# [OPE-L:5538] Re: Humbug Aggregate Price-Value Correlations

andrew kliman (Andrew_Kliman@classic.msn.com)
Sat, 27 Sep 1997 11:53:20 -0700 (PDT)

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I had written that "MAD's [mean absolute deviations] of 20-27 0mply that the
amount of value that is redistributed across sectors and across countries is
not negligible."

In ope-l 5529, Duncan writes: "I'm intrigued by the idea of measuring the
price/value deviation by the amount of surplus value actually redistributed
across sectors. This is on the trail of an economically meaningful measure of
the dispersion, which, as I said in an earlier post, seems very desirable in
this debate."

I think this is a very important point of agreement, both with respect to
measurement and with respect to hypothesis testing.

As far as measurement is concerned, the degree of dispersion can be measured
easily by means of a shift-share analysis. You simply take the absolute
values of the sectoral differences between surplus-value and profit. Then you
add them up. Then you halve that number, to eliminate the double counting of
each difference (otherwise a shift of 40 units of value from sector A to
sector B gets counted as a loss from A and as a gain to B, i.e., an absolute
value of 80 units). Then you take that number and divide by total
surplus-value in the economy.

The resulting number will basically range from 0 to 1. It will equal 0 when
no surplus-value is redistributed, and it will be close to 1 when some sector
that produces a negligible amount of surplus-value gets all the profit. (I
say "basically," because the possibility of negative profits means that the
upper bound is theoretically greater than 1.)

So what we get is a simple index with a quite transparent meaning. Its
statistical meaning is transparent, but also its economic meaning. This is a
major advantage as compared to the meaningless aggregate correlations, but
also as compared to MADs, etc.

There is another very important advantage to the shift-share index: the
numbers one gets seem to be independent of (a) the share of surplus-value to
total value and (b) the dispersion of the sizes of the sector. (I say "seem
to be" independent, because I've reached this conclusion from repeated
spreadsheet simulation rather than deduction, though I also can't think of a
reason why the numbers wouldn't be independent.) As I'll discuss in a moment,
this is in marked contrast to the MAD, the values of which are very sensitive
to both (a) and (b).

On hypothesis testing
=====================
Once one computes the shift-share index, or the MAD, or whatever, then the
issue is whether the number is small enough to say that values and prices are
close, so that values are good predictors of prices. The problem with some
work in this area, as I've noted, is that hypotheses don't seem to get tested.
A correlation or a MAD is reported, and the number is judged, arbitrarily, to
be small enough to permit the conclusion that values are "close" to prices.
Again, the question is: "close COMPARED TO WHAT?"

The implicit alternative hypothesis in some work seems to be that prices are
purely random. Thus *any* positive relationship between values and prices
seems to be a mark in favor of the hypothesis that relative values "explain"
relative prices.

Let's get real. Two components of value -- C and V -- are also components of
price (once one rejects dualism, but statistically, I doubt that even dualist
C's make much difference). So OF COURSE there will be SOME positive
relationship between value and price, and of course it will be stronger the
smaller the ratio of surplus-value to value is. I don't know of *any*
economist, living or dead, who has ever thought that prices are a completely
random variable that is determined without regard to two components of value,
C & V. So when the total lack of a positive relationship is used as the
implicit alternative hypothesis, the "relative values explain relative prices"
hypothesis is beating a straw man.

Hence, the real thing to concentrate on, as Duncan and I seem to agree, is
only the *remaining* component of value, S. The differences between prices
and values are the *same thing* as differences between surplus-values and
profits (given, again, the single-system interpretation). So the key issue
is how much surplus-value gets redistributed across sectors.

In accordance with this, I propose that those who do research in this area
employ the following hypothesis test:

HYPOTHESIS TEST:
Is the measure of relationship (shift-share, MAD, correlation, etc.) small
enough that one can reject (at the level of confidence of .01, .05, .10, etc.)
the "naive hypothesis" that surplus-value is randomly distributed across
sectors?

In other words, the naive hypothesis is that

Pj = (Cuj + Vj) + kj*(sum of S)

where P is sectoral price, Cu used used-up constant capital, S is summed
throughout the economy. kj is an evenly distributed continuous random
variable between 0 and 1, with the sum of kj's = 1. So the naive hypothesis
is that sectors recoup their paid costs and get a random cut of the
surplus-value on top of that.

If the naive hypothesis can be rejected, it means that the observed
price-value differences are sufficiently small that we can be confident that a
process of random distribution of surplus-value across sectors could not have
produced price-value differences as small as those observed. If anyone has
objections to this sort of test, I'd like to hear them.

So, for instance, let's say one computes a MAD of 20%, or a shift share index
of .22. Does this mean that prices and values differ rather significantly or
not? Well, it all depends on what kind of MAD or shift-share index one would
get were the naive hypothesis correct.

For instance, in the case of the shift-share analysis, I did a bunch of
simulations in which the share of S to value varied randomly across sectors
and the values produced in the sectors also varied randomly. I then
constructed the random kj's, and did runs of 5000 to see what shift-share
indexes would arise were surplus-value distributed in accordance with the
naive hypothesis. I varied the degree of dispersion of sector sizes, the
share of S to value, and the degree of dispersion of S across sectors to see
if these things mattered. They did not. This is important, because it means
that the shift-share indexes from different data sets can be compared --
something which is NOT the case with MAD's, as I'll discuss below.

In all cases, the expected value of the shift-share index resulting from
random distribution of S was very close to 0.33, with a standard deviation of
about .096-.098. The index seems to have a chi-square distribution. If
further simulation or analysis shows this to be a robust result, then everyone
can use the same tables to test the naive hypothesis.

Even if that isn't the case, one can always take the actual data and simulate
a random distribution of S enough times that one gets a probability
distribution for the shift-share index, assuming the naive hypothesis.

Now, to get to the point, does a shift-share index of 0.22 mean that prices
and values are close? Compared to what? Compared to the naive hypothesis.
0.22 is only 2/3 of the average value of the index assuming that the naive
hypothesis is correct. Yet the index has a *distribution*, and, looking at my
simulations, I see that more than 100f the random shift-shares are smaller
than 0.22. So the naive hypothesis *cannot* be rejected at the 10 0.000000e+00vel.
(The index needs to be below 0.214 to reject at the 10 0.000000e+00vel; 0.184 to reject
at the 5 0.000000e+00vel; and 0.131 to reject at the 1 0.000000e+00vel.)

MEAN ABSOLUTE DEVIATIONS
========================
Alejandro Valle (ope-l 5504) writes that "A MAD of 27 0oes not mean that
value and prices 'differ quite significantly' as Andrew assert[s]."

Tsoulfidis (ope-l 5510) writes that "a roughly 20 0eviation measured by MAD
or MAD weighted by the output of each sector are in the range of acceptance of
the proposition that labor values or prices of production are close enough to
the observed prices."

I responded to Alejandro in an earlier post. I now think my response, though
correct, was insufficient, because I was concentrating only on size without
regard also for hypothesis testing. What I now ask -- in response to both of
them -- is whether the reported MADs and MAWDs are small enough to reject the
naive hypothesis, and at what level?

Assume a 2-sector economy, in which the value of each sector's output is 100
and the surplus-value each produces is 20. Then, if we assume that profits
are nonnegative, the MAXIMUM MAD is 20% (again, according to the single-system
interpretation) since, if one sector gets all the surplus-value, the
price/value ratios are 1.2 and .8. So a reported MAD of 20 0n this case
would surely not allow us to say that values and prices are close.

The question is, how different are the MAD's computed from the actual data
sets from what a random distribution of surplus-value would yield? I do not
know. The problem is that, unlike the shift-share index, the size of the MADs
is very sensitive to the dispersion of industry sizes (or sectoral values) and
to the ratio of S to value. One can see this by studying what happens to the
maximum MAD: if you have a very small sector that receives all the profit,
its price/value ratio and will be much greater than if sectors are equal in
size, but the price/value ratios of other sectors will be about the same, so
the maximum MAD will be much larger. And clearly, the more S there is to
redistribute, the greater the spread between maximum and minimum price/value
ratios, and therefore the greater the maximum MAD.

Therefore I reach two conclusions:

(1) I don't think anything is learned just by reporting and comparing the
sizes of MADs *across* studies. The sizes of the MADs depend on
characteristics particular to the individual data set, including the
dispersion of industry sizes, which is arbitrary and conventional. So when
Alejandro judges a MAD of 27% to be "low," and Tsoulfidis reports on the
findings of a variety of studies, I don't think any conclusions can be drawn
without knowing also the key properties (dispersion of sizes, ratio of S to
total value) of the data sets in question, and even then I think
interpretation is not transparent.

(2) To determine the relevance of a particular MAD, one needs to see what the
distribution of MADs would look like FOR THAT PARTICULAR DATA SET assuming
that the naive hypothesis is correct, in a manner analogous to that which I've
described for the shift- share index. It is imperative that a probability
distribution be constructed for *each* data set because, again, the
distributions are very sensitive to the key properties of the data sets. Then
one needs to do a test of statistical significance, either by seeing whether
the naive hypothesis can be rejected, and at what level of significance, or by
testing the data against another REASONABLE alternative hypothesis. (Beyond
that, there are other questions, such as those of size, and other hypotheses.
I think the above is necessary but not sufficient.)

I don't think there's anything inherently wrong with MADS as a measure --
unlike the meaningless correlations of secotral aggregate prices and values --
but their meaning depends crucially on the properties of the particular sets
of data they come from. So, until we have additional information and testing
of the kind of just outlined, we have no way of knowing what the reported MADs
actually tell us. On the basis of some simulations, I have some rather strong
doubts that the MADs allow us to reject even the naive hypothesis in favor the
"relative values determine relative prices" hypothesis. But I'd like to be
sure. Until we get some hard data along the lines I've outlined, I suggest
that no be misled into thinking that the reported MADs support the "relative
values determine relative prices" hypothesis.

Andrew Kliman