour are exploited and which types exploit others. All the main propositions are valid irrespective of whether or not our model economy is in a sort of equilibrium.
Key words: exploitation, fundamental Marxian theorem, heterogeneous labour, von NeumannMorishima model.
The Fundamental Marxian Theorem (FMT), as formulated by Okishio (1963), asserts that in a capitalist economy exploitation is necessary to realize profits. After the original contribution by Okishio, there have appeared many contributions which generalize or criticize his result, notably by Morishima (1973, 1974) and Roemer (1981, 1982). In these it is assumed the economy is in equilibrium, or that labour is homogeneous. Some have claimed that the economy should be in a sort of equilibrium in order to reproduce the present system, i.e., production and class relationships. If, however, the FMT is valid only in equilibrium, its scope is so limited that the proposition becomes uninteresting. It has been also argued that Walrasian general equilibrium or micro foundation for the FMT is desirable. Again if the FMT is valid only when all the people maximize their utility function under some constraints, it is not applicable to the real world. We had better construct the robust concept of exploitation and consider the validity of the claim made by FMT in a general disequilibrium state for a snapshot period, say this year.
Our framework is similar to that of Morishima (1974). It is, however, considered independently of equilibrium conditions, and we allow for the existence of heterogeneous labour, and workers may save and have properties, which will bring them extra incomes. Besides proper joint production, thus durable capital goods, as well as durable consumption goods are admitted. The essence of our method is that various types of labour and normal goods are treated in a symmetric way, as expounded in Fujimoto and Opocher (2007).1 As in Morishima model, capitalists are not explicit, or simply they do not need to exist as those who are eager to exploit others. In our model there are only various types of workers, who render their services such as lathing, educational or managerial ones. When prices and wage rates are specified, it is possible to tell which types of labour are exploited, and some workers may well feel themselves as nonworkers, or even as celebrated capitalists. Our model does not specify how a particular type of labour is created from other types. Various types are just given at the beginning of a production period, and at the end of this period they may change to other types through onjobtraining or educational processes, all in the background. We simply need processes which reproduce each type of labour by consuming certain sets of commodities and services. These processes are called household activities in this paper, and we allow for the existence of alternative household activities to reproduce each type of labour. Our definition of values needs no price, and the propositions make clear some basic properties of the values defined. Our definition of exploitation, on the other hand, surely needs prices including wage rates. We argue that exploitation, or more precisely the exploiters and the exploited can in general exist among workers themselves, in models with heterogeneous labour even without typical capitalists.
The structure of this paper is as follows. Second section tells the reader a short history of the FMT and the merits of our model. In third section, in order to make this note easier to understand, we explain our definition of labour values and our approach, using a simple Leontief model, and then fourth section describes our model and defines the values of goods (and services) and various types of labour in terms of a particular type of labour. Fifth section defines the concept of exploitation. The main definitions and propositions are presented in fourth and fifth sections. Sixth section gives three numerical examples in order to show how our definitions work in a concrete way, though in abstract models. Finally concluding remarks, especially those concerning the links between the real world capitalist system and the new definitions as well as a new analytical method discussed in this paper, are stated in seventh section.
A SHORT HISTORY OF THE FUNDAMENTAL MARXIAN THEOREM
Okishio (1963) first established the FMT in a Leontief model of circulating capital and homogeneous labour. What he showed is that the existence of positive profits is equivalent to the existence of exploitation. Exploitation in Okishio (1963) means that the labour value of the consumption basket for one unit of labour force is less than unity. After the first contribution, Okishio (1977) continued to generalize his FMT to linear models with durable capital goods together with his students.
Morishima (1973, 1974), on the other hand, extended the FMT to a von Neumann model of joint production, thus incorporating durable capital goods. In these Morishima's contributions, exploitation exists when the minimum amount of labour force required to produce what has been consumed by the working class in a given production period is less than the actual amount of labour force used in the period. There appeared some papers claiming the finding of counterexamples: Steedman (1975) and Petri (1980). The former counterexample is related to whether one adopts the equality system or the inequality system in a model of joint production, while the latter is concerned with a model with so low a level of technology that no positive growth is possible.
Fujimoto (1978) established the FMT in disequilibrium employing a von Neumann model, and using the social average profit rate as well as the social average growth rate. Until this paper, the FMT had been dealt with in equilibrium only. Fujimori (1982) tried to generalize the FMT to models with heterogeneous labour by reducing various types of labour to a particular class of labour, e.g., unskilled one. Krause (1981) gave a method of reduction from various types of labour to a particular type. Heterogeneous labour has also been dealt with, e.g., in Bowles and Gintis (1977), Okishio (1977), and Steedman (1977). In their models, either single production is assumed or reduction to a particular type of labour is pursued. Moreover, workers' consumption basket is fixed depending on their type.
Then appeared a seemingly fundamental criticism against the FMT, i.e., the commodity exploitation theorem. This commodity exploitation theorem by Bowles and Gintis (1981) and Roemer (1982, 1986) attacks the FMT by insisting that only labour is given a special role in determining values. When commodities and labour are dealt with in a symmetrical way, any commodity would be exploited alike. The cause of this confusion comes from the misunderstanding that the existence of exploitation can be determined within the value world without introducing prices and wages. Even in the paper by Okishio (1963), it is implicitly assumed that prices are such that one unit of labour can obtain the designated basket of consumption goods by spending his/her wages. We cannot discuss on the existence of exploitation without knowing how much workers can buy using their wages. This is the point made explicit in Fujimoto and Fujita (2008).
In a recent paper, Matsuo (2008) has introduced utility functions for workers, in order to avoid the counterexample by Petri (1980) and to renounce the commodity exploitation theorem. It is desirable, however, to establish the FMT without borrowing the concept of individual optimization from the neoclassical world. Exploitation should be defined and grasped irrespective of the existence of utility functions, and, if at all, irrespective of whether workers maximize their utility or not.
Intuitively, in a model with homogeneous labour, if we choose that homogeneous labour as the standard of value, the value of the total net output of a period is the amount of labour rendered by the working class in the period. Then, it is natural to define the existence of exploitation to be the case where not all the total net products come into the workers' hands. Therefore, when there are positive profits, all or some portions of which escape from workers and are used to purchase some of net products, we can declare exploitation is going on. This natural way is in the background in fifth section (more precisely subsection 5.1) of this paper so long as a model with homogeneous labour is concerned.
In addition to the above intuitive approach to the phenomenon of exploitation, our model below has other merits. In the literature so far published on the FMT, none allow for durable consumption goods, nor for the existence of alternative household activities to reproduce labour. None treated values in a model with heterogeneous labour without reducing various types to simple labour, except for Bowles and Gintis (1977). Our model in section 4 can include these elements in a natural way, with no worry about the reduction problem. Besides, workers can save and have properties which bring them incomes other than wages. In a sense, some workers may be 'havea little's or simply 'haves': this fact has been neglected in the literature on the topic.
LABOUR VALUES FOR A LEONTIEF MODEL 2
We first explain our definition of values and our method by employing a Leontief model of circulating capital. Let us assume there are n kinds of commodities [goods and services (other than direct labour)] and one type of homogeneous labour. Let A be the nxn material (and service) input coefficient matrix, I the nxn or (n + 1)x(n + 1) identity matrix depending on the context. The symbol, , means the row nvector of labour input coefficients, and c the column nvector of workers' consumption basket which enables a household activity to reproduce one unit of labour in a production period. The row nvector of labour values are written as λ. The symbol R+n means the nonnegative orthant of the ndimensional Euclidean space, Rn. A prime to a vector indicates its transposition. A subscript attached to a vector, as in xi or (Ax)i , means the ith element of the vector. Following Okishio (1963), the labour values of normal commodities are determined by
while the labour value of one unit of labour (force), , is calculated as the inner product
Now we define an (n + 1)x(n + 1) matrix A as
which is termed the 'complete matrix' in Bródy (1970, p. 23). The last nth column of A may be called the input of the household activity to reproduce one unit of labour force. The above two equations  and  can be combined to lead to
It is important to note here that in the above equation, the direct input of labour is given unity as its value on the RHS, while the value of one unit of labour is calculated as on the LHS. Then equation  is rewritten as
where 0n is the row nvector whose elements are all zero. This leads to
where v is a positive scalar. Supposing (1 ) > 0, we normalize v so that v(1 ) = 1. By putting q (vλ, v), we finally obtain the equation
We make the following assumption.
Assumption A1 (productiveness assumption). There exists a nonnegative column (n + 1)vector x such that x > Ax. Thus, (I A)1 ≥ 0.3
Now equation  is solved for q as
On the other hand, from equation [3']
It is clear from equation  that q is the last row of the inverse (I A)1, thus the above equation gives
It naturally follows that 0 ≤ < 1 and λj ≥ 0 for j = 1, ..., n, i.e., the labour value of one unit of labour is less than one. When labour is indispensable to produce a basket c, we have 0 < . (See Fujimoto and Fujita, 2008). Therefore, when the productiveness assumption A1 is given, all we have to do is to solve equation  first, and then calculate labour values using equations  and .
Now even in a Leontief model we are using here, there exists a set of inefficient processes hidden, typically those which waste more or less resources. With these inefficient processes included, the input coefficient matrix A is enlarged to have more columns than rows: let us suppose A is now (n + 1)x(m + 1) with n ≤ m. We still assume there is only one homogeneous labour, and the household activity is on the last (m + 1)th column of A. Equation  becomes
So, equation  is now to be modified to an inequality system
The problem is how to define the labour values and also how to compute them. The answer has already been supplied by the wellknown nonsubstitution theorem.4 We first solve the following linear programming problem:
where , i.e., the (n + 1)column vector whose (n + 1)th entry is unity with all the remaining elements being zero. Then, the labour values are obtained by applying equations  and . Thus, our definition of labour values can be stated as:
Definition of values for a Leontief model: labour values in a Leontief model with alternative processes are nonnegative magnitudes assigned to commodities such that the labour value of labour itself be maximized under the condition that the total value of the output of each possible process should not exceed that of the input. When calculating the total value of the input of a process, unity is assigned to the direct input of the labour.5
And our assumption above is to be modified as:
Assumption A2. There exists a nonnegative column (m + 1)vector x such that x > Ax.
The nonsubstitution theorem has made it clear that a sort of efficiency on the output side is involved because of the duality in linear programming. That is,
This programming problem dictates that while producing one unit of labour force as net product, we should minimize the gross production of labour force.6 Because of our assumption A2 just stated, the problem (PL) has an optimal solution, and so does the problem (DL). We can now present:
Proposition 1. Given the assumption A2, the labour value of (homogeneous) labour is less than unity.
Proof. By the duality theorem, we know that the optimal values satisfy q*n+1 = x*m+l. On the other hand, it is clear from the constraint in (PL) that x*m+1 ≥ 1. Therefore we get q*n+1 ≥ 1, which gives 0 ≤ < 1 because of equation . QED.
It may be interesting to apply the principle of algebra in interpreting an optimal solution x*m+1 to the problem (PL). Suppose we knew the labour amount which has entered directly and indirectly into one unit of labour, and write this amount as , then the magnitude x*m+1 (1 ) shows the net residual which is not consumed in reproducing x*m+1 units of gross production of labour force. Since in the problem (PL), the system produces one unit of net labour force, we should have
which corresponds to equation .
LABOUR VALUES IN A GENERAL MODEL
Our model is a variant of Morishima model (Morishima, 1964) slightly generalized. That is, different from the original model in von Neumann (19451946), we explicitly deal with labour input coefficients, and moreover we allow for the existence of heterogeneous labour as well as proper joint production. Then, naturally we can deal with durable capital goods. Various types of labour are treated exactly like normal commodities, and so we use the symbol B and A as the output and input coefficient matrices, both of which now have n rows and m columns. There are altogether n kinds of goods, services, and various types of labour.7 On the other hand, there exist m production processes or household activities. This way of formulation enables us to take into consideration durable consumption commodities in household activities: a durable consumption commodity in a column of household activity of A will appear in the corresponding column of B as one period, say one year, older commodity. For each type of labour, there can be more than one household activities to reproduce that labour. Workers may save a part of their incomes, and may have properties. These complicating elements from the real world do not disturb our study while we deal with values and exploitation: this should be true in any model including Leontief models.
Now it is possible to select any commodity or a type of labour as the standard of value because goods and labour are treated in a completely symmetric way. Here, however, we choose a type of labour as the standard and let it be the ith labour commodity. We give our definition of labour values as follows:
Definition of values for our general model: values in a general inputoutput model are nonnegative magnitudes assigned to commodities (including services and various types of labour) such that the value of the standard commodity be maximized under the condition that the total value of the output of each possible process should not exceed that of the input. When calculating the total value of the input of a process, unity is assigned to the direct input of the standard commodity.8 Our productiveness assumption here is:
Assumption A3. There exists an x R+m such that
where e[i] is the ncolumn vector whose ith entry is unity with all the remaining elements being zero.9
Having defined values as above, we can now explain how to compute values in a way similar to the problem (DL) in section 3. Let us first define the following vectors:
The vector Λ[i] is the vector of values with ith labour being the standard of value, and the element λj[i] stands for the value of commodity j with icommodity as the standard of value. Our definition above is rewritten like this:
We can proceed as we have done for a Leontief model in third section. That is, the constraint in this problem can be transformed first through adding (1 λ i[ i]) · b(i) to both sides, then multiplying both sides by nonzero nonnegative v, yielding
where b(i) is the ith row of B. Then, we set
This normalization yields as our constraint
Since we have λ i[ i] = 1 1/v from our normalization, maximizing v is equivalent to maximizing λi[ i]. Writing v · simply as a variable vector q, thus qi v, we have the linear programming problem (DG):
Finally, the values can be calculated exactly as in equations  and , i.e.,
It is not difficult to establish:
Proposition 2. Given the productiveness assumption (A3), the ith labour value of ith labour is less than unity.
Proof. Consider the linear programming problem
By the duality theorem, we know that the optimal values satisfy qi* = b(i)x*. On the other hand, it is clear from the constraint in (PG) that b(i)x* ≥ 1. Therefore we get qi* ≥ 1, which gives 0 ≤ λi[ i] < 1 because of equation . QED. Since various types of labour and commodities are treated in a symmetric way, we can define the jth commodity values of goods and labour exactly as in the definition above, and establish:
Proposition 3. Given the assumption (A3) with e[i] replaced by e[j], they jth commodity value of jth commodity is less than unity.10
At the end of this section, let us briefly discuss a model in Morishima (1964) and that in Bowles and Gintis (1977). In Morishima's model, there is only one type of homogeneous labour, and a single household activity to reproduce this homogeneous labour, with no durable consumption goods involved. Thus, B and A are of the following special form:
where B and A are, respectively, the (n 1)x(m 1) output and input coefficient matrices of normal production processes. In this case, the problems (DG) designating labour as the numeraire coincides with Morishima's problem (DM) in Morishima (1974):
Therefore our model and definition of values include Morishima's as a special case.
When B is the identity matrix, the definition of values by Bowles and Gintis (1977) is again a special case of ours. They, however, regard propositions 2 and 3 above as the existence of exploitation. Each of them is merely an alternative expression of productiveness (A3) as is explained in Fujimoto and Fujita (2008). It is not difficult to recognize that the commodity values defined by Manresa et al. (1998) also form a special case of ours.
Our definition of values and our method of calculating them are useful to generalize the Fundamental Marxian Theorem as presented in Morishima (1974). Our model is surely more general than von NeumannMorishima model used by him, allowing for heterogeneous labour, a plural number of household activities to reproduce labour power of each type, and durable consumption goods, to say nothing of proper joint production in ordinary production processes. We do not need such concepts as "warranted rate of profit" or "capacity growth rate" defined in Morishima (1974). Moreover we need to define the concept of exploitation in a careful way because in our model there are various types of workers and they may have properties which increase their total income.
We now suppose the nth commodity is a type of labour, nth labour for short, and make the assumption (A3) in the preceding section for the vector e[n]. Let us suppose the household activity vectors to reproduce the nth labour are normalized so that they appear like this.
That is, each household activity produces one unit of nth labour by use of α unit of its own nth labour and a commodity (including other types of labour) basket c. We consider only those activities whose α satisfies 0 ≤ α < 1. Since some commodity in the basket c can be durable, we have on the output side a vector d, which may not be the zero vector. Both c and and d are a column (n 1)vector. One more assumption we make is:
Assumption A4. Any optimal vector x* for the problem (PG) uses at least one of the activities that reproduce the nth labour. We can prove:
Proposition 4. Let commodity i such that i ≠ n, ci > 0, be arbitrarily chosen. Given the assumptions A3 and A4, the ith commodity value of nth labour is always greater than the ith commodity value of the net consumption basket (c d) for any household activity used in an optimal vector of the programming problem (PG).
Proof. Take an optimal vector q* for the problem (DG). By the assumption (A4), we can find at least one pair of vectors, which is actually used, of the type (10) among the household activities which reproduce the nth labour. For these vectors, we have a strict equality in the constraint in (DG) because they are used. Thus, we get
Divide both sides of this equality by qi* , and we obtain, because of equation 
This equality yields
because ci > 0 and λ i[ i] < 1 by proposition 3. In sum,
This ends the proof. QED.
This proposition tells us that if workers of type n exchange one unit of their labour force for the basket c and leaves d after one period, they make a loss in terms of any commodity as the standard. Equation , when commodity i is in fact the nth labour, i.e., i = n, becomes
Thus, because of proposition 2, we get
From this we get:
Proposition 5. The n th labour value of the net consumption basket (c d) for any household activity used in an optimal vector of the programming problem (PG) is less than unity.
First, we define exploitation for the case where there exits single homogeneous labour. Let us denote by p (p1, p2, ... , pn) the row nvector of prices and wage rate. Specifically we write wn for the wage rate pn, supposing the nth commodity is the homogeneous labour. These prices and the wage rate are what we observe for a period under consideration, and are not necessarily some magnitudes which equilibrate certain markets. Let us denote by yn the total income of a typical worker for this particular period: yn wn·L + un, where L is the amount of labour supplied and un is the other incomes accruing to his/her properties or businesses as profits, both for a given period. Denote by ωn = yn/L, the income per unit of labour supplied. Clearly when un= 0, ωn = wn. In this subsection, the net consumption basket (c d) is the one actually used.
Definition of exploitation: a worker is exploited when his ωn satisfies
The meaning of this definition is like this. When the above inequality were an equality, we have
The left term on the LHS shows how many baskets (c d) could be bought by spending the total income yn. Hence the LHS as a whole means the amount of labour value which could be obtained in exchange for the total income yn, which is the reward for participating the production system and supplying labour force, L. Thus the above equality can be regarded as guaranteeing an equal exchange. Our definition requires that the income per unit of labour supplied be less than this level.
In the literature, the FMT is often discussed for models where there is single homogeneous labour, workers have no property to earn extra income, i.e., un = 0, and the wage rate is just enough to buy the consumption basket c with d = 0 due to the absence of durable consumption goods. Given these assumptions, i.e., ωn = wn = pc and d = 0, workers are exploited based upon our definition above, simply because
as is clear in the expression . In other cases, together with the assumption un = 0, a kind of equilibrium is stipulated in a Leontief model, e.g., a uniform positive rate of profit is prevailing among industries, whence follow the inequalities pj/wn > λj[n] for all j ≠ n, i.e., each wageunit price is larger than its corresponding labour value, which is enough to prove the existence of exploitation in our sense, provided the vector (c d) is nonnegative and nonzero. Profits are appropriated by the capitalists who are latent in our economic system.
The phenomenon of exploitation can be looked at from a different viewpoint. Let us consider the output, and assume that in a given period the economy's performance is described by
where x is an mcolumn activity level vector, and f an ncolumn net output vector.11 Both x and f are supposed to be nonnegative nonzero vectors, and in particular labour is not involved in f, i.e., fn = 0. Premultiplying by q the both sides of the output equation above, yields
On the other hand, postmultiplying by x both sides of the constraint in the programming problem (DG) with i = n, gives
Divide this inequality by qn, and we get
where α(i) is the ith row of A. This inequality means the following almost tautological statement.
Proposition 6. The total labour value of net output plus the labour value of reproduced labour cannot exceed the total labour utilized.
This proposition is valid even in the case of heterogeneous labour provided that no type of labour is involved in the net output vector f and that values are measured by a particular type of labour. This proposition 6 combined with our definition of exploitation implies
Proposition 7. Workers as a whole are exploited unless they have the income which would enable them to buy out the whole net output. And thus
Proposition 8 (A Fundamental Marxian Theorem). In an economy of homogeneous labour, the working class is exploited when there exist profits not all of which accrue to the working class, and some or all profits are used to buy a portion of the total product.
This is the most general FMT among those presented so far in models of homogeneous labour. Our economy may be in disequilibrium, and workers may have properties which give them extra income.
Heterogeneous labour and exploitation among workers themselves
Let us return to the case of heterogeneous labour. If we employ the definition of exploitation for the case of homogeneous labour, propositions 6 and 7 dictate that each type of workers should have the exclusive right over the whole net output; otherwise, that type is exploited. Thus each type is exploited when they share the net output without monopolizing the entire net output. Therefore, in the first instance, we have a proposition similar to proposition 8.
Proposition 9 (A Generalized Fundamental Marxian Theorem). In an economy of heterogeneous labour, the working classes as a whole are exploited when there exist profits not all of which accrue to them.
In the case of heterogeneous labour, however, the net output is a joint product by various types of labour. So, while it is natural that all the net output goes to the workers as a whole as expressed in proposition 9, it is selfish for workers of some type to claim the whole net output, insisting that its labour value is less or equal to their contribution to its production. Thus, there can be exploitation among workers themselves. To define such a concept, we need to devise out some fair division of the output.
Let us suppose there are k types of labour, 1 < k < m. We continue to consider the output equation
where x and f are supposed to be nonnegative nonzero vectors. As in section 5.1, we assume no type of labour is appearing in the net output vector f 12 Here we adopt a simple concept of "fairness" in the division of the net output among workers. The total labour force rendered by ith labour in the production expressed by equation  is
It is important to note that the labour force which has contributed to the production of net output is at the maximum13
And so, the contribution ratio of ith type to the total labour force is
where T is the set of indices for all the types of labour, its cardinality being k. On the other hand, the ratio of what the itype workers could buy to the total money value of the net output is
where wi is the wage rate for the itype workers and ui the other incomes accruing to individual properties of itype workers. The net consumption basket (ci di) is the one actually used for the type i. The numerator shows what we may call the net purchasing power of itype workers for the net output, after deducting the necessary expenses to sustain their life.
It should be remembered that if our economy is decomposable, we consider each decomposed part separately, and if that decomposed part of the economy involves only one type of labour, we merely apply the definition of exploitation for the homogeneous case.
Definition of exploitation among workers: when our economy is indecomposable, workers of type i are said to be exploited when his/her wage rate w, and the other income u, are such that
S[i] < r[i]
In the above definition, the reader may think that we have given an equal weight to each type of labour. This is not so. Various types are differentiated by their wage rates as well as consumption baskets. In other words, an equal weight is used only for calculating the shares S[i]and r[i]. Rich workers may lead a lavish life with little hesitation, and they may not want to be called "workers": they prefer "managers" or "entrepreneurs".
In this section we give three numerical examples in which there are two types of labour with no joint production. These examples are meant to tell the reader how to apply our definitions in a concrete manner, though the examples are highly abstract.
The first economy consists of hunters and fishermen, and the technological data are
The first row stands for beavers, the second for fish, the third for hunters' labour, and the fourth fishermen's labour. The first column means the process operated by hunters, the second by fishermen, while the third is the household activity of hunters and the fourth is that of fishermen. Assumption A3 is satisfied, e.g., by x = (1, 1, 1.1, 1.1)', to say nothing of assumption 4. When the standard of value is hunters' labour, the values are calculated as
Similarly when the standard is fishermen's labour, the values become
Since λ = 1/2 and λ = 1/3, propositions 2 and 5 are verified.
If we choose beaver and fish as the standard of value, we get, respectively,
These values shows proposition 4 is valid in this example.
Now suppose x = (1, 1, 1, 1)' and the quantity equation of the period is
where f = (1, 2, 0, 0)'. Proposition 6 can be verified using above values. Since the economy here is (completely) decomposable, we apply the definition of exploitation for the homogeneous case. Suppose the price vector for the normal commodities p = ($100, $100). For a hunter operating the first process, the income is $200. This hunter needs $100 to reproduce one unit of hunter's labour. Spending the remaining $100, he/she can buy out the net output of one beaver. Hence no exploitation is observed. In fact we have
The second example economy is indecomposable, though it consists of the same commodities and the same types of labour as in the first one. The complete inputoutput matrices are:
which satisfy assumptions 3 and 4. The labour values are computed respectively as
Suppose again p = ($100, $100), x = (1, 1, 1, 1)' and the quantity equation of the period is
where f = (0.8, 1, 0, 0)'. All the related propositions can be verified under these values. And so, we examine who exploits who. A hunter who runs the first process contributes to the net output as much as
Then let us examine how much workers can spend on the net output. A hunter gets $200 by operating the first process, uses $200 in his/her household activity to reproduce one unit of labour, and no money is left to buy a portion of the net output. On the other hand, a fisherman can secure the surplus money $180, which can exactly buy the whole net output. Thus we can judge, based on our definition, fishermen exploit hunters under these prices in spite of the fact a hunter requires more to survive than a fisherman. When the price vector p = ($100, $46), exploitation does not exist between two types of workers. This magnitude $46 can be obtained by solving the following equation for p2:
In the third example, there are fishermen and owners of fishingnets. Two objects are fish and services given by fishingnets. Fishermen do not have nets, and they have to borrow nets. We assume fishing nets last for ever. The complete inputoutput matrices are:
The first row stands for fish, the second for the service from fishingnet, the third for fishermen's labour, and the fourth is netowners' labour. The first column means the process ran by fishermen, the second ran by netowners, the third is the household activity of fishermen, and the fourth is that of netowners. The values with the fishermen labour as the standard, and those with netowners are respectively:
Suppose again p = ($100, $100), x = (1, 1, 1, 1)' and the quantity equation of the period is
where f = (3, 1, 0, 0)'. A fisherman who operates the first process contributes to the net output as much as
A fisherman gets $500 after subtracting the rental of nets by operating the first process, uses $100 in his/her household activity to reproduce one unit of labour, and so $400 is left to buy all the net output. On the other hand, a netowner has no surplus money Hence, fishermen exploit netowners under these prices. When the price vector p = ($100, $(2800/12) ($100, $233), exploitation does not exist between two types of workers. The balancing price, p2, can be found by the equation:
When the coefficient α42 = 1 reduces to 0.1, i.e., netowners can render their service with less work, and at the same time the coefficient b44 = 1 also decreases to 0.1 so that its net output remains zero at x = (1, 1, 1, 1)', then the values become
Workers' contributions to the net output change approximately to
When the price vector p ($100, $(2500/21) ($100, $119), no exploitation exists, and when the price of the service of fishingnet gets higher than $119, the netowners exploit the fishermen. It is noted that still the smaller is the coefficient α42 as well as b44, the smaller is the balancing price of the service from fishingnets (nearer to the lower limit $100), and the netowners are likely to be exploiters.
In modern developed economies, it seems difficult for workers to carry out a revolution to put an end to the capitalist system. Marx and Engels might think, in the 19th century, the days of proletarian dictatorship would soon be realized. While the working class has expanded since then, it has been polarized with a greater inequality of incomes among its members. As capital is accumulated more, a still higher degree of division of labour emerges, bringing about a possible cause of conflict among workers. When division of labour was in a primitive stage, e.g., as in Smith (1776), Marx could devise out abstract labour to commensurate labour contents of commodities in order to help the whole working classes to unite themselves. Workers have, however, been divided and conquered, at least so far, by the capitalist system. Division of labour was an impetus for capitalist systems, and now it seems to be a support for them. Children are today taught to compete with others, to work harder to get richer, moneyfetishism, and suggested that fellow workers are after all rivals, not friends. This article has tried to explain a possible conflict within the working classes using a linear model of heterogeneous labour, and to show an economic reason why workers will find it difficult to unite themselves to change their economic system. Certainly the degree of exploitation among workers matters here. Our definitions of values and exploitation are still tentative, and many variants and extensions can be made. A weak point of our method is that it is totally ahistorical, capturing a snapshot of one particular period. For example, in reality, workers migrate among various types, not simply reproducing themselves. We believe, however, this paper can be a useful starting line for further developments, just as d'Alembert's principle in mechanics is important to understand dynamics through static snapshots. Here in this concluding section, we make some remarks on our method.
In exsocialist countries, various services were excluded from the national account. What we have done in this article is an extreme opposite of this practice. We have subsumed every service available as rendered by workers or propertyowners. Therefore there needs not be an explicit profit rate as expressed by (1 + r) in markup ratios. There arises, however, a problem of how to determine the inputs required for those services, especially household activities to reproduce labour force. Since we allow for alternative methods, we may include every activity observed.
When we discuss labour values of commodities, it is not important whether a given economy is in equilibrium or not. In the real world, all the economies are out of equilibrium, with many sufferings from a long stagnant depression. We should be able to discuss values and surplus values also in such a disequilibrium state. Labour values depends solely on technological data. Prices as well as wage rates do play a part when we consider exploitation, unequal exchanges of labour services for what they could buy in markets. There is no need to presuppose the existence of exploiters or capitalists. This may not be accepted by Marx (1867) who supposed the grand two classes in a capitalist system, relegating landowners. A capitalist system without capitalists might be the highest stage of capitalism: capital only matters. Based on what they receive as wages, salaries or remunerations, they may be exploited by the economic system or can actively exploit the system.14 For our results to be applicable to the real world, we have to take into consideration unequal exchanges among nations. Some suggestions are made in Fujimoto and Opocher (2007) on how to define values when countries are engaged in international trade.
The reader may have been embarrassed by the net consumption vector (c d) in the definition of exploitation. When we discuss whether the exchanges of labour for a consumption basket with a given economic system is equal or unequal, we should subtract the vector d because this is not consumed after all and remain in the system, though this d is belonging to a certain worker, and this property right is protected by the system. In this respect, our treatment is again ahistorical.
The reader may also have an idea that a normal commodity can be exploited because normal commodities and various types of labour are dealt with in a symmetric way. This symmetry is only in the sphere of technological relations. Workers can refuse to participate in production when they regard the economic system as intolerably "unfair", while normal commodities do join the production as input obediently when workers thus operate.
It may also be desirable to define "unskilled labour" or "simple labour" using only value analysis. In Fujimoto and Opocher (2007), a method is presented. Besides, a method to distinguish necessities from luxuries is also mentioned.
Finally it is necessary to introduce the government to judge whether its functions are aggravating the degree of exploitation among the workers through fiscal policies. This could be a new addition to the cursed roles of the state described by Engels (1884).
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* This paper was written while the author was a visiting professor at Department of Economic Sciences, University of Padova. He is grateful to the Department for its hospitality, especially to Professor Arrigo Opocher for his help. Thanks are also due to professor Michael Rauck, who provided a cozy place where early versions of this paper were written; to the two anonymous referees for their suggestions, which are useful to make this paper more readable; and to professor Alejandro Valle Baeza, who introduced this journal to the author.
1 For simplicity each type of labour is assumed to be abundantly available, though heterogeneous labour can be dealt with in the same manner as various vintages of capital goods.
2 This section is a simpler version of what was presented in Fujimoto and Opocher (2007).
3 See Hawkins and Simon (1949) for their conditions concerning A.
4 See, for example, Fujimoto et al. (2003).
5 Maximizing qn+1 is equivalent to maximizing λR, which is obvious from equation .
6 The nonsubstitution theorem also asserts that the vector c can be any nonzero nonnegative vector, with no influence on the optimality of one and the same solution q. It is also easy to notice that the objective function can also be the total input of labour.
7 "Bads" are also allowed, though they are implicit. See Fujimoto and Opocher (2007).
8 Among a plural number of solutions, we adopt those which realize the maximum number of equalities in the constraints. And yet, a solution may not be unique.
9 We can further weaken the assumption to the existence of an activity level vector which realizes a simple reproduction. See Fujimoto and Opocher (2007).
10 This proposition, when labour is homogeneous, is called "commodity exploitation theorem". See Bowles and Gintis (1981), Roemer (1982, 1986), and Fujimoto and Fujita (2008).
11 The concept of net here is for the system of complete or extended matrices.
12 Positive elements in/are not excess supply, but are bought by some people and put to extra consumption or investment, or may be exported abroad.
13 More precisely, it is Λ[i]·f. See equation  with n replaced by i.
14 In practice, remunerations are seemingly from surplus. They can, however, be regarded as payments to services rendered, thus as costs.