Download - Integrating Sticky Prices and Sticky Information

INTEGRATING STICKY PRICES AND STICKY INFORMATION

Bill Dupor, Tomiyuki Kitamura, and Takayuki Tsuruga*

Abstract—Understanding the relationship between nominal and real vari-ables, most notably inflation and cyclical output, is one of the fundamentalquestions of economics. Toward this understanding, we develop a modelthat integrates sticky prices and sticky information—a dual-stickinessmodel. We find that both rigidities are present in U.S. data. We also showthat the dual-stickiness model’s closest competitor is the hybrid NewKeynesian model. For both models, current inflation depends in part onlast period’s inflation. The former model achieves this dependence endo-genously through the interaction of the two rigidities rather than throughbackward-looking behavior. U.S. data support the dual-stickiness modelover the hybrid model because lagged expectations terms appear in theformer’s inflation Euler equation. Finally, we show that it is quantitativelyimportant to distinguish between the two by simulating a dynamic equi-librium model under each of the two inflation equations.

I. Introduction

THE interaction of real activity and inflation is a corner-stone issue of macroeconomics. As with every other

major macro question, it has been placed under the lens ofrational expectations and microfounded dynamics in recentdecades. Approximately ten years ago, efforts to estimateexisting models of price rigidity intensified.1 These effortscenter on the aggregate Euler equation from a rationalexpectations sticky price model, often called the New Key-nesian Phillips curve (NKPC). A number of authors haveargued that the NKPC is empirically deficient. Fuhrer andMoore (1995) find that inflation is more persistent than themodels imply. Mankiw (2001) points out that the NKPC isinconsistent with the stylized fact of inflation inertia.Broadly speaking, experts have fallen into one of two campsin explaining the observed inflation inertia.

The first group contends that the original mechanism islargely successful once minor adjustments are made. Galıand Gertler (1999) and Galı, Gertler, and Lopez-Salido(2001) state that the so-called hybrid NKPC, which assumesthe presence of backward-looking firms, matches U.S. andEuropean data very well. Another adjustment toward im-provement of the NKPC is to introduce real rigidities toreduce the sensitivity of prices to real marginal cost.2 Whilereal rigidities are useful in obtaining estimated frequencies

of price changes consistent with micro evidence, mostempirical studies continue to find that backward-lookingfirms play an important role in accounting for the observedinflation inertia. Thus, the hybrid NKPC has been stronglysupported by the data, even though the assumption ofbackward-looking firms might be unappealing from thetheoretical viewpoint.

The second group advocates a major overhaul of theNKPC to account for inflation inertia. A few of the alterna-tives include imperfect common knowledge (Woodford,2003) and sticky information (Mankiw & Reis, 2002; Reis,2006). In Mankiw and Reis, only a fraction of firms chooseprices attentively with currently available information.Their sticky information economy replicates inflation inertiaextremely well from the theoretical viewpoint. As such, theypropose to replace sticky prices with sticky information.Unfortunately, however, recent empirical studies find thatempirical comparisons favor the sticky price model ratherthan the sticky information model.3

This paper proposes an alternative model for explaininginflation inertia. We develop a “dual-stickiness” model thatintegrates price stickiness and information stickiness. In ourmodel, each firm has two adjustment probabilities everyperiod: a chance to reset its price and an independentlydistributed chance to update its information. Among firmsthat reset their prices, a fraction of the firms choose theirnominal prices with new information, and the remainingdetermine prices with old information.

Remarkably, the dual-stickiness model’s log-linearizedinflation equation has a lagged inflation term. It endog-enously arises through the integration of the two types ofstickiness. First, price stickiness makes current inflationproportional to the average of newly set relative prices.Then information stickiness makes some of today’s price-setting firms behave similar to some of the last period’sprice-setting firms, creating dependence of the average ofnewly set relative prices on its own lag. The interaction ofthe two generates a lagged inflation term. In this sense, weargue that our dual-stickiness model provides a more plau-sible microfoundation for inflation inertia than the hybridmodel, which obtains a lagged inflation term from exog-enously assumed backward-looking firms.

The model’s log-linearized inflation equation also hascurrent and lagged expectations terms: forecasts of currentand future marginal cost based on current information andforecasts of current and future marginal cost growth andinflation based on old information. The lagged expectationsterm is empirically important in distinguishing the dual-stickiness from the hybrid model, because only the formerhas a lagged expectations term.

Received for publication October 11, 2006. Revision accepted forpublication August 22, 2008.

* Dupor: Ohio State University; Kitamura: Bank of Japan; Tsuruga:Kansai University.

We thank two anonymous referees, Paul Evans, Oleg Korenok, EijiOkano, Simon Price, Ricardo Reis, John M. Roberts, Julio Rotemberg,Mototsugu Shintani, Peter Sinclair, and seminar participants at variousinstitutions and conferences for helpful comments and discussions. Anearlier version of this paper was circulated as “Do Sticky Prices Need toBe Replaced with Sticky Information?” The views expressed in the paperare those of the authors and are not reflective of those of the Bank ofJapan.

1 The existing models at that time included Calvo (1983), Rotemberg(1982), and Taylor (1980).

2 Sbordone (2002) assumes firm-specific marginal cost under the as-sumption of inflexible capital movement. Christiano, Eichenbaum, andEvans (2005) and Tsuruga (2007) emphasize the importance of variablecapital utilization (and nominal wage rigidities) in reducing the sensitivityof prices to real marginal cost. 3 Examples include Coibion (2010), Kiley (2007), and Laforte (2007).

The Review of Economics and Statistics, August 2010, 92(3): 657–669

We take the model to U.S. data and simultaneouslyestimate the importance of price and information stickinessin a nested framework.4 We find that both rigidities arepresent in U.S. data. Hence, our empirical results contra-vene a wholesale replacement of sticky prices with stickyinformation. Instead, our results suggest that the integrationof price and information stickiness is extremely helpful inaccounting for U.S. inflation dynamics.

Our benchmark estimates are that in each quarter, 14% offirms reset prices and 42% update information. When weallow for a typical degree of strategic complementarity,these probabilities become 28% and 70%, respectively. Wealso measure the relative importance of each nominal rigid-ity and find that sticky prices are more important than stickyinformation in fitting U.S. inflation dynamics.

We then empirically compare the dual-stickiness andhybrid models. To distinguish the dual-stickiness from hy-brid model, we first focus on the correlations betweeninflation and lagged expectations, which feature the role ofsticky information. While both models account for inflationquite well in terms of goodness of fit, we find that laggedexpectations matter for inflation in a statistically significantway. Next, we present a further generalized specificationthat nests both the dual-stickiness and hybrid models. Wefind that the data support the dual-stickiness model over thehybrid model and thus argue that the latter may be mis-specified in explaining U.S. inflation dynamics. Finally,using a simple general equilibrium analysis, we show thatimpulse responses to a cost-push shock can be qualitativelydifferent between the two pricing frictions. This implies thatit is important to distinguish the two in understandingmacroeconomic dynamics.

The findings of this paper are broadly in line with thoseof recent papers by Klenow and Willis (2007) and Knotek(2006). They introduce sticky information into a state-dependent pricing model in general equilibrium and showcalibrations and estimation results emphasizing the role ofsticky information. Our time-dependent approach has anadvantage over the state-dependent approach in that we caneasily include dual-stickiness pricing into a wide class ofdynamic general equilibrium models with many state vari-ables.

An outline of the rest of the paper is as follows. SectionII describes the dual-stickiness and hybrid models. SectionsIII and IV present our empirical method and findings.Section V shows general equilibrium comparisons betweenthe two models. The final section concludes.

II. Two Pricing Problems

This section describes a firm’s problem under two differ-ent sets of frictions. After aggregation, we characterize twoinflation equations—one for each set of frictions. The dual-

stickiness model has both sticky prices and sticky informa-tion. This model also nests both the pure sticky price andpure sticky information cases. Our second set of frictions issticky prices and backward-looking firms, that is, the hybridmodel. This latter model has become a workhorse in em-pirical monetary economics.

A. The Dual-Stickiness Model

Consider a firm that is the monopolist producer and sellerof a particular good. The firm infrequently changes itsnominal price and also infrequently updates its information.With probability 1 � �, the firm may change its price;otherwise, its current period price equals its previous periodprice. With probability 1 � �, the firm updates its informa-tion set to include all current variables; otherwise, the firm’sinformation is the same as its previous period’s information.For tractability, these two random events—the opportunitiesto change a price and to update information—are uncorre-lated over time and with each other.

The economy is made up of the above types of firms witha measure of one, each producing and selling a distinctgood. Each faces the above probabilities of price changesand information updates.

We are interested in the behavior of inflation in thiseconomy. Let us define two nominal price indices. First, pt

denotes the log aggregate nominal price level in period t.Second, qt is a nominal price index for all newly set pricesin period t. We will say more about qt below.

Because a measure 1 � � of firms resets its price in eachperiod,

pt � �pt�1 � �1 � ��qt.

Or equivalently, subtracting pt from both sides and rearrang-ing yield

�t �1 � �

��qt � pt�, (1)

where �t is inflation rate. Intuitively, equation (1) states thatinflation is positive when the newly set prices are higherthan the overall price level. It also states that inflation isproportional to newly set relative prices qt � pt. Figure 1shows this proportionality diagrammatically. Note thatpt�1 � pt is the average relative prices for firms that are notallowed to change prices. Because the weighted sum of alllog relative prices must be 0 by definition, the two shadedareas in the figure must be equal.

Due to the sticky price assumption, a firm with zeroperiod old information (that is, current information) and theability to change its price chooses

ptf � �1 � ��

j�0

�

�jEt�mctjn �, (2)4 Note that our nested framework contrasts with the previous model

selection studies by Kiley (2007), Korenok (2008), and Laforte (2007).

THE REVIEW OF ECONOMICS AND STATISTICS658

where ptf is the full information optimal price and mct

n isnominal marginal cost in period t.5 Intuitively, equation (2)states that the firm sets its nominal price to the weightedaverage of current and future nominal marginal costs. Thisdecision is forward looking because of infrequent opportu-nities for price changes.

The decision of a price-resetting firm in period t with oneperiod old information is similar, except that this firm isrestricted to conditioning its optimal price on Et�1�, in-stead of Et�. Newly set prices based on older vintages ofinformation are similarly restricted.

Next, consider qt, the nominal price index for newly setprices. There will be newly set prices in period t based oninformation sets of various vintages: current, one period old,two period old, and so on. Thus, given the probability ofinformation updating 1 � �, qt is given by

qt � �1 � �� k�0

�

�kEt�k� ptf�. (3)

Thus, the formulation of the price index is identical to thesticky information model by Mankiw and Reis (2002),except that each individual price is determined in a forward-looking manner.

This equation can be rewritten as a first-order differenceequation. Using the fact that pt

f � ptf pt�1

f ,

qt � �qt�1 � �1 � �� ptf

(4)

��1 � �� k�0

�

�kEt�k�1�ptf�.

The intuition behind this structure is that some firms con-tinue to hold the same information between periods due toinformation stickiness, and so a similarity arises in thenewly set prices between two periods. To explore theintuition further, suppose that prices were initially stabilizedat 0 and that a positive shock occurs at period 0. The leftdiagram of figure 2 depicts a hypothetical path of qt as a

thick line. In period t � 1, some firms are inattentive to theshock. They set prices to 0 since they do not know that theshock occurred. In this sense, they stick to the initial state,and this stickiness is depicted in the diagram as an arrowmoving from qt�1 to q�1. In the next period t, some firmsremain inattentive to the shock (with a probability). They settheir prices to 0 by sticking to the initial state, as the arrowmoving from qt to q�1 indicates. As a result of the commonstickiness to the initial state, persistence of qt arises endo-genously, as the dotted arrow in the diagram shows.

The persistence of qt is carried over to its relative levelqt � pt. Using an identity pt � �pt�1 ��t (1 ��) pt, we can express qt � pt as a first-order differenceequation of the form

qt � pt � ��qt�1 � pt�1� � ��t � �1 � ��ptf � pt�

(5)

��1 � �� k�0

�

�kEt�k�1�ptf�.

Note that qt � pt is more persistent as � increases.Combining equations (2) and (5) with equation (1), we

can derive

�t � �D�t�1 � �1D�1 � ��

j�0

�

�jEt�mctjn � pt�

�2D�1 � ��

k�0

�

�k (6)

�1 � ��j�0

�

�jEt�k�1�mctj � �tj�,

where �D � ��/(� � � ��), �1D � (1 � �)(1 �

�)/(� � � ��), and �2D � �(1 � �)/(� � � ��).

Also, mct is real marginal cost given by mct � mctn � pt.

In the inflation equation (6), lagged inflation appearsendogenously. As equation (1) suggests, the sticky priceassumption generates a one-to-one relationship between �t

and qt � pt. As equation (5) suggests, the sticky informa-tion assumption generates persistent dynamics of qt � pt.This newly reset relative price persistence is transformedinto inflation persistence through the one-to-one relation-ship. Therefore, the combination of price and informationstickiness endogenously generates lagged inflation in theinflation equation.

Besides the lagged inflation, there are two other terms inequation (6). The second term of the right-hand side repre-sents the present discounted value of future nominal mar-ginal costs deflated by the current price level. This termcaptures the price-setting behavior of attentive firms. Also,5 We set the discount factor to unity for simplicity.

FIGURE 1.—RELATIONSHIP BETWEEN INFLATION AND RELATIVE RESET

PRICES

γ−1

1−tp

tp

1

tq

0

nominal prices

fraction of firms

tγπ

))(1( tt pq −− γ

INTEGRATING STICKY PRICES AND STICKY INFORMATION 659

the third term contributes to inflation through lagged expec-tations on future nominal marginal cost growth. It capturesthe price-setting behavior of inattentive firms.

B. The Hybrid Model

Galı and Gertler (1999) depart from the pure sticky pricemodel by assuming the presence of two types of firms. Afraction � of firms are backward looking and use a simplerule of thumb whenever they reset prices. The remainingfirms are forward looking and set their prices according toequation (2). Then the newly reset price index qt is rede-fined as a linear combination of the price set by backward-looking firms ( pt

b) and the price set by forward-lookingfirms ( pt

f):

qt � �ptb � �1 � �� pt

f, (7)

where ptb is given by

ptb � qt�1 � �t�1. (8)

Instead of following the standard derivation of the hybridNKPC, we derive a similar equation to equation (6). Com-bining equations (7) and (8) yields

qt � �qt�1 � �1 � �� ptf � ��t�1.

Substituting equation (2) into this equation, subtracting pt

from both sides, and then using equation (1), we obtain6

�t � �H�t�1 � �1H�1 � ��

j�0

�

�jEt�mctjn � pt�, (9)

where �H � �/(� � � ��) and �1H � (1 � �)(1 � �)/

(� � � ��).

The lagged inflation appears again in the inflation equa-tion (9) of the hybrid model. This appearance stems fromthe assumption that some price setters follow a backward-looking rule of thumb. The right panel of figure 2 diagram-matically depicts the dependence of qt on its own lag withan arrow directly moving from qt to qt�1.

We argue that the dual-stickiness model has more plau-sible microfoundations for explaining inflation inertia thanthe hybrid model. Under the dual-stickiness model, laggedinflation endogenously arises from the presence of inatten-tive firms that charge suboptimal price Et�kpt

f in a forward-looking manner. In contrast, under the hybrid model, laggedinflation arises from the presence of backward-lookingfirms. Another advantage of the dual-stickiness model is thatwhile the hybrid model nests only the pure sticky pricemodel (� � 0), the dual-stickiness model nests the puresticky information model (� � 0) as well as the pure stickyprice model (� � 0). In the following sections, we provideevidence that our dual-stickiness model is not only theoret-ically but also empirically more plausible than the hybridmodel.

III. Empirical Implementation

We estimate equations (6) and (9) using the two-stepapproach proposed by Sbordone (2002), Woodford (2001),and Rudd and Whelan (2005). In the first step, we run avector autoregression (VAR) to obtain the predicted seriesof a real marginal cost measure and inflation. Given theVAR process, the second step minimizes the variance of adistance between the model’s and actual inflation. Ourestimated parameters are the probability of no price change�, the probability of no information update �, and thefraction of backward-looking firms �. Furthermore, we canestimate the pure sticky price and pure sticky informationmodels by putting restrictions on equation (6): � � 0 for theformer and � � 0 for the latter. For example, if � � 0, then

6 Above, we use the fact that qt�1 � pt � (qt�1 � pt�1) � �t ��/(1 � �)�t�1 � �t.

FIGURE 2.—HYPOTHETICAL PATHS OF NEWLY SET PRICES qt UNDER THE TWO PRICING MODELS

1−tq

Sticky information

t1−t1−

tq

time

nominalprices

t1−t1−

tq

Backward-looking rule of thumb

1−tq

time

nominalprices

A B


equation (6) reduces to the sticky information Phillipscurve:

�t �1 � �

�mct � �1 � ��

k�0

�

�kEt�k�1�mct � �t�. (10)

Such a generalization allows us to compare the dual-stickiness model to alternative pricing models based on thestatistical significance of structural parameters.

Our two-step approach differs slightly from the previousstudies in that we do not estimate closed-form solutionsto the aggregate pricing Euler equations.7 We use the non-closed-form equations (6) and (9) to estimate structuralparameters.8 This is because it is generally impossible toderive a closed-form solution to the dual-stickiness model dueto infinitely many lagged expectations terms in equation (6).9

The details of our estimation procedure are as follows.First, we specify the forecasting model by introducing thevector Xt in the following VAR:

Xt � AXt�1 � εt. (11)

The vector Xt should include at least a real marginal costmeasure and inflation for estimation. The vector Xt mayinclude lags of variables. In general, for VAR( p) process, Xt

is given by a (3p 1) vector of [ x�t, x�t�1, . . . , x�t�p1]�,where xt is a vector of a set of variables in period t, and thefirst two elements of xt are mct and �t.

Next, we calculate a series of theoretical inflation giventhe forecasting process (11). Ordinary least squares pro-duces a consistent estimate of the coefficient matrix A. Letemc and e� denote the selection vectors with 3p elements.All elements are 0 except the first element of emc and thesecond element of e�, which are unity. Given the defini-tions, we express real marginal cost and inflation as e�mcXt

and e��Xt, respectively.For expositional purposes, consider a special case with

� � 0 (that is, equation [10]). Given the definitions ofselection vectors, Et�k�1(mct �t) � (e�mc( A � I) e��A) AkXt�k�1. Then equation (10) can be written as

�tm��, A� �

1 � �

�e�mcXt � �1 � ��e�mc�A � I�

(12)

� e��A) �k�0

�

�kAkXt�k�1,

where �tm(�, A) denotes the inflation predicted by the model

and � denotes the unknown parameter vector. In this par-ticular case, � � �. By introducing an arbitrary largetruncation value of K, we approximate this equation by

�tm��, A� �

1 � �

�e�mcXt � �1 � ��e�mc� A � I�

(13)

e��A) �k�0

K�1

�kAkXt�k�1.

When the model explains the data well, �tm(�, A) is close to

actual inflation. Using a consistent estimate A, we choosethe parameter � by

� � Argmin�

var ��t � �tm��, A��. (14)

We use the same procedure to estimate equations (6) and(9). Given the VAR process, the series of {Xt�k}k�0

� sufficesto express all discounted sums in equations (6) and (9). WebAppendix A shows that equation (6) can be expressed as10

�tm��, A� � �D�t�1 � �1

Db�Xt � �2Dc� �

k�0

�

�kAkXt�k�1, (15)

where b� � [(1 � �)e�mc �e��A][I � �A]�1 and c� �(1 � �)(1 � �)[e�mc( A � I) e��A][I � �A]�1.Similarly, equation (9) can be rewritten as

�tm��, A� � �H�t�1 � �1

Hb�Xt. (16)

The parameter vector here is � � [�, �]� for equation (15)and � � [�, �]� for equation (16). Once again, we choose anarbitrary truncation parameter K and minimize the varianceof the distance between the model’s and actual inflation.

To make statistical inferences, we use a bootstrap methodbecause the forecasted variables in the second step aregenerated regressors, and thus the usual asymptotic standarderrors are incorrect. A bootstrap method is more useful formaking statistical inferences than corrected asymptotic stan-dard errors because of the complicated estimation equation(15).

To conduct the bootstrap, we first generate 9,999 boot-strapped series of X*i,t from the empirical distribution of theresidual εt and the coefficient estimate A in equation (11).

7 Sbordone (2002) transforms the standard NKPC into a closed-formsolution for the logarithm of the price-unit labor cost ratio, taking nominalmarginal cost growth as given. Woodford (2001) and Rudd and Whelan(2005) rewrite the NKPC as a forward-looking solution for inflation andestimate parameters by minimizing a distance between the models’ andactual inflation.

8 Our estimation equation is not a closed-form solution for �t. Note thatEt(mctj

n � pt) in equation (6) can be written as Et(mctj �tj �tj�1 . . . �t1). Because these terms include the expectations offuture inflation, equations (6) and (9) are “nonclosed.”

9 While it is straightforward to derive a closed form under � � 0,transforming the pure sticky price model alone to a closed form changesthe estimation equation under � � 0 to a form incomparable to thedual-stickiness model. As such, we also use a nonclosed form to estimatethe pure sticky price model. Otherwise, comparisons between the dual-stickiness model and its alternatives could be unfair.

10 This Web appendix is available online at http://www.mitpressjournals.org/doi/suppl/10.1162/REST_a_00017.


Using the resampled X*i,t, we estimate structural parameters�i by minimizing the variance of �*i,t � �*i,tm(�i, A) for i �1, 2, . . . , 9,999, where �*i,t is the inflation from theresampled unconstrained VAR and �*i,tm(�i, A) is the re-sampled inflation predicted by the model. We compute theconfidence intervals of � from the bootstrapped distributionof �i.11

In our benchmark estimation, we use quarterly U.S. databetween 1960:Q1 and 2007:Q2. Inflation is measured as thelog difference of the NFB price deflator. Labor share, whichis a proxy for real marginal cost, is the log of (NFB unitlabor cost/NFB price deflator). In our estimation, the fore-casting power of the VAR is important in measuring theeffect of expectations on inflation. Rudd and Whelan (2005)find that the output gap has strong forecasting power forlabor share and inflation and that its inclusion into the VARhas a non-negligible effect on empirical performance of theNKPC. For this reason, we also include quadratically de-trended real GDP as an output gap measure yt. The VARcoefficient A is reported in table A1 in the Web appendix fora VAR(3), where we choose the VAR with three lags

according to the BIC. The table also reports that the inclu-sion of output gap is helpful in forecasting labor share andinflation. Finally, we choose a truncation parameter of K �12.

IV. Findings

Our two key findings are that both types of stickiness arepresent in the data and that the data favor the dual-stickinessmodel over the hybrid model.

A. The Dual-Stickiness Model and Its Alternatives

Table 1 reports the estimates from four models: dual-stickiness (Dual), hybrid, pure sticky price (SP), and puresticky information (SI) models. The 95% confidence inter-vals appear in brackets. For convenience, we restate thedual-stickiness and hybrid model equations in table 1A. Theother two models are special cases of the dual-stickinessmodel.

Are Both Types of Stickiness Present? First, both prob-abilities (� and �) differ from 0 significantly under thedual-stickiness model (row 1 of table 1B). Thus, both typesof stickiness matter in accounting for aggregate U.S. infla-tion. The 95% confidence intervals for � and � imply that9% to 19% of firms change prices every quarter, but 19% to

11 MacKinnon (2002) gives detailed explanations for the bias-correctedbootstrap intervals. To obtain 95% confidence intervals of estimates, wecompute the bias-corrected bootstrap interval [2� � ��* � 1.96s*�, 2� ��* 1.96s*�], where �, ��*, and s*� denote the original estimate from theactual data, the sample mean of the bootstrap estimates �*i, and thestandard deviation of �*i, respectively.

TABLE 1.—ESTIMATES OF THE FOUR INFLATION EQUATIONS

A. Models

Lag � Attentive Firms Inattentive Firms

Dual �t � �D�t�1 �1D�1 � ��

j�0

�


�2D�1 � ��

k�0

�

�k�1 � �� j�0

�

�j

� Et�k�1�mctj � �tj�

Hybrid �t � �H�t�1 �1H�1 � ��

j�0

�


B. Structural Parameters, Model Fit

� � � �R2 Var (�t � �t)

Dual 0.859 0.581 0.00 0.757 0.114[0.808, 0.910] [0.404, 0.814] –

Hybrid 0.875 0.00 0.516 0.757 0.114[0.837, 0.911] – [0.359, 0.725]

SP 0.882 0.00 0.00 0.608 0.184[0.842, 0.920] – –

SI 0.00 0.896 0.00 0.556 0.208– [0.857, 0.931] –

C. Reduced-Form Parameters

�D or �H �1D or �1

H �2D

Dual 0.531 0.063 0.087[0.389, 0.724] [0.027, 0.092] [0.039, 0.141]

Hybrid 0.550 0.065 0.00[0.392, 0.760] [0.028, 0.095] –

Note: Estimation is from 1960:Q1 to 2007:Q2. SP and SI stand for the pure sticky price and pure sticky information models, respectively. A VAR(3) using mct, �t, and yt is estimated over 1957:Q2–2007:Q2as the first-step VAR estimation. Panel A contains estimated equations for the dual-stickiness and hybrid models. Parameters �, �, and � denote the probability of price fixity and information fixity, and the fractionof backward-looking firms, respectively. The �R2 refers to the uncentered adjusted R2. The 95% bootstrap confidence intervals are in brackets. Definitions of �D, �H, �1

D, �1H, and �2

D are in the main text.


60% of these firms use the latest information to determineprices. Evaluated at the point estimates, the former is 14%and the latter is 42%, suggesting that only 5.9% in theeconomy choose the full information optimal price.12

Our estimate of the probability of information updating as42% under dual stickiness is somewhat high relative to ourSI model’s estimate and those in previous studies (whereindividual prices are completely flexible). The probability ofinformation updating ranges between 7% and 14% underpure sticky information (row 4 of table 1B). In comparisonwith ours, Khan and Zhu (2006) estimate the probability inthe range of 12% and 35%, using VAR forecasts. Usingdifferent estimation strategies from ours, Andres, Lopez-Salido, and Nelson (2005) and Korenok (2008) find theprobability is 15% and 30%, respectively.13

Given the statistical significance of price stickiness, ourfindings contravene the wholesale replacement of stickyprices with sticky information. However, it does not implythat information stickiness should be ignored. The estimateof the corresponding structural parameter � is statisticallyand economically significant.

Can We Distinguish between the Dual-Stickiness andHybrid Models? Next, we compare the dual-stickinessand hybrid models.14 The estimation equation of the hybridmodel (9) has the same terms as the dual-stickiness model(6), labeled “lag �” and “attentive firms” in table 1A.However, the dual-stickiness model has an additional term,labeled “inattentive firms.” Therefore, whether the laggedexpectations of nominal marginal cost growth are correlatedwith inflation distinguishes the dual-stickiness from hybridmodel. The statistical significance of the correspondingparameter �2

D is the decisive factor in distinguishing the two.We find that �2

D is significantly different from 0 in table1C. The point estimate is 0.087, and the lower bound on the95% confidence interval is 0.039. Later we estimate thedual-stickiness model using an alternative detrendingmethod, different marginal cost measures, and subsamples.For all but two specifications, the unique feature of that

model, lagged expectations term, enters the inflation equa-tion in a statistically significant way.

B. Comparing Goodness of Fit

Let us assess the models’ goodnesses of fit with the(uncentered) adjusted R2.15 It is not surprising that thedual-stickiness model dominates the SP and SI models interms of goodness of fit, because the dual-stickiness modelgeneralizes the other two. Still, the magnitude of improve-ment is impressive, and it occurs because the dual-stickinessmodel has lagged inflation. In terms of comparisons be-tween the SP and SI models, the SP model is comparable toor somewhat better than the SI model, which is consistentwith Korenok (2008) and Coibion (2010). The assessmentcan be done visually by looking at the path of the models’inflation. Figures 3 to 6 plot actual inflation and inflationpredicted by the four models between 1960:Q1 and 2007:Q2. The figures demonstrate that the inflation series gener-ated by the dual-stickiness model tracks actual inflationclosely, while those generated by the SP and SI models doso only roughly. Thus, the dual-stickiness model is success-ful in fitting inflation due to the presence of lagged inflation.

Using estimates of the variance of the distance betweenthe model’s and actual inflation, we can quantify the relativeimportance of price and information stickiness. We computethe percentage reduction in the variance of the distancebetween the model’s and actual inflation when we addanother type of stickiness into either the SP or SI model.First, we can see from table 1B that the variance of the SPmodel is 0.18. If one adds sticky information to the SPmodel, it becomes the dual-stickiness model, with varianceequal to 0.11. Hence, the percentage reduction in the vari-ance is �(0.11–0.18)/0.18 � 38%. In other words, addingsticky information contributes to a 38% reduction in thevariance of residuals in the SP model. Next, a similarcalculation shows that adding price stickiness to the SImodel reduces the variance of the SI model by about 45%.Therefore, adding sticky prices beats adding sticky infor-mation in terms of the percentage reduction in the varianceof residuals.

Next, table 1B also compares the dual-stickiness andhybrid models in terms of goodness of fit. Both modelsexplain inflation almost equally well. The adjusted R2 of thedual-stickiness and the hybrid models are the same up to thethird digit. The estimate of � under the dual-stickinessmodel is roughly equal to that under the hybrid model. Moreinterestingly, whereas the structural parameters � and �have different interpretations, the estimates of � and � arequite close to each other (� � 0.58 versus � � 0.52). Whydo the two models perform equally well? Why do theestimates of the different structural parameters take similarvalues?

12 Our estimates of probabilities of price changes are smaller than thosefound in empirical micro studies, for example, Bils and Klenow (2004),Nakamura and Steinsson (2008), and Klenow and Kryvtsov (2008). Wewill briefly explore this difference with their results in the robustnesssection.

13 There are several reasons that our estimates from the sticky informa-tion Phillips curve differ from those discovered by the previous studies,especially Khan and Zhu (2006), whose estimation strategy is the closestto ours. First, we use different specifications of VARs. Second, we uselabor share rather than the output gap. Third, we use in-sample forecastsof inflation and labor share rather than out-of-sample forecasts of inflationand the output gap.

14 Our hybrid model results are in line with Galı and Gertler (1999) andGalı, Gertler, and Lopez-Salido (2005). They emphasize that the keyparameters for assessing the importance of forward- versus backward-looking behavior are �f and �b, which are functions of � and �. (Specif-ically, �f � �/(� �) and �b � �/(� �) when the discount factorequals unity.) While they conclude �f � 0.65 and �b � 0.35, ourestimates imply �f � 0.63 and �b � 0.37 under the benchmark case evenwith our different estimation strategy.

15 Since the regressors in our estimation equation do not include aconstant, we use the “uncentered” adjusted R2 instead of the standard“centered” one.


To see why, rewrite equation (4) such that qt is decom-posed into the prices set by attentive and inattentive firms:

qt � �ptb � �1 � �� pt

f,

where ptb is given by

ptb � qt�1 � �1 � ��

k�0

�

�kEt�k�1�ptf�. (17)

In a comparison of equations (8) and (17), the differencearises in the second term of the right-hand side of theequations. While the second term of equation (8) is laggedinflation, that of equation (17) is the conditional expecta-tions on the change in the full information optimal price.Because there is no reason that pt

b is equal to ptb, the

estimated parameters are in general different. However, ifthe lagged expectations term in equation (17) happens to bevery close to �t�1, the empirical results of the two modelsbecome similar.

C. Robustness

This section establishes the robustness of our two mainfindings. Specifically, both types of stickiness are present inthe data, and �2

D is significant.

In the first-step VAR, our benchmark estimation used thequadratically detrended output gap. Since an HP-filteredoutput gap has been considered as an alternative measure ofoutput gap in the literature, we also use an HP-filteredoutput gap to check robustness. Row 1 of table 2 (HPY inVAR) shows the results when the first-step VAR replaces thequadratically detrended output gap with an HP-filtered out-put gap. Comparing this with the benchmark results in table1, our results remain robust to different measures of theoutput gap on the whole.16

Next, we may also use the output gap as a proxy for realmarginal cost in the second-step estimation. Rows 2 and 3of table 2 contain results under the assumption that mct �yt.17 That said, we replace labor share with alternative

16 We also did robustness analysis to alternative specifications of theVAR such as the lag length and the inclusion of the federal funds rate orthree-month Treasury bill rate in the benchmark VAR. These robustnesschecks revealed that parameter estimates and the goodness of fit remainessentially unaltered.

17 Suppose that household instantaneous utility function takes a simpleform of log Ct � �Lt and the production function is linear Yt � Lt, whereCt, Lt, and Yt denote the aggregate consumption, labor, and output. Alsosuppose that the market clearing condition is given by Ct � Yt. Then wecan express the marginal cost as mct � yt in terms of the log deviationfrom the steady state.

FIGURE 3.—INFLATION PREDICTED BY THE DUAL-STICKINESS MODEL

1960 1970 1980 1990 2000

0

5

10

15

actu

al/ t

heor

etic

al in

flatio

n

actualpredicted

%

FIGURE 4.—INFLATION PREDICTED BY THE HYBRID MODEL

1960 1970 1980 1990 2000

0

5

10

15

actu

al/ t

heor

etic

al in

flatio

n

actualpredicted

%

FIGURE 5.—INFLATION PREDICTED BY THE PURE STICKY PRICE MODEL

1960 1970 1980 1990 2000

0

5

10

15

actu

al/ t

heor

etic

al in

flatio

n

actualpredicted

%

FIGURE 6.—INFLATION PREDICTED BY THE PURE STICKY

INFORMATION MODEL

1960 1970 1980 1990 2000

0

5

10

15

actu

al/ t

heor

etic

al in

flatio

n

actualpredicted

%


output gap measures in the second-step estimation. Again,our main findings are robust to the alternative marginal costmeasures.18

We also estimate our dual-stickiness model over differentsubsamples. Our estimation strategy implicitly assumes thatthe unrestricted forecasting process of Xt (the coefficientmatrix A) is invariant over the whole sample. This assump-tion could be questionable if a shift in policy alters thedynamic path of a macroeconomic variable in its reducedform and affects the economic agents’ forecasts. Clarida,Galı, and Gertler (2000) and Orphanides (2004) argue thatU.S. monetary policy stance shifted after 1979. As such, wesplit samples to estimate the dual-stickiness model: 1960:Q1–1979:Q2 and 1984:Q1–2007:Q2. The first subsamplereflects Paul Volcker’s appointment as Federal Reservechairman in 1979. These results are shown in rows 4 to 7 oftable 2. As the table shows, our results are, on the whole,robust to splitting samples. The parameters � and � arestatistically different from 0, and �2

D is statistically signifi-cant at least in the first subsample.

Finally, we can consider the presence of strategic comple-mentarity. The presence of firm-specific capital or decreas-ing returns to labor leads to strategic complementarity,which reduces the sensitivity of prices to the average mar-ginal cost in the economy. Galı et al. (2001) and Walsh(2003) consider a continuum of firms indexed by i facingthe production function with decreasing returns to labor,Yt(i) � Lt(i)1�a, where a � 1. In this case, the fullinformation optimal price is

ptf � �1 � ��

j�0

�

�jEt��mctj � ptj�,

where � is a function of returns to labor 1 � a and theelasticity of substitution among differentiated goods �. Inparticular, � � (1 � a)/(1 a(� � 1)). Galı, Gertler, andLopez-Salido (2003) parameterize a � 0.27 and � � 11,resulting in � � 0.197 � 1. Thus, the parameter � reducesthe sensitivity of prices to aggregate real marginal costcompared to our benchmark case due to strategic comple-mentarity in price setting.

The bottom row of table 2 gives results under strategiccomplementarity of � � 0.2. Strategic complementarityshortens our estimate of the average duration between pricechanges to approximately 10.8 months, which is broadly inline with the findings that Nakamura and Steinsson (2008)obtained from micro-price data.19 Moreover, strategiccomplementarity reduces the degree of information sticki-ness. Our estimate of � becomes 0.30, implying the reduc-tion of the average duration between information updates upto approximately 4.3 months. However, even when weallow for strategic complementarity, �2

D remains significant.Therefore, the importance of sticky information in thedual-stickiness model remains robust to introducing strate-gic pricing complementarity.

D. Dual-Stickiness Pricing versus Hybrid Pricing in aNested Model

In this section, we show further evidence that the datasupport the dual-stickiness model over hybrid model. To doso, we extend the dual-stickiness model to allow for somebackward-looking firms.

18 We also varied the value of the truncation parameter K in thesecond-step estimation. The estimates changed very little.

19 Bils and Klenow (2004) first discovered the median durations betweenU.S. micro price changes are about 4.3 months. Nakamura and Steinsson(2008) argue that removing the effect of sales and product substitutionslengthens the median durations of regular price changes up to 8 to 11months, while Klenow and Kryvtsov (2008) continue to argue for frequentregular price changes of 7.2 months.

TABLE 2.—THE DUAL-STICKINESS MODEL UNDER ALTERNATIVE SPECIFICATIONS

� � �D �1D �2

D

HPY in VAR 0.877 0.443 0.417 0.073 0.058[0.829, 0.924] [0.231, 0.661] [0.233, 0.610] [0.040, 0.112] [0.019, 0.103]

mc � QDY 0.874 0.722 0.654 0.036 0.094[0.821, 0.911] [0.580, 0.922] [0.540, 0.812] [0.018, 0.054] [0.058, 0.148]

mc � HPY 0.838 0.449 0.413 0.098 0.080[0.787, 0.872] [0.267, 0.637] [0.263, 0.570] [0.077, 0.132] [0.038, 0.130]

Benchmark1960:Q1–1979:Q2 0.821 0.424 0.388 0.115 0.085

[0.717, 0.900] [0.070, 0.689] [0.092, 0.614] [0.058, 0.208] [0.009, 0.155]1984:Q1–2007:Q2 0.924 0.494 0.475 0.040 0.039

[0.797, 1.079] [0.260, 0.897] [0.298, 0.831] [�0.016, 0.075] [�0.107, 0.176]HPY in VAR

1960:Q1–1979:Q2 0.824 0.458 0.418 0.105 0.089[0.555, 0.994] [0.100, 0.769] [0.111, 0.680] [0.005, 0.284] [0.004, 0.201]

1984:Q1–2007:Q2 0.922 0.474 0.455 0.043 0.039[0.787, 1.080] [0.238, 0.887] [0.278, 0.822] [�0.010, 0.076] [�0.115, 0.185]

Strategic complementarity (� � 0.2) 0.723 0.300 0.269 0.241 0.103[0.653, 0.794] [0.099, 0.512] [0.110, 0.461] [0.116, 0.348] [0.028, 0.187]

Note: Estimation is from 1960:Q1 to 2007:Q2 in rows 1–3 and 8. Row 1 (HPY in VAR) uses an HP-filtered output gap instead of a quadratically detrended output gap in the first-step VAR estimation. Rows 2(mc � QDY) and 3 (mc � HPY) use the quadratically detrended output gap and HP-filtered output gap as a proxy for real marginal cost in the second-step estimation, respectively. In rows 4–7, the subsample1960:Q1–1979:Q2 is used for estimation based on a VAR(3) over 1957:Q2–1979:Q2, and the subsample 1984:Q1–2007:Q2 is used for estimation based on a VAR(3) over 1981:Q2–2007:Q2. Benchmark refers tothe case where the VAR uses a quadratically detrended output gap. Row 8 shows the estimates assuming that there is strategic complementarity. Other characteristics are explained in the note to table 1.


Suppose that a fraction � of firms are backward looking,that is, they charge prices according to equation (8). Theremaining firms are forward looking but have two constantprobabilities of price and information adjustment as in thedual-stickiness model. In this case, the newly reset priceindex qt becomes

qt � �1 � �� 1 � �� k�0

�

�kEt�k� ptf�� pt

b. (18)

The new definition of qt nests all models presented in thepaper. To see this, suppose that � � 0. Then this modelbecomes the dual-stickiness model because equation (18)reduces to equation (3). Since the dual-stickiness modelnests the SP and SI models, this generalized qt nests the SPand SI models. Also, if � � 0, equation (18) reduces toequation (7), implying the hybrid model.

Web Appendix B derives the following:

�t � �1�t�1 � �2�t�2 � �1G�1 � ��

�j�0

�

�jEt�mctjn � pt� � �2

G�1 � �� (19)

�k�0

�

�k�1 � �� j�0

�

�jEt�k�1�mctj � �tj�,

where �1 � [� �� (1 � �)]/�, �2 � ��/�, �1G

� (1 � �)(1 � �)(1 � �)/�, �2G � �(1 � �)(1 � �)/�

and � � � (1 � �)[� (1 � �)�]. The resulting equationadds only one additional lag of inflation. Hence, we canapply the same empirical procedure as before.

Table 3 shows estimates of equation (19). Our prelimi-nary estimates revealed that structural parameter � takes aneconomically nonsensical negative value. To avoid such adifficulty, we estimate the model with the restrictions that

� � (0, 1), � � (0, 1), and � � (0, 1).20 Then the obtainedpoint estimates are the same up to the third digit as those ofthe dual-stickiness model in table 1. That said, the datachoose the dual-stickiness model over the hybrid modelunder the generalized framework.

Summing up our empirical findings, we conclude that thedual-stickiness model is empirically more plausible than thehybrid model in two aspects. First, the lagged expectationsof nominal marginal cost growth are significantly correlatedwith current inflation, which the hybrid model does notpredict. Second, under the generalized framework, whichnests the dual-stickiness and hybrid models, the data choosethe dual-stickiness model over the hybrid model. Thesefindings suggest that the hybrid model may be misspecifiedin accounting for U.S. inflation. Nevertheless, researchersmight want to use the simpler hybrid model for their generalequilibrium analysis because the goodness of fit of thehybrid model is as good as the dual-stickiness model and thehybrid model looks like a good approximation to the dual-stickiness model. In the next section, we show that thissimplifying approximation is treacherous for understandingmacroeconomic dynamics.

V. General Equilibrium Comparisons

This section compares the dual-stickiness and hybridmodels by placing each inflation equation in an otherwiseidentical dynamic stochastic general equilibrium (DSGE)model. We then simulate impulse responses to a cost-pushshock to understand the differences between the effects ofthe two estimated inflation equations on macroeconomicdynamics.

20 In particular, we put the following restrictions. In estimating � suchthat � � (0, 1), we estimate � defined as � � 1/(1 exp(�)). Under thisrestriction, the estimated � always generate � � (0, 1). We impose thesame restrictions on � and � to obtain economically plausible structuralestimates.

TABLE 3.—THE NESTED HYBRID–DUAL-STICKINESS MODEL

A. Structural Parameters, Model Fit

� � � �R2 Var (�t � �t)

Benchmark 0.859 0.581 0.000 0.756 0.114[0.717, 0.957] [0.023, 0.988] [0.000, 0.000]

HPY in VAR 0.877 0.443 0.000 0.778 0.104[0.741, 0.959] [0.027, 0.950] [0.000, 0.000]

B. Reduced-Form Parameters

�1 �2 �1G �2

G

Benchmark 0.531 0.000 0.063 0.087[0.023, 0.945] [0.000, 0.000] [0.001, 0.382] [0.001, 0.280]

HPY in VAR 0.417 0.000 0.073 0.058[0.027, 0.913] [0.000, 0.000] [0.002, 0.337] [0.001, 0.249]

Note: The first row in each panel shows estimates of the generalized model where the first-step VAR estimation is the same as the benchmark case. The second row in each panel shows results of the generalizedmodel where the first-step VAR estimation uses an HP-filtered output gap instead of a quadratically detrended output gap.


A. Setup

We consider the following simple log-linearized model:

it � �yyt � ��t (20)

yt � Et� yt1� � ��it � Et��t1��, (21)

where it is the nominal interest rate between period t andt 1, and yt is the output gap. Equation (20) is an interestrate rule followed by the central bank. Equation (21) is astandard consumption Euler equation, which can be derivedfrom a representative household’s optimization problem.

We use inflation equations (6) and (9), from the dual-stickiness and hybrid models, respectively, appending acost-push shock ut to each. A positive cost-push shockcauses an exogenous rise in inflation. Holding everythingother than current inflation constant, a 1% cost-push shockraises current inflation by 1%. Here, we assume real mar-ginal cost equals the output gap: mct � yt.21

Next, we let �y � 0.5/4 and �� 1.5 following Taylor(1993), and �, the intertemporal elasticity of substitution,equal 1. For the two inflation equations, we use the pointestimates from the benchmark case in table 1. The resultsare qualitatively similar for a wide range of parametervalues for �y, ��, and �.

B. Impulse Responses

Figure 7 plots each model’s impulse response of inflation,the price level and output to a 1% independent and identi-cally distributed (i.i.d.) cost-push shock that occurs at t �0. We measure responses as percentage deviations from thesteady state.

Dynamic responses of the three variables are qualitativelydifferent between the dual-stickiness and hybrid models.The inflation and output responses are less persistent underthe dual-stickiness model than the hybrid model. The pricelevel increases in the impact period and monotonicallydecreases until it reaches to a new steady-state level. Incontrast, the price level under the hybrid model monotoni-cally rises and converges to a substantially higher steady-state level.

To understand the differences, we decompose firms inboth models into three types as shown in table 4. In thedual-stickiness model, a fraction � of firms are not allowedto change prices (type I), a fraction (1 � �)(1 � �t1) offirms know the shock and set prices to pt

f (type II), and theremaining fraction (1 � �)�t1 of firms do not know theshock and set prices to the initial steady-state level of 0(type III). In the hybrid model, a fraction � of firms are notallowed to change prices (type I), a fraction (1 � �)(1 � �)of firms set prices to pt

f (type II), and the remaining (1 �

�)� set prices according to the backward-looking rule ofthumb (type III). Because the estimates of (�, �) and (�, �)for the two models are close, the causes of the differentresponses lie in the difference in type III’s price-settingbehavior and the time-varying fractions of types II and III inthe dual-stickiness model.

Now we show how dynamics of variables are explained.Let us first focus on inflation. The inflation responses are notvery different between the two models at the impact period,because type III firms set prices to 0 in both models. Underthe dual-stickiness model, they do so because they think thatthe economy remains in the initial steady state. Under thehybrid model, they do so because both the last period’snewly set price and inflation are 0. Meanwhile, facing ashock of the same size, type II firms in both models raiseprices by approximately the same amount.22 Consequently,inflation responses in this impact period are approximatelythe same.

By the i.i.d. assumption, the shock to the economydisappears afterward. It follows that the optimal price p1

f ismuch lower than p0

f, reducing type II’s reset price. In thedual-stickiness model, type III firms continue to believe thatthe economy is unchanged and to choose prices of 0. Thecomposite effect of the reduction in the optimal price fortype II and inattentive pricing for type III reduces the price

21 See note 17 for an example of set of assumptions that justifies thisequation. Note that we assume neither strategic complementarity norstrategic substitutability.

22 This is only approximate because, due to the forward-looking behav-ior, type II firms in the hybrid model set slightly higher prices, taking intoaccount the higher future inflation rates.

FIGURE 7.—IMPULSE RESPONSES TO AN i.i.d. COST-PUSH SHOCK

0 5 10 15-0.5

0

0.5

1

1.5

π t

quarter

Dual StickinessHybrid

0 5 10 150.5

1

1.5

2

2.5

pt

quarter

0 5 10 15-2

-1

0

1

yt

quarter

%

%

%

Note: The figure displays the impulse responses of inflation �t, the price level pt, and the output gapyt to an i.i.d. cost-push shock. Responses are measured in percentage deviations from the steady state.


level and changes inflation to a slightly negative level. Onthe other hand, backward-looking type III firms in thehybrid model raise prices based on the previous period’spositive inflation. It causes inflation to remain high.

The same mechanism continues to work after t � 1. TypeIII firms in the dual-stickiness model remain inattentive tothe shock and continue to set 0 price. As a result, inflationshows negative responses until it converges to 0. Type IIIfirms in the hybrid model continue to raise prices as long asa positive last-period inflation is observed, causing inflationto be persistent. Therefore, after the impact period, inflationunder the dual-stickiness model shows quantitatively andqualitatively different dynamics from the hybrid model.

Output responses are almost mirror images of the infla-tion responses. Under our calibrated interest rate rule, thecentral bank responds strongly to inflation. Therefore, thenominal interest rate moves almost proportionally to theinflation response. Then the nominal interest rate movementnegatively affects output dynamics due to the consumptionEuler equation.

Before closing this section, it should be noted that thedynamics of the two models are not always different. Forinstance, figure 8 plots the impulse responses of the twomodels where we use an AR(1) cost-push shock with acoefficient of 0.9. In this case, the impulse responses arerelatively similar between the two models. In both models,the inflation responses are fairly persistent because type IIfirms in both models have a strong incentive to raise theirprices, anticipating the cost-push shock will remain positiveover many periods.

Summing up, our simple exercises show that the twomodels can exhibit notably different dynamics in a DSGEframework. We conclude that distinguishing between thetwo models can be important for understanding macroeco-nomic dynamics.

VI. Conclusion

We developed and estimated a dual-stickiness model thatintegrates sticky prices and sticky information. Our estima-tion results show that both rigidities are present in U.S. data.The model can explain U.S. inflation well, and its goodnessof fit is as good as that of the hybrid New Keynesian model.The empirical success of the two models is due to thedependence of current inflation on last period’s inflation.While the hybrid model achieves this dependence by as-

suming backward-looking firms, the dual-stickiness modeldoes so through the interaction of the two nominal rigidities.We consider this an important theoretical appeal of thedual-stickiness model. Moreover, our empirical results sug-gest that the dual-stickiness model is more plausible thanthe hybrid model in two aspects: (1) the data support theprediction of the dual-stickiness model that current inflationcorrelates with lagged expectations on current and futurenominal marginal cost growth; and (2) the data choose thedual-stickiness over hybrid model when a generalizedmodel that nests the two is estimated. Therefore, we con-clude that the dual-stickiness model has advantages over thehybrid model both theoretically and empirically. Finally, bysimple calibration exercises, we showed that dynamic re-sponses of inflation and output under the dual-stickinessmodel could quantitatively and qualitatively differ fromthose under the hybrid model. This finding implies thatdespite the almost identical goodnesses of fit, distinguishingbetween the two models is important when one examinesthe consequences of these pricing models on macroeco-nomic dynamics.

The analysis of this paper can be extended in a number ofdirections. First, although this paper adopted a limited-information approach that is relatively less subject to modelmisspecification, it would also be interesting to estimate themodel in a full-fledged DSGE framework. Besides effi-ciency gains, this approach would allow us to explicitly

TABLE 4.—THREE TYPES OF FIRMS IN THE DUAL-STICKINESS AND

HYBRID MODELS

Type I Type II Type III

Dual-stickinessFraction: � (1 � �)(1 � �t1) (1 � �)�t1

Price is: Fixed Set to ptf Set to 0

HybridFraction: � (1 � �)(1 � �) (1 � �)�Price is: Fixed Set to pt

f Set according torule of thumb

FIGURE 8.—IMPULSE RESPONSES TO A PERSISTENT COST-PUSH SHOCK

0 5 10 152

4

6

8

10

12

π t

quarter

Dual StickinessHybrid

0 5 10 150

50

100

150

pt

quarter

0 5 10 15-40

-20

0

yt

quarter

%

%

%

Note: The figure displays the impulse responses of inflation �t, the price level pt, and the output gapyt to a persistent cost-push shock (with an AR coefficient of 0.9). Responses are measured in percentagedeviations from the steady state.


model a variety of real rigidities, which are recently con-sidered as important sources of inflation persistence. Sec-ond, we focused only on aggregate inflation. Given thatrecent studies with sector-level data have found that infla-tion dynamics exhibit significant heterogeneity across sec-tors, estimating the dual-stickiness model using disaggre-gated data may be fruitful.23 Finally, this paper did notexplore welfare implications of dual stickiness. Examiningthe normative implications, especially for optimal monetarypolicy, is also an important step for future research.

23 For example, Leith and Malley (2007) discovered sector-level differ-ences of price stickiness under the hybrid sticky-price model. Boivin,Giannoni, and Mihov (2009) argue for the importance of sector-specificshocks on sector-level inflation using factor-augmented vector autoregres-sions.

REFERENCES

Andres, Javier, David Lopez-Salido, and Edward Nelson, “Sticky PriceModels and the Natural Rate of Hypothesis,” Journal of MonetaryEconomics 52:5 (2005), 1025–1053.

Bils, Mark, and Peter Klenow, “Some Evidence on the Importance ofSticky Prices,” Journal of Political Economy 112:5 (2004), 947–985.

Boivin, Jean, Marc Giannoni, and Ilian Mihov, “Sticky Prices and Mon-etary Policy: Evidence from Disaggregated U.S. Data,” AmericanEconomic Review 99:1 (2009), 350–384.

Calvo, Guillermo, “Staggered Prices in a Utility-Maximizing Frame-work,” Journal of Monetary Economics 12:3 (1983), 383–398.

Christiano, Lawrence, Martin Eichenbaum, and Charles Evans, “NominalRigidities and the Dynamic Effect of a Shock to Monetary Policy,”Journal of Political Economy 113:1 (2005), 1–45.

Clarida, Richard, Jordi Galı, and Mark Gertler, “Monetary Policy Rulesand Macroeconomic Stability: Evidence and Some Theory,” Quar-terly Journal of Economics 115:1 (2000), 147–180.

Coibion, Oliver, “Testing the Sticky Information Phillips Curve,” thisREVIEW 92:1 (2010), 87–101.

Fuhrer, Jeff, and George Moore, “Inflation Persistence,” Quarterly Jour-nal of Economics 110:1 (1995), 127–159.

Galı, Jordi, and Mark Gertler, “Inflation Dynamics: A Structural Econo-metric Analysis,” Journal of Monetary Economics 44:2 (1999),195–222.

Galı, Jordi, Mark Gertler, and David Lopez-Salido, “European InflationDynamics,” European Economic Review 45:7 (2001), 1237–1270.

“Erratum to European Inflation Dynamics,” European EconomicReview 47:4 (2003), 759–760.

“Robustness of the Estimates of the Hybrid New KeynesianPhillips Curve,” Journal of Monetary Economics 52:6 (2005),1107–1118.

Kahn, Hashmet, and Zhenhua Zhu, “Estimates of the Sticky-InformationPhillips Curve for the United States,” Journal of Money, Credit andBanking 38:1 (2006), 195–208.

Kiley, Michael, “A Quantitative Comparison of Sticky-Price and Sticky-Information Models of Price Setting,” Journal of Money, Creditand Banking 39:S1 (2007), 101–125.

Klenow, Peter, and Oleksiy Kryvtsov, “State-Dependent or Time-Dependent Pricing: Does It Matter for Recent U.S. Inflation?”Quarterly Journal of Economics 123:3 (2008), 863–904.

Klenow, Peter, and Jonathan Willis, “Sticky Information and StickyPrices,” Journal of Monetary Economics 54:S1 (2007), 79–99.

Knotek, Edward, “A Tale of Two Rigidities: Sticky Prices in a StickyInformation Environment,” Federal Reserve Bank of Kansas Cityresearch working paper no. 06-15 (2006).

Korenok, Oleg, “Empirical Comparison of Sticky Price and Sticky Infor-mation Models,” Journal of Macroeconomics 30:3 (2008), 906–927.

Laforte, Jean-Philippe, “Pricing Models: A Bayesian DSGE Approach forthe U.S. Economy,” Journal of Money, Credit and Banking 39:S1(2007), 127–154.

Leith, Campbell, and Jim Malley, “A Sectoral Analysis of Price-SettingBehavior in U.S. Manufacturing Industries,” this REVIEW 89:2(2007), 335–342.

MacKinnon, James, “Bootstrap Inference in Econometrics,” CanadianJournal of Economics 35:4 (2002), 615–645.

Mankiw, Gregory, “The Inexorable and Mysterious Tradeoff betweenInflation and Unemployment,” Economic Journal 111 (2001),C45–C61.

Mankiw, Gregory, and Ricardo Reis, “Sticky Information versus StickyPrices: A Proposal to Replace the New Keynesian Phillips Curve,”Quarterly Journal of Economics 117:4 (2002), 1295–1328.

Nakamura, Emi, and Jon Steinsson, “Five Facts about Prices: A Reeval-uation of Menu Cost Models,” Quarterly Journal of Economics123:4 (2008), 1415–1464.

Orphanides, Athanasios, “Monetary Policy Rules, Macroeconomic Stabil-ity and Inflation: A View from the Trenches,” Journal of Money,Credit and Banking 36:2 (2004), 151–175.

Reis, Ricardo, “Inattentive Producers,” Review of Economics Studies 73:3(2006), 793–821.

Rotemberg, Julio, “Sticky Prices in the United States,” Journal of PoliticalEconomy 90:6 (1982), 1187–1211.

Rudd, Jeremy, and Karl Whelan, “Does Labor’s Share Drive Inflation?”Journal of Money, Credit and Banking 37:2 (2005), 297–312.

Sbordone, Argia, “Price and Unit Labor Costs: A New Test of PriceStickiness,” Journal of Monetary Economics 49:2 (2002), 265–292.

Taylor, John, “Aggregate Dynamics and Staggered Contracts,” Journal ofPolitical Economy 88:1 (1980), 1–23.

“Discretion versus Policy Rules in Practice,” Carnegie-RochesterConference Series on Public Policy 39 (1993), 195–214.

Tsuruga, Takayuki, “The Hump-Shaped Behavior of Inflation and aDynamic Externality,” European Economic Review 51:5 (2007),1107–1125.

Walsh, Carl, Monetary Theory and Policy, 2nd ed. (Cambridge, MA: MITPress, 2003).

Woodford, Michael, “The Taylor Rule and Optimal Monetary Policy,”American Economic Review 91:2 (2001), 232–237.

“Imperfect Common Knowledge and the Effects of MonetaryPolicy” (pp. 25–58), in P. Aghion, R. Frydman, J. E. Stiglitz, andM. Woodford (Eds.), Knowledge, Information, and Expectations inModern Macroeconomics: In Honor of Edmund S. Phelps (Prince-ton, NJ: Princeton University Press, 2003).


Download - Integrating Sticky Prices and Sticky Information

Top Related