Distributional dynamics under smoothly state-dependent pricing

Finance and Economics Discussion Series Divisions of Research & Statistics and Monetary Affairs Federal Reserve Board, Washington, D.C. Distributional dynamics under smoothly state-dependent pricing James Costain and Anton Nakov 2011-50 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.

Distributional dynamics under smoothly state-dependent pricing James Costaina, Anton Nakovb,∗† a Banco de Espan˜a; b Federal Reserve Board Abstract Starting from the assumption that firms are more likely to adjust their prices when doing so is more valuable, this paper analyzes monetary policy shocks in a DSGE model with firm-level heterogeneity. The model is calibrated to retail price microdata, and inflation responses are decomposed into “intensive”, “extensive”, and “selection” margins. Money growth and Taylor rule shocks both have nontrivial real effects, because the low state dependence implied by the data rules out the strong selection effect associated with fixed menu costs. The response to sector-specific shocks is gradual, but inappropriate econometrics might make it appear immediate. Keywords: Nominalrigidity,state-dependentpricing,menucosts,heterogeneity,Taylorrule JEL classification: E31,E52,D81 ∗Corresponding author: James Costain, Banco de Espana, Calle Alcal´a 48, 28014 Madrid, Spain, +34-91-338-5732, james.costain@bde.es †An earlier version of this paper circulated as “Dynamics of the price distribution in a general model of statedependent pricing”. The authors thank R. Bachmann, M. Dotsey, O. Licandro, B. Mackowiak, V. Midrigan, A. Reiff, M. Reiter, R. Wouters, and K. Sheedy for helpful comments, as well as seminar participants at IAS Vienna, EUI, ECB, CERGE-EI, ESSIM 2008, CEF 2008, REDg 2008, SNDE 2009, SED 2009, Banco de Espan˜a “Workshop on Monetary Policy” (2009), and Banque de France “Understanding Price Dynamics” (2009), and also the editors and anonymous referees. They are especially grateful to V. Midrigan, E. Gagnon, and O. Kryvtsov for providing data. Anton Nakov thanks the Bank of Spain and the European Central Bank for their support and hospitality during the first drafts of this paper. The views expressed in this paper are those of the authors and do not necessarily coincide with those of the Bank of Spain, the European Central Bank, or the Federal Reserve Board.

Distributional dynamics under smoothly state-dependent pricing 2 1. Introduction Sticky prices are an important ingredient in modern dynamic general equilibrium models, including those used by central banks for policy analysis. But how best to model price stickiness, and to what extent stickiness of individual prices implies rigidity of the aggregate price level, remains controversial. Calvo’s(1983)assumptionofaconstantadjustmentprobabilityispopularforitsanalyticaltractability, and implies that monetary shocks have large and persistent real effects. However, Golosov and Lucas (2007, henceforth GL07) have argued that microfounding price rigidity on a fixed “menu cost”and calibrating to microdata implies that monetary shocks are almost neutral. This paper calibrates and simulates a general model of state-dependent pricing that nests the Calvo (1983) and fixed menu cost (FMC) models as two opposite limiting cases, with a continuum of smooth intermediate cases lying in between. As in Dotsey, King, and Wolman (1999) and Caballero and Engel (2007), the setup rests on one fundamental property: firms are more likely to adjust their prices whendoingsoismorevaluable. Implementingthisassumptionrequirestheselectionofaparameterized family of functions to describe the adjustment hazard; the exercise is disciplined by fitting the model to the size distribution of price changes found in recent US retail microdata (Klenow and Kryvtsov 2008; Midrigan 2011; Nakamura and Steinsson 2008).1 One of the calibrated parameters controls the degree of state dependence; matching the smooth distribution of price changes seen in microdata requires rather low state dependence. Therefore, impulse responses reveal substantial monetary nonneutrality, with real effects only slightly weaker than the Calvo model implies. The impulse response analysis considers a number of issues unaddressed by previous work on statedependent pricing. GL07 restricted attention to iid money growth shocks; this paper also considers the autocorrelated case, and shows that the shape and persistence of responses is primarily determined by the degree of state dependence, not by the autocorrelation of the driving process. Moreover, this paper also studies monetary policy governed by a Taylor rule, as opposed to an exogenous money growth process, which reinforces the conclusion that a calibrated model of state-dependent pricing implies nontrivial real effects. This paper also decomposes inflation into an “intensive margin” relating to 1A companion paper, Costain and Nakov (2011), discusses the calibration in greater detail, documenting the steadystate model’s fit to cross-sectional microdata on price adjustments, both for low and high trend inflation rates.

Distributional dynamics under smoothly state-dependent pricing 3 the average desired price change, an “extensive margin” relating to the fraction of firms adjusting, and a “selection effect” relating to which firms adjust. This decomposition corroborates the claim of GL07, which was challenged by Caballero and Engel (2007), that the selection effect is crucial for the behavior of the FMC model. A fourth contribution of this paper is to argue that prices respond slowly to sector-specific as well as aggregate shocks, despite some recent empirical claims to the contrary. The paper also implements an algorithm for computing heterogeneous-agent economies which is well-suited to modeling state-dependent pricing but has not yet been applied in this context. 1.1. Relation to previous literature Most previous work on state-dependent pricing has obtained solutions by limiting the analysis, either focusing on partial equilibrium (e.g. Caballero and Engel, 1993, 2007; Klenow and Kryvtsov, 2008), or assuming firms face aggregate shocks only (e.g. Dotsey et al., 1999), or making strong assumptions about the distribution of idiosyncratic shocks (e.g. Caplin and Spulber, 1987; Gertler and Leahy, 2005). But Klenow and Kryvtsov (2008) argue convincingly that firms are often hit by large idiosyncratic shocks. And while heterogeneity may average out in many macroeconomic contexts, it is hard to ignore in the debate over nominal rigidities, because firm-level shocks could greatly alter firms’ incentives to adjust prices. GL07 were the first to confront these issues directly, by studying a menu cost model in general equilibrium with idiosyncratic productivity shocks. They obtained a striking near-neutrality result. However, their model’s fit to price data is questionable, as our Figure 1 shows. A histogram of retail microdata displays a wide range of price adjustments, whereas their FMC model generates two sharp spikes of price increases and decreases occurring near the (S,s) bounds. Other micro facts have been addressed in more recent papers on state-dependent pricing. Eichenbaum, Jaimovich, and Rebelo (2011) and Kehoe and Midrigan (2010) modeled “temporary” price changes (sales), assuming that these adjustments are cheaper than other price changes. However, they ultimately conclude that the possibility of sales has little relevance for monetary transmission, which depends instead on the frequency of regular non-sale price changes. Guimaraes and Sheedy’s (2011) model of sales as stochastic price discrimination has the same implication. Thus, since the model developed in this paper has no natural motive for sales, it will be compared to a dataset of

Distributional dynamics under smoothly state-dependent pricing 4 “regular” price changes from which apparent sales have been removed. In another branch of the literature, Boivin, Giannoni, and Mihov (2009) and Mackowiak, Moench, and Wiederholt (2009) estimate that prices respond much more quickly to sectoral shocks than to aggregate shocks. However, the present paper performs a Monte Carlo exercise that shows that this finding should be treated with caution. Remarkably, even when the true response to a sector-specific shock is lagged and transitory, the estimation routine of Mackowiak et al. can erroneously conclude that sector-specific shocks have an immediate, permanent impact on prices. Our need to allow for firm-specific shocks complicates computation, because it implies that the distribution of prices and productivities across firms is a relevant state variable. This paper shows howtocomputeadynamicgeneralequilibriumwithstate-dependentpricingviathetwo-stepalgorithm of Reiter (2009), which calculates steady-state equilibrium using backwards induction on a grid, and then linearizes the equations at every grid point to calculate the dynamics. This approach avoids some complications (and simplifying assumptions) required by other methods. In contrast to GL07, it is not necessary to assume that aggregate output stays constant after a money shock. In contrast to Dotsey, King, and Wolman (2008), it more fully exploits the recursive structure of the model, tracking the price distribution without needing to know who adjusted when. In contrast to the method of Krusell and Smith (1998), used by Midrigan (2011), there is no need to find an adequate summary statistic for the distribution. In contrast to Den Haan (1997), there is no need to impose a specific distributional form. The nonlinear, nonparametric treatment of firm-level heterogeneity in Reiter’s algorithm makes it straightforward to calculate the time path of cross-sectional statistics, like our inflation decomposition; the linearization of aggregate dynamics makes it just as easy to analyze a variety of monetary policy rules or shock processes as it would be in a standard, low-dimensional DSGE model. Several closely related papers have also remarked that FMCs imply a counterfactual distribution of price adjustments, in which small changes never occur. They proposed some more complex pricing models to fix this problem, including sectoral heterogeneity in menu costs (Klenow and Kryvtsov, 2008), multiple products on the same “menu” combined with leptokurtic technology shocks (Midrigan, 2011), or a mix of flexible- and sticky-price firms plus a mix of two distributions of productivity shocks

Distributional dynamics under smoothly state-dependent pricing 5 (Dotsey et al., 2008). This paper proposes a simpler approach: we just assume the probability of price adjustment increases with the value of adjustment, and treat the hazard function as a primitive of the model. A family of hazard functions with just three parameters suffices to match the distribution of price changes at least as well as the aforementioned papers do. Our setup can be interpreted as a stochastic menu cost (SMC) model, like Dotsey et al. (1999) or Caballero and Engel (1999); under this interpretation the hazard function corresponds to the c.d.f. of the menu cost.2 Alternatively, our setup can be seen as a near-rational model, like Akerlof and Yellen (1985), in which firms are more likely to make mistakes when they are not very costly; in this case the hazard function corresponds to the distribution of error values.3 Under either interpretation, the key point is that the adjustment hazard increases smoothly with the value of adjusting, in contrast with the discontinuous jump in probability implied by the FMC model. An appropriate calibration of the smoothness of the hazard function yields a smooth histogram of price changes consistent with microdata; this smoothness is the same property that eliminates the strong selection effect found by GL07. Thus, none of the additions Dotsey et al. and Midrigan make to the FMC framework are crucial for their most important finding: a state-dependent pricing model consistent with observed price changes implies nontrivial real effects of monetary shocks, similar to those found under the Calvo framework. 2. Model This discrete-time model embeds state-dependent pricing by firms in an otherwise-standard New Keynesian general equilibrium framework based on GL07. Besides the firms, there is a representative household and a monetary authority that either implements a Taylor rule or follows an exogenous growth process for nominal money balances. The aggregate state of the economy at time t, which will be specified in Section 2.3., is called Ω . Time subscripts on aggregate variables will indicate dependence, in equilibrium, on aggregate t conditions Ω . For example, consumption is denoted by C ≡ C(Ω ). t t t 2For estimation purposes, Caballero and Engel usually assume that the probability of adjustment depends on the distanceofthechoicevariablefromsometargetlevel,butthisisjustanapproximationtoanunderlyingmodelinwhich the adjustment probability depends on the value of adjustment, as in our setup. 3The two interpretations imply slightly different Bellman equations: in the first case, but not in the second, a flow of menu costs is subtracted out of the firm’s flow of profits.

Distributional dynamics under smoothly state-dependent pricing 6 2.1. Household The household’s period utility function is 1 C1−γ−χN +νlog(M /P ), where C is consumption, 1−γ t t t t t N is labor supply, and M /P is real money balances. Utility is discounted by factor β per period. t t t Consumption is a CES aggregate of differentiated products C , with elasticity of substitution (cid:15): it (cid:26)(cid:90) 1 (cid:15)−1 (cid:27) (cid:15)− (cid:15) 1 C = C (cid:15) di . (1) t it 0 The household’s nominal period budget constraint is (cid:90) 1 P C di+M +R−1B = W N +M +T +B +U , (2) it it t t t t t t−1 t t−1 t 0 (cid:82)1 where P C di is total nominal consumption. B is nominal bond holdings purchased at t, paying 0 it it t interest rate R −1 at time t+1. T is a nominal lump-sum transfer consisting of seignorage revenues t t from the central bank plus dividend payments from the firms. Households choose {C ,N ,B ,M }∞ to maximize expected discounted utility, subject to the it t t t t=0 budget constraint (2). Optimal consumption across the differentiated goods implies C = (P /P )(cid:15)C , (3) it t it t where P is the price index t (cid:26)(cid:90) 1 (cid:27) 1− 1 (cid:15) P ≡ P 1−(cid:15)di , (4) t it 0 (cid:82)1 so total nominal spending can be written as P C = P C di. t t 0 it it Optimal labor supply, consumption, and money use imply the following first-order conditions: χ = C−γW /P , (5) t t t (cid:18) P C−γ (cid:19) R−1 = βE t t+1 , (6) t t P C−γ t+1 t νP (cid:18) P C−γ (cid:19) 1− t = βE t t+1 . (7) M C−γ t P C−γ t t t+1 t

Distributional dynamics under smoothly state-dependent pricing 7 2.2. Monopolistic firms Each firm i produces output Y using labor N as its only input, under a constant returns techit it nology: Y = A N . A is an idiosyncratic productivity process that is AR(1) in logs: it it it it logA = ρlogA +εa, (8) it it−1 it where 0 ≤ ρ < 1 and (cid:15)a ∼ i.i.d.N(0,σ2). Firm i is a monopolistic competitor that sets a price P , it a it facing the demand curve Y = C P(cid:15)P−(cid:15), and must fulfill all demand at its chosen price. It hires in a it t t it competitive labor markets at wage rate W , generating profits t (cid:18) (cid:19) W U = P Y −W N = P − t C P(cid:15)P−(cid:15) ≡ U(P ,A ,Ω ) (9) it it it t it it A t t it it it t it per period. Firms are owned by the household, so they discount nominal income between times t and t+1 at the rate βP(Ωt)C(Ωt+1)−γ , consistent with the household’s marginal rate of substitution. P(Ωt+1)C(Ωt)−γ Let V(P ,A ,Ω ) denote the nominal value of a firm at time t that produces with productivity it it t A and sells at nominal price P . Prices are sticky, so P may or may not be optimal. However, we it it it assume that whenever a firm adjusts its price, it chooses the optimal price conditional on its current productivity, keeping in mind that it will sometimes be unable to adjust in the future. Hence, the value function of an adjusting firm, after netting out any costs that may be required to make the adjustment, is V∗(A ,Ω ) ≡ max V(P,A ,Ω ). For clarity, it helps to distinguish the firm’s beginning-of-period it t P it t price, P(cid:101) ≡ P , from the end-of-period price at which it sells at time t, P , which may or may not it i,t−1 it be the same. The distributions of prices and productivities across firms at the beginning and end of t will be denoted Φ(cid:101) (P(cid:101),A) and Φ (P,A), respectively. t t The gain from adjusting at the beginning of t is: D(P(cid:101) ,A ,Ω ) ≡ maxV(P,A ,Ω )−V(P(cid:101) ,A ,Ω ). (10) it it t it t it it t P The main assumption of our framework is that the probability of price adjustment increases with the gain from adjustment. The weakly increasing function λ that governs this probability is taken as a primitive of the model. Invariance of λ requires that its argument, the gain from adjustment, be written in appropriate units. As was mentioned in the introduction, this setup can be interpreted as a stochastic menu cost model, or as a model of near-rational price decisions. In the SMC case,

Distributional dynamics under smoothly state-dependent pricing 8 the labor effort of changing price tags or menus is likely to be a large component of the cost; in the near-rational case, the adjustment probability should depend on the labor effort required to obtain new information or to recompute the optimal price. Under either interpretation, the most natural units for the argument of the λ function are units of labor time. Thus, the probability of adjustment (cid:16) (cid:16) (cid:17)(cid:17) (cid:16) (cid:17) will be defined as λ L P(cid:101) ,A ,Ω , where L P(cid:101) ,A ,Ω = D(P(cid:101)it,Ait,Ωt) expresses the adjustment it it t it it t W(Ωt) gain in time units by dividing by the wage. The functional form for λ will be specified in Sec. 2.2.1. The value of selling at any given price equals current profits plus the expected value of future production, which may or may not be sold at a new, adjusted price. Given the firm’s idiosyncratic state variables (P,A) and the aggregate state Ω, and denoting next period’s variables with primes, the Bellman equation under the near-rational interpretation of the model is (cid:18) (cid:19) W(Ω) V(P,A,Ω) = P − C(Ω)P(Ω)(cid:15)P−(cid:15) + (11) A (cid:26) (cid:20)(cid:18) (cid:19) (cid:21)(cid:12) (cid:27) βE P(Ω)C(Ω(cid:48))−γ 1−λ (cid:16) D(P,A(cid:48),Ω(cid:48)) (cid:17) V(P,A(cid:48),Ω(cid:48)) + λ (cid:16) D(P,A(cid:48),Ω(cid:48)) (cid:17) maxV (P(cid:48),A(cid:48),Ω(cid:48)) (cid:12) (cid:12)A,Ω . P(Ω(cid:48))C(Ω)−γ W(Ω(cid:48)) W(Ω(cid:48)) P(cid:48) (cid:12) Here the expectation refers to the distribution of A(cid:48) and Ω(cid:48) conditional on A and Ω. Note that on the left-hand side of the Bellman equation, and in the term representing current profits, P refers to a given firm i’s price P at the end of t, when transactions occur. In the expectation on the right, P it represents the price P(cid:101) at the beginning of t+1, which may (probability λ) or may not (1−λ) be i,t+1 adjusted prior to time t+1 transactions to a new value P(cid:48). The right-hand side of the Bellman equation can be simplified by using the notation from (9), and the rearrangement (1−λ)V +λmaxV = V +λ(maxV −V): (cid:26) (cid:12) (cid:27) (cid:12) V(P,A,Ω) = U(P,A,Ω)+βE P(Ω)C(Ω(cid:48))−γ [V(P,A(cid:48),Ω(cid:48))+G(P,A(cid:48),Ω(cid:48))](cid:12)A,Ω , (12) P(Ω(cid:48))C(Ω)−γ (cid:12) where (cid:18) D(P,A(cid:48),Ω(cid:48)) (cid:19) G(P,A(cid:48),Ω(cid:48)) ≡ λ D(P,A(cid:48),Ω(cid:48)). (13) W(Ω(cid:48)) The terms inside the expectation in the Bellman equation represent the value V of continuing without adjustment, plus the flow of expected gains G due to adjustment. Since the firm sets the optimal price

Distributional dynamics under smoothly state-dependent pricing 9 whenever it adjusts, the price process associated with (12) is  (cid:16) (cid:17)   P∗(A ,Ω ) ≡ argmax V(P,A ,Ω ) with probability λ D(P(cid:101)it,Ait,Ωt) P = it t P it t W(Ωt) . (14) it (cid:16) (cid:17)   P(cid:101) it ≡ P i,t−1 with probability 1−λ D(P(cid:101) W it, ( A Ω i t t ) ,Ωt) Equation (14) is written with time subscripts for additional clarity. 2.2.1. Alternative sticky price frameworks Our assumptions require the function λ to be weakly increasing and to lie between zero and one. The paper focuses primarily on the following functional form: λ λ(L) ≡ (15) λ+(1−λ) (cid:0) α (cid:1)ξ L with α and ξ positive, and λ ∈ [0,1]. This function equals λ when L = α, and is concave for ξ ≤ 1 and S-shaped for ξ > 1 (see the second panel of Fig. 1). The parameter ξ effectively controls the degree of state dependence. In the limit ξ = 0, (15) nests Calvo (1983), with λ(L) = λ, making the adjustment hazard literally independent of the relevant state variable, which is L. At the opposite extreme, as ξ → ∞, λ(L) becomes the indicator 1{L ≥ α}, which equals 1 whenever L ≥ α and is zero otherwise. This implies very strong state dependence, in the sense that the adjustment probability jumps from 0 to 1 when the state L passes the threshold level α. For all intermediate values 0 < ξ < ∞, the hazard increases smoothly with the state L. In this sense, choosing ξ to match microdata means finding the degree of state dependence most consistent with firms’ observed pricing behavior. ThecombinationofBellmanequation(12)with(13)isbasedonanear-rationalinterpretationofour setup; for 0 < ξ < ∞ this version of the model will be called “SSDP”, for “smoothly state-dependent pricing”. However, as Table 1 shows, (12) nests several other models under appropriate choices of the (cid:82)L gains function G and the hazard function λ. Subtracting a flow of menu costs E(κ|κ < L) ≡ κλ(dκ) 0 outofthegainsGconvertstheSSDPmodelintotheSMCmodel. TheFMCmodelsetstheadjustment probability to a step function, subtracting a constant menu cost α out of G; it is the limit of the SMC model as ξ → ∞. The Calvo model is derived both from SSDP and from SMC as ξ → 0.4 An alternative hazard function derived from Woodford (2009) is also considered. 4In the limit of SMC as ξ →0, the menu cost is zero with probability λ and infinite with probability 1−λ, which is when firms do not adjust. The flow of menu costs paid is therefore zero.

Distributional dynamics under smoothly state-dependent pricing 10 2.3. Monetary policy and aggregate consistency Two specifications for monetary policy are compared: a money growth rule and a Taylor rule. In both cases the systematic component of monetary policy is perturbed by an AR(1) process z, z = φ z +(cid:15)z, (16) t z t−1 t where 0 ≤ φ < 1 and (cid:15)z ∼ i.i.d.N(0,σ2). Under the money growth rule, which is analyzed first to z t z build intuition and for comparison with previous studies, z affects money supply growth: M /M ≡ µ = µ∗exp(z ). (17) t t−1 t t Alternatively, under a Taylor interest rate rule, which is a better approximation to actual monetary policy, the nominal interest rate follows R (cid:32) (cid:18) P /P (cid:19)φ π (cid:18) C (cid:19)φ c (cid:33)1−φ R (cid:18) R (cid:19)φ R t t t−1 t t−1 = exp(−z ) , (18) R∗ t Π∗ C∗ R∗ where φ ≥ 0, φ > 1, and 0 < φ < 1, so that when inflation P /P exceeds its target Π∗ or c π R t t−1 consumption C exceeds its target C∗, R tends to rise above its target R∗ ≡ Π∗/β. For comparability t t between the two monetary regimes, the inflation target is set to Π∗ ≡ µ∗, and the rules are specified so that in both cases, a positive z represents an expansive shock. Seigniorage revenues are paid to the household as a lump-sum transfer, and the government budget is balanced each period. Therefore total nominal transfers to the household, including dividend payments, are (cid:90) 1 T = M −M + U di. (19) t t t−1 it 0 Bond market clearing is simply B = 0. When supply equals demand for each good i, total labor t supply and demand satisfy (cid:90) 1 C (cid:90) 1 N = it di = P(cid:15)C P−(cid:15)A−1di ≡ ∆pC . (20) t A t t it it t t 0 it 0 Equation (20) also defines a measure of price dispersion, ∆p ≡ P(cid:15) (cid:82)1 P−(cid:15)A−1di, weighted to allow for t t 0 it it heterogeneous productivity. As in Yun (2005), an increase in ∆p decreases the goods produced per t unit of labor, effectively acting like a negative aggregate productivity factor.

Distributional dynamics under smoothly state-dependent pricing 11 At this point, all equilibrium conditions have been spelled out, so an appropriate aggregate state variable Ω can be identified. At time t, the lagged distribution of transaction prices Φ (P,A) is t t−1 predetermined. Knowing Φ , the lagged price level can be substituted out of the Taylor rule, using t−1 (cid:20)(cid:90)(cid:90) (cid:21)1/(1−(cid:15)) P = P1−(cid:15)Φ (dP,dA) . It can thus be seen that Ω ≡ (z ,R ,Φ ) suffices to define t−1 t−1 t t t−1 t−1 the aggregate state. Given Ω , equations (4), (5), (6), (8), (9), (10), (12), (13), (14), (16), (18), and t (20) together give enough conditions to determine the distributions Φ(cid:101) and Φ , the price level P , the t t t functions U(P,A,Ω ), V(P,A,Ω ), D(P,A,Ω ), and G(P,A,Ω ), and the variables R , C , N , W , and t t t t t t t t z . Thus they determine the next state, Ω ≡ (z ,R ,Φ ). t+1 t+1 t+1 t t Under a money growth rule, the time t state can instead be defined as Ω ≡ (z ,M ,Φ ). t t t−1 t−1 Substituting (7) for (6) and (17) for (18), knowing Ω ≡ (z ,M ,Φ ) suffices to determine Φ(cid:101) , Φ , t t t−1 t−1 t t U(P,A,Ω ), V(P,A,Ω ), D(P,A,Ω ), and G(P,A,Ω ), as well as P , C , N , W , z , and M . Thus t t t t t t t t t+1 t the next state, Ω ≡ (z ,M ,Φ ), can be calculated. t+1 t+1 t t 3. Computation The fact that this model’s state variable includes the distribution Φ, an infinite-dimensional object, makes computing equilibrium a challenge. The popularity of the Calvo model reflects its implication that general equilibrium can be solved up to a first-order approximation by keeping track of the average price only. Unfortunately, this result typically fails to hold if pricing is state-dependent; instead, computation requires tracking the whole distribution Φ. Equilibrium will be computed here following the two-step algorithm of Reiter (2009), which is intended for contexts, like this model, with relatively large idiosyncratic shocks and also relatively small aggregate shocks. In the first step, the aggregate steady state of the model is computed on a finite grid, using backwards induction.5 Second, the stochastic aggregate dynamics are computed by linearization, grid point by grid point. In other words, the Bellman equation is treated as a large system of expectational difference equations, instead of as a functional equation. 5Actually, Reiter’s algorithm permits calculation of the aggregate steady state using a variety of finite-element methods; we choose backwards induction on a grid since it is a familiar and transparent procedure.

Distributional dynamics under smoothly state-dependent pricing 12 3.1. Detrending Calculating a steady state requires detrending to make the economy stationary. Here it suffices to scale all nominal variables by the aggregate price level, defining the real wage and money supply w = W /P and m ≡ M /P , and the real prices at the beginning and end of t, p ≡ P(cid:101) /P and t t t t t t (cid:101)it it t p ≡ P /P . The beginning-of-t and end-of-t price distributions will be written as Ψ(cid:101) (p ,A ) and it it t t (cid:101)it it Ψ (p ,A ), respectively. At the end of t, when goods are sold, the real price level is one by definition: t it it (cid:26)(cid:90)(cid:90) (cid:27)1/(1−(cid:15)) 1 = p1−(cid:15)Ψ (dp,dA) . (21) t For this detrending to make sense, the nominal price level P must be irrelevant for real quantities, t whichmustinsteadbefunctionsofarealstatevariableΞ thatisindependentofnominalpricesandthe t nominal money supply. A time subscript on any aggregate variable must now denote dependence on the real state, implying for example w = w(Ξ ) =W(Ω )/P(Ω ) and C = C(Ξ ) = C(Ω ). While the t t t t t t t price level will cancel out, inflation Π ≡ P /P will still appear in the model. It must be determined t t t−1 by real variables, satisfying Π = Π(Ξ ,Ξ ) = P(Ω )/P(Ω ). t t−1 t t t−1 A similar property applies to the value function and profits, which must be homogeneous of degree one in prices. Thus, define real profits u and real value v as follows: u(p,A,Ξ) = u(P/P(Ω),A,Ξ) ≡ P(Ω)−1U(P,A,Ω), (22) v(p,A,Ξ) = v(P/P(Ω),A,Ξ) ≡ P(Ω)−1V(P,A,Ω). (23) To verify homogeneity, divide through the nominal Bellman equation (12) by P(Ω) to obtain (cid:26) C(Ξ(cid:48))−γ(cid:20) (cid:18) p (cid:19) (cid:18) p (cid:19)(cid:21)(cid:12) (cid:12) (cid:27) v(p,A,Ξ) = u(p,A,Ξ)+βE v ,A(cid:48),Ξ(cid:48) +g ,A(cid:48),Ξ(cid:48) (cid:12)A,Ξ , (24) C(Ξ)−γ Π(Ξ,Ξ(cid:48)) Π(Ξ,Ξ(cid:48)) (cid:12) using the definitions (cid:18) (cid:19) d(p,A,Ξ) (cid:101) g(p,A,Ξ) ≡ λ d(p,A,Ξ), (25) (cid:101) (cid:101) w(Ξ) d(p,A,Ξ) ≡ maxv(p,A,Ξ)−v(p,A,Ξ), (26) (cid:101) (cid:101) p

Distributional dynamics under smoothly state-dependent pricing 13 which satisfy g(p,A,Ξ) = G(P(Ω)p,A,Ω)/P(Ω) and d(p,A,Ξ) = D(P(Ω)p,A,Ω)/P(Ω).6 This de- (cid:101) (cid:101) (cid:101) (cid:101) trendingimpliesthatwhenafirm’snominalpriceremainsunadjustedattimet, itsrealpriceisdeflated by factor Π . Therefore the real price process is t  (cid:18) (cid:19)   p∗(A ,Ξ ) ≡ argmax v(p,A ,Ξ ) with probability λ d(Π− t 1pi,t−1,Ait,Ξt )  it t p it t w(Ξt) p it =   Π−1p with probability 1−λ (cid:18) d(Π− t 1pi,t−1,Ait,Ξt ) (cid:19) . (27)  t i,t−1 w(Ξt) To see that these definitions of real quantities suffice to detrend the model, define the real state as Ξ ≡ (z ,R ,Ψ ). Knowing Ξ , in the case of a Taylor rule, equations (5), (6), (8), (16), (18), (20), t t t−1 t−1 t (21), (22), (24), (25), (26), and (27), with substitutions of real for nominal variables where necessary, suffice to determine the distributions Ψ(cid:101) and Ψ , inflation Π , the functions u(p,A,Ξ ), v(p,A,Ξ ), t t t t t d(p,A,Ξ ), and g(p,A,Ξ ), and the variables C , w , N , R , and z . For a money growth rule, the t t t t t t t+1 real state can be defined as Ξ ≡ (z ,m ,Ψ ), and equation (18) is replaced by (7) and by t t t−1 t−1 m = µ∗exp(z )m /Π , (28) t t t−1 t which together determine R and m . Thus next period’s state Ξ can be calculated if Ξ is known. t t t+1 t 3.2. Discretization Priceprocess(27)isdefinedoveracontinuumofpossiblevalues, buttosolvethemodelnumerically, the idiosyncratic states must be restricted to a finite-dimensional support. Hence, the continuous model will be approximated on a two-dimensional grid Γ ≡ Γp ×Γa, where Γp ≡ {p1,p2,...p#p} and Γa ≡ {a1,a2,...a#a} are logarithmically-spaced grids of possible values of of p and A . Thus the it it time-varying distributions will be treated as matrices Ψ(cid:101) and Ψ of size #p × #a, in which the row t t j, column k elements, called Ψ(cid:101) jk and Ψjk, represent the fraction of firms in state (pj,ak) before and t t after price adjustments in period t, respectively. From here on, bold face is used to identify matrices and superscripts are used to identify notation related to grids. Similarly,thevaluefunctioniswrittenasa#p×#a matrixV ofvaluesvjk ≡ v(pj,ak,Ξ )associated t t t (cid:0) (cid:1) with the prices and productivities pj,ak ∈ Γ. The time subscript indicates the fact that the value 6Inderiving(24)from(12),initiallyatermoftheform C(Ω(cid:48))−γ V (P,A(cid:48),Ω(cid:48))appearsontheright-handside;using P(Ω(cid:48))C(Ω)−γ (cid:16) (cid:17) (23) this reduces to C(Ξ(cid:48))−γ v p ,A(cid:48),Ξ(cid:48) . Reducing the G term in the same way yields (24). C(Ξ)−γ Π(Ξ(cid:48),Ξ)

Distributional dynamics under smoothly state-dependent pricing 14 function shifts due to changes in the aggregate state Ξ . When necessary, the value is evaluated using t splines at points p ∈/ Γp off the price grid. In particular, the policy function p∗(A,Ξ ) ≡ argmaxv(p,A,Ξ ) (29) t t p∈R is selected from the reals (p ∈ R) instead of being chosen from the grid (p ∈ Γp), because the solution methodwillrequirepoliciestovarycontinuouslywiththeirarguments. Thepoliciesattheproductivity (cid:110) (cid:111) grid points ak ∈ Γa are written as a row vector p∗ ≡ p∗1...p∗#a ≡ (cid:8) p∗(a1,Ξ )...p∗(a#a,Ξ ) (cid:9) . Various t t t t t other equilibrium functions are also treated as #p × #a matrices. The adjustment values D , the t probabilities Λ , and the expected gains G have (j,k) elements given by7 t t djk ≡ maxv(p,ak,Ξ )−vjk, (30) t t t p∈R (cid:16) (cid:17) λjk ≡ λ djk/w , (31) t t t gjk ≡ λjkdjk. (32) t t t Given this discrete representation, the distributional dynamics can be written in a more explicit way. First, to keep productivity A on the grid Γa, it is assumed to follow a Markov chain defined by a matrix S of size #a ×#a. The row m, column k element of S represents the probability Smk = prob(A = am|A = ak). (33) it i,t−1 Also, beginning-of-t real prices must be adjusted for inflation. Ignoring grids, the time t−1 price p i,t−1 would be deflated to p ≡ p /Π at the beginning of t. Prices are forced to remain on the grid by (cid:101)it i,t−1 t a #p ×#p Markov matrix R in which the row m, column l element represents t Rml ≡ prob(p = pm|p = pl,Π = Π(Ξ ,Ξ )). (34) t (cid:101)it i,t−1 t t t−1 When the deflated price p /Π falls between two grid points, R rounds it up or down stochastically i,t−1 t t without changing its mean. Also, if p /Π drifts up or down past the largest or smallest grid points, i,t−1 t 7The max in (30), like the argmax in (29), ignores the grid Γp so that djk varies continuously in response to any t change in the value function.

Distributional dynamics under smoothly state-dependent pricing 15 then R rounds it down or up to keep prices on the grid. Thus the transition probabilities are t   1 if pl/Π ≤ p1 = pm  t         pl p / m Π − t− pm pm − − 1 1 if p1 < pm = min{p ∈ Γp : p ≥ pl/Π t } < p#p  Rml = pm+1−pl/Πt if p1 ≤ pm = max{p ∈ Γp : p < pl/Π } < p#p . (35) t pm+1−pm t      1 if pl/Π > p#p = pm  t      0 otherwise Combining the adjustments of prices and productivities, the beginning-of-t distribution Ψ(cid:101) can be t calculated from the lagged distribution Ψ as follows: t−1 Ψ(cid:101) = R ∗Ψ ∗S(cid:48), (36) t t t−1 wheretheoperator∗representsmatrixmultiplication. Twofactsexplainthesimplicityofthisequation. First, the exogenous shocks to A are independent of the inflation adjustment linking p with p . it (cid:101)it i,t−1 Second, productivity is arranged from left to right in the matrix Ψ , so productivity transitions t−1 are represented by right multiplication, while prices are arranged vertically, so price transitions are represented by left multiplication. Next, a firm with beginning-of-t state (p ,A ) = (pj,ak) ∈ Γ will adjust its price to p = p∗k with (cid:101)it it it t probability λjk, and otherwise leave it unchanged. If adjustment occurs, prices are kept on the grid t by rounding p∗k up or down stochastically to the nearest grid points, without changing the mean. To t be precise, let Γp be wide enough so that p1 < p∗k < p#p for all k ∈ {1,2,...#a}. For each k, define t l (k) so that plt(k) = min{p ∈ Γp : p ≥ p∗k}. Then the following #p×#a matrix governs the rounding: t t      plt p ( l k t ) ( − k) p − lt p ( ∗ t k k )−1 in column k, row l t (k)−1  P t ≡  p p lt ∗ t ( k k − )− p p lt l ( t k (k )− )− 1 1 in column k, row l t (k) . (37)     0 elsewhere NowletE andE bematricesofonesofsize#p×#p and#p×#a, respectively, and(asinMATLAB) pp pa let the operator .∗ represent element-by-element multiplication. Then the end-of-t distribution Ψ can t be calculated from Ψ(cid:101) as follows: t Ψ = (E −Λ ) .∗Ψ(cid:101) +P .∗(E ∗(Λ .∗Ψ(cid:101) )). (38) t pa t t t pp t t

Distributional dynamics under smoothly state-dependent pricing 16 The same transition matrices show up when the Bellman equation is written in matrix form. Let U be the #p ×#a matrix of current profits, with elements t (cid:16) w (cid:17) ujk = u(pj,ak,Ξ ) = pj − t C p−(cid:15) (39) t t ak t j (cid:0) (cid:1) for pj,ak ∈ Γ. Then the Bellman equation is simply (cid:26) C−γ (cid:27) V = U +βE t+1R(cid:48) ∗(V +G )∗S , (40) t t t C−γ t+1 t+1 t+1 t where G = Λ . ∗ D was defined by (32). Several comments may help clarify this Bellman t+1 t+1 t+1 equation. Note that the expectation E refers only to the effects of the time t + 1 aggregate shock t z , because multiplying by R(cid:48) and S fully describes the expectation over the idiosyncratic state t+1 t+1 (pj,ak) ∈ Γ. S has no time subscript, since the Markov productivity process is not subject to aggregate shocks, whereas the inflation adjustment represented by R(cid:48) varies with the policy shock. Also, while t+1 thedistributionaldynamicsiterateforwardintime, withtransitionsgovernedbyRandS(cid:48), theBellman equation iterates backwards, so its transitions are described by R(cid:48) and S. 3.3. Computation: steady state In an aggregate steady state, policy shocks z are zero, and transaction prices converge to an ergodic distributionΨ,sotheaggregatestateoftheeconomyisconstant: Ξ = (z ,R ,Ψ ) = (0,R,Ψ) ≡ Ξ t t t−1 t−1 undertheTaylorrule, orΞ = (z ,m ,Ψ ) = (0,m,Ψ) ≡ Ξunderamoneygrowthrule. Thesteady t t t−1 t−1 state of any aggregate equilibrium object is indicated by dropping the subscript t. The steady-state calculation nests the firm’s backwards induction problem inside a loop that determines the steady-state real wage w. Note first that Π = µ∗ = βR in steady state; hence the matrix R is known. Then, given w, (5) determines C, so all elements ujk of U can be calculated from (39). Then, backwards induction on the grid Γ can solve the Bellman equation V = U+βR(cid:48) ∗(V+G)∗S. (41) Solving (41) involves finding the matrices V, D, Λ, and G, so the matrix P can also be calculated from (37). Then (36) and (38) can be used to find the distributions Ψ(cid:101) and Ψ, and finally (4) serves

Distributional dynamics under smoothly state-dependent pricing 17 to check the guessed value of w. In discretized notation, equation (4) becomes #p #a 1 = (cid:88)(cid:88) Ψjk(cid:0) pj (cid:1)1−(cid:15) . (42) t j=1 k=1 If (42) holds at the ergodic distribution Ψ = Ψ, then a steady-state equilibrium has been found. t 3.4. Computation: dynamics Bellman equation (40) and distribution dynamics (36)-(38) are usually viewed as functional equations. However, under the discretization of Sec. 3.2, they can also be seen as two long lists of difference equations describing the values and probabilities at all grid points. Thus, Reiter (2009) proposes linearizing these equations around their steady state, calculated in Sec. 3.3. To do so, it first helps to reduce the number of variables by eliminating simple intratemporal relationships. Under a money growth rule, the model can be described by the following vector of endogenous variables: →− X ≡ (cid:0) vec(V )(cid:48), C , Π , vec(Ψ )(cid:48), m (cid:1)(cid:48) (43) t t t t t−1 t−1 →− Vector X , together with the shock process z , consists of 2#p#a + 4 variables determined by the t t following system of 2#p#a +4 equations: (40), (7), (42), (38), (28), and (16). Under a Taylor rule, m is replaced by R , and (7) and (28) are replaced by (6) and (18). Thus the difference equations t−1 t−1 governing dynamic equilibrium constitute a first-order system of the form (cid:16)→− →− (cid:17) E F X ,X ,z ,z = 0, (44) t t+1 t t+1 t whereE is an expectationconditional on z and all previous shocks.8 Next, systemF can be linearized t t numericallytoconstructtheJacobianmatricesA ≡ D−→ Xt+1 F, B ≡ D−→ Xt F, C ≡ D zt+1 F, andD ≡ D zt F. This yields the following first-order linear expectational difference equation system: →− →− E A∆X +B∆X +E Cz +Dz = 0, (45) t t+1 t t t+1 t where∆representsadeviationfromsteadystate. ThissystemhastheformconsideredbyKlein(2000), so it will be solved using his QZ decomposition method, though other linear rational expectations solvers would be applicable as well. →− →− 8Given (X ,X ,z ,z ), all other variables appearing in (40), (7), (42), (38), (28), and (16) can be substituted t+1 t t+1 t out using intratemporal equations. Given Π t and Π t+1 , R t and R t+1 are known; thus Ψ(cid:101)t = R t ∗ Ψ t−1 ∗S(cid:48) can be calculated too. The wage is given by (5), so U can be constructed. Finally, given V and V one can construct P , t t t+1 t D , and D , and thus Λ and G . Therefore the arguments of F suffice to evaluate the system (44). t t+1 t t+1

Distributional dynamics under smoothly state-dependent pricing 18 The virtue of Reiter’s method is that it combines linearity and nonlinearity in a way appropriate for thecontextofpriceadjustment, whereidiosyncraticshocksarelargerandmoreeconomicallyimportant to individual firms than aggregate shocks. To deal with large idiosyncratic shocks, it treats functions of idiosyncratic states nonlinearly (calculating them on a grid). But in linearizing each equation at each grid point, it recognizes that aggregate changes (policy shocks z, or shifts of the distribution Ψ) are unlikely to affect individual value functions in a strongly nonlinear way. On the other hand, it makes no assumption of approximate aggregation like that of Krusell and Smith (1998). 4. Results 4.1. Parameterization Our calibration seeks price adjustment and productivity processes consistent with microdata on pricechanges,likethoseinKlenowandKryvtsov(2008),NakamuraandSteinsson(2008),andMidrigan (2011). Since utility parameters are not the main focus, these are set to the values used by GL07. The discount factor is set to β = 1.04−1 per year; the coefficient of relative risk aversion of consumption is set at γ = 2. The coefficients on labor disutility and the utility of money are χ = 6 and ν = 1, respectively, and the elasticity of substitution in the consumption aggregator is (cid:15) = 7. The main price data that will serve as an empirical benchmark are the monthly AC Nielsen data reported by Midrigan (2011).9 Therefore, the model will be simulated at monthly frequency, with a zero steady state money growth rate, consistent with the zero average price change in that dataset. Midrigan reports the data after removing price changes attributable to temporary “sales”, so our simulation results should be interpreted as a model of “regular” price changes unrelated to sales. Conditional on these specification choices, the adjustment parameters (λ, α, and ξ) and productivity parameters (ρ and σ2) are chosen to minimize a distance criterion between the data and the model’s ε steady state.10 The criterion sums two terms, scaled for comparability: the first is the absolute 9However,wefitthemodeltoNakamuraandSteinsson’s(2008)measureofthemedianfrequencyofpriceadjustments. This is lower, but presumably more robust, than the mean adjustment frequency reported by Midrigan. 10The productivity process (8) is approximated on the grid Γa using Tauchen’s method; we thank Elmar Mertens for making his software available. The productivity grid has 25 points, and the price grid Γp has 31 points. Both grids are logarithmically spaced; steps in Γp represent 4% changes. Results are not sensitive to the use of this coarse grid, since the average absolute price adjustment is much larger (around 10%).

Distributional dynamics under smoothly state-dependent pricing 19 difference between the adjustment frequencies in the data and the simulation, while the second is the Euclidean distance between the frequency vectors associated with the histograms of nonzero price adjustments in the data and the simulation. Table 2 summarizes the steady-state behavior of the model under the estimated parameters, together with evidence from four empirical studies. The baseline specification, in which λ, α, and ξ are all estimated, is labelled SSDP. The table also reports Calvo (λ estimated, ξ ≡ 0, and α undefined) and FMC specifications (α estimated, ξ ≡ ∞, and λ undefined), as well as a version based on Woodford’s (2009) adjustment function and an SMC specification. All versions of the model match the target adjustment frequency of 10% per month. But the extreme cases of the model (Calvo and FMC) are less successful in fitting the size distribution of price adjustments than are the intermediate cases; the Calvo model understates the average size and standard deviation of price adjustments, whereas the FMC model overstates both. The trouble with the FMC model, as Fig. 1 shows, is that it only produces price changes lying just outside the (S,s) bands, whereas the adjustments observed in the data are very diverse.11 Thus the FMC model that best fits the data produces adjustments that are too large on average; no adjustments in the model are less than 5%, whereas one quarter of all adjustments are below the 5% threshold in the AC Nielsen data. The Calvo model errs in the opposite direction, with too many small price adjustments, though its fit statistics are better than those of the FMC model. In contrast, all three specifications with a smoothly increasing adjustment hazard (SSDP, SMC, and Woodford) match the data well, since they permit large and small price adjustments to coexist. In fact, there is so little difference between these models that only SSDP will be discussed from here on.12 Our estimates imply fairly strong frictions impeding price adjustment. The estimated function λ (see Fig. 1, right panel) rises quickly at zero but is thereafter very flat. It equals 10% per month at a loss of L = 0.0235 (6% of monthly labor input) and only reaches 30% per month at the highest loss that occurs with nonzero probability in the steady state equilibrium, which is L = 7.91, roughly 21 11Klenow and Kryvtsov (2008) document that large and small price changes coexist even within narrow product categories, and that the FMC model performs poorly even when menu costs are allowed to differ across sectors. 12Our companion paper, Costain and Nakov (2011), shows that the SSDP model performs somewhat better than Woodford’sspecificationathigh(e.g.70%annually)inflationrates. Butatlowinflationrates,theresponsestomonetary shocks (available from the authors) are indistinguishable across the SSDP, SMC, and Woodford specifications.

Distributional dynamics under smoothly state-dependent pricing 20 months’ worth of labor input. Of course, the λ function is flat in the Calvo model by construction. Our estimate of Woodford’s specification implies λ ≈ 1 at the most extreme losses that occur in equilibrium, but it is also very flat over the range of losses that occur frequently. For example, the cross-sectional standard deviation of λ is roughly 4% in SSDP and 3% in the Woodford setup, whereas it is 30% in the FMC model. Thus, in the models considered, firms do not adjust instantly even when faced with very large losses. Nonetheless, typical losses in equilibrium are more moderate, since firms usually adjust before reachingextremesituations. Acrossspecifications,thedecreaseinaverageprofitsduetopricestickiness (see Table 2) ranges from 1.5% of average revenues in the FMC case to 5.3% of average revenues in the Woodford specification. (The differences look larger when expressed as a fraction of average profits, since profits are a small fraction of revenues.) Clearly, these estimated adjustment frictions are nontrivial; whether they seem unrealistically large may depend on whether they are conceived literally as “menu costs” or as costs of managerial decision making, along the lines of Zbaracki et al. (2004). 4.2. Effects of monetary policy shocks Since all specifications are calibrated to the same observed adjustment frequency, the fact that only large, valuable price changes occur in the FMC model, whereas some changes in the SSDP and Calvo frameworks are trivial, has important implications for monetary transmission. Fig. 2 compares responses to several types of monetary shocks across these three adjustment specifications. All simulations assume the same utility parameters, and zero baseline inflation, and are calculated starting from the steady-state distribution associated with the corresponding specification. The first two rows show impulse responses to one percentage point money growth shocks, comparing the i.i.d. case with that of monthly autocorrelation φ = 0.8. The third row shows the responses to an i.i.d. interest rate z shock under a Taylor rule. Inallthreemodels, anincreaseinmoneygrowthstimulatesconsumption. Thefactthatsomeprices fail to adjust immediately means expected inflation rises, decreasing the ex ante real interest rate; it also means households’ real money balances increase; both of these effects raise consumption demand. However, as GL07 emphasized, the average price level adjusts rapidly in the FMC specification (lines

Distributional dynamics under smoothly state-dependent pricing 21 with circles), with a large, short-lived spike in inflation. This makes changes in real variables small and transitory, approaching the monetary neutrality associated with full price flexibility. At the opposite extreme, prices adjust gradually in the Calvo specification (lines with squares), leading to a large, persistent increase in output. The response of the SSDP model (lines with dots) mostly lies between the other two, but is generally closer to that of the Calvo model. ComparingthefirstandsecondrowsofFig.2showsthatwhiletheshapeoftheinflationandoutput responses differs substantially across models, it is qualitatively similar under iid and autocorrelated money growth processes. In the FMC model inflation spikes immediately regardless of the persistence of money growth. However, with autocorrelated money growth the initial spike exceeds 1% as firms anticipate that money growth will remain positive for some time. The rise in inflation is smaller but more persistent in the SSDP and Calvo cases. Note that the persistence of inflation does not differ noticeably depending on the autocorrelation of money growth, but instead appears to be determined primarily by the degree of state dependence. Thus the big difference between the impulse responses in the first and second rows is one of size, not of shape. The third row of Fig. 2 shows responses under a Taylor rule, assuming that the underlying shock z is i.i.d., and that the rule has inflation and output coefficients φ = 2 and φ = 0.5, and smoothing π c coefficient φ = 0.9. While money growth shocks are small, incremental changes to the level of the R nominal money supply, Taylor rule shocks involve large fluctuations in the level of nominal money. Nonetheless, the two types of monetary-policy shocks have similar real effects, and moreover, the finding that a micro-calibrated model of state-dependent pricing implies substantial monetary nonneutrality is strengthened in several ways by considering a Taylor rule. First, under the Taylor rule, the SSDP and Calvo impulse responses of inflation and consumption are even closer together than they were under the money growth rule. In fact, for consumption, both SSDP and FMC imply virtually the same effect on impact as that occurring in the Calvo model, though the effect is less persistent in the FMC case. Recall, though, that the Taylor rule responses in Fig. 2 suppose an initial drop in the nominal interest rate of 25 basis points. Since the interest rate is endogenous, the required underlying shock (cid:15)z varies across models, and it is particularly large in the FMC case. Therefore, it is useful to consider

Distributional dynamics under smoothly state-dependent pricing 22 additional ways of comparing the degree of monetary nonneutrality across models. Thus, Table 3 compares monetary policies that imply the same inflation variability, as in Sec. VI of GL07. The calculation asks the following question: if monetary-policy shocks were the only source of observed US inflation volatility, how much output variation would they cause? Under the SSDP specification, money growth shocks alone would explain 65% of observed output fluctuations; the figure rises to 116% under the Calvo specification, and falls to 15% in the FMC case.13 Assuming a Taylor rule, the differences across models are even stronger, and the monetary nonneutrality of the SSDP and Calvo specifications is even greater. Taylor rule shocks alone would explain 110% of US output fluctuation under the SSDP specification, rising to 306% in the Calvo case. The table also reports a “Phillips curve” coefficient, calculated by regressing log output on inflation, instrumented by the exogenous monetary policy shock. The conclusions are similar: the SSDP model implies large real effects of monetary shocks, closer to the Calvo specification than to the FMC specification, and the differences across models are larger under a Taylor rule than they are under a money growth rule. Next, Fig. 3 plots the response of price dispersion, ∆p, defined in (20). In our model, one reason t prices vary is that firms face different productivities. But additional price dispersion, caused by failure to adjust when necessary, implies inefficient variation in demand across goods that implies a decrease in aggregate productivity: C = N /∆p. In a representative agent model near a zero-inflation steady t t t state, ∆p is negligible because it is roughly proportional to the cross-sectional variance of prices, t a quantity of second order in the inflation rate.14 But cross-sectional price variance is not second order when large idiosyncratic shocks are present, so the dispersion wedge ∆p may be quantitatively t important, especially since (cid:15) = 7 magnifies variations in the ratio P /P . The first row of Fig. 3 it t shows that for SSDP and Calvo, increased money growth throws firms’ prices further out of line with fundamentals, increasing dispersion; raising consumption therefore requires a larger increase in labor in these specifications. In contrast, the variation in ∆p is smaller in the FMC case, because t all firms with severe price misalignments do in fact adjust. Interestingly, since the Taylor rule leans against inflationary shocks, there is much less variation in the price level for the SSDP and Calvo cases 13Thetableconsidersautocorrelatedmoneygrowthshocks. Theresultsfori.i.d.moneygrowthareverysimilar,since, as demonstrated in Figure 2, correlation mostly changes the scale of the impulse responses, rather than their shape. 14See for example Gal´ı (2008), p. 46 and Appendix 3.3.

Distributional dynamics under smoothly state-dependent pricing 23 in our Taylor rule simulation than there is under autocorrelated money growth. Thus, in all three specifications, a Taylor rule shock causes little variation in ∆p. t 4.3. Inflation decompositions Several decompositions can help illustrate the inflation dynamics implied by this model. To a firstorder approximation, inflation can be calculated as an average of log nominal price changes. Using our grid-based notation, and starting from the beginning-of-period distribution Ψ(cid:101) , t #p #a (cid:88)(cid:88) π = logΠ = xjkλjkΨ(cid:101) jk, (46) t t t t t j=1 k=1 (cid:16) (cid:17) where xjk ≡ log p∗(ak,Ξt) is the desired log price adjustment of a firm with price pj and productivity t pj ak. Klenow and Kryvtsov (2008) rewrite (46) as the product of the average log price adjustment x t times the frequency of price adjustment λ : t π = x λ , x ≡ (cid:80) j,k xj t kλj t kΨ(cid:101) j t k , λ ≡ (cid:88) λjkΨ(cid:101) jk. (47) t t t t (cid:80) λjkΨ(cid:101) jk t t t j,k t t j,k Dropping higher-order terms, this implies the following inflation decomposition: ∆π = λ∆x +x∆λ , (48) t t t where variables without time subscripts represent steady states, and ∆ represents a deviation from steady state.15 Klenow and Kryvtsov’s “intensive margin”, IKK ≡ λ∆x , is the part of inflation t t attributable to changes in the average price adjustment; their “extensive margin”, EKK ≡ x∆λ , is t t the part due to changes in the frequency of adjustment. Unfortunately, thisdecompositiondoesnotrevealwhetherariseintheaveragelogpriceadjustment x is caused by a rise in all firms’ desired adjustments, or by a reallocation of adjustment opportunities t from firms desiring small or negative price changes to others wanting large price increases. That is, IKK mixes changes in desired adjustments (the only relevant changes in the Calvo model) with the t “selection effect” emphasized by GL07. To distinguish between these last two effects, inflation can instead be broken into three terms: an intensive margin capturing changes in the average desired log 15Actually,KlenowandKryvtsov(2008)proposeatimeseriesvariancedecomposition,whereas(47)isadecomposition of each period’s inflation realization. But the logic of (47) is the same as that in their paper.

Distributional dynamics under smoothly state-dependent pricing 24 price change, an extensive margin capturing changes in how many firms adjust, and a selection effect capturing changes in who adjusts. These three effects are distinguished by rewriting (46) as (cid:88) (cid:16) (cid:17) (cid:88) π = x∗λ + xjk λjk −λ Ψ(cid:101) jk, x∗ ≡ xjkΨ(cid:101) jk. (49) t t t t t t t t t t j,k j,k Note that in (49), x∗ is the average desired log price change, and not an average of the actual t log price changes of those firms that do adjust (as was the case in 47). Thus, (49) says that inflation equals the mean desired adjustment times the adjustment frequency plus a selection term (cid:16) (cid:17) (cid:16) (cid:17) (cid:80) xjk λjk −λ Ψ(cid:101) jk = (cid:80) λjk xjk −x∗ Ψ(cid:101) jk that can be nonzero if some changes xjk are more j,k t t t t j,k t t t t t or less likely than the mean adjustment probability λ , or (equivalently) if firms with different probat bilities of adjustment λjk tend to prefer adjustments that differ from the mean desired change x∗. t t Equation (49) leads to the following inflation decomposition: (cid:88) (cid:16) (cid:17) ∆π = λ∆x∗ +x∗∆λ +∆ xjk λjk −λ Ψ(cid:101) jk. (50) t t t t t t t j,k Our intensive margin effect, I ≡ λ∆x∗, is the effect of changing all firms’ desired adjustment by the t t same amount (or more generally, changing the mean preferred adjustment in a way that is uncorrelated with the adjustment probability). I is the only nonzero term in the Calvo model, where λjk = λ t t for all j, k, t. Our extensive margin effect, E ≡ x∗∆λ , is the effect of changing the fraction of t t firms that adjust, assuming the new adjusters are selected randomly. Our selection effect, S ≡ t (cid:16) (cid:17) ∆ (cid:80) xjk λjk −λ Ψ(cid:101) jk, is the effect of redistributing adjustment opportunities across firms with j,k t t t t different desired changes xjk, while fixing the overall fraction that adjust. t An alternative decomposition, proposed by Caballero and Engel (2007), also differences (46): (cid:88) (cid:88) (cid:88) ∆π = ∆xjkλjkΨ(cid:101) jk + xjk∆λjkΨ(cid:101) jk + xjkλjk∆Ψ(cid:101) jk (51) t t t t j,k j,k j,k They further simplify this to (cid:88) ∆π = λ∆µ + xjk∆λjkΨ(cid:101) jk (52) t t t j,k under the assumption that all desired price adjustments change by ∆xjk = ∆µ when money growth t t increases by ∆µ , and by taking an ergodic average so that the last term drops out.16 Their first t 16Our equation (50) is intended to decompose each period’s inflation realization, so it allows for shifts in the current distribution Ψ(cid:101) jk. Caballero and Engel instead propose a decomposition (see their eq. 17) of the average impact of a t

Distributional dynamics under smoothly state-dependent pricing 25 term, ICE ≡ ∆µ λ, is the same as our intensive margin I , if their assumption that all desired t t t price adjustments change by ∆µ is correct. But therefore, their “extensive margin” term ECE ≡ t t (cid:80) xjk∆λjkΨ(cid:101)jk, confounds the question of how many firms adjust (our extensive margin E ) with the j,k t t question of who adjusts (our selection effect S ), which is the mechanism stressed by GL07. t The importance of identifying the selection effect separately becomes clear in Fig. 4, which illustrates our decomposition of the inflation impulse response to monetary shocks. The three components of inflation, I , E , and S , are shown to the same scale for better comparison. The graphs demonstrate t t t (in contrast to Caballero and Engel’s claim) that the short, sharp rise in inflation observed in the FMC specification results from the selection effect. This is true both under Taylor rule shocks, where inflation spikes to 1.5% on impact, of which 1.25% is the selection component, and under (autocorrelated) money growth shocks, where inflation spikes to 2.8%, with 2.25% due to selection. In contrast, inflation in the Calvo model is caused by the intensive margin only; in SSDP there is a nontrivial selection effect but it still only accounts for around one-third of the inflation response. Ontheotherhand, theextensivemarginE ≡ x∗∆λ playsanegligibleroleintheinflationresponse. t t This makes sense, because the simulation assumes a steady state with zero inflation, so steady state price adjustments are responses to idiosyncratic shocks only, and the average desired adjustment x∗ is very close to zero. Therefore E is negligible even though the adjustment frequency λ itself does t t vary.17 The extensive margin only becomes important when there is high trend inflation, so that the average desired adjustment x∗ is large and positive. As for the intensive margin, its initial effect after a money growth shock is similar across all adjustment specifications, but it is more persistent in the Calvo and SSDP cases than in the FMC case. The scale of the intensive margin depends on the autocorrelation of money growth: the mean desired price change rises roughly one-for-one after an i.i.d. money growth shock (not shown), and rises by roughly five percentage points when money growth has autocorrelation φ = 0.8 (first row z of Fig. 4). Thus, in the autocorrelated case, the intensive margin is initially I ≡ λ∆x∗ ≈ 0.5%. 1 1 monetary shock. Therefore they evaluate their decomposition at the ergodic distribution (the time average over all cross-sectional distributions, called f (x) in their paper). Since this is a fixed starting point of their calculation, they A do not need to include a ∆f (x) term. A 17Thefactthatthesteadystatehasexactlyzeroinflationisnotcrucialhere;E isquantitativelyinsignificantcompared t to the other inflation components at any typical OECD inflation rate.

Distributional dynamics under smoothly state-dependent pricing 26 In other words, firms wish to “frontload” price adjustment by approximately the same amount in all three specifications; but many of these changes occur immediately in the FMC case (showing up as a redistribution of adjustment opportunities, i.e., a selection effect), whereas they are realized gradually in the other specifications. Under a Taylor rule, the intuition is similar, bearing in mind that Fig. 4 is scaled to give an initial decline of 25 basis points in the nominal interest rate. This requires a larger underlying shock z in the FMC specification than in the other cases; thus the effect on the intensive margin is larger (but less persistent) for FMC than it is for Calvo and SSDP. 4.4. Comparing effects of sector-specific and aggregate shocks Another issue of intererest in recent empirical literature is how prices respond to sector-specific shocks. In particular, Boivin, Giannoni, and Mihov (2007) and Mackowiak, Moench, and Wiederholt (2009, henceforth MMW09) present evidence that sector-specific prices respond much more quickly to sector-specific shocks than they do to aggregate shocks. This is important, since it suggests that a Calvo model with a single adjustment rate may be inappropriate. Indeed, it might be interpreted as evidence for state dependence, and it suggests that the present model might be tested by assessing its ability to reproduce these empirical observations. To address these questions, this section investigates “sector-specific” shocks in our model, applying theestimationroutinesofMMW09toartificialpaneldataproducedbysimulatingtheSSDPcalibration under a Taylor rule. The data cover the price levels in 79 sectors over 245 months, as in the dataset of MMW09. Of course, the model defined here does not actually have a sectoral structure. Nonetheless, for a fixed integer #s > 0, one can simulate a panel of 79#s firms (each producing one specialized product), and call each block of #s consecutive firms a “sector”. Productivity innovations remain i.i.d. across firms, as they are elsewhere in the paper. However, since the number of firms per sector is finite, sampling error will cause average productivity to differ across sectors at each time. An innovation to average productivity in any artificially-defined sector can thus be regarded as a sector-specific shock. Two questions are then relevant. First, can empirical findings like those of MMW09 be reproduced by applying their methods to the model-generated data? Second, do their estimation methods correctly identify the effects of sector-specific shocks? The answers are yes and no, respectively.

Distributional dynamics under smoothly state-dependent pricing 27 The MMW09 statistical framework breaks inflation into aggregate and sector-specific components: π = µ +A (L)u +B (L)v (53) n,t n n t n n,t where π is the inflation rate in sector n at time t, µ is a sector-specific constant, A (L) and n,t n n B (L) are sector-specific lag polynomials, u is an aggregate shock, and v is a sector-specific shock. n t n,t Inflation π and the shocks u and v are all scaled to have unit variance. Fig. 5 shows estimated n,t t n,t impulse responses of sector-specific price levels to v (left column) and u (right column), identified n,t t by applying the Bayesian estimation programs of MMW09 to sectoral panel simulations from the SSDP model (one- and two-standard-error bands are shown too). All the impulse responses in the left column are consistent with the main finding of MMW09 (see Fig. 1 of their paper): the identified sector-specific shocks cause an immediate, permanent rise in prices, with little change thereafter. In other words, the inflation associated with sector-specific shocks is essentially white noise. In contrast, the right column shows that the reaction to aggregate shocks is more gradual. MMW09 also find that the impact of an sectoral shock on sectoral inflation is almost one-for-one, and that sectoral shocks account for around 90% of sectoral inflation variance. Fig. 5 shows that if simulated sectors are small (8 or 64 equally-weighted products), then the impact of a sectoral shock is indeed almost one-for-one; moreover, in these cases sectoral shocks explain 80% to 90% of sectoral inflation (see Fig. 6). However, if sectors consist of 512 equally-weighted products, then a one-standard deviation sectoral shock creates only 0.7 standard deviations of inflation, and sectoral shocks explain less than half of sectoral inflation variance. This is a consequence of the law of large numbers: with more firms per sector, sector-specific inflation stays closer to its conditional expectation, so aggregate shocks must explain a larger part of sectoral inflation (see the right-hand column of Fig. 5). Typical observationsofsectoralinflationinMMW09’sCPIdatainvolveseveralhundredindividualpricequotes, so sectors with 8 or 64 products are unrealistically small.18 But on the other hand, CPI weights vary greatly across products in each sector (Leaver and Folk, 2004). Therefore the fourth row of the figure reports a simulation with 512 products per sector, in which the CPI weights on the products are 18TheBLScollectsapproximately80000priceseachmonthtocalculatetheCPI;roughly70%ofthesepricescorrespond to the 79 sectors included in the MMW09 estimates. So the typical observation of sectoral inflation averages several hundred individual prices. We thank an anonymous referee for providing these details.

Distributional dynamics under smoothly state-dependent pricing 28 distributed according to Zipf’s law.19 This increases the importance of the highest-weighted goods, so 512 products with heterogeneous weights act like a much smaller number of equally-weighted products, with a contribution of sectoral shocks to sectoral inflation variance exceeding 85% (see Fig. 6.) Thus, running the estimation programs of MMW09 on simulated data from our model with 512 firms of heterogeneous size largely reproduces their empirical findings.20 However, this is rather puzzling, becausetheestimationresultsareinconsistent withtheknown propertiesofthesimulatedmodel. In the model, prices only adjust once in ten months on average, and the degree of state-dependence is low, so the true response of prices to any idiosyncratic or aggregate shock must be fairly slow. Moreover, all sector-specific behavior in the model is mean-reverting, whereas the estimates in Fig. 5 show a permanent effect of a sector-specific shock on prices. To demonstrate these facts numerically, we consider the sector-specific shock ¯(cid:15)a , defined as the weighted average of firm-specific productivity n,t shocks (cid:15)a across firms in sector n. We assume inflation can be written as a moving average of these i,t sectoral shocks and the aggregate monetary shock (cid:15)z: t π = µ +A (L)(cid:15)z +B (L)¯(cid:15)a +(cid:15)π (54) n,t n n t n n,t n,t The notation is the same here as in (53). However, sector-specific inflation will not generally equal its predicted value conditional on the underlying shocks, so this specification must allow for a sectorspecific inflation residual (cid:15)π . In contrast, (53) attributes any inflation unexplained by the aggregate n,t shock to the sector-specific shock, by construction. Fig. 7 reports the responses to the sector-specific shock ¯(cid:15)a (left column) and the aggregate shock n,t (cid:15)z, using the same simulated datasets analyzed in Fig. 5 (for accuracy, the length of each dataset is t extended to 5400 months.) Responses are estimated by OLS on a sector-by-sector basis; all sectoral estimates are shown in the same graph. For all four datasets, responses to sector-specific and aggregate shocks occur with a lag. The peak response to a sector-specific shock occurs five to ten months after the time of the shock; the sectoral price level thereafter reverts to steady state. The time of reaction to an aggregate shock is similar, but the effects are permanent. The only effect of increasing the number 19That is, the weight of the jth-largest firm in sector n in that sector’s CPI is proportional to j−1. 20Fig.2ofMMW09alsoreports“speedofresponse”statisticsshowingthatsector-specificinflationreactsmorequickly sector-specific shocks. The same result obtains in our simulations; the graphs are available from the authors.

Distributional dynamics under smoothly state-dependent pricing 29 of firms per sector is that aggregate shocks become more important for sectoral inflation, relative to sectoral shocks, for the reasons discussed previously. Why does the MMW09 estimation routine find effects of sectoral shocks so different from the response to the true sectoral shock, shown in Fig. 7? The problem is that true shocks in microdata are unknown to an econometrician, so Mackowiak et al. must identify sectoral shocks as residual price increases not explained by aggregate shocks. In the SSDP model, individual prices typically respond with a lag to true productivity shocks, so sectoral price levels do too. But in the MMW09 decomposition,themomentoftheshockcorrespondsby assumption tothemomentofthepriceincrease, so the response is estimated to be immediate. Mean reversion occurs by individual stochastic price jumps in the model, whereas MMW09 assumes past shocks decay deterministically (component Bv in their eq. 1). Hence their method interprets price movements back to the mean as a sequence of new sectoralshocksthathappentogointheoppositedirection(whichiswhytheinitialshockisinterpreted as permanent). Thus, results from their procedure (or others that identify sectoral shocks as inflation residuals, e.g. Boivin et al., 2007) should be treated with caution. Our Monte Carlo exercise shows that, at least in some cases, the procedure may exaggerate the speed of response to sectoral shocks, suggesting stronger state dependence than the data actually warrant. 5. Conclusions This paper has computed the impact of monetary policy shocks in a quantitative macroeconomic model of state-dependent pricing. It has calibrated the model for consistency with microeconomic data on firms’ pricing behavior, estimating how the probability of price adjustment depends on the value of adjustment. Given the estimated adjustment function, the paper has characterized the dynamics of the distribution of prices and productivities in general equilibrium. The calibrated model implies that prices rise gradually after a monetary stimulus, causing a large, persistent rise in consumption and labor. Looking across specifications, the main factor determining how monetary shocks propagate through the economy is the degree of state dependence. That is, raising the autocorrelation of money growth shocks just makes their effects proportionally larger, withoutanynotablechangeintheshapeorpersistenceoftheimpulseresponses. Incontrast,decreasing

Distributional dynamics under smoothly state-dependent pricing 30 state dependence from the extreme of fixed menu costs (FMC) to the opposite Calvo (1983) extreme strongly damps the initial inflation spike caused by a money shock and increases its effect on real variables. The parameterization most consistent with microdata (labelled “SSDP” throughout the paper) is fairly close to the Calvo model in terms of its quantitative effects. The conclusions are similar if the monetary authority follows a Taylor rule instead of a money growth rule, except that the degree of monetary nonneutrality differs more across adjustment specifications; in particular, the nonneutrality of the SSDP specification is increased. Thispaperalsodecomposestheimpulseresponseofinflationintoanintensivemargineffectrelating totheaveragedesiredpricechange,anextensivemargineffectrelatingtothenumberoffirmsadjusting, and a selection effect relating to the relative frequencies of small and large or negative and positive adjustments. Under the preferred (SSDP) calibration, about two-thirds of the effect of a monetary shock comes through the intensive margin, and most of the rest through the selection effect. The extensive margin is negligible unless the economy starts from a high baseline inflation rate. Under the FMC specification, a monetary shock instead causes a quick increase in inflation, driven by the selection effect, which eliminates most of its effects on real variables. Since the selection effect represents changes in the adjustment probability across firms, its strength depends directly on the degree of state dependence. We say that state dependence is strong in the FMC model because it makes λ a step function: at the threshold, a tiny increase in the value of adjustment raises the adjustment probability from 0 to 1. Therefore the histogram of price changes consists of two spikes: there are no small changes, and firms change their prices as soon as they pass the adjustment thresholds. Hence, in steady state, those firms that might react to monetary policy are all near the two thresholds; a monetary stimulus decreases λ from 1 to 0 for some firms desiring a price decrease, while increasing λ from 0 to 1 for others preferring an increase, making the inflation response quick and intense. That is, the same property that makes money nearly neutral in the FMC model is the one which makes that model inconsistent with price microdata. A model in which adjustment depends more smoothly on the value of adjusting fits microdata better and yields larger real effects of monetary policy. Our two other smooth specifications (SMC, and Woodford’s hazard function) yield results similar to the SSDP setup that was our main focus.

Distributional dynamics under smoothly state-dependent pricing 31 Low state dependence might seem inconsistent, at first, with recent empirical claims that prices react more quickly to sectoral shocks than to aggregate shocks. Indeed, our calibrated model implies thatpricesreactonlygraduallytosectoralshocks. However,thispaperdemonstratesthattheempirical methods of Mackowiak et al. (2009) may attribute an immediate, permanent impact to sectoral shocks even in a dataset where the true sectoral shocks have a lagged, temporary effect. The problem is that by treating any sector-specific change in inflation as a sectoral shock, they may confound sampling error in the timing of price adjustments with fundamental shocks. Applying the MMW09 estimation routines to simulated data from our model suggests that this problem may suffice to explain their empirical results, calling into question the evidence for price flexibility at the sectoral level. References Akerlof, G., and J. Yellen. 1985. “A Near-Rational Model of the Business Cycle with Wage and Price Inertia,” Quarterly Journal of Economics, vol. 100 (Supplement), pp. 823–38. Boivin, J., M. Giannoni, and I. Mihov. 2009. “Sticky Prices and Monetary Policy: Evidence from Disaggregated US Data,” American Economic Review, vol. 99, pp. 350–84. Caballero, R., and E. Engel. 1993. “Microeconomic Rigidities and Aggregate Price Dynamics,” European Economic Review, vol. 37, pp. 697–711. Caballero, R., and E. Engel. 1999. “Explaining Investment Dynamics in U.S. Manufacturing: a Generalized (S,s) Approach,” Econometrica, vol. 67, pp. 741–82. Caballero, R., andE.Engel.2007.“PriceStickinessinSsModels: NewInterpretationsofOldResults,” Journal of Monetary Economics, vol. 54, pp. 100–21. Calvo, G. 1983. “Staggered Prices in a Utility-Maximizing Framework,” Journal of Monetary Economics, vol. 12, pp. 383–98. Caplin, A., D. Spulber. 1987. “Menu Costs and the Neutrality of Money,” Quarterly Journal of Economics, vol. 102, pp. 703–26.

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Distributional dynamics under smoothly state-dependent pricing 33 Klenow, P., and O. Kryvtsov. 2008. “State-Dependent or Time-Dependent Pricing: Does it Matter for Recent US Inflation?,” Quarterly Journal of Economics, vol. 123, pp. 863–94. Krusell, P., and A. Smith. 1998. “Income and Wealth Heterogeneity in the Macroeconomy,” Journal of Political Economy, vol. 106, pp. 245–72. Leaver, S., and D. Solk. 2004. “Estimating Sampling Weights for the U.S. Consumer Price Index,” Proceedings of the Survey Research Methods Section, American Statistical Association, 3866–73. Mackowiak, B., E. Moench, and M. Wiederholt. 2009. “Sectoral Price Data and Models of Price Setting,” Journal of Monetary Economics, vol. 56, pp. S78–S99. Midrigan, V. 2011. “Menu Costs, Multi-Product Firms and Aggregate Fluctuations,” Econometrica, vol. 79, pp. 1139–80. Nakamura,E.,andJ.Steinsson.2008.“FiveFactsAboutPrices: aReevaluationofMenuCostModels,” Quarterly Journal of Economics, vol. 123, pp. 1415–64. Reiter, M. 2009. “Solving Heterogeneous-Agent Models by Projection and Perturbation,” Journal of Economic Dynamics and Control, vol. 33, pp. 649-55. Woodford, M. 2009. “Information-Constrained State-Dependent Pricing,” Journal of Monetary Economics, vol. 56, pp. S100–S124. Yun,T.2005.“OptimalMonetaryPolicywithRelativePriceDistortions,”American Economic Review, vol. 95, pp. 89–109. Zbaracki, M., M. Ritson, D. Levy, S. Dutta, and M. Bergen, 2004. “Managerial and Customer Costs of Price Adjustment: Direct Evidence from Industrial Markets,” Review of Economics and Statistics, vol. 86, pp. 514-33.

Distributional dynamics under smoothly state-dependent pricing 34 6. Appendix: notation Table N1: Exogenous parameters Symbol Definition Where Preferences of household β Utility discount factor Sec. 2.1. γ Coefficient of relative risk aversion Sec. 2.1. χ Disutility of labor Sec. 2.1. ν Coefficient on utility of money Sec. 2.1. (cid:15) Elasticity of substitution across differentiated goods Sec. 2.1. Technology of firms ρ Persistence of firm-specific productivity Sec. 2.2. σ2 Variance of firm-specific productivity shock Sec. 2.2. a ¯ λ Adjustment probability parameter Sec. 2.2.1. α Adjustment probability parameter Sec. 2.2.1. ξ Adjustment probability parameter Sec. 2.2.1. Monetary policy φ Persistence of monetary policy process Sec. 2.3. z σ2 Variance of monetary policy shock Sec. 2.3. z µ∗ Money growth target Sec. 2.3. Π∗ Inflation target in Taylor rule Sec. 2.3. C∗ Output target in Taylor rule Sec. 2.3. R∗ Interest rate target in Taylor rule Sec. 2.3. φ Interest smoothing parameter in Taylor rule Sec. 2.3. R φ Inflation weighting parameter in Taylor rule Sec. 2.3. π φ Output weighting parameter in Taylor rule Sec. 2.3. c

Distributional dynamics under smoothly state-dependent pricing 35 Table N2: Endogenous variables, nominal representation Symbol Definition Where Aggregate state Ω Nominal aggregate state Sec. 2.1. and 2.3. t Note: In the nominal representation, any aggregate variable indexed by t is determined, in equilibrium, as a function of the time t state Ω . Sometimes this functional relationship t will be written explicitly, e.g. W = W(Ω ). t t Variables appearing in household’s problem C Real household consumption Sec. 2.1. t N Labor supply Sec. 2.1. t M Nominal money supply Sec. 2.1. t P Nominal price level Sec. 2.1. t W Nominal wage Sec. 2.1. t R Nominal interest factor from t to t+1 Sec. 2.1. t B Nominal bonds held at t to pay off in t+1 Sec. 2.1. t T Nominal lump sum transfer to household at time t Sec. 2.1. t C Consumption of good produced by firm i Sec. 2.1. it P Price of good produced by firm i Sec. 2.1. it Other aggregate variables Φ(cid:101) Distribution of productivities and nominal prices at beginning of t Sec. 2.2. t Φ Distribution of productivities and nominal prices at end of t Sec. 2.2. t z Stochastic process driving monetary policy Sec. 2.3. t (cid:15)z Monetary policy shock Sec. 2.3. t µ Monetary growth factor from t−1 to t Sec. 2.3. t ∆p Price dispersion statistic Sec. 2.3. t

Distributional dynamics under smoothly state-dependent pricing 36 Table N3: Endogenous variables, nominal representation – continued Symbol Definition Where Variables specific to firm i Y Real output of firm i at time t Sec. 2.2. it A Productivity of firm i at time t Sec. 2.2. it (cid:15)a Productivity shock to firm i at time t Sec. 2.2. it N Labor input to firm i at time t Sec. 2.2. it U Nominal profits of firm i at time t Sec. 2.2. it P(cid:101) Nominal price of output of firm i at beginning of t Sec. 2.2. it Note: prior to selling (at end of period t), price P(cid:101) may or may not be adjusted. it The nominal price at which firm i sells in period t is called P it (hence P is the price that appears in the household’s problem.) it Functions describing firm behavior in equilibrium U(P,A,Ω) Nominal profits of firm with productivity A that sells at price P in state Ω Sec. 2.2. V(P,A,Ω) Nominal value of firm with productivity A that sells at price P in state Ω Sec. 2.2. V∗(A,Ω) Optimal value of firm with productivity A in state Ω Sec. 2.2. P∗(A,Ω) Optimal nominal price of firm with productivity A in state Ω Sec. 2.2. D(P(cid:101),A,Ω) Nominal gain from adjusting, given beginning-of-period nominal price P(cid:101) Sec. 2.2. L(P(cid:101),A,Ω) Real gain from adjusting, given beginning-of-period nominal price P(cid:101) Sec. 2.2. λ(L) Probability of price adjustment, given real gain L from adjusting Sec. 2.2. G(P(cid:101),A,Ω) Expected nominal gains from stochastic adjustment in current period Sec. 2.2.

Distributional dynamics under smoothly state-dependent pricing 37 Table N4: Endogenous variables, real representation Symbol Definition Where Aggregate state Ξ Real aggregate state Sec. 3.1. t Note: In the real representation, any aggregate variable indexed by t is determined, in equilibrium, as a function of the time t state Ξ . t Aggregate variables m Real money supply Sec. 3.1. t w Real wage Sec. 3.1. t Π Inflation factor from t−1 to t Sec. 3.1. t Ψ(cid:101) Distribution of productivities and real prices at beginning of t Sec. 3.1. t Ψ Distribution of productivities and real prices at end of t Sec. 3.1. t Note: C , N , R , z have the same meaning in the real and nominal representations. t t t t Some variables defined in the nominal representation are not mentioned in the real representation. Variables specific to firm i p Real price of firm i at beginning of period t Sec. 3.1. (cid:101)it p Real price of firm i at end of period t Sec. 3.1. it Functions describing firm behavior in equilibrium u(P,A,Ξ) Real profits of firm with productivity A that sells at real price p in state Ξ Sec. 3.1. v(P,A,Ξ) Real value of firm with productivity A that sells at real price p in state Ξ Sec. 3.1. p∗(A,Ξ) Optimal real price of firm with productivity A in state Ξ Sec. 3.1. d(p,A,Ξ) Real gain from adjusting, given beginning-of-period real price p Sec. 3.1. (cid:101) (cid:101) g(p,A,Ξ) Expected real gains from stochastic adjustment in current period Sec. 3.1. (cid:101) Note: Function λ has the same meaning in the real and nominal representations.

Distributional dynamics under smoothly state-dependent pricing 38 Table N5: Discretized real representation Symbol Definition Where Discretization Note: In the discretized real representation, superscripts indicate notation related to grids, and bold face indicates matrices and vectors. R The real numbers Sec. 3.2. Γa Finite grid of possible values of productivity Sec. 3.2. ak Element k of grid Γa Sec. 3.2. #a Number of elements of grid Γa Sec. 3.2. Γp Finite grid of possible values of real price Sec. 3.2. pj Element j of grid Γp Sec. 3.2. #p Number of elements of grid Γp Sec. 3.2. Γ Two dimensional grid of prices and productivities, Γ = Γp ×Γa Sec. 3.2. Endogenous variables Note: Firm-specific variables A , p , and p have the same meanings as in previous representations. it (cid:101)it it Aggregate variables Ξ , C , N , Π , R , m , and z have the same meanings as in previous t t t t t t t representations. Steady states of aggregate variables are indicated by dropping time subscripts. Matrix notation describing discretized problem of firm U Profits matrix, with elements ujk ≡ u(pj,ak,Ξ ) Sec. 3.2. t t t V Value matrix, with elements vjk ≡ v(pj,ak,Ξ ) Sec. 3.2. t t t p∗ Policy vector, with elements p∗k ≡ arg max v(p,ak,Ξ ) Sec. 3.2. t t p∈R t D Adjustment gains matrix, with elements djk ≡ max v(p,ak,Ξ )−vjk Sec. 3.2. t t p∈R t t Λ Adjustment probabilities matrix, with elements λjk ≡ λ(djk/w ) Sec. 3.2. t t t t G Matrix of expected gains from adjustment, with elements gjk ≡ λjkdjk Sec. 3.2. t t t t Note: Function λ has the same meaning it had in previous representations. Steady states of these matrices are indicated by dropping time subscripts.

Distributional dynamics under smoothly state-dependent pricing 39 Table N6: Discretized real representation, continued Symbol Definition Where Matrix notation describing distributional dynamics Ψ(cid:101) Beginning-of-period distribution matrix, with elements Ψ(cid:101) jk ≡ Ψ(cid:101) (pj,ak) Sec. 3.2. t t t Ψ End-of-period distribution matrix, with elements Ψjk ≡ Ψ (pj,ak) Sec. 3.2. t t t E , E Matrices of ones, of sizes #p ×#p and #p ×#a, respectively Sec. 3.2. pp pa S Markov productivity matrix, with elements Smk ≡ prob(A = am|A = ak) Sec. 3.2. it i,t−1 R Markov matrix for inflation adjustment and stochastic rounding to grid Γp, t with elements Rml ≡ prob(p = pm|p = pl), t (cid:101)it i,t−1 conditional on inflation Π = Π(Ξ ,Ξ ) Sec. 3.2. t t t−1 l (k) Index of least grid element above preferred real price: t plt(k) ≡ min{p ∈ Γp : p ≥ p∗k} Sec. 3.2. t P Matrix allocating newly adjusted prices to optimum value p∗k, t t with mean-preserving stochastic rounding to grid Γp Sec. 3.2. Note: Steady states of these objects are indicated by dropping time subscripts. Linearization of dynamics →− X Vector of variables in dynamic computation Sec. 3.4. t F Equation system linearized for dynamic computation Sec. 3.4. A, B, C, D Jacobian matrices appearing in linearized equation system Sec. 3.4. Note: Deviation between time t value and steady state is denoted by ∆.

Distributional dynamics under smoothly state-dependent pricing 40 Table N7: Inflation decomposition and sectoral shocks Symbol Definition Where Inflation decomposition π Inflation rate: π = logΠ Sec. 4.3. t t t λ Fraction of firms adjusting prices at time t Sec. 4.3. t xjk Desired log price adjustment, given real price pj and productivity ak Sec. 4.3. t x∗ Average desired log price adjustment at time t (across all firms) Sec. 4.3. t x Average log price adjustment at time t (across firms that adjust) Sec. 4.3. t I , E , S Intensive, extensive, and selection margins of inflation deviation Sec. 4.3. t t t IKK, EKK Intensive and extensive margins (Klenow and Kryvstov definition) Sec. 4.3. t t ICE, ECE Intensive and extensive margins (Caballero and Engel definition) Sec. 4.3. t t Note: Steady states of these objects are indicated by dropping time subscripts. Deviation between time t value and steady state is denoted by ∆. Sector-specific shocks π Inflation rate in sector n Sec. 4.4. n,t µ Sector-specific mean inflation Sec. 4.4. n A (L) Sector-specific lag polynomial on aggregate shocks Sec. 4.4. n B (L) Sector-specific lag polynomial on sectoral shocks Sec. 4.4. n u Aggregate inflation shock identified by MMW09 methodology Sec. 4.4. t v Aggregate inflation shock identified by MMW09 methodology Sec. 4.4. n,t (cid:15)z True shock to Taylor rule Sec. 2.3. t A Shock to average productivity in sector n Sec. 4.4. n,t (cid:15) Unexplained residual inflation in sector n Sec. 4.4. n,t

Tables for “Distributional Dynamics with Smoothly State-Dependent Pricing” James Costaina, Anton Nakovb a Banco de Espan˜a; b Federal Reserve Board Table 1: Adjustment specifications Specification Adjustment probability λ(L) Mean gains, in units of time: G(P,A,Ω)/W(Ω) ¯ ¯ Calvo λ λL(P,A,Ω) Fixed MC 1{L ≥ α} λ(L(P,A,Ω))[L(P,A,Ω)−α] ¯ ¯ (cid:0) ¯(cid:1) Woodford λ/[λ+ 1−λ exp(ξ(α−L))] λ(L(P,A,Ω))L(P,A,Ω) Stoch. MC λ ¯ /[λ ¯ + (cid:0) 1−λ ¯(cid:1) (α/L)ξ] λ(L(P,A,Ω))[L(P,A,Ω)−E(κ|κ < λ(L(P,A,Ω)))] SSDP λ ¯ /[λ ¯ + (cid:0) 1−λ ¯(cid:1) (α/L)ξ] λ(L(P,A,Ω))L(P,A,Ω) Note: λ(L) is the probability of price adjustment; L is the real loss from failure to adjust, as a function of firm’s price P and productivity A, and aggregate conditions Ω. G represents mean nominal gains from adjustment; dividing by the nominal wage W converts gains to real terms. λ¯, α and ξ are parameters to be estimated.

Tables for “Distributional Dynamics with Smoothly State-Dependent Pricing” 2 Table 2. Steady-state simulated moments for alternative estimated models and evidence Model Productivity parameters Adjustment parameters See eq. (8) for definitions See Table 1 for definitions ¯ Calvo (σ ,ρ) = (0.0850,0.8540) λ = 0.10 ε Fixed MC (σ ,ρ) = (0.0771,0.8280) α = 0.0665 ε (cid:0)¯ (cid:1) Woodford (σ ,ρ) = (0.0924,0.8575) λ,α,ξ = (0.0945,0.0611,1.3335) ε (cid:0)¯ (cid:1) Stochastic MC (σ ,ρ) = (0.0676,0.9003) λ,α,ξ = (0.1100,0.0373,0.2351) ε (cid:0)¯ (cid:1) SSDP (σ ,ρ) = (0.0677,0.9002) λ,α,ξ = (0.1101,0.0372,0.2346) ε Moments Calvo FMC Wdfd SMC SSDP MAC MD NS KK Frequency of price changes 10.0 10.0 10.0 10.0 10.0 20.5 19.2 10 13.9 Mean absolute price change 6.4 17.9 10.3 10.0 10.1 10.5 7.7 11.3 Std of price changes 8.2 18.4 13.6 12.2 12.2 13.2 10.4 Kurtosis of price changes 3.5 1.3 4.0 2.9 2.9 3.5 5.4 % price changes ≤5% in abs value 47.9 0.0 37.0 26.3 26.3 25 47 44 Mean loss in % of frictionless profit 36.8 10.6 37.4 25.6 25.6 Mean loss in % of frictionless revenue 5.2 1.5 5.3 3.6 3.6 Fit: Kolmogorov-Smirnov statistic 0.111 0.356 0.038 0.024 0.025 Fit: Euclidean distance 0.159 0.409 0.072 0.060 0.056 Note: Price statistics refer to non-sale consumer price changes and are stated in percent. The last four columns report statisticsfromMidrigan(2011)forACNielsen(MAC)andDominick’s(MD),NakamuraandSteinsson(2008)(NS),and Klenow and Kryvtsov (2008) (KK). To calibrate the productivity parameters ρ and σ2, together with the adjustment ε ¯ parametersλ,αandξ,weminimizeadistancecriterionwithtwoterms,(1)thedifferencebetweenthemedianfrequency of price changes in the model (fr) and in the data, and (2) the distance between the histogram of log price changes in the model (histM) and the data (histD): min(25(cid:107)fr−0.10(cid:107)+(cid:107)histM −histD(cid:107)).

Tables for “Distributional Dynamics with Smoothly State-Dependent Pricing” 3 Table 3. Variance decomposition and Phillips curves of alternative models Data SSDP model Calvo model FMC model Std of quarterly inflation (×100) 0.246 0.246 0.246 0.246 % explained by nominal shock 100 100 100 Money growth rule (see eq. 16-17) Std of money growth shock (×100) 0.174 0.224 0.111 Std of detrended output (×100) 0.909 0.586 1.053 0.121 % explained by money growth shock 64.5 115.9 13.3 Slope coeff. of the Phillips curve 0.598 1.069 0.134 Standard error 0.004 0.039 0.005 Taylor rule (see eq. 18) Std of Taylor rule shock (×100) 0.393 0.918 0.129 Std of detrended output (×100) 0.909 0.995 2.741 0.134 % explained by Taylor rule shock 109.6 301.6 14.7 Slope coeff. of the Phillips curve 1.055 2.785 0.126 Standard error 0.093 0.290 0.006 Note: for each monetary regime (Taylor or money growth rule) and each pricing model, the nominal shock is scaled to account for 100% of the standard deviation of inflation. The volatility of output in the data is measured as the standard deviation of HP-filtered quarterly log real GDP. The “slope coefficients” are the estimates of β in a 2SLS 2 regression of (log) consumption on inflation, instrumented by the exogenous nominal shock. The first stage regression is πq= α +α µq+(cid:15) , and the second stage is cq=β +β (4πˆq)+ε , where πˆq is the prediction for inflation from the t 1 2 t t t 1 2 t t t first-stage and the superscript q denotes conversion to quarterly frequency.

Figures for “Distributional Dynamics with Smoothly State-Dependent Pricing” James Costaina, Anton Nakovb a Banco de Espan˜a; b Federal Reserve Board

Figures for “Distributional Dynamics with Smoothly State-Dependent Pricing” 2 Price changes: models vs data 0.25 0.2 0.15 0.1 0.05 0 −0.5 0 0.5 ytisneD α=0.0311, λ=0.1089 1 AC Nielsen FMC Calvo 0.8 SSDP 0.6 0.4 0.2 0 0 0.02 0.04 0.06 0.08 0.1 Size of log price changes Loss from inaction tnemtsujda fo ytilibaborP ξ=50 ξ=0.05 ξ=1 ξ=0.23 (SSDP) Fig. 1. Price change distributions and adjustment function Note: size distribution of changes in log prices: data vs. models (left panel). Adjustment function λ for alternative values of state dependence ξ (right panel).

Figures for “Distributional Dynamics with Smoothly State-Dependent Pricing” 3 Money growth 1 0.5 0 0 5 10 15 20 elur htworg yenoM Inflation Consumption 0.6 0.6 0.4 0.4 0.2 0.2 0 0 0 5 10 15 20 0 5 10 15 20 Money growth 1 0.5 0 0 5 10 15 20 elur htworg yenoM Inflation Consumption 3 3 2 2 1 1 0 0 0 5 10 15 20 0 5 10 15 20 Nominal interest rate 0.25 0 −0.25 0 5 10 15 20 Months elur rolyaT SSDP Calvo Fixed MC Inflation Consumption 0.8 1.5 0.6 1 0.4 0.5 0.2 0 0 0 5 10 15 20 0 5 10 15 20 Months Months Fig. 2. The real effects of nominal shocks across models Note: responses of inflation and consumption to an iid money growth shock (top row); responses to a correlated money growthshock(middlerow); responsestoaTaylorruleshock(bottomrow). Inflationresponsesareinpercentagepoints; consumptionresponsesareinpercentdeviationfromsteady-state. Lineswithdots-benchmarkSSDPmodel; lineswith squares - Calvo; lines with circles - fixed menu costs.

Figures for “Distributional Dynamics with Smoothly State-Dependent Pricing” 4 Price dispersion 3 2.5 2 1.5 1 0.5 0 0 10 20 elur htworg yenoM Labor Consumption 3 3 2.5 2.5 2 2 1.5 1.5 1 1 0.5 0.5 0 0 0 10 20 0 10 20 Price dispersion 0.8 0.6 0.4 0.2 0 0 10 20 elur rolyaT SSDP Calvo Fixed MC Labor Consumption 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0 0 10 20 0 10 20 Months Months Months Fig. 3. Price dispersion across models Note: responses to a correlated money growth shock (top row); responses to a Taylor rule shock (bottom row). The responses are in percent deviation from steady-state. Lines with dots - benchmark SSDP model; lines with squares - Calvo; lines with circles - fixed menu costs.

Figures for “Distributional Dynamics with Smoothly State-Dependent Pricing” 5 Intensive margin 2.5 2 1.5 1 0.5 0 0 10 20 elur htworg yenoM Extensive margin Selection effect 2.5 2.5 2 2 1.5 1.5 1 1 0.5 0.5 0 0 0 10 20 0 10 20 Intensive margin 1.5 1 0.5 0 0 10 20 elur rolyaT SSDP Calvo Fixed MC Extensive margin Selection effect 1.5 1.5 1 1 0.5 0.5 0 0 0 10 20 0 10 20 Months Months Months Fig. 4. Inflation decomposition across models Note: decompositionoftheinflationresponseintoanintensivemargin,extensivemargin,andselectioneffect(seeeq.54). Top row: responses to a correlated money growth shock. Bottom row: responses to a Taylor rule shock. The responses are in percentage points and sum up to the total inflation response shown in Fig. 2. Lines with dots - benchmark SSDP model; lines with squares - Calvo; lines with circles - fixed menu costs.

Figures for “Distributional Dynamics with Smoothly State-Dependent Pricing” 6 1 0.5 0 5 10 15 20 25 smrif 8 Response to sectoral shock Response to aggregate shock 1 0.5 0 5 10 15 20 25 1 0.5 0 5 10 15 20 25 smrif 46 1 0.5 0 5 10 15 20 25 1 0.5 0 5 10 15 20 25 smrif 215 1.5 1 0.5 0 5 10 15 20 25 1 0.5 0 5 10 15 20 25 smrif fpiz 215 1 0.5 0 5 10 15 20 25 Months after shock Months after shock Fig. 5. Sectoral price responses to shocks identified from model-generated data Note: responses of sector-specific prices to “sector-specific” shocks (left column) and to “aggregate” shocks (right column), estimated from SSDP model-generated data. One- and two-standard-error bands shown. Simulated economy consistsof79sectorswith8,64,or512firmswithequally-weightedproducts(topthreerows),or512firmswithproduct weightssatisfyingZipf’slaw(fourthrow). SimulatedeconomyissubjecttoaggregateTaylorruleshocksandfirm-specific productivity shocks that are uncorrelated across firms; sectors are defined as fixed sets of unrelated firms. Shocks are identified by applying the procedure of Mackowiak, Moench, and Wiederholt (2009).

Figures for “Distributional Dynamics with Smoothly State-Dependent Pricing” 7 Estimated importance of sectoral shocks 95 90 85 80 75 70 65 60 55 50 45 100 200 300 400 500 Number of firms per sector )%( ecnairav noitalfni larotces ot noitubirtnoC Equal weights Zipf weights Fig. 6. Inflation variance contribution of sector-specific shocks identified from model-generated data Note: share of variance of sector-specific prices explained by “sector-specific” shocks, as a function of number of firms persector,estimatedfromSSDPmodel-generateddata. Simulatedeconomyconsistsof79sectorswithequally-weighted products (line with stars), or with product weights satisfying Zipf’s law (line with circles). Simulated economy is subject to aggregate Taylor rule shocks and firm-specific productivity shocks that are uncorrelated across firms; sectors aredefinedasfixedsetsofunrelatedfirms. ShocksareidentifiedbyapplyingtheprocedureofMackowiak, Moench, and Wiederholt (2009).

Figures for “Distributional Dynamics with Smoothly State-Dependent Pricing” 8 Responses to sectoral productivity shock 0.5 0 0 20 40 60 smrif 8 Responses to aggregate interest rate shock 1 0.5 0 0 20 40 60 0.5 0 0 20 40 60 smrif 46 1 0.5 0 0 20 40 60 0.5 0 0 20 40 60 smrif 215 2 1 0 0 20 40 60 0.5 0 0 20 40 60 Months after shock smrif fpiz 215 1 0.5 0 0 20 40 60 Months after shock Fig. 7. Sectoral price responses to true shocks in model-generated data Note: responsesofsector-specificpricesto“sector-specific”shocks(leftcolumn)andtoTaylorruleshocks(rightcolumn), estimated from SSDP model-generated data. Simulated economy consists of 79 sectors with 8, 64, or 512 firms with equally-weighted products (top three rows), or 512 firms with product weights satisfying Zipf’s law (fourth row). Simulated economy is subject to aggregate Taylor rule shocks and firm-specific shocks that are uncorrelated across firms; sectors are defined as fixed sets of unrelated firms. “Sector-specific” shock is the change in sector-specific weighted average productivity.