MENU COSTS AND THE NEUTRALITY OF MONEY 709 Hence, changes in the log of the firms real price are an integer multiple of D minus the log of the price level Two important requirements are necessary for(s, S)-type pe cies to be optimal. One requirement is stationarity of real balances over time-M(t)-P(t)=-P(0), so that demand ri is stationary We shall demonstrate that in equilibrium this requirement is atisfied. The other requirement concerns restrictions on the form of the anticipated infation process. Conditions for optimality of (s, s) pricing policies in a stochastic setting have been considered by Sheshinski and Weiss [1983], Danziger [1984], and more recently by Caplin and Sheshinski [1987]. Danziger considers a world with discrete inflationary shocks. He demonstrates that when inflation ary shocks arrive one at a time with exponentially distributed interarrival times, then the optimal pricing policy is of the(s, S) variety. With general inflationary processes, the optimal pricing take a more complex form The central qualitative feature of(s, s) pricing policies is that they make the time between successive price revisions endogenous prices change more frequently when inflation is rapid than when it slow. Alternative models of asynchronous price setting involve fixed decision times regardless of ensuing shocks to the economy. Seen in this light, one may be less concerned with the precise optimality of (s, S)pricing policies. Rather, they may be seen as a imple and tractable alternative to the assumption of a predeter mined pattern of price revisions Analysis of the time path of aggregate prices in our framework requires specification of the initial distribution of prices across firms in the economy. It is assumed that firms' initial real prices ri (0)are uniformly distributed over the range(s, S]. For ease of exposition we restate the uniformity assumption with a frequency distribution Fo(p) which defines the proportion of firms with the logs of their initial prices pi (O) no higher than p 12. While the discrete nature of Danziger's inflation process contradicts he neutrality proposition nevertheless 13. Even in the inventory literature, Arrow, Harris, and Marschak [19 and applied carf: [1959). Further, stationary(s, S) policies are frequently n situations wher warz, 1981)and in more general nonstationary environments [Karlin and Fabens, 1959- - MENU COSTS AND THE NEUTRALITY OF MONEY 709 Hence, changes in the log of the firm's real price are an integer multiple of D minus the log of the price level. Two important requirements are necessary for (s,S)-type poli￾cies to be optimal. One requirement is stationarity of real balances over time-M(t) -P(t) = -P(O), so that demand riis stationary. We shall demonstrate that in equilibrium this requirement is satisfied. The other requirement concerns restrictions on the form of the anticipated inflation process. Conditions for optimality of (s, S) pricing policies in a stochastic setting have been considered by Sheshinski and Weiss [1983], Danziger [1984], and more recently by Caplin and Sheshinski [1987].11 Danziger considers a world with discrete inflationary shocks. He demonstrates that when inflation￾ary shocks arrive one at a time with exponentially distributed interarrival times, then the optimal pricing policy is of the (s,S) variety.'' With general inflationary processes, the optimal pricing policy may take a more complex form. The central qualitative feature of (s,S) pricing policies is that they make the time between successive price revisions endogenous: prices change more frequently when inflation is rapid than when it is slow. Alternative models of asynchronous price setting involve fixed decision times regardless of ensuing shocks to the economy. Seen in this light, one may be less concerned with the precise optimality of (s,S) pricing policies.13 Rather, they may be seen as a simple and tractable alternative to the assumption of a predeter￾mined pattern of price revisions. Analysis of the time path of aggregate prices in our framework requires specification of the initial distribution of prices across firms in the economy. It is assumed that firms' initial real prices ri(0) are uniformly distributed over the range (s,S]. For ease of exposition we restate the uniformity assumption with a frequency distribution F,,(p) which defines the proportion of firms with the logs of their initial prices pi(0) no higher than p. 11. Sheshinski and Weiss [I9831 employ a special form of the stochastic inflation process. Caplin and Sheshinski [I9871 present a discrete time formulation with i.i.d. inflationary shocks. 12. While the discrete nature of Danziger's inflation process contradicts Assumption 1, our analysis including the neutrality proposition nevertheless applies. 13. Even in the inventory literature, Arrow, Harris, and Marschak [I9511 study (5,s)policies because of their relative simplicity. The first general proof of optimal￾ity is due to Scarf [1959]. Further, stationary (5,s)policies are frequently analyzed and applied in situations where they are undoubtedly suboptimal (such as in multi-echelon inventory systems [Schwarz, 19811 and in more general nonstationary environments [Karlin and Fabens, 19591