HIAS-E-50

One million miles to go: taking the axiomatic road todefining exploitation
Roberto Veneziani
School of Economics and Finance, Queen Mary University of London, Mile End Road, London E1 4NS, UK

Naoki Yoshihara
The Institute of Economic Research, Hitotsubashi University, Naka 2-1, Kuni- tachi, Tokyo 186-8603, Japan

October 2015

Hitotsubashi Institute for Advanced Study, Hitotsubashi University
2-1, Naka, Kunitachi, Tokyo 186-8601, Japan
tel:+81 42 580 8604
http://hias.ad.hit-u.ac.jp/
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One million miles to go: taking the axiomatic
road to deﬁning exploitation
Roberto Veneziani†

Naoki Yoshihara‡

October 29, 2015

Abstract
This paper analyses the Marxian theory of exploitation. The axiomatic approach standard in social choice theory is adopted in orderto study the concept of exploitation - what it is and how it should
be captured empirically. Two properties are presented that capturesome fundamental Marxian insights. It is shown that, contrary to thereceived view, there exists a nonempty class of deﬁnitions of exploitation that preserve the relation between exploitation and proﬁts - called
Proﬁt-Exploitation Correspondence Principle - in general economies
with heterogeneous agents, complex class structures, and production
technologies with heterogeneous labour inputs. However, among themain approaches, only the so-called ‘New Interpretation’ satisﬁes theProﬁt-Exploitation Correspondence Principle in general.
JEL: B51.
Keywords: Exploitation, proﬁts, axiomatic analysis.
We thank Gary Dymski, Don Katzner, Gil Skillman, Peter Skott, two anonymousreferees, the editors of this journal, and participants in the Analytical Political Economy
workshop and the American Economic Association conference (Philadelphia) for usefulcomments and suggestions. We worked on this paper while visiting the University of
Massachusetts, Amherst: their support and hospitality is gratefully acknowledged. The
usual disclaimer applies.
†(Corresponding author) School of Economics and Finance, Queen Mary University of
London, Mile End Road, London E1 4NS, United Kingdom (r.veneziani@qmul.ac.uk)‡
The Institute of Economic Research, Hitotsubashi University, Naka 2-1, Kunitachi,
Tokyo 186-0004, Japan (yosihara@ier.hit-u.ac.jp)

11Introduction

After a wave of intense research and debates in the 1970s and 1980s, theMarxian theory of exploitation has moved to the sidelines of academic economics.1 The notion of exploitation has never had much appeal for mainstream economists. But heterodox scholars, in the main, also seem sceptical
about the possibility of developing a logically coherent and empirically meaningful concept of exploitation. Two broad (and related) sets of objections
have been moved against the Marxian theory of exploitation.
First, it is unclear what Marxian exploitation actually is. At a generallevel, it can be deﬁned as the (forced) extraction of surplus labour, or asa power relation leading to systematic dierences between the labour contributed (in some relevant sense) by workers in productive activities and thelabour that they receive (in some relevant sense) in return. As intuitive as
these general formulations are, outside of stylised two-class economies withlinear technologies and homogeneous labour, the notions of surplus labour, orof labour contributed and received, have no obvious interpretation. Dierent
deﬁnitions of exploitation can be, and have in fact been proposed that rely ondierent interpretations of these concepts (e.g., Okishio [33]; Morishima [31];Roemer [37]; Flaschel [8]; Duménil [5]; Foley [10]). Actually, as Steedman[43] famously argued, it is not even clear that these concepts - and, a fortiori,
Marxian exploitation - can be coherently deﬁned.
Second, it is unclear what the concept of exploitation does. The conceptual problems, anomalies and counterexamples identiﬁed in the literature
have led to increasingly complex and often counterfactual deﬁnitions thathave progressively lost intuitive appeal, and also their applicability to actual capitalist economies, calling into question the usefulness of the notionof exploitation. At a methodological level, this is often attributed to themathematical turn taken by the debate on exploitation in the 1970s, which,according to critics, has shifted the focus from the broader conceptual issues
to relatively minor technical details.
This paper defends the relevance of Marxian exploitation in general production economies with heterogeneous labour, while adopting a formal ap1

To be sure, research on exploitation theory has not stopped. See, for example, recent debates on ﬁnancial exploitation and expropriation (Lapavitsas [18, 19]; Fine [7])and on the so-called Temporal Single-System Interpretation of value theory (Freeman andCarchedi [13]; Mohun [29]; Veneziani [46]). Nonetheless, these debates have had a relatively limited impact outside the conﬁnes of Marxist economics.

2

proach. First, a deﬁnition is proposed, which is conceptually related to the‘New Interpretation’ (Duménil [5]; Foley [10]; Duménil et al [6]). According
to this deﬁnition, an agent is exploited (resp., an exploiter) if and only if
the labour she contributes is greater (resp., lower) than the share of aggregate social labour that she receives via her income. This approach deﬁnes
exploitation as a feature of the competitive allocation of social labour rather
than as the result of productive ine!ciencies, or labour market imperfections.
Unlike the received deﬁnitions, it has a clear empirical content, for it is ﬁrmly
anchored to the actual data of the economy. Further, we show that it preserves some widely shared positive and normative intuitions concerning thenotion of exploitation, including the existence of a robust relation between
exploitation and proﬁts, in economies with heterogeneous labour. This is an
important and surprising result.
At a general, theoretical level, it is often argued that the concept of exploitation is central in Marxian economics because it links the visible, epiphenomenal characteristics of capitalist economies (as revealed by monetary variables) with deeper, structural forces and mechanisms. Whether proﬁts areindissolubly linked to exploitation is important, in this perspective, both if
exploitation is meant to uncover the fundamental forces underlying the dynamics of accumulation in capitalist economies; and if it is an essentiallynormative criterion to evaluate, and indict, capitalism. For, given private
ownership of productive assets, one should expect proﬁts to be a counterpartof the transfer of social surplus and social labour from workers to capitalists,and one of the causes of inequalities in well-being freedom.
The existence of a relation between exploitation and proﬁts has been famously proved by Okishio [34] in linear economies with homogeneous labour.
Given its relevance the result has been dubbed the Fundamental Marxian
Theorem (henceforth, FMT), and has sparked a substantial literature exploring its generality and relevance.2 While retaining the assumption ofhomogeneous labour, Steedman [43] famously proved that, under Okishio’s
standard deﬁnition, the FMT does not hold in von Neumann economies withjoint production. Morishima [31] and Roemer [37] proposed two alternative
deﬁnitions of exploitation that meet Steedman’s critique. Nonetheless, theseresults are far from conclusive: Yoshihara and Veneziani [53] have shown that2

The literature is too vast for a comprehensive list of references. In addition to theclassic contributions cited in the text, it is worth mentioning the more recent discussions
by Mohun [29]; Flaschel [8]; Yoshihara and Veneziani [53].

3neither Morishima’s [31] nor Roemer’s [37] deﬁnition preserves the FMT in
economies with a convex technology and homogeneous labour. Perhaps more
importantly, the robustness of the FMT in more general economies with heterogeneous skills and labour inputs is an open question.3 Indeed, it is often
argued that heterogeneous labour poses intractable problems for Marxian
exploitation theory.
As explained in section 4 below, our analysis bears only a broad conceptual relation with the FMT literature. Yet, in the light of the debates on
the FMT, it is remarkable that, if the New Interpretation is adopted, then it
is possible to establish a robust relation between exploitation and proﬁts ingeneral economic environments with heterogeneous labour. Indeed, the New
Interpretation is the only deﬁnition - among the main ones in the literature with this property. Veneziani and Yoshihara [49] established this conclusionin economies with homogeneous labour inputs, and focusing on a speciﬁc
set of equilibrium allocations. Here we signiﬁcantly generalise their analysis
by focusing on a larger set of economies and allocations, and tackling theconceptually fundamental and formally more complex case of heterogeneous
labour.4
The second contribution of the paper is methodological. An axiomatic
approach is developed, whereby the desirable characteristics of a deﬁnitionof exploitation are stated formally. The axiomatic approach is standard inseveral ﬁelds in economics, most notably in social choice theory. Yet, as thisis one of its ﬁrst applications to Marxian and classical themes,5 the next
subsection addresses some methodological issues related to the use of mathematics in economics. The aim is not to provide a general methodologicaldiscussion, but rather to explain why the axiomatic approach adopted inthis paper is both appropriate and insightful in the analysis of the conceptof exploitation. The uninterested reader may safely skip to section 2.
3

The FMT has been extended to consider heterogeneous labour (e.g., Fujimori [14];Krause [16]; Bowles and Gintis [2]), but only in the context of standard linear economies.
4
The axioms and deﬁnitions in Veneziani and Yoshihara [49] can be obtained as a
special case of those presented here, if one assumes homogeneous labour and focuses on
equilibrium allocations.
5
Notable exceptions include Yoshihara [51]; Veneziani and Yoshihara [49]; Flaschel etal. [9].

4

1.1

Taking the axiomatic road

This paper addresses the issue of the appropriate deﬁnition of exploitationand measure of exploitative relations in capitalist economies. The motivationof our analysis is not the lack, but the wealth of plausible candidates: as noted
above, many dierent deﬁnitions can be, and have in fact been proposed,
which incorporate dierent positive and normative intuitions.
The fundamental question is how to choose among all of the existing andthe conceivable deﬁnitions. Thus far, the debate has largely been reactive:
new deﬁnitions have often emerged as the product of a process of adjustmentof the theory to various anomalies and counterexamples identiﬁed in theliterature. We adopt a dierent approach. Rather than proposing anotherdeﬁnition, and comparing it with the existing alternatives, we develop an
axiomatic framework to analyse what exploitation is, and how it should bemeasured. The axiomatic method is used to rigorously and explicitly state
the normative and positive foundations of the notion of exploitation.6
The starting point of our analysis is that the concept of exploitation has aquantitative dimension. This is not to say that exploitation can or should be
reduced to a quantitative phenomenon. Purely distributive approaches that
reduce exploitation to an inequality in productive assets (such as Roemer’s[37] property relations approach), for example, are ultimately unsatisfactory.
For exploitative social relations arguably involve some form of power, force,or coercion, which need not be clearly measurable (Veneziani [47, 48]).
Yet the concept of exploitation also has an inherently quantitative dimension, such that it is meaningful to say that “economy A has become signiﬁcantly more exploitative over the past four decades”, or that “exploitation isworse in country B than in country C”. Exploitation diagnoses the process
6

A point of language should be clariﬁed here. The term ‘axiomatic’ is sometimes used
to deﬁne any formal, deductive reasoning starting from given premises (axioms). From
this perspective, all mathematical Marxian economics is ‘axiomatic.’ We use the term
axiomatic in a narrower sense to denote the method that is standard in social choice theory
and inequality analysis (and therefore a strict subset of formal techniques). In his analysisof the ‘axiomatic method’ in social choice theory, Thomson ([45], p.332) puts it thus: “Anaxiomatic study has the following components: 1. It begins with the speciﬁcation of adomain of problems, and the formulation of a list of desirable properties of solutions forthe domain. 2. It ends with ... descriptions of the families of solutions satisfying various
combinations of the properties.” The earlier literature on Marxian exploitation theory(formal or otherwise) was not axiomatic in this sense, as it moved from the speciﬁcation ofa domain of problems (economies) directly to the proposal of a given solution (deﬁnitionof exploitation).

5through which “certain inequalities in incomes are generated by inequalities
in rights and powers over productive resources: the inequalities occur, inpart at least, through the ways in which the exploiters, by virtue of their
exclusionary rights and powers over resources, are able to appropriate laboureort of the exploited” (Wright [50], p.1563). This paper focuses precisely
on the quantitative dimension of exploitation and on the most appropriate
way of capturing this aspect of exploitative social relations.
It may be objected that the possibility of measuring a certain phenomenon does not imply that formal tools should be used to study it.7 Criticalrealism, for example, has long argued against the mainstream insistence thatmathematical methods be used always and everywhere in economics. According to critical realists, social reality is most plausibly construed as an
open, structured, dynamic and internally related system and mathematicaldeductivist methods are inappropriate in the causal-explanatory analysis ofopen systems (Lawson [22, 24]).
A thorough discussion of the role of formal tools in social theorising goesbeyond the scope of this paper. But it is worth explaining why the main
objections against the use of mathematics in economics do not apply here.
This paper examines the category of “exploitation”, in order to understand what it means, and how it can be captured empirically. Our theoreticaleort can be conceived of as philosophical underlabouring a central conceptof Marxian economics and we use the axiomatic method typical of social
choice theory in order to identify the desirable properties of a deﬁnition (andmeasure) of exploitation. The analysis is not based on any assumptions whether explicit or implicit - about the nature of social reality as a closed
system, or about the pervasiveness of strict regularities and constant conjunctions of events.8 Neither closures of causal sequence - where some event
y is causally conditioned by some other event(s) x (Lawson [22], p.15) - nor
the weaker closures of concomitance - “where the events [x and y] are correlated, but where neither causally conditions the other” (ibid.) - underlie
the use of formal tools in our analysis.9 As we are trying to uncover neither
7
Conversely, however, the fact that some variables are not measurable does not ruleout a priori the use of formal tools (Katzner [15]).
8
It is worth stressing that ours are not general methodological claims about the intrinsic
properties of the axiomatic method (as a subset of the set of possible mathematical techniques) in the analysis of social reality. Rather, the point is that the axiomatic approach
is appropriate in this speciﬁc context, given the nature of the problem investigated.
9
If one stipulates a property of a deﬁnition (and measure) of exploitation whereby,6causal mechanisms nor correlations of events, our formal analysis does notpresuppose the existence of either type of closure.10
Ours is an investigation in scientiﬁc ontology and in this respect it issimilar in nature to Sen’s [39] analysis of the concepts of functionings andcapabilities (see Martins [25, 26]).11 Because of the nature of our inquiry,
the axiomatic framework does not incorporate any hypotheses concerningcausal laws (or even tendencies) within a predictionist perspective (Lawson[22], p.60). Rather, it aims to clarify the social category of exploitation andto capture exploitative relations a posteriori, focusing on the state of theeconomy at a given point in time. It is therefore conceptually analogous to
the approach used to identify appropriate measures of poverty and inequality(Foster [12]), or labour productivity (Flaschel et al [9]).
The axioms are thus abstract in that they incorporate relevant philosophical views about the nature and the positive and normative foundations of
Marxian exploitation. But they are also empirically oriented, in that they focus on observable magnitudes, and do not rely on “claims that are believed to
be false of our world, and of any really possible counterfactual world” (Lawson [23], p.766). Indeed, the key condition for the relevance of our axiomatic
analysis is that the main variables (prices, wages, output, labour time, andso on) be conceptually well-deﬁned, meaningful, and empirically measurable
within the appropriate temporal and spatial context.12
The axiomatic framework also precisely identiﬁes the domain and scope of
the analysis by focusing on a set of economic agents and on a class of economic
exploitation (y) is said to exist whenever aggregate proﬁts are positive (x), then this isneither a claim about proﬁts ‘causing’ exploitation, nor a claim about the existence of anempirical regularity whereby x and y occur together. It is a deﬁnitional claim stating that
if we observe x then an appropriate measure of exploitation should yield y.
10
Nor does it presuppose the existence of “conditional closures” (Setterﬁeld [41]), “quasiclosures” (Downward et al. [4]) or “local closures” (Lawson [21]).
11“[A]n approach which is centred on the characterisation of objects for ethical valuation, such as ‘functionings’ and ‘capabilities’, is essentially an ontological approach, which
provides the objects that we may then use when engaging in democratic ethical valuation.”(Martins [26], p.150).
12
The same condition is necessary for the use of descriptive statistics in empirical analyses advocated by Lawson ([20], p.221), or for the adoption of econometric techniques
within a triangulation strategy consistent with a critical realist outlook (see, for example,
Part II of Downward [3]). As Lawson ([20], p.69) puts it, “the determination of summary
statistics of a body of data such as means or growth rates of some phenomenon can be
included under the rubric of econometrics. Such activities are not being questioned here... . The speciﬁc realist emphasis that I am pursuing is not an anti-empirical one.”7

environments and allocations. But this by no means entails a reductionist
perspective. On the one hand, no strong restrictions are imposed on agent
behaviour or on the institutional mechanisms (market-based or otherwise)
regulating economic interactions. The axioms incorporate no assumptions
concerning individuals’ ontology (including their selﬁsh or ethical motivations, the origin of their preferences, or the notion of individual rationality),
the nature of social interactions, and so on. On the other hand, although
the axioms focus on the exploitation status of individuals at a given point
in time, they do not entail a commitment to atomism, to a static view ofsocial reality, or to the existence of a ﬁxed, given unit of analysis. Dependingon the object of analysis, the axiomatic system can be modiﬁed to incorporate aggregate properties relating, for example, to the exploitation status
of social classes, or to the overall level of exploitation in the economy, andwithout reference to individuals; or relational properties incorporating theidea that exploitation is a social relation such that an agent is exploited if
and only if there is someone exploiting her. But one can also introduce dynamic properties focusing, for example, on the relation between exploitation
and accumulation, or speciﬁcally concerning the measurement of exploitation when agents save; or distinguishing between the exploitation status ofindividuals within each period and over the course of their entire lives.
To be sure, the intuitive appeal of a deﬁnition is fundamental, and a certain deﬁnition should provide the right answers in situations in which we feelthat intuition is a reliable guide. The axiomatic method is not “a substitutefor intuition ... but instead ... a way of articulating [the intuitions that holdin speciﬁc situations] into operationally useful conditions pertaining to an
entire class of cases” (Thomson [45], p.356). Indeed, as shown by the often
surprising impossibility results obtained in social choice theory, or by the very
di!culty in providing a deﬁnition of exploitation that preserves key Marxian
insights, intuition alone can be insu!cient. When delineating the propertiesof the appropriate deﬁnition of exploitation, “Informal insights, important asthey are, cannot replace the formal investigations that are needed to examine the congruity and cogency of combinations of values and of apparently
plausible demands” (Sen [40], p.353).
Of course, the axiomatic method does not necessarily lead to univocal
conclusions: one can reject any of the axioms below, and propose a new one,possibly leading to an alternative deﬁnition of exploitation. This indeterminacy is not a property of axiomatic analysis per se: it is inherent in allsocial theorising and it simply reﬂects dierent positive and normative in8

tuitions.13 If anything, the axiomatic method has the advantage of making
such intuitions explicit, thus forcefully directing research and debate to thefoundational issues concerning the nature and measurement of exploitation.
This is an important point. Unlike in much of the mainstream, in thispaper the axiomatic method is not used in a purely instrumental way, forexample in order to generate predictions that match empirical data, and regardless of the actual relevance of the axioms (Lawson [22, 24]). The content
of the axioms is central to our analysis, and the relevance and meaningfulness of the conclusions ultimately depends on it. For, “The relevance of anaxiomatic result depends entirely on the acceptability or usefulness of its
constituent properties” (Foster [12], p.367).

2

Economic states

The aim is to analyse exploitation without imposing signiﬁcant restrictions
on agents’ behaviour, market structure, and so on. Therefore we keep thedescription of the economic environment to a bare minimum.14 Consider an
economy with N agents, n commodities, and T types of labour. Let N be
the set of agents with generic element D and let T be the set of types oflabour with generic element .
Technology is described by a production set P with elements - activities- of the form k = (kl , k, k) where kl  (kl ) MT is the (nonnegative)
proﬁle of the T labour inputs used in production, measured in hours;15 kis the (nonnegative) proﬁle of the n inputs of produced goods; and k is the(nonnegative) proﬁle of the outputs of the n goods.16 By measuring labour
inputs in terms of time, this description of the technology includes standard
economies as a special case.
For example, suppose that the production of one unit of each good requires a ﬁxed amount of the n commodities and of the T types of labour. As
13

As Sen ([38], p.187) noted in his discussion of the labour theory of value, “it is notreally surprising that dierent conventions for calculating aggregate labour magnitudes
would exist and also appear natural in dierent contexts. The source of these ambiguities. . . [rests] in the basic multiplicity of motivation underlying the labour theory.”
14
The model and notation are a generalisation of Roemer’s [36] classic framework.
15
Throughout the paper, all variables and vectors are assumed to belong to a Euclidean
space Rk of appropriate dimensionality k.
16
All variables should be dated. However, because we analyse exploitative relations at
a given point in time, we drop the time subscript for notational convenience.

9

in standard input-output theory, production techniques can be represented
by a pair (A, L), where A is a n × n non-negative (productive and indecomposable) Leontief matrix of material coe!cients and L  [L1 , ..., Ln ] isa T × n non-negative matrix of direct labour coe!cients (with at least one
strictly positive element in each row and column), where Li is a vector describing the amount of each type of labour necessary to produce one unitof good i. Then, P  {k = (kl , k, k) | <x = 0 : (Lx, Ax, x) = k} andwe say that P is representable by a Leontief production technique (A, L).
The net output vector arising from k is denoted as kb  k  k and theset of e!cient activities in P is denoted as YP .17 In the rest of the paper, we
assume that: technology displays constant returns to scale; ﬁrms can decide
not to activate any process; and the production of any output requires somelabour and some capital.18 These restrictions are rather mild and standard
in heterodox (and even mainstream) approaches.
Each agent D is endowed with a (nonnegative) vector of n productive
assets, / D , and a nonempty set of types of labour T D  T that can beused in production. The total amount of time that each D can use either
productively (possibly in dierent types of labour) or in leisure activities, isnormalised to one. Let C be the set of all conceivable choices of each agent
with generic element (cD , bD ), where cD is a nonnegative vector describing
D’s consumption of the n goods and bD = (bD1 , ..., bDT ) describes the amountof time of each type of labour spent by D in productive activities, whereD
bD 5 [0, 1] for all types of labour 5 TP
that D is endowed with; bD = 0 forDD
all other labour types 5 T \T ; andMT D b 5 1.
Let (p, w) be the (row) vector describing the (positive) prices of the ncommodities and the (nonnegative) wages of the T types of labour. Let Z max
be the maximum proﬁt rate that can be obtained from production activities
at prices (p, w), and let P Z (p, w) be the set of production processes that yield
the maximum proﬁt rate at (p, w).19
Our analysis does not depend on any speciﬁc assumptions on individual behaviour, or on the institutional framework in which agents interact.
However, both the key axioms and the main results apply only to a class of
economic scenarios which may, or may not, turn out to be true ex post. Yet
they are su!ciently general and theoretically relevant to warrant investiga17
18
19

Formally: CP  { 5 P | @0 5 P such that 0i > i for all i}.
These assumptions on P are stated formally in appendix
A.1.
n

o
l
l
Formally, max = max5P pew
and P  (p, w) =  5 P |  max = pew
.
p
p

10

tion and to cover a large set of possible cases.
An economic environment is a set of agents, N ; a production set, P ;a consumption space, C; a proﬁle of agents’ labour endowments, (T D )DMN ;and a proﬁle of agents’ endowments of productive assets, (/D )DMN . Given an
economic environment, an economic outcome is a price vector, (p, w), a proﬁleof consumption and labour decisions (cD , bD )DMN , and a proﬁle of production
activities
by agents (kD )DMN , with aggregate production activity,
P operatedp,w
D
20
k = DMN k , such that:
P (i) Daggregate net output can at least provide
p,w
for consumption, kb=
DMN c ; (ii) aggregate proﬁts are nonnegative,
pb
kp,w  wkp,w
=
0;
(iii)
aggregate
production maximises the proﬁt rate,
l
P
p,w
D
p,w
Z
k 5 P (p, w) with kl = DMN b ; and (iv) individual expenditure does
not exceed individual (capital and wage) income, pcD = Z max pkD + wbD foreach agent D. A pair of an economic environment and an economic outcome
is an economic state (henceforth, ES).
The concept of ES is very general, with no substantive restrictions on
behaviour, technology or institutions. Conditions (i) and (ii) are hardly demanding. Condition (iii) is only slightly more restrictive in that it postulates
that producers activate proﬁt-rate-maximising processes. This is consistent
with Marx’s ([28], ch.10) analysis of capitalist behaviour and is common inheterodox approaches (including Sra!an price theory). It is also theoretically justiﬁed because it is desirable to obtain a deﬁnition that can capture
exploitation under optimal capitalist behaviour. In Marxian theory, exploitation is the product of capitalist social relations, rather than mistakes, technical ine!ciencies, or market imperfections.21 Condition (iv) postulates that
individual expenditure is subject to the budget constraint. This yields no
signiﬁcant loss of generality and it does not rule out savings, as the bundle
cD is not necessarily restricted to consumption goods and services. Further,
it is theoretically justiﬁed because Marxian exploitation is the product of
capitalist social relations, and in particular of the wage relation, rather thancredit markets and individual life-cycle decisions.
The notion of ES does not incorporate any of the standard featuresof mainstream models such as utility functions, selﬁsh optimising (or evenboundedly rational) consumers, and dierentiable production functions. The
20
For all vectors x, y 5 Rn , x = y if and only if xi = yi (i = 1, . . . , n); x  y if and only
if x = y and x 6= y; x > yP
if and only if xi > yi (i = 1, . . . , n).

21
The equality p,w=
5N  does not imply any form of labour market equilibriumlor optimising behaviour. It simply states that the labour used in production is performedby the agents in the economy.

11

production set is su!ciently general to allow for the existence of a dierentiable production function, but it does not postulate it and nothing depends
on neoclassical assumptions of any sort. Nor does the deﬁnition of ES rely on
the concept of equilibrium. According to Lawson [24], a focus on equilibrium
is a requirement of mathematical models proceeding in deductive mode to
specify causal inﬂuences, but our inquiry is of a dierent, ontological nature.
Perhaps more importantly, the concept of ES is not a description of thefunctioning of the economy: how it is structured, what agents do or how theychoose, and so on. Rather, it acts formally as a domain condition: it deﬁnes
the scope and boundaries of the analysis. Social scientiﬁc theorising is always
context-speciﬁc, and the deﬁnition of ES makes the context explicit. It is
important to note, however, that such boundaries are rather wide indeed (andsigniﬁcantly wider than in the literature), for conditions (i)-(iv) can obtainin a range of economies and under many dierent assumptions concerning,e.g., institutions and behaviour.22 Further, although it is an open question
whether our conclusions hold even more generally, the key insights of thepaper are robust to several changes in the deﬁnition of ES.23

3

Deﬁning Marxian exploitation

In Marxian theory, exploitation can be seen in two related but slightly different ways. One can focus on the dierence between the labour performed
by workers and the labour socially necessary to produce the goods they consume. Or, one can conceptualise exploitative social relations as characterised
by systematic dierences between the labour that agents ‘contribute’ to theeconomy and the labour they ‘receive’. As soon as one moves away from thesimplest linear economies, none of these concepts is clearly deﬁned and, as
already noted, various deﬁnitions have been proposed which reﬂect dierent
views concerning the concept of exploitation.
Most (though not all) of the previous debates have revolved around theappropriate deﬁnition of socially necessary labour time, or of the labour received by agents. In order to focus on this issue, contributors have analysed
22

The set of economic states is nonempty. It contains, for example, generalisations of
Roemer’s [36, 37] economies as special cases (see the Addendum).
23
For example, all results continue to hold if condition (i) is weakened to require that
aggregate net output is at least as high as the consumption of any individual agent (bp,w =c for each ), or, indeed, that it is strictly positive.

12

economies with homogeneous labour, in which the labour performed by workers, or the labour that they contribute to the economy is uncontroversial: itcorresponds to the labour time spent in productive activities. In more general
economies, however, these concepts are not trivial.
In this section, we extend some of the main deﬁnitions to economies withheterogeneous labour. The key step is to acknowledge that although thenotion of labour performed, or contributed is theoretically important, thedierences between alternative approaches lie elsewhere. Therefore we adopt
a common deﬁnition of the labour performed, or contributed as the value oflabour time spent in production. Formally, at any ES, the amount of labour
contributed by agent D who supplies bD is equal to wbD .24
This approach is consistent with the classical economists’ view on how
to convert dierent types of labour into a single unit, whereby “the dierent
kinds of labour are to be aggregated via the (gold) money wage rates” (Kurzand Salvadori [17], p.324). According to Smith, for example,“It is often di!cult to ascertain the proportion between twodierent quantities of labour. The time spent in two dierent
sorts of work will not always alone determine this proportion.
The dierent degrees of hardship endured, and of ingenuity exercised, must likewise be taken into account. There may be morelabour in an hour’s hard work, than in two hours easy business; orin an hour’s application to a trade which it cost ten years labour
to learn, than in a month’s industry, at an ordinary and obvious
employment. But it is not easy to ﬁnd any accurate measure
either of hardship or ingenuity. In exchanging, indeed, the dierent productions of dierent sorts of labour for one another, someallowance is commonly made for both. It is adjusted, however,not by any accurate measure, but by the higgling and bargainingof the market” (Smith [42], ch.V, pp.34-35).
One can similarly interpret Ricardo’s ([35], ch.I, section II, p.11) arguments that “The estimation in which dierent quantities of labour are
held, comes soon to be adjusted in the market with su!cient precision ... ,24

It is worth stressing that w can be interpreted either as the vector of nominal wages,
or as expressing the ratios of the wages of each type of labour relative to some benchmark(for example, the wage of simple labour or the average wage). All of our deﬁnitions,axioms, and conclusions remain unchanged under either interpretation.

13

and depend much on the comparative skill of the labourer, and intensity of
the labour performed”. Despite some debates on the concept of “abstract
labour”, our approach is also consistent with Marx’s ([27], pp.51-2) views on
the conversion of complex labour into simple labour.25
Indeed, although exploitation theorists often do not provide a complete
extension of their deﬁnitions to economies with heterogeneous labour, in themain they do endorse (albeit sometimes implicitly) the classical economists’view on the use of wages for the reduction of dierent types of labour to asingle unit (Morishima [30], ch.14; Duménil et al. [6]).26
Consider ﬁrst Morishima’s [30, 31] classic deﬁnition. At any ES, according to Morishima, the amount of labour received by agent D, who consumes
cD , corresponds to the minimum amount of labour necessary to produce cD as
net output. Following Roemer [36, 37], we denote this amount as l.v. (cD ; w),that is, the labour value of cD at wages w.27 Therefore:
Deﬁnition 1: Consider any ES. Agent D 5 N , who supplies bD and purchases cD , is exploited if and only if wbD > l.v. (cD ; w) and an exploiter if andonly if wbD < l.v. (cD ; w).
Deﬁnition 1 has some desirable characteristics, according to Morishima([31], pp.616-618): the notion of exploitation is well-deﬁned because l.v. (c; w)
is unique, well-deﬁned and nonnegative.28 Furthermore, although wages arenecessary to convert dierent types of labour into a single unit, exploitation
status can be determined prior to and independent of the prices of commodities, as in classical Marxian theory.29
According to Roemer [36, 37], however, Deﬁnition 1 is conceptually ﬂawed
because it identiﬁes exploitation status based on production techniques that
may never be used by proﬁt-maximising capitalists. Like Morishima [31],Roemer [37] focuses on the bundle cD actually consumed by agents but argues
that its labour content is given by the minimum amount of labour necessary
to produce it as net output using proﬁt-rate-maximising production processes,
25
It is also worth noting that Yoshihara and Veneziani [55] have proved that in economies
with heterogeneous labour the wage-additive measure is the unique measure of labour
content that satisﬁes a small set of theoretically robust and intuitive axioms.
26
For a dierent approach see Okishio [32, 34]; Fujimori [14]; Krause [16].
27Formally, for any ES and any c 5 Rn+ , l.v. (c; w)  min {wl |  5 P & 
b = c}.
28
This follows from assumptions A0A2 in Appendix A.1 (see Roemer [36], Proposition
2.1). The same holds for l.v. (c; p, w) below.
29
Deﬁnition 1 reduces to Morishima’s [30, 31] classic deﬁnition in economies with homogeneous labour where  is a scalar. The same holds for Deﬁnitions 2 and 3 below.

14

for only the latter are activated by capitalists. Following Roemer [36, 37], we
denote this amount as l.v. (cD ; p, w), that is, the labour value of cD at prices
p and wages w.30 Then:
Deﬁnition 2: Consider any ES. Agent D 5 N , who supplies bD and purchases cD , is exploited if and only if wbD > l.v. (cD ; p, w) and an exploiter if
and only if wbD < l.v. (cD ; p, w).
Although they preserve some important Marxian insights, Deﬁnitions 1
and 2 have been criticised because exploitation status depends on counterfactual amounts of labour content (Flaschel [8]). For the production activities
yielding l.v. (cD ; w) or l.v. (cD ; p, w) may be dierent from those actually usedin the economy. According to critics, this use of counterfactuals is theoretically undesirable and makes exploitation an empirically vacuous notion,
since the computation of l.v. (cD ; w) and l.v. (cD ; p, w) requires informationthat is normally unavailable, including, in Morishima’s own words, “information about all the available techniques of production, actually chosen or
potentially usable” ([31], p.617, italics added).31
An alternative approach has been recently proposed by Yoshihara and
Veneziani [52, 51, 49]. Consider any ES with aggregate production activity
kp,w . For any nonnegative vector c, such that pc 5 pb
kp,w , the labour content
cof c is equal to c wkp,w
5 [0, 1] is such that c pb
kp,w = pc.32 Thus,l , where
p,w
the labour content of aggregate net output, kb , is equal to the value of totalp,w
social labour, wkl , and the labour contained in any bundle c (whose value
does not exceed national income) is equal to the fraction c of social labour
necessary to produce a fraction of aggregate net output, c k
b p,w , that has thesame value as c. We denote this amount as l.v. (c; p, w, kp,w ), that is, thelabour value of c at prices p, wages w and aggregate production, kp,w . Then:
Deﬁnition 3: Consider any ES. Agent D 5 N , who supplies bD and purchases cD is exploited if and only if wbD > l.v. (cD ; p, w, kp,w ) and an exploiter
if and only if wbD < l.v. (cD ; p, w, kp,w ).
Deﬁnition 3 is conceptually related to the ‘New Interpretation’ of Duménil [5]D
and Foley [10]. In fact, for any agent D, c represents D’s share of national30

Formally,
for any ES
and any c
5Rn+ ,min {wl |  5 P (p, w) & 
b = c}.
31
See Yoshihara and Veneziani [53] for a detailed discussion.
32
If pb
p,w = 0, we set  c = 0 by deﬁnition.

15

l.v. (c; p, w)D

income, and so c wkp,wis the share of (the value of) social labour that Dlreceives by earning the income necessary to buy cD . This is equivalent to “theamount of average social labor workers receive a claim to in the wage for eachhour they actually work - that is, ... the average wage multiplied by the valueof money” (Foley [11], p.43), or the value of labour power, according to the
New Interpretation. Then, “Exploitation through the wage labor relation
occurs when a worker expends more labor hours than he or she receives anequivalent for in wages” (Foley [11], p.122).
As in Roemer’s [37] approach (and unlike in Morishima’s), exploitation
status in Deﬁnition 3 can be identiﬁed only if goods’ prices are known, butsocial relations play a more central role, because the deﬁnition of exploitation
requires knowledge of the social reproduction point, and it is related to the
production and distribution of national income and social labour. Unlike Deﬁnitions 1 and 2, Deﬁnition 3 depends exclusively on empirically observablemagnitudes. Nonetheless, the New Interpretation has been criticised because,
unlike Deﬁnitions 1 and 2, the agents’ actual consumption choices are only
indirectly relevant to determine exploitation status, and unlike Deﬁnition 1,
exploitation status depends on information about commodity prices.
This brief (and admittedly partial) survey shows that there are many
possible approaches to explain what exploitation is and how it should be
captured. The question then is how to adjudicate between them. Somepreliminary answers are provided in the next section.

4

Axiomatising Marxian exploitation

In this section, we discuss two axioms incorporating some key properties
that a deﬁnition of exploitation, and measure of exploitative relations, should
satisfy and analyse their implications.
©
ª
For any ES, let W+  D 5 N | / D = (0, 0, ..., 0)0 & wbD > 0 : W+ is
the set of agents with no initial endowments who supply some (productive)
labour. Our axiomatic analysis focuses on the exploitation status of agentsin W+ . Theoretically, this set is of focal interest from a Marxian perspective
even in societies with a complex class structure: if any agents are exploited,propertyless agents who supply wage labour should be among them. Formally, as argued below, focusing on a strict subset of the set of agents makes
the axiomatic restrictions weak and undemanding.
The ﬁrst axiom is a domain condition capturing some minimal intuitions
16

that represent the core of Marxian exploitation theory and that are sharedby all of the main approaches.
Labour Exploitation with Heterogeneous Labour (LEH): Consider
any ES. Given any deﬁnition of exploitation, there exists a proﬁle of non|W |negative vectors (c1e , ..., ce + ) such that, for any D 5 W+ , pcDe = wbD , andDDD
for some kce 5 YP with kb ce = cDe and kb ce  cDe :cD

D is exploited if and only if wkl e < wbD .

In order to interpret LEH, recall that the exploitation status of agent D
is determined by the dierence between the amount of labour ‘contributed’and ‘received’ by D. As argued in section 3, in the main approaches consistent with the classical economists’ view, the former quantity is given by thevalue of the labour supplied by D, wbD . But there are many possible views
concerning the latter quantity. As a domain condition, LEH provides someminimal, key restrictions on the deﬁnition of the amount of labour that atheoretically relevant subset of agents receives.33
To be speciﬁc, LEH requires that the exploitation status of each propertyless worker D 5 W+ be determined by identifying a nonnegative vector
cDe - call it an exploitation reference bundle (hereafter, ERB) - whose labour
content - the amount of labour that D receives - isD the value of the labourcnecessary to produce the ERB as net output, wkl e . If D supplies wbD , andDcwbD is more than wkl e , then D is regarded as contributing more labour thanD receives, and is thereby exploited.
The ERB must have two properties. First, it must be (just) aordable, atprices p, by a propertyless worker D 5 W+ , who supplies bD units of labour atwages w: pcDe = wbD . This embodies the idea that the amount of labour that
D 5 W+ receives depends on her income, or more precisely, it is determinedby some reference bundle that D can purchase. In standard approaches, the
ERB is the bundle actually chosen by the agent. LEH is weaker in that it
only requires that the ERB be potentially chosen.
Second, the ERB must be technically feasible with an e!cient productionDDprocess: kce 5 YP with kb ce = cDe . This embodies the intuition that theamount of labour received by an agent is related to production conditions.
33

LEH only applies to labour-based deﬁnitions of exploitation. It is not relevant, forexample, for Roemer’s [37] property-relations deﬁnition. Related axioms are analysed in
Yoshihara and Veneziani [52]; Yoshihara [51]; Veneziani and Yoshihara [49].

17

More precisely, LEH states that the ERB be technologically feasible as net
output, and its labour content is the amount of labour socially necessary to
produce it. Observe that the axiom requires that the amount of labour associated with each ERB be uniquely determined with reference to production
conditions, but it does not specify how such amount should be chosen. Theremay be in principle many (e!cient) ways of producing the ERB, cDe , and thuscDof determining its labour content wkl e .34
LEH incorporates the view that the sphere of production is central indetermining exploitation status. ForD technology and production conditionscdetermine the amount of labour kl e socially necessary to produce cDe , andcD
technical innovations increasing labour productivity, and reducing kl e , tend
to increase exploitation. But LEH does not stipulate the primacy of thesphere of production, and it is consistent with approaches that allow for
distribution and circulation to aect exploitation status. For prices andwages explicitly appear in the axiom, which allows one to capture the eectof changes in market conditions on exploitation. Thus, for example, thecondition pcDe = wbD is consistent with an emphasis on workers’ purchasing
power (and thus on workers qua consumers) such that a generalised decreasein wages, or increase in prices, would
tend to increase exploitation for allcD
D 5 W+ - e.g., via a decrease in kl e associated with a decrease in cDe tomaintain pcDe = wbD . Given the focus on individual agents, however, LEH
can also capture the (possibly divergent) eects of changes in relative prices(and wages) on the exploitation status of dierent groups of workers (andcD
consumers) via changes in cDe , and the associated kl e .35
LEH imposes weak and theoretically reasonable restrictions on the appropriate deﬁnition of exploitation, and all of the main approaches, including
Deﬁnitions 1-3, satisfy it.36 LEH does not provide comprehensive conditions
for the determination of exploitation status: it only focuses on the strict subset of agents who own no physical assets and supply some labour in return for
34
Observe that LEH allows the ERB to be variable and a function of (p, w). Further,
once ce is identiﬁed, the existence of ce is guaranteed by assumptions A0 and A2 in
Appendix A.1.
35
The role of exchange processes has been emphasised, for example, by Sweezy [44] andBaran and Sweezy [1] in their classic analysis of monopoly capitalism and of the eects ofnoncompetitive practices on prices, wages and the production and distribution of surplus
value. We thank the Editors of this journal for this suggestion.
36
For a proof of this claim, see the Addendum. It can also be proved that the deﬁnitionof exploitation proposed in Flaschel [8] satisﬁes LEH.

18

a wage and imposes no restrictions on the set of exploiters. More generally,
LEH does not incorporate any assumptions on individual behaviour or on
the structure of economic interactions. Nor is it based upon any (classicalor neoclassical) equilibrium notion. It does not capture any causal mechanisms and embodies no assumptions on the nature of individuals or of social
reality. It captures the properties of the concept of exploitation in a purely
a posteriori - rather than predictionist - perspective, by focusing on the data
emerging from economic processes at a given point in time.
Because LEH is a domain condition that captures some aspects of Marxian exploitation theory shared by all of the main approaches, further restrictions must be imposed in order to discriminate among alternative deﬁnitions.
A key tenet of Marxian theory is the idea that in capitalist economies proﬁts
are closely related to the existence of exploitation. Given private ownershipof productive assets, proﬁts should be a counterpart of the transfer of social
surplus and social labour from asset-poor agents to wealthy ones, and a general correspondence should exist between positive proﬁts and the exploitationof at least the poorest segments of the working class. This is formalised in
the next axiom.
Proﬁt-Exploitation Correspondence Principle (PECP): Given any
ES such that W+ 6= B,> 0 if and only if every agent in W+ is exploited] .
[pb
kp,w  wkp,wl
In other words, at any ES, aggregate proﬁts are strictly positive if andonly if propertyless workers are exploited. This incorporates a key intuitionof Marxian exploitation theory at the centre of the debates on the Fundamental Marxian Theorem discussed in the Introduction. Yet, PECP isboth conceptually and formally distinct from the FMT. Conceptually, theFMT is conceived of as a causal statement: it is meant to prove that proﬁts
emerge from (are caused by) the exploitation of workers. Thus in the standard literature it is a result that may or may not hold in certain economies
under a given deﬁnition of exploitation. PECP is instead conceived of as
a fundamental, deﬁnitional property of Marxian exploitation theory. It isa statement about what exploitation is, or what intuitions it incorporates,and so it is formulated without specifying any deﬁnition: the appropriate
19

deﬁnition should be such that propertyless workers are exploited if and only
if proﬁts are positive.37
Formally, unlike the FMT, PECP establishes a connection between theexistence of aggregate proﬁts and the exploitation status of a subset of the setof agents, rather than the aggregate rate of exploitation. Thus, PECP allows for the possibility that propertyless workers are a strict subset of the setof exploited agents, and when proﬁts are zero it does not require that there
be no exploitation in the economy, but only that some propertyless workers
are not exploited. Moreover, like LEH, PECP focuses only on propertylessagents who supply some wage labour and so it imposes no constraints on
the deﬁnition of exploitation whenever all agents possess some wealth, orpropertyless agents are all unemployed (W+ = B). This restriction is theoretically appropriate, because the exploitation status of agents who do not
engage in any economic activities is unclear.
PECP establishes a rather weak link between exploitation and proﬁts.38
It is therefore surprising that, in conjunction with LEH, it characterises afamily of deﬁnitions of exploitation.39
Theorem 1: For any deﬁnition of exploitation satisfying LEH, the following
two statements are equivalent at any ES:(1) PECP holds under this deﬁnition;cD(2) if Z max > 0, then for all D 5 W+ with wkl e > 0, there³ is kDZ 5 YP ´suchDD
cD
that kb DZ 5 Rn+ , pb
kDZ > wkDZl = wbD and (kDZl , kDZ , kDZ ) = # D kl e , kce , kce forsome # D > 1.

Theorem 1 is mainly a technical result: it provides a condition that canbe used to check whether a given deﬁnition satisﬁes PECP. As such, it does
not identify a unique deﬁnition of exploitation that meets PECP, but rather
a class of deﬁnitions satisfying condition (2). Yet it has relevant implications
for the main received approaches. For there are economies in which condition(2) never holds, if Deﬁnitions 1 and 2 are adopted. In contrast, Deﬁnition 3
satisﬁes condition (2), and thus the PECP, in general.40

37
Observe that PECP does not stipulate the primacy of the sphere of production and
it is in principle consistent with theories of exploitation allowing for exploitative relations
to be inﬂuenced at the point and in the process of exchange.
38
Oberve also that PECP is silent on the set of exploiters.
39
The proofs of all formal results are in Appendix A.2.
40
It can also be proved that if a deﬁnition of exploitation satisﬁes LEH and PECP,
then no agent in W+ is exploited whenever proﬁts are zero.

20

Corollary 1: There exist ES’s such that neither Deﬁnition 1 nor Deﬁnition
2 satisﬁes PECP. Instead, Deﬁnition 3 satisﬁes PECP at any ES.
Theorem 1 and Corollary 1 show that, contrary to a widespread belief,
general technologies with heterogeneous labour do not necessarily pose insurmountable problems for Marxian exploitation theory. They prove theexistence of a non-empty class of deﬁnitions that preserve a robust relation
between exploitation and proﬁts in general economies with heterogeneous
labour.
Corollary 1 also provides support to the New Interpretation, as the only
deﬁnition, among the main ones in the literature, that satisﬁes PECP in general. It is worth emphasising, however, that Deﬁnitions 1 and 2 fail to satisfy
PECP because of some deep formal and conceptual issues that are unrelated to the treatment of heterogeneous labour. To see this, consider an economic environment with two commodities (n = 2), one type of homogeneous
labour (T = {1}), Nc > 0 capitalists who own some physical endowments
and Nw > 0 propertyless agents, with N = Nc + Nw . Suppose that there is
only one production process where both commodities are jointly producedwith ﬁxed input-output proportions and homogeneous labour, such that the
production set is P = {k = kk0 | k  0}, where k0  (1,  (1, 0) , (3, 1)).
Then it is easy to show that there exist ES’s with p1 > 0 and p2 > 0,
p1 + p2 = 1, and w = 1 in which each propertyless agent D supplies one unitof (homogeneous) labour and purchases a bundle cD = (1, 1), while capitalists
do not work at all. At these ES’s, aggregate proﬁts are pb
kp,w  wkp,w
>0
l
p,w
0
where k= Nw k is the unique proﬁt rate maximiser. Nonetheless, according to both Deﬁnition 1 and Deﬁnition 2, the labour received by each
propertyless worker D is equal to one - l.v. (cD ; w) = l.v. (cD ; p, w) = 1 - andtherefore none of them is exploited.41
This example shows that the inability of Deﬁnitions 1 and 2 to capture
the relation between exploitation and proﬁts does not derive from the presence of heterogeneous labour. Rather, the problem lies in their notion oflabour received by agents in economies with general technologies allowing
for joint products, ﬁxed capital, choice of techniques, and so on. Indeed, it ispossible to prove that in production economies with only circulating capitaland a single production technique available to produce each commodity withﬁxed coe!cients, a large subset of the set of deﬁnitions of exploitation satis1
In contrast, l.v. (c ; p, w, p,w ) = 1+p< 1 and all propertyless workers are exploited1
according to Deﬁnition 3.
41

21

fying LEH (including all of Deﬁnitions 1-3) preserves the relation between
exploitation and proﬁts. Formally:
Proposition 1: Suppose that P is representable by a Leontief production
technique (A, L). Then at any ES, PECP holds under any deﬁnition ofD
exploitation satisfying LEH with kb ce ¤ cDe .

5Conclusions

This paper explores a novel axiomatic approach to Marxian exploitation theory. Two properties - a domain axiom and the Proﬁt-Exploitation Correspondence Principle - are analysed, which incorporate some widely shared
intuitions concerning the normative and positive foundations of the conceptof exploitation. Contrary to the received view, a nonempty class of deﬁnitions of exploitation is characterised, which preserve the relation between
proﬁts and the exploitation of propertyless workers in general economies witha complex class structure, heterogeneous agents, complex technologies withheterogeneous labour inputs, general market structures, and so on. Interestingly, however, among the main approaches, only the ‘New Interpretation’ isshown to satisfy PECP in general. Given the theoretical relevance of PECP
in Marxian theory, this provides strong support for Deﬁnition 3 above.
To be sure, the relation between exploitation and proﬁts is only one albeit important - aspect of Marxian theory and the results in this paper do
not exhaust the analysis of the concept of exploitation. Yet, they do show
the potential of the axiomatic method and in closing the paper we brieﬂymention some lines for further research.
First of all, the concept of exploitation is meant to be a diagnostic of thecharacteristics of the social structure, both in its power dimension and in its
inequality dimension. In this paper, we have focused on the latter. It would
be important to extend our analysis to incorporate power, and coercive social
relations into the axiomatic framework.
Further, Theorem 1 does not identify a single deﬁnition that meets PECP,
but rather a class of deﬁnitions. It would be interesting to analyse whether
a unique deﬁnition can be characterised by imposing further properties, andif so, whether such deﬁnition is indeed the ‘New Interpretation’.
This is an open question, but two points are worth making that suggestthat the key insights of the paper are indeed robust. First, in the standard
22

Okishio-Morishima approach, the existence of exploitation is just a numericalrepresentation of the existence of surplus products. Thus, the FMT establishes the equivalence between positive proﬁts and the productiveness of theeconomy measured in terms of the labour numéraire. Yet, it has been provedthat a similar result holds when productiveness is measured in terms of any
other good, thus raising doubts on the signiﬁcance of the relation between
exploitation and proﬁts (Roemer [37]). Yoshihara and Veneziani [54] have
proved that this is not true if the ‘New Interpretation’ is adopted: no equivalence between proﬁts and exploitation holds if another commodity is used
to deﬁne exploitation.
Second, in Marxian theory, the social positions identiﬁed by the notionof exploitation are internally related: ‘they are what they are ... by virtue ofthe relation to other in which they stand’ (Lawson [22], p. 17). The existenceof an exploiter is inextricably linked to the existence of exploited agents and‘you cannot have the one without the other’ (ibid.). Perhaps surprisingly,
Yoshihara and Veneziani [52] have proved that the ‘New Interpretation’ isthe only conceivable deﬁnition that possesses this relational property.
In summary, this paper provides a general theoretical framework to analyse
the notion of exploitation. The results derived shed light on some important(and vexed) issues. More importantly, they show that it is fruitful to take theaxiomatic road to exploitation theory, even though there are still a million
miles to go.

A

Formal analysis

A.1

Assumption on technology

Let 0 be the null vector. The following assumptions on P hold throughout
the paper.
Assumption 0 (A0). P is a closed convex cone in R2n+T and 0 5 P .
Assumption 1 (A1). For all k 5 P , if k  0 then kl  0 and k  0.
b = c.
Assumption 2 (A2). For all c 5 Rn+ , there is a k 5 P such that k

23

A.2

Proofs

Proof of Theorem 1: Consider any ES. If W+ = B, the equivalence isimmediately established, for both PECP and condition (2) are vacuously
satisﬁed. Therefore, in the rest of the proof, suppose that W+ 6= B. Let
N ted  N denote the set of exploited agents.
cD(2),(1): Suppose that if Z max > 0, then for each D 5 W+ with wkl e > 0,
there
kDZ ´5 YP such that pb
kDZ > wkDZl = wbD and (kDZl , kDZ , kDZ ) =³ existsD
DcD#D kl e , kce , kce for some # D > 1.
Let pbkp,w  wkp,w= 0. Then by the deﬁnition of ES, Zmax = 0 andlcondition (2) is vacuously satisﬁed. By LEH, for each D 5 W+ , cDe 5 Rn+ ,D
D
pcDe = wbD > 0 and kce 5 YP with kb ce = cDe . Therefore, noting thatDcD
pb
kce = pcDe = wbD > 0, Z max = 0 implies that wkl e = wbD . Hence, by LEH,
D5
/ N ted holds for all D 5 W+ .
Let pbkp,w  wkp,w> 0 so that Z max > 0. Consider any D 5 W+ . If
l
cDe
cD
cD
wkl = 0, then by deﬁnition wkl e < wbD . If wkl e > 0, then by conditioncD(2), kDZ 5 YP with wkDZl = wbD > 0 and #D > 1 together imply wkl e < wbD .
Thus, by LEH, D 5 N ted holds for any D 5 W+ .
In summary, (2) implies that PECP holds under any deﬁnition of exploitation satisfying LEH.
(1),(2): Suppose that pb
kp,w  wkp,w> 0 / N ted  W+ .
l
Suppose that Z max > 0. By the deﬁnition of ES, pb
kp,w  wkp,w> 0lholds, and by LEH and PECP, for each D 5 W+ , there exist cDe 5 Rn+
DDD
and kce 5 YP with kb ce = cDe and kb ce  cDe such that pcDe = wbD > 0DccD
and wkl e < wbD . Consider D 5 W+ such that wkl e > 0. Then let #D 5
D
DcD
R+ be such that # D wkl e = wbD and let kDZ  # D kce . Since kce 5 YP
Dn
D
with kb ce = cDe = 0, then kb DZ 5 R³ + and, by A0,´ kZ 5 YP . Moreover, bycD

construction, (kDZl , kDZ , kDZ ) = #D kl e , kce , kce
DDD

for some # D > 1. Finally,D

kce > pb
kce = pcDe = wbD = wkDZl holds.
pb
kDZ = # D pb
In summary, if PECP holds, then (2) holds under any deﬁnition of exploitation satisfying LEH.

Proof of Corollary 1: For a proof that Deﬁnitions 1 and 2 do not satisfy
PECP, see Veneziani and Yoshihara [49], which proves the result in a subsetof the economic environments considered here.
We prove that Deﬁnition 3 satisﬁes condition (2) of Theorem 1. Consider
any ES with W+ 6= B.
24

Suppose Z max > 0. By the deﬁnition of ES, thiskp,w wkp,w> 0;l
Pimplies pb
p,w
DDand since wb > 0 for all D 5 W+ then wkl = DMW+ wb > 0. Then, for
wbDp,w: kp,w 5 YP holds by condition (iii) of ES andall D 5 W+ , let kDZ = wk
p,w k
l

D

wb
b DZ = wk
b p,w , and so kb DZ 5 Rn+ , and since
so by A0, kDZ 5 YP . Moreover kp,w kl> 0, it follows that pb
kDZ > wkDZl = wbD .
pbkp,w  wkp,w
l
D
D
D
D
Finally, under Deﬁnition 3, kce = c kp,w holds, where c = pekpcp,w for
³ D
´
D
D
c
all D 5 W+ . Hence, (kDZl , kDZ , kDZ ) = # D kl e , kce , kce for some #D > 1 if and
D

p,w
wb
p.w
only if wk, kp.w , kp,w ) = #D pekpcp,w (kp,w, kp,w ) for some # D > 1, andp,w (kll ,kl
the latter inequality holds for all D 5 W+ whenever pb
kp,w  wkp,w> 0, sincelDD
pc = wb for all D 5 W+ by condition (iv) of ES.
In summary, condition (2) of Theorem 1 holds at any ES.
D

Proof of Proposition 1: Suppose P is representable
by a Leontief producPtion technique (A, L). Consider any ES. If DMN cD = 0, then condition(iv) of ES implies W+ =P
B, and so PECP is vacuously satisﬁed. Therefore,
in what follows, assume DMN cD  0.
By LEH, for any D 5 W+ , there exists cDe  0 such that pcDe = wbD and
¡
£
¤ ¢
D
kce = vcDe , A (I  A)31 cDe , I + A (I  A)31 cDe

where v  L (I  A)31 > 0 by the indecomposability of A. By LEH, D is
cD
exploited if and only if wkl e = wvcDe < wbD .
Suppose Z max > 0. Then at any ES, it must be p > pA + wL. (To see
this, let x be the aggregate activity level at the ES. If pj 5 pAj + wLj ,for some j, then xj = 0 must hold by partP(ii) of the deﬁnition
P of ES.
D
However, by part (i) it must be (I  A) x = DMN c and since DMN cD ¡P¢
D0, the indecomposability of A implies x = (I  A)31
> 0, aDMN c
31contradiction.) Then, p > wv holds by (I  A) > 0. Therefore by LEH,
cD
for any D 5 W+ , it follows that wbD = pcDe > wvcDe = wkl e and D is exploited.
Suppose Zmax = 0. Then at any ES, p 5 pA+wL and p 5 wv. Hence, by
cD
LEH, for any D 5 W+ , wbD = pcDe 5 wvcDe = wkl e , and D is not exploited.

References[1] Baran, P.A., Sweezy, P., 1966. Monopoly capital. Monthly Review Press,
New York.
25

[2] Bowles, S., Gintis, H., 1977. The Marxian Theory of Value and Heterogeneous Labour: A Critique and Reformulation. Cambridge Journal of
Economics 1, 173-192.
[3] Downward, P., (ed.), 2003. Applied Economics and the Critical Realist
Critique. Routledge, London.
[4] Downward, P., Finch, J.H., Ramsay, J., 2003. Seeking a role for empiricalanalysis in critical realist explanation. In: Downward, P., (ed.). Applied
Economics and the Critical Realist Critique. Routledge, London.
[5] Duménil, G., 1980. De la Valeur aux Prix de Production. Economica,
Paris.
[6] Duménil, G., Foley, D.K., Lévy, D., 2009. A Note on the Formal Treatment of Exploitation in a Model with Heterogeneous Labor. Metroeconomica 60, 560-567.
[7] Fine, B., 2010. Locating Financialisation. Historical Materialism 18, 97116.
[8] Flaschel, P., 2010. Topics in Classical Micro- and Macroeconomics.
Springer, New York.
[9] Flaschel, P., Franke, R., Veneziani, R., 2013. Labor Productivity andthe Law of Decreasing Labor Content. Cambridge Journal of Economics
37, 379-402.
[10] Foley, D.K., 1982. The Value of Money, the Value of labour Power,
and the Marxian Transformation Problem. Review of Radical Political
Economics 14, 37-47.
[11] Foley, D.K., 1986. Understanding Capital. Harvard University Press,
Cambridge, MA.
[12] Foster, J.E., 1994. Normative Measurement: Is Theory Relevant? American Economic Review 84, 365-370.
[13] Freeman, A., Carchedi, G., (eds) 1996. Marx and Non-equilibrium Economics. Edward Elgar, Aldershot.

26

[14] Fujimori, Y., 1982. Modern Analysis of Value Theory. Springer-Verlag,
Berlin.
[15] Katzner, D.W., 2009. Analysis without Measurement. Cambridge University Press, Cambridge.
[16] Krause, U., 1982. Heterogeneous Labour and the Fundamental Marxian
Theorem. Review of Economic Studies 48, 173-178.
[17] Kurz, H.D., Salvadori, N., 1995. Theory of Production: A long-Period
Analysis. Cambridge University Press, Cambridge.
[18] Lapavitsas, C., 2009. Financialised Capitalism: Crisis and Financial
Expropriation. Historical Materialism 17, 114-148[19] Lapavitsas, C., 2013. Proﬁting without producing. Verso, London.
[20] Lawson, T., 1997. Economics and Reality. Routledge, London.
[21] Lawson, T., 1999. Connections and Distinctions: Post Keynesianism
and Critical Realism. Journal of Post-Keynesian Economics 22, 3-14.
[22] Lawson, T., 2003. Reorienting Economics. Routledge, London.
[23] Lawson, T., 2009. The current economic crisis: its nature and the courseof academic economics. Cambridge Journal of Economics 33, 759-777.
[24] Lawson, T., 2013. What is this ‘school’ called neoclassical economics?
Cambridge Journal of Economics 37, 947-983.
[25] Martins, N.O., 2006. Capabilities as causal powers. Cambridge Journalof Economics 30, 671-685.
[26] Martins, N.O., 2012. Sen, Sraa and the revival of classical political
economy. Journal of Economic Methodology 19, 143-157.
[27] Marx, K., 1954 [1867]. Capital, Vol.I. London, Lawrence & Wishart.
[28] Marx, K., 1959 [1894]. Capital, Vol.III. London, Lawrence & Wishart.
[29] Mohun, S., 2003. On the TSSI and the Exploitation Theory of Proﬁt.
Capital and Class 81, 85-102.
27

[30] Morishima, M., 1973. Marx’s Economics. Cambridge University Press,
Cambridge.
[31] Morishima, M., 1974. Marx in the Light of Modern Economic Theory.
Econometrica 42, 611-632.
[32] Okishio, N., 1957. Theory of Reproduction (in Japanese). Sobun Publisher.
[33] Okishio, N., 1963. A mathematical note on Marxian Theorems.
Weltwirtschaftliches Archiv 91, 287-299.
[34] Okishio, N., 1965. Fundamental Theory of Capitalist Economy (inJapanese). Sobun Publisher.
[35] Ricardo, D., 2004 [1819]. The Principles of Political Economy and Taxation. Dover Publication.
[36] Roemer, J.E., 1981. Analytical foundations of Marxian economic theory.
Harvard University Press, Cambridge, MA.
[37] Roemer, J.E., 1982. A General Theory of Exploitation and Class. Harvard University Press, Cambridge, MA.
[38] Sen, A.K., 1978. On the labour theory of value: some methodologicalissues. Cambridge Journal of Economics 2, 175-190.
[39] Sen, A.K., 1985a. Commodities and Capabilities. North-Holland, Amsterdam.
[40] Sen, A.K., 1999. The Possibility of Social Choice. American Economic
Review 89, 349-378.
[41] Setterﬁeld, M., 2003. Critical realism and formal modelling: incompatible bedfellows? In: Downward, P., (ed.). Applied Economics and the
Critical Realist Critique. Routledge, London.
[42] Smith, A., 2000 [1776]. The Wealth of Nations. Modern Library, NewYork.
[43] Steedman, I., 1977. Marx after Sraa. New Left Books, London.

28

[44] Sweezy, P., 1970 [1940]. The Theory of Capitalist Development. Monthly
Review Press, New York.
[45] Thomson, W., 2001. On the axiomatic method and its recent applications to game theory and resource allocation. Social Choice and Welfare
18, 327-386.
[46] Veneziani, R., 2004. The Temporal Single-System Interpretation ofMarx’s Economics: A Critical Evaluation. Metroeconomica 55, 96-114.
[47] Veneziani, R., 2007. Exploitation and Time. Journal of Economic Theory 132, 189-207.
[48] Veneziani, R., 2013. Exploitation, Inequality, and Power. Journal of
Theoretical Politics 25, 526-545.
[49] Veneziani, R., Yoshihara, N., 2015. Exploitation in economies with heterogeneous preferences, skills and assets: An axiomatic approach. Journal of Theoretical Politics 27, 8-33.
[50] Wright, E. O., 2000. Class, Exploitation, and Economic Rents: Reﬂections on Sorensen’s “Sounder Basis”. The American Journal of Sociology
105, 1559-71.
[51] Yoshihara, N., 2010. Class and Exploitation in General Convex Cone
Economies. Journal of Economic Behavior & Organization 75, 281-296.
[52] Yoshihara, N., Veneziani, R., 2009. Exploitation as the Unequal Exchange of Labour: An Axiomatic Approach. WP N.655, Queen Mary
University of London.
[53] Yoshihara, N., Veneziani, R., 2012. Proﬁts and Exploitation: A Reappraisal. Advances in Mathematical Economics 16, 85-109.
[54] Yoshihara, N., Veneziani, R., 2013. Exploitation of Labour and Exploitation of Commodities: a ‘New Interpretation’. Review of Radical Political
Economics 45, 517-524.
[55] Yoshihara, N., Veneziani, R., 2014. The Measurement of Labour Content: A General Approach. DP 2013-5, University of Massachusetts,
Amherst.
29