On the zeros of Cauchy polynomials with application to optimal depreciation

Abstract

A complete mathematical solution of a problem of periodic reinvestment equivalent to depreciation allowances is given. This problem was posed by Ijiri (1967) and further investigated by Buchner (1970). Mathematically it has been formulated by both authors in terms of a linear difference equation of the resulting book values. Since the adjoined characteristic polynomial is of Cauchy type, we have to find - fixing the first depreciation coefficient - a polynomial with minimal spectral radius. We present an explicit solution for this problem. Furthermore, the convergence of the sequence of book values is analyzed in the general case. Our result provides asymptotically the fastest increase of production capacity induced by periodic reinvestments, still permitting to some extent an individual policy of depreciation, in particular of increasing degressive character.