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Classical labour values -properties of economic reproduction

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We attempt to clarify and generalize the meaning of economic value as conceptualized in classical political economy. Using a modern formalism , we show that value can be derived as a basic property of systems of economic reproduction. The applicability of the concept is discussed and its relation to inequality, productivity, employment, and unproductive activities are demonstrated.
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Classical labour values — properties of economic
David Zachariah and Paul Cockshott
June 8, 2020
We attempt to clarify and generalize the meaning of economic value
as conceptualized in classical political economy. Using a modern for-
malism, we show that value can be derived as a basic property of
systems of economic reproduction. The applicability of the concept is
discussed and its relation to inequality, productivity, employment, and
unproductive activities are demonstrated.
1 Introduction
What is the basis of economic value? This question repeatedly crops up in
practical political discourse, cf. [Bacon and Eltis, 1978, Mazzucato, 2018].
In this paper, we show that the process of economic reproduction gives rise to
a characteristic accounting structure in which value is assigned uniquely to
goods and services, using the formalism of [Schwartz, 1961, Pasinetti, 1979].
After deriving value from this structure, we show that it corresponds to
the conception found in the classical approach to political economy, as well
as in the early labour movement, namely, social labour requirements, see
[Smith, 1776, Ricardo, 1817, Marx, 1867].
It has often been assumed that the notion of value is only applicable to
market-based economies. Conventional economic theory denies valuation is
possible outside the relations of commodity-exchanging agents.1But also
some heterodox economic theories claim that value and hence rational com-
parison of economic alternatives can only exist through market relations.2
Our conceptualization, however, shows that value can be derived as a prop-
erty of a self-reproducing economic system that can redeploy labour across
a range of production processes. This includes capitalist market economies,
planned economies and mixed state-regulated economies. We proceed to
show that this generalized conception addresses central questions that con-
cerned classical political economy and the early labour movement: the or-
ganization of production, productivity of an economy, extraction of an eco-
nomic surplus, real-economic class inequalities, unproductive activities, and
so on.
The derivation of all results are provided in the footnotes.
2 Economic reproduction and value
The real price of everything, what everything really costs to the man
who wants to acquire it, is the toil and trouble of acquiring it.
[Smith, 1776, book 1, ch. 5, emph. added]
Classical labour values 2
We consider an interconnected economic system that is capable of re-
producing itself. It produces distinct types of outputs for use3, that can
be represented by an ordered list of names, such as (iron, corn, sugar, ...).
Since the list is ordered, we can equivalently represent each output-type by
a number4, which leads to an efficient representation to describe ddistinct
kinds of outputs, numbered as 1,2, . . . , d. Associated with output-types
there are socially defined units of measure: metric tons of steel, bushels
of corn, kilograms of sugar, etc. Once the lists of output-types and their
units of measure are fixed, they permit representing quantities of products
in terms vectors.
Example 2.1 (Bundles of products).Consider a simple economy with d= 3
outputs-types: iron, corn and sugar, and two different bundles:
and b0=
The set of product bundles is a vector space that allows operations such
as addition, b+b0, and multiplication by a scalar, 2 ·b. The most basic
point about economic value is that it permits also the ordering of product
bundles, bb0, as one being greater than the other. That is, value renders
heterogeneous bundles of products into commensurable units.
But this raises a series of questions: When and why do distinct products
have any value? What indeed does it mean to have economic value? Or
to put it another way, what are the formal properties of value? We will be
arguing in what follows that both its formal properties – the value form if
you will – and its quantitative properties are derived from something more
fundamental than commodity exchange, namely, the technical and social
structure of reproduction.
Value maps bundles of products to scalar numbers that enable relative
comparisons between different bundles. Monetary prices may seem to pro-
vide such a valuation, since they assign a quantity of money to each unit
of a product. Market prices observed in commodity exchange, however,
randomly fluctuate from one transaction to the next and, as the classical
economists understood well, the very notion of goods being over- and under-
priced implies that value is more fundamental than prices. Suppose there
exists some nonnegative row vector vthat contains values for each unit of
Example 2.2 (Valuation).In the simple economy above, the vector
v=viron vcorn vsugar
encodes the (yet unspecified) amount of value per unit of iron, corn and
sugar, respectively. Then the value of bundles band b0above equal
vb = 5 ·viron and vb0= 2 ·vcorn + 1 ·vsugar,
Below we will deduce vusing two basic assumptions:
(i) Value is a real cost that only changes with the structure of economic
(ii) Labour can be trained and redeployed across economic activities.
Classical labour values 3
2.1 Production and consumption by workforce
After deducting the intermediate inputs consumed over a given period in the
overall production process, the economy produces a net product of goods and
services that are consumed, invested or hoarded. We shall denote this net
product bundle as n.
Example 2.3 (Net product).Suppose the net output of the economic system
,then vn = 10 ·viron + 100 ·vcorn + 50 ·vsugar
is the value of the net product, or total value added, yet to be defined.
Economic reproduction requires work so that one part of nis necessarily
consumed by the workforce and its dependents. The remainder is a surplus
product consisting of investment goods, luxuries, and so on. That is,
surplus product = net product consumption by workforce (1)
We now turn to specifying consumption of the workforce, which occurs in
conjunction with production. For a given period let κdenote the consump-
tion rate vector, which records the total amount of each output consumed
by the workforce divided by the number of units of labour deployed.5
Example 2.4 (Real-consumption rates).During a given period, suppose the
average consumption rate in the simple economy is one unit of corn per
person week deployed. Then we can write
The average technical conditions of production can be described by the
direct labour requirements per unit of output in each sector, denoted by the
row vector `, and a square matrix A={aij}, which records the amount of
output jdirectly required to produce a unit of output i. Both quantities
can be estimated in real economies using data from national accounts.6
Definition 2.1 (Consumption requirement matrix).The total workforce
consumption requirement equals κmultiplied by the total person weeks of
labour deployed to reproduce n. The total consumption can be expressed
as Rn, where
is a matrix of workforce consumption requirements.7Thus vRn equals the
value of the total consumption requirement in the economy.
Example 2.5 (Reproduction of simple economy).Let iron, corn and sugar
be indexed as {1,2,3}.8Then the average input requirements to reproduce
output iis given by the ith columns of
`=0.6 0.2 0.3and A=
0 0.20 0.30
0 0.02 0.10
0 0 0
Classical labour values 4
so that to reproduce one unit of iron requires 0.6 units of labour, and 0
units of corn or sugar. Similarly, to reproduce one unit of corn requires 0.2
units of labour, 0.2 units of steel, 0.02 units of corn and 0 units of sugar.
As a consequence, the net product nin Example 2.3 requires a total worker
consumption of
Rn =
0.60 0.33 0.51
where Ris computed using (2) and κin Example 2.4. Thus the workforce
consumes a total of 64 units of corn, with an unknown value of vRn =
64 ·vcorn.
2.2 Deriving economic value
We now study the net output of the economic system in terms of value. The
net product ncan be decomposed into the total consumption by workforce
and a surplus product (1). The value of the surplus is then the difference
between vn and vRn.
Definition 2.2 (Share of surplus value).In value terms, the surplus (1)
forms a share of the total value of the net product vn, which we denote
σ=vn vRn
vn ,(4)
and is bounded between 0% and 100%.9
At first sight, (4) is a mere accounting identity that is applicable to any
reasonable choice of valuation vector v. However, the following result shows
that this fundamental identity is deeper than that.
Result 2.1 (Determination of value).Economic value for each of output-
type, v, is determined uniquely by (4), up to a unit of choice. Specifically,
vis a left-eigenvector of Rin (2).10 Thus value naturally arises when
production requires a workforce that consumes part of the net product.
Example 2.6 (Values).In the simple economy considered above, the relative
values of iron, corn and sugar are determined by computing the nonnegative
left eigenvector of R. Using the consumption requirement matrix in (3), we
v=viron vcorn vsugar
=0.60 0.33 0.51
Thus a unit of iron is roughly twice as valueable as a unit of corn.
The value of a commodity [...] depends on the relative quantity of
labour which is necessary for its production, and not on the greater or
less compensation which is paid for that labour.
[Ricardo, 1817, ch. 1, emph. added]
Result 2.2 (Field property).Economic value vderived in Result 2.1 can
be obtained by integrating all coexisting labour requirements in production,
vλ(0) + λ(1) + λ(2) + · · · ,(5)
Classical labour values 5
where λ(k) = `Akis the vector of labour requirements of the kth inter-
mediary inputs in production.11 Value is therefore invariant to changes in
workers’ consumption or the distribution of the net product.12 The sum in
(5) converges for an economy capable of reproducing itself.
Value is not an intrinsic property of products, but is rather a field prop-
erty that reproducible goods and services acquire from the economic system.
That is, the value of a product bundle bis obtained by an integration over
the field, (λ(0) + λ(1) + λ(2) + · · · )b=vb.13
Example 2.7 (Vector field in simple economy).The coexisting labour re-
quirements for corn and sugar are illustrated in Figure 1.
0 0.1 0.2 0.3
(a) λ(0)
0 0.1 0.2 0.3
(b) λ(1)
0 0.01 0.02 0.03
(c) λ(2)
Figure 1: Illustration of coexisting labour requirements for corn and sugar,
described as a vector field. (a) Direct labour requirements. (b) Indirect
labour requirements via the inputs to corn and sugar. (c) Indirect labour
requirements via the subsequent set of inputs. Note that the magnitudes
decrease for each set of inputs k= 0,1,2, . . . and their sums are proportional
to value (5). The value of a bundle of corn and sugar b0in Example 2.2
equals vb0= 0.700 + 0.448 + 0.017 + · · · = 1.165.
The production of value is therefore inseparable from socially organized
production of real goods and services. It follows from the derivation above
that value production is distinct from monetary income generation; money
and prices are symbolic means by which value is claimed and distributed in
market economies. The natural unit of value is worker-time and using such
units we shall call vb the ‘labour value’ of product bundle b.14
3 Applicability of concept
When deriving vfrom (4), it appears that value is an economic property
that is invariant to the social institutions under which production takes
place. We present here a few remarks on the applicability of the concept.
Every child knows [...] that the masses of products corresponding to the
different needs required different and quantitatively determined masses
of the total labour of society. That this necessity of the distribution
of social labor in definite proportions cannot possibly be done away
with by a particular form of social production but can only change the
mode of its appearance, is self-evident. No natural laws can be done
away with. What can change in historically different circumstances is
only the form in which these laws assert themselves.
[Marx, 1868, emph. added]
Using our notation, nrepresents the ‘masses’ of different products and
the corresponding elements of vquantitatively determine the masses of to-
tal labour of society required. The derivation of vassumes that (i) value
Classical labour values 6
only changes with the structure of a viable interconnected economic system
that (ii) is capable of training and redeploying its finite amount of available
labour time across different production processes. This latter assumption
implies the existence of a vector `representing labour inputs across differ-
ent production processes in commensurable units of time, while the former
assumption implies to a matrix Awith a maximum eigenvalue less than
It is in the continual process of training and redeployment of labour-
ing capacity across production that an economic system renders concrete
work tasks as an expenditure of a commensurable abstract labour resource,
quantified in units of time.15 This would include a range of self-reproducing
economic systems. Did, for instance, value and the disposition of labour
time matter to the slave lords of antiquity? According to Cato, it appears
that they did:
When [the master of a farmstead] has learned the condition of the
farm, what work has been accomplished and what remains to be done,
let him call in his overseer the next day and inquire of him what part
of the work has been completed, what has been left undone; whether
what has been finished was done betimes, and whether it is possible to
complete the rest; and what was the yield of wine, grain, and all other
products. Having gone into this, he should make a calculation of the
labourers and the time consumed. [Hooper and Ash, 1935, p.9].
In the slave plantations described above by Cato, the disposition of
labour is self evident and ‘natural’, it is not obscured by monetary indi-
rection. But it is still labour in the abstract, albeit of a given group of
slaves, being distributed between concrete tasks: meadow clearing, faggot
bundling, road-work, etc.
The necessity to take into account the usage of labour time, whether that
be the time of slaves, wage labourers, citizens of a socialist commonwealth,
is a natural necessity that could not be abolished, only change its historical
form. By contrast, in economies with institutions that prevent the redeploy-
ment of workers across tasks, e.g. rigid forms of caste hierarchies, there can
be no general labour resource quantifiable in commensurable units.
3.1 Capitalist market economies
In a capitalist economy, the necessity to distribute labour appears as simply
expenditures of money on wages to top-level managers in decentralized firms.
So the wage budget allocated to different branches of a firm provides an
indirect representation of the needed allocation of labour.
As one descends the management hierarchy, the simple monetary view of
things becomes insufficient. The subsidiary managers have to allocate spe-
cific people to specific tasks just as the slave overseer had to. By contrast,
as one moves further away from the production process, the representation
of labour becomes increasingly obscure and monetary. Indeed, when the
products of the economy are allocated between agents as commodities, the
monetary calculations are based on market prices which randomly fluctu-
ate from one transaction to the other. The relation between market prices
of commodities and their labour values is necessarily a statistical one, see
[Farjoun and Machover, 1983].16 To an individual, money appears to be
freely disposable between different products, but in reality such choices are
limited by macroeconomic constraints set by v, which represent real costs
Classical labour values 7
irrespective of random market prices.
Nevertheless, firms in a capitalist market economy do solve labour allo-
cation problems via decentralized monetary calculations. The feasibility of
this monetary accounting mechanism rests on the fact that human labour
is flexible and can be redirected, either within the firm or on the employ-
ment market, between activities. In capitalism, the redeployment of labour
between concrete tasks across the production system occurs through the
transfer, hiring and firing of workers within and across decentralized firms.
This allows single scalar measure like money to function as a system of social
3.2 Planned economies
Planned economies too have to grapple with the finite nature of their labour
supplies, and the need to expend effort for any worthwhile effect. This im-
plies that they too will have to have social forms in which this necessity will
be expressed. The necessity for the labour force to be allocated in a man-
ner determined by technical conditions took in the planned Soviet-socialist
economies the form of a directive plan of n. This plan involved drawing
up material and labour balances for the overall economy. We know that
Soviet-socialist economies continued to use monetary calculations, which, to
a greater or lesser degree of adequacy, allowed indirect calculations to be
done on social labour requirements. While monetary calculation and allo-
cation in capitalist market economies redeploys a certain amount of labour
via the recreation of a pool of unemployed, the Soviet-socialist economies
did not develop the kind of labour time accounting, planning and regulation
that would be required to carry out reallocations of labour within a fully
employed workforce.
In capitalist war economies, production, by and large, still took place in
privately owned firms. There were state munitions factories like the Royal
Arsenal or the Oak Ridge and Los Alamos atomic weapons plants, but
these were exceptions. The state directed labour, by conscripting it into
the army, and by conscripting women and men in key trades into essential
war work. It also rationed the supply of key materials, fuels, and foodstuffs.
Firms were subject to negotiated direction to produce only munitions, or re-
stricted ranges of ‘utility’ products [Edgerton, 2011]. Money was still used
to pay for the munitions delivered, and to pay workers. Buying food re-
quired both money and ration cards. Money alone was not enough either
for the consumer or for firms. In peace, money as the universal ration con-
strains everything. Shortage of it constrains the working class consumers
and uncertainty about future revenue constrains even those firms who have
good cash reserves. Because the constraint on production comes via market
exchange in price units rather than directly in units of products, peace-time
capitalist market economies typically operate somewhat below full capacity.
In war, national survival dictates that every available resource be put to use.
The economy operates at the limits of its physical resources in materials,
people and machines.
The state as primary purchaser has to look not just at the projected
costs of ships, aircraft, etc., that it is ordering, but at all sorts of material
constraints. In deciding what type of destroyers to order, the Navy first
took into account the requirements of their admirals for the ships to carry
Classical labour values 8
guns of different types, torpedoes and anti-submarine weapons: all techni-
cal not financial issues. They then had to take into account the number of
ship yards in the country able to build ships of different sizes, the delivery
schedules for different kinds of projected weapons and ships machinery, the
availability of metals and alloys of different weights and strengths. They
then had to ask whether the demands on skilled labour would require the
cancellation or postponement of other orders.17 Money was a relatively sec-
ondary concern. The availability of state credit, that, at least within the
domestic economy was effectively unlimited, removed money as a constrain-
ing resource [Keynes, 2010]. The same point about money applied a fortiori
to the Soviet-socialist economies. Money was never a constraint for them.
Labour, plus available plant and equipment, however, were.
If we imagine a planned economy that does away with money altogether,
then it would still need value. Marx presented a version of this [Marx, 1970]
where labour time was to be used to allocate consumer goods. But value
accounting would also be needed to set budgets for public projects. Research
programs to develop vaccines, or to explore the moons of Jupiter would
need some limits posed on the amount of social labour that they could use.
The same applies to general democratic decisions about long-term structural
investment. Society as a whole could not meaningfully decide what portion
of its output should be devoted to investment and research, that is to say
σin our notation, unless the surplus was measurable. By Equation (4) any
consistent measure of this surplus implies labour value.
3.3 Fully automated economies?
Choosing to evaluate the replacement cost of products in terms of labour
time reflects not merely a concern for human beings but also the fact that
humans possess a capacity to be allocated across a wide range of concrete
tasks. This general capacity is then realized in economies that redeploy
labour and gives rise to the abstract representation `of direct labour re-
quirements. However, it may be objected that some future society may
have at its disposal a race of robots, so skilled and dexterous, so intelli-
gent and adaptable, that these beings may come to supplant us in our toils.
Would that not invalidate our system of calculation of values?
Not at all, it would merely substitute the time of these general purpose
robots for our own time. The equations of value would still apply, but with
this simple proviso, that the labour input time is to be understood as rede-
ployable general robot time. The consumption requirement matrix Rwould
at that point be translated into the ‘robot construction and maintainence’
requirement matrix, for these robots too will need energy, will require repair
and will absorb the effort of other robots in their initial construction.
Humans, in this hypothetical society, would be in the position of slave-
owning ancients: idlers depending on the surplus labour of others.
4 Implications for labour
We now proceed to apply the generalized conception of value to central
questions that concerned classical political economy and the early labour
movement: Economic inequality, productivity, employment and the utiliza-
tion of surplus economic capacity.
Classical labour values 9
4.1 Extraction of surplus labour
The specific economic form, in which unpaid surplus-labour is pumped
out of direct producers, determines the relationship of rulers and ruled,
as it grows directly out of production itself and, in turn, reacts upon it
as a determining element.
[Marx, 1894, ch. 47, emph. added]
Result 4.1 (Surplus labour-time).The share of surplus value σin (4) equals
the fraction of work in the economy to replace the surplus product. The share
is given directly by the structure of R, which has a unique nonzero eigenvalue
Example 4.1 (Share of surplus).Consider the real-consumption requirement
matrix Rin (3). Its eigenvalue is readily computed, and yields the following
share of surplus value σ= 67%. Thus 67% of the work in the economy is
materialized in the form of surplus outputs.
In economic systems in which the composition and distribution of the
net product nis not determined by the workforce, their surplus labour is
extracted, controlled and consumed by a distinct economic class.
4.2 Total productivity and employment
The values of commodities are directly as the times of labour employed
in their production, and are inversely as the productive powers of the
labour employed.
[Marx, 1865, sec. IV, emph. added]
Technical and organizational changes in the economy alter the average
production requirements, and therefore the vector field in (5).18 Let ˙
dt v
denote the change in labour values per unit of time. This quantity has
profound effects on both production and employment.
Result 4.2 (Productivity).Suppose the labour value of output-type iis re-
duced at the relative rate ρi ˙vi/vi. Then, for a fixed level of employment,
the net output of ican grow at the relative rate ρi. Thus labour values are
(inverse) measures of total productivity in the economy.
Example 4.2 (Labour value and total productivity growth).Suppose the
simple economy produces a net output of 100 units of corn. The labour
value of corn can be lowered by decreasing the direct labour input and/or
by decreasing the amount of coexisting inputs required. Thus technical im-
provements in the production of iron affect the labour value of corn. Suppose
its unit value vcorn decrease by the rate ρcorn = 5% per annum. Then the
net output of corn can increase exponentially as shown in Figure 2.
Result 4.3 (Employment).Suppose the final demand for output-type i
grows at the relative growth rate gi. Then the total demand for labour
changes by the rate giρi. Thus labour values are employment multipli-
ers in the economy.19
We see that economies with institutions that progressively lower the
labour values of the outputs are capable of increasing material living stan-
dards and/or leisure time exponentially. At the same time, economies that
lack coordination between technical change and changes in consumption
Classical labour values 10
0 5 10 15 20 25 30
Units of corn
Figure 2: Growth of capacity to produce corn as labour value of corn de-
creases by ρcorn = 5% per annum. Starting with an annual output of 100
units of corn, the amount increases fourfold within 30 years.
and investment demands can give rise to both persistent unemployment and
chronic labour shortages. When gi< ρi, the total demand for labour de-
clines exponentially and must be compensated by increased demand among
other outputs to prevent the rise of unemployment.
4.3 Productive and unproductive activities
A man grows rich by employing a multitude of manufacturers: he
grows poor by maintaining a multitude of menial servants.
[Smith, 1776, book II, ch. III, emph. added]
This remark may merely seem to apply to an individual employer but
in fact generalizes into a macroeconomic property: The surplus product (1)
is by definition the residual of the net product after deducting the outputs
consumed by the workforce, i.e., s=nRn. The relations between the
products of the economy are in general not symmetric: some outputs may
enter directly or indirectly as inputs to all goods and services, while other
outputs may not. This implies certain consequences in the production of s
which we deduce below.
Definition 4.1 (Basic and nonbasic outputs).Basic outputs are directly
and indirectly required in the production of all outputs, while the nonbasic
outputs are not. More formally, let e
A=A+κ` denote the augmented
input-output matrix, where the order of the outputs is arranged to form a
upper-block triangular structure:
The upper-left block corresponds to outputs indexed i= 1, . . . , b, which we
denote as ‘basic’. The remaining u=dboutputs are ‘nonbasic’.20
Classical labour values 11
Example 4.3 (Basics and nonbasics).For the simple economy we have
0 0.20 0.30
0 0.02 0.10
0 0 0
0.6 0.2 0.3
0 0.20 0.30
0.60 0.22 0.40
which is upper block-triangular as in (6). Therefore corn and iron (i= 1,2)
are basic outputs, while sugar is a nonbasic output.
The production of basic outputs forms a self-reproducing sector of the
economy which is critical in determining the share of surplus labour σ.
Result 4.4 (Determinants of the surplus).The share of surplus labour is
determined by productivity in and the workers’ consumption from the basic
sectors of the economy. That is,
σ= 1 vbκb,
where vband κbare the values of basic outputs and corresponding worker’s
consumption rates, respectively. Luxuries and other nonbasic outputs do not
affect σ.21
In other words, the share of surplus labour σincreases when the workers’
consumption rates decrease and/or the labour values of basic outputs de-
crease. The consumption rates κbcan be reduced by extending the number
of working hours without compensation, while the lowering of labour values
vbrequire technical changes in the basic sector.22
Result 4.5 (Dependence on surplus labour).Production of nonbasic outputs
is predicated on the extraction of surplus labour. More formally, if the share
of surplus labour is σ= 0 then the production of nonbasics outputs equals
Production of luxuries and other nonbasic outputs drains the surplus in
the basic sector. Activities involved such production impedes the expansion
of the basic sectors and are ‘unproductive’ in the sense of classical political
economy. In modern capitalist economies, this includes the arms industry
and finance sector. Conversely, many socialized goods and services, such as
public health care and education, are basic outputs and thus ‘productive’.
Result 4.6 (Drain on the basic sectors).The surplus of basic outputs is
impeded by the production of nonbasic outputs. More formally, let band
b0denote the net production of basic and nonbasic outputs, respectively, so
that n=b+b0. By redeploying labour from nonbasic to basic sectors, the
surplus product in the latter sectors can be increased by
vb (IR)b0,(8)
where the first term is the workers’ consumption freed up from the nonbasic
sectors and the second term is due to the increased production in the basic
sectors.24 Thus (8) represents a drain on the surplus capacity of the basic
sectors incurred through the production of nonbasic outputs.25
Classical labour values 12
Example 4.4 (Redeployment to basic sectors).Consider the net product in
Example 2.3, where 50 units of sugar constitutes the nonbasic outputs. That
Then the surplus product equals
s=nRn =
0.60 0.33 0.51
Suppose the total labour devoted to sustain the nonbasic sugar is redeployed
to expand net output of basic iron and corn uniformly. Using (8), the form
of the surplus product then changes into
That is, an increase of surplus iron and corn by +66% and +185%, respec-
5 The form of value
The analysis above is based on the premise that the value of product bun-
dles bhas the form of a linear function vb. This function is encoded by
the valuation vector v, where the ith element has the dimension value per
unit of output type i. This form of value gives rise to an ordering among
product bundles, bb0, defined as vb vb0.26 The value form enables
therefore rational comparisons between economic alternatives. In market-
based economies, this form is a necessity or else any agent could through a
sequence of exchanges with product bundles end up with greater quantities
of each output-type than they started out with, see also [Cockshott, 2009,
Cockshott et al., 2004].
The linear value form appears economically sound and intuitive, since
we are used to prices working this way. But in more general terms of prod-
uct bundles, one could conceive of other, nonlinear value forms that enable
an ordering of bundles. Indeed, it is not clear whether the linear form is
necessary in a non-commodity producing economic system. We leave it as
a challenge to the readers to either prove that this form of value arises as a
necessary property of social reproduction, or conversely to demonstrate that
other, nonlinear forms would be appropriate in post-capitalist economy.
6 Conclusion
We have shown that when production requires a workforce that consumes
a part of the net product, economic value can be derived as a fundamen-
tal property of a self-producing economic system. This derivation led us
back to the classical conception of value, with implications for inequality,
productivity, employment and distribution of surplus.
Classical labour values 13
The classical economists assumed social labour as the basis of value on
the ground that it was the original social cost of all produced commodities,
see [Ricardo, 1817]. An early attempt to instead deduce this relation can be
found in [Marx, 1867, ch. 1] and was based on the observation that labour is a
universal input that enters directly or indirectly into the production of every
output-type. This line of reasoning was, however, necessarily incomplete
since there are several other inputs that are universal in the same sense
[Sraffa, 1960, ch. 2].
By contrast, [Marx, 1885, pt. III] pioneered an analysis of economic re-
production that forms the foundation of this paper. Embedded in this anal-
ysis is the necessary consumption of the workforce and, as we have shown,
consequently the classical conception of value in its general form.
1[Steele, 1981] provides a review of criticisms against the idea that economic value
could exist outside markets.
2In the analysis of the Russian Marxist economist Rubin,
We moved from physiologically equal labour to socially equated labour, and
from socially equated to abstract universal labour. We enriched our defini-
tion of labour by new characteristics in the three stages of our investigation
and only when we moved on to the third stage and defined labour as ab-
stract universal, from which the category of value must necessarily follow,
was it possible for us to move from labour to value. We could define abstract
labour approximately as follows: Abstract labour is the designation for that
part of the total social labour which was equalised in the process of social
division of labour through the equation of the products of labour on the
market. [Rubin, 1978]
This line of thought has been advanced by the so called value-form school [Heinrich, 2012].
3What Marx called ‘use values’ [Marx, 1867].
4This is systematised in the international system of bar codes which associates a 12
digit number with each product kind.
5The consumption rate vector κcan be estimated from national accounts data using
the inputs to the household sector and the total wage bill.
6See product-product input-output matrices from national statistics agencies.
7The definition can be extended to the case of joint production with ddistinct sectors
operating with activity levels q, in which the net product equals n= (BA)q, where
Aand Bare input and output matrices. We assume that each output iis seperately
reproducible so that the exists a set of activity levels qi, for which niei= (BA)qi.
Therefore the inverse of (BA) exists and nrequires `q=`(BA)1nunits of labour,
which consumes a total bundle κ(`q) = κ`(BA)1n=Rn, where R=κ`(BA)1.
In the case of single outputs, considered in the text, we have B=I.
8The example is adapted from [Wright, 2017] but is designed to resemble the structure
of reproduction schemes considered in [Marx, 1885] where Departments I, IIa and IIb
correspond to ‘iron’, ‘corn’ and ‘sugar’, resepectively.
9More specifically, σ[0,1). Note that Marx’s unbounded ‘rate of surplus value’
vRn =σ
1σ[0,) is a mere transformation of σ.
10Eq. (4) can be rearranged into [(1 σ)vvR]n=0, which holds irrespective of the
composition of n. This corresponds to an eigenequation λv=vR, where λ1σis an
eigenvalue that is obtained as the solution to det(λIR) = 0. Using the matrix determi-
nant lemma, this is equivalent to (1 `(IA)1κλ1)λd= 0. Since σ= 1 corresponds to
a workforce that does not consume anything, λ= 1σ= 0 is an economically meaningless
eigenvalue and only λ=`(IA)1κ>0 is a meaningful solution. Next, after inserting
(2) into the eigenequation, we have that
λv= (vκ)`(IA)1(9)
so that v`(IA)1is a nontrivial solution after dividing (9) by λ > 0. Note that
the solution is invariant to the consumption-rate vector κ, which may vary with the
Classical labour values 14
distribution of net production. This is of course the classical unit labour values as defined
in the standard literature [Pasinetti, 1979].
11From the eigenequation (9) we derived the eigenvector v=`(IA)1. Using the
series expansion (IA)1=P
k=0 Ak, it follows that v=P
k=0 λ(k) and proves (5).
12The derivation of vfrom (4) is based on the decomposition of the net product, and is
not interpreted by evaluating ‘inputs and outputs’ in production (whether ‘simultaneous’
or ‘sequential’ evaluation), see Section 9 in the review by [Foley, 2000].
13Thus classical labour values can be understood as a field theory of value rather than
a substance theory of value, contrary to the characterization in [Mirowski, 1989].
14That is, a standard choice of numeraire is vn =L, where L= (λ(0) + λ(1) + λ(2) +
· · · )nis the total units of labour required.
15Classical labour values therefore differ radically from the concept of ‘value’ developed
by the so-called value-form school. In the latter conception, there can be no abstract
labour measured in hours nor can it be measured before the act of market exchange
[Heinrich, 2012, pp.50, 55, 65].
16Since market prices are randomly fluctuating quantities they do not form a ‘dual
system’ with repect to labour values.
17[Friedman and Baker, 2009] gives several examples of scheduling constraints on new
gun mountings, and slip sizes affecting UK destroyer construction plans in WWII. [Friedman, 2015]
gives the example of construction of the Admiral class capital ships being postponed due to
there not being enough shipbuilding labour to both build them and destroyers in 1917. For
large scale shipbuilding programmes, even in peace, similar forward planning of physical
constraints has to be done by the state [Arena et al., 2005].
18“The value of a commodity would therefore remain constant, if the labour time re-
quired for its production also remained constant. But the latter changes with every vari-
ation in the productiveness of labour. This productiveness is determined by various cir-
cumstances, amongst others, by the average amount of skill of the workmen, the state
of science, and the degree of its practical application, the social organisation of produc-
tion, the extent and capabilities of the means of production, and by physical conditions.”
[Marx, 1867, ch. 1]
19The total employment requirement for producing niunits of output iis Li=vini.
Therefore the relative change of employment is given by the identity ˙
where gi= ˙ni/ni. If the actual employment is fixed, then the left-hand side is 0 and
correspondingly gi=ρi.
20An equivalent definition, which does not require rearranging the sectors, is that output
iis basic if e>
A2+· · · +e
Ad)>0. We are naturally assuming that all consumption
goods require some amount of direct labor. The concept is a slight generalization of Sraffas
‘basic goods’ and includes the production of the workers’ consumption bundle. Note that
the outputs that are basic and nonbasic may change over time as the structure of the
economy changes, see [Cockshott and Zachariah, 2006].
21Using the inverse of the upper block triangular matrix (IA), we have that
Then it follows that σ= 1 `(IA)1κ= 1 vbκb.Note that we also have
vR =κ`(IA)1
0 0
Then the left eigenequation λv=vR yields two equations: λvb=vbRband λvu=vbRbu,
and consequently σ= 1λis determined by the basic outputs Rb. It is seen that the theory
of surplus value is only completed in the analysis of input-output relations [Marx, 1885,
pt. III] rather than the presentation in [Marx, 1867].
Classical labour values 15
22This corresponds to distinction between ‘absolute’ and ‘relative’ surplus value de-
scribed in [Marx, 1867]. Note that the nonbasic sector therefore cannot produce ‘relative’
surplus value, see [Cockshott and Zachariah, 2006].
23Using (4), we have vs =vn vRn =σvn = 0, when σ= 0. Since v>0and s0it
follows that s=0. By definition, s=nRn = (IR)n. Using the partitioning of Rin
(11), it follows that the net production of nonbasic outputs is a surplus product, that is,
nu=su. Using q= (IA)1n, we have that the activity levels in the nonbasic sectors
equal qu= (IAu)1nu= (IAu)1su=0.
24Consider redeploying the resources devoted to support the nonbasic sector to the basic
sector alone. Let the net product before and after the change be nand n0, respectively,
where total employment remains the same, i.e. vn0=vn. Suppose the redeployment is
such that the net product in the basic sector is increased uniformly by a factor α, i.e.,
n0=(1 + α)nb
Then it follows that the factor is α=vunu
vbnb. The resulting change in the surplus product
of the economy is
= (IR)n0(IR)n
= (IR)αnb
where the top rows correspond to the basic sectors.
25A early step towards such an analysis can be found in [Marx, 1885, ch. 20].
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Conference Paper
Full-text available
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This is a discussion of how best to reconcile the demands of war and the claims of private consumption.
Of the Division of LabourOf the Principle which Gives Occasion to the Division of LabourOf the Natural and Market Price of CommoditiesNote
Britain's war machine: weapons, resources, and experts in the Second World War
  • D Edgerton
[Edgerton, 2011] Edgerton, D. (2011). Britain's war machine: weapons, resources, and experts in the Second World War. Oxford University Press.
The British Battleship: 1906-1946
  • N Friedman
[Friedman, 2015] Friedman, N. (2015). The British Battleship: 1906-1946. Naval Institute Press.
An Introduction to the Three Volumes of Karl Marx's Capital
  • M Heinrich
[Heinrich, 2012] Heinrich, M. (2012). An Introduction to the Three Volumes of Karl Marx's Capital. Monthly Review Press.