Value & Capital, CHAPTER XVII, Section 5

 In this section, the author provides a fairly straightforward proof of his assertion, made in the previous section, that a fall in the rate of interest lengthens the average period of a stream of surpluses.

His proof begins by defining the concept of the marginal stream, based on the streams of surpluses before and after the change in the interest rate.  In his notation, (S0, S1, S2, ..., Sn) is the stream of surpluses under the production plan corresponding to the old interest rate, and (S'0, S'1, S'2, ..., S'n) is the stream that would be planned at the new rate of interest.  The marginal stream then consists of the stream of differences

S'0 – S0 , S'1 – S1 , S'2 – S2 , ..., S'n – Sn

where each difference could be positive or negative.

The new stream equals the old stream plus the marginal stream.  The author explains that the average period of the new stream is the average of the average period of the old stream and the average period of the marginal stream. He expresses the average period of the new stream as

(CP + cp) / (C + c)

where P and p are the average periods of the old and marginal streams, respectively, and C and c are the capital values of the old and marginal streams, respectively.  He then considers the particular marginal stream at an interest rate for which the capital value is arbitrarily close to zero.  He argues that the quantity cp is still positive.

We saw in an earlier chapter that the product of the average period of a stream by its capital value equals the capital value of an auxiliary stream formed by capitalizing, in each successive week, the items in the stream of surpluses which remain over after that week.  We saw too that every item in this auxiliary stream must be positive (otherwise it would never pay to go through with the production plan implied in the stream);  consequently the capital value of the auxiliary stream must be positive.

Thus in his formula above, c can be neglected, "but the term cp must not be neglected."  The expression then becomes

(CP + cp) / C  =  P + (cp / C)

which is strictly greater than P.  The author also gives a (much) more mathematical proof in the book's Appendix