IN TERMS OF FORMULAS

of interest. The rate of interest connecting the first yearwith the second will be called i', that connecting thesecond year with the third, i", and so on to i (m_1) . Underthis principle, the rates of time preference for all thedifferent individuals in the community for each year willbe reduced to a level equal to the rate of interest. Thiscondition, algebraically expressed, is contained in severalcontinuous equations, of which the first is:

f = fi' = V=--..=U-

This expresses the fact that the rate of time preferenceof the first year’s income compared with next is the samefor all the individuals, and is equal to the rate of interestbetween the first year and the next. A similar continuousequation may be written with reference to the time pref-erences and the rate of interest as between the secondyear’s income and the next, namely:

i" = fi" — U” —

Since the element of risk is supposed to be absent, it doesnot matter whether we consider these second-year ratesof interest and time preference as the ones which areexpected, or those which will actually obtain, for, underour assumed conditions of no risk, there is no discrep-ancy between expectations and realizations.

A similar set of continuous equations applies to time-exchange between each succeeding year and the next,up to that connecting year (m — 1) with year m. Therewill therefore be m — 1 continuous equations of the

X aa the principle of maximum desirability. The equivalence of theprinciple whether stated with reference to a maximum or to a mar-ginal equality is obvious, but the mathematical reader may care to seeit put in formulas as a “maximum” proposition as in the Appendix tothis chapter (Chapter XII), §2.

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