Andrew takes the model originally presented by Shaikh (and
adopted by Paul Cockshott and myself, among others), namely
Pj = a*Vj^(b)*exp(uj)
(or in log form, lnPj = lna + b*lnVj + uj), where Pj and Vl are,
respectively, sectoral prices and values and uj is an
sector-specific error term. He argues that if the restrictions
on the parameters suggested by the labour theory of value
(namely a = 1, b = 1) are valid, then the following ought to be
true (I'll label the proposition for reference):
A1: If we create two new variables, Yj = Pj/Sj and Xj = Vj/Sj,
where Sj is some measure of the "size" of the sectors, then we
should find (a) a high correlation between lnXj and lnYj and (b)
we should retrieve a good approximation to a = 1 and b = 1 when
we carry out a regression of lnYj on lnXj (and not only when we
regress lnPj on lnVj).
It would then follow that if we do not find these things (as
Andrew does not), the theory in question is rejected by the
data.
I pointed out earlier that if Sj = Vj for all j (i.e. if one
uses value itself as the measure of scale), proposition A1 is
definitely false: the correlation between Xj and Yj is then
undefined, regardless of the validity of the Shaikh model.
Andrew has not disputed this, but, if I understand him
correctly, he's saying that A1 holds good for all cases short of
actual degeneracy (zero variance) of Xj.
That is not the case. I have simulated the econometric
situation as follows.
I first generated dummy data using the Shaikh model with a = b =
1. That is, I took real sectoral value figures from the UK
economy and constructed dummy price figures by (i) generating a
set of dummy ujs by drawing randomly from the standard normal
distribution and scaling to get a roughly "realistic" variance,
(ii) raising "e" to the power of these ujs to construct a
multiplicative error, and (iii) multiplying these errors into
the values data. The resulting data set conforms to the labour
theory of value, as specified by Shaikh, by construction.
Not suprisingly, a log regression of the Pj thus generated on Vj
produces a good fit and a close approximation to a = 1 and b =
1. The correlation is also high.
I then simulated Andrew's scaling by constructing a dummy scale
series, using Sj = 0.85 * Vj + ej, where the ejs were again
drawings from a normal distribution. (The 0.85 coefficient was
designed to simulate roughly Andrew's cost measure of scale,
which will be smaller than value.)
What happens when you regress ln(Pj/Sj) on ln (Vj/Sj)? It all
depends on the variance of ej. If that variance is sufficiently
small, i.e. if the scale measure is very highly correlated with
the values, then, as I stated earlier, the "original"
correlation is destroyed. On repeated runs (independent
drawings of the ujs and ejs) I obtained typical results very
similar to those Andrew has reported: low or essentially no
correlation between the logs of the scaled variables, and slope
coefficients that were indistinguishable from zero -- I even got
"significant" negative slopes in some cases, as Andrew did.
Let me stress: these results came from data sets that, by
construction, had precisely the statistical properties claimed
by the Shaikh specification of the labour theory of value. I
conclude that, whatever Andrew is testing with his regressions
(the "labour theory of markup"?) it is not the theory that he is
claiming to refute.
Allin Cottrell.