This is a reply to Andrew's (2506) - continuing our recent discussion about
whether or not my interpretation of Marx's theory of prices of production
and the rate of profit leads to the same quantitative results as the
Sraffian interpretation.
I continue to argue that Andrew's argument substitutes a Sraffian
determination of the rate of profit for a Marxian determination of the rate
of profit, and is therefore invalid. As we shall see, this substitution is
even clearer in Andrew's most recent post.
I have argued that Marx's equation for the determination of the rate of
profit was:
(1) R = S / (C + V)
where C and V are the aggregate money quantities of constant capital and
variable capital taken as given in the Volume 1 analysis of the total amount
of surplus-value, and S is this total surplus-value determined in Volume 1.
On the other hand, the Sraffian equation for the determination of the rate
of profit is:
(2) p = (pA + pbL) (1 + r)
These very different methods of determination of the rate of profit yield
different quantitative results (more on this below).
In response, Andrew has argued that, once R has been determined as in (1),
it can be substituted into an equation like (2). I agree that such a
substitution can be made as a way of deriving individual unit prices from
the given r, a, b, and L, although this derivation is not an essential part
of Marx's theory.
However, Andrew then argues further that, if the rate of profit is
determined according to (2), then there is a unique r (with non-negative
prices). Therefore, Andrew concludes that R must be equal to r. To quote
Andrew (in 2506):
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...
[T]here 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)...
The maximum eigenvalue, which gives the nonnegative, non-zero solution,
is the maximum solution for 1/(1+r), which equals Fred's 1/(1 + S/(C+V)).
However, this last step in Andrew's argument is invalid. This last step
substitutes the Sraffian determination of the rate of profit for the Marxian
determination of the rate of profit. This is seen most clearly in the
middle sentence quoted above: "there is ONLY ONE SOLUTION TO THESE EQUATIONS
SYSTEMS, BOTH OF THEM ..." From this sentence, we can see that Andrew
interprets BOTH of his versions of equation (2) as equations 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.
In other words, equation (2) is interpreted according to the Sraffian method
of the determination of the rate of profit. If one interprets equation (2)
as a Sraffian equation, as Andrew does, then it is true that the solution to
this equation will indeed yield a unique r. However, the uniqueness of
this r determined according to Sraffian theory does not necessarily imply
that the R determined according to Marxian theory by equation (1) must be
equal to this Sraffian r. It only means that, WITHIN Sraffian theory, there
will be a unique r. It does not mean that the R determined according to
Marxian theory (and indeed every other theory) must be equal to the r
determined according to Sraffian theory.
On the other hand, if equation (2) is interpreted instead as a Marxian
equation, i.e. as an equation in which R is NOT an unknown, but is instead a
GIVEN which is determined prior to individual prices by equation (1) (i.e.
determined by the prior aggregate analysis of Volume 1), then THERE WILL NOT
BE A UNIQUE R WHICH SATISFIES EQUATION (2). Instead, equation (2) can be
satisfied by an infinite number of R's, or an infinite number of
combinations of R's and prices. One of these possible R's will be determined
by Marx's equation (1), and it will not necessarily be equal to the r
determined by the Sraffian interpretation of equation (2). The bare
mathematical fact that R can be included in an equation like (2) does not
mean that R is determined by (2) and thus that R = r. Therefore, Andrew's
argument is invalid. The very different relations of determination that are
assumed in equation (2) in Marxian theory and Sraffian theory result in
different quantitative results.
I have already argued in recent posts that an important indication that R
NOT = r is the different effects that technological change in "luxury goods"
industries (or Dept. 3) has on R and r. Technological change in Dept. 3
affects R determined by Marxian theory, but does not affect r determined by
Sraffian theory. Andrew's argument against this depends on his argument
just discussed that R = r. But if this latter argument is wrong, as I have
argued, then his argument concerning Dept. 3 is also wrong.
Another indication that R NOT = r is that Andrew's equation (2) - and his
whole argument - assumes only circulating capital, i.e. no fixed capital.
Andrew assures us that "it doesn't matter". But the treatments of fixed
capital in Marx's theory and in Sraffa's theory are entirely different. In
Sraffa's theory, fixed capital is treated as a "joint product"; prices of
"partially used machines" are determined along with the prices of output and
the rate of profit. In Marx's theory, fixed capital is taken as given as a
sum of money-capital, which is depreciated over the life of the machine. It
is well known that the two different treatments of fixed capital lead to
different depreciation patterns. Therefore, the different treatments of
fixed capital will also result in different prices and a different rate of
profit.
Another indication that the results of Sraffian theory and my interpretation
of Marx's theory are very different is that the prices determined in
Sraffian theory are RELATIVE prices (relative to an arbitrary numeraire),
whereas the prices determined in Marx's theory are ABSOLUTE prices (in terms
of money). Absolute prices can be determined in Marx's theory precisely
because the rate of profit is taken as a predetermined given and is NOT
assumed to be an unknown which is determined by equation (2).
Therefore, I conclude that my "monetary-macro" interpretation of Marx's
theory of prices of production does not lead to the same quantitative
results as Sraffian theory. The assumption of long-run equilibrium (i.e.
stationary prices) does not by itself turn Marx's theory (and every other
theory) into Sraffian theory. This assumption is not as "obtrusive" as
Andrew thinks.
Comradely,
Fred