More important than the formula is the basic notion of the rate of
return on investment. In the accounting literature the idea
itself appeared around 1890. Suppose you have five capitalists
who are combining their capitals to make a particular investment.
They predict that the investment will yield 5 annual payments of a
constant amount, A. Capitalist 1 gets the 1st payment, 2 the
second, and so on. How much should each invest to receive
the same rate of return, r?
1 invests A/(1+r)
2 invests A/(1+r)^2
3 invests A/(1+r)^3
4 invests A/(1+r)^4
5 invests A/(1+r)^5
The sum becomes the total investment. These days we see these
figures as present values. Nothing that complex. But note what
is happening here. The annual payment is a mix of profit and
depreciation when we focus only on the fixed capital investment.
But the depreciation portion represents both the sum of moral and
physical depreciation which Marx tells us is the total depreciation
charge for capitalists.
But now depreciation and profit have become inseparable. How do
we pull them apart? To begin the process we note that the initial
investment of each the five capitalists is a depreciation figure
for a particular year. That is, the depreciation figure for
year 1 is A/(1+r), for year 2 a/(1+r)^2 and so on. What the
capitalists would see as profits for each year is thus
P(N)=A-A/(1+r)^n
Since A/(1+r)^n decreases as n increases, profits seem to grow as
the investment ages. Can we recover the notion of value from
this appearance? I think so. That is, we know that if the
total investment over 5 years is C, then the total profit over
that period of time is 5*A-C. Now if we assume that the same
amount of labor is added in each year, then we have a problem
since profits seem to be increasing and the depreciation charges
decreasing as the investment ages. In other words, by
using the RRI, dead and living labor cannot be readily seen
in any given period. Yet, it's obvious that a new machine
worth $1000 which wears out after, say, 5 years can only account
for $1000 of the output in that time span. If the total annual
payments are $1200, then the $200 difference is surplus value or
profit. Thus, on average the profit created by living labor is
$40. By averaging the depreciation charges as well so that
the annual charge is $200, we see that the payment amounts are
$240 per year. With constant annual payments we would thus not
only have to transform surplus value into profit but also separate
depreciation from profit.
So far we've held A constant over the course of an investment. Yet
capitalists may well anticipate and get declining payments as time
passes. Let the first payment be A(1), the second A(2) and the nth
A(n). Then we can write
C = A(1)/(1+r)+A(2)/(1+r)^2+...+A(n)/(1+r)^n
They may also anticipate increases in the RRI, r, as the invested
capital ages. Thus, we would have
C = A(1)/[1+r(1)]+A(2)/[1+r(2)]^2+...+A(n)/[1+r(n)]^n
where r(n) is the rate of return in the nth period.
Clearly, a lot needs to be worked out in terms of how surplus value
is allocated on the basis of the RRI. Yet one thing is clear, the
RRI is based upon an unchanging amount of "C". You can't compute
a return on a given investment and keep changing that given simply
because what you invested in is now cheaper. Given that the total
of all payments (sum of A(n)'s) is greater than C, then you will
find a positive RRI.
John