MENU COSTS AND THE NEUTRALITY OF MONEY some positive demand even though prices are dispersed This may arise if the commodities are imperfect substitutes. It may also be that consumer search across firms is costly and that consumers do not recall prices posted by firms in earlier periods(see Benabou [1985b]) Costs are assumed to be fixed in real terms. Production at rate Xi(t) gives rise to real flow costs, C(X (t)). This assumption rules out stickiness in nominal input prices, including contractual wages This prevents us from addressing the relationship between price stickiness and wage stickiness, a topic of independent interest(see Blanchard [1983]). Additional study of the present model with input price stickiness is clearly desirable. All profits are distributed to consumers and firm costs accrue to consumers as income. The good is assumed to be nonstorable, so that the firms output is supplied at the same date it is produced. This removes intertemporal linkages embodied in inventories. As a result, the only variables that influence the firms flow rate of real profits B (t) are the instantaneous real price and the level of real money B, (t)= B[ri(t), M(t)-P(t) (5) max [e0X(t)-C(X(t)) x(t)≤rt) Thus, the output of firm i, X, (t), is a function of its real price and the level of real money balances which solves the problem in equation(5) X,(t)=X(r(t), M(t)-P(t)) Let X(t) represent the constant dollar value of aggregate output X(t)=1(q:(t)/Q(+)X, (t)di=Jo'e(,(t)di In the absence of menu costs, the firm picks its instantaneous price r (t)to maximize flow profits b(ri (t), M(t)-P(t)).Nominal price stickiness is introduced into the model in the form of a real 6. Gordon [1981] finds evidence for price stickiness for periods with wi diffe i址mn281 mple, Rote modity market implies market 7. By Walras'law, market clearing in he money market customers. The 9. with standard real money balances that increase demand for the commodity will also raise the firm s optimal real price
-- MENU COSTS AND THE NEUTRUITY OF MONEY 707 some positive demand even though prices are dispersed. This may arise if the commodities are imperfect substitutes. It may also be that consumer search across firms is costly and that consumers do not recall prices posted by firms in earlier periods (see Benabou [1985b]). Costs are assumed to be fixed in real terms. Production at rate Xi(t) gives rise to real flow costs, C(Xi(t)). This assumption rules out stickiness in nominal input prices, including contractual wages. This prevents us from addressing the relationship between price stickiness and wage stickiness, a topic of independent interest (see Blanchard [1983]).6 Additional study of the present model with input price stickiness is clearly desirable. All profits are distributed to consumers, and firm costs accrue to consumers as income.' The good is assumed to be nonstorable, so that the firm's output is supplied at the same date it is produced. This removes intertemporal linkages embodied in inventories. As a result, the only variables that influence the firm's Bow rate of real profits B,(t) are the instantaneous real price and the level of real money balances:' Thus, the output of firm i, Xi(t), is a function of its real price and the level of real money balances which solves the problem in equation (5): Let X(t) represent the constant dollar value of aggregate output: In the absence of menu costs, the firm picks its instantaneous price ri(t) to maximize flow profits B(r,(t), M(t) - P(t)).' Nominal price stickiness is introduced into the model in the form of a real 6. Gordon [I9811 finds evidence for price stickiness for periods with widely different forms of labor contract. This suggests that there are important sources of price stickiness other than the behavior of input prices. 7. By Walras' law, market clearing in the commodity market implies market clearing in the money market; see, for example, Rotemberg [1982]. 8. The present formulation allows the firm to ration its customers. The case without rationing can also be handled by the model; see Sheshinski and Weiss 119831. 6 With standard assumptions, increases in real money balances that increase demand for the commodity will also raise the firm's optimal real price
708 TARTERLY JOURNAL OF ECONOMICS ri FIGURE I menu cost, B, which is incurred each time the firm changes its nominal price. 0 This fixed transaction cost results in price sticki ness at the level of the individual firm. Rather than responding smoothly and continuously to changes in the overall price level the firm responds only occasionally, and with discrete price jumps We consider a firm that continuously monitors the price level, and pursues an(s, s) pricing policy, as introduced by Sheshinski and Weiss. The impact of this policy on the dynamics of the firms real price is illustrated in Figure I. The instant the log of the real price r(t)hits the fixed lower limit s, the firm adjusts its nominal price, returning the log of the real price to its upper limit S. Let D= S-s represent the size of the firms price increase. Then, the changes in the firms nominal price within any time period [0, t ]are always an integer multiple of the price range, p(t)-p(0)-k(t)D here k(t)20 is an integer. Noting that ri (0)-p ( o)and using the definition of the firms real price in equation(1), we may formally characterize the(s, S)pricing policy as follows: r (t)E(s, s] and (7)r:(t)-r(0)-(p(t)-p2(0)-(P(t)-P(0) k(t)D-(P(t)-P(0) of menu costs. If dedicated to the production of menus. This is ignored in
QUARTERLY JOURNAL OF ECONOMICS menu cost, 0,which is incurred each time the firm changes its nominal price.'' This fixed transaction cost results in price stickiness at the level of the individual firm. Rather than responding smoothly and continuously to changes in the overall price level the firm responds only occasionally, and with discrete price jumps. We consider a firm that continuously monitors the price level, and pursues an (s,S) pricing policy, as introduced by Sheshinski and Weiss. The impact of this policy on the dynamics of the firm's real price is illustrated in Figure I. The instant the log of the real price r(t) hits the fixed lower limit s, the firm adjusts its nominal price, returning the log of the real price to its upper limit S. Let D = S -s represent the size of the firm's price increase. Then, the changes in the firm's nominal price within any time period [O,t] are always an integer multiple of the price range, p (t ) -p(0) = k (t)D, where k(t) r 0 is an integer. Noting that ri(0) = pi(0) and using the definition of the firm's real price in equation (I),we may formally characterize the (s,S) pricing policy as follows: r,(t) E (s,S] and 10. There is an issue here concerning the proper treatment of menu costs. If these are indeed real costs, they should be explicitly included as part of output. Hence a closed model of the economy should properly include a sector of variable size dedicated to the production of menus. This is ignored in our formulation
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 policies 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 inflationary 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 predetermined 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 optimality 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
710 QUARTERLY JOURNAL OF ECONOMICS ASSUMPTION 3. Uniformity. The frequency distribution over initial real prices satis 0forp≤s, FP)={b/ d for p=s+b,with0≤b≤D rp≥ The uniform initial distribution of prices across the price range (s, S] is the analogue in prices of the standard assumption of uniformly staggered price changes over time. Indeed, Assumption 3 special case where inflation is constant at some rate X>0. However, it will be apparent that in a stochastic setting a uniform distribution of initial prices has significantly different implications In a fundamental sense Assumption 3 may be viewed as a statement about the endogenous tendency of prices to become uniformly distributed after a long history of inflationary shocks and pursuit of fixed(s, S)policies. This lies outside the current frame work, since firms pursuing identical (s, S)policies in the face of inflation retain forever the initial difference in their real prices. However, if firms pursue slightly distinct(s, S) policies, or random ize on their trigger price s(as in Benabou [ 1985a)), their real prices become statistically independent of one another with the passage of time. a related result for inventories states that, absent degenera cies, firms that pursue(s, S)inventory policies have inventory levels that are independent in the long run [Caplin, 1985 III. NEUTRALITY We address the connection between asynchronous price deci sions and aggregate price stickiness. To what extent is the individ ual firm stickiness in nominal prices reflected in aggregate price inertia? The central result of the paper is that real balances and aggregate output are invariant to monetary shocks. Price sticki ness disappears in the aggregate. Given (s, S) pricing rules, the initial distribution of real prices is invariant and remains uniform The aggregate nominal price index exactly reflects nominal money balances remains stationary. This results in constant aggregate In the absence of real shocks to the economy, money neutral Is appropri defined as follows
710 QUARTERLY JOURNAL OF ECONOMICS ASSUMPTION3. Uniformity. The frequency distribution over initial real prices satisfies (8) F,,(p) = * 0 b/D 1 i for p 5 s, for p = s + b, with 0 5 b 5 D, for p 2 S. The uniform initial distribution of prices across the price range (s,S] is the analogue in prices of the standard assumption of uniformly staggered price changes over time. Indeed, Assumption 3 is equivalent to an assumption of uniform staggered timing in the special case where inflation is constant at some rate X > 0. However, it will be apparent that in a stochastic setting a uniform distribution of initial prices has significantly different implications. In a fundamental sense Assumption 3 may be viewed as a statement about the endogenous tendency of prices to become uniformly distributed after a long history of inflationary shocks and pursuit of fixed (s,S) policies. This lies outside the current framework, since firms pursuing identical (s,S) policies in the face of inflation retain forever the initial difference in their real prices. However, if firms pursue slightly distinct (s,S) policies, or randomize on their trigger price s (as in Benabou [1985a]), their real prices become statistically independent of one another with the passage of time. A related result for inventories states that, absent degeneracies, firms that pursue (s,S) inventory policies have inventory levels that are independent in the long run [Caplin, 19851. We address the connection between asynchronous price decisions and aggregate price stickiness. To what extent is the individual firm stickiness in nominal prices reflected in aggregate price inertia? The central result of the paper is that real balances and aggregate output are invariant to monetary shocks. Price stickiness disappears in the aggregate. Given (s,S) pricing rules, the initial distribution of real prices is invariant and remains uniform. The aggregate nominal price index exactly reflects nominal money shocks. Consumer demand as a function of real prices and real balances remains stationary. This results in constant aggregate output. In the absence of real shocks to the economy, money neutrality is appropriately defined as follows