§13. Different Rates of Interest for Different Years

The system of equations thus involved when n persons instead of three and m years instead of two are considered introduces very few features of the problem not already contained in the simpler set of equations for two years and three persons. The new feature of chief importance is that, instead of only one rate of interest to be determined, there are now a large number of rates. It is usually assumed, in theories of interest, that the problem is to determine "the" rate of interest, as though one rate would hold true for all time. But in the preceding equations we have m - 1 separate rates of interest, viz., i', i'',..., i(m-1).

Under the hypothesis of a rigid allotment of future income among different time intervals, which is the hypothesis of the first approximation, there is nothing to prevent great differences in the rate of interest from year to year, even when all factors in the case are foreknown and there is every opportunity for arbitrage. By a suitable distribution of the values of c1, c2,..., cm, there may be produced any differences desired in the magnitudes of i', i'',..., i(m-1). Thus if the total enjoyable income of society should be foreknown to be 10 billion dollars in the ensuing year, 1 billion in the following year, and 20 billion in the third year, and if there were no way of avoiding these enormous disparities in the social income, it is very evident that the income of the middle year would have a very high valuation compared with either of its neighbors, and therefore that the rate of interest connecting that middle year with the first year would be very low, whereas that connecting it with the third year would be very high. It might be that a Edition: current; Page: [300] member of such a community would be willing to exchange $100 of the plentiful 10 billions for the first year, for only $101 out of the scarce 1 billion of next year, but would be glad to give, out of the third year's still more plentiful 20 billions, $150 for the sake of $100 in the middle and lean year.

In actual markets we find some influence of such differences between future years (as looked at today) in the differences between short term and long term interest rates.

The reason why, in actual fact, no abrupt or large variations in the rate of interest, such as from 1 per cent to 50 per cent, is ordinarily encountered is that the supposed sudden and abrupt changes in the income stream seldom occur. The causes which prevent their occurrence are:

(1) The fact that history is constantly repeating itself. For instance, there is regularity in the population, so that, at any point of time, the outlook toward the next year is similar to what it was at any other point of time. The individual may grow old, but the population does not. As individuals are hurried across the stage of life, their places are constantly taken by others, so that, whatever the tendency in the individual life for the rates of preference to go up or down with age, it will not be cumulative in society. Relatively speaking, society stands still.

Again, the processes of nature recur in almost ceaseless regularity. Crops repeat themselves in a yearly cycle. Even when there are large fluctuations in crops, the variations are seldom world-wide, and a shortage in the Mississippi Valley may be compensated for by an unusually abundant crop in Russia or Asia. The resultant regularity Edition: current; Page: [301] of events is thus sufficient to maintain a fair uniformity in the income stream for society as a whole.

(2) The tendency toward uniformity is also favored in real life by the fact that the income stream is not fixed, but may be modified in other ways than by borrowing and lending as in accordance with investment opportunities. The significance of these modifications is algebraically considered in the next chapter.

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