On the Generation of Discount Rates and Implicit Prices Through a Replacement Model

The purpose of this paper is to generate discount rates and implicit prices by applying theorems of Frobenius and Bellman to non-negative matrices. The model which is developed may be useful in situations in which management decisions must be made regarding the replacement of aging and obsolescing equipment for some specified period of time. One version of the model allows choices among alternatives and another version leads to a deterministic form. The discount rate which is derived is the reciprocal of the maximal growth rate which is obtained as the largest (positive) root of the non-negative matrix introduced in the first version of the model, while the implicit prices arise from an associated eigenvector. The model may be particularly useful in situations where market rates and market prices do not properly reflect values as is the case for some public goods.