A reply to Fred's ope-l 2435. Glad to see you're back, Fred.
In response to my proof that Fred's interpretation of MArx's transformation
gives a profit rate and relative prices equal in magnitude to those of
the Sraffians, because input and output prices are equal, Fred first
says that my proof requires acceptance that equations don't imply relations
of determination.
No, I do think equations imply relations of determination. By convention,
there's a differnce of determination implied when we write A = B from
when we write B = A. What my "position" is, is simply that, if A = B
then B = A. That, and transitivity of the equality relation, are the
only mathematical propositions one must accept to accept my proof.
The real issue seems to be that Fred says that, because his prices
differ from the Sraffian prices, so must his profit rate. The logic of
this I agree with. But Fred's relative prices are equal to those of
the Sraffians. Fred takes up one of my three proofs--the last of the
three he has not commented on--and says the proof is not legitimate,
basically because it assumes what it has to prove, that the prices of the
two equation systems (his and theirs) are identical.
I would agree, were my proof what Fred represented it to be. He, however,
omitted the crucial last paragraph of the proof! I won't reproduce it
verbatim, just indicate the key point. This is the proof in ope-l 2301.
The Sraffa system can be written as p[I - (A + bl)(1+r)] = 0. Fred's
can be written as p'[I - (A + bl)(1+ S/(C+V))] = 0. What I'm doing
now, which I didn't do explicitly before is use a different symbol to
indicate Fred's prices From the Sraffian prices. So p' does NOT mean
p-transpose. We'll see that it doesn't matter. Why? Because, as I
indicated in the last paragraph of my proof, there is one and only one
solution to these equation systems, both of them, that do not result in
either all the p's (or all the p's) being zero [or, as I forgot to
mention, some of them being negative). And that is when the determinant of
the expressions in square brackets are zero. They both must be zero, so they
both must have the same eigenvalue, so for the same A, b, and L, we must
have S/(C+V) = r. The maximum eignevalue, which gives the nonnegative,
non-zero solution, is the maximum solution for 1/(1+r), which equals
Fred's 1/(1 + S/(C+V)).
What *follows* from this is that the relative prices are equal. For all
commodities i and j, pi/pj = p'i/p'j. I'm not *presupposing* that they
are the same. There is only one profit rate that lets all the Sraffian
prices be nonnegative and some positive, and that is the one equal to
Fred's S/(C+V). That, again, is my prrof concerning the profit rate.
I've assumed only, as the Sraffians do, that their prices are never
negative and not all zero, not that their prices equal Fred's.
It is nevertheless true that the relative prices are equal. This is a
conclusion, not a presupposition.
Fred is absolutely right about what I'm claiming. Assume uniform
profitability (or equal mark-up over costs) AND input prices = output
prices. "[T]hen Andrew's logic 'proves' that EVERY THEORY of the rate
of profit leads to the same rate of profit as in Sraffian theory ... that
the rates of profit in the different theories are ... necessarily equal."
Exactly. This is what the "unobtrusive postulate" of input prices =
output prices does.
The voice may be the voice of Marx, but the hands are the hands of Sraffa.
(Apologies to the authors of Genesis.)
Andrew Kliman