The idea of exchangeability as an equivalence relation applies not at
the level of individual exchanges, but at the level of a
socially-determined system of exchangeability. It's the basis of the
notion of a SINGLE price-structure, independent of the numeraire... and
that's why it's an abstraction that Marx had to use in a discussion of
what is conserved in exchange.
The transitivity of exchangeability depends on abstracting from
arbitrage, it implies a consistency in the system of exchange ratios, so
that, whatever the choice of numeraire, there is no way to increase the
stocks of a given commodity through a series of exchanges. (In practice,
of course, this process of arbitrage from which we are abstracting does
occur, having as one of its results a tendency to reduce inconsistencies
in the system of exchange ratios.)
This abstraction is necessary to be able to speak of a single price for
a commodity, otherwise, depending on which commodity was chosen to
express prices, the relative prices of different commodities would
differ. This is not consistent with the approach that Marx took, in
seeking a conserved substance to explain "the" exchange ratios.
Steve>
> Consider the following example: 2 apples "equal" 3 oranges.
>
> An immediate question poses itself: In what sense are the 2 apples
> "equal"
> to the 3 oranges. They are qualitatively distinct commodity bundles,
> so to
> pose the question of equality means that we must find a common
> denominator
> in which to express the equality, whcih would enable us to say that
> they
> are equal in therms of this common denominator.
>
Brendan>
I don't think this follows. If the particular exchange of commodities is
embedded in a system of exchangeability which is an equivalence
relation, then that's all they need to be "equal". So if 2 apples are
exchangeable for 3 oranges, AND 3 oranges are exchangeable for 2 apples,
AND 2 apples are exchangeable for 2 apples, AND 3 oranges are
exchangeable for 3 oranges AND 3 oranges are exchangeable for something
else if and only if that something is exchangeable for 2 apples... then
3 oranges and 2 apples are equal with respect to exchangeability.
Nothing is required for equality (or implied in scalability) outside of
the relation itself.
In particular, an equivalence relation does not imply the existence of a
scalar quantity associated with each equivalence class.
Consider this experiment. I take an arbitrary collection of household
objects, and 6 boxes labeled 1 to 6. Now I go through the collection,
and for every element I toss a die and put the object into the box that
matches the number I have thrown. Now consider the relation "went into
the same box as" between members of my collection. A little thought will
show that it's an equivalence relation. A potato peeler and a cat are
"equal" with respect to the relation if they went into box number 2, but
there is no "common denominator" that links the objects in a box, and
the numbers of the boxes, into a system. There is no "common substance"
to the equivalence outside the relation itself. The numbers on the boxes
are just labels.
That's why Paul's argument starts with exchangeability as an equivalence
relation but doesn't stop there. It isn't until he considers the metric
within the space defined by the exchangeability relation, that notions
of "value" as a scalar have a place. And it's only after that that a
putative "substance" of value could have a place in his argument, it
seems to me.
Steve>
> In contrast to the route of thinking about exchange as a relation of
> equality, I have been suggesting that in the first instance we think
> about exchange as a transference of property rights. That is when two
> apples exchange for 3 oranges, we represent that exchange as: 2 apples
> <------> 3 oranges, where <------> represents the exchange relation.
> This
> is not an equivalence relation, because as Gil and Duncan have pointed
> out,
> apples do not exchange for apples and thus relexivity is violated.
> Thus,
> in particular, this exchange relation is not a relation of equality.
>
Brendan>
I don't accept that exchangeability is not reflexive. All the objections
to it amount to a claim that exchanging equal amounts of the same
commodity would be senseless. But this is not the case.
When are two commodities "the same"? The "sameness" of two concrete
instances of a commodity is a socially-constructed abstraction. No two
objects are identical, and any differences could be enough to motivate
an exchange, even though the two objects are socially constructed as the
"same" commodity.
My brother and I go into the shop and buy a jersey each, taken from the
same pile, paying the same price. I get a green one, he gets a grey one.
But when we get back to the office I decide I like the grey one and he
decides he likes the green one. We exchange jerseys; at what ratio? :-)
I gave another example earlier today, of gold exchanging for gold. My
brother (who actually exists outside of the above example) suggested
this... and he reckons it's taken from Karl Marx. :-)
But I think there's a stronger case to be made for reflexivity based on
a distinction between exchangeability and exchange in a narrow sense.
Someone (I think it was Ajit) gave an example of three commodities, of
which A regularly exchanged for B, and B for C but A was not exchanged
for C. This was because the actual industrial structure did not require
the transfer of those particular use-values between those particular
producers.
This was supposed to show the intransitivity of exchange... and it does
if exchange is considered as the actual set of exchanges that concretely
take place in a given time. But the system of exchange, exchangeability,
is more than this, it's a system of social relationships mediated by
objects.
Suppose that as a result of a technological change, industry C does need
to get commodity A. Can they do so? Of course they can... because
there's a system of exchangeability that allows any commodity to
exchange for any other. Someone at firm C gets on the phone to firm A,
and an order is placed with no fuss. The exchange ratio A/C won't be too
far off (A/B)/(C/B).
This broader notion of exchange validates reflexivity in itself. It is
not required that A ever actually exchanges for A (although I believe it
does), so long as it is true that IF someone ever did negotiate such a
deal the socially-constructed relation of exchangeability would produce
an expectation that the identical commodities would exchange at one for
one. That is, the opportunity to exchange the same commodity at par
exists, and at any other ratio does not exist.
Steve>
> Nor, in general does exchange satisfy transitivity. If we had full
> arbitrage of course, then exchange would satisfy transitivity.
>
Brendan>
Not once you've rejected reflexivity. A symmetric relation that is not
reflexive is not transitive.
Well, that's it... that was fun. I'm late leaving work now.
Cheers all,
Brendan