Groucho Marxism

The labour theory of value (LTV) posits that the exchange value of a commodity is proportional to the socially necessary labour time required to produce it. Marx was the greatest champion of the LTV but many Marxists have since abandoned it, raising the question of whether the LTV is a necessary component of Marxism or whether it can be discarded. In this blog post I will attempt to answer this question. To fix ideas, consider an economy which produces m commodities. For such an economy we can define a commodity vector as an mx1 column vector with positive elements. The economy is defined by an activity set, with the interpretation that an element (X,u,Y) of this set represents a possible configuration of commodity inputs X, labour inputs u, and commodity outputs Y.

According to Japanese Marxist economist Michio Morishima, the socially necessary labour time associated with the commodity vector M may be defined as the minimum value of u subject to the constraint that there exists commodity vectors X and Y such that (X,u,Y) is an element of the activity set and Y-X is greater than or equal to M. This is quite a technical definition so let us try to unpack it a bit. Minimizing u can then be thought of as giving us the ‘socially necessary’ part of socially necessary abstract labour. The constraint in the minimization is there to ensure that the commodity vector M can be produced by the economy; to see this, note that Y-X represents the net output of the production process defined by the triple (X,u,Y).

Thus, under this definition, to find the socially necessary labour time associated with the commodity vector M we must find the minimum value of u subject to the constraints that (X,u,Y) is in the activity set and Y-X ≥ M. Let u* be a solution to this problem, so that u* is the socially necessary labour time for M. Then given another labour input u that satisfies these constraints, the surplus labour time is given by u-u* and the rate of exploitation is given by (u-u*)/u. The ‘dual’ problem involves finding the maximum value of pM subject to the constraints that (X,u,Y) is in the activity set and p(Y-X) ≤ u. Let p* be a solution to this problem, so that p* represents the profit-maximizing price for M subject to these constraints. Then we would expect that p*M = u*.

As firms maximize profits, we would therefore expect that the exchange value of a commodity vector is equal to the socially necessary labour time associated with this commodity vector. This provides an intuitive justification for why the LTV should hold. This can be made rigorous in the case where we have X = Aq, u = Lq, and Y = Bq for some fixed mxn matrices A ≥ 0 and B ≥ 0, some fixed 1xn row vector L ≥ 0, and some variable nx1 ‘intensity’ vector q ≥ 0; such an economy is referred to as a von Neumann economy, after the Hungarian mathematician and physicist Jon von Neumann. Then the primal problem is to minimize Lq subject to (B-A)q ≥ M and the dual problem is to maximize pM subject to p(B-A) ≤ L, and if q* and p* are solutions to these problems it can be shown that p*M = Lq*.

This result can be restated succinctly as follows. Let (A,B,L) be a von Neumann economy, where A ≥ 0 is the mxn commodity input matrix, B ≥ 0 is the mxn commodity output matrix, and L ≥ 0 is the 1xn row vector of labour inputs. Then we may define the socially necessary labour time associated with the mx1 commodity vector M ≥ 0 as the minimum of Lq subject to (B-A)q ≥ M, and the exchange value of the commodity vector M as the maximum of pM subject to the constraint p(B-A) ≤ L. If q* and p* are solutions to these problems then, by the duality theorem of linear programming, we have p*M = Lq*. The exchange value of M is therefore equal to the socially necessary labour time associated with it. This provides a logical justification for why the LTV should hold.