ABSTRACT
As is well known, there are two basic approaches in contemporary economic theory: the Classical approach (which the Marxian and the Sraffian ones belong to) and the Postclassical (or Neoclassical) approach. One important feature of the former, which is relevant in the present context, is that the prices and the quantities appearing in the corresponding frameworks are respectively determined by distinct forces, instead of their being determined by the same set of forces, as within the Postclassical framework. The most important consequence of that specific feature of the Classical approach is that it allows the coming into being of the notion of ‘viability’ of an economic system, a fundamental property – as will be shown in the sequel – which indicates the possibility of the system to reproduce itself over time. As a preliminary, let us suppose an extremely simple economy producing wheat by means of wheat, part of the latter being used as means of sustenance for the workers and the rest as means of production. Let us call a ∈ (0, 1) the quantity of wheat used as means of production per unit of wheat produced, per unit of time, and c ∈ (0, 1) the quantity of wheat used as means of sustenance for the λ units of labours employed per unit of wheat produced per unit of time. Thus, the economy considered could be represented through these alternative compact forms:
a → 1 (20.A)
A + c → 1 (20.B)
a ⊕ λ → 1 (20.C)
A ‘viability’ condition for the economy which refers to representation (20.A) is:
1 – a > 0 (20.1)
an alternative ‘viability’ condition to (20.1), referring to representation (20.B), is:
1 – (a + c) ≥ 0 (20.2)
The ‘viability’ conditions (20.1) and (20.2) differ one from each other very profoundly. In fact, taking into account representation (20.A), for the economy to be ‘viable’ it is sufficient that condition (20.1) be verified; in other words, it is sufficient that from each unit of the wheat produced per unit of time any quantity left over for consumption (and, in the case, for investment and exports also) would do. In this case one could affirm that the economy is ‘productive’. For example, if a = 0.8, the economy is ‘viable’, for 1 – 0.8 > 0. By contrast, taking as a reference representation (20.B), and supposing c = 0.4, formula (20.2) leads to 1 – (0.8 + 0.4) < 0, and therefore the economy is obviously ‘non-viable’, simply because each unit of the wheat produced per unit of time would not be sufficient to replace all the quantity of the wheat used for its own production, that is quantity a (used as means of production) plus quantity c (used as means of sustenance for the labourers). The crucial difference between the two alternative criteria for ascertaining ‘viability’ – by comparing (20.1) and (20.2) – lies in the inclusion or not of the workers’ subsistence in defining the condition of the economy ‘viability’, or, which turns out to be the same thing, in considering essential or not the quantity of labour necessary for the production of wheat – as can be seen at a glance by comparing (20.A) with (20.B) or (20.C). At the basis of these alternative approaches there exist radically different conceptions of the economy and of its functioning – as will be seen in the sequel. The above preliminary considerations might be put also in a different way. By supposing condition (20.1) verified, the equation (1 – a)x = y has a unique solution x ≥ 0 for any y ≥ 0, where x = total quantity of wheat produced, y = net quantity of wheat produced. The practical problem to solve, in this respect, consists in the choice of the net quantity of wheat y that society would like to have at its disposal. In general, such a quantity could be fixed at any level whatsoever, and the above equation would continue to have an economic meaningful solution. The fixing of this quantity, however, while it encounters the obvious limit of the total amount of the labour force as well as of the available ‘productive capacity’ of the economy at any given point of time, does appear absolutely unconnected with the quantity of labour required per unit of product and, as a consequence, it does appear unconnected also with the corresponding subsistence for the workers. In fact, the ‘viability’ condition expressed by (20.1), by making reference to a representation of the economy which straightforwardly puts out of the picture the quantity of labour required per unit of product, puts out of the picture at the same time the subsistence for the workers. As the net product y could be fixed at a level independently of these subsistence, the former might in general be fixed at a level inferior to the latter, with the uneasy consequence that the economy could be considered ‘viable’ according to condition (20.1) but, at the same time, ‘unable’ to secure workers’ subsistence – as is plainly evident from the simple numerical example given above. In such a circumstance workers’ subsistence should be forced to be reduced within the limits imposed by the technological
constraint – as expressed by the quantity a of wheat used as means of production. This result is the mere reflection of having taken into account the technological constraint only, leaving out the notion of ‘viability’ the social constraint, given by the subsistence for the workers. On reflection, however, the exclusion of workers’ subsistence appears quite strange – suffices it to notice that labour is needed in each and every process of production. The ‘viability’ condition (20.1) could formally be modified by simply substituting a + c for a in the corresponding expression. In this way, one might be induced to think that workers’ necessary productive consumption has found its proper place. However, this is an illusion. In fact, in such a circumstance condition (20.1) would not be a general condition of ‘viability’ any more, for it would ipso facto exclude from the set of ‘viable’ systems those producing with no surplus – let alone the fact that all the formulas considered above are based on the implicit assumption of constant returns to scale. The notion of ‘viability’ – in the form which will be considered in the present chapter – has been explicitly and consistently put forward by Sraffa (1960). Before examining it in some detail, however, it may worth considering the ana logous notions, originally given at different times, by others, specifically by Hawkins and Simon (1949), by Gale (1960) and by Pasinetti (1977). Hawkins and Simon’s paper (1949) can conventionally be taken as the starting point in the study of the notion of ‘viability’ in economic analysis. The publication of that paper originated from a mathematical mistake contained in a lemma stated by Hawkins in his previous (1948) paper. The proof of that lemma, in fact, was ‘defective’. It referred to a system of n linear homogeneous equations of the type Bx = 0, which was actually the ‘closed’ Leontief system (1941),2 with the square matrix B of rank n – 1 and determinant equal to zero, the elements of the matrix B being bij = aij, for i ≠ j, and bii = aii – 1, for i = j, with the ‘technical’ coefficients of production aij ≥ 0, ∀ i, j, with 1 – aii > 0, ∀ i; whereas vector x represented the quantities of the commodities produced. The lemma asserted that the system of equations Bx = 0 could be satisfied by a solution with all values of equal sign. This was true, however, only for systems composed of three equations at most, but it failed, in the most general case, with systems composed by more than three equations – as was demonstrated through a counter-example produced by Hawkins and Simon (1949: 246). A fundamental feature of the ‘closed’ Leontief model – which Hawkins and Simon in their joint paper were referring to – should be noted: the nth column and the nth row of the matrix of the production coefficients refer to ‘final demand’ (consumption, investment, exports) and to labour coefficients respectively. Thus, each of them is formally assimilated, respectively, to any column representing input and to any row representing output. The transition to the ‘open’ Leontief model is operationally made by ‘detaching’ from the original matrix of the ‘closed’ model the column of the ‘final demand’ coefficients and the row of the labour coefficients.3 It should also be noted that this transition implies not only a formal manipulation of the type referred to now, but it also implies a substantive change in the representation of the economy, to the effect
that in the ‘open’ model a neat separation is definitely established between the ‘technical’ coefficients of production, which reflects the actual technical knowledge existing in the economy at a given point of time, and the heterogeneous elements composing the ‘final demand’, such as exports, investments and consumption, each one of them essentially influenced by distinct forces. At the basis of that separation – it is worth noticing – it lies a well defined theoretical approach, in which the objective data of the technology are given the crucial role of representing the essential ‘core’ of the productive system. The ‘open’ Leontief model can then be formally treated as a non homogeneous system of n – 1 linear equations of the type (I – A)x = y, where I = identity matrix, A = (n – 1) × (n – 1) matrix of the ‘technical’ coefficients of production, x = (n – 1) × 1 output vector, y = (n – 1) × 1 ‘final demand’ vector. Within this context, Hawkins and Simon (1949: 247), established a necessary and sufficient condition for the solution to the equation (I – A)x = y to be positive,4 with the immediate implication for the economy, characterized by a ‘technical’ matrix satisfying the condition referred to now, of being able to produce any list of goods, and, in particular, any list of consumption goods. In the simple economy of the first section of the chapter, above, condition (20.1) corresponds to the Hawkins and Simon ‘viability’ condition. It is worth noticing, therefore, the exclusive priority attributed by Hawkins and Simon to the ‘technical’ coefficients of production in determining the ‘viability’ of the system, in contrast with the subordinate role attributed instead to consumption. This is plainly evident from the structure of the equations given by Hawkins and Simon and explicitly admitted by themselves in stating the last equation of the ‘closed’ Leontief system being ‘linearly dependent on the first m equations’ (ibid.: 246) – these m equations actually correspond to the n – 1 equations of the system (I – A)x = y, due to the symbols used by Hawkins and Simon in the transition from the ‘closed’ to the ‘open’ model. (It must be added that if it is necessary that the determinant of the matrix of the ‘closed’ Leontief model must be equal to zero for the system to have a solution, it is also true that any row (or any column) can be considered linearly dependent on the remaining ones.)5 The ‘open’ Leontief model remains also the basic reference in the far more refined analysis subsequently made by Gale (1960), though he introduces – as will be presently seen – the crucial distinction between a ‘simple Leontief model’ characterized by the ‘technical’ coefficients of production only and independently of the labour coefficients, and a ‘simple Leontief model’ with labour explicitly considered. Gale (1960: 294) initially takes into account a linear single production model having equal number of goods and processes. He then puts the feasibility question: given a non negative square matrix of technical coefficients A, does a program exist which actually makes it possible for the production of given quantities of goods? The problem consists in ascertaining whether the equation
xT (I – A) = yT
has a meaningful solution; in other words, if there exist levels of activity (given by the vector x) which allow for the net production of quantities of goods (given by the vector y ≥ 0). As a preliminary, Gale defines as productive the above ‘simple linear production model’ if there exists a non negative vector x′ such that x′T > x′TA, and consequently he defines as productive the same matrix A. He is then in a position to prove (ibid.: 296-7), the following theorem:
If the matrix A is productive, then for any vector y ≥ 0 the equation
xT (I – A) = yT
has a unique non-negative solution.
