APPENDIX

From this, it follows directly that

dy x "

dyx!

= 1 +

But the left-hand member of this equation is by definition oneplus the marginal rate of return over cost. Since we havedesignated the rate of return over cost by n' we may substitute

du "

1 + Ty for the expression — and write the above equationthus:

1 + n' = 1 + t',

or thus:

_ / _ .7

Ty = t .

In other words, the condition that the marginal rate of returnover cost is equal to the rate of interest follows as a consequenceof the general condition that the present value of the incomestream must be a maximum. This proposition and its proofare analogous to those in regard to desirability or wantability,which have already been discussed in the Appendix to ChapterXII, that the condition of maximum wantability is equivalentin the condition that the marginal rate of preference is equal tothe rate of interest.

The same reasoning may be applied to any pair of successiveyears. Thus, if we assume variations in y" and y'", withoutany variations in the other elements of the income stream,y', y l °, ., y™, the original differential equation becomes

dy{

+

dyi'‘

or

or

or

i+i' 1 d + mi+t")

d Vl " 1 + 1 ’

1 + n" = l + i",

= 0 ,

r" = i".

All this reasoning implies, in using the differentiation process,that there is continuous variation, and that, at the margin, itis possible to make slight variations in any two successiveyears’ incomes without disturbing the incomes of the otheryears.

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