The Skewness of the Price Change Distribution: A New Touchstone for Sticky Price Models

Finance and Economics Discussion Series Divisions of Research & Statistics and Monetary Affairs Federal Reserve Board, Washington, D.C. The Skewness of the Price Change Distribution: A New Touchstone for Sticky Price Models Shaowen Luo and Daniel Villar 2017-028 Please cite this paper as: Luo, Shaowen and Daniel Villar (2017). “The Skewness of the Price Change Distribution: A New Touchstone for Sticky Price Models,” Finance and Economics Discussion Series 2017-028. Washington: Board of Governors of the Federal Reserve System, https://doi.org/10.17016/FEDS.2017.028. NOTE: Staff working papers in the Finance and Economics Discussion Series (FEDS) are preliminary materials circulated to stimulate discussion and critical comment. The analysis and conclusions set forth are those of the authors and do not indicate concurrence by other members of the research staff or the Board of Governors. References in publications to the Finance and Economics Discussion Series (other than acknowledgement) should be cleared with the author(s) to protect the tentative character of these papers.

The Skewness of the Price Change Distribution: A New Touchstone for Sticky Price Models∗ ShaowenLuo† DanielVillar‡ DepartmentofEconomics FederalReserveBoardof VirginiaTech Governors March10,2017 Abstract Wepresentanewwayofempiricallyevaluatingvariousstickypricemodels used to assess the degree of monetary non-neutrality. While menu cost modelsuniformlypredictthatpricechangeskewnessanddispersionfallwith inflation,intheCalvomodelbothrise. However,CPIpricedatafromthelate 1970’s onwards shows that skewness does not fall with inflation, while dispersion does. We develop a random menu cost model that, with a menu cost distribution that has a strong Calvo feature, can match the empirical patterns found. Themodelthereforeexhibitsmuchmoremonetarynon-neutralitythan existingmenucostmodels. JELclassificationcodes: E31,E32,E47,E52 ∗WewouldliketothankEmiNakamura,Jo´nSteinsson,RicardoReisandMichaelWoodfordfor theirinvaluableadviceandsupport. WealsothankJenniferLa’O,Mart´ınUribe,Ste´phaneDupraz, JorgeMej´ıa-Licona,SavitarSundaresan,ErickSager,TimothyErickson,andotherseminarparticipantsatColumbiaandtheBureauofLaborStatisticsforvaluablefeedbackandsuggestions. The data was made accessible to us by the BLS, and we thank Ted To and John Molino for their help asourBLScoordinators. Allremainingerrorsareourown. Theviewsexpressedinthispaperare those of the authors and do not necessarily reflect those of the Board of Governors or the Federal ReserveSystem. †sluo@vt.edu,3016PamplinHall,880WestCampusDrive,Blacksburg,VA24060 ‡daniel.villar@frb.gov,20th&ConstitutionAve. NW,Washington,DC20551 1

1 Introduction Thedynamicsofpricechanges(when,how,andwhyfirmschangethepricesofthe goodsandservicesthattheysell)havebeenamajorfocusofthestudyofmonetary economics for the past several decades. It is indeed well known that monetary variables have no influence on real economic activity (monetary neutrality) if all prices can be freely re-set at any point in time. Much work has therefore been done incorporating frictions in price-setting models, and using detailed price data to measure how sticky prices really are. One important finding in this literature is that the degree of monetary neutrality will depend not only on how often prices change, but also crucially on which prices change. If the prices that change are those most mis-aligned from their optimal level (as they would if firms must pay price adjustment, or menu costs), money will be much more neutral than if they were randomly selected (as in a model in the style of Calvo (1983)). In this paper, wefollowinthelineofworkthathasattemptedtodeterminetheextenttowhichthis selectionoccurs. We presenta newmethod fortesting thestrength ofthis selection effect, based on empirical patterns that have not been previously considered and usinganewdatasetofpricesinhighinflationperiods. Caplin and Spulber (1987) and Golosov and Lucas (2007) made the point that, in the presence of menu costs, only relatively large price changes will justify the payment of the cost and occur at all, which makes the aggregate price level considerably more responsive to nominal shocks than in the Calvo mode (reducing monetary non-neutrality). Understanding this selection mechanism is necessary to determine the extent of monetary non-neutrality due to price rigidity, and has received considerable attention in the monetary literature. This task is made challenging by the fact that the selection effect is a mechanism that cannot be observed directly. It would be very difficult to observe whether the prices that change are 2

thosepredictedbytheselectioneffect,soitspresenceandstrengthmustbeinferred indirectlyfromobservablepricechangestatistics. Theexistingworkinthefieldhas done this primarily by bringing quantitative price setting models together with the price data that has become available in the past decade. These studies have, for the mostpart,usedunconditionalmomentsofthepricechangedistribution(suchasthe frequency or size of price changes, averaged over time) to discipline the models in question. Inthispaper,weshowthatconditionalhighermomentsofpriceschanges are extremely informative and yield new insights on the selection effect. In particular, we find that the selection effect makes very strong predictions about how the shapeofthepricechangedistributionshouldchangewithaggregateinflation. Inmenucostmodels,thepresenceofafixedadjustmentcostinducesaselection effect: onlypricechangesthatarelargeenoughtojustifythecostoccur,leavingan inaction region of changes (centered at zero) that are too small to be justified. A positivemonetaryshock(raisingnominaldemand)willinducepricesthatwereotherwisealreadystronglymis-alignedtochange(leavingothersunchanged),meaning that average price changes would respond relatively strongly to such a shock. This implies, in turn, that the aggregate price level will be very responsive to monetary shocks, eliminating much of the effect of the monetary shock on real activity (money is close to neutral). We exploit the fact that this logic also has strong implications for how the distribution of price changes responds to such shocks: an inflationary shock will push more price changes out of the inaction region to the positive side, and into the inaction region from the negative side. There will thereforebemorepricechangesconcentratedonthepositivesideoftheinactionregion, leavingapricechangedistributionthatislessdispersedandmoreasymmetric(negatively skewed). Indeed, all existing menu cost models, because of the selection effect created by the presence of an adjustment cost, imply a very strong negative correlation between inflation and both dispersion and skewness of price changes, 3

andtheseareimplicationsthatcanbeempiricallytested. A limitation to studying these implications has been that the main source of price data in this line of work, the micro data underlying the Consumer Price Index, was, until recently, only available going back to 1988 (while other commonly used data sets go back even less far), covering periods of low and stable inflation.1 However, we use the data set recently presented in Nakamura et al. (2016), which extends the C.P.I micro data back to 1977, to evaluate whether the dispersion and skewness of price changes do indeed fall with inflation. Since the newly recovered period includes the highest inflation episodes in the post-war U.S., as well as the disinflationperiodinitiatedbytheFederalReserveunderPaulVolcker,ourdataset isparticularlywellsuitedfortheteststhatwepropose. We find that while the dispersion of price changes does go down considerably in high inflation periods, the skewness does not. This latter result is contrary to the predictionsofmenucostmodels,andisthereforeinconsistentwithaverystrongselectioneffect,whilethedispersionresultisconsistentwithmenucosts. Todevelop a model consistent with both results, we modify the menu cost model in a way that weakens the selection effect: introducing random, heterogeneous menu costs that add randomness to whether the firm will have an opportunity to change its price. The model therefore include some of the features of the Calvo model, and can be thoughtofasahybridbetweenstate-andtime-dependentmodels. Byworkingwith random menu costs, we follow the example of Dotsey et al. (1999), and we adjust the distribution of menu costs to fit the new correlations that we report, and find that,especiallytomatchthenon-negativeinflation-skewnesscorrelation,thedistri- 1Although some studies (such as Alvarez et al. (2016a); and Gagnon (2009)) have used price data from countries that experienced high inflation, they used this data to determine how the frequency of price change behaves at high inflation, without considering the higher moments of the price change distribution. Notably, Alvarez et al. (2016a) look at the dispersion of prices (within narrowproductcategories),butnotofpricechanges. 4

bution of menu costs needs to feature a positive probability of price changes being free,andahighprobabilityofmenucostsbeingveryhigh. Thesecorrelationsallow ustorestrictthemenucostdistributioninawaythatDotseyetal.(1999)couldnot, with important implications for monetary non-neutrality. Indeed, our model featuresamuchhigherlevelofmonetarynon-neutralitythananyoftheexistingmenu cost models: around six times higher than in a standard menu cost model, higher eventhaninMidrigan(2011)and70%ashighasinaCalvomodel. Our work builds on a number of earlier papers that investigate the effect of price setting dynamics on monetary non-neutrality. While a few empirical studies of price stickiness in certain industries have been around for some time (e.g. Cecchetti (1986); Carlton (1986); Kashyap (1995)), it is only starting with Bils and Klenow (2004) that monetary economists have been able to start measuring statistics related to price stickiness for the economy as a whole. The facts established by Bils and Klenow and the subsequent empirical studies on price stickiness (most notably, Klenow and Kryvtsov (2008); and Nakamura and Steinsson (2008)) have enriched the discussion on monetary non-neutrality by providing the models that evaluate monetary non-neutrality with a standard by which to be measured. Since Golosov and Lucas (2007), the literature has continued to combine quantitative, micro-founded,pricesettingmodelswithempiricalfactsfrommicropricedatasets, andinthiswaythenon-neutralitydebatehasadvanced(forexample,Nakamuraand Steinsson (2010); Midrigan (2011); Alvarez et al. (2016b)).Nakamura and Steinsson(2010)andMidrigan(2011)hadalreadypointedoutproblemswithsomeofthe predictions of the Golosov and Lucas model, and shown that changes to the model thatcorrectedtheseproblemsoverturnedtheresultoflowmonetarynon-neutrality. However, we show that even these modifications to the Golosov and Lucas model, though they reconcile the menu cost framework with the data in some ways, are alsoinconsistentwiththefactsthatwepresent. 5

In a slightly different style, Vavra (2013) showed that the frequency and dispersion of price changes are counter-cyclical in the U.S., and introduced countercyclicaldispersionshockstomatchthis. Gagnon(2009)andAlvarezetal.(2016a) use price data from high inflation episodes in Mexico and Argentina, respectively, to show that the frequency of price change rises with inflation, which is consistent with menu cost models. Our paper confirms this result, but documents more patterns based on other statistics that paint a more nuanced picture: changes in the shape of the price change distribution (measured by its dispersion and skewness) are also informative to distinguish between the models. Our work is also in some waysrelatedtoBallandMankiw(1998),whohadarguedthatchangesintheskewness of the distribution of desired price changes could, in a menu cost framework, drive fluctuations in inflation. We are instead considering how changes in inflation (driven by aggregate, or first moment shocks, in the models) will affect the skewnessofrealized(andobserved)pricechangesindifferentmodels. Anotherpaperthathastriedtoinferthedegreeofmonetarynon-neutralityfrom the shape of the price change distribution is Alvarez et al. (2016b). They present a price setting model that nests many of the models that we consider, and show that in this model, the kurtosis of price changes (along with the frequency of price change) is a sufficient statistic for the real effect of monetary shocks. Using price micro data from the French CPI and from Dominick’s supermarkets, they then find thatthemeasuredvalueofthekurtosisimpliesadegreeofmonetarynon-neutrality between those of the standard menu cost and Calvo models. Although their paper focuses on the kurtosis instead of the skewness, the logic behind their theoretical result is related. Indeed, the kurtosis captures the relative importance of very small and very large values in the price change distribution. A high kurtosis is unlikely to be consistent with a strong selection effect, because if selection were strong, price changes would not be concentrated at small values. In our case, the extent to 6

which the asymmetry of the price change distribution (as measured by the skewness)changeswithinflation(astheunderlyingdistributionofdesiredpricechanges moves) is also determined by presence or absence of inaction regions, which determine the strength of the selection effect. Overall, we view both of these papers as complementary ways to get at the question of monetary non-neutrality, and find similarresults. The rest of the paper is organized as follows. In Section 2, we present the predictionsofalargeclassofstickypricemodels,andexplainwhytime-andstatedependent models give such different predictions. Section 3 describes the data set that we use and evaluates the predictions of the different models based on the data. Section4presentsthegeneralizedmenucostmodel,comparingpredictionstowhat is observed in the data and shows the degree of monetary non-neutrality exhibited bythedifferentmodels. Finally,Section5providessomeconcludingremarks. 2 The Skewness of Price Change in Sticky Price Models In this section, we explain and illustrate how the co-movement between inflation and the higher moments of the price change distribution provides information on the strength of the selection effect, and therefore on the degree of monetary nonneutrality. First, we provide an intuitive explanation based on the mechanics of menu cost models, and then present simulations from various sticky price models toillustrateourpoint. 2.1 Intuition for the Menu Cost Model Price change dynamics in the menu cost model can be thought of in the following way: bothidiosyncraticandaggregatenominalshockstofirms’optimalpricesyield a distribution of desired price changes (the price change a firm would choose if it 7

changed its price, or in the absence of price change frictions). The presence of a menu cost means that only desired price changes above a certain size (positive and negative)willactuallyoccur,asonlythosewillyieldabenefittothefirmbigenough to compensate for the menu cost. The realized price change distribution in this modelisthereforetheunderlyingdistributionwithabandcontaining0removed,as illustratedinFigure1. Figure1: IntuitionfortheMenuCostModel Note: In the first three panels, the black curve represents the distribution of the desired price change. The dashed lines represent the Ss band. The grey shaded area represents the distribution of realized price changes. The last panel plots skewness (blue curve) and dispersion(orangecurve)oftherealizedpricechangedistributionasafunctionofthelevel ofinflation. DesiredpricechangesfollowN(µ,0.052),“S”bandat0.01,“s”bandat−0.01, whilevaryingµ. The presence of idiosyncratic shocks implies variation in firms’ desired price changes, and nominal aggregate shocks move the position (average) of the underlying distribution. For example, a positive aggregate shock moves the distribution to the right, which also leads to realized prices being higher on average, resulting in higher inflation (the reverse is true for negative aggregate shocks). As a positive aggregate shock raises the average desired price change and the average realized price change, some negative price changes (to the left of the inaction region) remain and form the left tail. Consequently, skewness, a measure of the asymmetry of a distribution, or the relative sizes of the right and left tails, becomes negative. 8

The resulting distribution has a left tail (price decreases relatively distant from the averagepricechange,whichispositive),withoutacorrespondingrighttail(asprice increases are to the right of the inaction region and relatively close to each other). As inflation rises (due to larger positive aggregate shocks), these negative price changesformalefttailinthepricechangedistributionthatisfurtherandfurther(to the left) of the average of the price change distribution, leading to a skewness that is more negative. This implies that the correlation between skewness and inflation is negative, as presented by the blue curve in the last panel of Figure 1.2 This does notoccurinaCalvomodel: insuchamodeleverydesiredpricechangehasafixed probability of being realized, so as the desired price changes rise, the shape of the realizedpricechangedistributiondoesnotchangeinameaningfulway. Another implication is that positive aggregate shocks reduce the dispersion of price changes because a bigger fraction of them are on one side of the inaction region, and therefore relatively close to each other. It is when the share of price changes on either side of the inaction region is equal that the dispersion is highest,andbythesamelogic,higherthanwheninflationisnegative(whenmoreprice changes are decreases). The last panel of Figure 1 shows that dispersion decreases with inflation in the positive region, and increases in the negative region, with the maximum attained at zero inflation. The intuition for this relationship has been appliedbyVavra(2013)toexplainwhy,instandardmenucostmodels,thefrequency of price change and dispersion will move in opposite directions in response to aggregate shocks. What we show here is that the same logic leads to an observable relationshipwithinflation,andthatitalsoappliestotheskewnessofpricechanges. 2Notably, the relationship between skewness and inflation is non-monotonic during extreme inflationscenarios: whileinflationapproachesinfinity,theskewnessincreasesandapproacheszero, as the selection effect plays little role when the desired price change distribution shifts far to the right and almost all prices change. However we do not observe this kind of hyperinflation in our sample. 9

Whatmakesthesecorrelationsinterestingisthattheyhavetodowiththecentral mechanism of the menu cost model: the selection effect. When firms face a fixed cost to changing their price, only relatively large price changes will occur, leading tothepresenceoftheinactionregion. Astheaverageoftheunderlyingdistribution rises (moved by aggregate shocks), there is a large response of inflation because there is a large share of price increases at the extensive margin, which leads to a relatively large rise in inflation, muting the real effect of the aggregate shock. This is the logic for why state-dependent models are known to imply low levels of monetary non-neutrality relative to a Calvo model. Indeed, both types of models cancapturethefactthatpricesdonotchangeineveryperiod. However,becausethe pricesthatchangearenotselectedinaCalvomodel(sothatmanypricechangeswill besmall),anominalshockofthesamesizewillleadtoamuchsmallerresponsein inflation,andalargerresponseofrealactivity. Naturally, the selection effect has received much attention in recent research on sticky prices, as it makes a crucial difference to the degree of monetary nonneutrality. However,thefundamentaldifficultyinempiricallyevaluatingthestrength oftheselectioneffectisthatitinvolvesthedesiredpricechangeoffirms. Sincemost firms’ prices do not change in any given month, the desired price change is unobserved in most cases. This makes it impossible to directly test whether the prices that change are those that are most mis-aligned, in line with the selection effect. Instead,onemustmakeaninferencebasedontheimplicationsmadebymodelsfor realized (and therefore observable) price changes. In this paper, we are presenting and implementing a new way of testing for the strength of the selection effect: the presence of selection in menu cost models implies the negative skewness and dispersion correlations (which are observable) that are the focus of our analysis, and thismotivatesourfocusonthesestatistics. 10

2.2 Existing Models We consider the empirical implications of the selection effect in the existing sticky price models, including the Calvo model, the Golosov and Lucas menu cost model and the variants of it that have appeared since. To do this, we consider models that can be separated into four categories: 1) Calvo, 2) Menu cost, 3) Observation costs, and 4) Rational Inattention. We choose six models in those categories to evaluate, namely the standard Calvo model, Golosov and Lucas (2007), Nakamura andSteinsson(2010),Midrigan(2011),Alvarezetal.(2011)andWoodford(2009). The menu cost models that we consider have a common basic structure: firms produce a differentiated output with labor and a production technology subject to idiosyncratic shocks. In addition, they face constraints on changing their nominal price. Different models introduce different constraints, and in some cases different processes for the idiosyncratic shocks. All models, however, include aggregate nominal demand shocks. By shifting marginal costs, the aggregate shocks shift the desired price of all firms. However, since the constraints to changing prices are different across models, the response of prices (both of inflation, the average price change, and of the distribution of price changes more generally) will also be differentacrossmodels. Thisiswhatwearedocumentinginthissection,andbelow weprovideaformalset-upofthemodels. First,householdsmaximizeexpecteddiscountedutilityofthefollowingform: ∞ (cid:88) E βτ−t[logC −ωL ]. t τ+t τ+t τ=t There is a continuum of monopolistically competitive firms, indexed by z, producingadifferentiatedproduct,andaggregateconsumptionisgivenbyaconstantelasticity of substitution aggregator, meaning that each firm faces the standard demand 11

functionforitsgood: (cid:18) p (z) (cid:19)−θ t c (z) = C , t t P t where θ is the elasticity of demand, and P is the CES price aggregator. Firms t produceoutputbasedonalinearproductionfunction,withlaborastheonlyinput: y (z) = A (z)L (z). t t t Productivity is subject to idiosyncratic shocks, which have been an important feature of sticky price models since Golosov and Lucas (2007). Large idiosyncratic shocks make it possible for such models to match the large heterogeneity and high average size of price changes observed in the data, which was documented notably by Nakamura and Steinsson (2008) and Klenow and Kryvtsov (2008). Following Midrigan (2011) and Vavra (2013), we assume that idiosyncratic shocks arrive infrequently with a Poisson probability p , and model the process in the following (cid:15) way:   ρlogA (z)+(cid:15) , withprobabilityp logA (z) = t−1 t (cid:15) , (cid:15) i ∼ id N(0,σ2). t t (cid:15)  logA (z), withprobability1−p t−1 (cid:15) As Midrigan (2011) had noted, this Poisson set-up allows the model to imply a distribution of price changes with fatter tails than the standard AR(1) productivity (used by Golosov and Lucas (2007) and Nakamura and Steinsson (2010), for example), which is closer to what is seen in the data. However, it nests the AR(1) set upwhentheprobabilityofashockoccurring(p )issetto1. Sincewewillconsider (cid:15) various models with AR(1) productivity, as well as Midrigan’s model with Poisson shocks, we maintain this set-up, and cover the different models by adjusting the relevantparameters. Inordertogenerateaggregatefluctuations,thestickypricemodelsthatwelook 12

at incorporate a stochastic process for nominal aggregate demand. Again, we stick to what is most often used in the literature by modelling nominal output as a log randomwalkwithdrift: logP C = logS = µ+logS +η , η i ∼ id N(0,σ2). t t t t−1 t t η This process stands in for monetary policy in these models: nominal output is determined exogenously, and firms’ price responses to these shocks determine how inflation, and how real output respond. We will use the same parameter values for this process (to match the behavior of US aggregate activity) across the different models, and we define monetary non-neutrality as the variation in aggregate real consumption induced by the nominal shocks. This has become the main way of introducing monetary variables in the menu cost literature because it lends itself muchmoreeasilytotheglobalsolutionmethodsthatareusedforsuchmodelsthan explicitly incorporating systematic monetary policy. Although Blanco (2016) developed a menu cost model with a Taylor-type policy rule, we do not attempt this for the models in this section. Our goal is to show how the price change distribution changes with inflation under different sticky price models, and the aggregate demandprocessthatweuseenablesustodothis. The general price-setting constraint takes the form of a (potentially time- and firm-varying) cost in terms of units of labor that must be paid for a firm to change itsnominalprice. Specifically,theperiodprofitfunctionthereforetakestheform: Π (z) = p (z)y (z)−W L (z)−χ (z)W I{p (z) (cid:54)= p (z)}. t t t t t t t t t−1 In the standard Golosov and Lucas (2007) menu cost model, the cost χ is fixed for allfirmsandperiods,andcanbecalibratedtomatchthefrequencyofpricechanges observedinthedata. TheidiosyncraticshockprocessisNormalAR(1),sop isset (cid:15) 13

to 1, and the standard deviation of shocks is calibrated to match the average size of price changes. This is, in a way, the most “state-dependent” model, as under the fixed menu cost firms are fully in control of the decision of when to change the priceforeachgood(subjecttotheconstantmenucost). The first extension to the menu cost model that we consider is the Nakamura and Steinsson (2010) multi-sector menu cost model, in which firms are separated into sectors. Firms in different sectors face a different menu cost and variance of idiosyncratic shocks. Second, we also analyze the model in Midrigan (2011), who introduced other modifications to the standard menu cost model: first by changing the idiosyncratic shock process so that it would feature fat tails (which we described above), and giving firms a motive to make small price changes3. In his model, multi-product firms can change the prices of all their products by paying the menu cost. This enables the model to match the considerable fraction of small pricechangesthatareobservedinthedata,butitalsomakesthemodelmuchmore difficult to solve. We follow Vavra (2013) in simplifying the Midrigan model by assuming that, instead of producing multiple products, firms each period are randomlygiventhepossibilityofchangingtheirpriceforfree(withalowprobability), orbypayingamenucost. Therandommenucoststructureyieldssimilarresultsfor monetarynon-neutralityasintroducingmulti-productfirms. Thisisalsoavariation oftheCalvoPlusmodelpresentedbyNakamuraandSteinsson(2010),andaddsthe probability of drawing a zero menu cost (free price change, p ) as an additional z parameter to calibrate. With the additional parameters in this model, we target the 3In Midrigan’s model, firms can also carry out temporary price changes, or sales, by setting regularpricesandpostedpricesthatcanbedifferentfromeachother. However,thisfeatureofthe model does not have a major effect on monetary non-neutrality, and we abstract from temporary pricechangesinouranalysis 14

fractionofpricechangesthataresmall,asinMidrigan(2011).4 We also consider a Calvo model, which has the set-up described above, except thatfirmshaveafixedprobabilityeveryperiodofreceivingtheopportunitytofreely changetheirprice(otherwise,theydonotgettochangeprice). Thisisequivalentto the simplified Midrigan model that we describe, but with the high menu cost set to infinity,andtheprobabilityofafreepricechangesettoequaltheaveragefrequency of price change in the data. This model includes idiosyncratic shocks to obtain a distributionofpricechanges,andwealsosetthevarianceoftheseshockstomatch theaveragesizeofpricechanges. Finally, we also include two models involving imperfect information: the Alvarezetal.(2011)modelofobservationandmenucosts,andtherationalinattention modelofWoodford(2009). Intheformer,firmsmustpayafixedcosttoobservethe relevant state (or conduct a “price review”), and a menu cost to change their price. Facing such costs, firms conducting a price review choose the date of the next review, and a price plan until that date. Because the Alvarez et al. (2011) model includes a menu cost, it features a high degree of selection. Woodford (2009) considers the same type of price-setting problem, but within the rational inattention framework proposed by Sims (2003): firms face a cost based on how much information they process, and therefore choose to receive limited information based on which they choose when to review prices. In this model, the cost of processing informationisacrucialparameter,andboththeCalvomodelandstandardmenucost modelarenestedasextremecasesoftheinformationcostinthisset-up(infiniteand zero, respectively). Furthermore, intermediate values of the information cost result inwhatisdescribedasa“generalizedSsmodel”: whileasimpleSsmodelinvolves 4Midrigan(2011)definesasmallpricechangeasapricechangethatislessthanhalf,inabsolute value,oftheaveragesizeofpricechange. Duetothevariationintheaveragesizeofpricechanges overtimeandacrosssectors,weprefertouseanabsolutemeasure,andfocusinsteadonthefraction ofpricechangesthataresmallerthan1%inabsolutevalue. 15

a threshold rule for price adjustment, a generalized Ss model features a probability of price adjustment as a function of the degree of price mis-alignment. This is the kind of model that we work with in Section 4, and we view the rational inattention frameworkasapotentialmicro-foundationforthis. As mentioned in the introduction, the studies that have examined price change statistics in high inflation environments have mostly focused on whether the frequency of price change rises with inflation, as the menu cost model predicts. Motivated by the logic explained above about the implications of the selection effect for the shape of the price change distribution in menu cost models, we will also consider the dispersion and skewness of price changes. We do this in two different ways: by analyzing short-run fluctuations in inflation, and changes in the value of steady-state inflation. Notably, the kind of analysis that we can carry out with Alvarez et al. (2011) and Woodford (2009) is more restricted than the perfect information models. We provide details on the simulation procedure for these two modelsinAppendixA. To analyze short-run fluctuations (the first case), we solve each model with a fixedvaluefortheparametersofthenominalaggregatedemandprocess(µandσ ), η andsimulateeachforalargenumberoffirmsandperiods. Fromthesimulatedprice series,wethencomputethevariouspricechangemomentsforeachperiod(obtainingatimeseriesforeachmoment),andlookattherelationshipwiththetimeseries for inflation endogenously derived. Our steady-state analysis (the second case) is more in line with what is doneby other papers, such as Alvarez et al. (2016a). Because much of the variation in inflation throughout our sample period is generally understood to reflect regime changes caused by systematic changes to the conduct of monetary policy, it is important to consider whether the correlations in question are the same when it is steady-state inflation that changes. For this analysis, we solve each model with different values for the steady-state inflation parameter (µ, 16

keeping all other parameters fixed), and for each solution computing the values of the price change moments from the model’s stationary distribution. We find that the relationships between inflation and price change moments are qualitatively the sameinbothcases(thatis,withrespecttoshortorlongrunchangesininflation). Figure2: Simulatedmomentsandinflationfromdifferentmodels Golosov & Lucas MC Midrigan MC Multi-Sector MC Calvo Woodford (2009) Corr = 0.73 Corr = 0.66 Corr = 0.76 Corr = 0.02 Corr = 0.09 0.2 0.2 0.25 0.2 0.2 y c0.15 0.15 0.2 0.15 0.15 n e u q e 0.1 0.1 0.15 0.1 0.1 rF 0.05 0.05 0.1 0.05 0.05 -0.01 0 0.01 -0.01 0 0.01 -0.01 0 0.01 -0.002 0 0.0020.004 -0.002 0 0.002 Corr = -0.88 Corr = -0.71 Corr = -0.82 Corr = 0.18 Corr = -0.05 )z 0.05 0.05 0.05 0.05 0.1 ( : v z 0.1 0.1 0.1 0.1 e D 0.15 d tS 0.05 0.05 0.05 0.05 0.1 -0.01 0 0.01 -0.01 0 0.01 -0.01 0 0.01 -0.002 0 0.0020.004 -0.002 0 0.002 Corr = -0.997 Corr = -0.93 Corr = -0.96 Corr = 0.62 Corr = 0.27 1 1 1 1 1 )z ( 0 0 0 0 0 : w z e -1 -1 -1 -1 -1 k S -2 -2 -2 -2 -2 -0.01 0 0.01 -0.01 0 0.01 -0.01 0 0.01 -0.002 0 0.0020.004 -0.002 0 0.002 Corr = -0.65 Corr = -0.85 Corr = -0.87 Corr = 0.56 Corr = 0.49 0.5 0.5 0.5 0.5 )z : ( 0.5 w z e 0 0 0 0 0 k S y lle -0.5 K -0.5 -0.5 -0.5 -0.5 -0.01 0 0.01 -0.01 0 0.01 -0.01 0 0.01 -0.002 0 0.0020.004 -0.002 0 0.002 : : : : : t t t t t In order to further illustrate these results, we present scatter plots between inflation and the different moments from the simulations (based on 1,000 months and 50,000 firms) corresponding to the short-run analysis. Figure 2 shows the correlations for the frequency of price change, the dispersion and skewness of price 17

changes, with a point representing a time period in the simulations.5 These bring out the fact that in the menu cost models, the relationships between inflation and dispersion and skewness are very clear and strong (especially in the Golosov and Lucas (2007) model for the dispersion): the skewness of price change falls very sharply with inflation in menu cost models, as does the dispersion for positive values of inflation (as explained above, the inflation-dispersion relationship is nonmonotonic). In contrast, the same relations in the Calvo and imperfect information models are not so strong. However, the Calvo and rational inattention models feature weakly positive relationships for price change skewness and dispersion. That is because price changes are not selected in the Calvo model, so the mechanism describedearlierisentirelyabsent. The intuition for the correlations is easiest to explain in the case of the “standard”GolosovandLucasmodel,asinsubsection2.1,yetitalsoappliestotheother menu cost models. In the multi-sector menu cost model, different sectors face different menu costs, and this can be thought of as sectors facing different inaction regions, with each sector behaving as described for the standard menu cost model. Therefore, the aggregate price change distribution behaves similarly to how each sector’s distribution does. Our simplified version of the Midrigan model involves firms randomly facing either a positive or zero menu cost. This weakens the selection effect, because there is now a positive probability that a firm will change its price even if it will be a small change, so that price changes are not entirely “selected” based on how out of line the original price is. However, the selection effect is still present to a certain extent, because it is only relatively large price changes 5The Alvarez et al. (2011) model contains no aggregate shocks. Therefore, the “short-run” analysisofthismodelisexcluded. Strictlyspeaking,theWoodford(2009)modelcannotbesolved withaggregatenominaldisturbances. Nonetheless,wetakeasimplifiedapproachfollowingSection 5ofWoodford(2009). Wesimulatethemodelwiththedynamicsofaggregatenominalexpenditure beingi.i.d. andmeanzerotoconductthe“short-run”analysis(refertoappendixAfordetail). The “long-run”analysisofthismodelisexcluded. 18

that will happen with certainty (as those will be the only ones for which a firm will be willing to pay the positive menu cost, when it is faced). The tails of the price change distribution will therefore be very sensitive to the aggregate shocks that drive inflation in the model, leading to the same relationships for price change dispersionandskewnessasintheGolosovandLucasmodel. Although the relationships come out very clearly in these simulations, it could be a concern that the higher moments that we are estimating might not be well defined in the distributions that we are working with. In addition, estimates of higher momentsareverysensitivetooutliers,whichwouldbeofconcernparticularlywhen weestimatefromthedata. Thatiswhywealsoconsideralternativemeasuresforthe dispersion and skewness of price change: the inter-quartile range (for dispersion) and Kelly’s coefficient of skewness (as opposed to “moment skewness”, which is what we have been estimating so far).6 Since these statistics are quantile-based, they are well-defined for any distribution, and they are also less sensitive to outliers. The correlations are similar for all the models (inter-quartile range compared withstandarddeviation,andmomentskewnesswithKellySkewness). Thelastrow ofFigure2showsscatterplotsofKellySkewnessinthedifferentmodels7. In Figure 3, we plot the results for the long-run analysis, in which we vary the valueofsteady-stateinflation. Foreachmodelsolution,wecanconstructastationary distribution of price changes, from which we can then compute the stationary valueforthedifferentpricechangemoments,andthesearethevaluesplottedinthe 6These statistics are defined as follows, with Q representing the ith percentile. Inter-quartile i range=Q −Q . KellySkewness= (Q90−Q50)−(Q50−Q10). Kellyskewnessessentiallymeasures 75 25 Q90−Q10 thedegreeofasymmetryinadistribution,comparingthesizeoftherightandlefttails. 7ThereisadiscontinuousjumpintheKellySkewnessvaluesfortheGolosovandLucasmodel becausethemedianpricechange(whichisusedtocomputeKellyskewness)jumpsdiscretelyfrom the left to the right band of the inaction region. The jump also corresponds to a value of approximately0inflation,asthatisconsistentwithanequalshareofpriceincreasesanddecreases. However,withinthepositive(ornegative)inflationperiods,therelationshipbetweeninflationandKelly skewnessisnegativeheretoo. 19

Figure3: Simulatedlongrunstatisticsfromdifferentmodels Golosov & Lucas Calvo Alvarez et al. 0.007 0.012 0.14 e g n a h C e c n o is r e0 0 . . 0 0 0 0 5 6 0 0 . . 0 0 1 1 0 1 0 0 . . 1 1 0 2 ir P .g p s iD0.004 0.009 0.08 v A 0.003 0.008 0.06 0 0.005 0.01 0 0.005 0.01 0 0.1 0.2 7 7 7 0 1.5 0 e g n a -0.5 1 -0.2 hs Cs e e c n w -1 0.5 -0.4 ir P e k .g S-1.5 0 -0.6 v A -2 -0.5 -0.8 0 0.005 0.01 0 0.005 0.01 0 0.1 0.2 7 7 7 figure. Whatthescatterplotsshowisthat,asinthe“short-run”analysis,thedispersion and skewness of price changes fall with trend inflation in the menu cost model (we areonlyplottingresultsfortheGolosovandLucas(2007)model,butthesamepattern holds for the other menu cost models). As in the short-run analysis, the Calvo model predicts weak positive relations for both moments with respect to steadystateinflation. Thiswillbeimportantwhencomparingtheskewnessofpricechange betweenthelowandhighinflationperiodsinthedata. Toconcludeourtheoreticalanalysis,weemphasizethatthecorrelationsthatwe consider all have the same sign in the four menu cost models (Golosov and Lucas (2007), Nakamura and Steinsson (2010), Midrigan (2011), and Alvarez et al. (2011)). Thescatterplotsshowthatthevaluestakenbymomentswereportdovary across the models (for example, in the Golosov and Lucas (2007) model the skewness of price changes takes a wider range of values than in the other models), but the fact that the sign and strength of the correlations across the models are similar 20

is notable. Indeed, the Nakamura and Steinsson (2010) and Midrigan (2011) menu cost models were developed as extensions of the Golosov and Lucas (2007) model to make it match new empirical facts, and the changes made considerably weakenedtheselectioneffectthatreducestheimportanceofmonetaryshocks. However, what we find here is that, despite the important changes made, they all have the sameimplicationsalongthedimensionsthatweareconsidering. 3 Empirical Evidence from High Inflation Periods Intheprevioussection,wedocumentedthepredictionsmadebyvariousstickyprice modelsonthebehaviourofpricechangesatdifferentinflationrates. Inthissection, we present the data set that we use, and the empirical results that test the model predictionsoftheprevioussection. 3.1 Data Set and Construction of Statistics Along with much of the sticky price literature, we make use of the micro data that underlies the U.S. Consumer Price Index (CPI). The CPI Research Database collected and maintained by the U.S. BLS contains about 80,000 monthly prices collected from around the U.S, classified into about 300 categories called Entry Level Items (ELI’s). As mentioned before, the data going back to 1988 has been availableforalittleoveradecade. Thedatagoingbackto1977hasrecentlybecome available,andthisisthenovelpartofthedatasetthatweuseextensively. Thisnew data set has thus far only been used by Nakamura et al. (2016), and that paper also describes in detail just how the data set was re-constructed. We have access to thevariablesthatidentifyspecificproducts,andthatrevealwhenasubstitutionhas occurred (when a new version of a product has replaced the old one). In addition, thedatasetcontainsinformationonwhenanygivenpriceisatemporarysale,oran 21

imputation (not properly collected). Because of this, we are confident that we are observingthepricechangesofidenticalproductsandservices,withthepricebeing actually observed; and all of this with the same standards throughout the sample period. In order to test the predictions that we presented in the previous section, we construct distributions of price changes for each month, from which the different moments of interest can be estimated period by period. We calculate the log price change for all the goods and services in our sample, and then construct the distributions subject to a few restrictions. We keep only non-zero price changes to compute the dispersion and skewness (while the frequency measures the fraction of non-zero price changes), and exclude temporary sales, substitutions, and price changes that are implausibly large in absolute value. We provide further details on theserestrictionsintheappendix. Nakamura and Steinsson (2008 and 2010) have shown that there is significant heterogeneity of price change statistics across sectors. We use their method to report the average overall frequency of price change: estimate the frequency of price change for each ELI, and then take a weighted average of the ELI frequencies (usingtheexpenditureweightsthatgointotheCPI).Forthefrequencyofpricechange we consider both the aggregate weighted median and mean frequency.8 For the dispersion and skewness, we follow a similar approach: we first estimate each moment by sector-month. However, as ELI’s are fairly narrow categories, most of them have a handful of price change observations in any given month, fewer than 8Nakamura and Steinsson (2008) highlight the difference between the mean and the median, arising from the fact that the distribution of frequencies by ELI is very skewed to the right, with a few ELI’s having very high frequencies. They argue that the median is a better measure of the averagefrequencyinthesensethatasingle-sectormenucostmodelcalibratedtomatchthemedian frequencyisamuchbetterapproximationofamulti-sectormodel,ofthekinddescribedinSection 2. Inthisway,themedianfrequencyisastatisticthatbetterdescribesthedegreeofpricestickiness (asitrelatestomonetarynon-neutrality). Thisisalsowhywecalibrateallthesinglesectormodels tomatchthemedianfrequency. 22

would be necessary to estimate higher moments with any precision. We therefore do not use ELI’s as our definition of sectors, but instead separate products into 13 “major groups”, which are listed in the appendix. While this sectoral classification is fairly broad, it allows us to separate goods and services into similar categories, while leaving enough observations in each sector-month to obtain good estimates ofthedispersionandskewness,andthenforeachmonthtakeweightedaveragesof thestatistics. This approach has another advantage for testing the model predictions that we focus on. Indeed, the models do not allow for differences across sectors, such as sector-specific shocks. These have the potential to strongly affect the shape of the overall price change distribution (when all price changes across sectors are pooled together), in turn affecting the higher moments of the distribution. Because of this, we might see the moments of the “pooled” distribution of price changes vary over time due to such sector-specific shocks, which would be unrelated to the mechanisms that are behind the predictions of the models that we described in the previous section. For this reason, we attempt to control for these types of effects by computingstatisticssectorbysector. 3.2 Results The goal of our empirical work is to determine whether the theoretical patterns documented above are borne out by the data. As in the theoretical section, we focus on the correlations between aggregate inflation and price change dispersion, and between inflation and price change skewness. The price change moments are calculated as described above, and our preferred measure for aggregate inflation is monthly core PCE inflation. Sharp changes in headline inflation tend to be driven by the global market prices of food and commodities, which would not be well de- 23

scribedbytheprice-settingmodelsthatweareworkingwith,makingcoreinflation preferable for us. However, we also compute correlations with headline inflation as a robustness check (as well as using estimates of the moments excluding price changes from food and energy categories). Finally, to control for seasonality in the inflation and moment series, we calculate the correlations after removing month dummies from the series, and after applying a moving average smoother to them. AlloftheseadditionalresultscanbefoundinAppendixC. The price data is monthly, and inflation series are monthly, so we can compute the correlations at a monthly frequency. However, the drawback of using monthly series is that each period’s moment estimates are based on relatively few observations, making them less precise (this is especially important for higher moments such as the dispersion or skewness). The alternative is to group price change observations by quarters or years (but still separating them by sector) and to estimate the moments from these samples, which gives us more precise estimates (as they are based on distributions with more observations), but only quarterly or annual moment series. Quarterly and annual inflation averages also have the advantage of containing less noise than monthly inflation series, so we will focus on presenting results using quarterly series (although we include all the monthly and annual results in the appendix). Figure 9 in Appendix plots the quarterly time series that we constructfortheInter-QuartileRangeandSkewnessofpricechanges. In the next subsection, we present the correlation results in two ways: first, withraw correlationsand scatterplots (whichare reportedin Figure4), aswith the models. Secondly, we estimate these relationships with regressions (allowing us to testforsignificanceandtoincludecontrols,whicharereportedinTable1). 24

Figure4: MomentsofPriceChangeandInflation,Quarterly Source: Authors’calculationsfromBLSCPIResearchDatabase ycneuqerF 52. 2. 51. 1. 50. Median Frequency; Corr = 0.643 0 .02 .04 .06 .08 .1 Inflation Pre 1984 Post 1984 RQI 41. 21. 1. 80. 60. Inter-Quartile Range; Corr = -0.734 0 .02 .04 .06 .08 .1 Inflation Pre 1984 Post 1984 ssenwekS 5. 0 5.- 1- Skewness; Corr = 0.336 0 .02 .04 .06 .08 .1 Inflation Pre 1984 Post 1984 3.2.1 Correlations We first verify that the frequency of price change rises with inflation, as found by Gagnon (2009) and Alvarez et al. (2016a). We present scatter plots using the quarterly moment and inflation series (the empirical counterpart to the simulation scatter plots from the previous section). Correlation values are reported in Tables 9-12 in Appendix C. Figure 4 confirms that there is a positive association between the frequency and inflation. As argued in the previous studies that had looked into this relation, this provides strong evidence against the Calvo assumption of timedependentpricesetting. Next, we look at the results for the moments that our discussion has focused on: the dispersion and skewness of price changes. Our main results is that while theredoesseemtobeaclearnegativerelationshipbetweeninflationanddispersion, thereisnosuchrelationbetweeninflationandskewness. Indeed,forbothmeasures of skewness (moment skewness and Kelly skewness; “Skewness” in the tables and graphs refers to moment skewness), the correlation is either strongly positive (over the whole sample period) or close to zero (post-1984). Skewness, while varying overtime,doesnotchangewithinflationinasystematicwayforlowlevelsofinflation (although there does seem to be a positive relationship when inflation is high). We see this from the different correlations for the different sample periods (which 25

roughlycorrespondtothehighandlowinflationperiods). Finally,allthesepatterns holdtrueregardlessofwhetherweexcludepotentiallyspurioussmallpricechanges (asdefinedbyEichenbaumetal.(2013))orapplyseasonaladjustmentandsmoothingtothedataseries. Next,weformalizethisanalysiswithlinearregressions. 3.2.2 Regressions Table1: CoefficientsonInflationforPriceChangeMoments-UsingCPIData Source: Authors’calculationsfromBLSCPIResearchDatabase 1977-2014 1985-2014 All FedDummies InflationOnly All FedDummies InflationOnly Frequency 0.708 0.728 0.771 0.777 0.810 0.587 (0.071) (0.095) (0.237) (0.224) (0.208) (0.252) IQR -0.296 -0.186 -0.257 -0.428 -0.414 -0.222 (0.042) (0.038) (0.089) (0.070) (0.077) (0.086) Skewness 3.936 4.309 2.665 1.732 1.541 3.634 (0.827) (1.012) (2.788) (1.641) (1.857) (3.279) KellySkewness 2.499 2.439 1.658 0.320 0.710 0.942 (0.354) (0.363) (0.948) (0.454) (0.423) (0.595) Theregressionsarerunusingquarterlyseries,wherequarterlyinflationisdefinedthemeanofthe12-monthlogchangesinthe CPIforthethreemonthsineveryquarter.Thedifferentcellsindicatedifferentspecifications,whichchangewithrespecttothe sampleperiodusedandwhatcontrolsareused.Standarderrorsthatareconsistentforheteroskedasticityandauto-correlation oftheresiduals(Newey-West)arereported. We now turn to regressions to determine whether these correlations are statisticallysignificant,andtoconsiderdifferentcontrolvariables. Thequestionofinterest about the coefficients on inflation is not merely whether they are statistically significantlydifferentfromzero,butalsowhethertheyaresignificantlydifferentfrom what the models predict. To do this, we estimate regressions of the frequency, dispersion (inter-quartile range) and skewness (both moment and Kelly skewness) of the price change distribution on inflation, with different specifications allowing for different sets of controls and sample periods. As before, we run the regressions both on the whole sample period and on only after 1984. This allows us to see if the relationship looks different between the low and high inflation periods. The 26

regressionsalltakethefollowingform: y = α+βπ +γControls +e , t t t t where y denotes the different price change moments (frequency, dispersion, and t skewness). Controls are included to address the fact that many important changes occurred in the U.S. monetary environment over our sample period, which could conceivably have a direct effect on the price change distribution. For example, expected inflation could affect firms’ price setting decisions separately from present realized nominal shocks, so we include expected inflation (measured by the University of Michigan Survey of Consumers) as a control. We also include dummy variables for the different Federal Reserve chair’s times in office, to control for differences in the conduct of monetary policy. The different specifications cover different combinations of controls (no controls, Fed dummies only, or Fed dummies with expected inflation) and the different periods. Table 1 show the estimates forβ fromthesedifferentspecifications.9 These results support what the correlations showed: the frequency of price change rises with inflation and the relationship between dispersion and inflation isnegativeandstatisticallysignificantinallspecificationsandsampleperiods. The skewness correlation, however, is significantly positive for the whole sample, but not significantly different from zero when the early, high-inflation period is excluded(andthisappliesforbothmeasuresofskewness). Theseresultsconfirmthat this relation is close to flat for low inflation periods, but clearly positive for high inflation periods. The fact that the skewness of price change is higher on average in high inflation periods is important, because it also goes against the menu cost models’predictionsathighvaluesofsteady-stateinflation,asshowedinFigure3. 9Regression results excluding certain small price changes based on Eichenbaum et al. (2013) arepresentedinTable21inappendixC. 27

Table2: CoefficientsonInflationforPriceChangeMoment-UsingSimulatedData Model Frequency IQR Skewness KellySkewness Golosov&Lucas 0.139 -0.937 -17.7 -0.40 MultisectorMenuCost 0.143 -0.218 -5.39 -4.33 Midrigan 0.348 -0.896 -9.84 -6.53 Calvo -0.003 0.040 2.93 1.00 RationalInattention 0.020 0.029 2.87 1.00 BLSCPIData 0.708 -0.296 3.94 2.50 Table 2 presents the coefficients on inflation from regressions of the same type, butrunonsimulateddatafromthedifferentmodels. Thelastrowpresentsthecoefficients using CPI data, which replicates the first column of Table 1. The first four models (menu cost models) have negative coefficients for the inter-quartile range, although for all but the multi-sector model, they are outside the 95% confidence intervals of the coefficients that we estimate. However, the disagreement with the data is much starker with the skewness coefficients. These are all very far outside the confidence intervals that we estimate for the skewness coefficients under all specifications,andthesameistrueforKellyskewness10. To summarize our results so far, in the broad class of state-dependent price settingmodelsthatweconsider,nonematchthedatainallthedimensionsthatwehave presented. As we have already argued, menu cost models make a counter-factual prediction on the skewness of price changes because of the state-dependence that underlies them. In the next section, we consider a menu cost model that weakens state-dependenceandcanbereconciledwiththeempiricalcorrelationsthatwefind. 10TheoneexceptionisthecoefficientfortheGolosovandLucasmodel,whichismuchsmaller inmagnitudethanintheothermenucostmodels,andismarginallyacceptedinthespecificationthat restrictsthesampletothepost-1984periodandusesonlyFedchaircontrols.Itappearsthatthevalue of the Kelly Skewness is extremely sensitive to the unusual shape of the price change distribution (bi-modal)inthismodel,leadingtothisweakrelationship. Themodel’sKellySkewnesscoefficient isstillrejectedinalltheotherspecifications,however. 28

4 A Generalized Menu Cost Model In this section, we present a menu cost model that has a similar setup as the menu cost models presented in Section 2: the demand system and technology faced by the firm are the same, but we generalize the price setting problem in the following way: themenucostfacedbyeachfirmeveryperiodisrandom. Formally,theperiod profitfunctionofthefirmtakesonthisform: iid Π (z) = p (z)y (z)−W L (z)−χ (z)W I{p (z) (cid:54)= p (z)}, χ (z) ∼ G(χ). t t t t t t t t t−1 t The difference with the Golosov and Lucas model is that here the menu cost can vary over time and across firms, the difference with the Midrigan model is that the distribution of menu costs is generalized, and as opposed to the Nakamura and Steinsson model, the menu cost for any given firm here varies over time.11 The assumption of random menu costs is similar to that made by Dotsey et al. (1999), butwepresentitwithintheframeworkwehavebeenusinguntilnow.12 4.1 Background on Random Menu Costs In addition to nesting the existing menu cost models considered thus far, our approachhasacloserelationtoanother,evenmoregeneralapproachalreadypursued by Caballero and Engel in a series of papers (1993, 2006a, 2006b). They propose thinkingaboutpriceadjustmentthroughthepriceadjustmenthazardfunctionofthe 11Thisset-upcanreplicatetheGolosovandLucasmodel,ifthemenucostdistributionisdegenerate,andtheMidriganmodel,ifthedistributionisdiscretewithtwosupportpoints(onebeingzero, the other being positive). The Calvo model is replicated when the higher support point is infinite. Since the Nakamura and Steinsson model involves different firms facing different menu costs that arefixedovertime,itisnotencompassedbyourset-up. 12ThekeydifferenceswithDotseyetal.(1999)arethattheirmodeldoesnotincludeidiosyncratic shocks, that it does include capital as an input to production, and that they did not have a way of usinginformationfrompricemicrodatatoplacerestrictionsonthemenucostdistribution,whichis whatthepresentexerciseisabout. 29

deviationofthecurrentpricefromitsoptimalvalue(p∗): H(x) = P(∆p|p∗ −p = x). Any of the models we have considered will imply a price adjustment hazard function. Inourrandommenucostmodel,aparticularmenucostdistributionwillimply aparticularhazardfunction,andwillthereforedetermineaggregateflexibility(and monetary non-neutrality) as shown by the expression above. In this way, there is a verytightrelationbetweentheseapproaches,andweshowinaseparatepaper(Luo andVillar(2016))thatthesamedataandempiricalpatternscanbeusedtoestimate thepriceadjustmenthazardfunction. A more structural approach to price stickiness that is also related to ours is Woodford (2009)’s model of rational inattention. He shows that by varying the cost of processing information, price setting under rational inattention in the style of Sims (2003) can also nest, as extreme cases, the single menu cost model (free information) and the Calvo model (infinitely costly information), as well as the spectrum in between, which he also describes with the adjustment hazard function implied by different information costs. Although not provided, we believe that a decision-theoretic justification for this random menu cost model can be derived based on the rational inattention framework. A menu cost model with inattention as a source of randomized discrete adjustment is observationally equivalent to a random-menu-costmodel(see(Woodford,2008,2009)).13. 13AsWoodford(2009)alsopointsout,thedirectempiricalevidenceontheactualcostsofprice adjustementputforthbyZbarackietal.(2004)indicatesthatthemostimportantpartofthosecosts arerelatedtotheprocessofgatheringthenecessaryinformationforapricereview. Inaddition,AndersonandSimester(2010)giveevidenceonhowpricechangescanantagonizeconsumers,which introducescoststochangingprices. Totheextentthatthemenucostsinthemenucostframework representthesecosts,webelievethatitisplausiblethatthemenucostsarerandomtosomeextent, and vary across firms and time. This lends plausibility to our random menu costs assumption, althoughweleavetheexplicitlymodellingoftheinformationalconstraintsorconsumerconsiderations thatunderlyittofutureresearch. 30

4.2 The Distribution of Menu Costs Introducingrandommenucostsallowsustodeterminetheextentofstate-dependence present in the model, or to what extent firms choose when to change their prices. Anextremecaseisperfectpriceflexibility,orfirmsbeingfreetochangetheirprices every period without facing any kind of cost for doing so (this is ruled out for being inconsistent with the fact that most prices do not change in any given month). After this comes a menu cost environment such as the one in Golosov and Lucas: firms are still able to choose when to change their prices, but are subject to a fixed cost (that is small in typical calibrations, to match the frequency of price change in the data). Adding randomness to the menu cost makes the price change decision more exogenous to the firm, as an additional dimension of the problem (how much changing the price will cost) is now outside the firm’s control (with the extreme beingtheCalvomodel,wheretheopportunitytochangepriceiscompletelyexogenous). The Midrigan model (both in Midrigan (2011), and the simplification of it that we present) goes in this direction, and as a result the degree of monetary nonneutralityinthatmodelismuchhigher. Weinterpretourresultssofarasindicating that a model would need even more exogeneity (but less than the Calvo model) to match the empirical facts that we have presented. Therefore, we parametrize the distributionofmenucostsinawaythatenablesustosetthedegreeofexogeneity. Thedistributionofmenucostswillneedtwoimportantfeatures: first,apositive probability of the menu cost being zero (of a free price change), which eliminates the inaction region in the price setting problem, as some firms, facing a free price change, will choose to change their prices even if it is by a small amount. However, the Midrigran model already includes this, and also predicts a counterfactual inflation-skewnesscorrelation. Theotherfeatureisthattheremustalsobeapositive andconsiderableprobabilitythatthemenucostwillbeveryhigh,sohighthatfirms 31

will not choose to change their price when faced with these menu costs. Indeed, in the existing models, the skewness of price changes falls with inflation because a positiveaggregateshockinducesmorefirmsthatfaceapositivemenucosttopayit, effectivelypushingthemoverathreshold,leadingtoanimportantshiftintheshape of the distribution. Having a positive probability of very high menu costs means that fewer firms will be pushed over this threshold, weakening this effect. It is also helpful to note that the Calvo model contains both of these features in the extreme, as it gives a positive probability of a free price change, and in all other cases the menu cost is infinite. Because of this, we say that the menu cost distribution in our generalized model will incorporate a strong “Calvo feature”, without going all the waytotheCalvoextreme. In order to achieve this, we present a relatively flexible distribution for menu costs. Weassumethatmenucostsareiidacrosstimeandfirms,sothateveryperiod each firm draws a menu cost χ from a mixed distribution. First, with a certain probability, the menu cost is zero, and otherwise it is drawn from a continuous distribution:   0, withprobabilityp χ = z , where F(k) = P(χ˜ ≤ k) = 1−e−λkα.  χ˜, withprobability1−p z This distribution is a transformation of the exponential distribution (it is the same when α = 1), and shares the important feature that the random variable is always positive. Thedifferenceisthatα governsthecurvatureofthedistributionfunction, which roughly corresponds to the fatness of the tails. Figure 20 in appendix D showshowtheshapeofthecumulativedistributionfunctionchangeswithα. For our purposes, what is important is that for low values of α, the probability of very low menu costs is relatively high, but the probability of very high menu costsisalsoquitehigh. Whenαishigh,theseextremeprobabilitiesarelow,andas 32

α rises, the density concentrates on one value, approximating the case of a unique menucost. 4.3 Calibration and Results Our set-up has introduced new parameters, relative to the models we have been considering: theinverseoftheaveragemenucost(λ),andthecurvatureofthemenu cost distribution (α). The other parameters important for the firm’s price setting problem are the variance of the idiosyncratic shocks (σ2), the arrival probability of (cid:15) shocks (p ), and the probability of a free price change (p ) which was used in the (cid:15) z Midriganmodel. Wesettheseparameterssothatthemodelcanmatchtheempirical factsthatwehavediscussedsofar. First,ourmodelwillmatchtheunconditionalpricechangemomentsmatchedby existingmodels. Theseincludetheaveragemonthlyfrequencyofpricechangeand the average size of price change. These have not been the focus of our discussions so far, but in order to compare the degree of monetary non-neutrality implied by the different models, it is necessary that they be calibrated to the same values for these moments. Our model therefore matches the (expenditure-weighted) median ofthesestatisticsmeasuredinourdata. Second, and in line with the focus of our paper, we will target the signs of the correlationsbetweeninflationandthedifferentpricechangemoments. Asprevious studies had shown (and we confirmed), the correlation between inflation and the frequency of price change is positive, so our model also matches this fact. In addition, our model will imply a strongly negative correlation between inflation and the dispersion of price changes (as menu cost models do). The novelty will be that theimpliedcorrelationbetweeninflationandtheskewnessofpricechangeswillbe non-negative,asinthedata. 33

Table 3 presents the parameter values that we choose to match these moments, andTable4showsthemomentsattainedbythemodel,comparedtotheirempirical values. The first two moments are matched almost exactly. For the empirical value ofthecorrelations(illustratedbythescatterplotsinFigure5,wepresenttheresults forthequarterlycorrelationsinvolvingtherawdata,includingalltimeperiods,and excluding suspicious small price changes (for dispersion and skewness), and the weighted median for the frequency. The model matches the dispersion and frequency correlations quite closely. However, the skewness correlation in the model isclosetozero,whileitisclearlypositiveinthedataforthewholesample. Table3: Parametervalues Table4: Simulationresults Parameter Description Value Moment Model Data λ Inv. averagemenucost 0.177 Avg. Frequency 10.7% 10.7% α FatnessoftailsofMC 0.27 Avg. Size 7.6% 7.5% p P(zeroMC) 0.056 Corr(IQR,π) -0.67 -0.70 z p P(idio. shock) 0.345 Corr(Skew,π) 0.19 0.39 (cid:15) σ Sizeofidio. shocks 0.0967 Corr(Freq,π) 0.67 0.63 (cid:15) Figure5: Scatterplots,GeneralizedMCModel Frequency and Inflation Dispersion and Inflation Skewness and Inflation Corr = 0.67 Corr = -0.67 Corr = 0.19 0.115 0.14 0.3 f o y c n e u q e r F e g n a h c e c ir p 0 0 .1 0 . 0 1 . 1 1 5 : R ) Q z I ( z 0 0 0 . . . 1 1 1 1 2 3 : w e ) k z S ( z 0 0 . . 0 1 2 0.095 0.1 -0.1 -0.002 0 0.003 0.006 -0.002 0 0.003 0.006 -0.002 0 0.003 0.006 : : : t t t While the skewness correlation in this model is lower than in the data, for the 34

range of inflation that occurs in the simulations (0-6%)14, the correlation also appearstobeclosetozerointhedata. Wecarryoutthesame“long-run”analysisasin Figure 3: solving the model for different values of trend inflation. We find that for higher steady-state inflation, the average level of skewness in the price change distributionrises,andthecorrelationbetweenperiod-by-periodpricechangeskewness andinflation(thesamecorrelationswehavebeenfocusingonsofar)alsorises. This result makes our model even more consistent with the data, as it shows that when steady-state inflation is higher (as it surely was in the early, high-inflation part of our sample), we should expect to see the skewness rising with inflation. This also makes our model stand out even more from the existing ones, as the other menu cost models feature a declining average price change skewness as steady-state inflation rises (and a period-by-period skewness correlation that is always negative). Figure6belowshowsthisclearlybyplottingthesteady-stateskewnesscorrelations for the Midrigan model (as an example) and our heterogeneous menu cost model separately. Figure6: Steady-StateSkewnessCorrelation Midrigan Model Generalized MC Model 0 0.4 e g -0.2 0.3 n a h C sse-0.4 0.2 e c n w irP e k-0.6 0.1 .g S v A -0.8 0 -1 -0.1 0.0 0.005 0.01 0.0 0.005 0.01 7 7 This pattern highlights how trend inflation plays an important role behind our model’s non-negative skewness correlation. Indeed, positive trend inflation leads 14Inflation is less volatile and moves within a narrower range in our generalized model than in theothermenucostmodels,eventhoughtheparametersofthenominalaggregatedemandprocess arethesame. Thisisadirectresultofthedifferencesinmonetarynon-neutralityinthemodels,as highernon-neutralitymeansthatthesamenominalshockshaveagreatereffectonrealconsumption (andinducegreaterrealvariation),leadingtolessvariationininflation. Thisisshownbelow. 35

firms to expect positive future inflation when considering whether to re-set their prices. This will lead them to be less likely to cut their prices, even when facing an idiosyncratic (or aggregate) shock that would reduce their current desired price. This asymmetry in firms’ willingness to cut prices also means that the left tail of thepricechangedistributionwillbelessresponsivetoaggregateshocks,weakening themechanismthatledtothenegativeskewnesscorrelationintheexistingmodels. What these results and figures make clear is that the generalized menu cost model that we presented, in making menu costs random in a way that weakens the selection effect, matches the important empirical facts that have been the focus of previous work on sticky prices as well as the existing models, and overturns the counter-factualpredictionofthesemodelsthatwehaveemphasized. Wenowshow whatthismeansforthedegreeofmonetarynon-neutrality. 4.4 Monetary Non-Neutrality Monetary non-neutrality in these models is defined as the variation in real consumption induced by the nominal aggregate demand shocks, which are the only aggregate shocks, and we compare this statistic for the Calvo model, the Golosov and Lucas and Midrigan menu cost models, and our generalized menu cost model. Aswehaveexplained,makingthemenucostsrandominthewaythatwehaveproposedweakenstheselectioneffectthatisatworkinmenucostmodels,soitistobe expected that this model would imply a greater degree of monetary non-neutrality. Table5belowprovidesaquantitativeillustrationofthis. As Golosov and Lucas (2007) had famously shown, their model features a trivial amount of monetary non-neutrality compared to the Calvo model. Between the menu cost models, the major difference is between the baseline (Golosov and Lucas)andtheothers. Allowingforsmallpricechanges,astheMidriganmodeldoes, 36

Figure7: ImpulseResponsesinModels 0.9 Calvo k c 0.8 Random MC Table5: MonetaryNon-Neutrality o h Midrigan s d 0.7 Golosov Lucas n Model Var(C t )∗104 a m 0.6 e GolosovandLucas 0.05704 d Midrigan 0.17718 .g 0.5 g GeneralizedMenuCost 0.35094 a p 0.4 p Calvo 0.52517 1 o 0.3 C t t0.2 fo R 0.1 I 0 0 1 2 3 4 5 6 Months after shock leads to a very large increase in monetary non-neutrality, and this was emphasized by Midrigan (2011). However, our generalized model goes further, and yields an even higher level of non-neutrality. The Calvo model still has a higher degree of monetarynon-neutrality,butourmodelgetssignificantlycloserthantheothers. To further illustrate the differences between the models, in Figure 7 we plot the impulse response of real aggregate consumption to a one percentage point increase in nominalaggregatedemandinthesamefourmodels. The effect on real activity is not only large, but also quite persistent in our model,andmuchmoresothaninthemenucostmodels. Inthissense,ourmodelis alsomuchclosertotheCalvomodel. 5 Conclusion The literature on sticky prices has paid considerable attention to the role of selection in price setting in determining the size of the real effects of monetary policy. Ourpaperhascontributedtothedebateontheimportanceoftheselectioneffectby 37

usingnewhistoricaldatafrommoderatetohighinflationenvironmentsintheU.S., and by focusing on statistics that have previously not been considered. Our main findingisthatthemenucostmodelsthathavebeenmostusedintheliteraturefailto matchthepositiverelationshipbetweeninflationandtheskewnessofpricechanges in the data, because they uniformly predict a sharp negative relationship. In addition,wearguethatthisrelationship,althoughnotobviousatfirstsight,followsvery intuitively from the selection effect that is central to menu cost models, and that makes these models imply relatively low monetary non-neutrality. We also show how a model with random menu costs can overcome this problem when the distribution of menu costs features a significant probability of very high and very low menu costs, making it resemble a Calvo model and weakening the selection effect. Finally,thismodelpredictsadegreeofmonetarynon-neutralitythatisconsiderably higherthanwhatispredictedbytheGolosovandLucasmodel,andhigherstillthan theMidriganmodel. In the context of the debate between time-dependent and state-dependent pricingmodels,wefollowWoodford(2009)inpresentingthedistinctionbetweentimeand state-dependent models as a continuum, or spectrum. Woodford (2009) shows howdifferentvaluesforthefirm’scostofprocessinginformationleadstoadifferent point on this spectrum. In contrast, our approach is agnostic as to what ultimately underliestherandomnessofmenucoststhatallowsourmodeltospanthetimeversus state dependent spectrum. Instead, our contribution is to determine what point on the spectrum is most consistent with the data. Future research could combine thesetwoapproachestogainabetterunderstandingintothenatureandimportance oftheinformationalconstraintsthatunderlypricerigidity. 38

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A Computational Procedure and Calibration of Sticky Price Models We solve the standard Calvo model, the Golosov and Lucas (2007) model, the Nakamura and Steinsson (2010) model, and the Midrigan (2011) model mentioned above by value function iteration, following themethoddescribedinNakamuraandSteinsson(2010). Themaindifficultywiththismethodappliedto thistypeofproblemisthatanimportantvariableenteringthefirm’sprofitfunctionistheaggregateprice level. Since its future evolution depends on each firm’s price, every firm’s current state is, in principle, a state variable for all firms, making the problem intractable. To get around this, we follow the example of Krusell and Smith (1998) and approximate the law of motion of the price level with a finite number of moments, as in Nakamura and Steinsson (2010). In particular, we impose that firms perceive future inflation to depend only on future nominal aggregate demand (S , which is exogenous), and the current t pricelevel: P S t t π ≡ log( ) = Γ( ). t P P t−1 t−1 Under this assumption, the state space can be reduced to three dimensions: the firm’s idiosyncratic productivity (exogenous), the firm’s relative price (choice variable), and real aggregate demand (C , which t determinestherealwageinequilibrium). Thelatterisendogenouslydetermined,buttheprobabilitydistribution of its future value is known fully with the law of motion of nominal aggregate demand, and the assumedlawofmotionofinflation. Thefirm’sproblemcanthereforebewrittenrecursivelywiththefollowingBellmanequation: (cid:26) (cid:20) (cid:21)(cid:27) p (z) S p (z) S V(A (z), t−1 , t ) = max ΠR(z)+E DR V(A (z), t , t+1 ) , t P t P t pt(z) t t t,t+1 t+1 P t+1 P t+1 where V(·) is firm z’s value function, ΠR(z)15 is firm z’s real profits at time t, and DR is the real t t,t+1 stochastic discount factor between time t and t+1. Our procedure to solve the model then closely follows Nakamura and Steinsson (2010): First, we discretize the state variables and propose a guess for the function Γ( St ) on the grid. Then, we solve for the firm’s policy function, F16, by value function Pt−1 iteration, using the proposed Γ(·) function, the stochastic processes for the exogenous variables (applied using the Tauchen (1986) method), and the menu cost structure of the firm’s problem. We then check whether F and Γ are consistent, by computing the price level (and inflation) implied by F for each value on the St grid and comparing it to the value given by Γ. If they are consistent, we stop and use F to Pt−1 simulate the models. If they are not consistent, we update Γ and go back to the value function iteration step and continue. To determine whether they are consistent, we compare the inflation values, grid point bygridpoint,andconsiderthattheyareconsistentwhenthedifferenceissmallerthedifferenceinvalues 15ItcanbeshownthattheprofitfunctionunderCESpreferencesandlinearproductionusingonlylaborcanbewrittenas ΠR(A,p˜,C)=Cp˜−θ[p˜− ωC] A 16Becausethevalueofthemenucostinourgeneralmodelisstochastic,thepolicyfunctionisalsoafunctionofthemenu cost. However,becauseweassumethatthemenucostsareiidovertime,theyarenotastatevariable. 41

betweengridpoints. The method described above applies to all the menu cost models (including the Calvo model). However, the imperfect information models are markedly different in several ways, and therefore require different methods. We simulate Alvarez et al. (2011) and Woodford (2009)) using the replication files providedbytheauthors. Weusethesamemethodsandparametervaluesusedintheoriginalpapers(Alvarez et al. (2011) for the observation costs model; Woodford (2009) for the rational inattention model), and use the policy functions to simulate the models. The kind of analysis that we can carry out with thesemodelsismorerestrictedthanfortheperfectinformationmodels. Indeed,theAlvarezetal.(2011) model contains no aggregate shocks (which in the other models drive period-by-period inflation movements). Therefore, we exclude this model from the “short-run” analaysis, in which the trend inflation parameterisfixedbutthereisnoaggregatedisturbanceinthemodel. Instead,weconductthe“long-run” analaysis by varying the parameters of trend inflation (from µ = 0 to µ = 0.2).17 For each level of trend inflation, we compute the average dispersion and skewness of price change and plot them against the level of trend inflation. Finally, the Woodford (2009) model contains no trend inflation. Strictly speaking, the model cannot be solved with aggregate nominal disturbances. Nonetheless, we take a simplified approachfollowingSection5ofWoodford(2009): anaggregatenominalshock,whichshiftsthedesired priceoffirmsbythesameamount,wouldaffecteachfirm’sprice-reviewdecisionthesamewayasinthe stationary equilibrium with only idiosyncratic shocks.18 Therefore, we take the hazard function for the case θ = 5 (unit information cost) as given and simulate the dynamics of price change for 1,000 periods and40,000firmswiththedynamicsofaggregatenominalexpenditurebeingi.i.d. andmeanzero. Weuse thesimulateddatatoconductthe“short-run”analysis. The“long-run”analysisofthismodelisexcluded. AsmentionedinSection2,theexistingmenucostmodelsandtheCalvomodelarecalibratedtomatch the median frequency of price change and the median average size of price change in the data. The way we compute these moments is by first calculating the frequency of monthly price changes and the mean absolute value ofprice change by ELI-year. We then computethe median across theELI frequencies for each year (to obtain an annual series for the median frequency) and to then take the mean across years. The average frequency that we obtain is 10.7%, and the average size of price change is 7.5%. For the Midrigan model (as well as our random menu cost model), we also target the fraction of price changes that are small (less than 1% in absolute value). We compute this as with the frequency and average size: evaluatefractionsbyELI-year,andtakeweightedmediansacrossELI’s. Wefindavalueof13.2%. Table 6 below shows the model-implied moments for the Golosov and Lucas, Midrigan, and Calvo models, as wellastherandommenucostmodelfromsection4,andcomparesthemtotheirempiricalvalues: All the models match the frequency and size moments almost exactly, and the Midrigan and random menu cost models match the fraction of small changes very closely. The Calvo and Golosov and Lucas 17The range of the trend inflation is much wider in this “long-run” study (from 0 to 0.2) than in the study of the other models(from0to0.01),becausetheAlvarezetal.(2011)modelislesssensitivetotheleveloftrendinflationthantheother models. 18Woodford(2009)usesthissimplifiedapproachtostudythemonetarynon-neutralityofthemodel. 42

Table6: Modelimpliedmoments Model AverageFrequency(%) AverageSize(%) FractionSmall(%) GolosovandLucas 10.7 7.6 0 Midrigan 10.6 7.6 12.3 Calvo 10.7 7.6 8.3 RandomMC 10.7 7.6 12.6 Data 10.7 7.5 13.2 models over- and undershoot the empirical value, respectively, as they do not target it. Table 7 below showstheparametervaluesthatwechooseforthesemodels. Table7: Parametervaluesformodels Parameter GolosovandLucas Value χ Menucost(asshareofsteadystaterevenue) 0.0178 σ Std.dev.ofidiosyncratictech.shocks 0.038 (cid:15) Midrigan χHigh Menucost(whenpositive) 0.034 σ Std.dev.ofidiosyncratictech.shocks 0.076 (cid:15) p Probabilityoffreepricechange 0.037 z p Probabilityofreceivingidio.shock 0.153 (cid:15) Calvo α Probabilityofpricechange 0.107 σ Std.dev.ofidiosyncratictech.shocks 0.194 (cid:15) For the multi-sector model, we use the same parameter values as in Nakamura and Steinsson (2010), whichmakethemodelmatchtheaveragefrequencyandsizeofpricechangeforeachof14sectors. B Data Set and Statistics Asmentionedinthemaintext,thedatasetweuseforourempiricalanalysisisthemicrodataunderlying the U.S. CPI for the period 1977-2014, with the previously unavailable period being 1977-1986. Daniel Villarworkedintensivelyintheprocessofre-constructingthisdatasetfromthemicrofilmmadeavailable by the Bureau of Labor Statistics. This process is described in detail in Appendix A.2., and it leaves us with a large data set that tracks the prices of individual, narrowly-defined products in a monthly or bimonthly frequency. We then combine this data set with the existing CPI data (1987 onwards), and that forms the data set for our analysis. Figure 8 below shows the size of our sample month by month. We plot both the number of non-missing available prices each month, as well as the number of price change observations available. The distinction is important, because we are always interested in price change statistics. The number of price observations is greater than the number of price change observations because for the price change to be observed in a particular month, we need both the current month’s 43

price, and last month’s price. So when a product has a missing price for some month, the price change willbemissingforthatmonthandthefollowingmonth. Figure8: Numberofobservationsbymonth 000051 000001 00005 0 1980 1990 2000 2010 Price Obs Price Change Obs Weprovidehereanexplanationfortherestrictionsthatwemakeonoursampleofpricechanges. The empiricalliteratureonpricesettinghasemphasizedtheimportanceofidentifying“pure”orregularprice changes, as opposed to price changes coming from temporary sales or substitutions. The reason is that sales and substitutions have features that make them different in terms of their relevance for the study of the role of monetary policy and aggregate shocks. Indeed, when a product goes on sale, its price will change, but it is not clear that this happens in response to any changes in aggregate conditions. What’s more, products on sale tend to revert back quickly to their pre-sale price. This distinction was pointed outnotablybyNakamuraandSteinsson(2008),andAndersonetal.(2015)documentthewaysinwhich salepricesbehavedifferentlyfromregularprices. In a similar way, the distinction between regular price changes and substitutions is made because a price change coming from a product substitution could reflect the changes in product characteristics or in quality that could be behind the substitution. Although it is possible in some cases to estimate the contribution of quality or characteristic changes to a substitution price change (and the BLS does for certain products), we prefer to use the product identifiers to focus on price changes involving identical products. The BLS also identifies whenever a product substitution occurs, or when a new “version” of a particular product is introduced. We treat a new version as an entirely new product, and only compute pricechangesbycomparingpricechangeswithinidenticalversions. The BLS makes a considerable effort to ensure that the prices of individual products are tracked, so thatthepricechangescannotbeattributabletochangesinanyproductcharacteristics. Thisconformswith our goals very well, as we are also only interested in price changes of identical products. An individual 44

product could be, for example, a two quart bottle of Diet Coke in a particular supermarket location in NewYorkCity,oraspecificfutonmodelinaparticularfurniturestoreinLosAngeles. Wecomputeprice changesasthedifferenceofthelogprice,or: P it ∆p = log( ). it P it−1 As discussed previously, we exclude observations for which there is any indication that the price was not actually observed but imputed, and for which the product was on sale. There are therefore missing observations in the price spells that we use. To compute the price change for any given month, we compare the price for that month to the previous month’s price, when it is available. When the previous month’spriceisnotavailable,wecomparethecurrentpricetothepricefromtwomonthsbefore. Without this, we would have to drop a significant amount of data, as many prices are only sampled every two months. Since price changes are relatively infrequent, we believe that it is overwhelmingly likely that if a price changed between any two months, it only changed once, which means that we are observing the true price change, whether it occurred in the current or previous month. This is then not extremely important,asformuchofouranalysiswecombinethepricechangesbyquarteroryear. With the price change observations, we then form distributions of these price changes, keeping only the non-zero changes, for each period (either month, quarter, or year). A few observations on how these are constructed are in order. First, since the vast majority of prices do not change in any given month, these distributions only include non-zero price changes (which corresponds to what we look at in the theoretical results). Second, because estimates of higher moments are very sensitive to outliers, we follow other empirical work in excluding price changes whose absolute value is above a certain value(e.g. KlenowandKryvtsov(2008);Alvarezetal.(2016b)),(ourthresholdisonelogpoint). Third, Eichenbaumetal.(2013)haveshedlightonproblemswiththemethodsofreportingandcollectingprices in some of the product categories of data sets such as the CPI. They show that this leads to erroneous smallpricechangesappearinginthedata,pricechangesthatcomefromthepricecollectionmethods,and that do not reflect actual price changes. This is particularly important for us, as estimates of dispersion andskewnesswillbesensitivetotherelativeamountsofsmallandlargepricechanges. Wedealwiththis byconstructingstatisticsthatexcludeverysmallpricechanges(< 1%inabsolutevalue)intheELI’sthat Eichenbaumetal. flaggedasproblematicasarobustnesscheck. Welabelestimatesconstructedwiththis restrictionwtih”EJRS”. Forthedispersionandskewnessstatistics,wefirstseparateobservationsintocategoriesthatwelabel major groups. There are thirteen of these, and table 8 below provides a list, along with the share of expenditureweightthattheyrepresent. Services represent the lion’s share of the weight. We then compute the dispersion and skewness statisticsfromeachmajorgroup,andforeachtimeperiodwethentakeanexpenditure-weightedaverage ofthestatistics,whichrepresentsthevalueofthestatisticsthatwewilluse. If,forexample,Skew isthe kt 45

Table8: CPIgroupweight MajorGroup Weight(%) ProcessedFood 8.2 UnprocessedFood 5.9 HouseFurnishings 5.0 Apparel 6.5 Transportation 8.3 MedicalCare 1.7 Recreation 3.6 Edu.Supplies 0.5 Miscellaneous 3.2 Services 38.5 Utilities 5.3 Gasoline 5.1 TravelServices 5.5 skewness of the distribution of price changes in major group k and period t, then the value of skewness thatweuseinouranalysis,Skew ,isgivenby: t (cid:88) Skew = w Skew . t k kt k Wefollowthesamemethodforthedispersion,andthusobtaintimeseriesfortheskewnessanddispersion of price changes. This also applies for the frequency, but there we calculate the frequency first by ELI, which is a much narrower category. That is because the frequency is merely an average of the dummy variable indicating whether a price has changed or not, and it is calculated based on the number of price change observations (zero or non-zero), while the other moments are only calculated based on the non-zero changes (which gives fewer observations). This means that the frequency can be estimated with reasonable precision by ELI. Finally, the expenditure weights that we use are those from the 1998 revision of the CPI, which are the latest ones available. Different weights were used for 1977-1987 and 1988-1997, but we keep the weights constant throughout the sample so that changes in the weights do notinducechangesinthestatisticsthatweestimate. C Additional Empirical Results In Section 2, we presented results on the empirical result between inflation and various price change moments, using both scatter plots and regressions. We provide additional empirical results that support the main message of 2: that the dispersion of price change falls with inflation, and that price change skewness does not. We start with the correlation values between inflation and the different moments, at variousfrequencies,andforexcludingandincludingthehighinflationperiod. Next, we present scatter plots in which the dispersion and skewness measures were computed by 46

Table9: Corr(Frequency,Inflation) WeightedMedian Monthly Quarterly Annual 1977-2014 1985-2014 1977-2014 1985-2014 1977-2014 1985-2014 Raw 0.575 0.399 0.671 0.536 0.764 0.618 Smoothed 0.769 0.552 0.785 0.628 - WeightedMean Raw 0.311 -0.019 0.314 -0.216 0.374 -0.243 Smoothed 0.371 -0.337 0.36 -0.295 - Table10: Corr(IQR,Inflation) AllObservations Monthly Quarterly Annual 1977-2014 1985-2014 1977-2014 1985-2014 1977-2014 1985-2014 Raw -0.602 -0.446 -0.716 -0.665 -0.776 -0.751 Smoothed -0.675 -0.706 -0.719 -0.742 - EJRS Raw -0.666 -0.434 -0.711 -0.689 -0.775 -0.779 Smoothed -0.792 -0.701 -0.709 -0.769 - Table11: Corr(Skewness,Inflation) AllObservations Monthly Quarterly Annual 1977-2014 1985-2014 1977-2014 1985-2014 1977-2014 1985-2014 Raw 0.265 0.084 0.345 0.067 0.473 0.122 Smoothed 0.506 0.136 0.474 0.133 - EJRS Raw 0.272 0.068 0.327 0.053 0.447 0.102 Smoothed 0.462 0.144 0.452 0.105 - Table12: Corr(KellySkewness,Inflation) AllObservations Monthly Quarterly Annual 1977-2014 1985-2014 1977-2014 1985-2014 1977-2014 1985-2014 Raw 0.584 0.069 0.674 -0.106 0.744 -0.165 Smoothed 0.696 -0.067 0.697 -0.199 - 47

Figure9: IQRandSkewnessofPriceChangeDistribution,Quarterly Source: Authors’calculationsfromBLSCPIResearchDatabase egnaR elitrauQ-retnI 41. 21. 1. 80. 60. 1980 1990 2000 2010 ssenwekS 5. 0 5.- 1- 1980 1990 2000 2010 excludingsmallpricechangesintheELI’spointedoutbyEichenbaumetal.(2013). ThemeasureofinflationthatwehadusedinthescatterplotsandregressionswasCorePCEinflation, which excludes food and energy prices that tend to be quite volatile (and that could be influenced by sectoral shocks that we do not consider in the models). In addition, since the PCE index is chained, it tendstoyieldalowervalueforinflationthantheCPI.However,fortheregressions,weusedCPIinflation because we include expected inflation as a control, and the survey of inflation expectations asks about expectations of CPI inflation specifically. We therefore used CPI inflation to make the two variables more comparable. In Figure 10 below, we plot the twelve month log change for both indexes. They both co-moveverystrongly,althoughthepeakismuchhigherfortheCPI. Figure10: Inflation 51. 1. 50. 0 50.- Figure: Inflation 1980 1990 2000 2010 CPI Core PCE In this section we show that our results do not depend on which inflation measure we use, so we 48

present scatter plots with CPI inflation, and regression results with Core PCE inflation as the regressor. The only difference that this makes is that in the regressions, the absolute value of the coefficients on inflation are slightly larger, because core PCE inflation does not attain as high a value, so the estimated slope of the moments on inflation is smaller. We also present results using series filtered by a moving average smoother and seasonally adjusted by removing quarterly dummies. Again, the the same results hold, but they come out a bit more clearly. For all of these results, we focus on using the quarterly inflationandmomentseries,althoughthesameresultswouldholdwiththemonthlyandannualseries. Figures11-14belowpresentscatterplotsofthesmoothedmomentandinflationseries. Figure11: Frequencyofpricechange&inflationsmoothed,quarterly naideM dethgiew-erutidnepxE ,ycneuqerF ylhtnoM 1. 50. 0 50.- Weighted Median; Corr = 0.907 -.02 0 .02 .04 .06 Year-on-year inflation, Quarterly Average Pre 1984 Post 1984 naeM dethgiew-erutidnepxE ,ycneuqerF ylhtnoM 1. 80. 60. 40. 20. 0 Weighted Mean; Corr = 0.566 -.02 0 .02 .04 .06 Year-on-year inflation, Quarterly Average Pre 1984 Post 1984 Figure12: IQRofpricechange&inflationsmoothed,quarterly naem dethgiew ,RQI egnahC ecirP 40. 20. 0 20.- All Observations; Corr = -0.721 -.02 0 .02 .04 .06 Year-on-year inflation, Quarterly Average Pre 1984 Post 1984 naem dethgiew ,RQI egnahC ecirP 40. 20. 0 20.- EJRS; Corr = -0.711 -.02 0 .02 .04 .06 Year-on-year inflation, Quarterly Average Pre 1984 Post 1984 49

Figure13: Skewness&inflationsmoothed,quarterly naem dethgiew ,ssenwekS egnahC ecirP 6. 4. 2. 0 2.- 4.- All Observations; Corr = 0.559 -.02 0 .02 .04 .06 Year-on-year inflation, Quarterly Average Pre 1984 Post 1984 naem dethgiew ,ssenwekS egnahC ecirP 6. 4. 2. 0 2.- 4.- EJRS; Corr = 0.544 -.02 0 .02 .04 .06 Year-on-year inflation, Quarterly Average Pre 1984 Post 1984 Figure14: Kellyskewness&inflationsmoothed,quarterly,corr=0.734 naem dethgiew ,ssenwekS ylleK 3. 2. 1. 0 1.- 2.- -.02 0 .02 .04 .06 Year-on-year inflation, Quarterly Average Pre 1984 Post 1984 50

Figure15: Frequencyofpricechange&CPIinflation,quarterly naidemqerf 52. 2. 51. 1. 50. Weighted Median; Corr = 0.785 -.05 0 .05 .1 .15 qinfl_cpi Pre 1984 Post 1984 naemqerf 53. 3. 52. 2. 51. Weighted Mean; Corr = 0.588 -.05 0 .05 .1 .15 qinfl_cpi Pre 1984 Post 1984 Figure16: IQR&CPIinflation,quarterly RQI 41. 21. 1. 80. 60. All Observations; Corr = -0.692 -.05 0 .05 .1 .15 qinfl_cpi Pre 1984 Post 1984 msnRQI 41. 21. 1. 80. 60. EJRS; Corr = -0.674 -.05 0 .05 .1 .15 qinfl_cpi Pre 1984 Post 1984 51

Figure17: Skewness&CPIinflation,quarterly wekS 1 5. 0 5.- 1- All Observations; Corr = 0.354 -.05 0 .05 .1 .15 qinfl_cpi Pre 1984 Post 1984 msnwekS 5. 0 5.- 1- EJRS; Corr = 0.335 -.05 0 .05 .1 .15 qinfl_cpi Pre 1984 Post 1984 Figure18: Kellyskewness&CPIinflation,quarterly,corr=0.674 weksk 4. 2. 0 2.- -.05 0 .05 .1 .15 qinfl_cpi Pre 1984 Post 1984 52

Figures15-18arescatterplotsusingCPIinflation. The patterns in these scatter plots are the same as in the ones presented in Section 3. We further confirmtheseresultswiththeregressiontablesbelow. Table13: Coreinflationasregressor-frequency CoefficientsforFrequencyRegressions WeightedMedian WeightedMean Specification 1977-2014 1985-2014 1977-2014 1985-2014 All 0.906*** 1.362*** -0.046 -0.231 (0.271) (0.313) (0.244) (0.305) FedDummies 1.248*** 1.503*** 0.978*** 0.281** (0.220) (0.214) (0.223) (0.258) InflationOnly 0.877*** 1.083*** 0.374** -0.580** (0.122) (0.253) (0.173) (0.296) Table14: Smoothedandseasonaladjustedseries-frequency CoefficientsforFrequencyRegressions WeightedMedian WeightedMean Specification 1977-2014 1985-2014 1977-2014 1985-2014 Fed&ExpectedInfl 0.711*** 0.796*** 0.462 0.326* (0.125) (0.210) (0.138) (0.189) FedDummies 0.778*** 0.889*** 0.723*** 0.284* (0.075) (0.207) (0.109) (0.163) InflationOnly 0.716*** 0.824*** 0.437*** -0.178 (0.062) (0.223) (0.105) (0.240) Table15: Coreinflationasregressor-IQR CoefficientsforIQRRegressions AllObservations EJRS Specification 1977-2014 1985-2014 1977-2014 1985-2014 InflationOnly -0.412*** -0.676*** -0.461*** -0.803*** (0.060) (0.081) (0.068) (-0.086) FedDummies -0.354*** -0.686*** -0.401*** -0.824*** (0.082) (0.095) (0.095) (0.099) Fed&ExpectedInfl -0.366*** -0.485** -0.429*** -0.594*** (0.127) (0.117) (0.142) (0.128) 53

Table16: Smoothedandseasonaladjustedseries-IQR CoefficientsforIQRRegressions AllObservations EJRS Specification 1977-2014 1985-2014 1977-2014 1985-2014 InflationOnly -0.301*** -0.493*** -0.330*** -0.561*** (0.043) (0.073) (0.047) (0.086) FedDummies -0.241*** -0.495*** -0.249*** -0.556*** (0.048) (0.084) (0.054) (0.097) Fed&ExpectedInfl -0.164** -0.377** -0.178** -0.431*** (0.069) (0.073) (0.075) (0.083) Table17: Coreinflationasregressor-skewness CoefficientsforSkewnessRegressions AllObservations EJRS Specification 1977-2014 1985-2014 1977-2014 1985-2014 InflationOnly 4.537*** 2.131 4.315*** 1.658 (1.306) (2.062) (1.285) (1.895) FedDummies 7.546*** 3.716 6.997*** 3.396 (1.686) (2.270) (1.572) (2.087) Fed&ExpectedInfl 4.683 6.224* 4.039* 5.991 (2.870) (3.316) (2.657) (3.136) 54

Table18: Smoothedandseasonaladjustedseries-skewness CoefficientsforSkewnessRegressions AllObservations EJRS Specification 1977-2014 1985-2014 1977-2014 1985-2014 InflationOnly 3.656*** 1.208 3.263*** 0.699 (0.776) (1.222) (0.776) (1.148) FedDummies 3.683*** 0.925 3.404*** 0.688 (0.689) (1.349) (0.680) (1.245) Fed&ExpectedInfl 0.969 0.453 0.785 0.152 (1.206) (1.504) (1.182) (1.367) Table19: Coreinflationasregressor-Kellyskewness CoefficientsforKellySkewnessRegressions AllObservations Specification 1977-2014 1985-2014 InflationOnly 2.973*** -0.603 (0.537) (0.512) FedDummies 4.035*** 0.504 (0.713) (0.606) Fed&ExpectedInfl 2.066** 0.136* (1.047) (0.721) Table20: Smoothedandseasonaladjustedseries-Kellyskewness CoefficientsforKellySkewnessRegressions AllObservations Specification 1977-2014 1985-2014 InflationOnly 2.465*** -0.088 (0.342) (0.394) FedDummies 2.479*** 0.282 (0.329) (0.435) Fed&ExpectedInfl 1.636** 0.204 (0.731) (-0.430) Whatthesetablesshowisthatwhilethesizeofthecoefficientsvariessomewhatacrossspecifications, theresultspresentedinSection2stillhold: thefrequencyofpricechangeriseswithinflation,thedispersion falls, and the skewness does not fall with inflation (the relationship is positive but not significant in thelowinflationperiod,andpositiveandmostlysignificantinthewholesample). 55

Figure19: MomentsofPriceChangeandInflation,Quarterly naeM dethgiew-erutidnepxE ,ycneuqerF ylhtnoM 53. 3. 52. 2. 51. Mean Frequency; Corr = 0.503 0 .02 .04 .06 .08 .1 Year-on-year inflation, Quarterly Average Pre 1984 Post 1984 naem dethgiew ,RQI egnahC ecirP 41. 21. 1. 80. 60. Inter-Quartile Range; Corr = -0.727 0 .02 .04 .06 .08 .1 Year-on-year inflation, Quarterly Average Pre 1984 Post 1984 naem dethgiew ,ssenwekS egnahC ecirP 5. 0 5.- 1- Skewness; Corr = 0.319 0 .02 .04 .06 .08 .1 Year-on-year inflation, Quarterly Average Pre 1984 Post 1984 56

Table 21: Coefficients on Inflation for Price Change Moments - Using CPI Data Excluding Small Price Changes 1977-2014 1985-2014 All FedDummies InflationOnly All FedDummies InflationOnly Frequency 0.164 0.686*** 0.438*** 0.018 0.339** -0.087 (0.203) (0.104) (0.108) (0.196) (0.167) (0.236) IQR -0.327*** -0.204*** -0.261*** -0.491*** -0.476*** -0.224*** (0.046) (0.044) (0.095) (0.082) (0.089) (0.092) Skewness 3.501*** 3.928*** 1.947 1.108 1.130 2.963 (0.828) (0.966) (2.538) (1.534) (1.705) (2.985) Note: Significant***at1%level(**at5%level;*at10%level). Thistablereportstheregressioncoefficientsoninflation fromregressionsoftheweightedaveragemeanfrequencyofpricechanges,aswellasweightedmeanpricechangeIQRand skewness,excludingcertainsmallpricechangesbasedonEichenbaumetal.(2013). Theregressionsarerunusingquarterly series,wherequarterlyinflationisdefinedthemeanofthe12-monthlogchangesintheCPIforthethreemonthsinevery quarter. Thedifferentcellsindicatedifferentspecifications,whichchangewithrespecttothesampleperiodusedandwhat controlsareused. exclusionofsmallpricechanges. Standarderrorsthatareconsistentforheteroskedasticityandautocorrelationoftheresiduals(Newey-West)arereported. 57

D Random Menu Cost Figure20: ShapeofMenuCostCDFforDifferentα Figure 17: Shape of Menu Cost CDF for Different , 1 0.9 0.8 0.7 0.6 )@0.5 (F 0.4 0.3 0.2 , =0.25 , =0.5 0.1 , =1 , =5 0 0 0.5 1 1.5 2 2.5 3 3.5 4 @ 58