[OPE-L:1004] Axioms and models

Allin Cottrell (cottrell@wfu.edu)
Thu, 8 Feb 1996 07:30:22 -0800

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I feel we _may_ be getting somewhere, in understanding each other's positions
if not in moving towards agreement. I'd like to build on this if possible.

I said, in response to Alan's claim that the definition v=vA+L "applies only
to the special case of both sale and purchase at values"

I'm now re-confused on something I thought we had agreed. The
_definition_ of value via the equation given above (whether or not it
represents what Marx meant) is surely independent of the issue of whether
sale and purchase take place at prices corresponding to values...

Alan has now offered a lengthy reply. He refers back to a previous
agreement, namely

Marx's derivation of the category of value and its magnitude is
independent of the assumption of price-value equivalence.

and says he will "try to show that you don't need the equation v = vA + L in
order to sustain our agreement."

My first reaction here is that Alan has mistaken my query. I can readily
accept that we don't _need_ this equation to sustain our agreement. The
agreement was sufficiently general as not to imply that equation. My concern
was that Alan seemed to be saying that the agreement _ruled out_ this
equation -- and my claim is that the agreement and the equation are perfectly
consistent. Perhaps I mistook Alan on this.

Anyway, Alan now says:

My interpretation - and, I think, Marx's - of agreement 1 is as
follows: with each commodity is associated a unique magnitude, its
value, which is defined independent of its price and is equal to
the labour time socially necessary to produce it.

Fine, I can accept that -- though for reference below I'd like to underline
the last clause, "and is equal to the labour time socially necessary to
produce it".

Alan continues by saying that

Nothing in this definition says that socially necessary
labour time is given by the equation v = vA+L.

Again, I agree. There may be alternative formulations. But things seem to
me to be slipping in Alan's next formulation:

The definition of value only says that it is equal to
a magnitude independent of price ... and that the
substance, and measure, of this magnitude, is socially
necessary abstract labour time.

Similar to the previous version, but lacking the definiteness of stating that
the value of a commodity is "the labour time socially necessary to produce

We then get a third version, labelled Axiom 1:

associated with each quantity of use-value is a unique
magnitude, its value; the dimension of this magnitude is
time and it is a linear function of use-value.

Here we are definitely on a slide. Now all we get is that the dimension of
value is time (actually, it should be person-time), plus the odd-looking
statement about its linearity as a function of use value (this would seem to
rest on the assumption not only of linearity in production, but that
use-value is itself a linear function of physical quantity -- but that's a
side issue).

Alan goes on to give four models for axiom 1:

v(t) = v(t)A(t) + L(t) (1),

v(t+1) = v(t)A(t) + L(t) (2),

the system

v = pA + L (3a)
p = (pA+L)(1+r) (3b)

and (his favoured version)

v(t+1)=p(t)A(t) + L (t) (4).

As Alan says, "each such equation allows us to define a magnitude, x labour
hours, which is associated with y use-values. Each such equation satisfies
axiom 1." He adds

I suspect that this will not satisfy many people. The central
point is therefore the following: chapter 1 of Volume I *only*
says that the magnitude of value is a quantity of labour time.
It does *not* say how this magnitude is determined.

OK, so now where do we stand? I'm afraid his suspicion is right: I'm not
satisfied. If we take as axiomatic the _first_ formulation above:

"with each commodity is associated a unique magnitude, its
value, which is defined independent of its price *and is equal to
the labour time socially necessary to produce it*" --

then, I claim, only equations (1) and (2) will serve as models (with the
labour-time socially necessary to produce the commodity taken on a current
or a historic basis respectively). The magnitude of value is not just any
old quantity of time. Note that if Alan's minimal "axiom 1" is our basis,
then the following would do equally well as a model:

v = dA + L

where d is an n-vector holding the times it takes to walk from the
British Museum to the n nearest pubs.