On the zeros of Cauchy polynomials with application to optimal depreciation (Q810349)

A complete mathematical solution of a problem of periodic reinvestment equivalent to depreciation allowances is given. This problem was posed by \textit{Y. Ijiri} [Manag. Sci. 13, 321-335 (1967)] and further investigated by \textit{R. Buchner} [The German Economic Review, 295-308 (1970)]. Mathematically it has been formulated by both authors in terms of a linear differential equation of the resulting books values. Since the adjoined characteristic polynomial is of Cauchy type, one has to find - fixing the first depreciation coefficient - a polynomial with minimal spectral radius. The authors of the present paper give an explicit solution for this problem. Furthermore, the convergence of the sequence of book values is analyzed in the general case. The result provides asymptotically the fastest increase of production capacity induced by periodic reinvestments, still permitting to some extend an individual policy of depreciation, in particular of increasing degressive character.