Abstract
Within the first 33 pages of his book, Sraffa (1960) changes his numéraire twice and settles with the third one for the rest of the book. This, by itself, had broken from the classical tradition in a significant way, but with the final numéraire came another momentous rupture from the classical tradition in the shift from taking wages as ‘given’ from outside to taking the rate of profit as ‘given’. Both the cause and the theoretical significance of this move have not been well understood in the orthodox Sraffian literature. In this note, I try to clarify this issue.
Introduction
In my previous article in the issue dedicated to Krishna Bharadwaj by this journal (Sinha, 2021), I made the point that Sraffa’s crucial assumption of at least one basic good in the system of production, or the idea of an economy with interconnected industries, rules out the possibility of a pure labour theory of value, as, in this case, no process of production can be completely reduced to only direct and indirect labour inputs working on free nature—production of all commodities must always include commodities as their means of production. This gives rise to the notion of a finite maximum rate of profits, 1 R—a rate of profits that must hold when wages are hypothetically put to zero. R is simply the inverse of what is loosely defined in general economics literature as the capital/output ratio of an economy. Since both capital and output are a collection of disproportionate heterogeneous goods, the value of R seems not to be well-defined until we know the prices. Furthermore, if prices change with variations in the value of wages or the wage rate, then R becomes a function of the wage rate. Once classical economists, particularly Ricardo and Marx, realised that variation in wages must affect their ‘natural prices’, they knew they had lost all means of determining the rate of profit. Sraffa, in 1942, hypothesised that R must remain constant with respect to variations in wages (or the rate of profits, given that wages and the rate of profits are inversely related). To prove his hypothesis, Sraffa developed an ingenious Standard system and showed that every system of basic goods can be rearranged into its Standard counterpart by simply rescaling the original equations, and that the Standard system is unique to every given system of basic goods. Two important consequences follow from here: (a) if the wage rate and all the prices are measured in terms of the Standard net output, or what Sraffa called, the Standard commodity, then it can be shown that the value of R is fixed with respect to variations in wages, and the weighted average rate of profit for any given level of wages is also determined without the knowledge of prices, and (b) the condition of a uniform rate of profits for all the industries associated with the classical ‘natural prices’ does not need the assumption of an equilibrium in a free and competitive market. One problem with this, however, is that it is not at all practical or convincing to think that wages could be ‘given’ in terms of the Standard commodity, and if the Standard commodity is only a unit of measure for a given wage, either in terms of a basket of real goods or some monetary unit, then its value in terms of the Standard commodity cannot be known without the knowledge of prices. Thus, to overcome this problem, Sraffa proposed that instead of wages being given from outside the system, he would take the rate of profits as given from outside, as the rate of profits is a number in terms of percentage and is not measured by the unit of the Standard commodity. This shift from taking wages as given from outside to taking the rate of profits as given from outside could be made only after demonstrating that the rate of profits can be determined without the knowledge of prices. The details of this proposition are described below.
The Story of Changes in the Numéraire and Its Relation to What Must Be Taken as ‘Given’
Sraffa’s enigmatic classic, Production of Commodities by Means of Commodities, begins with a brief chapter on ‘Production for Subsistence’, where arbitrarily one commodity is taken as the numéraire for measuring other prices: ‘One commodity is taken as standard of value and its price made equal to unity’ (Sraffa, 1960, p. 5). The second chapter introduces ‘Production with a Surplus’. Both in the case of ‘Production for Subsistence’ and in the early part of ‘Production with a Surplus’, wages are treated as physical subsistence for workers and are incorporated within the methods of production as physical inputs. This was more or less in the classical tradition—though classical economists allowed for real wages to be higher than the physical subsistence, they regarded it as a physical consumption basket of ‘wage goods’ and treated it as a part of capital investment (see Sinha, 2018). From page 9 (para. 8), Sraffa (1960), however, begins to break from the classical tradition. First of all, wages are removed from the set of capital investments or inputs and placed as a share of national income, paid post factum to the workers. Thus, wages are no longer considered to be determined by the physiological and social requirements of the workers, since national income may contain all sorts of ‘capital goods’ that cannot be conceived as ‘wage goods’. Furthermore, he quickly changes his numéraire from any arbitrary commodity to the composite commodity formed by the value of the net output or the national income:
The value of this set of commodities, or ‘composite commodity’ as it may be called, which forms the national income, we make equal to unity. It thus becomes the standard in terms of which the wage and the k prices are expressed (taking the place of the arbitrarily chosen single commodity in terms of which k – 1 prices, besides the wage, were expressed). (Sraffa, 1960, p. 11)
This move was ostensibly made to allow the analysis of the effects on prices of changes in wages, given the methods of production. If wages were given in terms of a money-commodity (the numéraire commodity), then the value of national income could not be known before all the prices are determined and, more importantly, its value would constantly change with changes in wages. If wages were given in terms of physical ‘wage goods’ then wages could never reach their maximum value if national income contained some capital goods. If, however, wages were given in terms of the ‘composite commodity’ constituting the national income, then it could move from 0 to 1, that is, its maximum value that absorbs the whole of national income, without any fetter.
Chapter III, titled ‘Proportion of Labor to Means of Production’, gets down to the business of analysing changes in prices with respect to changes in wages. Sraffa finds that if the proportion of labour to means of production were equal for all industries, then a continuous change in wages from 0 to 1 will have no impact on prices, as its impact on the rates of profit of all industries will be the same and so the condition of a uniform rate of profit for all industries will not be disturbed by the changes in wages. If, however, the proportion of labour to means of production of all industries were not equal, then the relative prices of commodities must change to ensure the condition of a uniform rate of profits for all industries. 2 He further analyses the nature of those price changes and finds that they can be highly complex and counter-intuitive due to the interconnections of ‘basic good’ industries. He, however, does not leave it at that. For some unspecified reason, Sraffa begins to look for the condition of production of a commodity that will not be affected by a continuous variation in wages from 0 to 1. This leads to Chapters IV and V, where the Standard system and the Standard commodity are developed, and their uniqueness to any given system of production with at least one ‘basic good’ is proved. It is in Chapter IV that we find, once more, another change in the numéraire. Now, the ‘composite commodity’, the Net National Income, is dropped from being the numéraire and is replaced by another ‘composite commodity’, the Net Standard Income: ‘From this we derive the Standard national income which henceforward we shall adopt as unit of wages and prices in the original system of production’ (Sraffa, 1960, p. 24). An interesting aspect of this numéraire is that it contains only ‘basic goods’ and thus, most likely, no ‘wage-goods’, as ‘wage goods’ are conceived as pure consumption goods and therefore will be relegated to the set of non-basics: ‘The drawback of this course is that it involves relegating the necessaries of consumption to the limbo of non-basic products’ (Sraffa, 1960, p. 10)—this is how far we have come from the classical tradition already! 3
Yet again, at the end of Chapter V, a more perplexing and significant move away from the classical economics takes place. Sraffa argues that the proposition that wages are given from outside in terms of the Standard commodity does not make sense, and therefore, one needs to shift from the classical position of taking wages as ‘given’ to taking the uniform rate of profits as ‘given’:
The last steps of the preceding argument have led us to reverse the practice, followed from the outset, of treating the wage rather than the rate of profits as the independent variable or ‘given’ quantity. The choice of wage as the independent variable in the preliminary stages was due to its being regarded as consisting of specified necessaries determined by physiological or social conditions that are independent of prices or the rate of profits. But as soon as the possibility of variations in the division of the product is admitted, this consideration loses much of its force. And when the wage is to be regarded as ‘given’, in terms of a more or less abstract standard, and does not acquire a definite meaning until the prices of commodities are determined, the position is reversed. The rate of profits, as a ratio, has a significance which is independent of any prices and can well be ‘given’ before prices are fixed. It is accordingly susceptible to being determined from outside the system of production, in particular by the level of the money rates of interest.4 In the following sections, the rate of profits will therefore be treated as the independent variable (Sraffa, 1960, p. 33).
The Reason for Reversing the Practice
So, what were the last steps that led Sraffa to reverse the practice? After establishing the fundamental structural relation, or the core of his theory, that is, r = R(1 –
But why could not Sraffa treat the Standard commodity as the money-commodity in terms of which wages could be ‘given’ or the ‘wage-bargain’ could take place? The reason for this appears to be simple. One of the essential aspects of ‘money’ is that it is a means of deferred payment, so that a wage contracted in its terms could be paid after the harvest. But a Standard commodity is specific to only one set of inputs and outputs; with any change in the total employment of labour or the technique of any basic-good industry, the Standard commodity must change. In other words, the Standard commodity is well defined only when the data after the harvest are available—hence, it cannot be conceived as a money-commodity. This brings us to a fundamental question: What was at stake that led Sraffa to abandon the composite commodity constituted by the national income as the numéraire, and to adopt the Standard commodity instead?
The answer to this question lies in what Sraffa had called ‘My Hypothesis’ in his working notes of the period 1942–1945:
What is demanded of the model is that it should show a constant (constant with respect to variations of r) ratio between quantity of capital & quantity of product. If this can be constructed, and proved to be general, a number of important ‘consequences’ follow. (Sraffa, n.d., D3/12/16: 14th August 1942)
During the early phase of his research, that is, 1927–1931, Sraffa had already come to the conclusion that attempts to calculate the quantity of capital in terms of ‘period of production’ by Jevons, Böhm-Bawerk and Wicksell were fundamentally flawed if the industries were interconnected. In this context, no matter how far back in the ‘period of production’ one goes, there still must remain a commodity residue. In other words, the ‘period of production’ is not an objective fact—at what stage the commodity residue becomes negligible depends upon the level of wages—if wages are high, then the commodity residue will become negligible earlier than if wages were low, and it will never become negligible if wages were zero. This immediately suggests that the relationship between wages,
Why the ‘Hypothesis’ was important to Sraffa, and what important consequences follow if it is proven to be true? To understand the importance of the ‘Hypothesis’, we should go back to Chapter III, with the numéraire in terms of the real national income as the composite commodity. In this case, we see that the variations in relative prices due to changes in r can be highly complicated and counter-intuitive because of the interconnected nature of the industries. However, as worked out by Sraffa (1960) in Chapter VI, para 49, no matter how complicated those price changes might be, a rise in the rate of profits must lead to a fall in the wages measured by the numéraire, made up of the real national income as the composite commodity (or any arbitrary commodity)—thus, an inverse relation between wages and the rate of profits could be established in Chapter III itself without the aid of the Standard commodity as the numéraire. Still, however, a problem with this numéraire remains; it is that the value of R, that is, the maximum rate of profits of the system, keeps changing as r and
Now, the discovery of the Standard commodity establishes exactly that. Sraffa shows that when the Standard commodity is used as the numéraire to measure wages and the prices, then the value of R remains constant throughout the range of r from 0 to R—the relationship between the three variables now becomes linear, which is given by r = R(1 – {…} the rate of profits at the various levels of
Hence, the most important consequence of establishing the ‘Hypothesis’ by discovering the Standard commodity as the numéraire was to establish the proposition that the distribution of income, or the determination of both the rate of profits and wages, can be made independently of prices and can be taken as ‘given’ prior to price determination. Another important consequence of the discovery of the Standard system and the Standard commodity, and particularly the proof of its uniqueness for any given system of basic goods, is that it establishes that the weighted average rate of profits of any given system of basic goods can be discovered without the knowledge of prices; and since the average must remain constant with respect to rescaling of the original equations, it proves that the industrial rates of profits of the system must be equal irrespective of equilibrium or disequilibrium of industrial supplies with their effectual demands—this explains why, while introducing the rate of profits in his system of k equations, Sraffa (1960, p. 6, para 4) could assert, without any qualification, that ‘([the rate of profits] must be uniform for all industries)’ (see Sinha, 2016, 2022 for details on this point).
Classical economists used the competitive mechanism of gravitation of ‘market prices’ to ‘natural prices’ as a ‘force’ that equilibrates the industrial supplies with their effectual demands, that brings the industrial rates of profits to uniformity. 7 In this case, the rate of profits could not be taken as given independently of prices, as the ratio of the equilibrium Net Output to Capital in general remains a ratio of heterogeneous goods arranged disproportionately—hence, it had to be determined either after knowing the prices or simultaneously with prices. Wages, on the other hand, could be taken as given from outside since they are not a ratio and therefore could be simply conceived as a bundle of ‘wage goods’, determined by physiological and sociological considerations, as inputs for the production of labour and therefore could be substituted for labour as means of production. This is the reason why Garegnani-led interpretation of Sraffa still prefers to use wages as ‘given’ from outside in Sraffa’s equations, since they believe in the classical gravitation mechanism and maintain that the prices associated with an equal industrial rate of profits of Sraffa’s system of equations must be determined simultaneously with prices—Gehrke and Kurz (2006) go to the extent of claiming that ‘Sraffa upon resuming his work on his book in the summer of 1942 adopted for good a share concept of wages in his third equations, with wages, w, expressed as a proportion of the net product’ (p. 107, emphasis added). This, of course, denies the fact that Sraffa did move to taking wages in terms of the Standard commodity, and that this does not express itself as ‘a proportion of the net product’.
Sraffa’s discovery of the Standard system, which is simply a rescaled system of equations derived directly from the observed system of inputs and outputs, however, shows that the weighted average rate of profit could be derived directly from the physical equations; since in this case the net output, as well as the capital, are made up of the same composite commodity, the Standard commodity, and hence, their ratio is well defined independently of prices. Therefore, Sraffa could dispense with the classical gravitation mechanism altogether, since the uniform rate of profit now turns out to be a structural property of the equation system of any given basic goods, as if ‘it is embedded in the things’; hence, it is no longer tied to the condition of equilibrium where ‘market prices’ turn out to be equal to the ‘natural prices’. This is what allows Sraffa to shift from using wages as given from outside to taking the rate of profits as ‘given’, since it can be determined without the knowledge of prices. The point to note, however, is that one cannot be eclectic in terms of taking either the wages or the rate of profits as given from outside, given one degree of freedom in the set of equations. If one chooses the real national income as the numéraire, then one is obliged to take wages as given from outside and not the rate of profits, since the rate of profits cannot be known before the determination of prices; but if one chooses the Standard commodity as the numéraire, then one is obliged to take the rate of profits as given from outside and not wages, as wages given in terms of the Standard commodity has no economic meaning—wages are contracted before the production commences, and the Standard commodity cannot be constructed before the production is completed. As a matter of fact, if one accepts the truthfulness of the structural relationship of Sraffa’s system of production and distribution given by r = R(1 –
To reiterate, Sraffa needed to first establish that the rate of profits of his system of equations can be determined without the knowledge of prices, to prove the core classical conjecture that the distribution of income is determined prior to the determination of prices, which he succeeds in doing in Chapters IV and V by developing the Standard system and proving its uniqueness to any given system of basic goods. But this proposition requires that wages and prices must be measured by the Standard commodity—this obliges the theoretician to take the rate of profits as ‘given’ from outside, since taking wages as ‘given’ in terms of the Standard commodity is economically meaningless. This should clarify more than 60 years of confusion in the Sraffian literature on the question of numéraire, and the choice of an independent variable in Sraffa’s system of equations.
