Bellino, Enrico - On Sraffa's Standard Commodity as Invariable Measure of Value

Abstract The necessity to express the relative price of a commodity in terms of another commodity makes it impossible to distinguish, within a variation of its relative price, that part of the change that can be ascribed to the characteristics of the commodity itself from that part that is to be ascribed to the characteristics of the commodity of reference, i.e. the numeraire. Ricardo (1817) pointed out this problem and the necessity to find an “invariable measure of value”, but he was not able to solve this problem. Sraffa (1960) suggested to use a composite commodity (that is, a bundle of commodities) to accomplish this function. Within his framework of production of commodities by means of commodities he built the “Standard commodity”, which is a composite commodity which he claims to be a standard of value invariant with respect to changes in the distribution of income. But in Sraffa’s book there is no explicit proof of this claim. This gave rise to a lot of misunderstandings about the standard commodity and its rˆ ole as invariable measure of value. In several contributions Sraffa’s solution to the Ricardo’s problem was questioned. In this work I shall try first to clarify what it means, for a numeraire, to be an invariable measure of value. Then I shall show that Sraffa’s standard commodity does satisfy this condition. On this basis I will re-examine the function of the standard commodity within the analysis of income distribution. A survey of the literature on the problem is presented at the end of the paper.

KEYWORDS: standard commodity, invariable measure of value, price theory, Sraffa price system, value theory, distribution theory.

J.E.L. CLASSIFICATION: B12, D33, D46, E11 .

On Sraffa’s Standard Commodity as Invariable Measure of Value Enrico Bellino1 Universit`a Cattolica del Sacro Cuore (Milano) C.O.R.E. (Louvain-la-Neuve) English not accurately checked

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I would like to thank professor Luigi Pasinetti for his stimulus to go deep into some issues concerning the standard commodity and for his detailed comments on previous versions of this work. I would like to thank also Christian Bidard, Flavia Cortelezzi, Pierangelo Garegnani, Marco Piccioni, Fabio Ravagnani, Angelo Reati, Neri Salvadori, Ernesto Savaglio, Ian Steedman, Paolo Varri and the participants to a seminar in Catholic University for useful discussions on this topic. Usual caveats apply.

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[Introduction

Introduction

The problem to isolate within a variation of the relative price of a commodity that part of it that can be ascribed to the price of commodity itself from that part that is to be ascribed to the commodity used as numeraire was emphasized at least two centuries ago by Ricardo. It is useful to start by quoting those passages where Ricardo states the main points of the problem. Two commodities vary in relative value, and we wish to know in which the variation has really taken place. If we compare the present value of one, with shoes, stockings, hats, iron, sugar, and all other commodities, we find that it will exchange for precisely the same quantity of all these things as before. If we compare the other with the same commodities, we find it has varied with respect to them all: we may then with great probability infer that the variation has been in this commodity, and not in the commodities with which we have compared it. If on examining still more particularly into all the circumstances connected with the production of these various commodities, we find that precisely the same quantity of labour and capital are necessary to the production of the shoes, stockings, hats, iron, sugar, &c.; but that the same quantity as before is not necessary to produce the single commodity whose relative value is altered, probability is changed into certainty, and we are sure that the variation is in the single commodity: we then discover also the cause of its variation. Ricardo (1817, pp. 17–18) When commodities varied in relative value, it would be desirable to have the means of ascertaining which of them fell and which rose in real value, and this could be effected only by comparing them one after another with some invariable standard measure of value, which should itself be subject to none of the fluctuations to which other commodities are exposed. Of such a measure it is impossible to be possessed, because there is no commodity which is not itself exposed to the same variations as the things, the value of which is to be ascertained; that is, there is none which is not subject to require more or less labour for its production. Ricardo (1817, pp. 43–44)

But along with technological change (the change in the quantity of labour necessary to produce a commodity) Ricardo considers another source of variation of relative prices: the change in income distribution.

Introduction]

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But if this cause of variation in the value of a medium could be removed—if it were possible that in the production of our money for instance, the same quantity of labour should at all time be required, still it would not be a perfect standard or invariable measure of value, because, as I have already endeavoured to explain, it would be subject to relative variations from a rise or fall of wages, on account of the different proportions of fixed capital which might be necessary to produce it, and to produce those other commodities whose alteration of value we wished to ascertain. Ricardo (1817, p. 44)

Thus Ricardo was looking for a standard of value that were ‘invariant” to technical change as well as with respect to changes in the distribution of income.1 And he concludes: If, then, I may suppose myself to be possessed of a standard so nearly approaching to an invariable one, the advantage is, that I shall be enabled to speak of the variations of other things, without embarrassing myself on every occasion with the consideration of the possible alteration in the value of the medium in which price and value are estimated. Ricardo (1817, p. 46)

Nobody has been able to solve Ricardo’s problem in its entirety. Sraffa (1960) offered a partial solution to this problem by building, within his framework of production of commodities by means of commodities, a numeraire, called “Standard commodity”, that he claims to be an invariable measure of value with respect to exogenous changes in the distribution of income.2 But the notion of standard commodity and its rˆ ole within the Sraffa’s framework has always been one of the most discussed and often misunderstood in Sraffian and in anti-Sraffian literature. Actually Sraffa explains very clearly the Ricardo’s problem. He writes: The necessity of having to express the price of one commodity in terms of another which is arbitrarily chosen as standard, complicates 1

An attempt of reconstruction of the Ricardo’s search for an invariable measure of value has been done by Kurz and Salvadori (1993). In that paper they also refer to an invariance property with respect to interspacial comparisons that the standard that Ricardo was looking for should have had to exhibit (see Kurz and Salvadori (1993, pp. 96– 98)). 2 The solution of the other side of the problem – that is, a unit of value invariant with respect to technical change – has been performed by Pasinetti (see (1981, 1993)); he called such a unit ‘dynamic standard commodity’.

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[Introduction the study of the price-movements which accompany a change in distribution. It is impossible to tell of any particular price-fluctuation whether it arises from the peculiarities of the commodity which is being measured or from those of the measuring standard. The relevant peculiarities, as we have just seen, can only consist in the inequality in the proportions of labour to means of production in the successive layers into which a commodity and the aggregate of its means of production can be analyzed; for it is such an inequality that makes it necessary for the commodity to change in value relative to its means of production as the wage changes. Sraffa (1960, p. 18)

But, at the same time, Sraffa is not equally clear in showing why his standard commodity solves the requirement of invariance with respect to change in income distribution. He gives some intuitive hints in § 21 before building the standard commodity; later he concentrates on the building of the standard commodity (§ 23–28 and § 33–35), on the properties of the standard system (chap. V) and on the fact that if in a single production model this commodity is used as numeraire then the relationship between the wage rate and the profit rate becomes independent on prices (§ 29–32). But after the building of the standard commodity there is no explicit discussion about if and why the standard commodity is a measure of value invariant with respect to changes in income distribution. And all those scholars that dealt with and discussed the Sraffa’s standard commodity offered very few hints to understand this point. 3 Only Baldone (1980, pp. 274–277) and Kurz and Salvadori (1993, pp. 121-122, n. 16) sketch two proofs of the invariance of the standard commodity with respect to changes in the distribution of income. In this work I present a proof of this result in a way that seems more suitable to understand the whole topic from the economic point of view and that is easier to be connected with the economic intuitions suggested by Sraffa in his § 21. The paper is organized as follows: in section 2 the essentials of the Sraffa’s single product price framework will be recalled; in section 3 the capability of the standard commodity to be an invariable measure of value with respect to changes in income distribution will be dealt and in section 4 some observations concerning the analysis of distribution will be drawn in 3

A quick survey of the literature on the standard commodity can be found, later, in Section 6.

Basic framework]

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light of the properties of invariance of the standard commodity. In section 5 we will see some generalizations and extensions of the obtained results and in section 6 we will present a synthetic survey of the existing literature on the standard commodity, emphasizing the most common objections, criticisms and misunderstanding about this notion.

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Review of the basic framework

The reference framework is the single product Sraffa’s price system with circulating capital: pT = (1 + π)pT A + waT0 pT b = 1,

(1a) (1b)

where p is the (n, 1) vector of prices, A is the (n, n) non-negative inputoutput matrix, π and w are two scalars indicating the rate of profit and the wage rate, respectively, a0 is the (n, 1) non-negative vector of labour input coefficients and b is an (n, 1) non-negative vector representing the commodity bundle used as numeraire. Symbol ‘T ’ denotes the transpose of a vector. System (1a) is constituted by n equations in n + 2 unknowns, i.e. the n prices, the profit rate and the wage rate. System (1) determines relative prices once one of the two distributive variables is fixed from outside. Following Sraffa, we fix the profit rate exogenously with respect to the price system. By solving equation (1a) with respect to p we obtain: pT = waT0 [I − (1 + π)A]−1 ;

(2)

thanks to Perron-Frobenius theorems on non-negative matrices the inverse matrix in (2) exists and is non-negative for 0 ≤ π < Π, where Π := 1/λM −1 and λM is the dominant eigenvalue of A. In order to assure Π > 0 we assume λM < 1, that is, that technique A is “viable”. By substituting vector p given by (2) in equation (1b) we obtain the expression of the relationship between the profit rate and the wage rate,

6

[Basic framework

this latter being expressed in terms of numeraire b:4 w(b) (π) :=

aT0 [I

1 . − (1 + π)A]−1 b

(3)

Again, thanks to Perron-Frobenius theorems, the elements of the inverse at the denominator are non-decreasing functions of π for 0 ≤ π < Π; hence the wage rate is a non-increasing function of the rate of profit. (If matrix A is indecomposable the wage rate comes to be a strictly decreasing function of the rate of profit.) Re-substituting this expression into equation (2) we obtain the expression of the vector of prices as a function of the profit rate only: pT(b) (π) =

aT0 [I

1 · aT [I − (1 + π)A]−1 . − (1 + π)A]−1 b 0

(4)

As [I − (1 + π)A]−1 is non-negative the solutions with respect to the wage rate (3) and the price vector (4) are non-negative for any π within the interval [0, Π).5 Turning to the quantity-side the ‘standard system” is an economic system in which “the various commodities are represented among its aggregate means of production in the same proportions as they are among its products.” (Sraffa 1960, p. 19; emphasis in the original). Let q the (n, 1) vector of the total quantities to be produced of the various commodities; in the standard system q must satisfy the following conditions: q = (1 + R)Aq aT0 q = 1,

(5a) (5b)

where R is the uniform physical rate of surplus and the total quantity of labour employed has been normalized to unity. Let us indicate by q∗ the nonnegative vector that satisfies system (5); mathematically it is the right-hand 4

In what follows we will use te convention to indicate by index (b) the (composite) commodity, b, in terms of which the wage rate, w(b) , and the vector of relative prices, p(b) = [p(b)i ], i = 1, . . . , n, are expressed. In the case in which the commodity used as numeraire is a single commodity, j, we will write w(j) and p(j) = [p(j)i ], i = 1, . . . , n. Obviously we have pT(b) b = 1 or p(j)j = 1. (We will indicate explicitly the numeraire in terms of which the wage rate and prices are expressed every time there is the need to recall the attention on this point.) 5 For the details of this analytical formulation of the Sraffa’s price system see Pasinetti (1977, chap. 5) or Kurz and Salvadori (1994, chap. 4).

Theory of value]

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eigenvector of matrix A correspondent to its dominant eigenvalue λ(A) = 1/(1 + R) = 1/(1 + Π). Vector q∗ is called gross standard product. The net standard product is defined by:

y∗ := (I − A)q∗ =

R q∗ ; 1+R

thus y∗ is proportional to q∗ , hence

Ay∗ =

1 y∗ 1+R

(6)

holds; moreover as q∗ satisfies equation (5b) we obtain:

aT0 y∗ =

R . 1+R

(7)

The standard net product can be considered a composite commodity; it is what Sraffa calls the ‘standard commodity’.

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The standard commodity within the theory of value

The key to an understanding of the sense in which the standard commodity is an invariable measure of value is an analysis of the reason why relative prices change when distribution is varied.6 Consider singularly the price equations of the various commodities and 6

As recalled in the Introduction, the property of “invariance” of a commodity can be intended at least in two ways: with respect to technical changes and with respect to change in income distribution. For brevity here and in what follows the property of “invariance” is to be intended, unless differently specified, with respect to changes in income distribution.

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[Theory of value

express prices in term of whatever (composite) numeraire, b:7   T 1 p(b)1 = (1 + π)p(b) a + w(b) a01       ..   .    T i p(b)i = (1 + π)p(b) a + w(b) a0i    ..    .      T n p(b)n = (1 + π)p(b) a + w(b) a0n  pT(b) b = (1 + π)pT(b) Ab + w(b) aT0 b = 1,

(8a)

(8b)

where ai is the i−th column of matrix A, representing the input coefficients of the various commodity used in industry i and a0i is the i−th element of vector a0 , representing the input coefficient of labour used in industry i, i = 1, · · · , n. Suppose now that a variation of the rate of profit, for example an increase, takes place. How should the other variables, i.e. the wage rate and relative prices vary? Obviously the whole reasoning is quite complex, as there is full interdependence among all variables. To throw light on the argument, Sraffa carries out a causal argument. We shall follow Sraffa in this attempt. Suppose for the moment that we keep all prices unchanged. Then a uniform reduction (whatever it may be) of the wage rate would not be sufficient to restore the balance in all industries: in fact in those industries which employ a sufficiently high proportion of labour to means of production there would arise a surplus, while in those industries which employ a sufficiently low proportion of labour to means of production there would arise a deficit. If we want to eliminate the surpluses and the deficits caused by such a change in distribution it is necessary that the prices of the various commodities, p(b)i , i = 1, · · · , n, vary.8 In general this possibility is 7

The case of a numeraire constituted by a single commodity, i, can be obtained as a particular case by setting b = ei , where ei is the i−th elementary vector. 8 Sraffa crucial claim is that this necessity does not arise for that commodity – if it exists – which is produced by employing labour and the means of production in that “critical proportion” which marks the watershed between ‘deficit’ and ‘surplus’ industries.” Sraffa (1960, p. 13). We will see later (§ 6.1) that this does not mean that the price of such a commodity remains constant; it does vary, but the “causes” of such a change are to be ascribed to the necessity to restore the balance in other industries, not in the industry characterized by the “critical proportion”.

Theory of value]

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available for all commodities with the exception of the commodity used as numeraire, as its price, by definition, is equal to 1. Yet the overall decrease of the wage rate will not be sufficient in general to eliminate the surplus or the deficit originated in the (aggregate) industry that produces the numeraire, because of the differences in the proportions between labour and the means of production that characterize the various industries. On the other hand, by observing equation (8b), which fixes the price of numeraire at 1, we see that the prices of all commodities, p(b) , appear in it as variables. So for the various levels of π equation (8b) imposes a constraint on p(b) .9 Hence when the rate of profit is varied, the price system, p(b) , will have to vary also in order to restore the balance in the industry that produces the numeraire. Then when distribution changes, there are not one but two sorts of pressures on the price of each commodity; we shall call them ‘own Industry effect” and ‘Numeraire effect”: (I) own Industry effect: the variation of the price of a commodity arising from the necessity to restore the balance within the corresponding industry; (N) Numeraire effect: the variation of the price of all commodities arising from the necessity to restore the balance in the industry that produces the numeraire. This second sort of push, undergone by all prices, is at the root of the Ricardo’s problem, as it makes impossible to tell of any particular price-fluctuation whether it arises from the peculiarities of the commodity which is being measured or from those of the measuring standard. Sraffa (1960, p. 18) By contrast, we could define ‘invariable measure of value” a commodity that, if used as numeraire, renders effect (N) null, that is, a numeraire that does not engender pressures on the prices of the various commodities in order to restore the balance in its own industry. 9

System (8) is, in fact, a fully interdependent system of n + 1 equations in n + 1 unknowns, i.e., the n prices and the wage rate (the rate of profit is to be considered here as an exogenous parameter).

10

[Theory of value

At this point we have all elements to verify that Sraffa’s standard commodity satisfies such requirement of invariance. Consider again the price equations of the various commodities and express all prices and the wage rate in terms of standard commodity, y∗ :   T 1 p(y∗ )1 = (1 + π)p(y∗ ) a + w(y∗ ) a01       ..   .    T i (9a) p(y∗ )i = (1 + π)p(y∗ ) a + w(y∗ ) a0i    ..    .      T n p ∗ = (1 + π)p ∗ a + w ∗ a0n  (y )n

(y )

(y )

pT(y∗ ) y∗ = (1 + π)pT(y∗ ) Ay∗ + w(y∗ ) aT0 y∗ = 1.

(9b)

Let us focus the attention on the last equation of this system, (9b). By using equations (6) and (7) equation (9b) becomes: (1 + π)

R 1 pT(y∗ ) y∗ + w(y∗ ) = 1. 1+R 1+R

But as pT(y∗ ) y∗ = 1 we have R 1+π + w(y∗ ) = 1, 1+R 1+R i.e. w(y∗ ) = 1 −

π . R

(9b0 )

Hence system (9) reduces to: p(y∗ )1 = (1 + .. .

π)pT(y∗ ) a1

+ w(y∗ ) a01

p(y∗ )i = (1 + π)pT(y∗ ) ai + w(y∗ ) a0i .. . p(y∗ )n = (1 + π)pT(y∗ ) an + w(y∗ ) a0n w(y∗ ) = 1 −

π . R

                        

(9a)

(9b0 )

Analysis of distribution]

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In this case we see that the vector of prices has disappeared from the last equation of system (9). Equation (9b0 ) has now become an equation in the variable w(y∗ ) only (π is here considered as an exogenous parameter). If w(y∗ ) decreases according to this rule the variations of the ‘value of capital plus profit’ component and of the wage component entirely compensate each others within the industry of the numeraire, and thus the prices of all other commodities have not to vary in order to restore the balance in this industry – the standard commodity industry. Hence the standard commodity, if used as numeraire, makes effect (N) null. It is precisely in this sense that it can be claimed that the standard commodity is an invariable measure of value. This property descends from the particular proportions between labour and the means of production that characterize its (aggregate) industry and that assure that each variation of the wage component is always exactly offset by an opposite variation of the ‘value of capital plus profit’ component. In this situation the variation of each relative price pi arises only from the necessity to restore the balance in the respective industry i. Hence the standard commodity, when used as numeraire, permits to observe the variations of the relative price of each commodity in response to changes in the rate of profit in isolation (“as in a vacuum”, Sraffa (1960, p. 18), without the disturbances arising from the “peculiarities [...] of the measuring standard” (Sraffa 1960, p. 18).

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The standard commodity within the analysis of distribution

It is worth to recall here briefly the rˆole played by the standard commodity within the analysis of distribution. Relationship (3) between the profit rate and the wage rate can be written in an alternative form, more suitable to understand the underlying argument. Suppose to express all prices and the wage rate in term of commodity bundle b; substitute equation (1a) into (1b); by solving with respect to w we obtain: w(b) (π) =

pT(b) b − (1 + π)pT(b) (π)Ab aT0 b

=

1 − (1 + π)pT(b) (π)Ab aT0 b

.

(10)

As the wage rate and the profit rate are uniform across sectors we see that, once the numeraire has been chosen, the level of the wage rate – in terms

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[Analysis of distribution

of that numeraire – can be calculated, for any given rate of profit, from the price equation of the industry that produces the numeraire, b, whose technical coefficients are given by vector Ab and by scalar aT0 b.10 Relationships (10) or (11) show the strict interdependence between prices and distribution. They permit to single out the three phenomena that take place simultaneously when π varies: (D) change in Distribution: the variation of the wage rate due to the a change in the distribution of income – i.e. due to a different way of dividing of the ‘pie’ of net income between workers and capitalists –; (C) change in the value of Capital: the variation of the (relative) value of capital required to produce the numeraire, pT(b) (π)Ab or pT(i) (π)ai , variation due to the change of the whole price system, p(b) (π) or p(i) (π), in response to a change in distribution; (W) Wage-numeraire effect: as the wage rate is expressed in terms of the numeraire, b or i, part of the variation of w(b) or of w(i) , in response to a change in π, are caused by the peculiarities of the industry of the numeraire, i.e. by the necessity to restore the balance in this industry. Effects (C) and (W) overlap effect (D) – which is the main goal of the analysis of distribution – and make it unobservable in isolation. The use of the standard commodity as numeraire permits to isolate effect (D) from effects (C) and (W). In fact: • effect (W) is eliminated because – as we saw in section 3 – the standard commodity is an invariable measure of value; 10

As a particular case if we want to express all prices and the wage rate in terms of commodity i we set b = ei . The corresponding wage rate-profit rate relationship reduces to w(i) (π) =

p(i)i − (1 + π)pT(i) (π)ai 1 − (1 + π)pT(i) (π)ai = . a0i a0i

(11)

Also in this case we see that the wage rate w(i) can be calculated, for any given rate of profit, from the price equation of the industry that produces the numeraire i. (The technical coefficients involved in this relationship are ai and a0i but the production processes of all other basic commodities are however indirectly involved through the vector of prices at the numerator.)

Analysis of distribution]

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• effect (C) is eliminated as the vector of prices disappears from the relationship between the wage rate and the profit rate if we use the standard commodity as numeraire; in fact, if we set b = y∗ in equation (1b), that is if we set pT(y∗ ) y∗ = 1 we obtain w(y∗ ) (π) = =

pT(y∗ ) (π)y∗ − (1 + π)pT(y∗ ) (π)Ay∗ aT0 y∗ 1 − (1 + π)/(1 + R) , R/(1 + R)

=

(12) (13)

that is, w(y∗ ) (π) = 1 −

π , R

(14)

which is the well-known Sraffa’s relationship. It is easy to see the economic reason why the vector of prices disappears from the wage-profit relationship. As said before the wage rate, for any given rate of profit, can be obtained from the price equation of the industry of the numeraire, that, in this case, is the standard commodity, y∗ . y∗ , by definition, has the property that the various single commodities that appear in it are represented in the same proportions in the set of capital goods necessary to produce it, Ay∗ (from equation (6) we have that y∗ = (1 + R)Ay∗ ); in other words, the standard commodity and the set of capital goods necessary to produce it are the same (composite) commodity. Hence the wage rate, expressed in terms of this commodity, can be calculated simply by a subtraction between quantities of the same commodity, without the need to use the price vector to evaluate them. The economic interest of the result contained in equation (14) is the fact that the price system p cancels out and disappears from the relationship between the rate of profit and the wage rate. This finding gives a rigorous basis to the possibility of treating the problem of distribution of income independently of the price system.11 12 It should be noted that prices do not 11

On this see Sraffa (1960, Appendix D), Broome (1977), Garegnani (1984) or Lippi (1998). 12 Analytically this independence of distributive relationships from the price system can also be seen by comparing our previous systems (8) and (9). System (8) is a fully interdependent system in p(b) and w(b) , while system (9) is a causal system, in which,

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[Analysis of distribution

disappear completely from the problem of distribution: in fact the independence of the wage-profit relationship from the price system is obtained if all prices and the wage rate are expressed in terms of standard commodity. But this latter does not constitute, in general, the bundle of commodities consumed by workers, bw ; when workers spend their wages to buy bundle bw the “number” of such bundles they can buy turns out to depend on the price system, that is, the interdependence between prices and distribution reappears: w(y∗ ) 1 − π/R w(bw ) = T = T . p(y∗ ) bw p(y∗ ) bw But by re-writing the above expression in the form  π R w(bw ) = 1 − · , T R (R − π)a0 [I − (1 + π)A]−1 bw

(15)

we could see that the use of the standard commodity has permitted to separate a physical kernel of the distributive relationships (our previous effect (D)), described by the linear factor of (15), from the complications arising from the variation of the whole price system (our previous effects (C) and (W)), described by the non-linear factor of (15).13 This separation permits to individuate – in analogy to what happens into a one-commodity economy – the physical aspect of the distribution problem for a multi-commodity economy, notwithstanding profits, wages and outputs must be expressed in value terms by using the price system. Ricardo caught this point very clearly, disproving thus the ‘illusion’, deriving by the definition of prices as sum of wages, profits (and rents), that the trade-off between the distributive variables can be accommodated by ‘suitable’ variations of prices.14 (For further details on this aspect see Garegnani (1984, in particular sections III–VII).) These conclusions reflect one reconstruction of Ricardo’s thought. An alternative line of interpretation is expressed in Porta (1982). I will not enter into these issues here. firstly, equation (9b0 ) determines w(y∗ ) , then, given w(y∗ ) , equations (9a) determine p(y∗ ) . The notion of “causality” here used is to be intended in the sense given by Luigi Pasinetti in Pasinetti (1965) 13 I owe this observation and the analytics to present it to Neri Salvadori. 14 ‘Suitable’ variations of prices do not break down the trade-off between wages and profit as any increase of prices would affect at the same time both sides of price equations (1a), that is either the revenues or the costs of each industry.

Extensions and generalizations]

5

15

Extensions and generalizations

5.1

Plurality of standard commodities

In the second part of the seventies there appear some independent contributions in which it was proved that, in some cases, the standard commodity is not the only commodity that makes the relationship between the wage rate and the profit rate linear.15 It has been shown (see Miyao (1977), AbrahamFrois and Berrebi (1978), Bidard (1978)), that under certain circumstances there exist other composite commodities, that Miyao called “generalized standard commodity”, that make the job. Consider the following matrix, called by Miyao “labour profile matrix”:   aT0  aT A   0   T 2   . a A K = 0  (n,n) ..     . aT0 An−1

Miyao (1977, Theorem 3, pp. 158–159), proved that each composite commodity defined by y = y∗ + z ≥ o, where y∗ is the Sraffa’s standard commodity,  is a sufficient small scalar and z satisfies Kz = o,

(16)

is a generalized standard commodity. Let r(K) = H(≤ n). If H < n.

(17)

we can find up to n − H linearly independent vectors zh , h = 1, · · · , n − H, satisfying system (16). Thus by choosing  sufficiently small we can build 15

6.

On the faultiness of this way to characterize the standard commodity see our Section

16

[Extensions and generalizations

up to n − H generalized standard commodities,

yh = y∗ + zh ≥ o,

h = 1, · · · , n − H.

There are two extreme cases. i) H = n: in this case system (16) has only the trivial solution z = 0 and the Sraffa’s standard commodity is the unique standard commodity. ii) H = 1: in this case we have n − 1 generalized standard commodities; this latter case corresponds to the assumption of uniform capital intensity among sectors, and in this case each commodity is a generalized standard commodity.

The possibility of existence of a plurality of standard commodities is thus linked to the drop of rank of matrix K. This condition has no practical interest from the economic point of view, as the elements of K are given by technology. Notwithstanding as this case has (someway inexplicably) attracted the attention of many economists we could ask whether the generalized standard commodities yh , when they exist, are or not invariable measure of value. The response is positive. In fact if we use a generalized standard commodity as numeraire we set:

pT yh = 1

⇔

pT y∗ + pT zh = 1,

h = 1, . . . , n − H.

Extensions and generalizations]

17

As it is easy to prove that pT zh = 0 for h = 1, . . . , n − H 16 we have that pT yh = 1 ⇔ pT y∗ = 1, ⇔ (1 + π)pT(yh ) A + w(yh ) aT0 y∗ = 1 ⇔w=

R−π R

h = 1, . . . , n − H,

that is, the price system disappears also from the equation that sets the price of this numeraire, yh , at 1. As before in this way the price equation of each generalized standard commodity does not impose any further constraint on the variations of the price system in response to changes in π. Thus each generalized standard commodity yh is an invariable measure of value. As a by-product we can observe that there is no difference in expressing prices and the wage rate in terms of the Sraffa’s standard commodity y∗ or 16

Miyao defines the generalized standard commodity as that composite commodity that makes the wage rate-profit rate relationship linear, that is, that composite commodity y that satisfies w 1  r = 1 − . (18) pT (I − A)y R aT0 y Miyao (1977, theorem 1, p. 154) proves that equation (18) is equivalent to aT0 At y = (1 + R)aT0 At+1 y, h

∗

t = 0, 1, 2, . . . ,

(19)

h

hence y = y + z satisfies the following recurrence conditions on labour inputs: aT0 At (y∗ + zh ) = (1 + R)aT0 At+1 (y∗ + zh ),

h = 1, . . . , n − H,

t = 0, 1, 2, . . . .

Thus we have that aT0 At zh = (1 + R)aT0 At+1 zh ,

h = 1, . . . , n − H,

t = 0, 1, 2, . . . .

(20)

Moreover as Kz = o we have aT0 At zh = 0,

h = 1, . . . , n − H,

t = 0, 1, . . . , n − 1.

Hence thanks to (20) and (21) we have aT0 At zh = 0,

h = 1, . . . , n − H,

t = 0, 1, 2, . . . .

Returning to pT zh we have: pT zh = waT0 [I − (1 + π)A]−1 zh = =w

+∞ X t=0

aT0 At zh = 0,

h = 1, . . . , n − H

.

(21)

18

[Extensions and generalizations

in terms of the generalized standard commodity yh : in fact as pT yh = 1 ⇔ pT y∗ = 1, the solutions with respect to prices and to the wage rate of the two systems ( ( pT = (1 + π)pT A + aT0 pT = (1 + π)pT A + aT0 and T h T p y = 1 p y∗ = 1 coincide not only in relative terms but also in absolute value. From the economic point of view it could be objected that this equivalence is only formal as prices p(yh ) are expressed in terms of commodity yh while prices p(y∗ ) are expressed in terms of commodity y∗ . But as pT yh = pT y∗ (= 1) each unit of yh can “command” one unit of y∗ , hence the equivalence is substantial.

5.2

Joint production

It is possible to extend our previous conclusions to those cases in which the introduction of joint production does not raise problems for the existence of an economic meaningful standard commodity. Consider a square system (i.e. in which there are as many processes as many commodities). The standard gross product of the system is defined by: Bq = (1 + R)Aq aT0 q = 1,

(22a) (22b)

where B is an (n, n) non-negative matrix of outputs of the various processes. Suppose that system (22) has a real non-negative solution with respect to q and to R. Let q∗ be this non-negative vector. The standard commodity is defined, as usual, as the net product of the standard system: y∗ := (B − A)q∗ = RAq∗ .

(23)

Consider now the system of prices expressed in terms of the standard commodity: pT B = waT0 + (1 + π)pT A

(24a)

pT y∗ = = 1.

(24b)

Literature]

19

Equation (24b) entails pT (B − A)q∗ = RpT Aq∗ = 1

(25)

By combining equations (24a) with equation (24b) we get: 1 = pT (B − A)q∗ = waT0 q∗ + πpT Aq∗ . Thanks to (22b) and (25) the price vector p disappear from equation (24b) that set the price of numeraire equal to 1 and we yield: w(y∗ ) = 1 −

π , R

that is, • prices have not to vary in order to restore the balance in the industry that produces the numeraire; this entails that the composite commodity y∗ is an invariable measure of value; • prices disappear from the relationship between the wage rate and the profit rate: this permits to separate the analysis of distribution from the price system.

6

A quick survey of the literature

The literature that focused upon the standard commodity is enormous. Yet most part of it has not been very helpful in shedding light on this topic. In particular it is possible to single out some common misunderstandings arisen about the standard commodity. As in the previous sections it turns out to be useful to distinguish whether we are considering aspects involving the standard commodity within the theory of value or within the analysis of distribution.

6.1

The standard commodity within the theory of value

The standard commodity within the theory of value is used to isolate – when chosen as numeraire – the variations of the price of each commodity i originating exclusively from the peculiarities of industry i from those arising from the industry of the numeraire. As said in the Introduction, Sraffa does not provide a satisfactory proof of this property for the standard commodity.

20

[Literature

Almost all those authors that accepted the property of invariance of the standard commodity limited themselves to re-phrase – and in some cases just to quote – what Sraffa said in his § 21, without any further clarification: see, for example, Napoleoni (1962, sect. 8 and 9), Newman (1962, sect. IV), Bharadwaj (1963, pp. 1451–1452), Levine (1974, pp. 875–876), Bacha, Carneiro, and Taylor (1977, pp. 44–48), Harcourt and Massaro (1964, sect. 1). Only Baldone (1980, pp. 274–277), Mainwaring (1984, chap. 7), Kurz and Salvadori (1993, pp. 121-122, n. 16) and Abraham-Frois and Berrebi (1989) gave some hints or sketched out a proof of this invariance property, but they did not clarify the issue satisfactorily.17 The rest of authors rejected the property of invariance of the standard commodity considering it as a non-sense (see, for example, Johnson (1962), Catz and di Ruzza (1978), Flaschel (1986), Woods (1987)).18 It is worth to see in some details the two main objections raised against the invariance in value of the standard commodity, as they permit to bring to light some common misconceptions concerning the requirements that an invariable measure of value should have to satisfy. Objection 1 The standard commodity is not an invariable measure of value since its value – obviously expressed in terms of another (composite) commodity, b, – is not constant with respect to π. In fact, if we calculate it we obtain: pT(b) (π)y∗ = w(b) (π)aT0 [I − (1 + π)A]−1 y∗ ; 17

In particular the conclusions reached in Abraham-Frois and Berrebi (1989) are subject to all criticisms raised by Catz and di Ruzza (1990). 18 It is curious the attitude undertaken by Joan Robinson, that in her book review of Sraffa’s Production of Commodities considers the standard commodity as an “ingenious and satisfying” solution to “the problem that flummoxed Ricardo” Robinson (1961, p. 10); subsequently she softens her enthusiasm by saying that “Sraffa takes great trouble to provide a foolproof numeraire in which prices can be expressed, but the Keynesian wage unit serves as well” Robinson (1979, p. xx), till to conclude that “The definition of the standard commodity takes up a great part of Sraffa’s argument but personally I have never found it worth the candle. [...] This is not the unit of value like a unit of length or of weight that Ricardo was looking for.” Robinson (1985, p. 163). An attempt to reconstruct the Joan Robinson’s position on the standard commodity has recently been presented by Gilibert (1996); see also Porta (1995).

Literature]

21

by developing the inverse in a power series (we can do this for 0 ≤ π < Π) and by using (6) and (7) we have: pT(b) (π)y∗ =

aT0 [I

1 R · . −1 − (1 + π)A] b R − π

But, it is not the constancy of the expression pT(b) (π)y∗ that qualifies y∗ as invariable a measure of value; the key element – we saw – is the fact that when the standard commodity is chosen as numeraire the prices of other commodities do not undergo pressures in order to restore the balance in the industry that produces the numeraire.19 It is true that, as wages fell [the rate of profit rose, E. B.] such a commodity [the commodity produced by that industry which employs the proportion of labour to means of production which constitutes the borderline between ‘deficit’ industries and ‘surplus’ industries E. B.] would be no less susceptible than any other to rise or fall in price relative to other individual commodities; but we should know for certain that any such fluctuation would originate exclusively in the peculiarities of production of the commodity which was being compared with it, and not with its own.” (Sraffa 1960, p. 18)

Objection 2 The standard commodity is not an invariable measure of value since the prices of the other commodities, expressed in terms of the standard commodity, vary when distribution changes. In fact, in this case, thanks to (9b0 ), we have:  π T a [I − (1 + π)A]−1 . pT(y∗ ) (π) = 1 − R 0 19

(26)

The misunderstanding of this point seems to be at the basis of some confusion in Flaschel (1986) criticism: in particular see his comment to his numerical example at p. 590, or later, at p. 593 where he states “We already know that 0 = p0 q = ...” – where p0 indicates the derivative of vector p with respect to π and q indicates, in Flaschel’s notation, the output of the industry that satisfies the Sraffa’s ‘critical’ proportion. Schefold (1986) in his reply to Flaschel does not seem himself very clear on this point. On the contrary Abraham-Frois and Berrebi (1989, p. 127), in order to show the invariance of the standard commodity, make the same mistake, as they say: “Por que la marchandise composite [y] ait une valeur invariante quand la r´epartition varie, il faut ed il suffit que pT y reste constant quand π ´evolve de 0 ` a R, c’est-` a-dire que la diff´erentielle d(pT y) soit nulle” (my notation).

22

[Literature Again, the standard commodity is an invariable measure of value as, when distribution changes, no pushes for the prices of other commodities arise from equations (9b), that is, to restore the balance in the standard commodity industry. But the prices of the other commodities can vary (or have to vary) to restore the balance in their own industries. Thanks to the invariance of numeraire y∗ , equation (26) just describes the fluctuations of prices we are interested in, that is, those fluctuations that arise “from the peculiarities of the commodity which is being measured” (Sraffa 1960, p. 18).20

Finally it is to be recalled the view expressed by two eminent mainstream economists that gave some contributions in linear production theory and its linkages with Sraffian analysis: Edwin Burmeister and Paul A. Samuelson. According to them the standard commodity is a tool to restore the labour theory of value (Samuelson) or a “generalization ” of it (Burmeister).21 But as they rightly observe that the standard commodity cannot pursue these 20

It is curious to note that this erroneous conception of invariance of the standard commodity is used on one side by Flaschel and Woods and on another side by Blaug to maintain opposite conclusions: • Flaschel, in building his equation (5), assumes ∆p = o (see Flaschel (1986, p. 592)) and concludes against the invariance of the standard commodity (Flaschel 1986, pp. 592–595). (Again Schefold (1986), in replying to Flaschel, does not seem to have caught clearly this error.) Also Woods intends invariance in the same sense; in fact he considers “the Standard Commodity at two distinct rates of profit. When π2 > π1 , w2 < w1 and prices constant at p(π1 ) (to satisfy Sraffa’s hypothesis) ...” (Woods 1987, pp. 79 and ff.; our notation; emphasis added) and concludes against the invariance of the standard commodity. • Blaug, on the contrary, starting on the same notion of invariance concludes in favour of the invariance of the standard commodity: “[i]t is also obvious, at least intuitively, that an exogenous change in wages unconnected in changes in productive techniques alters the rate of profit but has no effect on relative prices measured in terms of the standard commodity for the simple reason that the change alters the measuring rod in the same way as it alters the pattern of prices being measured. The ‘standard commodity’ therefore provides an ‘invariable measure of value’” (Blaug 1987, p. 436; emphasis added). 21

Samuelson (1990, p. 271) says: “Sraffa (1960) establishes the concept of a standard (or reference) basket of commodities. This helps legitimize Ricardo’s hankering for a labour theory of value”. Burmeister (1984, p. 83) says: “Sraffa’s celebrated contribution was to invent a different “labor theory of value” by constructing a new wage measure such that the linearity [... of the profit rate-wage rate relationship, E. B.] remains valid even when

Literature]

23

purposes they conclude for the non-usefulness of the notion of standard commodity (Samuelson (1987, p. 456), Samuelson (1990, pp. 271-2), Burmeister (1980, pp. 91–92), Burmeister (1984, pp. 84–87)). It is not easy to understand the origin of such interpretation, but there is no evidence in Sraffa’s work to think that the standard commodity was conceived to restore the labour theory of value.

6.2

The standard commodity within the analysis of distribution

The standard commodity in the theory of value is used to separate the analysis of distribution from the theory of value. We have seen in our previous section 3 that when the standard commodity is used as numeraire, prices disappears from the expression of the wage rate-profit rate relationship before obtaining it. This point emerges clearly only from few contributions (see, for example, Meldolesi (1966), Pasinetti (1977) Broome (1977) Garegnani (1984) and Mainwaring (1984)). In general the main point that is presented in explaining the rˆ ole of the standard commodity within the analysis of distribution is the fact that it – when used as numeraire – makes the wage rate-profit rate relationship linear.22 This statement is formally correct but it is devoid of any economic content. It does not express the spirit of the Sraffa’s result. The property of linearity of this relationship is simply a consequence of the assumption concerning the date in which wages are paid: in system (1) prices are determined by supposing that wages are paid at the end of the production process. As proved by Pasinetti (1977, pp. 131–132) – or later by Bidard (1991, pp. 38-39) – if wages are paid at the beginning of the production period the price equation becomes pT = (1 + π)(waT0 + pT A),

(27)

and the relationship between the profit rate and the wage rate expressed in terms of the standard commodity is no longer a straight line but it is an there is not equal organic composition of capital.” A similar view was firstly presented by Napoleoni (1962, pp. 111–112) 22 See, for example, Napoleoni (1962, p. 111), Domined` o (1962, in particular p. 723), Bharadwaj (1963, p. 1452), Burmeister (1968), Miyao (1977), Abraham-Frois and Berrebi (1978), Flaschel (1986, p. 591), Samuelson (1987, p. 456) and Samuelson (1990, p. 272)

24

[Literature

equilateral hyperbole:23 w(y∗ ) =

R−π . R(1 + π)

(28)

The point to stress is that, in spite of its non-linearity, the wage-profit relationship is obtained independently on the price system (prices cancel out from its expression); thus it continues to make possible the separation of the analysis of distribution from the complications arising from the price system. In fact if we re-express the wage rate w(y∗ ) in terms of the commodity bundle consumed by workers, w(bw ) =

w(y∗ ) pT(y∗ ) bw

=

R−π R(1 + π) · , T R(1 + π) (R − π)a0 [I − (1 + π)A]−1 bw

we can isolate as before a factor, (R − π)/[R(1 + π)], which is no more linear but is independent on prices; thus it describes the physical aspect of the distribution separately from the complications arising from the price system. It is also possible to verify, with a reasoning analogous to that carried out in our previous section 3, that the standard commodity, y∗ , continues to constitute an invariable measure of value also with reguard to price system (27). In fact, if we consider system pT = (1 + π)(waT0 + pT A) pT y∗ = (1 + π)(waT0 y∗ + pT Ay∗ ) = 1 and we focuse upon its last equation – which is the price equation of the standard commodity – we see that, thanks to equations (6) and (7), it becomes 1 = (1 + π)[w(y∗ ) R/(1 + R) + 1/(1 + R)], that is, w(y∗ ) =

R−π . R(1 + π)

Again the fact that the vector of price has disappeared from the balance equation of the industry that produces the numeraire means that the wage By substituting into equation pT y∗ = 1 the price system given by equation (27) we obtain (1 + π)(w(y∗ ) aT0 y∗ + pT Ay∗ ) = 1; thanks to equations (6) and (7) prices disappear and we obtain relationship (28). 23

References]

25

component and the ‘value of capital plus profit’ component compensate each other for any variation of the distribution. Hence prices need not to vary in order to restore the balance in the industry of the numeraire. Thus the standard commodity continues to be an invariable measure of value also with reference to price system (27).

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