Discussion Paper No.310
On the Ricardian Invariable Measure of Valuein General Convex Economies: Applicability of
the Standard Commodity
Kazuhiro Kurose & Naoki Yoshihara

December 12, 2013

TOHOKU ECONOMICS RESEARCH GROUPGRADUATE SCHOOL OF ECONOMICS AND
MANAGEMENT TOHOKU UNIVERSITY
27-1 KAWAUCHI, AOBA-KU, SENDAI,
980-8576 JAPANOn the Ricardian Invariable Measure of Value in General Convex Economies:
Applicability of the Standard Commodity
Kazuhiro Kuroseyand Naoki Yoshiharaz

December 12, 2013

Abstract
The purpose of this paper is to examine the critical arguments made by Burmeister, Samuelson, and others, with respect to Sra¤a (1960). Sra¤a did not address thesearguments, but they are relevant from the viewpoint of modern economic theories. In
his arguments about the standard commodity, Sra¤a assumed that a change in income distribution has no e¤ect on the output level and choice of techniques. However,modern economic theories allow interdependence among changes in income distribution, output level, and choice of techniques. Therefore, it is interesting to consider the
existence of an invariable measure of value and linearity of income distribution in amodel where such interdependence is discussed. We assume general convex economies
with non-increasing returns to scale. In this model, we obtain the conditions underwhich the existence of an invariable measure of value and the validity of the linearity
of income distribution are assured.

Keywords: Ricardo’s invariable measure of value, Sra¤a’s standard
commodity, General convex economies, Linear relation of income distribution
JEL Classi…cations: B51, D30, D51
This paper was written when the …rst author was a visiting fellow of the Catholic University of
Milan from 2012 to 2013. He wishes to express his gratitude to the university. An earlier version ofthis paper was presented at the 17th ESHET conference, the Japanese Society for Post Keynesian
Economics, Japan Society of Political Economy, and a seminar in Milan. The authors thank all
the participants, especially Enrico Bellino, Luigi Pasinetti, Neri Salvadori, Bertram Schefold, Paolo
Varri, and Takashi Yagi, for their helpful comments and suggestions. The research is …nancially
supported by the Nomura Foundation. The usual disclaimer applies.
yGraduate School of Economics and Management, Tohoku University, Kawauchi 27–1, Aoba-ku,
Sendai 980-8576, Japan. E-mail: kurose@econ.tohoku.ac.jpz
The Institute of Economic Research, Hitotsubashi University, Naka 2–1, Kunitachi, Tokyo 1860004, Japan. E-mail: yoshihara@ier.hit-u.ac.jp1

1

Introduction

As is well known, in his later years, David Ricardo intensively searched for an invariable measure of value.1 His struggle to …nd it is shown in his Principles, his papers
entitled ‘Absolute Value and Exchangeable Value’, which were written in the last few
weeks of his life (Ricardo, 1951D, pp. 361–412), and a set of his letters to Malthus,
McCulloch, and others.
An invariable measure of value can be de…ned as a measure that is invariable withrespect to changes in both income distribution and technique (Ricardo, 1951A, chap.
1). Without an invariable measure of value, we cannot in general distinguish betweenchanges in the price of the numéraire and that of the commodity measured, whenrelative prices change. The advantage of the invariable measure of value, if it exists,
is that we can distinguish between the variations which belong to the commodity
itself and those which are occasioned by a variation in the medium by which values
or prices are expressed (Ricardo, 1951A, p. 48). For Ricardo, the pursuit of theinvariable measure of value is directly related to the completion of the embodied
labour theory of value,2 although the importance of the measure of value was notadequately understood by his contemporaries such as Malthus.3
Even though it is true that the embodied labour theory of value cannot generally hold when the rate of pro…t is positive, it does not mean that the invariablemeasure of value is no longer signi…cant. The purpose of Ricardo’s construction of
the invariable measure of value is to build a solid foundation not only to measuresuch important variables as national income or national wealth precisely, but also tocompare those variables intertemporally. Therefore, no one can deny the importanceof the invariable measure of value even today.
In the 20th century, Sra¤a (1960) revived the concept of the invariable measure of
1
Ricardo’s concern about an invariable measure of value appeared as early as in his contributions
to the ‘bullionist’controversy. As a bullionist, he wrote some notes on the stability of the generalprice level during the Napoleonic wars, which are collected in Ricardo (1951C). He had alreadypointed out the need for an invariable measure of value, which would enable an intertemporalcomparison of values, and argued that such a measure did not exist in reality; however, money
could be regarded as an invariable measure of value at least as the …rst approximation (see especially
Ricardo, 1951C, p. 65). However, his arguments at this stage were not rigorously based on thetheory of value. See Kurz and Salvadori (1993) concerning the conceptual transition of Ricardo’s
invariable measure of value.
2
Ricardo (1952C, p. 358) said, ‘As soon as we are in possession of the knowledge of the circumstances which determine the value of commodities, we are enabled to say what is necessary to give
us an invariable measure of value.’ See also Sra¤a (1951) for the transition of Ricardo’s theory ofvalue in detail.
3
See Porta (1992) concerning debates on the measure of value between Ricardo and Malthus.

2

value, which had fallen into oblivion since the so-called Marginal Revolution in the1870s. Unlike Ricardo, he divided the problem of identifying an invariable measureof value into two parts: the …rst is to search for a measure of value that is invariablewith respect to changes in technique, left aside the change in income distribution,and the other is to search for a measure of value that is invariable with respect to thechange in income distribution, left aside the change in technique. Sra¤a exclusively
concentrated on the latter by constructing a special, composite commodity termed
the standard commodity.4 As we will see, he also demonstrated an interesting relationship with respect to income distribution if the standard commodity is adopted
as the numéraire: the linear relationship of income distribution. Although many
economists have paid great attention to the results obtained by Sra¤a, it seems thatthey have not reached a consensus on evaluating Sra¤a (1960). Some economists
appreciate him, whereas others do not unconditionally admit the signi…cance of thestandard commodity and the linearity of income distribution. In particular, those
who are critical of Sra¤a (1960) regard the assumption of a …xed technique without
constant returns to scale as being too restrictive, and thus, downgrade the relevanceof Sra¤a (1960). As we shall see, Burmeister (1968, 1975, 1977, 1980, 1984), Samuelson (2000, 2008), Samuelson and Etula (2006), and others claimed that Sra¤a’s
analyses are irrelevant without the assumption of constant returns to scale.
We think that the views of Burmeister, Samuelson, and others are worth examining, because they point out relevant problems from the viewpoint of modern economic
theory, which Sra¤a had not addressed. As we will see, in his arguments about thestandard commodity, Sra¤a (1960) assumed that a change in income distribution has
no e¤ect on the output level and choice of techniques. Of course, such an assumption
is just an analytical device to construct a model; it is plausible that the change inincome distribution is related to changes in output level or choice of techniques in
actual economies. In fact, almost all modern economic theories admit interdependence among changes in income distribution, output level, and choice of techniques,
even though the logical consequences of such interdependence are di¤erent among
theories. Even those who are favourable to Sra¤a would not be able to deny thisinterdependence. Curiously enough, there is little literature on whether or not aninvariable measure of value and linearity of income distribution can be obtained in
4

Pasinetti’s dynamic standard commodity is one of the examples that pay attention to the former.
According to Pasinetti (1981, p. 105, n. 13), the economic system is, so to speak, ‘frozen’at a given
technique where Sra¤a constructed the standard commodity whose value is invariant with respect
to the change in income distribution; whereas the economic system is ‘frozen’ at a given income
distribution where Pasinetti constructed the dynamic standard commodity which always requires
the same quantity of ‘augmented’labour through time. Therefore, we cannot analyse a change inincome distribution by using the dynamic standard commodity.

3

models in which the above-mentioned interdependence is allowed.5 Therefore, we
attempt to examine their critical arguments with respect to Sra¤a (1960). We assume general convex economies with non-increasing returns to scale and attempt tospecify the conditions under which the invariable measure of value and linearity ofincome distribution can be obtained.
This paper is organised as follows: In Section 2, we present a brief review of theconcept of Ricardo’s invariable measure of value and Sra¤a’s standard commodity.
Subsequently, we brie‡y review the history of the debates on the standard commodity and the linear relation of income distribution that Sra¤a derived. In Section 3,
we discuss a generalisation of the standard commodity to more general models thanSra¤a’s (1960). As a result, we obtain the conditions under which the invariable measure of value and linearity of income distribution are maintained even in a rigorous
general convex economy. In Section 4, we present our concluding remarks.

2

The Invariable Measure of Value and Debatesconcerning Sra¤a (1960)

In this section, we brie‡y review the concept of Ricardo’s invariable measure ofvalue and Sra¤a’s standard commodity. We also review the linear relation of income
distribution, which is obtained in Sra¤a (1960), and the debates concerning thesigni…cance of the standard commodity and linearity of income distribution.

2.1

Ricardo’s invariable measure of value

A measure of value is indispensable for exchanging commodities e¢ ciently. Ricardo
asserted that the conditions necessary to make a measure of value perfect are that itshould itself have a value, and that value should itself be invariable (Ricardo, 1951D,
p. 361). Concerning the …rst condition, he clearly argued that the labour content
embodied in such a commodity represents the exchange value of the commodity. The
second condition, the invariance of the value of such a commodity, perplexed him
throughout his life.
Why is it di¢ cult to obtain an invariable measure of value? First, the techniqueto produce it must remain unchanged. In other words, a commodity eligible tobecome the invariable measure of value is one ‘which now and at all times required5

One of the exceptions is Yagi (2012). Following Pasinetti (1981, 1993), he constructed a model inorder to compare two di¤erent economic systems (called Period 1 economy and Period 2 economy).
In addition, he investigated the invariable measure of value and linearity of income distribution.

4

precisely the same quantity of labour to produce it.’ However, Ricardo realised, ‘Of
such a commodity we have no knowledge, and consequently are unable to …x on
any standard of value’ (Ricardo, 1951A, p. 17, n. 3). In fact, Ricardo regarded
money (that is, gold and silver) as the invariable measure of value, but it is just‘as near as approximation to a standard measure of value as can be theoreticallyconceived’ (Ricardo, 1951A, p. 45). The justi…cation is based on his recognition
that the techniques of production of gold and silver are subject to fewer variations(Ricardo, 1951A, p. 87).
With respect to the second condition, even though the technique to produce
gold and silver is unchanged, gold and silver cannot be the invariable measure ofvalue. This is because all industries have di¤erent proportions of capital and labour,di¤erent proportions of circulating and …xed capital, di¤erent degrees of durabilityof …xed capital, and di¤erent time-periods necessary to bring the commodity tomarket. In this situation, the change in the level of wage rates causes changes inrelative prices. Furthermore, as already mentioned, we cannot precisely measure thechange in prices of the commodities measured, because the prices of gold and silverthemselves are subject to the relative variations. Therefore, the invariable measureof value never existed in reality. According to Ricardo (1951A, p. 45), however, thee¤ect of a change in income distribution on relative prices is smaller than the e¤ectof a change in technique. Therefore, Ricardo thought of the deviation of value fromthe embodied quantity of labour as su¢ ciently slight (Ricardo, 1951B, p. 66), andthus, he was reluctantly content to say that money can be regarded as the invariablemeasure of value at the …rst approximation.

2.2

Sra¤a’s standard commodity and income distribution

Sra¤a (1960) revived the concern about the invariable measure of value, which in turnled to intensive and comprehensive controversies about capital and income distribution.6 As already mentioned, Ricardo had de…ned the conditions that the invariablemeasure of value should satisfy: the invariance of the measure of value with respect
to changes in both income distribution and technique. Ricardo was perplexed by
the conditions, because he attempted to solve the two simultaneously. In contrast,
Sra¤a concetnrated on …nding a measure of value that is invariable with respect toa change in income distribution, left aside the chnage in techniques. Furthermore,
it is Sra¤a’s breakthrough idea to …nd a special, composite commodity, termed thestandard commodity, which plays the role of the invariable measure of value; whereas
Ricardo attempted to …nd a single commodity that plays the role.
6

See, for example, Harcourt (1972) for more detail on these controversies.

5Let us brie‡y review the concept of the standard commodity in a single production
system. The price system is de…ned as follows:p = (1 + ) pA + wL;(1)

where p, L, and A denote the price vector, the labour coe¢ cient vector, and the
input coe¢ cient matrix, respectively. For the sake of simplicity, A is assumed to be
an indecomposable and productive matrix.
and w denote the rate of pro…t andthe wage rate, respectively. In order to escape from the impasse that Ricardo faced,Sra¤a attempted to …nd an (imaginary) industry that has a value-ratio of the net
product to means of production such that the increase in pro…t is exactly o¤set by
the decrease in wage when the wage rate is reduced. The value-ratio can be obtainedby solving the following system:(1 + ) Aq = q;
Lq = 1;
where q is the vector denoting the output level of the industry that has the valueratio. Since we assume the indecomposability of A, the above system of equations has
the solution of> 0 and q > 0 from the Perron-Frobenius theorem.7 The above
system of equations is called the standard system.
is the value-ratio termed thestandard ratio, which is related to the Frobenius root A as A = 1+1 . Furthermore,
q is the corresponding eigenvector, termed the standard commodity. By substituting
the solution into the system, we obtain:
(1 +

) Aq
Lq

= q,
= 1.

(2)
(3)

From formula (2), we obtain:p [I A] q
=
pAq

;

(4)

where I denotes the identity matrix. Formula (4) means that the ratio of the net
product to means of production, measured by the standard commodity, is always
7

See Pasinetti (1977, pp. 95–7) in detail.

6constant, irrespective of price variations.8 Therefore,
is a real ratio that is independent of prices. Sra¤a de…ned the standard net product and chose it as thenuméraire as follows:p [IA] q = 1:(5)Note that the standard commodity q does not need to be actually produced; it isa “purely auxiliary construction”(Sra¤a, 1960, p. 31). Sra¤a (1960, p. 26) said thatany actual economic system can always be transformed into the standard system.
The ratio that keeps the net product to means of production constant irrespective of
prices is always, so to speak, ‘hidden’within any actual economic system. In Sra¤a’s
model, nothing except income distribution ever changes; the technique in use, output
level, and proportion of means of production to labour are all …xed. Therefore, noassumption on returns to scale needs to be made here, as Sra¤a (1960, p. v) himself
said. Under such assumptions, he exclusively analysed the change in relative prices
caused by the change in income distribution. Owing to formulae (4) and (5), there
is no need for a variation in the price of q to restore the surplus or de…cit in the(imaginary) industry which produces the commodity, when the wage rate is reduced.
Therefore, the variation in relative prices caused by a change in income distribution
is solely attributed to the variation in prices measured on the basis of the invarianceproperty of the numéraire de…ned by the standard commodity.
The advantage of adopting the standard commodity as the numéraire is not onlythat it plays the role of the invariable measure of value, but also that it shows us theuseful relation of income distribution. From (1) and (5), we obtain:
=

(1

w) :

(6)

Here, w denotes the wage rate or the wage share in terms of the standard commodity,
8

According to Schefold (1986, 1989), the watershed condition and the recurrence condition must
hold in order for the standard commodity to serve as the invariable measure of value. The former
means that the industry producing the standard commodity must adopt that proportion of meansof production to labour in which there is no need to change its price when income distributionchanges, because the industry neither earns a surplus nor incurs a de…cit. The latter conditionmeans that the proportion ‘recurs in all the successive layers of the industry’s aggregate meansof production without limit’ (Sra¤a, 1960, p.16). In other words, the conditions imply that not
only the industry producing the standard commodity, but also all the industries that produce themeans of production necessary for the production of the standard commodity, must adopt the‘watershed’ proportion of means of production to labour. The industry producing q obviously
satis…es both conditions. As is easily shown, the proportion is 1 . In fact, only the watershed
condition is su¢ cient for the existence of the standard commodity, insofar as the proof is based on
the Perron-Frobenius theorem.

7

whereas is the actual rate of pro…t. The distributional relation is expressed by thestraight line in the case where the ‘organic composition’ of capital is not equal in
all industries, if and only if the standard commodity is adopted as the numéraire.
The important implication of function (6) is that the rate of pro…t can be obtained
without knowing prices, once we know the wage in terms of the standard commodity.
In other words, the standard commodity enables us to deal with income distribution
independently of prices. As Pasinetti (2006, p. 154) pointed out, the relevance offunction (6) does not lie in its linearity, but in the fact that it is independent ofprices.
We can conclude that Sra¤a resolved the problem that Ricardo could not, but theresolution was partial, because Sra¤a did not consider another problem. This is the
problem of the measure of value invariable with respect to the change in technique.

2.3

After Sra¤a (1960)

There have been many reactions to Sra¤a (1960) since its publication and somedebates on the results that Sra¤a derived. The debates focused not only on theinvariable measure of value, but also on the usefulness of the standard commodity andthe distributional relation given by function (6). Some arguments appreciate Sra¤a’sachievements, especially his contribution of constructing the standard commodity as
the invariable measure of value (for example, Roncaglia, 1978). Other arguments are
critical of Sra¤a (1960). First, some economists argued that the standard commodity
does not play the role of the Ricardian invariable measure of value. Those arguments
pointed out the ‡aw in Sra¤a’s analysis. Flaschel (1986) is a typical example. The
second critical argument was that the standard commodity and the distributional
relation shown by function (6) are so restrictive that they are not too helpful for
relevant analyses. Those arguments were mainly raised by neoclassical economists,
who were interested in variations in output and proportions of means of production.
Let us examine Flaschel (1986) …rst. According to him, there is a speci…c andcomplete solution to the problem of determining the conditions for the invariablemeasure of value, but Sra¤a’s standard commodity does not ful…l those conditions.
It seems to us that his de…nition of invariance is di¤erent from those of Ricardo andSra¤a. He de…ned that given e Ae as the numéraire, where e is a vector, all theelements of which are units, an arbitrary composite commodity b has the invarianceproperty if and only if pb = 1 holds for any non-negative and non-zero p, withp [I A] e 1 (Flaschel, 1986, pp. 597–8).9 Certainly Sra¤a (1960, p.11) adopted
9

In Flaschel (1986, p. 597), it is explicitly written as ‘the problem of invariance cannot be
described unless a measure of value has already been assumed. This fact is implicitly taken into8

[I A] e as the numéraire, but the numéraire adopted in the context is irrelevantto the issue of the invariable measure of value, and his arguments on the standard
commodity have nothing to do with the numéraire of [I A] e. Flaschel’s critiqueof the standard commodity, therefore, seems pointless.10
As for the second argument critical of Sra¤a, the typical and early example isBurmeister (1968), in which the results obtained by Sra¤a (1960) were rigorously
formulated. The conclusions he derived are summarised as follows:
1) It is dubious what economic signi…cance can be attached to the standard
commodity.
2) The linearity of the distributional relation does not hold if wages are paid atthe beginning of the production period rather than at the end.
3) Without the assumption of constant returns to scale and a …xed coe¢ cientsmatrix, Sra¤a’s analysis is meaningless if the quantity produced by an arbitrary
industry changes.
After Burmeister (1968), he published a set of papers related to Sra¤a’s analysis andrepeated conclusions similar to those above (Burmeister, 1975, 1977, 1980, 1984).
However, his conclusions show that he misunderstood some aspects of Sra¤a (1960).
The …rst conclusion made by him is a serious misunderstanding. Burmeister
regarded the standard commodity as the actual consumption basket by which thereal wage rate w in function (6) is measured.11 Therefore, he argued that the standard commodity has no economic signi…cance; ‘Sra¤a’s weights used to construct
his basket of goods are seen to be determined completely from the technology without regard for consumption preferences’(Burmeister, 1984, p. 509). However, theadoption of the standard commodity as the numéraire does not imply that people
must actually consume each commodity in the same proportion as that given by thestandard commodity.12 Moreover, it does not imply that each commodity is actually
produced in the same proportion as that given by the standard commodity (see Kurz
and Salvadori, 1987, pp. 876–7).
The second conclusion is obviously correct. Using the same notations as before,
account by Sra¤a ([1960], Ch. 3) in his assumption p (e Ae) 1.
the search for (conditionsfor) a “measure of value”relative to an already given measure of value! But what can be expectedfrom the solution of such a problem?’
10
Concerning Flaschel’s (1986) critique, see also Baldone (2006) and Bellino (2004), in which
the conditions for the invariable measure of value in Ricardo’s and Sra¤a’s sense are adequately
formulated.
11
Samuelson (2008) also blundered into the same misinterpretation as Burmeister.
12
See Pasinetti (1977, p. 116, n. 40). Any proportion is feasible in the actual consumption
basket, even when the standard commodity is adopted as the numéraire.

9the distributional relation in this case is expressed as follows:(1 w) :13
(7)
1+ w
Although it is a hyperbolic function, the basic property obtained when the standard
commodity is chosen as the numéraire is still intact: the distributional relation shownby function (7) is entirely independent of prices.
The third conclusion is controversial. Samuelson (2000) and Samuelson and Etula(2006) also argued that constant returns to scale is an indispensable assumption inorder to retain the signi…cance of Sra¤a’s analysis. Samuelson (2000, p. 123) statedthat ‘if a Sra¢ an denies constant returns to scale, the one-hundred-page 1960 classic
evaporates into a few paragraphs of vapid chit-chat.’ Against these arguments, some
proponents of Sra¤a argued that the assumption on returns to scale is unnecessary in
Sra¤a’s analysis. The characteristic of the analysis is that it is based on the classicalsurplus approach, which is clearly expressed in Ricardo (1951A). In the approach,the analysis of the distribution of physical surplus comes …rst. Therefore, Eatwell(1977) emphasised the di¤erence in the analytical basis between classical and neoclassical economics. In the former, the size and composition of output, technique in
use, and real wage are the data, on the basis of which the distribution of surplus,
price formulation, and quantities of input and labour employed are obtained. In
the latter, on the contrary, the preferences of individuals, initial endowment of commodities and/or factors of production, distribution of the initial endowments amongindividuals, and technology are the data, and all variables are determined by the interaction between supply and demand. It is based on the marginal method, and thus
the assumption on returns to scale is necessary in neoclassical economics.14 Eatwell(1977) thus argued that the assumption of constant returns to scale is irrelevant in
Sra¤a’s analysis, because it is based on the classical surplus approach.
However, Burmeister and Samuelson considered what happens to the model when
the output level changes. If constant returns to scale are not assumed, the change in
output level would cause a change in technique. Therefore, Burmeister (1975) argued
that function (6) becomes useless if constant returns to scale are not assumed. As
is already mentioned, unless constant returns to scale are assumed, the techniquegenerally changes as the output level changes. Since each coe¢ cient matrix has
the speci…c standard ratio, the standard ratio also changes when the technique in
use changes. Therefore, function (6) no longer gives us any useful information on=

13

See Pasinetti (1977, p. 131).
See also Pasinetti (1977, chap. 1; 1981, chap. 1) concerning the contrast between classical andneoclassical economic thoughts and methods.
14

10

income distribution when a change in the output level causes a change in techniquein economies without constant returns to scale.15 Burmeister (1977, pp. 69–70) thus
replied to Eatwell: ‘I conclude that constant returns to scale is irrelevant for Sra¤a’s
analysis only if one is content to pose irrelevant questions.’
Although it is true that Burmeister’s interpretation of Sra¤a included the misunderstanding, it is also true that he raised important questions on income distribution
which Sra¤a had not addressed. The questions are whether or not the invariablemeasure of value exists in economies where not only income distribution but also
technical change are available; and if it exists, what kind of relationship between theinvariable measure of value and income distribution holds. We think it worthwhile
to examine them. From the viewpoint of modern economic theories, these are, infact, natural questions, because nearly all modern economic theories allow for interdependence among changes in income distribution, output level, and techniques. In
fact, Sra¤a (1925, 1926) himself had considered the relationship between returns toscale and choice of techniques, although his consideration was related to the critiqueof Marshallian partial equilibrium analysis.

3

The Standard Commodity and Income Distribution

In this section, we focus on the questions raised by Burmeister, Mainwaring, andSamuelson. In other words, we investigate the conditions for obtaining the invariablemeasure of value and maintaining the linearity of income distribution in generalconvex economies with non-increasing returns to scale.

3.1

Preliminary

Let R+ be the set of all non-negative real numbers, and R++ be the set of all positive
numbers. Let Rn+ (resp. Rn++ ) be the n-fold Cartesian product of R+ (resp. R++ ).
For any x; y 2 Rn+ , we write x = y to mean [xi = yi for all i = 1; : : : ; n], x y tomean [xi = yi for all i = 1; : : : ; n and x 6= y], and x > y to mean [xi > yi for alli = 1; : : : ; n].
Let there be N agents in the economy, and let us use N as the population in
the economy with generic element 2 N . Let there be n commodities which are
reproducible. Let 0 denote the null vector. Production technology is freely available
15

Mainwaring (1979), a non-neoclassical economist, also argued that Sra¤a’s analysis is not valid
without the assumption of constant returns to scale or other restrictive assumptions.

11to all agents, who can operate any activity in the production set P , which haselements of the form = ( l ;; ), where l 2 R+ is the e¤ective labour inputof the process; 2 Rn+ are the inputs of the produced goods used in the process; and2 Rn+ are the outputs of the n goods. Thus, elements of P are vectors in R2n+1 .
Let P satisfy the following assumptions.
Assumption 0 (A0). P is closed in R2n+1 and 0 2 P . Moreover, for any ;with = 0 , and for any t 2 [0; 1], t + (1 t) 0 2 P .
Assumption 1 (A1). For all 2 P , if
0 , then l > 0 and
0.
Assumption 2 (A2). For all c 2 Rn+ , there is a 2 P such that b
0Assumption 3 (A3). For all 2 P , and for all ( 0l ;; 0) 2 R
0
0
0
0
0
0
; ) 5 , then ( l ;; ) 2 P.
if ( l ;Assumption 4 (A4). There exists r > 0 such that for all( k l ; k ; kr ) 2 P .

R

n

0

2P

= c.
Rn+ ,

2 P , and for any k > 0,

For each production possibility set P , let us denote @P f 2 P j @ 0 2 P : 0 > g
and SPf 2P j@ 02P : 0g, where the former and the latter are respectively the boundary and the e¢ ciency frontier of the production set P . Moreover,given k > 0, let P ( l = k) f 2 P j l = kg and@P ( l = k)

f 2 P ( l = k) j @ 0 2 P ( l = k) : (

0

;

0

)>(

; )g :The model of production sets with A0~A4 covers a broad class of production
technologies as follows:
Example 1: Given a von Neumann technology (A; B; L), where A and B are n m
non-negative matrices and L is a 1 m positive vector. Suppose that for each sector
j = 1; : : : ; m, there exists at least one commodity i = 1; : : : ; n such that aij > 0. we
can de…ne a production set P(A;B;L) as
P(A;B;L)

2R

Rn

Rn+ j 9x 2 Rm
+ :5 ( Lx; Ax; Bx) .

Note that for each 2 SP(A;B;L) , there exists x 2 Rm
= ( Lx; Ax; Bx).
+ such that
The set P(A;B;L) satis…es all of A0~A4. As a special case of the von Neumann technology, we can consider the case that m = n and B = I, which implies a Leontief
technology (A; I; L). Then, we can de…ne P(A;L)
P(A;I;L) as in the de…nition ofP(A;B;L) .
Example 2: Let us consider a class of Leontief technology Ak ; Lk k=1;:::;m , wherefor each k = 1; : : : ; m, Ak is a n n non-negative, productive, and indecomposable
12matrix and Lk is a 1 n positive vector, such that for any k; k 0 = 1; : : : ; m, and0000for any non-negative n 1 vectors xk and xk , Ak xk = Ak xk implies xk = xk and00as
Lk xk = Lk xk . Given this, we can de…ne a production set P(Ak ;Lk )k=1;:::;mRn

2R

P(Ak ;Lk )k=1;:::;m

9 x
By the supposition of

ks
ks 2S

Ak ; Lk

Rn+ j 9S
Rn+ :5

k1; : : : ; kS
X

ks

ks

L x ;

ks 2S
k=1;:::;mf1; : : : ; mg ;
X

ks

ks

A x ;

ks 2S

X

x

ks

ks 2S

, the production set P(Ak ;Lk )!)satis…es

k=1;:::;m

A0~A4.
Suppose that each agent can supply at most one unit of labour per production
period. Moreover, let ! 2 Rn+ be the social endowments of commodities. Then, one
economy is represented by a list, hN ; P ; !i.

3.2

The standard commodity in general convex economies

We are now ready to de…ne the standard commodity.
De…nition 1: Given an economy hN ; P ; !i, a standard commodity is a vector y 2yRn++ , such that there exists 2 @P ( l = 1) with (i) b = y; (ii) yi = j for any i; j =1; : : : ; n; and (iii) there is no

and

b 0i
0
i

0

> yi for any i = 1; : : : ; n.

2 @P ( l = 1) with

b 0i
0
i

=

b 0j

0
j

i

j

for any i; j = 1; : : : ; niThe standard commodity de…ned here is a generalisation of Sra¤a’s de…nition. Firstly,
the condition 2 @P ( l = 1) with (i) b = y implies that the standard commodity
can be produced as a net output via a production activity associated with one unit oflabour input, which corresponds to equation (3) in Section 2.2. Secondly, condition(ii) of De…nition 1 is a generalisation of the condition represented by equation (2)in Section 2.2. Indeed, it can be interpreted as the uniform rate of surplus that is
obtained when the wage rate is zero and the whole of the net product is distributed
to pro…ts. Thirdly, condition (iii) of De…nition 1 is a generalisation of the maximality condition of the uniform surplus rate represented byof equation (2). Becauseof these, we can see that De…nition 1 is an extension of the Sra¢ an de…nition of
the standard commodity characterised by equations (2) and (3) to a more generalproduction technology P with A0~A4.
13.

Given this de…nition, the general existence of the standard commodity can beshown, as the following theorem shows.
Theorem 1: Let r 5 1. Then, under A0~A4, there exists a standard commodity
2 @P ( l = 1) and b = y .
y 2 Rn++ associated with

The proof of Theorem 1 is relegated to the Appendix.
The above theorem does not necessarily imply the unique existence of the standard commodity, though its uniqueness is not essential in the following analysis. In
contrast, if the production set P is more suitably speci…ed, then the unique existence of the standard commodity can be shown, which will be brie‡y discussed in
the Appendix.

3.3

Linearity of distributive relation and the standard commodity

In this section, we will examine whether or not the standard commodity of theeconomy hN ; P ; !i can function as the invariable measure of value in the economy
hN ; P ; !i. De…ne the set of price systems by the simplex
(
)
n
X
Mn+1
(p; w) 2 Rn+1
jpi + w = 1+
i=1

with generic element (p; w). The simplex is su¢ cient to provide all the necessary
information about price systems, since we are only concerned about the relative
prices of all commodities and labour, which are invariant with respect to changes in
the numéraire commodity. Furthermore, it is su¢ cient for the main purpose of ouranalysis to focus on the change of price systems normalised in the simplex. Indeed, if
the price of a commodity bundle is invariable with respect to any change in the pricesystems normalised in the simplex, it is also invariable with respect to any change
in price systems measured by any numéraire. Finally, any commodity bundle would
not constitute an n-dimensional simplex as the set of price vectors when it is chosen
as the numéraire. This is because the n-dimensional simplex presumes n + 1 numberof components, whereas any commodity bundle is constituted by at most n typesof commodities, since it does not contain labour by de…nition. Hence, without lossof generality, we can analyse any price system independently of the issue of which
commodity (bundle) is chosen as the numéraire by focussing on the normalised pricevector in the simplex.
14

Consider a situation where a price system changes from (p; w) 2Mn+1 to (p0 ; w0 ) 2Mn+1 .
Let (resp. 0 ) be the maximal pro…t rate associated with the price system (p; w)0(resp. (p0 ; w0 )). Then, let 4p p0 p, 4w w0 w, and 4. In this case,n
if 4py = 0 holds for a commodity bundle y 2 R+ n f0g, it is not because y is chosen
as the numéraire, but because its value is invariable with respect to this change.16
Then, the following de…nitions are a generalisation of Baldone (2006):
De…nition 2: Given an economy hN ; P ; !i, let (p; w) 2Mn+1 and (p0 ; w0 ) 2Mn+1 be
two di¤erent equilibrium prices, and and 0 the respectively associated maximal
pro…t rates. Then, a commodity bundle y 2 Rn+ n f0g serves as the invariable measureof value with respect to change from (p; w) to (p0 ; w0 ), if and only if there exist
x 2 Rn+ n f0g and k > 0, such that ( k; x; x + y) 2 P and 4py = 0 holds wheneverthis price change involves a redistribution between pro…t and wage, namely, 4 px +4wk = 0.
De…nition 3: Given an economy hN ; P ; !i, a commodity bundle y 2 Rn+ n f0g serves
as the invariable measure of value, if and only if for any di¤erent equilibrium prices(p; w) 2Mn+1 and (p0 ; w0 ) 2Mn+1 , it serves as the invariable measure of value withrespect to change from (p; w) to (p0 ; w0 ).
That is, a commodity bundle serves as the invariable measure of value, if and only
if for any change in the price system involving a redistribution of pro…t and wage, theprice of this commodity bundle is invariable. More precisely speaking, let us consider
counterfactually a change in factor prices from ( ; w) to ( 0 ; w0 ), while keeping thecommodity price vector p constant, such that the increase (resp. decrease) in pro…tis exactly equal to the decrease (resp. increase) in wage in the production processof the targeted commodity bundle y. Such a change may be derived from a purely
political con‡ict on the income distribution between capital and labour, or it may
involve a change in technique. In any case, however, it may result in a change
in commodity prices from p to p0 . Then, the commodity bundle y can serve as theinvariable measure of value with respect to the change from (p; w) to (p0 ; w0 ) wheneverpy = p0 y. Furthermore, if the commodity bundle satis…es such an invariable property
for any change in price systems with its corresponding redistribution between wage
and pro…t, it can serve as the invariable measure of value.
It is worth emphasising that in the above de…nitions, the invariable property
must hold regardless of the causality of such a price change. For instance, even if theprice change is generated owing to technical change so that the selected production
16

Remember that, since price systems are normalised in the simplex, no commodity (bundle) is
selected as the numéraire in this analysis.

15

activity is changed in equilibrium, the value of the commodity bundle is required
to be invariable. Note that the possibility of price change due to technical change
should not be a concern whenever the production set is given by a simple Leontief
technology P = P(A;L) .
Now, we are ready to discuss the necessary and su¢ cient condition for the standard commodity to serve as the invariable measure of value.
Theorem 2: Let r 5 1. Then, under A0~A4, for any economy hN ; P ; !i, let us
take any equilibrium prices (p; w) 2Mn+1 and (p0 ; w0 ) 2Mn+1 . Then, the standard
commodity y associated with= ( 1; x ; x + y ) 2 @P serves as the invariablemeasure of value with respect to change from (p; w) to (p0 ; w0 ), if and only if there exist
non-negative numbers ; 0 2 R+ such that py = px +w , p0 y = 0 p0 x +w0 0 ,and = 0 hold.
Proof. By Theorem 1, there exists a standard commodity yx with some
> 0 in the economy hN ; P ; !i. Let (p; w) 2Mn+1 (resp. (p0 ; w0 ) 2Mn+1 ) be an
equilibrium price with the associated maximal pro…t rate (resp. 0 ). By de…nition,y = x . Since l = 1, py 5 px + w and p0 y 5 0 p0 x + w0 generally hold.
Therefore, there are non-negative numbers ; 0 2 R+ such that py = px + w
0and p0 y = 0 p0 x + w0
hold. Then,
4p (x + y ) = (1 +Since y =+ 4 ) 4px + (4 px + 4w)

( 0

).

x , the above equation can be reduced to
4py =

1

( + 4 ) 4py + (4 px + 4w)

( 0

).

( 0

)) .

Thus, we have
4py = 1

1

1

( +4 )((4 px + 4w)

Suppose that y serves as the invariable measure of value with respect to a change
from (p; w) to (p0 ; w0 ). Then, by De…nition 2, 4 px + 4w = 0 implies 4py = 0.
Then, by the above second equation, 0= 0 must hold.
0
Conversely, let there be ; 2 R+ such that py = px + w, p0 y = 0 p0 x +0w0, and = 0 hold. Then, the above last equation implies that 4py = 0 follows
from 4 px + 4w = 0. Thus, by De…nition 2, y serves as the invariable measureof value with respect to a change from (p; w) to (p0 ; w0 ).
16

Theorem 2 provides the necessary and su¢ cient condition for the standard commodity to serve as the invariable measure of value in general convex economies.
Remember that (resp. 0 ) represents the loss of pro…ts generated by operating the
production processwith one unit of labour input at the equilibrium price (p; w)00(resp. (p ; w )). Given this, the standard commodity can serve as the invariablemeasure of value with respect to a change from (p; w) to (p0 ; w0 ), if and only if theloss of pro…ts generated by producing this commodity does not change due to such achange in prices. Then, it obviously follows that the standard commodity can serveas the invariable measure of value, if and only if the loss of pro…ts generated by producing this commodity is invariable with respect to any change in equilibrium prices.
Therefore, the theorem suggests that the Ricardian invariable measure of value cannot exist in general convex economies, since the necessary and su¢ cient condition
can hardly be satis…ed. It follows that even the standard commodity cannot serveas the invariable measureof value in such economies.
onp 0 p 0 w 0l
. Then, the following corolLet P (p; w)
2 P j = arg max 0
p 0lary gives us a typical situation where the standard commodity serves as the invariable measure of value.
Corollary 1: Let r 5 1. Then, under A0~A4, for any economy hN ; P ; !i, let us
take any equilibrium prices (p; w) and (p0 ; w0 ), such that
2 P (p; w) \ P (p0 ; w0 )holds. Then, the standard commodity y serves as the invariable measure of value
with respect to a change from (p; w) to (p0 ; w0 ).
Proof. Note that
= ( 1; x ; x + y ) 2 P (p; w) \ P (p0 ; w0 ) implies py =00px + wand p y = 0 p0 x + w0
hold for = 0 = 0 . Then, by Theorem 2,
the desired result immediately follows.
Note that when the production set is represented by a simple Leontief technology, P = P(A;L) , then for any equilibrium price vectors (p; w), it follows that
2 P(A;L) (p; w). Thus, y can be the invariable measure of value with respect toany change in equilibrium prices. However, in this case, the change in equilibrium
prices is generated simply by a change in income distribution with no involvementof technical change, because P(A;L) does not allow producers any technical choice.
From now on, let us assume that the standard commodity y is selected as thenuméraire, though it would not serve as the invariable measure of value in general. Then, by de…nition, any price vector p 2 Rn+ of commodities is normalised aspy = 1. De…ne the set of price vectors measured by the standard commodity asMy(p; w) 2 Rn+1
j py = 1 . Given such a situation, let us analyse whether and+17

under what condition the linear distributional relationship between pro…t and wageis preserved in general convex economies.
Given P , let r 5 1. Then, P represents a non-increasing returns to scale production technology. Thus, we can obtain the following theorem.
Theorem 3: Given P with A0~A4, let r 5 1. Let (p0 ; w0 ) 2My be an equilibrium
price vector associated with a production activity 0 2 @P and a maximal pro…t rate0
= 0 such thatp0 0 = (1 +p0
5 (1 +0) p0 0 + w0 0l and0) p0 + w0 l for all

2 P.

Then, the linear function of income distribution, 0 = (1 w0 ) is derived by using
the standard commodity y , if and only if p0 y = 0 p0 x + w0 holds.
Proof. By de…nition of (p0 ; w0 ), it is generally true that p0 (x + y ) 5 (1 + 0 ) p0 x +w0 . If p0 (x + y ) < (1 + 0 ) p0 x + w0 , then 0 = (1 w0 ) does not hold. Indeed,from x + y = (1 + ) x , it follows that p0 y = p0 x < 0 p0 x + w0 . Since p0 y = 1,
then 0 > (1 w0 ) holds. Conversely, let p0 (x + y ) = (1 + 0 ) p0 x + w0 . Then,w0= 0 holds,
since p0 y = 1, p0 y = 1 = 0 p0 x + w0 . Thus, since p0 x = 1,p0 xwhich is equivalent to 0 = (1 w0 ).
The crucial point for the above analysis is whether e¢ cient production of thestandard commodity is feasible at the actual equilibrium prices. Note that this point
is irrelevant in single-product systems such as Leontief production economies and as
in Sra¤a (1960), where the standard commodity is always produced e¢ ciently at any
equilibrium price system. In contrast, our model of general convex economies allows
for the possibility of joint production as well as of technical choices, under which thestandard commodity may be produced only ine¢ ciently at some equilibrium pricesystem. In such a case, Theorem 3 suggests that the linearity of income distribution
no longer holds. Combined with Theorem 2 and Corollary 1, it implies that topreserve the value-invariance property and the linear income distribution propertyof the standard commodity, e¢ cient production of the standard commodity is anotherindispensable presumption, which is hidden in the model of single-product systems.
Furthermore, y = x , which is obtained from Theorem 1, corresponds to equation (2). Therefore, as is pointed out in footnote 8, the standard commodity de…ned
here also satis…es both the watershed and recurrence conditions (Schefold, 1986,1989). Indeed, here, the ‘watershed’proportion of means of production to labour is1, given normalisation of l = 1. This ratio is preserved not only by the industries
18

producing y , but also by all the industries producing the means of production, x ,which are necessary for the production of y . As is shown by Theorem 2, however,the standard commodity de…ned here cannot always serve as the invariable measureof value, even though both the conditions are satis…ed. This is a striking di¤erence
between our standard commodity and Sra¤a’s (1960).

4

Concluding Remarks

The purpose of this paper is to examine the applicability of the standard commodity
to more general cases than Sra¤a (1960) assumed. In particular, the paper examines the problems which were not addressed by Sra¤a (1960), but by Burmeister,
Samuelson, and others. That is, the invariable measure of value and the linearity
of income distribution are investigated in a model that allows for interdependence
among changes in income distribution, output level, and choice of techniques. To
this end, we construct general convex economies with non-increasing returns to scalein which the existence of the standard commodity is guaranteed. The implicationof Theorems 2 and 3 can be summarised as follows: as Sra¤a (1960, p. v) himself
suggested, in economies with Leontief production technology (that is, the case of constant returns to scale without joint production nor technical choice), the standard
commodity perfectly serves as the invariable measure of value and the linear relationof income distribution is preserved. Otherwise, the standard commodity cannot generally serve as the invariable measure of value and the linearity of income distribution
cannot be preserved. This is because, unlike in the case of Leontief economies, e¢ cient production of the standard commodity is not necessarily possible at the actual
equilibrium prices.
According to Kurz (2012, p. 1556), the existence ofin formula (4) con…rms thecorrectness of Ricardo’s conjecture: ‘The great questions of Rent, Wages, and Pro…ts
must be explained by the proportions in which the whole produce is divided between
landlords, capitalists, and labourers, and which are not essentially connected with thedoctrine of value’(Ricardo, 1952A, p. 194). It implies that the rate of pro…t can beobtained without knowing the structure of prices. However, our results demonstratethat Ricardo’s conjecture is not generally valid. The conjecture is valid only under
the same assumptions as imposed on Theorem 3, namely, e¢ cient production of thestandard commodity is feasible at the actual equilibrium prices. Otherwise, the rateof pro…t is not independent of the price structure.

19

5

Appendix

5.1

Proof of Theorem 1

Proof of Theorem 1: Given P ( l = 1) which is convex by r 5 1, let P l =1 be theminimal closed convex cone containing P ( l = 1). By de…nition, P l =1 is a closed
convex conen with P l =1P( l = 1) = P (o l = 1). If r = 1, P l =1 = P . Given P l =1 , let
P l =12 P l =1 j i=1;:::;n i = 1 . Let F : P l =1 ! R+ be such that for each
2 P l =1 , F ( ) = mini=1;:::;n

i
i

where

0
+1 if

i
i

if i = 0=0 and i > 0.
iThis mapping is continuous and well-de…ned by A1. Note that, by A2 and A4,0there exists 0 2 @P ( l = 1) such that b 0 > 0. Hence, for P
0 2 @P l =1 ,i=1;:::;nFP

0i=1;:::;n0i> 0. This implies sup 2P

l =1

i

F ( ) > 0. Suppose that sup 2P

l =1

F( )=

+1. Then, there exists a sequence k
P l =1 such that k !
with limk!+1 F k =F ( ) = sup 2P =1 F ( ). By de…nition of F , F ( ) = +1 implies that=l
( l; 0; ) for some l = 0 and some
> 0. Since P l =1 is closed,
2 P l =1 .
By construction, P l =1 satis…es A1, which is a contradiction of
2 P l =1 . Thus,2
sup 2P =1 F ( ) < +1. Then, sup 2P =1 F ( ) = max 2P =1 F ( ). Letlllarg max 2P =1 F ( ). Then, by the cone property,
2 P ( l = 1) and2lll
arg max 2P ( l =1) F ( ). Hence, without loss of generality, let
2 arg max 2P ( l =1) F ( ).
Then,
2 @P ( l = 1). Since there exists 0 2 @P ( l = 1) such that F ( 0 ) > 0,
> 0.
max 2P ( l =1) F ( ) > 0 holds, which implies that
F ( ) j ( 1;; ) 2 P ( l = 1)g. Then, V is a closed conDe…ne Vfvex set with V \Rn++ = ?. Then, there exists p 2 Rn+ n f0g such that p [F( ) ]5n0 for all 2 P ( l = 1) and p z > 0 for all z 2 R++ . This implies that if there exists i 2 f1; : : : ; ng with i > F ( ), then pi = 0. By p 2 Rn+ n f0g, there existsiF ( ) ] = 0. Hence, p
i 2 f1; : : : ; ng with i = F ( ) and pi > 0. Thus, p [iis a supporting vector of
2 @P ( l = 1). Let
2 P ( l = 1) be such that for each2ii 2 f1; : : : ; ng with> F ( ), ( i ; i ) 2 R++ with i
F ( ). (Note thatiisuch a construction is possible by A3.) Furthermore, for each i 2 f1; : : : ; ng withi
= F ( ), ( i ; i ) ( i ; i ). Then, by construction, p [F( )] = 0,iwhich implies that
2 @P ( l = 1). Note that
> 0 and
=F( )
.
0
Let y
. Remember that there exists
2 @P ( l = 1) such that20

b 0 > 0 and F ( 0 ) > 0, which implies F ( ) = F ( 0 ) > 1. Therefore, y > 0. Then,
there exists a positive number> 0 such that x = y for x> 0. By
De…nition 1, y > 0 is a standard commodity of the economy hN ; P ; !i.

5.2

A note on the unique existence of the standard commodity

As discussed in section 3.2, if the general production set P is properly restricted,
then the unique existence of the standard commodity can be shown by applying thenon-linear Frobenius Theorem à la Fujimoto (1979, 1980). Let us brie‡y see whatkinds of restrictions are required.
Firstly, let us introduce the following additional assumption:
andAssumption 5 (A5). For any 2 @P ( l = 1) and any 0 2 Rn+ with 0
0
0
0
0
0
0
n
0
, there exists ( l ; ) 2 R
R+ with l = 1 and ( l ;; ) 2 @P , suchthat 0; and there exists at least one commodity i = 1; : : : ; n with 0i = i , suchthat 0i > i .
A5 is a generalisation of the indecomposability assumption formulated by Fujimoto(1979, 1980) in the context of the non-linear Frobenius Theorem. The indecomposability of the economy implies that any commodity is directly and/or indirectly used
as an input in producing other commodities. Though A5 simply focuses on the
production points with one unit of labour input, such an interpretation is sensible.
Indeed, the indecomposability property stipulated by A5 is preserved not only inthe case of production points in @P ( l = 1), but also in the case of any production points in @P ( l = k) with any k > 0. This is guaranteed by the property ofhomogeneity of degree r > 0 of each production set assumed by A4.
Secondly, for each commodity i = 1; : : : ; n, let Hi : Rn+ ! R+ be a continuousand quasi-concave function, such that for each x 2 Rn+ , Hi (x) 2 R+ with Hi (0) = 0
and Hi (x) = Hi (x0 ) if x = x0 . Moreover, for each commodity i = 1; : : : ; n, Hi
is homogenous of degree r where 0 < r. Let H(H1 ; : : : ; Hn ) be such that forneach x 2 R+ , H (x) = (H1 (x) ; : : : ; Hn (x)). Suppose that for any x 2 Rn+ , x
0
implies H (x)0, and for any x; x0
0 with x
x0 but xx0 , it follows thatH (x)H (x0 ) and there exists at least one commodity i = 1; : : : ; n with xi = x0i
such that Hi (x) > Hi (x0 ). Let L : Rn+ ! R+ , which is a continuous and strongly
increasing function, such that for each x 2 Rn+ , L (x) 2 R+ with L (0) = 0 and
L (x) > 0 for any x 0. Furthermore, for any x 0 and any k > 0, L (kx) = kL (x)21holds. Then, given a pair of (H; L), we can de…ne a production set P(H;L) as
P(H;L)

2R

Rn

Rn+ j 9x 2 Rn+ :5 ( L (x) ; x; H (x)) .

This P(H;L) satis…es the above (A0)~(A5) since H is homogenous of degree r. Thus,
if r < 1, then P(H;L) exhibits a production set of decreasing returns to scale.
When the production set P is speci…ed by P = P(H;L) , it is possible to show thatthe standard commodity uniquely exists in the economy N ; P(H;L): ; ! by means of
the non-linear Frobenius Theorem à la Fujimoto (1979, 1980).
Note that it is possible to show that the production set P(Ak ;Lk )discussed
k=1;:::;min Example 2 of Section 3.2 satis…es (A0)~(A5) and can be represented by a pro…le(H; L) as P(H;L) = P(Ak ;Lk ).
k=1;:::;m

It is also possible to …nd a von Neumann production set P(A;B;L) , which is represented by a pro…le (H; L) with (A0)~(A5).
Example A1: Let us take a von Neumann technology (A; B; L) with m = n,whose associated P(A;B;L) satis…es (A0)~(A5). Then, for the linear mapping A,
1( )Rm
the inverse mapping A 1 : Rn+
x 2 Rm= Ax+ j+ is given by A
1nfor each 2 R+ . Then, since A is single-valued, A is a linear mapping by Berge(1963, p. 135, Theorem 7.2). Moreover, assume that A 1 is single-valued, which
implies that n = m and A 1 is the inverse matrix of A. Then, de…ne H B A 1
and L
L A 1 . Thus, for each 2 SP(A;B;L) , H ( ) = and L ( ) = l hold.
This implies that P(H;L) = P(A;B;L) . Then, if B 6= I, P(H;L) represents a production
set where joint production is possible.
Moreover, let the matrix (B A) not have the inverse matrix. In this case, for
any c 0, Rank (B A; c) < n holds, so that there are in…nite number of solutionsx
0 satisfying (B A) x = c. Then, for a given net output vector c, the set ofalternative (e¢ cient) techniques to produce c as a net output is given by
Rn j 9x 2 Rn+ : (B@P (A;B;L) (c) = ( Lx; Ax) 2 R

A) x = c .

This is equivalent to the following set:
@P (H;L) (c) = ( L ( ) ;Rn j 9 2 Rn+ : H ( )

)2R

=c ;

by P(H;L) = P(A;B;L) . Thus, the system (H; L) can represent economies with joint
production as well as those with alternative production techniques.
Instead, if A and B are n m matrices with n < m, then we cannot de…ne
H as a single-valued mapping, though n < m is a natural case in that the number22

of alternative production processes is su¢ ciently large. Indeed, if n < m, thenRank (A) < m. Thus, A 1 : Rn+
Rm+ , the inverse of the linear mapping A, is not
single-valued. Thus, if for some x; x0 2 A 1 ( ), Bx 6= Bx0 holds, then we cannot
de…ne H as a single-valued mapping. Further, if for some x; x0 2 A 1 ( ), Lx 6= Lx0
holds, then we cannot de…ne L as a single-valued mapping.
For instance, let
A=

1 0 1
0 1 1
0

;B =

0 2 1
2 0 2
0

; L = (1; 1; 1) , and consider

=

2
1

.

1
1
2
1
= Ax0 , but l = Lx = 3 >Then, for x = @ 1 A and x0 = @ 0 A, Ax =01
2
1
2 = Lx0 = 0l , = Bx == Bx0 = 0 , and b = (B A) x =4401= (B A) x0 = b 0 . This numerical example has the property
3
3that Bx 6= Bx0 and Lx 6= Lx0 hold for some x; x0 2 A 1 ( ).

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to changes in income distribution?, Cambridge Journal of Economics, vol. 30, 313–9.
Bellino, E. 2004. On Sra¤a’s standard commodity, Cambridge Journal of Economics,
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Berge, C. 1963. Topological Spaces: Including a Treatment of Multi-Valued Functions, Vector Spaces and Convexity, Edinburg, Oliver & Boyd, reprinted in 1997,
New York, Dover.
Burmeister, E. 1968. On a theorem of Sra¤a, Economica, vol. 35, 83–7.
Burmeister, E. 1975. A comment on ‘This age of Leontief . . . and who?’, Journal of
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Burmeister, E. 1977. The irrelevance of Sra¤a’s analysis without constant returns toscale, Journal of Economic Literature, vol. 15, 68–70.
Burmeister, E. 1980. Capital Theory and Dynamics, Cambridge, Cambridge University Press.
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