strategic interaction among heterogeneous price …

STRATEGIC INTERACTION AMONG HETEROGENEOUS PRICE

SETTERS IN AN ESTIMATED DSGE MODEL

Olivier Coibion and Yuriy Gorodnichenko*

Abstract—We consider a dynamic stochastic general equilibrium modelin which firms follow one of four price-setting regimes: sticky prices,sticky information, rule of thumb, or full-information flexible prices. Theparameters of the model, including the fractions of each type of firm, areestimated by matching the moments of the observed variables of themodel to those found in the data. We find that sticky price firms andsticky information firms jointly account for over 80% of firms in themodel. We compare the performance of our hybrid model to pure stickyprice and sticky information models along various dimensions, includingmonetary policy implications.

I. Introduction

THE nature of firms, price-setting decisions has longplayed a pivotal role underlying controversies in macro-

economics. Whereas real business cycle models assume thatfirms with full information are free to set prices optimally atall times, New Keynesian models are typically defined bydepartures from the assumption of flexible prices. Recentwork has also emphasized the implications of deviating fromthe assumption of full information in price setting.1 Thispaper is motivated by the idea that a single assumption aboutfirms’ price-setting decision processes may be insufficient toadequately capture macroeconomic dynamics by missingpotentially important interactions among heterogeneousfirms. Indeed, firm-level evidence indicates striking hetero-geneity in price setting, as well as significant informationcosts.2 We develop and estimate a dynamic stochastic gen-eral equilibrium model that allows four commonly assumedprice-setting sectors to coexist and interact via their price-setting decisions. Our results indicate that (a) the hybridmodel fits the data substantially better than any of the modelsconsisting solely of one type of firm, (b) sticky-price andsticky information firms account for more than 80% of allfirms in the hybrid model, (c) neither rule-of-thumb nor flex-ible-price full information firms are important to match the

moments of the data and (d) strategic interaction among dif-ferent price setting practices is qualitatively and quantita-tively important.

To assess the relative importance of heterogeneity infirms’ price-setting behavior, we consider a continuum ofmonopolistic producers of intermediate goods, divided intofour segments, each of which uses a different price-settingapproach: sticky prices, sticky information, rule of thumb,and full information flexible price firms.3 This setup isnested in an otherwise standard New Keynesian model witha representative consumer and a central bank. The para-meters of the model, including the share of each type offirm, are estimated jointly using a method-of-momentsapproach. This delivers a set of predicted moments for theobservable variables that can be directly compared to thoseof the data.

Because we allow these four types of firms to coexist,our model nests many price-setting models considered inthe literature. For example, sticky price models are fre-quently augmented with rule-of-thumb firms to better matchthe inflation inertia observed in the data, but the relativeimportance of forward-looking versus backward-lookingbehavior has been much debated.4 Our result that stickyprice firms account for approximately 60% of firms is con-sistent with the findings of much of this literature, but weargue that sticky price firms should be modeled along withsticky information firms rather than rule-of-thumb firms togenerate inflation inertia.

Flexible price, full information firms are included to cap-ture the potential importance of heterogeneity in rates atwhich prices and information are updated. Bouakez, Cardia,and Ruge-Murcia (2006), Carvalho (2006), and Aoki(2001) demonstrate that heterogeneity in price stickinessacross sectors affects the dynamics and optimal monetarypolicy of a sticky price model, respectively. By includingflexible price firms, our model nests a simple case of suchheterogeneity. The fact that these types of firms receive anestimated share of only 8% indicates that heterogeneity ofthis sort is relatively unimportant to match the moments ofthe data.

The presence of sticky price, sticky information, and rule-of-thumb firms also nests empirical work to assess the fac-tual support for the New Keynesian Phillips curve (NKPC)versus the sticky information Phillips curve (SIPC). Whileresults have been either ambiguous or favored the NKPC(Korenok, 2008; Kiley, 2007; Coibion, 2010), most of this

Received for publication September 4, 2008. Revision accepted for pub-lication March 19, 2010.

* Coibion: College of William and Mary; Gorodnichenko: Universityof California, Berkeley and NBER.

We thank Bob Barsky, Angus Chu, Bill Dupor, Chris House, Ed Kno-tek, Peter Morrow, Serena Ng, Phacharapot Nuntramas, Oleg Korenok,Matthew Shapiro. Eric Sims, Mark Waston, an anonymous referee, andSeminar Participants at the University of michigan and North AmercianEcnonometric Socity Meeting for helpful comment. We also thank theRobert V. Roosa Fellowship and Rackham Dissertation Fellowship forfinancial support, and the SciClone Computational Cluster (College ofWilliam and mary) and Center for Advanced Computing (University ofMichigan) for computational suppot.

1 Models in Sims (2003), Woodford (2001), and Mankiw and Reis(2002) are prime examples.

2 Empirical work typically finds a large amount of heterogeneity in thefrequency of price changes by firms, as well as in the source of costs tochanging prices. For example, Bils and Klenow (2004) find large differ-ences in durations between price changes across sectors, while Zbarackiet al. (2004) report significant information costs.

3 Sticky price firms are modeled a la Calvo (1983), sticky informationfirms are as in Mankiw and Reis (2002), and rule-of-thumb firms alwaysupdate prices by last period’s inflation rate, as in Barsky and Kilian(2001).

4 See Gali and Gertler (1999) and Rudd and Whelan (2006).

The Review of Economics and Statistics, Month Year, 00(00): 000–000

� 2011 The President and Fellows of Harvard College and the Massachusetts Institute of Technology

literature has assumed that either the NKPC or SIPC (or theirweighted average) forms the true models without allowingcoexistence of different price-setting mechanisms. We buildon this approach by allowing for both sticky price and stickyinformation firms to coexist and interact via strategic com-plementarities in price setting. Our finding that both types offirms are required to best match the data thus calls into ques-tion much of this previous work focused on only one modelor the other.

By considering a hybrid model with sticky prices andsticky information, this paper is most closely related torecent work by Dupor, Kitamura, and Tsuruga (2010), Kno-tek (2009), and Klenow and Willis (2007), each of whichsuperimpose delayed information updating as in Mankiwand Reis (2002) upon firms already facing nominal rigid-ities: menu costs in Knotek (2009) and Klenow and Willis(2007) and time-dependent updating in Dupor et al. (2010).Each finds empirical evidence for sticky prices and stickyinformation. Thus, our results complement their findings.However, our approach differs from theirs in three impor-tant aspects. First, whereas each of these papers considersmodels in which all firms are subject to both sticky pricesand sticky information, our model allows sticky price andsticky information firms to coexist and interact via strategiccomplementarities in price setting, but does not allow anyfirm to have both sticky prices and sticky information.While we view our approach as a better approximation tothe fact that the relative importance of pricing and informa-tional rigidities varies across firms, and thus are likely to bebest modeled with different pricing assumptions, whethersticky prices and sticky information are best integrated ver-tically (as in Klenow & Willis, 2007, Knotek, 2009, andDupor et al., 2010) or horizontally is an as-of-yet unex-plored empirical question. Second, our model is more gen-eral since it nests sticky price, sticky information, and rule-of-thumb firms, as well as flexible price full informationfirms, whereas Klenow and Willis (2007), Knotek (2009)and Dupor et al. (2010) exclude either rule-of-thumb orflexible price full information or both types of firms. Third,neither Knotek (2009) nor Dupor et al. (2010) use fully spe-cified DSGE models for their empirical results and thus arenot able to explore the implications of heterogeneous pricesetting for the sources of business cycles and optimal mone-tary policy.

To estimate our DSGE model, we use the dynamic auto-and cross-covariances of observable variables. Thesemoments provide important insights about the lead-lagstructure of economic relationships. By comparing the abil-ity of the estimated hybrid model and estimated pure mod-els to match these moments of the data, one contribution ofthe paper is being able to assess why the data prefer ourhybrid model over pure sticky price or sticky informationmodels. For example, the moments of the data indicate thatinflation leads output growth and interest rates. This sty-lized fact is the primary reason that sticky price firmsaccount for such a large fraction of firms since sticky prices

induce more-forward looking behavior than alternativeprice-setting setups do.

We also consider the implications of our results for opti-mal monetary policy. While much work has been devotedto studying optimal monetary policy for sticky price mod-els, and some work has extended this type of analysis tosticky information, Kitamura (2008) is the only other paperthat considers optimal monetary policy in a hybrid stickyprice and sticky information model and does so using thevertically integrated hybrid model of Dupor et al. (2010).5

Based on our estimated DSGE model, we find that therecould be gains in welfare if the central bank used policyrules different from the estimated Taylor rule. In particular,our simulations indicate improvements when the centralbanker has a more aggressive response to inflation or incor-porates an element of price-level targeting in his or herreaction function. We show that using pure sticky price orsticky information models can mislead the central bankerabout potential gains from using alternative policy rules inthe presence of heterogeneous price setting. The fact thatKitamura (2008) reaches a similar conclusion using analternative integration of price and informational rigiditiessupports the notion that accounting for both types of rigid-ities has important monetary policy implications that arenot adequately addressed in either pure sticky price or puresticky information models. Finally, we find that there is lit-tle penalty from using a policy with a response to inflationthat is uniform across sectors relative to policy rules withdifferential responses.

The structure of the paper is as follows. In section II, wepresent the model. Section III discusses the empirical meth-odology. Our benchmark estimates, discussion, and robust-ness analysis are in section IV. and Section V considers theimplications of our results for optimal monetary policy, andsection VI concludes.

II. Model

The model has three principal types of agents: consu-mers, firms, and the central bank. The consumer’s problemis modeled as a representative agent with internal habit for-mation. Production is broken into final goods and inter-mediate goods. Production of the final goods is perfectlycompetitive, whereas the intermediate goods are producedby a continuum of monopolistic producers that, follow dif-ferent price-setting rules. Finally, the central bank setsinterest rates according to a Taylor (1993) rule.

A. Consumer’s Problem

The representative agent seeks to maximize the presentdiscounted value of current and future utility levels,

5 See Woodford (2003) for optimal monetary policy based on stickyprices and Ball, Mankiw, and Reis (2005) for the sticky informationmodel.

2 THE REVIEW OF ECONOMICS AND STATISTICS

maxfCtþk ;NtþkðiÞ;Htþkg1k¼0

Et

X1k¼0

bk

�egtþk lnðCtþk � hCtþk�1Þ

� 1

1þ 1=g

Z 1

0

N1þ1=gtþk ðiÞdi

�;

where Ct is consumption at time t, Nt(i) is labor supplied tointermediate goods firm i, h is the degree of internal habitformation, g is the Frisch labor supply elasticity, b is thediscount factor, and gt is a shock to the marginal utility ofconsumption. We allow labor to be supplied individually tospecific firms to generate strategic complementarity in pricesetting. Consumption enters in a logarithmic form to beconsistent with a balanced-growth path. Each period, theconsumer faces the following budget constraint,

Ctþk þHtþk

Ptþk�Z 1

0

NtþkðiÞWtþk

Ptþkdi

þ Htþk�1

PtþkRtþk�1 þ Ttþk;

where Ht is the stock of risk-free bonds held at time t, Rt isthe gross nominal interest rate earned on bonds in the subse-quent period, Wt(i) is the nominal wage earned from laborsupplied to intermediate goods firm i, and Tt consists ofprofits returned to the consumer. Finally, Pt is the price ofthe consumption good at time t.

Defining Lt to be the shadow value of wealth, the first-order conditions with respect to each control variable are

Consumption Kt ¼egt

Ct � hCt�1

� bhEtegtþ1

Ctþ1 � hCt; ð1Þ

Labor Supply N1=gt ðiÞ ¼ KtðWtðiÞ=PtÞ; ð2Þ

Bonds Kt ¼ bEt½Ktþ1RtðPt=Ptþ1Þ�: ð3Þ

B. Production

The final good is produced by a perfectly competitiveindustry using a continuum of intermediate goods through a

Dixit-Stiglitz aggregator Yt ¼ ðR 1

0YtðjÞðh�1Þ=hdjÞh=ðh�1Þ

. This

yields the following price level, Pt ¼ ðR 1

0PjðjÞ1�hdjÞ1=ð1�hÞ

.

The demand facing an intermediate producer j is then given

by YtðjÞ ¼ ðPtðjÞ=PtÞ�hYt.We assume that intermediate goods producers have a

production function that is linear in labor, Yt (j) ¼ At Nt (j).Despite the presence of firm-specific labor supply, weassume that firms treat wages as exogenously determined.The optimal frictionless price (P#

t ) is a markup l : y/(y � 1)over firm-specific nominal marginal costs, where the latterare given by MCt (j) ¼Wt (j)/At. Eliminating the firm-speci-fic elements of marginal cost by substituting in the labor

supply condition and the firm-level demand yields the fol-lowing relationship between real firm-specific marginalcosts and aggregate marginal costs,

MCtðjÞPt

¼ ðPtðjÞ=PtÞ�xh

Dt

MCt

Pt;

where x : g�1, MCt/Pt : (Yt/At)x Dt/[Lt At (1 � sN,t)],

and Dt �R 1

0ðPtðiÞ=PtÞ�xhdi is a measure of the dispersion

of prices across firms. We can then write a firm’s instanta-neous optimal desired relative price as

P#t

Pt

!1þxh

¼ lDt

� �MCt

Pt: ð4Þ

Since there is no capital, government spending, or inter-national trade in the model, the goods market–clearing con-dition is simply Yt ¼ Ct.

C. Price-Setting Behavior

Intermediate-good-producing firms are assumed to be inone of four price-setting sectors: sticky prices, sticky infor-mation, rule of thumb, or flexible prices. Without loss ofgenerality, firms of the same pricing sector are grouped intosegments so that the price level can be written as

Pt ¼"Zs1

0

Pspt ðjÞ

1�hdjþZs2þs1

s1

Psit ðjÞ

1�hdj

þZs3þs2þs1

s2þs1

Prott ðjÞ

1�hdjþZ11

s3þs2þs1

Pflext ðjÞ1�hdj

#1=ð1�hÞ

;

where sp, si, rot, and flex are indices for sticky price, stickyinformation, rule-of-thumb, and flexible price firms, respec-tively. Importantly, firms are otherwise identical in thesense that a firm in a given sector is the same competitor toall other firms symmetrically regardless of whether they arein the same sector. The weighting parameters s1, s2, and s3

are the fractions of firms that belong to the sticky price,sticky information, and rule-of-thumb sectors, respectively.Flexible price firms are assigned the remaining mass ofs4 ¼ 1-s1-s2-s3. Firms cannot switch sectors. Defining

the price level specific to sector d as Pdt �

s�1d

R sdþsd�1þ...sd�1þsd�2þ... PtðjÞ1�hdj

h i1=ð1�hÞwith s0 ¼ 0, we can

rewrite the aggregate price level as

Pt ¼ ½s1ðPspt Þ

1�h þ s2ðPsit Þ

1�h þ s3ðPrott Þ

1�h

þ s4ðPflext Þ1�h�1=ð1�hÞ:

ð5Þ

Sticky price firms. These firms face a constant probabil-ity 1�dsp of being able to change their price each period. Afirm with the ability to change its price at time t will choose

3STRATEGIC INTERACTION AMONG HETEROGENEOUS PRICE SETTERS

a reset price Bt to maximize the expected present dis-counted value of future profits,

BtðjÞ ¼ age maxBX1k¼0

dkspEtfKt;tþkðB�MCtþkðjÞÞYtþkðjÞg;

where Lt,tþk is the nominal stochastic discount factorbetween times t and t þ k and firm-specific marginal costsand output are as before. Taking the first-order conditionand replacing firm-specific marginal costs and output withtheir corresponding aggregate terms yields the optimalitycondition,X1

k¼0dk

spEt

nKt;tþkYtþkPh

tþk

hB1þxh

�lMCtþkPhx

tþk=Dtþk

io¼ 0;

ð6Þ

so that all firms with the opportunity to reset prices choosethe same value of Bt. The price level for sticky price firmsobeys

Pspt ¼ ð1� dspÞB1�h

t þ dspPsp1�h

t�1

h i1=ð1�hÞ: ð7Þ

Sticky information firms. These firms face a Poisson pro-cess for updating their information sets, with the probabilityof getting new information in each period given by 1�dsi. Inevery period, firms set prices freely given their informationset. The profit-maximization problem at time t for firm j,which last updated its information set at time t� k, is then

Psitjt�kðjÞ ¼ age maxPEt�k½ðP�MCtðjÞÞYtðjÞ�;

where firm-specific marginal costs and demand are deter-mined as before. This yields the first-order condition that

Et�k ðP=PtÞ�hYt ðP#t =PÞ

� �1þxh�1h i

¼ 0; ð8Þ

which implies that all sticky information firms with thesame information set will choose the same price. The pricelevel for sticky information firms is

Psit ¼ ½ð1� dsiÞ

X1k¼0

dksiðPsi

tjt�kÞ1�h�1=ð1�hÞ: ð9Þ

To have a finite state space of the model, we will truncatethe sum in equation (9) to only p lags, where p is chosensufficiently large to not affect our results.

Rule-of-thumb firms. These firms always change theirprices by the previous period’s inflation rate.6 Hence, theprice level for the rule-of-thumb sector follows

Prott ¼ Prot

t�1ðPt�1=Pt�2Þ: ð10Þ

Flexible price/information firms. These firms arealways free to change prices and have complete informa-tion. They thus always set prices equal to the instanta-neously optimal price. The price level for flexible pricefirms is then just Pflex

t ¼ P#t .

D. Shocks

We assume the following shock processes. First, technol-ogy shocks follow a random walk with drift

log At ¼ log aþ log At�1 þ ea;t;

where ea,t are independently distributed with mean zero andvariance r2

a. Preference shocks follow a stationary AR(1)process,

gt ¼ qggt�1 þ eg;t;

where eg,t are independently distributed shocks with meanzero and variance r2

g.

E. Log-Linearizing around the Balanced-Growth Path

To ensure stationarity, we log-linearize the model aroundthe balanced-growth path in which Yt/At is stationary. Notethat equation (1) ensures that LtAt is also stationary. Defin-ing yt and kt to be the log deviations of Yt/At and LtAt fromtheir balanced-growth paths, respectively, we can rewriteequation (1) in log-linearized form as

1� ha

� �1� bh

a

� �kt ¼ 1� h

a

� �1� qgb

ha

� �gt

þ ha bEtytþ1 þ yt�1 � ea;t

� �� 1þ b h

a

� �2� �

yt

ð11Þ

and the Euler equation as

kt ¼ Etktþ1 þ ðrt � Etptþ1Þ; ð12Þ

where pt � logðPt=Pt�1Þ � logðpÞ and p ¼ Pt=Pt�1 alongthe balanced-growth path. The log deviation of the interestrate rt is defined as rt � logðRt=RÞ.

We allow the log of steady-state inflation to differ fromzero, as in Hornstein and Wolman (2005), Cogley and Sbor-done (2008), and Ascari, and Ropele (2009). The log devia-tion of inflation from its steady-state value is a weightedaverage of sector specific inflation rates,

pt ¼X

dsCPI

d pdt for d ¼ f1; 2; 3; 4g;

ð13Þ

where pdt � logðPd

t =Pdt�1Þ� logðpÞ, pd � Pd=P is the steady-

state relative price level of sector j, and sCPId � sdpd

1�his the

effective share of sector d in the aggregate price index.

6 Technically this implies that the relative price level of rule-of-thumbfirms is indeterminate in a stationary steady state. This can be avoided byassuming a Poisson probability 1�drot that each firm is allowed to set itsprice equal to P#

t . Taking the limit as drot goes to 1 leads to a well-definedrelative price level equal to P#/P. We omit this in the text for simplicitybut assume it implicitly later when we characterize the steady state.


Because inflation is not zero on average, sticky pricefirms have to take into account the fact that prices will tendto rise on average. From Equation (6), we can find thesteady-state relative reset price to be

ðB=PÞ ¼ ½ð1� c1Þ=ð1� c2Þ�1=ð1þxhÞðP#=PÞ > ðP#=PÞ;

where c1 � dspaR�1

ph and c2 � dspaR�1

p1þhð1þxÞ. But thenonzero rate of inflation also affects the steady-state levelof the optimal relative price. Specifically, one can show that

P#

P

!¼ s1

1� dsp

1� dspph�1

!"

� 1� c1

1� c2

� � ð1�hÞð1þxhÞ

þð1� s1Þ#1=ðh�1Þ

:

To the extent that the extra weight attached to s1 will ingeneral not be equal to 1, the optimal relative price will alsothen differ from 1. In particular, when p > 1, there exists aunique b�ðh;x; dsp; pÞ such that if b > b*, the steady-stateaverage relative price level of sticky price firms is greaterthan 1, while the optimal relative price is less than 1.7 Thisis because when trend inflation is positive, the relative resetprice chosen by sticky price firms declines over time as theaggregate price level rises. If firms care enough about futureprofits, then they must choose a high reset price today toavoid the relative reset price being too low in the distantfuture. This will cause the average relative price level ofsticky price firms to be greater than 1.

If the steady-state average relative price level of a sectord is greater than 1, then its share in the final good will belower than implied by its mass in the output index, that is,sCPI

d < sd. Consequently, price changes in this sector willhave a smaller effect on aggregate inflation than would bethe case if it had a steady-state relative price of 1, as can beseen in Equation (13). Because b > b* in all of our esti-mates, we have sCPI

1 < s1.From Equation (4) and the definition of the real marginal

cost, the log-linearized deviation of the instantaneouslyoptimal relative price is given by

p#t � logðP#

t =PtÞ � logðP#=PÞ¼ ð1þ xhÞ�1ðxyt � k1Þ:

ð14Þ

Defining bt as the log deviation of Bt/Pt from its stationarysteady-state value and log-linearizing equation (6), expressedin stationary variables, around the balanced-growth pathleads to the following expression for the reset price,8

bt ¼ ð1� c2ÞX1k¼0

ck2Etp

#tþk þ

1

1þ hx

X1k¼1

ðck2 � ck

1Þ

� Et½gytþk � rtþk�1� þ1

1þ hx

�X1k¼1

½ck2ð1þ hð1þ xÞÞ � hck

1�Etptþk;

ð15Þ

where gyt : log (Yt/Tt�1) � log a is the log deviation of thegrowth rate of output from its mean.

Denoting the log deviation of the relative price levelin sector d from its steady-state value as pd

t �logðPd

t =PtÞ � logðPdt =PtÞ, the log-linearized relative price

level of sticky price firms follows

pspt ¼ ð1� dspÞðb=pspÞ1�hbt þ dspp

h�1ðpspt�1 � ptÞ;

where the steady-state ratio of reset prices to the sticky-price level is given by9

ðb=pspÞ ¼ ½ð1� dspÞ=ð1� dspph�1Þ�1=ðh�1Þ:

For sticky information firms, the log-linearized optimal re-lative price at time t conditional on information dated t � kis psi

tjt�k ¼ Et�kp#t so the log-linearized relative price level

for sticky information firms can be expressed as

psit ¼ dsip

sit�1 þ ð1� dsiÞp#

t þ ð1� dsiÞdsi

�X1

k¼0dk

si ðEt�1�kpt � ptÞ þ Et�1�kDp#t

:ð16Þ

Since the inflation rate for rule-of-thumb firms is

prott ¼ pt�1; ð17Þ

the log-linearized relative price level of rule-of-thumb firmsfollows

prott ¼ prot

t�1 þ pt�1 � pt ¼ prott�1 � Dpt: ð18Þ

Inflation of flexible-price full information firms is

pflext ¼ p#

t � p#t�1 þ pt: ð19Þ

F. Central Bank

To close the model, we assume that the central bank setsinterest rates according to a Taylor (1993) type rule withinterest smoothing such that

rt ¼ ð1� q1 � q2Þ½/ppt þ /gygyt�þ q1rt�1 þ q2rt�2 þ er;t;

ð20Þ7 We formally prove this result in Coibion and Gorodnichenko (2008).8 For these sums to be well defined in the steady state requires that c2 <

1. Note that we express the reset price in terms of optimal prices ratherthan real marginal costs. The reason is that real marginal costs are also afunction of the price dispersion Dt. With non zero trend inflation, this dis-persion term has first-order effects. By expressing price-setting decisionsin terms of desired optimal prices, we reduce the state space of the modelby eliminating the need to keep track of the dynamics of price dispersion.

9 For the relative reset price to be well defined in equilibrium requiresthe additional condition that dsp�p1�h < 1.


which allows the central bank to respond to inflation andthe growth rate of output.10 We include two lags of theinterest rate on the right-hand side of equation (20) becausein our previous work, we document (e.g., Gorodnichenko &Shapiro, 2007; Coibion & Gorodnichenko, 2011) that twolags appear to be the appropriate statistical description ofserial correlation in the policy rule. The policy innovationser,t are assumed to be independently distributed with meanzero and variance r2

r .

III. Estimation Approach

Our log-linearized model has three variables that corre-spond directly to observable macroeconomic series: theinflation rate, the growth rate of output, and the nominalinterest rate. The advantage of focusing on output growthrather than the output gap, as traditionally done, is that out-put growth is directly observable, whereas the output gap isnot.11 To estimate the underlying parameters of the model,we use a method-of-moments approach that seeks to matchthe contemporaneous and intertemporal covariances of theobservable variables from the data to those of the model.After solving our model for the unique rational expectationsequilibrium and letting C denote the vector of parametersin the model, we can rewrite it in reduced form as

Xt ¼ AðWÞXt�1 þ BðWÞUt

with the measurement equation

Yt ¼ !Xt þ Nt;

where Ft � iid(0,P

F (C)) is the vector of structural shocks,Xt is the vector of variables in the model, Yt ¼ [gyt pt rt]

0 isthe vector of observable variables, U is the appropriateknown fixed selection matrix, and Xt � iid(0,

PX (C)) is the

vector of serially and contemporaneously uncorrelated mea-

surement errors withP

N ¼ diagfr2me;gy;r

2me;p;r

2me;rg.

12

Using this state-space representation of the model, wecan extract the corresponding moments of the model for theobservable variables and denote the resulting matrix withDY,n (C) : [vech(XY,0 (C))0 vec(XY,1 (C))0 �� vec(XY,n

(C))0]0, where XY,j : cov(Yt, Yt�j) is the jth autocovarianceof Yt. On the other hand, we can compute the sample auto-covariance matrix for the observed variables,

DY;n� ½vechðXY;0Þ0 vecðXY;1Þ0 � � � vecðXY;nÞ0�0, where XY; j is

the sample estimate of XY,j in the data. Our method-of-moments estimator of the parameters is then given by

W ¼ arg min ðDY;nðWÞ � DY;nÞ0n

�W � ðDY;nðWÞ � DY;NÞo;

where W is a weighting matrix. Following Abowd and Card(1989), Altonji and Segal (1996), and others, we use theidentity weighting matrix in the estimation of the covar-iance structure.13

Most work on estimating DSGE models relies on maxi-mum likelihood or Bayesian approaches. We follow ouralternative method-of-moments approach for several rea-sons. First, Ruge-Murcia (2007) compares method-of-moments estimators with other popular methods such asmaximum likelihood for estimating DGSE models and findsthat it performs well in simulations. Second, a particularlyappealing feature of our method-of-moments approach isthat the moments of the data used in the estimation have aneconomic interpretation. Comparing the predicted momentsof the model to those of the data highlights which features ofthe data can and cannot be matched by the model. As we dis-cuss in section IV B, our method-of-moments estimator thusallows us to shed light on why the pure models are rejectedin favor of the hybrid model. Thus, we interpret our empiri-cal approach as one way to get inside the black box of esti-mated DSGE models.14 Finally, we use simulation-basedmethods to estimate structural parameters without requiringthe researcher to take a stand on priors. Our simulation-basedmethod illustrates how medium- and large-scale models canbe estimated within the classical statistical framework.

IV. Results

We use U.S. data from 1984:Q1 until 2008:Q2.15 Thegrowth of output is measured as 400 � log(RGDPt/RGDPt�1), where RGDP is chained real GDP. Inflation ismeasured using the CPI by 400 � log(Pt/Pt�1). The interestrate is 400 � log(1þRt), where Rt is the effective federalfunds rate (at a quarterly rate). We focus on the contem-

10 We follow Ireland (2004) and allow the central bank to respond tooutput growth rather than some measure of the output gap. Our qualitativeresults are insensitive to the inclusion of an additional output gap term inthe Taylor rule, as shown in section IV D.

11 In addition, the theoretically motivated output gap would tend to bepoorly approximated by standard detrending methods (see Andres,Lopez-Salido, & Nelson, 2005). Gorodnichenko and Ng (2010) also showthat using growth rates of variables could lead to better statistical esti-mates than using levels of persistent variables.

12 Sargent (1989), Watson (1993), and others emphasize the importanceof measurement errors in reported macroeconomic variables as well as inimproving the fit of dynamic stochastic general equilibrium models. Weintroduce measurement errors to absorb those short-term fluctuations inmacroeconomic variables that are unrelated to structural shocks.

13 These authors find that W equal to the identity matrix performs betterthan the optimal weighting matrix in the context of estimating covariancestructures. The optimal weighting matrix, which contains high-ordermoments, tends to correlate with the moments, and this correlation under-mines the performance of the method-of moments-estimator. We investi-gate the robustness of our results to the weighting matrix in section IV D.

14 One could, of course, study the same predicted moments as we dobased on alternative estimation procedures. However, it is well knownthat when one estimates a model using one set of moments, the resultingestimates may then fare poorly on an alternative set of moments (seeDupor, Han, and Tsai, 2009). Thus, we use moments that have an intuitiveeconomic interpretation in the estimation procedure to derive greaterinsight into why the estimated parameters come out the way they do. Thistask is harder to accomplish with alternative estimators such as MLE.

15 We focus on this period rather than the full sample because of thestructural break in the monetary policy reaction function, as well as trendinflation that occurred in the early 1980s.


poraneous covariances and the first three cross-autocovar-iances of these series to estimate the parameters of themodel. We restrict the number of autocovariances to mini-mize the computational burden and sharpen inference as theplethora of weakly informative moments tends to deterio-rate the estimator’s performance.16

Our model contains the following set of parameters:W ¼ fa; R; pb; g; h; h; dsi; dsp; s1; s2; s3; /p; /gy; q1; q2; qg;rr; ra; rme;gy; rme;p; rme;rg. We calibrate the balanced-growth inflation rate, interest rate, and growth rate of outputto those observed in our sample, p ¼ 1:0077, a ¼ 1.0076,R ¼ 1:0130, and we impose b ¼ 0.99 to guarantee that theconsumer’s problem is well bounded.. We set g ¼ 1, afairly typical calibrated value for the Frisch labor supplyelasticity, and set y ¼ 10 such that the steady-state markupis about 11%. We experiment with alternative values of gand y in robustness checks. We choose to calibrate theseparameters rather than estimate them because these para-meters have known identification problems. For example,Del Negro and Schorfheide (2008) and Canova and Sala(2009) report that standard monetary models have difficul-ties in distinguishing real (which is governed by g and y)and nominal (which is governed by dsp and dsi) rigidities.All other parameters are estimated using Markov chainMonte Carlo (MCMC) methods, with details provided in

the appendix. We set the truncation of past expectations forthe sticky information firms to p ¼ 12. We restrict thedegree of habit formation, the shares of firms, and the per-sistence of the preference shock to be between 0 and 1.Price and informational rigidities (dsp and dsi, respectively)are restricted to be between 0.3 and 0.95.17

A. Baseline Estimates

Table 1 presents our baseline estimates for the hybridmodel, as well as estimation results for restricted models.For our baseline model, the degree of habit persistence, at0.79, is well within the range of estimates found in otherstudies. Our Taylor rule estimates imply strong responses bythe central bank to both inflation and the growth rate of out-

TABLE 1.—ESTIMATION RESULTS

Hybrid Model No ROT Firms No SI Firms Only SP and SI Firms

Estimate (s.e.) Estimate (s.e.) Estimate (s.e.) Estimate (s.e.)

FundamentalsLabor supply elasticity (g) 1 1 1 1Elasticity of substitution across goods (y) 10 10 10 10Habit formation (h) 0.79 (0.09) 0.80 (0.16) 0.75 (0.23) 0.77 (0.16)

Taylor ruleInflation response (/p) 2.80 (0.66) 3.15 (0.93) 3.30 (0.92) 2.92 (0.84)Output growth response (/gy) 2.61 (0.77) 3.38 (1.18) 2.45 (1.06) 2.88 (0.97)Interest smoothing (q1) 1.52 (0.09) 1.33 (0.15) 1.32 (0.28) 1.46 (0.16)Interest smoothing (q2) �0.59 (0.08) �0.40 (0.13) �0.42 (0.21) �0.53 (0.14)

Price settingSticky price sector (s1) 0.62 (0.12) 0.63 (0.15) 0.66 (0.22) 0.75 (0.13)Sticky information sector (s2) 0.21 (0.10) 0.23 (0.11) 0.00 0.25 (0.13)Rule-of-thumb sector (s3) 0.09 (0.06) 0.00 0.32 (0.18) 0.00Price rigidity (dsp) 0.81 (0.07) 0.81 (0.08) 0.80 (0.14) 0.80 (0.10)Information rigidity (dsi) 0.95 (0.13) 0.95 (0.13) 0.75 0.95 (0.14)

ShocksPersistence preference shock (qg) 0.85 (0.15) 0.88 (0.13) 0.87 (0.12) 0.87 (0.12)Standard deviation: policy shocks (rr) 0.21 (0.05) 0.25 (0.12) 0.30 (0.21) 0.23 (0.10)Standard deviation: preference shocks (rg) 9.26 (3.19) 10.33 (3.06) 9.33 (3.70) 9.54 (3.06)Standard deviation: technology shocks (ra) 3.26 (1.35) 2.92 (1.12) 2.84 (1.45) 3.20 (1.09)

Measurement ErrorStandard deviation: Output growth (rme,gy) 1.44 (0.38) 1.47 (0.31) 1.43 (0.52) 1.44 (0.31)Standard deviation: Inflation (rme,p) 1.23 (0.12) 1.22 (0.16) 1.27 (0.34) 1.22 (0.14)Standard deviation: Interest rate (rme,r) 0.09 (0.07) 0.11 (0.07) 0.09 (0.07) 0.10 (0.05)

Value of objective function 51.2 52.4 59.0 52.3Minimum v2 statistic 22.49 25.33 39.48 25.03p-value DD test - 0.19 0.01 0.23

The table presents estimates of the baseline model, using a truncation of past expectations for sticky information firms of p ¼ 12, as well as estimates of restricted models. We use contemporaneous covariancesand cross-autocovariances up to three lags. Data are from 1984:Q1 to 2008:Q2. Standard errors are constructed using nonparametric bootstrap. Bootstraps are done by running a four-lag VAR on our data and usingthe VAR coefficients and residuals to generate, 2000 replications of the data, which are used to reestimate the model in each bootstrap replication. Minimum v2 statistic is computed as in Newey (1985). P-value DDtest is the simulated probability value for the distance difference test (based on the difference in minimum v2 statistics) of imposed restrictions. See the text and appendix for details on estimation approach.

16 We consider the effect of using more moments in the robustness sec-tion, IV D.

17 The lower bound on pricing and information rigidities is imposed toavoid identification issues, since when these rigidities are low, firms in thesesectors behave very much like flexible price, full information firms, makingidentification of shares of firms tenuous. Likewise, we set an upper bound toavoid scenarios where firms reset prices very infrequently. In our estimationprocedure, we also restrict parameters to be consistent with a unique deter-minate rational expectation equilibrium. We calculated that for the baselinespecification, the MCMC chain generated less than 0.05% of draws, whichled to nonuniqueness or nonexistence. Thus, most of our draws were awayfrom the indeterminacy region. In addition, when we ran multiple long (2million draws or more) chains, we observed that the averages across chainsconverged to very similar values as our baseline estimates, which is consis-tent with the chains exploring the parameter space sufficiently well. We alsoreran chains while fixing close-to-boundary parameters and found similarpoint estimates and standard errors for other parameters. Results are avail-able on request.


put, with substantial inertia apparent in the interest rate. Theweight assigned to sticky price firms is 62%. Sticky informa-tion firms receive a weight of 21%. Thus, sticky prices andsticky information jointly account for over 80% of firms inthe model. Rule-of-thumb firms account for 9% of firms,while flexible price firms receive a share of 8%. If we adjustthe shares to reflect the fact that sticky price firms chargehigher prices on average than other sectors, the effectiveshare of sticky price firms falls to 59%, while the effectiveshare of sticky information firms rises to 23%. The estimateddegree of price rigidity dsp is 0.81, which implies that stickyprice firms update their prices every five quarters on average.Note that while this is higher than typical estimates of pricerigidities (Bils & Klenow, 2004, Nakamura & Steinsson,2008), the average price duration across all firms is on theorder of three quarters, which is consistent with the litera-ture. Sticky information firms, with an estimated degree ofinformational rigidity of 0.95, update their information setsonly infrequently, which is consistent with estimated degreesof informational rigidities in Khan and Zhu (2006) and Kno-tek (2009) over the post-1982 period.

Because no single firm type receives a share of 100%, thefirst implication of our results is that our nested model bestmatches the data when more than a single type of firm is pre-sent. However, sticky price and sticky information firmsjointly account for most of the firms in the economy. Toassess the relative importance of each type of firm, we con-sider restricted estimates of our models in table 1. One ver-sion eliminates rule-of-thumb firms. The share of sticky pricefirms rises to 63%, while that of sticky information goes to23% of firms. The model achieves only a slightly highervalue of the objective function than the baseline case, indicat-ing that rule-of-thumb firms contribute little to the ability ofthe model to match the data. When one eliminates stickyinformation firms, on the other hand, the model fares worsein matching moments, and we can reject this restriction at the1% significance level. Thus, while rule-of-thumb firms donot appear to play a significant role in matching moments ofthe data, the sticky information firms certainly do. Finally,we consider a specification in which both rule-of-thumb andflexible firms are set to 0. This version of the model yields adistribution of firms of about three-quarters sticky price firmsand one-quarter sticky information firms, and we cannotreject this restriction at 10% significance level.

Thus, the most striking result from our estimation is thatsticky price and sticky information firms play the mostimportant role in matching the moments of the data. Whenone accounts for these two types of firms, there is little needto allow for rule-of-thumb behavior or flexible price fullinformation firms. This result is particularly noteworthy fortwo reasons. First, much of the literature on sticky pricesand sticky information has focused on testing one modelagainst the other (Korenok, 2008; Kiley, 2007; Coibion,2010). Our results imply instead that both are needed tomatch the moments of the data. Second, sticky price modelsare commonly augmented with rule-of-thumb firms to intro-

duce more inflation inertia (Gali & Gertler, 1999). How-ever, when one allows for both rule-of-thumb and stickyinformation firms, the data favor sticky information as acomplement to sticky price models.

B. How Does the Hybrid Model Differ from the NestedPure Models?

In this section, we study why the data prefer a hybrid stickyprice, sticky information model over the pure models. First,we reestimate the structural parameters of the model underthe assumption that only one type of firm exists and constructvariance decompositions for each estimated model. Second,we compare the predicted moments of the hybrid and puremodels to those of the data. Third, we contrast the impulseresponse functions of each estimated model.

Estimates of pure models. To get a sense of how thehybrid model differs from pure models, we first reestimatethe parameters while imposing that the model be entirelycomposed of sticky price, sticky information, or flexibleprice full information firms. The results are presented intable 2. Note first that the sticky price model achieves thelowest value of the objective function after the hybrid model,the sticky information model comes second, and the flexibleprice model does much worse. For all three restricted mod-els, p-values for the restrictions imposed by these models areless than or equal to 5%. In addition, there are some notabledifferences in estimated parameters across models. Thesticky price model points to somewhat stronger responses bythe Fed to inflation and output growth than in the hybridmodel, while the sticky information model yields a much lar-ger response to output growth but a smaller response to infla-tion. For estimated shock processes, the biggest difference isthat the standard deviation of technology shocks is muchlower in the sticky information model than in other models.

The differences in the estimates have important implica-tions for the relative importance attributed to each shock inexplaining macroeconomic dynamics. Table 3 presents theone-year-ahead variance decompositions of output growth,inflation, and interest rates due to structural shocks in eachmodel.18 For output growth, all of the models yield the con-

18 The share of variance attributed to measurement error is 51%, 68%,and 0.2% for output growth rate, inflation rate, and interest rate, respec-tively. Although the measurement errors soak up a relatively large frac-tion of contemporaneous variation in output growth and inflation, theyhave no effect on (auto)covariances, which the model can match well.The main reason we have to rely on measurement errors is that there is aclear break in the size of the autocovariances of output growth rate andinflation rate. These estimates are in line with previous studies. For exam-ple, Watson (1993) reports that the share of errors in the statistical modelof the economy should be in the 40% to 60% range of total variationwhen one uses a basic real business cycle model. These estimates are alsoconsistent with the amount of sampling uncertainty in macroeconomicseries. For example, Broda and Weinstein (2010) report that the standarddeviation of the sampling error alone in the CPI quarterly inflation rate isabout 0.5. The standard deviation of the inflation rate in the sample theyanalyzed is 0.68, so that the measurement error can easily account for50% of variation in the inflation rate.


clusion that most of the variance is due to preferenceshocks. For inflation, there is much more variation acrossmodels. The hybrid and sticky price models attribute muchof the variance of inflation to technology and preferenceshocks. Both sticky information and flexible models, on theother hand, attribute much more importance to monetarypolicy innovations. With respect to interest rate fluctua-tions, all of the models attribute much of the variation topreference shocks, although the sticky information model

again attributes some role (25%) to monetary policy inno-vations, while the flexible model assigns a sizable weight totechnology shocks. Overall, the sticky price model yields avariance decomposition of macroeconomic variables thatclosely mirrors that of the hybrid model, with preferenceshocks being most important but with technology shocksplaying a key role in explaining inflation. The sticky infor-mation model places little weight on technology shocks andinstead assigns a much larger role to monetary policy inno-vations.

Comparing predicted moments. To further contrast thepure and hybrid models, we consider which features of thedata each model can match. Figure 1 presents the autoco-variances of the observable variables implied by the modelsand those found in the data, as well as 95% confidenceintervals derived from a nonparametric bootstrap.19 Firstand most dramatic, the flexible price model is unable toreproduce the high autocovariance of interest rates and out-put growth rate observed in the data. Second, all other mod-els adequately reproduce the autocorrelation of output

TABLE 3.—VARIANCE DECOMPOSITION

Model Policy Preference Technology

Source of Variance of Growth in OutputHybrid 5 85 10Sticky price 5 72 23Sticky information 20 80 0Flexible 0 67 33

Source of Variance of InflationHybrid 5 25 70Sticky price 5 41 54Sticky information 50 44 7Flexible 57 29 14

Source of Variance of Interest RatesHybrid 10 90 0Sticky price 5 95 0Sticky information 25 75 0Flexible 0 67 33

The table presents variance decompositions from structural shocks given the parameter estimates for

the hybrid model from T1table 1 and the pure models from T2table 2. Horizon is four quarters.

TABLE 2.—ESTIMATES OF THE PURE MODELS

Sticky Price Model Sticky Information Model Flexible Price Model Weighted Model

Estimate (s.e.) Estimate (s.e.) Estimate (s.e.) Estimate (s.e.)

FundamentalsLabor supply elasticity (g) 1 1 1 1Elasticity of substitution across goods (y) 10 10 10 10Habit formation (h) 0.69 (0.21) 0.82 1.00 (0.03) 0.71 (0.11)

Taylor ruleInflation response (/p) 3.46 (0.92) 1.70 (0.85) 1.74 (0.71) 3.35 (1.00)Output growth response (/gy) 2.44 (1.19) 4.52 (1.31) 0.76 (0.44) 1.89 (0.64)Interest smoothing (q1) 1.29 (0.22) 1.45 (0.14) 1.15 (0.46) 1.44 (0.17)Interest smoothing (q2) �0.39 (0.19) �0.50 (0.14) �0.67 (0.24) �0.55 (0.13)

Price-SettingSticky price sector (s1) 1.00 0.00 0.00 0.53 (0.14)Sticky information sector (s2) 0.00 1.00 0.00 0.45 (0.12)Rule-of-thumb sector (s3) 0.00 0.00 0.00 0.00Price rigidity (dsp) 0.80 (0.09) 0.84 (0.07)Information rigidity (dsi) 0.90 (0.07) 0.48 (0.17)

ShocksPersistence preference shock (qg) 0.88 (0.11) 0.86 (0.14) 1.00 (0.28) 0.86 (0.08)Standard deviation: policy shocks (rr) 0.27 (0.17) 0.34 (0.15) 0.29 (0.20) 0.14 (0.06)Standard deviation: preference shocks (rg) 9.54 (3.08) 9.29 (4.25) 9.87 (4.68) 8.79 (3.08)Standard deviation: technology shocks (ra) 3.05 (1.27) 0.57 (0.42) 3.54 (1.65) 3.47 (1.45)

Measurement errorStandard deviation: output growth (rme,gy) 1.37 (0.43) 1.44 (0.40) 1.91 (0.25) 1.39 (0.43)Standard deviation: Inflation (rme,p) 1.24 (0.14) 1.17 (0.16) 0.93 (0.49) 1.28 (0.34)Standard deviation: Interest rate (rme,r) 0.09 (0.07) 0.12 (0.09) 1.52 (0.15) 0.13 (0.05)

Value of objective Function 60.0 92.3 1431.3 61.0Minimum v2 statistic 34.7 34.1 >100 >100p-value DD test 0.05 0.04 <0.01 -

The table presents estimates of the pure models where firms use only one pricing mechanism and of the weighted model where aggregate dynamics is the sum of noninteracting economies with a single type ofprice setting. In the weighted model, s3 is set to 0 since inflation does not respond to shocks when only ROT firms are present. We use contemporaneous covariances and cross-autocovariances up to three lags. Thetruncation of past expectations for sticky information firms is p ¼ 12. Data are from 1984:Q1 to 2008:Q2. Standard errors are constructed using nonparametric bootstrap. Bootstraps are done by running a four-lagVAR on our data and using the VAR coefficients and residuals to generate 2,000 replications of the data which are used to reestimate the model in each bootstrap replication. Minimum v2 statistic is computed as inNewey (1985). P-value DD test is the simulated probability value for the distance difference test (based on the difference in minimum v2 statistics) of imposed restrictions. See the text and appendix for details onestimation approach.

19 The bootstrap is done by running a VAR(4) on our measures of GDPgrowth, inflation, and interest rates over the same time period as our sam-ple. We then use the VAR to simulate new data of the same length andcalculate the auto- and cross-covariances from the simulated data. We use2,000 bootstraps to generate 95% confidence intervals.


growth and interest rates, largely because this is driven bythe estimates of internal habit formation and high interestrate smoothing in the central bank’s reaction function.Third, the sticky information model tends to somewhatoverstate the persistence of inflation.

Figure 2 presents the cross-covariances of inflation withrespect to leads and lags of output growth and interest rates,as well as that of output growth to leads and lags of interestrates. The moments of the data indicate that inflation leadsoutput growth and interest rates, such that high inflationtoday is associated with higher interest rates and lower out-put growth in subsequent quarters. In addition, outputgrowth leads interest rates. The fully flexible model is lar-gely incapable of reproducing these lead-lag characteristicsof the data. The sticky information model has difficultyreproducing the fact that inflation leads output growth andinterest rates: in the case of output growth, the sticky infor-mation model predicts that the highest covariance (in abso-lute value) is contemporaneous, while in the case of interestrates, the sticky information model predicts that inflationshould lag interest rates. The sticky price model, on the

other hand, replicates these lead-lag patterns more pre-cisely. This reflects the forward-looking behavior embodiedin the reset-price decisions of sticky price firms. The hybridmodel overall yields dynamics that are very similar to thoseof the sticky price model.

Impulse responses. To understand why the sticky infor-mation model places more weight on monetary policyshocks but less weight on technology than either the hybridor pure sticky price model, we consider in figure 3 theimpulse responses of the observable variables to one-unitshocks in the estimated hybrid, sticky price, and stickyinformation models.20

Consider first the effects of preference shocks, sincethese shocks account for the brunt of the variance decompo-sition of macroeconomic variables across models. In

FIGURE 1.—AUTOCOVARIANCES OF OBSERVABLE VARIABLES

The figure plots autocovariances of the observable variables in the data (1984:Q1–2008:Q2), as the solid lines, as well as those predicted by the hybrid model and pure models (using estimates in tables 1 and 2), asthe bold dashed lines. The gray-shaded areas are bootstrapped 95% confidence intervals. Bootstraps are done by running a four-lag VAR on our data and using the VAR coefficients and residuals to generate 2,000replications of the data, from which we generate a distribution of autocovariances. The horizontal axis indicates the timing in quarters of the lagged variable used in the autocovariance.

20 We omit responses from the flexible price model because flexibleprice and rule-of-thumb firms account for a small fraction of firms in thehybrid model and the responses of flexible firms are very large on impactand dwarf those of the other models. The estimates for each model aretaken from table 2.


response to preference shocks, output growth jumps up andreturns monotonically back to 0 over time. This response issimilar across models and is driven by the estimated habitformation parameter and the persistence of the shock. Therapid decline in output growth helps match the autocorrela-tion function of output growth for all models. The gradualincrease in the interest rate helps replicate the observationthat output growth leads interest rates in the data. Becauseinflation is positive after this shock (albeit with a lag forsticky information), this shock can also help replicate thepositive correlation between inflation and interest ratesobserved in the data. However, it cannot explain the con-temporaneous negative correlation between inflation andoutput growth. For the sticky information model, thedelayed response of inflation to the preference shock causesinflation to lag output growth and interest rates, a result atodds with the data.

When we turn to technology shocks, the key findings forsticky price and hybrid models are the contemporaneousdecrease in inflation and increase in output growth. Thisresponse allows these two models to replicate the uncondi-tional negative correlation between inflation and output

growth observed in the data. In addition, because inflationjumps down on impact and returns rapidly to the steadystate while output growth converges only slowly after thispermanent shock, this shock allows the sticky price andhybrid models to replicate the finding that inflation leadsoutput growth. This accounts for the substantial weightassigned to this shock by the sticky price and hybrid modelsin accounting for inflation dynamics. For the sticky infor-mation model, the permanent nature of the technologyshock yields a very delayed response of inflation, whichagain tends to counterfactually imply that inflation lags out-put growth.

In response to monetary policy shocks, the increase inthe interest rate leads to a decrease in output growth andinflation across models. Since this tends to imply a negativecorrelation between output growth and interest rates, aswell as between inflation and interest rates, the sticky priceand hybrid models assign almost no weight to this shock, asthe key lead lag relationships are already accounted for bythe preference and technology shocks. However, we can seefrom this impulse response why monetary policy shocksplay such an important role for the sticky information

FIGURE 2.—CROSS-COVARIANCES OF OBSERVABLE VARIABLES

The figure plots cross-autocovariances of the observable variables: output growth (gy), inflation (p), and interest rates (r) in the data (1984:Q1–2008:Q2) and those predicted by the hybrid model, as well as thosepredicted by the pure models, (using estimates in table 1). Solid lines are from data, while dashed lines are those of each model. The gray-shaded areas are bootstrapped 95% confidence intervals. Bootstraps are doneby running a four-lag VAR on our data and using the VAR coefficients and residuals to generate 2,000 replications of the data, from which we generate a distribution of cross-autocovariances. The horizontal axisindicates the timing of the variable used in the cross-autocovariances (negative numbers indicate lags; positive numbers are leads).


model. Note that inflation declines for a number of quartersafter a monetary policy shock under the sticky informationmodel, a point Mankiw and Reis (2002) emphasized. Theinterest rate, on the other hand, peaks in the second quarterand then returns monotonically back to 0. Thus, after thefirst period, the correlation between inflation and the inter-est rate is positive in the sticky information model as infla-tion and interest rates decline simultaneously. In addition,because inflation falls in the first period while the interestrate starts to decline only in the second period, this shockhelps deliver a lead of inflation over interest rates, whichwas the feature of the data that the sticky information modelcould not match with preference and technology shocks.

C. How Important Is Strategic Interaction among DifferentPrice-Setting Firms?

To see how the behavior of firms within the hybrid modelcompares to their behavior when they are the only type offirm, we plot in figure 4 the response of inflation in eachsector to structural shocks, as well as the response of aggre-gate inflation in a model consisting of only this type of firm.For the latter, we use the estimated parameters of the hybrid

model and simply alter the share of firms to isolate the stra-tegic interaction effect.21

Focusing first on sticky price firms, in response to mone-tary policy, technology, and preference shocks, inflationamong sticky price firms within a hybrid model is substan-tially dampened (by about 30% on impact) relative to whatit would have been had these been the only type of firm inthe model. For sticky information firms, the effect isreversed: their inflation response is more rapid within thehybrid model than in a pure sticky information model. Thisis strategic complementarity at work: the resulting inflationresponses in each sector are much more similar than the

FIGURE 3.—IMPULSE RESPONSE FUNCTIONS

The figure plots impulse responses (percentage deviation from steady state) of baseline (hybrid), pure sticky price, and pure sticky information models (based on estimates reported in tables 1 and 2) to a unit inno-vation to monetary policy, preference shock, and technology. Time is in quarters on the horizontal axis.

21 With policy responding to endogenous variables, the behavior offirms in pure models should differ from the hybrid model even in theabsence of pricing complementarities. To address this issue, we consid-ered a version of the model with exogenous money supply and a moneydemand curve. The results were almost identical to those reported in thepaper, so we can argue that the dynamics in figure 4 are driven largely bystrategic complementarity in price setting rather than by the endogenousresponse of monetary policymakers. However, we cannot eliminate theindirect interaction of firms by aggregate demand, which is present in thehybrid model and absent in the weighted sum of pure models. Thus, bycomparing the hybrid and weighted sum of pure models, we observe thejoint effect of direct (pricing complementarities) and indirect (aggregatedemand) interaction between firms. We are grateful to an anonymousreferee for pointing these issues out.


inflation responses of the pure models. The effect of strate-gic complementarity is even more striking in the case offlexible price full information firms. Whereas inflation forthese firms would be substantial on impact, but virtually nilin subsequent periods, within the hybrid model, their infla-tion response is severely dampened. This reflects how muchmore sensitive these firms are to the behavior of other firmsbecause they are unconstrained in their actions, whereas allother firms face some kind of constraint, which is similar inspirit to Haltiwanger and Waldman (1991). In the bottomrow of figure 4 we contrast the dynamics of aggregate infla-tion in the hybrid model and the dynamics of the weighted-sum of inflation in the pure models. We interpret theweighted sum dynamics as a case where consumers have atwo-tier utility function with very low elasticity of substitu-tion across sectors and y ¼ 10 elasticity of substitutionwithin sectors.22 The hybrid model exhibits more gradualand persistent dynamics than the weighted sum over pure

models, suggesting that ignoring strategic interactionbetween firms with different price setting may considerablydistort the aggregate dynamics.

To assess more formally the quantitative importance ofstrategic interaction in price setting across sectors, we rees-timate all of the parameters of our model but applied to aweighted sum of sector-specific models (the weightedmodel). In other words, for a given set of parameters, wefeed these parameters into three separate economies whereeach economy is characterized by a single price-setting type(sticky price, sticky information, and flexible-price fullinformation) and use a weighted sum of these economies toconstruct the aggregates, from which we construct the pre-dicted moments for the weighted model.23 Although theestimates are broadly similar to our baseline results, we findthat the share of sticky price firms falls while the share ofsticky information firms increases so that each type consti-tutes approximately 50% (see table 2). However, the fit ofthe weighted model is considerably worse than the fit of thebaseline model. Based on tests of the overidentifyingrestrictions, we can reject the validity of the weightedmodel but not that of the baseline model at any standard

FIGURE 4.—SECTOR-SPECIFIC VERSUS PURE MODEL INFLATION

The figure displays the response of inflation (percentage deviation from steady state) to 1 standard deviation shocks (labeled on top). Solid lines indicate the response of inflation in each sector (labeled at left)within the hybrid model (using estimates of table 1) while the dashed lines indicate the response of a pure model consisting only of that sector’s type of firms (sd ¼ 1 for sector d). The bottom row compares theresponse of aggregate inflation in the hybrid model (in black solid lines) to a weighted sum of inflation rates from the pure models (dashed lines), where the weights are the effective weights (sCPI) of each sector fromthe baseline estimates. Baseline parameter estimates are used in each case. Time is in quarters on the horizontal axis.

22 In other words, the economy is split into four islands, each populatedwith a single type of price-setting firm; there is no direct interactionacross islands by pricing complementarities, input and output markets aswell as any macroeconomic variable; aggregate behavior is a (weighted)sum of dynamics across islands. We then compare these aggregatedynamics with the dynamics in the hybrid model when different types offirms are allowed to interact.

23 Note that we dropped rule-of-thumb firms because this model yieldsa zero response of inflation to all shocks in the absence of other firms.


significance level. This result illustrates that strategic inter-action across sectors within the hybrid model is a key ele-ment to match the data.

D. Robustness Analysis

In this section, we consider the robustness of our esti-mates to several potential issues. The first issue we addressis the set of moments used in the estimation. Our baselineresults relied on the autocorrelations of our observable vari-ables over three quarters and dynamic cross-correlations atmaximum leads and lags of three quarters as well. The pur-pose of focusing on such a restricted set of moments was toconcentrate on those moments that are most precisely esti-mated. As a robustness check, we consider the use of a lar-ger set of moments, specifically using autocovariances overtwo years, and report results in table 4. Most of the para-meters are similar to baseline estimates. The estimatedlevels of price and informational rigidities are almost iden-tical to our baseline estimates, and the estimated shares offirms continue to imply that more than 80% of firms aresticky price or sticky information firms.

An alternative approach to dealing with the precision ofthe moments used in the estimation is to allow a nonidentityweighting matrix. Although the optimal weighting matrix

would seem an ideal candidate, many studies report poorperformance of this weighting matrix in applications(Boivin & Giannoni, 2006) and Monte Carlo simulations(Altonji & Segal, 1996) that involve estimation of covar-iance structures.24 A practical compromise is a diagonalweighting matrix with estimated variances of the momentson the diagonal and zeros for off-diagonal entries. Replicat-ing our baseline estimation procedure with the diagonalweighting matrix, we find that sticky price firms account forapproximately 50% of firms, while rule-of-thumb andsticky information firms account for 18% and 16%, respec-tively,. Essentially, the use of the diagonal weight matrixdownplays some informative moments and does not allowus to clearly separate rule-of-thumb and sticky informationfirms. Most other parameter estimates are broadly similar tothe estimates based on the identity weight matrix.

We also consider sensitivity to the elasticity of labor sup-ply. While most empirical work has found low elasticitiesof labor supply, some of the RBC literature has focused onthe case with infinite labor supply (as in Hansen, 1985). In

TABLE 4.—ROBUSTNESS OF ESTIMATES

More MomentsDiagonal

Weighting MatrixIndivisible

LaborResponse toOutput Gap

AR(1) InterestSmoothing

Truncationp ¼ 24

Estimate (s.e.) Estimate (s.e.) Estimate (s.e.) Estimate (s.e.) Estimate (s.e.) Estimate (s.e.)

Fundamentalsg 1 1 ? 1 1 1y 10 10 10 10 10 10h 0.87 (0.09) 0.75 (0.18) 0.81 (0.06) 0.81 (0.12) 0.81 (0.19) 0.82 (0.10)

Taylor Rule/p 2.58 (0.93) 3.04 (0.80) 2.13 (0.52) 2.67 (0.73) 3.48 (0.98) 2.72 (0.71)/gy 1.87 (0.87) 2.75 (1.09) 2.83 (0.86) 2.73 (0.90) 4.05 (1.18) 2.32 (1.00)/x 0.00 0.00 0.00 0.02 (0.01) 0.00q1 1.74 (0.09) 0.66 (0.16) 1.50 (0.20) 1.60 (0.17) 0.91 (0.11) 1.61 (0.19)q2 �0.79 (0.08) 0.14 (0.09) �0.57 (0.17) �0.66 (0.15) 0.00 �0.67 (0.13)

Price settings1 0.57 (0.17) 0.48 (0.18) 0.49 (0.14) 0.61 (0.15) 0.60 (0.14) 0.63 (0.12)s2 0.24 (0.12) 0.16 (0.10) 0.00 (0.00) 0.23 (0.10) 0.15 (0.07) 0.22 (0.09)s3 0.11 (0.07) 0.18 (0.12) 0.51 (0.14) 0.08 (0.05) 0.09 (0.05) 0.08 (0.06)dsp 0.82 (0.11) 0.76 (0.15) 0.85 (0.02) 0.80 (0.12) 0.81 (0.09) 0.81 (0.10)dsi 0.95 (0.15) 0.65 (0.24) 0.52 (0.16) 0.95 (0.15) 0.94 (0.16) 0.95 (0.09)

Shocksqg 0.79 (0.15) 0.85 (0.11) 0.85 (0.23) 0.84 (0.15) 0.89 (0.08) 0.83 (0.10)rr 0.02 (0.01) 0.89 (0.49) 0.39 (0.13) 0.15 (0.08) 0.47 (0.22) 0.13 (0.13)rg 11.70 (4.31) 10.61 (3.75) 8.66 (3.25) 9.52 (3.99) 10.63 (3.35) 9.37 (3.36)ra 3.46 (1.54) 1.56 (0.95) 0.85 (0.39) 3.32 (1.32) 2.49 (1.00) 3.39 (1.37)

Measurement errorrme,gy 1.28 (0.48) 1.30 (0.47) 1.42 (0.53) 1.45 (0.37) 1.50 (0.32) 1.47 (0.49)rme,p 1.33 (0.19) 1.19 (0.20) 1.20 (0.44) 1.24 (0.15) 1.20 (0.18) 1.25 (0.32)rme,r 0.20 (0.13) 0.51 (0.29) 0.10 (0.04) 0.12 (0.08) 0.10 (0.06) 0.12 (0.06)

value of objective function 123.4 4.3 73.1 50.9 56.8 51.6Minimum v2 statistic 221.8 24.0 31.2 21.7 27.5 20.1p-value DD test - - - 0.37 0.02 -

The table presents robustness estimates of the baseline model. The truncation of past expectations for sticky information firms is p ¼ 12. We use contemporaneous covariances and cross-autocovariances up to threelags unless otherwise specified. Data are from 1984:Q1 to 2008:Q2. In the scenario ‘‘More Moments,’’ we use contemporaneous covariances and cross-autocovariances up to eight lags. Standard errors are constructedusing nonparametric bootstrap. Bootstraps are done by running a four-lag VAR on our data and using the VAR coefficients and residuals to generate 2,000 replications of the data, which are used to reestimate themodel in each bootstrap replication. Minimum v2 statistic is computed as in Newey (1985). P-value DD test is the probability value for the distance difference test (based on the difference in minimum v2 statistics)of imposed restrictions. See the text and appendix for details on estimation approach.

24 In Monte Carlo simulations (available on request), we found that esti-mates based on the identity-weighting matrix have better statistical prop-erties than estimates based on the diagonal weighting matrix and the opti-mal weighting matrix in time series of the same length as ours.


table 4, we present estimates of the hybrid model under theassumption of indivisible labor (g ¼ ?), which impliesthat x ¼ 0 so that there is no strategic complementarity inprice setting, although (in contrast to the weighted model)there is some interaction via aggregate demand. Eliminatingstrategic complementary has a dramatic effect on theresults. Specifically, the shares of both sticky informationand flexible price firms go to 0, while that of rule-of-thumbfirms rises to 50%. This change in outcome leads to a sub-stantial deterioration in the model’s ability to match thedata, and we can reject the model at any standard signifi-cance level using tests of the overidentifying restrictions.The reduced share of sticky information firms reflects thefact that in the absence of strategic complementarity inprice setting, sticky information firms fail to produce infla-tion inertia (Coibion, 2006). Because sticky price firms tendto induce excessive forward-looking behavior in inflation,the model needs other types of firms to slow the adjustmentof inflation to shocks. With sticky information firms unableto achieve this role in the absence of strategic complemen-tarity, the estimation instead places a significant weight onrule-of-thumb firms.

As a check, we explore the effects of using alternativepolicy reaction functions. First, we consider the followingTaylor rule, rt¼ð1�q1�q2Þ½/pptþ/gygytþ/xxt�þq1rt�1þq2rt�2þer;t, where xt is the log deviation between actualoutput and the level of output that would occur in theabsence of price and informational rigidities. The estimatedresponse to the output gap is very low and not statisticallydifferent from 0, while the other parameters are largelyunchanged. Second, we integrate the following Taylor ruleinto our model, rt ¼ ð1� q1Þ½/ppt þ /grgyt� þq1rt�1 þ er;t,which restricts interest smoothing to be an AR(1) process.With this restriction, the main results are broadlyunchanged, with sticky price firms accounting for 60% offirms and sticky information firms accounting for 15%. Theshare of flexible price firms rises to 16%. However, we canreject imposing an AR(1) specification at the 5% signifi-cance level.

The estimated degree of information rigidity dsi impliesthat firms update their information sets infrequently.Although the estimate of dsi is consistent with previous stu-dies, a large value of dsi may imply that the results are sensi-tive to the truncation lag p. To verify that our results areinsensitive to the choice of p ¼ 12, we reestimate the modelwith p¼ 24 and find very similar results, as illustrated in table4. In particular, the estimated degree of information rigidityand the share of sticky information firms are unchanged.

Our final robustness check is with respect to the elasticityof substitution across intermediate goods y. We fixed thisparameter in our baseline estimation because previous workhas shown that it is difficult to differentiate empiricallybetween nominal and real rigidities, making the joint identi-fication of y and the shares of firms tenuous. To assess howsensitive our results are to y, we redid our baseline estima-tion procedure for values of y ranging from 7 to 15. Our

results for the key parameters of interest, shares of firmsand the degree of price and information rigidities, are in fig-ure 5. In panel A, we can see that lower values of y have asubstantial effect on estimated shares of firms. Specifically,the share of sticky information firms declines rapidly, whilethat of sticky price firms rises. As lower values of y reducestrategic complementarity in price setting, we get higherestimates of price rigidity to keep the persistence of infla-tion high (panel B). However, the fit of the model worsenssubstantially as strategic complementarity decreases moder-ately (panel C). With higher values of y, the estimatedshares of firms are very similar to our baseline estimatesunder the assumption of y ¼ 10. As y rises, the degree ofstrategic complementarity increases ((1 þ x)/(1 + xy)decreases and prices become less sensitive to changes inoutput; see panel D), as does the inherent persistence ofinflation, and hence we obtain lower estimates of pricerigidity. The fit of the model actually improves with highervalues of y, indicating that even more strategic complemen-tarity is desirable to match the data. Thus, one could inter-pret our baseline results as a lower bound on the importanceof strategic complementarity in price setting across hetero-geneous price-setting firms.

V. Implications for Optimal Monetary Policy

The presence of different types of firms in the modelraises the issue of what kind of monetary policy is optimalin such a setting. To assess the effect of different policies,we follow much of the literature and assume that the centralbanker has the following loss function,

L1 ¼ varðptÞ þ xyvarðxtÞ þ xrvarðrtÞ; ð21Þ

where xy and xr show the weight on output gap and interestrate volatility relative to inflation volatility so that the varia-bility in the output gap (and later the output growth rate)and the interest rate are converted to their inflation-varianceequivalents. We also consider an alternative loss function,which penalizes the volatility of output growth instead ofthe volatility of the output gap:

L2 ¼ varðptÞ þ xyvarðgytÞ þ xrvarðrtÞ: ð22Þ

This alternative loss function may be interesting for ouranalysis because, as Amato and Laubach (2004) show, habitformation introduces a concern for the volatility in thechange of consumption and, hence, the loss function shouldinclude a term that captures the volatility of output growth.

In principle, the parameter xy can be derived from the Phil-lips curve. However, because we have different interactingprice-setting mechanisms as well as nonzero steady-stateinflation, we could not find a closed-form solution for thePhillips curve and xy, so objective functions (21) and (22) arenot necessarily model-consistent for welfare calculations.Consequently, we are agnostic about the relative weight of


output gap variability, and we experiment with differentvalues of xy. In the baseline scenario, we set xy¼ 1. The lastterm in the loss function is the penalty for the volatility of thepolicy instrument (interest rate), which helps to keep the opti-mal responses to output growth and inflation bounded. Wefollow Woodford (2003) and calibrate xr¼ 0.077.

We constrain our analysis to simple rules with fixed coeffi-cients (undercommitment) similar to the estimated interestrate rule (20) for reasons highlighted in Williams (2003).First, simple rules can often closely approximate fully opti-mal rules. Second, simple rules tend to be more robust. Third,with many sectors and the complicated structure of themodel, we could not find a closed-form solution of the objec-tive function and hence could not derive fully optimal rules.

The first question we pose is whether the central bankcould have achieved lower losses by responding differently

to aggregate inflation and output growth than what is impliedby our estimates of the Taylor rule. In the exercise, weassume that the estimated shares of firms, the degrees ofnominal and informational rigidities, and other estimatedparameters do not change with the policy rule. Panel A in fig-ure 6 presents the isoloss maps for different combinations of/p and /gy in the Taylor rule. Generally there are substantialgains from increasing the response to inflation, whichreduces the volatility of inflation, the interest rate, and theoutput gap. Holding everything else constant, a more aggres-sive response to inflation decreases the volatility of inflation,the interest rate, and the output gap and weakly increases thevolatility of output growth. In contrast, a stronger responseto the output growth rate has the opposite effect on the volati-lity of relevant macroeconomic variables. Since the volati-lity of output growth is fairly insensitive to changes in /p

FIGURE 5.—ROBUSTNESS OF ESTIMATES TO ELASTICITY OF SUBSTITUTION (h)

The panels display estimation results of the baseline model for different values of y, as indicated on the horizontal axis of each panel. For each value of y, we ran a chain of 500,000 iterations, dropping the first100,000 iterations. A: The estimated shares of sticky price (s1), sticky information (s2), and rule-of-thumb firms (s3) for different values of y. B: The estimated levels of price rigidity (dsp) and informational rigidity(dsi) for different values of y. C: Values of the objective function (averaged across chains) for each value of y. D: The sensitivity of prices to change in output. A lower value on the vertical axis corresponds to ahigher degree of strategic complementarity.


and /gy in the Taylor rule, social welfare generally improveswith larger /p and somewhat smaller /gy regardless of whatvalues we use for xy in the loss functions.

The second question we ask is whether the optimal poli-cies in pure sticky price and sticky information models aresimilar to those found in the hybrid model. In particular, one

FIGURE 6.—WELFARE ISOLOSS MAPS


may be concerned that using pure models to design policyrules can misguide policymakers about trade-offs. Becausescales of the social loss maps vary across models, we com-pute isoloss maps for pure sticky price (PSP) and pure stickyinformation (PSI) models and normalize these maps by thecorresponding values of the loss function evaluated at theestimated Taylor rule parameters. These rescaled isolossmaps, which we call relative welfare maps, can be inter-preted as losses relative to the loss incurred when a policy-maker uses the estimated Taylor rule. We also scale the iso-loss map for the hybrid model and then divide the relative

welfare for the PSP and PSI models by the relative welfaremap for the hybrid model. In summary, we consider maps

LPSPk ð/p;/gyÞ=LPSP

k ð/p; /gyÞLHYBR

k ð/p;/gyÞ=LHYBRk ð/p; /gyÞ

" #

andLPSI

k ð/p;/gyÞ=LPSIk ð/p; /gyÞ

LHYBRk ð/p;/gyÞ=LHYBR

k ð/p; /gyÞ

" #; k ¼ 1; 2;

ð23Þ

where ð/p; /gyÞ are the estimated values of the policy reac-tion function reported in table 1.

FIGURE 6.—(CONTINUED)

The figure plots isoloss maps for two welfare functions L1 and L2 for various combinations of the policy reaction function (Taylor rule). Volatilities of the variables are computed using the parameter estimates ofthe hybrid model. The star indicates the position of the estimated Taylor rule. A, B, and C: /gy on the horizontal axis shows the long-run response of the policy instrument (interest rate) to a unit increase in the outputgrowth rate. On the vertical axis, /p shows the long-run response of the policy instrument (interest rate) to a unit increase in inflation. Other parameters in the Taylor rule (interest rate smoothing, volatility of theinterest rate shock) are held constant. D: The figures in brackets show the value of the social loss function evaluated at the estimated Taylor rule. On the horizontal axis, /PLT shows the long-run response of the policyinstrument (interest rate) to a unit increase in the deviation of the price level from its target. On the vertical axis, /p shows the long-run response of the policy instrument (interest rate) to a unit increase in inflation.Other parameters in the Taylor rule (interest rate smoothing, volatility of the interest rate shock, output growth rate response) are held constant E: /p:SPþROTþFLEX on the horizontal axis shows the long-run responseof the policy instrument (interest rate) to a unit increase in aggregate inflation in the sticky information, rule-of-thumb, and flexible price sectors. On the vertical axis, /p:SP shows the long-run response of the policyinstrument (interest rate) to a unit increase in inflation in the sticky price sector. Other parameters in the Taylor rule (interest rate smoothing, volatility of the interest rate shock, output growth rate response) are heldconstant. The shaded region shows the Taylor rule parameter combinations associated with equilibrium indeterminacy.


The resulting maps (23) show to what extent using PSPand PSI models misinforms policymakers about trade-offsrelative to the hybrid model.25 Specifically, if the ratio ofrelative welfare maps for PSP or PSI to the relative welfaremap for the hybrid model is close to 1 uniformly in (/p,/gy)space, there is no distortion in the trade-offs. If the ratio isgreater than 1 (smaller than 1) with deviations of (/p,/gy)from ð/p; /gyÞ, then using the PSP or PSI model understates(overstates) the gain in welfare. Panels B and C in figure 6show the ratio of relative welfare maps (23) for PSP andPSI models, respectively. These maps demonstrate thatusing pure models instead of the hybrid model can misleadpolicymakers about potential gains from using alternativepolicy rules. For example, a policymaker who uses the PSImodel to design policy underestimates the benefits fromstronger responses to inflation relative to gains implied bythe hybrid model because the ratio of relative welfare mapsrapidly falls as /p increases. Hence, we conclude that usingpure models can provide a distorted picture of trade-offsactually faced when price setting is heterogeneous.

Given that PSP and PSI models have different implica-tions for whether the central bank should target the pricelevel or inflation, the third question we ask is whether ourhybrid model predicts an important role for price-level tar-geting. To answer this, we augment the Taylor rule with aterm that corresponds to price-level targeting (PLT):

rt ¼ð1� q1;r � q2;rÞ/ppt þ ð1� q1;r � q2;rÞ/PLTpt

þ ð1� q1;r � q2;rÞ/gygyt þ q1;rrt�1

þ q2;rrt�2 þ er;t;

where pt is the price level linearized around p�t ¼ p0pt. Inthis exercise, we fix /gy at the estimated value, vary /p and/PLT, and plot the resulting isoloss maps in Panel D of fig-ure 6. In general, there are significant welfare gains fromhaving an element of PLT in the Taylor rule. In fact, evensmall, positive responses to deviations from the price-leveltarget dramatically reduce the volatility of the output gap,the interest rate, and inflation. At the same time, similar tothe inflation response, a more aggressive PLT responsetends to weakly increase the volatility of output growth.However, because this increase is very small, the changesin welfare are strongly dominated by declines in var(xt),var(pt) and var(it), so that PLT is generally desirable for allreasonable values of xy. Importantly, introducing PLTeliminates a region of equilibrium indeterminacy (comparewith panel A, figure 6) and therefore PLT could reduce thevolatility of macroeconomic variables in other ways.

Finally, having the central bank respond to aggregateinflation imposes the restriction that a 1% increase in infla-tion in a sector leads to an increase in the interest rate pro-

portional to that sector’s effective share of inflationdynamics, as defined in Equation (20). The fourth questionwe ask is whether there are gains to be had by respondingdifferently to inflation in each sector. For this purpose, wecompute optimal policy rules using

r1 ¼ ð1� q1;r � q2;rÞ/ðSPÞp pðSPÞ

t

þ ð1� q1;r � q2;rÞ/ðSIþROTþFLEXÞp pðSIþROTþFLEXÞ

t

þ ð1� q1;r � q2;rÞ/gygyt þ q1;rrt�1 þ q2;rrt�2 þ er;t;

where we assume that the central banker can differentiatebetween sectors that have prices fixed for some time (SP)and those that have prices changing every period (SI, ROT,and FLEX). Here, we again fix /gy at the estimated value,vary /ðSPÞ

p and /ðSIþROTþFLEXÞp , and plot the resulting isoloss

maps in panel E of figure 6.26 We find a striking result: theisoloss maps are approximately linear in /ðSPÞ

p and/ðSIþROTþFLEXÞ

p in the neighborhood of the estimatedresponse to inflation. Hence, the policymaker does not facean increasing marginal penalty for targeting only one of thesectors. In addition, we find that responding to inflationonly in the sticky price sector is generally more stabilizingthan responding to inflation only other sectors, which isconsistent with Aoki (2001) and with the notion that stickyprice firms play a disproportionally large role in governinginflation dynamics through strategic complementarity inpricing setting (see section IV C). At the same time, the pol-icymaker can generally achieve a lower level of social lossby having a less aggressive response to inflation when tar-geting inflation in all sectors rather than in just one sector.

VI. Conclusion

Empirical work has documented a striking amount of het-erogeneity in pricing practices in both the frequency atwhich firms update prices and the source of costs underly-ing firm decisionmaking processes. We present a model inwhich four commonly used representations of how firms setprices are allowed to coexist and interact using their price-setting decisions. This model nests many specifications pre-viously considered in the literature. We find that the twomost important types of price-setting behavior are describedby sticky prices and sticky information, while rule-of-thumb and flexible pricing are quantitatively unimportant.This finding suggests that sticky information firms may bemore important than previously thought.

In addition, because the dynamic cross-covariancesreveal important insights about the leadlag structure of eco-nomic relationships, we can provide intuitive explanationsfor how the hybrid model outperforms pure sticky price orsticky information models. For example, we argue that apure sticky information model tends to underpredict the

25 Alternatively, one can interpret the ratio of the relative welfare mapsas the difference-in-difference estimator for the changes in the welfarechanges when the policymaker considers alternative values of /p and /gy

in the Taylor rule.

26 Note that moving along the 45 degree line in panel E of figure 6 cor-responds to moving along the vertical line that passes the estimated Tay-lor rule parameter combination in panel A.


degree of forward-looking behavior in inflation. In contrast,previous work that emphasized the time-series representa-tion of the data could not readily provide an economic ratio-nale for why one model is preferred to others.

Heterogeneity in price setting poses important issues forpolicymakers. We demonstrate that focusing on modelswith a single price-setting mechanism can misinform cen-tral bankers about tradeoffs they face. Our simulations sug-gest that a more aggressive response to inflation, whichmay include an element of price-level targeting, could sub-stantially improve social welfare functions. At the sametime, we do not find large benefits from targeting sectorswith some particular form of price setting so that targetingaggregate inflation is a reasonable strategy.

While we focus on the possibility of important differ-ences in how firms set prices, this approach could be natu-rally extended to wage-setting decisions. Christiano,Eichenbaum, and Evans (2005), for example, argue thatsticky wages with indexation are a particularly importantelement in matching macroeconomic dynamics. Yet, aswith prices, allowing indexation cannot reproduce the factthat wages often do not change for extended periods oftime. A more natural approach could be to allow heteroge-neity in wage-setting assumptions for different sectors ofthe economy to capture the fact that some sectors havehighly flexible wages, others have sticky wages withoutindexation, and some sectors, particularly those under unioncontracts, choose time paths for future wages infrequently.Even with small sticky wage or union wage sectors, thebehavior of the flexible wage sector could be substantiallyaltered if there is strategic complementarity in wage setting.

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APPENDIX:

Technical Details on Estimation

To estimate models, we use a Markov chain Monte Carlo (MCMC)method developed in Chernozhukov and Hong (2003; henceforth CH).

We employ the Hastings-Metropolis algorithm to implement CH’s esti-mation method. Specifically our procedure to construct chains of length Ncan be summarized as follows:

Step 1: Draw Y(n), a candidate vector of parameter values for the chain’sn þ 1 state, as Y(n) ¼ C(n) þ (n), where C(n) is the current nstate of the vector of parameter values in the chain, C(n) is avector of i.i.d. shocks taken from N(0,� ), � is a diagonalmatrix.

Step 2: Take the n þ 1 state of the chain as

Wðnþ1Þ ¼ HðnÞ with probability minf1; exp½JðWðnÞÞ � JðHðnÞÞ�g;WðnÞ otherwise

�

where J(C(n)) is the value of the objective function at the currentstate of the chain and J(Y(n)) is the value of the objective functionusing the candidate vector of parameter values.

The initial Xw is calibrated to about 1% of the parameter value and thenadjusted on the fly for the first 100,000 draws to generate 0.3 acceptancerates of candidate draws, as proposed in Gelman et al. (2004).

CH show that W ¼ 1N

PNn¼1 WðnÞ is a consistent estimate of C under

standard regularity assumptions of GMM estimators. CH also provethat the covariance matrix of the estimate of C is given by T�1VXV,where X ¼ DWGWD0, W in the weighting matrix, D is the Jacobian of themoment conditions, T is sample size, G is the covariance of moment con-

ditions, and V ¼ 1N

PNn¼1 ðWðnÞ �WÞ2 ¼ varðWðnÞÞ. Note that if W is the

optimal weight matrix, the covariance matrix is given by V. Given theshort samples and highly nonlinear optimization, we employ bootstrap-based standard errors, which we find to have better coverage rates. Ourbootstrap procedure can be summarized as follows: (a) we estimate aVAR, (b) resample the residuals, (c) construct new series using theresampled residuals and estimated VAR, (d) estimate the parameters onnewly created data, (e) repeat steps b to d many times, and (f) computestandard errors based on bootstrap replications. We found in simulationsthat this procedure has superior statistical properties.

We use 500,000 draws for our baseline and robustness estimates anddrop the first 100,000 draws (‘‘burn-in’’ period). We run a series of diag-nostics to check the properties of the resulting distributions from the gen-erated chains. We find that the simulated chains converge to stationarydistributions and that simulated parameter values are consistent with goodidentification of parameters. More details are available in Coibion andGorodnichenko (2008).


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