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Sraffa, Money and Distribution

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Part of the Palgrave Studies in the History of Economic Thought book series (PHET)

Abstract

Sraffa’s early work on monetary Economics, his contributions to the theory of capital and his critique of neoclassical tenets are often seen as disjoint contributions. In contrast, we suggest that Sraffa’s contributions have to be viewed as a coherent whole and offer a classification of his scholarly contributions. We point to the unifying thread between his early and later work, which concerns the insufficiency of economic mechanisms or market forces to exclusively determine values (prices, profit rates and wage rates). Their simultaneous determination and consequently the overall indeterminacy of the system are important. We outline the motivations for incorporating money (in the form of credit and debt) inside the traditional Sraffian schemes, to expand the original system and harness its potential. We believe that there is both a need and scope for incorporating deferred means of payments, which are essential for the functioning of any evolved economic system.

Keywords

  • Sraffa
  • Means of payment
  • Money
  • Capital
  • Panico
  • PCMC

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Fig. 15.1

Notes

  1. 1.

    In the correspondence with Garegnani, Sraffa makes this point quite forcefully, see Sect. 3.1

  2. 2.

    It is also important to note in this context that Sraffa has spent time learning the practical workings of the wheels of banking from the inside as an apprentice at the provincial bank Banca di Legnano e Busto Arsizio post his graduation.

  3. 3.

    Sraffa’s Italian translation of A Tract on Monetary Reform was also published in 1925. Sraffa also published brief book reviews in the Giornale degli Economisti that concerned monetary and banking issues.

  4. 4.

    Although the title of Hayek’s book is Prices and Production, it is really about a monetary theory of the business cycle. The title of Sraffa’s review in fact is Dr. Hayek on Money and Capital. One of Sraffa’s points was that a disequilibrium scenario requires the use of deferred means of payments—i.e., money or credit-debt contracts.

  5. 5.

    This point made by Sraffa is often misunderstood or misinterpreted. Zambelli (2018b) takes the numerical values used by Sraffa and demonstrates that the value of capital and the industry capital/output ratios are all functions of distribution (prices). The values of aggregate capital, or the value of capital per industry do not remain constant as the distribution changes. Furthermore, their changes are not monotonic with respect to changes in distribution.

  6. 6.

    In his paper discussing Sraffa’s early work Sraffa (1920, 1922a) on money and banking, Panico (1988b, p. 7) underlines that:

    Sraffa started his research working on applied monetary and banking problems, making some relevant contributions to the area. The theoretical problems that made his fame as an outstanding theoretician came to dominate his interests only at a later stage.

    Panico also argues that:

    … there is a close link between [Sraffa’s] earlier writings on applied problems and his later theoretical works, a link which clarifies the origins of his theoretical interests, showing the error of considering his contributions as ‘merely abstract exercise in pure theory’.

  7. 7.

    In his thesis, for instance, he takes the quantity theory of money as a given and applied it to monetary issues of his time (Panico 1988b, p. 10).

  8. 8.

    His critique of Hayek’s book was made alongside Keynes’ own critique of Hayek (Keynes 1931; Hayek 1931b, 1932b; Sraffa 1932a, b). There is evidence of Sraffa’s involvement in the corrections of ‘A Treatise on Money’ (Keynes 1930) as well. It is worth noting that Sraffa translated (Keynes 1923, ‘A Tract on Monetary Reform’) into Italian and was involved in the discussions surrounding the writing of Keynes’ ‘General Theory’. In this context, it is also important to bear in mind that Chapter 17 of the General Theory , (Keynes 1936, Ch: The Essential Properties of Interest and Money) may be interpreted as addressing the relation between interest and money using some of the arguments put forward by Sraffa in his review of Hayek. In fact, Keynes does acknowledge that the idea of the own rate of interest was taken from Sraffa’s critique (Sraffa 1932a) of Hayek’s ‘Prices and Production’ (Keynes 1936, p. 223, fn. 1). All these attest to the interpretation that money and monetary issues were not in anyway insignificant in Sraffa’s academic interests.

  9. 9.

    This was an important point already put forward in Sraffa (1922a) and subsequently in his debate with Hayek (Sraffa 1932a). Chapter 17 of Keynes (1936) may be interpreted to be addressing this problem using arguments put forward by Sraffa.

  10. 10.

    Text in the original:

    […] Lei opta […] per il saggio del profitto “determinato da influenza esterne del sistema …e particolarmente dal livello dei tassi monetari dell’interesse” […] sono restio ad avventurarmi, anche solo per accenni, sul terreno di una teoria del profitto determinato dai saggi monetari dell’interesse, col rischio di attribuire a lei opinioni che non ha.

    Questo per la recensione. La questione peró mi interessa molto […] Espresso in termini molto rozzi il modo con cui vedo il problema è questo. Da Ricardo a Wicksell l’idea è stata che il saggio di interesse si adeguasse al saggio del profitto ottenuto nella produzione.

    La giustificazione di quella teoria sta, mi sembra, nell’idea di una domanda di prestiti di elasticità pressoché infinita (dato il tempo richiesto per formulare i piani di investimento) intorno ai livelli dell’interesse corrispondenti al saggio di profitto ottenuto nella produzione. Se infatti tale elasticità esistesse, il saggio di interesse di mercato dovrebbe sempre tendere al suo livello “naturale” (l’eventuale tentative della autorità monetarie di fissare un diverso saggio creando o distruggendo credito fallirebbe, allora, perché il risultante processo di inflazione o deflazione costringerebbe a ritornare al saggio “naturale”) [Sraffa Papers, D3/12/111/150, emphasis added]. To this question, Sraffa replies in his letter as below:

  11. 11.

    Text in the original:

    Dal punto di vista dei suoi studi mi pare che sia difficile unificare i punti di vista di Ricardo e Wicksell, per quanto entrambi facciano dipendere il saggio dell’interesse dal saggio del profitto. Ma quanto al “saggio del profitto” hanno in comune solo il nome: per il primo è l’eccedente al si sopra del salario dato, per il secondo il prodotto marginale del capitale! Come si fa a metterli in pariglia? Io vorrei sapere in che senso si possa dire che “la giustificazione di quella teoria” sta nella “domanda di prestiti ad elasticità infinita a saggio di interesse uguale al saggio di profitto”. Giustificazione un corno! Io la chiamo “ripetizione identica in parole difficili”, grave tautologia.

    Non vedo la difficoltà della determinazione del saggio del profitto mediante un saggio dell’interesse controllato o convenzionale, a condizione che non si presupponga il saggio del profitto determinato da circostanze ineluttabili esterne. [Sraffa Papers, D3/12/111/150, our translation, emphasis added]

  12. 12.

    The text in Italian:

    … sono convinto che il mantenimento del saggio d’interesse da parte della banca o della borsa abbia avuto la sua parte nel determinare la distribuzione del reddito fra le classi sociali: perché è un passaggio obbligato per chi dà e per chi prende a prestito.

    […] io non ho inteso dir niente di molto impegnativo, e in generale ho solo voluto metter fuori qualche segnale per evitare che si creda che il sistema viene presentato come “fondamenta” per una teoria delle offerte relative di capitale e lavoro! E’ la negazione che mi sembra importante: quanto alla affermativa non ho nessuna intenzione di mettere avanti un’altra teoria meccanica che, in una forma o nell’altra, ribadisca l’idea che la distribuzione sia determinata da circostanze naturali, o tecniche, o magari accidentali ma comunque tali da rendere futile qualsiasi azione, da una parte o dall’altra, per modificarla (Sraffa Papers, D3/12/111/149, in bold emphasis added).

  13. 13.

    If our interpretation is correct, it is puzzling that many Sraffians have placed a strong emphasis on the mechanics of the economic determination of the long term uniform profit rate .

  14. 14.

    On the notion of the period of production and on Sraffa’s personal notes on it, see Sinha (2016, Chap. 5, pp. 111–151). On Sraffa’s critique of the period of production as being independent of the rate of profits , see PCMC, Chapter 6, and a rather harsh comment of Sraffa (1962) on the review of PCMC by Harrod (1961).

  15. 15.

    See also Zambelli (2018a) for an empirical demonstration of this point.

  16. 16.

    Hahn (1982), Hahn and Petri (2002), Fratini (2018), Garegnani (2000), Mandler (1999a, b, 2002a, b, 2005), Negishi (2014, p. 7), Parrinello (2008), Schefold (2005, 2008), Sinha and Dupertuis (2009) are exam - ples where Sraffian schemes are studied in relation to Walrasian (intertemporal) general equilibrium . In these contributions there is a clear acknowledgment that Sraffian schemes are equivalent to the aggregate budget constraints of the producers, workers and consumers that compose the economic system.

  17. 17.

    There is evidence from Sraffa’s diaries that they met frequently while Hicks was in Cambridge (from 1935 to 1938). There is also the draft of the letter from Sraffa to Hicks in which Sraffa makes a precise quote in relation to Hicks’ “The Theory of Wages”, first Edition, Hicks (1932). In a personal conversation, Massimo Di Matteo at the University of Siena, who has been a student of Hicks, told the second author that he had indeed asked Sraffa about what the ‘useful criticisms’ were. He recalls Hicks’ answer that they discussed matters related with Walrasian General Equilibrium and Pareto.

  18. 18.

    See also Panico (1988b, p. 26) in support of this interpretation:

    A well-defined thread, going throughout his writings up to Production of Commodities by Means of Commodities, can thus be found in Sraffa’s work. His theoretical interests were enhanced by the need to provide firm analytical foundations for his original view of the working of the economic system. This view emphasised the role of state intervention in determining income distribution and the equilibrium position of the economy, and the fact that policy decisions are influenced by the pressures of the dominant groups, and are not the result of purely technical considerations. His earlier analyses on the influence of monetary policies on income distribution were closer to the classical and the Marxist tradition than to the neoclassical one. The study of these analyses, together with that of his contacts with Keynes, thus provides useful insights in the evolution of Sraffa’s thought, underlining that his later work is not ‘merely abstract exercise in pure theory’.

  19. 19.

    After twelve chapters, defining and proving theorems about Walrasian general equilibrium, Arrow and Hahn (1971) state this clearly:

    Of course, our model is in no shape to give a satisfactory formal account of the role of money. In particular it would be hard to “explain” the holding of money or why it mediates in most acts of exchange. (p. 338)

  20. 20.

    The quotation we refer to in Sect. 3.2 is made at the end of section 48 in PCMC. Figure 2 in page 36 of Sraffa’s book shows the value of different ‘labour terms’ which when summed up make the value in terms of ‘embodied labour’ of capital, which varies as the profit rate (i.e. distribution of the surplus) varies. This result is also stressed with the aid of Fig. 3, which is found in the same page as the above quotation.

  21. 21.

    There is a possibility here for a misleading conclusion that this critique of economic theory that is based on the impossibility of measuring capital, or the values in general, is restricted to the adherence of a labour theory of value . This would be incorrect. Zambelli (2018b), for example, has shown that the values, even when the prices and values are determined by taking the surplus as the numéraire , are a function of distribution. Furthermore, when the agricultural sector (or any other sector) is taken to be the numéraire , there is no simple, unique measure of capital that is independent of distribution. This is a ‘modern’ critique and does not depend on the labour theory of value and the Austrian notion of the ‘period of production’ (Zambelli 2018b).

  22. 22.

    For a contemporary discussion of the transformation problem see also, among others , Dobb (1967), Laibman (1973), Samuelson (1971), Seton (1957), Shaikh (1984), Steedman (1977), Winternitz (1948).

  23. 23.

    Pilling (1972); de Brunhoff (1973, 1975) discuss the difference between Ricardo (and a Ricardian interpretation of the work of Sraffa made by the neo-Ricardians) and Marxian theory of value as it is dealt with in Marx’s Capital. “For Marx did not attempt to construct a pure economie theory, his field of problems and his point of departure is quite different from that of Ricardo and that of pure economics” (de Brunhoff 1973, p. 423). Crucial points are (1) Marx’s distinction between ‘labour-time’, ‘abstract labour’, ‘labour force power’ and ‘socially determined labour’ and (2) the different, unique meaning given to money.

  24. 24.

    Marx’s critique to Ricardo on this point may be summarized by the following:

    With him [Ricardo ], however, wage labour and capital are again conceived as a natural, not as a historically specific social form [Gesellschaftsform] for the creation of wealth as use value; i.e. their form as such, precisely because it is natural, is irrelevant, and is not conceived in its specific relation to the form of wealth, just as wealth itself, in its exchange- value form, appears as a merely formal mediation of its material composition; thus the specific character of bourgeois wealth is not grasped precisely because it appears there as the adequate form of wealth as such, and thus, although exchange value is the point of departure, the specific economic forms of exchange themselves play no role at all in his economics. Instead, he always speaks about distribution of the general product of labour and of the soil among the three classes, as if the form of wealth based on exchange value were concerned only with use value, and as if exchange value were merely a ceremonial form, which vanishes in Ricardo just as money as medium of circulation vanishes in exchange. Therefore, in order to bring out the true laws of economics, he likes to refer to this relation of money as a merely formal one. This does not mean that commodities do not have use values, but it is money which is the form that the commodities take. Hence also his weakness in the doctrine of money proper. (de Brunhoff 1973, p. 429, author translation from Marx‘s Grundrisse)

  25. 25.

    Zambelli (2018b) discusses this assumption. The uniform rate of profits is obviously a special case of Eqs. 15.5. This assumption is justified in PCMC, which is meant as a Prelude to a Critique of Economic Theory.

  26. 26.

    en × 1 is the summation vector.

  27. 27.

    I possess a copy of Keynes (1923) and both of Panico’s references to it, on p. 9 of his article, are incorrect (at least with regard to page numbers). Moreover, the quotation on the quantity theory of money is not entirely faithful to the original, in addition to the fact that neither Keynes, nor Sraffa accepted ‘the form dominant at that time’, which was Irving Fisher’s version of the quantity theory; as a matter of fact Keynes went into great trouble, in chapter III, § 1, of the same book, to give details of the version he (and Sraffa) worked with.

  28. 28.

    I shall explain, below, what is meant by a monolithic story.

  29. 29.

    This may be one reason for Sraffa’s disapproval of the famous chapterm17 of Keynes’s GT.

  30. 30.

    This does not mean I subscribe to a Darwinian-Mendelian theory of evolution. Even without reading Sraffa’s unpublished writings, I am sure he did not subscribe, either, to a conventional theory of evolution.

  31. 31.

    In our remarks concerning Keynes and Sraffa’s works during the early 1920s on the monetary theory of their time, we do not imply that they approved Fisher’s (or even that of Wicksell, or Hayek) version of the quantity theory of money. We emphasise that they both did, inside that framework, use the quantity theory to speculate on the relationship, for example, between prices and distribution.

  32. 32.

    We both, as well as many others, have consulted Sraffa’s archives, which are now available online. The material found in the archives confirms Sraffa’s statement that

    [...] the central propositions [of PCMC] had taken shape in the late 1920’s [...] (PCMC, p. vi)

    This and the archival material are evidence in support of our view that Sraffa’s research is to be seen as a coherent (although surely not monolithic!) whole.

  33. 33.

    In passing, we conjecture that this might have been the reason why Sraffa did not like or approve of Ch. 17 of the General Theory. Keynes (1936, Ch. 17, §II and §III ) develops an argument that there would be a ‘natural’ tendency for the own rates of interest of commodities to converge towards the own rate of interest of money:

    Thus with other commodities left to themselves, ‘natural forces,’ i.e. the ordinary forces of the market, would tend to bring their rate of interest down until the emergence of full employment had brought about for commodities generally the inelasticity of supply which we have postulated as a normal characteristic of money (Keynes 1936, Ch. 17, §III, p. 235)

    According to Keynes, full employment would not be reached, and hence the ‘natural forces’ of the markets would not fully operate, because money has a higher liquidity premium with respect to the carrying costs than any other commodity. Keynes does not challange or consider the direction of motion of prices. On the contrary, Sraffa is clearly concerned with this and argues that when the distribution changes the prices could change in all different directions (Lutz and Hague 1961, see Sraffa’s remark, p. 325). Even when the system is made determinate by assuming uniform rate of profits (or the wage rate), prices could move in different directions and consequently, the commodity own rates of interst could also do so.

    The similarity of arguments in Garegnani’s letter to Sraffa (see our chapter), with those of the converging mechanisms described by Keynes in Ch. 17 is striking. By analogy and induction, it may be reasonable to suppose that Sraffa’s response to Keynes’ Ch. 17 would be similar to that of his repsonse for Garegnani’s justification.

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Appendices

Appendix A: Sraffian Schemes and Distribution

1.1 Production and Methods of Production

In PCMC (p. 3, §1 and p. 10, §9) Sraffa assumes that there is ‘an annual cycle of production with an annual market’.

At the end of the production cycle there are n produced commodities:

$$ \mathbf{b}={\left[{b}_1,{b}_2,?{b}_i,?,{b}_n\right]}^T $$
(15.1)
$$ i=1,\kern0.5em 2,\kern0.5em ?,\kern0.5em n. $$

The method of production (PMCM, p. 3, §1 and p. 6 §4) producing bi is a linear combination of means of production and labour:

$$ {a}_i^1,\kern0.5em {a}_i^2,?,\kern0.5em {a}_i^j,?,\kern0.5em {a}_i^n,\kern0.5em {l}_i?{b}_i $$
(15.2)

where \( {a}_i^j \) denotes the means of production produced by industry j used in the production of commodity i and li the labour used in the production of bi.

In a more compact form, the method of production i may be written as ai, li → bi.

1.2 Surplus to Be Distributed

The ‘economy produces more than the minimum necessary for replacement and there is a surplus to be distributed’ (PCMC, p. 6, §4).

That is,

$$ {\displaystyle \begin{array}{ccccc}{s}_1& =& {b}_1& -& \sum \limits_{i=1}^n{a}_i^1\\ {}{s}_2& =& {b}_2& -& \sum \limits_{i=1}^n{a}_i^2\\ {}\vdots & =& \vdots & -& \vdots \\ {}{s}_i& =& {b}_j& -& \sum \limits_{i=1}^n{a}_i^j\\ {}\vdots & =& \vdots & -& \vdots \\ {}{s}_n& =& {b}_n& -& \sum \limits_{i=1}^n{a}_i^n\end{array}} $$
(15.3)

where si is the surplus of commodity i available for distribution after the quantities \( \left\{{a}_i^j\right\} \) have been put aside for the next year’s production. Alternatively, it is the quantity produced in the previous period which is left over, once the inputs used in production have been removed.

In compact matrix notation, we have

$$ \mathbf{S}={\left(\mathbf{B}-\mathbf{A}\right)}^T\mathbf{e} $$
(15.4)

where: e is the n × 1 unit or summation vector (each element is 1); T is the transpose operator; S is the n × 1 Physical Surplus vector or Physical Net National Product ; B is the n × n diagonal matrix composed of gross production b as its diagonal elements.

1.3 Self-Replacing Prices

Sraffa also assumes, for most of his book, that wage is paid post factum (PCMC, 9–10, §89). Given the knowledge of the methods of production used during the previous annual production cycle (and assuming that the same methods will be used), Sraffa searches for uniform prices that would allow the system to replicate these commodities during the next production cycles. These prices, following the classical tradition, could have different names (PCMC, pp. 7–8, §8). Here we refer to them as self-replacing prices.

If this is our problem setup, the accounting balance would require that

$$ {\displaystyle \begin{array}{rcl}\left(1+{r}_1\right){\mathbf{a}}_1\mathbf{p}+{l}_1w& =& {b}_1{p}_1\\ {}\left(1+{r}_2\right){\mathbf{a}}_2\mathbf{p}+{l}_2w& =& {b}_2{p}_2\\ {}\cdots & =& \dots \\ {}\left(1+{r}_i\right){\mathbf{a}}_i\mathbf{p}+{l}_iw& =& {b}_i{p}_i\\ {}\dots & =& \dots \\ {}\left(1+{r}_n\right){\mathbf{a}}_n\mathbf{p}+{l}_nw& =& {b}_n{p}_n\end{array}} $$
(15.5)

where: p = [p1, p2, …, pn]T is the price vector, r1, r2, …, rn are the sectoral profit rates and w is the uniform wage rate.

In matrix notation, Eq. 15.5 become

$$ \left(\mathbf{I}+\mathbf{R}\right) Ap+\mathbf{L}w= Bp $$
(15.6)

where: I is the identity n × n matrix; R = diag(r) is the diagonal matrix, whose diagonal elements are the rate of profits in each single industry, r1, r2, …, ri, …rn (vector r); A is the n × n matrix denotes the means of production \( \left\{{a}_i^j\right\} \); L is the n × 1 vector whose elements {li} are the labour used in production.

1.4 Number of Equations and Number of Variables

The system of Eqs. 15.5 (or 15.6) is indeterminate. There are at least two cases that we can consider.

  1. (1)

    Non-uniform rates of profits : in this case, there are 2 × n + 1 variables (n prices, n profit rates and the wage rate w) and n equations. This is a more general case and it is examined in Zambelli (2018b).

  2. (2)

    Uniform rate of profits : this is case that Sraffa examined in PCMC. Sraffa simplifies the problem by assuming that the rates of profits are uniform. In that case, the n rates of profits are assumed to be equal to a single rate of profits : r = r1 = r2 = ⋯ = rn (PMCM, p. 11, §11).Footnote 25 Here there are n equations and the number of variables are reduced to n + 2 (i.e., n prices, a wage rate w and a uniform rate of profits r). Hence the system is still indeterminate.

1.5 Net National Income in a Self-replacing System

1.5.1 The Share between Workers and Producers of the Surplus in Value Terms

‘The national income of a system in a self-replacing state consists of the set of commodities which are left over when from the gross national product we have removed item by item the articles which go to replace the means of production used up in all industries [the surplus vector S]. The value of this set of commodities […] we make equal to unity’ (PCMC, p. 11, §12).

Formally, the value of the Net National Product (\( {Y}^{\mathrm{NNP}} \)) is:

$$ {Y}^{\mathrm{NNP}}={\mathbf{S}}^{\mathbf{T}}\mathbf{p}=1 $$
(15.7)

If the value of the Net National Product is unity, this implies that STp = 1 is the numéraire .

In value terms, the value of the surplus that goes to producers is \( \sum \limits_{i=1}^n\left({r}_i{\mathbf{a}}_i\mathbf{p}\right)={\mathrm{e}}_{n\times 1}^T\mathbf{RAp} \). Similarly, the value of the surplus that goes to the workers is \( \sum \limits_{i=1}^n\left(w{l}_i\right)={\mathrm{e}}_{n\times 1}^Tw\mathbf{L}: \)Footnote 26

Y N N P = e n × 1 T RAp Share to Producers value + e n × 1 T w L Share to Workers value
(15.8)

For the special case in which the rate of profits is uniform, as in PCMC, we have:

Y N N P = e n × 1 T r A p Share to Producers value + e n × 1 T w L Share to Workers value
(15.9)

1.5.2 Distribution of the Surplus to Industries and Workers: Value and Physical Terms

The value of the Net National Product, STp is distributed to the n industries and workers:

  • d1STp is the share going to industry 1, d2STp is the share to industry 2, …, dnSTp is the share to industry n

  • dwSTp is the share that goes to the workers

Clearly \( \sum \limits_i^n{d}_i+{d}_w=1 \).

d S T p= d 1 S T p d 1 S T p d 1 S T p d 1 S T p Distribution of Surplus or N N P = b 1 p 1 a 1 p l 1 w b 1 p 1 a 2 p l 2 w b n p n a n p l n w e T L w Purchasing Capacity Income = r 1 a 1 p r 2 a 2 p r n a n p e T L w Purchasing Capacity Income
(15.10)

Given that STp = 1, the vector d = [d1, d2, …, dn, dw]T denotes distribution of the surplus S both in value and physical terms.

Recall that STp = 1 (see Eq. 15.7), the above accounting identity, Eq. 15.10, may be written in a compact form as

$$ \mathrm{d}=\left[\begin{array}{c}{d}_{n\times 1}\\ {}{d}_w\end{array}\right]-\left[\begin{array}{cc}\left(\mathrm{B}-\mathrm{A}\right)& -\mathbf{L}\\ {}{0}_{1\times n}& {\mathrm{e}}^T\mathbf{L}\end{array}\right]\ \left[\begin{array}{c}\mathbf{p}\\ {}w\end{array}\right]=\left[\begin{array}{cc}\mathbf{RA}& {0}_{n\times 1}\\ {}{0}_{1\times n}& {\mathrm{e}}^T\mathbf{L}\end{array}\right]\ \left[\begin{array}{c}\mathbf{p}\\ {}w\end{array}\right] $$
(15.11)

Clearly, when the self-replacing prices p and wage rate w are given, the distribution is uniquely determined and the profit rates r are determined as well. Also, when the distribution d is given, prices p and the wage rates w are also determined. In this case we have

$$ \left[\begin{array}{c}\mathbf{p}\\ {}w\end{array}\right]={\left[\begin{array}{cc}\left(\mathbf{B}-\mathbf{A}\right)& -\mathbf{L}\\ {}{0}_{1\times n}& {\mathrm{e}}^T\mathbf{L}\end{array}\right]}^{-1}\mathrm{d}=\left[\begin{array}{cc}{\left(\mathbf{RA}\right)}^{-1}& {0}_{n\times 1}\\ {}{0}_{1\times n}& {\left({\mathrm{e}}^T\mathbf{L}\right)}^{-1}\end{array}\right]\ \mathrm{d} $$
(15.12)

From Eqs. 15.11 and 15.12, we see that distribution and prices are determined simultaneously.

The vector d is the physical distribution of the surplus S to producers and workers, but it is also the distribution in value terms, that is, the share in value of the net national product shown in Eq. 15.8. The share going to the producers is given by eTRAp and that which goes to the workers is given by eTLw.

Furthermore , Zambelli (2018b, Appendix A) shows an equivalence between the actual physical distribution and the equivalent distribution in terms of a proportion of the surplus vector S. In other words, if di > dj, it implies that diSTp > djSTp and it as if diS > djS. This allows for a physical comparison of the distribution of the surplus.

1.6 Indeterminacy or Exogeneity

In both cases (1) and (2) mentioned earlier, the system can be closed only by adding some additional exogenous elements or additional theories. Here we have chosen the physical surplus S as the numéraire . The economic system is described in terms of n equations (Eqs. 15.5) and the numéraire equation (Eq. 15.7).

This implies that when considering a more general case with non-uniform rates of profits , we have n + 1 equations and 2n + 1 variables. Therefore, there are n degrees of freedom: that is, all the self-replacing prices, rates of profits and wage rate cannot be computed by the knowledge of the quantities and the methods of production alone. In order to solve the system of equations, n variables among {p1, p2, …, pn, r1, r2, …, rn, w} have to be given exogenously, that is, from outside the economic system.

When we consider Sraffa’s special case with uniform rate of profits (case ii), we have n + 1 equations (i.e., Eqs. 15.5 is restricted to the case in which the rates of profits are uniform and the numeraire Eq. 15.7) and n + 2 variables {p1, p2, …, pn, r, w}. In this case, there is only one degree of freedom and one of the n + 2 variables have to be exogenously given.

Appendix B: Incorporating Credit and Debt Inside a Sraffian Framework: An Outline

We now outline the way in which we can introduce money, credit and debt, inside the Sraffian schemes (for details, see our companion paper Venkatachalam and Zambelli (2021)).

When we discuss an economic system, whether real or virtual, we often deal with a world of bilateral or multilateral exchanges. In such a world, when someone buys a good, there is always a counterpart who sells. In the traditional Sraffian schemes, the possibility of deferred means of payment have not been adequately explored. For the aims and arguments developed in PCMC, the introduction of deferred means of payments are not strictly necessary and their omission may be understandable. But as claim in Sect. 3, money and deferred means of payments are indeed important and that there is a clear scope to extend the framework presented in PCMC by incorporating money, credit and debt.

In Appendix A, we provided a formal treatment of generalized Sraffian schemes, where the rates of profits are non-uniform (system presented in PCMC with uniform rate of profits is a special case). As we have discussed above, when the system is indeterminate, there is ample space in theory for including elements that are not typically considered as belonging to the realm of economics.

Consider the more general case where the rates of profits are allowed to be nonuniform. We have n + 1 equations and (2n + 1) the variables (see Appendix A.6). If we now assume that the commodity prices \( \overline{\mathbf{p}} \) and wage rate \( \overline{w} \) are exogenously given we have the following inequalities:

a 1 1 p ¯ 1 + + a 1 j p ¯ j + + a 1 n p ¯ n + l 1 w ¯ Virtual Expenditures = < > b 1 p ¯ 1 Virtual Revenues a 2 1 p ¯ 1 + + a 2 j p ¯ j + + a 2 n p ¯ n + l 2 w ¯ = < > b 2 p ¯ 2 + + + + + = < > a i 1 p ¯ 1 + + a i j p ¯ j + + a i n p ¯ n + l i w ¯ = < > b i p ¯ i + + + + + = < > a n 1 p ¯ 1 + + a 1 j p ¯ j + + a n n p ¯ n + l n w ¯ = < > b n p ¯ n
(15.13)

In matrix notation, this could be written as:

A p ¯ + L w ¯ Virtual Expenditures < = > B p ¯ Virtual Revenues
(15.14)

The prices and the wage rate in the above equations are to be seen as virtual or bookkeeping prices, which may or may not be the actual exchange prices. In fact, they cannot be the actual exchange prices when there is at least one industry for whom the bookkeeping expenditures (left-hand side) would be greater than bookkeeping revenues (right-hand side). In this case, the rate of profits of this industry is obviously negative.

This is a situation where there would not be enough purchasing power for the exchanges to take place and some industries would be left with unsold commodities. This, in turn, would mean that the economic system as a whole would not be in the self-replacing condition.

Clearly, the industries with virtual expenditures higher than their virtual revenues would not have the necessary purchasing power to buy the means required to replicate production of the previous period. These industries, therefore, are in a condition where there is a potential financial deficit. Concurrently, there would be industries which would not be able to sell all of their product. The industries in ‘financial deficit’ would be able to purchase the necessary means of production only by agreeing to a deferred payment to take place during the years to follow. At the same time, the industries in potential ‘financial surplus’ would be able to sell all of their product by agreeing to deferred payments by the borrowers. The traditional Sraffian schemes outlined in Appendix A can be extended or generalised to accomodate this situation.

We can conveniently view these deferred means of payments as a form of an I Owe You (IOUs). This can be seen to include all forms of financial contracts, where there is a promise to return goods in the future. These obligations may be with an explicit delivery date (as in the case of forward contracts associated with real goods), or with a relatively loose delivery date (as in the case of standard means of exchange such as cash, checks, debt and credit accounts, bonds etc.).

We can consider a situation wherein a commodity is exchanged for a promise to pay back at a future point in time (deferred payment). The entity selling this commodity would see its credit increasing and correspondingly, the counterpart buying this commodity would have its debt increase. This might take place by the writing off of means of payments previously generated or by issuing new means of payments. Furthermore, this could take place through an institution, like the banking system, or through direct contracts. A full-fledged treatment of Sraffian schemes with deferred means of payments can be found in our companion piece Venkatachalam and Zambelli (2021).

Given this structure, we can ask several interesting questions and perform thought experiments. To start with, we can explore the set of prices and possible financial conditions that would allow the system to reproduce itself, if one allows for transfer of purchasing power in the form of credit and debt. In other words, identifying the set of prices that would allow self-replacing. Note that IOUs may be generated and transferred from one period to another, which in turn might influence the set of self-replacing prices in the subsequent periods. These may offer us insights concerning the potential paths that are available for the system to evolve in the future. To answer questions of this nature, it is also important to study the effects that already existing future promises to pay may have on determining the set of self-replacing prices. All of this require keeping a close track of the credit and debt structure with careful accounting built in to the framework.

We also note that in our attempts to introduce money, we follow Sraffa’s approach in PCMC and do not provide a theory of prices and distribution. Instead, the objective would be to determine, for given past methods of production, usage of labour and output, those set of prices and the possible financial conditions that would allow the system to reproduce itself. We hope that this provides some pointers to the direction in which Sraffian schemes can be generalised.

Comment A Comment on ‘Sraffa, Money and Distribution’

It is a daunting prospect, to envisage comments from an old dilettante, but who nevertheless thinks very highly of Sraffa’s contribution to economics, in general; it is especially daunting to be asked to provide—what will turn out to be innocuous and inexpert—remarks on the comprehensive studies of Sraffa’s published and unpublished material and its resultant outcome in a highly competent paper on Sraffa, Money and Distribution by two scholars of integrity and originality. Moreover, the senior author has studied the unpublished writings and notes of Sraffa, deposited in the Wren Library, at Trinity College, Cambridge. All this makes my comments take on a perspective that strengthens my regrettable reputation of being a glorified dilettante! However, my respect for the editor of this volume means that I have to give it a try.

Sraffa, Money and Distribution is a competently written, impeccably researched—in a paper of 17 pages of text, with two appendices of 6 dense pages, the references numbering 77, are spread over 4 pages—closely and, to the extent I can gauge, correctly reasoned, fine contribution. My remarks, confined to a reading of the 17 pages of main text, except for two, very minor dissenting points, on the main arguments in the two appendices (even although a substantiation of the theme in Appendix B is said to be ‘work in progress’, by the two distinguished authors).

As I read this fine paper, (partly) based on the extensive reading (by the senior author) of the unpublished material of the Sraffa papers, primarily at the Wren Library of Trinity College, Cambridge—and ruminations by both authors—I understood the main theme to be that Sraffa’s earlier work in money, banking, returns-to-scale, all under some kind of equilibrium condition, is one with the propositions in PCMC. This main theme is due to his—Sraffa’s—increasing awareness of ‘the insufficiency of market forces to determine values.’

As the authors state, already in the Abstract, but strengthened in the main text:

In contrast, we suggest that Sraffa’s contributions have to be viewed as a coherent whole …. We point to the unifying thread between his early and later work, which concerns the insufficiency of economic mechanisms or market forces to exclusively determine values (prices, profit rates and wage rates). (Italics added)

The authors seem to accept, to substantiate the above theme, that ‘Sraffa’s contributions to economics can be broadly divided into two parts’ (p. 2, italics added)—one being monetary and banking theory and the other, for simplicity, can be referred to as capital (or value) and distribution theory (although the authors are more careful in their nomenclature).

They seem to want to buttress their stance on the ‘two part’ theory by appealing to, for example, Panico (1988a).Footnote 27 I don’t think they need this support, particularly because it is dubious in its theoretical underpinnings. I, as a non-reader of Sraffa’s unpublished works, do not subscribe to this monolithicFootnote 28 story. For one, there was the returns-to-scale issue, strongly dependent on various determinate and indeterminate concepts of equilibria; the other is, in Adarkar’s felicitous words, Mr. Sraffa’s Commodity Rate (chapter VII, in Adarkar 1935)—also known as the own rates of interest. This, in turn, via Sraffa’s Keynes inspired critique of Hayek (Sraffa 1932a, b), leads to the dichotomy between the short-run and long-run,Footnote 29 on the one hand, and what came to be the construction of the standard commodity .

When he constructed the arguments, the economics and the mathematics of PCMC, he gradually divested himself of all dependence on money, banking, returns-to-scale, competitive vs. non-competitive, short-run versus long-run and, above all, any notion of equilibrium. PCMC is a non-equilibrium, eternal-run, theory of production and distribution which cannot determine, uniquely or otherwise, values—even after taking into consideration the exogenously determined profit rate (§44, PCMC).

I think, largely based on Professors Chiodi and Zambelli’s reporting to me of the contents of the unpublished material in Sraffa’s works, there is no mileage in working with a monolithic story; I think scholars evolve,Footnote 30 and sweep aside what they think are irrelevant accoutrements to a developing theory of a system.

I come, now, to the two dissenting points, that I wish to state:

  • There is no justification for Nuti to state, what the authors report on p. 18, as being quoted by Panico; in particular, no one can determine what is the ‘most appropriate way’ for anything.

  • I have difficulties of giving up the Sraffian assumption of uniform rates of profit ; this is why I refer, now, to PCMC dealing with an economic system in the eternal-run of non-equilibrium configurations.

PCMC is, if anything, a manifesto of David—without Ricardo’s equilibrium, long-run, accoutrements. That is why Sraffa was able to write Keynes, on 20 December, 1932:

[D]on’t treat too ill my David.

Reply A Reply to Velupillai

Vela Velupillai has provided thoughtful comments on our piece and this allows us to provide some clarifications that may be necessary. Velupillai writes:

I understood the main theme to be that Sraffa’s earlier work in money, banking, returns–to–scale, all under some kind of equilibrium condition, is one with the propositions in PCMC. This main theme is due to his—Sraffa’s—increasing awareness of ‘the insufficiency of market forces to determine values’.

Based on this, he seems to interpret our position as one that characterises Sraffa’s research program as being monolithic. Such an interpretation would be incorrect and hence we want to clarify this further to avoid confusion. First, we argue that Sraffa’s research contributions are coherent and that his views on money are not necessarily incompatible with that of PCMC. We do not, however, make a much stronger claim that it is monolithic. Second, the only equilibrium notion that we attribute to Sraffa (from his early work onwards) is the idea of accounting or bookkeeping equilibrium.Footnote 31 The continuing thread that we trace across his body of work does not have any reliance on equilibrium. We do not imply at any point that Sraffa assumed some kind of equilibrium condition, except for the accounting equilibrium. Third, Sraffa’s production prices or self-replacing prices are accounting equilibrium prices, that is, prices for which the quantities sold multiplied by the prices (Revenues) are equal to the quantities bought multiplied by the prices (Expenditures) plus what is left or residual (Profits or Losses, which are Revenues minus Expenditures).

The prices considered by Sraffa, in PCMC and in the preparatory work,Footnote 32 are never actual or market prices . Instead, they are virtual (uniform) prices which, if realized during the annual market, would allow the system to be in a self-replacing state. This seems to be the thought/imaginary experiment that Sraffa conducts for a thorough critique of economic theory. However, it is not Sraffa’s theory of distribution , as Bharadwaj (1963, p. 1450, quoted in our chapter) has elegantly summarised.

When we attempt to relax or modify some of the assumptions of PCMC, we have not used any notion of equilibrium, except that of the bookkeeping or accounting equilibrium and search for the prices that would guarantee it. In the Appendix B and in Venkatachalam and Zambelli (2021), we conduct a thought experiment where we search for the prices and the nature of credit and debt contracts that would allow the system to remain in the self-replacing state. We believe that this is coherent with Sraffa’s method of investigation, although our goals may be different.

PCMC, as we have stressed in our contribution, is a Prelude to a Critique of Economic Theory. This is often acknowledged by the scholars working with PCMC, but very few take it as such. The major critique, which is much more than a ‘Prelude’, is the impossibility of computing the self-replacing prices based solely on information concerning the methods of production.

Once the major critique concerning indeterminacy has been put forward, Sraffa continues by considering the imaginary case in which one of the distributive variables r (or the wage rate w) is exogenously given and prices are computed as a function of the distributive variable. This critique is strong and it is based on the fact that as the profit rate changes, prices may not change monotonically (PCMC, p. 38, Fig. 3)Footnote 33 and this allows him to conclude that

The reversal in the direction of the movement of relative prices, in the face of unchanged methods of production, cannot be reconciled with any notion of capital as a measurable quantity independent of distribution and prices. (PCMC, p. 38)

Supported by textual evidence, we think that Sraffa’s view or research program, from the beginning of the 1920s to 1960, is a coherent whole. It concerns the insufficiency of economic mechanisms or market forces to exclusively determine values (prices, profit rates and wage rates ). His contributions are all critical of the standard notions of equilibrium and we wholeheartedly embrace this.

As we have emphasised in our contribution (see. p. 15), non-uniform rate of profits , money or notions of credit or debt are not necessary for Sraffa’s prelude to a critique in PCMC. However, this does not imply that there is no place for them in Sraffa’s system or that it is illegitimate to introduce them. We argue that there is such a scope and that this will open up new avenues for research.

We take note of Velupillai’s difficulties in giving up the Sraffian assumption of the uniform rates of profit. However, we have difficulties in understanding the connection between the uniform rates of profits assumption and Velupillai’s view that “PCMC [is] dealing with an economic system in the eternal–run of non-equilibrium configurations”. We are sure that he is not suggesting that the rules of accounting are broken, because this would dismantle PCMC. Sraffa has shown that the system of equations in PCMC is indeterminate and this could be seen as a non-equilibrium configuration. If this is what Velupillai means, we are in complete agreement, however, with the difference that we do not find it difficult to give up the assumption of uniform profit rates. The system remains indeterminate even when the uniform rate of profits assumption is dropped.

We are indeed sympathetic to the idea that an economic system ought to be studied without the shackles of equilibrium and as non-equilibrium configurations. In fact, we believe that our work is a research agenda in this direction. We are also convinced that the constructive and algorithmic method, one that Velupillai has advanced over the years, is a promising way to achieve this. We attempt to do so by combining constructive methods, Sraffa’s core insights and by removing simplifying assumptions which we do not consider to be essential in PCMC and by adding other elements.

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Venkatachalam, R., Zambelli, S. (2021). Sraffa, Money and Distribution. In: Sinha, A. (eds) A Reflection on Sraffa’s Revolution in Economic Theory. Palgrave Studies in the History of Economic Thought. Palgrave Macmillan, Cham. https://doi.org/10.1007/978-3-030-47206-1_15

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