Small and large price changes and the propagation of monetary shocks

Transcription

1 Small ad large price chages ad the propagatio of moetary shocks Ferado Alvarez Uiversity of Chicago Fracesco Lippi EIEF & Uiversity of Sassari Hervé Le Biha Baque de Frace Jue, 4 Abstract We documet the presece of both small ad large price chages i idividual price records from the CPI i Frace ad the US. After correctig for measuremet error ad cross-sectio heterogeeity, the size-distributio of price chages has a positive excess kurtosis. We propose a aalytical meu cost model that ecompasses several classic models, as Taylor 98), Calvo 983), Reis 6), Golosov ad Lucas 7) ad accouts for observed cross-sectioal patters. We show that the ratio of kurtosis to the frequecy of price chages is a sufficiet statistics for the real effects of moetary policy i a large class of models. JEL Classificatio Numbers: E3, E5 Key Words: price settig, micro evidece, size-distributio of price chages, kurtosis of price chages, meu-cost, Calvo pricig rule, output respose to moetary shocks We beefited from the commets of Mauel Amador, Saroj Bhattarai, Ricardo Caballero, V.V. Chari, Mike Dotsey, Jordi Galí, Aubik Kah, Ricardo Caballero, Virgiliu Midriga, Daiel Levy, Nicola Pavoi, Giorgio Primiceri, Jo Steisso, Silvaa Tereyro, Julia Thomas, Michael Waterso, Iva Werig, ad semiar participats at the 4 NBER EF&G meetig i NY, Baque de Frace, Chicago Fed, 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 provided excellet research assistace.

2 Itroductio This paper provides a aalytical characterizatio of the steady state equilibrium ad of the respose of output to a uexpected moetary shock for a class of 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 of them as a optimal mechaism. The mai aalytical result of the paper is that, i a large class of models that exteds 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 give by 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, albeit ot i such a stark fashio as i equatio ), 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 review ad exted the evidece o its measuremet. 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 size of the price chages after the shock is ot take at radom i the cross sectio of firms: i their meu cost model the price chages immediately after a shock are also the largest, so that the CPI respose is fast. Istead, i Calvo the size of the average price chage followig a shock is costat, so that the CPI rises more slowly ad the real effects are six times larger. Surprisigly, these features are fully captured by kurtosis, as i equatio ), sice it is the mass i the proximity of the adjustmet barriers

3 that determies the ature of the price chages followig a moetary shock. We show that a selectio effect also operates relative to the timig, ot just the size, of price chages. For istace i the models of Taylor ad Calvo, calibrated to the same frequecy N p i ), the size of the price chages is the same so there is o selectio i the Golosov-Lucas sese). Yet the real cumulative output effect i Calvo is twice the effect i Taylor. This happes because i Taylor the distributio of times util adjustmet is uiform, but i Calvo it is expoetial, with a thicker right tail of firms that adjust very late. This selectio effect cocerig the times of adjustmet is also captured by equatio ), sice a more dispersed distributio of times to adjust produces a distributio of price chages that is a mixture of ormals with differet variaces, ad hece a higher kurtosis. The mai advatage of equatio ) is robustess: the formula allows us to discuss the output effects of small moetary shocks without havig to solve for the whole geeral equilibrium or providig 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 is, at least i priciple, observable i the data. Of course equatio ) does ot hold i all models. For istace, it does ot hold i models where the steady state distributio of price chages is ot iformative about the cost ad beefits of chagig prices. The simplest example is a model with o meu costs i which with zero iflatio) the distributio of price chages mirrors the distributio of the uderlyig cost shocks. Yet, if there is a moetary shock, all prices chage immediately, ad there is o effect o output. I those models additioal iformatio is eeded to idetify the parameters ad assess the effects of moetary shocks. Overview We begi the paper with a review of the micro evidece o price settig behavior, across several micro datasets, ad the develop a theoretical model that is able to qualitatively replicate the observed patters. The empirical cotributio documets the presece of small ad large price chages, i.e. a very peaked distributio of price chages, usig a dataset of price records uderlyig the Frech CPI as well as several US datasets. This fidig, remiiscet of Kashyap 995) semial ivestigatio of selected catalog retail goods, persists eve at a very disaggregate level of product-outlet-type, rulig out a explaatio based o While this example is extreme, the same logic applies i a model with a small meu cost ad ifrequet shocks to productio costs that are similar to those observed i empirical distributio of price chages see Appedix H for details).

4 pure cross-sectio heterogeeity, ad it is similar to Kleow ad Kryvtsov 8) aalysis of the US CPI data. We also ackowledge that the CPI data may cotai measuremet error that teds to distort the measure of peakedess of the distributio of price chages, a issue emphasized by Eichebaum et al. 4). We estimate that after takig ito cosideratio measuremet error ad cross-sectioal heterogeeity the shape of the size-distributio of price chages is i betwee a Normal ad a Laplace distributio, with a kurtosis that is about 4 for the US ad about 5 for Frace. We develop a aalytical model that matches these patters, featurig 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, i.e. lack of a commo drift. 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 a 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 four 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 a Calvo model. I this case the size of the iactio set chages with 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). This result ca be used to aalyze the effects of ucertaity shocks which have received a lot of attetio recetly, as i e.g. Vavra 3). Secod, the model provides a mappig betwee the cost of price adjustmet ad the model parameters: we give a complete aalytical characterizatio of the meu cost implied by observable statistics such as the frequecy ad the variace of price chages. This mappig ca be used to quatify a value of ψ/b cosistet with the Abset radom meu cost the multi-product model ca at most produce a zero excess kurtosis. 3

5 evidece o the costs of price adjustmet, as measured by e.g. Levy et al. 997). It thus provides a ew dimesio to assess the plausibility of previous models, such as Calvo pricig, by ispectig the magitude of their implied adjustmet costs. Third, 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 ). To summarize, the parameterizatio of the model ca be thought as follows: the parameters ψ/b ad σ are pied dow by the observatios o N p i ), Std p i ), while the parameters, l) are pied dow by the shape of the distributio of price chages, e.g. by its kurtosis, ad by the data o the costs of price adjustmet. 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 or Reis 6) 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. For ay give l the level of kurtosis is icreasig i. Likewise, for a give the level of kurtosis is icreasig i l. 3 Fourth, we use the model to characterize aalytically the impulse resposes of the aggregate ecoomy to a oce-ad-for-all uexpected permaet icrease of the moey supply. The aggregate effect of a moetary shock depeds o the shock size, the frequecy, ad the kurtosis of price adjustmets. The depedece o the size of the shock is a hallmark of meu cost models: moetary shocks that are large relative to the size of price adjustmets) lead to almost all firms adjustig prices ad hece imply eutrality. We provide a detailed aalysis of the miimum size of the shock that delivers this eutrality full price flexibility). Istead, small shocks yield real output effects whose size crucially depeds o N p i ) as well as o the kurtosis of price chages, Kur p i ), a result that is ew i the literature ad was discussed i equatio ). 4 Although the model assumes a zero steady-state iflatio, we show aalytically that a 3 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. 4 Furthermore, our decompositio of the determiats of the real output effect shows how to measure ad aggregate across heterogeous sectors, see Appedix R. 4

6 small iflatio has oly a secod order effect o both the kurtosis of the price chages, the frequecy of price chages as well as o the cumulative real output effect. Hece the model ca be applied to ecoomies with a positive but small) iflatio rate, as observed i most idustrialized ecoomies. 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. 5 We show that itroducig a radom adjustmet costs serves a similar role as that of fat-tailed shocks i Midriga ), icreasig the real effect of moetary shocks ad brigig the model behavior closer to a Calvo model yet there are some differeces, see the cocludig remarks i Sectio 5 for a discussio). Our model is related to CalvoPlus model of Nakamura ad Steisso ) who cosider firms facig idiosycratic cost shocks as well as a meu cost that oscillates radomly betwee a high 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 aalysis of Dotsey, Kig, ad Wolma 999) who study umerically the propagatio of shocks whe firms face a meu cost that is draw from a distributio with a smooth desity, but face o idiosycratic shocks ad more recetly the work by Vavra 3). 6 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 theirs. 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 parameters/statistics ad the size of the output effect of moetary shocks. The paper is orgaized as follows: the ext sectio presets the cross sectio evidece o price settig behavior for Frace ad the US. Sectio 3 presets the theoretical model ad its cross sectio predictios: it is show that the model has fudametally four parameters ad 5 A selected list is Chari, Kehoe, ad McGratta ), Kiley ), Caballero ad Egel 7), Golosov ad Lucas 7). Chari, Kehoe, ad McGratta ) ad Kiley ) obtai that, cotrollig for the frequecy of price chages, the respose of output is stroger ad more persistet uder Calvo tha uder Taylor cotracts. Golosov ad Lucas 7) compare a moetary shock i a meu cost ad a Calvo model, with similar frequecies of price chage, ad fid that the half-life of the respose to the shock i the Calvo set-up is about five times larger tha i the meu cost model. 6 Dotsey, Kig, ad Wolma 9) exted their previous cotributio of a radom meu cost model to icorporate idiosycratic shocks. 5

7 we discuss the mappig betwee those ad observable measures of price settig behavior. Sectio 4 derives the model predictios o the effect of a uexpected moetary shock. Sectio 5 discusses the robustess ad scope of our mai results. The distributio of price chages: micro-evidece A vast amout of research has ivestigated the patters of price chages at the microecoomic level i the past decade. A recurrig fact that emerges from those studies is that the size distributio of price chages exhibits a large amout of small price chages, as oted by Kleow ad Mali ); Cavallo ); Kleow ad Kryvtsov 8); Che et al. 8) ad Midriga 9, ) usig selected samples of micro data from the US as well as may other idustrial coutries. This sectio revisits this evidece usig a detailed dataset of price quotes uderlyig the Frech Cosumer Price Idex about 65% of the CPI weights from 3 to ). We discuss measuremet error by comparig the CPI data with other sources, presumably less affected by measuremet error. Fially, we compare our evidece with comparable evidece for the US. Two issues that are ivestigated i detail cocer heterogeeity ad measuremet error. Heterogeeity across type of goods ad of outlets is pervasive i price data. A well kow result is that a mixture of distributios with differet variaces ad the same kurtosis will have a larger kurtosis. For this reaso we stadardize the data at levels at which we suspect there is heterogeeity i the variaces ad focus o the kurtosis of the pooled data. We defie the stadardized price chages, z, by demeaig ad dividig by the stadard deviatio of price chages at fie cell levels. A cell is a category of good ad of outlet type. We the compute the statistics for the pooled stadardized data. 7 The ature of the correctio for measuremet error is to compare the CPI statistics with data for similar goods ad outlet types that are less affected by measuremet error, as i the iteret store scraped data from Cavallo ), ad the scaer data sets used by Midriga ), Eichebaum et al. 4) amog others. We also aalyze the effect of outliers by lookig at the differetial effect of trimmig across datasets. The practice of ormalizig the data as well as removig outliers has bee used before, as i e.g. Kleow ad Kryvtsov 8); Midriga 9). We fid it useful to compare the empirical distributio of price chages to three parametric distributios ordered i terms of icreasig frequecy of extreme price chages: the biomial, the Normal, ad the Laplace distributio. Our aalysis shows that, after removig the time ivariat) cross sectio heterogeeity ad correctig for measuremet error, the size distributio of price chages still features a cosiderable mass of large as well as small 7 The model of Sectio 3 allows for sector ad/or good or outlet) heterogeeity ad discusses aggregatio. 6

8 price chages, relative to the biomial distributio implied by the stadard meu cost model. Overall we coclude that, after takig ito accout heterogeeity ad measuremet error, the shape of the empirical distributio of price chages lays i betwee a Normal ad a Laplace distributio. To quatify the presece of extreme price chages we focus o statistics that are iformative about the shape of the size distributio, that are appropriate for symmetric, zero-mea, distributios ad that are scale-free. These statistics measure the frequecy of extreme i.e. large ad small) observatios relative to the stadard deviatio of the distributio. Because of its promiet role i our theoretical aalysis we will focus especially o kurtosis whose value, for the bechmark Biomial, Normal ad Laplace distributio, is, 3 ad 6 respectively.. The Frech microecoomic data uderlyig the CPI The data are a logitudial dataset of mothly price quotes collected by the INSEE i order to compute the Frech CPI, over the period 3:4 to :4. 8 Each record relates to a precisely defied product sold i a particular outlet i a give moth. It cotais the price level of the product, as well as limited additioal iformatio such as a outlet idetifier, a idex whe relevat) for package size say liter) ad flags idicatig the presece of sales. The raw dataset cotais aroud millio price quotes ad covers about 65% of the CPI weights. 9 The dataset also icludes CPI weights, which we use to compute aggregate statistics. Price chages are computed as times the log-differece i prices per uit. To miimize the presece of measuremet errors we discarded observatios with item substitutios which might give rise to spurious price chages) ad removed outliers which, i our baselie aalysis, we defied as price chages whose absolute value is smaller tha. percet, or larger tha about) percet. A importat issue with the data o price chages is the treatmet of sales. The relevace of dealig with sales i aalyzig price stickiess was emphasized by Nakamura ad Steisso 8); Kehoe ad Midriga 7) ad Midriga ) iter alia. The INSEE dataset cotais a idicator variable that idetifies whether a give observed price correspods to a sales promotio discout either seasoal sale or temporary discouts). Price chages that 8 The dataset is documeted i details i Berardi, Gautier, ad Le Biha 3). 9 Some categories of goods ad services are ot available i our sample: fresh foods, rets, ad prices cetrally collected by the statistical istitute - amog which car prices ad admiistered ad public utility prices e.g. electricity). Note that, while rets are out of our dataset, cost of ower-occupied housig is ot icorporated i the Frech CPI, so the share of housig i the CPI is lower tha i some other coutries. Some sales ivolve large discouts, up to 7%, e.g. i clothig. The upper threshold log/3), which is about percet log poits, allows us to accommodate the after-sale price to retur to the origial level. See Appedix B for more iformatio ad several robustess checks. The flag is documeted by the field aget rather tha costructed usig a statistical filter. Baudry et al. 7

9 Figure : Histogram of Stadardized Price Adjustmets: Frech CPI 3- All data Excludig sales Data Stadardized Normal Stadardized Laplace Data Stadardized Normal Stadardized Laplace The figures use the elemetary CPI data from Frace 3-). 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 upper pael uses about.5 millio data poits, the lower pael about. millio. result from sales icludig price chages from a sales price to a regular oe) accout for approximately 7% of all the price chages. Overall, the icidece of sales o the frequecy of price chage is less importat tha i the US where accordig to Nakamura ad Steisso 8) the share of price chage due to sales is.5%. I the followig, as a robustess check, we report results both with ad without sales observatios. We ow documet the patters o the peakedess ad thick tails of the distributio of price chages. As those patters vary cosiderably across sectors ad outlet types, a cocer already metioed is that a large variace ad kurtosis of price chages may essetially reflect that observatios of price chages are draw from a mixture of distributios, ad thus may be artefacts.we cosider a breakdow of the data ito J cells, where each cell is defied by a good category ad outlet type for istace, oe cell will be bread i supermarkets). We will here use the fiest partitio possible i our data each cell is a COICOP category at the 6-digit level i a outlet type) ad have aroud,5 cells. I each cell j the stadardized price chage at date t for good i is defied as z jit = p jit m j )/σ j where m j ad σ j are the mea ad stadard deviatio of price chages i cell j, ad price chages equal to zero are disregarded idex i reflects that there is typically more tha oe store or good type per cell 7) ivestigate the extet of udetected sales ad coclude this is a limited cocer. There are outlet types ad 7 CPI categories. Not every category of good is sold i every outlet type so there are about half of the potetial,99 cells. 8

10 e.g., differet types of bread or differet stores sellig bread i the sample at a give date). Figure is 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 the empirical distributio of stadardized price chages is closer to the Laplace distributio tha to the Normal. 3 We also cosider the statistic E[ p i ]/Std p i ) as a reverse) measure of the frequecy of extreme price chages. The mai differece of this statistic with respect to kurtosis is that it is less sesitive to extreme outliers. For the Biomial, Normal ad Laplace distributios the referece values for this statistic are:,.8 ad.7. Table reports the frequecy of price chages as well as selected momets of the distributio of price chages. The frequecy of price chage is aroud 7% per moth, or about price chages per year. The fractio of price decreases amog price chages is aroud 4%. The average absolute price chage ot reported i the table) is sizable 9.%), as is the stadard deviatio of price chages 6.6%). These patters match those documeted by Alvarez et al. 6) for the Euro area. With the qualificatio that the frequecy of price chages is typically foud to be smaller i the Euro area tha i the US, they also broadly match the US evidece by e.g. Nakamura ad Steisso 8). The kurtosis ad peakedess of the distributio of price chages have ot bee quatitatively documeted so far o Europea data. The kurtosis of o-stadardized price chages is very high:.8. This level of kurtosis is of same order of magitude as that documeted by Kleow ad Mali ) for the US. As argued above, a high kurtosis is likely the cosequece of a mixture of observatios take from distributios with differet variaces. Cosiderig the stadardized price chages, i.e. correctig for cross-sectio heterogeeity i the variaces, reduces kurtosis to 8.9 also, kurtosis is halved o data excludig sales). Ufortuately the iformatio available i the CPI data prevets us from correctig heterogeeity at a fier level e.g., we do ot kow the UPC of the product or the store where it is sold). I other databases where more iformatio is available the reductio i the measured kurtosis is eve more promiet, as show usig US scaer dataset i Appedix B.. It is show that kurtosis falls by aother 3 to 4% whe movig to the product-store level. The fractio of extreme small or large) price chages is oticeable. The fractio of absolute stadardized price chages lower tha oe fourth of the mea is. percet. Also.9 percet of absolute ormalized price chages are larger tha times the mea of the absolute stadardized price chage. Overall, it appears that these figures are very close to the 3 I Appedix B. we provide similar histograms by groups of good at a disaggregated level. Most of them have the same patter as Figure, that is a distributio that is more peaked tha the gaussia, ad ofte more peaked tha the Laplace. 9

11 Table : Selected momets from the distributio of price chages Data Bechmarks all records exc.sales Normal Laplace Frequecy of price chages Fractio of price chages that are decreases Momets for the size of price chages: p i Average.3. Stadard deviatio Kurtosis.8.9 Momets of stadardized price chages: z Kurtosis Momets for the absolute value of stadardized price chages: z Average: E z ) Fractio of observatios <.5 E z ) Fractio of observatios <.5 E z ) Fractio of observatios > E z ) Fractio of observatios > 4 E z ) Number of obs. with p i,544,89,8,83 Source is INSEE, mothly price records from Frech CPI, data from 3:4 to :4. Coverage is aroud 65% of CPI weight sice rets, ad prices of fresh food ad cetrally collected items e.g. electricity, trai ad airplae tickets) are ot icluded i the dataset. Frequecy of price chage is the average fractio of price chages per moth, i percet. Size of price chage is the first-differece i the logarithm of price per uit, expressed i percet. Observatios with imputed prices or quality chage are discarded. Observatios outside the iterval. p i log/3) are removed as outliers. Exc. sales exclude observatios flagged as sales by the INSEE data collectors. Momets are computed aggregatig all prices chages usig CPI weights at the product level. The third ad fourth paels report momets for the stadardized price chage z ijt = pijt mj σ j where m j ad σ j are the mea ad stadard deviatio of price chages i category j see the text). The Normal ad Laplace distributios used i the last two colums have a zero mea ad stadard deviatio equal to oe. oes that would be produced by a stadardized) Laplace distributio. Cosistetly, the size of the average absolute stadardized price chage i the data is equal to.7, the same value that obtais for the statistic E[ p i ]/Std p i ) if p i follows a Laplace distributio. There is a mass of large price chages, say larger tha 4 stadard deviatios, that are virtually abset i the Gaussia case. Removig sales has a large effect o the variace of price chages, as idicated by the results reported i the secod colum of Table. 4 However, removig sales does ot affect our fidigs o the peakedess of the distributio. Kurtosis actually icreases whe sales 4 We remove the observatios flagged as sales as well as the subsequet icrease back to the regular price. To compute the stadardized o-sales price chages, we first discard sales-related observatios, the stadardize the data.

12 observatios are removed both i the raw data as well as i the stadardized data. This is also visible i the right pael of Figure which plots the distributio of stadardized ex-sales) price chages.. Measuremet error ad the estimatio of kurtosis I this sectio we discuss evidece o the magitude of both very large ad very small price chages due to measuremet error, ad discuss its effect o measures of kurtosis. Eichebaum et al. 4) wared that the small price chages recorded i the data may reflect measuremet error. Appedix B.3 explores the cocers raised by Eichebaum et al. 4) ad cocludes that they also apply to the Frech data, albeit to a lesser extet. We aalyze here the cosequeces of oe particular type of measuremet error ad derive a simple correctio for kurtosis. We show that a small amout of this measuremet error, icosequetial for measurig the aggregate the cost of livig, may have sizeable cosequeces for the measuremet of the descriptive statistics displayed i Table, such as kurtosis, ad suggest a procedure to correct for it. Let the observed price chages p m be give by a mixture of two distributios: p u p m = ɛ with prob. ζ with prob. ζ where we iterpret ɛ as measuremet error ad p u as a true price chage. This assumptio aims to capture that, eve at the fiest level of disaggregatio, some price chages i the CPI data are the cosequece of small product substitutios e.g. differet brads for a give good beig recorded) which do ot reflect a actual chage i the good s price. Likewise, i scaer dataset, spuriously small price chages may origiate from the weekly ature of the prices beig recorded, which e.g. averages customers with ad without discout coupos. Assume the distributio of p u has stadard deviatio σ u ad kurtosis k u. Likewise the distributio of ɛ has stadard deviatio σ e ad kurtosis k e. Both distributios are assumed to have zero expected value. Oe iterpretatio is that quality chages ot recorded by the statistical office) geerate artificial price chages. We assume that these price chages are small, i.e. that σ e is small, ad that the process for the ureported chage i quality is idepedet of the true chages i prices. The kurtosis of the observed price chages, Kur p m ), is ζσu the equal to: Kur p m ) = k 4+kɛ/ku)σ4 e u. Lettig σ ζ σu 4+ ζ) σe 4 e go to zero we obtai that +ζ ζ)σ e σ u kurtosis measured over the observed) price chages is: lim σe Kur p m ) = k u /ζ. Thus, if the sample icludes a fractio ζ of true price chages ad the rest are spuriously imputed small price chages the kurtosis will icrease by a factor /ζ, relative to the kurtosis of the

13 true distributio. 5 The limit Kur p m ) = k u /ζ suggests to quatify ζ by comparig the observed kurtosis across a sample with measuremet error ad oe without. We ow tur to addressig this issue empirically. We match a subset of our Frech CPI data with the prices for several Frech retailers take from the Billio Price Project BPP) dataset see Cavallo )). The BPP data are scraped o-lie, thus they are arguably less cotamiated by measuremet errors. 6 We compare the results obtaied usig the scraped BPP data from two large retailers with our results based o the CPI data for a similar type of outlet: to this ed we restrict our dataset to CPI price records i hypermarkets, excludig gasolie. We also compare with the BPP data from a large Frech retailer specialized i electroic ad appliaces. I that case we restrict the CPI dataset to goods i the category of appliaces ad electroic usig the Coicop omeclature, collected i outlets type hypermarkets, supermarkets, ad large area specialists. Comparig the values of kurtosis from both data sets suggests that ζ =.5. We ca apply this magitude to the full sample of CPI data of Table, for which o measuremet error-free couterpart like the BPP exists, to obtai a corrected kurtosis. The umber thus obtaied for the kurtosis rages betwee 4 ad 5 usig the kurtosis of 8.9 of stadardized price chages), so it lays i betwee the kurtosis of the Normal ad the Laplace distributio. Sesitivity to trimmig. To assess the hypothesis that large price chages are due to measuremet error we compare the differetial effect o kurtosis of trimmig large absolute value) price chages i CPI vs trimmig them i scaer data. For the Frech CPI we fid that excludig log percet) price chages for which p i log), istead of just excludig those that are p i log/3), decreases kurtosis from 8.9 to 7. see lies 4 ad 8 of Table 6). We iterpret this as evidece of measuremet error uder the hypothesis that large price chages i the CPI are due to trascriptio data errors, which are virtually abset i iteret scraped data. 7 scaer data is preseted i the ext sectio. Similar evidece o the effects of trimmig usig US 5 Uder this iterpretatio the umber of measured price chages, deoted by N am will be higher tha the umber of true price chages per uit of time, say N au. Let s deote N aɛ the expected umber of icorrectly imputed price chages. We have: N am = N au + N aɛ = ζn am + ζ)n am. Thus if we have two estimates of Kur p i ) ad of N p i ) ad we assume that oe has o measuremet error ad the other has a fractio ζ of small imputed price chages as described above, ca estimate ζ usig either the ratio of the two estimates of kurtosis or the ratio of the two estimates of the umber of price chages per uit of time. 6 We are extremely grateful to Alberto Cavallo for sharig part of his data with us. 7 I the Frech CPI we have observed cases where a sigle large price chage which reverts to its origial value, which correspods to cosecutive prices that are equal except for the traspositio of a digit, which is almost surely a clerical error.

14 Table : Compariso of the CPI vs. the BPP data i Frace Statistic BPP BPP CPI BPP CPI retailer retailer 5 Hypermarkets retailer 4 Large ret. electr. duratio moths) Statistics for stadardized price chages: z mea z % below.5 mea z % below.5 mea z kurtosis of z Note: The BPP data are documeted i Cavallo ). Results were commuicated by the author. For CPI data source is INSEE, mothly price records from Frech CPI, data from 3:4 to :4. Sub-sample i colum 3) is price records i outlet type hypermarkets. Sub-sample i colum 5) is goods i the category of appliaces ad electroic, as idetified usig the Coicop omeclature, collected i the followig outlets type: hypermarkets, supermarkets, ad large area specialists. Data are stadardized withi each subsample usig Coicop categories..3 A compariso with the US data This sectio compares the Frech evidece with the US evidece preseted, respectively, i Kleow ad Kryvtsov 8) ad i the scaer data used by Eichebaum, Jaimovich, ad Rebelo ), as well as other scaer datasets. Figure plots four histograms: two are price chages from the US ad Frech CPI data, while the other two are theoretical bechmarks. The first oe i red) is the distributio of stadardized weighted) price chages excludig sales) for the US based o Figure 3 of Kleow ad Kryvtsov 8). 8 Sice the distributio is trucated at -3 ad +3, its stadard deviatio is.83 istead of. Its kurtosis is The secod histogram i blue) is the distributio of the stadardized price chages excl. sales) for the Frech CPI, costructed usig the trimmig criteria used for the US. This distributio has a stadard deviatio.95 ad a kurtosis of 4.4. The smaller stadard deviatio ad much smaller kurtosis tha i Table are due to the discretizatio ad trucatio. To see the effect of these treatmet of the data, ote that Kleow ad Mali ) report a kurtosis of for posted prices ad 7.4 for regular prices, without discretizig, cesorig, or stadardizig the data. For compariso Vavra 3) fids that after trimmig the data i a way similar to our treatmet of the Frech data, but without stadardizig it, the kurtosis of US CPI price chages is 6.4. O the other had, Vavra fids that the kurtosis of stadardized price chages is The figure also reports the stadardized Normal ad Laplace distri- 8 The histogram has twety four bis, spaced every.5 uits, of the distributio of stadardized regular price chages excl. sales). The stadardizatio was doe by ELI, the arrowest categories of goods. After stadardizatio the distributios are weighed accordig to the CPI weight. 9 See Table IV ad footote 6 of Vavra 3) for the specifics o the trimmig. We thak Vavra for providig this statistics which is ot available i his paper. 3

15 butios discretized ad trucated). The mai message of Figure is that the histogram of stadardized, o-sales, price chage are rather similar i Frace ad the US. Furthermore, i both cases the shape is closer to that of a Laplace distributio tha to a Gaussia oe ad cosistetly with previous sub-sectio, i both cases we cojecture measuremet error explais why these distributios are actually more peaked tha the Laplace). Figure : Histogram of Stadardized Price Adjustmets: US ad Frech CPI.5 US CPI Frace CPI Laplace Normal..5 Fractio Size of price chages Sales data are excluded. Data for Frace are from the CPI as i Figure. The CPI data for the US are take from Figure 3 i Kleow ad Kryvtsov 8). Price chages equal to zero are discarded. Table 3 provides a further compariso based o datasets presumably less subject to measuremet errors. For Frace we use data from the BPP, ad those from hypermarkets i the CPI dataset. For the US we use the scaer data evidece reported by Eichebaum, Jaimovich, ad Rebelo ), Midriga ), as well as our ow aalysis of the Symphoy- IRI scaer dataset. As reflected by the summary statistics, the distributio is somewhat more peaked i Frace; for istace the kurtosis is 5 i the BPP agaist values betwee 3 ad 4 i US scaer datasets described below. Overall, we fid that the share of small price chages is o-egligible i both coutries. For completeess we report complemetary evidece take from scaer data sets, described i more detail i the olie Appedix B.. While geerally immue to measuremet errors due to trascriptio ad product substitutios, eve scaer price data are affected by measuremet error sice they report average weekly prices for a item, so that observed prices are a average of prices paid by customers with ad without discouts. We hypothesize, as Midriga ) ad Eichebaum et al. 4

16 4) amog others, that the weekly averagig itroduces spurious small price chages. The EJR colum of Table 3 uses the data from Eichebaum, Jaimovich, ad Rebelo ): after stadardizig at the good-store level ad removig very extreme price chages smaller tha.% or larger tha % log poits) as i our bechmark for Frace), yields a kurtosis of 3.. Alteratively, trimmig price chages smaller tha oe dollar cet, certaily due to measuremet error, the kurtosis falls little below the oe of the Normal. Likewise, usig the larger Symphoy-IRI dataset see Broeberg, Kruger, ad Mela 8)), shows that removig extreme price chages ad stadardizig the data reduces kurtosis from a level of about 3 to 4.3. Fially, we otice that the effect of extreme price chages o the measuremet of kurtosis is also preset i the CPI. For the Frech CPI excludig log percet) price chages for which p i as opposed to oly excludig those price chages for which p i /, reduces kurtosis from 7. to 6.3 see lies 4 vs of Table 6). For the Norvegia CPI Wulfsberg ) fids that kurtosis is also sesitive to both large ad small price chages, removig the ad 99 percetiles decreases kurtosis from 8. to 5.7. Table 3: Compariso across datasets for large Hypermarkets i Frace ad the US Frace US scaer data) CPI BPP EJR Symphoy IRI Midriga 9) Statistics for stadardized price chages: z mea of z % below.5 mea z % below.5 mea z kurtosis of z All price chages icludig sales. The BPP statistics for Frace are a average of the oes reported i Table. The EJR scaer data are from Eichebaum, Jaimovich, ad Rebelo ), the IRI scaer data are from the Symphoy-IRI database described i Appedix B.. For both scaer data sets idividual observatios are average weekly prices. We remove price chages outside the iterval. p i log/3) ad stadardize the data at the good upc) store level see Appedix B.). The data from Midriga 9) are take from his Table ad b, usig simple averages of the AC Nielse ad Domiick s scaer data. Midriga stadardizes price chages by dividig by the mea absolute value of the price chages at the product-store-moth level. He also removes price chages smaller tha cet or larger tha %. Overall we coclude that, after accoutig for heterogeeity ad measuremet error, the presece of both small ad large price chages appears relevat i Frace as well as i the US. The shape of the stadardized empirical distributio of price chages lays i betwee a Normal ad a Laplace distributio. The distributio appears close to a Normal i the US ad closer to Laplace i Frace. We summarize the results of this sectio by statig that the stadardized distributio of price chages, after takig measuremet error ad heterogeeity ito accout, has kurtosis of about 4 for US ad of about 5 for Frace. 5

17 3 A tractable meu cost model 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 ); 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) the radom meu costs ad that 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 oe or all goods at o extra cost. Each of these assumptios idividually is capable 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 discussed above. 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 shape of the size distributio that is comparable to the oe i the data oe eeds A radom meu cost problem for a firm sellig = good. Cosider a firm whose profit-maximizig price at time t, p t), follows the process dp t) = σ dw t) where W t) is a stadard browia motio with o drift, ad σ is the stadard deviatio of the iid iovatios to p. The techology to chage prices is as follows: to chage the price at will the firm eeds to icur a fixed meu cost of size ψ. However, with some probability the firm receives a opportuity to adjust the price for free. Assume this probability is Poisso, i.e. that the free-adjustmets have a costat hazard rate per uit of time, equal to λ. Let pt) deote the price gap at time t, i.e. the differece betwee the actual sale price P t) ad the profit maximizig price p t), i.e. pt) P t) p t). The istataeous firm losses i.e. reductio i profits) created by the price gap are give by the quadratic fuctio: B p t). Let V p) be the preset-value cost fuctio for a firm with price gap p. Upo the arrival of a free adjustmet opportuity the firm optimally resets the price 6

18 gap to zero, 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 Appedix O 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 there is o first order derivative of the value fuctio sice the price gaps have o drift). The value-matchig ad smooth-pastig coditios are give by V p) = V ) + ψ ad V p) =. Two properties of the optimal threshold p are worth oticig proved later for ay ): the value fuctio, ad the optimal decisio rules, are a fuctio of λ + r, as opposed to each of them separately. Ituitively this is because whe a free adjustmet opportuity occurs the price gap is adjusted, so that λ acts as a additio to the discout factor. Secod, for a small value of ψ/b or a small value of λ + r, the value of p is isesitive to λ + r. More precisely, the derivative of p with respect to λ + r is zero as ψ/b or λ + r ted to zero. A Taylor expasio of the value fuctio yields ) 4 the followig approximate optimal threshold p = which is accurate whe ψ/b is small. 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. The distributio of price chages w p i ) is symmetric aroud p i =. This distributio has a mass poit at p i = ± p with probability l, i.e. this is the fractio of price chages that occurs because the price gap reaches the boudaries of the iactio regio. The remaiig fractio of price chages, l, occurs whe a free adjustmet opportuity arrives, at which time the price gap is set to zero. Price chages i the rage p p, p) have a desity 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 the case i which ψ is large. 7

19 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]. ) Thus the distributio of price chages is give by Pr p i = p) = Pr p i = p) = l) Pr p i dp) = l gp)dp w p i )dp for p p, p) which is a symmetric tet shaped distributio i the p, p) iterval with the two mass poits at the boudaries ± p. As detailed below the kurtosis of this distributio is icreasig i λ, ad i particular the distributio of price chages is more peaked tha that of a stadard meu cost model where λ =. We make two remarks about this simple model which will hold, ad be geeralized, i the more geeral model developed ext. The first oe is 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 of this property the ratio of momets from the size distributio of price chages, such as kurtosis, are scale free ad ca be used to retrieve iformatio o φ. The secod property, which we state here ad prove below for the more geeral ecoomy, is that the shape of the impulse respose fuctio of this ecoomy to a oce ad for all) moetary shock depeds oly o l. We will show how oe ca simply scale or relabel) oe or both axes of a impulse fuctio to aalyze ecoomies with the same l that differ i either N p i ) or Std p i ). 3. Extedig the model to 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 8