A reply to Duncan's ope-l 2287.
There was a typo in my derivation of the Incremental IRR with "money as
the numeraire," to use Duncan's terminology, or with price determined by
labor-time, to use my terminology.
I wrote that the
Incremental IRR = (return per period)/(initial $ investment)
which is correct, given the fixed capital lasts forever. I also derived
the social return per period correctly (in my view) as N(t+1) - N(t)
= No(d-1)d^t for the new investment of period t+1. And I derived the
initial $ investment of that period correctly (again, in my view) as
p(t+1)*{F(t+1) - F(t)] = (Fo/Xo)No(c-1)^dt.
The error crept in when I plugged these into the top equation. I
wrote it as
Incremental IRR = [Xo(d-1)]/[Fo(b-1)]
It SHOULD have read:
Incremental IRR - [Xo(d-1)]/[Fo(c-1)]
Sorry.
What Duncan calls the "internal rate of return to the capitalists
who invested in t = 1, ... taking output as the numeraire, Rx[1],"
I called the "Simultaneist Incremental IRR" (3d paragraph from the
end of my post). Duncan thought we might be getting different
results, but they agree: Rx[t] = SIIRR[t] = [Xo(b-1)]/[Fo(c-1)]
for all periods t > 0.
A couple of other minor technical points. At time 1, the new entrants
invest F[1] - F[o] = Fo(c-1); at time 2, F[2] - F[1] = Fo(c-1)b.
Similarly, the output that emerges at the end of period 1 = beginning
of period 2, produced by the new entrants, is X[1] - X[o] = Xo(b-1);
the new entrants of the next period produce X[2] - X[1] = Xo(b-1)b.
Again, I think this has nothing to do with the differences between us,
because the ratios come out the same with Duncan's expressions.
I also agree with Duncan's way of computing the IRR when output is
numeraire (except that there's a typo in his formula. It should be
Xo(b-1), not Xo(c-1) or Xo(c-1)b.
As for the period 0 capitalists' IRR, with money as the numeraire,
we get slightly different results because I assumed p[o] = p[1].
In other words, p[t] = (No/Xo)(d/b)^(t-1) *except* when t = 0.
I assumed this in order to start from a static equilbrium and
have the temporal and simultaneist profit rates start out the same.
The real difference seems to come when we calculate the incremental
IRR's in subsequent periods, with money as numeraire. I do not
understand how Duncan derives his expression. This may or may not be
related to his disagreement with my statement that "the social
return per period is the increment to labor extraction ... for the
entrant of period t+1." Duncan remarks:
"The problem is that although 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 labor employed on
any vintage of factory has to be discounted by this change in price."
Marx holds that the same amount of labor always adds the same amount of
value. He says this in Ch. 13 of Vol. III and many other places.
(This is not an appeal to authority; my purpose is to represent the
theory accurately.) So I don't understnad what it means for the
labor to be devalorized, unless labor is not the substance of value
and the immanent measure of value. Of course, the value of money can
change, or the monetary expression of value, or what have you, but
the present calculations assume it is constant. Hence, each "hour"
of labor adds $1 of value, no matter when that labor is extracted.
This is also the basis of my price equation. Calling p(t+1) the
unit output price of period t, the equation is
p(t+1)*X(t) = N(t),
total price of output of period t = total living labor extracted in period
t. The price of each *unit* of the commodity becomes devalorized,
because any given sum of labor-time is spread over more commodities as
productivity rises. But each unit of labor adds just as much value as
any other, whenever it is extracted.
Perhaps I'm not getting something, though.
Andrew Kliman