[OPE-L:3950] Re: Frank Thompson's Theorem

andrew kliman (Andrew_Kliman@msn.com)
Wed, 8 Jan 1997 13:15:42 -0800 (PST)

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A response to Gil's ope-l 3946.

First, let me say that I *would* indeed like to read Gil's and David's papers.
Too much occurs at once at these conferences and I lose track of what I'm
doing. I simply forgot to ask Gil and David for copies of their papers.
Please do send me copies, guys, okay? I should also note that I was also
under the impression that Gil's paper was not on the same topic as the
Thompson and Laibman papers.

Gil writes: "I----------
have repeatedly found that TSS advocates on this list have made claims about
supposedly intrinsic features of the 'simultaneist' approach which are
spurious at best."

Gee, spurious *at best*? I hate to think about what the worst may be. It is
of course impossible to reply to this claim of Gil's without knowing what
exactly he thinks is spurious at best. All I can do is comment on his
objections to my interpretation of Frank Thompson's theorem, which I'll do in
a moment, after getting some other stuff out of the way.

It is also difficult to respond to Gil's comment that "The 'sequentialist'
aspect, where it is not logically suspect, is as far as I can tell essentially
beside the point, at least with respect to the question of establishing a
tendency for the rate of profit to fall." I know that the notion that
sequentialism or successivism is "logically suspect" is an old one, shared by
Marshall, Walras, and Bortkiewicz, but I'm not aware of Gil or anyone else on
this list having made such a claim before, so I don't know what Gil's talking
about and therefore can't respond. In general, my view is that simultaneism
is logically suspect, because it permits the price of a good to have two
prices at the same moment (as output of one period and as input of the next).

As far as "sequentialism" being beside the point with respect to the FRP, let
me note that in a one-sector circulating capital model, or in a multisector
circulating capital model in which relative prices aren't changing, the TSS
and simultaneist profit rates, RT and RS, are related in the following way:

1+ RT = (1+RS)*(P[out]/P[in])

where P is the price of the one good, or the numeraire good. Thus the fact
that our profit rate in these cases is lower is due precisely to the fact that
input and output prices aren't determined simultaneously.

Gil writes: "David and my models also have the virtue, not shared by Andrew's
1988 account, of incorporating Marx's explicitly stated presumption that the
rate of profit is self-regulating via *endogenous* variations in the rate of
capital accumulation."

Gil's absolutely right about the endogenity of accumulation in Marx and its
exogeneity in my examples (not only 1988, also 1996). It is a good thing, I
agree, to understand growth as endogenous. Inasmuch as I've been interested
in refuting the Okishio theorem and not modeling the accumulation process as
such, I decided to make the examples as simple as possible and show that
there's nothing up my sleeve.

Now on to the Thompson theorem. Gil writes:

"Contrary to Andrew's claim, Frank's result has nothing whatsoever to do with
simultaneism _per se_ .... . Using somewhat different analytical procedures,
our [David Laibman and Gil Skillman] results show that the connection between
the composition of capital and the rate of profit depends not on simultaneism
vs. sequentialism, but rather static vs. dynamic contexts.

"Employing absolutely simultaneist, but dynamic, models of an accumulating
capitalist economy, we show that a trend of (viable) VCC-raising technical
change leads to a tendential decline in the steady-state (my terminology) or
'consistent-path' (David's language) rate of profit under standard assumptions
about production conditions."

I note, first, that this second paragraph does NOT contradict the Thompson
theorem. The paper of Frank's which I read is quite clear that --- and the
math is consistent with the proposition that --- there can be Okishio-viable
technical change that both raises the VCC and leads to a lower equilibrium
(uniform) profit rate. What Frank says, however, and what his theorem
implies, is that, in such cases, the rise in the VCC is not a CAUSE of the
fall in the profit rate. Instead the rising VCC tends to COUNTERACT the fall
caused by the rise in the real wage. (In a similar manner, the Okishio
theorem doesn't say that a falling profit rate is incompatible with viable,
labor-saving, technical change. If real wages rise sufficiently along with
the technical change, a falling profit rate is possible. But the technical
change itself is not a CAUSE of the FRP. Instead it tends to COUNTERACT the
fall caused by the rise in the real wage.)

I hope to disprove the claim that statics vs. dynamics, rather than
simultaneism vs. successivism, is the issue. Of course, Gil may be using
these terms differently, so let me define what I mean. Simultaneism refers to
the postulate that input and output prices of a production period are
necessarily equal (in the given context); successivism (or sequentialism)
refers to the denial of this postulate. Static refers to the situation in
which the things under consideration are not changing, including comparative
statics, wherein one compares the magnitudes of variables in two or more
unchanging situations and draws inferences from them. Dynamic refers to the
situation in which the things under consideration are changing.

Now, it is true that Frank's proof is comparative static. But I doubt that
this is what causes his results. His "model" has the following additional
properties: (a) one-sector, (b) no fixed capital, (c) Okishio-viable
technical change, (d) the Minimal Assumption that falling or constant labor
demand does not lead to a rise in the real wage, (e) the VCC rises, and (f)
SIMULTANEISM (as defined above). He shows that the SIMULTANEIST profit rate
will either rise or remain unchanged --- unless labor demand rises (i.e.,
unless the percentage change in "capital" outstrips the percentage change in
the VCC). Moreover, if the SIMULTANEIST profit rate does fall, this fall can
be due only to the rise in labor demand, which has pushed up the real wage,
with the rising real wage being the proximate cause of the FRP. The rise in
the VCC is not a cause, but a counteracting factor.

I would be absolutely flabbergasted if anyone could produce even a single
dynamic example that satisfies (a) through (f), that exhibits nonincreasing
labor demand, and that results in a falling SIMULTANEIST profit rate. In
fact, let me prove its impossibility. Let K be the amount of means of
production, L the amount of living labor, w the real wage rate, X the amount
of output, and v the SIMULTANEIST unit value of the single good. The
SIMULTANEIST profit rate is defined as r(t) = v(t)X(t)/[v(t){K(t) + w(t)L(t)}]
- 1 = X(t)/[K(t) + w(t)L(t)] - 1. Now, Okishio-viable technical change
requires that any new technique in which K(t+h) and L(t+h) produce X(t+h)

v(t)X(t+h) > v(t)*[K(t+h) + w(t)L(t+h)]*(1+ r(t))

so that, canceling v(t) and rearranging:

X(t+h)/[K(t+h) + w(t)L(t+h)] > 1 + r(t).

Now, if labor demand is not increasing, then the minimal assumption implies
that w(t+h) < or = w(t), so that

X(t+h)/[K(t+h) + w(t+h)L(t+h)] > or = X(t+h)/[K(t+h) + w(t)L(t+h)]

and therefore that

X(t+h)/[K(t+h) + w(t+h)L(t+h)] > or = 1 + r(t).

But the left hand side is nothing other than 1 + r(t+h), so that

1 + r(t+h) > or = 1 + r(t),

and the profit rate cannot fall. The proof has employed (a), (b), (c), (d),
and (f), and has shown that NO dynamic pattern of technical change that
satisfies these conditions can lead to a falling profit rate. Rather, *all*
must lead to a rising (or constant, in the special case of a nondecreasing
real wage) profit rate, including dynamic patterns of technical change that
raise the VCC.

The only question that remains is whether there exist any patterns of viable
technical changes that do raise the VCC. There are. What is necessary is
that the rate of growth of output, X, outstrip the rate of growth of the wage
bill, wL. It is a bit tedious to show this, so I won't, but one simply takes
the terms of the VCC, plugs them into the viability condition, and
manipulates. (Note that the conditions of the problem imply that wL actually
falls or remains constant.)

Dynamics have absolutely nothing to do with any of this. The reason is quite
simple. The SIMULTANEIST profit rate depends solely on the physical and real
wage coefficients of the current period, or moment, in question; it is
path-independent. This is true in a static model, but it's equally true in a
dynamic model. Whether the context is static or dynamic is just not relevant.

What *is* relevant is SUCCESSIVISM vs. SIMULTANEISM. The crux of Frank's
proof is that, if labor demand is not increasing, if the real wage is a
nondecreasing function of labor demand, AND IF the only thing that can cause
the profit rate to fall is a rising real wage, then
VCC-increasing/labor-demand-nonincreasing technical changes can't lead to an
FRP. Given viability and all that, then it is indeed the case that the only
thing that can cause the SIMULTANEIST profit rate to fall is a rising real
wage, so the Thompson theorem does indeed show that
VCC-increasing/labor-demand-nonincreasing technical changes can't lead to a
falling SIMULTANEIST profit rate. On the other hand, it is not true that the
only thing that can cause the SUCCESSIVIST profit rate to fall is a rising
real wage, so the Thompson theorem simply doesn't apply.

In sum, I think I have shown that, contrary to Gil's claim, Frank's result has
everything to do with simultaneism _per se_, and nothing to do with statics
_per se_. I reiterate my view that his theorem "is a beautiful example of the
incompatibility of Marx's value theory and simultaneism."

Andrew Kliman