APPENDIX

dx"

dF( ) /

dx" /

m )

dx 1

In other words, 1 + i" = 1 + or i" = /". Similarly, t = /for every other pair of successive years.

We have here, in mathematical language, the reason that thepoint of maximum total wantability is also the point at whichthe marginal rate of time preference for a unit of each year’sincome over that of next year’s income is equal to the rate ofinterest connecting these two years. 2

APPENDIX TO CHAPTER XIII§ 1 (to Ch. XIII, § 9)

Rate of return over cost expressed in the notation of the calculus

In the notation of the calculus, the rate of return over cost,called in the text r/, is defined in terms of the partial differentialquotient with the opposite sign of next year’s income withrespect to this year’s income of Individual 1. That is, by

definition 1 + rf = — ~j- and 1 + r" = — jjpr- Analogous

formulas express the remaining r’s for Individuals 1,2 . n.

§ 2 (to Ch. XIII, § 9)

Rate of return over cost derived by differential equationsThe magnitudes of 1 + r/, 1 -| - r", ., 1 + r/” 0 ,

’ “ dy"(m-i) > ma y ^ expressed in

terms of yi, y", ., yf” 0 by differentiating the equation

for the effective range of choice, <pi (yf, y", ., yf m) ) — 0.

2 The mathematical reader will note that the function F here representingtotal wantability W is vitally related to the functions F in Chapters XIIand XIII, representing(the marginal rate of time preference /, since 1 + /is the ratio of the differential quotient of W relatively to this year’s incometo the corresponding differential quotient for next year’s income.

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