Toward Closing a Century Old Debate Transformation of Values into Prices in a World of Heterogeneous Labors

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by  Ajit Chaudhury

Rethinking Marxism
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Toward Closing a Century-Old Debate:
Transformation of Values into Prices in a World of
Heterogeneous Labors
Ajit Chaudhury
To cite this article: Ajit Chaudhury (1998) Toward Closing a Century-Old Debate: Transformation
of Values into Prices in a World of Heterogeneous Labors, Rethinking Marxism, 10:1, 120-132,
DOI: 10.1080/08935699808685521
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Rethinking MARXISM Volume 10, Number 1 (Spring 1998)
Rernarx
Toward Closing a Century-Old Debate:
Transformation of Values into Prices in a World
of Heterogeneous Labors
Ajit Chaudhury
This essay deals with primarily a technical problem in Marx’s algebra: the problem
of transforming values into prices in a world of heterogeneous labors. But more is at
issue than simply this algebra. The essay foregrounds two competing methods of
reading Marx’s Cupirul: one inscribed in an essentialist philosophy searching for a
“source” of profit (and exploitation) outside the terrain of exchange, and the other
informed by the idea of overdetermination, jettisoning the notion of source in all its
variants. The concluding section of the essay shuns algebra and highlights these points
implicit in the “transformation problem.”
The issue of transforming values into prices has remained a problem ever since
the publication of Marx’s Cupirul. In his solution Marx claims that (1) the quantity
of surplus-value = the quantity of profit (for the economy as a whole), and (2) the
sum of prices = the sum of values. Critics of Marx point out that the two equalities
do not generally hold. It is only recently Wolff, Callari, and Roberts (1984) have
provided a solution to the transformation problem that satisfies the two equalities;
all the earlier solutions required restrictive conditions to ensure these equalities.
What distances their work from others’ is their conceptualization of the Marxian
notion of values. They invoke Althusser’s notion of overdetermination and point
out that values and prices are overdetermined by each other; in other words, prices,
Remarx 121
too, constitute values. This entails a fundamental rethinking of the problem of
value. Conven’tional approaches do presume that values can be solved for inde-
pendent of prices, whereas Wolff, Callari, and Roberts view values as dependent
on prices.
In this paper I argue that Wolff, Callari, and Roberts do not work out the full con-
sequences of their imaginative argument. While values in their system are over-
determined by prices, they chose to abstract from the issue of skill differences among
workers to focus on the traditional, narrow specification of the transformation prob-
lem. However, this discursive strategy does not leave enough space for analysis of
some of the interesting implications of overdetermination in the context of value theory:
specifically, how prices overdetermine the very notion of labor-time itself. This essay
extends their argument to the case of heterogeneous labors and shows how they can
be converted into homogeneous (equal, abstract) labor in terms of a notion of abstract
labor-time overdetermined by prices. As we shall see, Wolff, Callari, and Roberts’s
(Marx’s) results carry over also in this model: the mass of profit is equal to the mass
of surplus-value and the sum of prices is equal to the sum of values. I counterpose this
solution to that proposed by others- in particular, by Steedman (1989)-which does
not satisfy these equalities.
I present the solution in terms of elementary algebra in order to make it as acces-
sible as possible for nonspecialists. However, the central focus remains the implica-
tions of abstract labor in the context of Marx’s value theory. While for Steedman
abstract labor is just “a pair of words,” I argue that his (mis)conception presumably
follows from his premise: that Marx must prove that surplus (abstract) labor is the
source of profit just as he must prove that values are the underlying determinant of
prices. Since in Steedman’s solution values and prices are determined in separate
systems of equations and therefore there is no significant sense in which values de-
termine prices, he concludes that “Marx’s concept of abstract labor is of little or no
use, other than as a metaphor” (1989, 14). On the contrary, our commitment to the
notion of overdetermination rejects the very notion of a source, the idea of prior
determination.
We therefore find it quite in keeping with Marx’s spirit to write the expression for
abstract labor (as also the other quantities in value equations) in price of production
terms and proceed to follow through its logical consequences. It then turns out that
Wolff, Callari, and Roberts’s argument carries over into a world of heterogeneous
labor. More important, it reaffirms a method of reading Capital informed by the idea
of overdetermination with additional (con)textual support from Marx’s writings in
the context of abstract labor.
The Traditional Approach to the TransformationP roblem
In the traditional approach, value is labor-time embodied in one unit of the com-
modity. If vj represents the value of thejth commodity, then
122 Chaudhury
(1) vj = Cviaij+ aoj
where aiji s the amount of the ith commodity required on average to produce one unit
of the jth commodity and aoji s the (implicitly homogeneous) labor required to pro-
duce one unit of thejth commodity.
Let bi represent the amount of the ith commodity required to produce one unit of
labor-power. Then the value of one unit of labor-power-that is, its reproduction cost
in labor hours-is Cvibi.T he value of thejth commodity can then be broken into its
constituent parts as
(2) vj = C viaij+ aoj C vibi + aoj (1 - C vibJ
or
(3) vj = C vjcjj + sj
where
c.. = a..+ b.a .
‘J IJ I OJ
Sj = aoj(1 - C vibj)
= surplus-value per unit of the jth commodity.
The representative price equation is
+
(4) pj = (1 r) Epicjj
(5) = cpicij+ Pj
where
Pj = r C pjcij
= profit per unit of the jth commodity.
Equation (5) determines r and pj. But prices here are determined only up to ratios.
To fix prices we set up the normalizing equation
(6) C p,Xj = C v,Xj
where X, is total production of thejth commodity.
However, Marx’s statements make it clear that an acceptable solution of the trans-
formation problem requires the simultaneous fulfillment of the two equalities
and
(7) C PTj = C STj.
It is now well known that (6) and (7) hold simultaneously only under very restnc-
tive conditions. Note that
(8) I: PJj - C S,Xj = C (pi - vi) (Xi - C ~i,Xj).
Remarx 123
Therefore, if pi # virt hen
c PJi - c SJi = 0
if
In this case, (8) reduces to
c PJi - c SJi = c (pi - Vi) mi = k c (pi - Vi)Xi= 0
(by the normalizing condition [6]).
It is interesting to examine the economic implications of the condition Xi- cCi,Xj
= Hi.N ote that Xi is the gross production level of the ith commodity, while cCiJj
shows the aggregate quantity of the ith commodity used up as an input (either as
constant or variable capital). Therefore, Xi - cCi,Xj shows the aggregate surplus of
the ith commodity, the amount available for investment. If we assume that the entire
surplus is reinvested so that the economy experiences maximal growth, the aggre-
gate growth rate (equal to the rate of profit) is here given by
xi - LiJj
=i,Xj
which also expresses the maximal rate of growth for the ith sector. It can be verified
that the condition Xi- CciJj = kXi implies
Xi -c~i,Xj k
xci,Xj = I-k
which states that all sectors must attain the same maximal growth rate w. ,he
economy is on the so-called maximal balanced growth path. Morishima (1973) pointed
out that the two Marxian equalities do hold if the economy is on the maximal bal-
anced growth path. Obviously, this is a very restrictive condition.
Wolff, Callari, and Roberts’s Solution of the Transformation Problem
Wolff, Callari, and Roberts (1984) point out that the two much discussed Marxian
equalities are not falsifiable propositions, but are definitionally true. What is at stake
is not mathematics, but a proper conceptualization of the problem. They establish
this by positing a distinction between Marx’s method and that of Ricardo-specifi-
cally between their value theories. What distances them from the others is their argu-
ment that values are overdetermined (i.e., constituted) by prices.
In their construction, Marx’s value system and the price system are two alterna-
tive but jointly necessary modes of accounting in a capitalist economy: one from the
standpoint of production (let us say, by the laborer) and the other from the stand-
point of circulation (let us say, by the capitalist).’ Both the laborer and the capitalist
124 Chaudhury
objectively face a situation in which the reproduction of the system requires the use
of commodities (functioning as constant and variable capital) that they can measure
only in terms of the objectively given market (production) prices. However, they
employ different accounting frames for the measurement of the gross outputs. While
the capitalist evaluates them as the sum of constant capital and variable capital (costs
+
measured at market prices) marked up by (1 r), the laborer employs a different
accounting system. For the laborer, the gross product is the sum of constant capi-
tal-expressed as labor-time by means of production prices-plus the quantity of
abstract labor time he adds to it. Marx’s value magnitudes emerge as the weights
attached to unit commodities in this accounting system.
Therefore, in Wolff, Callari, and Roberts’s construction
(10) vj = cpiaii + a,j.
Like the traditional value equation (l), equation (10) can also be written as
(1 1) Vj = epicii+ sj
where cii and Sj retain their previous meanings with the difference that Sjs are now to
be specified in terms of production prices (with Sj = aoj(l- Cpibi))w hereas in the
conventional approach outlined earlier in the essay, Sj is independent of prices. As
in the traditional approach the representative price equation is
+
pj = (1 r)cpicii
+
(12) = epicii Pi
where Pi, as before, is profit per unit of the commodity. It immediately follows from
equations ( 11 ) and (12 ) that
c
CPJj = VJj
implies
c
PJj = )’ SC
Thus, the two Marxian equalities follow definitionally in Wolff, Callari, and
Roberts’s solution of the transformation problem. Indeed, in Marx’s original solu-
tion the point of departure is the equation
( 13) ~ ’ j= Cpi~ijX+j S j’
from which he derives
1. The language of dual “standpoints” is my own formulation of Wolff, Callari, and Roberts’s ap-
proach. But I believe that this is a convenient way of presenting the problem to begin with. Later, par-
ticularly in the concluding part of the essay, we shall see the other implications of this accounting sys-
tem. We shall discover that the laborers are the objects of this accounting, not the subjects.
Remarx 125
xS’j
(14) r=-
CCPiCiJj
and
(15) p,Xj = (1 + r) Cpjcj7j
Starting from the value equation (13), in just two successive steps [( 14) and (15)]
Marx derives the rate of profit and the prices. The algebra of the solution of the
transformation problem is as simple as that. In the light of Wolff, Callari, and Roberts’s
solution of the transformation problem we understand that the issue of transforming
values into prices is, after all, not a problem. Only a group of Marxists-and bour-
geois economists-have transformed this into a problem. For Sweezy (1991), Marx’s
original point of departure4quation (13 )-is represented as
(16) W,= (Cj+ 5)+ S)
where Wj = gross product of thejth industry in value terms; Cj = total constant capi-
5
tal used in the jth industry at value terms; = total variable capital used in the jth
industry in value terms; and S) = surplus-value generated in the jth industry.
From (16), Marx derives r as
CS;
(17) r= x(cj+ 5)’
Given r, prices are determined as
(18) p,Xj = (1 + r) (C, + 5).
Sweezy shares Bortkiewicz’s objection that (18) is not a valid equation inasmuch as
5
C, and are not expressed in terms of prices.
+
In light of Wolff, Callari, and Roberts’s solution, we understand that each (Cj
5)
is already expressed in terms of prices in the value equation (16) and that there-
fore there is no need for a further transformation (into price terms) in the price equa-
tions. Properly understood, Marx’s algebra is simple and correct. Only a group of
+ 5)
Marxists do not understand this because they express (C, in value terms and not
at production prices.
Wolff, Callari, and Roberts in a World of Heterogeneous Labors
In this section, I push forward Wolff, Callari, and Roberts’s argument to the case
of heterogeneous labors. In other words, the focus will be on the abstract labor is-
sue-the determination and properties of value and surplus-value in the context of
heterogeneous labor. We may recall that they abstract from this issue to focus on the
traditional narrow specification of the transformation problem. I press forward their
argument and point out that the very representation of abstract labor time-its con-
cept-presupposes exchange-that is, prices.
126 Chaudhury
In fact, that is how labor is represented in classical political economy-presup-
posing and invoking the categories of exchange. Ricardo, for example, is aware of
the implications of the heterogeneity of labor; he is aware that labor needs to be rep-
resented in terms of a common conceptual (i.e., homogeneous) unit. This, however,
does not pose much of a problem in Ricardo’s model inasmuch as he could assume,
on the basis of empirical observations, that relative wages were fixed at levels ex-
pressing the comparative skills of different types of labor in terms of their real wage.
Thus, Ricardo’s value theory as well as his distribution theory presuppose real wages;
the skills of the laborers and therefore the value they create in a given period of time
are in proportion to their relative wages.
The estimation in which different qualities of labor are held comes soon to be ad-
justed in the market with sufficient precision for all practical purposes, and depends
much on the comparative skiil of the laborer and intensity of the labor performed. The
scale, when once formed, is liable to little variation. If a day’s labor of a working jew-
eler be more valuable than a day’s labor of a common laborer it has long ago been
adjusted and placed in its proper position in the scale of values. (Ricardo 191 1, 11)
Smith treated the problem of the representation of labor in the same spirit. He wrote:
“It is not easy to find any accurate measure either of hardship or ingenuity . . . It is
adjusted, however, not by any accurate measure, but by the haggling and the bar-
gaining of the market” (Smith 1963, quoted from Ricardo 1911 , 12).
In contrast, one current marxist tradition favors a cost of production approach to
skilled labor and its value-creating power. Presuming that one unit of unskilled labor
creates one unit of value, this approach presents each unit of skilled labor as creating
k units of value, where the values of kj > 1 are endogeneously determined by the
production requirement of each type of skilled labor.2 Besides being economistic,
this approach misses the simple point that even completely unskilled concrete labors
in different occupations need to rendered equivalent, in some way, before labor quan-
tities can be defined and compared. This was pointed out by Rubin (1973, translated
from 1928 edition) and stressed more recently by Bowles and Gintis (1977), who
argue that some of the observed wage differentials (for example, those due to sex
and race differences) cannot be explained in terms of differences in skill. Steedman
(1989) notes these points and argues that Marx’s theory requires that the vector of
abstract labor per unit of concrete labor be proportional to the wage-rate vector. He
also points out that Marx was well acquainted with classical writings in which the
concept of a “quantity of labor” was explicitly based on relative wages.
So let us make Marx classical-“Ricardian”-as far as the problem of represen-
tation of labor goes. Labor is homogenized in terms of relative wages; without mak-
ing use of market categories-here, relative wages-one cannot represent labor-time.
In other words, prices constitute not only values (as in Wolff, Callari, and Roberts)
but also labor-time.
2. This is the way the problem is formulated in standard textbooks (for example, Sweezy [1991]).
Remarx 127
Thus, labor becomes homogeneous if different types of labor are weighted in
terms of their relative market wages. The theory does not explain how these rela-
tive wages are determined; they are exogenous to the system. Now each unit of
homogeneous labor definitionally produces surplus-value at the same rate (say, s);
otherwise, the definition of homogeneous labor is contradicted. But this does not
give us any information about the value of s; we only know that s is conceptually
the same for each unit of homogeneous labor. Therefore, the quantity of surplus-
value also remains unknown. As we shall see, there is no way of knowing value,
surplus-value, and the rate of exploitation without knowing the prices because prices
constitute them.
Let us pose the problem algebraically. I consider, for simplicity, only two indus-
tries producing two commodities (XiXz) with values (vivz)a nd production prices
(pip2).T he first industry employs labor of type 1 (unskilled) labor and the second
industry employs labor of type 2 (skilled) labor. Suppose (up2)a nd (blbz)a re the
subsistence baskets for one unit of unskilled and skilled labor, respectively. Retain-
ing the previous meanings aVa nd uoj( ij= 1,2), it follows that C saolp,aia nd Z saogibi
are the surplus-values produced in the two industries, s being unknown. Similarly,
the total abstract labor time necessary to produce one unit of each commodity is given
by X (1 + s) aolpiaia nd C ( 1 + s) aogibir espectively. In other words, production prices
constitute (overdetermine) abstract labor time.
Let us now turn to the value and price equations. Given the above, we can write
the value equations as
(19) vi =PI a//+ PZ~Z+/ ( 1 + s)%/piat
(20) vz =PI aiz + P ~ ~+Z (1Z + s)&ogibi.
Rearranging these equations we get
(21) vi = PI dil + ~2 d21 + S ~ OQIi ai
(22) vz = PI d,2 + ~2 d22 + S~OQZi bi
where
(23) 41= a11 + a01
(24) d2, = a2i + QOi a2
(25) di2 = a12 + ao2b1
(26) 4 2= a22 + soh.
Similarly, the price equations can be written as
(27) Pi = (1 + r) (Pi dil + P2 dZi)
(28) Pz = (1 + r) (Pi di2 + p2 42).
Equations (27) and (28) determine pI/pza nd r.
128 Chaudhury
It is important to stress that Wolff, Callari, and Roberts’s (Marx’s) equalities hold
even when their analysis is extended to include heterogeneous labors. If we insist
with Marx that the sum of values equals the sum of prices, then CvJ;= Xp,Xi,w here
Xir efers, as before, to the total output produced in the ith industry. From this, as before,
it follows that
(29) rCCpjdiJ, = s [Cpjaiao+lX XIp ;biao2X2]
or, the mass of profit equals the mass of surplus-value. Thus, the two much-discussed
Marxian equalities are fulfilled. Alternatively, one can start from equation (29)
to get a value of s that satisfies it. The other equality XvjXj= CpJi then follows
definitionally, as it involves adding the same quantity (the sum of total means of
production and means of subsistence at pI/p2)t o both sides of (29).
We conclude this section by showing how the system determines the value mag-
nitudes. We note that price equations (27) and (28) determine rand relative prices
@,/p2);e quation (29) can then determines. The two value equations then determine
vI and v2 in terms of p2.I n a system with more than two commodities, the last com-
modity can be considered to be gold, the money commodity; in this case, assigning
gold a (money) price of 1 allows the determination of all values, prices and quanti-
ties of abstract labor as absolute magnitudes denominated in units of gold. One could,
of course, proceed further to define a relation between gold units and some measure
of actual labor performed so that all variables could be expressed in terms of labor-
time, but the above suffices for our purposes here.
So, the task of transforming values into prices is not much of a problem: values,
prices, the rate of profit, and the rate of exploitation are such that the two much-
debated equalities follow (the sum of prices equals the sum of values and the mass of
profit equals the mass of surplus-value). Old Marx was right. Neither is the task of
aggregating concrete labors into abstract labor very problematic: aggregate them via
their relative wages.
Why, then, have so many wise men foundered on these two rocks over a century?
Because all of them are prisoners of a philosophy-ssentialism-that dies hard. They
all share the primordial religious belief that there is an “origin” and look for a source
of profit outside the terrain of exchange, a “value system” untouched by the vices of
exchange. And therefore they specify the value equations incorrectly, without any
reference to prices. Their algebra gets mixed up with their ideology, and the task of
transforming values into prices itself is transformed into a problem.
Perhaps it is Steedman who, among Marx’s critics, has best understood what is at
issue in the transformation problem. He has correctly appreciated that Marx aggre-
gates different kinds of concrete labors via their relative wages and therefore the traces
of exchange are always already there in the value equations. Only he does not follow
through the mathematical and philosophical consequences of his profound observa-
tion: that means of production are to be measured at gold prices (as distinct from
gold values) and the whole baggage of essentialism-with its age-old belief in ori-
gin-has to be thrown into the ocean. And that is precisely what we have done in