[OPE-L:3952] Re: Frank Thompson's Theorem

Gil Skillman (gskillman@wesleyan.edu)
Thu, 9 Jan 1997 09:50:40 -0800 (PST)

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Andrew writes:
>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?

Will do, Andrew. To what address?

>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.

Here's one example among many, from last February:

"The usual justification for this [simultaneist value] equation [v = vA + L]
is the following: (i) Values are defined in Volume I, as the prices which
goods would sell for, if they exchanged at values."

Not only is this characterization spurious--nobody has or needs to justify
the simultaneist approach to value on this basis--but it's circular (thus
worse than spurious).

> 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.

But I have on at least two counts: the question of measurement, and a
question of the absence or presence of markets to equate prices across time
(possible subject to a discount factor). There's also a question remaining
in my mind of logical consistency with Marx's own comments, but since I
don't think this is resolvable on the net I'll just treat this as a personal

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).

I don't understand Andrew's point here, since the simultaneist definition of
value makes no reference to prices, so I can't see how "it permits the price
of a good to have two prices at the same moment." But let me anticipate:
suppose I buy some lumber for use in my production process, pay the going
price for it, and immediately use it in production. That price
simultaneously represents the price of an output (somebody had to produce
the lumber for me to buy it) and the price of my input.

>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.

This makes a certain (unspecified) claim about futures markets. If these
exist, input and output prices would certainly be determined simultaneously.
There is no uncertainty in this model to preclude this.

>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.

OK, but this "simplifying assumption" rules out the central point at issue:
the self-regulating nature of accumulation, and therefore of the profit
rate. It strikes me that a central conundrum that must be answered in
addressing Marx's account of the FRP is how the profit rate might be found
to fall tendentially given this self-regulating aspect of capital accumulation.

>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.)

Untrue. Our results show that a time trend of VCC-raising technical change
is a CAUSE of the fall in the steady-state (my language) or consistent-path
(David's language) rate of profit, in the sense that under the stated
conditions, there would be no such fall in the absence of that trend of
technical change. Thus in particular there can be NO sense in which a time
trend of VCC-raising technical change can be said to be a "counteracting"

>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.

My model certainly assumes this, so it is certainly simultaneist. So is

> 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.

To state the problem in that way is to miss the central point. Taking
Marx's starting point of an *accumulating* capitalist economy as given, it
is meaningless to
speak of "non-increasing labor demand." Of course it's increasing. The key
question in a *dynamic* economy concerns its *rate* of increase. Andrew is
insisting on a *comparative static* assessment of an essentially *dynamic*
phenomenon, which is exactly what David's and my approaches reject.

Again, the issue is not simultaneist vs. sequentialist. In particular, my
model satisfies Andrew's conditions (a), (c), (e) and (f) directly, and
subsumes conditions (b) and (d) as special cases.

> 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

But this stipulation is nonsensical in the context of a dynamic economy,
vitiating all that follows.

>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.

I'll repeat: the problem here is that Andrew is applying an essentially
static methodology in a dynamic context. The true reference point in a
*dynamic* economy is the *steady-state* (David would say "consistent path")
rate of profit, the rate of profit toward which the economy would tend.
This rate of profit declines over time given the stated economic conditions
and a time trend of (viable) VCC-increasing technical change.

>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.

Indeed it is. See above.

>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."

I must disagree, for the stated reasons.