A reply to Duncan's ope-l 2344. I really do appreciate Duncan's taking
the time to slog through these equations with me, and I can definitely
sympathize from experience with the annoyance of all the little things
that can go wrong. But I think it has been helpful, because I too
sense that we're on the verge of clarifying the technicalities and
can soon get down to brass tacks.
My results match Duncan's for the "output as numeraire" IRRs, both for
the initial, period 0, investment, and for each of the subsequent
period 1, ... investments.
My results match Duncan's for the "money as numeraire" IRR on the
period 0 investment, where the returns each period, following
Duncan, are the returns *received* by the investor, and computed
on the basis of the commodity's social price (or value).
Calculating the IRR on subsequent investments according to the
same conception, I get a *slightly* different answer:
R$[1] = R$[2] = ... = {(d/b)Xo(b-1)/[Fo(c-1)]} - (1 - d/b).
This differs from Duncan only because he has (b/d) instead of the
first (d/b).
I also get R$[1] = R$[2] = ... = R$[0] + {(d/b)Xo(b-c)/[Fo(c-1)]},
which differs again slightly from Duncan's calculation.
But the differences are really not important, I think. I can agree
(and have to, given my result) that the IRR rises, in the sense
that the IRR to those investing after period 0 is greater than the
IRR of period 0 investors.
Yet I think this approach to the question of the tendency of the
profit rate contains one basic ambiguity--are the subsequent
investors' IRRs higher than the period 0 investors' IRRs because
they are generating higher profits, or because they are appropriating
superprofits in the market by selling at a social price higher than
their individual price (monetary expression of individual value)?
If it is the latter which is true, as I think it is, then it is
quite possible (and I think it is the case) that in fact the period
0 investors are *producing* more profit (surplus-value) each period
relative to their initial investment than the subsequent periods'
investors are.
Hence, a rising IRR based on profit *received* by the individual
firm does not permit us to conclude that the general rate of profit
in the economy rises. If indeed the later investors are getting
a higher IRR only because of superprofits, and in the absence of
superprofits their IRRs would be lower, then the general rate of
profit must be falling. (This follows from the notion that value
can only be redistributed, not altered in magnitude, in exchange.)
I, for one, can think of only 2 ways to ascertain whether the
higher IRRs of the later investors are masking a falling general
rate of profit:
(1) Directly calculate the general rate of profit. I would compute
it as r[t] = S[t]/K[t], where S is surplus-value and K is capital
advanced. Given the assumptions Duncan and I are working with, we
have S[t] = N[t] = No(d)^t. K[t] is given by K[t+1] = K[t] + I[t],
where I is investment, and I[t] = p[t+1]{F[t+1] - F[t]} =
(No/Xo)Fo(c-1)(d)^t. Doing these computations gives a falling rate
of profit as time goes on, given c > d.
This way of approaching things if what started the discussion,
however, because it is not at all obvious that the surplus-value of
one period, divided by the historical value of capital, is a relevant
profit rate. It certainly does not look like the IRR, the latter
being widely used and accepted in the real financial and business
world.
Thus, to get at the issue of the tendency of the profit rate by looking
explicitly at IRRs, I have tried:
(2) Calculate the IRR on each particular investment. To be sure I'm
not letting redistribution of value get in the way, I measure the
additional profit (surplus-value) each new investment generates
instead of the profit the investor receives.
Now, Duncan raises questions about the interpretation of the story
that goes along with this calculation. I want to address this in
a moment. But first, I want to point out that methods (1) and (2)
are *equivalent*.
Given the particular assumptions of my example, each investment's
returns *generated* per period are constant, though unequal to
those of other periods' investments. Calling the per period return
on the jth investment DjS, and the amount invested Ij, the IRR of
the jth investment is Rj = DjS/Ij.
Now, if we want to compute the (weighted) average economy-wide IRR
in period t, av.R[t], it is
av.R[t] = sum from 0 to t of wj*Rj
where wj is the weight attached to the jth IRR. Weight each IRR
according to the share of this investment in total investment:
wj = Ij/(sum from 0 to t of Ij).
But this means that
av.R[t] = sum from 0 to t of (DjS/[sum from 0 to t of Ij]).
But the sum of DjS = S[t] and the sum of Ij = K[t], so by this
weighting method, av.R[t] = S[t]/K[t].
Note also that the sum of weights is unity.
Hence, at least in the example(s) Duncan and I have been working with,
the ratio of per period profit to the historical value of capital is
*equivalent* to the average internal rate of return, where the returns
are those generated, not those received.
I'm not sure I understand Duncan's question about the scenario I
assume in computing the IRR by using returns received instead of
returns generated (sorry--that's BACKWARDS; I can't fix it now). I'll
try to answer it with a numerical example. But I think several
different stories might be told. The real crux of the matter is,
I think, that however one tells the story, any new investment raises
surplus-value by the difference between what the per-period amount
of labor extracted is, given this investment, and what it would
be without this investment.
Ok, here's the example:
new entrants
of period F X N P(out) I incr IRR IRR
------------ ------ ---- ---- ------ ------ -------- ----
0 16,000 8000 8000 1 16,000 50 %
1 1600 1600 400 84/96 1600 25 % 47.73
2 1920 1920 420 882/1152 1680 25 % 45.75
P(out) is the output price of the period = the sum of the N's thru
that period divided by the sum of X's thru that period. (Each period's
entrants *continue* to produce, at the same level, in all subsequent
periods. I is the money investment undertaken once and for all
by the entrants of that period, = to the F they acquire (once and
for all) times the output price of the last period. incr IRR is the
incremental *social* IRR of the particular investment = N/I. IRR
is the weighted average IRR, computed on the basis of the particular
IRRs of all entrants thru this period.
The example conforms to my earlier ones. It assumes b = 1.2, c = 1.1
and d = 1.05. It can be continued ad infinitum by letting each
subsequent new F and X be 1.2 time the previous one, and each new
N be 1.05 times the previous one. All subsequent incr IRRs will be
25%, and IRR will gradually fall to 25%. (Stationary price calculations
will give the same period 0 figures, but each subsequent incr IRR will
be 100%, so the (Okishian) profit rate will rise from 50% to 1000ver
time.)
To the extent that I understand Duncan's question, I will say no, the
existing firms do not benefit from the technical changes--their output,
labor used, and F used remain the same each period. They continue to
produce with the same technique. I don't know if this clarifies
anything, but maybe the discussion will continue more easily with
sum numbers instead of a bunch of weird symbols.
That's all I can think of for now.
Andrew