# [OPE-L:41] [OPE-L:270] Re: RE: Re: Re: Chapter 1

Stephen Cullenberg (Stephen.Cullenberg@ucr.edu)
Sat, 31 Oct 1998 12:18:16 -0800 (PST)

>The important thing about equivalence relations is the partition they
>produce. Consider the relation "has the same price as" for pairs of
>bundles of commodities. This relation partitions the set of bundles of
>commodities into a set of discrete classes. Each bundle is a member of
>one and only one class. All the elements of each equivalence class are
>related to each other element in their class, and to no element of a
>different class.
>
>The reason to "put that structure on exchange" is to allow for the
>notion of a single price for a commodity. A single price means that for
>a particular amount of a commodity, there is exactly one sum of money
>for which it is exchangeable. This price is the particular amount of
>money which is a member of the equivalence class (with respect to the
>equivalence relation of exchangeability) of the bundles of commodities
>which it prices.
>
>This is an abstraction, true, (from e.g. arbitrage), but a sensible one
>all the same.
>
>Cheers
>
>Brendan

Brendan,

I would like to differentiate between a "price", which I take to be an
exchange ratio, or an expression of terms of trade, and a "value", which
links the exchange ratio to something else. Any two commodities which are
exchanged establishes two prices, each in terms of the other. As I have
been arguing, there is no a priori reason to presume that this exchange can
or should be understood as an equality, or as a member of an equivalence
class.

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.

The typical candidates are (a) utility, (b) a money unit of account, say \$,
and (3) labor. Now, to say that 2 apples are equal to 3 oranges in terms
of utility, gives an indifference relation, which makes sense for an
individual's indifference curve, but is no basis for trade between two
individuals. That is, if two apples give me exactly the same amount of
utility as 3 oranges, what could possibly motivate me to trade?

To say that the \$ value of 2 apples = \$ value of 3 oranges, is just to
express the exchange of apples for oranges and both for dollars in a common
unit of account by deciding on the dollar as the numeraire. One could, of
course, just as easily speak of the apple price of dollars and the apple
price of oranges, etc.

Now, with regard to abstract labor, one could of course identify the
abstract labor embodied in the 2 apples or 3 oranges and express the
equality as follows: the amount of abstract labor embodied in 2 apples
equals the amount of abstract labor embodied in 3 oranges. And, if we
define, for a moment, the amount of abstract labor embodied in a commodity
as its value, then we could say that the value of 2 apples equals the value
of three oranges.

I have no problem with this as far as it goes. My concern is why should we
expect the process of exchange to be one of equal amounts of abstract labor
time. That is, do the 2 apples exchange for the 3 oranges "because" they
have equal amounts of abstract labor time contained in them? This is an
interesting theoretical proposition, but one which has very little
robustness as Gil has argued here, and a vast literature has argued
elsewhere. Or, is all that is being claimed is that we can always find
exchange ratios between commodities such that the amount of abstract labor
contained in each commodity is equal. I agree this can done. But, then,
one needs to ask: why set up this equality? What happens if this equality
doesn't hold? Is the equality a "simplfying" one, like assuming linearity,
or is it critical for subsequent qualitiative results. If subsequent
propositions about surplus value, etc., are based on, or derive necessarily
from, such an arbitrary equality, then it seems to me a not very persuasive
basis for a theory of surplus value.

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.

Nor, in general does exchange satisfy transitivity. If we had full
arbitrage of course, then exchange would satisfy transitivity. But we are
no where near such a condition, for all the reasons that Gil has pointed
out, among many others. One only has to consider the complete failure of
PPP theories to realize how incomplete and complicated arbitrage is.

To my mind this is a critically important issue because it affects one's
research focus. If the exchange process need not adhere to the analytical
structure of equality, then the research question becomes how do actual
markets (from labor to stock to various commodity markets) work in all
their particularity, and how do they afect the issues Marxists have always
been concerned with, asymmetries of various kinds, power relations,
ideologies (endogenous preferences in differnt words), class relations and
production organizations, etc.

Steve

***********************************************
Stephen Cullenberg
Department of Economics office: (909) 787-5037, ext. 1573
University of California fax: (909) 787-5685
Riverside, CA 92521 http://www.ucr.edu/CHSS/depts/econ/sc.htm