Thanks very much for your responses. I took a quick look at Debreu's
book that you cited, and it was helpful. This is the way I understand it
now:
In Chapter 4 ("Consumers"), Debreu first defines an "indifference"
relation as a preference relation that it reflexive, symmetrical, and
transitive, and defines the "indifference class" of X as the set of all
consumption bundles for which are consumer is indifferent to X. Then
Debreu shows that preferences can be represented by utility functions, in
which different levels of utility are represented by real numbers. In this
representation, commodity bundles within a given indifference class are
assigned THE SAME NUMBER. This means that the utilities derived
from these different bundles ARE EQUAL. This relation of EQUAL
utilities necessarily follows from a representation of the preference relation
of INDIFFERENCE in terms of utilities
Now perhaps one could say that preferences don't have to be represented
~by utility functions. I would say OK, but then a lot of neoclassical
economics goes out the window (including utility maximization).
Whenever preferences are represented as utility functions, then
INDIFFERENCE is necessarily represented by EQUAL utilities.
If preferences are NOT represented by real numbers, then perhaps one
could say (as Steve does) that indifference per se is a relation that satisfies
RST but is not a relation of equality.
However, whenever one is dealing with relations of real numbers (as in
utility functions), the satisfaction of the conditions of RST necessarily
implies relations of equality. That is what Russell (whom Chris cites) was
talking about - relations of real numbers that satisfy the conditions of RST
and therefore are equals.
Finally, Marx's argument in Section 1 is about real numbers, the exchange-
values of commodities (1 quarter of wheat, x boot-polish, etc). Marx
assumes that these real numbers, these exchange-values of commodities,
satisfy the conditions of RST (i.e. that the exchange of commodities is
mutually consistent). Hence, Marx concludes, and rightly so, that these
exchange-values must necessarily be EQUAL to one another.
Therefore, x boot-polish, y silk, z gold, etc. must, as exchange
values, be mutually replaceable or of identical magnitude.
(C.I. 127)
I look forward to further discussion.
Fred