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International Review of Applied Economics
Vol. 22, No. 6, November 2008, 707–724

Price–value deviations: further evidence from input–output
data of Japan
Lefteris Tsoulfidis*
University of Macedonia, Thessaloniki, Greece
of Applied

This paper subjects to empirical testing the labour theory of value using input–output
data from the economy of Japan for the years 1970, 1975, 1980, 1985 and 1990. The
results of the analysis show that labour values and prices of production are extremely
good approximations to market prices. In fact, the proximity of prices of production
to market prices is closer than that of labour values, a result which suggests that prices
of production constitute more concrete centres of gravitation for market prices.
Furthermore, we find that prices of production change as a result of variations in
income distribution more often than not in a monotonic way and that in fewer cases
they display curvatures, which may even reverse the order between prices of production
and values.
Keywords: values; prices of production; income distribution; input–output tables
JEL classifications: B12, B14, C67, D57

1. Introduction
The objective of this paper is to subject to empirical testing the labour theory of value and
the relationship between labour values and prices of production using input–output data
from the economy of Japan for the years 1970, 1975, 1980, 1985 and 1990. The central
claim that this paper makes is that labour values and prices of production are centres of
gravitation for market prices. This result derives from the fact that the three kinds of prices
are close to each other as this closeness can be ascertained by a number of relevant statistics. Furthermore, by using a polynomial expression suggested by Steedman (1999) it is
shown that under certain realistic conditions prices of production and values are strictly
connected to each other and that prices of production can be broken down to labour values
multiplied by a polynomial expression, where the effects of higher order terms of this
expression become progressively weaker. Lastly, this paper examines the movement of
prices of production induced by changes in income distribution in order to investigate
the extent to which the feedback effects of revaluation of inputs (capital) according to the
new prices of production may lead to complex trajectories in prices of production of a real
The remainder of the paper is organized as follows: section 2 briefly reviews the relevant
literature and puts our discussion into the appropriate theoretical context. Section 3 critically
evaluates the hitherto empirical analysis on the question of price–value deviations. Section
4 sets up the linear model of production with circulating capital and shows the details of the
*Email: lnt@uom.gr
ISSN 0269-2171 print/ISSN 1465-3486 online
© 2008 Taylor & Francis
DOI: 10.1080/02692170802407668


L. Tsoulfidis

methods of estimation. Section 5, presents the results of the empirical analysis using actual
input–output data for five benchmark years. Section 6 evaluates the results of the analysis.
Finally, section 7 concludes with some remarks about future possible extensions of the
2. Theoretical context and issues
We know that for classical economists the relative labour content of commodities is the
main determinant of relative prices. David Ricardo in particular (1821, ch. 1) argued that
the presence of fixed capital, turnover time, and variations in income distribution add
further detail to the principal relationship. Karl Marx also accepted this proposition, arguing
however that the passage from the labour content of commodities to market prices requires
a series of mediating steps. First of all, he differentiates between value and exchange value.
More specifically, value is defined as the socially necessary abstract labour time needed for
the production of a commodity. Since commodities are produced by different capitals
(firms) that form an industry, the social (or average) value is defined as the ratio of the
industry’s total abstract labour time to the number of commodities produced. Naturally,
individual and social values will differ and this difference will be responsible for the accrual
of higher profits to the more efficient capitals, that is, to those capitals with an individual
value lower than the industry’s average. The monetary expression of value, which Marx
(1867, chs 1 and 2) calls direct price, is defined as the ratio of the value of the commodity
to the value of money commodity (gold). The average or the social direct price of an industry is a sort of centre of gravity for the market price. Direct prices form, in very general
terms, the basis of the analysis in volumes I and II of Capital, while in volume III Marx
introduces the notion of prices of production, as more concrete centres of gravitation for
market prices. Marx also tried to show (although not completely) that prices of production
originate in and, therefore, are derived from direct prices which means that the two types of
prices are intrinsically related, a relationship that stems from the operation of competition
between industries. In fact, prices of production are derived as a generalization of values
(direct prices). More specifically, starting from the hypothetical case of zero rate of profit
where prices of production coincide with values and as the rate of profit increases to its
theoretical maximum (rmax) the deviation of the prices of production from values is
expected, in general, to rise. Figuratively speaking the movement of the three kinds of prices
over time is portrayed in Figure 1a, while Figure 1b displays the general relation between
prices of production and direct prices, when the rate of profit varies from zero to its theoretical maximum.
Marx’s analysis would not be complete if it stopped at the average prices of production
as regulators of market prices. We know that descending to an even lower level of abstraction (Marx 1867–94: vol. III, chs. 8–10, also pp. 640–737) from the average magnitudes he
arrives at the regulating magnitudes, which may not necessarily coincide with the average
ones. A characteristic example is in agricultural production, where the price of production
of the marginal (not the average) land really constitutes the immediate centre of gravity of
market prices. The idea of marginal conditions are generalized in Marx by discussing for
example the case of manufacturing, where the regulating price of production will in general
be different from the average. In fact, the regulating conditions and, therefore, the regulating
prices of production are identified with those firms or capitals of an industry, where there is
acceleration or deceleration of capital accumulation. This aspect of Marx’s work has
received very little attention precisely because the concept of regulating capital and the
associated notions of regulating value and price of production are extremely difficult to
Figure 1. Labour values, prices of production and market prices.

International Review of Applied Economics


price of




price of

price of





Figure 1.


relative rate of profit


Labour values, prices of production and market prices.

operationalize.1 As a consequence, the empirical analysis up until now has been restricted
exclusively to average direct prices and average prices of production, and although these are
very good approximations to market prices they are nevertheless not the most appropriate
in Marxian theoretical terms.
3. Empirical research on price value deviations
The empirical research on the relationship between labour values (direct prices) and market
prices using input–output data for the economies of the USA, former Yugoslavia, Italy,
England and Greece has shown that the two sets of prices are very close to each other. The
same is also true for prices of production and market prices. More specifically, Shaikh
(1984) reports that the deviations of direct prices or prices of production from market prices
are in the order of 17–19% for Italy, while using crude input–output data for the USA and
a circulating capital model he finds that the order of price–value deviations is in the range
of 20–25%. This original research was continued by Ochoa (1984, 1989) by using Shaikh’s
methodology and expanding it to include the matrices of fixed and circulating capital
advanced, found that the mean absolute deviation (MAD)2 of direct prices (i.e. prices
proportional to values) and market prices for a set of six benchmark input–output tables
spanning the period 1947–1972 is 12.2%, the MAD of market prices and prices of production is 13.6%, while the MAD of direct prices and prices of production is almost 17%,
results which render the so-called ‘transformation problem’ of limited empirical significance. The research for the economies of the UK (Cockshott, Cottrell, and Michaelson
1995, Cockshott and Cottrell 1997) former Yugoslavia (Petrovic 1987) and Greece
(Tsoulfidis and Maniatis 2002) gave results similar to those derived for the US economy.
More recently, Zachariah (2006) using input–output data from a number of OECD countries
finds support for the hypothesis that labour values and prices of production constitute very
good attractors for market prices.
Negative results for the labour theory of value are reported in the study by Steedman
and Tomkins (1998) for the UK, Ireland and for a sample of Australian regions whose
price–value deviations were much higher than those usually found. This study, however,
is not without its problems which may account for its rather negative results. First,
Steedman and Tomkins (1998) do not include the matrix of depreciation coefficients


L. Tsoulfidis

presumably because of the lack of appropriate data. We know, however, that depreciation
often significantly affects the estimations of values; for example, Chilcote’s (1997) analysis of the US economy finds that ‘the inclusion of depreciation improves the predictive
powers of the estimators by 20% to 25%’ (Chilcote 1997, 166). Second, Steedman and
Tomkins (1998) do not address the issue of widespread self-employment especially for
Ireland and the agricultural regions of Australia and we presume that they do not adjust
the labour input coefficients for the number of self-employed. Such an adjustment, in our
view is critical to a theory that is based so crucially on labour time. As a consequence,
their research and the results they derive (for Ireland which in the year 1985 certainly
does not possess the features of a well-developed capitalist country and to various
Australian mainly agricultural regions3) cannot be used directly as evidence for or against
the idea of proximity of labour values to market prices. While their results with respect to
the UK, given the high level of industry detail (123 industries) and the other two limitations (i.e. lack of depreciation matrix and the issue of self-employment) do not show a
significant divergence from those reported in the above-mentioned studies. For example,
they note that the usual measures of deviation for the case of UK ‘are not “wildly” out of
line with those presented by Petrovic, Ochoa and Bienenfeld’ (Steedman and Tomkins
1998, 384).
While all of the above studies are based on input–output data, this is not the case with
some recent papers, where there is an effort to compute direct prices and prices of production with the use of national income accounts data. For example, Kliman (2002, 2004) in
his studies of the US economy for the period 1977–1997 derives estimates of labour values
that are much closer to market prices than those reported in Ochoa (1984). However,
Kliman argues that these results are ‘artefacts’ and should not be taken as validation of the
labour theory of value. More specifically, Kliman, by focusing on the cross-section regressions between inter-industry output evaluated at labour values (independent variable) and
market prices (dependent variable), finds that the correlation coefficient is artificially high
(usually above 95%) as a result of the industry size. Large industries use large amounts of
labour and so their sales are also large, therefore, a high correlation coefficient is trivially
expected.4 For this reason Kliman proposes to scale down, or in his wording to ‘deflate’,
both variables by the total (labour and non-labour) cost of production in his effort to reduce
the ‘size-induced bias’. The regressions that he ran in the ‘deflated’ variables gave correlation coefficients near zero.
Diaz and Osuna (2006) following Kliman’s methodology and using data from the
national income accounts of the Spanish economy for the period 1986–1994 also found relatively small price–value deviations (p. 352n) and extremely high correlations (an R2 in the
range of 97%).5 Diaz and Osuna, however, took issue with Kliman’s claims arguing that
controlling for the size of an industry by total cost is inappropriate, because it reduces
severely the variability of the independent variable (sales evaluated in terms of values) and
given that the dependent variable is sales; that is, the quantity sold at market prices, which
are by definition equal to one and, therefore, display no variation and so their co-variation
with the vector of labour values or prices of production will be nil. If the size of industries
is ‘deflated’ differently then, Diaz and Osuna conclude, any results are possible. More
specifically, when the size of industries was ‘deflated’ by just the non-labour cost the R2
increased from near zero to about 40% and when finally the size of industries was ‘deflated’
by the gross capital stock the high correlations were restored, with the R2 lying in the range
of 90%. From these findings one would expect that Diaz and Osuna would, at a minimum,
reject Kliman’s ‘deflation’ process and propose an alternative one. However, this is not the
case since Diaz and Osuna (2006) point out a more fundamental problem that in their view

International Review of Applied Economics


is the lack of precise knowledge of physical units of measurement that renders market prices
as unknown magnitudes. As a consequence, they are led to the conclusion that no inference
what so ever can be drawn about the closeness of computed prices (direct prices or prices
of production) and the (unknown) market prices.
In our view, Diaz and Osuna (2006) correctly showed the arbitrariness in Kliman’s
‘deflation’ process. The ‘deflation’ process has also been criticized by Cockshott and
Cottrell (2005) who argued that if labour values are close to market prices as the regressions
show and if this is an ‘artefact’ as Kliman claims then ‘alternative value bases’ such as, for
example, the electricity or agricultural content of commodities should also give high correlations. Such correlations, however, have been shown to be false in the studies by Cockshott,
Cottrell, and Michaelson (1995), Chilcote (1997), Cottrell and Cockshott (1997, 2005),
Tsoulfidis and Maniatis (2002) and more recently Zachariah (2006). Thus, the agnostic position of Diaz and Osuna (2006) which claims that we do not know the exact physical units
of measurement and, therefore, we cannot draw any inference about the actual market prices
is at least questionable.
It is important to stress at this juncture that in input–output analysis we cannot identify
the actual market prices by examining the data of any single year because we need to know
the exact physical units of measurement, however such information is not available and also
it would be meaningless in itself because each sector we are dealing with produces a variety
of goods. In the input–output literature, however, it has long been established that we do
not need to know the exact physical units of measurement; we only need to assume that
whatever the physical units of measurement are they remain fixed during our analysis. By
way of an example (cf. Ochoa 1984, 58), suppose that corn is sold for 25 cents a kilogram.
By adopting the evaluation of the product in terms of euro we essentially say that 4 kg of
corn will be equal to 1 euro. The physical unit of measurement of corn becomes 4 kg.
Consequently, in input–output analysis market prices are set equal to one because it is
impossible (and also meaningless) to collect data on the physical output produced, we only
assume that whatever these units of measurement they remain constant. This is the reason
why virtually all input–output tables are cast in terms of sales rather than in physical units
of measurement.6 Nevertheless, it is possible to show that calculations such as those of
values (direct prices) and prices of production can be carried out and the conclusions drawn
are the same regardless of the type of data, sales or physical output. Ochoa (1984, 58–70)
constructs a realistic numerical example of input–output data of three sectors and carries out
the computation of direct prices, prices of production in terms of physical data and market
data and shows that the estimations are not affected. For a comprehensive discussion of the
same issue and an appropriate numerical example in the context of Leontief’s price model
see Miller and Blair (1985, 351–357). Once we stipulate such an assumption then the direct
prices and prices of production are derived as a proportion to market prices whatever these
might be.7
It is important to note that the OLS regressions between market prices and values can
only show that values are close to market prices and not necessarily that the labour content
of commodities determines their market prices. This is the reason why we bring additional
empirical evidence about the relationship between values, prices of production and market
prices from the economy of Japan. Implicit in our analysis is the assumption that by establishing the proximity of direct prices and prices of production to actual prices as a regularity
in the input–output data of more and more countries, we will be also dealing with the issue
of determination, which to our view can only run from labour values and prices of production to market prices since the converse would be contrary to the historical development of
economic theory.


L. Tsoulfidis

4. Methodology
The above studies indicate that in most real economies that have been examined empirically
up until now the differences in the capital intensity (e.g. as measured by the capital–labour
ratio) across sectors of the economy does not give rise to substantial differences in terms of
prices. As a consequence, all the categories of prices be they direct prices (where the differences in the intensity of capitals of the individual sectors play no role whatsoever) or prices
of production (prices with a uniform rate of profit) will be very close to each other as well
as with the observed market prices. The question pursued in this paper is to what extent
similar results can be derived for the economy of Japan. In order to answer such a question
we calculate below, on the basis of input–output data for Japan, labour values and their
monetary expressions, i.e. direct prices as well as prices of production using data on circulating capital.
Starting with the labour values (λ) we estimate the total (direct and indirect) labour
requirements per unit of output produced. More specifically, the labour values of each of
the 33 industries of the Japanese economy are derived from the solution of the following
system of equations:

λ = ao + λA and λ = ao ( I + A ) −1


Where, λ is the row vector of labour values, A is the square matrix of input–output coefficients, a0 is the row vector of adjusted for skills direct labour coefficients and I is the identity matrix.8 Furthermore, we scale the so-estimated labour values to prices proportional to
values, that is, we equate the sum of labour values expressed in money terms (direct prices)
to the sum of market prices according to the usual condition of the transformation problem.
That is,


(2 )

Where v is the row vector of direct prices, e is the row unit vector identified with the market
prices and x is the column vector of gross output. With this normalization, the equality
between the gross output evaluated in direct prices (vx) to the gross output evaluated in
market prices (ex) will always hold true. In other words, the normalization of prices
maintains the value of money constant.
The prices of production on circulating capital are estimated from the following
P = ( wa0 + PA )(1 + r )

( 3)

where P is a row vector of relative prices of production, w is a scalar representing the money
wage, r is a scalar representing the economy’s uniform rate of profit and A the matrix of
input–output coefficients. Following Pasinetti’s (1973) notion of vertical integration, we
rearrange the above equation, which gives:
p = (1 + r )wλ + rPH

Where λ ≡ a0 [I − A]−1 represents the vertically integrated labour input coefficients from
which we derive the vector of values by invoking the normalization condition (2). Finally,

International Review of Applied Economics


H ≡ A [I − A]−1, represents the matrix of the vertically integrated technical coefficients of
production. Thus by taking into account the labour values the above equation can be restated
as follows:
p = (1 + r )wv + rpH

( 4)

where p ≡ P(ex/λx). Hence, if we assume that w = 0 we get r = R, where R is the maximum
rate of profit of the economy, and the corresponding price system will be:
p = R pH


The RHS eigenvector of matrix H defines the standard commodity q, whose scale is fixed
in terms of the actual output vector (x), such that s=q(vx)/(vq), where s is the normalized
standard commodity (Shaikh 1998).
Prices of production are estimated starting from equation (3) and by taking into account
that w = Pb, that is, the money wage is equal to the vector of the basket of goods (b) evaluated in terms of prices of production, we form the following Eigen equation:
P[1 / (1 + r )] = P ( A + ba0 )

(6 )

From (6) and the normalization with the standard commodity, that is ps = vs = vx = ex,
we derive a ‘wage–profit rate frontier’ which is not linear given that wages are paid ex ante
(Pasinetti 1977, ch. 5, appendix). Furthermore, from equation (4) post-multiplied by the
standard commodity s, we derive:
ps = (1 + r )wvs + rpHs or vs = (1 + r )wvs + ( r / R ) vs

Finally, by solving the above equation for the wage rate we get:
w = ( R − r ) / [(1 + r )R ]

(7 )

In our effort to show that direct prices and prices of production are intrinsically
connected to each other and that prices of production are derived from labour values we
invoke Steedman’s (1999) approximation of prices of production through labour values.
This approximation is derived starting from equation (4) which can be restated as follows:
p = (1 + r )wv[ I − rH ]−1

(8 )

We define the relative rate of profit ρ ≡ r/R (where, 0 ≤ ρ ≤ 1) and we replace in equation
(7) which gives w(1 + r) = 1 − ρ. Consequently, equation (8) can be rewritten:
p = (1 + ρ ) v[ I + HRρ ]−1

(9 )

We know that the maximum eigenvalue of HR equals one and, therefore, the matrix HRρ
has a maximal eigenvalue less than one and so is convergent. Consequently, equation
(9) can be restated:


L. Tsoulfidis

p = (1 − ρ ) v[ I + HRρ + ( HRρ )2 + ( HRρ )3 + ...]

(10 )

The above equation is a polynomial of the vector of prices in terms of (1 − ρ)ρn showing
that prices of production can be derived starting with labour values. More specifically,
prices of production are equal to labour values times a multi-term mark-up, and according
to the number of terms that we include in this mark-up we approximate the prices of production through values to the desired degree of accuracy. It is important to stress that for a relatively good approximation there is no need to include many terms provided that the value
of ρ is less than 0.5 (Steedman 1999, 315–316). Consequently, it is interesting to test the
reliability of Steedman’s approximation of prices of production through labour values with
the data of a real economy.
Finally, equation (9) helps us in addressing the issue of changes in prices of production
as a result of variations in income distribution. The latter are reflected in the movement of
the relative rate of profit ρ which we allow to take on values ranging from zero to one. The
resulting vectors of prices of production for each ρ are divided by direct prices. For ρ = 0
the ratio of the two sets of prices is one, as ρ increases prices of production differ from direct
prices, a difference that in general increases with ρ. The significance of these deviations is
that they convey some useful information about the transfers of value between industries.
As ρ increases the gains in profits that accrue to the various industries as a result of the fall
in wages are consistent with a uniform rate of profit only if relative prices change in accordance to their capital intensity. More specifically, the capital intensive industries have a
lower increase in profits than that of the labour intensive industries, so the restoration of a
uniform rate of profit requires an increase in the relative price of the capital intensive industries and a fall in the relative prices of the labour intensive industries so both types of industries gain the same (higher) rate of profit. As a consequence, the sign of the deviation of
prices of production from values as income distribution changes may be very important to
know a priori.
There are two conjectures about the exact movement of prices of production, the first is
that of Ricardo (1951, 30–43) and Marx (vol. III, ch. 11), who argued that the movement of
prices of production will be monotonic either in an upward or in a downward direction
depending on the intensity of capital of each industry if it is above or below to the economy’s average. The second is Sraffa’s (1960, 37–38) who challenged this claim by arguing
that prices of production are affected in complex ways and so they are expected to change
in a nonlinear and certainly non-monotonic way. Furthermore, the revaluation of capital
according to the new prices of production may change the rank of capital intensity (Pasinetti
1977, ch. 5). At the theoretical level, there is no doubt, that there is much complexity in the
movement of prices of production and that very little progress has been achieved so far in
resolving this complexity. At the empirical level, one wonders about the actual movement
of prices of production in the data of a real economy.
5. Results
The results on direct prices and prices of production are displayed in Table 1 below,
together with the usual measures of deviation, i.e. the MAD defined above and the Mean
Absolute Weighted Deviation (MAWD), whose difference from the MAD is that the row
vector of absolute deviations is multiplied by the column vector of each sector’s share in the
total gross output. Although, the MAWD is appropriate for this kind of estimation nevertheless one cannot rule out completely the possibility of some bias owing to the normalization

International Review of Applied Economics
Table 1.


Measures of deviations and profit rates.





Prices of production vs market prices 0.268
Prices of production vs labour values 0.112
ρ ≡ r/R





Labour values vs market prices

Rate of profit
Maximum rate of profit
Relative rate of profit

condition. For this reason the d-statistic proposed by Steedman and Tomkins (1998) –
defined as d = [2(1 − cosθ)]1/2, where θ is the angle between the two vectors under comparison – is also employed. The major advantage of the d-statistic over the MAD and other
related measures of deviation is that it does not depend on the normalization condition. Thus
the inclusion of the d-statistic is expected on the one hand to give us unbiased estimates of
deviations for each year of our analysis and on the other to give us an approximate idea of
the possible magnitude of the bias that is expected from the other measures of deviations
displayed in Table 1 above.
Table 2 below presents the results derived through Steedman’s approximation (equation
10) for the year 1980. The first two columns show the deviations of labour values and
prices of production from market prices, whereas the remaining columns refer to
Steedman’s successive approximation of prices of production through the polynomial
expression (10). More specifically, column A gives the term (1–ρ)v of that polynomial
expression, while column B the term (1 − ρ)v[I + HRρ], column C the term (1 – ρ)v[I +
HRρ + (HRρ)2] and so forth for the column D. The deviation of these approximations from
the estimated prices of production, as measured by the MAD, is shown in the last row of
Table 2.
We also tried the same exercise for the other years with similar results, something that
was expected from the low values of ρ (see Table 1). We decided not to include all these
approximations for reasons of clarity of presentation and economy of space.
Table 3 below reports only the results of labour values as they are estimated from equation (1) and (2) and prices of production from equation (6) after the appropriate normalization with the standard commodity derived from (5). The estimations are carried for each of
the five benchmark years of our analysis and are reported in Table 3 below.
The last hypothesis that we subjected to empirical testing is the changes in prices of
production induced by changes in income distribution. For this purpose we experimented
with equation (9) and input–output data of Japan. In these experiments we changed the
value of ρ from zero to one in intervals of tenths, and we estimated the ratio of prices
of production for each particular industry to its respective value. We restricted our analysis to the years 1980 and 1990 provided that the results for the other three years are
pretty much the same. The trajectories of prices of production are depicted in Figures 2
and 3.

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