Some comments on Andrew's reply:
Andrew:
A reply to Duncan's ope-l 2344....
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.
Duncan:
The model, to remind us, is one in which output grows according to the
rule X[t] = Xob^t, labor as N[t] = Nod^t, and fixed capital as F[t] =
Fo((b-c)/(b-1)) + Fo((c-1)/(b-1))b^t. We agree that the price of output,
assuming one unit of labor remains equivalent to $1 will follow the path
p[0] = p[1] = No/Xo, p[t] = (No/Xo)(d/b)^t for t = 1,2,.... We are
assuming b > c > d > 1. Below Andrew presents an example where Xo/Fo = .5,
b = 1.2, c = 1.1, d = 1.05.
My results for the internal rate of return to investments in period 0 and
period 1 and thereafter, taking output as the numeraire, were Rx[0] =
(Xo/Fo), Rx[1] = Rx[2] = ... = (Xo/Fo)((b-1)/(c-1). Since b > c, this
internal rate of return rises (for Andrew's example numbers, from 50% to
100%).
Andrew:
My results match Duncan's for the "money as numeraire" IRR on the
period 0 investment, where the returns each period, following
Duncan, are the returns *received* by the investor, and computed
on the basis of the commodity's social price (or value).
Duncan:
This was R$[0] = (Xo/Fo)-(1-(d/b)). For Andrew's example, this works out
to 37.5%.
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).
Duncan:
On reviewing my computation, I agree with this result of Andrew's. For
Andrew's example this works out to 75%.
Andrew:
I also get R$[1] = R$[2] = ... = R$[0] + {(d/b)Xo(b-c)/[Fo(c-1)]},
which differs again slightly from Duncan's calculation.
Duncan:
When I go over this once again, I get R$[1] = R$[2] = ... = R$[0] +
(Xo/Fo)((d/b)((b-1)/(c-1))-1). This is of some interest, since it
indicates that the money internal rate of return to the later investments
might rise or fall, depending on the relative size of d and c. As we have
seen, in Andrew's example this rate of return is actually higher than for
the period 0 investments.
Andrew:
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)?
Duncan:
This is a tricky question to answer without specifying exactly what is
happening to the allocation of labor across the different vintages of
capital. In the table Andrew presents below, he assumes that there is no
change in the productivity of labor in each factory after it is built. But
there might be "disembodied" labor-augmenting technical change, reducing
the required labor force in each vintage of factory uniformly.
Andrew:
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.
Duncan:
Well, they're employing more labor, but this labor produces less output
per worker. Marx is pretty explicit that workers employed on obsolete
capital produce less value than those employed on the current good
practice capital (and refers to this qualification as "socially necessary"
labor).
Andrew:
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.
Duncan:
I agree with this conclusion for the "general rate of profit", which we
seem to be defining as the profit of a given period divided by the
historical cost of the capital in existence in that period. For the
record, I get
GR$[t] = (Xo/Fo)(d-1)(1/((c-1)-(d^-t)(c-d))),
which clearly declines over time if c > d. This formula reproduces
Andrew's calculations in the Table below.
We can also calculate the "general rate of profit" with output as the
numeraire. For this I get:
(Xo/Fo)(b-1)(1/((c-1)+(b^-t)(b-c)))
which clearly rises over time if b > c > 1.
Andrew:
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.
Duncan:
I think this is the crux of the matter. We might observe that the "general
rate of profit" for every vintage of factory declines in this scenario,
since each factory continues to produce the same output and prices are
constantly falling. We might also observe that this is quite common in
real capitalist economies, whenever a capitalist invests in producing a
product that is anticipated to fall in money value over time. As a result
the internal rate of return turns out to depend most heavily on the
returns in the first periods of operation of the facility, and capitalists
horizons appear to be "short" in these markets, quite reasonably, given
the anticipated fall in the price of the output.
Here some of Jerry's questions about the appropriateness of the
assumptions in this model are of interest. If the capital had any finite
fixed lifetime, it would eventually depreciate and go off the books, which
would keep the "general rate of profit" closer to the anticipated IRR.
(Dumenil and Levy calculate the IRR for the US economy in "Economics of
the Profit Rate" precisely to check how large a bias this effect might
introduce, and it turns out to be small.) Furthermore, if the workers did
have to be paid some real wage, even a very small one, the older factories
would eventually be closed because they couldn't cover their variable
costs represented by the wage bill, so this would also limit the degree to
which the "general rate of profit" would diverge from the IRR.
(Andrew then presents an alternative, equivalent method of calculating the
general rate of profit in money terms. I agree with his conclusion that
the two methods, at least in this setting, are equivalent, and won't copy
this part of his posting.)
Andrew:
I'm not sure I understand Duncan's question about the scenario I
assume in computing the IRR by using returns generated instead of
returns received (correcting Andrew's typo: DKF). 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
Duncan:
This makes matters clear, but this isn't the only scenario consistent with
the basic assumptions of this model, since the labor might be spread
uniformly across the factories of all the different vintages. Here the
assumption that the real wage is zero does make a qualitative difference,
since, as I've pointed out above, even a small real wage would prevent the
older vintages of factories from operating indefinitely.
Andrew:
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.
Duncan:
I think it is confusing to call this the IRR, (internal rate of return),
instead of "GPR" (general profit rate).
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.)
Duncan:
We now have pretty much agreement on all the computational details. The
questions to be addressed are: (1)is a falling general rate of profit in
this type of scenario a problem for the capitalist society? (2)is it the
falling rate of profit Ricardo and Smith referred to? (3)is it the falling
rate of profit that Marx studied in Volume III of Capital?
My tentative answers to these questions are: (1) No. This fall in the
"general rate of profit" corresponds to new and better techniques
devaluing old ones. This is a potential problem for the owners of old
factories, if they haven't correctly anticipated the path of prices, but
not for the system as a whole, since this type of technical change
actually improves the conditions for the expanded reproduction of capital.
(2) No, though neither Smith nor Ricardo are very explicit about whether
or not they are taking technical change into account. Smith seems to be
talking about the fall in the rate of profit in a sector as capital enters
in pursuit of higher profits. Ricardo is talking about the fall in profits
due to rising rents as capital accumulation and population growth occur.
(3) No. Marx understood that innovation devalued existing capital, but
also understood that this effect was limited by the lifetime of capital
facilities which, while long, is not infinite. He was comparing the
profitability of whole capitalist systems at very different stages of
development, over a time scale in which this effect would disappear, and
his insight concerned the historical pattern of labor-augmenting,
capital-using technical change characteristic of much of capitalist
development.
Yours,
Duncan