[OPE-L:7408] Interpretations of the "numeraire"

[Gil Skillman <gskillman@xxxxxxxxxxxxxxxxx>]

> Paulo writes in 7404
>
> > Gary, it seems to me that Marx`s argument is that gold can not have a price
> > because its price would be it own self. By definition price has to be the
> > expression of value in something different from the thing which is being
> > measured.
> > It  seems to me therefore that one can not interpret that phrase as being
> > something equivalent to considering gold as the numeraire.

But one can, as I suggest in 7403:  the "1"  associated with the money
commodity in the augmented price vector can, with no change in the
mathematics whatever, be interpreted as simply a value accounting
placeholder, indicating in effect that a unit of gold is worth a unit of
gold--"its price would be its own self," in Paulo's words.  Gold is the
numeraire in this case in the precise sense (consistent with Marx's usage),
that prices of non-money commodities are expressed in gold.

> > The numeraire is
> > tipically a way of closing a system which presents more variables than
> > equations.

Again, that's one interpretation of what it means to recognize or define a
numeraire good, but not a necessary one.  Another is simply the recognition
that, in a monetized capitalist exchange system, prices of non-money
commodities are expressed in terms of the "numeraire" money
commodity.  Mathematically, there is absolutely no difference.

> > Chosing a numeraire allows for the closure of the system.

It may do that as well (but not necessarily:  Fred argues that merely
defining a numeraire good may still leave a Sraffian system
underdetermined, while in the scenario of capitalist exchange I lay out in
7399, defining the numeraire good is associated with *overdetermining* the
system, once you add Fred's assertion concerning the "prior" determination
of the profit rate), but again, that's not the only possible interpretation
of what it means to recognize or define a numeraire good.

> >  Marx does not seem to be
> > closing any system!
...Yes, perhaps,  giving rise to the possibility that he advanced logically
inconsistent analytical claims, with no means *within* his analytical
system to uncover possible inconsistencies.


Now Rakesh writes in 7407

> Yes indeed Marx attempted to demonstrate why capitalism was not a closed
> but an open system susceptible to historical change by its very contradictions.

"Closed" and "open" are being used in two different ways here.  Paulo was
using the term "closure" in a purely mathematical sense relating the number
of variables to the number of (independent) linear equations; Rakesh seems
here to be using "open" to characterize social relations.

Thus, I see no necessary inconsistency with what Gary or I wrote.  One can
grant that capitalism is a system characterized by *social* contradiction
while still legitimately criticizing *logical* contradictions in someone's
*analysis* of that social system.  The fact that capitalism is a
contradictory social system doesn't grant the license to insist on the
equivalent of "2+2= 7."

> So yes isn't there a big difference between a theory which must select
> (and construct) a numeraire with (as Blaug emphasises) no real world
> significance in order to close a system of equations in order to throw
> into analytical relief a wage/profit frontier and a theory which attempts
> to explain why in the bourgeois mode of production the form of appearance
> of economic magnitudes has to be money which as a result of this function
> carries the seeds of economic contradiction and crisis?

But as I explained in 7403, there is no necessary formal difference between
the two systems on this score.  The money commodity in the latter theory
can be represented in a manner that is *mathematically identical* to the
numeraire good in the former system--suggesting the potential relevance of
such things as "wage/profit frontiers" for the system Marx intended to analyze.

Gil