The real effects of monetary shocks in sticky price models: a sufficient statistic approach

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1 The real effects of moetary shocks i sticky price models: a sufficiet statistic approach Ferado Alvarez Uiversity of Chicago Hervé Le Biha Baque de Frace Fracesco Lippi EIEF & Uiversity of Sassari Jauary, 6 Abstract We aalytically solve a meu cost model that ecompasses several models, such as Taylor 98, Calvo 983, Reis 6, Golosov ad Lucas 7, Nakamura ad Steisso, Midriga ad Alvarez ad Lippi 4. The model accouts for the positive excess kurtosis of the size-distributio of price chages that appears i the data. We show that the ratio of kurtosis to the frequecy of price chages is a sufficiet statistics for the real effects of moetary shocks, measured by the cumulated output respose followig a moetary shock. We review empirical measures of kurtosis ad frequecy ad coclude that a model that successfully matches the micro evidece produces persistet real effects that are about 4 times larger tha the Golosov-Lucas model, about 3% below the effect of the Calvo model. We discuss the robustess of our results to chages i the setup, icludig small iflatio ad leptokurtic cost shocks. JEL Classificatio Numbers: E3, E5 Key Words: price settig, meu cost, Calvo pricig, micro evidece, kurtosis of price chages, sufficiet statistic, output respose to moetary shocks We beefited from the commets of semiar participats at the 4 NBER EF&G meetig i NY, Chicago Fed, Federal Reserve Board, Mieapolis Fed, ad the followig uiversities: Boccoi, Bologa, Fireze, MIT, Ohio State, HECER ad Lausae. We are grateful to Alberto Cavallo, Pete Kleow, Oleksiy Kryvtsov, ad Joseph Vavra for providig us with several statistics ot available i their papers. We thak David Argete ad Fredrik Wulfsberg for provig us with evidece for the US scaer data ad Norway s CPI respectively. We thak the INSEE ad the SymphoyIRI Group, Ic. for makig the data available. All estimates ad aalysis i this paper, based o data provided by SymphoyIRI Group, Ic. are by the authors ad ot by SymphoyIRI Group, Ic. We thak the Fodatio Baque de Frace for supportig this project. Part of the research for this paper was sposored by the ERC advaced grat 348. The views expressed i the paper are of the authors ad do ot ecessarily reflect those of the Baque de Frace. Roberto Robatto, David Argete ad Jea Flemmig provided excellet research assistace. A previous versio of the paper was titled Small ad large price chages ad the propagatio of moetary shocks.

2 Itroductio This paper provides a aalytical characterizatio of the steady state cross-sectioal momets, ad of the respose of output to a uexpected moetary shock, i models with sticky prices. By combiig the assumptios of multiproduct firms ad radom meu costs the model is able to produce, i various degrees, both the small ad large price chages that have bee documeted i the micro data startig with Kashyap 995. Differet sticky price set-ups, spaig the models of Taylor 98, Calvo 983, Golosov ad Lucas 7, some versios of the CalvoPlus model by Nakamura ad Steisso, the ratioal iattetiveess model by Reis 6, as well as the multi-product models of Midriga, Bhattarai ad Schoele 4 ad Alvarez ad Lippi 4, are ested by our model. This uified framework allows us to uveil which assumptios are required to obtai each model as a optimal mechaism. The mai aalytical result of the paper is that, i a large class of models that icludes those listed above, the total cumulative output effect of a small uexpected moetary shock depeds o the ratio betwee two steady-state statistics: the kurtosis of the size-distributio of price chages Kur p i ad the average umber of price chages per year N p i. Formally, give the labor supply elasticity /ɛ ad a small moetary shock δ, we show that the cumulative output M, amely the area uder the output impulse respose fuctio, is M = δ 6 ɛ Kur p i N p i. The impact of the frequecy N p i o the real output effect is uderstood i the literature ad motivates a large body of empirical literature. The mai ovelty is that the effect of Kur p i is equally importat, ad motivates our iterest to discuss its measuremet ad report ew evidece o it. For a symmetric distributio, kurtosis is a scale-free statistic describig its shape, specifically its peakedess: the extet to which large ad small observatios i absolute value appear relative to itermediate values. We show that this statistic embodies the extet to which selectio of price chages occurs. The selectio effect, a termiology itroduced by Golosov ad Lucas 7, idicates that the firms that chage prices after the moetary shock are the firms whose prices are i greatest eed of adjustmet, ot a radom sample. Selectio gives rise to large price adjustmets after the shock, so that the CPI respose is fast. Such selectio is abset i Calvo where the adjustig firms are radomly chose ad, after a shock, the size of the average price chage across adjustig firms is costat, so that the CPI rises more slowly ad the real effects are more persistet. Surprisigly, the kurtosis of the steady-state distributio of the size of price

3 chages fully ecodes the selectio effect. Ituitively, i the Golosov-Lucas model steady state price chages are cocetrated aroud two values: very large ad very small, which imply the smallest value of kurtosis equal to oe. I cotrast, the size distributio of price adjustmets uder a Calvo mechaism is very peaked, featurig a large mass of very small as well as some very large price chages, which is exactly what is captured by the high kurtosis predicted by the Calvo model equal to six. I additio to selectio i the size of price chages, recet cotributios have highlighted a related selectio effect i the timig of price chages, see e.g. Kiley ; Sheedy ; Carvalho ad Schwartzma 5; Alvarez, Lippi, ad Paciello 5. This paper shows that the selectio cocerig the timig is also ecoded i the kurtosis of price chages. For istace, i the models of Taylor ad Calvo, calibrated to the same mea frequecy of price chages N p i, the size of the average price chage across adjustig firms is costat after a moetary shock, so there is o selectio cocerig the size. Yet the real cumulative output effect i Calvo is twice the effect i Taylor. This happes because i Taylor the time elapsed betwee adjustmets is a costat T = /N p i, while i Calvo it has a expoetial distributio with mea T, with a thick right tail of firms that adjust very late. Notice how these features are captured by kurtosis: i Taylor the costat time betwee adjustmets T implies that price chages are draw from a ormal distributio, hece kurtosis is three. I Calvo, istead, the expoetial distributio of adjustmet times implies that price chages are draw from a mixture of ormals with differet variaces, ad hece a higher kurtosis equal to six. The mai advatage of our theoretical result is its robustess: equatio allows us to discuss the output effect of moetary shocks i a large class of models without havig to solve for the whole geeral equilibrium or provide details about several other modelig choices. We see this result i the spirit of the sufficiet statistic approach itroduced i the public fiace literature by Chetty 9 ad applied to the ew trade literature by Arkolakis, Costiot, ad Rodriguez-Clare : i a utshell, the idetificatio of a robust relatioship that cotais useful ecoomic iformatio idepedetly of may details of the model ad ca be measured i the data. A key assumptio for equatio to hold is that the distributio of the cost shocks faced by the firms is ormal ad that iflatio is small. I Sectio 5 we discuss the realism of these assumptios ad explore the robustess of our model to the assumptio of o-ormal cost shocks that match actual cost data. We show that equatio remais accurate i predictig the real effects of moetary shocks. The paper is orgaized as follows: we coclude the itroductio with a syopsis of the model ad a brief overview of the related literature. Sectio discusses the measuremet of the kurtosis of price chages, a cetral statistic i our theory. Sectio 3 presets the

4 theoretical model ad its cross sectio predictios. Sectio 4 characterizes aalytically the effect of a uexpected moetary shock. Sectio 5 discusses the robustess of our mai result ad its applicability to actual ecoomies. Sectio 6 cocludes. Overview of mai results Sice kurtosis is a sufficiet statistic for the real effect of moetary policy i the class of models we aalyze, we begi by discussig its estimatio. We idetify two potetial sources of upward bias: heterogeeity across types of goods ad outlets ad small measuremet errors icorrectly imputig small price chages whe there was oe. We the aalyze the size distributio of price chages usig two datasets, the Frech CPI ad US Domiick s data. We show that, after cotrollig for cross-sectioal heterogeeity ad correctig for measuremet error, measured kurtosis is i the viciity of 4 i a large a sample of lowiflatio coutries. We develop a aalytical model that features both the small ad large price chages which lead to excess kurtosis. The model exteds the multi-product setup developed i Alvarez ad Lippi 4, where the fixed meu cost applies to a budle of goods sold by each firm. Each good is subject to idiosycratic cost shocks that create a motive for price adjustmet. The shocks are ucorrelated across goods ad we assume zero iflatio both assumptios ca be relaxed: we show aalytically i Sectio 4 that the results for a small iflatio rate are virtually idetical to the oes for zero iflatio. The multi-product assumptio geerates the extreme price chages, both small ad large. We exted that setup by itroducig radom meu costs, a feature that produces a positive excess kurtosis of the size-distributio of price chages. I particular, we assume that at a exogeous rate λ each firm receives a opportuity to adjust its price at o cost, as i a Calvo setup. The model has four fudametal parameters: the size of the meu cost relative to curvature of the profit fuctio ψ/b, the volatility of idiosycratic cost shocks σ, the umber of products ad the arrival rate of free adjustmets λ. The model yields three ew theoretical results. First we characterize how the iactio set behaves as a fuctio of the parameters. For a small meu cost ψ/b the model behaves as i Barro 97, Dixit 99 ad Golosov ad Lucas 7: the size of the iactio set displays the usual high sesitivity i.e. a quartic root with respect to the cost ad the volatility of the shocks σ the optio value effect. Iterestigly, the decisio rule is uaffected by the presece of the free adjustmets as log as the meu cost is small. The decisio rule chages substatially for large meu costs, a assumptio that is useful to geerate behavior that approaches that of the Calvo model. I this case the size of the iactio set chages with Abset radom meu cost the multi-product model ca at most produce a zero excess kurtosis. 3

5 the square root of the meu cost ad the arrival rate, ad somewhat surprisigly it becomes uresposive to the volatility of idiosycratic shock σ, so that chages i the ucertaity faced by firms iduce o chage i behavior i.e. there is o optio value. Secod, by aggregatig the optimal decisio rules across firms we characterize the frequecy N p i, stadard deviatio Std p i, ad shape of the distributio of the price chages, e.g. its kurtosis: Kur p i. We show that for ay pair of parameters {, λ}, the two remaiig parameters {ψ/b, σ} map oe-to-oe oto the observables N p i, Std p i. This mappig is coveiet for the aalysis because it allows us to freeze the two observables N p i ad Std p i, which oe ca take from the data, while retaiig the flexibility to accommodate various shapes for the size-distributio of price chages as well as various data o the cost of price adjustmet. I particular, we show that the shape of the distributio of price chages ca be writte exclusively i terms of ad the fractio of free-adjustmets l λ/n p i. I our model the shape of the distributio of price chages rages from bimodal for the model where l = ad = as i the Golosov-Lucas model to Normal for = ad l =, our versio of Taylor s model, ad Laplace i the case l = for ay, our versio of the Calvo model. I those three models the kurtosis of price chages is, respectively,, 3 ad 6. I our set-up a give kurtosis may be obtaied by differet combiatios of ad l, yet we argue that models with high yield a better represetatio of the cross-sectioal data because it elimiates the predomiat mass of large price chages that arises i models where is small. The model is thus able to match several cross sectioal features of the micro data, such as the frequecy, stadard deviatio ad kurtosis of price chages, usig realistic small values for the meu-cost of price adjustmets. Third, we use the model to characterize aalytically the impulse respose of the aggregate ecoomy to a oce-ad-for-all uexpected permaet icrease of the moey supply i Sectio 4. The effect of a moetary shock depeds o the shock size. Large shocks relative to the size of price adjustmets lead to almost all firms adjustig prices ad hece imply eutrality. We fully characterize the miimum size of the shock that delivers this eutrality. Istead, small shocks, such as those foud i empirical impulse resposes, yield real output effects whose cumulative effect is completely ecoded by the frequecy ad kurtosis of price chages. This result is ew i the literature ad was discussed i equatio. To further illustrate our result, we also preset the impulse respose fuctio associated with the total cumulative output effect. We compute this impulse respose by a calibratio which shows that a model that successfully matches the cross-sectioal micro evidece produces persistet real effects that are ot too differet from what is see i several empirical studies of the propagatio of moetary shocks. I particular, we fid that the half life of a moetary shock We show that kurtosis is icreasig i both ad l i.e. both partial derivatives are positive. 4

6 is about 4 times more persistet tha i the Golosov-Lucas model, but 3% less tha i the Calvo model. Other related literature I additio to the papers cited above, our aalysis relates to a large literature o the propagatio of moetary shocks i sticky price models, uifyig earlier results that compare the propagatio i the Calvo model with the propagatio i either the Taylor or the meu cost model of sticky prices. Kiley showed that, cotrollig for the frequecy of price chages, the respose of output is more persistet uder Calvo tha uder Taylor cotracts. Golosov ad Lucas 7 compared a meu cost ad a Calvo model, with the same frequecy of price chages, ad fid that the half-life of the respose to the shock i Calvo is about five times larger tha i the meu cost model. Our model is related to the CalvoPlus model of Nakamura ad Steisso who cosider firms facig idiosycratic cost shocks as well as a meu cost that oscillates radomly betwee a large ad a small value. I their model, like i ours, the radom meu cost makes the adjustmet decisio state depedet, a feature that dampes substatially the real effect of moetary shocks relative to the Calvo model. Our paper also relates to the radom meu cost model with idiosycratic shocks by Dotsey, Kig, ad Wolma 9, as well as to Vavra 4, who studies the propagatio of shocks i the presece of aggregate volatility shocks. Give the umerical ature of these cotributios, these papers do ot provide a explicit map betwee the model fudametals, the steady state statistics ad the propagatio of shocks. We see our results as complemetary to these umerical aalyses. Our model allows for a aalytical characterizatio of the firm s decisio rule, the ecoomy s steady state statistics, the idetificatio of the key model parameters, as well as a characterizatio of the relatioship betwee these statistics ad the size of the output effect of moetary shocks. More recetly, Karadi ad Reiff 4 have explored meu cost models where the iovatio to cost shocks are assumed to be leptokurtic istead of ormal. Their model ests special cases, such as the model with ormal cost shocks, as well as models a la Midriga or Gertler ad Leahy 8 where shocks are either large or zero ad thus cost shocks have a large kurtosis, ad a cotiuum of itermediate cases. They show that leptokurtic cost shocks may boost the output effect of moetary shocks i priciple, eve though i their calibratio to actual cost data they fid ear moey eutrality. While our bechmark model features ormal cost shocks, Sectio 5 uses a data-cosistet parametrizatio to explore the cosequeces of leptokurtic cost shocks o our mai results. We show that equatio cotiues to covey a reliable approximatio for the real effects of moetary shocks. 5

7 Measurig the kurtosis of price chages A vast amout of research has ivestigated the size of price chages at the microecoomic level i the past decade e.g. Bils ad Kleow 4, Nakamura ad Steisso 8, Kleow ad Mali. A robust empirical patter is that the size distributio of price chages exhibits a large amout of small, as well as large, price chages. This feature of the distributio is reflected i a kurtosis that is above the oe of the ormal distributio i.e. larger tha 3. This patter does ot oly reflect cross sectioal differeces betwee goods types, it also appears at the very disaggregated product level, i.e. a give good typically records both small ad large price chages. Most theoretical models fail to produce a sizedistributio for price chages with such features. The model we preset i the ext sectio will be able to match such patters ad will assig kurtosis a cetral role i the trasmissio of moetary shocks. Because kurtosis is a sufficiet statistic for the real effect of moetary policy i the class of models we aalyze, we discuss two importat sources of upward bias that arise i estimatio: heterogeeity across types of goods ad outlets ad measuremet errors icorrectly imputig small price chages whe there was oe. We the aalyze the size distributio of price chages usig two datasets, the Frech CPI ad US Domiick s data. We fid that, after cotrollig for cross-sectioal heterogeeity ad correctig for measuremet error, the kurtosis is substatially smaller tha what is measured i the raw data. We coclude with a overview of the existig evidece from other sources, which shows that the value of kurtosis is i the viciity of 4 i a large a sample of low-iflatio coutries. I other words, the distributio lays betwee a Normal ad a Laplace distributio. The other key sufficiet statistic for the real effects of moetary policy i our model is the average frequecy of price chages, a feature shared with may models. We do ot explore the measuremet of the frequecy i detail sice this is the subject of may papers. We do however highlight that measuremet error also affects the estimates of price duratios, leadig to overestimatig the frequecy of price chages, as documeted recetly by Cavallo 5 o US data.. Accoutig for measuremet error ad heterogeeity Two cocers arise about the measuremet of kurtosis usig the micro data: heterogeeity ad measuremet errors. Heterogeeity refers to the fact that price data i geeral combie a wide variety of goods. A well kow statistical result is that a mixture of distributios with differet variaces ad the same kurtosis has a larger kurtosis tha each sub-populatio. Likewise, measuremet error may bias the estimatio of kurtosis as well as of the frequecy of price chages. We use a simple statistical model to illustrate these poits. Let the observed 6

8 price chages p m be give by a mixture of three idepedet zero-mea distributios, with a commo kurtosis k ad stadard deviatios σ j. I particular, let p m = I p + I p + I e p e where I j is a idicator variable that refers to the distributio that origiates the price chage j = {,, e}. The distributio idexed by j = e describes observatios that are spurious, i.e. due to measuremet error. The distributios idexed by j = {, } refer to two subpopulatios of goods, with stadard deviatios σ ad σ. Let ζ be the probability that the observatio is a measuremet error, ad π the probability that the price chage is draw from populatio coditioal o a true price chage beig observed. The objective is to compute the kurtosis ad frequecy of price chages as they appear i a sample geerated by this mixture. We focus o a ecoomy i which the size of the measuremet error is small, ispired by Eichebaum et al. 4, so that we cosider the limitig case σ e. 3 Some algebra shows that measured kurtosis is: lim Kur p m = k σ e Ω ζ where Ω π σ 4 + π σ 4 π σ + π σ This formula that for the estimatio to be ubiased, i.e. Kur p m = k, it is ecessary that there is o measuremet error ζ = ad o heterogeeity σ = σ. If either of these coditios fail, the measured kurtosis is upward biased. I the empirical aalysis we address the heterogeeity-iduced bias by stadardizig the price chages at a disaggregate cell level by demeaig the price chage observatios ad dividig them by their stadard deviatio. A cell is, for istace, a category of good ad of outlet type, such as a baguette i supermarkets. Midriga. A similar approach was followed by Kleow ad Kryvtsov 8 ad Notice that, abset measuremet error, the stadardizatio procedure would deliver a ubiased estimate of kurtosis k. To address the small measuremet error issue we discard observatios with tiy price chages e.g. smaller tha cet, a commo practice i the empirical studies of price chages. We the compute the statistics for the pooled stadardized data. The model also illustrates the differetial impact of measuremet error ad heterogeeity o the measured frequecy of price chages. Measuremet error makes the expected umber of measured price chages, N m, greater tha the umber of true price chages, N p, so that N m = N p + N e where N e is the umber of icorrectly imputed price chages all measured per period. This predictio is supported by recet evidece by Cavallo 5, who shows 3 The web Appedix G. explores the cocers about the spurious small price chages raised by Eichebaum et al. 4, cocludig that they also apply to the Frech CPI data, albeit to a lesser extet. Eve at the fiest level of disaggregatio some price chages i the CPI data reflect product substitutios e.g. differet brads for a give good beig recorded rather tha a actual chage i the good s price. Likewise, spurious small price chages may origiate i scaer dataset from the uit value problem of weekly prices, whereby the recorded prices average the prices paid by customers with ad without discout coupos. 7

9 that price duratios estimated over scaer datasets are affected by measuremet issues that produce a dowward bias of measured duratios. We have that N e = ζn m ad N p = ζ N m. Notice that heterogeeity has o effect o the measuremet of the frequecy of price chage, ulike i the case of kurtosis. If two samples are observed, oe measuremet error free ad the other oe with a fractio ζ of small spurious price chages, the ζ ca be estimated usig the ratio of the two estimated frequecies of price chages. If kurtosis k ad k m is also measured o the two samples, the oe ca ifer what part of the bias i the measuremet ca be attributed to measuremet error usig the bias factor /ζ ad what part is due to heterogeeity the factor Ω.. Evidece We use two large datasets to provide evidece o the kurtosis of the size distributio, accoutig for measuremet error ad heterogeeity. The first oe is the Domiick s dataset, featurig weekly scaer data from a large supermarket chai i Chicago. 4 Aroud 5, UPCs are available, belogig to 9 product categories such as beer or shampoo, over 4 weeks from September 989. Followig Midriga we focus o oe particular store, the oe with most observatios store #. The secod piece of evidece is a logitudial dataset of mothly price quotes uderlyig the Frech CPI, over the period 3:4 to :4, cotaiig aroud millio price quotes, documeted i details i Berardi, Gautier, ad Le Biha 5. Each record relates to a precisely defied product sold i a particular outlet i a give moth. Oe mai advatage of CPI data is the broader coverage of household cosumptio. The raw dataset covers about 65% of the CPI basket some categories of goods ad services are ot available i our sample. The dataset also icludes CPI weights, which we use to compute aggregate statistics. I both datasets price chages are computed as times the log-differece i prices price per uit i the case of CPI where package size may vary, ulike with UPCs. As a first hedge agaist measuremet error we discard observatios with item substitutios i CPI data which might give rise to spurious price chages. We apply the followig trimmig to the data: for Domiick s we disregard chages that are smaller tha cet due to measuremet error ad drop observatios with price levels smaller tha cets or larger tha 5 dollars deemed implausible i view of the type of items sold ad the distributio of price levels. I the CPI data we drop price chages whose absolute value is smaller tha. percet. I both datasets, we remove as outliers observatios with log-price chages larger i absolute value tha the 99 th percetile of absolute log price chages. To hadle the issue of heterogeeity, 4 Data at the uiversal product code UPC level are provided by James M. Kilts Ceter, Uiversity of Chicago Booth School of Busiess. 8

10 Figure : Histogram of Stadardized Price Chages: Frace CPI ad US Domiick s CPI data Frace Domiick s US Data Stadardized Normal Stadardized Laplace Data Stadardized Normal Stadardized Laplace The figures use the elemetary CPI data from Frace 3-, ad the Domiick s data set. Price chages are the log differece i price per uit, stadardized by good category 7 ad outlet type ad pooled. Price chages equal to zero are discarded. The pael with Frech CPI uses about.5 millio data poits, the pael with Domiick s about.3 millio. we stadardize the data at the cell level. A cell is defied by a good category ad outlet type i the CPI data, ad a UPC-store i scaer data. We use the fiest partitio possible i our data: for CPI each cell is a COICOP category at the 6-digit level i a outlet type, ad we have aroud,5 cells. As metioed we compute the stadardized price chages by subtractig the cell mea for all o zero price chages ad dividig by the cell-specific stadard deviatio. Figure summarizes our mai fidigs with a weighted histogram of the stadardized price chages. O the same graph we superimpose the desity of the stadard Normal distributio as well as the stadardized Laplace distributio both have uit variace. The Laplace distributio has a kurtosis of 6 ad is thus more peaked tha the Normal. It is apparet that both empirical distributios of stadardized price chages are more peaked tha the Normal. The kurtosis of the stadardized price chages measured o the Frech CPI is large, equal to 8. removig sales has a mior effect: kurtosis icreases to 8.9. This estimate is still likely upward biased because of the remaiig measuremet errors ad the remaiig heterogeeity, which icrease estimated kurtosis as idicated by Ω > ad /ζ > i equatio. Notice that our cotrol for heterogeeity i the CPI data is partial, as the iformatio available prevets us from correctig heterogeeity at a fier level: e.g., 9

11 we do ot kow the UPC of the product or the store where it is sold. To get a sese of how this residual heterogeeity impacts results, we performed a computatio i the Domiick s datasets by stadardizig price chages at the product level i.e. beer, or shampoo which roughly matches the fiest level of disaggregatio available i the CPI as opposed to the UPC level. Kurtosis is the 4.9, agaist 4. i the baselie UPC-level stadardizatio. Notice that i additio to this % bias, the CPI has the additioal problem that CPI prices are ot collected i a sigle store or area, givig rise to yet aother source of heterogeeity that is ot preset i Domiick s data where we focus o a sigle store ad area. Moreover, measuremet error is likely preset, as our trimmig of the data is quite coservative: usig a % to idetify spurious small price chages as suggested by Eichebaum et al. 4 reduces estimated kurtosis to 7 see the web Appedix G. for more evidece. As a further assessmet of measuremet error, we matched a specific subset of the Frech CPI data with data take from the Billio Price Project BPP dataset, see Cavallo 5. These data were costructed with the specific itet of addressig heterogeeity ad measuremet error issues, ad so they provide a ideal eviromet to assess the relevace of the issues discussed above. 5 We matched the BPP data from 3 retailers with the correspodig items i the CPI see Appedix A. We fid that for these retailers kurtosis computed o the BPP data is aroud / the value computed from usig the CPI data. This suggests that the factor Ω/ζ i equatio, is aroud i this sample. Extrapolatig this fidig to the CPI idex, yields a estimated kurtosis of about 4. As discussed i Sectio. measuremet error but ot heterogeeity also affects the measuremet of the frequecy of price chages. Ideed, the data o duratio i the BPP vs the CPI is also cosistet with the presece of substatial measuremet error. I two of the three outlets cosidered the frequecy of adjustmet is half the frequecy detected for the correspodig outlet i the CPI data, suggestig that the bias i kurtosis is mostly due to measuremet error. I the third outlet by cotrast the duratio of price adjustmet betwee the BPP vs the CPI is similar, suggestig that the discrepacy i kurtosis maily reflects residual heterogeeity i the CPI data. Our aalysis of the Domiick s data reveals that kurtosis of the stadardized price chages is equal to 4.. This fidig is cosistet with the hypothesis that the more graular ad more precise ature of the iformatio i the dataset allows for better cotrol over heterogeeity as well as measuremet errors. Fially, Table provides a overview of available estimates of the kurtosis of stadardized price chages for the US ad some other coutries. Most estimates are located i the viciity of 4. I particular, the estimates by Cavallo ad Rigobo 5 based o the BPP iteret scraped data discussed above stad out as the most 5 Cavallo 5 ad Cavallo, Neima, ad Rigobo 4 documet that olie prices are represetative of offlie prices for a selected sample of large retailers.

12 Table : Overview of estimates of kurtosis US Other coutries Source: M NS8 V3 CR5 KR4 W CR5 Kurtosis: The kurtosis is computed usig stadardized price chages. The labels for the various studies are: M: Midriga, NS8: Nakamura ad Steisso 8, V3: Vavra 4, KR4: Karadi ad Reiff 4 Hugary, W:Wulfsberg Norway, CR5: Cavallo ad Rigobo 5 US ad media value across 3 low-iflatio coutries. comprehesive across coutries ad least affected by measuremet errors, although the coverage of the goods is ot as comprehesive as the CPI. These authors use data from 4 coutries, stadardizig price chages at the UPC level to deal with heterogeeity. Their Table shows that Kurtosis is 4 i the US ad i Frace values are rouded to itegers. The media value i the 4 coutry sample is 5, while i the subsample of 3 coutries with iflatio below 5 percet the media is 4. 3 A tractable model with a radom meu cost This sectio presets a meu cost model aimed at qualitatively matchig the patters documeted above. I the caoical meu cost model price adjustmets occur whe a threshold is hit, so that the implied distributio of price chages fails to geerate the small chages that appear i the data see the discussio i Midriga ; Cavallo 5; Alvarez ad Lippi 4. The model that we propose here is able to produce a large mass of small price chages ad the positive excess kurtosis that we documeted above. Two igrediets are key to this ed: i radom meu costs ad ii the meu cost faced by the firm, ψ, applies to a budle of goods, so that after payig the fixed cost the firm ca reprice all goods at o extra cost. Each of these assumptios idividually is able to geerate some small price chages ad higher kurtosis tha i a caoical model where = ad where meu costs are costat. The assumptio of radom meu costs is key to geerate a positive excess kurtosis i the distributio of price chages. The combiatio of the two is importat: i the models where = with or without radom meu costs the distributio of price chages has a mass poit at the adjustmet threshold, a feature that is i stark cotrast with the evidece. The promiece of large price chages i.e. a U shaped distributio persists eve i a model with =, as i Midriga where the distributio of price chages asymptotes ear the adjustmet threshold, or = 3 as i Bhattarai ad Schoele 4. We show below that i order to geerate a size distributio whose shape is comparable to the data oe eeds

13 6. Geeral Equilibrium Setup. The geeral equilibrium set up is essetially the oe i Golosov ad Lucas 7, adapted to multi-product firms see Appedix B i Alvarez ad Lippi 4 for details. Households have a costat discout rate r ad a istataeous utility fuctio which is additively separable: a CES cosumptio aggregate, log i real balaces, liear i leisure, with costat itertemporal elasticity of substitutio /ɛ for the cosumptio aggregate, so that the labor supply elasticity to real wages is /ɛ. A coveiet implicatio of this setup is that omial wages are proportioal to the moey supply i equilibrium, so that a moetary shock icreases the firms margial costs proportioately. Each firm produces goods, each with a liear labor-oly techology, subject to idiosycratic productivity shocks idepedet across products, whose log follows a browia motio with istataeous variace σ. The firm faces a demad with costat elasticity η > for each of its products, comig from the household s CES utility fuctio for the cosumptio aggregate. To keep the expediture shares statioary across goods i the face of the permaet idiosycratic shocks, we assume offsettig preferece shocks as i Woodford 9, Boomo, Carvalho, ad Garcia Midriga, Alvarez ad Lippi 4. The frictioless profit-maximizig price for good i at time t is thus give by a costat markup η/η over the margial cost. Let P i t be the log of the frictioless profit maximizig price which follows the process dp i t = σ dw t where W t is a stadard browia motio with o drift, ad σ is the stadard deviatio of productivity. The techology to chage prices is as follows: each firm is subject to a radom meu cost to simultaeously chage the price of its products. I a period of legth dt this cost amouts to ψ L uits of labor with probability λdt, or zero with probability λdt. Let p i t deote the price gap for good i at time t, i.e. the differece betwee the actual log sale price P i t ad the log profit maximizig price P i t, i.e. p i t P i t P i t. The firm flow profit ca be approximated, up to a secod order, by a quadratic loss fuctio i Bp it where the scale parameter B = /ηη is related to the demad elasticity. 6 Uder this approximatio it is coveiet to express the meu cost ψ L i uits of flow profits at the optimal price, a costat value that we deote by ψ. To keep the model simple we assume that all goods i the ecoomy are ex-ate symmetric ad subject to shocks with a commo variace σ, ad all firms share the same fudametal parameters. A extesio to a case with ex-ate heterogeeous firms, ivolvig differet frequecies ad kurtosis of price adjustmet, is preseted i Appedix E. Our empirical 6 This approximatio is quite accurate for the typical small values of the meu cost used i the literature. See Alvarez ad Lippi 4 for a quatitative illustratio of the accuracy ad for a survey of several papers that use the quadratic formulatio.

14 aalysis of heterogeeity ad the stadardizatio of data employed i the previous sectio are fully cosistet with the theoretical framework discussed i this extesio provided that the shocks that hit the idividual goods are ormally distributed but possibly with differet variaces. 3. A simple case with = good. The simplest illustratio of our radom meu cost model obtais for the case where =, so the price gap p is scalar. Let V p be the preset-value cost fuctio for the firm. Upo the arrival of a free adjustmet opportuity, i.e. a zero meu cost, the firm optimally resets the price gap to zero give the symmetry of the loss fuctio ad the law of motio of price gaps, hece the Bellma equatio for the rage of iactio reads: r V p = Bp + λ V V p + σ V p, for p p, p, where p is the threshold rule defiig the regio where iactio is optimal see the web Appedix H for the calculatios of this sectio. This equatio states that the flow value of the Bellma equatio is give by the istataeous losses, Bp, plus the expected chage i the value fuctio, which is due either to a free adjustmet with rate λ i which case the price gap is reset to zero or to the volatility of shocks σ. The value-matchig ad smooth-pastig coditios are give by V p = V + ψ ad V p =. A Taylor expasio 4 of the value fuctio yields the followig approximate optimal threshold p = which is accurate for small values of the meu cost ψ. 7 Computig the expected time betwee adjustmets yields a expressio for the average umber of adjustmets per period, N p i, which we use to measure the fractio of free adjustmets over the total umber of adjustmets, a variable we call l, as l λ N p i = e φ + e φ e φ + e φ 6ψσ B, where we defie φ λ p σ which shows that the fractio of free adjustmets l depeds oly o the parameter φ. The parameter φ ca be iterpreted as the ratio betwee λ, the umber of free adjustmets, ad σ / p, the umber of adjustmets i a model where λ = ad the threshold policy p is followed. Let p i = p deote the price chage implemeted by a firm that adjusts whe its price 7 The approximatio obtais as ψr+λ Bσ, see Propositio 3 of Alvarez ad Lippi 4. Exactly the same expressio was established by Barro 97 ad Dixit 99 for the case i which λ =. Below we discuss a approximate threshold for large values of ψ that is useful to iterpret the Calvo model. 3

15 gap is p. The distributio of price chages is symmetric aroud p i =. This distributio has two mass poits at p i = ± p. The two poits, which accout for l of the mass, are due to the price chages that occur whe the price gap hits the boudaries of the iactio regio. The remaiig price chages, a fractio l of the mass, occur whe a free adjustmet opportuity arrives, at which time the price gap is set to zero. Price chages i the rage p i p, p have a desity l gp where gp deotes the desity of the ivariat distributio of price gaps give by gp = φ p e φ e φ p p p e φ p for p [ p, p]. 3 This desity is a symmetric i the p, p iterval ad its shape is high-peaked tetshaped. The full distributio features two mass poits at the boudaries ± p. As show below the kurtosis of this distributio is icreasig i λ, ad i particular the distributio of price chages is more peaked tha that of the stadard meu cost model where λ =. We otice that the shape of the distributio of price chages depeds oly o the fractio of free adjustmets l or, equivaletly, o φ. This meas that two ecoomies, or sectors, that differ i the stadard deviatio of price chages Std p i ad/or i the frequecy of price adjustmet N p i will display a distributio of price chages with exactly the same shape oce its scale is adjusted provided that they have the same value of l. This property is useful to aggregate the sectors of a ecoomy that are heterogeous i their steady state features N p i, Std p i. Because kurtosis is a scale free statistic, i.e. idepedet of Std p i, it is completely determied by φ i this model. The geeral model adds oe parameter,, as a determiat of the shape of the size distributio of price chages ad hece of kurtosis. 3. The model with multi-product firms This sectio icorporates the model with free adjustmet opportuities discussed above ito the model of Alvarez ad Lippi 4 where the firm is sellig goods, so that p is ow a vector i R, but pays a sigle fixed adjustmet cost to chage the prices. We icorporate this feature for several reasos. First, as explaied above, i the model with = good there is a mass poit o price chages of size p i = p. For = the mass poit disappears but the distributio of price chages still features the largest mass of observatios the highest desity ear the adjustmet thresholds. There is o evidece of this i ay data set we ca fid. Values of 6 produce a bell-shaped size distributio of price chages that is much closer to what is see i the data. Secod, the model with λ = has a kurtosis that icreases with, hece providig a alterative to radom meu costs. Third, for large 4

16 ad λ = the distributio of price chages teds to the Normal distributio, which is both a ice bechmark ad a accurate descriptio of the price chages for some sectors. Fially, the multi-product model with > has a alterative, broader, ratioal iattetiveess iterpretatio for the adjustmet cost ψ. I particular, oe ca assume that the firm freely observes its total profits but ot the idividual oes for each product, uless it either pays the cost ψ or a free observatio opportuity arrives, i which case it is able to set the optimal price to each of them. This allows a broader iterpretatio of the meu cost, icludig ot oly the physical cost of chagig prices but also the cost related to gatherig ad processig the iformatio for idividual products. 8 the ratioal iattetiveess model of Sectio 4.3 i Reis 6. For istace, as the model coverges to We ow briefly describe the setup of the firm problem with products. As before the free adjustmet opportuities are idepedet of the drivig processes {W i t} for price gaps i =,,...,, ad arrive accordig to a Poisso process with costat itesity λ. I betwee price adjustmets each of the price gaps evolves accordig to a Browia motio dp i t = σ dw i t. It is assumed that all price gaps are subject to the same variace σ ad that the iovatios are idepedet across price gaps. 9 We assume that, whe the free opportuity arrives, the firm ca adjust all prices without payig the cost ψ. The aalysis of the multi product problem ca be greatly simplified by usig the sum of the squared price gaps, y p as a state variable, istead of the vector p = p,..., p, as doe i Alvarez ad Lippi 4. The scalar y summarizes the state because the period objective fuctio ca be writte as a fuctio of it ad because, from a applicatio of Ito s lemma, oe ca derive a oe dimesioal diffusio which describes its law of motio, amely where W is a stadard Browia motio. dy = σ dt + σ y dw Usig N p i ad V ar p i to deote the frequecy ad the cross sectioal variace of the price chages of product i, the ext propositio establishes a useful relatioship that holds i a large class of models for ay policy for price chages, which we describe by a stoppig time rule: Propositio Let τ describe the time at which a price chage takes place, so that all price gaps are closed. Assume the stoppig time treats each of the price gaps symmetrically. For 8 As a example, see Chakrabarti ad Scholick 7 who argue that for stores such as Amazo or Bares ad Noble physical meu cost are small, yet prices chage ifrequetly, ad thus coclude that the cost may be of a differet ature. Iterestigly, they fid that for such retailers price chages are sychroized across products, which is a implicatio of the multi-product model. 9 Alvarez ad Lippi 4 discuss the case with correlated price gaps. Ituitively, as correlatio icreases the model becomes more similar to the = case, sice the price gaps of a firm become more similar. 5

17 ay fiite stoppig time τ we have: N p i V ar p i = σ. 4 The propositio highlights the trade-off for the firm s policy: more frequet adjustmets are required to have smaller price gaps. See Appedix B for the proof, where the reader ca verify that the key assumptios are radom walks ad symmetry. We uderlie that equatio 4 holds for ay stoppig rule, ot just for the optimal oe. Ideed it holds for a larger class of models, for istace those with correlated price gaps ad a richer class of radom adjustmet cost. Upo the arrival of a free adjustmet opportuity the firm will set the price gap to zero, hece the Bellma equatio for the rage of iactio reads: r vy = B y + λ v vy + σ v y + σ y v y, for y, ȳ, 5 where B y is the sum of the deviatio from the optimal profits from the goods. Give the symmetry of the problem after a adjustmet of the prices the firm will set all price gaps to zero, i.e. will set p = y =. The value matchig coditio is the v + ψ = vȳ, which uses the fact that whe y reaches a critical value, deoted by ȳ, the firm ca chage the prices by payig the fixed cost ψ. The smooth pastig coditio is v ȳ =. The ext lemma establishes how to solve for ȳ usig the solutio of the problem with λ = discussed i Alvarez ad Lippi 4. I particular, usig r + λ as a modified iterest rate i the solutio of the problem with λ =, allows us to immediately compute the solutio for the case of iterest i this paper. We have: Lemma Let vy; r, λ ad ȳr, λ be the optimal value fuctio ad adjustmet threshold for a problem with discout rate r ad arrival rate λ. The vy; r, λ = vy; r + λ, + λv; r + λ, for all y ad thus ȳr, λ = ȳr + λ,. r The proof of this lemma follows a straightforward guess ad verify strategy. The lemma allows us to use the characterizatio of ȳ with respect to r give i Propositio 4 of Alvarez ad Lippi 4 to study the effect of r + λ o ȳ. The ext propositio summarizes that result ad exteds the characterizatio of the optimal threshold to the case where ψ is large, a case that is useful to uderstad the behavior of a ecoomy with a lot of free adjustmets opportuity as i a Calvo mechaism see Appedix B for the proof. The web Appedix J gives the aalytical solutio for the value fuctio ad provides more details. 6

18 Propositio Assume σ >,, λ + r > ad B >, ad let ȳ be the threshold for the optimal decisio rule. We the have that:. As ψ the ȳ +σ ψ B. As ψ we have ȳ ψ or ȳ r + λ/b or + σ ψ B. large ad large ψ, amely lim ψ/ lim ȳ/ ψ/ ȳ ψ r + λ. Moreover this also holds for B ȳ = r + λ/b or r + λ. ψ/ B The propositio shows that ȳ is approximately costat with respect to λ for small values of ψ, so that for small meu costs the result is the well kow quartic root formula recall that y has the uits of a squared price gap ad the iactio regio is icreasig i the variace of the shock, due to the higher optio value. Iterestigly, ad ovel i the literature, the secod part of the propositio shows that for large values of the adjustmet cost the rule becomes a square root ad that the optimal threshold does ot depeds o σ, which shows that for large adjustmet costs the optio value compoet of the decisio becomes egligible. Moreover, whe the meu costs are large the threshold ȳ is icreasig i λ: the prospect of receivig a free adjustmet tomorrow icreases iactio today. We ow tur to the discussio of the model implicatios for the frequecy of price chages. We let N p i be the expected umber of adjustmets per uit of time of a model with a give λ ad ȳ. We establish the followig see Appedix B for the proof: Propositio 3 Let Γ deote the gamma fuctio. The fractio of free adjustmets is l = λ/n p i, where l = i= i= Γ [ ] i i! Γ +i φ i Γ [ ] i i! Γ +i φ i Lφ,, where φ λȳ σ 6 The propositio shows that l is a fuctio oly of two variables: ad φ, ad that it is icreasig i φ. As was the case for the model with =, the parameter φ ca be iterpreted as the ratio betwee λ, the umber of free adjustmets, ad σ / p, the umber of adjustmets i a model where λ = ad the threshold policy ȳ is followed. For a give there is a oe to oe ad oto mappig betwee φ ad l: as φ the l, ad as φ the l. Fially we characterize the ivariat distributio of y for the case where λ >, a key igrediet to compute the size-distributio of price chages. The desity of the ivariat λ distributio solves the Kolmogorov forward equatio: fy = f yy f y for σ y, ȳ, with the two boudary coditios fȳ = ad ȳ fydy =. It is clear from 7

19 these coditios that f is uiquely defied for a give triplet: ȳ >, ad λ/σ. The geeral solutio of this ODE is λy 4 λy λy fy = C σ I ν + C σ K ν σ where I ν ad K ν are the modified Bessel fuctios of the first ad secod kid, C, C are two arbitrary costats ad ν =, see Zaitsev ad Polyai 3 for a proof. The costats C, C are chose to satisfy the two boudary coditios. While the desity i equatio 7 depeds o 3 costats, φ ad ȳ, its shape depeds oly o costats, amely ad φ, as formally stated i Lemma 3 i Appedix B. The lemma shows that oe ca ormalize ȳ to ad compute the desity for the correspodig φ. We deote the margial distributio of price chages by w p i. 7 Recall that a firm chages all prices whe y first reaches ȳ or whe a free adjustmet opportuity occurs eve though y < ȳ. Therefore to characterize the price chages p i of good i belogig to the vector of price gaps p we eed three objects: the fractio of free adjustmets l, the ivariat distributio fy ad the margial distributio of price chages coditioal o a value of y, ω p i ; y which, followig Propositio 6 of Alvarez ad Lippi 4 whe, is ω p i ; y = 3/ p i y if p i y Beta, y if p i > y where Beta, deotes the Beta fuctio. I this case the cross-sectioal stadard deviatio of the price chages is Std p i ; y = y/. The margial distributio of price chages w p i is give by ȳ w p i = ω p i ; ȳ l + 8 ω p i ; yfydy l for. 9 which shows that the distributio w p i is a mixture of the ω p i, y desities. These desities are scaled versios of each other with differet stadard deviatios. The mixture icreases the kurtosis of the distributio of price chages compared to the case where λ =, for the reasos discussed i Sectio. Before illustratig the shapes produced by this We ote that both modified Bessel fuctios are positive, that I ν y is expoetially icreasig with I ν, ad that K ν y is expoetially decreasig with K ν = +. The web Appedix N gives a closed form expressio for f i terms of power series. I particular Propositio 6 i Alvarez ad Lippi 4 shows that the variace ad kurtosis of ω p i, y are give by y/ ad 3/ + respectively. 8

20 distributio, the ext propositio shows that those shapes are completely determied by ad l ad o other parameters see Appedix B for the proof: Propositio 4 Let w p i ;, l, be the desity fuctio for the price chages p i i a ecoomy with goods, a share l of free adjustmets, ad a uit stadard deviatio of price chages Std p i =. This desity fuctio is homogeous of degree - i p i ad Std p i, which implies w a p i ;, l, a = a w p i;, l, for all a >. The propositio implies that we ca aggregate firms or idustries that are heterogeous i terms of frequecy N p i ad stadard deviatio of price chages Std p i provided that ad l are the same. Notice i particular that the frequecy of price chages N p i does ot have a idepedet effect o the distributio of price chages as log as l remais costat. Figure shows the shapes of the distributio of price chages p i i equatio 9 obtaied for differet combiatios of ad l. It is evidet that for small values of the shape of the distributio does ot match the tet-shaped patters that are see i the data as i e.g. Figure. For the case whe = the desity of the price chages diverges at the boudaries of the domai where p i = ± ȳ/. This feature echoes the two mass poits that occur i the = case where a o-zero mass of price chages occurs exactly at the boudaries. For 6 the shape of the desity takes a tet-shape, similar to the oe that is see i the data. As the fractio of free adjustmets approaches the desity fuctio coverges to the Laplace distributio. Usig that p i is distributed as a mixture of the ω p i, y, we ca compute several momets of iterest, such as V ar p i = l ȳ + l Kur p i = 3 + ȳ y fydy l ȳ + l ȳ y fydy l ȳ + l ȳ y fydy > 3 + As stated i Propositio 4 the value of the kurtosis Kur p i depeds oly o two parameters: ad l. Moreover, kurtosis is icreasig i both ad l, as ca be see i Figure 4 which plots the value of Kur p i /6 for various combiatios of ad l, the oly two parameters determiig kurtosis. For small values of l kurtosis is icreasig i up to a level of 3. For istace, if l = ad, the kurtosis coverges to 3 sice the distributio of price chages at the time of adjustig for each firm becomes ormal; this value is the highest that the purely multi-product model with l = ca produce. For ay, 9