[OPE-L:2971] Math Theorems, Economic Theorems

andrew kliman (Andrew_Kliman@msn.com)
Thu, 5 Sep 1996 14:25:43 -0700 (PDT)

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Some further thoughts on whether Okishio and Roemer define "equilibrium profit

rate" in a way that immunizes their theorem from refutation. This is largely
a further response to Gil and Duncan, but I want to start by agreeing with
something Allin wrote in ope-l 2937. He notes that the Okishio theorem
rightly demolished a lot of pro-FRP thinking. Allin, however, uses this fact
to reject the claim that, if the theorem is true, it is nothing other than the

Perron-Frobenius theorem.

I don't think this rejection follows. *Some* valid inferences can be drawn
from the
exercise; among them is the inference that *if* there is full adjustment to a
post-innovation stationary-price/equal-profit-rate state, *then* the new
profit rate
cannot be lower than the initial one (given the theorem's assumptions). This
sufficient to disabuse Allin of what he had thought before reading Steedman.
It does not, however, make the theorem true, because the theorem is a theorem
about what *will* happen (at least what will happen if profit rates are
after the innovation [there are different versions of the theorem out there]).

The claim that, if true, the theorem is reduced to Perron-Froebenius is this:

once one attempts to immunize the theorem from refutation by turning all the
economic terms of the theorem into *definitions*, one deprives them of any
economic content, and indeed of any reference to the non-mathematical world.
The terms are no longer economic terms, but superfluous English words attached

to mathematical entities. r* > r is true. It was proved by P & F. Now, I
will attach the labels "post-innovation yellow logarithm" to r* and
"pre-innovation yellow logarithm" to r, and conclude that the post-innovation
yellow logarithm must be greater than the pre-innovation yellow logarithm.
And this conclusion is absolutely true. But not only is it meaningless
outside the given context; more importantly, it is not an *additional* truth
above and beyond r* > r. It is the *exact same* truth, couched in different
words. It *is* the P-F theorem, no more and no less.

Hence, for mathematical results to have any implications for the
nonmathematical world, the mathematical terms must be understood as
representations of nonmathematical entities, and the latter must be defined
*independently*, not by the mathematics itself. (Note that I am not claiming
that the nonmathematical entities must exist physically, refer to observable
facts, etc.)

I of course do not think that Okishio or Roemer were doing anything as silly
as comparing the size of yellow logarithms. But the only reason I don't is
because I think, and have some strong evidence, that they were *not* defining
the pre- and post-innovation equilibrium profit rates as r and r*. Rather,
they were letting r and r* represent externally defined economic concepts, and

making an argument concerning the relative magnitudes of these nonmathematical

entities. Their theorem thus fails because of a possible difference between
r* and the economic concept it was intended to represent. I.e., they failed
to prove that the externally defined, post-innovation, equilibrium profit rate

will equal r*).

The P-F theorem, however, refers only to mathematical entities, and is true.

While the difference between mathematical theorems and economic "theorems"
expressed in mathematical symbols is perhaps not something most economists
have thought about, it is a readily explained and not too mindcracking
difference. Therefore, although I agree with Duncan that some people will
conclude that I don't understand the nature of a theorem, or the P-F theorem,
etc., when I say that I have refuted the Okishio theorem, I'm willing to make

the claim anyway and then explain how the Okishio theorem differs from the P-F


Andrew Kliman