Note on discount rates and present values

Abstract

It is a long-established proposition that, under profit maximising conditions, the interest elasticity of investment will vary positively with the length of life of projects. Thus, Shackle ( i , Chap. XI) proves that the present value of an income stream is more sensitive to changes in the rate of discount the further into the future that stream extends. But Shackle's proof is in terms of continuous time, with present value expressed as an integral of a time function. Since, however, the literature on the application of discounted-cash-flow techniques to investment decisions (and its public sector analogue, cost-benefit analysis) treats time as a discrete variable, it is perhaps worthwhile proving the theorem in terms more in accord with this literature. We shall do this for an income stream, though of course the proof is equally valid for a cost stream, and shall take two cases, one where the income flow is constant from period to period, and the other where it varies.