A reply to Fred's (ope-l 2298) reply to my (ope-l 2256) "Moseley's
Interpretation, Pt. I."
I stand corrected; I'll accept that Fred means by prices of production
"long-run equilibrium" prices, which imply stationary prices. But
I don't think this is at all what Marx had in mind in Ch. 9 of Vol.
III. He repeatedly notes that he is examining a period of one
year.
Fred seems to accept my 3-dept. examples, as well as the quantitative
conclusions that I draw, and must be drawn, from them. But he tells
a story in which the change in the profit rate, due to tech. change
in Dept. III, leads to tech. change in the other depts. as "spillover
effects." So he wants to argue that the tech change in Dept. III
*causes* the change in the profit rate, and that the tech. change
elsewhere in the economy is not a *cause* of the change in the
profit rate, but rather a result.
One cannot conclude anything of this sort from the numbers. We have
a static equilibrium comparison of two value/price tableaux. We
cannot infer any *causal* relations from such an exercise, as is well
known.
But it really doesn't matter. I was disputing Fred's claim that the
rate of profit in his interpretation can be *quantitatively* different
from the "Sraffian" profit rate, because (he claims), tech. change in
Dept. III *alone* can affect the profit rate. My point was that
tables of the sort I presented do not support this conclusion, that
there *is* tech. change elsewhere in the economy in these examples, and
that Fred's profit rate will *always* be the same as the Sraffian
rate. At the end of my post I even gave the input/output relations
corresponding to these particular tables and noted that if one uses
the Sraffian formula for computing the profit rate, one will get
the exact same profit rate as Fred's S/(C+V).
Strangely, Fred does not dispute this point. Anyone who wants to
confirm it can take my expressions, do the calculations to find
the input/output coefficients, plug them into the Sraffian formula,
and ascertain that the "Sraffian" profit rate is exactly the same as Fred's
S/(C+V). This will always be the case. I'll accept even one single
counterexample from Fred (or anyone else) as a refutation of my
claim (assuming 3 depts., C and V equal to the price of means of
production and real wages, and a uniform profit rate).
I also provided a general proof of this proposition. Fred doesn't like
the order of steps of the proof. But it doesn't matter. The nature
of a proof doesn't change when you change the order of the steps. If
each step is valid, the conclusion is valid. Period.
Fred also doesn't like the tautology I employed in the proof, saying that
it leads to a "logical flaw," "assumes what is to be proved," and that
"an arbitrary tautology is the basis of Andrew's conclusion that my
rate of profit is the same as the Sraffian rate of profit and does not
depend on technical conditions in Dept. 3."
Unfortunately, it seems that Fred has forgotten what a tautology is.
It is, by definition, true by definition. It cannot result in
a logical flaw, it does *not* assume anything, other than certain
laws of mathematics, and is not the basis for any conclusion--because
it lacks determinate content.
Fred points to another alleged "logical flaw," which is that supposedly
I represented his profit rate incorrectly. In fact, I did say how
Fred thinks it is determined--R = (sum of S)/(sum of C + sum of V),
just as he replies to me. But supposedly I did something wrong by
representing it within Fred's production price equations. According to
Fred, this "implicitly assumes that R is detewrmined" by these
equations. It does not. Again, this evinces a misunderstanding of
mathematical proofs of the sort I put forth. They work in terms of
*equalities*, not relations of determination.
But I'll accept Fred's challenge to construct a proof that begins with
his general rate of profit. Here goes.
The general rate of profit = S/(C+V). The price of production
equations are
Pi = (Ci + Vi) (1 + S/(C+V)).
Fred does not tell us what the input/output coefficients or unit prices
are. But, given a 3-dept example, we must still always have--IF PRICES
ARE STATIONARY, WHICH FRED REQUIRES--
Pi = pi*Xi Ci = p1*ai*Xi, Vi = p2*bi*li*Xi. Substituting these
into the above, we have, in matrix form
p = p(A + bl)(1 + S/(C+V))
after cancelling out the Xi's in each equation.
Now, as is well known, the Sraffian price equations are written as
p = p(A + bl)(1 + r),
where r is the Sraffian profit rate.
Now, the only nonzero solutions for r in the bottom equations are
given by a zero determinant for [I - (A + bl)(1+r)]. In the top
equations, we can write p[I - (A + bl) (1 + S/(C+V))] = 0, and the
only nonzero solutions for p (sorry, I meant "p" not "r" in the
1st sentence of this paragraph, too) occur if the determinant of
the expression in square brackets is zero.
Thus, S/(C+V) = r.
Fred asks: "How can a theory that assumes that the rate of profit is
determined by aggregate magnitudes prior to the determination of
individual prices and depends in part on technical conditions in
luxury goods industries be transformed by simple algebraic manipulation
into a theory that assumes the opposite?" Answer: (a) logical
priority is irrelevant to mathematical equalities and identities;
(b) the "depends in part ... industries" is either wrong, or not
denied by Sraffian theory in the sense in which Fred asserts it
(i.e., no one has said Fred's causal "spillover effects" story
is mathematically impossible, including Sraffians); and (c) Fred's
interpretation makes the technical coefficients conform to the
Sraffian technical coefficients when he postulates stationary
prices, and thus he makes the profit rate also conform.
Finally, Fred writes:
"Andrew is so opposed to stationary prices ...." Well, I don't like
always having to pay more for things, so I'd love to have some
stationary prices. :-)
Andrew Kliman