It will not have escaped readers notice that there is a `phsyicalist' flavour to our article. Mirowski (1989) has recently had a good deal of innocent fun with the propensity of economists to emulate the queen of the sciences. His satire is directed mainly against the utility theorists, but he does devote some attention to Marx, accusing him of oscillating between a field and a substance theory of value, and in the transformation problem of having `one conservation principle too many'.

We think the last accusation is valid, but contrary to the Sraffians, we think that on both empirical and theoretical[Farjoun and Machover, 1983] grounds it is the equalisation of the rate of profit that must go. His accusation with regard to the contradiction between field and substance theories is relevant to our formulation however, since it would appear that we have used a substance definition of value and a field theory for our empirical test. We will attempt to show both that the distinction between field and substance theories is more subtle than Mirowski presents, and that the emprical tests in the literature are not invalidated by this distinction.

By the field version of value theory, Mirowski means the definition of value as socially necessary, as opposed to embodied, labour. He takes as his formal model what has become the standard mathematical account of the determination of labour values by athors like Morishima (1973) or Steadman (1977). But it is a little unfair to project back these 20th century formulations based upon the mathematics of input/output tables, onto Marx. Marx gave no precise mathematical formulation to socially necessary labour. The standard formulation of the I/O table method is only one of possible definitions of socially necessary labour, and involves some very unrealistic assumptions. If these assumptions are dropped and the model made more realistic the distinction between field and substance theories vanishes.

The standard method of deriving labour values from solving the linear I/O equations is based upon the unrealistic assumption that production takes no time to occur. Marx did not assume this, and devoted much of Capital II to analysing the turnover times of capital. Any process of determination of prices must operate in time through actual production processes. It is `socially necessary' that the steel used in the keel a ship completed today was produced a year or two earlier. The socially necessary labour in steel produced today may differ from that which went into the steel a year ago, but only the former can affect the value of the ship. No real process allows instantaneous information transfer and market economies are no exception to this rule. If one were to look for a physical analogy, applying the Morishima equations under technological change would be to try to solve an electrodynamic problem with electrostatics.

The value of the ship will be affected by the value of steel when it was purchased ( assuming it was not purchased unnecessarily early ). It will also indirectly be affected by the value of steel at a still earlier period when steel was purchased to make the tools used to build the ship. Generalising the v_j value of commodity j, is affected by the v_i the value of commodity i at a series of times in the past. These affects will be mediated by coupling coefficients k₁, k₂,... corresponding to what fraction of the v_j is made up of v_is at times t-1, t-2,.... Thus if by v_ji we mean the component of v_j determined by v_i we have a difference equation of the form:

v_{j_ti} = k₁ v_{i_t-1} +k₂ v_{i_t-2}+ .....

Expressed in continuous terms this will obviously give us a differential equation of the form

[(dv_j)/( dv_i)] = k₁ v_i - k₂ [(dv_i)/ dt] +k₃ [(d²v_i)/( dt²)] - ..., which gives us a highly non-linear field theory. There is no contradiction between this and the substance theory.

Are we then justified in using what are basically the Morishima equations, i.e., the wrong field equations, in our verification of a substance theory of value?

Yes, because for the k the constants 1 >> k₁ >> k₂ >> k₃... and similarly, k_2..n will all be very much smaller than k₁. One can get a feel for what their likely scale is by looking at the leontief inverse of the i/o table. Even when we take two highly cross-linked industries like steel and shipbuilding, we find 0.07 > å_i k_i. We can thus assume that although there will be some errors in our estimation of values due to using linear field equations, these will be small relative to noise. As a conservation law, the law of value is stochastic and obviously does not hold to the same precision as natural conservation laws, though it should be noted that on an appropriate scale these too are stochastic.

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