§10. Zero or Negative Rates of Interest

We have already seen (Chapter XI, §9), that zero or negative rates of interest are theoretically possible. In terms of formulas all that is needed to make the rate of interest zero is that the forms of the F and f functions shall be such as to produce this result. This implies that these functions shall have solution values equal to zero.

Of course it would be possible that interest, impatience, and return over cost for one particular year might be zero or negative without this being true for other years. If they were zero for all the years, we should have the interesting result that the value of a finite perpetual annuity (greater than zero per year) would be infinity. No one could buy a piece of land for instance, expected to yield a net income forever, for less than an infinite sum. A perpetual government bond from which an income forever was assured would have an infinite value. Since this is quite impracticable, we thereby reduce to an absurdity the idea that it is possible to have at one and the same time:

• 1. A zero rate of interest for each year forever; and
• 2. a perpetual annuity greater than zero per year.

But the absurdity is lessened or disappears altogether if either:

• 1. The zero rate of interest is confined to one year; or
• 2. no perpetual annuity greater than zero per year is possible.

Unusual conditions may easily reduce the rate of interest for one year to zero. As to an unproductive or barren world, like the hard-tack island, only a finite totality of income would be possible; a perpetual annuity even of one crumb of hard-tack a year would be impossible.

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