Discounted and Per Unit Net Revenues and Costs of Product Warranty: The Case of Phase-Type Lifetimes

In a paper published in this journal, Mamer (1987) studies the expected costs to producers and benefits to consumers of three types of product warranty: the ordinary free replacement warranty, the unlimited free replacement warranty, and the pro-rata warranty. In addition to assuming a random product lifetime, Mamer's models allow for randomness in consumer repurchase behavior and for the possibility of an independent damage process acting on products sold under warranty. The criteria for evaluating warranty policies are discounted costs and net revenues over an infinite horizon, and per unit costs and net revenues. In this paper, we consider the case of phase-type product lifetimes. Under this assumption, we simplify the computation of the expected cost and revenue functions. Since most distributions can often be represented by phase-type distributions, our simplifications enhance the applicability of Mamer's useful yet computationally unwieldy results. For the per-unit-net-revenue criterion, our results supplement those of Mamer's. There, we also establish an interesting relation pertaining to expected net revenues as a function of consumer loyalty. We find that under the per-unit-net-revenue criterion, from the producer's perspective, repeated purchases by the same customer are not as profitable as one would intuitively conjecture. Such a counter-intuitive relation makes the need for being able to compute various cost and revenue functions more compelling---particularly when lifetimes are not exponential.