Optimal Monetary Policy with State-Dependent Pricing

Finance and Economics Discussion Series Divisions of Research & Statistics and Monetary Affairs Federal Reserve Board, Washington, D.C. Optimal Monetary Policy with State-Dependent Pricing Anton Nakov and Carlos Thomas 2011-48 NOTE: Staff working papers in the Finance and Economics Discussion Series (FEDS) are preliminary materials circulated to stimulate discussion and critical comment. The analysis and conclusions set forth are those of the authors and do not indicate concurrence by other members of the research staff or the Board of Governors. References in publications to the Finance and Economics Discussion Series (other than acknowledgement) should be cleared with the author(s) to protect the tentative character of these papers.

Optimal Monetary Policy with State-Dependent Pricing (cid:3) Anton Nakov Carlos Thomas Federal Reserve Board Banco de Espaæa 2nd November 2011 Abstract In an abstract economic model, we study optimal monetary policy from the timeless perspective under a general state-dependent pricing framework. We (cid:133)nd that when (cid:133)rms are monopolistic competitors subject to idiosyncratic menu cost shocks, households have isoelastic preferences, and there is no government spending, strict price stability is optimal bothinthelongrunandinresponsetoaggregateshocks. Keytothis(cid:133)ndingisan(cid:147)envelope(cid:148) property: At zero in(cid:135)ation, a marginal increase in the rate of in(cid:135)ation has no e⁄ect on (cid:133)rms(cid:146) pro(cid:133)ts and therefore it has no e⁄ect on the probability of price adjustment. Our results lend support to more informal statements about the suitability of the Calvo model for studying optimal monetary policy despite its apparent con(cid:135)ict with the Lucas critique. We o⁄er an analytic solution that does not require local approximation or e¢ ciency of the steady state. Keywords: monetary policy, state-dependent pricing, monopolistic competition JEL Codes: E31 1 Introduction A key normative question in monetary economics is the design of optimal monetary policy. An extensive amount of literature studies this question under the assumption that the timing of price changes is given exogenously, typically using the Calvo (1983) model with a constant adjustment rate.1 Useful as it is as a (cid:133)rst approximation, this literature nevertheless is subject to the Lucas We are grateful for comments and suggestions to John Roberts, Jordi Gali, Luca Dedola, Oreste Tristani, (cid:3) GianniLombardo,seminarparticipantsatECBandtheFederalReserveBoardandconferenceparticipantsatSCE 2011. Anton Nakov thanks the European Central Bank for its hospitality and support during the (cid:133)rst drafts of this paper. The views expressed in this paper are those of the authors and should not be interpreted as coinciding with the views of the Federal Reserve System or the Eurosystem. Corresponding author: Carlos Thomas, Servicio de Estudios, Bank of Spain, Alcala 48, 28014 Madrid, Spain. E-mail: carlos.thomas@bde.es 1For example, Clarida, Gal(cid:237), and Gertler (1999); Woodford (2002, 2003); Yun (2005); Benigno and Woodford (2005). 1

(1976) critique: In principle, the frequency of price changes should not be treated as a parameter which is independent of policy. Many economists, therefore, have argued against the use of the Calvo model, claiming that it provides a poor approximation to more elaborate models of price adjustment. For example, Golosov and Lucas (2007) show that the behavior of (cid:133)rms in the Calvo model is very di⁄erent from that in a (cid:147)menu cost(cid:148)model, when (cid:133)rms are subject to idiosyncratic productivity shocks as well as aggregate money growth shocks. This paper studies optimal monetary policy in a model of state-dependent pricing by monopolistically competitive (cid:133)rms. In models of this sort the frequency of adjustment is a statistic determined in equilibrium, not an exogenous parameter. In particular, we will work with a model in which individual prices are sticky because (cid:133)rms are subject to random idiosyncratic lump-sum costs of adjustment (cid:224) la Dotsey, King, and Wolman (1999). Each (cid:133)rm would change its price only if the increase in the (cid:133)rm(cid:146)s value due to adjustment exceeds the (cid:147)menu cost.(cid:148)As a result, the probability with which (cid:133)rms reoptimize prices depends on the gains from adjustment. This framework is very (cid:135)exible because it nests a variety of pricing speci(cid:133)cations, including the Calvo model and the (cid:133)xed menu cost model as extreme limiting cases (Costain and Nakov, 2011). Asidefrompricingbeingstate-dependent,oursetupfollowscloselythestandardNewKeynesian model with Calvo pricing (for example, Benigno and Woodford, 2005). Inparticular, the monetary authority is assumed to set the nominal interest rate, with money(cid:146)s role being only that of a unit of account. An important distinction with Clarida, Gal(cid:237), and Gertler (1999), Woodford (2002), and Yun (2005), is that we assume no production subsidy to o⁄set the markup distortion due to monopolistic competition. This implies that the steady state level of output is ine¢ ciently low. Hence, the central bank has a constant temptation to in(cid:135)ate the economy so as to bring output closer to its e¢ cient level. We derive the optimal plan from the timeless perspective, as in Woodford (2003).2 We demonstrate analytically that, if preferences are isoelastic and there is no government spending, it is optimal to commit to zero in(cid:135)ation both in the long run and in reaction to shocks. Importantly, this result holds for a general speci(cid:133)cation of the menu cost distribution. In the optimal allocation, price markups are positive but constant, output is at its natural ((cid:135)exible-price) level, and price dispersion is minimized. Perhaps surprisingly, this prescription coincides with the one obtained under Calvo pricing (Benigno and Woodford, 2005). Thereasonwhyzeroin(cid:135)ationisoptimalinourmodelofstate-dependentpricingisthefollowing. Relative to Calvo pricing, our stochastic menu costs model implies two additional welfare e⁄ects of in(cid:135)ation. First, (cid:133)rms must spend real resources (menu costs) on adjusting nominal prices. This distortion is minimized at zero in(cid:135)ation because under such a policy all (cid:133)rms are at their optimal 2That is, the plan ignores policymakers(cid:146)incentives to behave di⁄erently in the initial few periods, exploiting the private sector(cid:146)s expectations that had formed prior to the plan(cid:146)s starting date. 2

price. The second e⁄ect is somewhat more subtle. The main di⁄erence between exogenous-timing and state-dependent pricing models is that price adjustment frequencies are endogenous in the latter. A priori, the monetary authority could have an incentive to use in(cid:135)ation so as to a⁄ect the rate at which (cid:133)rms change their prices. If (cid:133)rms adjusted their prices faster in reaction to shocks, in principle one would expect that to have bene(cid:133)cial welfare e⁄ects. However, the fact that adjusting (cid:133)rms set optimally their prices implies that, in the timeless perspective regime with zero in(cid:135)ation, a marginal increase in the rate of in(cid:135)ation has no e⁄ect on (cid:133)rms(cid:146)pro(cid:133)ts, and therefore it has no e⁄ect on the rate of adjustment. This envelope property implies that the monetary authority has no incentive to deviate from zero in(cid:135)ation in order to a⁄ect the speed of adjustment. We also show that the same reasons for which zero in(cid:135)ation is optimal under Calvo pricing continue to hold under state-dependent pricing. First, ine¢ cient price dispersion is minimized at zero in(cid:135)ation. Second, in the timeless perspective regime with zero in(cid:135)ation, the marginal welfare gainfromraisingoutputtowarditssociallye¢ cientlevel(i.e. amovementalongthePhillipscurve) exactly cancels out with the marginal welfare loss from committing to and generating expectations of future in(cid:135)ation (i.e., an upward shift of the Phillips curve). This (cid:133)nding echoes Kydland and Prescott(cid:146)s (cid:147)rules versus discretion(cid:148). However, we (cid:133)nd that it is independent of whether pricing is time- or state-dependent. Our results thus lend support to more informal statements about the suitability of the Calvo model for studying optimal monetary policy despite its apparent con(cid:135)ict with the Lucas (1976) critique. In particular, we provide su¢ cient conditions under which, even though pricing is statedependent and so the adjustment frequency is endogenously determined, it turns out that the probability of adjustment remains constant under the optimal policy. When these conditions are satis(cid:133)ed, which is what the literature usually assumes, the distinction between time-dependent and state-dependent pricing frameworks vanishes, provided that monetary policy is set optimally.3 The following section lays out the model and derives the conditions for equilibrium. Section 3 sets up the optimal monetary policy problem and obtains the main result regarding the optimality ofzeroin(cid:135)ationfromthetimelessperspective; italsoformalizesthemainintuitionwithasimpli(cid:133)ed version of the model (with the full proof in the Appendix). Section 4 analyzes numerically the case with positive government expenditure; for a plausible calibration of the model, we (cid:133)nd that the optimal deviations from strict price stability in response both to productivity and to government spendingshocksareindistinguishablefromzero.4 Section5concludeswithadiscussionofapossible extension. 3Independently, Lie (2009) studies numerically the optimal monetary policy in a New Keynesian model with stochastic menu costs and a monetary friction. 4Existing studies of optimal monetary policy with monopolistic distortions prove analytically the existence of a short-run tradeo⁄ between in(cid:135)ation and output stabilization in the presence of positive government spending (Benigno and Woodford, 2005; Woodford, ch. 6, section 5). However, they do not quantify the importance of the tradeo⁄. We show that in a model such as ours the tradeo⁄is negligible. 3

2 Model There are three types of agents: households, (cid:133)rms, and a monetary authority. We begin by describing the behavior of households and (cid:133)rms. 2.1 Households A representative household maximizes the expected (cid:135)ow of period utility u(C ) x(N ;(cid:31) ); dist t t (cid:0) counted by (cid:12), subject to 1 (cid:15)=((cid:15) 1) (cid:0) C t = C i ( t (cid:15) (cid:0) 1)=(cid:15)di (cid:18)Z0 (cid:19) and 1 P C di+R 1B = W N +B +(cid:5) ; it it t(cid:0) t t t t 1 t (cid:0) Z0 where C is a basket of di⁄erentiated goods i [0;1]; of quantity C and price P ; N denotes t it it t 2 hours worked and W is the nominal wage rate; (cid:31) is an exogenous shock to the disutility of labor;5 t t B are nominally riskless bonds with price R 1, and (cid:5) are the pro(cid:133)ts of (cid:133)rms owned by the t t(cid:0) t household, net of lump-sum taxes. The (cid:133)rst order conditions are u (C )w = x (N ;(cid:31) ); (1) 0 t t 0 t t u (C ) R 1 = (cid:12)E 0 t+1 ; (2) t(cid:0) t (cid:25) u (C ) t+1 0 t where w W =P is the real wage, (cid:25) P =P is the gross in(cid:135)ation rate, and the aggregate t t t t t t 1 (cid:17) (cid:17) (cid:0) price index is given by 1 1=(1 (cid:15)) (cid:0) P P1 (cid:15)di : t (cid:17) it(cid:0) (cid:18)Z0 (cid:19) 2.2 Firms There is a continuum of (cid:133)rms on the unit interval. Firm i(cid:146)s production function is y = z n ; it t it where z is an exogenous aggregate productivity process. The (cid:133)rm(cid:146)s labor demand thus equals t n = y =z and its real cost function is w y =z . The real marginal cost common to all (cid:133)rms is it it t t it t 5Our results hold also in the case when the utility of consumption is a⁄ected by a preference shock; here we ommit such a shock for simplicity. 4

therefore w =z . Optimal allocation of expenditure across product varieties by households implies t t that each individual (cid:133)rm faces a downward-sloping demand schedule for its good, given by y = it (P =P ) (cid:15)y : it t (cid:0) t Following Dotsey et al. (1999), we assume that (cid:133)rms face random lump sum costs of adjusting prices ((cid:147)menu costs(cid:148)), distributed i.i.d. across (cid:133)rms and over time. Let G((cid:20)) and g((cid:20)) denote the cumulativedistributionfunctionandtheprobabilitydensityfunction,respectively,ofthestochastic menu cost (cid:20) 0. We assume that a positive random fraction of (cid:133)rms draw a zero menu cost, so (cid:21) that G(0) > 0.6 Assuming that (cid:20) is measured in units of labor time, the total cost paid by a (cid:133)rm changing its price is w (cid:20).7 t Let v denote the value of a (cid:133)rm that adjusts its price in period t before subtracting the menu 0t cost. Let v (P) denote the value of a (cid:133)rm that has kept its nominal price unchanged at the level jt P in the last j periods. This (cid:133)rm will change its nominal price only if the value of adjustment, v w (cid:20), exceeds the value of continuing with the current price, v (P). Therefore, from the set 0t t jt (cid:0) of (cid:133)rms that last reoptimized j periods ago (which we henceforth refer to as (cid:147)vintage-j (cid:133)rms(cid:148)), only those with a menu cost draw (cid:20) (v v (P))=w will choose to change their price. The 0t jt t (cid:20) (cid:0) real value of an adjusting (cid:133)rm is given by u (C ) v v (P) 0 t+1 0;t+1 1;t+1 v = max (cid:5) (P)+(cid:12)E G (cid:0) v (cid:4) (P) 0t t t 0;t+1 1;t+1 P (cid:26) u 0 (C t ) (cid:20) (cid:18) w t+1 (cid:19) (cid:0) (cid:21) u (C ) v v (P) 0 t+1 0;t+1 1;t+1 +(cid:12)E 1 G (cid:0) v (P) ; t 1;t+1 u (C ) (cid:0) w 0 t (cid:20) (cid:18) t+1 (cid:19)(cid:21) (cid:27) where (cid:12)u (C )=u (C ) is the stochastic discount factor between periods t and t+s t, 0 t+s 0 t (cid:21) P w P (cid:15) t (cid:0) (cid:5) (P) Y t t (cid:17) P (cid:0) z P (cid:18) t t (cid:19)(cid:18) t(cid:19) is the (cid:133)rm(cid:146)s real pro(cid:133)t as a function of its nominal price P, and (v0;t+1 vj+1;t+1(P))=wt+1 (cid:0) (cid:4) (P) w (cid:20)g((cid:20))dk j+1;t+1 t+1 (cid:17) Z0 is next period(cid:146)s expected adjustment cost for a (cid:133)rm currently in vintage j. The real value of a 6We make this technical assumption to ensure a unique stationary distribution of (cid:133)rms over price vintages in the case of zero in(cid:135)ation. See the Appendix for details. 7Alternatively, we can assume that (cid:20) is measured in terms of the basket of (cid:133)nal goods, in which case the total cost paid by a (cid:133)rm changing its price is simply (cid:20). The results are not dependent on this assumption. 5

(cid:133)rm in vintage j, as a function of its current nominal price P, is given by u (C ) v v (P) 0 t+1 0;t+1 j+1;t+1 v (P) = (cid:5) (P)+(cid:12)E G (cid:0) v (cid:4) (P) jt t t 0;t+1 j+1;t+1 u (C ) w (cid:0) 0 t (cid:20) (cid:18) t+1 (cid:19) (cid:21) u (C ) v v (P) 0 t+1 0;t+1 j+1;t+1 +(cid:12)E 1 G (cid:0) v (P): (3) t j+1;t+1 u (C ) (cid:0) w 0 t (cid:20) (cid:18) t+1 (cid:19)(cid:21) We assume that J periods after the last price adjustment, (cid:133)rms draw a zero menu cost.8 This meansthat(cid:133)rmsinvintageJ 1 knowthatinthefollowingperiodtheywill adjusttheirpricewith (cid:0) probability one at no cost. Therefore, expression (3) holds for vintages j = 1;:::;J 2, whereas (cid:0) for vintage-(J 1) (cid:133)rms the corresponding value function is (cid:0) u (C ) 0 t+1 v (P) = (cid:5) (P)+(cid:12)E v : (4) J 1;t t t 0;t+1 (cid:0) u 0 (C t ) The optimal price setting decision is given by u (C ) v v (P ) 0 = (cid:5) (P )+(cid:12)E 0 t+1 1 G 0;t+1 (cid:0) 1;t+1 t(cid:3) v (P ); (5) 0t t(cid:3) t u (C ) (cid:0) w 10;t+1 t(cid:3) 0 t (cid:20) (cid:18) t+1 (cid:19)(cid:21) where w P (cid:5) (P) = (cid:15) t ((cid:15) 1) (P) (cid:15) 1P(cid:15)Y : 0t z (cid:0) (cid:0) P (cid:0) (cid:0) t t (cid:20) t t(cid:21) Iterating (5) forward, and using the implications of (3) and (4) for the terms v (P ), j = j0;t+j t(cid:3) 1;:::;J 1, we can express the pricing decision as (cid:0) P = (cid:15) j J = (cid:0) 0 1(cid:12)jE t j k=1 (1 (cid:0) (cid:21) k;t+k )u 0 (C t+j )P t (cid:15) +j Y t+j (w t+j =z t+j ) ; t(cid:3) (cid:15) 1 J 1(cid:12)jE j (1 (cid:21) )u (C )P(cid:15) 1Y (cid:0) P j= (cid:0) 0Q t k=1 (cid:0) k;t+k 0 t+j t+(cid:0)j t+j P Q where v v 0t jt (cid:21) G (cid:0) (6) jt (cid:17) w (cid:18) t (cid:19) denotes the period-t adjustment probability of (cid:133)rms in vintage j = 1;:::;J 1, and we de(cid:133)ne v jt (cid:0) (cid:17) v (P ) for short. As emphasized by Dotsey et al. (1999), this pricing decision is analogous to the jt t(cid:3) j (cid:0) one in the Calvo model. In particular, the term j (1 (cid:21) ) is the endogenous probability k=1 (cid:0) k;t+k that the price chosen at t survives for the next j periods, thus replacing the exogenous probability Q 1 (cid:21)C j where (cid:21)C is the constant adjustment probability in the Calvo model. We can rewrite (cid:0) (cid:0) 8This(cid:1)is a tractability assumption which ensures a (cid:133)nite state space under zero in(cid:135)ation or when the support of the menu cost distribution is unbounded from above. 6

the price decision in terms of stationary variables as (cid:15) J 1(cid:12)jE j (1 (cid:21) ) j (cid:25) u (C )Y (w =z ) (cid:15) j= (cid:0) 0 t k=1 (cid:0) k;t+k k=1 t+k 0 t+j t+j t+j t+j p = ; (7) (cid:3)t (cid:15) 1 (cid:16) (cid:17) (cid:15) 1 (cid:0) P J j= (cid:0) 0 1(cid:12)QjE t j k=1 (1 (cid:0) (cid:21) k;t+ Q k ) j k=1 (cid:25) t+k (cid:0) u 0 (C t+j )Y t+j (cid:16) (cid:17) P Q Q where p P =P is the optimal relative price and j (cid:25) = P =P is accumulated in(cid:135)ation (cid:3)t (cid:17) t(cid:3) t k=1 t+k t+j t between periods t and t+j. Q 2.3 Market clearing Labor input is required both for the production of goods and for changing prices. Labor demand for production by (cid:133)rm i is n = y =z = (P =P ) (cid:15)y =z . Thus, total labor demand for proit it t it t (cid:0) t t duction purposes equals (cid:1) y =z , where (cid:1) 1 (P =P ) (cid:15)di denotes relative price dispersion. t t t t (cid:17) 0 it t (cid:0) At the same time, the total amount of labor used by vintage-j (cid:133)rms for pricing purposes equals R (v0t (cid:0) vjt)=wt(cid:20)g((cid:20))dk, where is the mass of (cid:133)rms in vintage j. Equilibrium in the labor jt 0 jt market therefore implies R Y (cid:1) J 1 (v0t vjt)=wt N = t t + (cid:0) (cid:0) (cid:20)g((cid:20))dk: (8) t z jt t j=1 Z0 P Also, equilibrium in the goods market requires that Y = C +G ; (9) t t t where G denotes government expenditure, which follows an exogenous process. t 2.4 In(cid:135)ation, price dispersion, and price distribution dynamics All (cid:133)rms adjusting at time t choose the same nominal price, P . Given that no nominal price t(cid:3) survives for longer than J periods by assumption, the (cid:133)nite set of beginning-of-period prices at any time t is P ;P ;:::;P . Let denote the time-t fraction of (cid:133)rms with beginning-oft(cid:3) 1 t(cid:3) 2 t(cid:3) J jt (cid:0) (cid:0) (cid:0) period nominal price P , for j = 1;2;:::;J, with J = 1. The price level evolves according (cid:8) t(cid:3) j (cid:9) j=1 jt (cid:0) to P J J 1 P t 1 (cid:0) (cid:15) = (P t(cid:3) )1 (cid:0) (cid:15) (cid:21) jt jt + (cid:0) P t(cid:3) j 1 (cid:0) (cid:15) (1 (cid:0) (cid:21) jt ) jt ; j=1 j=1 (cid:0) P P (cid:0) (cid:1) 7

where adjustment probabilities (cid:21) J 1 are given by (6), and where (cid:21) = 1. Rescaling both sides f jt gj=(cid:0)1 J;t of the above equation by P , we obtain t 1 (cid:15) J J 1 p (cid:0) 1 = (p (cid:3)t )1 (cid:0) (cid:15) j=1 (cid:21) jt jt + j= (cid:0) 1 j k (cid:0) = (cid:3)t 1 0 (cid:0) (cid:25) j t k! (1 (cid:0) (cid:21) jt ) jt : (10) (cid:0) P P Q This equation determines the in(cid:135)ation rate (cid:25) t , given p (cid:3)t (cid:0) j J j= (cid:0) 0 1 and f (cid:25) t (cid:0) j g J j= (cid:0) 1 2. Similarly, price dispersion follows (cid:8) (cid:9) (cid:15) J J 1 p (cid:0) (cid:1) t = (p (cid:3)t ) (cid:0) (cid:15) j=1 (cid:21) jt jt + j= (cid:0) 1 j k (cid:0) = (cid:3)t 1 0 (cid:0) (cid:25) j t k! (1 (cid:0) (cid:21) jt ) jt ; (11) (cid:0) P P Q where again (cid:21) = 1. The distribution of beginning-of-period prices evolves according to J;t = (1 (cid:21) ) (12) j;t j 1;t 1 j 1;t 1 (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) for j = 2;:::;J, and = 1 J = (cid:21) +(cid:21) +:::+ : (13) 1t (cid:0) j=2 j;t 1;t (cid:0) 1 1;t (cid:0) 1 2;t (cid:0) 1 2;t (cid:0) 1 J;t (cid:0) 1 P 2.5 Equilibrium There are 8+2J +(J 1) = 7+3J stationary endogenous variables: C , N , Y , R , (cid:25) , p , w , (cid:0) t t t t t (cid:3)t t (cid:1) , J , v J 1; and (cid:21) J 1. The equilibrium conditions are (1), (2), the J 1 equations t jt j=1 f jt gj= (cid:0) 0 f jt gj= (cid:0) 1 (cid:0) (6), equations (7) to (11), the J laws of motion (12) and (13), the value functions (cid:8) (cid:9) (cid:15) p w p (cid:0) (cid:3)t j t (cid:3)t j v jt = j k (cid:0) = 1 0 (cid:0) (cid:25) t k (cid:0) z t ! k j (cid:0) = 1 0 (cid:0) (cid:25) t k! Y t (cid:0) (cid:0) Q u 0 (C t+1 ) Q (v0;t+1 (cid:0) vj+1;t+1)=wt+1 +(cid:12)E (cid:21) v +(1 (cid:21) )v w (cid:20)dG((cid:20)) t j+1;t+1 0;t+1 j+1;t+1 j+1;t+1 t+1 u (C ) (cid:0) (cid:0) 0 t " Z0 # for j = 0;1;:::;J 2, and (cid:0) (cid:15) p (cid:3)t (J 1) w t p (cid:3)t (J 1) (cid:0) u 0 (C t+1 ) v J (cid:0) 1;t = ( k J = (cid:0) 0 (cid:0) 1) (cid:0) (cid:0) 1(cid:25) t k (cid:0) z t ! ( k J = (cid:0) 0 (cid:0) 1) (cid:0) (cid:0) 1(cid:25) t k ! Y t +(cid:12)E t u 0 (C t ) v 0;t+1 ; (cid:0) (cid:0) Q Q plus a speci(cid:133)cation of monetary policy. If we were to close the model with a Taylor rule, this would give us a total of 2+(J 1)+5+J+J+1 = 7+3J equations. Instead, we will study the optimal (cid:0) state-contingent monetary policy plan, which will essentially double the number of equations and 8

variables. 2.5.1 Flexible-price equilibrium It is instructive to derive the (cid:135)exible-price equilibrium in this framework. In such an equilibrium, menu costs are zero and all (cid:133)rms choose the same nominal price P = (cid:15) wtP in each period t. t(cid:3) (cid:15) 1 zt t (cid:0) All relative prices are one: p = P =P = 1. The equilibrium conditions simplify to (cid:3)t t(cid:3) t u(Cfp)wfp = x(Nfp;(cid:31) ); 0 t t 0 t t z Nfp = Yfp; t t t Yfp = Cfp +G ; t t t (cid:15) z = wfp; t (cid:15) 1 t (cid:0) and so we obtain the classical decoupling of real and nominal variables. The (cid:135)exible-price output Yfp derived above is used in de(cid:133)ning the output gap as the ratio between actual output and its t (cid:135)exible-price counterpart. 3 Optimal monetary policy 3.1 The general problem For the purpose of deriving the optimality conditions of the Ramsey plan, it is useful to de(cid:133)ne P (cid:25)acc j 1 (cid:25) = t ; j = 1;:::;J 1; jt (cid:17) k (cid:0) =0 t (cid:0) k P t j (cid:0) (cid:0) Q that is, the accumulated in(cid:135)ation between periods t j and t. This implies j (cid:25) = (cid:25)acc . (cid:0) k=1 t+k j;t+j We also de(cid:133)ne Q (cid:18) j 1 (1 (cid:21) ); j = 1;:::;J 1; jt (cid:17) k (cid:0) =0 (cid:0) j (cid:0) k;t (cid:0) k (cid:0) that is, the probability that aQprice chosen at t j survives until t, which in turn implies (cid:0) j (1 (cid:21) ) = (cid:18) . These de(cid:133)nitions allow us to express the optimal pricing decision k=1 (cid:0) k;t+k j;t+j in equation (7) in a more compact form, Q (cid:15) J j= (cid:0) 0 1(cid:12)jE t (cid:18) j;t+j (cid:25)a j; c t c +j (cid:15) u 0 (C t+j )Y t+j (w t+j =z t+j ) p = : (cid:3)t (cid:15) (cid:0) 1 P J j= (cid:0) 0 1(cid:12)jE t (cid:18)(cid:0)j;t+j (cid:25)(cid:1) a j; c t c +j (cid:15) (cid:0) 1 u 0 (C t+j )Y t+j P (cid:0) (cid:1) 9

Similarly, we replace j 1 (cid:25) by (cid:25)acc in the laws of motion of in(cid:135)ation and price dispersion, and k (cid:0) =0 t k jt (cid:0) in the (cid:133)rms(cid:146)value functions. It is useful to express the variables (cid:25)acc and (cid:18) recursively, Q jt jt (cid:25)acc = (cid:25) (cid:25)acc ; j = 1;:::;J 1; jt t j 1;t 1 (cid:0) (cid:0) (cid:0) (cid:18)acc = (1 (cid:21) )(cid:18)acc ; j = 1;:::;J 1; jt (cid:0) jt j 1;t 1 (cid:0) (cid:0) (cid:0) wheretherecursionsstartwith(cid:25)acc = 1and(cid:18)acc = 1,respectively. Weusew = x (N ;(cid:31) )=u (C ) 0;t 1 0;t 1 t 0 t t 0 t (cid:0) (cid:0) to substitute for the real wage in the equilibrium conditions. In addition, we use the constraint Y = C +G to substitute for C . Finally, we de(cid:133)ne v~ v u (C ), j = 0;1;:::;J 1, such that t t t t jt jt 0 t (cid:17) (cid:0) (v v )=w = (v~ v~ )=x (N ;(cid:31) ). At time 0, the central bank chooses the state-contingent 0t jt t 0t jt 0 t t (cid:0) (cid:0) path for all endogenous variables, which maximizes the following Lagrangian: = E (cid:12)t u(Y G ) x(N ;(cid:31) ) L 0 0 1 t=0 f t (cid:0) t (cid:0) t t J 1 (cid:15) J 1 x N ;(cid:31) +(cid:30)p (cid:3) p P(cid:0) (cid:12)j(cid:18) ((cid:25)acc )(cid:15) 1Y u (Y G ) (cid:0) (cid:12)j(cid:18) ((cid:25)acc )(cid:15)Y 0 t+j t+j t (cid:3)t j;t+j j;t+j (cid:0) t+j 0 t+j (cid:0) t+j (cid:0) (cid:15) 1 j;t+j j;t+j t+j z " j=0 j=0 (cid:0) t+j (cid:1)# (cid:0) P P +(cid:30)N t N t (cid:0) Y t (cid:1) t =z t (cid:0) J j= (cid:0) 1 1 jt 0 (v~0t (cid:0) v~jt)=x 0 (Nt;(cid:31) t ) (cid:20)g((cid:20))d(cid:20) +(cid:30)(cid:25) t h (p (cid:3)t )1 (cid:0) (cid:15) J j=1 (cid:21) jt P jt + j J = (cid:0) R 1 1 p (cid:3)t j =(cid:25)a jt cc 1 (cid:0) (cid:15) (1 (cid:0) (cid:21) jt ) i jt (cid:0) +(cid:30)(cid:1) t h (p (cid:3)t ) (cid:0) (cid:15) PJ j=1 (cid:21) jt jt + P j J = (cid:0) 1 1 (cid:0)p (cid:3)t j =(cid:25)a jt cc(cid:1) (cid:0) (cid:15) (1 (cid:0) (cid:21) jt ) jt (cid:0) i (cid:1) t (cid:0) h i J 1 P v~ v~P (cid:0) J (cid:1) J + (cid:0) (cid:30) (cid:21)j (cid:21) G 0t (cid:0) jt + (cid:30) j (1 (cid:21) ) +(cid:30) 1 + j=1 t (cid:20) jt (cid:0) (cid:18) x 0 (N t ;(cid:31) t ) (cid:19)(cid:21) j=2 t j;t (cid:0) (cid:0) j (cid:0) 1;t (cid:0) 1 j (cid:0) 1;t (cid:0) 1 t " 1t j=2 j;t # P P (cid:2) (cid:3) P J 2 p x (N ;(cid:31) ) p (cid:15) + (cid:0) (cid:30) v t j (cid:25) (cid:3)t a (cid:0) cc j u 0 (Y t (cid:0) G t ) (cid:0) 0 z t t (cid:25) (cid:3)t a (cid:0) cc j (cid:0) Y t (cid:0) v~ jt j=0 " (cid:18) jt t (cid:19)(cid:18) jt (cid:19) # P + J (cid:0) 2 (cid:30) vj(cid:12) (cid:21) v~ +(1 (cid:21) )v~ x N ;(cid:31) (v~0t+1 (cid:0) v~j+1;t+1)=x 0 (Nt+1;(cid:31) t+1 ) (cid:20)g((cid:20))d(cid:20) t j+1;t+1 0;t+1 (cid:0) j+1;t+1 j+1;t+1 (cid:0) 0 t+1 t+1 0 j=0 (cid:20) (cid:21) P (cid:0) (cid:15) (cid:1)R +(cid:30) v t J (cid:0) 1 " p (cid:25) (cid:3)t (cid:0) a J ( c J c 1 (cid:0) ;t 1) u 0 (Y t (cid:0) G t ) (cid:0) x 0 (N z t t ;(cid:31) t ) ! p (cid:25) (cid:3)t (cid:0) a J ( c J c 1 (cid:0) ;t 1) ! (cid:0) Y t (cid:0) v~ J (cid:0) 1;t +(cid:12)v~ 0;t+1 # (cid:0) (cid:0) +(cid:30) (cid:25) t a 1 cc [(cid:25)a 1t cc (cid:0) (cid:25) t ]+ J j= (cid:0) 2 1(cid:30) (cid:25) t a j cc (cid:25)a jt cc (cid:0) (cid:25) t (cid:25)a j cc 1;t 1 (cid:0) (cid:0) +(cid:30) t (cid:18)1[(cid:18) 1t (cid:0) (1 (cid:0) (cid:21) 1t ) P ]+ j J = (cid:0) 2 1(cid:30)(cid:2) (cid:18) t j [(cid:18) jt (cid:0) (1 (cid:0) (cid:21) jt )(cid:3)(cid:18) j (cid:0) 1;t (cid:0) 1 ] g : (14) P Since the nominal interest rate only appears in the consumption Euler equation, the latter is excluded from the set of constraints on the Ramsey problem. Instead, this equation is used residually to back out the nominal interest rate path consistent with the optimal allocation. The (cid:133)rst-order conditions of the above problem are derived in the Appendix. Our object of interest is optimal monetary policy from a (cid:147)timeless perspective.(cid:148)As explained 10

by Woodford (2003), this type of policy does not exploit the private sector(cid:146)s expectations that formed prior to the particular date on which the plan was implemented. Instead, the central bank commits itself to behave, from date 0, in a way consistent with the way it would have chosen to behave had it committed to the optimal policy in the in(cid:133)nite past. The interest is thus in optimality in the long run, once the economy has converged to its ergodic distribution. The Appendix proves the following result: Proposition 1 Let functional forms for preferences be of the constant elasticity type and government expenditure be zero. Then the zero in(cid:135)ation policy ((cid:25) = 1) is optimal from the timeless t perspective. There are two important aspects of the above proposition. The (cid:133)rst is that optimal trend in(cid:135)ation is zero. Therefore, the presence of monopolistic distortions does not justify a positive rate of trend in(cid:135)ation, and the optimal policy involves a commitment to eventually eliminating any ine¢ cient price dispersion due to staggered price setting. This normative prescription is the same as the one implied by the standard New Keynesian model with Calvo price setting, as shown by BenignoandWoodford(2005).9 ThemaininsightoftheCalvoframework, aboutthedesirabilityof zero long-run in(cid:135)ation, thus continues to hold in a general model of state-dependent pricing. The key di⁄erence between exogenous-timing models of price adjustment such as Calvo(cid:146)s and statedependent pricing models is the endogeneity of the timing of price adjustment in the latter. A priori, the central bank could have an incentive to use trend in(cid:135)ation to in(cid:135)uence the speed at which (cid:133)rms change prices, if such a policy were to have bene(cid:133)cial e⁄ects on society. The above result implies that the endogeneity of price adjustment frequencies does not a⁄ect the optimality of zero trend in(cid:135)ation. To understand the intuition for this result, let us consider the di⁄erent channels through which trend in(cid:135)ation a⁄ects welfare. Two of these channels are common to exogenous-timing models such as Calvo or Taylor. One is that, in the presence of staggered prices, in(cid:135)ation increases the extent of price dispersion, distorting the economy(cid:146)s pricing system. This leads to ine¢ cient allocation of resources across product lines, and increases the total amount of (labor) resources needed to produce a given consumption basket; hence, it lowers welfare. Notice that ine¢ cient pricedispersionattainsaglobalminimumatzeroin(cid:135)ationbecause, undersuchapolicy, allrelative prices end up being equal. The other common channel, through which trend in(cid:135)ation a⁄ects welfare, works through its two opposing e⁄ects on the in(cid:135)ation-output tradeo⁄: On the one hand, holding constant in(cid:135)ation expectations, a rise in current in(cid:135)ation allows the central bank to raise output toward its socially 9The same result holds for another prominent exogenous-timing model of price adjustment, namely the Taylor model, where adjustment probabilities are zero for a number of periods after a price change and one afterwards. A proof of the latter result is available upon request from the authors. 11

e¢ cient level, thus reducing the monopolistic distortion and improving welfare; intuitively, the economy moves along the New Keynesian Phillips curve (NKPC).10 On the other hand, choosing higher in(cid:135)ation raises the in(cid:135)ation expectations of price-setters; the latter produces an upward shift of the NKPC, thus worsening the short-run tradeo⁄ between in(cid:135)ation and output. As it turns out, at zero in(cid:135)ation, the marginal welfare cost of raising in(cid:135)ation expectations exactly o⁄sets the marginal welfare bene(cid:133)t of exploiting the short-run in(cid:135)ation-output tradeo⁄. Whiletheformertwowelfaree⁄ectsof trendin(cid:135)ationarecommontoexogenous-timingmodels, ourframeworkwithidiosyncraticmenucostshocksincludestwoadditionalchannelsthroughwhich trendin(cid:135)ationa⁄ectswelfare. Oneisthatin(cid:135)ationforces(cid:133)rmstospendrealresources(menucosts) on adjusting their nominal prices; this distortion is minimized at zero in(cid:135)ation, because eventually all (cid:133)rms end up being at their optimal price. The other extra channel is more subtle; Namely, in the stochastic menu costs model, adjustment frequencies are endogenous. In particular, trend in(cid:135)ation a⁄ects the relative prices of di⁄erent cohorts of (cid:133)rms (p = j 1 (cid:25) , j = 0;:::;J 1), t (cid:0) j k (cid:0) =0 t (cid:0) k (cid:0) which has an e⁄ect on their pro(cid:133)ts, on their value functions, and ultimately on the gains from Q adjustment. A priori, the central bank may be tempted to use trend in(cid:135)ation to in(cid:135)uence the speed of price adjustment, so as to shift the NKPC in a way that improves the in(cid:135)ation-output tradeo⁄. However, the fact that adjusting (cid:133)rms choose their prices in an optimal way implies that, at zero in(cid:135)ation, a marginal increase in the in(cid:135)ation rate has no e⁄ect on (cid:133)rms(cid:146)pro(cid:133)ts, and therefore it has no e⁄ect on adjustment probabilities. This envelope property implies that the monetary authority has no incentive to create trend in(cid:135)ation to in(cid:135)uence the speed with which (cid:133)rms change their prices. The second important aspect of proposition 1 is that the optimal deviations from zero in(cid:135)ation in response to technology or preference shocks are zero as well. Therefore, the occurrence of these exogenous disturbances to preferences or technology does not justify temporary departures from strict price stability.11 The intuition for this result is as follows. There are four potential ine¢ ciencies in the present model, related to: (1) the level and volatility of price dispersion; (2) the volatility of the average markup; (3) the waste of resources due to menu costs; and (4) the level of the average markup due to monopolistic competition. Distortions (1) through (3) are directly related to the friction in price setting, and(cid:150)absent idiosyncratic shocks to desired prices(cid:150)a policy of 10The (cid:147)New Keynesian Phillips curve(cid:148)is the structural relationship between in(cid:135)ation (current and expected) and output that arises in the standard New Keynesian model. Here, the optimal price decision (equation 7) and the relationship between in(cid:135)ation and the optimal relative price (equation 10) can be combined into a dynamic relationship between in(cid:135)ation and real marginal costs, where the latter can also be expressed in terms of aggregate output by using equations (1), (8), and (9). The resulting dynamic relationship between in(cid:135)ation and output may be interpreted as a (cid:147)New Keynesian Phillips curve.(cid:148)Notice that the endogenous price adjustment frequencies, (cid:21) , jt a⁄ect both the intercept and the slope of that curve. 11Benigno and Woodford (2005) reach the same conclusion about the standard model with Calvo pricing. While they derive their result for a linear-quadratic approximation to the actual optimal monetary policy problem, our (cid:133)nding is based on the exact non-linear welfare function and equilibrium conditions. 12

strict price stability eliminates all three. It does so by replicating the (cid:135)exible-price equilibriumand eliminatingtheincentivesforpriceadjustment. Ine¢ ciency(4)isastaticmarkupdistortiondueto monopolistic competition. As we have just seen, the optimal plan does not involve a correction of this ine¢ ciency because it is outweighed by the gains of committing to zero in(cid:135)ation and achieving the minimum possible price dispersion in the long run, independently of the price-setting policies followed by (cid:133)rms. The aforementioned envelope property, by which a marginal increase in in(cid:135)ation leavespriceadjustmentfrequenciesuna⁄ected, continuestoholdastheeconomyishitbyaggregate shocks. 3.2 An illustration with two cohorts While the appendix provides the proof of the optimality of zero in(cid:135)ation in the full-blown model, it isillustrativetoformalizetheaboveintuitionswithasimpli(cid:133)edversionof themodel. Inparticular, we consider the case of J = 2 cohorts, such that (cid:133)rms that adjust their nominal price today may or may not adjust in the following period, but adjust with certainty two periods after the last price change. To further simplify, we assume functional forms u(C ) = log(C ) and x(N ;(cid:31) ) = (cid:31) N , t t t t t t such that the real wage is w = x (N ;(cid:31) )=u (C ) = (cid:31) C . As in proposition 1, we also assume t 0 t t 0 t t t away government spending, G = 0, such that C = Y . To simplify the notation, let and t t t t 1t (cid:17) (cid:21) (cid:21) denotethemeasureandadjustmentprobabilityof(cid:133)rmsinvintage1. Themeasureof(cid:133)rms t 1t (cid:17) invintage 2is then = 1 , andthe lawof motionof is simply = 1 (1 (cid:21) ) . Let 2t t t t t 1 t 1 (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) also v v denote the value of (cid:133)rms in vintage 1. Finally, we de(cid:133)ne v~ v =Y and v~ v =Y , t 1t 0t 0t t t t t (cid:17) (cid:17) (cid:17) such that (v v )=w = (v~ v~)=(cid:31). Taking all these elements, the central bank maximizes the 0t t t 0t t (cid:0) (cid:0) 13

following Lagrangian: Y (cid:1) (v~0t v~t)=(cid:31) = E (cid:12)t log(Y ) (cid:31) t t (cid:31) (cid:0) (cid:20)g((cid:20))d(cid:20) L 0 0 1 t=0 t (cid:0) t z (cid:0) t t ( t Z0 P (cid:15) (cid:31) Y (cid:31) Y +(cid:30)p (cid:3) p 1+(cid:12)(1 (cid:21) )(cid:25)(cid:15) 1 t t +(cid:12)(1 (cid:21) )(cid:25)(cid:15) t t+1 t (cid:3)t (cid:0) t+1 t+(cid:0)1 (cid:0) (cid:15) 1 z (cid:0) t+1 t+1 z (cid:20) (cid:0) (cid:18) t t+1 (cid:19)(cid:21) (cid:0) (cid:1) p 1 (cid:15) +(cid:30)(cid:25) (p )1 (cid:15)((cid:21) +1 )+ (cid:3)t 1 (cid:0) (1 (cid:21) ) 1 t (cid:3)t (cid:0) t t (cid:0) t (cid:25) (cid:0) (cid:0) t t (cid:0) " (cid:18) t (cid:19) # p (cid:15) +(cid:30)(cid:1) (p ) (cid:15)((cid:21) +1 )+ (cid:3)t 1 (cid:0) (1 (cid:21) ) (cid:1) t (cid:3)t (cid:0) t t (cid:0) t (cid:25) (cid:0) (cid:0) t t (cid:0) t " (cid:18) t (cid:19) # v~ v~ +(cid:30)(cid:21) (cid:21) G 0t (cid:0) t +(cid:30) +(1 (cid:21) ) t (cid:20) t (cid:0) (cid:18) (cid:31) t (cid:19)(cid:21) t t (cid:0) t (cid:0) 1 t (cid:0) 1 (cid:2) (cid:3) (cid:31) Y (v~0;t+1 v~t+1) =(cid:31) +(cid:30)v t 0 p (cid:3)t (cid:0) z t t (p (cid:3)t ) (cid:0) (cid:15) (cid:0) v~ 0t +(cid:12) (cid:21) t+1 v~ 0;t+1 +(1 (cid:0) (cid:21) t+1 )v~ t+1 (cid:0) (cid:31) t (cid:0) (cid:20)g((cid:20))d(cid:20) " (cid:18) t (cid:19) Z0 !# p (cid:31) Y p (cid:15) +(cid:30)v (cid:3)t 1 t t (cid:3)t 1 (cid:0) v~ +(cid:12)v~ : t (cid:25) (cid:0) (cid:0) z (cid:25) (cid:0) (cid:0) t 0;t+1 " (cid:18) t t (cid:19)(cid:18) t (cid:19) #) For the present analysis, it su¢ ces to di⁄erentiate the Lagrangian with respect to in(cid:135)ation and the optimal relative price for a particular state at time t. While the derivative of the Lagrangian with respect to (cid:25) captures the direct marginal e⁄ect of in(cid:135)ation on welfare, the derivative with t respecttop capturesitsindirecte⁄ectthroughitsstructuralrelationshipwiththeoptimalrelative (cid:3)t price. That relationship is given by the equation multiplied by (cid:30)(cid:25) in the Lagrangian. Indeed, if t we use the latter equation to solve for the optimal relative price as a function of current and past in(cid:135)ation, and then use the resulting expression to substitute for p in the optimal price decision (cid:3)t (theequationmultipliedby(cid:30)p (cid:3)), weobtainadynamicrelationshipbetweenin(cid:135)ationandaggregate t activity. The latter may be interpreted as a (cid:147)New Keynesian Phillips curve.(cid:148)The derivatives with respect to (cid:25) and p are given by t (cid:3)t @ p (cid:15) (cid:31) Y @ L (cid:25) 0 = (cid:30)p t (cid:3) 1 (cid:25) (cid:3)t (cid:0) 1 ((cid:15) (cid:0) 1) (cid:0) (cid:15) 1 z t t (cid:15) (cid:25)(cid:15) t(cid:0) 1(1 (cid:0) (cid:21) t ) t (cid:0) (cid:20) t (cid:0) t (cid:21) p + (cid:30)(cid:25) t ((cid:15) (cid:0) 1) (cid:25) (cid:3)t (cid:0) 1 +(cid:30)(cid:1) t (cid:15) p (cid:3)t 1 (cid:0) (cid:15) (cid:25)(cid:15) t(cid:0) 1(1 (cid:0) (cid:21) t ) t (15) (cid:20) t (cid:21) (cid:0) p (cid:31) Y (cid:0) (cid:1) +(cid:30)v t ((cid:15) (cid:0) 1) (cid:25) (cid:3)t (cid:0) 1 (cid:0) (cid:15) z t t p (cid:3)t 1 (cid:0) (cid:15) (cid:25)(cid:15) t(cid:0) 1; (cid:20) t t (cid:21) (cid:0) (cid:0) (cid:1) 14

@ @ L p 0 = (cid:30)p t (cid:3) 1+(cid:12)(1 (cid:0) (cid:21) t+1 )(cid:25)(cid:15) t+(cid:0)1 1 (cid:0) (cid:30)(cid:25) t ((cid:15) (cid:0) 1)p (cid:3)t +(cid:30)(cid:1) t (cid:15) (p (cid:3)t ) (cid:0) (cid:15) (cid:0) 1((cid:21) t t +1 (cid:0) t ) (cid:3)t (cid:2) p (cid:3) (cid:2) (cid:3) (cid:12)E (cid:30)(cid:25) ((cid:15) 1) (cid:3)t +(cid:30)(cid:1) (cid:15) (p ) (cid:15) 1(cid:25)(cid:15) (1 (cid:21) ) (16) (cid:0) t t+1 (cid:0) (cid:25) t+1 (cid:3)t (cid:0) (cid:0) t+1 (cid:0) t+1 t+1 (cid:20) t+1 (cid:21) (cid:31) Y (cid:31) Y p +(cid:30)v t 0 (cid:15) z t t (cid:0) ((cid:15) (cid:0) 1)p (cid:3)t (p (cid:3)t ) (cid:0) (cid:15) (cid:0) 1 +(cid:12)E t (cid:30)v t+1 (cid:15) z t t+1 (cid:0) ((cid:15) (cid:0) 1) (cid:25) (cid:3)t (p (cid:3)t ) (cid:0) (cid:15) (cid:0) 1(cid:25)(cid:15) t+1 ; (cid:20) t (cid:21) (cid:20) t+1 t+1(cid:21) respectively.12 We now conjecture that the central bank commits to follow a policy of zero net in(cid:135)ation,or(cid:25) = 1. Itisstraightforwardtoshowthatundersuchapolicytheeconomyconvergesto t an equilibrium in which p = (cid:1) = 1. That is, both (cid:133)rm vintages have the same relative price, and (cid:3)t t price dispersion is eliminated. Thus, both vintages end up having the same value, v = v , which 0t t (cid:22) (cid:22) (cid:22) in turn implies (cid:21) = G(0) (cid:21) > 0. The vintage distribution converges to = 1= 2 (cid:21) . t t (cid:17) (cid:0) (cid:17) Finally, the real marginal cost equals the inverse of the monopolistic mark-up, (cid:31) Y =z = ((cid:15) 1)=(cid:15), t t t(cid:0) (cid:1) (cid:0) implying that output equals its (cid:135)exible-price level of section 2.5.1 at all times. Imposing the latter conjecture in expressions (15) and (16), we obtain @ L 0 = (cid:30)p (cid:3) 1 (cid:21) (cid:22) + (cid:30)(cid:25)((cid:15) 1)+(cid:30)(cid:1)(cid:15) 1 (cid:21) (cid:22) (cid:22) ; (17) @(cid:25) (cid:0) t 1 (cid:0) t (cid:0) t (cid:0) t (cid:0) (cid:0) (cid:1) (cid:2) (cid:3)(cid:0) (cid:1) @ L 0 = (cid:30)p (cid:3) 1+(cid:12) 1 (cid:21) (cid:22) (cid:30)(cid:25)((cid:15) 1)+(cid:30)(cid:1)(cid:15) (cid:22) (cid:12) 1 (cid:21) (cid:22) E (cid:30)(cid:25) ((cid:15) 1)+(cid:30)(cid:1) (cid:15) (cid:22) ; (18) @p t (cid:0) (cid:0) t (cid:0) t (cid:0) (cid:0) t t+1 (cid:0) t+1 (cid:3)t (cid:2) (cid:0) (cid:1)(cid:3) (cid:2) (cid:3) (cid:0) (cid:1) (cid:2) (cid:3) (cid:22)(cid:22) (cid:22) (cid:22) where we have also used the fact that (cid:21) +1 = . The (cid:133)rst e⁄ect to notice is that, under our (cid:0) conjecture, all terms involving the Lagrange multipliers (cid:30)v0 and (cid:30)v in expressions (15) and (16) t t have disappeared. Such terms capture the marginal welfare e⁄ect of both variables through their e⁄ect on the value of both (cid:133)rm cohorts (v ;v ). Therefore, once the economy has converged to 0t t the timeless perspective regime with zero in(cid:135)ation, a marginal deviation of in(cid:135)ation from zero has no e⁄ect on the gains from adjustment, and hence it has no e⁄ect on the adjustment frequency either. This is the (cid:147)envelope property(cid:148)that we referred to before. In equation (17), the term involving (cid:30)p (cid:3) captures the marginal welfare e⁄ect from an increase t 1 (cid:0) in time (t 1) expectations of in(cid:135)ation at time t, whereas the term involving (cid:30)p (cid:3) in equation t (cid:0) (18) re(cid:135)ects the marginal welfare e⁄ect from an increase in the optimal relative price (and thus in in(cid:135)ation) at time t. We show in the Appendix that, in the full-blown model, the multiplier (cid:30)p (cid:3) converges to a constant value (cid:30) (cid:22)p (cid:3) in the timeless perspective regime, which is also true in this t simpli(cid:133)ed version. Using this in (18), setting the resulting expression equal to zero (as required by 12Both derivatives have been rescaled by (cid:12)t times the probability of reaching the particular state at time t conditional on the state at time 0. 15

the (cid:133)rst-order optimality condition), and solving for (cid:30)(cid:25), we obtain t (cid:30)(cid:25) = (cid:30) (cid:22)p (cid:3)= (cid:22) (cid:30)(cid:1)(cid:15) =((cid:15) 1): t (cid:0) t (cid:0) (cid:16) (cid:17) Using this to substitute for (cid:30)(cid:25) in (17), the latter becomes t @ L 0 = (cid:30) (cid:22)p (cid:3)= (cid:22) (cid:30)(cid:1)(cid:15)+(cid:30)(cid:1)(cid:15) 1 (cid:21) (cid:22) (cid:22) (cid:30) (cid:22)p (cid:3) 1 (cid:21) (cid:22) @(cid:25) (cid:0) t t (cid:0) (cid:0) (cid:0) t h i = (cid:30)(cid:1)((cid:15) (cid:15)) 1 (cid:21) (cid:22) (cid:22) + (cid:0) (cid:30) (cid:22)p (cid:3)(1 (cid:1) 1) 1 (cid:0) (cid:21) (cid:22) (cid:1) (19) t (cid:0) (cid:0) (cid:0) (cid:0) = 0+0 = 0:(cid:0) (cid:1) (cid:0) (cid:1) Therefore, once the economy has converged to the timeless perspective regime with zero in(cid:135)ation, the central bank has no incentive to create positive or negative in(cid:135)ation at the margin, because the potential welfare costs cancel out the potential gains. The term involving (cid:30)(cid:1) in (19) captures the t marginal welfare e⁄ect of in(cid:135)ation through its e⁄ect on price dispersion, which disappears under the timeless perspective regime with zero in(cid:135)ation. Finally, the term involving (cid:30) (cid:22)p (cid:3) is the di⁄erence between the positive marginal e⁄ect stemming from a movement along the NKPC, (cid:30) (cid:22)p (cid:3) 1 (cid:21) (cid:22) , (cid:0) and the negative marginal e⁄ect due to the shift in the NKPC, (cid:30) (cid:22)p (cid:3) 1 (cid:21) (cid:22) . Under the zero (cid:0) (cid:1) (cid:0) (cid:0) in(cid:135)ation policy, both e⁄ects exactly cancel each other out. (cid:0) (cid:1) The speci(cid:133)c example above is intended to formalize the main intuition; more generally, the optimality of zero in(cid:135)ation from the timeless perspective holds for any number of cohorts and for standard (isoelastic) preferences, as shown in the Appendix. 4 Optimal policy with positive government expenditure Theprevioussectionderivedtheoptimalpolicyundertheassumptionthatgovernmentexpenditure iszero. Wenowbrie(cid:135)yanalyzethemoregeneralcasewithpositivegovernmentexpenditure. Inthis case, we no longer have a closed-form analytical solution, so we illustrate the results by simulation. We show the optimal dynamic responses of several key variables to two types of shocks: aggregate productivity and government consumption. Our main (cid:133)nding is that, under a (cid:133)rst- or secondorder approximation to the general equilibrium dynamics of the model, the optimal deviations of in(cid:135)ation from zero are negligible. Thus, the optimal stabilization policy is basically equivalent to strict in(cid:135)ation targeting, and all real variables follow closely their (cid:135)exible-price counterparts. We also (cid:133)nd that the responses are virtually identical to the ones obtained in the Calvo model. 16

4.1 Calibration To produce impulse responses, we must (cid:133)rst choose functional forms and assign values to the model(cid:146)sparameters. WetakemostoftheparametersfromGolosovandLucas(2007). Inparticular, u(C ) = C1 (cid:13)=(1 (cid:13))with(cid:13) = 2;andx(N ) = (cid:31)N1+’=(1+’)with(cid:31) = 6and’ = 1:Thediscount t t(cid:0) t t (cid:0) factor is (cid:12) = 1:04 1=4 and the elasticity of substitution among product varieties is (cid:15) = 7. (cid:0) We further assume that the cumulative distribution function of menu costs takes the form (cid:24) +(cid:20) G((cid:20)) = ; (cid:11)+(cid:20) where both (cid:24) and (cid:11) are positive parameters. Therefore, from equation (6) the fraction of vintage-j (cid:133)rms that adjust their price in a given period equals v v (cid:24) +(v v )=w 0t jt 0t jt t (cid:21) = G (cid:0) = (cid:0) : jt w (cid:11)+(v v )=w (cid:18) t (cid:19) 0t (cid:0) jt t As in Costain and Nakov (2011), this function is increasing in the gain from adjustment v v 0t jt (cid:0) and is bounded above by 1. Unlike Costain and Nakov (2011), the function is bounded below not by0 but by(cid:24)=(cid:11) > 0: We make this technical assumptionto ensure a unique stationary distribution of (cid:133)rms over the ((cid:133)nite number of) price vintages in the case of zero in(cid:135)ation. Any arbitrarily small (cid:24) would work and so we pick the value 10 10. We then set (cid:11) = 0:0006 so that, under a (cid:0) policy targeting 2% annual in(cid:135)ation (broadly consistent with the average observed rate in the United States since the mid-1980s), the model produces an average frequency of price changes of once every three quarters (broadly consistent with the micro evidence found, for example, by Nakamura and Steinsson, 2008). With these settings, the model implies virtually zero probability of adjustment when the gain from adjustment is zero. Finally, we set the maximum price duration to J = 24 quarters, a number that is much greater than any observed price duration in recent U.S. evidence. Figure 1 shows the adjustment hazard function and the distribution of (cid:133)rms by price vintage with 2% trend in(cid:135)ation. In the left panel, the adjustment probability increases rapidly with price age, reaching 90% after 10 quarters. As shown in the right panel, this implies that virtually no price survives more than eight quarters. We focus on two types of shocks. One is an aggregate productivity shock with persistence (cid:26) = 0:95 and the other is a government expenditure shock with persistence (cid:26) = 0:9: Government z g expenditure is calibrated so that it accounts for roughly 17% of GDP in steady state, consistent with U.S. postwar experience. 17

4.2 Impulse responses under the optimal policy We use a (cid:133)rst-order Taylor expansion to approximate the equilibrium dynamics of our model. Figure 2 plots the responses of several variables of interest to two independent shocks: a 1% improvement in aggregate productivity, and a 1% increase in the level of government spending. Characteristically, four variables (cid:150)the optimal reset price, in(cid:135)ation, price dispersion (shown in the last row of the (cid:133)gure), and the output gap, de(cid:133)ned as the ratio between actual output and its (cid:135)exible price counterpart (and shown in the third panel on the top row), remain constant in responsetoeachoftheshocks. Thisispreciselywhathappensinresponsetothesameshocksinthe Calvo model (not shown due to the overlap, but available upon request). Moreover, the responses of the interest rate, consumption, hours worked, and wages, all coincide with their counterparts in the Calvo model. Hence, the central bank(cid:146)s incentives to deviate from zero in(cid:135)ation to reduce monopolistic distortions are virtually nonexistent in response to the two real shocks. In passing, we note that a second-order accurate solution of the model yields virtually identical impulse responses, both under Calvo and under stochastic menu costs, at least for small aggregate shocks.13 We thus (cid:133)nd that the simple linear Calvo framework o⁄ers a very good approximation to thebehaviorofacashlessstate-dependentpricingeconomyundertheoptimalmonetarypolicyfrom the timeless perspective, even though the two economies behave quite di⁄erently under suboptimal policies. 5 Conclusion We have shown that the main lessons for optimal monetary policy derived in the canonical Calvo model carry over to a more general setup in which (cid:133)rms(cid:146)likelihood of adjusting prices depends on the state of the economy. In particular, the optimal long-run rate of in(cid:135)ation is zero, and the optimal dynamic policy is strict in(cid:135)ation targeting. This (cid:133)nding means that the central bank should not use in(cid:135)ation to try to o⁄set the static distortion arising from monopolistic competition. We show that, under conditions typically assumed in the literature, the probability of adjustment remains constant even if pricing is state-dependent, provided that monetary policy is set optimally. Thus, when the su¢ cient conditions are met, any di⁄erence between time-dependent and state-dependent pricing vanishes under the optimal policy. These results lend support to more informal statements about the suitability of the Calvo model for studying optimal monetary policy despite its apparent con(cid:135)ict with the Lucas (1976) critique. Our analysis is a step toward a fuller model that would include (cid:133)rm-level shocks not only to the price adjustment costs, but also to desired prices, for example, due to idiosyncratic productivity 13We use 24 vintages when approximating the solution to (cid:133)rst order, and 8 vintages when approximating it to second order. When plotted, the two sets of impulse responses are indistinguishable to the naked eye. 18

shocks.14 In this extended model, monetary policy would not be able to replicate the (cid:135)exible-price equilibrium(cid:147)forfree,(cid:148)because(cid:133)rmswithdi⁄erentproductivitieswouldwanttosetdi⁄erentprices. Instead, deviations from price stability would a⁄ect the balance between price increases and price decreases, with potential welfare gains coming from this rebalancing. The magnitude of such a welfare e⁄ect of in(cid:135)ation is an intriguing question, which we leave for future research. 14A simple extension with (cid:133)rm-level shocks to desired prices is to assume that such shocks happen with a constant probability. If the shocks to desired prices are so large that adjustment to them brings gains exceeding some maximum menu cost, then prices would be (cid:135)exible with respect to the micro-level shocks, but sticky with respect to aggregate shocks. By construction, in this environment, our analysis from section 3 would remain true. 19

References [1] Benigno, Pierpaolo, and Michael Woodford (2005).(cid:147)In(cid:135)ation Stabilization and Welfare: The Case of a Distorted Steady State,(cid:148)Journal of the European Economic Association, December 2005, vol. 3 (6), pp.1185(cid:150)236. [2] Calvo, Guillermo (1983). (cid:147)Staggered Prices in a Utility-Maximizing Framework,(cid:148)Journal of Monetary Economics vol. 12, pp. 383(cid:150)98. [3] Clarida, R., Jordi Gali, and Mark Gertler (1999).(cid:147)The Science of Monetary Policy: a New Keynesian Perspective,(cid:148)Journal of Economic Literature, vol. 37 (4), pp.1661(cid:150)707. [4] Costain, James, and Anton Nakov (2011). (cid:147)Price Adjustments in a General Model of State- Dependent Pricing,(cid:148)Journal of Money, Credit and Banking, vol. 43 (2-3), pp. 385(cid:150)406. [5] Dotsey, Michael, Robert King, and Alexander Wolman (1999).(cid:147)State-Dependent Pricing and the General Equilibrium Dynamics of Money and Output,(cid:148)Quarterly Journal of Economics 114 (2), pp. 655(cid:150)90. [6] Golosov, Mikhail, andRobertE. Lucas, Jr. (2007).(cid:147)MenuCostsandPhillipsCurves,(cid:148)Journal of Political Economy vol. 115 (2), pp.171(cid:150)99. [7] Hamilton, James (1994).Time Series Analysis. Princeton, N.J.: Princeton University Press. [8] Lie, Denny (2009).(cid:147)State-Dependent Pricing and Optimal Monetary Policy,(cid:148) unpublished manuscript, Boston: Boston University. [9] Lucas Jr., Robert (1976).(cid:147)Econometric Policy Evaluation: a Critique,(cid:148)Carnegie-Rochester Conference Series on Public Policy, vol. 1, pp.19(cid:150)46. [10] Poole, David (2006).Linear Algebra. Belmont, Calif.: Thomson Brooks/Cole. [11] Woodford, Michael (2002).(cid:147)In(cid:135)ation Stabilization and Welfare,(cid:148)Contributions to Macroeconomics, vol. 2 (1), article 1. [12] Woodford, Michael (2003). Interest and Prices. Princeton, N.J.: Princeton University Press. [13] Yun, Tack (2005).(cid:147)Optimal Monetary Policy with Relative Price Distortions,(cid:148)American Economic Review, vol. 95 (1), pp.89(cid:150)109. 20

Appendix In this appendix, we obtain the solution to the optimal monetary policy problem from the timeless perspective. The central bank maximizes the Lagrangian given by expression (14) in the main text. The (cid:133)rst-order conditions are as follows (all expressions are equal to zero): J 1 p (cid:15) x (N ;(cid:31) ) (cid:1) u 0 (C t )+ (cid:0) (cid:30)p t (cid:3) j (cid:25) (cid:3)t a (cid:0) cc j [u 0 (C t )+Y t u 00 (C t )] (cid:0) (cid:15) 1 0 z t t (cid:18) jt ((cid:25)a jt cc)(cid:15) (cid:0) (cid:30)N t z t j=0 (cid:0) (cid:20) jt (cid:0) t (cid:21) t J 1 Pp x (N ;(cid:31) ) p (cid:15) + (cid:0) (cid:30) v t j (cid:25) (cid:3)t a (cid:0) cc j [u 0 (C t )+u 00 (C t )Y t ] (cid:0) 0 z t t (cid:25) (cid:3)t a (cid:0) cc j (cid:0) ; (Y t ) j=0 (cid:20) jt t (cid:21)(cid:18) jt (cid:19) P J 1 J (cid:30)p t (cid:3)E t (cid:0) (cid:12)j(cid:18) j;t+j ((cid:25)a j; c t c +j )(cid:15) (cid:0) 1Y t+j u 0 (C t+j ) (cid:0) (cid:30)(cid:25) t ((cid:15) (cid:0) 1)p (cid:3)t +(cid:30)(cid:1) t (cid:15) (p (cid:3)t ) (cid:0) (cid:15) (cid:0) 1 (cid:21) jt jt j=0 j=1 J P1 p (cid:2) (cid:3) P E (cid:0) (cid:12)j (cid:30)(cid:25) ((cid:15) 1) (cid:3)t +(cid:30)(cid:1) (cid:15) (p ) (cid:15) 1((cid:25)acc )(cid:15)(1 (cid:21) ) (cid:0) t t+j (cid:0) (cid:25)acc t+j (cid:3)t (cid:0) (cid:0) j;t+j (cid:0) j;t+j j;t+j j=1 (cid:20) j;t+j (cid:21) P J 1 x N ;(cid:31) p +E (cid:0) (cid:12)j(cid:30) vj (cid:15) 0 t+j t+j ((cid:15) 1) (cid:3)t (p ) (cid:15) 1((cid:25)acc )(cid:15)Y u (C ); (p ) t t+j z u (C ) (cid:0) (cid:0) (cid:25)acc (cid:3)t (cid:0) (cid:0) j;t+j t+j 0 t+j (cid:3)t j=0 " t(cid:0)+j 0 t+j (cid:1) j;t+j# P p (cid:15) x (N ;(cid:31) ) (cid:30)p t (cid:3) j (cid:25) (cid:3)t a (cid:0) cc j ((cid:15) (cid:0) 1) (cid:0) (cid:15) 1 (cid:15) z 0 u ( t C ) t (cid:18) jt ((cid:25)a jt cc)(cid:15) (cid:0) 1Y t u 0 (C t ) (cid:0) (cid:20) jt (cid:0) t 0 t (cid:21) p + (cid:30)(cid:25) t (cid:25) (cid:3)t a (cid:0) cc j ((cid:15) (cid:0) 1)+(cid:30)(cid:1) t (cid:15) (p (cid:3)t j ) (cid:0) (cid:15)((cid:25)a jt cc)(cid:15) (cid:0) 1(1 (cid:0) (cid:21) jt ) jt (cid:20) jt (cid:21) (cid:0) p x (N ;(cid:31) ) +(cid:30) v t j (cid:25) (cid:3)t a (cid:0) cc j ((cid:15) (cid:0) 1) (cid:0) z 0 u ( t C ) t (cid:15) (p (cid:3)t j ) (cid:0) (cid:15)((cid:25)a jt cc)(cid:15) (cid:0) 1Y t u 0 (C t ) (cid:20) jt t 0 t (cid:21) (cid:0) (cid:25)acc (cid:25)acc +(cid:30) j (cid:12)E (cid:30) j+1(cid:25) ; ((cid:25)acc ) t (cid:0) t t+1 t j=1;:::;J 2;t (cid:0) (cid:30)p t (cid:0) (cid:3) (J (cid:0) 1) " p (cid:25) (cid:3)t (cid:0) a J ( c J c 1 (cid:0) ;t 1) ((cid:15) (cid:0) 1) (cid:0) (cid:15) (cid:0) (cid:15) 1 (cid:15) x z 0 t ( u N 0 ( t ; C (cid:31) t ) t ) # (cid:18) J (cid:0) 1;t ((cid:25)a J c (cid:0) c 1;t )(cid:15) (cid:0) 1Y t u 0 (C t ) (cid:0) p + " (cid:30)(cid:25) t (cid:25) (cid:3)t (cid:0) a J ( c J c 1 (cid:0) ;t 1) ((cid:15) (cid:0) 1)+(cid:30)(cid:1) t (cid:15) # (p (cid:3)t (cid:0) (J (cid:0) 1) ) (cid:0) (cid:15)((cid:25)a J c (cid:0) c 1;t )(cid:15) (cid:0) 1(1 (cid:0) (cid:21) J (cid:0) 1;t ) J (cid:0) 1;t (cid:0) +(cid:30) v t j p (cid:25) (cid:3)t (cid:0) a ( c J c (cid:0) 1) ((cid:15) (cid:0) 1) (cid:0) x z 0 ( u N ( t ; C (cid:31) ) t ) (cid:15) (p (cid:3)t (J 1) ) (cid:0) (cid:15) (cid:25)a J cc 1;t (cid:15) (cid:0) 1 Y t u 0 (C t )+(cid:30) (cid:25) t a J c (cid:0) c 1;((cid:25)a J cc 1;t ) " J 1;t t 0 t # (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:1) p (cid:15) x (N ;(cid:31) ) (cid:30)p t (cid:0) (cid:3) j (cid:20) (cid:25) (cid:3)t a j (cid:0) t cc j (cid:0) (cid:15) (cid:0) 1 z 0 t u 0 ( t C t ) t (cid:21) ((cid:25)a jt cc)(cid:15)Y t u 0 (C t )+(cid:30) (cid:18) t j (cid:0) (cid:12)E t (cid:30) t (cid:18) + j+ 1 1(1 (cid:0) (cid:21) j+1;t+1 ); ((cid:18) j=1;:::;J (cid:0) 2;t ) 21

(cid:30)p t (cid:0) (cid:3) (J (cid:0) 1) " p (cid:25) (cid:3)t (cid:0) a J ( c J c 1 (cid:0) ;t 1) (cid:0) (cid:15) (cid:0) (cid:15) 1 x z 0 t ( u N 0 ( t ; C (cid:31) t ) t ) # ((cid:25)a J c (cid:0) c 1;t )(cid:15)Y t u 0 (C t )+(cid:30) t (cid:18)J (cid:0) 1; ((cid:18) J (cid:0) 1;t ) (cid:0) p 1 (cid:15) p (cid:15) (cid:30)(cid:25) t (p (cid:3)t )1 (cid:0) (cid:15) (cid:0) (cid:25) (cid:3)t a (cid:0) cc j (cid:0) jt +(cid:30)(cid:1) t (p (cid:3)t ) (cid:0) (cid:15) (cid:0) (cid:25) (cid:3)t a (cid:0) cc j (cid:0) jt " (cid:18) jt (cid:19) # " (cid:18) jt (cid:19) # +(cid:30) (cid:21)j +(cid:12)E (cid:30) j+1 +(cid:30) vj 1(v~ v~ )+(cid:30) (cid:18)j(cid:18) ; ((cid:21) ) t t t+1 jt t (cid:0) (cid:0)1 0;t (cid:0) jt t j (cid:0) 1;t (cid:0) 1 j=1;:::;J (cid:0) 1;t p 1 (cid:15) (cid:0) (cid:30)N t 0 (v~0t (cid:0) v~1t)=x 0 (Nt;(cid:31) t ) (cid:20)g((cid:20))d(cid:20)+(cid:30)(cid:25) t (p (cid:3)t )1 (cid:0) (cid:15)(cid:21) 1t + (cid:25) (cid:3)t a (cid:0) c 1 c (cid:0) (1 (cid:0) (cid:21) 1t ) " (cid:18) 1t (cid:19) # R p (cid:15) +(cid:30)(cid:1) t (p (cid:3)t ) (cid:0) (cid:15)(cid:21) 1t + (cid:25) (cid:3)t a (cid:0) c 1 c (cid:0) (1 (cid:0) (cid:21) 1t ) (cid:0) (cid:12)E t (cid:30) t+ 2 1 (1 (cid:0) (cid:21) 1t )+(cid:30) t 1; ( 1;t ) " (cid:18) 1t (cid:19) # p 1 (cid:15) (cid:0) (cid:30)N t 0 (v~0t (cid:0) v~jt)=x 0 (Nt;(cid:31) t ) (cid:20)g((cid:20))d(cid:20)+(cid:30)(cid:25) t (p (cid:3)t )1 (cid:0) (cid:15)(cid:21) jt + (cid:25) (cid:3)t a (cid:0) cc j (cid:0) (1 (cid:0) (cid:21) jt ) " (cid:18) jt (cid:19) # R p (cid:15) +(cid:30)(cid:1) t " (p (cid:3)t ) (cid:0) (cid:15)(cid:21) jt + (cid:18) (cid:25) (cid:3)t a j (cid:0) t cc j (cid:19) (cid:0) (1 (cid:0) (cid:21) jt ) # ( j=2;:::;J (cid:0) 1;t ) +(cid:30) j (cid:12)E (cid:30) j+1(1 (cid:21) )+(cid:30) 1; t (cid:0) t t+1 (cid:0) jt t (cid:30)(cid:25) t (p (cid:3)t )1 (cid:0) (cid:15) +(cid:30)(cid:1) t (p (cid:3)t ) (cid:0) (cid:15) +(cid:30) t J +(cid:30) t 1; ( J;t ) J 1 J 1 1 J 2 (cid:30)N (cid:0) jt L g(L ) (cid:0) (cid:30) (cid:21)jg(L ) (cid:30)v0+ (cid:0) (cid:30) vj [(cid:21) L g(L )]+(cid:30) vJ 1; (cid:0) t x (N ;(cid:31) ) jt jt (cid:0) t jt x (N ;(cid:31) )(cid:0) t t 1 j+1;t (cid:0) j+1;t j+1;t t (cid:0)1 j=1 0 t t j=1 0 t t j=0 (cid:0) (cid:0) P P P (v~ ) 0t 1 1 (cid:30)N L g(L )+(cid:30) (cid:21)jg(L ) (cid:30) vj+(cid:30) vj 1[1 (cid:21) +L g(L )]; (v~ ) t jtx 0 (N t ;(cid:31) t ) jt jt t jt x 0 (N t ;(cid:31) t )(cid:0) t t (cid:0) (cid:0)1 (cid:0) jt jt jt j=1;:::;J (cid:0) 1;t J 1 (cid:15) x (N ;(cid:31) ) J 1 x (N ;(cid:31) ) x (N ;(cid:31) ) (cid:0) (cid:30)p (cid:3) (cid:18) ((cid:25)acc)(cid:15)Y 00 t t +(cid:30)N 1+ (cid:0) 00 t t (L )2g(L ) (cid:0) 0 t t (cid:0) t j(cid:15) 1 jt jt t z t jtx (N ;(cid:31) ) jt jt j=0 (cid:0) t " j=1 0 t t # (cid:0) J 1 P x (N ;(cid:31) ) J 1 x (N ;(cid:31) ) p (cid:15) P + (cid:0) (cid:30) (cid:21) t jg(L jt )L jt x 00 (N t ;(cid:31) t ) (cid:0) (cid:0) (cid:30) v t j 00 z t t (cid:25) (cid:3)t a (cid:0) cc j (cid:0) Y t j=1 0 t t j=0 t (cid:18) jt (cid:19) JP2 P + (cid:0) (cid:30) vj x (N ;(cid:31) ) (L )2g(L ) Lj+1;t(cid:20)g((cid:20))d(cid:20) ; (N ) t 1 00 t t j+1;t j+1;t (cid:0) 0 t j=0 (cid:0) h i P R Y (cid:30)N t (cid:30)(cid:1); ((cid:1) ) (cid:0) t z (cid:0) t t t 22

(cid:0) (cid:30) (cid:25) t a 1 cc (cid:0) J j= (cid:0) 2 1(cid:30) (cid:25) t a j cc (cid:25)a j cc 1;t 1 ; ((cid:25) t ) (cid:0) (cid:0) where we have de(cid:133)ned the adjustment gainPL (v v )=w = (v~ v~ )=x (N ;(cid:31) ) for comjt 0t jt t 0t jt 0 t t (cid:17) (cid:0) (cid:0) pactness. Wenowconjecturethatthetimelessperspectiveoptimalpolicyinvolveszeronetin(cid:135)ation at all times, (cid:25) = 1. Under such a policy, in the timeless perspective regime (that is, after all transt itional dynamics have disappeared) the economy converges to the following equilibrium: (cid:15) x (N ;(cid:31) ) (cid:25) = p = (cid:1) = 0 t t = 1 = (cid:25)acc; j = 1;:::;J 1 t (cid:3)t t (cid:15) 1 z u (C ) jt (cid:0) t 0 t (cid:0) v = v L = 0; j = 1;:::;J 1 0t jt jt ) (cid:0) (cid:22) (cid:22) j (cid:21) = G(0) (cid:21) > 0 (cid:18) = 1 (cid:21) ; j = 1;:::;J 1 jt jt (cid:17) ) (cid:0) (cid:0) (cid:22) j 1 1 (cid:21)(cid:0) (cid:0) (cid:1) (cid:22) = (cid:0) ; jt J 1 1 (cid:21) (cid:22) k (cid:17) j (cid:0)(cid:0) (cid:1) k=0 (cid:0) Y C +G NP= t (cid:0)= t (cid:1) t t z z t t for all t. Thus, all (cid:133)rms end up having the same relative prices. Price dispersion is eliminated; the average price markup is constant at the level (cid:15)=((cid:15) 1), such that output, employment and (cid:0) consumption equal their (cid:135)exible-price levels of section 2.5.1 at all times; adjustment gains are zero and the vintage distribution converges to a stationary distribution. Imposing our conjecture in the (cid:133)rst-order conditions, we obtain Y u (C ) J 1 (cid:30)N J 1 u (C )Y (cid:15) 1 0 = 1+ t 00 t (cid:0) (cid:30)p (cid:3) 1 (cid:21) (cid:22) j t + (cid:0) (cid:30) vj 1+ 00 t t (cid:0) ; (20) u (C ) t j (cid:0) (cid:0) z u (C ) t u (C ) (cid:0) (cid:15) 0 t j=0 (cid:0) t 0 t j=0 (cid:20) 0 t (cid:21) P (cid:0) (cid:1) P J 1 J 1 0 = (cid:30)p (cid:3)E (cid:0) (cid:12)j 1 (cid:21) (cid:22) j Y u (C ) (cid:30)(cid:25)((cid:15) 1)+(cid:30)(cid:1)(cid:15) (cid:21) (cid:22) (cid:0) (cid:22) + (cid:22) t t (cid:0) t+j 0 t+j (cid:0) t (cid:0) t j J j=0 j=1 ! P (cid:0) (cid:1) (cid:2) (cid:3) P J 1 E (cid:0) (cid:12)j (cid:30)(cid:25) ((cid:15) 1)+(cid:30)(cid:1) (cid:15) 1 (cid:21) (cid:22) (cid:22) ; (21) (cid:0) t t+j (cid:0) t+j (cid:0) j j=1 P (cid:2) (cid:3)(cid:0) (cid:1) 0 = (cid:30)p (cid:3) 1 (cid:21) (cid:22) j Y u (C )+ (cid:30)(cid:25)((cid:15) 1)+(cid:30)(cid:1)(cid:15) 1 (cid:21) (cid:22) (cid:22) +(cid:30) (cid:25)a j cc (cid:12)E (cid:30) (cid:25)a j+ cc 1; (22) (cid:0) t j (cid:0) t 0 t t (cid:0) t (cid:0) j t (cid:0) t t+1 (cid:0) 0 = (cid:0) (cid:30)p t (cid:0) (cid:3) (cid:0) (J (cid:0) 1) 1(cid:1) (cid:0) (cid:21) (cid:22) J (cid:0) 1 Y t u 0 (cid:2)(C t )+ (cid:30)(cid:25) t ((cid:15) (cid:0) 1)(cid:3)+(cid:0) (cid:30)(cid:1) t (cid:15)(cid:1) 1 (cid:0) (cid:21) (cid:22) (cid:22) J (cid:0) 1 +(cid:30) t (cid:25)a J c (cid:0) c 1; (23) (cid:0)0 = (cid:30)(cid:1)(cid:18)j (cid:12)E (cid:30) (cid:18)j+1 1(cid:2) (cid:21) (cid:22) , j = 1;:::(cid:3);J(cid:0) 2;(cid:1) (24) t (cid:0) t t+1 (cid:0) (cid:0) 0(cid:0)= (cid:30) (cid:18)J (cid:1)1; (25) t (cid:0) 0 = (cid:30) (cid:21) t j +(cid:12)E t (cid:30) t+ j+ 1 1 (cid:22) j +(cid:30) (cid:18) t j 1 (cid:0) (cid:21) (cid:22) j (cid:0) 1 ; j = 1;:::;J (cid:0) 1; (26) (cid:0) (cid:1) 23

0 = (cid:30)(cid:25) +(cid:30)(cid:1) (cid:12) 1 (cid:21) (cid:22) E (cid:30) 2 +(cid:30) 1; (27) t t (cid:0) (cid:0) t t+1 t 0 = (cid:30)(cid:25) +(cid:30)(cid:1) +(cid:30) j (cid:12) 1 (cid:21) (cid:22)(cid:0)E (cid:30) j(cid:1)+1 +(cid:30) 1; j = 2;:::;J 1; (28) t t t (cid:0) (cid:0) t t+1 t (cid:0) 0 = (cid:0)(cid:30)(cid:25) +(cid:30)(cid:1)(cid:1) +(cid:30) J +(cid:30) 1; (29) t t t t J 1 g(0) J 2 0 = (cid:0) (cid:30) (cid:21)j (cid:30)v0 +(cid:21) (cid:22) (cid:0) (cid:30) vj +(cid:30) vJ 1; (30) (cid:0) t x (N ;(cid:31) ) (cid:0) t t 1 t (cid:0)1 j=1 0 t t j=0 (cid:0) (cid:0) P P g(0) 0 = (cid:30) (cid:21)j (cid:30) vj + 1 (cid:21) (cid:22) (cid:30) vj 1; (31) t x (N ;(cid:31) ) (cid:0) t (cid:0) t (cid:0)1 0 t t (cid:0) (cid:0) (cid:1) J 1 (cid:15) N x (N ;(cid:31) ) (cid:30)N J 1 x (N ;(cid:31) )N 0 = 1 (cid:0) (cid:30)p (cid:3) 1 (cid:21) (cid:22) j t 00 t t + t (cid:0) (cid:30) vj 00 t t t ; (32) (cid:0) (cid:0) t j(cid:15) 1 (cid:0) x (N ;(cid:31) ) x (N ;(cid:31) ) (cid:0) t x (N ;(cid:31) ) j=0 (cid:0) 0 t t 0 t t j=0 0 t t (cid:0) P (cid:0) (cid:1) P Y 0 = (cid:30)N t (cid:30)(cid:1); (33) (cid:0) t z (cid:0) t t 0 = (cid:0) (cid:30) (cid:25) t a 1 cc (cid:0) J j= (cid:0) 2 1(cid:30) (cid:25) t a j cc : (34) We now use equations (20) to (33) to solve for thPe Lagrange multipliers. From (25) and (24), it follows immediately that (cid:30) (cid:18)j = 0; j = 1;:::;J 1: (35) t (cid:0) Equations (27) to (29) allow us to solve for the (cid:30) j multipliers, obtaining t (cid:30) 1 = (cid:30)(cid:25) +(cid:30)(cid:1) ; t (cid:0) t t (cid:0) (cid:1) (cid:30) j = 0; j = 2;:::;J: (36) t Using (35) and (36) in equations (26), we obtain (cid:30) (cid:21)j = 0; j = 1;:::;J 1: (37) t (cid:0) Using the latter, equations (30) and (31) can be expressed compactly as (cid:30)v = A(cid:30)v , where t t 1 (cid:30)v = [(cid:30)v0;(cid:30)v1;:::;(cid:30) vJ 1] and (cid:0) t t t t (cid:0) 0 (cid:22) (cid:22) (cid:22) (cid:22) (cid:22) (cid:21) (cid:21) (cid:21) ::: (cid:21) (cid:21) 1 (cid:22) 21 (cid:21) 0 0 ::: 0 0 03 (cid:0) (cid:22) 0 1 (cid:21) 0 ::: 0 0 0 A = 6 (cid:0) 7: 6 7 J J 6 ::: ::: ::: ::: ::: ::: :::7 (cid:2) 6 7 6 0 0 0 ::: 1 (cid:21) (cid:22) 0 07 6 (cid:0) 7 6 (cid:22) 7 6 0 0 0 :: 0 1 (cid:21) 07 6 (cid:0) 7 4 5 24

The matrix A has J 1 eigenvalues with modulus equal to 1 (cid:21) (cid:22) < 1 and one unit eigenvalue.15 (cid:0) (cid:0) The system is thus stable, and the elements in (cid:30)v converge to (cid:133)nite values that depend on initial t conditions. Therefore, in the timeless perspective regime, in which all transitional dynamics have disappeared, the multipliers (cid:30) vj converge to constant values (cid:30) (cid:22)vj, j = 0;:::;J 1. We then use (32) t (cid:0) to solve for (cid:30)N, obtaining t (cid:15) N x (N ;(cid:31) ) J 1 x (N ;(cid:31) )N J 1 (cid:30)N = x (N ;(cid:31) ) 1+ t 00 t t (cid:0) 1 (cid:21) (cid:22) j (cid:30)p (cid:3) + 00 t t t (cid:0) (cid:30) (cid:22)vj : t 0 t t (cid:15) 1 x (N ;(cid:31) ) (cid:0) t j x (N ;(cid:31) ) " 0 t t j=0 (cid:0) 0 t t j=0 # (cid:0) P (cid:0) (cid:1) P Using the latter in (20), we obtain Y u (C ) N x (N ;(cid:31) ) J 1 1 1 u (C )Y (cid:15) 1x (N ;(cid:31) )N J 1 t 00 t + t 00 t t (cid:0) 1 (cid:21) (cid:22) j (cid:30)p (cid:3) = + 00 t t (cid:0) 00 t t t (cid:0) (cid:30) (cid:22)vj; ( )u (C ) x (N ;(cid:31) ) (cid:0) t j (cid:15) (cid:15) (cid:0) ( )u (C ) (cid:0) (cid:15) x (N ;(cid:31) ) (cid:20) (cid:0) 0 t 0 t t (cid:21)j=0 (cid:0) (cid:20) (cid:0) 0 t 0 t t (cid:21)j=0 P (cid:0) (cid:1) P where we have used the fact that, under our conjecture, x (N ;(cid:31) )=[z u (C )] = ((cid:15) 1)=(cid:15). At this 0 t t t 0 t (cid:0) point, we assume away government spending, G = 0, such that Y = C . We also assume that t t t functional forms for preferences are of the constant elasticity type. Let (cid:27) ( )C u (C )=u (C ) t 00 t 0 t (cid:17) (cid:0) and ’ N x (N ;(cid:31) )=x (N ;(cid:31) ) denote the constant elasticities of marginal consumption utility t 00 t t 0 t t (cid:17) and marginal labor disutility, respectively. Then we have J 1 1=(cid:15) 1=(cid:15) (cid:27) ’((cid:15) 1)=(cid:15) J 1 (cid:0) 1 (cid:21) (cid:22) j (cid:30)p (cid:3) = + (cid:0) (cid:0) (cid:0) (cid:0) (cid:30) (cid:22)vj (cid:4): (cid:0) t j (cid:27) +’ (cid:27) +’ (cid:17) j=0 (cid:0) j=0 P (cid:0) (cid:1) P It can be shown that all J 1 roots of the characteristic polynomial J 1 1 (cid:21) (cid:22) j xJ 1 j have (cid:0) j= (cid:0) 0 (cid:0) (cid:0) (cid:0) (cid:22) modulus equal to 1 (cid:21) < 1, hence they all lie inside the unit circle. Therefore, in the timeless (cid:0) P (cid:0) (cid:1) perspective regime, the multiplier (cid:30)p t (cid:3) converges to the constant value (cid:30) (cid:22)p (cid:3) (cid:17) (cid:4)= J j= (cid:0) 0 1 1 (cid:0) (cid:21) (cid:22) j . Using this in equation (21), together with (cid:21) (cid:22) J j= (cid:0) 1 1 (cid:22) j + (cid:22) J = (cid:22) 1 and (cid:22) j = 1 (cid:0) (cid:21) (cid:22) j (cid:0) P 1 (cid:22) 1 , t (cid:0) he latt (cid:1) er equation can be expressed as P (cid:0) (cid:1) J 1 0 = E (cid:0) (cid:12)j 1 (cid:21) (cid:22) j (cid:30) (cid:22)p (cid:3)Y u (C ) (cid:30)(cid:25) ((cid:15) 1)+(cid:30)(cid:1) (cid:15) (cid:22) t (cid:0) t+j 0 t+j (cid:0) t+j (cid:0) t+j 1 j=0 n o JP1 (cid:0) (cid:1) (cid:2) (cid:3) = E (cid:0) (cid:12)j 1 (cid:21) (cid:22) j (cid:6) ; (38) t t+j (cid:0) j=0 P (cid:0) (cid:1) where we have de(cid:133)ned (cid:6) (cid:30) (cid:22)p (cid:3)Y u (C ) (cid:30)(cid:25)((cid:15) 1)+(cid:30)(cid:1)(cid:15) (cid:22) . All J 1 roots of the polynomial t (cid:17) t 0 t (cid:0) t (cid:0) t 1 (cid:0) J 1(cid:12)j 1 (cid:21) (cid:22) j xJ 1 j have modulus equal to (cid:12) 1 (cid:21) (cid:22) < 1 and are thus inside the unit circle. j= (cid:0) 0 (cid:0) (cid:0) (cid:0) (cid:2) (cid:0) (cid:3) P15Every(cid:0)column(cid:1)of A sums to unity, which implies that un(cid:0)ity is a(cid:1)n eigenvalue of A (Hamilton, 1994, p. 681), but A is also a Leslie matrix, hence it has only one positive and dominant eigenvalue (Poole, 2006, p. 328). Hence, all other eigenvalues must lie inside the unit circle. 25

Therefore, equation (38) has a unique solution given by (cid:6) = 0, or equivalently t (cid:30) (cid:22)p (cid:3)Y u (C ) (cid:30)(cid:25)((cid:15) 1)+(cid:30)(cid:1)(cid:15) (cid:22) = 0; (39) t 0 t (cid:0) t (cid:0) t 1 (cid:2) (cid:3) which pins down the multiplier (cid:30)(cid:25) as a function of the variables Y u (C ) and (cid:30)(cid:1). The latter t t 0 t t multiplier is in turn determined by equation (33). (cid:25)acc Equation (23) can be solved for (cid:30) J 1, obtaining t (cid:0) (cid:30) (cid:25) t a J c (cid:0) c 1 = 1 (cid:0) (cid:21) (cid:22) J (cid:0) 1 (cid:30) (cid:22)p (cid:3)Y t u 0 (C t ) (cid:0) (cid:30)(cid:25) t ((cid:15) (cid:0) 1)+(cid:30)(cid:1) t (cid:15) (cid:22) 1 = 0; n o (cid:0) (cid:1) (cid:2) (cid:3) (cid:22) (cid:22) J 2 (cid:22) where we have used = 1 (cid:21) (cid:0) and where the second equality follows from (39). Using J 1 1 E t (cid:30) (cid:25) t+ a J c 1(cid:0) c 1 = 0 and (cid:22) J (cid:0) 2 = (cid:0) 1 (cid:0) (cid:0) (cid:21) (cid:22) (cid:0) J (cid:0) (cid:1) 3 (cid:22) 1 in equation (22) for j = J (cid:0) 2, the latter im (cid:25)a p c l c ies (cid:30) t (cid:25)a J c (cid:0) c 2 = 0. Operating in the same fashion, equations (22) for j = 1;:::;J 3 imply that (cid:30) j = 0 for j = (cid:0) (cid:1) t (cid:0) 1;:::;J 3. (cid:0) Itonlyremainstoverifythatequation(34)holdsgiventhesolutionoftheLagrangemultipliers. (cid:25)acc This is obvious, as we have already shown that (cid:30) j = 0 for j = 1;:::;J 1. t (cid:0) 26

Probability of price adjustment 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 5 10 15 20 Vintage (j) λ ss,j Fig.1: Price adjustment probability and firm distribution by vintage Firm distribution by price vintage 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 5 10 15 20 Vintage (j) ψ ss,j

Fig.2: Responses to a technology and a government spending shock Shocks Real interest rate Output gap 1.5 0.05 0.1 Tech. process Gov’t spending 0.05 1 0 0 0.5 −0.05 −0.05 0 −0.1 −0.1 5 10 15 20 5 10 15 20 5 10 15 20 Consumption Hours worked Real wage 1 0.5 1.5 1 0.5 0 0.5 0 0 −0.5 −0.5 −0.5 5 10 15 20 5 10 15 20 5 10 15 20 Optimal price Inflation Price dispersion 0.5 0.5 0.5 0 0 0 −0.5 −0.5 −0.5 5 10 15 20 5 10 15 20 5 10 15 20 Quarters Quarters Quarters