Interpreting the Nature of Sraffa’s Equations: A Critique of Garegnani’s Interpretation

Abstract

This chapter argues that Sraffa broke away from both the classical idea of rooting theory in the notion of ‘ultimate cause’ as well as mechanical causation of equilibrium of forces. Sraffa’s idea of a system of basic goods denies the possibility of reducing production to labour and Nature alone. His system is composed of well-defined structural properties, an implication of which is that the rate of profits in all the industries must be uniform. Garegnani and his associates missed out on this insight of Sraffa and therefore imposed the classical mechanism of gravitation of ‘market prices’ to ‘natural prices’ in order to make sense of the condition of the uniform rate of profits in the equations of Sraffa’s system. The chapter shows that this produced contradictions and incoherence in our understanding of Sraffa’s theoretical achievement.

Appendices

Comment

1.1 Sraffa’s Protoeconomics—A Comment on Ajit Sinha’s ‘Interpreting the Nature of Sraffa’s Equations: A Critique of Garegnani’s Interpretation’

• Bertram Schefold 

Sinha’s chapter opens with an interesting question. What is new in the way of how Sraffa leads back to classical economics? According to Sinha, Sraffa disliked that Smith used subjectivist hypotheses like self-love. There are notes in the archive where Sraffa calls Smith the first of the vulgar economists. Smith’s adding-up theory of value may be one of the reasons why Sraffa chose this denomination. Another could be that Sraffa strove to eliminate subjectivist foundations of economic thought from his analysis, and Sinha suggests that Smith’s use of self-love constituted such a subjectivism. We know from Smith’s Theory of Moral Sentiments that his psychology was more sophisticated than that, although it is true that, in the Wealth of Nations, not much remains of the more complex notions except this self-love. No economist would deny the importance of profit maximization as an explanation of capitalist behaviour. Smith’s notion of self-love is broader than that, hence the ease with which neoclassical economists identify with Adam Smith in the belief that, where there is self-love, there is also the maximization of utility. Sraffa, according to Sinha, wished to eliminate such subjectivist notions. To be more precise, Sraffa did not deny the reality of profit maximization, but he sought to found his theory on another basis. It is true that profit maximization is not invoked explicitly in Production of Commodities by Means of Commodities, but there is the uniform rate of profits. Sraffa says, where he introduces this rate, that it “must” be uniform, without explaining the “must” (Sraffa 1960, p. 6). Garegnani does not hesitate to invoke competition, driven by the maximization of profits, to explain this “must”, according to Sinha (2021). Ajit Sinha believes that here the pupil misunderstood his master. Sinha cannot seriously deny that Sraffa accepted the reality of profit maximization, but he is of the opinion that Sraffa did not make such an assumption. Hence he claims that Sraffa’s theory of prices can be founded without it. How can we interpret such a claim?

We ask more generally how a new academic discipline can be founded or on which grounds a new theory can be erected. One difficulty of founding economic theory as a scientific discipline derives from the fact that we are part of the economy, our behaviour is shaped by our economic experiences and our language is full of economic terms. Their significance should be explained and not be taken for granted.

The conceptual framework of the new theory must be founded on something outside the subject under consideration; otherwise, one moves in a circle, because explanans and explanandum get mixed up.² The concepts thus introduced must refer to a universe of which we have an intuitive understanding. We have, in a Kantian tradition, to reflect on the conditions of cognition. Kant thought that the instruments for our intuitive apprehension of the empirical world were given to us, as far as space was concerned, by our vision of Euclidian space, and the conceptual apparatus for its description was given prior to experience in the form of mathematics. The foundations of physics therefore could not be anything. The movement of masses in mechanics could occur only according to laws compatible with Euclidian space, with our understanding of time and our consequent analysis of movement. Kant therefore undertook to derive the fundamentals of physics, for him the Newtonian axioms, from his philosophy in his Metaphysische Anfangsgründe der Naturwissenschaft (Kant 1787), and the result seemed convincing until it was understood that there were also other geometries beyond the Euclidian one, that the speed of light and communication had a specific limit and that elementary particles were quantized.³ Kant had hoped thus to get to fundamental statements that would be true a priori and yet be more than tautologies (that would be “synthetic” and not only “analytical” truths). In this he failed, but the postulate that the instrument of the description should be given prior to the description has remained as a methodological principle for deductive reasoning.

But the postulate cannot be fulfilled completely. What could be the fundamental concepts outside economics which one might use to found economic theory? How shall we introduce concepts such as good, commodity, labour without a prior understanding of economic matters, basing ourselves exclusively on a non-economic discourse? The neoclassicals, at least in the Austrian variant (Mises 1940), reply that all action is economic and everything economic has to do with human action. The essential propositions of economics then follow from the understanding of action, by gradually filling up this deductive conception with intuitive content. The economic agents, in possession of their faculties and of goods, can produce for themselves or exchange. They exert their freedom to act in the act of exchange. While Weber wanted to understand and order different rationalities, there are not different rationalities but different degrees of understanding of what is going on in the Austrian approach. A rain dance is rational action, insofar as one tries to influence superhuman forces to make it rain, but one is in error regarding the causes, which make the weather change. Rationality then implies utility maximization along with other rules.

Marx, with his dialectical materialism, chose a different procedure. He took a number of elementary economic concepts such as labour and exchange and defined plausible relations from which he could deduce a theoretical framework, such that more complex notions could be derived. He assumed that there was labour, distinguished between abstract and concrete forms, and postulated that goods are exchanged according to their content of abstract labour—a proposition, which seemed roughly plausible, but of which Marx knew that it was only a provisional approximation to the true state of things in a competitive capitalist world. He got from a theory of value to a theory of exchange, money and production, from there to exploitation and accumulation. The drive of capital was to chase after surplus value, but it first had to be explained that profit was a form of surplus value, and before there could be talk of a maximization of the rate of profit, the rate of profit had to be defined. Competition was not any form of human rivalry, but it showed in the equalization of the rates of profit and the tendency for a general rate of profit to form. The uniform rate of profit was therefore not simply assumed as a result of competition, but, on the contrary, competition had first to be explained in terms of the process of the equalization of rates of profit, and these resulted from the drive to amass surplus value by exploiting labour.

As soon as this has been understood, one can see (cf. Schefold 2019), looking at the beginning of the Marxian construction, that it is really the opposite of what Austrian neoclassicism assumes: The value ratios do not result from the free intercourse of independent agents who may decide to swap or not to swap, but the exchange takes place between interdependent producers according to value relationships that are given to them. The material circumstances determine what they can do; Marx sticks to his materialist determinism even where he deals with entrepreneurial action. The capitalist is only the “character mask”; the process of surplus value creation is impersonal and anonymous.

Sinha is right that Sraffa has his own way of founding economics, at any rate that section of it which is concerned with value, price and distribution. I used to call this Sraffa’s protoeconomics. He begins with his famous consideration of an “extremely simple society”. Not the logic of action, not predisposed labour times, but the structure of the reproduction of commodities determines the price ratios. The author leads the reader from assumptions to conclusions. The homogeneity of the commodities is clearly simply assumed, and it is simply assumed that the structure of production is such that self-replacement is possible.

Various metaphors have been used to visualize how these magnitudes become known. Sinha prefers to speak of an observer. This is reminiscent of Einstein’s thought experiments, and Sraffa, in taking up this particular metaphor, probably was influenced by contemporary discussions of modern physics, as Kurz and Salvadori (2005) argue. Thought experiments involve strong assumptions and abstractions. Sinha is in the danger of taking the metaphors too literally. Sraffa begins his protoeconomics with

$$ \mathbf{Ap}=\mathbf{p}\ \mathrm{and}\ \mathbf{eA}=\mathbf{e} $$

where e is the summation vector, so that the second formula is the otherwise self-explanatory formalization of “self-replacement”. If this is assumed and A is an indecomposable semi-positive input output matrix, the second equation shows that e is a Frobenius eigenvector of A, with an eigenvalue of 1, so that the right-hand side Frobenius eigenvector p > 0 must also exist: The existence of prices is guaranteed. It is a foundation of prices, independent of a labour theory of value or of utility theory.

There follows the model of production with a surplus, where, instead of labour inputs, wage goods consumed by the labourers, whatever their social status, slaves, serfs or wage labourers, are treated on the same footing as the inputs to production. The formula summarizing this model is

$$ \left(1+R\right)\mathbf{Ap}=\mathbf{p} $$

Here we have the uniform rate of profit R, of which Sraffa said that it “must” be uniform, without telling us why. Sinha sees two possibilities to explain the “must”. There is his own attempt to provide a structural reason and Garegnani’s reference to competition, which Sinha criticizes extensively. Sinha’s procedure is curious. The structural reason, which he tries to give, is nowhere to be found in Sraffa’s published writings. The opposition between Garegnani and Sraffa, which he tries to reconstruct, is a priori not plausible. Garegnani was without question Sraffa’s most important pupil, his friend and support over many years to the end. Garegnani may have had his own agenda, but there was trust between them, and Garegnani was appointed Sraffa’s literary executor. Finally, it does not occur to Sinha that a structuralist explanation of the “must”, if one really exists, and Garegnani’s interpretation do not necessarily exclude each other; they might be complementary on two different levels of abstraction. Garegnani’s interpretation is indispensible, if one wants to get from protoeconomics to economics. For instance, the importance of competitive processes shows in classical theory in the treatment of market prices. If one moves from Sraffa’s protoeconomic prices to the classical theory, whether one speaks of the Ricardian theory of rent or of the Marxian theory of accumulation, whether one deals with Malthus and effective demand or with John Stuart Mill and the stationary state: One will always use natural prices to conceptualize different states of the economy in evolution, and one will know that market prices deviated from them, in ways that depend on competitive processes, because of harvest cycles, crisis and political disturbances. And what is the meaning of the natural prices, if the market prices do not somehow gravitate to them?

Sinha nowhere explains how he wants to get from protoeconomics to economics and lose his innocence, as it were. The diversity of his observations forces me also to make diverse remarks that cannot be ordered without some arbitrariness.

We can start from market prices, since Sinha begins with Garegnani’s interpretation of the formation of market prices in Adam Smith. He speaks of demand and supply schedules in this context, as if Smith had a visualization of supply and demand as in neoclassical theory. Here Sinha seems to me to be uninformed about the state of discussion of supply and demand in the eighteenth century. In particular, reference should be made to James Steuart. In the eighteenth century, and still in Marx, demand and supply were seen as “forces”. This visualization of the market process is fundamentally different from that expressed in terms of schedules. Such schedules are absent in Smith; they first appear in Rau and in Cournot. But this is another story which it would take longer to explain (Schefold 1997).

An example that we find in the earlier literature concerns the arrival of a ship in a colony (e.g. Steuart 1967 [1767, 1805], pp. 270–273). If people in the town can expect that other ships will follow soon, the merchant will sell his wares at more or less habitual prices, that is, prices that reflect normal costs, because the buyers will otherwise prefer to wait. If the arrival of other ships is more uncertain, moral suasion will play a role. The examples given by Adam Smith comprise the conditions in a beleaguered city, where prices can rise very high because of extreme scarcity. The gravitation of market prices to natural prices therefore is a matter of anecdotal circumstances. There is no theory of gravitation in Adam Smith, but he has an idea what it is and where it can lead, and we, of course, can construct modern models of gravitation.

In doing so, we have many possibilities. The natural price may be realized as an average over many years. This applies in particular to agricultural commodities. Or market prices can be close to natural prices, because charging mark-ups on direct costs such that normal profits are obtained in the long run is a good strategy for producers of industrial commodities. The adaptation of prices can be formalized as a convergent process, if the quantities are rigidly given; an example is the following self-explanatory sequence which, if it starts from any prices, will converge to natural prices in terms of the wage rate:

$$ {\mathbf{p}}_{t+1}=\left(1+r\right)\mathbf{A}{\mathbf{p}}_t+l $$

There are more sophisticated gravitation models with cross-dual dynamics (the result, whether gravitation obtains, depends on assumptions). One can even regard intertemporal general equilibrium theory as the gravitation process of market prices to prices characterized by a uniform rate of profit, if conditions for the supply of unproduced factors on the one hand and demand conditions on the other are stationary, as I have discussed elsewhere. Sinha is right that Garegnani has made different attempts to formalize gravitation processes, which were not all equally convincing. My conclusion is that gravitation is an axiom of classical theory, not a result, and many and diverse models have been invented to justify this axiom.

His misconception of what gravitation is about leads Sinha to a number of unhelpful statements. It is true that Smith was not interested in the repercussions of gravitation in one market on gravitation in other markets. But that does not mean that Smith was generally unaware of the interdependence of markets; he was simply arguing, as one later would say, ceteris paribus. He was right to be cautious. If one talks about the interdependence of markets, one should not too easily postulate the law of one price. The same commodity can have different prices in different sectors in the real world. A detailed model of the gravitation process then becomes very complex; yet, the tendency remains theoretically and empirically plausible.

Why does Sraffa in Production of Commodities by Means of Commodities not need assumptions about returns to scale? One reason is this: Because the analysis of gravitation is not part of protoeconomics, although it may help to justify the assumptions. Sraffa hints however at some applications where constant returns to scale are the most convenient hypothesis for getting an intuition of what he discusses. An example is his rising supply schedules in the case of intensive rent.

I do not agree that the contrast between natural prices and market prices could not be introduced in the system without a surplus, as Sinha seems to believe, but there is no room to introduce such a special gravitation model here.

I hope we can agree that Sraffa’s system is so interesting not the least because of its open character with regard to distribution, the composition of output and accumulation (Schefold 1987, Part Three). His theory is strong and rigorous, as far as the logic of the formation of natural prices is concerned, if distribution is given, but the influences on distribution may be varied and quite different, depending on whether one is dealing with an economy experiencing strong growth or one which is stagnant. The structure of the economy (the protoeconomics) does not suffice to explain how the economy will behave regarding changes of distribution, demand and employment. The question then is how the step from protoeconomics to economics is to be made. Sinha seems to be tempted to simplify too much, when he uses the image of the observer in this context. He believes that mere observation could reveal the protoeconomic structure of an actual economy. Can one not see what factories there are, how many workers there operate and how many consumption goods are being delivered?

But the observer of the real economy will not see homogeneous commodities, for competition regarding the quality of commodities will accompany competition regarding prices. The observer does not see one technique in use in any industry, but usually techniques that rival with each other and the dominant technique cannot easily be identified. Actual production is not based on single-product industries, but on a certain degree of joint production in most cases, and often it will be difficult to ascribe outputs and even inputs to specific processes. The observer, in trying to catch the underlying system, will not be able to avoid subjective interpretations, attempts to understand the intentions of producers and the desires of consumers. The bold simplicity of Sraffa’s assumptions cannot be defended by the ingenuous postulate of the observer. I think it is one of the best and most original insights in Production of Commodities by Means of Commodities that Sraffa characterizes long period positions by an equality of the number of commodities produced and the number of processes used. There are not many economists out there who have understood what a long period position is, and the number of those who have understood this condition is even smaller. It is quite naive to believe that this condition could be verified in reality in a simple and straightforward manner. And yet I believe that phenomena, by which this law imposes itself, can be observed. I tried to show it using examples of energy production and consumption (Schefold 1985). It is a domain where the observer seems to be confronted with objective data: Household a uses gas, household b uses electricity for heating, country A uses nuclear power for electricity production and country B uses wind. But which system will dominate? What is “socially necessary” here depends on a political process, which needs to be understood.

We now turn to Sinha’s assertion that the uniformity of the rate of profit can be described as a structural property of the system. In a quite formal sense this is trivial and a matter of definition. But if unequal rates of profit are admitted as a possibility, the question becomes whether a meaningful structural property can be found that imposes itself as a plausible assumption, that would imply uniformity and that would be in some sense more fundamental than the recourse to competition.

As we already indicated, Sraffa (1960, p. 6) introduces “Production with a Surplus” in the second chapter of his book (fourth paragraph) and says: “The distribution of the surplus must be determined through the same mechanism and at the same time as are the prices of commodities”. And he continues: “We add the rate of profits (which must be uniform for all industries) as an unknown”. What is the “mechanism”, and what does “must” mean? According to the standard interpretation, rejected by Sinha and associated by him with Garegnani, the “mechanism” is competition.

It is clear, however, that there can be homogeneity of commodities, of prices and a structure of an economy producing a surplus such that unequal rates of profit result. Consider the following model of physiocracy, based on a suggestion by Jean Cartelier (1976): There is a physical surplus in both of two sectors. The artisans produce one unit of steel by means of 1/2 of steel and 1/4 of corn. Farmers produce one unit of corn by means of 1/4 of steel and 1/2 of corn, so that we get the following table:

	Steel	Corn
Artisans	1/2	1/4
Farmers	1/4	1/2

1.2 Diagram 1: A Model of Physiocracy

If we interpret this table as a Sraffa system, it is clear that we have standard proportions 1:1 and a maximum rate of profit of 1/3. Relative prices are also given 1:1. For the maximum rate of profit, we may write R = 1/3 and 1 + R = 4/3. If p is the price of steel in terms of corn, we get

$$ \left(1+R\right)\left(\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.p+\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.\right)=p $$

$$ \left(1+R\right)\left(\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.p+\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\right)=1 $$

But there is no full-scale capitalism if we interpret the table as a model of physiocracy. The wages of the workers are part of the means of production. The surplus is appropriated not by capitalists but by the monarchy in the form of a tax (impôt unique). We denote the price of steel in terms of corn by u and the tax rate by q; the tax is to be levied in agriculture alone by virtue of the physiocratic doctrine: Agriculture is based on nature, nature is productive, the productive sector should be taxed. Hence we have the price system:

$$ \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.u+\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.=u $$

$$ \left(1+q\right)\left(\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.u+\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\right)=1 $$

Obviously, one obtains u = 1/2 and 1 + q = 8/5, hence q = 3/5, so that q−R = 4/15.

This means that, with the tax rate q = 3/5, the entire surplus is levied in agriculture.

q−R = 4/15 means that the tax rate exceeds the rate of profit (if it is uniform) by 4/15. If physiocracy rules, the onus of surplus extraction is shifted to the farmers.

Historically, or, rather, in the history of ideas, one goes from the physiocratic system with the surplus levied in agriculture to the capitalist system with the uniform rate of profit, therefore from the model which we have introduced as the second to the first. Really, institutions change, if the surplus is now, other things being equal, appropriated by the capitalists. Marx would have said that they share the booty equally.

Instead of making competition and institutions responsible for the equalization of the rates of profit, Sinha has in his book a model with several sectors and profit differentials. He wants to argue that profit differentials must disappear thanks to a transformation into a standard system, but here we have, for the model of physiocracy, a standard system from the start, and the differentials persist. That the differentials become zero in Sinha is nothing but an assumption for which he provides a formal reasoning, but he has no economic argument to substantiate his formal procedure; he does not even state his assumption in clear and explicit form. I do not see why his formalism should be better than the simple assumption that the rate of profit is uniform; rather, I think it is confusing.

I do not deny that there is a deeper motivation. Sraffa, according to various notes in the archive quoted by Sinha, wanted to show that the rate of profit was not a price phenomenon. In a sense, this is obvious from the start, since the rate of profit is expressed in percentage terms, and the percentage can be announced before prices are made explicit. As it is a percentage that is given, it can be given independently of a numéraire. To visualize it, one may refer to the corn model and to the standard commodity as a physical analogue, as was once suggested by Eatwell (1975). The question then is how to express the wage, for the real wage depends on the standard of prices, and so Sraffa left theories of distribution behind which are based on a somehow given real wage. Instead, he advanced the hypothesis that it is the rate of profit, which is “determined from outside the system of production”. He suggested that it might be determined in particular “by the level of the money rates of interest” (Sraffa 1960, p. 33). This has remained controversial. Distribution does depend on prices, if the theory of distribution must start from a real wage. Hence there is an ambiguity in the statement that distribution is not a price phenomenon, and I do not feel at ease when the attempt is made to remove this ambiguity by referring to the rate of profit instead. Is distribution primarily determined by the subsistence needs of workers, by effective demand, by the rates of interest or by marginal productivity? This is partly a theoretical and partly an empirical and historical question. To have laid the stress mainly on the rate of profit is for me not Sraffa’s main achievement. By referring to the rate of profit, he posed a problem, but he did not solve it.

Sraffa’s protoeconomics remain fascinating. In the end, the question is where we get when the step to economics is made.

Reply

1.1 A Response to Professor Schefold

• Ajit Sinha 

Professor Schefold (2021) has done me honour by painting such a wide canvas while commenting on my chapter—its significance goes much beyond the narrow confines of my chapter. In my response, I will stick to his main criticisms of my thesis. It is noteworthy that Schefold does not find any flaw in my structuralist interpretation of or the proof of Sraffa’s proposition regarding uniformity of the rate of profits in his system of equations. He, nevertheless, claims, without giving any reason, that it is ‘trivial’ and a matter of ‘definition’. Now my argument that the uniformity of the rate of profits in Sraffa’s system of equations of basic goods is a structural property of his system crucially depends upon or rather follows from Sraffa’s earlier demonstration that for any given system of basic goods there exists a corresponding Standard system and that the Standard system is unique to the given system. Take, for example, my equation systems (vi) and (vii): if no Standard system existed or if there were more than one such Standard systems, then there will be no grounds for me to claim that R’ must be equal to R and, therefore, there will be no way of establishing the proposition that all r’s must be equal. Thus my proof can be called ‘trivial’ if and only if one could characterise Sraffa’s discovery of the Standard system and the demonstration of its existence and uniqueness for all systems of basic goods as ‘trivial’—I suspect Professor Schefold will not go so far. As far as its being a matter of ‘definition’ is concerned, it depends on what one means by definition—all mathematical proofs are analytical in nature and to that extent a matter of ‘definition’.

The root of the problem lies in the most crucial aspect of the differentia specifica of Sraffa’s system of basic goods that gives rise to commodity residue and the classical idea of ‘labour’ as the sole primary factor of production along with free nature, which leaves no commodity residue. Schefold’s comment does not even touch upon this matter—as a matter of fact, the concept of ‘commodity residue’ does not even once appear in his comment. Therefore, his criticism of my chapter remains peripheral and misses the point. As I have highlighted in my chapter (Sinha 2021), the existence of commodity residue in a system of production with basic goods does not conceptually allow for any industry to be completely vertically integrated. Thus it does not allow any industry to be independent—a consequence of this is that prices of all industries must be determined simultaneously—applying arbitrary prices (such as market prices) from outside the equation system effectively destroys the integrity of the interconnected nature of the system; it amounts to treating every industry as completely vertically integrated industries exchanging only their ‘final goods’; in this context, the gravitation mechanism does not require the so-called ceteris paribus assumption, and Adam Smith, in my understanding, never made any such assumption as Schefold suggests.

Schefold (2021) presents a counter-example to my thesis that the rate of profits in Sraffa’s system of equations must be uniform. He provides a simple example of a two-sector model with surplus production. Since his system is already in the Standard proportion, its average (and in this simple case also maximum) rate of profits can be calculated as a ratio of its net output to input, which turns out to be 1/3 and the prices turn out to be 1:1. He then introduces an institutional variable from outside the equation system—this institution is the State, which appropriates the total surplus (equal to ¼ steel + ¼ corn) by imposing taxes. Now, from the equations it is quite clear that the all-powerful State could directly tax both the sectors at the rate of 1/3 of their total outputs and appropriate the total surplus. In this case, the sectors would be reduced to their subsistence, leaving the prices intact. But Schefold suggests that the State wants to appropriate the total surplus by taxing only one sector, that is the agricultural sector. In that case, the State has to find out the terms of trade or the price ratio between the two sectors that would transfer all the surplus of the system to the agricultural sector alone. To solve this problem, the State solves the two equations given by: q(1/4u + ½) = 1/4u + ¼ and (1 + q)(1/4u + ½) = 1, where q is the rate of taxation and u is the price of steel in terms of corn. This gives us the value of u = ½ and q = 3/5. Schefold then claims that ‘here we have, for the model of physiocracy, a standard system from the start, and the differentials persist’. But all this is just a convoluted exercise of the case I have already taken care of in my chapter—as I have argued earlier, taking prices as ‘market prices’ from outside of the equation system and solving for unequal rates of profits amounts to breaking Sraffa’s system of equations as each industry could solve for its rate of profits independently given the set of prices. In this case as well, the State must first decree the price ratio equal to 1:1/2 to ensure that all surplus is transferred to the agricultural sector. Imposition of these prices breaks the system in the same manner as imposition of the so-called market prices. Otherwise, 3/5 rate of taxation on agricultural sector with old prices given at 1:1 would contradict the two-equation system. My argument may be better illustrated by an analogy—a soap bubble has a natural shape, which is spherical, and the structural property of it can be described mathematically, but this does not mean that by putting outside pressure on it one cannot distort its shape—a State by putting taxes and subsidies on various commodities can always twist the prices; it can also destroy or break the economy by imposing taxes more than the surplus produced by the system just as we all know a sufficient pressure on a soap bubble breaks it too. Here I would also like to add that Schefold’s characterisation of his example as a ‘model of physiocracy’ is rather problematic. In physiocracy, it is the rent income that is considered surplus and taxed. But rent is earned on land and is not determined as a percentage on the value of the means of production, which is the case in Schefold’s example.

Now we come to Schefold’s defence of classical gravitation of market prices to natural prices. On this account, he appears to be all over the place—I have not been able to find a central thread running through it except for an insistence that there must be natural prices, and market prices must gravitate to it. He first asserts that gravitation process is ‘indispensable’ if one wants to get from ‘protoeconomics’ to economics—this is argued in the context of evolution of an economy, where conceptualisation of the two states could be made only in terms of ‘natural prices’—‘And what is the meaning of the natural prices, if the market prices do not somehow gravitate to them?’ (Schefold 2021). Then we are told that ‘the gravitation of market prices to natural prices therefore is a matter of anecdotal circumstances.’ So already we have based an ‘indispensable’ aspect of economic theory on something that is a matter of anecdotal circumstances! He further goes on to assert that Adam Smith had no theory of gravitation, but ‘we, of course, can construct modern models.’ He makes three suggestions in this regard. (i) In the case of agricultural goods, natural prices can be conceptualised as the average price of market prices over many years—but what if market prices turn out to be rising or falling over the years? It still will have an average but in what sense it could be considered the ‘realized natural price’? And, of course, over the years techniques of some basic good could be changing as well—how does average market price take care of that? (ii) In the case of industrial goods, he suggests that one could conceive their pricing policies to be such that they use a mark-up on their cost equal to the normal rate of profits—in this case there is no gravitation mechanism as we are simply assuming that industries set their prices equal to their natural prices irrespective of whether their supplies are equal to their effectual demands or not. And (iii) if quantities are rigidly given then adaptation of market prices to natural prices can be formulised by the equation system given by: P_t+1 = P_t(1 + r) + l, where wage is taken as the numeraire—this, of course, is an alternative method of solving a simultaneous equation problem by assuming that the rate of profits must be equal. This simply cannot describe the gravitation mechanism since gravitation mechanism works precisely by adjusting quantities supplied to the given effectual demands—keeping supplies rigidly fixed takes away the very presupposition of the whole process. Interestingly, Schefold never asks: what if the gravitation process does not work? He finally goes on to claim that ‘gravitation is an axiom of classical theory, and not a result.’ But what good is an axiom that can be shown to be wrong, as it is shown in Dupertuis and Sinha (2009)? And even if it is an axiom of classical theory, the relevant question for us is thus: is it an axiom of Sraffa’s theory? And the answer to that question is ‘definitely not’. As Sraffa explicitly states that his equations represent ‘the actual economic system of observation’ (Sraffa 1960, p. 22) and also claims that ‘the economic systems of reality are not self-replacing. They are in a constant state of transition and obsolescence, due to changes both in the types of (kind of) commodities produced and in methods of their production’ (Sraffa n.d. 1955, D3/12/75, p. 16). In other words, his equation system is like a photograph of the existing economic system that is not in equilibrium or the centre of gravitation of any sort. Schefold criticises me for suggesting that Adam Smith’s gravitation argument implicitly assumes a demand schedule (but not a function). He argues that ‘[h]ere Sinha seems to me to be uninformed about the state of discussion of supply and demand in the 18th century. In particular, reference should be made of James Steuart. In 18th century, and still in Marx, demand and supply were seen as “forces”.’ In this context, I will only like to point out that Sraffa in 1931 interpreted Marshall’s demand and supply schedules as generating forces and clearly stated that he does not assume any forces—I have not found any evidence in Sraffa’s archive that suggests that he repudiated his 1931 position on the question of ‘not assuming any forces’: ‘The real point is that it is believed that Marshall’s curves provide ‘forces’ which, in case the price falls below or above AB {equilibrium prices} by ‘chance’ will restore it to AB. Now I am not assuming any forces’ (Sraffa n.d., D3/12/7: 65).

Schefold also criticises me for taking Sraffa’s allusion to ‘observation’ too literally. He argues that Sraffa’s equations assume homogeneous goods and a well-defined technique—both these aspects of his theory, according to Schefold, cannot be observed. As a matter of fact, there aren’t two apples in the world that are identical or homogeneous. But this does not stop sellers and buyers of apples to buy and sell them in terms of either dozens or kilograms. If a kind of apples has a price then this very fact tells us that it has a homogeneous unit of measure as price is quoted in terms of a unit of measure. Whether all apples in a dozen or in a kilogram are physically and chemically identical or not is not the relevant question here—the observer simply observes that apples are socially measured as homogeneous commodity in certain socially accepted unit, and this information is available from the producers of apples. So where is the problem? Similarly, the spectre of a ‘dominant technique’ and how to establish it is a problem for Garegnani and his associates because they need to think in terms of supply adjustments for their gravitation mechanism but not for my interpretation of Sraffa’s equations. Sraffa’s equations are industrial equations, and they do not represent some sort of technique of production of a firm. If an economy uses coal, hydro, nuclear and wind technologies to produce electricity, then the observer will simply take down all the inputs used by various production units on the side of inputs and the total output of electricity on the side of output—this will not represent any particular technique. There is no reason for the data of inputs and outputs of Sraffa’s equations to represent a technique for solving his equations as these data are ex-post data and do not require us to think in terms of adjustments of outputs.

Now we come to the fundamental distinction between Garegnani and his associates’ interpretation of Sraffa’s equations and Sraffa’s own description of it. As I have pointed out in my chapter, Sraffa repeatedly claims that ‘profit is a non-price phenomenon.’ Schefold tries to underplay this significant theoretical claim by suggesting that it only refers to the absence of the unit of prices in the rate of profits, as it is a percentage. But this is not the case. Sraffa comes to this conclusion only after determining his fundamental equation, r = R(1−w), where the measure of w is in terms of the Standard commodity. This immediately determines the value of r if we know w since it is already established that R is determined and remains fixed with respect to variations in w or r. This shows that both r and w are determined independently of prices. Now the question is which one should be taken as ‘given’ from outside? Since wages in the real world are given in terms of ‘money’ and not in terms of the Standard commodity, the given ‘money’ wages could be translated into w in the Standard commodity units only after knowing the prices but r, which is a percentage and is not given in terms of the Standard commodity, can always be taken as ‘given’ from outside independently of prices. This is the reason why Sraffa moves from taking wages as given to taking the rate of profits as given—but it crucially depends on the equation, r = R(1−w), which shows that r is a non-price phenomenon and therefore can be taken as given before the prices are determined. This, however, is not possible if you insert gravitation mechanism into Sraffa’s system. The conceptual framework of the gravitation mechanism requires the rate of profits to be determined by prices. Schefold claims that ‘Distribution does depend on prices, if the theory of distribution must start from a real wage’—this apparently is argued on the grounds that ‘the real wage depends on the standard of prices.’ But this is not true. If one takes real wages as the standard of prices then this problem will be taken care of or, for that matter, take Sraffa’s equation system given in his book (Sraffa 1960, p. 6); in this case real wages are either zero or incorporated in the inputs data. The question is this: can we determine the equal rate of profits of this system of equations independently of prices without the help of the Standard system? And the answer is, no. Since the Standard system is not an integral element in the analysis of the system of equations in the gravitational framework, the rate of profits cannot be determined independently of the determination of prices—they must be simultaneously determined; and therefore, not only distribution turns out to be dependent on prices but also the rate of profits cannot be taken as given from outside. This reveals the fundamental dissonance between Sraffa’s own interpretation or rendition of his theory and the interpretation given to it by Schefold or Garegnani and his associates.

Let me now take up the main concern of Professor Schefold. He characterises Sraffa’s theory as a representative of ‘protoeconomics’ and argues that one needs to insert the idea of gravitation if one wants to get to economics from ‘protoeconomics’. It seems to me that Schefold is largely in agreement with me or at least not in serious disagreement in so far as Sraffa’s theory is characterised as a structure with well-defined relational properties among its variables, but he thinks that it only represents a ‘protoeconomics’. For him, an economic theory must take into account changes in those variables based on ‘causation’. For example, Sraffa’s equation, r = R(1−w), represents the relation in which the three variables must always be related given any input-output data of basic goods. In this case, empirically if we find that the value of R = 20% and the value of r = 10%, then the value of w must turn out to be ½ in terms of the Standard commodity of the system. But this does not mean that if for some reason r changes to 5% then w must necessarily rise to ¾ in the empirical or the real world. This is because a real change in r could lead to choice of other techniques and many other changes in production decisions leading to a new set of data with a new R and a new Standard commodity with a new w. All Sraffa’s equation tells us is that the new set of data will again represent the same relationship between R, w and r. Schefold seems to argue that an economic theory must be able to explain how a change in r brings about the change in the structure itself. This is a problem of causation. In my understanding, Sraffa refrains from developing a theory based on causation because he thinks that a relation of causation is dependent on several contingent factors and can be discussed only in terms of particular cases—a general predictive theory based on causation is not possible—that’s why the story of changes should be left to historians where hindsight is 20/20. This is a general problem with structuralist theories—they are good at explaining the relations of the structure and the conditions of its reproduction but do not have much to say when it comes to predicting changes in the structure itself—Marx tried to do it on a grand scale by introducing dialectics but could not get much beyond pointing to the internal contradictions in the conditions of the reproduction of the structure. Finally, if we need to develop even some tentative theories based on causation, we will have to introduce time dimension explicitly in it and I do not understand the insistence that all dynamic theory must have a well-defined equilibrium position as its reference point. If Sraffa is right in describing economic change as ‘a constant state of transition and obsolescence, due to changes both in the types of (kind of) commodities produced and in methods of their production’ (op. cit.), then the ‘centre of gravitation’ is of no help here and we must try to develop our dynamic theories without any reference to equilibrium. My interpretation of Sraffa’s prices foots the bill as it makes the condition of equilibrium irrelevant.

It is a general refrain among many Sraffians that Garegnani’s interpretation cannot be challenged given his close relationship with Sraffa. Schefold also repeats this trope: ‘The opposition between Garegnani and Sraffa, which [Sinha] tries to reconstruct, is a priori not possible. Garegnani was without question Sraffa’s most important pupil, his friend and support over many years to the end. Garegnani may have had his own agenda, but there was trust between them and Garegnani was appointed Sraffa’s literary executor’ (Schefold 2021). In this case, I would only like to point out that both James Mill and McCulloch were close friends of Ricardo (one in the role of a master and the other in the role of a pupil), but how many Sraffians would accept their interpretation of Ricardo over Sraffa’s?—even though Sraffa was born 75 years after Ricardo’s death and therefore did not even belong to Ricardo’s intellectual era. The intellectual history of the world is full of such examples where close associates have been found to be wanting in understanding the works of geniuses they were closely associated with. I rest my case.