A reply to Fred's ope-l 2545.
One of my proofs that Fred's relative prices and profit rate are identical
in magnitude to the Sraffians' showed that the Sraffian equations are
written as
p[I - (A + bl)(1+r)] = 0 (1)
and Fred's price of production equations can be written as
f[I - (A + bl)(1 + S/(C+V))] = 0 (2)
I have here changed the notation slightly for clarity--f is the vector of
Fred's unit prices.
Let us call the condition in which no prices are negative and some are
positive NNSP. The last step of my proof noted that for this to be
true of vector p, the expression in square brackets in (1) must have a
zero determinant, and for NNSP to be true of vector f, the expression in
square brackets in (2) must have a zero determinant. Hence, given the
same economy (same A, b, and l), r = S/(C+V). And hence, pi/pj = fi/fj
for all commodities i, j.
Fred now disputes this by saying that "Andrew interprets BOTH of his
[equation systems as ones] that are SOLVED FOR THE RATE OF PROFIT; i.e.
as equations in which the rate of profit is treated as an unknown and
which are solved to determine the rate of profit."
Not so. As the above makes clear, I am referring to a unique solution
for UNIT PRICES that satisfies NNSP. In ope-l 2506, which Fred quotes
in ope-l 2545, I also thought I had made this clear: "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 [f's]) being zero" or some of them being
negative. Even if I wasn't clear there, it is nonetheless the case that
the proof does NOT depend on S/(C+V) being unknown. No, I understand
very well that it is a given magnitude "determined"--i.e., stipulated
--before, and independently of--the determination of f, A, b, and l.
Therefore Fred's objection to my proof doesn't hold up, but the proof does.
Fred also says that his equations permit the determination of unit prices
of production from the given S/(C+V) and the technical and real wage
coeffficents. This isn't so. What is already given in Fred's interpre-
tation are S/(C+V) and all the sectoral aggregate C's, V's, S's, and aggregate
prices. These constrain the technical and real wage coefficents--Fred
is not free to select them. For instance, in a 2 or 3 department model with
only circulating capital, I've already shown that a1= p1*a1*x1/(p1*x1) =
C1/P1, which we see is determined because C1 and P1 are given. Similarly,
I've shown that b2*l2 = p2*b2*l2*x2/(p2*x2) = V2/P2, and so it too is
not free to be selected. ... What does this have to do with the main
point under dispute? Well, it may help Fred recognize why his results are
quantitatively identical to the Sraffians': given the stipulation that
input prices equal output prices (which enables (2) to be written in the
first place), the condition NNSP, and the pre-given S/(C+V), Fred's
A, b, and l are THEN (mostly) determined. As I've noted in prior posts,
enough is determined so that we always know that Fred's S/(C+V) is the
same function of technical coefficients as the Sraffian r is. Fred,
please note: to say that S/(C+V) is a function of technical coefficients
DOES NOT, DOES NOT, DOES NOT, imply in the least that S/(C+V) is constrained
by these coefficients, or is not given logically prior to them. As I've
just noted, your procedure constrains the technical and real wage
coeffcients on the basis of value and production price aggregates.
Fred continues to try to answer questions concerning magnitude with statements
that have nothing to do with magnitude. The key confusion that pervades
everything is a mixing up of two wholly different meanings of the word
"determination." Fred thinks that because the order of his procedures,
or the order of what he thinks were Marx's procedures, differs from the
order of Sraffian procedures, that therefore the relative prices and the
profit rate will not be equal. But the order of procedures concerns the
"logical method" of determination, while the fact that the *functional
determinants* of the profit rate and relative prices are identical in Frd's
interpretation of Marx and in Sraffian theory is what makes their magnitudes
the same.
Here's a simple analogy. Karl Marx determines total revenue through the
prior analysis of capital in general. Pierro Sraffa starts with the
quantity of the good bought/sold. Marx uses TR to determine P and Q.
Sraffa uses Q to determine P and TR. Now, imagine someone proves that
there is only one possible price, P = P*. Then, for Marx, we can
determine the magnitude of Q: Q = TR/P*. For Sraffa, we can determine
the magnitude of TR: TR = P* x Q. Very different sets of procedures.
Nonetheless, the simple fact is that, if we are referring to the same
Q (the same economy), Marx's TR and Sraffa's TR will be identical in
magnitude. The logical method of determination does not matter here.
Whether the fundamental given is the given money sum of total revenue
or the fundamental given is the physical condition (Q), the fact reamins
that given P* and the same Q, Marx and Sraffa have TR's identical in size.
Even if I haven't yet convinced Fred that my proofs are valid, this
argument should suffice to show that talk of givens and logical method
simply does not imply that two different sets of givens and two different
logical methods in any way give different quantitative results.
Fred is right that his interpretation can lead to different quantitative
results regarding absolute prices--simply because the Sraffians have no
absolute prices. Should they choose, however, they can select a numeraire
that would give them the same absolute prices as Fred.
Fred is also right that if we introduce fixed capital, and conceive Sraffian
theory narrowly, so that fixed capital is treated as a joint product, then
Fred can come up with a truly different set of relative prices and a
different profit rate. So I modify my claim to that extent. But the
irrelevance of Dept. III to the magnitude of the general profit rate,
the impossibility of mechanization itself lowering that rate, the fact that
the general rate is a function solely of technical and real wage coeffi-
cients--all these results, contrary to Marx's own, remain the case under
Fred's interpretation, as long as we are considering uniform profitability.
Andrew Kliman