This includes the many people, for example, who supposes Volume I assumes
exchange at values. It also includes all those who argue that Marx is
attempting to infer equality from equivalence in chapter V. He isn't trying
to do that, and so there's really very little mileage in endless
discussions on his failure to do it.
My point is not however entirely irrelevant to other points which Gil has
not responded to, which I'll call 'non-non-sequiturs'
Point 1: the law of one price
=============================
I fortunately found a precise reference in [OPE-L 746 of 1996] so it's
obvious the issue is an ongoing one. Brendan also introduces this. Gil
writes:
>Referring to 'the' exchange value of a quarter of wheat begs a
>serious question. Unless the "law of one price" holds, i.e.
>something like perfectly competitive markets exist, it is meaningless
>to speak of "the" exchange value of a quarter of wheat, or any other
>good. It may simultaneously have multiple exchange values conditional on the
>idiosyncratic transaction conditions faced by the various possible
>pairs of exchangers. For example, a transaction of the silk owner
>with some other owner of a quarter of wheat may yield an entirely
>different exchange ratio."
My point is straightforward and Gil hasn't answered it: the 'law of one
price' holds whenever we are studying commodities, because that's what a
commodity is; something with one price. If Gil wants to drop this
assumption, then he isn't studying a commodity economy, or he is at best
re-defining commodity in a very different way to Marx and, I think it's
fair to say, most of economics.
If he wants to say that a more general enquiry is needed into non-commodity
exchange, which I think on the whole is his general plan, then who can
disagree? I support this enquiry, which leads to interesting results.
However the criticism of Marx is in turn a non-sequitur, because Marx's
plan was a different one. You can't really criticise Marx for not doing
what you want to do; all you can say is that you are trying to do different
things.
Point 2: an exchange-relation is more than just an equivalence relation
=======================================================================
The second point which Gil doesn't reply to has cropped up in all these
exchanges and I think it's time to address it frontally. In my view, I
repeat, some very sloppy math enters some of these discussions. An
exchange-relation is *not* just an equivalence relation. It's a *linear*
equivalence relation. Straight away, this rules out most of the
counter-examples that Gil and also Brendan provide.
To repeat the point in my last post, here's what Gil says again:
>As a counter example, consider a preference ordering R. A preference
>relationship among bundles which satisfies reflexivity, transitivity, and
>symmetry establishes a relationship of indifference, not equality. To say
>that I am indifferent between two bundles in no way implies the two bundles
>are equal in the sense required by Marx.
Here's what I replied, and Gil hasn't answered:
But if I am indifferent between bundle A and bundle B, this does not imply
that I am indifferent between two As and two Bs. If A exchanges with B,
then two As exchange for two Bs; this is why an exchange relation is
not the same as an indifference relation. Don't forget that; it makes all
the difference. We have more than just
(1) aRb->bRc
(2) aRa
(3) aRb -> bRa
we *also* have
(4) aRb -> (ka)R(kb) where k is any scalar.
More prosaically we can explain the same point with reference to Brendan's
example of a grey sweater swapping for a green sweater. The point is that
we are not entitled to deduce, because a grey sweater can be swapped by
brothers for a green sweater, that the same brothers (or anyone else in
society, don't forget) would exchange two grey sweaters for two green
sweaters. If one brother is a risk-avoider and worries about losing
sweaters, and the other brother doesn't like to clog up his wardrobe, they
wouldn't swap two for two, even though they would swap one for one.
Moreover you can't go to a shop and say 'You must sell me two sweaters for
the price of one-and-a-half, because I'm not over-fond of grey sweaters.'
Postulate (4) is an independent postulate. It follows from the
commodity-relation, not from equivalence.
The entire discussion about equivalence classes and magnitude amounts to
this: (1), (2) and (3) don't imply (4). Sure they don't, but Marx never
says they do. Marx says, since we're discussing in a commodity economy, all
four rules apply, and that's what I, Marx, am analysing, as I said on page
one of my work.
My reading is that Marx deduces from the fact that a linear function of
use-value is conserved in private exchange, that it is also conserved in
the totality of exchanges among all members of society. I don't think
that's trivial. If it was, more people would have recognised the argument.
A postscript on tautologies
===========================
Gil hasn't dropped a position which I think militates against clear
discussion: he says either Marx is tautological, or he is wrong. I
challenged him at the time and he came off the fence; he seems to have
gotten back on it. Gil, you have to say which. Either he is tautological,
or he is false. He can't be both, and if no-one can reply to you adequately
unless you say which, or at least deal with one at a time.
Moreover, I think the accusation of tautology is in any case a very dubious
and rather sloppy one. In logic, every valid deduction is a tautology. It's
just another word for valid deduction. So therefore, Goedel's theorem,
Church's theorem, the proof of Fermat's last theorem, etc... are all 'mere
tautologies'.
I suspect what Gil means is that it is obvious or trivial. This reminds me
of a true story about G.H.Hardy, the famous mathematician who taught at UCL
where I did math. One day, Hardy was in the middle of a proof. He wrote a
line of deduction on the board and underneath wrote the justification,
'obvious'. Then he stopped and thought. He turned to his class and said 'is
it obvious?'. No-one replied. Then he went to his chambers for three hours,
during which time the very polite class of Edwardian toffs sat and fiddled.
Then he came back to his class, said 'yes, it is obvious', and carried on
the proof.
Alan