THE THEORY OF INTEREST

§5. The Reasoning not “Circular ”

At first sight it may appear to those not familiar withthe mathematics of simultaneous equations and variablesthat the reasoning is circular; the rate of interest de-pends on individual rates of impatience; these rates ofimpatience depend on the time shapes of individual in-come streams; and the choice of these time shapes ofincome streams depends, as we have just seen, on therate of interest itself.

It is perfectly true that, in this statement, the rate ofinterest depends in part on a chain of factors which finallydepend in part on the rate of interest. Yet this chain isnot the vicious circle it seems, for the last step in thecircle is not the inverse of the first.

To distinguish between a true and a seeming exampleof a circular dependence we may cite simple problems inalgebra or mental arithmetic. Suppose we wish to find theheight of a father who is known to be three times astall as his child. To solve this we need to know somethingmore about these two heights. If we are told in additionthat the child’s height differs from his father’s by twiceitself, the problem is really circular and insoluble, for theadditional condition is really reducible to the first, beingmerely a thinly veiled inversion of it. The problem essen-tially states (1) that the father’s height is three timesthe child’s and (2) that the child’s is one-third of thefather’s—an obvious circle.

But if the dependence of the father’s height on thechild’s is essentially different from—independent of—the dependence of the child’s on its father’s, there is. nocircle. Thus supposing, as before, that the father is threetimes as tall as the child, let us stipulate in addition that

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