2. THE STRUCTURE OF UEL THEORY

In order to illustrate the key conceptual issues, consider the baseline case of a two-class economy in which one good, say corn, is produced by means of itself and labour. To be specific, a < 1 units of corn seed and L > 0 units of (a single type of homogeneous) labour are necessary to produce one unit of corn. Assume that all agents have the same skills. Suppose further that capitalists save and accumulate, while all workers earn a subsistence wage and consume the same (subsistence) amount of corn c. Under these rather stringent assumptions, the definition of UEL exploitation is relatively uncontroversial. Let v denote the amount of labour necessary to produce one unit of corn: this comprises both the amount of labour L directly necessary in production and the amount of labour contained in the corn seed, va. Formally, v = va + L and so v = L(1 − a)⁻¹. Then agent ν is exploited (resp., an exploiter) if and only if the labour she expends in production, λ^ν, is greater (resp., lower) than the labour contained in c, vc.Footnote ⁹

In this paper, we argue that the definition of an agent’s exploitation status requires the identification of three separate normative benchmarks. The first is the amount of labour that the agent gives, or contributes – which, in this setting, is simply equated to the labour time performed in economic activities, λ^ν. The definition of the amount of labour received, instead, requires the definition both of a relevant bundle (or, more generally, bundles) received by the agent, which can be called the exploitation reference bundle(s) (henceforth, ERB(s)), and of the labour associated with, or contained in it – here, respectively, c and vc.

In the simple economy considered, all three benchmarks are relatively uncontroversial.Footnote ¹⁰ As soon as the restrictive assumptions on technology, behaviour and endowments are relaxed, however, the definition of the labour ‘given’ and ‘received’ by agents is not obvious, and several approaches have indeed been proposed. For example, if agents have heterogeneous skills, then perhaps one should focus on effective (i.e. skill-adjusted) labour contributed, rather than time expended in production. But maybe not.

Similarly, if more general technologies are considered (allowing, for example, for joint products), the naïve generalization of the standard definition of labour received can yield paradoxical results – such as bundles containing a negative amount of labour, as famously shown by Steedman (Reference Steedman1977) – and so various definitions of the labour contained in a given bundle have been proposed. As Roemer (Reference Roemer1982b: 283) puts it, ‘If the production set is complicated in various ways, then it is not possible to give an unambiguous definition of what ‘labor-embodied’ means’.

Finally, if agents do not consume a given, equal subsistence bundle, then the choice of the ERB(s) is not obvious: it could be either the bundle actually consumed by an agent, or some normatively relevant bundle(s) unrelated to individual consumption decisions.

Different choices reflect different normative and positive intuitions. In the next section, we illustrate the range of possible approaches to UEL exploitation in an abstract economic environment by focusing on some of the classic definitions proposed in the literature.

3. THE FRAMEWORK

We analyse UEL exploitation theory at an abstract level. Hence, in what follows, we impose only minimal restrictions on endowments, technology and preferences, and make no assumptions concerning agents’ behaviour – agents may be standard utility maximizers, or they may be characterized by some form of bounded rationality.

Let $\mathcal {N}=\left\lbrace 1,...,N\right\rbrace$ be the set of agents, with generic element $\nu \in \mathcal {N}$, and let $\mathcal {M}=\left\lbrace 1,...,M\right\rbrace$ be the set of produced commodities in the economy. Let $\mathbb {R}$ (resp., $\mathbb {R}_{+}$) denote the (resp., non-negative) real numbers. Let P be the production set, describing the technologically feasible production techniques. Elements of P, activities, are vectors denoted as α, which describe the amount of effective labour α_l and the vector of produced goods $\underline{\alpha }$ used as inputs in order to produce a vector of outputs $\overline{\alpha }$.Footnote ¹¹ We assume that technology displays constant returns to scale.Footnote ¹² Denote the vector of net outputs (outputs minus inputs) arising from α as $\widehat{\alpha }\equiv \overline{\alpha }-\underline{\alpha }$.

For each agent $\nu \in \mathcal {N}$, let s ^ν > 0 be ν’s skill level. If $\overline{L}>0$ is the amount of time each agent is endowed with, then ν’s effective labour endowment is $l^{\nu }=s^{\nu ^{{}}}\overline{L}$.Footnote ¹³ Similarly, if λ^ν is the amount of time spent by ν in production, then Λ^ν = s ^νλ^ν is the effective labour performed by ν. Let $\omega ^{\nu }\in \mathbb {R}_{+}^{M}$ denote ν ’s endowment of productive assets, and let $\omega =\sum _{\nu \in \mathcal {N} }\omega ^{\nu }$ be the vector of social endowments. Let p, w, and r, denote, respectively, the vector of the prices of produced commodities, the wage rate, and the rate of return on capital.

Each economy has a list $\left( \mathcal {N},\mathcal {M},P,\left( \omega ^{\nu }\right) _{\nu \in \mathcal {N}},\left( l^{\nu }\right) _{\nu \in \mathcal {N}}\right)$. Given this list, a vector $\boldsymbol{z}\equiv \left( \boldsymbol{x},\boldsymbol{\Lambda }\right) \in \mathbb {R}_{+}^{NM+N}$ represents an allocation $\boldsymbol{x}=\left( x^{\nu }\right) _{\nu \in \mathcal {N}}$ of the M commodities to the N agents in the economy and a profile $\boldsymbol{\Lambda }=\left( \Lambda ^{\nu }\right) _{\nu \in \mathcal {N}}$ of (effective) labour supplied by each agent. This vector is called an economically feasible allocation at prices (p, w, r) if and only if there exists a production activity α ∈ P satisfying $\underline{\alpha }\leqq \omega$, $\widehat{ \alpha }=\sum _{\nu \in \mathcal {N}}x^{\nu }$, $\alpha _{l}=\sum _{\nu \in \mathcal {N}}\Lambda ^{\nu }$, 0≦Λ^ν≦l ^ν, and px ^ν = wΛ^ν + rpω^ν for each agent $\nu \in \mathcal {N}$.Footnote ¹⁴ In other words, an allocation is economically feasible if the aggregate amount of each commodity allocated to agents can be produced given the aggregate endowments of physical and human capital in the economy, and each agent’s budget constraint is satisfied at the given prices.Footnote ¹⁵ Denote the set of economically feasible allocations at prices (p, w, r) by $\mathcal {Z}^{\left( p,w,r\right) }\subseteq \mathbb {R}_{+}^{NM+N}$. If x ^ν is the commodity bundle actually purchased by agent $\nu \in \mathcal {N}$ at an allocation $\boldsymbol{z}\in \mathcal {Z}^{\left( p,w,r\right) }$ then we shall denote it as c ^ν.

In this general economic environment, there is no obvious or widely shared definition of UEL exploitation. For any bundle $c\in \mathbb {R}_{+\text{ } }^{M}$, let l.v.(c) denote the minimum amount of (effective) labour necessary to produce c as net output.Footnote ¹⁶ In his classic definition, Morishima (Reference Morishima1974) focuses on the bundle actually purchased by an agent, c ^ν, and defines its labour content as l.v.(c ^ν):

Definition 1 has some desirable characteristics, according to Morishima (Reference Morishima1974: 616–618): the notion of exploitation is well-defined because, under mild assumptions on technology, l.v.(c) is unique, well-defined and positive whenever c ≠ (0, . . ., 0) and exploitation status is determined prior to and independent of price information, as in the standard Marxian approach, focusing only on production data.

According to Roemer (Reference Roemer1982a: 152), however, Definition 1 is conceptually flawed because it embodies a merely technological concept of value and is independent of the social relations of production, as it identifies exploitation status based on production techniques that may never be used by profit-maximizing capitalists. Like Morishima (Reference Morishima1974), Roemer (Reference Roemer1982a) focuses on the bundle actually bought by agents, c ^ν, but argues that its labour content should be determined focusing on profit-rate-maximizing activities, for only the latter production processes will be activated in a capitalist economy.

Let an allocation $\boldsymbol{z}=\left( \boldsymbol{\Lambda },\boldsymbol{x} \right) \in \mathcal {Z}^{\left( p,w,r\right) }$ with a price vector (p, w, r) be an equilibrium allocation whenever for any $\nu \in \mathcal {N}$, (x ^ν, Λ^ν) is preferred by ν to any other (x′^ν, Λ′^ν) satisfying $x^{\prime \nu }\in \mathbb {R}_{+}^{M}$, 0≦Λ′^ν≦l ^ν, and px′^ν = wΛ′^ν + rpω^ν.Footnote ¹⁷ For any bundle $c\in \mathbb {R} _{+\text{ }}^{M}$, let l.v.(c; p, w, r) be the minimum amount of (effective) labour necessary to produce c as net output with a profit-rate-maximizing activity at given equilibrium prices.Footnote ¹⁸ Then:

While individual consumption choices are central in Definition 2, Roemer (Reference Roemer1982a) has also proposed an alternative approach in which agents’ exploitation status is independent of their preferences over bundles of produced goods, focusing on the maximum and the minimum amounts of labour embodied in bundles that they can purchase. For all $\nu \in \mathcal {N}$ and all Λ^ν≦l ^ν, let B(ω^ν, Λ^ν; p, w, r) be the set of consumption bundles that agent ν can purchase at prices (p, w, r), if she supplies Λ^ν units of labour, given her endowment ω^ν.Footnote ¹⁹ Then:

Although they preserve many standard insights of exploitation theory, Definitions 1–3 have been criticized because exploitation status depends on counterfactual amounts of labour, as the production activities yielding l.v.(c ^ν) or l.v.(c ^ν; p, w, r) may be different from those actually used (Flaschel Reference Flaschel1983, Reference Flaschel2010). According to critics, this use of counterfactuals is theoretically undesirable and makes exploitation an empirically vacuous notion, since the computation of l.v.(c ^ν) and l.v.(c ^ν; p, w, r) requires information that is not available, including, in Morishima’s (Reference Morishima1974: 617) own words, ‘information about all the available techniques of production, actually chosen or potentially usable’.

The last definition considered here is an extension of the so-called ‘New Interpretation’ (Duménil Reference Duménil1980; Foley Reference Foley1982).Footnote ²⁰ Let α^{p, w, r} ∈ P be the aggregate (profit-rate-maximizing) production activity at an equilibrium allocation $\boldsymbol{z}$ with prices (p, w, r). Definition 4 identifies the labour associated with, or contained in any non-negative bundle of goods.

By Definition 4, the labour content of aggregate net output, $\widehat{\alpha }^{p,w,r}$, is equal to total social labour, α^{p, w, r}_l, and for any bundle c whose value does not exceed national income $p\widehat{\alpha }^{p,w,r}$, the labour contained in c is equal to the fraction τ^c of social labour necessary to produce a fraction of aggregate net output, $\tau ^{c}\widehat{\alpha }^{p,w,r}$, that has the same value as c.Footnote ²¹ As in Roemer’s (Reference Roemer1982a) approach, in Definition 4 the labour content of a bundle can be identified only if the price vector is known. Yet social relations play a more central role, because the definition of labour content requires a prior knowledge of social production α^{p, w, r} and labour content is explicitly linked to the redistribution of total social labour. Then:

In other words, for any agent $\nu \in \mathcal {N}$, τ^{c ν} represents ν’s share of national income, and so τ^{c ν}α^{p, w, r}_l is the share of social labour that ν receives by earning income barely sufficient to buy c ^ν. Thus, in the New Interpretation, the notion of exploitation is related to the production and distribution of national income and social labour, and depends on empirically observable magnitudes. Yet, Definition 5 has been criticized because, unlike Definitions 1 and 2, the actual bundles purchased by agents are only indirectly relevant to determine exploitation status, and unlike Definition 1, the notion of exploitation depends on price information.

In summary, our brief, and admittedly partial review shows that various definitions can be, and have in fact been, proposed which incorporate very different normative and positive intuitions. In the next sections, we show that in spite of these differences there exists a common core defining UEL theory.

4. UNEQUAL EXCHANGE OF LABOUR: FOUNDATIONS

In this section, and the next, we identify the foundations of UEL exploitation theory that are shared by all of the main approaches. Our purpose is not to discriminate among competing views, but rather to define the theoretical boundaries of UEL theory. Formally, we aim to identify a general domain for all admissible UEL exploitation forms.

Let $\mathcal {N}^{ter},$$\mathcal {N}^{ted},\mathcal {N}^{n}$ be three subsets of the set of agents $\mathcal {N}$ denoting, respectively, the set of exploiters, the set of exploited agents, and the set of agents that are neither exploited nor exploiters. At the most general level, a definition of exploitation can be interpreted as a rule that identifies the sets $\mathcal {N}^{ter},$$\mathcal {N}^{ted},\mathcal {N}^{n}$ at each feasible allocation – that is, as a mapping from $\mathcal {Z}^{\left( p,w,r\right) }$ to the power set of $\mathcal {N}$.

The first axiom of UEL theory requires that a definition of exploitation unambiguously identifies the exploitation status of every agent at any feasible economic allocation.

According to Axiom 1, even though exploitation may be a property of aggregates, such as social classes (or nations, see Roemer Reference Roemer1982a: chapter 1), such aggregates are ultimately made of individuals with a well-defined exploitation status. Axiom 1 thus excludes radically holistic approaches, which identify the existence of exploitation only at the aggregate level, without any reference to (coalitions of) individuals. It does not rule out approaches viewing ‘exploitation as a social phenomenon, and the existence of exploitation need not imply, in principle, the existence of individual exploiters or exploited’ (Roemer Reference Roemer1982a: 136). But it does require the exploitation status of each agent to be clearly defined: at any feasible allocation, for every agent $\nu \in \mathcal {N}$, either $\nu \in \mathcal {N}^{ter}$ and ν is an exploiter; or $\nu \in \mathcal {N}^{ted}$ and ν is exploited; or $\nu \in \mathcal { N}^{n}$ and ν is exploitation-neutral.Footnote ²²

Axiom 1 defines exploitation as a property, not primarily as a relation (Elster Reference Elster1985: 174). It requires exploitation status to be defined at an overall allocation, and is silent about exploitation in individual transactions (unlike in approaches focusing on just acquisition, such as Steiner (Reference Steiner1984), or disadvantage, such as Goodin (Reference Goodin and Reeve1987) and Sample (Reference Sample2003)). This does not exclude the possibility that exploitation be diagnosed relationally, at the level of individual transactions, and agents may be exploited in some transactions, and exploiters in others (e.g. Fleurbaey Reference Fleurbaey2014: 661). According to Axiom 1, however, the agents’ overall exploitation status should also be identified taking into account all of their economic activities.Footnote ²³

In principle, Axiom 1 may apply to any definitions of exploitation, including those focusing on fair pricing (e.g. Robinson Reference Robinson1933; and Reiff Reference Reiff2013) or property relations (e.g. Roemer Reference Roemer1982a: chapter 7). In UEL theory, however, at any allocation, the exploitation status of an agent ν is identified by comparing the labour that ν ‘gives’, in some relevant sense, and the labour that ν ‘receives’, in some relevant sense. Thus, in any UEL approach, two variables must be specified for each agent: the labour given and the labour received. As a shorthand notation, let L ^ν_G and L ^ν_R denote, respectively, the labour ‘given’ and the labour ‘received’ by agent $\nu \in \mathcal {N}$ at allocation $\boldsymbol{z}\in \mathcal {Z}^{\left( p,w,r\right) }$. Our task is to identify the restrictions on L ^ν_G and L ^ν_R common to all UEL approaches.

In all UEL approaches, the labour ‘given’ by agent ν is just a scalar. In simple economies with homogeneous labour, it is equal to the labour time expended by ν in production.Footnote ²⁴ In more general settings, however, various definitions can be adopted, which incorporate different normative intuitions. In order to focus on the core of UEL exploitation theory shared by all approaches, we leave L ^ν_G unspecified while assuming that it is a scalar for all $\nu \in \mathcal {N}$. We return to this issue in Section 7 below.

The definition of the labour that each agent ‘receives’ is more controversial, and many different proposals can be (and have been) made, as shown in Section 3. The labour ‘received’ by an agent at an allocation is not necessarily captured by a single number (see, for example, Definition 3) and each L ^ν_R could be a (possibly infinite) set of labour amounts related, for example, to an agent’s budget set. In all UEL approaches, however, exploitation status is monotonic in the sense that for an agent to be classified as exploited, she must ‘give’ more labour – have a higher L ^ν_G – than would be necessary to classify her as an exploiter, everything else equal (that is, for given consumption, income, preferences, and so on). Therefore we submit that in UEL theory, at any allocation $\boldsymbol{z}\in \mathcal {Z}^{\left( p,w,r\right) }$, the exploitation status of each agent ν can be defined by identifying at most two cut-off values L ^ν_min, L _max^ν which may be interpreted as the upper and lower bounds of the labour ‘received’ by ν, and can be called the reference labour amounts.Footnote ²⁵

The next axiom defines the core of the theory of exploitation as the unequal exchange of labour.

Axiom 2 incorporates an ‘idea of exploitation, as a certain kind of lack of reciprocity’ (Cohen Reference Cohen1979: 343), whereby an agent is exploited whenever ‘the labor he contributes in one form does not return to him in another form’ (Husami Reference Husami1978: 44). The lack of reciprocity may be normatively relevant per se,Footnote ²⁶ or because differences in labour ‘given’ and ‘received’ reflect normatively significant inequalities. Axiom 2 holds in approaches that define exploitation as the (forced) extraction of surplus, or unpaid labour (Holmstrom Reference Holmstrom1977; Elster Reference Elster, Nielsen and Ware1997; Reiman Reference Reiman1987), but it is more general than that and it does not imply a commitment to an objective theory of value. For the labour ‘given’ and ‘received’ by agents need not be determined objectively, and may reflect agents’ preferences and beliefs. Indeed, Axiom 2 does not imply a commitment to any theory of value, or to the claim that labour is the only thing that produces value.Footnote ²⁷ It only identifies labour as the key normative variable of interest – the main unit of account of exploitation theory – and thus rules out approaches based on wealth, income, utility, and so on. In exploitation theory, labour accounting is the ‘way of characterizing what it is that people give one another . . . (where ‘give’ is understood very broadly to refer to any way in which some person undergoes a loss that ends up a gain to another)’ (Reiman Reference Reiman1987: 9).Footnote ²⁸

Axiom 2 provides a general statement of the basic principle of UEL theory. In order to have an operational definition, however, and setting aside for the moment the definition of L ^ν_G, it is necessary to identify the labour ‘received’ by agents. This is by no means trivial. For there are potentially up to 2N reference labour amounts at any allocation. So the question is; how are $\left( L_{\min }^{\nu },L_{\max }^{\nu }\right) _{\nu \in \mathcal {N}}$ determined? Or, equivalently, how do agents ‘receive’ labour?

As Roemer (Reference Roemer1985: 31) puts it, ‘exploitation theory views goods as vessels of labor, and calculates labor accounts for people by comparing the ‘live’ labor they expend in production with the ‘dead’ labor they get back in the vessels’. Axiom 3 formalizes this view and states that agents receive labour via certain commodity bundles.

Axiom 3 implies that, for each agent ν, the reference labour amounts, L ^ν_min, L _max^ν, can be interpreted as the amounts of labour associated with, or contained in some theoretically relevant reference commodity bundles, the ERBs. In the standard approach outlined in Section 2, the labour received by workers is equal to the labour embodied in the goods required for their subsistence: c ^ν_min, c _max^ν are equal and coincide with a given subsistence basket, and the labour associated with them, f ^ν, is computed by identifying employment multipliers which uniquely identify the unit labour embodied in each commodity, and a fortiori in any bundles of goods.Footnote ²⁹ Axiom 3 is more general, as it allows c ^ν_min, c _max^ν to be different and unrelated to any subsistence notion (or indeed to agents’ consumption choices) and it does not require the adoption of the standard notion of labour embodied. Indeed, it does not require a general theory for the determination of the labour contained in any bundles of goods: it only states that any definition of UEL exploitation should have a well-defined (albeit possibly implicit) notion of labour associated with, or contained in, the ERBs. Thus, Axiom 3 also captures the intuitions of approaches that do not focus on the labour embodied in given consumption bundles, such as the ‘New Interpretation’ (Duménil Reference Duménil1980; Foley Reference Foley1982), according to which the labour ‘received’ by agents is equivalent to the amount of social labour agents receive a claim to via their income.

Axiom 3 says nothing about the relation between (c ^ν_min, c _max^ν) and (L ^ν_min, L _max^ν). For each agent ν and each (c ^ν_min, c _max^ν), f ^ν is a purely normative (and possibly agent-specific) construct that identifies the labour ‘received’ by ν based on some reference bundles. Axiom 3 does not specify how f ^ν translates the ERBs into amounts of labour, as its properties are left unspecified. Some theoretically relevant restrictions on f ^ν, and on the choice of the ERBs (c ^ν_min, c _max^ν), are identified in the next section.

5. THE EXPLOITATION REFERENCE BUNDLES

Axioms 1–3 are not a complete theory of exploitation. They provide a rigorous framework to conceptualize UEL exploitation, and the theoretical choices of different approaches, in terms of restrictions on $\left( L_{G}^{\nu },L_{R}^{\nu }\right) _{\nu \in \mathcal {N}}$. Concerning L ^ν_R, Axiom 3 implies that in order to define the exploitation status of each agent, it is necessary both to select the relevant ERBs, c ^ν_min, c ^ν_max, and to identify their labour content – that is, the (possibly implicit) function f ^ν that associates a reference labour amount with each ERB. In this section, we explore some restrictions on c ^ν_min, c _max^ν and f ^ν that aim to incorporate formally weak, theoretically robust, and widely shared intuitions in UEL theory.

Axiom 4 requires c ^ν_min, c ^ν_max to be in the set of economically feasible choices for agent ν.

Axiom 4 stipulates that the ERBs be at least potentially affordable (recall that l ^ν is agent ν’s labour endowment). By focusing on potential, rather than actual income, it allows for many different views: the ERBs may coincide with the bundle actually purchased by agents, or they may be some normatively relevant commodity bundles that are actually or only potentially affordable.Footnote ³⁰ The former approach is relevant if one believes that the ERBs should be related to agents’ actual consumption choices, because the way in which revenues are actually allocated (e.g. between consumption and investment) may be relevant to the moral status of their distribution (e.g. Elster Reference Elster1985: 177). The latter approach is important, for example, in modal definitions in which exploitation status cannot be established just by looking at actual behaviour (Elster Reference Elster1985: 173ff).

Thus, Axiom 4 holds in UEL approaches that take the ERB to be unique and equal to the workers’ subsistence basket.Footnote ³¹ But it also holds in approaches that abandon a subsistence view and define the labour received by worker ν focusing on ‘commodities whose value is equivalent to the wages [ν ] receives’ (Buchanan Reference Buchanan1982: 37; see also Morishima Reference Morishima1973: 46); or even more generally, in those approaches that determine the labour received by any agent by looking at the amount of labour contained ‘in the goods he can purchase with his revenues from production (which may come from wages, profits, or the sale of commodities)’ (Roemer Reference Roemer1989: 90).

Axiom 4 imposes an economic feasibility condition on the ERBs. Following classic UEL theory, Axiom 5 requires that c ^ν_min, c ^ν_max be also technically feasible as net output of some production process.

The emphasis on production conditions is standard in the traditional Marxist view according to which the amount of labour received by workers is equal to the value of labour power. Because labour power is a produced commodity, its value is given by the amount of labour necessary to produce it, which is identical with the amount of time required to produce the means of subsistence of the worker. More generally, in UEL theory, exploitation is a specific wrong rooted in the sphere of production (Cohen Reference Cohen1979: 345; Elster Reference Elster1985: 167; Fleurbaey Reference Fleurbaey1996: 184), and formal UEL definitions ‘map from the fundamental relations of production into the labour market and from this to the market of commodities, and then invert the mapping so as to go from the market of the final goods back to the production relationship’ (Morishima and Catephores Reference Morishima and Catephores1978: 43).

Axiom 5 requires the ERBs to be technically feasible as net output of some production process.Footnote ³² This is because the direct labour used to produce a bundle c as net output allows one to capture the total amount of labour – direct and indirect – contained in producing c from scratch. A focus on net output is also natural in those UEL approaches that define embodied labour such that total labour performed in the economy in a given period is equal to the labour contained in net national product, and conceive of exploitation as measuring (using a labour metric) how net national product is parcelled out to individuals.Footnote ³³

While Axioms 4 and 5 impose restrictions on the ERBs, the next two properties focus on the function f ^ν that associates a reference amount of labour to each ERB. Let $\boldsymbol{s}=\left( s^{1},...,s^{N}\right)$. Axiom 6 is a natural complement to Axiom 5 in that it requires that the reference labour amounts also be related to production conditions.

The notion of labour contained in Axiom 6 is by no means metaphysical and bears no conceptual relation with the disputes on the labour theory of value. The labour contained in the ERBs is a well-defined amount related to current technical conditions,Footnote ³⁴ and to the direct amount of (effective) labour used in production. Contra Steedman (Reference Steedman1977), Axiom 6 rules out negative reference labour amounts, because the labour contained in the ERBs, and thus the labour received by agents should be non-negative by definition (Morishima and Catephores Reference Morishima and Catephores1978: 32; Flaschel Reference Flaschel2010: 18).

According to Axiom 6, the reference labour amounts may be quantities of effective labour, but this is not a requirement: for each $\nu \in \mathcal {N}$, the positive scalar $\kappa ^{\nu }\left( P,\boldsymbol{s}\right)$ is meant to linearly transform effective labour units into units of labour, and different specifications of the linear transformations $\left( \kappa ^{\nu }\left( P,\boldsymbol{s} \right) \right) _{\nu \in \mathcal {N}}$ correspond to different notions of the labour ‘received’ (as well as ‘given’) by agents. That is, the appropriate choice of $\left( \kappa ^{\nu }\left( P,\boldsymbol{s}\right) \right) _{\nu \in \mathcal {N}}$ may depend on the normative use of the concept of exploitation (Fleurbaey Reference Fleurbaey1996: 176–177). We return to this issue in Section 7 below.

A focus on production and on the notion of labour contained in the ERBs may seem restrictive in that it rules out, for example, approaches focusing on exchange whereby the reference labour amounts are determined based on labour commanded – that is, on the amount of labour power that an agent may purchase with her income. This should not be a major concern. Considerations of purchasing power enter our framework explicitly in Axiom 4. Further, Axiom 6 allows for the possibility that price magnitudes and market relations enter the definition of labour contained in the ERBs, and although it does impose some restrictions on f ^ν, they are extremely weak and the choice of reference labour amounts remains significantly open, as there may be very many ways of producing the ERBs.Footnote ³⁵

Given that our aim is to identify the foundations of UEL theory shared by all of the main approaches, there is no reason to impose any strong restrictions on f ^ν. Different restrictions may appropriately reflect alternative views, or may be desirable in different economic environments.Footnote ³⁶ Yet, a common assumption in the UEL literature is that, however determined, the reference labour amounts be well-defined and unique.

According to Axiom 7, if the ERBs of two agents are equal, then the associated reference labour amounts should be the same: the labour contained in the ERBs should be uniquely identified based on production conditions and not on idiosyncratic subjective factors. Although there are many conceivable ways of determining L ^ν_min and L ^ν_max, Axiom 7 states that such differences should be at the level of the theory, and not of arbitrary individual characteristics. This does not imply that subjective preferences and individual choices are irrelevant in determining exploitation status. It only means that the labour contained in the ERBs should not depend on the identity of the agents’ receiving them.Footnote ³⁷

Axiom 7 holds in the standard Marxian approach, whereby the labour content of a bundle, however defined, is uniquely determined and exploitation status is independent of agents’ subjective characteristics. According to Morishima (Reference Morishima1973: 181), for example, a fundamental property of a definition of exploitation is that ‘each worker has to be shown to be equally exploited by the capitalists’. Yet, Axiom 7 also seems reasonable more generally and it is widely shared in UEL approaches: as argued by Yoshihara and Veneziani (Reference Yoshihara and Veneziani2011), one of the essential, if not defining characteristics of exploitation as a normative construct in political philosophy is its (weakly) objectivist thrust. ‘If an agent could change from being exploited into being an exploiter simply as a result of a change of tastes, some of the moral connotations of exploitation would be lost’ (Elster Reference Elster1985: 174; see also Roemer Reference Roemer1982a: 110). Axiom 7 seems the weakest, and most easily defensible, part of a general objectivist theory of UEL exploitation. For if labour is ‘received’ via some commodity bundles, then it seems natural to require that a single reference labour amount be associated with any ERB.

Axioms 4–7 complete the analysis of the basic structure of UEL approaches. In UEL theory, the exploitation status of an agent ν is characterized by the labour ‘given’ and ‘received’ by her. The labour received by ν is determined by identifying some relevant commodity bundles, the ERBs, which are potentially affordable (Axiom 4) and technically feasible (Axiom 5), and the labour contained in them, which is (a linear transformation of) the amount of labour necessary to produce them (Axiom 6), and is uniquely determined independently of subjective factors (Axiom 7).

6. LABOUR EXPLOITATION

Axioms 1–7 define the common structure underlying different approaches to UEL theory. In this section, we formalize a domain condition, called Labour Exploitation: it summarizes Axioms 1–7 and so, we argue, identifies the domain of admissible exploitation forms.Footnote ³⁸

Labour Exploitation (LE): Given any definition of exploitation, the sets $\mathcal {N}^{ted},\mathcal {N}^{ter}$should have the following property. At any allocation $z\in \mathcal {Z}^{\left( p,w,r\right) }$with a price system (p, w, r), there exist a non-negative vector $\left( L_{G}^{\nu }\right) _{\nu \in \mathcal {N}}$, a positive vector $\left( \kappa ^{\nu }\left( P,\boldsymbol{s}\right) \right) _{\nu \in \mathcal {N}}$, a profile of commodity bundles $\left( c_{\min }^{\nu },c_{\max }^{\nu }\right) _{\nu \in \mathcal {N}}$and a profile of activities $\left( \alpha _{\min }^{\nu },\alpha _{\max }^{\nu }\right) _{\nu \in \mathcal {N}}$ that satisfy production objectivism such that for each agent $\nu \in \mathcal {N}$: c ^ν_min, c _max^ν ∈ B(ω^ν, l ^ν; p, w, r), α^ν_min, α_max^ν ∈ P with $\widehat{\alpha }_{\min }^{\nu }\geqq c_{\min }^{\nu }$, $\widehat{\alpha }_{\max }^{\nu }\geqq c_{\max }^{\nu }$, and α^ν_lmax≧α_lmin^ν, and

$$\begin{eqnarray*} \nu &\in &\mathcal {N}^{ted}\text{ {\it if and only if} }\kappa ^{\nu }\left( P,\boldsymbol{s}\right) \alpha _{l\max }^{\nu }<L_{G}^{\nu }, \\ \nu &\in &\mathcal {N}^{ter}\text{ {\it if and only if} }\kappa ^{\nu }\left( P,\boldsymbol{s}\right) \alpha _{l\min }^{\nu }>L_{G}^{\nu }\text{.}\vspace*{-1pt} \end{eqnarray*}$$

According to LE, at any feasible allocation, the exploitation status of every agent ν is determined by the difference between the labour that ν ‘gives’ to the economy, and the labour she ‘receives’. Whereas the former quantity is left unspecified, the labour received by ν is determined by identifying at most two (potentially) affordable and technically feasible bundles, c ^ν_min, c ^ν_max, and their labour content which is equal to (a linear transformation of) the labour necessary to produce them as net output, $\kappa ^{\nu }\left( P,\boldsymbol{s}\right) \alpha _{l\min }^{\nu ^{{}}}$, $\kappa ^{\nu }\left( P,\boldsymbol{s}\right) \alpha _{l\max }^{\nu ^{{}}}$. The amount of labour that ν receives is the (possibly degenerate) interval $\left[ \kappa ^{\nu }\left( P,\boldsymbol{s}\right) \alpha _{l\min }^{\nu ^{{}}},\kappa ^{\nu }\left( P,\boldsymbol{s}\right) \alpha _{l\max }^{\nu ^{{}}}\right]$, and so, for any $\nu \in \mathcal {N}$, if L ^ν_G is more (resp., less) than $\kappa ^{\nu }\left( P, \boldsymbol{s}\right) \alpha _{l\max }^{\nu ^{{}}}$ (resp., $\kappa ^{\nu }\left( P,\boldsymbol{s}\right) \alpha _{l\min }^{\nu ^{{}}}$) then ν is regarded as ‘giving’ more (resp., less) labour than ν ‘receives’ and therefore a member of $\mathcal {N}^{ted}$ (resp., $\mathcal {N}^{ter}$).

The domain condition LE summarizes the content of Axioms 1–7. Indeed, the next result proves that LE is logically equivalent to them.Footnote ³⁹

Theorem 1. At any allocation $z\in \mathcal {Z} ^{\left( p,w,r\right) }$, a definition of exploitation satisfies LE if and only if it satisfies Axioms 1–7.

In the previous sections, we have argued that Axioms 1–7 describe the structure and the fundamental intuitions common to all of the main approaches in UEL theory. Then, by Theorem 1, LE also captures the key intuitions of UEL theory. In order to substantiate this claim, we now show that in spite of all the differences highlighted in Section 3, all of the main definitions in the literature satisfy LE, and thus Axioms 1–7.

To see this, note that in all of the approaches examined in Section 3, at all $\boldsymbol{z}\in \mathcal {Z}^{\left( p,w,r\right) }$, L ^ν_G = Λ^ν and $\kappa ^{\nu }\left( P,\boldsymbol{s} \right) =1$, for all $\nu \in \mathcal {N}$. In Definitions 1 and 2, at all $\boldsymbol{z}\in \mathcal {Z}^{\left( p,w,r\right) }$, the ERB is unique and corresponds to the bundle actually purchased by agents, c ^ν_min = c _max^ν ≡ c ^ν ∈ B(ω^ν, l ^ν; p, w, r), for all $\nu \in \mathcal {N}$. However, in Definition 1, the labour content of the ERB is the minimum amount of labour necessary to produce it among all conceivable production activities, whereas in Definition 2 only profit rate maximizing processes are considered.Footnote ⁴⁰ In Definition 3, given an equilibrium allocation $\boldsymbol{z}$ with prices (p, w, r), using the same notion of labour content as in Definition 2, c ^ν_max is the bundle whose labour content is maximal among all the bundles that the agent can purchase at (p, w, r), while c ^ν_min is the bundle whose corresponding labour content is minimal.Footnote ⁴¹ Note that, in general, c ^ν_min ≠ c _max^ν ≠ c ^ν under Definition 3. Finally, in Definition 5, given an equilibrium allocation $\boldsymbol{z}$ with prices (p, w, r) and aggregate profit-rate-maximizing production activity α^{p, w, r}, the ERB is unique but, for all $\nu \in \mathcal {N}$, it is defined counterfactually as the proportion of net aggregate output that the agent may purchase with her income$\ c_{\min }^{\nu ^{{}}}=c_{\max }^{\nu ^{{}}}=c_{e}^{\nu ^{{}}}\equiv \tau ^{c^{\nu ^{{}}}}\widehat{\alpha }^{p,w,r}\in B\left( \omega ^{\nu },l^{\nu };p,w,r\right)$, where $\tau ^{c^{\nu ^{{}}}}=\frac{pc^{\nu ^{{}}} }{p\widehat{\alpha }^{p,w,r}}$, and its labour content is given by the corresponding share of aggregate social labour, α^{c_e ν}_l ≡ τ^{c ν}α^{p, w, r}_l.

Similarly, it is possible – albeit notationally intensive – to show that other definitions satisfy LE, including the input-output approach proposed by Flaschel (Reference Flaschel1983, Reference Flaschel2010); the variant of Definition 2 that focuses on any activities in P ^π(p, w, r) such that $\widehat{\alpha }\geqq c$, and not only labour-minimizing ones (Roemer Reference Roemer1982a: 164–168); and even the subjectivist approach developed by Matsuo (Reference Matsuo2008), according to which the labour received by an agent who consumes a bundle c corresponds to the minimum amount of labour necessary to produce another bundle c′ as net output, which gives at least as much utility as c.

This confirms that LE incorporates the main insights of UEL exploitation theory shared by all of the main approaches.

7. ETHICAL RESTRICTIONS

Despite the seemingly irreconcilable differences between definitions focusing on physical or monetary magnitudes, actual choices or potential consumption, objective data or subjective preferences, actual or counterfactual production processes, and so on, Axioms 1–7, and therefore LE, identify a fundamental structure that defines UEL exploitation theory in all of its variants – even though this structure, and the notions of ERBs and their labour content are usually left implicit. Hence they can be interpreted as defining the boundaries of the admissible class of UEL exploitation forms.

Of course, the boundaries of UEL theory delineated by Axioms 1–7 are extremely wide and potentially include an infinite number of conceivable definitions – i.e. an infinite number of ways of specifying the ERBs, the reference labour amounts, and L ^ν_G. It is therefore natural to ask how our framework could be extended to determine exploitation status within a (possibly core) class of definitions. To be precise, can one identify additional restrictions on the definitions of labour given and labour received that characterize a core class of approches within the admissible set defined by LE?

In this section, we show that, by identifying the common foundations of UEL theory, LE also provides the formal and conceptual framework to answer this question and to examine the normative and positive implications of different approaches. We illustrate this point by showing how the differences between alternative definitions can be conceived of as different additional ethical restrictions imposed on LE. We return to the more general issue of the possible extensions of our approach in the Conclusions.

A first important issue concerns the relevance of individual choices in determining exploitation status. In LE, the labour received by agent ν is determined by some reference bundles that ν can in principle purchase. In Definitions 1 and 2, however, the ERB should be the bundle c ^ν actually chosen by agent ν at allocation $\boldsymbol{z}$. Formally:

The theoretical justification for Axiom 8 rests on the idea that disregarding agents’ consumption choices implies identifying their exploitation status based on information that is potentially at odds with their actual situation and well-being.

This subjectivist view is not uncontroversial. Following the standard Marxian approach, for example, one may insist that exploitation status depend on production decisions, and not on possibly idiosyncratic consumer choices. This intuition can be formalized as follows:

Axiom 9 is called ‘Consumer objectivism’ because it does not rule out the possibility that subjective factors matter in other aspects of UEL theory – for example, by defining L ^ν_G as the agents’ actual labour supply or even by introducing subjective elements in the determination of the labour ‘content’ of the ERBs.

A second question concerns the kind of production processes that are normatively relevant to identify exploitation status. LE requires that both the ERBs and their labour content be determined with reference to production, but otherwise imposes no restrictions. In UEL approaches, however, it is often required that only efficient production processes be considered in order to determine exploitation status. Let ∂P denote the set of efficient production activities.Footnote ⁴² The next axiom states that the ERBs should be producible as net output of efficient production activities.

The efficiency view is consistent with Marx’s emphasis on the notion of socially necessary labour time and it embodies the idea that exploitative relations do not emerge from productive inefficiencies and mistakes. According to Morishima (Reference Morishima1973: 180), for example, in the computation of labour amounts one should focus on ‘those techniques by the use of which the amount of labour needed to produce given amounts of commodities can be minimized’. From a normative viewpoint, these are the techniques that maximize the productivity of human labour (Morishima 1973: 184).Footnote ⁴³ From a positive viewpoint, one may argue that inefficient techniques are at best a transient epiphenomenon in a competitive economy.

Yet, one may object that if the normative emphasis of UEL theory is on participation in productive activities, then it is misleading to focus on possibly counterfactual processes, and exploitation status should be identified based on production processes that are actually activated in the economy (Flaschel Reference Flaschel1983, Reference Flaschel2010). Let $\alpha ^{\boldsymbol{z}}\in P$ be the actual aggregate production activity at a feasible allocation $\boldsymbol{z}\in \mathcal {Z}^{\left( p,w,r\right) }$. One way to formalize this view is as follows.

Axiom 11 states that the reference labour amounts should be identified based on the production processes actually activated in the economy – more precisely, they are equal to some fraction of aggregate social labour performed. Thus, it incorporates the idea that actual labour spent in production is what matters in defining exploitative relations because it describes actual participation (Sen Reference Sen1978).

So far, we have left L ^ν_G unspecified because there is no widely shared definition of the labour ‘given’ by agents in economies with heterogeneous labour. In particular, two main approaches can be distinguished which focus either on the effective labour contributed, or on the labour time expended by agents. The former approach can be formalized as follows:

By Axiom 12, both labour ‘given’ and labour ‘received’ are measured in terms of effective labour. As noted in Section 3, this is the natural extension of all of the classic definitions of exploitation, and it is the approach adopted in much of the literature on exploitation in economies with heterogeneous labour (see e.g. Krause Reference Krause1982; Duménil et al. Reference Duménil, Foley and Lévy2009). Moreover, the ‘contribution view’ incorporates an important normative intuition: an efficient and UEL exploitation-free allocation coincides with the proportional solution, a well-known fair allocation rule whereby every agent’s income is proportional to her contribution to the economy (Roemer and Silvestre Reference Roemer and Silvestre1993). Proportionality is a strongly justified normative principle, whose philosophical foundations can be traced back to Aristotle, and it can be justified in terms of the Kantian categorical imperative (Roemer Reference Roemer2010, Reference Roemer2018).Footnote ⁴⁴

Alternatively, one may argue that UEL theory captures some inequalities in the distribution of material well-being and free hours that are – at least prima facie – of normative relevance (Fleurbaey Reference Fleurbaey1996, Reference Fleurbaey2014).Footnote ⁴⁵ For example, they may be deemed relevant because material well-being and free hours are key determinants of individual well-being freedom (Rawls Reference Rawls1971; Sen Reference Sen1985). But they are also relevant in approaches that link exploitation and the Marxian notion of alienation in production (Buchanan Reference Buchanan1982). From this perspective, the key variable of normative interest is labour time:Footnote ⁴⁶

Axiom 13 does not require that $\kappa ^{\nu }\left( P,\boldsymbol{s}\right) =1$ for all agents ν, because the positive vector $\left( \kappa ^{\nu }\left( P,\boldsymbol{s}\right) \right) _{\nu \in \mathcal {N}}$ is necessary to transform units of effective labour into labour time, and various theoretically relevant specifications are possible. For example, if $\left( L_{R}^{\nu }\right) _{\nu \in \mathcal {N}}$ captures the distribution of social labour and α^ν_min, α_max^ν are the outcomes of social production activity, then, for all $\nu \in \mathcal {N}$, $\kappa ^{\nu }\left( P,\boldsymbol{s}\right)$ can be defined as the average skill level at allocation $\boldsymbol{z}$: $\kappa ^{\nu }\left( P,\boldsymbol{s} \right) \equiv \frac{\sum _{\nu \in \mathcal {N}}\lambda ^{\nu }}{\sum _{\nu \in \mathcal {N}}s^{\nu }\lambda ^{\nu }}$. Alternatively, one may argue that L ^ν_R should be equal to the labour time that would be necessary for each agent to produce the ERBs autarkically and set $\kappa ^{\nu }\left( P,\boldsymbol{s}\right) \equiv \frac{1}{s^{\nu }}$, for all $\nu \in \mathcal {N}$.

Axioms 8–9, 10–11, and 12–13 incorporate different normative intuitions and have rather different implications, and in this paper we do not adjudicate them. The previous discussion shows, however, that our domain axiom provides the formal and conceptual framework to rigorously examine alternative approaches to exploitation theory and to identify a core class of definitions of exploitation within the admissible set identified by LE.