Groucho Marxism

Consider an economy with m sectors that each specialize in the production of one commodity type. The economy is determined by an m×m input-output matrix of inter-sector coefficients A = (a_ij), where a_ij ≥ 0 is the quantity of commodity i required to produce one unit of commodity j; a 1xm vector of direct labour coefficients L = (L_i), where L_i > 0 is the quantity of labour directly required to output 1 unit of commodity i; and an mx1 subsistence consumption vector b = (b_i), where b_i ≥ 0 is the quantity of commodity i required to sustain a worker unit time. The price vector associated with a profit rate Ris the 1xm vector determined by: p = (pA+wL)(1+R), where w = pb represents the subsistence wage.

Rearranging the above equation gives: p[I-A(1+R)] = wL(1+R). Assuming that the matrix I-A(1+R) is invertible, we can write: p = wL(1+R)[I-A(1+R)]^-1. Right multiplying by b and using the fact that w = pb gives: wL(1+R)[I-A(1+R)]^-1b = w; dividing by w gives: L(1+R)[I-A(1+R)]^-1b = 1. In 1973, the Japanese economist Michio Morishima showed that the profit rate R determined by this above equation is positive if and only if L[I-A]^-1b < 1. This result is referred to as the Fundamental Marxian Theorem (FMT). To understand what this theorem tells us, we need to make a couple more definitions. Let T denote the number of time units in a working day, and let B = Tb. Then B_i represents the quantity of commodity i required to sustain a worker for one working day.

The equation determining the profit rate r can then be written as: L(1+R)[I-A(1+R)]^-1B = T. This gives a relationship between the profit rate R and the length of the working day T; it is clear from inspection that as R goes up, T goes up, and vice-versa. Thus, one way capitalists can increase their profits is by extending the length of the working day. According to the FMT, R is greater than zero if and only if L[I-A]^-1B < T. The theorem therefore gives a condition on the length of the working day that must hold in order for profits to be positive. This is the most neutral form of the theorem as it involves no concepts such as ‘value’ or ‘exploitation’. However, it raises the question of what the critical value T* = L[I-A]^-1B actually stands for.

One way to give this critical value meaning is to appeal to the labour theory of value. To see this, note that the vector L[I-A]^-1 is the solution to the equation V = VA+L. If V is the solution to this equation then V_i represents the labour time required to produce 1 unit of commodity i. Therefore, T* = VB = L[I-A]^-1B represents the total labour time required to sustain a worker for one working day. In our derivation we have assumed that the profit rate R, wage w, and length of working day T are the same across all sectors of the economy. The implicit assumption is that in a competitive economy, these values will be equalized across sectors through competitive arbitrage. For example, if the profit rate was higher in one sector than others, this would attract capital to that sector which would drive the profit rate down.

The value of variable capital required to produce the ith element of the subsistence vector b is given by the equation: v_i = L_iVb = L_iVB/T; and the associated surplus value is given by: s_i = L_i-v_i. Hence, the rate of surplus value is given by: r = s_i/v_i = (I-Vb)/Vb = (T-VB)/VB. As T* = VB, it is clear that r is positive if and only if T > T*. Thus, by the FMT, the rate of profit is positive if and only if the rate of surplus value is positive, and vice-versa. The FMT therefore also provides a link between the rate of profit and rate of surplus value, two key concepts in Marxian economics. Note also that this formulation makes clear the distinction between prices, which are determined by p = p(A+bL)(1+R), and values, which are determined by V = VA+L. In general, these will not be equal.

It should be noted, however, that our formulation is unrealistic for several reasons. First, it does not allow for joint production, whereby several outputs from single productivity necessarily emerge together. Second, it does not allow for the existence of durable capital goods, instead implicitly assuming that all capital goods are used up in production (such goods are often referred to as ‘circulating capital’). And third, it does not allow for capitalists to choose different production techniques. It can be shown that the FMT still holds after relaxing these conditions, but this comes at the price of having to discard the notion of labour values. This suggests that it is the FMT rather than the labour theory of value which forms the true core of Marx’s economic theory.