Andrew's reply on the Okishio theorem, the FRP and related matters of
7/26/96 raises several issues. I'd like to address them separately.
Duncan wrote:
"I don't think you understand "refutation" in the same way as I do, as the
remarks above show. The theorem states its own premises and conclusion,
and as
far as I can see has not been refuted yet by the production of a
counterexample that satisfies the premises and contradicts the
conclusion."
Andrew:
What do you mean by "states its own premises and conclusion"? Are you
saying
that the Okishio theorem is the same as the relevant Perron-Frobenius
theorem?
Duncan now:
The Okishio types of theorem consider an input/output economy in a state
where input prices are equal to output prices and profit rates are
equalized. "Viable" techniques are those alternative techniques of
production that lower costs at the current (stationary) prices. The
theorem considers what happens to the profit rate if a viable technique is
introduced into the economy and the system again achieves a stationary,
profit-rate equalizing price system at the same real wage characterized as
a specific basket of goods representing workers' consumption, and
concludes that the profit rate with the viable technique cannot be lower
than the original profit rate. I haven't yet seen a counterexample to this
assertion.
Andrew:
Do you disagree that one "conclusion" of the theorem, and/or of its
proponents-interpreters, is that Marx was wrong to claim that
profit-maximizing firms could adopt new techniques that would lead the
"equilibrium" (equalized) profit rate to fall?
Duncan now:
This is certainly the interpretation placed on the theorem by some people,
though not by me. I argue that Marx was holding the value of labor-power
in the sense of the money wage multiplied by the value of money constant,
not the real wage in the sense of the basket of commodities consumed by
workers, and it is easy to demonstrate cases where the adoption of viable
techniques can lower the profit rate (again assuming that the system finds
a stationary, profit-rate equalizing price system) with a constant value
of labor-power.
But in the mathematical sense statements about whether or not Marx was
"right" or "wrong" are not part of the theorem, only part of the
discussion it engenders.
Andrew:
Have I introduced any premises
that the theorem precludes? If so, which one(s)? Haven't I contradicted
the
conclusion that "the equilibrium rate of profit" cannot be lower than the
original one? If not, why not?
Duncan now:
Your examples never allow the economy to return to a stationary,
profit-rate equalizing system of prices, because of your premise of
continuing technical change, and/or the existence of infinitely lived
fixed capital.
The point is that the Okishio theorem does not in fact address the case
you raise, which is what happens when there is continuing technical
change, so that the system cannot equilibrate at a stationary price
system. I think this is an interesting case, and very much worth
understanding. But it requires a more sophisticated mathematical treatment
than the Okishio model can provide. In this sense your examples, as
interesting as the questions they raise may be, don't seem directly
relevant to the mathematics of the Okishio theorem.
Duncan wrote:
What you are arguing is that the Okishio theorem is not relevant to
understanding or evaluating Marx's discussion of the falling rate of
profit on
your reading and interpretation of Marx, which is a different point. The
crux
of this argument seems to be the definition of the profit rate under
conditions of technical change.
Andrew:
I'm certainly arguing what you say, but not only that. I'm also
arguing that (a) if the Okishio theorem is not relevant to Marx's law, it
does not refute it;
Duncan now:
This is exactly the position I take, on the different grounds involving
the value of labor-power I described above.
Andrew:
(b) the theorem's proponents must *prove* that it is
relevant to his law if they wish validly to claim that the theorem refutes
it;
Duncan now:
The phrase "the theorem's proponents" is slippery in this context. I
accept the theorem, but I do not accept it as refuting what I understand
to have been the substance of Marx's discussion of the FRP. Clearly
"proving" that the theorem "refutes" Marx is not a mathematical issue at
all, but one of exegesis of Marx's text. As far as I can see, however, the
great majority of people who understand anything at all about the FRP
discussion seem to believe that Marx thought the profit rate could fall
with a constant real wage in an economy that finds its way to a
stationary, profit rate-equalizing price system due to individual
capitalists adopting cost-reducing techniques at existing stationary
profit rate-equalizing prices, and that Okishio's theorem shows that this
belief is wrong. So if Roemer, van Parijs et al. haven't "proved" their
point, they surely have succeeded in persuading most people of it.
Andrew:
(c) I have another defensible interpretation of Marx's argument in which
Okishio/Roemer's conclusion is not relevant (because adjustment to a
post-mechanization stationary price scenario need not occur, and will not
occur, given my premises), so that (d) the theorem's proponents haven't
proved
that their interpretation is accurate, and therefore haven't proved that
Marx's law is false. Do you agree or disagree with this?
Duncan now:
I still don't completely understand your interpretation of Marx's
argument, so I am reserving judgment on part (c). I agree on the different
grounds I outlined above with (d).
Andrew:
Moreover, I'm arguing that, *irrespective of its relationship to Marx's
law*,
a necessary step in the proof of the theorem is omitted, which invalidates
it.
Okishio/Roemer claim to *show* that the post-mechanization "equilibrium"
profit rate cannot be lower than the original one, but they only show that
this is the case *if* prices are stationary in the post-mechanization
"equilibrium." If prices are not stationary, they've proved nothing.
Duncan now:
My problems here are largely semantic. Since I take the Okishio theorem to
assume a stationary profit rate-equalizing price system both before and
after the technical change, I don't see that a necessary step in the proof
of their theorem is missing. You disagree with the premise: it's true of
any theorem that it says nothing about the cases ruled out in the
hypotheses.
Andrew:
And
thus for their claim to be valid generally, they must prove that a
stationary
price scenario will result, but fail to do so. (Especially in his
Australian
Econ Papers article, Roemer shows he's aware that the proof depends on
dynamic
adjustment to this stationary price equilibrium. He does some hand-waving
to
argue that competition will equalize profit rates, but doesn't even
produce an
argument that, much less prove that, stationary prices will result.) In
short, the theorem requires a proof of dynamic adjustment, which it lacks.
Do
you agree?
Duncan now:
I disagree that the theorem as stated requires a proof of dynamic
adjustment, since I agree that it assumes the dynamics away.
It would certainly be desirable to have a more general treatment of this
problem with a persuasive and general model of dynamic adjustment.
Although a lot of work has been done on dynamic adjustment (including some
interesting papers on "cross-dual" dynamics by Dumenil/Levy, Flaschel, and
others) I wouldn't say that a consensus model of dynamics has emerged
within which one could address the FRP problem. It might be a good step to
look at "Okishio-type" questions within the models of dynamic price
adjustment that have been advanced.
Andrew:
Is my definition of the profit rate different from theirs? I think not.
If
there is no fixed capital, the definitions are the same, but my (and
Alan's
and John's) rate can fall when theirs rises, due to the output prices
being
lower than the input prices.
Duncan now:
One of the points on which I'm still unclear is exactly what your
definition of the profit rate is when prices are changing. The problem is
that when prices change over the life of an investment the ratio of the
profit to the invested capital in each period will also be changing,
leading to an inherent ambiguity in the concept of "the profit rate". In
our previous discussion I tried to approach this by using the IRR as a
summary profit rate over the whole life of an investment, but I'm still
not sure this corresponds to some of the computations you make, since your
language sometimes suggests that you consider "the profit rate" on a given
investment to be changing over the life of the investment, which the IRR
cannot do. What can happen with the IRR is that it can change for
different "vintages" of investment. But as far as I can see in your
examples the IRR rises with vintage. You then refer to some concept of
"average IRR" which I don't understand.
More to come,
Duncan