[OPE-L:7120] [OPE-L:622] Relational properties of exchange with money [OPE-L 608]

Alan Freeman (a.freeman@greenwich.ac.uk)
Mon, 08 Mar 1999 08:55:55 +0000

Mainly a response to Brendan but also dealing with one point of Ajit's ("do
you have a theory of price"?)

I'm very rushed but I think I'm quite happy with Brendan's axiomatisation in
which money is assumed from the outset. In particular I'm very pleased to see
that the relations he defines apply for any possible set of prices, which I
regards as the decisive issue, because my point all along is that what Marx
means by 'equality' is merely the fact of belonging to the same
equivalence-class, of being mutually-exchangeable, and that he does *not*
claim that goods which are mutually exchangeable must have the same value or
the same amount of abstract labour in them.

If we can agree on an axiomatisation that establishes this point, we can then
go on to study the controversial aspects of Marx's chapter 1 and chapter 5
argument in the light of this axiomatisation, namely:

(a) the chapter 1 argument: does exchangeability demand a 'third property';
some quantifiable property that exchangeable objects possess, in terms of
which we may explain the fact that they are exchangeable. Is it the case, if
Brendan's relations (or mine) hold, that we may 'explain' this in terms of a
predicate of commodities -- Marx's "third" property-- that does not depend on
the precise proportions in which goods exchange; and is it the case that we
are obliged to do so, or can we dispense with this explanatory 'accessory' and
cast the argument without reference to any independent predicate of objects
that find themselves in the same equivalence class? Moreover, if we choose to
do things this second way, what explanatory power do we lose as a result?

Having thought about this some more, and in particular having revisited and
restudied some of the key issues in foundational mathematics, I am more than
ever convinced that we are at least philosophically obliged to conclude that
such a predicate must exist, even if we accept Frege's redefinition of
predicates in terms of sets, since for Frege (and the 'method of abstraction
by definition') every class actually corresponds to a predicate.

The complexity of the issue arises precisely because the ratios in which goods
exchange does *not depend* on this third predicate. That is, the equivalence
classes defined by the predicate 'value' are not the same as the equivalence
classes defined by the predicate 'price'.

Attempts to visualise the relation as, for example, something like a balance,
in which the weights on both sides of the pan are actually equal, as opposed
to having a distinct quantitative relation to each other which nevertheless
regulates a qualitatively single substance, are misconceived. A better analogy
might be the relations entered into by massive objects exercising a
gravitational force on each other. These would form pairwise equivalence
classes based on exercising the same force, for example, but the same force
can be exercised by many different pairs of masses. The category of mass is no
less necessary to explain gravitational interaction.

The sense in which this third predicate provides an 'explanation' of prices is
therefore missed if we read Marx as saying that the *magnitude* of prices is
determined by their values. I believe he says something entirely different;
namely, we may *represent* the ratios in which they exchange as a ratio
between their values. We may establish a correspondence between the ratios in
which goods exchange, and the ratios between the values on each side of an
exchange, such that wemay reduce all prices, all ratios between heterogenous
and disparate use-values, to ratios between a single predicate of each
commodity that is quantitatively and qualitatively defined independent of, and
prior to, the exchange-relations into which they enter.

(b) the chapter 5 argument: is it the case, again if Brendan's axioms hold (or
mine), and if we define the predicate referred to in point (a) immediate
above, under what circumstances will this predicate, when quantified, yield a
'substance' that is conserved in exchange at arbitrary prices? My belief, of
which I think I can construct a proof, is that the predicate must itself be a
linear function of use-values, if it is to be conserved in exchange. The
predicate is thus a function u(a) of a use-value (a) with a 'distributive'
property: v(a U b) = v(a) + v(b).

Approximately speaking, then, the 'theorem' of exchange that appears as Marx's
first and second equalities can then be translated as follows: the following
three propositions about a value relation are equivalent:

(i) value is a linear/distributive function of use-value;
(ii)value is conserved in exchange
(iii)any relative price may be represented as a ratio between the values of
the baskets on each side of an exchange-relation between baskets that can be
found in the same exchange-class. In particular we may in this way define
'unit prices' in value terms for any given use-value, or basket of use-values
of constant proportions. We may thus 'explain' price in terms of value.

In consequence I think any function of use-value satisfying the above axiom
will yield the following:

(i) u is conserved in any complete system of exchange relations, that is, any
set of exchanges in which the basket of use-values is the same on both sides
of the equation;

(ii)we may establish a correspondence between sets of relative prices and
transfers of u; any vector of prices p induces a set of transfers of value
between agents or, dually, between stocks of use-values, such that these
transfers sum to zerp.

In brief reply to Ajit, then, a 'theory' of prices is the above
correspondence; it is an explanation for prices in terms of a redistribution
of a conserved magnitude, value.

I believe, though I would not claim this categorically, that it can be proven
that such an explanation can be constructed, and only constructed, on the
basis of the above axioms.

I'd like time (which I don't have!) to inspect Brendan's axiomatisation and
see if we can establish that it is equivalent, or equivalent under certain
well-defined conditions, to my own axiomatisation, for which we would have to
prove the following thesis:

Let Brendan's axiom-system be B
Let Alan's axiom-system be A

Theorem: For any possible system of commodities obeying B, A must hold;
conversely for any possible system of commodities obeying A, B must hold.

If this is the case the two systems are formally equivalent. However, I think
it is useful to begin without supposing money. There are two reasons for this;
analytically, I don't like to begin by supposing something whose properties
one seeks to derive. I think this was Marx's reasoning and I think his
correspondence, writings and remarks on this question establish that he
considered important to *derive* money from the commodity and not to
presuppose it

The second reason is a practical one; there is not, in most markets, a single
money. There are at least three competing world-currencies, setting aside the
many national monies, and each of these monies has many different forms. The
category of 'money', supposed as if there was only one, is therefore a bit

But in general I think, from a quick reading, that I'm in solid agreement with
the whole drift of Brendan's