Abstract
Production prices have often been conceived as those exchange rates among commodities which allow their reproduction in the same quantities (or along a proportional growth path). A deeper investigation of Sraffa’s price equations reveals that this characterization is not well grounded. The notion of ‘viability’ of a system is thus re-defined here in such a way as to allow the determination of production prices for economies that are not in a self-replacing state (i.e. displaying positive net products in some industries and negative net products in others). Viability is here connected to the possibility for each industry to reintegrate the value of the means of production and to obtain a uniform non-negative rate of profit. It appears, thus, as a notion that mainly impacts the value side. This specification is relevant for an approach where the quantities are determined separately from the relations between value and income distribution. The focus on non-self-replacing systems opens the question on how to handle this and all cases of systematic changes in output levels without losing the property of persistence of production prices, which permits them to be regarded as centres of gravitation. This issue is explicitly linked to possible specific assumptions that can be made on returns. The various views on this point are discussed here in relation to the main features of the modern classical approach.
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Notes
Piccioni (2000) and Ravagnani (2001) argue in detail against the foundations of this association. In Ravagnani’s paper one may find a set of references where this view is advanced. For example, regarding the system presented by Sraffa in his 1st Chapter, Schefold writes:
This model serves the purpose of, and derives its value from, clarifying in general the function of relative prices in Sraffa’s system. Relative prices … represent the exchange ratios between physical goods that make reproduction within a technical (methods of production) and social (distribution) framework possible (Schefold 1989, p. 285, emphasis in the original).
Sraffa’s original example has not been considered here for reasons that will be clarified later.
This change of the unit of measure allows one to represent technical processes by using production coefficients without entailing specific assumptions on returns.
The Perron-Frobenius theorems state that a semi-positive indecomposable square matrix, A, has an eigenvalue, λ M , with the following characteristics:
-
i)
λ M is real and positive and the modulus of all other eigenvalues are lower than λ M ;
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ii)
The right-hand and the left-hand eigenvectors associated to λ M are positive, while the right-hand and the left-hand eigenvectors associated to all other eigenvalues have at least one negative component;
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iii)
λ M is included between the minimum and the maximum of all the sums of the columns (or of the rows) of matrix A.
Eigenvalue λ M is called the dominant eigenvector of A. For details see, for example, Pasinetti (1977a, Mathematical appendix, § 12.2).
-
i)
In input–output analysis, where the price system is normally paired with the quantity system, the notion of viability appears essentially as a notion regarding quantities (see, for example, Kurz and Salvadori 1995, chs. 2–4, which develop in detail the case of systems with a surplus). On the contrary, in the present paper the emphasis is placed on the consequences of the fulfillment of condition (5) for the relations concerning value. The physical dimension, however, is not lost: (5) is a condition that concerns matrix A, whose elements are quantities.
It is curious to observe that the numerical example provided in Sraffa's Chapter I cannot be transformed into a non-self-replacing system through a mere simplification of the price equations by a whole number, in contrast to our example in Sect. 2: in fact, since both equations in this example,
\(\begin{aligned} 2 80p_{w} + { 12}p_{i} & = { 4}00p_{w} , \\ 1 20p_{w} + { 8}p_{i} & = { 2}0p_{i} , \\ \end{aligned}\)
can be simplified by the same factor (i.e. by the number 4), we obtain a system which, after simplification, is still in a self-replacing state:
\(\begin{aligned} 70p_{w} + { 3}p_{i} & = { 1}00p_{w} , \\ 30p_{w} + { 2}p_{i} & = { 5}p_{i} , \\ \end{aligned}\)
where 70 + 30 = 100 and 3 + 2 = 5. Is this a coincidence?
It is clear from the quoted passage that Sraffa’s definition of viability is not restricted to the case of systems with zero surplus. We will see in the next section how these arguments extend to the case of systems with a positive surplus.
By result iii) of the Perron-Frobenius theorems, condition (10) entails λ M ≤ 1; in order to exclude the equal sign observe that (10) implies that there is a positive surplus for at least one commodity. Since A is indecomposable, all prices are positive; thus the value of this surplus is positive, and it is distributed (in proportion to the value of the means of production of each industry) in the form of profits. Thus the rate of profit is positive, i.e. R > 0 and hence λ M = 1/(1 + R) < 1. In more formal terms, suppose, by contradiction, that λ M = 1; hence, the price vector would satisfy Ap = p which, after being pre-multiplied by vector u T, entails
\({\mathbf{u}}^{\text{T}} {\mathbf{Ap}} = {\mathbf{u}}^{\text{T}} {\mathbf{p}}.{ \quad\quad(}* )\)
On the other hand, as A is indecomposable, p > o; post-multiplying both sides of (10) by p obtains
\({\mathbf{u}}^{\text{T}} {\mathbf{Ap}} < {\mathbf{u}}^{\text{T}} {\mathbf{p}};{ \quad\quad (}** )\)
but (*) contradicts (**), so the assumption λ M = 1 is untenable.
An exception is represented by the ‘freak’ case considered by Sraffa (1960, p. 25, fn 1 and Appendix B) of a non-basic product with an own-rate of reproduction so unusually low as to be lower than that of basic commodities. However, this case is not relevant from the economic point of view.
For appreciating the reasons why Sraffa could avoid to introduce assumptions on returns, see Eatwell (1977).
It is to be noted that the gravitation process is supposed to take place in real time (not in virtual time, as with Walrasian tâtonnement) through the movements of capital among industries. Consequently, if capital has to move from one industry to another, it must remain employed in each industry for at least one production period. The achievement of a long period position, characterized by uniform rates of profit, thus takes more than one production period.
A similar point is discussed in Chiodi (1998, in particular § 4).
On this see, for example, Garegnani (1990), where the same technical coefficients are employed in the system defining the ‘natural prices’—his Eq. (1) —and the equations defining the ‘market rates of profits’—his Eq. (3)—which clearly refer to actual sectoral output levels, that in general do not coincide with the ‘normal’ configuration.
On this see Kurz and Salvadori (2005).
The discriminatory element of Fig. 1b with respect to Fig. 1a is the constant supply curve. This requires:
M1) Constant returns to scale;
M2) Constant rewards of non-produced factors as long as the output level changes.
The same situation is guaranteed in the non-substitution theorem by assuming:
S1) Constant returns to scale;
S2) The existence of just one primary (scarce) factor, so that factors’ rewards cannot express their relative scarcity;
S3) A given rate of interest.
Assumptions M1 and S1 are equivalent; assumptions S2 and S3 entail assumption M2.
Similarly, Sraffa vehemently rejects the idea that his prices are independent of demand. In a famous letter addressed to Arun Bose (classified in Sraffa’s archive as SP, C32/3), Sraffa writes:
Trinity College,
Cambridge.
9th December, 1964.
Dear Arun,
I am sorry to have kept your MS so long - and with so little result.
The fact is that your opening sentence is for me an obstacle which I am unable to get over. You write: “It is a basic proposition of the Sraffa theory that prices are determined exclusively by the physical requirements of production and the social wage-profit division, with consumers demand playing a purely passive role.”
Never have I said this: certainly not in the two places to which you refer in your note 2. Nothing, in my view, could be more suicidal than to make such a statement. You are asking me to put my head on the block so that the first fool who comes along can cut it off neatly.
Whatever you do, please do not represent me as saying such a thing.
This initial and to me quite maddening obstacle has prevented me, in spite of many attempts, from reading understandingly your article. You must find a more detached reader to advise you about it. I am very sorry to seem so unhelpful, but I have spent quite a lot of time upon your work, to no purpose. I do not think that it would be any good keeping it longer, so I now return it to you.
Yours sincerely,
By the way, without constant returns, a change in the level or in the composition of final demand does affect production coefficients, so that prices as well as the rate of profit change; but, these effects have nothing to do with the monotonic relations between the proportions in which final goods are demanded and the prices of the factors more intensively employed in producing those final goods [on this, see Pasinetti (1977b)].
This is a distinction that releases Sraffa’s prices from any direct link to an equilibrium between quantities: Sraffa’s prices do not originate from equality between demand and supply. Thus, it seems of no immediate utility to equip Sraffa’s price equations with a set of quantity equations like in the von Neumann model, where one set of variables is ‘dual’ to the other. This does not prevent that there are situations where Sraffa’s price equations can usefully be paired with a set of conditions involving outputs and quantities demanded. For example, this is the case of formal models of gravitation, where the long-period position is characterized by:
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a)
equality between the output level of each commodity and the corresponding demand,
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b)
the possibility of indefinitely replicating the same output levels (steady state) or increasing these levels, leaving the proportions among industries unchanged (balanced growth).
But, as regards (a), the main force pushing actual (‘market’) prices toward production prices and, thus, the rates of profit toward a uniform level, is competition among capitalists, i.e. capital movements in search of the highest rate of profit, not a tâtonnement process driven directly by the excesses of quantities demanded with respect to the quantities supplied. As regards (b), the representation of the long period position as a steady state or a balanced growth path is merely a simplification acceptable as long as the process of structural change takes place at a slower pace with respect to capital mobility induced by the differences in the rates of profit of the various industries [on this see Cesaratto (1995), and Duménil-Lévy (1995)].
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a)
A similar position seems to be expressed by Ravagnani (2001, p. 361, fn. 2).
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Acknowledgements
I wish to thank Christian Bidard, Antonia Campus, Guglielmo Chiodi, Roberto Ciccone, Saverio Fratini, Heinz Kurz, Sergio Levrero, Sergio Nisticò, Sergio Parrinello, Luigi Pasinetti, Fabio Ravagnani, Andrea Salanti, Neri Salvadori, Paolo Trabucchi and Paolo Varri for the discussions on the topics here presented. I also thank three anonymous referees of this Journal for their comments and criticisms to a previous version of this paper. Finally, I am grateful to Micaela Tavasani for revising the English.
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Bellino, E. Viability, reproducibility and returns in production price systems. Econ Polit 35, 845–861 (2018). https://doi.org/10.1007/s40888-017-0082-2
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DOI: https://doi.org/10.1007/s40888-017-0082-2