Groucho Marxism

In Marxian economics, the ‘transformation problem’ refers to the problem of finding a general rule by which to transform the values of commodities – based on their socially necessary labour content, according to Marx’s labour theory of value – into the prices of commodities seen in the marketplace. Here I will provide a mathematical treatment of the problem based on a 1974 paper by the Japanese Marxist economist Micho Miroshima. Let (A,L) be a Leontief economy where A ≥ 0 is the mxm commodity input matrix and L ≥ 0 is the 1xm labour input row vector. The labour value vector for such an economy is the 1xm row vector v satisfying v = vA+L. The ith component of this vector represents the socially necessary labour contained in 1 unit of commodity i.

Given scalar profit and wage rages r,w ≥ 0, the equilibrium price vector is the 1xm row vector p* satisfying p* = (1+r)(p*A+wL). The ith component of this vector represents the equilibrium price of 1 unit of commodity i. It is assumed that wages are set at subsistence level, so that if the price vector 1xm price vector is p then w = pD for some mx1 column vector D ≥ 0. The ith component of this vector represents the quantity of commodity i required to keep one labourer working per unit time. The equilibrium price equation can then be written as p* = (1+r)p*(A+DL); setting M = A+DL, this can be written as p* = (1+r)p*M. Converting this into a recursive equation gives us a model of price dynamics: p’ = (1+r)pM, where p and p’ denote the price vector at the current and next time step respectively.

The product vD represents the socially necessary labour contained in, or value of, the column vector D. Fix an mx1 output row vector q. Then the scalar vAq is interpreted as the value of constant capital C and the scalar vDLq is interpreted as the value of variable capital V. Since the total value of output is given by vq, the surplus value is given by s = vq-C-V = vq-vAq-vDLq; substituting in the expression for the value vector v above gives S = (I-vD)Lq, where I is the mxm identity matrix. The rate of exploitation is therefore given by e = S/V = (I-vD)Lq/vDLq; and dividing through by Lq (a scalar) gives: e = (1-vD)/vD. Now let q* be the output vector corresponding to the equilibrium price vector p*, so that q* = (1+r)Mq*. Then it can be shown using a bit of algebra that r = evDLq*/vMq*.

The formula r = evDLq*/vMq* transforms the vector of values, v, into a rate of profit, r. On the other hand, the formula p’ = (1+r)pM transforms the rate of profit, r, into a vector of prices, p’. These two formulae can therefore be seen as a solution to the transformation problem. Starting from an initial price vector, the second formula will generate a sequence of price vectors which, under certain conditions, will converge to the equilibrium price vector p*. However, this is incomplete as an algorithm as it assumes that the commodity vector q* is known. To get around this, Miroshima suggests replacing q* with the iteration q’ = vqMq/vMq, which he then demonstrates converges to q* under fairly weak assumptions.

Miroshima proposes the following as a complete algorithm to get from values to prices: (1) calculate sequence of commodity vectors according to the recursion q’ = vqMq/vMq until a stationary solution q* is obtained; (2) calculate r = evDLq*/vMq*; (3) calculate the sequence of price vectors according to the recursion p’ = (1+r)pM until a stationary solution p* is obtained. He then proves that, provided the price sequence begins from the value vector v, the aggregate output in terms of prices p*q* is equal to the aggregate output in terms of values, vq*, and that the aggregate profit, P* = p*(I-M)q*, is equal to the aggregate surplus value, S* = v(I-M)q*. These conclusions are close to, although not identical with, conclusions Marx reached in Das Capital.