I am way behind on the latest phases of the Okishio theorem discussion.
Unfortunately, it seems that I'll have to make discrete replies here and
there, due to lack of time.
For now, I want to address just one point that Duncan made in ope-l 3056:
"Consider a purely circulating capital, one-good model in which it requires
a(t) units of output in period t-1 to produce 1 unit of output in period t.
(Suppose for simplicity that the goods consumed by workers are included in the
coefficient a(t).) ... the path of the stationary solutions is not the same as
the dynamical path of the system, and ... the money numeraire profit rate will
be lower than the commodity numeraire profit rate. ...
"But unless the rate of technical change (and hence the rate of fall of money
prices of output) is accelerating, this model does not yield a falling rate of
profit. The money rate of profit is lower than the output rate because of the
falling money value of output, but the difference does not diverge, and the
money rate of profit does not decline just because there is technical change."
I have spent a lot of time playing with such examples, and I don't think
Duncan's assertions are right, in general. To take one extreme example,
assume that
1. a(t) = a(t+1) = 0.8, a constant
2. workers' consumption is zero
3. the living labor extracted is period is constant
4. the growth rate of output is 25%
5. $1 = 1 hour of living labor
6. in the initial period, period 0, 4 units of the commodity and 100 units
of living labor are used to produce 5 units of the commodity
7. in the initial period, the input and output prices are equal
Assumptions 1. and 2. imply that the simultaneist or "commodity numeraire"
profit rate will equal 1/a - 1 = 1/0.8 -1 = 25%, throughout all time.
If value is determined by labor-time (as understood in the TSS interpretation
of Marx's value theory), then assumption 1., together with assumptions 3.-6.
imply that the path of the unit price will be determined by
P(t)*5*(1.25)^t = P(t-1)*4*(1.25)^t + 100
and using assumption 7., this implies that P(0) = P(1) = 100.
Running this on the spreadsheet, and calculating the profit rate as r(t) =
[P(t)/P(t-1)](1/a) -1, about which Duncan and I agree, we get the following
figures (where P is the *input* price of the period):
t P r (%)
0 100 25
1 100 20
2 96 16.7
3 89.60 14.3
6 65.54 10
16 14.07 5
21 5.76 4
36 0.32 2.5
46 0.04 2
96 1e-6 1
196 5e-16 0.5
and the profit rate eventually gets extremely close to zero.
{This example can also be solved analytically. The solution for the path of
the unit price, remembering that P(0) = 100, is
P(t) = [100 + 25*t]*(0.8)^t .
Plugging this into the profit rate formula, one gets, after some manipulation,
r(t) = 1/(4 + t). }
I would not say that technical change is accelerating in this example, but
that the rate of change is constant. Nonetheless, with value determined by
labor-time, the money rate of profit does fall and there is an increasing
difference between it and the simultaneist or "commodity numeraire" rate of
profit.
It is also easy to relax a lot of the assumptions, and still produce examples
in which the temporalist and simultaneist profit rates diverge systematically.
One can, for instance, let the extraction of living labor increase or
decrease over time, and/or introduce positive wages, and/or permit some
surplus-value to be consumed (which assumptions 4. and 6., together,
precluded). For example, assume that extraction of living labor increases but
at a slower rate than output increases, that the real wage rate is constant
and positive, and that a(t) is constant *or* falling. It is possible to
produce an infinite number of examples (maybe uncountably infinite, though,
since I don't understand Cantor, I can't say), which satisfy these conditions,
in which the simultaneist profit rate approaches a limit larger than the
starting profit rate while the temporalist profit rate approaches a limit
smaller than the starting profit. This refutes the Okishio theorem, even in
the absence of fixed capital.
For one such example, assume that living labor increases by 0.80er period,
and that output increases by 5%. Let the real wage rate (per unit of living
labor) equal 1/3100 and be constant. Retain assumptions 1., 5., 6. and 7.
Both the temporalist and the simultaneist profit rates in period 0 equal 24%.
The simultaneist rate eventually converges on the limit of 25%, but the
temporalist profit rate converges on the limit of 20%.
Duncan's inference that a constant rate of technical change implies a constant
dynamic profit rate assumes that, if the rate of technical change is constant,
the rate of change in money prices will also be constant. I agree that a
constant rate of change in prices would yield the results he describes. Yet I
think the above example exemplifies (as an example should do :) ) that a
constant rate of technical change is compatible with a *variable* rate of
change in prices, and thus with profit rate dynamics different from those he
describes. {In the above example, the rate of change of the unit price is
[P(t) - P(t-1)]/P(t-1) = -1/[20*(1/t) + 5], for t > 0, which is
time-dependent.}
Duncan's inferences may have been based on an implicit assumption that no
constant capital-value is transferred to the value of the product (as in some
of my examples we have analyzed). If value is determined by labor-time, then
this assumption would imply that a constant rate of technical change leads to
a constant rate of change in the unit value of the commodity. But his
discussion of the profit rate does assume circulating constant capital, which,
along with depreciation of fixed capital, is always present in capitalism.
Another possibility is that Duncan's inferences were based on the path of the
stationary solution to the dynamic price equation, rather than system's actual
dynamic path. If I had constrained input and output prices to be equal, in
the traditional manner, I would have gotten the path of the simultaneist price
Ps(t) = 100*(0.8)^t ,which does decline at a constant rate (- 20%).
The situation Duncan poses thus mirrors very closely Alan's cobweb model with
variable slope coefficients on the supply and demand functions; there's a
systematic and growing difference between the actual time path of the price
and the time path of the steady-state "solution" for the price (given the TSS
price equation).
I agree 100% with Duncan that use of the method of comparative statics is
justified when (and I'd say only when) the actual time-path of a dynamical
system converges on the path of its steady-state or stationary counterpart, or
at least does not diverge and stays "close." As Duncan notes, the only way to
tell is to examine the behavior of the dynamical system. What the present
examples show is that the conditions needed to generate
constant-rate-of-change prices seem to be rather stronger than might have been
thought, and that therefore the probability of nondivergence between the
actual profit rate and its simultaneist counterpart is rather smaller than
might otherwise have been thought. The present examples thus disclose the
perils of failing to examine the behavior of the actual dynamical system.
This is why I think Alan was correct to insist that people write down and
study their own dynamic equations. Inferences based on intuition can be
wrong, particularly because the intuition of almost all of us, myself
included, is informed by years of training in comparative statics.
Andrew Kliman