In Andrew's new version of the "spurious correlation" charge
he says
"(3) Next, assume that the price/value ratios (r) are wholly
random. In any individual case, price can be very high
relative to value or very low. To approximate this, assume
that r is a random number evenly distributed between, say, 0
and 1,000,000."
A rectangular distribution on (0, 10^6) has an expectation
of 500,000. That is, Andrew is assuming that total price is
500,000 times greater than total value for the economy as a
whole. He's free to choose units to make that the case, but
this is eccentric and, more importantly, masks a serious
problem with assuming a uniform distribution. Suppose we
take the standard normalization which makes total price
equal total value. The distribution of price/value ratios
then has an expectation of 1. Its minimum can't be less
than zero (capitalists don't pay you to take their
products). It follows that -- if it is rectangular -- it
has a maximum of at most 2. But we know that can't be
right. Rent-rich sectors such as oil sell their product at
a price/value ratio substantially in excess of 2. So the
probability distribution can't be rectangular.
The rectangular distribution may be the easiest one to
simulate; that doesn't mean it can be gaily assumed in
relation to any and every phenomenon.
Thinking along the lines indicated above but in much more
depth, Farjoun and Machover develop a theoretical argument,
based on very general facts about capitalist economies,
leading to the conclusion that price/value ratios should be
distributed normally, with a standard deviation of something
like 1/3 of the mean. Their argument is not watertight, but
-- broadly speaking -- Paul C and I find it to be borne out
by the data. Had we found that actual sectoral p/v ratios
followed a rectangular distribution we would have taken this
as strong evidence against the theory, but we found no such
thing.
Paul and I have argued in various publications that the
"proper" object of study is precisely the shape of the
distribution of price/value ratios. Is this approximately
normal? Is its dispersion wide or narrow? (Wide or narrow
compared with other relevant probability distributions, such
as those of profit rates, ratios of prices to prices of
production, organic compositions, ratios of prices to oil,
steel or electricity content, etc.)
To my mind the correlation coefficient between sectoral
prices and values has some use as a descriptive statistic.
Other things equal, it will be higher, the narrower is the
distribution of price/value ratios across the sectors.
Andrew is right, though, that even if that distribution is
very wide, the correlation coefficient will still tend to be
positive (and even "statistically significant"). It would
be a mistake to argue, "We found a significant positive
correlation between prices and values therefore the labour
theory of value is supported." As Andrew says, the implied
null hypothesis here is obscure and probably senseless. But
that has not been our argument.
Alan has recently made some interesting additional points; I
hope to get to those soon.
Allin Cottrell
Department of Economics
Wake Forest University