[OPE-L:1533] Re: Temporality vs simultaneity

akliman@acl.nyit.edu (akliman@acl.nyit.edu)
Fri, 22 Mar 1996 14:00:11 -0800

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A reply to Gil's ope-l 1492, with reference to John Ernst's reply (ope-l
1493). My reply will also take up Fred's objection, similar to Gil's
about historical cost measurement of the profit rate.

Gil writes that the TSS interpretation is neither necessary or sufficient
for "rehabilitating Marx's theory of the falling rate of profit," and that
"A number of alternative postulate sets re-establish Marx's result." My
question here is very similar to John's: what do you mean, Gil, by
"Marx's theory" of the FRP and "Marx's result"? If you mean that one can
show conditions under which the profit rate will fall without the TSS
interpretation of Marx's value theory, well, of course. But that is not
what the debate has been about.

*Marx's* law of the FRP is that the profit rate can fall, and under certain
circumstances, will fall, due to mechanization ITSELF. The Okishio
theorem was never meant as a proof that the profit rate can't fall. It
was meant as a proof that the cause can't be mechanization ITSELF; i.e.,
given profit-maximization, blah-blah-blah, only a rise in the real wage
rate can cause a falling profit rate. Roemer, in _Analytical Foundations
of Marxian Economic Theory_ is *very* clear about this: "Although the
real wage in fact does not remain fixed, the problem has been to understand
whether a FRP can be construed to be due to technical innovation *itself*,
independent of changes in the real wage [p. 113, my emphasis]." "The
argument of this chapter is that there is *no hope for producing a FRP
theory in a competitive, equilibrium environment with a constant real
"It must be reiterated that this does not mean that the rate of profit
does not fall. ... The general point is this: If the rate of profit falls
in such a changing real wage model, it is a consequence of the class
struggle that follows [sic] technical innovation, not because of the
innovation *itself*" [p. 132, my emphases].

I don't fully understand what Gil's Metroeconomica paper is about, and it
sounds interesting, but it seems to belong to the class of rising real
wage models of the FRP. If so, it doesn't "rehabilitate" *Marx's*
theory of the FRP.

Roemer also states a common belief: for "a theory of a falling rate of
profit in capitalist economies [to exist, it is necessary to] relax some of
the *assumptions* of the stark models discussed here [Roemer's extensions
of Okishio's model]" [Roemer AFMET, p. 132, my emphasis]. Gil has a
similar view: "What's really at stake, it seems to me, is the legitimacy
of the *assumptions*" [my emphasis]. What Gil is implying is that I
(and John and Alan) haven't refuted the Okishio theorem on value-theoretic
grounds because "Given the postulates of the theorem, the conclusion is
logically valid. Absent a demonstration that the proof is deductively
invalid, the theorem itself isn't refuted."

I think I HAVE demonstrated that the proof is deductively invalid, on the
basis of, i.e., without altering, the postulates of the theorem. The
conclusion does NOT follow from the postulates, as I understand them.

Let me be clear: I do not claim to refute the relevant Frobenius-Perron
theorem upon which the Okishio theorem is based. Nor do I claim to
find an error in Okishio's or Roemer's algebraic manipulations. I DO
claim that they "deduce" invalid conclusions from the math.

To explain this, I'll turn to Roemer's particularly amusing proof of the
theorem in his "The Effect of Technological Change on the Real Wage and
Marx's Falling Rate of Profit," _Austrailian Economic Papers 17, June
1978. On pp. 152-53, Roemer writes: "if technical change is introduced
when it is cost-reducing, the final general equilibrium effect will be
to _increase_ the rate of profit, assuming the real wage-consumption
bundle of workers remains unchanged. (This was first shown by Okishio ...."

Later on p. 153, Roemer begins his "proof." "The equal-profit rate equations
for this economy are: ...

p = (1+r)pM (2.4)

[I'm using r instead of Roemer's Greek pi for the profit rate. M = A + bl.]


"We shall assume that technical change is _cost-reducing at current prices_."
[Roemer then states this symbolically in inequality (2.5)]. ... "If such a
technique appears and is adopted, the profit rate will *immediately rise*
in sector 1. This will encourage more firms to *enter* sector 1 from
sector 2; prices *will be cut in competition and *eventually* a new
equilibrium tableau *will emerge* [my emphases]:

p* = (1+r*)p*M*" (2.4')

[Roemer calls (2.4') "Tableau 2"]. Then he states immediately:
"It is a theorem that is (2.5) holds then r* > r: WHICH IS TO SAY that if
the real wage (b) remains fixed then cost-reducing technical innovations
GIVE RISE, EVENTUALLY, to a rise in THE equilibrium rate of profit in a
competitive situation" [p. 154, my emphases].

Now the math simply DOES NOT support any such conclusion. In other words,
a comparison of (2.4) and (2.4') and their associated r's permits absolutely
no conclusion regarding the dynamic process that follows adoption of the
new technique. Will the profit rate in sector 1 immediately rise? If so,
Roemer doesn't show it. Can we then be sure that firms will enter sector
1 from sector 2? *Far* more importantly, how do we know that "eventually"
Tableau 2 "will emerge," so that the innovation "give[s rise, eventually"
to a rise in the equilibrium rate of profit? The rigorous Roemer certainly
proves nothing like this. He merely substitutes a lot of dynamic language
that conjures up an unproved adjustment to a stationary price equilibrium
in place of a harmless and rather trivial static equilibrium comparison,
via the untrue words "which is to say."

I would suggest that a valid English translation of the demonstration would
be: "Assume a static equilibrium at stationary prices, with an equalized
profit rate. Assume a technical change that lower's some producer's costs
at current prices. Use the new input coefficents to compute an imaginary
set of prices and the imaginary profit rate associated with it, given the
further assumptions that the profit rate is again equalized and that
stationary prices then prevail. The latter, imaginary, profit rate must be
greater than the initial one."

Lest any of this seem to be quibbling, it is very well known that the
stability properties of (2.4) are rather shaky. Moreover, the investiga-
tions of stability, to my knowledge, do NOT permit technical change during
the process of adjustment or non-adjustment to a new stationary price
equilibrium. It is rather obvious that if techniques are *continually*
changing, (2.4) will never be reached and thus r* and the subsequent
rates associated with Tableaux 3, 4, ... will *remain* imaginary. And
the TSS refutations of the Okishio theorem do rely on continual technical
change--which the theorem DOES NOT prohibit. Indeed, to have any relevance
to Marx's law of the FRP at all, the theorem *cannot* exclude the possibility
of continual technical change, since Marx's Vol. III, Ch. 13 statement of
the law clearly states that it feres to a continual rise in productivity,
a continual rise in the composition of capital, etc ["feres" in line above
should be "refers"].

Once one DOES examine the dynamic set off by the technical change, one needs
a theory of the determination of value and the profit rate. One can no
longer *postulate* that input and output prices must be equal. So the
physical quantities, even with a uniform profitability assumption, no longer
are sufficient to determine the profit rate. So, in keeping with Marx--
since, again, the Okishio theorem is supposed to be a refutation of *Marx's*
law of the FRP--John, myself, and Alan all (independently) invoked Marx's
theory of the determination of value by labor-time as we understand it.
Together with continual mechanization, this can lead to a fall in the
profit rate under conditions in which the imaginary (which Roemer calls
"actual" at least 13 times in Ch. 5 of AFMET) Okishio profit rate must rise,
even if EQUILIBRIUM exists in the sense that the profit rate is equalized
each period. (Note that Roemer, in the above, *never* defines exactly
what he means by "equilibrium." While there are good reasons to think that
intercapitalist competition leads to a tendency for the profit rate to be
equalized, there is really no good reason to think that any real process
leads to stationary prices, given the results of the abovementioned
investigations of this issue. I do not consider iterating a matrix to
be a meaningful indication of convergence to stationary prices.)

The most amusing aspect of Roemer's discussion in the AEP, however, is
his almost self-parodying use of the Method of Substitutionism to try to
link the mathematical results to Marx's law. Roemer begins (pp. 154-55)
by quoting from Vol. III, p. 264 (Progress):

"[Sentence 1:] No capitalist ever voluntarily introduces a new method of
production ... so long as it reduces the rate of profit. [But he gets
superprofit. This is because] [Sentence 6:] His method of production
stands above the social average. [Sentence 7:] But competiton makes it
general and subject to the general law. [Sentence 8:] There follows a
fall in the rate of profit perhaps first in this sphere of production,
and eventually it achieves a balance with the rest which is, therefore,
wholly independent of the will of the capitalist."

Roemer then begins substituting his interpretation for what Marx said,
without even acknowledging that interpretation is involved or that
other interpretations are possible: "Sentence seven points out that
eventually, through competition, the new equilibrium Tableau 2 is estab-
lished. And, finally [in sentence 8], Marx's intuition fails him, when in
the last sentence he says that the new rate of profit r* will be less than

Now one can argue that it is an utterly arbitrary use to language to think
that sentence 7 can mean anything else but the stationary price equations
(2.4') of Tableau 2, but I would find this suggestion laughable, as I would
the related suggestion that "fall in the rate of profit" can only mean that
r* < r, if language is being used rationally. This is particularly because,
as Roemer notes in concluding this paragraph (p. 155): "since the general
proof that the [sic] equilibrium profit rate r* is greater than r relies on
the Froebenius-Perron theorems, not discovered until a generation after
Marx's death, we may not fault him too much for his mistaken intuition."

In other words, in order to reduce Marx to the level of a third-rate graduate
student, Roemer says Marx was not too much at fault because he didn't know
the math. But to attempt to tie what Marx actually claimed about the FRP
to an otherwise irrelevant matrix algebra theorem, well, then, certainly,
Marx was simply translating Tableau 2 into German in sentences 7 & 8.

To summarize: the TSS refutations of the Okishio theorem are indeed
refutations. They do NOT relax Okishio's or Roemer's *assumptions*. They
show that Oksihio's and Roemer's *conclusions* are deductively invalid.
And they show that, WITHOUT invoking alternate assumptions about capital-
ists' behavior or the nature of technical change, the rate of profit can
fall under conditions in which the Okishio/Roemer simultaneist profit rate
must rise. These demonstrations rely crucially on Marx's theory of the
determination of value by labor-time because, as I show, if value is not
determined by labor-time and instead, only relative prices affect the
profit rate, then we get the Okishio/Roemer results in the one-commodity
case (at least).

Finally, let me briefly address Gil's point that "as I understand Andrew's
argument, it requires that capitalists calculate rates of profit on the
basis of *historical* costs of capital (i.e., that capitalists fail to
ignore sunk costs). ... Now, I doubt that this is the case ...."

My refutation of Okishio's theorem *requires* nothing concerning how
capitalists calculate. The Okishio theorem is meant to refute *Marx's*
law of the FRP. My work refutes the refutation. So what is at issue
is how MARX calculates the profit rate. Second, the TSS refutations of
the Okishio theorem do not actually require historical cost valuation
at all, because they do not require fixed capital. With circulating
capital valued at pre-production reproduction (not historical) cost, and
input prices and output prices that differ, the profit rate can fall due
to excess of pre-production over post-production prices (values).

But I DO think that capitalists calculate their rate of return on investment
by using historical costs (which is not to say that they use only this
measure). At least that's what they necessarily do if they use the
internal rate of return formula, which is simply an application of a
present value formula. In both versions of my refutation of Okishio, I
follow Roemer in assuming fixed capital that lasts forever. For rate
of return calculations, this is analytically equivalent to a consol or
perpetual bond; it never matures, and yields a never-ending stream of
interest payments.

It is well known that in this case (taking the limit of the sum of the
infinite series), the present-value determined price of the bond is
given by

P = C/r

where P is price, C is the coupon interest each year, and r is the annual
rate of return.

Now, let's imagine a financial community of simultaneists. One of them
buys a consol for $1000, which pays interest of $100 each year. Hence,
r = 10%. But after a few years, the bond's price in the market has
collapsed; its "replacement cost" is now only $500. It still gives an
infinite stream of interest payments, so the above formula still applies.
Being a good simultaneist, and having read Ch. 14 of Vol. III of
_Capital_, but especially having read the Robinson-Sweezy-Okishio-et al.
interpretations of that chapter, our investor figures that the cheapening
of his/her asset raises his/her profit rate ("cheapening of the elements
of constant capital"). To compute the exact amount, s/he now puts $500
instead of $1000 for P on the left-hand side. C remains $100. And, to
his/her delight, our simultaneist investor finds that his/her profit rate
has doubled to 20%.

S/he is overjoyed. And since the rest of the financial community is made
up of simultaneists, all of them continually cheer whenever bond prices
take a dive. They continually cheer good economic news that tends to make
bond prices fall. And they put pressure on the central banks to inflate
the economy, in the belief that this will lower bond prices. Since the
central banks suck up to the financial community, we find them continually
trying to inflate.

And all of the simultaneist investors secretly dream at night that their
assets become totally worthless, giving them an infinite rate of return!

Sweet dreams

Andrew Kliman