CONDI: A Cost-Of-Nominal-Distortions Index

Transcription

1 CONDI: A Cost-Of-Nomial-Distortios Idex Stefao Eusepi Federal Reserve Bak of New York Bart Hobij Federal Reserve Bak of Sa Fracisco Adrea Tambalotti Federal Reserve Bak of New York Jauary 2009 Abstract We costruct a price idex for moetary policy, with weights o the prices of di eret PCE products chose to miimize the welfare costs of omial distortios: a cost-of-omial-distortios idex (CONDI). We compute these weights i a multi-sector New-Keyesia model with time-depedet price settig, calibrated usig U.S. data o the sectorial dispersio of price stickiess, demad elasticity ad the returs to labor. We d that the CONDI weights mostly deped o price stickiess ad are less a ected by the other sources of heterogeeity we cosider. Moreover, CONDI stabilizatio leads to egligible welfare losses compared to the optimal policy ad is better approximated by core rather tha headlie i atio targetig. A eve better approximatio of the CONDI ca be obtaied with a adjusted core idex that covers total expeditures excludig autos, clothig, eergy, ad food at home, but that icludes food away from home. Keywords: core i atio, omial rigidities, optimal moetary policy, price idexes. JEL-code: E31, E52, E58 We would like to thak Carlos Carvalho, Vasco Cúrdia, Gauti Eggertsso, Adrea Ferrero, Mark Gertler, Marc Giaoi, Marvi Goodfried, Zheg Liu, Emi Nakamura, Federico Ravea, Robert Rich, Ayşegül Şahi, Jó Steisso ad Michael Woodford for their commets ad suggestios ad Mark Bils for providig the data o markups from Bils ad Kleow (2004). Ted Rosebaum, Kristy Mayer ad Peter Fieldig provided excellet research assistace. Adrea Tambalotti thaks IGIER Uiversità Boccoi for its hospitality while coductig part of this research. The views expressed i this paper solely re ect those of the authors ad ot ecessarily those of the Federal Reserve Bak of New York, the Federal Reserve Bak of Sa Fracisco, or those of the Federal Reserve System as a whole. 1

2 1 Itroductio Core i atio is at the ceter of may cetral baks e orts to moitor ad pursue price stability. At the Federal Reserve, this focus is well re ected by the iclusio of core PCE i atio the chage i the persoal cosumptio expeditures (PCE) price idex ex food ad eergy amog the four macroecoomic variables featured i the Summary of Ecoomic Projectios published by the FOMC four times a year. 1 O a iformal level, the ratioale for focusig o core i atio is that the prices of food ad eergy are amog the most volatile compoets of headlie i atio. Therefore, attempts to stabilize headlie i atio i the face of shocks to o-core prices would require sharp movemets i real activity. Moreover, this icreased real volatility might also result i a icrease, rather tha a reductio, i i atio volatility, if the shocks to o-core prices ted to dissipate faster tha the time it takes moetary policy to a ect overall prices. 2 This argumet has bee formally articulated i at least two ways. First, curret core i atio is a better predictor of future headlie i atio tha curret headlie i atio itself (e.g. Blider ad Reis, 2005). This is a statistical statemet of the iformal idea that o core prices are volatile. If this statistical statemet is true, cetral baks ruig a explicit or implicit form of i atio forecast targetig should pay close attetio to core as a idicator of future i atioary pressures. 3 The secod argumet i favor of focusig o core i atio as a guide for moetary policy comes from New Keyesia theory. I a ecoomy i which prices chage oly ifrequetly, ad do so at di eret rates for di eret goods, the cetral bak should cocetrate more o the stabilizatio of i atio i the goods with stickier prices, sice it is i their productio that the real distortios caused by price dispersio are larger. This priciple was rst illustrated by Aoki (2001) i a two-good ecoomy i which oe good has perfectly exible prices. I this case, the moetary authority should focus exclusively o stabilizig i atio i the sticky price (core) good. Beigo (2004) showed that a similar result holds i a multi-good case with a arbitrary distributio of price stickiess across goods. 4 I practice, these theoretical results are usually iterpreted as implyig that cetral baks should 1 The other variables are headlie PCE i atio, GDP growth ad uemploymet. 2 For a extremely clear statemet of this reasoig from the perspective of a policymaker see Mishki (2007) 3 This statistical uderpiig for the role of core i atio i policymakig has recetly received much scrutiy i the literature ad i the policy debate (Rich ad Steidel, 2007; Croe et al., 2008; Kiley, 2008 ad Buiter, 2008). 4 Beigo (2004) casts his aalysis i a iteratioal cotext, with may heterogeous Coutries i a moetary uio, rather tha may.heterogeous sectors i a closed ecoomy. The two iterpretatios of his aalysis are formally idetical, as observed by Woodford (2003). 2

3 target core i atio (e.g. Mishki, 2007; Plosser, 2008), sice the prices of o core goods ted to be more exible tha those of other goods ad services. 5 I this paper, we revisit quatitatively the theoretical argumet i support of core i atio targetig, i light of the recet detailed microecoomic evidece o the frequecy of price adjustmet preseted by Nakamura ad Steisso (2008a, NS i what follows). We also study the robustess of this argumet to the presece of heterogeeity i labor shares across goods-producig sectors, aother potetially importat source of asymmetric distortios, eve whe all sectors share the same degree of omial rigidity. 6 Our aalysis proceeds i three steps. First, we costruct a database with measures of price stickiess ad labor shares across PCE categories, at two levels of aggregatio. At the coarser level of aggregatio, we oly distiguish betwee o-core goods, which iclude food ad eergy, ad core goods, which iclude everythig else. At the er level of aggregatio, we cosider ftee major types of products, such as motor vehicles ad parts, food at home ad away from home, housig, ad medical care. We also cosider the baselie case of oe homogeeous good. The costructio of this database is oe of the cotributios of the paper, sice comprehesive measures of the degree of heterogeeity i the productio of persoal cosumptio goods ad i their price exibility were ot previously available. For price stickiess, our primary source is NS, whose data refers to the frequecy of price adjustmet for the 273 etry level items (ELIs) i the o-shelter compoet of the Cosumer Price Idex (CPI). This data covers about 70 percet of CPI expeditures, but it excludes etirely housig services (ret ad ower s equivalet ret) ad a large fractio of PCE medical care. To exted this partial evidece o CPI items to cover all the ftee major PCE products at our er level of aggregatio, we supplemet it with evidece from Geesove (2003) o the degree of omial rigidity i housig rets. Moreover, we use evidece o medical care services i the Producer Price Idex to re e the estimate of price stickiess i medical care implied by NS umbers. Fially, we re ate the CPI expediture shares of the products we cosider to re ect those i the PCE. As for labor shares, we compute them applyig the method proposed by Valetiyi ad Herredorf (2008) to the major PCE products i our database. This method is particularly suitable to the task, 5 I the New Keyesia model based o Calvo (1983) pricig, more exible prices result i a more volatile i atio rate, at least uder plausible assumptios o the behavior of margial cost (Bils ad Kleow, 2004). I this respect, the theoretical uderpiig for core i atio targetig just described could be iterpeted as aother variat of the iformal volatility argumet we sketched above. 6 I the model we preset, each sector produces oe (composite) good. Therefore i what follows we use the words sector, good ad product iterchageably. 3

4 because it allows to costruct iput shares for the compoets of al demad, such as cosumptio, takig ito accout the iput-output structure of the U.S. ecoomy. The secod step of our aalysis is the costructio of a Cost-of-Nomial-Distortios Idex (CONDI). The CONDI is a Torqvist (1936) price idex a weighted average of i atio rates that weighs i atio i di eret goods as a fuctio of the share of overall omial distortios associated with the productio of each good. This is i cotrast to a cost-of-livig idex (COLI), such as the PCE, which weighs goods by their expediture share. To quatify the cotributio of each cosumptio sector to overall distortios, we calibrate a multi-sector extesio of the textbook New Keyesia model to the evidece o sectorial heterogeeity discussed above. I this framework, we de e the CONDI as the liear combiatio of i atio rates whose stabilizatio maximizes the welfare of the model s represetative aget, as i Beigo (2004). Fially, the third step of the aalysis is to compare the performace of CONDI stabilizatio to that of the ucostraied optimal policy, as well as to other, more familiar, approaches to moetary policy. I particular, we focus our attetio o two strict i atio targetig strategies, the stabilizatio of headlie ad of core PCE i atio (i.e. PCE ex food ad eergy). This compariso provides a quatitative theoretical uderpiig for a discussio of the relative merits of moetary policies that aim to stabilize di eret types of i atio. Three mai results emerge from our quatitative aalysis. First, the optimal weights i the CONDI deped largely o sectorial heterogeeity i price stickiess, ad oly margially o variatio i prefereces ad techology, as re ected by labor shares. This dig co rms the robustess of the basic priciple that moetary policy should put more emphasis o the stabilizatio of i atio i sectors with more rigid prices. More speci cally, amog o-core expeditures, the CONDI attributes almost o weight to the very exible prices of eergy goods ad of food purchased for cosumptio at home, but a large weight to food away from home. At the same time, two categories that are part of core, but whose prices are very exible, receive little weight i the CONDI: motor vehicles ad clothig ad shoes. Secod, CONDI stabilizatio provides a excellet approximatio to the ucostraied optimal policy. I fact, the outcomes of the two policies are virtually idistiguishable i terms of welfare. Moreover, core PCE stabilizatio is a better policy tha headlie PCE stabilizatio, because core i atio o et readjusts the expediture weights o sectorial i atio rates i a directio similar to 4

5 the CONDI. I fact, the ex-post time series of CONDI i atio, built with the optimal weights ad the historical realizatio of sectorial prices, is highly correlated with core PCE i atio over the period from 1998 to 2006, but oly moderately so with headlie i atio. Third, core i atio targetig is oly a very rough approximatio of CONDI stabilizatio i terms of welfare. However, this approximatio ca be improved through a simple reclassi catio of major products across the core ad o-core aggregates. I particular, this reclassi catio would etail movig motor vehicles ad clothig ad shoes to o-core ad food away from home to core. This latter adjustmet is i fact slated to happe as part of the 2009 bechmark revisio of the Natioal Icome ad Product Accouts. The welfare loss from a policy that stabilized this adjusted core PCE i atio, compared to the optimal policy, is equivalet to a permaet icrease of aual i atio i the optimal equilibrium by 0.5 percetage poits (Jese, 2002). I compariso, the i atio equivalets of core ad total PCE stabilizatio are 0.8 ad 1.3 percet respectively. This paper is related to a fairly large literature o the welfare costs of price distortios i New Keyesia models, which icludes the work of Aoki (2001) ad Beigo (2004), as well as Erceg, Hederso ad Levi (2000), who cosider the case of distortios i the goods ad labor markets, ad Huag ad Liu (2005), who focus istead o the presece of omial rigidities i the productio of itermediate iputs. The key lesso of this literature is that i atio stabilizatio is most importat i the sectors i which omial rigidities are more proouced, sice these are the sectors with larger real distortios. The cotributio we add to this ormative literature is the detailed quatitative dimesio of our aalysis, which was made possible by the data collectio work of Bils ad Kleow (2004), Kleow ad Krytsov (2008) ad Nakamura ad Steisso (2008a). This empirical work i tur spurred a rich literature o the positive evaluatio of macroecoomic models of price rigidity, started by Kleow ad Krytsov (2008) ad Golosov ad Lucas (2007) ad ow icludig work by Midriga (2008), Burstei ad Hellwig (2007), Gertler ad Leahy (2008), ad Woodford (2008), as well as by Carvalho (2006) ad Nakamura ad Steisso (2008b) i a multi-sector eviromet similar to ours. To our kowledge, oe of this work icludes a ormative dimesio to its aalysis, which is istead the focus of this paper. 7 The paper closest i spirit to ours is Makiw ad Reis (2003). These authors ask the same broad questio we address i this paper what measure of i atio should a cetral bak target? ad do so 7 But see Burstei ad Hellwig (2008) for the ormative implicatios of the presece of meu costs i a oe-sector model. 5

6 i the cotext of a model of price settig with several dimesios of sectorial heterogeeity. However, their approach to the aswer is sigi catly di eret from ours, i several respects. First, they cosider a model with sticky iformatio, rather tha sticky prices. Secod, they adopt a ad hoc, ad uusual, objective for moetary policy. The cetral bak wats to miimize the volatility of real activity, with o regard to that of i atio. Third, their quatitative applicatio is oly meat to be suggestive, sice the ceterpiece of the paper is a theoretical aalysis of the e ect of heterogeity o the optimal i atio target i a two-sector versio of their model. 2 Measurig Heterogeeity Across PCE Categories I this sectio, we preset a dataset that icludes measures of two importat forms of heterogeeity i the productio ad pricig of PCE goods. The rst, ad most commoly studied, is the frequecy of price adjustmet, a idicator of di ereces i the degree of omial rigidities across goods. The secod is the reveue share of labor, which we iterpret as evidece of di ereces i the productio techology ad i the markups charged by rms. We focus o these sources of heterogeeity, because they give rise to a asymmetry across goods i the distortios stemmig from omial rigidities. I the New Keyesia framework we adopt, these asymmetries might justify distortig the weights of a CONDI with respect to those of a COLI. The quati catio of these distortios is the mai objective of this paper. 2.1 Price Stickiess The empirical study of the price-settig process at the microecoomic level is oe of the most active areas of macroecoomic research of the last few years. Studies such as Bils ad Kleow (2004), Kleow ad Krytsov (2008) ad Nakamura ad Steisso (2008a) for the Uited States ad Dhye et al. (2006) for the Euro Area have cotributed to the dissemiatio of a wealth of detailed evidece o the stickiess of prices, especially for cosumptio goods. For the Uited States, the primary source of this evidece is the CPI Research Database at the Bureau of Labor Statistics, which cotais the product level price data used to costruct the CPI. However, the mai i atio gauge for moetary policy purposes i the Uites States is the PCE de ator. Therefore, we also adopt the PCE de ator as the referece for our aalysis, sice we wish to costruct a price idex, the CONDI, whose mai purpose is to be a useful iput for moetary policy. 6

7 As a result of this choice, we must covert the available CPI-based evidece o price stickiess ito measures that are de itioally cosistet with the PCE de ator. We use NS data as the startig poit for this coversio because it is readily available ad focuses o a period ( ) i which i atio was low ad stable. NS report the average fractio of prices that chage each moth for 257 Etry Level Items (ELIs) i the o-shelter compoet of the CPI, which covers about 70 percet of total expeditures. 8 They distiguish betwee chages i actual ad regular prices. Actual price chages iclude chages due to sales ad chages due to substitutios of discotiued items with closely matchig oes. Regular price chages, o the cotrary, oly iclude chages i o-sale prices from oe moth to the ext for the same item. NS argue that sales ad product substitutios are maily drive by cosideratios other tha the desire of rms to chage their prices ad thus result i far less macroecoomic price exibility tha regular price chages. 9 with oe exceptio. For this reaso, we focus here o the frequecy of regular price chages, We use the frequecy of actual price chages for clothig ad shoes (31 percet), because the media frequecy of regular price chages is a very low 3.5 percet. This implies a average life for the price of a article of clothig of more tha two years, which seems ureasoable give the high turover i apparel due to seasoal purchasig patters ad fashio chages discussed by Liegey (1994). Aother major PCE product for which we do ot follow NS is medical care. A large fractio of medical care prices i the PCE do ot refer to the out-of-pocket expeses covered by the CPI, but rather to services cosumed by idividuals ad paid by isurace compaies. Therefore, the frequecy of price adjustmet for the medical care ELIs reported by NS is ot a accurate measure of the degree of price stickiess i this sector. However, aecdotal evidece suggests that the prices for medical services result from bargaiig betwee the isurers ad the health providers, which usually take place oce a year. This frequecy of price adjustmet is also cosistet with the behavior of the o seasoally adjusted producer prices for medical services. As a result, we calibrate the average duratio of PCE medical care prices to be a year, which implies that 8.3 percet of these prices adjust o average every moth. Figure 1 summarizes the evidece o the distributio of price stickiess across goods by way of expediture weighted CDFs. O the horizotal axis is the average fractio of prices that chage i a moth, from the stickiest to the most exible. For each poit o the CDFs, the vertical axis represets 8 This ELI-level data is part of the supplemetary material for the published versio of NS, available at vefactselitablesup1.xls. (URL last accessed: Jauary 25th 2009) 9 For the impact of sales o moetary eutrality see i particular Kehoe ad Midriga (2008). 7

8 the fractio of expeditures o goods whose prices adjust as or less frequetly tha the correspodig frequecy o the horizotal axis. The price adjustmet CDF for the ELI-level CPI data is labeled CPI i the gure. The frequecy of adjustmet o the horizotal axis is for NS regular prices, except for the ELIs withi clothig ad shoes ad medical care. For the former we use posted prices, while for the latter we use our estimate of 8.3 percet. The weights are the expediture shares for each ELI as reported by NS. They are re ated to sum to 100 percet of CPI expeditures. The resultig expediture-weighted media mothly frequecy of price chage is 10.6 percet. We eed to covert this evidece ito measures of price stickiess for the ftee major goods ad services categories i our PCE database. The coversio ivolves three steps. First, we re ate each of the ELI weights so that the sum of the weights of all the ELIs withi a particular PCE major product is equal to the average PCE expediture share o that product over the period The resultig CDF is labeled CPI - with PCE weights i Figure 1. The implied media frequecy of price chage is 8.3 percet. This shift of the distributio towards less exible prices is explaied by the fact that some services, most otably medical services, receive less weight i the CPI tha i the PCE, due to the di erece i scope betwee the two price idices. The prices of these services ted to be stickier tha the CPI media. I the secod step of the coversio, we fold ito NS data evidece from Geesove (2003) o price adjustmet i housig services. Expeditures o housig services represet a very large fractio of total expeditures i the Uited States: close to 30 percet i the CPI ad about 15 percet i the PCE. Ufortuately, the oly evidece o price stickiess for these services i NS is o lodgig away from home, while the bulk of housig expeditures i the PCE is o teat ad ower-occupied housig. 11 Geesove (2003) estimates from the Aual Housig Survey that rets o 29 percet of apartmets do ot chage i a year. 12 Assumig a costat probability of price adjustmet i each moth, this umber implies that the ret o 10.3 percet of retal uits chages mothly. We assume that this estimate of omial rigidity would hold also if owers reted out the dwelligs they curretly occupy. 10 This re atio requires a mappig from the CPI ELIs ito the PCE major products, which are somewhat di eret from their equivalet i the CPI. To the best of our kowledge, or the BEA or the BLS provide a o cial versio of this mappig, although buildig oe is fairly straightforward. The details of the mappig we adopted are available upo request. 11 Expeditures o ower-occupied housig are based o imputed owers equivalet rets. 12 Geesove (2003) uses AHS data over the period This is a very di eret sample tha the used by NS ad it covers a period of relatively high i atio, which might lead to a overstatemet of the frequecy of price adjustmet i rets. However, this is by far the most reliable evidece o price stickiess of housig services we were able to d. 8

9 Hece, we attribute a 10.3 percet mothly frequecy of price chage to teat ad ower-occupied housig ad readjust the weights of the other ELIs withi housig to be cosistet with its PCE expediture share. The resultig CDF is labeled PCE disaggregated i Figure 1. This CDF tracks the previous two very closely for the stickiest half of expeditures, but the iclusio of the data o housig, whose price exibility is slightly higher tha the weighted media, shifts it higher i its more exible half. The resultig media frequecy of price adjustmet goes from 8.3 percet to 8.6 percet, but this small chage hides i part a shift of the right side of the distributio with frequecies betwee 10 ad 40 percet towards stickier prices. This is a good illustratio of the di culty to capture the richess of actual distributios of price stickiess with oly oe measure of cetral tedecy. Fially, i the third step of the coversio, we propose three levels of aggregatio for the evidece we have collected: (i) a baselie with oe good, (ii) a two-good case, i which we separate core ad o-core goods (i.e. food ad eergy) ad (iii) a 15-good case by major type of product. For each of these three cases, we take the expediture-weighted media of the frequecy of price chage withi the relevat category as its measure of price stickiess. At the last level of aggregatio, our dataset icludes the thirtee major types of product used by the Bureau of Ecoomic Aalysis (BEA) i the PCE NIPA tables, plus a distictio betwee food at home ad away from home (rather tha just food) ad betwee electricity ad gas ad other household operatios (rather tha household operatios aloe). 13 The reaso for icludig these slightly er distictios is that the BEA categories we have split are very heterogeous i terms of price exibility. The prices of food away from home, for example, are amog the stickiest, while food at home is at the other had of the exibility spectrum. Table 1 icludes a complete list of the product categories icluded i our dataset. The CDF associated with our er level of aggregatio is labeled PCE 15 category aggregates i Figure 1. The aggregatio usig medias shifts the CDF further towards more sticky prices amog the relatively exible oes (i.e. to the right of the media). As a result, the media frequecy of price adjustmet across the 15 aggregates is 9.0 percet. The fourth colum of Table 1 lists the resultig frequecy of price adjustmet for each of the categories i the three levels of aggregatio. For the baselie, we use the mothly frequecy of price 13 I NIPA Table (Price Idexes for Persoal Cosumptio Expeditures by Major Type of Product), for example, we d a distictio betwee durable goods, odurable goods, ad services, as well as amog thirtee more detailed categories at the ext level of disaggregatio. 9

10 adjustmet obtaied from the disaggregated PCE categories, 8.6 percet. Lookig at the two-category case, we see that core prices adjust about two-thirds as frequetly as o core prices. This di eretial is largely due to the exibility of eergy prices. I fact, food prices as a whole are about as sticky as core prices, sice the prices of food away from home, essetially a service, are amog the stickiest i the ecoomy. 2.2 Reveue Share of Labor The secod form of heterogeeity across cosumptio goods we wish to measure is i the reveue share of labor. To costruct these shares, we eed to match data o cosumptio goods, which are part of al demad, with data o factor iputs at the idustry level. The problem is that there is o direct mappig of idustries ito al goods. I the literature, there are two mai approaches to the solutio of this problem. The rst approach, followed for example by Hu ma ad Wye (1999) ad Bouakez, Cardia, ad Ruge-Murcia (2005), is to use a reasoable groupig of idustry data ad de e al goods, icludig cosumptio goods, accordig to this groupig. This approach is ot suitable for our purposes, because it results i cosumptio goods that are ot cosistet with the categories i the PCE. The secod approach, followed by Valetiyi ad Herredorf (2008), uses iter-idustry relatioships to recostruct which idustries produce the value added embodied i cosumptio goods. This approach allows to costruct cosumptio-good-speci c aggregates that are cosistet with the PCE classi catios ad that accout for the whole structure of the U.S. supply chai. For this reaso, this is the methodology we follow here. This approach ivolves the maipulatio of U.S. iput-output tables ad of idustry data o value added ad factor costs. These maipulatios are summarized i Appedix A.1. I our applicatio, we focus o the major products i the PCE, rather tha o the broader compoets of al demad cosidered by Valetiyi ad Herredorf (2008). Our data sources are the iput-output tables published i Chetres (2007) ad data o idustry factor paymets from Bureau of Ecoomic Aalysis (2008). With these iputs, we calculate a time series of aual labor shares for the PCE categories at our three levels of aggregatio over the period The resultig average labor shares are reported i the fth colum of Table 1. We d that the average reveue share of labor i total PCE is 70.3 percet, somewhat higher tha the 65 percet reported by Valetiyi ad Herredorf (2008). Labor shares vary substatially across 10

11 cosumptio goods. Eergy goods have the lowest labor share, 61 percet, while, at 83 percet, the labor share of medical services is the highest amog all 15 PCE categories. At rst glace, the variatio i labor shares appears smaller tha that i the frequecies of price chages. To study the implicatios of these two kids of heterogeeity for moetary policy, we icorporate them ito our model, which we preset i the ext sectio. 3 A Multi-Sector Model with Price Rigidities I this sectio, we sketch a multi-sector geeralizatio of the textbook New-Keyesia model, alog the lies of Beigo (2004) ad Woodford (2003). 14 The model ecoomy is populated by a cotiuum of worker-producers idexed by j 2 [0; 1]. Each of these agets produces a sigle di eretiated good i a speci c sector. = 1; :::; N, ad cosumes a composite of all goods. Sectors produce a composite cosumptio good that we idetify with oe PCE category i the data. determied by the fractio a of producers that belog to the sector, with PN a = 1. The size of each sector is The productio process di ers across sectors i three dimesios. First, the frequecy with which producers are allowed to chage their prices (1 ), i.e. price stickiess, as i Beigo (2004). Secod, the elasticity of output with respect to chages i the labor iput 1, i.e. the returs to labor. Third, the elasticity of demad faced by each producer ( ), which determies their desired (or steady state) markup. These last two parameters joitly determie the steady state reveue share of labor i each sector. They are also importat determiats of the welfare costs of price dispersio, give ay level of price stickiess. We itroduce these two particular sources of sectorial heterogeeity to cofrot the evidece o labor shares across cosumptio sectors preseted above. To our kowledge, we are the rst to model this particular form of heterogeeity i a New Keyesia framework, ad to study its ormative implicatios. 3.1 Worker-Producers Aget j i sector maximizes lifetime utility 1X U j t = E t "log T t C j T T =t # h T (j) 1+ ; Details o the model are available i a olie appedix. 11

12 subject to the ow budget costrait E t Q t;t+1 B j t+1 + P t C j t = Bj t + (1 ) p t (j)y t (j) + T t ; where B j t+1 is a portfolio of omial assets with state cotiget price Q t;t+1 the stochastic discout factor. With complete markets, agets ca isure agaist idiosycratic shocks ad thus all have the same level of cosumptio, if their iitial itertemporal budget costrait is the same. Therefore, we drop the superscript j o cosumptio from ow o. Aget j 2 produces ad sells a di eretiated good y t (j) at the price p t (j), accordig to the productio fuctio y t (j) = Z ;t h t (j) 1 where Z ;t is a sector-speci c productivity process ad 1 1 is the elasticity of output with respect to chages i the labor iput. This parameter is idexed by ; sice it di ers across sectors. It represets the rst form of sectorial heterogeeity we itroduce i our model. The productivity process is AR(1) i logs l Z ;t = l Z ;t 1 + ;t ; with ;t N(0; ) a sector-speci c shock that is i:i:d. across sectors ad time. Fially, deotes a sector-speci c sales tax (or subsidy) ad T t lump-sum trasfers from the govermet Cosumptio Aggregates ad Price Idexes Fial cosumptio is a Cobb-Douglas aggregate of goods produced i each sector where C ;t is itself a composite de ed by C ;t = C t " 1 a NY (C ;t =a ) a ; =1 1 Z j2 c t (j) 1 dj # 1 ad c t (j) is the amout cosumed of the good produced by etrepreeur j i sector. The parameter govers the elasticity of substitutio amog the cotiuum of varieties withi the cosumptio 15 We assume that the sector-speci c subsidy o sets the gross markup charged by rms i steady state, so that the ecoomy uctuates i a eighborhood of the e ciet equilibrium. This assumptio sigi catly simpli es the derivatio of a secod order approximatio to the utility of the represetative aget, which is the welfare criterio i our ormative aalysis. See Woodford (2003) for details o this approach to optimal moetary policy aalysis ad Beigo ad Woodford (2005) for a alterative approach that does ot rely o the e ciecy of the steady state. 12

13 aggregate that de es sector. These aggregates are ormalized so that, i steady state, a represets the share of expeditures directed to the purchase of composite good : We calibrate these shares to be cosistet with the evidece, although we do ot focus o them i the ormative aalysis, sice the ecoomic implicatios of this form of heterogeeity are ot particularly iterestig. The miimum expediture overall price idex is therefore P t NY =1 P a ;t; a fuctio of the sectorial price idexes Z 1 P ;t p t (j) 1 a j2 1 1 dj : 3.2 First Order Coditios Demad Fuctios The cosumer s itratemporal problem yields the followig demad fuctios for each di eretiated good produced by j 2 as a fuctio of the sectorial demad C ;t pt (j) 1 c t (j) = C ;t a ad for each sectorial cosumptio aggregate as a fuctio of total cosumptio 1 P;t C ;t = a C t : P t Combiig the two, we obtai P ;t pt (j) c t (j) = P ;t P R ;t 1 Ct where P R ;t deotes the relative price of sector with respect to the overall price idex. From these formulas, we observe that is the elasticity of demad faced by each producer i sector : This is the secod dimesio of sectorial heterogeeity we icorporate i our model Aggregate Cosumptio The path of cosumptio for the aggregate good is described by the usual Euler equatio 1 = E t uc (C t+1 ) u C (C t ) 13 R t t+1 ;

14 where R t Rt [EtQt;t+1] 1 is the gross omial iterest rate paid o oe period bods ad t is the gross i atio rate i the geeral price level Pricig Each producer j 2 faces a xed per-period probability (1 which varies across sectors, is the third source of heterogeeity we model. ) of re-settig her price. This probability, Whe give the chace, producer j chooses a price p ;t to maximize utility, takig as give the demad fuctio she faces ad the behavior of the other agets i the ecoomy. The pricig problem ca therefore be writte as " # 1X max E t ( ) T t (y t;t =Z ;T ) (1+) T (1 ) p ;t y t;t p ;t 1 + where T =t y t;t p;t P ;T P;T P T Y T, ad T = u C (C T ) u C (C t ) is the level of demad faced at time T by a producer who last set her price i t ad Y T C T is a idex of aggregate output. The secod term i the square bracket is the disutility su ered from producig a level of output y t;t. This disutility is sector speci c, due to the di erece i productio fuctio across sectors, although we assume a commo Frisch elasticity of labor supply : The rst order coditio with respect to the optimal price i sector the gives 1 (" # ) X E t ( ) T t p ;t (y t;t =Z ;T ) ~ y t;t = 0; P T ( 1) (1 ) U C (C T ) T =t P t P T where ~ (1 + ) 1 is the elasticity of real margial cost with respect to output. 3.3 Log-liearized Dyamics Log-liearizatio of the rst order coditios ad of the aggregate price idex described above yields h a set of expectatioal di erece equatios i the edogeous variables ^Yt ; t ; f ;t g ; ^P R ;t o ; ^R i t, where hats deote log-deviatios from steady state, ;t log P ;t log P ;t 1 is sectorial i atio ad t log P t log P t 1 is aggregate i atio. These equatios iclude the Euler equatio ^Y t = ^Rt E t t+1 + E t ^Yt+1 ; 14

15 ad a set of Phillips curves for the determiatio of i atio i each sector (1) ;t = E t ;t+1 + k h ^Yt ^Y f t i ^P R Rf ;t ^P ;t where ^Y f t = ^Z t X a log Z ;t ad ^P Rf ;t = ^Z;t ^Zt are the levels of output ad the relative price that would prevail uder exible prices ad the slope is k = 1 + ~ 1 + ~ = (1 ) 1 : To these we add a equatio for aggregate i atio t = NX a ;t =1 ad the de itio of the log-chage i relative price (2) ^P R ;t = ^P R ;t 1 + ;t t : We close the model with a descriptio of moetary policy. 3.4 Moetary Policy I the cashless ecoomy with omial rigidities preseted above, moetary policy ca a ect allocatios by choice of a state cotiget path for the omial iterest rate. This choice ca be modeled as a simple feedback rule, i which the iterest rate is set as a fuctio of some edogeous variables, or as the result of maximizatio of a objective fuctio. 16 This latter approach is at the ceter of the ormative part of this study, but we follow the former whe calibratig the model, sice a policy rule has the best chace to provide a satisfactory empirical characterizatio of the observed behavior of moetary policy i the Uited States. 16 Svesso (2000) discusses i detail various approaches to the implemetatio of moetary policy i this class of models. 15

16 However, we do ot write the policy rule explicitly i terms of the iterest rate, but rather implicitly, as that rule that would result i a certai state cotiget path of omial icome. I particular, we assume that omial icome, Y t P t Y t ; which i our model is equal to cosumptio expeditures, follows the uit root process (3) l Y t = Y t ; as i Nakamura ad Steisso (2008b), for example, where Y t is i.i.d. with stadard deviatio Y The Policy Objective The mai objective of this paper is to compute a price idex that miimizes the cost of omial distortios: CONDI. The criterio we adopt for the evaluatio of this cost is the ucoditioal expectatio of the utility fuctio of the represetative, or average, worker-producer i the ecoomy " 1X Z # 1 E t h t (j) 1+ log C t 1 + dj ; t=0 which we approximate to secod order as 0 W u cc 1 2 E X t L t t=0 with 8 P < (4) L t a ~ 2 ;t + P a (1 + ~ ) ^P R ;t : (1 + P 2 a ~ ) ^Yt 2 P a ~ ^P R Rf ;t ^P ;t ^Y f t ^P Rf 2 ;t + ^Yt This approximate loss fuctio depeds o the i atio rate i each sector, o the deviatios of relative prices ad of aggregate output from their exible price couterparts, as well as from the cross-product of these two deviatios. This latter term appears i the approximatio because of the heterogeeity i productio fuctios across sectors, which is re ected i ~ : If this parameter were costat across sectors, the weighted log-deviatios of relative prices from their steady state value would be zero, both i the actual as well as i the exible price equilibrium, so that the cross term would disappear. I this case, ad with = ; we would recover a approximate loss fuctio idetical to that i Beigo (2004). Through its parameters, the loss fuctio depeds crucially o all the sources of heterogeeity i the model. Focusig o the coe ciet o i atio variability, which is the largest cotributor to welfare ^Y f t : 16

17 losses, we see that a more elastic demad (higher ), a more cocave productio fuctio (higher ad thus higher ~ ) ad a lower frequecy of price adjustmet, which results i a atter Phillips curve (higher ad thus lower ), all amplify the losses from a give path of sectorial i atio. The period loss fuctio i (4), together with the sectorial Phillips curves (1), highlights fairly clearly the ature of the policy tradeo facig the moetary authority. I the rst best, output ad the relative price i each sector are equal to their exible price couterparts. This i tur results i zero i atio i every sector. However, this equilibrium is ot withi the reach of a moetary policy authority with oly oe istrumet. 17 The reaso is that the productivity shocks speci c to each sector iduce movemets relative prices, ad thus i sectorial i atio, eve uder exible prices. Such relative price movemets caot all be o set by the choice of the output gap. Coversely, a choice of the output gap that resulted i zero i atio o average would ot imply zero i atio i each sector, give a radom draw of productivity shocks The CONDI At the ceter of our ormative aalysis is a class of strict targetig rules that perfectly stabilize a weighted average of good-speci c i atio rates a xed weight Törqvist (1936) idex of the form (5) target t = NX ;t = 0, with 0 ad =1 NX = 1. The CONDI is the idex that correspods to the best policy withi this class. More formally NX ;t ; CONDI t =1 where the set of weights f g is chose to maximize W; uder the costraits that embed the optimal behavior of the private sector, equatios (1) ad (2). We also cosider two alterative targetig rules, headlie PCE targetig ad core PCE targetig. PCE targetig is de ed by the stadard expediture weights PCE = a for = 1; : : : ; N; =1 while core targetig has weights 8 < core = : 0 for =2 core P 0 2core a 1 0 a for 2 core, 17 I this model, we ca thik of the output gap as the policy istrumet. The evolutio of the omial iterest rate ecessary to achieve a certai path for the output gap (or output) ca always be iferred from the cosumptio Euler equatio, sice this relatioship is ot a bidig costrait o the optimal policy problem. 17

18 where the set of core goods cosists of all types of expeditures except for those o food ad eergy. The compariso of the welfare implicatios of the three targetig rules just described forms the basis for our discussio of the relative merits of moetary policies that focus o the stabilizatio of core rather tha headlie i atio. Of course, a strict i atio target of ay kid headlie, core, or CONDI caot be a practical recommedatio for policy. Nevertheless, the compariso amog targetig rules we propose ca provide useful idicatios o the type of i atio idex that cetral baks should moitor most closely as a gauge of the distortioary e ects of i atio. The referece poit for the evaluatio of the relative performace of the three targetig rules we cosider is the ucostraied optimal policy. This is the solutio to the liear-quadratic Ramsey problem de ed by the welfare fuctio W ad by the costraits (1) ad (2). Uder our assumptios, this solutio provides a rst order approximatio of the optimal equilibrium, as well as a secod-order approximatio of welfare uder this equilibrium (Woodford, 2003). For this evaluatio, we follow Jese (2002) ad Deis ad Söderström (2006) ad compute a i atio equivalet for each targetig rule. The i atio equivalet for ay suboptimal policy is a simple mootoic trasformatio of the welfare di eretial betwee the optimal ad the suboptimal policy. As we show i Appedix A.3, it ca be iterpreted as the costat amout of i atio that would eed to be added exogeously to the path of i atio uder the optimal policy to make the represetative aget idi eret betwee this distorted equilibrium ad the suboptimal oe. We adopt this particular measure of the distace betwee two policies, rather tha a cosumptio equivalet, for example, because it results i a direct compariso of the costs of stabilizig the wrog kid of i atio to those of stabilizig i atio aroud the wrog level. The optimal level of i atio ad the costs of deviatig from it have bee widely debated i the literature ad amog policymakers at least sice Friedma (1969) ad thus they provide a useful bechmark for our discussio Calibratio I this sectio, we use the evidece preseted i Table 1 to disciplie the choice of the model parameters that gover the degree of sectorial heterogeeity. For the parameters without a cross-sectioal dimesio, we use stadard values to the extet possible. The calibratio assumes that the PCE categories 18 See for example Kah, Kig ad Wolma (2003) for a very detailed study of the e ect of several frictios o the optimal level of i atio i a model with price stickiess. Billi (2008) is a recet treatmet focused o the role of the zero boud o omial iterest rates ad icludes detailed refereces, while Billi ad Kah (2008) cotais a discussio of the related policy debate i the Uited States. Fisher ad Modigliai (1978) is a classic treatmet of the costs of i atio ad their sources. 18

19 i Table 1 correspod to the = 1; : : : ; N cosumptio composites/sectors i the model ad that time, t, is measured i moths Homogeous Parameters Our choice of the parameters that are costat across sectors is guided by Carvalho (2006) ad Nakamura ad Steisso (2008b), who calibrate multi-sector models with price rigidity similar to ours. We set the discout factor so that the steady state aual real iterest rate is 4 percet ad pick a iverse Frisch elasticity of labor supply,, equal to 0:5. This value is a compromise betwee the liear speci catio, = 0; adopted by Nakamura ad Steisso (2008b) ad typical of the RBC literature (Hase, 1985) ad the low elasticities of labor supply usually estimated by the empirical labor literature, which might suggest values for aroud 2, as i oe of the speci catios i Carvalho (2006). We choose the parameters of the distributio of the productivity shocks, the stadard deviatio ad autocorrelatio ; to replicate the stadard deviatio ad autocorrelatio of mothly PCE i atio over the period 1998 to 2006, which are equal to 0:20 ad 0:19 percet respectively. This procedure implies di eret values for ad across di eret quatitative reditios of the model, depedig for example o the umber of sectors cosidered ad o the kids of heterogeeity icluded i the speci catio. This is because, as i ay DSGE model, the mappig from the distributio of the primitive shocks to the momets of the edogeous variables depeds o the speci catio of the rest of the model. For this momet matchig exercise, we assume that moetary policy is coducted so that omial icome Y t P t Y t ; which i the model is equal to cosumptio expeditures, follows the uit root process (3), as i Nakamura ad Steisso (2008b). We calibrate the stadard deviatio of the iovatio to this process, Y t ; to match the stadard deviatio of the mothly growth rate of cosumptio expeditures over the period , which is equal to 0:49 percet Heterogeous Parameters The calibrated values for the parameters that are heterogeous across sectors are summarized i Table 1. The rst, ad least iterestig, form of heterogeeity we must take ito accout i the calibratio is the size of each sector. I the model, this size is govered by the parameters fa g ; which determie the steady state expediture share directed to each sector. We calibrate these parameters to match the average expediture shares of the relevat PCE categories over the period The secod dimesio of heterogeeity we calibrate is the frequecy with which producers ca adjust 19

20 their prices. I the time-depedet price settig model we cosider, every moth a fractio (1 ) of the goods that belog to composite have their price adjusted. We match this fractio to the frequecy of price chage data listed i the fourth colum of Table 1. The last dimesio of heterogeeity we calibrate is the reveue share of labor. I the model, the steady state share of labor i the reveue geerated by sales i sector, s l, equals s l = 1 1. This share depeds o the elasticity of output to chages i the labor iput, 1 ; ad o the iverse of the gross desired markup of rms i sector ; 1 ; which i tur is a fuctio of the elasticity of demad faced by each rm, : The data do ot allow us to distiguish betwee variatios i the labor share due to di ereces i demad or i labor elasticities. For this reaso, we preset results for three parameterizatios, which are all cosistet with the observed labor shares. The rst parameterizatio, which we deote by (I), attributes all the variatio i labor shares to di ereces i demad elasticities,. The secod parameterizatio (II) attributes all the variatio i labor shares to di ereces i labor elasticities,. The third parameterizatio (III) is the itermediate case i which half of the variatio i labor shares comes from ad the other half from. 19 If the demad elasticity does ot vary across goods, as i the oe-sector baselie model ad uder parameterizatio (II), we set = = 5; as i oe of Carvalho s (2006) speci catios, which implies a steady state markup of 25%: This is very close to the average wholesale markup from the 1997 Cesus of Wholesale Trade amog the idustries that Bils ad Kleow (2004) were able to match to cosumer goods i the CPI. 20 A value of 5 for the elasticity of demad is itermediate betwee the low elasticities i the rage of 3 to 4 typically foud i the IO literature ad used for example by Nakamura ad Steisso (2008b) ad Midriga (2008), ad the higher values more ofte adopted i the macroecoomic literature i the rage betwee 7 ad 10 based o the implicatios of these elasticities for steady state markups (Woodford, 2003; Golosov ad Lucas, 2007). If the labor elasticity of output does ot vary, as i the oe-sector baselie model ad uder parameterizatio (I), we choose = = 0:88. Give the baselie elasticity of demad = 5, this is the degree of decreasig returs to labor that is cosistet with the average reveue share of labor i total PCE of 70 percet over the period 1998 to This parameterizatio is explaied i more detail i Appedix A We thak Mark Bils for providig us this matched dataset. The average markup i this dataset is 24%, which implies a elasticity of demad of 5.1, while the weighted average markup (weigthed by CPI expediture shares) is 20%, with a implied elasticity of demad of

21 The three parameterizatios correspodig to the observed heterogeeity i labor share are explaied i more detail i Appedix A.2. The resultig parameter values are listed i the last four colums of Table 1. 4 Results I this sectio, we discuss the properties of the CONDIs implied by the calibratios discussed above. We start from a two-good, two-sector versio of the model, which distiguishes betwee core ad o-core goods. The simplicity of this speci catio highlights the qualitative relatioship betwee the CONDIweights ad the parameters that are heterogeous across sectors ad the ecoomic ituitio behid it. We the move o to a empirically more realistic 15-good versio of the model, i which we ca study i more detail the allocatio of CONDI weights withi the core ad o-core sectors, where a sigi cat amout of heterogeeity remais. Fially, we cosider the practical implicatios of our results for moetary policy. 4.1 The Two-Good Model: Core ad No-Core The results for the two-good calibratio of the model are reported i Table 2. For ease of referece, the rst group of colums reports the calibrated values of the parameters that chage across sectors, which we already discussed i the previous sectio. The colum labeled 1 reports the CONDI weights i the case i which the frequecy of price adjustmet ad the labor share i both sectors are set to their baselie homogeous values. These weights are the same as the PCE expediture weights: the CONDI ad the COLI coicide. Whe the two sectors are structurally idetical, there is o reaso to twist the CONDI weights with respect to the expediture weights because the distortios caused by omial rigidities are the same across sectors. Moreover, i this case, PCE stabilizatio is a way of implemetig the optimal policy, as co rmed by the fact that its i atio equivalet is zero. I fact, the sectorial Phillips curves ca be aggregated i a ecoomy-wide Phillips curve with o tradeo betwee output gap stabilizatio ad (headlie) i atio stabilizatio. Therefore, the latter also delivers the former, at uchaged relative prices, thus reproducig the e ciet equilibrium. Blachard ad Galí s (2007) divie coicidece holds uder this particular parameterizatio, as origially show by Beigo (2004). Stabilizig core i atio is ot a good policy i these circumstaces, sice it would imply igorig 21

22 the distortios i the o-core sector. Whe the two sectors have the same price stickiess ad other parameters, these distortios are just as large as those i the core sector, although the core sector accouts for a much larger part of expeditures. As a result, the i atio equivalet for this policy is 0.5 percet per moth, a large loss compared to the optimal policy. Colum 2 cosiders the case i which sectors di er oly i the frequecy of price adjustmet. Accordig to our calculatios, 11.8 percet of o-core prices chage every moth, as opposed to 8.3 percet i the core sector. As a result, core i atio receives a weight of 89.9 percet i the CONDI, compared to a PCE weight of 81.3 percet. As expected, the CONDI puts more emphasis o the stabilizatio of i atio i the stickier sector, sice this is where the distortios due to price dispersio are larger. However, the o core sector still receives a o-egligible weight of 10.1 percet, give that its prices are far from perfectly exible. I terms of weights, the, the CONDI is a almost perfect average of total ad core PCE. However, this result does ot imply that headlie ad core targetig are equivalet policies i terms of welfare, as we ca observe from the last two rows of Colum 2. Core targetig performs worse tha headlie stabilizatio uder this calibratio. The two policies have i atio equivalets of 0.51 ad 0.36 percet respectively. This result suggests that the mappig from the weights i the targetig criteria to their welfare implicatios is ot symmetric aroud the optimal weightig scheme: the losses icrease more steeply as we shift weight towards the core sector. The other remarkable result from Colum 2 is that the i atio equivalet of CONDI stabilizatio is virtually zero (0.005 percet per moth). I fact, CONDI stabilizatio delivers similarly low i atio equivalets across all the calibratios we cosider i Table 2. This result co rms the robustess of Beigo s (2004) coclusio regardig the ability of a policy that stabilizes a optimally weighted i atio rate to approximate the optimal equilibrium very closely. A importat implicatio of the excellet welfare performace of CONDI stabilizatio i our framework is that the CONDI weights we have computed would chage little if we embedded their optimal choice i a more exible policy rule, such as a iterest rate feedback rule. Eve the, i fact, the optimizatio would have to retur somethig very similar to the strict CONDI targetig rule we have assumed at the outset, ad with the same CONDI weights, sice there is very little room to improve o this rule s performace. I colums 3 through 5 of Table 2 we move o to cosider three alterative calibratios of the model, i which we allow the labor share to di er across sectors, but keep the degree of price stickiess at its 22