A reply to Andrew's OPE-L 2302:
I think Andrew and I are agreeing on a lot of the issues involved in
calculating the profit rate in his scenario, and that we can fruitfully
focus on the conceptual issues once we get over the few remaining puzzles.
I did find an error in one of my calculations, which may help to bridge
over the gap.
Just to try to nail everything down, here's what I'm assuming. At time 0
capitalists invest in factories that cost Fo units of output and continue
to produce Xo forever afterward without depreciation. The first outputs
appear at the end of period 0, that is, the beginning of period 1 so that
this output Xo is sold at period 1 prices. In period 1 capitalists invest
in another type of factory which cost Fo(c-1) units of output, and produce
output starting at the end of period 1, that is, the beginning of period
2, Xo(b-1), which continues forever. At the beginning of period t =
2,3,..., capitalists invest Fo(c-1)b^(t-1) in factories which yield a
constant stream of output Xo(b-1)b^t, but since these factories are
effectively just like the period 1 factories, the whole story can be
thought through by comparing periods 0 and 1.
Prices, as a result of the stipulation that the value of money is constant
at 1$ equals 1 unit of labor, follow the path p[0] = No/Xo, p[1] = No/Xo,
p[2] = (No/Xo)(d/b)^t-1 for t = 2,3,....
For clarity's sake, let's write out the equations determining the internal
rates of return calculated in various ways.
First, let us calculate the internal rate of return to capitalists who
invested in period 0, taking output as numeraire. For period 0, we have an
investment of Fo units of output, which is to be equated to the present
discounted value of a constant stream of output Xo starting in period 1,
to determine the period 0 rate of return in output terms, Rx[0]:
Fo = Xo((1+Rx[0])^-1 + (1+Rx[0])^-2 +...) which gives both Andrew and me
the same answer, Rx[0] = Xo/Fo.
Now, for the IRR to capitalists who invest in period 1 (and thereafter)
measured in commodity numeraire, we have an investment of Fo(c-1) in
period 1 to be equated to the present discounted value of the stream of
returns Xo(b-1) starting in period 2, discounted back to period 1:
Fo(c-1) = Xo(b-1)((1+Rx[1])^-1 + (1+Rx[1])^-2 +...), which gives both
Andrew and me Rx[1] = (Xo(b-1))/(Fo(c-1)) > Rx[0] if b > c, as we are
assuming.
I think we agree on these points.
Now let us consider the IRR to capitalists who invest in period 0 in the
money numeraire. Their money investment is p[0]Fo = (No/Xo)Fo, and they
get a stream of money returns p[1]Xo, p[2]X0,..., which constantly
declines due to the cheapening of output in labor terms. To calculate the
internal rate of return to period 0 investment with money as numeraire,
R$[0], we solve:
(No/Xo)Fo = (No/Xo)Xo((1+R$[0])^-1 + (d/b)(1+R$[0])^-2 +
(d/b)^2(1+R$[0])^-3...), which implies R$[0] = (Xo/Fo)-(1-(d/b))
= Rx[0] - (1-(d/b)). This is less than Rx[0] due to the constant fall in
the money price of output. I believe, in fact, that Andrew comes to the
same conclusion.
Now calculate the IRR to capitalists who invest in period 1 in the money
numeraire. Andrew approaches this by substituting directly the labor he
attributes to this vintage of capital, but let me approach it
symmetrically to the above calculations, which don't require any
assumption about exactly how the labor is distributed across the vintages,
since we work entirely in terms of output. We have a money investment of
p[1]Fo(c-1) = (No/Xo)Fo(c-1) yielding a constant stream of output Xo(b-1),
which is sold to generate a stream of money revenue p[2]X0o(b-1),
p[3]Xo(b-1),..... To find the IRR for period 1 investments in the money
numeraire we solve the equation:
(No/Xo)Fo(c-1) = (No/Xo)Xo(b-1)((d/b)(1+R$[1])^-1 + (d/b)^2(1+R$[1])^-2 +
(d/b)^3(1+R$[1])^-3...). This gives (keeping my fingers crossed that I
didn't make any math mistakes, which I too often do, and did last time
around on this problem, and that I don't introduce any inadvertent typos)
R$[1] = (b/d)(Xo(b-1))/(Fo(c-1)) - (1-(d/b)) = R$[0] - (1-(b/d))(Xo/Fo) >
R$[0] if b > d as we assume in this scenario. (The wrinkle here is that
the price change starts in period 2, intoducing the factor (b/d)).
Now here is where Andrew and I diverge, because he calculates this IRR
directly from considering the employed labor and its changes. I confess
that I'm not completely clear on Andrew's assumptions as to exactly how
the labor and factories interact to produce the macro-results which he
takes as axioms, since there are several different scenarios possible. One
is that there is disembodied labor-augmenting technical change that
applies to all the factories of whatever vintage, in which case the labor
employed in each factory in each period should be just proportional to the
output from that vintage in the period. In this case it seems to me that
calculating the IRR using labor should give the same result as I arrived
at above.
If we can clear this up, perhaps we can move to the more interesting
discussion of what concept of the profit rate is relevant both to the
actual fate of capitalist economies and to the interpretation of Marx's
discussion of the problem.
Yours,
Duncan