Heterogeneity in Price Stickiness and the Real Effects of Monetary Shocks

Transcription

1 Heterogeneity in Price Stickiness and the Real Effects of Monetary Shocks Carlos Carvalho Department of Economics Princeton University December 26 Abstract There is ample evidence that the frequency of price adjustments differs substantially across sectors. This paper introduces sectoral heterogeneity in price stickiness into an otherwise standard sticky price model to study how it affects the dynamics of monetary economies. Qualitative and quantitative results from a realistic calibration for the U.S. economy show that monetary shocks tend to have larger and more persistent real effects in heterogeneous economies, when compared to identical-firms economies with similar degrees of nominal and real rigidity. In the presence of strategic complementarities in price setting, sectors with lower frequencies of price adjustment have a disproportionate effect on the aggregate price level. In order to better approximate the dynamics of the calibrated heterogeneous economy, an identical-firms model requires a frequency of price changes that is up to three times lower than the average of the heterogeneous economy. I would like to thank Kevin Amonlirdviman, Roland Bénabou, Alan Blinder, Marco Bonomo, Vasco Cúrdia, Per Krusell, Jonathan Parker, Ricardo Reis, Felipe Schwartzman, Christopher Sims, Lars Svensson, Michael Woodford, and participants from the NBER Summer Institute 26 in Monetary Economics, Econometric Society NASM 26, ESWC 25, LAMES 24, EEA Meeting 24, Macroeconomics seminar at Princeton University, and EPRU seminar at the University of Copenhagen for comments. Earlier versions of this paper circulated under the titles Heterogeneity in Price Stickiness and the New Keynesian Phillips Curve, and Heterogeneity in Price Setting and the Real Effects of Monetary Shocks. It has been greatly improved by suggestions from John Leahy, two anonymous referees, and especially David Romer. Any remaining errors are my own. Financial support from Princeton University is gratefully acknowledged. Address for correspondence: Department of Economics, Princeton University, Fisher Hall, Princeton, NJ , USA. cvianac@princeton.edu.

2 Introduction There is ample evidence that the frequency of price adjustments differs substantially across sectors (Blinder et al., 998, and Bils and Klenow, 24, for the U.S. economy; Dhyne et al., 26, and references cited therein for the Euro area). However, most sticky price models do not account for heterogeneity in price setting behavior. Apart from analytical convenience, the only reason not to take heterogeneity explicitly into account would be if it did not matter for aggregate dynamics in any significant way. In this paper I show that this is not the case by introducing sectoral heterogeneity in the frequency of price changes into an otherwise standard sticky price model. I analyze the effects of heterogeneity through a set of analytical results that are applicable to arbitrary cross-sectional distributions of the frequency of price changes, and quantitative results based on a realistic calibration of such distribution for the U.S. economy. To obtain the latter, I use the statistics on price setting behavior in the U.S. economy reported recently by Bils and Klenow (24) (henceforth BK). To isolate the effects of heterogeneity on the dynamic properties of the model, I contrast the response of heterogeneous economies to monetary shocks with that of identical-firms economies under different calibrations. My main finding is that, for realistic calibrations of the model, heterogeneity in price stickiness leads monetary shocks to have larger and more persistent real effectsthaninidentical-firms economies with similar degrees of nominal and real rigidities. The differences are quantitatively important, to the extent that accounting for them with an identical-firms model requires lowering the frequency of price changes by a factor of up to three (relative to the actual average frequency of price changes in the heterogeneous economy). Heterogeneity in the frequency of price changes naturally leads to differences across sectors in the speed of adjustment to a shock. In turn, the resulting changes in the cross-sectional distribution of sectoral relative prices during the adjustment process have non-trivial aggregate effects. After a heterogeneous economy is hit by a shock, the initial phase of the adjustment process is driven mainly by sectors in which prices adjust relatively frequently, since the majority of price changes are undertaken by firms in these sectors. As time passes, the distribution of the frequency of price changes among firms which have yet to make the bulk of their adjustment becomes progressively dominated by firms in sectors with relatively low adjustment frequencies. As a result, the speed of adjustment in the heterogeneous economy slows down through time. I call this the frequency composition effect: high frequency sectors dominate the earlier part of the adjustment process, whereas low frequency sectors drive most of the dynamics subsequently. 2

3 In the presence of strategic complementarities in price setting, pricing decisions of firms in sectors with more frequent price changes are influenced by the existence of slower-adjusting sectors, since the former do not want to set prices that will deviate too much from the aggregate price in the future. On the other hand, firms in sectors in which prices change less often are also influenced by the pricing decisions of firms in the relatively more flexible sectors, but to a lesser extent. As a result, the former have a disproportionate effect on the aggregate price level. The mechanism at work is in many respects similar to the interaction between responders and non-responders in Haltiwanger and Waldman (99), or between firms with staggered price adjustments in the presence of Taylor s (98) contract multiplier. I refer to it as the strategic interaction effect due to heterogeneity in price stickiness. As a result of these mechanisms - the frequency composition and the strategic interaction effects - the dynamic response of a heterogeneous economy to a nominal disturbance can differ markedly from the response of an otherwise identical economy in which all firms change prices with the same frequency. In particular, those mechanisms endow the heterogeneous economy with the ability to display more persistent dynamics in response to monetary shocks. To explore this feature of heterogeneous economies, I contrast their response to shocks with that of their identical-firms (or one-sector) counterparts. By identical-firms counterparts I mean economies that are otherwise identical to the heterogeneous economy, except that all firms change prices with the same frequency. In making the comparisons, I focus on two benchmark identical-firms economies: one with a frequency of price changes equal to the average frequency of the heterogeneous economy, and another with a frequency of price changes such that the average durationofprice spellsequalsthat of the heterogeneous economy. I find that monetary shocks indeed tend to have larger and longer-lived real effects in heterogeneous economies, when compared to their identical-firms counterparts. Moreover, the differences are quantitatively important. This result has implications for the mapping between the microeconomic evidence on pricesetting behavior and the associated parameters in commonly used one-sector models. Calibrations of identical-firms models based on the average or the median frequency of price adjustments, or even the average duration of price rigidity, can understate the real effects of monetary shocks relative to the underlying heterogeneous economy in a quantitatively important way. Given the prominence of one-sector models in the literature and the ample evidence on heterogeneity in price stickiness, an important practical question is how to calibrate an identicalfirms model in order to best mimic the dynamics of a heterogeneous economy. A related issue refers to estimates of the frequency of price changes obtained with identical-firms models: how should we interpret them in light of the microeconomic evidence? 3

4 Motivated by those questions, I tackle the problem of finding the single frequency of price changes in an identical-firms model that best approximates the dynamic response of the calibrated heterogeneous economy to empirically plausible monetary shocks. I find that the best-fitting identical-firms economy features more nominal rigidity than what is implied by the average or the median frequency of price changes, or the average duration of price rigidity, of the heterogeneous economy. In order to better approximate the dynamics of the calibrated heterogeneous economy, an identical-firms model requires a frequency of price changesthatisuptothreetimeslowerthantheaverageoftheheterogeneous economy. The strategic interaction effect manifests itself in this exercise, in that the extent of additional nominal rigidity required to approximate the dynamics of the heterogeneous economy is increasing in the degree of strategic complementarities in price setting. In general, differences across sectors in the speed of adjustment to a shock lead the dynamics of output and inflation to depend on the whole cross-sectional distribution of sectoral output gaps (or relative prices). This is so because deflationary (inflationary) pressures are unevenly distributed across sectors after a contractionary (expansionary) monetary shock. In this paper, for simplicity I model heterogeneity using the price setting specification proposed by Calvo (983): in every period, each firm changes its price with a constant, sector-specific probability. As a result, heterogeneity in thefrequencyofpricechangesgivesrisetoageneralized new Keynesian Phillips curve that accounts explicitly for heterogeneity in prices stickiness. It differs from the standard new Keynesian Phillips curve (NKPC) in a fundamental way, in that heterogeneity produces a new, endogenous shift term that can be written as a weighted average of sectoral output gaps. Moreover, the coefficient on the aggregate output gap in the Phillips curve also depends on the sectoral distribution of price stickiness. The standard NKPC obtains as a special case when the frequency of price changes is the same across all sectors. From the analysis of the generalized NKPC, it is also clear that heterogeneity in price stickiness introduces dynamic features in the economy that cannot be captured by the standard NKPC. Moreover, the generalized NKPC sheds light on why identical-firms models need to be endowed with relatively more nominal rigidity in order to generate real effects of monetary shocks that can stand up to those obtained in the calibrated heterogeneous economy. I start in Section 2 by analyzing a multi-sector version of a familiar reducedform new Keynesian model. I use it to understand the basic features of the economy, and for that purpose I study its response to shocks to an exogenous nominal aggregate demand process. I describe the calibration of the cross-sectional distribution of price stickiness based on the BK data, and use the calibrated model to illustrate the frequency composition and the strategic interaction effects. Ex- 4

5 ploring the tractability of the reduced-form model, I also obtain analytical results that allow comparison of the real effects of nominal shocks in an arbitrary heterogeneous economy with those in their identical-firms counterparts, in the absence of strategic complementarities in price setting. I finish the section addressing the problem of how to calibrate an identical-firms model to approximate the dynamic response of the calibrated heterogeneous economy to several types of shocks to nominal aggregate demand. In Section 3 I present a fully specified multi-sector general equilibrium model with heterogeneity in the frequency of price adjustments. It is a standard new Keynesian model without capital accumulation to which I add heterogeneity in price stickiness across different sectors. Monetary policy is conducted under an interest rate rule, and is subject to shocks. The latter affect the economy through the intertemporal choices made by optimizing, forward looking consumers. Segmented labor markets introduce real rigidities in the economy (Ball and Romer, 99), which in turn can generate strategic complementarities in price setting. I present the generalized NKPC, and study the dynamic response of a calibrated economy to interest rate shocks. The results on heterogeneity in sectoral price setting behavior, presented in this paper in the context of the Calvo (983) model, extend to a large class of alternative price setting specifications. As shown in Carvalho (25) and Carvalho and Schwartzman (26), the latter includes Taylor (979, 98) staggered pricing, and sticky information models as in Mankiw and Reis (22). This suggests that heterogeneity in price setting behavior and its interaction with real rigidities may have an important role to play in models of monetary economies, irrespective of the nature of frictions to price adjustment. In the conclusion (Section 4), I discuss some of the implications of my findings for related research, as well as how to think about the role of heterogeneity in price setting behavior in the context of models in which the frequency of pricing decisions is chosen by firms. Many papers address issues that are related to the subject of this paper. Recently, some authors have allowed for heterogeneity in price stickiness in the context of time-dependent models (e.g. Ohanian et al., 995; Bils and Klenow, 22, 24; Bils et al., 23). In earlier work, Taylor (993) extended his original model (979, 98) to account for wage contracts of different durations. In a different framework, with state- rather than time-dependent pricing rules, Caballero and Engel (99, 993) also allow for heterogeneity in the frequency of price changes. However, these papers do not focus on isolating the role of heterogeneity in aggregate dynamics. This requires comparing models with heterogeneous firms with otherwise equivalent models in which all firms are identical. This kind of analysis is undertaken by Aoki (2) and Benigno (2, 24), who explore the effects of heterogeneity in price stickiness on optimal monetary policy in two-sector models. Dixon and Kara (25) study what I refer to as the strategic interaction effect in a 5

6 model with Taylor staggered wage setting. In work that is closely related to mine, Carlstrom et al. (26b) use a two-sector model with different degrees of nominal rigidity to study how sectoral relative prices affect aggregate dynamics. They find relative price effects that are qualitatively similar to the ones obtained by Aoki (2) in a two-sector economy featuring one sticky- and one flexible-price sector, and by Benigno (2, 24) in a two-country model with different degrees of nominal price stickiness. Barsky et al. (26) study a two-sector model with durable consumption goods and heterogeneity in the frequency of price changes, in which the degree of price stickiness in the durable goods sector turns out to be disproportionately important for aggregate dynamics. 2 A baseline reduced-form model 2. Assumptions In the economy there is a continuum of imperfectly competitive firms divided into sectors that differ in the frequency of price adjustments. Firms are indexed by their sector, k [, ], andbyj [, ]. The distribution of firmsacrosssectors is summarized by a density function f on [, ]. All firms set prices as in Calvo (983): in every period of length, each firm changes its price with a constant probability. The probabilities are sector specific, and denoted λ k. The occurrences of price changes are independent across all firms in the economy, and as a result in each period a fraction λ k of firms in sector k change their prices. In the absence of frictions to price adjustment, the optimal level of an individual firm s relative price, which is the same for all firms, is given by: p t p t = θy t, () where p is the individual frictionless optimal price, p is the aggregate price level and y is the output gap. 2 All lowercase variables should be interpreted as logdeviations from a deterministic, zero inflation steady state. In (), θ, whichis always positive, determines the degree of strategic complementarities in price setting. Prices are strategic complements (substitutes) if θ < (> ). In a fully specified model, strategic complementarities can arise as a result of large In addition, they allow for interest rate rules in which the monetary authority can respond to different sectoral inflation rates with different intensities. Carlstrom et al. (26a) use that model to study equilibrium determinacy. 2 This equation can be derived from first principles as in Blanchard and Kiyotaki (987) or Ball and Romer (989). 6

7 real rigidities (Ball and Romer, 99), such as firm-specific capital and/or labor inputs, or production chains, for example. 3 Theaggregatepricelevelisgivenby: p t = Z f (k) Z p k,j t djdk, (2) where p k,j t is the price charged by firm j from sector k at time t. Whenever a firm from sector k has a chance to change its price, it sets x k t according to: x k t =argmin x X β s ( λ k ) s E t x p 2 t+s (3) s= =( ( λ k ) β) X (( λ k ) β) s E t p t+s, s= where β is the per-period discount factor, and E t is the expectation conditional on time-t information. This optimization problem can be justified through a secondorder approximation to the profit loss that the firm incurs from not charging the frictionless optimal price p. Given this price setting behavior, the aggregate price level can be written as: p t = Z where the sectoral price indices, p k t,aregivenby: p k t = λ k X s= f (k) p k t dk, (4) ( λ k ) s x k t s. (5) To focus on the supply side of the model, I assume that nominal aggregate demand, m t = y t + p t, follows an exogenous stochastic process. For simplicity, I specify: A (L) m t = u t, where u t isazeromean,finite variance i.i.d. process assumed to be in the time-t information set, and A (L) is a polynomial in the lag operator L (Lm t = m t ). In what follows, I will focus on two specifications for this process: an AR() in levels, and an AR() in growth rates. 3 For a detailed exposition of different sources of strategic complementarities in price setting see Woodford (23, ch.3). 7

8 2.2 Calibrating the sectoral distribution of adjustment frequencies In general, the dynamics of the heterogeneous economy depend on the whole distribution of frequencies of price adjustment. In the next subsection I provide some analytical results that only rely on a few moments of such distribution, but other results still depend on its entirety. To address this issue I use the statistics on price setting behavior in the U.S. economy reported by Bils and Klenow (24). More specifically, I identify each sector in the model with one of the goods and services categories listed in their appendix, and set λ k equal to the monthly frequency of price changes reported for the category identified with sector k. Asaresulttheunitoftime equals one month. I set the sectoral weights equal to the CPI weights for these categories, renormalized to add up to one. This results in 35 sectors. To make it easy to refer to particular sectoral variables, I order the sectors so that sector (sector 35) displays the highest (lowest) frequency of price changes. To convert a per-period probability of price change λ into an expected duration of price rigidity d Iusetheformulad =. This is based on the ln( λ) relationship between the per-period probability of a price change λ, and the underlying rate of arrival of price changes in continuous time α: λ = e α.as a result, the expected duration of price rigidity d can be less than one period if the rate of arrival of price changes in continuous time is high enough. Based on this approach, I compute the sample statistics presented in Table. 4 Choosing an empirical distribution has the obvious advantage of making the calibration somewhat realistic. However, it is worth highlighting a few conceptual issues involved. First, the Bils and Klenow (24) data on which the calibration is based are still aggregated to some extent and therefore should understate the degree of heterogeneity that actually exists at a more disaggregated level. Moreover, it does not cover all sectors of the U.S. economy, and differs from the data used in other studies in some important dimensions. For example, it features less nominal rigidity than what had been documented in earlier work (e.g. Carlton, 986; Blinder et al., 998). More importantly, the purpose of this paper is to study the role of heterogeneity in price stickiness in shaping the dynamic response of economies to nominal shocks. Given those remarks, the quantitative results from the calibrated heterogeneous economy should be analyzed relative to their counterparts in identical-firms models calibrated with moments of the same 4 The sectoral rates of arrival of price changes are calculated as α k = ln ( λ k ) /. The sample statistics based on the assumption that the discrete time model holds strictly are: an inverse average frequency of price changes of 3.8 months, an inverse median frequency of price changes of 4.8 months, and an average duration of price rigidity of 7. months. The standard deviation of durations of price rigidity is unchanged at 7. months. 8

9 distribution of price stickiness. Table : Moments of the Cross-Sectional Distribution of the Frequency of Price Changes Description Formula Months Inverse average frequency duration of price ridigity, α = 35 P α k= f (k) α k 2.9 Median frequency based duration of price ridigity ln( λ med 4.3 ) Average duration of price d = rigidity Standard deviation of durations of price rigidity 35 P k= µ 35 P k= f (k) d k,d k = ln( λ k 6.6 ) f (k) d k d 2 /2 7. Obs: Based on the statistics reported by Bils and Klenow (24). λ med denotes the median frequency of price changes in their data. This is actually the inverse of the average rate of price change arrivals. Nevertheless, I will refer to it as the inverse average frequency, for short. Technically, d k is the expected duration of price spells in sector k. So, this is actually the cross-sectional average of the expected durations of price spells. A caveat as in applies. This is the cross-sectional standard deviation of the expected sectoral durations of price rigidity. A caveat as in applies. 2.3 Real effects without strategic complementarities I start by analyzing the effects of heterogeneity in price stickiness in the absence of strategic complementarities, and therefore set θ =. To illustrate the main features of the model, I first solve for the response of the economy to shocks to nominal aggregate demand assuming that it evolves according to an AR() in levels with autoregressive coefficient φ. I refer to those as level shocks: A (L) m t = u t, with A (L) = φ L. Figure presents the impulse response function (IRF) of the output gap to a permanent negative level shock (φ =). 5 For comparison purposes it also displays the output gap in identical-firms economies calibrated with the moments 5 Thesizeoftheshockonlyaffects the scale of the responses. The IRF for the price level is 9

10 reported in Table. Henceforth, I will refer to the economies calibrated with the inverse average frequency duration, the median frequency based duration, and the average duration of price rigidity as, respectively, the average-frequency, median-frequency, and average-duration economies. The qualitative features illustrated with this example are common to the other types of shocks. Figure : Permanent Level Shock - Output Gaps Output Gaps Heterogeneous economy Average-duration economy Median-frequency economy Average-frequency economy months The adjustment process in the average- and median-frequency economies is clearly too fast relative to the heterogeneous economy. The average-duration economy displays a more comparable (albeit still different) adjustment process. Aqualitativedifference between the identical-firms economies and the heterogeneous economy is that in the former the IRFs are characterized by a constant rate of decay, whereas in the latter they are not. In heterogeneous economies, adjustment is faster initially, because the majority of price changes are undertaken by firms in sectors with a relatively high frequency of price changes. As time passes, the distribution of the frequency of price changes among firms which have yet to make the bulk of their adjustment becomes progressively dominated by firms in sectors with relatively lower adjustment frequencies. As a result, the speed of adjustment in the heterogeneous economy slows down through time. I refertothisasthefrequency composition effect: high frequency sectors dominate the earlier part of the adjustment process, whereas low frequency sectors drive most of the dynamics subsequently. 6 the mirror image across the horizontal axis. Throughout the paper I assume a discount rate of 3% per year, except when stated otherwise. 6 The frequency composition effect is related to the effects that arise when aggregating

11 Figure 2 presents analogous results for the relatively more realistic specification in which m t follows an AR() in growth rates with autoregressive coefficient φ 2. Irefertothiscaseasgrowth rate shocks: A (L) m t = u t, with A (L) = ( + φ 2 ) L + φ 2 L 2. I set φ 2 =.89, so that shocks have a half-life of 6 months. The same pattern emerges in the dynamic response of the heterogeneous economy relative to the identical-firms economies, as a result of the frequency composition effect. 2.4 Some analytical results Given the differences in dynamics between heterogeneous economies and their one-sector counterparts, a natural question is whether we can make any general statements about the differences in the real effects of monetary shocks in these economies. The dynamics of a heterogeneous economy clearly depend on the whole distribution of price stickiness, and so it is hard to make statements about the exact shape of impulse response functions for an arbitrary distribution. Figure 2: Growth Rate Shock - Output Gaps 2 Output Gaps Heterogeneous economy Average-duration economy Median-frequency economy Average-frequency economy months heterogeneous hazard functions. In fact, in the case of a permanent level shock to nominal aggregate demand and no strategic complementarities, the two are essentially identical. This is no longer the case for more general monetary shocks, or when there are strategic complementarities in price setting. For an interesting application of results on aggregation of heterogeneous hazard functions to price-setting models see Alvarez et al. (25).

12 In the absence of strategic complementarities in price setting, however, it is possible to obtain general results about a sensible measure of the overall effects of a monetary shock, which takes into account both the intensity and the persistence of its real effects: the expected (normalized) cumulative effect on the output gap. 7 It turns out to be a useful indicator of the extent to which heterogeneous economies display more persistent dynamics then identical-firms economies, as will become clear in subsequent results. For analytical convenience I derive these results in the context of the underlying continuous time model, by letting. The derivation of the latter, as wellastheproofsoftheresults,areintheappendix.tointroducethenotation used below, I relate the parameters of the continuous time model with their perperiod counterparts in Table 2. With this notation, the expected (normalized) cumulative effect on the output gap of a time-zero monetary shock of size u is given by u E R y (t) dt. Table 2: Relating Continuous and Discrete Time Parameters and Variables Parameters Continuous time Discrete time ( -length periods) Arrival rate of price changes in sector k α k λ k = e α k Discount rate δ β = e δ Decay rate of level shocks Decay rate of growth rate shocks ρ γ φ = e ρ φ 2 = e γ Variables m (t),y(t),... m t,y t, Level shocks The first set of results refers to level shocks. Proposition (Level shocks in the continuous time model) When θ =,the expected (normalized) cumulative real effect of a level shock to nominal aggregate demand is equal to: Z f (k) α k + ρ + δ dk. 7 This measure is also discussed, for example, in Christiano et al. (25). 2

13 Corollary For an arbitrary heterogeneous economy, the expected (normalized) cumulative real effect of a level shock to nominal aggregate demand is always greater than in the corresponding average-frequency economy. Corollary 2 For an arbitrary heterogeneous economy, the expected (normalized) cumulative real effect of a level shock to nominal aggregate demand is never greater than in the corresponding average-duration economy. The effect is maximal in the limiting case of permanent shocks and no discounting (ρ =,δ =),in which case it equals the real effect in the corresponding average-duration economy: d = Z f (k) α k dk. The above results imply that if level shocks are temporary and/or the discount rate is positive (ρ and/or δ ), the total expected real effects of monetary shocks in an arbitrary heterogeneous economy are always bounded between the effects in the corresponding average-frequency and average-duration economies (in the absence of complementarities). This follows directly from Jensen s inequality: the expected cumulative real effect of a level shock in an identical-firms economy is convex in the frequency of price changes, and concave in the (expected) duration of price spells. The intuition as to why the average frequency of price adjustments can be quite misleading as an indicator of the overall degree of nominal rigidity can be developed from the following limiting case: imagine a heterogeneous economy with a non-negligible fraction of firms that adjust prices continuously. Then, irrespective of how low the frequencies of price adjustment of the remaining firms are, the average frequency in the economy will be infinite. Nevertheless, monetary shocks may still have large real effects due to firms with finite adjustment frequencies. The intuition of this extreme example carries through to more realistic distributions, and to discrete time as well: in heterogeneous economies, a high average frequency of price adjustment need not imply small monetary non-neutralities (note that the implication does hold in identical-firms economies). To get a first idea of how large these effects can be in quantitative terms, take the limiting case of permanent level shocks and zero discount rate. By this measure, using the BK data for the U.S. economy, the total real effects more than double when heterogeneity is accounted for: the inverse average frequency duration of price rigidity is 2.9 months, while the average duration of price rigidity is 6.6 months. 8 8 The results for the Euro area, based on the statistics reported by Dhyne et al. (26), are similar. With the statistics for the U.S. economy reported recently by Nakamura and Steinsson (26a) the results are even more pronounced. I use the data for what they refer to as regular 3

14 2.4.2 Growth rate shocks In the case of growth rate shocks, taking Jensen s inequality into account and using the average duration of price rigidity instead of the average frequency of price changes to summarize the extent of nominal rigidity in the heterogeneous economy does not suffice. The reason is that heterogeneity has an additional impact on cumulative real effects of shocks, as shown below. Once again, the results are shown in the absence of strategic complementarities in price setting (θ =). For analytical convenience, I assume no discounting (δ =). Proposition 2 (Growth rate shocks in the continuous time model) When θ = and δ =, the expected (normalized) cumulative real effect of a temporary (γ >) shock to the growth rate of nominal aggregate demand in an arbitrary heterogeneous economy is equal to: Z f (k) dk. γα k + α 2 k Inthecaseofpersistentshocks(γ ), it is approximately equal to the second moment of the cross-sectional distribution of (expected) durations of price rigidity in the economy: 9 Z f (k) dk = d 2 + σ 2 α d, 2 k where σ 2 d R ³ f (k) α k d 2 dk is the variance of the cross-sectional distribution of (expected) durations of price rigidity in the economy. In particular, this result implies that for shocks with enough persistence the expected normalized cumulative real effects in the heterogeneous economy exceed those in either the average-frequency or the average-duration economies. The intuition for why heterogeneity has a direct effect on cumulative real effects in the case of persistent growth rate shocks can actually be developed from the identical-firms case. A lower frequency of price changes increases the magnitude of real effects, and reduces the speed at which they fade away. Jointly, these two features lead total real effects to depend on the square of the frequency of price changes, where they exclude the effects of sales. The sample moments are calculated in the same way as in Table. The inverse average frequency duration of price rigidity is 3.5 months, while the average duration of price rigidity is 3 months. This leads to a ratio of The approximation error is of order O (γ). In the Appendix I present the results with discounting, discuss the reasons for this approximation, and show that the implied error is small for a wide range of parameter values. 4

15 price changes. With heterogeneity, the mechanism at work is qualitatively the same, and in the absence of complementarities the overall effect is the weighted average of the effect for each sector. It thus depends on the second moment of the distribution of adjustment frequencies. To give a firstideaofhowlargethiseffect can be in quantitative terms, I compute the ratio of the approximate expected (normalized) cumulative real effect of a persistent growth rate shock in the heterogeneous economy to the same measure in the average-frequency economy using once again the moments reported in Table. In that case, the standard deviation of the cross-sectional distribution of (expected) durations of price rigidity is σ d =7. months. Recall that the average duration of price rigidity is 6.6 months, while the inverse average frequency duration measure is 2.9 months. Therefore, the ratio referred to above is ( ) /2.9 2 =.2. Even correcting for Jensen s inequality and using the average duration as a measure of nominal rigidity produces cumulative effects which are less than half of those in the heterogeneous economy: in that case the ratio is ( ) /6.6 2 =2.2. From such sample moments one can also obtain an estimate of the duration of price rigidity required for an identical-firms economy to match the heterogeneous economy s response to a persistent growth rateq shock, in terms of its expected normalized cumulative real effects. It equals d 2 + σ 2 d. In the BK data this yields 9.7 months, which is more than three times the inverse average frequency duration, or one and a half times the average duration of price rigidity The strategic interaction effect When there are strategic complementarities in price setting (θ <), pricing decisions of a given firm depend on the behavior of other firms through the aggregate price level. As a result, the response of the economy to shocks becomes more sluggish. This result is well known in the context of identical-firms models(for a recent exposition see Woodford, 23, ch. 3). In this subsection I uncover an interaction between strategic complementari- With permanent level shocks, a change in the frequency of price adjustments affects the speed of the adjustment process, but not the magnitude of real effects on impact. Mathematically,thetotalrealeffect of the shock is convex in the frequency of price changes, but linear in the duration of price rigidity (when the discount rate is zero). In the statistics reported recently by Nakamura and Steinsson (26a), σ d =.7 months. So the ratio to the average-frequency economy using their data is /3.5 2 =25.The ratio to the average-duration economy is /3 2 =.8. 2 In the Nakamura and Steinsson (26a) data this results in 7.5 months, which is 5.5 times the inverse average frequency duration, and.35 times the average duration of price rigidity. 5

16 ties and heterogeneity in the frequency of price changes that generates even more sluggish responses to a shock. The intuition behind this interaction can be understood in the context of the framework of responders and non-responders proposed by Haltiwanger and Waldman (99). With strategic complementarities, the pricing decisions of firms in sectors with more frequent price changes are influenced by the existence of slower-adjusting sectors, since the former do not want to set prices that will deviate too much from the aggregate price in the future. On the other hand, sectors in which price adjustment is less frequent play, to some extent, the role of the non-responders: they do respond to the shock and are also influenced by the pricing decisions of firms in the relatively more flexible sectors, but naturally to a lesser extent. Note that for complementarities to have these effects there is no need for strictly non-responders to exist. Another way to see this is to recall the intuition for Taylor s (98) contract multiplier: strategic complementarities amplify the real effects of monetary shocks, despite thefactthatallfirms eventually get a chance to respond to the shock. The crucial feature is that price adjustments are staggered, so that at any point in time some firms behave as non-responders. As a result of the interaction between complementarities and heterogeneity, sectors in which prices are more sticky end up having a disproportionate effect on the aggregate price level. I refer to this result as the strategic interaction effect due to heterogeneity in price stickiness. To capture the dynamic effects arising from this interaction, I start by comparing the implications of strategic complementarity in the calibrated heterogeneous economy and in the corresponding average-frequency economy. 3 For that purpose, I need to specify the degree of strategic complementarities in the economy. If the reduced-form model is taken literally, θ is a free parameter. However, it can be regarded as a reduced form coefficient for the degree of real rigidity in the fully specified model presented in Section 3 (where this is shown to be the case). Large real rigidities correspond to small values of θ, and thus potentially to strategic complementarities. In the fully specified model, the degree of real rigidity depends on primitive parameters such as the elasticity of substitution between the varieties of the consumption good, the (Frisch) elasticity of labor supply, and the intertemporal elasticity of substitution in consumption. Here I adopt a value for θ that is consistent with the range of parameter values used to calibrate the fully specified model presented later. For concreteness, I set θ = Figure 3a displays the IRFs of the output gap to the same (negative) permanent level shock analyzed earlier. It includes IRFs with and without strategic 3 The comparison with the median-frequency and the average-duration economies yields qualitatively similar results. 4 For the curious reader, this results from unit intertemporal elasticity of substitution in consumption, a 6.7% desired markup over marginal cost, and unit (Frisch) elasticity of labor supply in the context of firm-specific labor. 6

17 complementarities. The results illustrate how the latter interact with heterogeneity in price stickiness to generate larger and more persistent real effects of monetary shocks: complementarities do increase the real effects of the shock in the identical-firms economy, but even more so in the heterogeneous economy. Figure 3a: Permanent Level Shock - Strategic Interaction Effect Output Gaps Heterogeneous economy Average-frequency economy months The strategic interaction effect is also evident in the analysis of the IRF of sectoral prices, based on Figure 3b. For the heterogeneous economy I plot the sectoral price index for the sector in which prices change most frequently (sector ; denoted p t ), while for the identical-firms economy I plot the aggregate price level (p t ). Without strategic complementarities (θ =) prices in the least sticky sector respond faster than the aggregate price level in the identical-firms economy. With strategic complementarities (θ =.25) prices respond more slowly in both economies, but even more so in the heterogeneous economy: because of the strategic interaction effect, even the sectoral price index in the fastest-adjusting sector becomes more sluggish than the aggregate price level in the identical-firms economy. The same pattern emerges in the case of growth rate shocks, the effect of which is illustrated in Figure 4 through the IRF for the output gap. To provide an additional, perhaps more subtle perspective on the strategic interaction effect I perform the following exercise: 5 given a degree of strategic complementarities, I solve for the equilibrium response of 35 identical-firms economies to a nominal shock, where each economy is identical to one of the 35 sectors of the heterogeneous economy (i.e., it has the same frequency of price changes). I compare the (weighted) average of the responses of these economies 5 I thank Marco Bonomo for this suggestion. 7

18 to the response of the heterogeneous economy with 35 sectors and the same level of complementarities. Figure 3b: Permanent Level Shock - Strategic Interaction Effect x Sectoral Prices Heterogeneous economy: p t Average-frequ. economy: p t months Figure 4: Growth Rate Shock - Strategic Interaction Effect 2 Output Gaps Heterogeneous economy Average-frequency economy months If complementarities are absent (θ =) the results of the two calculations are identical. With complementarities, the average of the 35 economies already incorporates the within sector effects, and any differences must be attributed to the interaction between firms in different sectors, once the 35 one-sector 8

19 economies are embedded into the same (multi-sector) economy. The results are presented in Figures 5a and 5b for, respectively, a permanent level shock and a growth rate shock (with φ 2 =.89). Figure 5a: Permanent Level Shock - Strategic Interaction Effect Output Gaps Heterogeneous economy Avg of 35 economies months Figure 5b: Growth Rate Shock - Strategic Interaction Effect 2 Output Gaps Heterogeneous economy Avg of 35 economies months 9

20 2.6 Fitting IRFs with an identical-firms model In this subsection I pose the question of which parameterization for an identicalfirms economy best approximates the dynamics of a given heterogeneous economy in terms of its impulse response functions. To address this question I perform the following exercise: given the empirical distribution of adjustment probabilities obtained from BK, and a degree of strategic complementarities in the heterogeneous economy (determined by θ), I find the adjustment probability λ id in an identical-firms economy that minimizes the sum of squared deviations of its IRFs from the heterogeneous economy s IRFs. The identical-firms economy is constrained to have the same degree of complementarities as the heterogeneous economy that actually generated the target IRF. Using the IRF for the output gap, I do these calculations for several degrees of complementarities, and for level and growth rate shocks with varying levels of persistence. The results are presented in Tables 3(a,b). Instead of reporting the best-fitting frequency λ id, I report the corresponding (expected) duration of price rigidity d id = / ln ( λ id), for which the results are easier to analyze. Table 3a reports the results for level shocks. It is clear that the higher the degree of complementarities (lower θ), and the more persistent the shock (higher half-life), the larger the duration of price rigidity required for an identical-firms economy to approximate the dynamics of the calibrated heterogeneous economy. Moreover, d id generally exceeds the inverse average frequency duration of price rigidity in the BK data (2.9 months), and in some cases even the average duration of price rigidity of the heterogeneous economy (6.6 months), if there are enough complementarities in price setting and if shocks are persistent enough. 6 The results for growth rate shocks are presented in Table 3b, and are even more pronounced. The best-fitting duration might exceed the average duration of price rigidity (6.6 months) even if prices are strategic substitutes, providedthat the shock is persistent enough. 7 6 Bils and Klenow (22) perform this exercise using a model with Taylor staggered price setting. They focus on permanent level shocks to the money supply, and find that the bestfitting identical-firms economy features contract lengths of 4 months, which is roughly the median-frequency based duration of price rigidity in their data. Note that with persistent level shocks and a large degree of strategic substitutability in price setting (large θ) the same result obtains here, despite the different price setting specification. 7 With yet more persistent shocks - half-lives of up to 5 years - and lower discount rates, the best fitting duration seems to converge to 9.7 months for all degrees of real rigidity. This is consistent with the evidence in Tables 3(a,b) that the more persistent the shock, the smaller the role of strategic complementarities. This limiting duration seems to coincide with the one obtained with the analytic approximation to the effect of a persistent growth rate shock in the absence of strategic complementarities, despite the different metric. Perhaps it can be shown analytically that this convergence does indeed occur. 2

21 Table 3a: Best-Fitting Duration d id - Level Shocks Half-life (years) θ Durations are reported in months. Table 3b: Best-Fitting Duration d id - Growth Rate Shocks Half-life (years) θ Durations are reported in months. 3 Heterogeneity in a new Keynesian model In this section I move beyond the simple reduced-form model analyzed previously and introduce heterogeneity in the frequency of price adjustments into an otherwise standard, fully specified new Keynesian sticky price model without capital accumulation. The demand side of the model consists of the intertemporal IS equation that results from consumers optimization, and an interest rate rule that specifies how interest rates react to inflation and the output gap. Real rigidities are generated by a firm-specific labor input. 8 The framework allows me to go beyond the convenient, but unfortunately unrealistic specification of monetary disturbances as shocks to an exogenous nominal income process, and study monetary shocks that are empirically more plausible. 8 The exact source of real rigidity is not important for the aggregate dynamics of the model in response to monetary shocks. However, different sources of real rigidities might have different implications for the response of endogenous variables to other types of shocks at disaggregated levels (Klenow and Willis, 26). 2

22 This is an important step because the aggregate effects of nominal rigidity in general depend on the nature of monetary shocks, and this is also true for heterogeneity in price stickiness, as the results of the previous section have shown. Therefore, I specify an interest rate rule that is consistent with results from the empirical literature. After deriving a loglinear approximation of the model around its deterministic zero inflation steady state, I present the underlying generalized new Keynesian Phillips curve. 9 I then calibrate the model with the sectoral distribution of price stickiness described in Subsection 2.2, and standard values found in the literature for the remaining structural parameters, in order to analyze the effects of heterogeneity in price stickiness. 3. The fully specified model A representative consumer derives utility from a Dixit-Stiglitz composite of differentiated consumption goods and supplies a continuum of differentiated types of labor to monopolistically competitive firms, which he owns. The latter set prices as in Calvo (983), and are divided into sectors that differ in the frequency of price adjustments. Firms are indexed by their sector, k [, ], andbyj [, ]. The probability of a price change by a firm in sector k in any given period is denoted λ k, and these events are independent across all firms in the economy. The distribution of firms across sectors is summarized by a density function f on [, ]. Eachfirm hires labor of a specific typeinacompetitivemarkettoproduce a likewise specific variety of the consumption good according to a linear technology. I assume a cashless economy with a one-period nominal bond in zero net supply. 2 The representative consumer solves: max E s.t. P t C t = X t= Z β t C σ f (k) t σ Z Z Z L + ϕ kj,t f (k) + ϕ djdk L kj,t W kj,t djdk + T t + I t B t B t, 9 For the effects of steady state inflation on the dynamics of related models see Ascari (24), and Cogley and Sbordone (25). 2 This framework is equivalent to assuming a continuum of consumers, each of whom supplies one of the labor varieties to firms, and who pool risks by trading a rich enough set of statecontingent assets so as to ensure that they face the same budget constraint at any point in time. Alternatively, one could use a consumer-producer ( yeoman farmer ) model with the same kind of risk-sharing possibilities. 22