# [OPE-L:4735] RE: max and actual profit rate

andrew klima (Andrew_Kliman@msn.com)
Fri, 11 Apr 1997 09:43:46 -0700 (PDT)

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A reply to Allin's ope-l 4712.

He wrote: "Andrew drew up an illustration of the point that, with a constant
workforce and V = 0, accumulation of capital (in value terms) is a necessary
and sufficient condition for a declining rate of profit (in value terms). It
just doesn't matter how you fill in the use-value side of the accounts (i.e.
what is happening to physical productivity). This seems right to me, but I'm
wondering about the relationship between this demonstration and the trajectory
of the 'actual' rate of profit. As I recall, Roemer makes the argument that
the trajectory of the _maximum_ rate of profit -- and it's the maximum we're
looking at, by setting V = 0 -- is 'irrelevant', in the sense that the max
rate can decline indefinitely while the actual profit rate (given V > 0) rises
indefinitely, with the two approaching a common asymptote from different
directions."

Good question!

Roemer, and before him Okishio, were completely right about the possibility of
the convergence of the actual and max profit rates from opposite directions,
so that the tendency of the max rate is insufficient to conclude anything
concerning the actual rate. I have noted this before on this list (ope-l
2970, 2979).

Now, the simple answer to Allin is to point out that, in my example (in which
V = 0; living labor, L = V+S = 5; and C[t+1] = C[t] + V[t] + S[t]), the
maximum profit rate, L/C, falls to ZERO as time proceeds. So if the actual
profit rate were positive initially, it must eventually fall. Note that the
convergence-from-opposite-directions argument requires that the initial actual
profit rate be lower than the asymptote.

The only problem with this answer is that Roemer and Okishio aren't talking
about the ACTUAL-maximum and ACTUAL-actual profit rates, but
STATIC-HYPOTHETICAL maximum (V = 0) and actual (V > 0) rates. So, in other
words, the decline in the maximum rate of which they speak is not a decline
brought on by a *process* of capital accumulation. It is, rather, a
hypothetical series in which the simultaneist profit rate is plotted against
the A-matrix -- NOT against TIME. In that context, it makes sense (as much
as anything makes sense in such a context) to say that if wages are positive,
the actual profit rate will be less.

But I was dealing with technical change and accumulation as a process. Here,
when workers' consumption is positive, the rate of accumulation slows down,
and the *whole trajectory* of the profit rate changes. The actual profit
rate is thus not simply the maximum rate adjusted for the fact that wages are
positive. So I don't consider the answer I gave Allin above to be strictly
correct.

Here is a more satisfactory answer:

First, for a bit of notation: I'll say that

W[t+1] = C[t] + L[t] (1)

where W is total value. Everything is being measured in labor-time. Note
that I'm not dating by *production period*, but by *point in time*. The
constant capital and living labor, expended at time t, result in commodities
having value W at time t+1.

In my example in the IWGVT at EEA panel, there was no fixed capital, L was
constant, and all value was reinvested. Let's retain these assumptions, to
make things conform as closely as possible to what I had, but allow some value
to be reinvested to purchase labor-power. This will make the list moderator
happy, as we know.

Hence,

W[t+1] = C[t+1] + V[t+1] (2)

But this means that

C[t] = W[t] - V[t],

so that

W[t+1] = W[t] - V[t] + L, (1')

or

W[t+1] = W[t] + S[t] (1'')

since S[t] = L - V[t].

Thus, we can define the (actual) rate of profit between times t and t+1 as

R[t,t+1] = S[t]/W[t] (3)

(since W[t] = C[t] + V[t]).

Now, let's investigate the conditions under which the (actual) profit rate can
rise or remain constant indefinitely.

We must have

R[t+1,t+2] > or = R[t,t+1]

i.e.,

S[t+1]/W[t+1] > or = S[t]/W[t].

Using (1''), we can write

S[t+1]/{W[t] + S[t]} > or = S[t]/W[t].

Assume that S[t] > 0. Then, cross-multiplying, we have:

S[t+1]/S[t] > or = {W[t] + S[t]}/W[t],

or

(S[t+1] - S[t])/S[t] > or = S[t]/W[t],

or

(S[t+1] - S[t])/S[t] > or = R[t,t+1],

or

the percentage change in the mass of surplus-value must exceed or equal the
existing profit rate (expressed as a percentage).

Since we have assumed S[t] > 0, then, likewise, R[t,t+1] > 0.

If R is rising over time, then clearly the percentage change in the mass of S
must be ever-increasing. This condition becomes harder to satisfy as time
proceeds, since L is constant, and S[t] &lt; L. So let's investigate the
easier-to-satisfy condition, in which the profit rate remains constant. Then
we would have

(S[t+1] - S[t])/S[t] = R[0,1],

or

S[t+1] = S[t]*(1+R[0,1]),

which means that

S[t] = S[0]*(1+R[0,1])^t .

Since R[0,1] > 0, by assumption, S[t] grows indefinitely large over time.

But this is not possible, since again L is constant, and S[t] &lt; L. Hence the
actual profit rate must eventually fall.

To recap: given

(a) no fixed capital,
(b) L is constant,
(c) W[t+1] = C[t+1] + V[t+1]
(d) W[t+1] = C[t] + L[t],
(e) R[t,t+1] = S[t]/{C[t] + V[t]}, and
(f) R[0,1] > 0,

R must eventually fall.

Note that this suffices to refute the Okishio theorem, since it is possible
for R to fall when real wages are constant and technical change is
Okishio-viable, as I showed at the EEA session. As Allin noted quite
correctly,

"It just doesn't matter how you fill in the use-value side of the accounts
(i.e. what is happening to physical productivity)."

Okay, so where's the trick? Well, the "vintage labor" version of the
simultaneist "labor theory of value" (Roberts, Foley), marginal productivity
theory, and demand-side theories of value all reject (d). Traditional
versions of the simultaneist "labor theory of value" reject the notion that
(c) is possible when unit values fall, and therefore reject (e) in such cases
as well.

Andrew Kliman (AX)