Value & Capital, Notes to Chapter XIV -- B. INTEREST AND THE CALCULATION OF INCOME, part 2

This section lays out the graphical construct alluded to at the conclusion of part 1 of Note B.  As explained there, the purpose of the construct is to study the relation between interest rates and the present value of actually expected receipts.  It plots capitalized values along the horizontal axis;  along the vertical axis, it plots the discount ratio, which is related to the interest rate.  Namely, if i is the interest rate, then the discount ratio β equals 1 / (1 + i).

For an assumed given stream of receipts, there is a capital-value curve RR (plotted as a solid curve in the figure below) that shows the capitalized value of that stream for a given discount ratio.  Also, as the author explains

Corresponding to any particular level of income, we have a capital-value curve (dotted in the diagram) which shows the present value of the standard stream corresponding to that particular level of income (according to the definition of income we are using).

The figure follows the usual convention in economics, of putting the independent variable on the vertical axis.

Therefore, "If the discount ratio is OH, the present value of the prospective receipts is HA, and the level of income is that represented by the dotted curve SS, which passes through A."  A change in the discount ratio will move the point A along the curve RR.  Whether a rise in the discount ratio means a rise in the level of income depends on whether SS is steeper than RR as it is drawn here (meaning that SS is less elastic than RR). 

The author then proceeds to discuss elasticity of income with respect to the discount ratio.  If the expected stream of receipts in the various time periods is (x0, x1, ... , xv), then the capital value of this stream is

x0 + β x1 + β2 x2 + … + βv xv.  

Since mathematically the x-elasticity of y is the product of the derivative of y with respect to x, and the ratio x / y, it is clear that the elasticity of the capital value with respect to the discount ratio is 

The author then goes on to explain that

... when we look at the form of this elasticity we see that it may be very properly described as  the Average Period of the stream;  for it is the average length of time for which the various payments are deferred from the present, when the times of deferent are weighted by the discounted values of the payments.

This is clear from looking at the individual terms of the numerator:  the numerical coefficients are the numbers of time periods for which the payment is deferred, and the rest of each term represents the discounted value of the payment; the (common) denominator scales each payment as a fraction of the whole capital value.

The author concludes by noting that a comparison of the above average period of the stream of receipts with the average period of the standard stream will determine whether a fall in the rate of interest will increase income.  If the above average period is greater than the standard, it will raise income;  if not, a rise in the interest rate will raise income.