Frontiers of Macroeconomics Volume 2, Issue 1

2006

Article 1

Heterogeneity in Price Stickiness and the Real Effects of Monetary Shocks Carlos Carvalho∗

∗

Princeton University, [email protected]

c Copyright 2006 The Berkeley Electronic Press. All rights reserved.

Heterogeneity in Price Stickiness and the Real Effects of Monetary Shocks∗ Carlos Carvalho

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. KEYWORDS: heterogeneity, price stickiness, aggregation, persistence, new Keynesian Phillips curve

∗

I would like to thank Kevin Amonlirdviman, Roland B´enabou, Marco Bonomo, Alan Blinder, Vasco C´urdia, Per Krusell, Jonathan Parker, Ricardo Reis, Felipe Schwartzman, Christopher Sims, Lars Svensson, Michael Woodford, and participants from the NBER Summer Institute 2006 in Monetary Economics, Econometric Society NASM 2006, ESWC 2005, LAMES 2004, EEA Meeting 2004, 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 08544-1021, USA. E-mail: [email protected].

Carvalho: Heterogeneity in Price Stickiness

1

Introduction

There is ample evidence that the frequency of price adjustments differs substantially across sectors (Blinder et al., 1998, and Bils and Klenow, 2004, for the U.S. economy; Dhyne et al., 2006, 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 (2004) (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 effects than in identical-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 Published by The Berkeley Electronic Press, 2006

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low frequency sectors drive most of the dynamics subsequently. 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 (1991), or between firms with staggered price adjustments in the presence of Taylor’s (1980) 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 identicalfirms 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 duration of price spells equals that 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 price-setting behavior and the associated parameters in commonly used onesector 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 identical-firms model in order to best mimic the dynamics of a http://www.bepress.com/bejm/frontiers/vol2/iss1/art1

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Carvalho: Heterogeneity in Price Stickiness

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? 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 changes that is up to three times lower than the average of the heterogeneous 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 crosssectional 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 (1983): in every period, each firm changes its price with a constant, sector-specific probability. As a result, heterogeneity in the frequency of price changes gives rise to a generalized 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 Published by The Berkeley Electronic Press, 2006

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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. Exploring 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, 1990), 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 (1983) model, extend to a large class of alternative price setting specifications. As shown in Carvalho (2005) and Carvalho and Schwartzman (2006), the latter includes Taylor (1979, 1980) staggered pricing, and sticky information models as in Mankiw and Reis (2002). 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., 1995; Bils and Klenow, 2002, 2004; Bils et al., 2003). In earlier work, Taylor (1993) extended his original model (1979, 1980) to account for wage contracts of different durations. In a different framework, with state- rather than time-dependent pricing rules, http://www.bepress.com/bejm/frontiers/vol2/iss1/art1

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Carvalho: Heterogeneity in Price Stickiness

Caballero and Engel (1991, 1993) 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 (2001) and Benigno (2001, 2004), who explore the effects of heterogeneity in price stickiness on optimal monetary policy in two-sector models. Dixon and Kara (2005) study what I refer to as the strategic interaction effect in a model with Taylor staggered wage setting. In work that is closely related to mine, Carlstrom et al. (2006b) use a two-sector model with different degrees of nominal rigidity to study how sectoral relative prices affect aggregate dynamics.1 They find relative price effects that are qualitatively similar to the ones obtained by Aoki (2001) in a two-sector economy featuring one sticky- and one flexible-price sector, and by Benigno (2001, 2004) in a two-country model with different degrees of nominal price stickiness. Barsky et al. (2006) 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

2.1

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 ∈ [0, 1], and by j ∈ [0, 1]. The distribution of firms across sectors is summarized by a density function f on [0, 1]. All firms set prices as in Calvo (1983): in every period of length ∆, each firm changes its price with a constant probability. The probabilities are sector specific, and denoted λk . The occurrences of price changes are independent across all firms in the economy, and as a result in each period a fraction λk 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 − pt = θyt ,

(1)

1

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. (2006a) use that model to study equilibrium determinacy. Published by The Berkeley Electronic Press, 2006

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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 log-deviations from a deterministic, zero inflation steady state. In (1), θ, which is always positive, determines the degree of strategic complementarities in price setting. Prices are strategic complements (substitutes) if θ < 1 (> 1). In a fully specified model, strategic complementarities can arise as a result of large real rigidities (Ball and Romer, 1990), such as firm-specific capital and/or labor inputs, or production chains, for example.3 The aggregate price level is given by: Z 1 Z 1 f (k) pk,j (2) pt = t djdk, 0

0

is the price charged by firm j from sector k at time t. where pk,j t Whenever a firm from sector k has a chance to change its price, it sets xkt according to: xkt = arg min x

∞ X s=0

¡ ¢2 ∆β s (1 − λk )s Et x − p∗t+s∆

= (1 − (1 − λk ) β)

∞ X s=0

(3)

((1 − λk ) β)s Et p∗t+s∆ ,

where β is the per-period discount factor, and Et is the expectation conditional on time-t information. This optimization problem can be justified through a second-order 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: Z 1 pt = f (k) pkt dk, (4) 0

where the sectoral price indices, pkt

= λk

pkt ,

are given by:

∞ X s=0

(1 − λk )s xkt−s∆ .

(5)

2

This equation can be derived from first principles as in Blanchard and Kiyotaki (1987) or Ball and Romer (1989). 3 For a detailed exposition of different sources of strategic complementarities in price setting see Woodford (2003, ch.3). http://www.bepress.com/bejm/frontiers/vol2/iss1/art1

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Carvalho: Heterogeneity in Price Stickiness

To focus on the supply side of the model, I assume that nominal aggregate demand, mt = yt + pt , follows an exogenous stochastic process. For simplicity, I specify: A (L) mt = ut , where ut is a zero mean, 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 (Lmt = mt−∆ ). In what follows, I will focus on two specifications for this process: an AR(1) in levels, and an AR(1) in growth rates.

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 (2004). More specifically, I identify each sector in the model with one of the goods and services categories listed in their appendix, and set λk equal to the monthly frequency of price changes reported for the category identified with sector k. As a result the unit of time ∆ 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 350 sectors. To make it easy to refer to particular sectoral variables, I order the sectors so that sector 1 (sector 350) displays the highest (lowest) frequency of price changes. To convert a per-period probability of price change λ into an expected −1 duration of price rigidity d I use the formula d = ln(1−λ) . This is based on the 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 = 1 − 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 1.4 Choosing an empirical distribution has the obvious advantage of making the calibration somewhat realistic. However, it is worth highlighting a few con4

The sectoral rates of arrival of price changes are calculated as αk = − ln (1 − λk ) /∆. 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 7.1 months. The standard deviation of durations of price rigidity is unchanged at 7.1 months. Published by The Berkeley Electronic Press, 2006

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ceptual issues involved. First, the Bils and Klenow (2004) 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, 1986; Blinder et al., 1998). 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 distribution of price stickiness. Table 1: Moments of the Cross-Sectional Distribution of the Frequency of Price Changes Description

Formula 1 , α

“Inverse average frequency” duration of price ridigity∗

Standard deviation of durations of price rigidity∗∗∗

Months

f (k) αk

d=

350 P

f (k) dk , dk =

k=1

µ 350 P

k=1

2.9

k=1

−1 ln(1−λmed )

“Median frequency based” duration of price ridigity Average duration of price rigidity∗∗

α=

350 P

4.3

−1 ln(1−λk )

6.6

¶1/2

7.1

¡ ¢2 f (k) dk − d

Obs: Based on the statistics reported by Bils and Klenow (2004). λmed 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.

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Carvalho: Heterogeneity in Price Stickiness

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 = 1. 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(1) in levels with autoregressive coefficient 0 ≤ φ1 ≤ 1. I refer to those as level shocks: A (L) mt = ut , with A (L) = 1 − φ1 L. Figure 1 presents the impulse response function (IRF) of the output gap to a permanent negative level shock (φ1 = 1).5 For comparison purposes it also displays the output gap in identical-firms economies calibrated with the moments reported in Table 1. 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 1: Permanent Level Shock - Output Gaps 1

Output Gaps 0

Heterogeneous economy Average-duration economy Median-frequency economy Average-frequency economy 0

5

10

15

20

months

25

30

35

40

5

The size of the shock only affects the scale of the responses. The IRF for the price level is the mirror image across the horizontal axis. Throughout the paper I assume a discount rate of 3% per year, except when stated otherwise. Published by The Berkeley Electronic Press, 2006

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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. A qualitative difference 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 refer to this as 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.6 Figure 2 presents analogous results for the relatively more realistic specification in which mt follows an AR(1) in growth rates with autoregressive coefficient 0 ≤ φ2 ≤ 1. I refer to this case as growth rate shocks: A (L) mt = ut , with A (L) = 1 − (1 + φ2 ) L + φ2 L2 . I set φ2 = 0.89, 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 state6

The frequency composition effect is related to the effects that arise when aggregating 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. (2005). http://www.bepress.com/bejm/frontiers/vol2/iss1/art1

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Carvalho: Heterogeneity in Price Stickiness

ments about the exact shape of impulse response functions for an arbitrary distribution. Figure 2: Growth Rate Shock - Output Gaps 2

Output Gaps 0

Heterogeneous economy Average-duration economy Median-frequency economy Average-frequency economy 0

5

10

15

20

months

25

30

35

40

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 ∆ → 0. The derivation of the latter, as well as the proofs of the results, are in the Appendix. To introduce the notation used below, I relate the parameters of the continuous time model with their per-period counterparts in Table 2. With this notation, the expected (normalized) cumulative effect on R ∞the output gap of a time-zero mon−1 etary shock of size u0 is given by u0 E0 0 y (t) dt.

7

This measure is also discussed, for example, in Christiano et al. (2005).

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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

αk ≥ 0

λk = 1 − e−αk ∆

Discount rate

δ≥0

β = e−δ∆

Decay rate of level shocks

ρ≥0

φ1 = e−ρ∆

Decay rate of growth rate shocks

γ≥0

φ2 = e−γ∆

m (t) , y (t) , ...

mt , y t , ...

Variables 2.4.1

Level shocks

The first set of results refers to level shocks. Proposition 1 (Level shocks in the continuous time model) When θ = 1, the expected (normalized) cumulative real effect of a level shock to nominal aggregate demand is equal to: Z 1 1 dk. f (k) αk + ρ + δ 0 Corollary 1 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 = 0, δ = 0), in which case it equals the real effect in the corresponding average-duration economy: Z 1 1 d= f (k) dk. αk 0 http://www.bepress.com/bejm/frontiers/vol2/iss1/art1

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Carvalho: Heterogeneity in Price Stickiness

The above results imply that if level shocks are temporary and/or the discount rate is positive (ρ ≥ 0 and/or δ ≥ 0), 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 identicalfirms 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 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 8

The results for the Euro area, based on the statistics reported by Dhyne et al. (2006), are similar. With the statistics for the U.S. economy reported recently by Nakamura and Steinsson (2006a) the results are even more pronounced. I use the data for what they refer to as “regular price changes,” where they exclude the effects of sales. The sample moments are calculated in the same way as in Table 1. The “inverse average frequency” duration of price rigidity is 3.5 months, while the average duration of price rigidity is 13 months. This leads to a ratio of 3.8. Published by The Berkeley Electronic Press, 2006

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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 = 1). For analytical convenience, I assume no discounting (δ = 0). Proposition 2 (Growth rate shocks in the continuous time model) When θ = 1 and δ = 0, the expected (normalized) cumulative real effect of a temporary (γ > 0) shock to the growth rate of nominal aggregate demand in an arbitrary heterogeneous economy is equal to: Z 1 1 f (k) dk. γαk + α2k 0 In the case of persistent shocks (γ ≈ 0), it is approximately equal to the second moment of the cross-sectional distribution of (expected) durations of price rigidity in the economy:9 Z 1 1 2 f (k) 2 dk = d + σ 2d , αk 0 ´2 where ≡ 0 f (k) − d dk is the variance of the cross-sectional distribution of (expected) durations of price rigidity in the economy. σ 2d

R1

³

1 αk

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. 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.10 9

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. 10 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. http://www.bepress.com/bejm/frontiers/vol2/iss1/art1

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To give a first idea of how large this effect 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 1. In that case, the standard deviation of the cross-sectional distribution of (expected) durations of price rigidity is σd = 7.1 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 (6.62 + 7.12 ) /2.92 = 11.2. 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 (6.62 + 7.12 ) /6.62 = 2.2.11 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 rate qshock, in terms 2

of its expected normalized cumulative real effects. It equals d + σ 2d . 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.12

2.5

The strategic interaction effect

When there are strategic complementarities in price setting (θ < 1), 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 identicalfirms models (for a recent exposition see Woodford, 2003, ch. 3). In this subsection I uncover an interaction between strategic complementarities 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 “nonMathematically, the total real effect 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). 11 In the statistics reported recently by Nakamura and Steinsson (2006a), σ d = 11.7 So the ratio to the average-frequency economy using their ¢data is ¡ 2 months. ¡ ¢ 13 + 11.72 /3.52 = 25. The ratio to the average-duration economy is 132 + 11.72 /132 = 1.8. 12 In the Nakamura and Steinsson (2006a) data this results in 17.5 months, which is 5.05 times the “inverse average frequency” duration, and 1.35 times the average duration of price rigidity. Published by The Berkeley Electronic Press, 2006

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responders” proposed by Haltiwanger and Waldman (1991). 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 (1980) contract multiplier: strategic complementarities amplify the real effects of monetary shocks, despite the fact that all firms 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.13 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 = 0.25.14 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 13

The comparison with the median-frequency and the average-duration economies yields qualitatively similar results. 14 For the curious reader, this results from unit intertemporal elasticity of substitution in consumption, a 16.7% desired markup over marginal cost, and unit (Frisch) elasticity of labor supply in the context of firm-specific labor. http://www.bepress.com/bejm/frontiers/vol2/iss1/art1

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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 1

Output Gaps 0

Heterogeneous economy Average-frequency economy 0

5

10

15

20

months

25

30

35

40

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 1; denoted p1t ), while for the identical-firms economy I plot the aggregate price level (pt ). Without strategic complementarities (θ = 1) prices in the least sticky sector respond faster than the aggregate price level in the identical-firms economy. With strategic complementarities (θ = 0.25) 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:15 given a degree of strategic complementarities, I solve for the equilibrium response of 350 identical-firms economies to a nominal shock, where each economy is identical to one of the 15

I thank Marco Bonomo for this suggestion.

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350 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 to the response of the heterogeneous economy with 350 sectors and the same level of complementarities. Figure 3b: Permanent Level Shock - Strategic Interaction Effect 1

Sectoral Prices

x 10

Heterogeneous economy: p1t Average-frequ. economy: pt

0

5

10

15

20

25

months

30

35

40

Figure 4: Growth Rate Shock - Strategic Interaction Effect 2

Output Gaps 0

Heterogeneous economy Average-frequency economy 0

5

10

15

20

months

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25

30

35

40

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If complementarities are absent (θ = 1) the results of the two calculations are identical. With complementarities, the average of the 350 economies already incorporates the “within sector” effects, and any differences must be attributed to the interaction between firms in different sectors, once the 350 one-sector 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 φ2 = 0.89). Figure 5a: Permanent Level Shock - Strategic Interaction Effect 1

Output Gaps 0

Heterogeneous economy Avg of 350 economies 0

10

20

30

months

40

50

60

Figure 5b: Growth Rate Shock - Strategic Interaction Effect 2

Output Gaps

Heterogeneous economy Avg of 350 economies 0

10

20

30

months

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40

50

60

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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 identicalfirms 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 = −1/ ln (1 − 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.16 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, provided that the shock is persistent enough.17 16

Bils and Klenow (2002) 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. 17 With yet more persistent shocks - half-lives of up to 15 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 http://www.bepress.com/bejm/frontiers/vol2/iss1/art1

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Table 3a: Best-Fitting Duration d∗id - Level Shocks∗ Half-life (years)

θ 0.10 0.25 0.50 0.75 1.00 1.50 2.50 ∗

0.25 5.47 4.81 4.33 4.07 3.89 3.65 3.37

0.50 6.12 5.42 4.88 4.56 4.33 4.02 3.66

0.75 6.48 5.77 5.19 4.83 4.58 4.22 3.80

1.00 6.72 6.00 5.39 5.00 4.73 4.34 3.88

2.00 7.20 6.46 5.77 5.33 5.00 4.55 4.02

3.00 7.42 6.65 5.92 5.45 5.11 4.62 4.06

5.00 7.63 6.83 6.05 5.55 5.19 4.68 4.09

∞ 7.95 7.05 6.20 5.66 5.27 4.73 4.12

Durations are reported in months.

Table 3b: Best-Fitting Duration d∗id - Growth Rate Shocks∗ Half-life (years)

θ 0.10 0.25 0.50 0.75 1.00 1.50 2.50 ∗

3

0.25 8.02 7.26 6.62 6.25 6.00 5.67 5.30

0.50 8.13 7.51 7.02 6.74 6.55 6.29 6.00

0.75 8.24 7.71 7.30 7.06 6.90 6.69 6.44

1.00 8.33 7.87 7.51 7.30 7.16 6.97 6.75

1.50 8.47 8.10 7.81 7.65 7.54 7.39 7.21

2.00 8.57 8.27 8.03 7.89 7.80 7.67 7.52

3.00 8.73 8.50 8.31 8.21 8.14 8.04 7.92

5.00 8.93 8.76 8.62 8.55 8.50 8.43 8.34

Durations are reported in months.

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.18 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. 18 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 Published by The Berkeley Electronic Press, 2006

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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. 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.19 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.

3.1

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 (1983), and are divided into sectors that differ in the frequency of price adjustments. Firms are indexed by their sector, k ∈ [0, 1], and by j ∈ [0, 1]. The probability of a price change by a firm in sector k in any given period is denoted λk , 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 [0, 1]. Each firm hires labor of a specific type in a competitive market to produce 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.20

at disaggregated levels (Klenow and Willis, 2006). 19 For the effects of steady state inflation on the dynamics of related models see Ascari (2004), and Cogley and Sbordone (2005). 20 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 state-contingent 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. http://www.bepress.com/bejm/frontiers/vol2/iss1/art1

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The representative consumer solves: ⎞ ⎛ Z 1 Z 1 1+ ϕ1 ∞ 1−σ X L −1 C kj,t ⎠ − βt ⎝ t f (k) max E0 1 djdk 1 − σ 1 + 0 0 ϕ t=0 s.t. Pt Ct =

Z

1

f (k)

0

Z

1

Lkj,t Wkj,t djdk + Tt + It−1 Bt−1 − Bt ,

0

where β is the discount factor, Ct is consumption of the composite good, Lkj,t is the supplied quantity of type kj labor, Wkj,t is the associated nominal wage, Tt are firms’ profits received by the consumer through lump-sum transfers, Bt denotes bond holdings that accrue (gross) interest at rate It , and Pt is a price index to be defined below. The parameters σ −1 and ϕ stand for, respectively, the intertemporal elasticity of substitution in consumption and the (Frisch) elasticity of labor supply. The composite consumption good is given by: Ct ≡

∙Z

1

ε−1

f (k)

0

Ck,t ≡ f (k)

∙Z

1

ε−1 ε

Ck,t dk

ε−1 ε

Ckj,t dj

0

ε ¸ ε−1

ε ¸ ε−1

,

,

(6) (7)

where Ck,t is the subcomposite of goods produced by firms in sector k, and Ckj,t is consumption of the variety of the good produced by firm j from sector k (henceforth “firm kj”). The normalization in the sectoral aggregators yields a symmetric equilibrium if prices are fully flexible. The elasticity of substitution between consumption varieties is ε > 1.21 Denoting by Pkj,t the price charged by firm kj in period t, the corresponding consumption price index is: Pt =

∙Z

1

f

1−ε (k) Pk,t dk

0

1 ¸ 1−ε

,

21

Allowing the elasticity of substitution between the sectoral subcomposites in equation (6) to differ from the within-sector elasticity of substitution (equation 7) changes some qualitative features of the model. In particular, sectoral price indices start to have a direct effect on pricing decisions of firms from the given sector, in addition to the indirect influence through the aggregate price level. I experimented with different calibrations for these two elasticities, including the case in which (6) is a Cobb-Douglas aggregator, and the substantive findings of the paper were essentially unchanged. Published by The Berkeley Electronic Press, 2006

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Pk,t =

∙Z

1 1−ε Pkj,t dj

0

1 ¸ 1−ε

,

where Pk,t is a sectoral price index. The first order conditions for the representative consumer’s optimization problem are: 1 ϕ

Lkj,t Wkj,t = −σ , kj ∈ [0, 1]2 , Pt Ct "µ # ¶−σ Pt Ct+1 = β −1 , Et It Ct Pt+1 Ck,t = f (k) Ct

µ

Pk,t Pt µ

Ckj,t = f (k)−1 Ck,t

¶−ε

, k ∈ [0, 1] , ¶−ε Pkj,t , kj ∈ [0, 1]2 . Pk,t

Firm kj hires labor of its specific type to produce its variety of the consumption good according to a linear technology: Ykj,t = Nkj,t , where Ykj,t is the production of its variety and Nkj,t is the specific labor input. All agents are price takers in the labor market. When setting its price Xkj,t at time t, firm kj solves: max Et

∞ X s=0

Qt,t+s (1 − λk )s [Xkj,t Ykj,t+s − Wkj,t+s Nkj,t+s ]

s.t. Ykj,t+s = Nkj,t+s , ¶−ε µ Xkj,t Yt+s , Ykj,t+s = Pt+s where I have used the market clearing condition in the goods markets. The stochastic nominal discount factor between periods t and t + s used to price firms’ profits, Qt,t+s , is given by: Qt,t+s = β

µ

Ct+s Ct

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¶−σ

Pt . Pt+s

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The first order condition yields:

Xkj,t

ε = ε−1

Et

∞ P

s=0

Et

ε Qt,t+s (1 − λk )s Pt+s Yt+s Wkj,t+s ∞ P

s=0

s

Qt,t+s (1 − λk )

.

ε Pt+s Yt+s

The distribution of future nominal wages, Wkj,t+s , conditional on time-t information is the same for all firms in sector k that change prices in period t, and therefore they choose the same nominal price. Thus, the sectoral price indices can be rewritten as: £ ¤ 1 1−ε 1−ε 1−ε Pk,t = λk Xk,t + (1 − λk ) Pk,t−1 ,

where Xk,t denotes the common price set by all firms from sector k which change prices in period t. To close the model, I specify the interest rate rule followed by the monetary authority. Motivated by the results in Rudebusch (2002), I assume an interest rate rule that is subject to persistent shocks:

¶φ Yt y υt It = β e , Y where φπ and φy are the standard parameters associated with Taylor-type interest rate rules, υ t = κυ t−1 + ξ t , and ξ t is a zero mean, finite variance i.i.d. process. Y is the output level associated with the deterministic zero inflation steady state.22 As in standard monetary models, given the interest rate rule the requirement of a unique equilibrium imposes restrictions on the policy parameters φπ and φy .23 These being satisfied, the equilibrium, conditional on an arbitrary initial nominal price level, is characterized by the optimality conditions for the consumer’s utility maximization problem and for every firm kj’s profit maximization problem, by market clearing conditions for all varieties of the consumption good (Ckj,t+s = Ykj,t+s ) and all labor types (Nkj,t+s = Lkj,t+s ), and by the bond market clearing (Bt = 0). The corresponding deterministic zero inflation steady state is obtained by setting υ t = 0 for all t in the interest rate rule, and solving for the equilibrium. −1

µ

Pt Pt−1

¶φπ µ

22

In the Appendix I also present results with a policy rule that features interest rate smoothing instead of persistent shocks. 23 Carlstrom et al. (2006a) study equilibrium determinacy in a two-sector model with different degrees of nominal rigidity. Published by The Berkeley Electronic Press, 2006

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3.2

A log-linear approximation

The Appendix presents the full set of equations that characterize the loglinear approximation of the model around its deterministic zero inflation steady state, which I use in what follows. Here I only present the equations needed to characterize the dynamics of output and inflation. These are (lowercase variables denote log deviations from the zero inflation steady state): ∙ ¸ X∞ ϕ−1 + σ s s xk,t = (1 − β (1 − λk )) Et yt+s , (8) β (1 − λk ) pt+s + s=0 1 + ϕ−1 ε pk,t = λk xk,t + (1 − λk ) pk,t−1 , (9) Z 1 f (k) pk,t dk, (10) pt = 0

yt = yk,t = it = υt =

Et yt+1 − σ −1 (it − Et π t+1 ) , yt − ε (pk,t − pt ) , φπ π t + φy yt + υ t , κυ t−1 + ξ t ,

(11) (12)

where π t ≡ pt − pt−1 . ϕ−1 +σ Note that by setting θ ≡ 1+ϕ −1 ε I recover exactly the equations of the supply side of the reduced-form model presented in Section 2. If the intertemporal elasticity of substitution in consumption is equal to one (log preferences), and the disutility of labor is linear, so that labor supply is infinitely elastic, then θ = 1.24

3.3

The generalized new Keynesian Phillips curve

Based on equations (8), (9), (10), and (12) I obtain the following generalized new Keynesian Phillips curve that accounts for heterogeneity in price stickiness (the derivation is in the Appendix): µ −1 ¶ ϕ +σ 1 ψ π t = βEt πt+1 + ψ − (13) yt + gt , −1 1 + εϕ ε ε where: ψ≡

Z

0

1

¶ λk f (k) − βλk dk, 1 − λk µ

24

It is straightforward to show that θ ≤ σ, with equality if and only if ϕ = ∞. This is why generating strategic substitutability in price setting in this model requires unrealistic parameter values. http://www.bepress.com/bejm/frontiers/vol2/iss1/art1

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gt ≡

Z

1

0

fb(k) ≡ R 1 0

fb(k) yk,t dk, λk 1−λk

f (k)

³

− βλk

λk 1−λk

´ f (k) . − βλk dk

The standard NKPC obtains as a particular case when firms in all sectors face the same probability of price adjustment, λ say, in which case (13) simplifies to: ¶ ¶ µ −1 µ ϕ +σ λ π t = βEt π t+1 + yt . − βλ (14) 1−λ 1 + εϕ−1 Heterogeneity in price stickiness changes the standard NKPC in two important ways. First, the coefficient on the aggregate output gap (yt ) differs from the standard NKPC. Second, heterogeneity produces an endogenous shift term that is proportional to a weighted average of the sectoral output gaps (gt ), where the weights fb(k) are adjustment-frequency-based transformations of the actual sectoral weights f (k). They are such that sectors with a higher frequency of price changes receive a relatively larger weight. Note that gt can have either sign in any given period, depending on the distribution of sectoral output gaps. Owing to the endogenous shift term in the Phillips curve, the coefficient on the output gap no longer summarizes the degree of stickiness in the heterogeneous economy. This invalidates the usual association between the overall degree of (nominal and real) rigidities and the slope of the Phillips curve. Moreover, the endogenous shift term can alternatively be written as a weighted average of sectoral relative prices instead of output gaps, which implies that the generalized NKPC has other, equivalent representations featuring different pairs (endogenous shift term, coefficient on the output gap).25 I pick the representation of the generalized NKPC presented above, in which the shift term is proportional to a weighted average of sectoral output gaps, to illustrate the effects of heterogeneity through the Phillips curve. Given this representation, heterogeneity has two opposite effects on the coefficient on the output gap, when compared with the standard NKPC of 25

Just replace the sectoral output gaps with sectoral relative prices, using the demand for the sectoral subcomposites of the varieties of the consumption good (equation 12). The result is a representation of the generalized NKPC with a “relative-price shift term,” akin to the one obtained by Aoki (2001) in a two-sector model with a sticky- and a flexibleprice sector, and by Benigno (2001) in a two-region model with different degrees of nominal rigidity. Carlstrom et al. (2006b) find similar results as the latter paper. Published by The Berkeley Electronic Press, 2006

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the corresponding average-frequency and average-duration economies. On the one hand, it increases the sensitivity of inflation to the output gap through the component of the slope coefficient associated with the degree of nominal rigidity, ψ.26 That is, absent the endogenous shift term, if this were the only effect on the slope of the Phillips curve the heterogeneous economy would actually display more flexibility at the aggregate level than either the averagefrequency or the average-duration economies. On the other hand, however, heterogeneity reduces the sensitivity of inflation to the output gap through the component of the slope coefficient associated with the degree of real rigidities in the economy, which is reduced by 1ε . Loosely speaking, because of this effect the economy with heterogeneity behaves as if it featured more real rigidities. The net result of these two opposite effects on the slope coefficient depends on the calibration of the model, and so I postpone a quantitative analysis until the next subsection. Irrespective of the net effect of heterogeneity on the slope of the Phillips curve, the dynamics of output and inflation in the economy depend further on the behavior of the endogenous shift term. To gain intuition on how the latter affects aggregate dynamics, suppose the economy is hit by a contractionary interest rate shock. The output gap falls as consumers adjust consumption, and that generates pressures for price cuts throughout the economy. Note however, that contrary to identical-firms economies, inflation is also subject to compositional effects in the cross-section of sectoral output gaps through the endogenous shift term. The latter attenuates the response of inflation, because the relatively fast-adjusting sectors feature a higher output gap, and such sectors receive a relatively higher transformed sectoral weight fb(k). So, inflation is actually “driven” by a combination of sectoral output gaps that can behave quite differently from the aggregate (average) output gap.27

3.4

Aggregate dynamics

In calibrating the model, I use the sectoral distribution of price adjustment frequencies described in Subsection 2.2. For the interest rate rule, I use parameter values in the range of estimates found in the empirical literature. For the policy responses to inflation and the output gap I set the point estimates λ This is a direct result of Jensen’s inequality and the fact that 1−λ − βλ is convex in λ. 27 Of course this combination of output gaps is not an exogenous driving process, given that all prices and quantities are jointly determined in the model. Nevertheless, given the equilibrium paths during the adjustment process it is instructive to conjecture how the adjustment of prices and inflation would be altered if instead the output gap were the “driving process,” as in the standard NKPC. 26

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reported by Rudebusch (2002) φπ = 1.24, and φy = 0.33/12.28 For the autoregressive parameter, I set κ = 0.95, which is an intermediate value in the range of estimates found in the literature.29 As before, I set β = 0.9975 (3% per year), and consider two sets of values for the (Frisch) elasticity of labor supply (ϕ), the intertemporal elasticity of substitution (σ −1 ) and the price elasticity of demand for the varieties of the consumption good (ε): in the first configuration I set ϕ = 0.5, σ = 1, and ε = 11; alternatively, I consider ϕ = 1.5, σ = 1, and ε = 5. The first configuration features a relatively low labor supply elasticity and a high demand elasticity, and implies a relatively high degree of real rigidities in the model, whereas the second set of parameters involve higher labor supply elasticity, and a lower elasticity of demand, leading to less real rigidities. In both configurations, the parameter values are well within the ranges found in the literature. Figures 6(a,b) display the response of the output gap and inflation to a contractionary interest rate shock. They present the results for the two sets of parameters, and for comparison purposes also include the response of the corresponding average-frequency economy to the same shock. The results illustrate the main aggregate effect of heterogeneity in price stickiness: the response of inflation is more muted in the heterogeneous economy, despite the larger real effects of the interest rate shock. This can only be reconciled with the intuition based on the standard NKPC if inflation is less responsive to the output gap than in the average-frequency economy. Put differently, heterogeneity generates more persistent dynamic responses to this type of monetary shock as well. In Figure 7 I display ³ −1 the IRFs ´ of the terms that “drive” inflation in the genϕ +σ 1 eralized NKPC, ψ 1+εϕ−1 − ε yt and ψε gt , as well as their sum, for the first set of parameter values. After the (positive) shock to the interest rate rule, the term associated with the output gap falls (solid blue line), and then reverts back to steady state. However, the term that is proportional to the weighted average of sectoral output gaps (red dash-dotted line) shoots up soon after the shock and continues to increase for a while before starting to revert back 28

Dividing the coefficient on the output gap by 12 corrects for the fact that the estimates obtained in the literature are typically based on annualized measures of inflation and interest rates, while the time unit in the calibrated model is 1 month. 29 Most papers that estimate interest rate rules use quarterly data. So, I adjust the estimated coefficients of the autoregressive error to make them compatible with the time unit of one month. The estimates of these autoregressive coefficients with quarterly data typically fall in the 0.75 - 0.95 range, which maps into the 0.91 - 0.98 range for the monthly frequency. Published by The Berkeley Electronic Press, 2006

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to steady state.30 The resulting IRF of the sum of these terms (black dashed line), which is what actually enters the generalized NKPC, indeed presents quite different dynamics when compared to the output gap. Figure 6a: Fully Specified Model - Output Gaps y

Output Gaps 0

Average-frequency economy

Heterogeneous economy

= 1, = 1.5, = 5 = 1, = 0.5, = 11 0

5

10

15

20

months

25

30

35

40

Figure 6b: Fully Specified Model - Inflation Rates y

0

Inflation Rates

x 10

Heterogeneous economy

Average-frequency economy

= 1, = 1.5, = 5 = 1, = 0.5, = 11 0

5

10

15

20

months

25

30

35

40

30 The shift term gt might actually move in the opposite direction of yt on impact, depending on the calibration of the model.

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Carvalho: Heterogeneity in Price Stickiness

Figure 7: Generalized New Keynesian Phillips Curve y

2

Terms in the Generalized NKPC

x 10

1 0

0

10

µ

-1

à _____-1

20

30

40

months

¶

1 y t

Ãg

t

50

60

sum

This decomposition helps in understanding how the endogenous shift term in the generalized NKPC affects the dynamic behavior of the economy. After the contractionary interest rate shock, the adjustment process requires lower output and falling prices in aggregate terms: loosely speaking, a “deflationary environment.” However, inflation is actually driven by a combination of output gaps that is not so deflationary for two reasons. First, because sectors that adjust prices more frequently eventually move to a situation of positive sectoral output gap. Second, those sectors are exactly the ones that are assigned a relatively higher weight in the transformed distribution of sectoral weights fb(k). This is why the response of inflation is muted relative to the averagefrequency economy, despite the larger (average) recession (Figures 6a,b). For completeness, in Table 4 I report the slope coefficients implied by the previous calibrations for the heterogeneous, average-frequency, and averageduration economies. Note that the coefficient on the output gap in the generalized NKPC is actually quite close to the coefficient in the average-frequency economy, but that nevertheless the former exhibits substantially larger and more persistent effects after the interest rate shock.

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∗

θ≡

ϕ−1 +σ . 1+εϕ−1

Based on the distribution of price stickiness described in Subsection 2.2. For each economy the coefficient on the output gap in the NKPC (Slope) is the product of the first two columns.

3.5

Fitting an identical-firms model to the heterogeneous economy

In this subsection I revisit the question of which parameterization for an identical-firms economy best approximates the dynamics of a heterogeneous economy, now in terms of the response to interest rate shocks. I keep the same set of parameters used in the previous subsection, but in addition consider shocks with different degrees of persistence, and different coefficients for the policy rule. For concreteness, I consider three values for κ: 0.91, 0.95, and 0.98,31 and also report the results with φπ = 1.24, and φy = 0. In analogy with Subsection 2.6, I fit the IRFs of an identical-firms model to those of the calibrated heterogeneous economy by finding the frequency of price changes that minimizes the sum of squared deviations between the two sets of IRFs of a given endogenous variable. I consider three different metrics, corresponding to the IRFs for output, prices, and inflation.32 The value of the remaining structural parameters in the identical-firms economy is set equal to the values used to generate the IRFs of the heterogeneous economy. The results with the calibrations described above are presented in Table 5. In all cases, I find that the best fitting identical-firms economy features substantially more nominal rigidity than the average-frequency economy. In 31

This covers parameter values in the estimated 0.75 - 0.95 range (with quarterly data; see n. 29). 32 I experimented with yet additional IRFs and combinations thereof, and found that the results are essentially unchanged. This is also the case for the results with the policy rule featuring interest rate smoothing, presented in the Appendix. One way around having to choose an arbitrary endogenous variable as the basis for constructing the metric for the exercise would be to take a somewhat different approach and use a formal measure of fit for calibrated models, such as the ones proposed by Watson (1993). This would require augmenting the model with additional disturbances. I leave this for future research. http://www.bepress.com/bejm/frontiers/vol2/iss1/art1

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almost all cases this is also true of the average-duration economy. The extent of additional nominal rigidity increases in the degree of real rigidities in the economy, as a result of the strategic interaction effect. Table 6 reports similar results with φπ = 1.24, φy = 0. Table 5: Best-Fitting Durations d∗id - φπ = 1.24, φy = 0.33/12∗

κ = 0.91 κ = 0.95 κ = 0.98 ϕ σ ε yt pt π t yt pt π t yt pt πt 0.50 1 11 8.5 8.7 7.2 8.9 8.8 7.7 9.1 9.0 8.6 1.50 1 5 7.8 8.0 6.7 8.2 8.2 7.2 8.7 8.6 8.1 ∗ Durations are reported in months; yt , pt , π t indicate the variable used to construct the metric; sample “inverse average frequency” duration is 2.9 months; sample average duration of price rigidity is 6.6 months.

Table 6: Best-Fitting Durations d∗id - φπ = 1.24, φy = 0∗

κ = 0.91 κ = 0.95 κ = 0.98 ϕ σ ε yt pt π t yt pt π t yt pt πt 0.50 1 11 9.7 8.3 7.0 9.5 8.4 7.4 9.2 8.4 7.8 1.50 1 5 8.4 7.4 6.3 8.5 7.5 6.6 8.6 7.6 6.7 ∗ Durations are reported in months; yt , pt , π t indicate the variable used to construct the metric; sample “inverse average frequency” duration is 2.9 months; sample average duration of price rigidity is 6.6 months.

These results are fully consistent with the analysis of the generalized new Keynesian Phillips curve presented in the previous subsection. For an identicalfirms model to generate a recession that is comparable to the one observed in the heterogeneous economy after a contractionary interest rate shock, and yet be consistent with the relatively muted inflation response, it must feature a relatively flat Phillips curve. Given the degree of real rigidities, this must be attained through a lower frequency of price changes.

4

Conclusion

Most sticky price models build on the assumption that firms change prices with the same frequency. This would be a good approximation either if empirically Published by The Berkeley Electronic Press, 2006

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the degree of heterogeneity in price stickiness were small or if, despite being significant, heterogeneity turned out not to matter. In this paper I show that this is not the case, and that heterogeneity in price stickiness should have a larger role in models used to analyze the real effects of monetary shocks. I present results showing that heterogeneity affects the dynamic response of economies to monetary shocks in important ways. In particular, relying on both analytical insights and results from a calibrated model, I find that empirically plausible monetary shocks tend to generate larger and more persistent real effects in the presence of heterogeneity in price stickiness, relative to otherwise identical models in which all firms change prices with the same frequency. In the presence of strategic complementarities in price setting, sectors in which prices are adjusted less frequently have a disproportionate effect on the aggregate price level. This interaction further amplifies the role of heterogeneity in price stickiness in generating richer dynamics after the economy is hit with a monetary shock.

4.1

Scope of the results

The results on heterogeneity in sectoral price setting behavior are presented in this paper in the context of the Calvo pricing model. However, it should be clear that the frequency composition effect is likely to be present in any model with heterogeneity in the frequency of price changes. In addition, models with complementarities in price setting will likely also feature the strategic interaction effect due to heterogeneity in price stickiness. In fact, as shown in Carvalho (2005) and Carvalho and Schwartzman (2006), the results of this paper extend to a large class of alternative price setting specifications, including Taylor (1979, 1980) staggered pricing, and sticky information models as in Mankiw and Reis (2002). Moreover, the reduced-form model analyzed in Section (2) can be obtained from different fully specified models, with different sources of real rigidity. This suggests that heterogeneity in price setting behavior and its interaction with strategic complementarities may have an important role to play in models of monetary economies, irrespective of the nature of frictions to price adjustment, and of the source of real rigidities.33 In this paper I take the cross-sectional distribution of the frequency of price changes as given. A natural step is to assess whether the findings in this 33

In a recent paper, Coibion and Gorodnichenko (2006) highlight the interactions between firms that follow different price-setting rules in a model with sticky-price, sticky-information, rule-of-thumb and flexible-price firms. Bonomo (1992) shows that interactions between timeand state-dependent pricing firms reinforce their roles in generating monetary non-neutrality. http://www.bepress.com/bejm/frontiers/vol2/iss1/art1

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paper extend to a setting in which the frequency of price changes is chosen by firms. This can be done with either endogenous time-dependent pricing rules (Bonomo and Carvalho 2004, 2006),34 or state-dependent pricing policies (e.g. Barro 1972, Sheshinski and Weiss 1977). In both classes of models, the cross-sectional distribution of the frequency of price changes will depend on how factors such as the frictions to price-setting, the incidence of sectoral- and firm-specific shocks, and the degree of competition, vary across sectors. With endogenous time-dependent pricing, the frequency of price changes is chosen optimally by firms, but nevertheless the latter do not react to shocks in between pricing decisions. As a result, endogenizing the cross-sectional distribution of the frequency of price changes with such models should leave the results of this paper essentially unchanged, if the differences in the frequency of price changes across sectors stem from factors that do not affect the structure of the underlying frictionless economy. This is the case with differences in information/adjustment costs, and in the law of motion of sectoral and firmspecific shocks, for example. With state-dependent pricing, firms can react to shocks at all times, which implies that price changes made in response to a monetary shock involve the so-called “selection effect:” firms that do adjust are the ones for which it pays to incur the menu cost and make a price change.35 The selection effect will be at work in a model with heterogeneity in state-dependent pricing. However, that need not affect the way in which heterogeneity influences aggregate dynamics. The critical issue here is whether the selection effect will vary systematically in the cross-section of frequencies of price adjustment chosen in equilibrium by firms in different sectors. Absent such a systematic relationship, there is no reason to expect the effects of heterogeneity to be attenuated or amplified in a 34

Such rules are optimal if firms face a joint information/adjustment cost, and as a result choose to set a fixed price until an optimally chosen date when they will again incur the cost to gather and process information about economic conditions, and change prices. Ball, Mankiw and Romer (1988) propose this type of pricing policy in the presence of menu costs as a suboptimal yet simpler substitute for state-contingent policies. In Romer (1990), an optimally chosen frequency of price adjustment in a Calvo-type model serves as a tractable (suboptimal) alternative to state-dependent policies. Reis (2006) in turn provides microfoundations for time-dependent pricing in the absence of adjustment costs. Information gathering and processing costs lead firms to update their pricing policies at optimally chosen dates, in between which prices follow a preset path. 35 In its extreme form, originally emphasized by Caplin and Spulber (1987), such an effect can be strong enough to lead to monetary neutrality, despite the frictions that prevent continuous price adjustment at the firm level. Such a selection effect was later shown to hold somewhat more generally, albeit usually in attenuated form, by Caballero and Engel (1991, 1993), and more recently by Danziger (1999), Golosov and Lucas (2005), Midrigan (2006), Gertler and Leahy (2006), and Caballero and Engel (2006). Published by The Berkeley Electronic Press, 2006

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state-dependent pricing context, relative to the effects obtained in this paper. The most straightforward way to extend the present analysis to a statedependent pricing context would be to introduce heterogeneity in menu costs and in the rate of incidence of idiosyncratic shocks in a model similar to Gertler and Leahy (2006). This would allow investigation of any systematic relationship between the selection effect and the distribution of price rigidity across sectors, and assessment of whether the results obtained here indeed extend to state-dependent pricing models as well.36 As a by-product, this approach should also deliver a generalized NKPC that accounts for heterogeneity in price stickiness, with a state-dependent pricing foundation. Its relationship to the generalized NKPC presented in this paper is likely to parallel the one between the standard NKPC based on Calvo pricing, and the (S,s)-based NKPC derived by Gertler and Leahy (2006).

4.2

Implications for related research

Heterogeneity in price stickiness affects aggregate dynamics in quantitatively important ways. From a normative perspective, given that it affects optimal monetary policy (Aoki, 2001, and Benigno, 2004), it might be worthwhile to assess its quantitative implications in a realistically calibrated model. From an empirical perspective, given the different nature of the dynamics of heterogeneous economies, the models can be formally tested against standard identicalfirms models in a statistical sense. In the context of estimation of dynamic, stochastic, general equilibrium models with nominal rigidity, this calls for the introduction of multiple sectors with different degrees of stickiness.37 For the large literature on single equation estimation of the NKPC, the results can 36

Actually, in important work on heterogeneity and (S,s) policies, Caballero and Engel (1991, 1993) obtain results that suggest the presence of the same kind of mechanisms due to heterogeneity that were highlighted in this paper. In particular, Caballero and Engel (1991, Section 6) find that allowing for differences in S,s bands (what they call “structural heterogeneity”) may slow the speed of adjustment of the economy back to steady state after it is hit by a shock. In addition, Nakamura and Steinsson’s (2006b) multi-sector menu cost model implies quantitative effects of heterogeneity that are very similar to the ones obtained in this paper. 37 Smets and Wouters (2003) mention the possibility that heterogeneity could bias their results (n. 3). Christiano et al. (2005) claim that “inference about nominal rigidities is sensitive to getting the real side of the model right,” but abstract from the possibility that the same applies to heterogeneity in the frequency of price changes. Coenen et al. (2004, 2006) and Boukaez et al. (2005) represent promising steps in the direction of incorporating heterogeneity in price rigidity into estimated macroeconomic models. Jadresic (1999) presents earlier econometric evidence that heterogeneity in price stickiness improves the performance of sticky price models when applied to U.S. data. http://www.bepress.com/bejm/frontiers/vol2/iss1/art1

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be regarded as rationalizing the inclusion of sectoral data in the regressions (relative prices or sectoral output gaps).38 Models like the one analyzed in this paper may also be able to shed light on the evidence of fast adjustment to sectoral conditions and slow response to aggregate conditions documented recently by Altissimo et al. (2004) and Boivin et al. (2006).39 Finally, when using identical-firms models and relating them to microeconomic evidence on pricing behavior, we should account for the fact that one potentially important source of persistence is missing. Knowledge of how heterogeneity affects aggregate dynamics should lead us to deal with the parameters of (misspecified) identical-firms models more appropriately.

Appendix I start by presenting the derivation of the reduced-form model in continuous time, and the proofs of Propositions 1-2 and of the associated Corollaries. Then, I present the log-linear approximation to the fully specified model, followed by additional results with a policy rule featuring interest rate smoothing, and the generalized NKPC. 1) The reduced-form continuous time model The counterpart of equation (1) is: p∗ (t) − p (t) = θy (t) ,

(15)

where the notation is evident by analogy. The aggregate price level is given by: Z 1 Z 1 Z 1 f (k) pk,j (t)djdk = f (k) pk (t) dk. (16) p(t) = 0

0

0

Taking the limit as ∆ → 0 in equations (3) and (5) yields: Z ∞ xk (t) = arg min e−(δ+αk )s Et [x − p∗ (t + s)]2 ds x

(17)

0

38

Some authors have argued for the inclusion of relative prices of goods such as crude oil and “food and energy,” or of imported goods. Their argument was that these terms would proxy for supply shocks (e.g. Roberts, 1995, and Gordon, 1997). 39 Using different methodologies, the two papers present results for sectoral prices, and the latter paper also presents results for the response of sectoral quantities to identified monetary policy shocks. Their findings are in the spirit of Imbs et al. (2005), who contrast the dynamics of deviations from the law of one price at disaggregated levels with the behavior of the real exchange rate. Published by The Berkeley Electronic Press, 2006

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= (δ + αk )

Z

∞

e−(δ+αk )s Et p∗ (t + s) ds,

0

pk (t) =

Z

t

αk e−αk (t−s) xk (s) ds,

(18)

−∞

where the mapping between the discrete and the continuous time models is given in Table 2. 2) Steady state and shocks To obtain the IRFs used subsequently in the proofs of Propositions 1-2, I analyze one-time shocks in the continuous time model, starting from a deterministic steady state. This approach is justified because of certainty equivalence. In the steady state, nominal aggregate demand, m (t) = y (t) + p (t), grows at a constant rate µ ≥ 0. This implies that, after a normalization, m (t) = µt. Firms in sector k set prices as: Z ∞ e−(δ+αk )s (θ (µt + µs) + (1 − θ) p (t + s)) ds. xk (t) = (δ + αk ) 0

The aggregate price level is given implicitly by: Z 1 Z t Z ∞ −αk (t−s) p (t) = θ f (k) αk e (δ + αk ) e−(δ+αk )r (µs + µr) drdsdk 0

+ (1 − θ)

Z

0

−∞

1

f (k)

Z

0

t −αk (t−s)

αk e

(δ + αk )

−∞

Z

∞

e−(δ+αk )r p (s + r) drdsdk.

0

Using the method of undetermined coefficients, it is straightforward to show that the aggregate price level also grows at rate µ, and is given by: Z 1 δµ dk, f (k) p (t) = µt − θ (δαk + α2k ) 0 while individual prices are set according to: µ ¶ 1 δ xk (t) = µt + µ − . αk θ (δαk + α2k ) R1

δµ dk. Inciden(δαk +α2k ) tally, notice that the usual result of non-superneutrality of money in simple reduced-form Calvo models extends naturally to the setting with heteroge-

Output is constant at the natural rate y (t) =

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f (k) θ

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neous firms. The distribution of adjustment frequencies interacts with the degree of strategic complementarities in price setting in determining the steady state output gap. In the context of level shocks, assume that µ = 0, and that at t = 0 nominal aggregate demand is hit by a shock of size u0 , which then decays exponentially at rate ρ ≥ 0. For t ≥ 0, the path for nominal aggregate demand is therefore given by m (t) = u0 e−ρt . After learning of the shock, whenever a firm from group k gets a chance to adjust its price it sets: Z ∞ xk (t) = (δ + αk ) e−(δ+αk )s Et p∗ (t + s) ds Z0 ∞ ¢ ¡ = (δ + αk ) e−(δ+αk )s θu0 e−ρ(t+s) + (1 − θ) p (t + s) ds. 0

The corresponding path for the aggregate price level is defined implicitly by: Z 1 Z t p (t) = f (k) αk e−αk (t−s) xk (s) dsdk (19) 0

Z

Z

1

−∞

Z

t

∞

f (k) αk e (δ + αk ) e−(δ+αk )r u0 e−ρ(s+r) drdsdk 0 0 Z t Z ∞ Z 01 −αk (t−s) f (k) αk e (δ + αk ) e−(δ+αk )r p (s + r) drdsdk, + (1 − θ)

= θ

−αk (t−s)

0

0

0

where the second integrals in each term of the last expression range from 0 (and not from −∞) to t because µ = 0 implies p (0) = 0. · In the case of a growth rate shock, m (t) jumps at t = 0 from µ to µ + u0 , where u0 is the size of the shock. Thereafter the shock decays exponentially at Rt · · −γt rate γ ≥ 0, so that m (t) = µ+u0 e−γt , and m (t) = 0 m (s) ds = µt+u0 1−eγ . After learning of the shock, whenever a firm from group k gets a chance to change its price it sets: Z ∞ xk (t) = (δ + αk ) e−(δ+αk )s Et p∗ (t + s) ds 0

= (δ + αk )

Z

0

∞

−(δ+αk )s

e

µ µ ¶ ¶ 1 − e−γ(t+s) θ µt + µs + u0 + (1 − θ) p (t + s) ds. γ

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The corresponding path for the aggregate price level is defined implicitly by: Z t Z 1 f (k) αk e−αk (t−s) xk (s) dsdk (20) p (t) = 0 −∞ µ µ ¶¶ Z 0 Z 1 1 δ −αk (t−s) µs + µ f (k) αk e − dsdk = αk θ (δαk + α2k ) 0 −∞ ¶ µ Rt Z 1 αk e−αk (t−s) (δ + αk ) 0 R dk +θ f (k) ∞ × 0 e−(δ+αk )r µ (s + r) drds 0 ! à Rt Z 1 −αk (t−s) α e (δ + α ) k k R ∞0 −(δ+α )r 1−e−γ(s+r) +θ dk f (k) k × e u0 drds 0 γ 0 ¶ µ Rt Z 1 −αk (t−s) α e (δ + α ) k k 0 R∞ dk. + (1 − θ) f (k) × 0 e−(δ+αk )r p (s + r) drds 0 3) Proof of Proposition 1 Set θ = 1 in (19). Then: Z 1 Z t Z ∞ −αk (t−s) p (t) = f (k) αk e (δ + αk ) e−(δ+αk )r u0 e−ρ(s+r) drdsdk 0 0 0 ¸ ∙ −ρt Z 1 (e − e−αk t ) αk (αk + δ) = u0 dk. f (k) (αk − ρ)(αk + ρ + δ) 0 R∞ The expected normalized cumulative real effects, u10 E0 0 m (t) − p (t) dt, are given by: ∙ ¸ Z 1 Z 1 ∞ (e−ρt − e−αk t ) αk (αk + δ) −ρt dk dt u0 e − f (k) u0 0 (αk − ρ)(αk + ρ + δ) 0 ∙ ¸ (e−αk t − e−ρt ) αk (αk + δ) −ρt f (k) e + = dkdt (αk − ρ)(αk + ρ + δ) 0 0 ¸ Z ∞∙ Z 1 (e−αk t − e−ρt ) αk (αk + δ) −ρt e + f (k) dtdk = (αk − ρ)(αk + ρ + δ) 0 0 Z 1 1 f (k) dk. = αk + ρ + δ 0 Z

∞

Z

1

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Carvalho: Heterogeneity in Price Stickiness

4) Proof of Corollaries 1 and 2 1 is convex in αk , Jensen’s inequality combined with Corollary 1: Since αk +ρ+δ Proposition 1 delivers the result. 1 Corollary 2: Since αk +ρ+δ is concave in α−1 k Jensen’s inequality implies that for ρ, δ > 0, the expected (normalized) cumulative real effect of a level shock in the heterogeneous economy is smaller than in anR identical-firms economy with 1 the same average duration of price rigidity d ≡ 0 f (k) α−1 k dk. For ρ = δ = 0, the result simplifies to d.

5) Proof of Proposition 2 Recall that I assume δ = 0. Set θ = 1 in (20). Then: ⎫ ⎧ R ³ ´ R Z 1 ⎨ 0 αk e−αk (t−s) µs + µ ds + t α2 e−αk (t−s) × ⎬ αk −∞ 0 k ³ ´ i hR dk p (t) = f (k) −γ(s+r) ∞ 1−e −α r ⎭ ⎩ k 0 µs + µr + u0 dr ds × 0 e γ Z 1 2 −γt 2 −αk t ) α (e − 1) + γ (1 − e = µt + u0 dk. f (k) k 2 2 γ (γ − αk ) 0 R∞ The expected normalized cumulative real effects, u10 E0 0 m (t) − p (t) dt, are given by: Z Z 1 1 − e−γt α2 (e−γt − 1) + γ 2 (1 − e−αk t ) 1 ∞ µt + u0 f (k) k − µt − u0 dkdt u0 0 γ γ (γ 2 − α2k ) 0 ¸ 1 − e−γt α2k (e−γt − 1) + γ 2 (1 − e−αk t ) = − dkdt f (k) γ γ (γ 2 − α2k ) 0 0 Z 1 1 = f (k) dk. γαk + α2k 0 Z

∞Z 1

∙

Denote the expected normalized cumulative real effects as a function of γ by: n (γ) =

Z

1

f (k)

0

α2k

1 dk. + γαk

A first order approximation around γ = 0 yields: µZ 1 ¶ ¡ ¢ 1 n (γ) = n (0) − f (k) 3 dk γ + O γ 2 . αk 0 Published by The Berkeley Electronic Press, 2006

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Therefore, for γ ≈ 0, the expected normalized cumulative real effects are approximately equal to: Z 1 1 2 f (k) 2 dk = d + σ 2d , n (0) = αk 0 where

σ 2d

≡

R1 0

f (k)

³

1 αk

´2 − d dk.

6) Effects of growth rate shocks with discounting and Proposition 2 The reason why the second result in Proposition 2 only holds approximately is that the expression derived for the expected normalized cumulative real effects is not valid for the case of permanent growth rate shocks (γ = 0). In that case, the output gap is identically zero in the model with heterogeneity, as well as with identical-firms. This is a feature of Calvo pricing which has not received much attention in the literature, although it has been documented elsewhere. Mankiw and Reis (2002), for example, find the same result with the NKPC in their “sudden disinflation” experiment. When they consider temporary shocks to the growth rate of the money supply, however, they find that there are real effects in the Calvo economy. The result is not due to the lack of real rigidities, because Mankiw and Reis’s (2002) experiments do account for them. So, treating the expected normalized cumulative effect as a function of the degree of persistence of the shock, the fact that they differ for a permanent and an arbitrarily persistent shock implies a discontinuity at γ = 0.40 Economically, what underlies this discontinuity is the fact that in the case of extremely persistent (but temporary) shocks, most of its real effects are spread out into the distant future. When the shock hits, firms know that nominal income growth will remain at the new level for a long time, but that it will, eventually, return to the previous level. Because of nominal rigidity, this generates small, but nevertheless non-zero real effects, which last for a long time. When the shock is permanent, however, the real effects are exactly zero. This discontinuity disappears, however, when there is discounting (δ > 0).41 In this case, the relevant equations for t ≥ 0 for a permanent shock (γ = 0)

40

Technically, it arises because, although y (t) → 0 pointwise as γ → 0, it is not uniformly integrable on [0, ∞), and the exchange of limit and integration yields a different result. 41 Mankiw and Reis (2002) abstract from discounting in their analysis of the Calvo model. http://www.bepress.com/bejm/frontiers/vol2/iss1/art1

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Carvalho: Heterogeneity in Price Stickiness

are: xk (t) = (δ + αk ) = (δ + αk )

Z

∞

Z0 ∞

e−(δ+αk )s Et p∗ (t + s) ds e−(δ+αk )s (µt + µs + u0 t + u0 s) ds,

0

h ³ ´i ⎧ R 0 1 δ −αk (t−s) ⎪ µs + µ ds α e − ⎨ 2 1 αk −∞ k δαk +αk R t f (k) p (t) = + α e−αk (t−s) (δ + αk ) × ⎪ 0 ⎩ R ∞ −(δ+α0 k )rk × 0 e (µs + µr + u0 s + u0 r) drds Z

⎫ ⎪ ⎬ ⎪ ⎭

dk.

Computing the expected normalized and discounted cumulative real effects yields: Z 1 Z ∞ 1 δµ + αk (u0 + µ) 1 −δt dk. E0 e (m (t) − p (t)) dt = f (k) u0 u0 αk (αk + δ) (αk + δ) 0 0 Note that besides considering discounting in the objective function of firms, I am also discounting the real effects of the shock. To depart from the zero inflation steady state, set µ = 0 to obtain: Z 1 1 f (k) dk. (21) (αk + δ)2 0 For a temporary shock, the equations are: Z ∞ e−(δ+αk )s Et p∗ (t + s) ds xk (t) = (δ + αk ) 0 µ ¶ Z ∞ 1 − e−γ(t+s) −(δ+αk )s = (δ + αk ) µt + µs + u0 e ds, γ 0 ¶¸ 1 δ µs + µ dsdk p (t) = f (k) αk e − αk δαk + α2k 0 −∞ Z 1 Z t Z ∞ −αk (t−s) + f (k) αk e (δ + αk ) e−(δ+αk )r (µs + µr) drdsdk 0 0 0 Z 1 Z t Z ∞ 1 − e−γ(s+r) + f (k) αk e−αk (t−s) (δ + αk ) e−(δ+αk )r u0 drdsdk. γ 0 0 0 Z

1

Z

0

−αk (t−s)

∙

µ

Computing the expected normalized and discounted cumulative real effects

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yields: 1 E0 u0

Z

0

∞

−δt

e

(m (t) − p (t)) dt =

Z

1

f (k)

0

1 (δ + γ) µ + αk (u0 + µ) dk. u0 αk (αk + δ) (αk + δ + γ)

Starting from the zero inflation steady state (µ = 0), expected normalized and discounted cumulative real effects are instead given by: Z 1 1 dk. f (k) (αk + δ) (αk + δ + γ) 0 As γ → 0, this clearly converges to: Z 1 f (k) 0

1 dk, (αk + δ)2

which is exactly the expected normalized (and discounted) cumulative real effect in the case of a permanent shock, as just derived (equation 21).42 This discontinuity result need not, however, undermine the usefulness of the approximation used in Proposition 2. Since the discontinuity disappears when there is discounting, the approximation derived for permanent shocks when δ = 0 may still be useful for realistic values of the discount rate. This is indeed the case for a heterogeneous economy calibrated from the BK data, as the results in Table A.1 show. As a matter of fact, the exact result for temporary shocks can also be viewed as an approximation to the more realistic case of δ > 0. For varying degrees of persistence of the shock,43 Table A.1 presents the expected normalized cumulative real effects as derived in Proposition 2, as well as the exact expected normalized and discounted cumulative effect, for discount rates ranging from 0.01 to 0.05 (per year). It is clear that the approximation is quite accurate, and that for very persistent shocks the real effects are indeed approximately proportional to the second moment of the distribution of expected durations of price rigidity.

42

Technically, the reason why the discontinuity disappears is that, with discounting, y (t) becomes uniformly integrable on [0, ∞). 43 I also present the half-lives. The time unit is one year.

−δt

e

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Carvalho: Heterogeneity in Price Stickiness

Table A.1: Expected Normal. Cumul. Real Effects of Growth Rate Shock∗ Discount Rate (δ ) Proposition 2 0.01 0.02 0.03 0.04 0.00 ∞ 0.65 0.62 0.59 0.57 0.55 0.07 10 0.56 0.54 0.52 0.50 0.49 0.14 5 0.50 0.49 0.47 0.46 0.44 0.23 3 0.45 0.43 0.42 0.41 0.40 0.35 2 0.39 0.38 0.37 0.36 0.36 0.46 1.5 0.36 0.35 0.34 0.33 0.32 0.69 1 0.30 0.29 0.29 0.28 0.27 1.39 0.5 0.21 0.20 0.20 0.20 0.19 ∗ Heterogeneous economy calibrated from the BK data. θ = 1.

γ

Half-Life (years)

0.05 0.53 0.47 0.43 0.39 0.35 0.32 0.27 0.19

7) The log-linear approximation of the fully specified model The equations characterizing the log-linear approximation of the fully specified model around the zero inflation steady state, some of which were presented in the main text, are (lowercase variables denote log deviations from such steady state): Z 1 ct = f (k) ck,t dk, 0 Z 1 ckj,t dj, ck,t = 0 Z 1 pt = f (k) pk,t dk, 0 Z 1 pkj,t dj, pk,t = wkj,t − pt ct ck,t ckj,t ckj,t bt xkj,t pk,t

0 −1

ϕ lkj,t + σct , kj ∈ [0, 1]2 , Et ct+1 − σ −1 (it − Et π t+1 ) , ct − ε (pk,t − pt ) , ck,t − ε (pkj,t − pk,t ) , ykj,t = nkj,t = lkj,t , 0, X∞ = (1 − β (1 − λk )) Et β s (1 − λk )s wkj,t+s , = = = = = =

s=0

= λk xk,t + (1 − λk ) pk,t−1 ,

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where xk,t denotes the common price chosen by all firms kj that change prices at t. The model is closed with a log-linear version of the interest rate rule. After firm kj sets its price xkj,t at time t it faces a demand for its variety given by ykj,t+s = yt+s − ε (xkj,t − pt+s ), until it changes its price again. This determines its demand for labor of its specific type, nkj,t = ykj,t . Equating it to the relevant labor supply yields an equation for the equilibrium wage for labor of type kj as a function of aggregate variables and xkj,t : ¡ ¢ ¡ ¢ wkj,t+s = 1 + ϕ−1 ε pt+s + ϕ−1 + σ yt+s − ϕ−1 εxkj,t . So, the reduced set of equations that characterize the dynamics of prices and output in the log-linear model simplifies to: ∙ ¸ X∞ ϕ−1 + σ s s yt+s , β (1 − λk ) pt+s + xk,t = (1 − β (1 − λk )) Et s=0 1 + ϕ−1 ε pk,t = λk xk,t + (1 − λk ) pk,t−1 , Z 1 pt = f (k) pk,t dk, 0

yt = yk,t = it = υt =

Et yt+1 − σ −1 (it − Et π t+1 ) , yt − ε (pk,t − pt ) , φπ π t + φy yt + υ t , κυ t−1 + ξ t .

8) Results with interest rate smoothing Despite the evidence presented by Rudebusch (2002), a large part of the empirical literature on interest rate rules allows for interest rate smoothing. Therefore, here I consider an alternative specification for the policy rule that features interest rate smoothing instead of the persistent shock: " µ ¶φπ µ ¶φy #1−ν Pt Yt −1 (It−1 )ν eqt , It = β Pt−1 Y where ν is the interest rate smoothing parameter, and qt is a zero mean, finite variance i.i.d. shock. In log-linear form, the policy rule becomes: £ ¤ it = (1 − ν) φπ πt + φy yt + νit−1 + qt .

In calibrating this policy rule I set φπ = 1.53, and φy = 0.93/12 (based on

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Carvalho: Heterogeneity in Price Stickiness

Rudebusch, 2002), and I also report the results with φπ = 1.53, and φy = 0. I consider three values for ν: 0.91, 0.95,and 0.981, and use the same calibration reported in Subsection 3.4 for the remaining structural parameters. The results are reported in Tables A.2 and A.3. Table A.2: Best-Fitting Durations d∗id - φπ = 1.53, φy = 0.93/12∗

ϕ σ 0.50 1 1.50 1 ∗

ε 11 5

ν yt 6.6 5.8

Durations are reported in

= 0.91 ν = 0.95 ν pt π t yt pt π t yt 8.4 6.6 7.0 8.5 6.8 7.8 7.5 6.0 6.1 7.7 6.1 6.6 months; yt , pt , π t indicate the variable used to

= 0.98 pt πt 8.6 7.3 7.9 6.4 construct the

metric; sample “inverse average frequency” duration is 2.9 months; sample average duration of price rigidity is 6.6 months.

Table A.3: Best-Fitting Durations d∗id - φπ = 1.53, φy = 0∗

ν = 0.91 ν = 0.95 ν ϕ σ ε yt pt π t yt pt π t yt 0.50 1 11 7.3 7.3 6.1 7.4 7.5 6.2 7.6 1.50 1 5 6.1 6.4 5.2 6.2 6.6 5.3 6.4 ∗ Durations are reported in months; yt , pt , π t indicate the variable used to

= 0.98 pt πt 7.8 6.4 7.0 5.5 construct the

metric; sample “inverse average frequency” duration is 2.9 months; sample average duration of price rigidity is 6.6 months.

9) The generalized NKPC Define p∗t = pt + θyt , with θ =

ϕ−1 +σ . 1+εϕ−1

Rewrite (8) as:

xk,t = (1 − (1 − λk ) β) = (1 − (1 −

∞ X

((1 − λk ) β)j Et p∗t+j

j=0 λk ) β) p∗t +

(22)

(1 − λk ) βEt xk,t+1 .

Sectoral price indices are given by: pk,t = λk

∞ X j=0

(1 − λk )j xk,t−j

(23)

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Multiplying by f (k) and integrating across sectors yields the aggregate price index: Z 1 f (k) pk,t dk (24) pt = 0 Z 1 = f (k) (λk xk,t + (1 − λk ) pk,t−1 ) dk. 0

Leading the above equation and taking expectations as of time t yields: Z 1 Z 1 Et pt+1 = f (k) λk Et xk,t+1 dk + f (k) (1 − λk ) pk,t dk. (25) 0

0

From (22), solve for Et xk,t+1 to get: Et xk,t+1 =

1 − (1 − λk ) β ∗ xk,t − p. (1 − λk ) β (1 − λk ) β t

Multiplying by f (k) λk and integrating across sectors yields: ¸ ∙ Z 1 Z 1 λk xk,t λk (1 − (1 − λk ) β) ∗ f (k) λk Et xk,t+1 dk = f (k) − pt dk (1 − λk ) β (1 − λk ) β 0 0 (26) Subtracting (24) from (25), and using (26) delivers: Et pt+1 − pt = =

Z

0

1

Z

f (k) λk Et xk,t+1 dk − 1

Z

1

f (k) λk xk,t dk

0

f (k) (1 − λk ) (pk,t − pk,t−1 ) dk + 0 Z 1 Z 1 λk λk (1 − (1 − λk ) β) ∗ = xk,t dk − pt dk f (k) f (k) (1 − λk ) β (1 − λk ) β 0 0 Z 1 Z 1 − f (k) λk xk,t dk + f (k) (1 − λk ) pk,t dk 0 0 Z 1 f (k) (1 − λk ) pk,t−1 dk − 0

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Carvalho: Heterogeneity in Price Stickiness

¶ Z 1 λk λk (1 − (1 − λk ) β) ∗ − λk xk,t dk − pt dk = f (k) f (k) (1 − λk ) β (1 − λk ) β 0 0 Z 1 Z 1 f (k) pk,t dk − f (k) λk pk,t dk + 0 0 | {z } Z

1

µ

=pt

Z 1 Z 1 − f (k) pk,t−1 dk + f (k) λk pk,t−1 dk, 0 0 | {z } =pt−1

so that:

Z

1 1 − (1 − λk ) β λk xk,t dk f (k) Et pt+1 − pt = pt − pt−1 + (1 − λk ) β 0 Z 1 λk (1 − (1 − λk ) β) ∗ − pt dk f (k) (1 − λk ) β 0 Z 1 − f (k) λk (pk,t − pk,t−1 ) dk.

(27)

0

Now, from (23): λk xk,t = pk,t − (1 − λk ) pk,t−1 ⇐⇒ 1 − (1 − λk ) β 1 − (1 − λk ) β λk xk,t = (pk,t − (1 − λk ) pk,t−1 ) (1 − λk ) β (1 − λk ) β µ ¶ 1 = − 1 (pk,t − (1 − λk ) pk,t−1 ) (1 − λk ) β ¶ µ 1−β 1 − 1 − λk − pk,t = (1 − λk ) β β ¶ µ 1−β + + λk (pk,t − pk,t−1 ) β λk (1 − (1 − λk ) β) (28) pk,t = (1 − λk ) β 1−β + (pk,t − pk,t−1 ) + λk (pk,t − pk,t−1 ) . β

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Replacing (28) into (27) yields: ¸ ∙ Z 1 λk (1 − (1 − λk ) β) Et pt+1 − pt = pt − pt−1 + pk,t dk f (k) (1 − λk ) β 0 ∙ ¸ Z 1 1−β (pk,t − pk,t−1 ) + λk (pk,t − pk,t−1 ) dk f (k) + β 0 µ ¶ Z 1 λk (1 − (1 − λk ) β) ∗ pt − λk (pk,t − pk,t−1 ) dk f (k) − (1 − λk ) β 0 # " Z 1 λk (1−(1−λk )β) (p − (p + θy )) k,t t t (1−λk )β dk f (k) = pt − pt−1 + (pk,t − pk,t−1 ) + 1−β 0 β Z 1 1 λk (1 − (1 − λk ) β) = (pt − pt−1 ) + (pk,t − pt ) dk f (k) β (1 − λk ) β 0 Z 1 λk (1 − (1 − λk ) β) − θyt dk. f (k) (1 − λk ) β 0 Multiplying by β, using the fact that pt −pk,t = ε−1 (yk,t − yt ), and rearranging yields: ¶ ¶ ¶ µZ 1 µ µ 1 λk pt − pt−1 = βEt (Et pt+1 − pt ) + θ − f (k) − βλk dk yt ε 1 − λk 0 ¶ µ Z 1 1 λk + f (k) − βλk yk,t dk. (29) ε 0 1 − λk Finally, let π t ≡ pt − pt−1 denote the inflation rate in period t. So: µ −1 ¶ ϕ +σ 1 ψ π t = βEt πt+1 + ψ − yt + gt , −1 1 + εϕ ε ε where: ψ ≡ gt ≡

Z

1

0

Z

1

0

fb(k) ≡ R 1 0

¶ λk f (k) − βλk dk, 1 − λk µ

fb(k) yk,t dk, λk 1−λk

f (k)

³

− βλk

λk 1−λk

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Carvalho: Heterogeneity in Price Stickiness

References [1] Altissimo, F., B. Mojon and P. Zaffaroni (2004), “Fast Micro and Slow Macro: Can Aggregation Explain the Persistence of Inflation?,” mimeo available at http://www.ecb.int/events/conferences/ html/inflationpersistence.en.html. [2] Alvarez, L., P. Burriel and I. Hernando (2005), “Do Decreasing Hazard Functions for Price Changes Make Any Sense?,” ECB Working Paper No. 461. [3] Aoki, K. (2001), “Optimal Monetary Policy Responses to Relative-Price Changes,” Journal of Monetary Economics 48: 55-80. [4] Ascari, G. (2004), “Staggered Prices and Trend Inflation: Some Nuisances,” Review of Economic Dynamics 7: 642-667. [5] Ball, L., N. G. Mankiw and David Romer (1988), “The New Keynesian Economics and the Output-Inflation Trade-off,” Brookings Papers on Economic Activity 1: 1-65. [6] Ball, L. and D. Romer (1989), “Equilibrium and Optimal Timing of Price Changes,” Review of Economic Studies 56: 179-198. [7]

(1990), “Real Rigidities and the Non-Neutrality of Money,” Review of Economic Studies 57: 183-203.

[8] Barro, R. (1972), “A Theory of Monopolistic Price Adjustment,” Review of Economic Studies 39: 17-26. [9] Barsky, R., C. House and M. Kimball (2006), “Sticky Price Models and Durable Goods,” forthcoming in the American Economic Review. [10] Benigno, P. (2001), “Optimal Monetary Policy in a Currency Area,” CEPR Working Paper No. 2755. [11]

(2004), “Optimal Monetary Policy in a Currency Area,” Journal of International Economics 63: 293-320.

[12] Bils, M. and P. Klenow (2002), “Some Evidence on the Importance of Sticky Prices,” NBER Working Paper #9069. [13]

(2004), “Some Evidence on the Importance of Sticky Prices,” Journal of Political Economy 112: 947-985.

Published by The Berkeley Electronic Press, 2006

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[14] Bils, M., P. Klenow and O. Kryvtsov (2003), “Sticky Prices and Monetary Policy Shocks,” Federal Reserve Bank of Minneapolis Quarterly Review 27: 2-9. [15] Blanchard, O. and N. Kiyotaki (1987), “Monopolistic Competition and the Effects of Aggregate Demand,” American Economic Review 77: 647666. [16] Blinder, A., E. Canetti, D. Lebow and J. Rudd (1998), Asking about Prices: A New Approach to Understanding Price Stickiness, Russell Sage Foundation. [17] Boivin, J., M. Giannoni and I. Mihov (2006), “Sticky Prices and Monetary Policy: Evidence from Disaggregated U.S. Data,” mimeo available at http://www.columbia.edu/~mg2190/. [18] Bonomo, M. (1992), “Time-Dependent and State-Dependent Pricing Rules: Interaction Reinforces Non-neutrality,” Ph.D. Dissertation, Chapter 1, Princeton University. [19] Bonomo, M. and C. Carvalho (2004), “Endogenous Time-Dependent Rules and Inflation Inertia,” Journal of Money, Credit and Banking 36: 1015-1041. [20]

(2006), “Imperfectly Credible Disinflation under Endogenous Time-Dependent Pricing,” mimeo available at www.princeton.edu/ ~cvianac/research.html.

[21] Boukaez, H., E. Cardia and F. Ruge-Murcia (2005), “The Transmission of Monetary Policy in a Multi-Sector Economy,” CIREQ Working Paper # 20-2005. [22] Caballero, R. and E. Engel (1991), “Dynamic (S, s) Economies,” Econometrica 59: 1659-1686. [23]

(1993), “Heterogeneity and Output Fluctuations in a Dynamic Menu-Cost Economy,” Review of Economic Studies 60: 95-119.

[24]

(2006), “Price Stickiness in Ss Models: Basic Properties,” mimeo available at http://econ-www.mit.edu/faculty/index.htm? prof_id=caball&type=paper.

[25] Calvo, G. (1983), “Staggered Prices in a Utility Maximizing Framework,” Journal of Monetary Economics 12: 383-98. http://www.bepress.com/bejm/frontiers/vol2/iss1/art1

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Carvalho: Heterogeneity in Price Stickiness

[26] Caplin, A. and D. Spulber (1987), “Menu Costs and the Neutrality of Money,” Quarterly Journal of Economics 102: 703-726. [27] Carlton, D. (1986), “The Rigidity of Prices,” American Economic Review 76: 637-658. [28] Carvalho, C. (2005), “Heterogeneity in Price Setting and the Real Effects of Monetary Shocks,” mimeo available at http://ideas.repec.org/ e/pca106.html. [29] Carvalho, C. and F. Schwartzman (2006), “Heterogeneous Price Setting Behavior and Aggregate Dynamics: Some General Results,” mimeo available at http://www.princeton.edu/~cvianac/research.html. [30] Carlstrom, C., T. Fuerst and F. Ghironi (2006a), “Does It Matter (for Equilibrium Determinacy) What Price Index the Central Bank Targets?,” Journal Economic Theory 128: 214-231. [31] Carlstrom, C., T. Fuerst, F. Ghironi and K. Hernandez (2006b), “Relative Price Dynamics and the Aggregate Economy,” mimeo available at http://copland.udel.edu/~kolver/research/index.html. [32] Christiano, L., M. Eichenbaum and C. Evans (2005), “Nominal Rigidities and the Dynamic Effects of a Shock to Monetary Policy,” Journal of Political Economy 115: 1-45. [33] Coenen, G. and A. Levin (2004), “Identifying the Influences of Nominal and Real Rigidities in Aggregate Price-Setting Behavior,” ECB Working Paper Series no. 418. [34] Coenen, G., A. Levin and K. Christoffel (2006), “Identifying the Influences of Nominal and Real Rigidities in Aggregate Price-Setting Behavior,” mimeo available at http://www.guentercoenen.com/. [35] Cogley, T. and A. Sbordone (2005), “A Search for a Structural Phillips Curve,” Federal Reserve Bank of New York Staff Report #203. [36] Coibion, O. and Y. Gorodnichenko (2006), “Strategic Interaction Among Heterogeneous Price-Setters in an Estimated DSGE Model,” mimeo available at http://www-personal.umich.edu/~ygorodni/. [37] Danziger, L. (1999), “A Dynamic Economy with Costly Price Adjustments,” American Economic Review 89: 878-901. Published by The Berkeley Electronic Press, 2006

53

Frontiers of Macroeconomics , Vol. 2 [2006], Iss. 1, Art. 1

[38] Dhyne, E., L. Álvarez, H. Le Bihan, G. Veronese, D. Dias, J. Hoffman, N. Jonker, P. Lünnemann, F. Rumler and J. Vilmunen (2006), “Price Changes in the Euro Area and the United States: Some Facts from Individual Consumer Price Data,” Journal of Economic Perspectives 20: 171-192. [39] Dixon, H. and E. Kara (2005), “Persistence and Nominal Inertia in a Generalized Taylor Economy: How Longer Contracts Dominate Shorter Contracts,” ECB Working Paper Series no. 489. [40] Gertler, M. and J. Leahy (2006), “A Phillips Curve with an Ss Foundation,” NBER Working Paper #11971. [41] Golosov, M. and R. Lucas (2005),“Menu Costs and Phillips Curves,” NBER Working Paper # 10187. [42] Gordon, R. (1997), “The Time-Varying NAIRU and its Implications for Economic Policy,” Journal of Economic Perspectives 11: 11-32. [43] Haltiwanger, J. and M. Waldman (1991), “Responders Versus NonResponders: A New Perspective on Heterogeneity,” Economic Journal 101: 1085-1102. [44] Imbs, J., H. Mumtaz, M. Ravn and H. Rey (2005), “PPP Strikes Back: Aggregation and the Real Exchange Rate,” Quarterly Journal of Economics 120: 1-43. [45] Jadresic, E. (1999), “Sticky Prices: An Empirical Assessment of Alternative Models,” IMF Working Paper # 99/72. [46] Klenow, P. and J. Willis (2006), “Real Rigidities and Nominal Price Changes,” Federal Reserve Bank of Kansas City Research Working Paper 06-03, available at http://www.kc.frb.org/Publicat/ Reswkpap/rwpmain.htm. [47] Mankiw, G. and R. Reis (2002), “Sticky Information Versus Sticky Prices: A Proposal to Replace the New Keynesian Phillips Curve,” Quarterly Journal of Economics 117: 1295-1328. [48] Midrigan, V. (2006), “Menu Costs, Multi-Product Firms and Aggregate Fluctuations,” mimeo available at http://web.econ.ohio-state.edu/ ~midrigan/research.html.

http://www.bepress.com/bejm/frontiers/vol2/iss1/art1

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[49] Nakamura, E. and J. Steinsson (2006a), “Five Facts About Prices: A Reevaluation of Menu Cost Models,” mimeo available at http:// www.people.fas.harvard.edu/~nakamura/papers.html. [50]

(2006b), “Monetary Non-Neutrality in a MultiSector Menu Cost Model,” mimeo available at http://www.people. fas.harvard.edu/~steinss/hu-papers.html.

[51] Ohanian, L., A. Stockman and L. Kilian (1995), “The Effects of Real and Monetary Shocks in a Business Cycle Model with Some Sticky Prices,” Journal of Money, Credit and Banking 27: 1209-1234. [52] Reis, R. (2006), “Inattentive Producers,” Review of Economic Studies 73: 793-821. [53] Roberts, J. (1995) “New Keynesian Economics and the Phillips Curve,” Journal of Money, Credit, and Banking 27: 975-984. [54] Romer, D. (1990), “Staggered Price Setting With Endogenous Frequency of Adjustment,” Economics Letters 32: 205-210. [55] Rudebusch, G. (2002), “Term Structure Evidence on Interest Rate Smoothing and Monetary Policy Inertia,” Journal of Monetary Economics 49: 1161-1187. [56] Sheshinski, E. and Y. Weiss (1977), “Inflation and Costs of Price Adjustment,” Review of Economic Studies 44: 287-303. [57] Smets, F. and R. Wouters (2003), “An Estimated Dynamic Stochastic General Equilibrium Model of the Euro Area,” Journal of the European Economic Association 1: 1123-1175. [58] Taylor, J. (1979), “Staggered Wage Setting in a Macro Model,” American Economic Review 69: 108-113. [59]

(1980), “Aggregate Dynamics and Staggered Contracts,” Journal of Political Economy 88: 1-23.

[60]

(1993), Macroeconomic Policy in a World Economy: From Econometric Design to Practical Operation, W. W. Norton.

[61] Watson, M. (1993), “Measures of Fit for Calibrated Models,” Journal of Political Economy 101: 1011-41. Published by The Berkeley Electronic Press, 2006

55

Frontiers of Macroeconomics , Vol. 2 [2006], Iss. 1, Art. 1

[62] Woodford, M. (2003), Interest and Prices: Foundations of a Theory of Monetary Policy, Princeton University Press.

http://www.bepress.com/bejm/frontiers/vol2/iss1/art1

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