Stationary optimal lengths for the plant renewal problem

We study the stylized plant renewal problem when the plant net present value is maximized through the choice of the renewal dates or plant's lives, assuming that the number of renewals – and then the horizon – is finite. We observe that, under various assumptions about the cash-flow profiles due to the plants, their optimal life lengths are almost constant: this simplifies the renewal decisions. First we provide numerical and analytical explanations of the phenomenon. Then we compare the constant plant life in the finite horizon case and the stationary length in the infinite horizon one. We show that a finite horizon problem cannot be approximated by an infinite one, since the optimality conditions which provide, respectively, the constant and the stationary length are inconsistent. The usual shortcut for assuming an infinite horizon instead of the true, finite one, is then incorrect: on the contrary simplifications in the renewal problem can come from the constancy of the finite horizon optimal plant life.