# [OPE-L:2167] Re: Empirical method

Allin Cottrell (cottrell@wfu.edu)
Sat, 11 May 1996 18:18:56 -0700

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Alan has recently expressed scepticism over the methodology of correlating
the price of the output of various industries and the labour-time embodied
in the output of those industries, as a means of empirically testing the
labour theory of value.

I think I understand Alan's objection, and would like to offer a rebuttal.
(If I haven't got the objection right, I can be sure I'll hear about that
soon enough).

Suppose we take three vectors of random numbers, x, y and z. The elements
of each vector are drawn independently, and the expected correlation
between any pair is zero. Now we form two new vectors, u and v, as
follows: the kth element of u is the product of the kth element of x and
the kth element of z, while v(k) = y(k)*z(k). Now, what is the expected
value of the correlation between u and v? It is positive, due to the
common factor provided by the action of z on both x and y. If x and y
started out positively correlated, this correlation will be magnified by
the transformation.

The above are simple statistical propositions; they are not in dispute.
Now, as I understand it, Alan's claim is that have an important
application to the "Shaikh methodology" for testing the labour theory of
value. What we really want to assess is the correlation between price (x)
and labour-content (y) at the level of the individual commodity. But what
we actually have to work with are aggregate price and aggregate
labour-content at industry level. But the latter variables correspond to
u and v above, where the z vector is the vector of "scales" of the various
industries. That is, the aggregate price for the car industry is the
price per car multiplied by the number of cars produced, while the
aggregate labour-content is the labour-time per car, also mutiplied by the
number of cars produced, and so on for every other industry. Thus,
according to the statistical reasoning above, even if there is no
correlation between labour-content and price at the level of the
individual commodity, there will be an induced correlation at industry
level; and if there is a positive correlation to start with it will be
spuriously amplified.

Do I have the objection right?

The objection looks quite plausible at first glance, but actually it is
spurious. The easiest way to see why is, I think, as follows. Suppose
for a moment that the objection were valid. Then, in principle at any
rate, there is an obvious correction one could apply. One could obtain
the z vector, i.e. the vector of scales of the various industries, and
divide the elements of the industry-level price and labour-content vectors
by the elements of z. Then we'd be back at what we "ought" to be
studying.

The trouble with this notion is that the z vector doesn't exist. I don't
mean it's not published by the BEA or the CSO -- it doesn't exist in
principle. What, for instance, is the scale of the electricity industry?
Is it the number of kilowatt-hours produced per year? The number of
watt-seconds? The number of terawatt-hours? What is the scale of the oil
industry? Is is measured in barrels, thimblefuls or metric tonnes? The
beer industry: bottles, fluid ounces, cases, gallons? If the elements of
the putative z vector all had the same dimension, then the choice of a
unit of measurement, while arbitrary, would not be problematic because it
would simply scale all the elements of z correspondingly. But the
putative "z" is composed of incommensurables. There's no such animal.

Paul C and I have argued (drawing on Farjoun and Machover) that the proper
object of study is simply the ratio of price to embodied labour across the
various industries, with each industry weighted according to the
proportion of total social labour it accounts for. We examine the
coefficient of variation of this variable, and assess whether its
dispersion is broad or narrow compared with other variables of interest in
Marxian economics. It turns out to follow a rather narrow distribution
(compared, e.g. to the rate of profit). And we also show that the
dispersion of price/labour-content is very much narrower than that of
price/steel-content, or price to electricity-content, or price to
oil-content -- which provides, if you wish, a more pragmatic refutation of
the idea that the "closeness" of price and labour-content is some sort of
statistical artifact. It's not; and I find it puzzling that some Marxists
seem so concerned to argue that it is.

Allin Cottrell