Exploitation as the Unequal Exchange ofLabour: An Axiomatic Approach�

Naoki Yoshiharay Roberto Venezianiz

This version: October 2011

Abstract

In subsistence economies with general convex technology and ratio-nal optimising agents, a new, axiomatic approach is developed, whichallows an explicit analysis of the core positive and normative intuitionsbehind the concept of exploitation. Three main new axioms, calledLabour Exploitation in Subsistence Economies, RelationalExploitation, and Feasibility of Non-Exploitation, are presentedand it is proved that they uniquely characterise a de�nition of ex-ploitation conceptually related to the so-called �New-Interpretation�(Duménil, 1980; Foley, 1982; Duménil at el., 2009), which focuses onthe unequal distribution of (and control over) social labour, and on

�We are grateful to Nizar Allouch, Amitava Dutt, Bill Gibson, Chiaki Hara, TakashiHayashi, Motoshige Ito, Atsushi Kajii, Kazuya Kamiya, Michele Lombardi, Marco Mar-iotti, Akihiko Matsui, Sripad Motiram, Shinichiro Okushima, Prasanta Pattanaik, NeriSalvadori, Dan Sasaki, Rajiv Sethi, Tomoichi Shinotsuka, Gil Skillman, Peter Skott, YukiYoshihara, and participants in the Public Economic Theory 2008 conference (Seoul, 2008),in the Far East and South Asia Meeting of the Econometric Society (Tokyo, 2009), in theAnalytical Political Economy workshop (London, 2009), and in seminars at HitotsubashiUniversity, the University of Tokyo, the Institute of Economic Research (Kyoto Univer-sity), Ritsumeikan University, Niigata University, and Keio University for comments andsuggestions. The usual disclaimer applies. This research started when Roberto Venezianiwas visiting Hitotsubashi University. Their generous hospitality and �nancial support isgratefully acknowledged.

yThe Institute of Economic Research, Hitotsubashi University, Naka 2-1, Kunitachi,Tokyo 186-0004, Japan. (yosihara@ier.hit-u.ac.jp)

zDepartment of Economics, Queen Mary, University of London, Mile End Road, Lon-don E1 4NS, United Kingdom. (r.veneziani@qmul.ac.uk)

1

individual well-being freedom and the self-realisation of men. Then,the main results of Roemer�s (1982, 1988) classical approach and allthe crucial insights of exploitation theory are generalised, proving thatevery agent�s class and exploitation status emerges in the competitiveequilibrium, that there is a correspondence between an agent�s classand exploitation status, and that the existence of exploitation is in-herently linked to the existence of positive pro�ts.

JEL: D63 (Equity, Justice, Inequality, and Other Normative Cri-teria and Measurement); D51 (General Equilibrium: Exchange andProduction Economies); C62 (Existence and Stability Conditions ofEquilibrium); B51 (Socialist; Marxist, Sra¢ an).Keywords: Justice, Exploitation, Class, Convex Economies.

2

1 Introduction

The notion of exploitation is prominent in the social sciences and in politicaldiscourse. It is central in a number of debates, ranging from analyses oflabour relations, especially focusing on the weakest segments of the labourforce, such as children, women, and migrants (see, e.g., ILO, 2005; 2005a;2006); to controversies on drug-testing and on the price of life-saving drugs,especially in developing countries;1 to ethical issues arising in surrogate moth-erhood (see, e.g., Field, 1989; Wood, 1995). The concept of exploitation isalso central in the politics of the Left. In the 2007 programme of the GermanSocial Democratic Party, for example, the very �rst paragraph advocates asociety �free from poverty, exploitation, and fear�(SPD, 2007, p.3), and the�ght against exploitation is repeatedly indicated as a priority for the biggestparty of the European Left. The notion of exploitation is arguably the cor-nerstone of Marxist social theory, but it is also extensively discussed in nor-mative theory and political philosophy (see, e.g., Wertheimer, 1996; Wol¤,1999; Bigwood, 2003; and Sample, 2003).2 Yet, there is little agreement con-cerning even the most basic features of exploitative relations, and both thede�nition of exploitation and its normative content are highly controversial.3

In general, agent A exploits agent B if and only if A takes unfair ad-vantage of B. Despite its intuitive appeal, this de�nition leaves two majorissues in need of precise speci�cation from a normative perspective, namelythe source of the unfairness and the structure of the relationship between Aand B that allows A to take advantage of B. There is considerable debate

1In a section devoted to �Ethical Issues,� the Investigation Committee on the clini-cal trial of the drug �Trovan�conducted by P�zer in 1996 in Kano (Nigeria) argue that�Compensations to the participants were minimal or non existent, as such a clear case ofexploitation of the ignorant was established�(Federal Ministry of Health of Nigeria, 2001,p.88).

2The notion of exploitation is relevant, for example, in Lockean or Neo-Lockean politicalphilosophy, whereby exploitation occurs if the principle of Just Acquisition of unownedland is violated by State intervention; or in Neoclassical economic philosophy, wherebyexploitation occurs in non-competitive markets if the distributive principle of marginalproductivity is violated.

3For a review of some of the debates in exploitation theory, see Nielsen and Ware(1997).

3

in the economic and philosophical literature concerning both issues. At oneextreme, in his seminal theory of exploitation, John Roemer (1982, 1988)argues that exploitation is a purely distributive concept which makes no ref-erence to the interaction between agents and identi�es an injustice stemmingfrom an unequal distribution of assets. At the other extreme, contra Roemer,various authors deny that exploitation involves a distributive injustice andclaim that the moral force of exploitation derives entirely from the objec-tionable features of the interaction between agents (e.g., Wol¤, 1999; Wood,2004).As shown by Veneziani (2008), and as acknowledged by Roemer himself in

later contributions, Roemer�s claim that exploitation is a purely distributiveconcept is not convincing.4 Some notion of unequal power, or dominance, isarguably crucial in any theory of exploitation and the positive and normativeanalysis of exploitative relations involves some consideration of the way inwhich A and B interact.5 It seems, however, equally reductive to assume thatinequalities deriving from exploitative relations between agents are immate-rial in the judgment of exploitation. As forcefully argued by Warren (1997,p.63), �exploitation involves inequality on both ends of exchange: inequalityde�ning the context of the exchange (that is, [di¤erential ownership of pro-ductive assets]) and inequality de�ning the outcome.�6 So, although power,force, or dominance, are arguably crucial elements of exploitation theory, theanalytical focus on this paper is on the source of the unfairness of exploitativerelations, and on the injustice involved in the concept of exploitation withineconomic relations.More speci�cally, this paper analyses the theory of exploitation as an

unequal exchange (hereafter, UE) of labour, according to which exploitativerelations are characterised by a di¤erence between the hours of labour that anindividual provides and the hours of labour necessary to produce commodi-ties that she can purchase with her income. There are at least two reasons

4For a thorough analysis of Roemer�s distributive approach, see Skillman (1995). Skill-man (1995) argues that Roemer�s approach is consistent with Marx�s own account.

5In fact, Yoshihara (1998) showed that exploitative status is linked with the degree oflabour-discipline, which re�ects power relations in capitalist economies.

6In this perspective, �it is not unequal power itself that is supposed objectionable, butrather the fact that one person gains unjustly through the exercise of power (whether coer-cive or uncoercive) over another�(Warren, 1997, p.62). According to Warren, the relevantoutcome inequality concerns indeed the unequal performance of labor, as suggested alsoin this paper.

4

to focus on labour as the measure of the injustice of exploitative relations.First, in a number of crucial economic interactions, the notion of exploita-tion seems inextricably linked with some form of labour exchange.7 Second,the UE de�nition of exploitation captures some inequalities in the distrib-ution of material well-being and free hours that are - at least prima facie- of normative relevance. As shown in this paper, for example, it is possi-ble to signi�cantly generalise the so-called Class-Exploitation CorrespondencePrinciple (hereafter, CECP; see Roemer, 1982, 1988), according to which ina private-ownership economy with positive pro�ts, class and UE exploitationstatus are strictly related, and they accurately re�ect an unequal distributionof assets. That is, in equilibrium, the wealthiest agents emerge as exploiters,and members of the capitalist class, whereas poor agents are exploited, andmembers of the working class. As in standard Marxist theory, then, exploita-tive relations are relevant in that they re�ect unequal opportunities of lifeoptions, due to unequal access to productive assets.Interestingly, however, the UE concept of exploitation can also be seen

as capturing inequalities in the distribution of well-being freedom. In fact,as argued by Rawls (1971) and Sen (1985a, 1985b), among the others, anindividual�s well-being freedom captures her ability to pursue the life shevalues.8 9 There are two crucial factors that determine the degree of indi-vidual�s well-being freedom, or self-realisation: one is the amount of incomeshe can spend to purchase the commodities necessary to achieve her goals,and the other is the amount of time she has to sacri�ce as labour supply inorder to purchase such commodities.10 Then, the rate of labour exploitationcan represent the degree of well-being freedom, or indeed unfreedom, of the

7Despite his initial criticism of the UE de�nition, Roemer has later acknowledged thatthe exchange of labour is essential in explotiative relations, and �the expenditure of e¤ortis characteristically associated with exploitation�(Roemer, 1989b, fn.11). See also Roemer(1989a, pp. 258-260; 1989b, p. 96).

8Interestingly, this notion of freedom is conceptually related to the Marxian notion ofself-realisation, and one can argue that both commodities and free time are essential foran individual�s self-realisation.

9In the Rawls-Sen theory, inequalities in the distribution of well-being freedom areformulated as inequalities of capabilities. The resource allocation problem, in terms ofequality of capability, is explicitly analysed in Gotoh and Yoshihara (2003).10In this view, labour only yields disutility and it reduces the possibility to self-realise.

To assume away completely the possibility that work itself may be a source of well-beingfreedom may be unrealistic, but it is appropriate at the level of analysis of this paper, andit is consistent with a Marxian analysis of capitalist relations of production.

5

agent, since it can be taken as an index of the relative attainment of thesetwo goods by using labour time as the numéraire: if an agent gains from theunequal exchange of labour, then she is exploiting the free hours which someother agents sacri�ced as labour supply for the production of the commodi-ties she can purchase, whereas if she su¤ers from the unequal exchange oflabour, she is exploited in the sense that some of the free hours she sacri�cedas labour supply to purchase the commodities are appropriated by somebodyelse.Granting the normative relevance of the unequal exchange of labour, there

are many possible ways of rigorously specifying the concept of UE exploita-tion and a number of alternative de�nitions have indeed been proposed inthe literature (for a thorough discussion, see Yoshihara, 2007). This pa-per provides the �rst rigorous axiomatic analysis of UE exploitation: thisis a completely new approach to exploitation theory and it provides a fullygeneral framework to compare the most important de�nitions of exploita-tion discussed in the literature. An axiomatic approach was long overdue inexploitation theory, where the proposal of alternative de�nitions have some-times appeared as a painful process of adjustment of the theory to the var-ious counterexamples and formal exceptions found in the literature. Thede�nitions of exploitation thus constructed have progressively lost the in-tuitive appeal, the normative relevance, and even the connection with theactual, observed variables emerging from a competitive mechanism. By tak-ing an axiomatic approach, this paper suggests to start from �rst principles,thus explicitly discussing the normative intuitions behind exploitation theory.Therefore, the approach proposed in this paper, and the analysis developedbelow should be interesting for all exploitation theorists, and indeed for allsocial scientists and political philosophers, even if the speci�c axioms pro-posed may be deemed unsatisfactory.To be precise, this paper analyses exploitation theory in a class of con-

vex subsistence economies which generalise Roemer (1982, 1988). In thisclass of economies, each producer can use a general convex technology andis assumed to minimize her labour supply under the constraint that she hasto earn enough to purchase a subsistence vector. First of all, an axiom isintroduced, called Labour Exploitation in Subsistence Economies (hereafter,LES), which restricts the way in which the sets of exploiters and exploitedagents should be identi�ed in equilibrium. This axiom is taken as the minimalnecessary condition to stipulate the normative intuitions behind exploitationtheory, and it is shown that all the main de�nitions of exploitation proposed

6

in the literature (see, for example, Morishima, 1974; Roemer, 1982; Yoshi-hara, 2007) do satisfy LES. Then, Theorems 1 and 2 provide a signi�cantgeneralisation of Roemer�s (1982, 1988) celebrated results: they derive theequilibrium class and exploitation structures of a general convex cone sub-sistence economy with optimising agents for a whole class of de�nitions ofexploitation satisfying LES. Further, Theorems 3 and 4 derive the neces-sary and su¢ cient conditions under which, for a whole class of de�nitionsof exploitation satisfying LES, and for any convex subsistence economy, theCECP holds and the existence of exploitation is synonymous with the exis-tence of positive pro�ts, respectively. These results are theoretically relevantbecause, as argued by Roemer (1982), although they are formally derivedas theorems, their epistemological status is as postulates: any de�nition ofexploitation should preserve them.Then, three additional axioms are introduced: the �rst states that any de-

�nition of UE exploitation should guarantee the feasibility of non-exploitativeallocations: this is a desirable property if exploitation is not to be consideredan evil that agents should learn to live with. The second axiom requiresthat the de�nition of the status of exploiter or exploited agent should beindependent of the distribution of productive endowments in the economy.The third axiom captures the relational nature of exploitative relations byruling out the possibility that there exists exploiters without any agent beingexploited, and vice versa. Interestingly, although almost all the formulationsof UE exploitation discussed in the literature satisfy LES, the feasibilityof non-exploitation, and the independence axiom, Theorem 5 proves thata generalised version of the so-called �New Interpretation�(Duménil, 1980;Foley, 1982; Duménil at el., 2009) is the unique de�nition of exploitationthat satis�es all axioms. As a corollary, it follows that the generalised �NewInterpretation� is the unique formulation of UE exploitation that satis�esthe four axioms and under which the CECP holds in the class of generalconvex cone subsistence economies.There are two main reasons to focus on static subsistence economies.

First of all, the analysis of subsistence economies with a labour market istheoretically crucial in that they provide the simplest institutional frameworkin which exploitation arises. In particular, in order to analyse the distributiveissues related to exploitation, it is appropriate to abstract from the role thatexploitation plays in the accumulation process.11 The model of a subsistence

11It is also worth noting that the subsistence vector can also be interpreted as a social

7

economy may not be a realistic representation of �actual economies,�but it isan abstract model suitable to illustrate some of the essential characteristics ofmarket economies with private ownership of productive assets. Besides, froma theoretical viewpoint, one may argue that the distinctive characteristic ofcapitalist economies is the existence of a labour market, in which labour isexchanged as a commodity, rather than accumulation and growth. Second,from a formal viewpoint, subsistence economies provide a particularly neatframework for the analysis of exploitation, which allows one to derive starkresults. However, the main conclusions can be generalised to accumulationeconomies (see Yoshihara, 2007) and to dynamic economies along the lines,e.g., of Veneziani (2007, 2008) and Veneziani and Yoshihara (2009), albeit atthe cost of a substantial increase in unnecessary technicalities.The rest of the paper is organised as follows. In section 2, the model of

a general convex subsistence economy is set up. In section 3, the notions ofexploitation and classes are de�ned and axiom LES is presented. It is thenshown that all the most important de�nitions of exploitation presented inthe literature satisfy LES. The complete class and exploitation structures ofthe economy for the class of de�nitions of exploitation satisfying LES arederived. In section 4, the necessary and su¢ cient conditions for the CECPto hold for a class of de�nitions of exploitation satisfying LES are derived. InSection 5 the three additional axioms, called Feasibility of Non-Exploitation,Independence of Endowment Structure, and Relational Exploitation are pre-sented and the main characterisation result of the paper is derived. Section6 concludes and the existence of a general equilibrium is proved in Appendix1, whereas all the proofs of the formal results are in Appendix 2.

2 A Model of General Convex SubsistenceEconomies

Let P be the production set. P has elements of the form � = (��0;��; �)where �0 2 R+ , � 2 Rm+ , and � 2 Rm+ . Thus, elements of P are vectors in

reference bundle of commodities, which represents a decent living standard, rather than asa consumption bundle necessary for survival. In this case, once the decent living standardis reached, each agent is free to determine how to use her spare time, if any. Then, theexistence of UE exploitation represents unequal allocations of free hours among agents,given that everyone reaches the decent living standard, which straightforwardly impliesunequal opportunities for self-realisation and well-being freeedom.

8

R2m+1. The �rst component, ��0, is the direct labour input of the process�; the next m components, ��, are the inputs of goods used in the process;and the last m components, �, are the outputs of the m goods from theprocess. The net output vector arising from � is denoted as b� � ���. Theset P is assumed to be a closed convex cone containing the origin in R2m+1.Moreover, let 0 2Rm be such that 0 = (0; :::; 0)0: it is assumed that12

A1. 8� 2 P s.t. �0 � 0 and � = 0, [� � 0) �0 > 0].A2. 8 c 2 Rm+ , 9� 2 P s.t. b� = c.A3. 8� 2 P , 8 (��0; �0) 2 Rm� �Rm+ , [(��0; �0) 5 (��; �)) (��0;��0; �0) 2 P ].A1 implies that labour is indispensable to produce any non-negative outputvector; A2 states that any non-negative commodity vector is producible as anet output; and A3 is a free disposal condition, which states that, given anyfeasible production process �, any vector producing (weakly) less net outputthan � is also feasible using the same amount of labour as � itself.Given P , it is possible to de�ne the set of net output vectors that can be

produced using exactly l units of labour, denoted as bP (�0 = l). Formally:bP (�0 = l) � fb� 2 Rm j 9� = (�l;��; �) 2 P s.t. �� � � b�g .Finally, for any set X � Rm, the boundary of X is de�ned as @X �fx 2 X j @x0 2 X s.t. x0 > xg, and coX is the convex hull of the set X.Consider a generalisation of Roemer�s (1982) subsistence economy. Let

N be the set of agents, with generic element �. All agents � 2 N have accessto the same technology P , but they possess di¤erent endowments !� , whosedistribution in the economy is given by (!� )�2N 2 RNm+ . An agent � 2 Nendowed with !� can engage in three types of economic activity: she can sellher labour power �0 , she can hire others to operate �

� =����0 ;��� ; �

��2

P , or she can work on her own operating �� = (���0 ;��� ; �� ) 2 P . Given aprice vector p 2 Rm+ and a nominal wage rate w, it is assumed that each agentchooses her activities, �� , �� , and �0 , in order to minimise the labour sheexpends subject to earning enough income to purchase a subsistence vectorof commodities b 2 Rm+ . Moreover, she must be able to lay out in advancethe operating costs for the activities she chooses to operate, either with herown labour or with hired labour, using her wealth, and she cannot work morethan her labour endowment.12For all vectors x = (x1; : : : ; xm) and y = (y1; : : : ; ym) 2 Rm, x = y if and only if

xi = yi (i = 1; : : : ;m); x � y if and only if x = y and x 6= y; x > y if and only if xi > yi(i = 1; : : : ;m).

9

Formally, given (p; w), every agent chooses (�� ; �� ; �0 ) in order to solveprogram MP � :

min��0 + �0

subject to

[p (�� � �� )] +hp��� � ��

�� w��0

i+ [w �0 ] = pb,

p�� + p�� 5 p!� ,

��0 + �0 5 1.

Given (p; w), let A� (p; w) be the set of actions (�� ; �� ; �0 ) 2 P �P � [0; 1],which solve ��s minimisation problem MP � at prices (p; w).Let a convex cone subsistence economy be given by a listE = hN ; (P; b) ; (!� )�2Ni.

Let E denote the set of all convex cone subsistence economies. Based on Roe-mer (1982), the equilibrium notion for an economy E 2 E can be de�ned.

De�nition 1: A reproducible solution (RS) for an economy E 2 E is a pair�(p; w) ; (�� ; �� ; �0 )�2N

�, where p 2 Rm+ and w = 0 such that:

(a) 8� 2 N , (�� ; �� ; �0 ) 2 A� (p; w) (individual optimality);(b) �+ � 5 ! (social feasibility),where � �

P�2N �

� , � �P

�2N �� , and ! �

P�2N !

� ;

(c) �0 = 0 (labour market equilibrium)where �0 �

P�2N �

�0 and 0 �

P�2N

�0 ; and

(d) b�+ b� = Nb (reproducibility),where b� �P�2N(�

� � �� ), b� �P�2N(�� � �� ), and �0 �

P�2N �

�0 .

In other words, at a RS (a) all agents optimise; (b) aggregate capital is su¢ -cient for production plans; (c) the labour market is in equilibrium; and (d) netoutput is su¢ cient for subsistence. For the sake of brevity, in what follows,the notation (p; w) is used to represent the RS

�(p; w) ; (�� ; �� ; �0 )�2N

�.

In order to avoid an excess of uninteresting technicalities, it is assumed,as in Roemer (1982), that agents who are able to reproduce themselves with-out working use just the amount of wealth strictly necessary to obtain theirsubsistence bundle b: in a subsistence economy, wealthy agents have no rea-son to accumulate or to consume more than b; hence, by stating that theydo not �waste� their capital, assumption NBC is consistent with capitalistbehaviour.

10

Non Benevolent Capitalists (NBC): If agent � has a solution to MP �

with ��0 + �0 = 0, then agent � chooses (�

� ; �� ; �0 ) to minimise p�� +p�� .

It is now possible to prove some preliminary results. Lemma 1 provesthat at a RS, the net revenue constraint binds for all agents.

Lemma 1: Assume NBC. Let (p; w) be a RS for E = hN ; (P; b) ; (!� )�2Ni.Then, pb�� + pb�� � w��0 + w �0 = pb for all �.The next Lemma proves that the wealth constraint binds for all agents

who work at the solution to MP � .

Lemma 2: Let (p; w) be a RS for E = hN ; (P; b) ; (!� )�2Ni such that pb��w�0 > 0. For any � 2 N , if ��0 + �0 > 0, then p�� + p�� = p!� .

For any (p; w) and any � 2 P , de�ne the pro�t rate � = pb��w�0p�

, and

let �max = max�2Ppb��w�0p�

. By optimality, it is immediate to prove that onlyproduction processes yielding the maximal pro�t rate will be activated. Thenext result proves an important property of the set of solutions of MP � .

Lemma 3: Let (p; w) be a price vector such that �max > 0. If (�� ; �� ; �0 )solves MP � , then (�0� ; �0� ; 0�0 ) also solves MP

� whenever �� + �� =�0� + �0� and 0�0 � �0�0 = �0 � ��0 .

The equilibrium price vector can be characterised.

Proposition 1: Let (p; w) be a RS for E = hN ; (P; b) ; (!� )�2Ni. Then (i)p � 0 with pb > 0; (ii) �max = 0; (iii) w > 0.

In general convex cone economies it is not possible to prove that at a RSthe price vector will be strictly positive. In fact, it is possible for some goodto be produced as a joint product without being used as an input.13

Proposition 2 derives aggregate net output in equilibrium.

13It is reasonable to conjecture that Roemer�s Independence of Production assumptionmay yield a strictly positive vector of commodities prices in equilibrium, by requiring thatfor any � 2 P with b� > c = 0, there is another vector �0 2 P such that b�0 = c and�0 > �00 (see Roemer, 1981, Assumption 7, p.47). However, all the arguments in thispaper hold in the more general case and therefore no restriction is needed to guarantee astrictly positive price vector.

11

Proposition 2: Let (p; w) be a RS for E = hN ; (P; b) ; (!� )�2Ni. If p > 0,then b�+b� = Nb. Conversely, if b�i+b�i > Nbi for some good i, then pi = 0.The next result derives the optimal amount of labour expended by every

agent at a RS.

Proposition 3: Let (p; w) be a RS for E = hN ; (P; b) ; (!� )�2Ni. Then,��0 +

�0 = maxf0; pb��

maxp!�

wg for all �.

By Proposition 3, it follows that agent � will not work at the optimal solutionif and only if p!� = pb

�max, which implies �max > 0.

3 Exploitation and Class in Convex Subsis-tence Economies

The concept of exploitation in a general convex economy can now be intro-duced. First of all, if exploitation is conceived of as involving an unequalexchange of labour (or simply as labour exploitation), it is necessary to iden-tify the normative benchmark, that is the normatively relevant amount oflabour involved in the exchange, which is usually de�ned as the labour valueof an agent�s �labour power.� Agent � is said to be exploited (resp. an ex-ploiter) if she performs more (resp. less) labour than the labour value of her�labour power.� The amount of labour expended by the agent is unambigu-ously �� � ��0 + �0 , but there are various ways of de�ning the value of labourpower, which is related to some reference bundle of commodities (e.g., thatthe agent does or can purchase). For any bundle c 2 Rm+ , the labour valueof c must be de�ned. Unlike in standard Leontief economies, the de�nitionof the labour value of c is not obvious, and various de�nitions have, in fact,been proposed (see Yoshihara, 2007, for a thorough discussion). Followingare the most relevant ones discussed in the literature.De�nition 2 has been proposed by Morishima (1974). It suggests that the

labour value of a given bundle of goods corresponds to the minimum amountof labour necessary to produce that bundle as net output. Given c 2 Rm+ , let� (c) � f� 2 P j b� = cg.De�nition 2 [Morishima (1974)]: Let c 2 Rm+ be a given nonnegative bundleof commodities. Then, the labour value of c is given by:

l:v:(c) = minf�0 2 R+j� 2 � (c)g.

12

The next De�nition has been proposed by Roemer (1982). It suggests thatthe labour value of a given bundle of goods corresponds to the minimumamount of labour necessary to produce that bundle as net output using apro�t-rate maximising technique. Given a price vector (p; w), let

P (p; w) ��� = (��0;��; �) 2 P j

p�� (p� + w�0)p�

= �max�.

De�nition 3 [Roemer (1982)]: Let c 2 Rm+ be a given nonnegative bundleof commodities. Then, the labour value of c is given by

l:v:(c; p; w) = minf�0 2 R+j� 2 P (p; w) \ � (c)g.

Instead of discussing the virtues and limitations of existing de�nitionsof exploitation, and possibly introducing a new one, this paper adopts anovel approach and suggests starting from �rst principles, by de�ning thedesirable properties that any de�nition of exploitation should satisfy. At themost general level, the UE notion of exploitation aims to describe a relationalproperty of a given social structure by focusing on the distribution of labourassociated to a given resource allocation. In principle, there are many possiblealternative de�nitions of exploitation, but one might argue that there shouldbe some common structure characterising all forms of exploitation as the UEof labour, which characterises an admissible class of de�nitions. The nextaxiom represents a domain condition which precisely identi�es the admissibledomain of all forms of exploitation, thus specifying the relevant frameworkfor the discussion of the properties of UE exploitation in the rest of thepaper. Let B (p; b) �

�c 2 Rm+ j pc = pb

: B (p; b) is the set of bundles that

cost exactly as much as the subsistence vector at prices p. Then:

Labour Exploitation in Subsistence Economies (LES): Let (p; w) bea RS for E = hN ; (P; b) ; (!� )�2Ni. Two subsets N ter � N and N ted � N ,N ter \ N ted = ?, constitute the set of exploiters and the set of exploitedagents if and only if there exist c; c 2 B (p; b) such that there exist �c 2 � (c)with b�c = c and �c 2 � (c) with b�c = c such that �c0 = �c0 and for any� 2 N ,

� 2 N ter , �c0 > �� ;

� 2 N ted , �c0 < �� .

13

Axiom LES requires that, at any RS, the sets N ter and N ted are char-acterised by identifying two (possibly identical) reference commodity vectorsc; c 2 Rm+ . Both reference bundles c and c can be purchased by the consumerand they identify the value of labour power. Thus, if an agent � 2 N op-timally works �� to earn the income necessary to purchase the subsistencebundle b, and �� is less (resp. more) than the labour expended to produce c(resp. c), then she is regarded as expending less (resp. more) labour than the�value of labour power.�According to LES, the set of such agents coincideswith N ter (resp. N ted).As the domain condition for the admissible class of exploitation-forms,

LES captures the essential insights of the UE theory of exploitation in con-vex subsistence economies.14 Given any de�nition of exploitation, the setsN ter and N ted are identi�ed: in the UE approach, the two sets, and the ex-ploitation status of each agent �, are determined by the di¤erence betweenthe amount of labour that � �contributes�to the economy, in some relevantsense, and the amount she �receives�, in some relevant sense. In convex subsis-tence economies, the former quantity is unambiguously given by the amountof labour performed, �� , whereas there are many possible UE views concern-ing the amount of labour that each agent receives, which incorporate di¤erentnormative and positive concerns. As a domain condition, LES incorporatesthe main features of UE theory that are shared by all the main approachesproposed in the literature.First, according to LES, the amount of labour that each agent receives

depends on their equilibrium income, or more precisely, it is determined inequilibrium by some reference consumption vectors that agents can purchase.In the standard approach, the reference vector is unique and it corresponds tothe bundle actually chosen by the agent. LES is much weaker in that allowsfor more than one reference vector and it only requires that the referencevectors be potentially a¤ordable, even if they are not actually purchased.Second, LES captures another key tenet of the UE theory of exploitation

by stipulating that the amount of labour associated with each reference bun-dle - and thus potentially �received�an agent - is related to the productionconditions of the economy. More precisely, LES states that the referencebundles be technologically feasible as net output, and it de�nes their labour

14It should be stressed that LES only applies to labour-based de�nitions of exploitation.It is not relevant, for example, for Roemer�s property-relations de�nition of exploitationthat emphasises inequalities in ownership of productive assets. Similar versions of LEScan be de�ned in di¤erent economies; see Yoshihara (2007).

14

�content� as the amount of labour necessary to produce them. Thus, theamount of labour �received�by each agent - or, in the standard terminology,the value of labour power - is a function of the amount of social labour thatis allocated to agents. It is worth noting that LES requires that the amountof labour associated with each reference bundle be uniquely determined withreference to production conditions, but it does not specify how such amountshould be chosen, and there may be in principle many alternative ways ofproducing c; c, and thus of determining �c0; �

c0.

To be sure, one might argue that an even weaker version of LES shouldbe imposed which allows for more than two reference bundles, and associatedlabour amounts, as well as for heterogeneous bundles across individuals. Forexample, one may argue that all a¤ordable bundles should be considered.Although the objection may be important in principle, the restrictions onreference bundles are arguably mild and reasonable in the economies con-sidered in this paper, and the axiom LES can be generalised. If individualexploitation status is monotonic in labour performed, ceteris paribus, then allthe relevant information to determine exploitation status can be summarisedin at most two reference bundles, and the associated amounts of labour. Sim-ilarly, although LES might be generalised to include agent-speci�c referencebundles, this is redundant in the context of convex subsistence economieswith agents of identical preference.Finally, it is worth noting that the vectors c and c in LES need not

be uniquely �xed, and may be functions of (p; w) and b. Further, once cand c are identi�ed, the existence of �c and �c is guaranteed by A3. Thecondition �c0 = �c0 is necessary for N

ter and N ted to be disjoint in everypossible economy.Various de�nitions of labour exploitation proposed in the literature are

discussed below, which satisfy LES. These de�nitions may be partitionedinto two main approaches, depending on how the value of labour power isde�ned. In what may be de�ned as the direct approach, the value of labourpower is de�ned as the labour value of the bundles that agents actuallyconsume, whatever the de�nition of labour value is. In the indirect approach,instead, the value of labour power is de�ned as the labour value of somereference bundle that agents can a¤ord with their subsistence income, eventhough they do not necessarily purchase it.Two de�nitions that follow the �rst approach are considered. The �rst one

is an application to subsistence economies (in which every agent consumesthe bundle b) of Morishima�s (1974) classical de�nition.

15

De�nition 4: Let (p; w) be a RS for E = hN ; (P; b) ; (!� )�2Ni. Agent �is exploited if and only if �� > l:v:(b); she is an exploiter if and only if�� < l:v:(b); and she is neither exploited nor an exploiter if and only if�� = l:v:(b).

De�nition 4 satis�es LES by choosing c = c = b, where �c = �c is cho-sen to satisfy �c0 = �c0 = l:v:(b). The second de�nition is a re�nement ofMorishima�s and is due to Roemer (1982).

De�nition 5: Let (p; w) be a RS for E = hN ; (P; b) ; (!� )�2Ni. Agent � isexploited if and only if �� > l:v:(b; p; w); she is an exploiter if and only if�� < l:v:(b; p; w); and she is neither exploited nor an exploiter if and only if�� = l:v:(b; p; w).

Denote the production vector � 2 P (p; w) with �0 = l:v:(c; p; w) by � (c).De�nition 5 satis�es LES by choosing c = c = b� (b) with �c = �c = � (b), orc = c = b with �c = �c = (�0 (b) ; � (b) ; � (b) + b).As an illustration of the second approach, two de�nitions are discussed.

The �rst has been proposed by Yoshihara (2007):

De�nition 6: Let (p; w) be a RS for E = hN ; (P; b) ; (!� )�2Ni. Agent � 2 Nis exploited if and only if �� > minc2B(p;b) l:v: (c; p; w); she is an exploiter ifand only if �� < minc2B(p;b) l:v: (c; p; w), and she is neither exploited nor anexploiter if and only if �� = minc2B(p;b) l:v: (c; p; w).

Let c� = argminc2B(p;b) l:v: (c; p; w). De�nition 6 satis�es LES by choosingc = c = b� (c�) with �c = �c = � (c�), or c = c = c� with �c = �c =(�0 (c

�) ; � (c�) ; � (c�) + c�).The second example of the indirect approach is an extension of the so

called �New Interpretation,�originally developed by Dumenil and Foley [Du-menil (1980); Foley (1982)]15 to convex cone economies, which has been pro-posed by Yoshihara (2007). Let (p; w) be a RS and let � + � be the corre-

sponding aggregate production point. Let t 2 [0; 1] be such that t�b�+ b�� 2

B (p; b).

De�nition 7 : Let (p; w) be a RS for E = hN ; (P; b) ; (!� )�2Ni. Agent� 2 N is exploited if and only if �� > t (�0 + �0); she is an exploiter if and

15See also Lipietz (1982); for a recent survey see Mohun (2004).

16

only if �� < t (�0 + �0), and she is neither exploited nor an exploiter if andonly if �� = t (�0 + �0).

De�nition 7 satis�es LES by choosing c = c = 1N

�b�+ b�� with �c = �c =1N(�+ �) and t = 1

N.

Although the most important de�nitions of exploitation proposed in theliterature satisfy LES, the axiom is by no means trivial. Consider for in-stance the de�nition of exploitation proposed by Matsuo (2008), accordingto which an agent who consumes c is exploited if there is a feasible bundlethat is at least as good as c, in utility terms, that can be produced using lesslabour than is actually expended by the agent. This de�nition does not seemimmediately relevant in the framework of this paper, because by the sub-sistence hypothesis, the agents�utility functions are not strictly increasingin consumption, thus contradicting one of Matsuo�s assumptions. However,one may de�ne labour value à la Matsuo counterfactually by asking whatwould be the minimum amount of labour necessary to reach a given level ofutility, if agents were endowed with a continuous, strictly increasing utilityfunction u : Rm+ ! R de�ned over consumption goods. In this case, it can beshown that Matsuo�s de�nition of exploitation does not satisfy LES. To seethis, consider the following example, which also illustrates the implicationsof LES.16

Example 1: Consider the following von Neumann technology

A =

�2 4 2 24 4 2 0

�, B =

�2 8 6 612 12 6 0

�, L = (1; 1; 1; 1) .

where, following usual notational conventions, A is the input matrix, B is theoutput matrix, and L is the direct labour vector. The corresponding produc-tion possibility set is P(A;B;L) �

�� 2 R� � R2� � R2+ j 9x 2 R4+ : � 5 (�Lx;�Ax;Bx)

.

Then, consider a convex cone subsistence economy E 2 E de�ned by P(A;B;L),b = (2; 2), and ! = (N;N). Let (!� )�2N be such that !

� = (�� ; �� ), where�� 5 2 for all � 2 N , and !�0 = (2; 2) for some � 0 2 N .16Alternatively, the function u may be interpreted as an objective measure of material

welfare deriving from consumption, rather than a representation of workers� subjectivepreferences. This interpretation is consistent with the equilibrium notion adopted in thispaper. A detailed analysis of Matsuo�s approach, and a discussion of recent developmentsin exploitation theory is in Yoshihara and Veneziani (2009).

17

Let ej 2 R4+ be a unit column vector with 1 in the j-th component and0 in every other component. Consider four reference production processes:�1 � (�Le1;�Ae1; Be1), �2 � (�Le2;�Ae2; Be2), �3 � (�Le3;�Ae3; Be3),and �4 � (�Le4;�Ae4; Be4), such that b�1 � (0; 8), b�2 � (4; 8), b�3 � (4; 4),and b�4 � (4; 0). Note that, bP (�0 = 1) = co�b�1; b�2; b�3; b�4;0.In this economy, p = (1; 0) and w = 2 constitute a RS, with �� = 0,

�� = ���3

2, and �0 = 1 � ��

2for all � 2 N . The corresponding aggre-

gate production is � + � = N�3

2. Note that, for each agent, B (p; b) =�

c 2 R2+ j 9x 2 R+ : c = (2; x).

According to Matsuo (2008), the commodity vector which serves to de�nethe value of labour power is given by:

cM � arg minc2R2+ , �2P

�0 subject to b� = c & u (c) = u (b) .But, by the strict monotonicity of u, cM 2 bP ��0 = 1

2

�n@ bP ��0 = 1

2

�, which

implies pcM < pb. Thus, cM =2 B (p; b), which implies that exploitation à laMatsuo does not satisfy LES.

Let W � � p!� . Theorem 1 characterises the exploitation status of everyagent, based on their initial wealth W � , for all de�nitions of exploitationsatisfying axiom LES.

Theorem 1: Let (p; w) be a RS for E = hN ; (P; b) ; (!� )�2Ni such that�max > 0. Then, for any formulation of labour exploitation satisfying LES:(i) agent � is an exploiter if and only if W � > 1

�max[pb� w�c0];

(ii) agent � is exploited if and only if W � < 1�max

[pb� w�c0]; and(iii) agent � is neither an exploiter nor exploited if and only if

1�max

[pb� w�c0] 5 W � 5 1�max

[pb� w�c0].

Theorem 1 is considerably more general than similar results derived byRoemer (1982), in that it applies to a whole class of de�nitions of exploita-tion, rather than a speci�c approach. Thus, for any de�nition of exploitationsatisfying axiom LES, Theorem 1 identi�es the wealth cut-o¤s that parti-tion the set of agents based on their exploitation status. Given the analysisin the previous section, one immediately notes that for each of the De�ni-tions analysed there is a unique wealth cut-o¤. Thus, by Theorem 1, underDe�nition 4, exploitation status depends on whether wealth is higher than,

18

lower than or equal to W � = 1�max

[pb� w (l:v: (b))]; similarly, under De�n-ition 5, the wealth cut-o¤ is W � = 1

�max[pb� w (l:v: (b; p; w))]; under De�-

nition 6, the wealth cut-o¤ is W � = 1�max

�pb� w

�minc2B(p;b) l:v: (c; p; w)

��;

�nally, under De�nition 7, the wealth cut-o¤ is W � = 1�max

hpb� w(�0+ 0)

N

i.

Actually, minc2B(p;b) l:v: (c; p; w) 5 l:v: (b; p; w) 5 t(�0 + �0) =�0+ 0N, and

l:v: (b) 5 l:v: (b; p; w) for all E 2 E , and thus the wealth cut-o¤s can beranked. Therefore, ceteris paribus, the set of exploited agents according toDe�nition 7 is (weakly) included in the set of exploited agents according toDe�nition 5, which is in turn (weakly) included in the set of exploited agentsaccording to De�nition 4.17

Following Roemer (1982), classes can also be de�ned in this economy,based on the way in which agents relate to the means of production. At aRS of the subsistence economy, an individually optimal solution for an agent� consists of a vector (�� ; �� ; �0 ). Therefore, let (a1; a2; a3) be a vector whereai = f+;0g; i = 1; 2, and a3 = f+; 0g, where �+�means a non-zero vector inthe appropriate place. Agent � is said to be a member of class (a1; a2; a3), ifthere is an individually optimal (�� ; �� ; �0 ) which has the form (a1; a2; a3).The notation (+;+; 0) implies, for instance, that an agent works in her own�shop�and hires others to work for her; (+;0;+) implies that an agent worksboth in her own �shop�and for others, etc. Although there are seven possibleclasses in the subsistence economy, it can be proved that at a RS, the set ofproducers N can be partitioned into the following �ve, theoretically relevantclasses.

C1 = f� 2 N j A� (p; w) has a solution of the form (0;+; 0)g ,C2 = f� 2 N j A� (p; w) has a solution of the form (+;+; 0) n (+;0; 0)g ,C3 = f� 2 N j A� (p; w) has a solution of the form (+;0; 0)g ,C4 = f� 2 N j A� (p; w) has a solution of the form (+;0;+) n (+;0; 0)g ,C5 = f� 2 N j A� (p; w) has a solution of the form (0;0;+)g .

The notation (+;+; 0) n (+;0; 0) means that agent � is a member of class(+;+; 0) but not of class (+;0; 0), and likewise for the other classes. Asa �rst step in the analysis of classes, Lemma 4 proves that (+;+;+) and(0;+;+) are indeed redundant.

17It is worth noting that it is not di¢ cult to construct economies E 2 E , in which thewealth cut-o¤s are indeed di¤erent and the inequalities are strict.

19

Lemma 4: Let (p; w) be a given price vector. Let � be such that W � > 0and �� > 0 at the solution of MP � . Let � belong to either (+;+;+) or(0;+;+). Then, exactly one of the following statements holds:

if �0 < ��0 for all optimal (�� ; �� ; �0 ) , then � 2 (+;+; 0)n(+;0; 0);

if �0 = ��0 for some optimal (�� ; �� ; �0 ) , then � 2 (+;0; 0);

if �0 > ��0 for all optimal (�� ; �� ; �0 ) , then � 2 (+;0;+)n(+;0; 0).

Given Lemma 4, it is now possible to prove a generalisation of Roemer�s(1982) core result concerning the correspondence between class status andwealth. Theorem 2 proves that classes C1 to C5 are pairwise disjoint andexhaustive, and wealthier agents belong to the upper classes.

Theorem 2: Let (p; w) be a RS for E = hN ; (P; b) ; (!� )�2Ni such that�max > 0. Then:(i) For any i 6= j, Ci \ Cj = ? and [5i=1Ci = N ;(ii) For any �; � 2 N , if � 2 Ci and � 2 Cj with i < j then W � > W � .

Theorems 1 and 2 identify two di¤erent partitions of the set of agentsbased, respectively, on their exploitation status and their relation to themeans of production (more precisely, their position in the labour market).In both cases, an agent�s wealth is the main determinant of her position inthe social structure, and therefore it is legitimate to ask whether class andexploitation status are related, as predicted in Marxian theory. This is themain object of analysis in the next section.

4 The Class-Exploitation Correspondence Prin-ciple

In classical Marxian theory, exploitation and classes are related: capitalistsare exploiters and proletarians are exploited. On the one hand, this corre-spondence of class and exploitation status gives a speci�c normative relevanceto the concept of class; on the other hand, it emphasises the importance ofrelations of production in the generation of exploitation. More formally, theClass-Exploitation Correspondence Principle (CECP) can be de�ned as fol-lows.

20

Class-Exploitation Correspondence Principle (CECP) [Roemer (1982)]:Given any economy E 2 E, at any RS with �max > 0,(A) every member of C1 [ C2 is an exploiter.(B) every member of C4 [ C5 is exploited.

A fundamental contribution of Roemer�s theory is the proof of theCECPin the private-ownership economy, which is derived as a result of the analy-sis, rather than being assumed. Epistemologically, though, Roemer forcefullyargues that the central relevance of the CECP in class and exploitation the-ory implies that it should be considered as a postulate, by requiring that anysatisfactory de�nition of exploitation (and classes) satis�es the CECP. Con-sistently with this approach, this section analyses the CECP under di¤erentde�nitions of exploitation satisfying LES.As a �rst step, it is useful to provide a characterisation of class status

in general convex cone subsistence economies. Let (p; w) be a RS such that

�max > 0. Let �min 2 P (p; w) be such that p�min�0min

= min�2P (p;w)

hp��0

iand

�maxp�min + w�0min = pb, and let �max 2 P (p; w) be such that p�max�0max

=

max�02P (p;w)

hp��0

iand �maxp�max + w�0max = pb. Note that p�min 5 p�max,

and that �min; �max are well-de�ned. The next proposition provides a precisecharacterisation of class status based on an agent�s wealth.

Proposition 4: Let (p; w) be a RS for E = hN ; (P; b) ; (!� )�2Ni such that�max > 0. Then,(i) � 2 C1 , W � = pb

�max;

(ii) � 2 C2 , p�max < W� < pb

�max;

(iii) � 2 C3 , p�min 5 W � 5 p�max; and(iv) � 2 C4 , 0 < W � < p�min;(v) � 2 C5 , W � = 0.

In other words, Proposition 4 derives four wealth cut-o¤ levels that parti-tion the set of agents into �ve classes. The richest agents are big capitalists,who can reproduce themselves without working, and the poorest, property-less agents are proletarians, who can only sell their labour in order to survive.All other agents fall into the intermediate classes, based on their wealth. Inprinciple, the wealth cut-o¤s identi�ed in Proposition 4 may be di¤erentfrom those characterising exploitation status identi�ed in Theorem 1, whichdepend on the actual de�nition of exploitation adopted. The next theorem

21

provides the general condition for the CECP to hold under any de�nitionof exploitation satisfying LES.

Theorem 3: Let (p; w) be a RS for E = hN ; (P; b) ; (!� )�2Ni such that�max > 0. For any de�nition of labour exploitation satisfying LES, theCECP holds if and only if c; c are such that

p�min 51

�max[pb� w�c0] 5

1

�max�pb� w�c0

�5 p�max.

Again, Theorem 3 is signi�cantly more general than the analogous resultsproved by Roemer (1982), in that it provides general necessary and su¢ cientconditions for the CECP to hold for an entire class of de�nitions of exploita-tion, rather than for a speci�c approach. Actually, there is another way ofstating Theorem 3, which may illustrate its signi�cance in terms of a de�n-ition of exploitation satisfying LES. Let (p; w) be a RS with �max > 0, andlet

� (p; w) � f� 2 P j pb� = pb and pb�� w�0 2 [�maxp�min; �maxp�max]g :Note that � (p; w) is the set of feasible production processes yielding a netoutput with the same monetary value as the subsistence vector and a pro�trevenue equal to the pro�t revenue under one of the maximal-pro�t-rateprocesses. Similarly, the set of net outputs associated with � (p; w) is de�nedas follows: b� (p; w) � �b� 2 Rm+ j � 2 � (p; w). Theorem 3 can be re-statedas proving that at a RS (p; w) such that �max > 0, for any de�nition of labourexploitation satisfying LES, the CECP holds if and only if c; c 2 b� (p; w).In other words, for any de�nition of labour exploitation satisfying LES,

the CECP holds if and only if the pro�t revenue associated with the pro-duction of the reference bundles c and c is equal to the pro�t revenue underone of the maximal-pro�t-rate processes. This characterisation holds regard-less of whether the amount of labour necessary to produce c; c is minimal ornot, and regardless of whether the net output is actually produced at the RS.Note that this does not exclude the possibility that the CECP holds under alabour value formulation with no maximal-pro�t-rate production process.18

This property is due to the assumption of subsistence economies. In fact,in the case of accumulation economies, as Yoshihara (2007) shows, if the

18For a proof of this claim see Example A.1 in Appendix 2 below.

22

CECP holds under a labour value formulation, then this formulation shouldbe associated with some maximal-pro�t-rate production process.19

Let (p; w) be a RS and let � (p; w) ��� 2 P (p; w) j pb� = pb: � (p; w)

is the set of feasible, maximal-pro�t-rate production processes yielding a netoutput with the same monetary value as the subsistence vector. Similarly,the set of net outputs associated with � (p; w) can be de�ned as follows:b� (p; w) � �b� 2 Rm+ j � 2 � (p; w). Then, noting that b� (p; w) � b� (p; w),the next Corollary immediately follows from Theorem 3.

Corollary 1: Let (p; w) be a RS for E = hN ; (P; b) ; (!� )�2Ni such that�max > 0. For any formulation of labour exploitation satisfying LES, if itscorresponding c; c are such that c; c 2 b� (p; w), then the CECP holds underthis formulation.

Corollary 1 provides su¢ cient conditions for the CECP to hold for anyde�nition of labour exploitation satisfying LES. In particular, it states thattheCECP holds if this de�nition evaluates the amount of labour necessary toproduce a net output, which has the same monetary value as the subsistencevector at the RS, by operating one of the maximal-pro�t-rate processes.Based on Theorem 3 and Corollary 1, it is possible to prove that the CECPholds under De�nitions 5, 6, and 7.

Corollary 2: Let (p; w) be a RS for E = hN ; (P; b) ; (!� )�2Ni such that�max > 0. Then the CECP holds under De�nitions 5, 6, and 7.

Instead, De�nition 4 does not preserve the CECP, because the commod-ity bundle used to de�ne the value of labour power under De�nition 4 neednot be produced with the same pro�t revenue as that of the maximal-pro�t-rate processes. This is shown formally in the following proposition.

Proposition 5: Let exploitation be de�ned according to De�nition 4. Thereis a convex cone, subsistence economy E 2 E in which there exists a RS with�max > 0, such that the CECP does not hold.

In the context of accumulating economies, Roemer (1982; Chapter 5, p.148) had already suggested that in accumulation economies, if Morishima�s

19Note that Roemer (1982; Chapter 5; p.164) also stated basically the same claim,though he only discussed the case of exploiters.

23

de�nition of exploitation is used, then �the CECP is false for the cone tech-nology� - a claim rigorously proved by Yoshihara (2007). The roots of thefailure of the CECP, however, are di¤erent in the cases of subsistence andaccumulation economies. In fact, as Roemer (1982) and Yoshihara (2007)show, in accumulation economies the failure of the CECP is due to theexistence of both non-exploiting capitalists and non-exploited workers. Incontrast, in subsistence economies, as the proof of Proposition 5 in the Ap-pendix shows, the failure of the CECP is due to the existence of exploitedcapitalists.The results presented in this section are quite relevant for exploitation

theory. Given the epistemological relevance of the CECP discussed above,Proposition 5 suggests that Morishima�s classical de�nition of exploitation isinadequate to capture the central intuitions of Marxian theory.20 Contraryto Roemer�s claims, however, Corollary 2 proves that, even in the generalsetting analysed in this paper, the CECP does hold for various de�nitions ofexploitation presented in the literature. More generally, Theorem 3 suggeststhat, for all possible convex subsistence economies and all equilibria, theCECP holds for a whole class of notions of exploitation as UE of laboursatisfying the arguably weak and reasonable axiom LES.21

Finally, the following equivalence relation can be proved:

Theorem 4: Consider any de�nition of exploitation which satis�es LES.Let (p; w) be a RS for E = hN ; (P; b) ; (!� )�2Ni and c; c 2 b� (p; w) for thisde�nition of exploitation. Then, the following statements are equivalent :(1) �max > 0;(2) the CECP holds;(3) every producer in C5 is exploited.

Theorem 4 has two important implications for exploitation theory. Firstly,the equivalence between (1) and (3) represents a signi�cant generalisationof the so-called �Fundamental Marxian Theorem� (FMT) [Roemer (1980,1981)], according to which the equilibrium maximal pro�t rate is positive ifand only if every worker is exploited. The FMT is proved to hold in general20Actually, Morishima�s de�nition has another arguably undesirable property. In fact,

it is not di¢ cult to construct an example of a subsistence economy in which all producershave the same endowment of capital goods, but they are all exploited in the sense ofDe�nition 4, a rather counterintuitive result.21As shown by Yoshihara (2007), a similar result holds in accumulation economies.

24

convex cone economies with a complex class structure and for a whole set ofde�nitions of exploitation satisfying axiom LES. This is important because,as for the CECP, although it is proved as a result, the epistemologicalstatus of the FMT in exploitation theory is usually that of an axiom, andalternative de�nitions of exploitation are often compared in terms of theirability to preserve it. Actually, one might discuss which axiom - the CECPor the FMT - is more relevant and whether imposing both signi�cantlydecreases the set of available de�nitions of exploitation. The second, andsomewhat more surprising, implication of Theorem 4 is that for the wholeclass of de�nitions of exploitation satifying LES with c; c 2 b� (p; w), theCECP and the FMT are equivalent, and therefore one need not chooseamong them: any de�nition preserving one, also preserves the other.

5 An Axiomatic Characterisation of Exploita-tion

The results presented in the previous sections are encouraging: there exist aset of de�nitions of exploitation as the unequal exchange of labour that satisfythe reasonable condition imposed by LES, and preserve the CECP (andthe FMT) as a result, in the equilibrium of the private-ownership economy.Furthermore, both LES and the requirement that theCECP holds in generalconvex economies are not trivial, and some of the de�nitions proposed inthe literature do not satisfy them. In terms of identifying and defendingone de�nition of exploitation, however, the analysis developed so far is notconclusive as it cannot discriminate between a number of competing notions.In line with the novel axiomatic approach adopted in this paper, the mainpurpose of this section is to propose and defend some additional theoreticalconditions that any de�nition of labour exploitation should satisfy and thento provide a characterisation result.Consider any economy E 2 E . For any de�nition of exploitation satisfying

LES, recall that the corresponding reference bundles c; c 2 Rm+ identifyingthe value of labour power depend on p: So, let

�(p; w) ; (�� ; �� ; �0 )�2N

�be

a RS for E = hN ; (P; b) ; (!� )�2Ni and let c; c be the corresponding referencebundles. Non-exploitative allocations can be de�ned as follows.

De�nition 8: Suppose that the de�nition of exploitation satis�es LES.Given a RS

�(p; w) ; (�� ; �� ; �0 )�2N

�and the corresponding reference bun-

25

dles c; c, the allocation (�0� ; �0� ; 0�0 )�2N 2 (P � P � [0; 1])N is non-exploitative

at (p; w) if(i) Nb 5

P�2N b�0� +P�2N

b�0� andP�2N �0�0 =

P�2N

0�0 ; and

(ii) �c0 5 �0�0 + 0�0 5 �c0 for all � 2 N .

De�nition 8 is a rather weak de�nition of non-exploitative allocations,because it only requires that subsistence be guaranteed and that all labourrequired to implement the allocations be available. No constraint is imposed,instead, on aggregate capital endowments, or on the way in which such allo-cation can be reached. Arguably, though, non-exploitative allocations thatare within the realm of social feasibility are of focal normative relevance. Thenext de�nition provides a precise notion of social feasibility.

De�nition 9: Given a RS�(p; w) ; (�� ; �� ; �0 )�2N

�, an allocation (�0� ; �0� ; 0�0 )�2N

is feasible non-exploitative w.r.t (p0; w0) in E = hN ; (P; b) ; (!� )�2Ni if:(i) the allocation (�0� ; �0� ; 0�0 )�2N is non-exploitative at (p; w) and(ii)

�(p0; w0) ; (�0� ; �0� ; 0�0 )�2N

�is a RS for hN ; (P; b) ; (!� )�2Ni.

Clearly, the feasibility of a non-exploitative allocation depends on the actualvalues of c; c 2 Rm+ , that is, on the notion of labour exploitation adopted.The next De�nition outlines the concept of e¢ ciency that is relevant in

convex subsistence economies.

De�nition 10: An allocation (�� ; �� ; �0 )�2N 2 (P � P � [0; 1])N is e¢ -

cient for E = hN ; (P; b) ; (!� )�2Ni if it satis�es:(i)

P�2N �

� +P

�2N �� 5

P�2N !

� , Nb 5P

�2N b�� + P�2Nb�� , andP

�2N ��0 =

P�2N

�0 ; and there is no other allocation (�

0� ; �0� ; 0�0 )�2N2 (P � P � [0; 1])N such that the above (i) is satis�ed and(ii) �0�0 +

0�0 5 ��0 + �0 for all � 2 N , and �0�

�0 + 0�

�0 < ��

�0 +

��0 for some

�� 2 N .

In other words, if an allocation is e¢ cient, it is impossible to reduce thelabour time of some agent without increasing the amount of labour expendedby someone else.22 It is also worth noting that if an allocation is e¢ cientaccording to De�nition 10, then even a benevolent social planner cannot

22Thus, De�nition 10 appropriately focuses on the socially relevant phenomenon, namelythe minimisation of labour, but it gives no weight to the behavioural assumption encom-passed in NBC, according to which big capitalists minimise capital outlay.

26

improve on it by choosing (�0� )�2N 2 PN such thatP

�2N �0� 5

P�2N !

� ,Nb 5

P�2N b�0� ; and �0�0 5 ��0 + �0 for all � 2 N , and �0��0 < ��

�0 +

��0 for

some �� 2 N .The next axiom requires that, whatever the notion of labour exploitation

is, its corresponding non-exploitative allocation should be feasible for eacheconomy.

Feasibility of Non-exploitation (FNE): Consider a de�nition of ex-ploitation satisfying LES. Given any economy E 2 E, let (p; w) be ane¢ cient RS and let c; c 2 Rm+ be the corresponding reference bundles. If(�0� ; �0� ; 0�0 )�2N is non-exploitative at (p; w) such that

P�2N �

0�+P

�2N �0� =P

�2N !� , then there exists (p0; w0) such that for some suitable redistribution

(!0� )�2N of (!� )�2N with

P�2N !

0� =P

�2N !� , (�0� ; �0� ; 0�0 )�2N is feasi-

ble non-exploitative with respect to (p0; w0) in hN ; (P; b) ; (!0� )�2Ni.

FNE is a rather weak and desirable property for all concepts of exploita-tion. For, �rst, all of the formulations of exploitation discussed in this papersatisfy it, and second, it requires that if a non-exploitative allocation existswith the current social endowment, then it must be possible to implementsuch an allocation with some appropriate distribution of social endowments.Arguably, any normatively relevant concept of exploitation should allow forthe existence of non-exploitative allocations. If exploitation could not beeliminated, then its relevance for the evaluation of capitalist societies wouldarguably be diminished, as an evil that agents must learn to live with. Then,given that an ideal (non-exploitative) allocation is technologically feasible interms of De�nitions 1(b), 1(c), and 1(d), FNE would require that it be ra-tional in terms of De�nitions 1(a). The appeal to markets as the mechanismto implement non-exploitative allocations may be controversial, but one mayargue that in the context of this paper the usual equivalence between marketallocations and the social planner problem holds. Furthermore, theoretically,FNE can be seen as requiring that non-exploitation is compatible with therational choices of individuals. To be precise, since the CECP shows thatthe emergence of class and exploitation is the consequence of rational choiceof individuals, the FNE guarantees that the emergence of class and exploita-tion is not a necessary condition for rational choice of individuals, becausenon-exploitation is also required to be compatible with rational choices. Morepragmatically, thus far, markets have proved to be the only sustainable, de-centralised decision-making allocation mechanism in the sense of Hurwicz

27

(1972), and, absent a viable alternative, this justi�es what may seem an ex-cessively narrow focus. Furthermore, FNE is interesting also in relation torecent proposals of market socialism as a mechanism to combine e¢ ciencyand social justice, including the elimination of exploitation (see, e.g., Roe-mer, 1994). Actually, as shown below, at the level of abstraction at whichthis analysis is conducted, the focus on markets allows one to introduce someimportant issues concerning the relation between exploitation and e¢ ciency.In addition to FNE, the following axiom is a natural requirement for any

de�nition of exploitation:

Independence of Endowment Structure (IES): Consider a de�nition ofexploitation satisfying LES. Given two economies E = hN ; (P; b) ; (!� )�2Niand E 0 = hN ; (P; b) ; (!0� )�2Ni, let

�(p; w) ; (�� ; �� ; �0 )�2N

�and�

(p; w) ; (�0� ; �0� ; 0�0 )�2N�be their corresponding RSs such that

P�2N (�

� + �� ) =P�2N (�

0� + �0� ). Moreover, let c; c 2 Rm+ and c0; c0 2 Rm+ be the correspond-ing reference bundles. Then, c = c0 and c = c0.

Axiom IES implies that the reference commodity bundles may depend onthe price vector (p; w) and the equilibrium social production point �p;w of thegiven RS, but they should be independent of the allocation of production ac-tivity vectors, which are determined by the underlying endowment structure.This condition seems uncontroversial and indeed all the main de�nitions ofexploitation discussed in the literature satisfy it.Finally, the following axiom captures an arguably essential feature of any

theory of exploitation.

Relational Exploitation (RE): Consider a de�nition of exploitation sat-isfying LES. At any RS for hN ; (P; b) ; (!� )�2Ni, the two subsets N terandN ted are such that N ter 6= ? if and only if N ted 6= ?.

From a formal viewpoint, axiom RE imposes a rather weak restriction onLES. From a theoretical viewpoint, it captures the crucial relational aspectinherent in exploitative relations, such that if an agent is exploited, she mustbe exploited by someone, and viceversa if an exploiter exists, she must beexploiting someone.Before proving the main Theorem of this section, which provides a com-

plete axiomatic characterisation of exploitation, some intermediate resultsare derived, which are also interesting in their own right. First of all, the

28

next Lemma proves that all RS�s in which the wealth constraints bind, aree¢ cient in the sense of De�nition 10.

Lemma 5: Let�(p; w) ; (�� ; �� ; �0 )�2N

�be a RS for E = hN ; (P; b) ; (!� )�2Ni:

IfP

�2N p��� + ��

�=P

�2N p!� , then (�� ; �� ; �0 )�2N is e¢ cient.

Lemma 5 suggests that the scarcity of capital is a su¢ cient condition fore¢ ciency in equilibrium. Yet, it does not rule out the possibility of ine¢ cientRS�s. In particular, if

�(p; w) ; (�� ; �� ; �0 )�2N

�is a RS with

P�2N p

��� + ��

�<P

�2N p!� , then (�� ; �� ; �0 )�2N is not necessarily e¢ cient. Formally, it is

not possible to prove a Marxist version of the First Welfare Theorem becauselocal nonsatiation may be violated in this economy for all those agents whodo not work at the optimum. In fact, a necessary condition for a RS to beine¢ cient is the existence of big capitalists who can reproduce themselveswithout working, and who maintain capital scarcity �arti�cially� by min-imising capital outlay, consistently with the assumption NBC. An exampleof an ine¢ cient RS in a convex subsistence economy is discussed in detail inAppendix 3 below and the role of big capitalists is shown. This is an inter-esting case, as it suggests a relation between ine¢ ciencies and extreme formsof injustice, viz. exploitative social relations. From a normative perspective,however, this is the least challenging case since there is no trade-o¤, at leastlocally, between an improvement in e¢ ciency and a reduction of social in-justices, namely exploitation. Even from a Marxian viewpoint, it may beargued that exploitation theory should focus on the amount of labour timenecessary to produce the subsistence bundle by adopting e¢ cient productiontechniques.Proposition 6 proves that the amount of labour performed is uniquely

determined at all e¢ cient equilibria of the subsistence economy.

Proposition 6: Let E = hN ; (P; b) ; (!� )�2Ni and E 0 = hN ; (P; b) ; (!0� )�2Nibe two economies such that

P�2N !

� =P

�2N !0� . Let

�(p; w) ; (�� ; �� ; �0 )�2N

�and

�(p0; w0) ; (�0� ; �0� ; 0�0 )�2N

�be two e¢ cient RSs for E and E 0; respec-

tively. Then �0 + 0 = �00 +

00.

According to Proposition 6, for a given vector of aggregate productive assets,the amount of labour performed is independent of the distribution of endow-ments and it is equalised across all e¢ cient RS�s. The next Lemma, instead,focuses on e¢ cient and resource-egalitarian equilibria and proves that if ane¢ cient RS exists, then the same price vector supports an egalitarian RS in

29

which all individuals have the same vector of productive endowments andactivate the same production process as self-employed agents.

Lemma 6: Let�(p; w) ; (�� ; �� ; �0 )�2N

�be a RS for E = hN ; (P; b) ; (!� )�2Ni

such thatP

�2N p��� + ��

�=P

�2N p!� . Then

�(p; w) ; (�0� ; 0; 0)�2N

�is

a RS for E 0 = hN ; (P; b) ; (!0� )�2Ni, where �0� =P�2N (�

�+�� )

Nand !0� = !

N

for all � 2 N .

Given the new axioms, FNE, RE, and IES, the following characterisa-tions can be derived.

Theorem 5: A formulation of labour exploitation satis�es LES, FNE, RE,and IES if and only if for all E 2 E and every RS (p; w), �c0 = �c0 =

�0+�0N.23

Theorem 5 can thus be taken as providing support to the extension ofDumenil-Foley�s �New Interpretation�(De�nition 7) as the appropriate de-�nition of exploitation, which is uniquely characterised from a rather smallset of arguably weak axioms.24 Actually, Theorem 5 has a rather strikingimplication: because, as shown above, virtually all of the most relevant de�-nitions of exploitation proposed in the literature such as De�nitions 4, 5, 6,and 7 satisfy axioms LES, FNE, and IES, it is the rather mild axiom RE -which requires the presence of exploiters, whenever some agent is exploited- that rules out all alternative de�nitions except De�nition 7. By Theorem5, not all de�nitions of exploitation satisfy these reasonable properties, andindeed our axiomatic analysis allows us to precisely identify the limits of thereceived de�nitions of exploitation: if Morishima�s celebrated de�nition isadopted, for example, it is not di¢ cult to construct examples in which REis violated and all agents are exploited.Interestingly, however, even the adoption of a more traditional stance em-

phasising the importance of the notions of surplus labour and socially neces-sary labour time in exploitation theory, rather than the relational propertyincorporated inRE, does not rescue the main received de�nitions of exploita-tion. In particular, following classical approaches to exploitation theory, it

23Such a characterisation by using RE still holds if RE is weakened so that its require-ment is imposed only on any RS with a positive pro�t rate.24It is worth noting that under De�nitions 4 and 5, there indeed exist some E 2 E and

some RS (p; w), such that l:v:(b) 6= �0+�0N and l:v:(b�(b); p; w) 6= �0+�0

N , so that Theorem5 does capture a fundamental di¤erence between alternative de�nitions.

30

may be argued that Socially Necessary Labour corresponds to the minimumamount of labour necessary to produce the subsistence consumption bundleand this quantity is an important theoretical benchmark, such that the refer-ence labour expenditures �c0; �

c0 that identify exploiters and exploited agents

should be no lower than socially necessary labour. More formally:

Exploitation as E¢ cient Use of Labour (EEUL): Consider a de�nitionof exploitation satisfying LES. Given any economy hN ; (P; b) ; (!� )�2Ni, let�(p; w) ; (�� ; �� ; �0 )�2N

�be a RS such that c; c 2 Rm+ are the corresponding

reference bundles. Then, �c0; �c0 = l:v: (b).

In the canonical Marxian theory of exploitation in the Okishio-Leontiefframework, socially necessary labour time is uniquely de�ned and, since �c0 =�c0 = l:v: (b), it identi�es the sets of exploiters and exploited agents. AxiomEEUL generalises such a theory to convex subsistence economies, and in fact,this axiom represents a condition which the classical Marxian theory of labourvalue and exploitation satis�es if a subsistence economy hN ; (P; b) ; (!� )�2Niis characterised by a simple Leontief technology P = P(A;I;L), where A and Iare square matrices. However, this is a rather weak requirement in the sensethat most of the formulations of exploitation discussed in this paper satisfyit.In order to derive the characterisation result, it is necessary to slightly

modify the axiom FNE.

Feasibility of Non-exploitation� (FNE�): Consider a de�nition of ex-ploitation satisfying LES. Given any economy E 2 E, let (p; w) be an ef-�cient RS and let c; c 2 Rm+ be the corresponding reference bundles. Then,for any non-exploitative allocation (�0� ; �0� ; 0�0 )�2N at (p; w), there exists(p0; w0) such that for some suitable redistribution (!0� )�2N of (!

� )�2N withP�2N !

0� =P

�2N !� , (�0� ; �0� ; 0�0 )�2N is feasible non-exploitative with

respect to (p0; w0) in hN ; (P; b) ; (!0� )�2Ni.

FNE� is slightly stronger than FNE, because it does not focus on non-exploitative allocations at the givel level of social endowments: implementabil-ity is imposed regardless of whether the non-exploitative allocation is physi-cally feasible in the current economy or not.Given FNE�, it is now possible to derive an alternative characterisation

of De�nition 7.

31

Theorem 6: A formulation of labour exploitation satis�es LES, FNE�,EEUL, and IES if and only if for all E 2 E and every RS (p; w), �c0 = �c0 =�0+�0N.

In other words, one may argue that there are two possible views on exploita-tion. One is that exploitation is a notion that characterises social relations inmarket economies, as described by axiom RE. Alternatively, one may insistthat exploitation theory is based on a notion of �surplus labour�and sociallynecessary labour. Quite surprisingly, Theorems 5 and 6 prove that the twoapproaches coincide in the rather general context analysed in this paper andour extension of Dumenil-Foley�s �New Interpretation�(De�nition 7) is theonly de�nition incorporating both perspectives.Finally, from Theorems 3 and 5, the next result follows:

Corollary 3: De�nition 7 is the sole formulation of labour exploitation sat-isfying LES, FNE, RE, and IES, and under which the CECP holds atevery RS (p; w) with �max > 0 for all E 2 E .

Thus, the extension of Dumenil-Foley�s �New Interpretation�(De�nition 7)is the sole formulation of exploitation which satis�es all axioms and whichpreserves an important property of exploitation theory, namely the CECPin general convex economies. Of course, a similar corollary can be derivedusing Theorem 6.

6 Concluding remarks

This paper provides a formal analysis of exploitation in convex cone subsis-tence economies with rational optimising agents. The distributive injusticeassociated with exploitative relations is rigorously explored and the relevanceof a notion of exploitation that emphasises the unequal exchange of labour isdefended. Firstly, a de�nition of exploitation is provided according to whichexploitative relations involve an unequal distribution of (and control over)social labour, consistently with a normative approach that focuses on indi-vidual well-being freedom and the self-realisation of men. The main resultsof Roemer�s classical theory are generalised and it is proved that this de�ni-tion - actually, a whole class of de�nitions of exploitation - preserves all theclassical Marxist insights on exploitation in a framework that is considerablymore general than the standard Leontief, or von Neumann economies: every

32

agent�s class and exploitation status emerges in the competitive equilibrium;there is a correspondence between an agent�s class and exploitation status;and the existence of exploitation is inherently linked to the existence of pos-itive pro�ts. Thus, many important properties of exploitation theory areshown to be considerably more robust than is generally thought.Secondly, and perhaps more importantly, unlike in the previous litera-

ture, this paper develops a new, axiomatic approach to exploitation theory.The core intuitions behind the concept of exploitation are thus directly ad-dressed, in order to derive the appropriate de�nition of exploitation from�rst principles, rather than from ad hoc attempts to eschew a number oftechnical or theoretical problems. The main positive and normative aspectsof exploitation theory are explicitly formalised in a set of axioms that everylabour-based de�nition of exploitation should arguably satisfy. The axiomsseem rather weak and reasonable, but they uniquely characterise the de�ni-tion of exploitation adopted in the paper, which focuses on the distributionof social labour and which, quite interestingly, is conceptually related to theso-called �New-Interpretation�(Dumenil, 1980; Foley, 1982).To be sure, the main results presented above hold in the rather large class

of convex cone, subsistence economies considered in the paper, but nothingis said, for example, concerning economies in which agents accumulate, orhave richer, and possibly heterogeneous preferences. The extension of theanalysis to di¤erent contexts, or under alternative assumptions concerningagents�behaviour, indeed represents an interesting line for further research,but two important points should be made concerning the contributions ofthis paper. Firstly, although the details of the analysis are likely to change ifdi¤erent economies are analysed, the main insights of the paper, concerningthe normative and positive relevance of exploitation and the appropriatede�nition of exploitation, would not be a¤ected: they seem independent ofthe speci�c (subsistence) framework and their relevance seems to extendbeyond the latter. Thus, for example, even if agents are allowed to consumedi¤erent bundles of goods, an appropriate de�nition may still be seen asidentifying a normatively relevant reference bundle - such as the �averageconsumption bundle�or the per-capita value of net domestic product - andrequiring that the value of labour power be measured accordingly. Secondly,and perhaps more importantly, the rigorous axiomatic analysis developed inthis paper suggests a fruitful methodological approach to exploitation thatcan be applied to di¤erent contexts, and that should frame any future debatesin exploitation theory.

33

7 Appendix

7.1 Appendix 1: The existence of a RS

This appendix provides a complete characterisation of reproducible solutions.Without loss of generality, let S �

�p 2 Rm+ j pb = 1

be the set of normalised

price vectors. Let �0 (!) � max f�0 j 9� = (��0;��; �) 2 P s.t. � 5 !g.Given the production set P , let eP � �(��0; �) 2 R� � Rm j (��0;��; �) 2 Pand let

C �n! 2 Rm+ j 9� 2 P : � 5 !, ��Nb = �, ��Nb � � & (��0; Nb+ �) 2 @ ePo .

That is, ! 2 C implies that there exists a production vector that is feasiblefrom ! and produces Nb as net output. Moreover, such a production point isweakly e¢ cient in the sense that (��0; Nb+ �) 2 @ eP . Note that by A1vA3,C is non-empty, closed, and convex.In order to provide a full characterisation of RS�s, two more sets must

be de�ned. Given the production set P and the scalar w = 1, let eP (w) ���� w�0b 2 Rm+ j (��0;��; �) 2 P & b�� w�0b 2 Rm+ and letC� �

�! 2 Rm+ j 9� 2 P & 9w = 1 : � 5 !, ��Nb = �, ��Nb � �,

& Nb+ �� w�0b 2 @ eP (w)o.That is, ! 2 C� implies that there exists a production vector that is

feasible from ! and produces Nb as net output. Moreover, its associated�surplus product�Nb�w�0b is a non-negative vector for some w = 1. Finally,such a production point is also weakly e¢ cient in the sense that Nb + � �w�0b 2 @ eP (w) for some w = 1. Note that C� � C and by A1vA3, C� is alsonon-empty, closed, and convex. Theorem A.1 proves a necessary conditionfor a RS to exist.

Theorem A.1: Let b 2 Rm++ , ! 2 Rm+ . Under A1vA3, if a reproduciblesolution (RS) exists for the economy E = hN ; (P; b) ; (!� )�2Ni then ! 2 C.Furthermore, if there is some agent � with p!� = 0, then ! 2 C�.

Proof. 1. Let (p; w) be a RS with its corresponding aggregate pro-duction vector � + � 2 P , and such that p 2 S. First, � + � 5 !

and � + � � Nb = � + � immediately follow from De�nition 1(b)-(c).

34

Next, Propositions 1(i) and 2 imply that � + � � Nb � � + � . Finally,

at the RS, �max = pNb�w(�0+�0)p(�+�)

, and by Proposition 2, it is possible to

choose a positive number � = 1 such that �p��+ �

�= p!, and therefore

1 + �max =pN�b+p�(�+�)�w�(�0+�0)

p!. This implies that pN�b + p�

��+ �

��

w� (�0 + �0) � pb�0+p�0�w�00 for all ���00; b�0 + �0� 2 eP such that p�0 = p!.Therefore

��� (�0 + �0) ; N�b+ �

��+ �

��2 @ eP and (p; w) is a supporting

price of it. Since eP is a convex cone, �� (�0 + �0) ; Nb+ ��+ ��� 2 @ eP and(p; w) is a supporting price of it.2. Suppose there is some agent � with p!� = 0. The �rst part of the proof

is as in step 1. Then note that since p 2 S and the upper bound of laboursupply is one, it follows that w = 1, for otherwise agent � with p!� = 0cannot survive even if she supplies the maximum amount of labour. Finally,choose a positive number � = 1 such that �p

��+ �

�= p! holds. Then,

1 + �max =pN�b+p�(�+�)�w�(�0+�0)

p!. This implies that pN�b + p�

��+ �

��

w� (�0 + �0) = pb�0 + p�0 � w�00 for all b�0 + �0 � w�00b 2 eP (w) such thatp�0 = p!. Moreover, since �max = 0 at a RS and Lemma 1 holds, it followsfrom b 2 Rm++ that Nb � w (�0 + �0) b = 0, which implies that N�b +���+ �

�� �w (�0 + �0) b 2 @ eP (w) and p 2 S is a supporting price of it.

Since eP (w) is a convex cone, Nb + ��+ �� � w (�0 + �0) b 2 @ eP (w) andp 2 S is a supporting price of it.

De�ne a domain of social endowments slightly smaller than C� as follows:

C�� ��! 2 Rm+ j 9� 2 P & 9w = 1 : � = !, ��Nb = �, ��Nb � �,

& Nb+ �� w�0b 2 @ eP (w)o.Within this domain, the existence of a RS can be proved. For any given !2 C��, de�ne the set

A (!) �n� 2 P& w = 1 j � = !; ��Nb = �; ��Nb � �;& Nb+ �� w�0b 2 @ eP (w)o.Theorem A.2: Let b 2 Rm++ , ! 2 Rm+ and �0 (!) 5 N . Under A1vA3,if ! 2 C�� then there exists (��; w) 2 A (!) such that there exists p 2 Sthat supports Nb+ �� � w��0b 2 @ eP (w) in eP (w). Furthermore, if (!� )�2N

35

is such thatP

�2N !� = ! and pb � �maxp!� = 0 for all � 2 N , then a

reproducible solution (RS) exists for the economy E = hN ; (P; b) ; (!� )�2Niat (p; w).

Proof. 1. By de�nition, if ! 2 C��, there exists �� = (���0;���; ��) 2 Psuch that �� = !, �� � Nb = �� with �� � Nb � ��, and Nb + �� �w��0b 2 @ eP (w) for some w = 1. Therefore the aggregate production vector�� satis�es De�nition 1, parts (b) and (d). It must be shown that there is aprice vector such that �� emerges from individually optimal choices and thatthe labour market clears. Note that since it is assumed that �0 (!) 5 N ,it follows that �� � ��0b = �� � Nb. Since Nb + �� � w��0b 2 @ eP (w),by the supporting hyperplane theorem, there exists p 2 S that supportsNb+���w��0b 2 @ eP (w) in eP (w). That is, pNb+p���w��0 = pb�+p��w�0for all b�+��w�0b 2 eP (w). Hence, given that �� = !, pNb+ p���w��0 =pb� + p� � w�0 holds for all b� + � � w�0b 2 eP (w) such that p� = p!.Therefore, since eP (w) is a convex cone, the previous argument shows that(���0;���; Nb+ ��) realises the maximal pro�t rate at (p; w). However,since �� � Nb = ��, �� also realises the maximal pro�t rate at (p; w), andpb�� = pNb. Furthermore, since b�� + �� � w��0b 2 eP (w), it follows thatpNb� w��0 = 0, which implies �max = 0.2. Let (!� )�2N be a distribution of initial endowments such that

P�2N !

� =

!. Consider the price vector (p; w) derived in step 1. Note that �� 2 P (p; w)and �� = !. Let �max � pb���w��0

p�� . Suppose that (!� )�2N satis�es the condi-tion in the second part of the statement, so that pb � �maxp!� = 0, for all� 2 N . Then, let �� � p!�

p�� for all � 2 N , and let �� � ���� for all � 2 N .

Then, by de�nition, p�� = p!� and �� 2 P (p; w) for all � 2 N . Moreover,let � � pb��maxp��

wfor all � 2 N . Then, by de�nition, pb��

maxp��

w= 0 for

all � 2 N . Since w = 1 and pb = 1, it follows that 1 = �0 = 0, all � 2 N .Therefore by Lemma 3, (0; �� ; �0 )�2N 2 ��2NA� (p; w) . FurthermoreX�2N

�0 =

P�2N

�pb� �maxp��

�w

=

�Npb� �max

P�2N p�

��

w=(Npb� �maxp��)

w.

This impliesP

�2N �0 = �

�0 =

P�2N �

�0 , which completes the proof.

Given Theorem A.2, it is possible to state a simpler su¢ cient condition forthe existence of a RS. Consider the following domain of social endowments:

36

C��� ��! 2 Rm+ j 9� 2 P & 9w = 1 : � = !, ��Nb = �, ��Nb � �,

�0 = N�1w& Nb+ �� w�0b 2 @ eP (w)o.

Corollary A.1: Let b 2 Rm++ , ! 2 Rm+ and �0 (!) 5 N . Under A1vA3,if ! 2 C��� then a reproducible solution (RS) exists for the economy E =hN ; (P; b) ; (!� )�2Ni.

7.2 Appendix 2: Proofs

Proof of Lemma 1: Suppose, contrary to the statement, that pb�� + pb�� �w��0 + w

�0 > pb for some �. If ��0 +

�0 > 0, then by the convex cone

property of the production set P , agent � can reduce either �0 or ��0 without

violating feasibility, which contradicts optimality. If ��0 + �0 = 0, then by

the convex cone property of the production set P , agent � can reduce ��

without violating feasibility, which contradicts NBC.

Proof of Lemma 2: Suppose, contrary to the statement, that p�� + p�� <p!� . Then, by increasing the investment of capital and hiring other agents,� can increase her pro�ts and reduce her labour expended, while reachingsubsistence, which contradicts optimality.

Proof of Lemma 3: By the convexity of MP � , it follows that (�0� ; �0� ) istechnically feasible with �0�0 +�

0�0 = �

�0 +�

�0 . This implies that labour expen-

diture is the same in both production plans, since ( 0�0 + �0�0 )� (�0�0 + �0�0 ) =

( �0 + ��0 )�(��0 + ��0 ). By Lemma 1, noting that only processes that yield the

maximal pro�t rate are operated, it is possible to write �max(p�� + p�� ) +w��0 + w

�0 = pb. But then, it is immediate to check that (�0� ; �0� ; 0�0 )

yields the same amount of net revenue and capital outlay.

Proof of Proposition 1: Part (i). Suppose, contrary to the statement,that p = 0, which implies pb = 0. Then, at the solution to MP � , it will be��0 +

�0 = 0, all �, and thus �0+ 0 = 0. However, by A1 �0+ 0 = 0 implies

that �+ � = 0, which contradicts part (d) of the De�nition of RS.Part (ii). Suppose, contrary to the statement, that �max < 0. By

Lemma 1, for all �, (pb�� � w��0 ) + �pb�� � w��0� + w��0 + w �0 = pb where(pb�� � w��0 ) < 0 and �pb�� � w��0� < 0. Hence, at the solution to MP � it

37

must be �� = �� = 0, all �, which contradicts part (d) of the De�nition ofRS.Part (iii). Suppose, contrary to the statement, that w 5 0. Then, at

the solution to MP � , it will be �0 = 0 for all �. Further, �max 5 0 can be

ruled out, because by part (i), pb > 0. However, if �max > 0, then at thesolution to MP � it will be ��0 > 0 for all � with p!

� > 0, contradicting part(c) of the De�nition of RS.

Proof of Proposition 2: By Lemma 1, the net revenue constraint of everyagent holds as an equality. Summing over �, one obtains pb� + pb� � w�0 +w 0 = pNb and using part (c) of the De�nition of RS, pb�+ pb� = pNb. Theresult then follows by part (d) of the De�nition of RS.

Proof of Proposition 3: By Lemma 1, pb��+pb���w��0�w��0+w( �0+��0 ) =pb holds for all �. Noting that only processes yielding the maximal pro�t ratewill be used, the latter expression can be written as �max(p�� +p�� )+w��0 +w �0 = pb for all �. The result then follows by Proposition 1(iii) and Lemma2.

Proof of Theorem 1: Let c; c 2 B (p; b) be the reference consumptionbundles of a given de�nition satisfying LES. For this de�nition, � 2 N ter ifand only if there is a �c 2 � (c) such that b�c = c and �c0 > ��0+ �0 , � 2 N ted ifand only if there is a �c 2 � (c) such that b�c = c and �c0 < ��0 +

�0 , and

� 2 Nn�N ter [N ted

�if and only if �c0 5 ��0 +

�0 5 �c0. Note that by

A1, �c0 > 0 and �c0 > 0. Moreover, by Proposition 1(iii), it follows thatw�c0 = w�c0 > 0.Consider agent � with ��0 +

�0 = 0 at the solution toMP

� . Since �c0 > 0,such an agent is an exploiter by the above characterisation of N ter. ByProposition 3, it follows that W � = pb

�max> 1

�max[pb� w�c0].

Next, consider any agent � with ��0 + �0 > 0 at the solution to MP � .

By Proposition 3, ��0 + �0 =

pb��maxW �

wand therefore, by LES, � will be an

exploiter if and only if pb��maxW �

w< �c0 , which holds if and only if W

� >1

�max[pb� w�c0]. The other two conditions follow in like manner.

Proof of Lemma 4: 1. First, note that by the convexity ofMP � , it followsthat if �0 < �

�0 for some optimal (�

� ; �� ; �0 ) and 0�0 > �

0�0 for some other

optimal (�0� ; �0� ; 0�0 ), then there is a solution to MP� such that 00�0 = �00�0 .

Therefore, the three cases in the statement are mutually exclusive and they

38

decompose agents with W � > 0 and �� > 0 at the solution of MP � intodisjoint sets.2. Suppose �0 < �

�0 for all optimal (�

� ; �� ; �0 ). If �0 = 0, then clearly

� 2 (+;+; 0) because by assumption �� > 0 and 0 = �0 < ��0 . If �0 > 0, thenconstruct (�0� ; �0� ; 0�0 ) such that

0�0 = 0, �

0�0 = �

�0 � �0 and �0�0 = ��0 + �0 .

By Lemma 3, (�0� ; �0� ; 0�0 ) is also optimal (it yields the same amount ofnet revenue, labour expenditure, and capital outlay). It is su¢ cient to showthat � =2 (+; 0; 0): Suppose, contrary to the latter statement, that � has anoptimal solution of the form (�� ; 0; 0). As in Lemma 3, it is possible to showthat � also has a solution (�0� ; �0� ; 0�0 ) such that �

0� = 0, �0� = �� and 0�0 = �

�0 . But this contradicts the assumption that all optimal solutions for

� are such that �0 < ��0 .

3. The other two cases are proved similarly.

Proof of Theorem 2: Part (i). First, � 2 C5 if and only ifW � = 0. Next,by Proposition 3, � 2 C1 if and only if W � = pb

�max. The result then follows

from Lemma 4.Part (ii). It is su¢ cient to prove that the wealth ordering in the state-

ment is true for agents �, � 2 N with 0 < W � < pb�max

and 0 < W � < pb�max

.By Lemma 4, it follows that �; � 2 [i=2;3;4Ci. Suppose � 2 C4 and �2 C3 but W � > W � . (W � = W � is ruled out by the disjointedness ofclasses.) Since � 2 C3, by reversing the reasoning in Lemma 4, it is pos-sible to show that � has an optimal solution such that �0� = 0, �0� = ��

and 0�0 = ��0 , with �0�0 = 0�0 . Next, by Lemma 3, consider �

0s solutionsof the form (0; �� ; �0 ). Since working time is strictly decreasing in wealth,W � > W � implies �� = �0 < �

� = 0�0 . However, �0�0 =

0�0 and ��0 <

�0

implies ��0 < �0�0 which is impossible given that W

� > W � and thus agent �can hire more labour than � by investing in sectors with the maximal pro�trate �max > 0. A similar argument proves that if � 2 C2 and � 2 C3 thenW � > W � .

Proof of Proposition 4: 1. By Lemma 4, Proposition 3, and Theorem2(ii), it is su¢ cient to prove part (iii) of the statement.2. Suppose p�min 5 W � 5 p�max. By Lemma 3, it is possible to consider

� 0s solutions of the form (0; �� ; �0 ), without loss of generality. By optimality,at the solution to MP � , pb�� � w��0 + w �0 = pb, or equivalently �maxp�� +w�� = pb, with p� = W � . But then, since p�min 5 W � 5 p�max, by theconvexity of P , it follows that there exists some � 2 P , such that �maxp� +

39

w�0 = pb, with p� = W � : The latter equation implies that �0 = �� , andthus �� has a solution of the form (+;0; 0).3. Conversely, suppose that � 2 C3, so that �� has a solution of the form

(+; 0;0) : This implies that there exists � 2 P such that �maxp�+w�0 = pb,with p� = W � , which implies p�min 5 W � 5 p�max.

Proof of Theorem 3: The result follows immediately from Theorem 1 andProposition 4.

Proof of Corollary 2: 1. First, consider De�nition 7. By Theorem 3, whatneeds to be shown is that p�min 5 1

�max

hpb� w

�(�0+�0)

N

�i5 p�max holds.

Let W � 1�max

hpb� w(�0+�0)

N

i. Then, �maxW + w(�0+�0)

N= pb. By de�nition,

W =p(�+�)N

holds, since by Lemma 1, p�b�+ b�� = Npb holds, and at

a RS, only processes yielding the maximal pro�t rate are operated. Then,�0maxp�max

5 �0+�0p(�+�)

5 �0minp�min

by the optimality of production plans. Note that

�maxp(�+�)�0+�0

= pb(�0+�0)=N

�w, �maxp�min�0min

= pb�0min

�w, and �maxp�max�0max

= pb�0max

�w,which imply that pb

�0min� w 5 pb

(�0+�0)=N� w 5 pb

�0max� w. Thus, �0max 5

(�0+�0)N

5 �0min holds. Given that �maxp�min+w�0min = �maxW+ w(�0+�0)N

=

�maxp�max + w�0max = pb, the last inequality implies that p�min 5 W 5p�max.2. Consider next De�nitions 5 and 6. As noted above,minc2B(p;b) l:v: (c; p; w) 5

l:v: (b; p; w) 5 t(�0 + �0) =�0+ 0N. Note that l:v: (b; p; w) 5 �0+ 0

Nfollows

from pb =p(b�+b�)N

. Hence, by Theorem 3 and step 1 of the proof, it followsthat p�min 5 1

�max

�pb� w�0+ 0

N

�5 1

�max

�pb� w

�minc2B(p;b) l:v: (c; p; w)

��.

Hence, what needs to be shown is 1�max

�pb� w

�minc2B(p;b) l:v: (c; p; w)

��5

p�max.Let c� � argminc2B(p;b) l:v: (c; p; w). By the de�nition of l:v: (c�; p; w),

there exists � (c�) 2 P (p; w) such that �0 (c�) = l:v: (c�; p; w). In this case,pc� = pb� (c�) holds. Let us show this. Suppose pc� < pb� (c�) if c� � b� (c�).Since pb� (c�) = �maxp� (c�) + w�0 (c

�), it follows that pc� < �maxp� (c�) +w�0 (c

�). In this case, for some 0 < t < 1, it is true that ptb� (c�) =pc� and t�0 (c�) = l:v: (tb� (c�) ; p; w) < �0 (c

�) = l:v: (c�; p; w). This is acontradiction, since tb� (c�) 2 B (p; b) and �0 (c�) = minc2B(p;b) l:v: (c; p; w).Thus, pc� = pb� (c�). The last equation implies that � (c�) 2 � (p; w) and

40

b� (c�) 2 b� (p; w). Note that , since c� 5 b� (c�) by de�nition, �(c�) 2 Pimplies (��0 (c�) ;�� (c�) ; c� + � (c�)) 2 P by A3. Since pc� = pb� (c�),b� (c�) 2 b� (p; w) implies that (��0 (c�) ;�� (c�) ; c� + � (c�)) 2 � (p; w) andc� 2 b� (p; w). Thus, since c� 2 b� (p; w) implies pc� � w�0 (c�) 5 �maxp�max,1

�max

�pb� w

�minc2B(p;b) l:v: (c; p; w)

��5 p�max holds from pc� = pb, which

completes the proof.

Proof of Proposition 5: Consider the following von Neumann technology:

A =

�2 2 102 2 0

�, B =

�2 6 1610 10 0

�, L =

�1; 1;

4

3

�.

Given this data, the production possibility set P(A;B;L) can be de�ned:

P(A;B;L) ��� 2 R� � R2� � R2+ j 9x 2 R3+ : � 5 (�Lx;�Ax;Bx)

:

Then, consider a convex cone subsistence economy de�ned by P(A;B;L), b =(2; 2), and ! = (N;N). Let (!� )�2N be such that !� = (�� ; �� ), where�� 5 2, all � 2 N , and !i = (2; 2), some i 2 N .Let ej 2 R3+ and �j, j = 1; 2; 3, be de�ned as in Example 1 above.

Then, b�1 � (0; 8), b�2 � (4; 8), and b�3 � (6; 0). Moreover, bP (�0 = 1) =co�b�1; b�2; 3

4b�3;0. In this economy, p = (1; 0) and w = 2 constitute a RS

with �� = 0, �� = ���2

2, and �0 = 1 � ��

2for all � 2 N . The corresponding

aggregate production is � + � = N�2

2. In such a case, � (�+ �; p; w) �

pb�2�w�20p�2

= 1, whereas � (�1; p; w) � pb�1�w�10p�1

< 0 and � (�3; p; w) � pb�3�w�30p�3

=13= �

�34�3; p; w

�. Thus, P (p; w) = f� 2 P j 9� > 0 : � = ��2g and �� 2

P (p; w) for all � 2 N .Hence, p�min = p�max = p

�2

2= 1. In contrast, l:v: (b) = 17

36= 1

4�20 +

16�30,

so that 1�max

[pb� w (l:v: (b))] = 2 � 1718= 19

18> 1. Therefore, every agent

� 2 C1 [ C2 with 1 < W � < 1918is exploited, and the CECP does not hold

in this economy if De�nition 4 is adopted.

Proof of Theorem 4: From Theorem 3, there only remains to prove that(3))(1). Suppose that �max = 0. Then, every producer earns the income pbsolely from the wage. Thus, to earn the same income, every producer sup-plies the same amount of labour which is equal to the labour input �0+�0

N

corresponding to the social production activity per capita. Note that if�max = 0, then c; c 2 b� (p; w) implies w�c0 = w�c0 = pb = w�0+�0

N. Thus,

41

�c0 =�0+�0N

= �c0, which implies that there is no exploiter nor exploited agentby LES.

Proof of Lemma 5: Suppose that (�� ; �� ; �0 )�2N is not e¢ cient. This im-plies that there exists another allocation (�0� ; �0� ; 0�0 )�2N such that

P�2N �

0�+P�2N �

0� 5P

�2N !� , Nb 5

P�2N b�0� +P�2N

b�0� , P�2N �0�0 =

P�2N

0�0 ,

and �0�0 + 0�0 5 ��0 + �0 for all � 2 N and �0�

�0 + 0�

�0 < ��

�0 +

��0 for some

�� 2 N . Then, premultiplying the �rst two inequalities by p, one obtainsX�2N

p�0� +X�2N

p�0� 5X�2N

p!� ,X�2N

pb�0� +X�2N

pb�0� = pNb =X�2N

pb�� +X�2N

pb�� ,X�2N

�0�0 +X�2N

0�0 <X�2N

��0 +X�2N

�0 .

Let �00 �P

�2N �0� +

P�2N �

0� and 000 �P

�2N �0�0 +

P�2N

0�0 . By de�-

nition, �000 = 000. Thus, since p�

00 5P

�2N p!� , pb�00 = pNb =

P�2N pb�� +P

�2N pb�� , and 000 <P�2N �

�0 +

P�2N

�0 , it follows that

pb�00 � w�000p�00

=pb�00 � w 000p�00

>pNb� w

�P�2N �

�0 +

P�2N

�0

�p�00

.

This impliesP

�2N p��+P

�2N p�� < p�00, since (�� ; �� ; �0 )�2N is a RS and

thus only processes yielding the maximum pro�t rate are operated. However,since

P�2N p�

� +P

�2N p�� =

P�2N p!

� , this implies p�00 >P

�2N p!� , a

contradiction.

Proof of Proposition 6: The result follows immediately noting that if�(p; w) ; (�� ; �� ; �0 )�2N

�is e¢ cient then �0 + �0 5

P�2N �

00�0 +

P�2N

00�0

for all feasible (�00� ; �00� ; 00�0 )�2N that satisfy De�nition 10(i). Therefore�0+�0 5 �00+ 00. A similar argument holds for

�(p0; w0) ; (�0� ; �0� ; 0�0 )�2N

�proving �0 + �0 = �00 + 00, which establishes the desired result.

Proof of Lemma 6: By the construction of (�0� )�2N ,�(p; w) ; (�0� ; 0; 0)�2N

�satis�es De�nition 1(b), (c), and (d). Therefore, only individual optimalityneeds to be proved. Since

P�2N p

��� + ��

�=P

�2N p!� , it follows thatP

�2N p�0� =

P�2N p!

0� , so that p�0� = p!0� holds for all � 2 N . By

42

Lemma 1,P

�2N p�b�� + b��� = Npb, and therefore

P�2N pb�0� = Npb, so

that pb�0� = pb holds for all � 2 N . Further, note that, by Lemma 5, the allo-cation (�� ; �� ; �0 )�2N is e¢ cient, which implies that

P�2N (�

�0 +

�0 ) is the

minimal amount of labour expenditure to produce Nb as a social net outputunder the capital constraint

P�2N !

� . This implies that �0�0 is the minimallabour expenditure to produce b under the constraint !0� for each � 2 N .Hence, �0�0 is the solution of the problem MP � given the price vector (p; w)in the economy hN ; (P; b) ; (!0� )�2Ni. Since

P�2N �

0� =P

�2N (�� + �� )

by de�nition, (p; w) supportsP

�2N �0� as a pro�t-rate maximizing pro-

duction point. Therefore,�(p; w) ; (�0� ; 0; 0)�2N

�is a RS for the economy

hN ; (P; b) ; (!0� )�2Ni.

Proof of Theorem 5:((): It is easy to see that De�nition 7 meets RE and IES. Thus, it

su¢ ces to show that De�nition 7 meets FNE. Let�(p; w) ; (�� ; �� ; �0 )�2N

�be an e¢ cient RS for the economy hN ; (P; b) ; (!� )�2Ni. Then, by De�nition7, there exists a non-exploitative allocation such that �0�0 +

0�0 =

�0+�0N

forall � 2 N , and �0 + �0 = � + � holds. Suppose that p

��+ �

�= p! holds.

Then, by Lemma 6, the non-exploitative allocation is feasible with respectto (p; w) in hN ; (P; b) ;

�!N; : : : ; !

N

�i with (�0� )�2N 2 PN such that b�0� = b�+b�

N

and �0�0 =�0+ 0N

for each � 2 N . Suppose that p��+ �

�< p! holds. Choose

(!0� )�2N such that p��+ �

�= p!0 with p

��� + ��

�= p!0� for all � 2 N

. Then,�(p; w) ; (�� ; �� ; �0 )�2N

�is still an e¢ cient RS for the economy

hN ; (P; b) ; (!0� )�2Ni. Then, by Lemma 6, the non-exploitative allocation isfeasible with respect to (p; w) in hN ; (P; b) ;

�!N; : : : ; !

N

�i with (�0� )�2N 2 PN

such that b�0� = b�+b�N

and �0�0 = �0+ 0N

for each � 2 N . Thus, in any case,De�nition 7 meets FNE.()): Consider any de�nition of labour exploitation satisfying LES,FNE,

RE, and IES.Case 1, �max > 0. Consider an economy hN ; (P; b) ; (!� )�2Ni with a RS�

(p; w) ; (�� ; �� ; �0 )�2N�such that �max > 0 and � + � =

P�2N !

� = !.By Lemma 5, (�� ; �� ; �0 )�2N is e¢ cient. Suppose the condition �

c0 = �

c0 =

�0+�0N

does not hold.1. Suppose �c0 <

�0+�0N. Let us consider another economy hN ; (P; b) ; (!0� )�2Ni

with !0� = !Nfor any � 2 N . Then, let us consider an allocation (�0� ; �0� ; 0�0 )�2N 2

(P � P � [0; 1])N such that �0� = �+�N, �0� = 0, and 0�0 = 0 for any � 2 N .

43

Then, by construction,�(p; w) ; (�0� ; �0� ; 0�0 )�2N

�constitutes a RS with

�max > 0 and �0 + �0 = � + �. Then, by IES, its corresponding referencebundles c0; c0 2 Rm+ meet c0 = c and c0 = c. Thus, since

�0+�0N

is the labour ex-

penditure of every agent in�(p; w) ; (�0� ; �0� ; 0�0 )�2N

�, �c

0

0 <�0+�0N

impliesthat every agent is exploited, which contradicts RE.2. Suppose �c0 >

�0+�0N. Let us consider the same economy and the same

allocation as in the above argument. Then, we can see that �c00 >

�0+�0N

implies that every agent is an exploiter in the RS�(p; w) ; (�0� ; �0� ; 0�0 )�2N

�for the economy hN ; (P; b) ; (!0� )�2Ni, which also contradicts RE.3. Suppose �c0 <

�0+�0N

< �c0. If�0+�0N

��c0 6= �c0�

�0+�0N, then we can use

the same argument as in steps 1 and 2 above to obtain a contradiction. Forinstance, if �0+�0

N� �c0 > �c0 �

�0+�0N, then we can construct an alternative

economy hN ; (P; b) ; (!0� )�2Ni and its RS�(p; w) ; (�0� ; �0� ; 0�0 )�2N

�such

that �0 + �0 = � + �, and there exists one agent � 0 2 N such that �0�0

0 + 0�

00 > �c0 and for any other � 2 Nn f� 0g, �0�0 + 0�0 = �c0. Then, the desiredcontradiction follows from IES and RE.If �0+�0

N� �c0 = �c0 �

�0+�0N, then there exists a non-exploitative alloca-

tion (�0� ; �0� ; 0�0 )�2N such that �0 + �0 = � + �, �00 +

00 = �0 + 0, and

�0�0

0 + 0�0

0 < �0+�0N

for some � 0 2 N . Then, consider (�00� ; �00� ; 00�0 )�2Nsuch that for any � 2 Nn f� 0g, (�00� ; �00� ; 00�0 ) = (�0� ; �0� ; 0�0 ) and for � 0,�00�

00 + 00�

00 = �0+�0

N,��00�

0; 00�

00

�=��0�

0; 0�

00

�, and

��00�

0; �00�

0�=��0�

0; �0�

0�.

By A3,��00�

00 ; �

00�0 ; �00�0� 2 P . Moreover, since

P�2N �

00� +P

�2N �00� =P

�2N �0� +

P�2N �

0� ,P

�2N �00� +

P�2N �

00�=P

�2N �0� +

P�2N �

0�,P

�2N �00�0 =

P�2N �

0�0 , and

P�2N

00�0 =

P�2N

0�0 , if (�

0� ; �0� ; 0�0 )�2N is anon-exploitative allocation at (p; w), then so is (�00� ; �00� ; 00�0 )�2N by De�n-ition 8. Moreover, since

P�2N �

00� +P

�2N �00� =

P�2N �

0� +P

�2N �0� = !,

FNE can be applied to (�00� ; �00� ; 00�0 )�2N , so that there exist (p0; w0) and

(!0� )�2N withP

�2N !0� = ! such that

�(p0; w0) ; (�00� ; �00� ; 00�0 )�2N

�is a

RS for the economy hN ; (P; b) ; (!0� )�2Ni. In this case, sinceP

�2N �00�0 +P

�2N 00�0 > �0 + �0 and (�

� ; �� ; �0 )�2N is e¢ cient, Proposition 6 im-plies that (�00� ; �00� ; 00�0 )�2N is ine¢ cient. The last statement implies that

p0�b�00 + b�00� > w (�000 + 000) (that is, �00max > 0), since any RS with �max = 0

is e¢ cient. Moreover, by Lemmas 2 and 5, there must be some agent � suchthat �00�0 + 00�0 = 0. But then, by assumption A1 on P , this implies that(�00� ; �00� ; 00�0 )�2N would only be non-exploitative if c = 0, which contra-dicts the assumption that c 2 B(p; b), because pb > 0 by Proposition 1.

44

4. Suppose �c0 =�0+�0N

< �c0. Then we can construct an alternativeeconomy hN ; (P; b) ; (!0� )�2Ni and its RS

�(p; w) ; (�0� ; �0� ; 0�0 )�2N

�such

that �0 + �0 = � + �, and there exists one agent � 0 2 N such that �0�0

0 + 0�

00 < �c0 and for any other � 2 Nn f� 0g, �

c0 = �0�0 + 0�0 . Then, the desired

contradiction follow from IES and RE. Next, suppose �c0 <�0+�0N

= �c0.Then, applying the same logic as in the case of �0+�0

N� �c0 6= �

c0 �

�0+�0N, we

can obtain a contradiction by IES and RE.Consider an economy hN ; (P; b) ; (!� )�2Ni with a RS

�(p; w) ; (�� ; �� ; �0 )�2N

�with �max > 0, such that � + � �

P�2N !

� = !. If p��+ �

�= p!

holds, then the same argument as for � + � = ! can be applied, because(�� ; �� ; �0 )�2N is e¢ cient. If p

��+ �

�< p! holds, then by Lemma 2,

there is a subset N� � N such that for each � 2 N�, pb�� � w��0 = pb <�maxp!� � w��0 holds. Thus, by selecting !0� � !� appropriately from each� 2 N�, it is possible to �nd !0 �

P�2NnN� !� +

P�2N� !0� such that

p��+ �

�= p!0 holds. Then, by Lemma 5,

�(p; w) ; (�� ; �� ; �0 )�2N

�consti-

tutes an e¢ cient RS for this economy hN ; (P; b) ; (!0� )�2Ni. By applying thesame argument as in the case of �+� = !, one can check that any formulationof exploitation satisfying LES, FNE, and RE is identical to De�nition 7 at�(p; w) ; (�� ; �� ; �0 )�2N

�in hN ; (P; b) ; (!0� )�2Ni. Then, by IES, the same

statement also holds for�(p; w) ; (�� ; �� ; �0 )�2N

�in hN ; (P; b) ; (!� )�2Ni.

Case 2, �max = 0. Consider an economy with a RS�(p; w) ; (�� ; �� ; �0 )�2N

�with �max = 0. Then, Proposition 3 implies that ��0+

�0 =

pbwholds for each

� 2 N . This implies that ��0+ �0 =�0+�0N

holds for each � 2 N and this RSis e¢ cient, according to De�nition 10.Let c; c 2 B (p; b) be the corresponding reference bundles. Since �max =

0, then pc � w�c0 5 pb � w�0+�0N

= 0 and pc � w�c0 5 pb � w�0+�0N

=

0. Thus, �c0; �c0 = �0+�0

Nby c; c 2 B (p; b). By LES, �c0 = �c0 holds, so

that either (i) every agent is neither exploiter nor exploited, or (ii) everyagent is exploiter. Note that (ii) corresponds to the case that �c0 = �c0 >�0+�0N. Then, by assumption A3, there exists a non-exploitative allocation

(�0� ; �0� ; 0�0 )�2N at (p; w) such thatP

�2N �0�0 +

P�2N

0�0 > �0 + �0 andP

�2N �0� +

P�2N �

0� = !, and the same argument can be applied as in step3 above to prove that (�0� ; �0� ; 0�0 )�2N would only be non-exploitative ifc = 0, which contradicts the assumption that c 2 B(p; b). Thus, �c0 = �c0 =�0+�0N.

Let �c0 > �c0 =�0+�0N. Then, by assumption A3, there exists another

45

non-exploitative allocation (�0� ; �0� ; 0�0 )�2N at (p; w) such thatP

�2N �0�0 +P

�2N 0�0 > �0+ �0 and

P�2N �

0� +P

�2N �0� = !, and the same argument

can be applied as in step 3 above to prove that (�0� ; �0� ; 0�0 )�2N wouldonly be non-exploitative if c = 0, which contradicts the assumption thatc 2 B(p; b).In sum, if a de�nition of labour exploitation satis�es LES, RE, FNE,

and IES, then the condition �c0 = �c0 =

�0+�0N

must hold.

Proof of Theorem 6:((): It is easy to see that De�nition 7 meets EEUL and IES. Therefore,

it su¢ ces to show that De�nition 7 satis�esFNE�. Let�(p; w) ; (�� ; �� ; �0 )�2N

�be an e¢ cient RS for the economy hN ; (P; b) ; (!� )�2Ni. Then, by De�nition7, at any non-exploitative allocation it must be �0�0 +

0�0 = �0+�0

Nfor all

� 2 N .Suppose that p

��+ �

�= p! holds. Then, by Lemma 6, the non-

exploitative allocation is feasible with respect to (p; w) in hN ; (P; b) ;�!N; : : : ; !

N

�i

with (�0� )�2N 2 PN such that b�0� = b�+b�N

and �0�0 =�0+ 0N

for each � 2 N .Suppose that p

��+ �

�< p! holds. Choose (!0� )�2N such that p

��+ �

�=

p!0 with p��� + ��

�= p!0� for all � 2 N . Then,

�(p; w) ; (�� ; �� ; �0 )�2N

�is still an e¢ cient RS for the economy hN ; (P; b) ; (!0� )�2Ni and by Lemma 6,the non-exploitative allocation is feasible with respect to (p; w) in hN ; (P; b) ;

�!N; : : : ; !

N

�i

with (�0� )�2N 2 PN such that b�0� = b�+b�N

and �0�0 =�0+ 0N

for each � 2 N .Thus, in either case, De�nition 7 meets FNE�.()): Consider any de�nition of labour exploitation satisfying LES,FNE�,

EEUL, and IES.Case 1, �max > 0. Consider an economy hN ; (P; b) ; (!� )�2Ni with a RS�

(p; w) ; (�� ; �� ; �0 )�2N�such that �max > 0 and � + � =

P�2N !

� = !.By Lemma 5, (�� ; �� ; �0 )�2N is e¢ cient. Suppose the condition �

c0 = �

c0 =

�0+�0N

does not hold. First of all, By LES and EEUL, �c0 = �c0 = l:v: (b).Then, there always exists a non-exploitative allocation at (p; w).1. Suppose �c0 >

�0+�0N. Let (�0� ; �0� ; 0�0 )�2N be a non-exploitative allo-

cation at (p; w). Then, by FNE�, this allocation is feasible non-exploitativewith respect to some (p0; w0) in hN ; (P; b) ; (!0� )�2Ni. By de�nition it mustbeP

�2N �0�0 +

P�2N

0�0 � �00 +

00 = N�c0 > �0 + �0, thus implying that

(�0� ; �0� ; 0�0 )�2N is ine¢ cient for hN ; (P; b) ; (!0� )�2Ni.Let us show the last statement. If �00 +

00 > �0 + �0, then another

allocation (�00� ; �00� ; 00�0 )�2N can be constructed such thatP

�2N �00� +

46

P�2N �

00� =P

�2N �� +

P�2N �

� andP

�2N �00�0 +

P�2N

00�0 =

P�2N �

�0 +P

�2N �0 , and moreover, �

00�0 +

00�0 5 �0�0 + 0�0 holds for all � 2 N , and strictly

for some � 2 N . By the above two equations,P

�2N �00�0 =

P�2N

00�0 holds,

sinceP

�2N ��0 =

P�2N

�0 . This construction implies that (�

0� ; �0� ; 0�0 )�2Nis ine¢ cient.The last statement implies that p0

�b�0 + b�0� > w0 (�00 + 00) (that is,

�0max > 0), since any RS with �max = 0 is e¢ cient. Moreover, by Lem-mas 2 and 5, there must be some agent � such that �0�0 +

0�0 = 0. But then,

by assumption A1 on P , this implies that (�0� ; �0� ; 0�0 )�2N would only benon-exploitative if c = 0 which contradicts the assumption that c 2 B(p; b),because pb > 0 by Proposition 1.2. Suppose �c0 <

�0+�0N. Let (�0� ; �0� ; 0�0 )�2N be a non-exploitative allo-

cation at (p; w). Then, by FNE�, this allocation is feasible non-exploitativewith respect to some (p0; w0) in hN ; (P; b) ; (!0� )�2Ni. By de�nition it mustbeP

�2N �0�0 +

P�2N

0�0 � �00 +

00 5 N�c0 < �0 + �0, thus implying that

the original allocation (�� ; �� ; �0 )�2N was ine¢ cient, a contradiction.3. Suppose �c0 <

�0+�0N

< �c0. Let (�0� ; �0� ; 0�0 )�2N be a non-exploitative

allocation at (p; w): by FNE�, (�0� ; �0� ; 0�0 )�2N is feasible with respect tosome (p0; w0) in some economy hN ; (P; b) ; (!0� )�2Ni. If

P�2N �

0�0 +P

�2N 0�0 <

�0 + �0, then an argument similar to �c0 <

�0+�0N

is applied. SupposeP�2N �

0�0 +

P�2N

0�0 > �0 + �0. Since � + � = ! by construction, the

RS allocation (�� ; �� ; �0 )�2N is e¢ cient for hN ; (P; b) ; (!� )�2Ni. However,since

P�2N �

0�0 +

P�2N

0�0 > �0 + �0, then (�

0� ; �0� ; 0�0 )�2N is ine¢ cient,as shown in the case of �c0 >

�0+�0N. Thus, as shown in the case of �c0 >

�0+�0N,

(�0� ; �0� ; 0�0 )�2N would only be non-exploitative if c = 0, which contradictsthe assumption that c 2 B(p; b). If

P�2N �

0�0 +

P�2N

0�0 = �0+�0, then con-

sider (�00� ; �00� ; 00�0 )�2N such that for any � 2 Nn f� 0g, (�00� ; �00� ; 00�0 ) =

(�0� ; �0� ; 0�0 ) and for �0, �00�

00 + 00�

00 > �0�

00 +

0�00 ,��00�

0; 00�

00

�=��0�

0; 0�

00

�,

and��00�

0; �00�

0�=��0�

0; �0�

0�. By A3,

��00�

00 ; �

00�0 ; �00�0� 2 P . Moreover, sinceP

�2N �00� +

P�2N �

00� =P

�2N �0� +

P�2N �

0� ,P

�2N �00� +

P�2N �

00�=P

�2N �0� +

P�2N �

0�,P

�2N �00�0 =

P�2N �

0�0 , and

P�2N

00�0 =

P�2N

0�0 ,

if (�0� ; �0� ; 0�0 )�2N is a non-exploitative allocation at (p; w), then so is(�00� ; �00� ; 00�0 )�2N by De�nition 8. However,

P�2N �

00�0 +

P�2N

00�0 >

�0 + �0 induces a contradiction as discussed in the above.4. Suppose �c0 =

�0+�0N

< �c0, and let (�0� ; �0� ; 0�0 )�2N be a non-

exploitative allocation at (p; w) such that for some � 0; � 00 2 N , �0�00 + 0�0

0 = �c0

47

and �0�00

0 + 0�00

0 = �c0. Then, sinceP

�2N �0�0 +

P�2N

0�0 > �0+ �0, we apply

the same argument as in the last paragraph. Suppose �c0 <�0+�0N

= �c0, andlet (�0� ; �0� ; 0�0 )�2N be a non-exploitative allocation at (p; w). By EEUL, itfollows that �0+�0 > l:v: (Nb). But then (�

0� ; �0� ; 0�0 )�2N can be such thatP�2N �

0�0 +

P�2N

0�0 < �0 + �0, so that an argument similar to �

c0 <

�0+�0N

is applied. Thus, by EEUL and LES, �c0 = �c0 =

�0+�0N

should hold, whichis a contradiction.Consider an economy hN ; (P; b) ; (!� )�2Ni with a RS

�(p; w) ; (�� ; �� ; �0 )�2N

�with �max > 0, such that � + � �

P�2N !

� = !. If p��+ �

�= p!

holds, then the same argument as the case of � + � = ! can be applied,since (�� ; �� ; �0 )�2N is e¢ cient. If p

��+ �

�< p! holds, then by Lemma

2, there is a subset N� � N such that for each � 2 N�, pb�� � w��0 =pb < �maxp!� � w��0 holds. Thus, by selecting !0� � !� appropriately fromeach � 2 N�, it is possible to �nd !0 �

P�2NnN� !� +

P�2N� !0� such

that p��+ �

�= p!0 holds. Then, by Lemma 5,

�(p; w) ; (�� ; �� ; �0 )�2N

�constitutes an e¢ cient RS for this economy hN ; (P; b) ; (!0� )�2Ni. By ap-plying the same argument as in the case of � + � = !, one can checkthat �c0 = �

c0 =

�0+�0N

for�(p; w) ; (�� ; �� ; �0 )�2N

�in hN ; (P; b) ; (!0� )�2Ni.

Then, by IES, �c0 = �c0 =�0+�0N

holds for�(p; w) ; (�� ; �� ; �0 )�2N

�in

hN ; (P; b) ; (!� )�2Ni.Case 2, �max = 0. Consider an economy with a RS

�(p; w) ; (�� ; �� ; �0 )�2N

�with �max = 0. Then, Proposition 3 implies that ��0+

�0 =

pbwholds for each

� 2 N . This implies that ��0+ �0 =�0+�0N

holds for each � 2 N , and this RSis e¢ cient, according to De�nition 10.Let c; c 2 B (p; b) be the corresponding reference bundles. Since �max = 0,

then pc � w�c0 5 pb � w�0+�0N

= 0 and pc � w�c0 5 pb � w�0+�0N

= 0.Thus, �c0; �

c0 = �0+�0

Nby c; c 2 B (p; b). By LES, �c0 = �c0 holds, so that

either (i) every agent is neither an exploiter nor exploited, or (ii) every agentis an exploiter. Note that (ii) corresponds to the case that �c0 = �c0 >�0+�0N. Then, by assumption A3, there exists a non-exploitative allocation

(�0� ; �0� ; 0�0 )�2N at (p; w) such thatP

�2N �0�0 +

P�2N

0�0 > �0+�0, which

implies that (�0� ; �0� ; 0�0 )�2N is ine¢ cient, as shown in the case of �max > 0.

Thus, as shown in step 1 of the proof above, (�0� ; �0� ; 0�0 )�2N would only benon-exploitative if c = 0, which contradicts the assumption that c 2 B(p; b).Thus, �c0 = �c0 = �0+�0

N.

Let �c0 > �c0 =�0+�0N. Then, by assumption A3, there exists another

48

non-exploitative allocation (�0� ; �0� ; 0�0 )�2N at (p; w) such thatP

�2N �0�0 +P

�2N 0�0 > �0+�0, to which the same argument can be applied as in the case

of �c0 = �c0 > �0+�0N. Thus, (�0� ; �0� ; 0�0 )�2N would only be non-exploitative

if c = 0, which contradicts the assumption that c 2 B(p; b).In sum, if a de�nition of labour exploitation satis�es LES,EEUL, FNE�,

and IES, then the condition �c0 = �c0 =

�0+�0N

must hold.

7.3 Appendix 3: Additional Claims

In this Appendix, some additional claims made in the paper are rigorouslyproved. First, Example A.1 proves that, if the de�nition of exploitationsatis�es LES, the CECP may hold if a de�nition of labour value is adoptedwhich does not focus on pro�t-maximising processes.

Example A.1: Consider the following von Neumann technology:

A =

�2 4 2 24 4 2 0

�, B =

�2 8 6 612 12 6 0

�, L = (1; 1; 1; 1) ,

where the notation is the same as in Example 1 above and the production pos-sibility set is P(A;B;L) �

�� 2 R� � R2� � R2+ j 9x 2 R4+ : � 5 (�Lx;�Ax;Bx)

.

Then, consider a convex cone subsistence economy de�ned by P(A;B;L), b =(2; 2), and ! = (N;N). Let (!� )�2N be such that !� = (�� ; �� ), where�� 5 2, all � 2 N , and !i = (2; 2), some i 2 N .Let ej and �j, j = 1; 2; 3; 4, be de�ned as in Example 1 discussed in

Section 3 above. Then b�1 � (0; 8), b�2 � (4; 8), b�3 � (4; 4), and b�4 �(4; 0). Also, bP (�0 = 1) = co�b�1; b�2; b�3; b�4;0. In this economy, p = (1; 0),w = 2 constitute a RS, with �� = 0; �� = ���3

2; and �0 = 1 � ��

2; all

� 2 N . The corresponding aggregate production is � + � = N�3

2. At this

RS, � (�+ �; p; w) � pb�3�w�30p�3

= 1, whereas � (�1; p; w) � pb�1�w�10p�1

< 0;

� (�2; p; w) � pb�2�w�20p�2

= 12; � (�4; p; w) � pb�4�w�40

p�4= 1. Thus, P (p; w) =

f� 2 P j 9� > 0 : �� 2 co f�3; �4gg.Choose c = c = b�2

2, so that c; c 2 B (p; b). Since pb�2�w�20

2= 1 =

�maxp�min = �maxp�max = �

maxp�32, Theorem 3 states that the CECP holds

if a de�nition of exploitation is adopted which satis�es LES with c = c = b�22.

However, since ���2

2; p; w

�= 1

2, it follows that �c = �c = �2

2=2 P (p; w).

49

Example A.2 instead proves the existence of ine¢ cient RS�s, and it high-lights their structure and the role of big capitalists in generating ine¢ ciencies.

Example A.2: Consider the following von Neumann technology:

A =

�1 21 1

�, B =

�3 43 4

�, L = (1; 0:8) ,

where the notation is the same as in Example 1 above and the production pos-sibility set is P(A;B;L) �

�� 2 R� � R2� � R2+ j 9x 2 R2+ : � 5 (�Lx;�Ax;Bx)

.

Consider a subsistence economy E = hN ; (P; b) ; (!� )�2Ni de�ned by N =f1; 2g, P = P(A;B;L), b = (1; 1), and (!1; !2) = ((2; 1) ; (0; 0)).To begin with, it is shown that the price vector (p; w) = ((1; 0) ; 1), and the

allocation��1; �1; 10

�= (0; (�1;� (1; 1) ; (3; 3)) ; 0), and

��2; �2; 20

�=

(0; 0; 1) constitute a RS for this economy. First, given this (p; w), the activity�1 = (�1;� (1; 1) ; (3; 3)) 2 P(A;B;L) is a maximal pro�t-rate production

point. Note that �1 (p; w) � pb�1�w�10p�1

= 1. Let �0 � (�0:8;� (2; 1) ; (4; 4)) 2

P(A;B;L). Then, �0 (p; w) � pb�0�w�00p�0 = 0:6. Thus, �1 (p; w) > �0 (p; w). By the

property of P(A;B;L), any other production point � 2 P(A;B;L) is representedas � 5 t�1 + t0�0 for some suitable non-negative values t; t0 = 0. Thus,�1 (p; w) > �0 (p; w) implies that �1 is a maximal pro�t-rate point. Second,since pb = 1 and w = 1, agent 2�s optimal solution is 20 = 1, since !

2 = (0; 0).

Third, for agent 1,�0; �1; 0

�is the optimal solution, since pb�1 �w�10 = pb,

�1 (p; w) = �max, and p�1 < p!1. Note that b�1 = 2b and �10 = 20 = 1. Since�1 � !1, which implies �1 � ! � !1+!2,

�(p; w) ; (�� ; �� ; �0 )�2N

�is a RS

with p� < p!.The allocation (�� ; �� ; �0 )�2N is ine¢ cient. Consider �0 as the alter-

native social production point. Note that �0 = !, b�0 = (2; 3) � 2b, and�00 = 0:8 < �10. Construct an alternative allocation (�

0� ; �0� ; 0�0 )�2N as��01; �01; 010

�= (0; �0; 0) and

��02; �02; 020

�= (0; 0; 0:8). Since 020 < 20,

this implies that the RS allocation (�� ; �� ; �0 )�2N is not e¢ cient.

As noted in Section 5 above, the source of the ine¢ ciency is the violationof the assumption of local nonsatiation: an increase in wealth does not makeagent 1 better o¤. Thus, it is individually rational for agent 1 to use thelabour intensive activity �1 because it yields the maximum rate of pro�tat (p; w) = ((1; 0) ; 1), instead of the capital-intensive, and socially optimaltechnique �0 which yields a lower pro�t rate.

50

8 References

Bigwood, R. (2003): Exploitative Contracts, Oxford, Oxford University Press.

Duménil, G. (1980): De la Valeur aux Prix de Production, Economica, Paris.

Duménil, G., Foley, D. K., and D. Lévy (2009): �A Note on the Formal Treat-ment of Exploitation in a Model with Heterogenous Labor,�Metroeconomica60, pp. 560-567.

Federal Ministry of Health of Nigeria (2001): Report of the InvestigationCommittee on the Clinical Trial of Trovan by P�zer in Kano (1996), availableat www.washingtonpost.com/wp-srv/world/documents/Clinical_Trial_Report.pdf.

Field, M.A. (1989): Surrogate Motherhood, Cambridge, Harvard UniversityPress.

Foley, D. K. (1982): �The Value of Money, the Value of Labor Power, and theMarxian Transformation Problem,�Review of Radical Political Economics14, pp. 37-47.

Gotoh, R. and N. Yoshihara (2003): �A Class of Fair Distribution Rules a laRawls and Sen,�Economic Theory 22, 63-88.

Hurwicz, L. (1972): �On Informationally Decentralized Systems,�inDecisionand Organization: A Volume in Honor of J. Marschak (R. Radner and C.B.McGuire, eds.), Amsterdam: North-Holland.

International Labour O¢ ce (2005): Human tra¢ cking and Forced LabourExploitation, Geneva, International Labour O¢ ce.

International Labour O¢ ce (2005a): Forced Labour: Labour Exploitation andHuman Tra¢ cking in Europe, Geneva, International Labour O¢ ce.

International Labour O¢ ce (2006): The end of child labour: Within reach,Geneva, International Labour O¢ ce.

Lipietz, A. (1982): �The So-Called �Transformation Problem�Revisited,�Journal of Economic Theory 26, pp. 59-88.

Matsuo, T. (2008): �Pro�t, Surplus Product, Exploitation and Less thanMaximized Utility,�Metroeconomica 59, pp. 249-265.

Mohun, S. (2004): �The Labour Theory of Value as Foundation for EmpiricalInvestigations,�Metroeconomica 55, pp. 65-95.

51

Morishima, M. (1973): Marx�s Economics, Cambridge University Press, Cam-bridge.

Morishima, M. (1974): �Marx in the Light of Modern Economic Theory,�Econometrica 42, pp. 611-632.

Nielsen, K. and R. Ware (1997) (eds): Exploitation, New Jersey, HumanitiesPress.

Roemer, J. E. (1980): �A general equilibrium approach to Marxian eco-nomics,�Econometrica 48, pp. 505-530.

Roemer, J. E. (1981): Analytical Foundations of Marxian Economic Theory,Cambridge University Press, Cambridge.

Roemer, J. E. (1982): A General Theory of Exploitation and Class, HarvardUniversity Press, Cambridge.

Roemer, J. E. (1989a): �Second Thoughts on Property Relations and Ex-ploitation,�Canadian Journal of Philosophy Suppl. Vol. 15, 257-266.

Roemer, J. E. (1989b): �What is Exploitation? Reply to Je¤rey Reiman,�Philosophy and Public A¤airs 18, pp. 90-97.

Roemer, J. E. (1994): A Future for Socialism, Harvard University Press,Cambridge.

Sample, R.J. (2003): Exploitation, What it is and Why it is Wrong, Lanham,Rowman and Little�eld.

SPD (2007): Hamburg Programme. Principal guidelines of the Social Demo-cratic Party of Germany. http://parteitag.spd.de/servlet/PB/menu/1727812/index.html.

Skillman, G. L. (1995): �Ne Hic Saltaveris: the Marxian Theory of Exploita-tion after Roemer,�Economics and Philosophy 11, 309-31.

Veneziani, R. (2007): �Exploitation and Time,�Journal of Economic Theory132, pp. 189-207.

Veneziani, R. (2008): �Exploitation, Inequality, and Power,�COE/RES Dis-cussion Paper Series, No. 254, Graduate School of Economics and Instituteof Economic Research, Hitotsubashi University, March 2008.

Veneziani, R. and N. Yoshihara (2009): �Globalisation and Exploitation,�mimeo, Queen Mary, University of London, and The Institute of EconomicResearch, Hitotsubashi University.

52

Yoshihara, N. (1998): �Wealth, Exploitation and Labor Discipline in theContemporary Capitalist Economy,�Metroeconomica 49, pp23-61.

Yoshihara, N. (2007): �Class and Exploitation in General Convex ConeEconomies,�forthcoming, Journal of Economic Behavior and Organization.

Yoshihara, N. and R. Veneziani (2009): �Strong Subjectivism in the MarxianTheory of Exploitation: A Critique,�CCES Discussion Paper Series, No.21,Hitotsubashi University.

Warren, P. (1997): �Should Marxists Be Liberal Egalitarians?,�Journal ofPolitical Philosophy 5, pp. 47-68.

Wertheimer, A. (1996): Exploitation, Princeton, Princeton University Press.

Wol¤, J. (1999): �Marx and exploitation,�The Journal of Ethics 3, pp.105-120.

Wood, A. W. (1995): �Exploitation,�Social Philosophy and Politics 12, pp.135-158, reprinted in, Nielsen, K. and R. Ware (1997), pp. 2-26.

53