Groucho Marxism

A central question of Marx’s Das Capital is: why is capitalism profitable and productive? Marx’s answer to this question boils down to: because capitalists exploit workers. But why does worker exploitation result in positive profits and positive growth? To tackle this problem scientifically, Marx had to go it alone as much of the mathematical apparatus economists use today had not yet been invented. He first used the classical labour theory of value to calculate the value or the labour-time directly or indirectly necessary to produce a unit of each commodity. He then divided the total supply of labour by a worker, T, into a paid part T* and an unpaid part T-T*, both measured in terms of labour time; and he defined the ‘rate of exploitation’ by (T-T*)/T*.

Using this definition, Marx established a theorem to the effect that the equilibrium profit rate and the equilibrium growth rate are positive if and only if the rate of exploitation is positive. It is important to note that his proof relied on the labour theory of value. As soon as durable capital goods, joint production, and choice of techniques are admitted, we must discard the labour theory of value, at least in the way Marx formulated it. This raises the question of whether Marx’s theorem still holds in under these more general assumptions. In 1974, the Japanese Marxist economist Michio Miroshima tackled this question and found the answer to be ‘yes’ in the case where we allow joint production, whereby several outputs from a single activity can emerge together.

Consider an economy which produces m types of commodities using n processes, where in general m is not equal to n. Assume that the economy employs N workers, where each workers works on average T hours per day and is paid wages at a subsistence level. Let us denote the mx1 subsistence-consumption column vector (per worker) by C, so that N units of C are required to keep the N workers alive for one day. We can then define the ‘necessary labour time’ for the economy as the minimum labour time necessary to produce consumption goods, CN, and the ‘surplus labour’ as the total labour time per day, TN, minus the necessary labour time. This definition of necessary labour was introduced by Miroshima in his 1974 paper.

In each process j, quantities A_ij ≥ 0 are used up and quantities B_ij ≥ 0 are produced of commodity type i, and a quantity L_j of labour time is also used. Let A and B denote the mxn matrices with elements A_ij and B_ij, and let L denote the 1xn row vector with elements L_j. Each commodity i will have an associated price p_i, and each process j will be used with an ‘intensity’ q_j. Given a 1xm price vector p = (p_i), the vector pA represents the total cost of operating the different processes and the vector pB represents the total revenue obtained from the different processes. Similarly, given an nx1 intensity vector q = (q_j), the vector Aq represents the total commodities of different types used up, the vector Bq represents the total commodities of different types produced, and the scalar Lq represents the total labour time used.

Under Miroshima’s definition, the necessary labour time required to produce consumption goods C is given by the minimum value of Lq subject to the constraints q ≥ 0 and Bq ≥ Aq+ CN. This is referred to as a ‘linear programming’ problem, as it involves minimizing a linear function subject to linear constraints. Let q* be a solution to this problem, so that the necessary labour time is given by Lq*. Then the surplus labour is given by TN-Lq* and the rate of exploitation by (TN-Lq*)/Lq*. Next, consider the ‘dual’ linear programming problem of finding the maximum of vCN subject to the constraints v ≥ 0 and vB ≤ vA+L. Let p* be a solution to this problem. Then by the so-called ‘duality theorem’ of linear programming, we have p*CN = Lq*, and the rate of exploitation is therefore given by (T-p*C)/p*C.

Let w ≥ 0 be the wage rate. As wages are assumed to be set at subsistence level, we have wT = pC, therefore w = pD where D = C/T. Let r_j be the rate of profit for process j. The 1xm price vector p satisfies (pB)_j = (1+r_j)p(A+DL)_j for each j. Letting r* = max r_j, we must then have pB ≤ (1+r*)p(A+DL). This inequality says that the revenue at the next time step – i.e. pB – cannot be more than was spent at the previous time step multiplied by 1 + the maximum profit rate – i.e. (1+r*)p(A+DL). Now let g_i be the growth rate of commodities of type i. The nx1 intensity vector satisfies (Bq)_i = (1+g_i)(Aq+DL)_i for each i. Letting g* = min g_i, we must then have Bq ≥ (1+g*)(pA+DL). This inequality says that what the economy to consumes at the next time step – i.e. (1+g*)(pA+DL) – cannot be more than was produced at the previous time step – i.e. Bq.

The profit rate that is guaranteed in this economy is given by the minimum value of r* satisfying the inequality pB ≤ (1+r*)p(A+DL) for some p > 0. We may refer to this minimum value as the ‘warranted profit rate’. Similarly, the growth rate that is guaranteed by this economy is the maximum value of g* satisfying the inequality Bq ≥ (1+g*)(pA+DL) for some q > 0. We may refer to this minimum value as the ‘warranted growth rate’. Miroshima proved that under some relatively weak assumptions, the following statements are equivalent: (1) the warranted profit and warranted growth rate are greater than zero; (2) the rate of exploitation is greater than zero. This result is referred to as the ‘Generalized Fundamental Marxian Theorem’.

The theorem states that the propositions (1) the economy is profitable and productive, and (2) capitalists exploit workers, are equivalent. It demonstrates that Marx’s key insight in Das Capital – that capitalism is profitable and productive because capitalists exploit workers – holds in more general economies with joint production.