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

The literature on the aggregation of (S, s) policies has ignored the impact of aggregate behavior on the individual's optimization problem. In the case of pricing, the feedback effects are clear. Not only do pricing strategies determine the evolution of the price level, but the evolution of the price level also influences the optimal pricing strategies. In this paper, we provide a consistent treatment of aggregation and optimization. We use this model to analyze three issues in the menu cost pricing literature: the relationship between strategic complementarity and the real effects of money; the relationship between the variance of the money supply and the correlation between money and output; and the relationship between the cost of price adjustment and the size of price adjustment. QUESTIONS CONCERNING THE DYNAMICS of aggregate variables such as prices, employment, investment, and consumption represent the core of business cycle analysis. One striking feature of these variables is the radical difference between properties of these aggregates and the nature of the individual behavior that underlies them. The behavior of a firm's prices, investment and employment, and the behavior of an individual's consumption all involve frictional elements that lead to discrete adjustment at the microeconomic level. Heterogeneity among individuals, however, tends to smooth the behavior of the corresponding aggregates. In recent years, a large body of research has developed to examine the manner in which microeconomic frictions influence aggregate dynamics. One of the centerpieces of this research has been the (S, s) model developed by Arrow, Harris, and Marschak (1951). The key element of this model is the state dependence of individual decisions. Agents act when a state variable crosses some critical threshold which balances the cost and benefits of adjustment. The aggregate implications of this form of microeconomic behavior have been analyzed by Blinder (1981), Caplin (1985), and Mosser (1991) in the context of inventory dynamics; by Caplin and Spulber (1987), Caballero and Engel (1991, 1993), and Caplin and Leahy (1991) in the context of prices; and by Bertola and Caballero (1990), Caballero (1993), and Eberly (1994) in the context of con- sumer durables. One of the most limiting aspects of these models is that they focus exclusively on the impact that microeconomic inertia has on aggregate dynamics. They 1We would like to thank Michael Harrison, loannis Karatzas, John Leahy Sr., Andreu Mas-Colell, a co-editor and four anonymous referees for helpful discussions and comments, and the National Science Foundation and the Sloan Foundation for financial support.

We present a new approach to the theory of imperfect competition and apply it to study price competition among differentiated products. The central result provides general conditions under which there exists a pure-strategy price equilibrium for any number of firms producing any set of products. This includes products with multi-dimen- sional attributes. In addition to the proof of existence, we provide conditions for uniqueness. Our analysis covers location models, the characteristics approach, and probabilistic choice together in a unified framework. To prove existence, we employ aggregation theorems due to Prekopa (1971) and Borell (1975). Our companion paper (Caplin and Nalebuff (1991)) introduces these theorems and develops the application to super-majority voting rules. WE PRESENT A NEW APPROACH to the theory of imperfect competition and apply it to study price competition among differentiated products. The central result is that there exists a pure-strategy price equilibrium for any number of firms producing any set of products. In addition to the proof of existence, we provide conditions for uniqueness. Our model both unites diverse strands of the earlier literature and opens up uncharted areas for future analysis. In particular, we expand the traditional one-dimensional framework to allow for multi-dimen- sional product differentiation. Our approach involves twin restrictions on consumer preferences: one on individuals' preferences, the other on the distribution of preferences across society. These are generalizations of the restrictions supporting 64%-majority rule presented in Caplin and Nalebuff (1988). To prove existence, we apply a new technique of aggregation. This technique is valuable in a variety of other problems. In the companion paper, we use the aggregation result to generalize our earlier work on 64%-majority rule and to characterize the relationship between the distribution of human capital and the distribution of income (Caplin and Nalebuff (1991)). There are additional applications in statistics and in search theory. We begin with a brief review of the early literature on imperfect competition, describing in more detail the existence problem and previous solutions. Section 3 presents our twin assumptions, and shows that they cover many standard cases. In Section 4, we introduce the aggregation theorem and use it in the analysis of demand functions. The proof of existence of equilibrium is in Section

A celebrated result of Black (1948a) demonstrates the existence of a simple-majority winner when preferences are single-peaked. The social choice follows the preferences of the median voter: the median voter's most-preferred outcome beats any alternative. However, this conclusion does not extend to elections in which candidates differ in more than one dimension. This paper provides a multi-dimensional analog of the median voter result. We provide conditions under which the mean voter's most preferred outcome is unbeatable according to a 64%-majority rule. The conditions supporting this result represent a significant generalization of Caplin and Nalebuff (1988). The proof of our mean voter result uses a mathematical aggregation theorem due to Prekopa (1971, 1973) and Borell (1975). This theorem has broad applications in economics. An application to the distribution of income is described at the end of this paper; results on imperfect competition are presented in the companion paper, Caplin and Nalebuff (1991).

Many electoral rules require a super-majority vote to change the status quo. Without some restriction on preferences, super-majority rules have paradoxical properties. For example, electoral cycles are possible with anything other than 100 percent majority rule. The auth ors show that these problems do not arise if there is sufficient simi larity of attitudes among the voting population. Their definition of social consensus involves two restrictions on domain: one on individu al preferences, the other on the distribution of preferences. When th is consensus exists, 64 percent majority rule has many desirable prop erties, including the elimination of all electoral cycles. Copyright 1988 by The Econometric Society.