§ 2 (to Ch. XII, § 3)
Equality of marginal rate of time preference and rate of interestimplies that desirability of income stream is made a maximum
Assume at first that only two years are considered. The factthat total desirability or wantability of the individual, asreckoned at the beginning, depends on the amount of incomethis year and next year may be represented by the equationW" = F(c' + s', c" + x"),
where W" represents his total wantability, and the equationrepresents this W" as a function of his income stream consistingof c' + x' this year and c" + x" next year. This W" is repre-sented in Chapters X and XI by the numbers attached to theseveral Willingness or Wantability lines, each representing acertain level of wantability of Individual 1. But as we shallhere consider only one individual, we omit the subscript num-bers, 1, 2,., n. The individual under consideration will
attempt to adjust x' and x" so as to maximize W. We are toprove algebraically that the condition that W shall be a maxi-mum implies also that the rate of interest i shall be equal to theindividual’s rate of preference /. The condition 1 that W shallbe a maximum is that the total differential of W or of its equalF(c' + x ', c" + x"), called below F( ) shall be zero; thus
dw ~^7 1 < k; + ^ dx ”
0 ,
where the 3’s represent the partial differentials with respect tox' and x".
From this equation it follows that
3F( ).
dx" 3F( ) /dx' dx' / '
dx'‘
The left-hand number of this equation is 1 + i, as may be seenby differentiating the equation of the loan as originally stated,viz.:
1 See any text book on the calculus, e.g. Wilson, E. B. Advanced Calcu-lus. Boston, Ginn & Co., 1912, pp. 118-125.
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