# [OPE-L:5497] Re: Humbug Aggregate Price-Value Correlations

Allin Cottrell (cottrell@wfu.edu)
Sun, 21 Sep 1997 15:02:31 -0700 (PDT)

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Andrew has recently recycled an objection that Alan Freeman
made some time ago. Here is a slightly edited version of
the response I made the first time around, which I still
believe is valid...

"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.

Here is a rebuttal.

Suppose we take three vectors of non-negative 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. Alan's claim is that have an important
application to the methodology for testing the labour theory
of value developed by Anwar Shaikh and since used by many
researchers including Paul Cockshott and myself. What we
really want to assess is the correlation between price (x)
and labour-content (y) at the level of the individual
commodity; 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

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 [note Sept 1997: this demonstration is
published in the current issue of the Cambridge Journal of
Economics] -- 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
Department of Economics
Wake Forest University