IN TERMS OF FORMULAS

§13. Different Rates of Interest for Different Years

The system of equations thus involved when n personsinstead of three and m years instead of two are consid-ered introduces very few features of the problem notalready contained in the simpler set of equations for twoyears and three persons. The new feature of chief impor-tance is that, instead of only one rate of interest to bedetermined, there are now a large number of rates. It isusually assumed, in theories of interest, that the problemis to determine “the” rate of interest, as though one ratewould hold true for all time. But in the preceding equa-tions we have m — 1 separate rates of interest, viz., i',

«// 4*(m—1)

6 >••••> 6 •

Under the hypothesis of a rigid allotment of future in-come among different time intervals, which is the hy-pothesis of the first approximation, there is nothing toprevent great differences in the rate of interest fromyear to year, even when all factors in the case are fore-known and there is every opportunity for arbitrage. By

a suitable distribution of the values of Ci, c 2 ,-, c m ,

there may be produced any differences desired in themagnitudes of i', i", _, Thus if the total enjoy-

able income of society should be foreknown to be 10billion dollars in the ensuing year, 1 billion in the fol-lowing year, and 20 billion in the third year, and if therewere no way of avoiding these enormous disparities inthe social income, it is very evident that the income ofthe middle year would have a very high valuation com-pared with either of its neighbors, and therefore that therate of interest connecting that middle year with the firstyear would be very low, whereas that connecting it withthe third year would be very high. It might be that a

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