Groucho Marxism

Marxists often claim that workers are exploited under capitalism. In fact this is usually considered a central tenet of Marxism. But what exactly do we mean by ‘exploited’? In this blog post I will attempt to provide a rigorous definition (actually three rigorous definitions). To kick things off, consider an economy which produces m commodities using n types of labour. For such an economy we can define a commodity vector as a vector with positive elements of length m, and a labour as a vector with positive elements of length n. The economy is defined by an activity set, with the interpretation that an element (x,u,x’) of this set represents a possible configuration of commodity inputs x, labour inputs u, and commodity outputs x’ for the economy.

For our economy, a price vector is a vector with positive elements of length m, and a wage vector is a vector with positive elements of length n. The labour value of the commodity vector y, given a wage vector w, may be defined as the minimum value of the product wu subject to the constraint that there exists commodity vectors x and x’ such that (x,u,x’) is an element of the activity set, and x’-x is greater than or equal to y. This is quite a technical definition, so let us try to unpack it a bit. In our economy we have n types of labour, and we want to define the labour value of a commodity as a single number. The obvious way to do that is by weighting each type of labour according the wage that can be earned for doing that type of labour. This is where the product wu comes in.

The product wu can be thought of as a quantity which represents ‘abstract labour’ in the Marxist sense. Minimizing this value can then be thought of as giving us the ‘socially necessary’ part of ‘socially necessary abstract labour’. The constraint in the minimization is simply there to ensure that the commodity vector q can be produced by the economy; to see this, note that x’–x represents the net output of the production process defined by the triple (x,u,x’). Let us denote the labour value of the commodity vector y given the wage vector w by v(y). Now consider an agent i in the economy who provides labour input L_i and receives (or purchases) net output y_i. We can say that this agent is exploited if wu_i greater than v(y_i), and an exploiter if wu_i is less than v(y_i).

Again, this is quite a technical definition, so let us unpack it. The quantity wu_i represents the agent’s labour input, weighted by their wage, and the quantity v(y_i) represents the labour value of the net output they receive. Thus, we say that the agent is exploited if their weighted labour input is greater than the value they receive, and conversely we say that they are an exploiter if their weighted labour input is less than the value they receive. This definition of exploitation was first put forward by the Japanese Marxist economist Michio Miroshima in 1973. According to the American Marxist economist John Roemer, however, this definition is flawed as it identifies exploitation based on activities that may never be used by profit-maximizing capitalists.

The gross profit associated with the activity (x,u,x’), price vector p, and wage vector w, is given by the equation: H = p(x’-x)-wu; and the associated profit rate is given by the equation: r = H/px. Then, according to Roemer, the labour value of the commodity vector y, given the price vector p, and wage vector w, may be defined as the minimum value of the product wu subject to the constraint that there exists commodity vectors x and x’ such that (x,u,x’) maximizes the profit rate, and x’-x is greater than or equal to y. This is very similar to Miroshima’s definiton; the only difference is that activities are now constrained to be profit-maximizing.

The problem with both of the above definitions of labour value is that they are based activities that might not actually be used in practice. An alternative approach which gets around this problem was recently put forward by the Marxist economists Naoki Yoshihara and Roberto Veneziani. Given an activity (x,u,x’), a price vector p, and a commodity vector y, let s(y) denote the share of py in total nominal net output p(x’-x), so that s(y) = py/p(x’-x). Then, given a wage vector w, we can define the labour value of the commodity vector y as v(y)= s(y)wu. We can then say that agent i is exploited if wu_i is greater than v(y_i), and an exploiter if wu_i is less than v(y_i). This gives us a third definition of labour values and exploitation.

Yoshihara and Veneziani also introduce something they refer to as the ‘profit-exploitation correspondence principle’. This principle states that aggregate profits are positive if and only if propertyless workers – that is, agents with no initial endowment of commodities who provide some labour – are exploited. They then show that of the three definitions of exploitation given above, only theirs is guaranteed to adhere to this principle. Given the theoretical relevance of profit-exploitation correspondence principle in Marxian theory, this provides strong support for their definition of exploitation being the most appropriate one from a Marxist perspective.