I think I understand Andrew's example, but I still don't understand his
profit
rate conclusion.
We are dealing with an economy that has nondepreciating factories of
different "vintages", t = 0,1,.... (I assume that we measure the fixed
capital in
each factory in the same units as output.) At time 0 some capitalists
invested
Fo in factories which produce output Xo in t = 1 and forever after. At
time 1
there is a technical innovation, and other capitalists invest Fo(c-1)b in
factories, which produces output Xo(b-1)b in t = 2 and forever after. We
assume that c<b, so that the new factories are more productive. At time 2
still
other capitalists invest Fo(c-1)b^2 in factories, which produce output
Xo(b-
1)b^2. Thus these factories are just as productive as the ones installed
in time
1, though there are more of them.
Assuming that 1$ is the equivalent of 1 unit of social labor, Andrew says
that
money prices will follow the path p[0] No/Xo, and p[t+1] = (No/Xo)(d/b)^t
for
t = 0,1,.... I have no problem with this.
First, I calculated the internal rate of return to the capitalists who
invested in t
= 0, taking output as the numeraire, Rx[0]. I get Rx[0] = Xo/Fo, which
seems to
agree with Andrew's result. Then I calculated the internal rate of return
to
the capitalists who invested in t = 1, again taking output as the
numeraire,
Rx[1]. I get Rx[1] = (Xo/Fo)((b-1)/(c-1)), by the same method. This does
not
seem to agree with Andrew's result, which appears to be
(Xo/Fo)((d-1)/(b-1)).
Since this may be a point of misunderstanding, let me summarize my
calculation here:
Investment = Fo(c-1)b = Present discounted value of returns = Xo(c-
1)bSum[t=1, Infinity](1+Rx[1])^-t, which yields Rx[1] =
(Xo/Fo)((b-1)/(c-1)).
On my calculation, then, the internal rate of return on the period 1 and
later
investments is higher, given Andrew's assumption b > c > d> 1 than the
internal rate of return on the investments in period 0.
Then I repeated these calculations using money as the numeraire, with
Andrew's price series. For the internal rate of return to the investments
of
the period 0 capitalists calculated in money, R$[0], I get R$[0] =
(Xo/Fo)-(1-
(d/b)) = Rx[0] - (1-(d/b)) < Rx[0] (on the assumption that b > d) because
of the
fall in the money value of output. The parallel calculation for R$[1] =
Rx[1]
-(1-(d/b)) < Rx[1]. The internal rate of return calculated using money as
the
numeraire has risen by exactly the same amount between period 0 and period
1 (and later periods) as the internal rate of return calculated with
output as
the numeraire.
Perhaps the divergence between my conclusions and Andrew's lies in
Andrew's remark "...the social return per period is the increment to labor
extraction...for the entrant of period t+1." The problem is that although
these
capitalists continue to employ the same amount of labor in all periods
after
their initial investment, this labor is devalorized by the technical
change, as
reflected correctly in Andrew's price series. As a result, the future
employed
labor on any vintage of factory has to be discounted by this change in
price.
I think there is an important insight into capitalist reality behind this
scenario, which is that the profits of innovating capitalists come partly
at the
expense of the devaluation of existing capital. This is a problem for the
owners of existing capital, and may be a very big problem in some
circumstances (for example, if they have financed this capital with fixed-
interest debt). It's not clear that it is a systemic problem, however,
since
technical change of the kind posited in Andrew's example actually improves
the long-run conditions of reproduction of capital, or that this
corresponds to
the fall in the general rate of profit Marx discussed in Volume III of
Capital.
Duncan