THE THEORY OF INTEREST

from 4 mn + m — 2n — 1, we have, as the final net num-ber of unknowns,

3 mn + m — n — 1

which is the same as the total net number of independentequations. 4 Thus the problem is fully determinate underthe assumptions made.

‘Instead of thus banishing the r’s, an alternative reconciliation isto retain them but to add for each an equation of definition of 1 + r.Thus 1 + ri (corresponding to the slope of the Opportunity Curve)is a derivative from the y ”s and y"’s of the two successive yearscalled this year and the next (in other words, a partial derivative)making 1 + ri' dependent upon (in other words, a function of) the y’a.That is,

ri — <pi(yi, y ",..., yi m) ),

which function is empirical and derivable from the opportunity function<p already given.

Analogously we may express the equations of definition for ri, ri .. r n 'and likewise for the corresponding r" ’s, r'" ’s, etc., up to r (m) 's makingn(rra-l) equations of definition. In this way, retaining the r’s we have4 mn + m — 2 n — 1 independent equations and the same number ofunknowns.

The complication mentioned in Chapter VII, §10, that the incomestream itself depends upon the rate of interest, does not affect thedeterminateness of the problem. It leaves the number of equations andunknowns unchanged, but merely introduces the rate of interest into theset of equations expressing the Opportunity principles. These equationsnow become

q>i(yi,yi' . Vi (m) ; i", i (m) ) = 0

etc., and their derivatives, the functions, are likewise altered in formbut not in number.

The mathematical reader will have perceived that I have studiouslyavoided the notation of the Calculus, as, unfortunately, few economicstudents are, as yet, familiar with that notation, and as it has seemedpossible here to express the same results fairly well without its use. See,however, the Appendix to this chapter (Chapter XIII), §1-5.

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