《宏观经济学 Macroeconomics》课外读物：Menu Costs and the Neutrality of Money

Menu Costs and the Neutrality of money OR。 Andrew S Caplin; Daniel F Spulber The Quarterly Journal of Economics, Vol. 102, No 4.(Nov,, 1987), pp. 703-726 Stable url: http:/inks.jstororg/sici?sici=0033-5533%28198711%29102%03a4%03c703%3amcatno%3e2.0.c0%03b2-6 The Quarterly Journal of Economics is currently published by The MIT Press Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/about/terms.htmlJstOr'sTermsandConditionsofUseprovidesinpartthatunlessyouhaveobtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the jsTOR archive only for your personal, non-commercial use Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at Each copy of any part of a JSTOR transmission must contain the same copyright notice that ap on the screen or printed page of such transmission STOR is an independent not-for-profit organization dedicated to and preserving a digital archive of scholarly journals. For more information regarding JSTOR, please contact support @jstor. org http://www.jstor.org Tue may1511:40212007

Menu Costs and the Neutrality of Money Andrew S. Caplin; Daniel F. Spulber The Quarterly Journal of Economics, Vol. 102, No. 4. (Nov., 1987), pp. 703-726. Stable URL: http://links.jstor.org/sici?sici=0033-5533%28198711%29102%3A4%3C703%3AMCATNO%3E2.0.CO%3B2-6 The Quarterly Journal of Economics is currently published by The MIT Press. Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/about/terms.html. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www.jstor.org/journals/mitpress.html. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is an independent not-for-profit organization dedicated to and preserving a digital archive of scholarly journals. For more information regarding JSTOR, please contact support@jstor.org. http://www.jstor.org Tue May 15 11:40:21 2007

THE QUARTERLY JOURNAL OF ECONOMICS Vol. CII November 1987 Issue 4 MENU COSTS AND THE NEUTRALITY OF MONEY* ANDREW S. CAPLIN AND DANIEL F SPULBER A model of endogenous price adjustment under money growth is presented Firms follow(s, S)pricing policies, and price revisions are imperfectly synchronize In the aggregate, price stickiness disappears, and money is neutral. The connection etween firm price adjustment and relative price variability in the presence of onetary growth is also investigated. The results contrast with those obtained in models with exogenous fixed timing of price adjustment. INTRODUCTION Historically determined nominal prices can lead to inertia in the aggregate level of prices, leaving room for monetary shocks to influence real variables. Formal models connecting the microeco nomic behavior of nominal prices with aggregate price stickiness lude models with staggered price and decisions [Fischer 1977; Taylor, 1980; Blanchard, 1983; Parkin, 1986], models with partial adjustment of prices(e. g, Rotemberg [1982]), and the more recent"menu cost"models of Akerlof and Yellen [1985], Blanchard and Kiyotaki 1985], and Mankiw [1985]. We present an alternative aggregate model with microeconomic price stickiness that empha sizes the importance of endogenous timing of price adjustments The model provides conditions under which money shocks have no real effects a number of macroeconomic models of price stickiness have a common microeconomic base: infrequent but large changes in *We thank a o. SES-82-19121. The paper was nted at the WBER Progress o the noc metrist society, Cambridge, M A 1 985, and at the e 1987 by the President and Fellows of Harvard College and the Massachusetts Institute of The quarterly Journal of Economics, ovember 1987

THE QUARTERLY JOURNAL OF ECONOMICS Vol. CII November 1987 Issue 4 MENU COSTS AND THE NEUTRALITY OF MONEY* ANDREWS. CAPLINAND DANIELF. SPULBER A model of endogenous price adjustment under money growth is presented. Firms follow (s,S) pricing policies, and price revisions are imperfectly synchronized. In the aggregate, price stickiness disappears, and money is neutral. The connection between firm price adjustment and relative price variability in the presence of monetary growth is also investigated. The results contrast with those obtained in models with exogenous fixed timing of price adjustment. Historically determined nominal prices can lead to inertia in the aggregate level of prices, leaving room for monetary shocks to influence real variables. Formal models connecting the microeco￾nomic behavior of nominal prices with aggregate price stickiness include models with staggered price and wage decisions [Fischer, 1977; Taylor, 1980; Blanchard, 1983; Parkin, 19861, models with partial adjustment of prices (e.g., Rotemberg [1982]), and the more recent "menu cost" models of Akerlof and Yellen [1985], Blanchard and Kiyotaki [1985], and Mankiw [1985]. We present an alternative aggregate model with microeconomic price stickiness that empha￾sizes the importance of endogenous timing of price adjustments. The model provides conditions under which money shocks have no real effects. A number of macroeconomic models of price stickiness have a common microeconomic base: infrequent but large changes in *We thank Andrew Abel, Roland Benabou, Olivier Blanchard, Dennis Carlton, Stanley Fischer, Benjamin Friedman, Barry Nalebuff, William Nordhaus, David Romer, Julio Rotemberg, Eytan Sheshinski, John Veitch, and an anonymous referee for valuable comments. Spulber's research was supported by the National Science Foundation under Grant No. SES-82-19121. The paper was presented at the Fifth World Congress of the Econometric Society, Cambridge, MA, 1985, and at the NBER Program in Economic Fluctuations Conference, October, 1985. o 1987 by the President and Fellows of Harvard College and the Massachusetts Institute of Technology. The Quarterly Journal of Economics, November 1987

QUARTERLY JOURNAL OF ECONOMICS nominal variables are assumed to be more economical than frequent small changes. The models also share the assumption that the time between successive price revisions is preset, and hence unresponsive to shocks to the economy. This assumption is questionable both at the microeconomic level and in the aggregate. Formal microeco nomic models(e. g, Sheshinski and Weiss [1983 ])strongly suggest that more rapid inflation will shorten the time between price revisions. Empirical evidence against the fixed timing assumption is presented by Cecchetti [1986] and Liebermann and Zilbefarb [1985]. At the aggregate level large monetary shocks may increase the number of agents revising their nominal prices in a given period This in turn reduces the extent of price level inertia. An important open question remains: what are the real effects of monetary shocks with endogenous timing of price revisions? The present paper assumes that individual firms adjust their prices using(s, S)pricing policies of Sheshinski and Weiss [1977, 183]. To model asynchronization, we make a cross-sectional assumption on initial prices. The price level is derived endoge nously by aggregating across firms. aggregate price stickiness then vanishes despite the presence of nominal price rigidity and imper fectly synchronized price revisions. The presence of relative price variability as a consequence of inflation is also observed endogenously through aggregation of cross-sectional price data. A simple formula is derived linking nominal price adjustment by firms with cross-sectional variability of inflation rates The basic model is outlined in Section II. The neutrality proposition is presented in Section III. In Section iv the model is applied to study relative price variability. Section V provides further discussion of the model and its assumptions. Conclusions are given in Section VI. II. THE MODEI IIA. The Aggregate Setting We provide an aggregate model of price dynamics with individ ual firms pursuing asynchronous(s, S) pricing policies. The struc ture of the aggregate model is kept as simple as possible to highlight the distinction between our model and others with asynchronous ts of nomina ption is rotemberg [1983]who considers instead increasing marginal 1. An e

704 QUARTERLY JOURNAL OF ECONOMICS nominal variables are assumed to be more economical than frequent small changes.' The models also share the assumption that the time between successive price revisions is preset, and hence unresponsive to shocks to the economy. This assumption is questionable both at the microeconomic level and in the aggregate. Formal microeco￾nomic models (e.g., Sheshinski and Weiss [1983]) strongly suggest that more rapid inflation will shorten the time between price revisions. Empirical evidence against the fixed timing assumption is presented by Cecchetti [I9861 and Liebermann and Zilbefarb [1985]. At the aggregate level large monetary shocks may increase the number of agents revising their nominal prices in a given period. This in turn reduces the extent of price level inertia. An important open question remains: what are the real effects of monetary shocks with endogenous timing of price revisions? The present paper assumes that individual firms adjust their prices using (s,S) pricing policies of Sheshinski and Weiss [1977, 19831. To model asynchronization, we make a cross-sectional assumption on initial prices. The price level is derived endoge￾nously by aggregating across firms. Aggregate price stickiness then vanishes despite the presence of nominal price rigidity and imper￾fectly synchronized price revisions. The presence of relative price variability as a consequence of inflation is also observed endogenously through aggregation of cross-sectional price data. A simple formula is derived linking nominal price adjustment by firms with cross-sectional variability of inflation rates. The basic model is outlined in Section 11. The neutrality proposition is presented in Section 111. In Section IV the model is applied to study relative price variability. Section V provides further discussion of the model and its assumptions. Conclusions are given in Section V1. IIA. The Aggregate Setting We provide an aggregate model of price dynamics with individ￾ual firms pursuing asynchronous (s,S) pricing policies. The struc￾ture of the aggregate model is kept as simple as possible to highlight the distinction between our model and others with asynchronous 1. An exception is Rotemberg [I9831who considers instead increasing marginal costs of nominal price revisions

MENU COSTS AND THE NEUTRALITY OF MONEY price and wage decisions. These alternative models frequently assume a staggered pattern of timing(e. g, Akerlof [1969 Fischer [1977, Taylor 1980), and Blanchard [1983]) Money growth is subject to continuous shocks. The stochastic process governing monetary growth is taken as exogenous by all firms in the economy. Let M()denote the logarithm of the money supply at time t, where time is measured continuously We assume that the money supply process is increasing over time and does not ASSUMPTION 1. Monotonicity and Continuity. The money supply does not decrease over time, M(t2)2M(t,) for t2 2t1. Also, the money supply process is continuous in the time parameter t Normalize such that M(0)=0 The monotonicity assumption will rule out periods of deflation. The continuity assumption allows a simple characterization of firm pricing policies. The assumption also plays a role in analyzing the cross-sectional behavior of prices. This issue is taken up below.The monetary process is sufficiently general as to accommodate feed back rules. We shall consider particular examples of monetary es bel There is a continuum of firms in the economy indexed by i E [0, 1]. All firms face identical demand and cost conditions. The assumed microeconomic structure is based on the menu cost model of Sheshinski and Weiss [1977, 1983]. Let qi (t)and Q (t)represen firm i's nominal price and the aggregate price index, respectively with pi (t)and P(t)their respective logarithms. The aggregate price index, P(t), is derived endogenously below from individual firm prices. It is convenient to express firm is real price, q(t)/Q(t),in log form, ri (t), t)-ln[q(t)/?(t)], for all E [o, 1]. We take ri (O)as given The aggregate price index Q(t)is determined endogenously by aggregating individual firms' nominal prices qi(t). The index is assumed to depend only on the frequency distribution over nominal prices. Because firms have menu costs of price adjustment, prices may remain dispersed in the long run. Thus, the set of observed prices at any date may be described by a time-dependent frequency distribution function, say G (q). The index is assumed also to 2. In general, the money growth process may be set as a feedback rule based

MENU COSTS AND THE NEUTRaITY OF MONEY 705 price and wage decisions. These alternative models frequently assume a staggered pattern of timing (e.g., Akerlof [1969], Fischer [1977], Taylor [1980], and Blanchard [1983]). Money growth is subject to continuous shocks. The stochastic process governing monetary growth is taken as exogenous by all firms in the e~onomy.~ Let M(t) denote the logarithm of the money supply at time t, where time is measured continuously. We assume that the money supply process is increasing over time and does not make discrete jumps. ASSUMPTION1. Monotonicity and Continuity. The money supply does not decrease over time, M(t,) rM(t,) for t, 2 t,. Also, the money supply process is continuous in the time parameter t. Normalize such that M (0) = 0. The monotonicity assumption will rule out periods of deflation. The continuity assumption allows a simple characterization of firm pricing policies. The assumption also plays a role in analyzing the cross-sectional behavior of prices. This issue is taken up below. The monetary process is sufficiently general as to accommodate feed￾back rules. We shall consider particular examples of monetary processes below. There is a continuum of firms in the economy indexed by i E [0,1]. All firms face identical demand and cost conditions. The assumed microeconomic structure is based on the menu cost model of Sheshinski and Weiss [1977, 19831. Let qi(t) and Q(t) represent firm i's nominal price and the aggregate price index, respectively, with pi(t) and P(t) their respective logarithms. The aggregate price index, P(t), is derived endogenously below from individual firm prices. It is convenient to express firm i's real price, q(t)lQ(t), in log form, ri(t), for all i E [0,1]. We take ri(0) as given. The aggregate price index Q (t) is determined endogenously by aggregating individual firms' nominal prices qi(t). The index is assumed to depend only on the frequency distribution over nominal prices. Because firms have menu costs of price adjustment, prices may remain dispersed in the long run. Thus, the set of observed prices at any date may be described by a time-dependent frequency distribution function, say G,(q). The index is assumed also to 2. In general, the money growth process may be set as a feedback rule based on the history of output

QUARTERLY JOURNAL OF ECONOMICS satisfy homogeneity; when nominal prices double, so does the ASSUMPTION 2. Symmetric Price Index. The aggregate price index Q(t)depends only on the frequency distribution of nominal prices and satisfies homogeneity: (2)Q(t)=Q(G,(q)), where G (q)is the proportion of firms i∈[0,1 such that q:(t)≤q, 3)if G, (q)-GL (q) for all q, then xQ(t,)-Q(t2), for any ti, t2 20 This condition is satisfied by a wide variety of common price indices.An example of a price index that satisfies Assumption 2 is a simple average of nominal prices based on their frequency distribu tion,Q(t)=adG (q). More generally, let Q(t) w(q, G ())qdG (q), where w(, G)represents weights as a function of prices q and the distribution of nominal prices G. The assump tion requires the weights to satisfy w(q, G, )-w(g, Gi, ) when G, ( q)-G (q) for all q. An example of such a set of weights is w(q, G)-ql/ adG(q) IIB. The Market Setting Consumer demand is assumed to depend only on the firm s real price and on real money balances. Writing the arguments in log form, consumer demand faced by firm i, Ti, is defined by Ti (t)=r(r: (t), M(t)-P(t)), where r (t)and M(t)-P(t)are the log of firm i's price and the lo of real balances, respectively. One rationale for this is to assume that real balances enter consumer utility functions, as in, for example, Rotemberg [1982, 1983]. Note also that all firms can have Individual firms set s and S taking the price level as exogenous S, the index endogenously determines P(O O)relative to the exe 4. Blanchard and Kiyotaki [1985]and Ball and Romer 1986] derive symmetric price indices based on an underlying symmetric utility framework dependent of future prices rules out Benabou [1985a real money balances may also influence real demand. For present purposes, Proposition 1 will allow us to ignore this potentially complex depender

706 QUARTERLY JOURNAL OF ECONOMICS satisfy homogeneity; when nominal prices double, so does the index.3 ASSUMPTION2. Symmetric Price Index. The aggregate price index Q(t) depends only on the frequency distribution of nominal prices and satisfies homogeneity: (2) Q (t) = Q (G,(q)), where Gt(q) is the proportion of firms i E [0,1] such that q,(t) i q, (3) if G,,(q) = Gt2(hq) for all q, then XQ(t,) = Q(t,), for any t,, t, r 0. This condition is satisfied by a wide variety of common price in dice^.^ An example of a price index that satisfies Assumption 2 is a simple average of nominal prices based on their frequency distribu￾tion, Q(t) = fqd~t(q).More generally, let Q(t) = f w (q,Gt(. ))qdGt (q), where w (q,G) represents weights as a function of prices q and the distribution of nominal prices G. The assump￾tion requires the weights to satisfy w(q,GtI) = w(hq, G,,) when Gtl(q) = Gt2(Xq) for all q. An example of such a set of weights is w(q,G) = q/f qdG(q). IIB. The Market Setting Consumer demand is assumed to depend only on the firm's real price and on real money balances. Writing the arguments in log form, consumer demand faced by firm i, ri, is defined by (4) r,(t) = r(ri(t), M(t) -P(t)), where ri(t) and M(t) -P(t) are the log of firm i's price and the log of real balances, re~~ectively.~ One rationale for this is to assume that real balances enter consumer utility functions, as in, for example, Rotemberg [1982,1983]. Note also that all firms can have 3. Individual firms set s and S taking the price level as exogenously given. However, for given levels s and S , the index endogenously determines P(0):will the exogenous and endogenous indices be consistent? The answer is generally no: however, if we associate higher real balances with higher levels of s and S , there will be some initial specification of real balances guaranteeing this static consistency, since higher real balances raise the desired average real price, raising the endogenous level of P(0)relative to the exogenous level. 4. Blanchard and Kiyotaki [I9851 and Ball and Romer [I9861 derive symmetric price indices based on an underlying symmetric utility framework. 5. The assumption that demand is independent of future prices rules out consumer speculation. Benabou [1985a] presents an analysis of optimal pricing policies in the face of consumer storage and speculation. In principle, the future path of real money balances may also influence real demand. For present purposes, Proposition 1will allow us to ignore this potentially complex dependence

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 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 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 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

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 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, 19851. 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 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

MENU COSTS AND THE NEUTRALITY OF MONEY 711 DEFINITION 1. Money is neutral if aggregate real output is invariant to monetary shocks, X(t)=X(O), for allt20 Monetary policy may influence the distribution of real prices across firms in our model as will be seen in Section Iv. However, these distributional effects cancel out in the aggregate Suppose that firms follow s, S)policies in anticipation of constant real balances. That is, firms expect that P(t)-M(t) Then, by the description of (s, S)pricing policies in equation(7),we nay calculate each firms nominal price as a function of cumulative money growth and the firms initial price (t)=k(t)D+p2(O), where ki (t)is an integer determined by the requirement that (t)E(s, S]. Proposition 1 verifies that aggregation of these nomi- nal prices yields a price level equal to cumulative money growth at each time t, so that money is neutral The neutrality result may be understood by observing that the (s, S)policy moves real prices around a circle The method of proof r: t FIGURE II

MENU COSTS AND THE NEUTRALITY OF MONEY 711 DEFINITION1. Money is neutral if aggregate real output is invariant to monetary shocks, X(t) = X(O), for all t 2 0. Monetary policy may influence the distribution of real prices across firms in our model as will be seen in Section IV. However, these distributional effects cancel out in the aggregate. Suppose that firms follow (s,S) policies in anticipation of constant real balances. That is, firms expect that P(t) = M(t). Then, by the description of (s,S) pricing policies in equation (7), we may calculate each firm's nominal price as a function of cumulative money growth and the firm's initial price: where hi(t) is an integer determined by the requirement that ri(t) E (s,S]. Proposition 1verifies that aggregation of these nomi￾nal prices yields a price level equal to cumulative money growth at each time t, so that money is neutral. The neutrality result may be understood by observing that the (s,S) policy moves real prices around a circle. The method of proof