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

7/28/2019 Bellino, Enrico - On Sraffa's Standard Commodity as Invariable Measure of Value

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Abstract

The necessity to express the relative price of a commodity in terms of an-

other commodity makes it impossible to distinguish, within a variation of

its relative price, that part of the change that can be ascribed to the char-

acteristics of the commodity itself from that part that is to be ascribed to

the characteristics of the commodity of reference, i.e. the numeraire. Ricar-

do (1817) pointed out this problem and the necessity to find an invariablemeasure of value, but he was not able to solve this problem. Sraffa (1960)

suggested to use a composite commodity (that is, a bundle of commodi-

ties) 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 Sraf-

fas book there is no explicit proof of this claim. This gave rise to a lot of

misunderstandings about the standard commodity and its role as invariable

measure of value. In several contributions Sraffas solution to the Ricardos

problem was questioned. In this work I shall try first to clarify what itmeans, for a numeraire, to be an invariable measure of value. Then I shall

show that Sraffas 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 the-

ory, Sraffa price system, value theory, distribution theory.

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


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On Sraffas Standard Commodity as Invariable

Measure of Value

Enrico Bellino1

Universita Cattolica del Sacro Cuore (Milano)

C.O.R.E. (Louvain-la-Neuve)

English not accurately checked

1I 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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2 [Introduction

1 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 com-

modities, 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 commod-

ity: we then discover also the cause of its variation. Ricardo (1817,pp. 1718)

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 commodi-

ties 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 forits production. Ricardo (1817, pp. 4344)

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.


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Introduction] 3

But if this cause of variation in the value of a medium could be

removedif it were possible that in the production of our money forinstance, 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 sub-

ject 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 embar-

rassing 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 Ricardos problem in its entirety. Sraf-

fa (1960) offered a partial solution to this problem by building, within his

framework of production of commodities by means of commodities, a nu-

meraire, 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 r ole within the

Sraffas 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 Ricardos 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

1An attempt of reconstruction of the Ricardos search for an invariable measure of

value has been done by Kurz and Salvadori (1993). In that paper they also refer toan 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)).2The 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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4 [Introduction

the study of the price-movements which accompany a change in dis-

tribution. It is impossible to tell of any particular price-fluctuationwhether it arises from the peculiarities of the commodity which is be-

ing 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 pro-

duction 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 stan-

dard commodity solves the requirement of invariance with respect to changein income distribution. He gives some intuitive hints in 21 before building

the standard commodity; later he concentrates on the building of the stan-

dard commodity ( 2328 and 3335), 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 ( 2932). 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 Sraffas standard commodity offered very few hintsto understand this point. 3 Only Baldone (1980, pp. 274277) 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

Sraffas single product price framework will be recalled; in section 3 thecapability 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 in3A quick survey of the literature on the standard commodity can be found, later, in

Section 6.


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Basic framework] 5

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.

2 Review of the basic framework

The reference framework is the single product Sraffas price system with

circulating capital:

pT = (1 + )pTA + waT0 (1a)

pTb = 1, (1b)

where p is the (n, 1) vector of prices, A is the (n, n) non-negative input-

output matrix, 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 ofa 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 + )A]1; (2)

thanks to Perron-Frobenius theorems on non-negative matrices the inversematrix in (2) exists and is non-negative for 0 < , where := 1/M1

and M is the dominant eigenvalue ofA. In order to assure > 0 we assume

M < 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,


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6 [Basic framework

this latter being expressed in terms of numeraire b:4

w(b)() :=1

aT0 [I (1 + )A]1b

. (3)

Again, thanks to Perron-Frobenius theorems, the elements of the inverse at

the denominator are non-decreasing functions of for 0 < ; 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 expres-

sion of the vector of prices as a function of the profit rate only:

pT(b)() =1

aT0 [I (1 + )A]1b

aT0 [I (1 + )A]1. (4)

As [I (1 + )A]1 is non-negative the solutions with respect to the

wage rate (3) and the price vector (4) are non-negative for any within the

interval [0, ).5

Turning to the quantity-side the standard system is an economic sys-

tem in which the various commodities are represented among its aggregate

means of production in the same proportions as they are among its prod-

ucts. (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 thestandard system q must satisfy the following conditions:

q = (1 + R)Aq (5a)

aT0 q = 1, (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 non-

negative vector that satisfies system (5); mathematically it is the right-hand4In 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 asnumeraire 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.)5For the details of this analytical formulation of the Sraffas price system see Pasinetti

(1977, chap. 5) or Kurz and Salvadori (1994, chap. 4).


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Theory of value] 7

eigenvector of matrix A correspondent to its dominant eigenvalue (A) =

1/(1 + R) = 1/(1 + ). Vector q is called gross standard product. The net

standard product is defined by:

y := (I A)q =R

1 + Rq;

thus y is proportional to q, hence

Ay

=

1

1 + R y

(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.

3 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

6As 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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8 [Theory of value

express prices in term of whatever (composite) numeraire, b:7

p(b)1 = (1 + )pT(b)a

1 + w(b)a01...

p(b)i = (1 + )pT(b)a

i + w(b)a0i...

p(b)n = (1 + )pT(b)a

n + w(b)a0n

(8a)

pT(b)b = (1 + )pT(b)Ab + w(b)a

T0 b = 1, (8b)

where ai is the ith column of matrix A, representing the input coefficientsof 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 in-

crease, 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 wouldnot 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

7The 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 ith 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.


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Theory of value] 9

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 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 pres-

sures 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

Ricardos 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 thatdoes not engender pressures on the prices of the various commodities in

order to restore the balance in its own industry.

9System (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).


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10 [Theory of value

At this point we have all elements to verify that Sraffas standard com-

modity 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:

p(y)1 = (1 + )pT(y)a

1 + w(y)a01...

p(y)i = (1 + )pT(y)a

i + w(y)a0i...

p(y)n = (1 + )pT(y)an + w(y)a0n

(9a)

pT(y)y = (1 + )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 + )1

1 + RpT(y)y

+ w(y)R

1 + R= 1.

But as pT(y)y = 1 we have

1 +

1 + R + w(y)R

1 + R = 1,

i.e.

w(y) = 1

R. (9b)

Hence system (9) reduces to:

p(y)1 = (1 + )pT(y)a

1 + w(y)a01...

p(y

)i= (1 + )pT

(y

)ai + w

(y

)a

0i...

p(y)n = (1 + )pT(y)a

n + w(y)a0n

(9a)

w(y) = 1

R. (9b)


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Analysis of distribution] 11

In this case we see that the vector of prices has disappeared from the

last equation of system (9). Equation (9b) has now become an equation in

the variable w(y) only ( 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 notto 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).

4 The standard commodity within the analysis of

distribution

It is worth to recall here briefly the role 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)() =pT(b)b (1 + )p

T(b)()Ab

aT0 b=

1 (1 + )pT(b)()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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12 [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 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 ofcapital required to produce the numeraire, pT(b)()Ab or p

T(i)()a

i,

variation due to the change of the whole price system, p(b)() or

p(i)(), 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 , 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;

10As 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)() =p(i)i (1 + )p

T(i)()a

i

a0i =

1 (1 + )pT(i)()ai

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 tech-

nical 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.)


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Analysis of distribution] 13

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)() =pT(y)()y

(1 + )pT(y)()Ay

aT0 y

= (12)

=1 (1 + )/(1 + R)

R/(1 + R), (13)

that is,

w(y)() = 1

R, (14)

which is the well-known Sraffas 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 samecommodity, 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 in-

dependently of the price system.11 12 It should be noted that prices do not

11On this see Sraffa (1960, Appendix D), Broome (1977), Garegnani (1984) or Lippi

(1998).12Analytically 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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14 [Analysis of distribution

disappear completely from the problem of distribution: in fact the indepen-

dence 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(bw) =w(y)

pT(y)

bw=

1 /R

pT(y)

bw.

But by re-writing the above expression in the form

w(bw) =

1

R

R

(R )aT0 [I (1 + )A]1bw

, (15)

we could see that the use of the standard commodity has permitted to

separate a physicalkernel 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-linearfactor 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-commodityeconomy, 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

IIIVII).) These conclusions reflect one reconstruction of Ricardos thought.

An alternative line of interpretation is expressed in Porta (1982). I will not

enter into these issues here.

firstly, equation (9b) 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)13I owe this observation and the analytics to present it to Neri Salvadori.14Suitable 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.


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Extensions and generalizations] 15

5 Extensions and generalizations

5.1 Plurality of standard commodities

In the second part of the seventies there appear some independent contribu-

tions 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), Abraham-

Frois and Berrebi (1978), Bidard (1978)), that under certain circumstances

there exist other composite commodities, that Miyao called generalized s-

tandard commodity, that make the job. Consider the following matrix,

called by Miyao labour profile matrix:

K(n,n)

=

aT0aT0 A

aT0 A2

...

aT0 An1

.

Miyao (1977, Theorem 3, pp. 158159), proved that each composite com-

modity defined by

y = y + z o,

where y is the Sraffas 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 nH linearly independent vectors zh, h = 1, , nH,

satisfying system (16). Thus by choosing sufficiently small we can build

15On the faultiness of this way to characterize the standard commodity see our Section

6.


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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 Sraffas 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 prac-

tical interest from the economic point of view, as the elements of K are

given by technology. Notwithstanding as this case has (someway inexplica-

bly) 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:

pTyh = 1 pTy + pTzh = 1, h = 1, . . . , n H.


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Extensions and generalizations] 17

As it is easy to prove that pTzh = 0 for h = 1, . . . , n H16 we have that

pTyh = 1 pTy = 1,

(1 + )pT(yh)A + w(yh)aT0 y

= 1

w =R

Rh = 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 . 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 Sraffas standard commodity y or16Miyao 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

pT(IA)y=

1

aT0 y

1

r

R

. (18)

Miyao (1977, theorem 1, p. 154) proves that equation (18) is equivalent to

aT0A

ty = (1 + R)aT0A

t+1y, t = 0, 1, 2, . . . , (19)

hence yh = y + zh satisfies the following recurrence conditions on labour inputs:

aT0A

t(y + zh) = (1 + R)aT0At+1(y + zh), h = 1, . . . , n H, t = 0, 1, 2, . . . .

Thus we have that

aT0A

tzh = (1 + R)aT0A

t+1zh

, h = 1, . . . , n H, t = 0, 1, 2, . . . . (20)

Moreover as Kz = o we have

aT0A

tzh = 0, h = 1, . . . , n H, t = 0, 1, . . . , n 1. (21)

Hence thanks to (20) and (21) we have

aT0A

tzh = 0, h = 1, . . . , n H, t = 0, 1, 2, . . . .

Returning to pTzh we have:

pTzh = waT0 [I (1 + )A]

1zh =

= w

+t=0

aT0A

tzh = 0, h = 1, . . . , n H .


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18 [Extensions and generalizations

in terms of the generalized standard commodity yh: in fact as pTyh = 1

pTy = 1, the solutions with respect to prices and to the wage rate of the

two systemspT = (1 + )pTA + aT0

pTyh = 1and

pT = (1 + )pTA + aT0

pTy = 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 pTyh = pTy(= 1)

each unit of yh

can command one unit of y

, hence the equivalence issubstantial.

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 (22a)

aT0 q = 1, (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 com-

modity:

pTB = waT0 + (1 + )pTA (24a)

pTy = = 1. (24b)


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Literature] 19

Equation (24b) entails

pT(B A)q = RpTAq = 1 (25)

By combining equations (24a) with equation (24b) we get:

1 = pT(B A)q = waT0 q + pTAq.

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

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 commod-

ity 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.


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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 clarifi-

cation: see, for example, Napoleoni (1962, sect. 8 and 9), Newman (1962,

sect. IV), Bharadwaj (1963, pp. 14511452), Levine (1974, pp. 875876),

Bacha, Carneiro, and Taylor (1977, pp. 4448), Harcourt and Massaro (1964,

sect. 1).

Only Baldone (1980, pp. 274277), 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 .

In fact, if we calculate it we obtain:

pT(b)()y = w(b)()a

T0 [I (1 + )A]

1y;

17In particular the conclusions reached in Abraham-Frois and Berrebi (1989) are subject

to all criticisms raised by Catz and di Ruzza (1990).18It is curious the attitude undertaken by Joan Robinson, that in her book review of

Sraffas 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 wageunit serves as well Robinson (1979, p. xx), till to conclude that The definition of the

standard commodity takes up a great part of Sraffas 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 Robinsons position on the standard commodity has recently been presented by

Gilibert (1996); see also Porta (1995).


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Literature] 21

by developing the inverse in a power series (we can do this for 0

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