Profit Maximization Models for Exponential Decay Processes

A number of real world processes can be modelled as exponential decay processes. Examples are: machine replacement, oil well extraction, advertising goodwill, repair and cleaning activities, etc.. In this paper we analyze a series of discounted or undiscounted, deterministic or stochastic exponential decay models. We characterize finite and infinite horizon optimal solutions for each model. We show that the solution can be characterized for the oil drillers problem in the following way: Once a well of sufficient capacity is drilled, oil is pumped from it until the oil remaining decreases to a fixed cut-off level; then the well is abandoned, and a new well is drilled. The resulting process when repeated over time appears to be the same as an oil source which produces oil revenue continuously at the fixed cut-off level. In other words, the excess revenue received from an oil well when its capacity is greater than the cut-off level is just sufficient to pay for drilling costs for a new well. Although the models analyzed in this paper are idealized, the results presented should provide a guide to the analysis of more realistic situations