§ 3 (to Ch. XIII, § 7, also § 9)
Mathematical proof that the principle of maximum present valueis identical with the principle that the marginal rate of returnover cost is equal to the rate of interest.
The mathematical proof that the principle of maximumpresent value of optional income streams is identical with theInvestment Opportunity Principle B or that the rate of marginalreturn over cost is equal to the rate of interest is as follows:
The present value V\ of any income stream yi, yi', .,
yi m) , of Individual 1 or their combined discounted value is
v ,f' | y" I A _^_
Kl ~ 2/1 + 1 + i' + . + (l+z v ) (1+i").(!+*■<—“)’
The condition that this expression shall be a maximum is thatthe first differential quotient shall be zero. That is,
dV 1 = dy 1 '-t
dyi"
1+t'
+
dyi m)
(1+*') (!+»").(1+i ( ’"- 1) )
= 0 .
This last equation expresses the relations which must exist
between dyi, dyi', ., dyi m) , in order that the income
stream, yi, yi', ., yi m) , may have the maximum present
value.
This condition contains within itself a number of subsidiaryconditions. To derive these, let us consider a slight variationin the income stream, affecting only the income items pertainingto the first two years, yi, and yi' (the remaining items, y"',
., yi tm) , being regarded for the time being as constant) and
let us denote the magnitudes of dyi and dyi', under this as-sumption of restricted variations, by dyi and dyi'. Then, under
the condition assumed of constancy of yi", yi v , ., yi m} ,
dyi", dyi”, ., dyi m) , are equal to zero, and the equation
becomes
dyi +
dyi’
1 + i'
[ 514 ]
0 .