A general survey was undertaken of the techniques used in determining loss functions to be minimized for the purpose of finding optimal s,S values in the Arrow, Harris, and Marschak dynamic inventory model. A brief discussion of the cost functions and the aspects of demand, backlogging, lag time, and discount rate are presented in the interest of better understanding of their use in the real world. Analytic solutions to the model were derived by two different, but related, methods involving stationary costs. The first method involves direct use of the cost functions in a Markov process to arrive at an integral equation of renewal which is developed into a stationary loss function by an Abelian limit theorem. The second method involves the determination of the stationary distribution of stock level through renewal theory.