The Variability of Aggregate Demand with (S, s) Inventory Policies

This paper develops a general theory of the aggregate implications of (S, s) inventory policies. It is shown that (S, s) policies add to the variability of demand, with the variance of orders exceeding the variance of sales. Overall, the (S, s) theory contradicts the widely held notion that retail inventories act as a buffer, protecting manufacturers from fluctuating sales. In 1951, Arrow, Harris, and Marschak 3 introduced the (S, s) form of inventory policy. The policies are designed for retailers of finished goods, who face economies of scale when placing orders with their suppliers. To pursue an (S, s) inventory policy, the retailer establishes a lower stock point s, and an upper stock point S. No order is placed until inventories fall to s or below, whereupon they are restored to the maximum of S. A general proof of the optimality of these (S, s) inventory policies was provided by Scarf 13. At the microeconomic level, the model has been extensively investigated. Formulae are available to compute optimal policies (e.g., Ehrhardt 6), and these policies are xidely used in industry (e.g., Schwartz (ed.) 14). In addition, the model has been extended to increasingly complex demand environments (e.g., Karlin and Fabens 11). In contrast, little is known about the macroeconomic implications of (S, s) policies. Several recent papers have begun to correct this deficiency. Akerlof has suggested that pursuit of constant threshold money holding policies of the (S, s) variety might be responsible for the observed low short-run income elasticity of the demand for money (Akerlof 1 and Akerlof and Milbourne 2). In the operations research literature, Ehrhardt, Schultz, and Wagner 7 analyzed the demand environment of a wholesaler supplying several retailers. They required that the distinct retailers have independent sales, ruling out the analysis of common factors in sales. Finally, simulation results of Blinder 4 suggested a role for the (S, s) model in understanding retail sector inventories. However the theoretical difficulties with the model remained unresolved. Blinder commented: If firms have a technology that makes the S, s rule optimal, aggregation across firms is inherently difficult. Indeed it is precisely this difficulty which has prevented the S, s model from being used in empirical work to date (Blinder 4, p. 459). In this paper we present a general theory of the aggregate implications of (S, s) policies. Our central finding is that (S, s) policies add to the variability of demand, with the variance of orders exceeding the variance of sales. This result holds even in the presence of common factors in retail sales. In addition, a close connection