Kyoto University,
Graduate School of Economics
Research Project Center Discussion Paper Series

How Well Do the Sticky Price Models Explain the
Disaggregated Price Responses to Aggregate Technology andMonetary Policy Shocks?
Jouchi Nakajima, Nao Sudo and Takayuki Tsuruga

Discussion Paper No. E-10-007

Research Project Center
Graduate School of Economics
Kyoto University
Yoshida-Hommachi, Sakyo-ku
Kyoto City, 606-8501, Japan

October, 2010

How well do the sticky price models explain the
disaggregated price responses to aggregate technology
and monetary policy shocks?
Jouchi Nakajima y, Nao Sudozand Takayuki Tsurugax
This version: October, 2010

Abstract
This paper documents empirically and analyzes theoretically the responses of disaggregated prices to aggregate technology and monetary policy shocks. Based on theprice data of US personal consumption expenditure, we …nd that disaggregated priceresponses have features across shocks and across sectors that are di¢ cult to explain using standard multi-sector sticky price models. In terms of shocks, a substantial fractionof disaggregated prices initially rise in response to a contractionary monetary policy
shock, while most prices fall immediately in response to an aggregate technological improvement. In terms of sectors, the disaggregated price responses are correlated weakly
with the frequency of price changes. We extend the standard model to reconcile theseobservations. We …nd that the cost channel of monetary policy and cross-sectional
heterogeneity in real rigidity could pave the way for explaining these facts.
We thank Kosuke Aoki, Jing Han, Tsutomu Watanabe, and seminar participants at the Bank of Japanand Kansai University for helpful discussions and comments. We also acknowledge Miles S. Kimball andJohn G. Fernald for supplying the updated data on total factor productivity. Takayuki Tsuruga gratefully
acknowledges the …nancial support of a Grant-in-aid for Scienti…c Research. Views expressed in this paper
are those of the authors and do not necessarily re‡ect the o¢ cial views of the Bank of Japan.
yBank of Japan and Duke University (E-mail: jouchi.nakajima@stat.duke.edu ).
z
Associate Director, Institute for Monetary and Economic Studies, Bank of Japan (E-mail:nao.sudou@boj.or.jp).
x
Associate Professor, Graduate School of Economics, Kyoto University (E-mail: tsuruga@econ.kyotou.ac.jp).

1Keywords: Sticky prices; Disaggregated in‡ation; Vector autoregressions; Monetary policy; Total factor productivity
JEL classi…cation: E31; E32; E52

1

Introduction

Recent empirical studies on structural vector autoregressions (SVAR) have shown that aggregate prices respond di¤erently, depending on the type of shock. While the di¤erencein the responses may be explained by the di¤erence in the exogenous stochastic processes
across shocks per se, some argue that the di¤erence occurs in a theoretically inconsistentway. This was pointed out originally by Altig, Christiano, Eichenbaum and Evans (2005,
ACEL). They show that aggregate in‡ation responds rapidly to an aggregate technology
shock compared with a monetary policy shock. Later, Dupor, Han, and Tsai (2009) …nd
that the estimated degree of price stickiness changes substantially when they match the theoretical impulse response functions to each shock with the data. Also, recent studies haveargued that disaggregated prices respond di¤erently across sectors. With the stylized fact ofsigni…cant cross-sectional heterogeneity in the frequency of price changes in the micro data,
recent empirical studies examine the cross-sectional relationship between the price dynamics
and the degree of nominal price rigidity.1 Along this line of research, however, little attention
has been paid to the question of how di¤erently disaggregated prices respond to aggregate
technology and monetary policy shocks and of how disaggregated price responses are relatedwith the frequency of price changes.2
In this paper, we study empirically and theoretically the disaggregated price responses
across shocks and across sectors. We estimate the responses of highly disaggregated prices
1

Examples of recent papers include: Blis and Klenow (2004), Klenow and Kryvtsov (2008), and Nakamura
and Steinsson (2008), who measured the frequency of price changes using micro-price data; and Balke andWynne (2007), Bils, Klenow, and Kryvtsov (2003), Boivin, Giannoni, and Mihov (2009), and Mackowiak,
Moench, and Wiederholt (2009), who used VAR analysis to investigate sector-speci…c impulse response
functions.
2Bils, Klenow, and Kryvtsov (2003) studied disaggregated in‡ation responses to two aggregate shocks.
Using a two-sector model consisting of a sticky price sector and a ‡exible price sector, they evaluated theirsticky price model by examining the price response of sticky price goods relative to that of ‡exible pricegoods to the two shocks. In contrast, we evaluate multi-sector models with much more disaggregated prices
by comparing simulated responses with the data.

2

of personal consumption expenditure (PCE) items to a positive shock to the aggregate
technological growth rate and to a contractionary shock to the monetary policy rule. The
estimated disaggregated price responses indicate qualitative di¤erences between shocks andlarge variations across sectors. We show that a standard multi-sector Calvo-type sticky pricemodel cannot replicate these features for disaggregated price responses. We then explore
possible explanations for the di¤erence across shocks and across sectors by extending themodel.
Using a standard SVAR, we estimate the impulse response functions of disaggregatedprices to the two types of shocks. A comparison across shocks suggests that the bulk of
disaggregated prices show a quicker decline to a positive shock to technological growth rate
than to a contractionary monetary policy shock. In particular, nearly all of the disaggregated prices fall immediately in response to technological improvement, whereas a substantial
number of disaggregated prices initially rise and then decline in response to a contractionary
monetary policy shock. On the other hand, a comparison across sectors indicates thatcross-sectional variations in nominal price rigidity play a limited role in accounting for thevariations in the disaggregated price responses. In particular, we observe statistically significant negative correlations between the disaggregated price response and frequency of pricechanges, in most periods after the shocks, but we also observe that the cross-sectional link
is weak quantitatively, regardless of the length of the period after the shocks.
We …nd that these features in the data cannot be explained well by a standard multisector Calvo-type sticky price model à la Carvalho (2006) with heterogeneous price stickinessacross sectors. In the model, disaggregated prices exhibit qualitatively symmetric responses
to both shocks. That is, they fall immediately in response to both shocks. Furthermore, thecross-sectional variation in price responses is strongly correlated with the frequency of priceadjustment.
To reconcile our …ndings with the data, we extend the multi-sector sticky price modelby introducing: (i) a cost channel of monetary policy; and (ii) heterogeneity of real rigidity
across sectors. The …rst helps generate the asymmetric responses of disaggregated prices
to the two shocks. The second helps weaken the cross-sectional correlation between the
disaggregated price response and the frequency of price adjustment.
3

When there is a cost channel of monetary policy, nominal marginal cost depends on thenominal interest rate. In response to an unexpected rise in the federal funds rate, nominal
marginal cost increases initially. Consequently, disaggregated prices rise temporarily. In
contrast, in response to a positive technology shock, nominal marginal cost decreases substantially, because the federal funds rate is lowered by the monetary policy. As a result,disaggregated prices fall more quickly in response to a positive technology shock than to amonetary policy shock.
Heterogeneity in real rigidity weakens the e¤ect of nominal price rigidity on variations in
disaggregated price responses. However, it is not necessarily the case that any real rigidities
can produce this outcome. We consider two types of real rigidity discussed by Chari, Kehoeand McGrattan (2000): (i) sector-speci…c …xed factor in the production function; and (ii)sector-speci…c kinked demand curve à la Kimball (1995). Our simulation exercise suggeststhat the former is successful in explaining the data while the latter is not. This implies that
distinguishing real rigidities is important for understanding prices.
The rest of the paper is as follows. Section 2 describes our data and econometric methodology. Section 3 presents our empirical results. Section 4 discusses the predictions of thestandard multi-sector sticky price model and its extensions and argues for the importanceof heterogeneity in real rigidity. Section 5 concludes.

2

Econometric Methodology and Data

In this section, we estimate the e¤ect of aggregate technology and monetary policy shocks on
disaggregated prices. To this end, we …rst use macroeconomic variables to identify shocks totechnological growth rate and monetary policy by SVAR similar to ACEL. We then regress
disaggregated in‡ation on its own lags and macroeconomic variables to estimate the impulse
response functions of disaggregated prices to the aggregate shocks.33This approach is essentially the same as Balke and Wynne (2007).

4

2.1

Identifying aggregate technology and monetary policy shocks

In the macro VAR, we formulate a 10-variable VAR with four lags. Let Yt be a vector ofmacroeconomic variables that includes

log (Relative price of investmentt ),log(GDPt /Hourst ),log(GDP de‡atort ), Capacity utilizationt , log(Hourst ), log(GDPt /Hourst ) - log(Real wagest ),log(Consumptiont /GDPt ), log(Investmentt /GDPt ),log(Commodity price indext ) and Fed-

eral funds ratet .4 We use the same set of macroeconomic variables as ACEL except that (i)we add the commodity price index into the system of equations; and (ii) we omit the velocity
of circulation from the system.5 The sample period is 1959:Q3–2008:Q3.
We identify aggregate technology shocks using long-run restrictions and monetary policy
shocks using short-run restrictions in the same spirit as ACEL. In particular, we identify theaggregate technology shock by assuming that only innovations to the growth of total factor
productivity (and capital embodied technology) a¤ect the long-run level of labor productivity. When imposing this long-run restriction, we use the instrumental variable method of
Shapiro and Watson (1988). Furthermore, we identify the monetary policy shock using theblock recursive restrictions of Christiano, Eichenbaum, and Evans (1999): monetary policy
shocks do not contemporaneously a¤ect the …rst nine variables in Yt .

2.2
Let

Disaggregated in‡ation equation
j;t be the quarterly change in the (log) price index for sector j. Our estimation equation

is designed to assess the dynamic e¤ects of the macroeconomic variables in Yt on

j;t =

4
X

j;` j;t ` +

`=1

4
X

Yt0 ` j;` + "j;t ;

j;t

(1)

`=0

4

While ACEL employed the price of investment used in Fisher (2006), we construct the price series ofinvestment, using the price de‡ators and weights for durables, structures, equipment and software, residential
investment, and government investment in the National Income and Product Accounts. Furthermore, we
use the commodity price index from the three-month average of the monthly CRB spot index, published by
the Commodity Research Bureau.
5
We choose this particular set of macroeconomic variables for two practical reasons. First, a number of
empirical studies have included a commodity price index in a VAR to identify the monetary policy shock,following the suggestion by Sims (1992). Second, when we include the velocity of circulation, the subsample
analysis using the period 1984:Q1–2008:Q3 reveals that the VAR system becomes explosive.

5

where "j;t is a regression error, which can be interpreted as an idiosyncratic shock. Furthermore,j;` denotes the `-th autoregressive parameter of disaggregated in‡ation, andj;`

is a (10 1) parameter vector for the macroeconomic variables Yt ` . The constant term is
suppressed for expositional purposes. In this regression, we allow for the possibility that
disaggregated in‡ation responds to shocks at the impact period. Furthermore, we assume
that the e¤ect of disaggregated in‡ation on the macroeconomic variables is negligible.6
The disaggregated price data used in our estimation are the PCE price series for 1959:Q3–
2008:Q3, which are published by the Bureau of Economic Analysis (BEA). Because our macro
VAR is based on quarterly data, we use the quarterly price series, which is the three-monthaverage of the monthly price series that the BEA releases every month. Among the 363 priceseries from the BEA’s underlying table for PCE prices, we choose highly disaggregated priceseries to the extent possible. We thus remove price indices that overlap categories as a resultof aggregation (e.g., durables, nondurables, and services).
Because we are interested in the relationship between the disaggregated price responses
and the degree of nominal price rigidity, we also drop some price series to match PCE
price series with the entry-level items (ELIs) in the Consumer Price Index. We use thefrequencies of price changes excluding sales and product substitutions measured by Nakamura
and Steinsson (2008). They measure the good-speci…c frequency of price changes from the
CPI Research Database gathered by the Bureau of Labor Statistics over 1988–1997 and
1998–2005. In matching their frequencies over 1998–2005 with the PCE price series, wetake the weighted average of frequencies based on the expenditure weights when a price in
the PCE data set corresponds to more than one ELI in their data set of frequency of pricechanges. When a price in the PCE data set does not correspond to any ELI in their dataset, we drop the price series. Using this sample selection process, we obtain 134 price series
for estimation.
6Note that the endogeneity problem arises if disaggregated prices have a nonnegligible e¤ect on themacroeconomic variables. However, the price series we use are highly disaggregated so that it would bereasonable to assume that they barely a¤ect aggregate variables.

6

3

Empirical Results

In this section, we aim to establish that aggregate technology and monetary policy shocks
have qualitatively asymmetric dynamic e¤ects on disaggregated prices and that disaggregatedprice responses have a weak relationship to heterogeneity in nominal price stickiness.

3.1

Response of disaggregated prices

Figure 1 plots the impulse response functions to aggregate technology and monetary policy
shocks. The upper left panel shows the (unweighted) mean and median responses of disaggregated prices to a one percent increase in the aggregate technology growth rate, andthe lower left panel shows those to a one percent increase in the federal funds rate. The
dashed lines represent cross-sectional variability in disaggregated price responses using the
10th–90th percentile ranges. The right panels plot the aggregated price responses to the two
shocks for comparisons.
The …gure has two notable features. First, while a shock to aggregate technology growthleads to an immediate decline in most disaggregated prices, a monetary policy shock appears
to have the delayed e¤ect on the disaggregated prices. In particular, unlike the response to theaggregate technology shock, a large number of prices are above zero after the contractionary
monetary policy shock. Second, the 10th–90th percentile ranges appear to be wide, whichsuggests considerable cross-sectional di¤erences in the disaggregated price responses. We
now look at these features in more detail.
3.1.1

How di¤erent are disaggregated prices across shocks?

Focusing on the …rst feature of the disaggregated price responses, we compute the shares ofthe positive price responses to the total number of price responses for each of the shocks. The
shares are the white bars in Figure 2, while the shares of the signi…cantly positive responses
at the …ve percent signi…cance level are shown by the shaded bars. Here, the horizontal axis
measures the quarters after each of the shocks. The upper panel is for a positive aggregate
technology shock, whereas the lower panel is for a contractionary monetary policy shock.
In response to the technology shock, about 9.7–20.2 percent are positive for the …rst
7

four quarters, evaluated at the point estimate, but none of the responses is signi…cantly
positive at the …ve percent signi…cance level. In contrast, in response to the monetary policy
shock, about 45.5–55.2 percent of the 134 price series are positive for the …rst four quarters,
while only 1.5–9.7 percent of the 134 price series are signi…cantly positive at the …ve percent
signi…cance level.
This asymmetry across shocks observed at the sector level has an implication for theasymmetry of the aggregated price responses between the two shocks, discussed in ACEL,
Dupor, Han, and Tsai (2009) and Paciello (2009a,b). In particular, the fact that mostdisaggregated prices fall at once in response to a positive aggregate technology shock and alarge number of prices increases in response to a contractionary monetary policy shock implies
that the asymmetry of aggregated price responses stems from responses at the disaggregatedlevel and is not an artifact of the weights used for aggregating prices. Thus, the asymmetry
should be explained in the multi-sector sticky price model at the disaggregated level as wellas the aggregate level.
3.1.2

How di¤erent are disaggregated prices across sectors?

We next evaluate variations in the disaggregated price responses across sectors, the second
feature of our impulse response analysis. The multi-sector sticky price model predicts thatfrequently adjusted prices should respond more quickly to any shock than infrequently adjusted prices. Motivated by this prediction, we examine to what extent the frequency of
price changes can be cross-sectionally associated with the disaggregated price responses. In
other words, we examine the correlations between the disaggregated price responses estimated from our SVAR and frequencies of regular price changes reported by Nakamura andSteinsson (2008).
Let

k
j(

) be the impulse response of the disaggregated price of sector j inquarters

after a price-reducing shock k. Using the monthly frequency of price changes f rj , we calculate the sample correlation coe¢ cients between
by Corrj

k
j(

k
j(

) and f rj across sectors, denoted); f rj , where we have 134 sectors for sector index j, and shock index kcorresponds to a positive shock to aggregate technology growth or a positive shock to thefederal funds rate. Furthermore, we consider quarters after shock8

up to 24 quarters (i.e.,

= 0; 1; 2; :::; 24). Note that the correlation should be negative because more frequently adjusted prices decrease by a larger amount in response to shock k. Thus, we examine whether
the signs of the correlation coe¢ cients are negative and see how the correlation coe¢ cients
evolve overfor each k.7

Overall, the correlation coe¢ cients are negative for both shocks, which is consistent withthe prediction of the standard sticky price model in terms of direction. The circular markers
in Figure 3 present the correlation coe¢ cients for various periods offor each shock. Herethe length of each bar attached to a circular marker represents the 95 percent con…dence
intervals of Corrjkj(); f rj . As shown in the upper panel of the …gure, the correlationcoe¢ cients take negative values over all periods of , when the shock is a positive technology
shock. Moreover, these are signi…cantly negative in all quarters except for the …rst twoquarters. In contrast, the lower panel of the …gure shows that the correlation coe¢ cients fora positive federal fund rate shock are negative in most periods although they are positive in
the …rst four quarters. These are signi…cantly di¤erent from zero only for

9:

However, the correlations are weak over the entire period after the shocks. The correlationcoe¢ cients for the technology shock range between -0.32 for= 6 and -0.13 for= 0, andthe average over the entire period is -.029. For the monetary policy shock, the correlationcoe¢ cients range between -0.31 for= 24 and 0.08 for= 2 and the average over the entire

period is -0.19.
The weak correlation might come from the fact that we do not consider broad categorizations in the PCE items such as durables, nondurables and services. To explore thispossibility, we regress the disaggregated responses for eachon f rj together with a constantand dummy variables for durables and services. The R-squareds for technology and monetary policy shocks are 15.1 and 7.1 percent, respectively, when the average is taken across .
Without dummies for durables and services, the average of the R-squareds over 24 quarters
is only 8.7 percent for the technology shock and 5.0 percent for the monetary policy shock,
implying that the increments in R-squared are marginal. Therefore, the role of the frequency
of price changes in accounting for variations in disaggregated price responses is limited, and7

As a robustness check, we also use the frequency of price changes including sales reported in Nakamura
and Steinsson (2008). We …nd that the results are unaltered qualitatively even if we change the frequency
of price changes.

9most variations remain unexplained even after consideration of broad categorizations in the
PCE items.

3.2

Robustness

This subsection conducts sensitivity analysis based on di¤erent identi…cation schemes ofaggregate shocks and subsample analysis.
3.2.1

Identi…cation schemes

Our empirical results are obtained based on an SVAR. However, some argue that the SVAR
approach is problematic because identifying assumptions may not hold. Thus, we employ themeasure of aggregate shocks obtained under alternative identi…cation schemes: the monetary
policy shock developed by Romer and Romer (2004) and the quarterly version of the puri…ed
total factor productivity series in Basu, Fernald, and Kimball (2006).8 These aggregate
shocks are convenient because the use of these “exogenous” shocks allows us to avoid thedi¢ cult task of …nding what identi…cation assumptions in the SVAR are plausible. Moreover,
we also use the factor-augmented VAR by Boivin, Giannoni, and Mihov (2009) to identify
the monetary policy shock.
To obtain the impulse responses of disaggregated prices based on the …rst two shocks,
we follow Romer and Romer’s (2004) approach. We regress disaggregated in‡ation on its
own lags and contemporaneous and lagged values of the identi…ed aggregate shock. Ourregression is given by

j;t =

8
X

k
j;` j;t ` +

`=1

where

16
X

k
k
k
j;` St ` + "j;t ;

(2)

`=0

k
j;` denotes the coe¢ cient of lagged disaggregated in‡ation,kj;` is the coe¢ cient of

the shock series and "kj;t is the error term. Again, the constant term is suppressed. The
number of lags ` is chosen according to Romer and Romer (2004). Here, Stk is either themonetary policy shock measure by Romer and Romer (2004) or the quarterly measure of
8

These quarterly data of the puri…ed total productivity series were kindly provided by Miles S. Kimball.
This data set is produced by John Fernald and roughly matches the original annual data set in Basu, Fernald,and Kimball (2006) when converted from quarterly to annual data.

10

puri…ed technology growth of Basu, Fernald and Kimball (2006). The former measure isoriginally monthly data between January 1969 and December 1996. In running the above
regression, we convert the monthly series into a quarterly series by taking the sum of thethree monthly values of the original series. This results in con…ning the sample period to theperiod over 1969:Q1 to 1996:Q4 to obtain the impulse responses of disaggregated in‡ation.
The sample period of the puri…ed total factor productivity series is from 1959:Q3 to 2008:Q3,
which is the same as that of disaggregated in‡ation.
Overall, the results are robust qualitatively to the use of the two aggregate shocks mentioned above. First, in terms of the di¤erence between shocks, the shares of the positive
responses to the total price responses for the monetary policy shock are quite similar to theresults based on the SVAR. The share of positive responses for the technology shock is somewhat larger than the baseline empirical results. For example, the share increases to 23.1–38.1
percent from the benchmark SVAR results of 9.7–20.2 percent in the fourth quarter after
the shock. However, only a few of the responses are signi…cantly positive at the …ve percent
signi…cance level.
Second, the role of the frequency of price changes in explaining variations in price responses across sectors is also limited when we use the alternative measure of aggregate shocks.
Figure 5 indicates that the correlation coe¢ cients are negative for the technology shock, but
the absolute value of the correlation coe¢ cient is again low. The correlation coe¢ cients forthe price responses to the monetary policy shock are not signi…cantly di¤erent from zero up
to three years after the shock and then turn out to be signi…cantly negative.
Lastly, we report the estimation results when the factor-augmented VAR of Boivin, Giannoni, and Mihov (2009) is used for identifying the monetary policy shock. To makecomparison easier with our previous results, we use the disaggregated price responses estimated by Boivin, Giannoni, and Mihov (2009) and select the 134 price responses out of their
191 price responses. The price data are monthly from January 1976 to June 2005. Figure 6
displays the share of positive price responses in the 134 price responses to a contractionary
monetary policy shock (the upper panel) and correlation coe¢ cients with the frequency of
price changes (the lower panel). The upper panel shows that the shares of positive priceresponses range between 16.4 and 68.7 percent for the …rst year after the shock and that11

the shares of signi…cantly positive price responses at the …ve percent signi…cance level range
between 1.5 and 14.2 percent. Again, a large number of disaggregated prices rise temporarily
after a monetary tightening shock, consistent with the results based on the SVAR. Turning
to the correlation coe¢ cients, the point estimates are negative from the impact period, but
the absolute values remain small.
3.2.2

Subsample analysis

Up to this point, our results have been based on the period from 1959:Q3. However, recent
studies have pointed out changes in the time series properties of in‡ation and the e¤ectivenessof monetary policy since the early 1980s.9 To see the e¤ect of a change in the sample period
on our results, we reestimate (1) using data from 1984:Q1 to 2008:Q3.
The di¤erence across shocks in terms of the share is less clear than the full sample
estimation as shown in Figure 7. However, when we focus on the share of signi…cantly
positive price responses, the asymmetry across shocks is still observable. The share ranges
between 5.2 and 22.4 percent for the …rst year after the monetary policy shock while almost
no prices show a signi…cantly positive response after the technology shock.
The correlation coe¢ cients shown in Figure 8 again suggest that the cross-sectional link
is weak between the disaggregated price response and the frequency of price change. Furthermore, compared with the full sample analysis, the correlation coe¢ cients display clearer
asymmetry across shocks. While the coe¢ cients under the technology shock are signi…cantly
negative, ranging between -0.54 and -0.33, the coe¢ cients under the monetary policy shock
are not signi…cantly di¤erent from zero for all :4

Multi-sector Sticky Price Models

In this section, we examine sticky price models for the two empirical features of the disaggregated price responses. We …rst study the baseline multi-sector sticky price model. In thebaseline model, …rms in the economy are identical except that the degree of price stickiness
9
See, for example, Boivin and Giannoni (2006) and Stock and Watson (2002) for evidence on the reductionin the volatility of in‡ation. Clark (2006) also documented the structural break in the disaggregate in‡ation
series in the early 1980s.

12di¤ers across sectors. The simulation exercises suggest that the baseline model needs to bemodi…ed in accounting for the two empirical features of the disaggregated prices. We thendiscuss some extensions to make the model …t the data better.

4.1

The baseline model

4.1.1

Households

Consider a continuum of households, indexed by h 2 [0; 1]: The in…nitely lived households
are monopolistic suppliers of di¤erentiated labor services and set their nominal wage rates
in a staggered manner as in Erceg, Henderson, and Levin (2000). Their preferences are over
the aggregate consumption Ct , di¤erentiated labor service Lt (h), and real money balances
Mt =Pt ; as described in the following expected utility function

max Et

1
X
t=0

where

t

"

log(Ct )Lt (h)1+
+
L
1+

m log

denotes the discount factor of households satisfying

inverse of the Frisch labor-supply elasticity, and

L

0 and

Mt
Pt

#

;

2 (0; 1),m

(3)
0 denotes the

0 are utility weights onlabor disutility and the utility of real money balances, respectively. Aggregate consumption
is a composite aggregated over N goods

Ct

N
Y

1
N
;
Cj;t

j=1

where aggregation weights are the same across sectors and Cj;t is the household’s consumptionof goods produced in sector j: The aggregate price index Pt is given by

Pt =

N
Y

1
N
;
Pj;t

j=1

where Pj;t is the disaggregated price of good j:
The budget constraint for household h isNX
j=1

Pj;t Cj;t +

Bt
+ Mt
Rt

Wt (h) Lt (h) + Bt 1 + Mt 1 +13

t + Tt :(4)

In the right-hand side of the equation, the household earns the nominal wage rate Wt (h) per
unit of labor supply Lt (h) and carries the nominal one-period bond Bt 1 and the nominal
money balances Mt 1 from the previous period to the current period. Households also receive
the total pro…ts of …rms

t and transfers Tt from the monetary authority.

In the left-handside of (4), households purchase N consumption goods and hold the nominal bond discounted
by the gross nominal interest rate on one-period bonds and cash for the next period. We
assume complete state-contingent markets and identical initial conditions for all households
so that we can drop the household index h from variables except for Wt (h) and Lt (h).
i w =( w 1)hR
1
Let Lt be the composite of di¤erentiated labor service : Lt = 0 Lt (h)( w 1)= w dh
;

where

w

> 1 is the elasticity of substitution. The demand curve for di¤erentiated labor

services Lt (h) is Lt (h) = [Wt (h)=Wt ] w Lt , where Wt denotes the aggregate wage index
i1=(1 w )hR
1:
de…ned as Wt = 0 Wt (h)1 w dh
In each period, the household can choose its nominal wage optimally with probability

1w

to maximize expected lifetime utility. When the household is allowed to reset its

nominal wage, its optimal wage rate Wt satis…es
Et
Wt =

w
w

1

P1

s

s=0 (

Et

w)

P1

s=0 (

L
s
w)

h

h

Wt
Wt+s

Wt
Wt+s

i

i

w

w

L1+
t+s
:

Lt+s

Under Calvo-type wage stickiness, the law of motion for the nominal aggregate wage index
is given by

Wt =
4.1.2

1 w
w Wt 1 + (111w ) Wt

1

w

w

:

(5)

Firms

The economy has N sectors. In each sector, there is a continuum of …rms indexed by
f 2 [0; 1], each of which produces di¤erentiated products Yj;t (f ). Let Yj;t be a composite ofdi¤erentiated goods produced in sector j; for j = 1; :::N , that is de…ned as

Yj;t =

Z 1

p
1

p

Yj;t (f )

0

14

p

p

df

1

;

(6)

wherep > 1 denotes the elasticity of substitution between di¤erentiated products in eachsector. The demand function for di¤erentiated product Yj;t (f ) is given by

Yj;t (f ) =p

Pj;t (f )
Pj;t

(7)

Yj;t :The disaggregated price index Pj;t is de…ned as Pj;t =hR

1
Pj;t (f )1
0

p

df

i1=(1

p)

:

Each …rm in sector j produces output using the following technology:

Yj;t (f ) = Zt Lj;t (f )(8)

F Zt :

Here Zt represents the aggregate technology that is common to all …rms in the economy.10
Furthermore, Lj;t (f ) is the labor demand used to produce output Yj;t (f ), and F is the…xed cost calibrated to guarantee the zero pro…ts of all …rms at the steady state. Given the
production function (8), the nominal marginal cost function M Ct is
M Ct =

Wt
;
Zt

(9)

which is common across all …rms and sectors.
In each period, …rms are allowed to reset prices optimally with the probability of 1j

under monopolistic competition in the product market and their prices remain …xed otherwise. Given the demand function (7), the optimal reset price Pj;t solves the maximization
problem:max Pj;t (f ) Et

1
X

(

s
j)

t+s Dj;t;t+s (f )ts=0s:t: Dj;t;t+s (f ) = Pj;t+s (f ) Yj;t;t+s (f )

;

(10)

Wt+s Lj;t;t+s (f ) ;(11)

Pj;t+swhere Dj;t;t+s (f ), Yj;t;t+s (f ); and Lj;t;t+s (f ) are the current period pro…ts of the …rm, theoutput, and labor demand, conditional on the optimal reset price Pj;t , respectively.

t+s isthe Lagrange multiplier associated with the household’s budget constraint (4). The optimal
10

Here we do not introduce sector-speci…c technology and …rm-speci…c technology in the production function because our interest is in the disaggregated price responses to aggregate shocks.

15

reset price Pj;t satis…es

Pj;t =

Et

p
p

1

P1

s
j)

s=0 (

Et

P1

t+s

Yj;t;t+s (f ) M Ct+s =Pj;t+s

t

:

s
j)

s=0 (

t+s

(12)

Yj;t;t+s (f ) =Pj;t+stUnder Calvo-type price stickiness, the price index of the goods j evolves according to

Pj;t =
4.1.3

h

1 p
j Pj;t 1 + (1j ) Pj;t

1

p

i1 1

p

:

(13)

Aggregate technology and monetary policy rule

We assume that the growth rate of the aggregate technology follows an AR(1) process of the
form
Zt
=
Zt 1
where ez;t is i.i.d. and

Zt 1
Zt 2

z

(14)

exp(ez;t );z 2 [0; 1).

The nominal interest rate Rt is determined by the lagged nominal interest rate and theaggregate in‡ation rate
Rt = Rt r 1
where

r

(1

Pt
Pt 1

r)

(15)

exp (er;t ) ;is the autoregressive parameter of the policy rate,> 1 is a policy weight onin‡ation and er;t is an i.i.d. monetary policy shock.
4.1.4

Equilibrium and market clearing conditions

The market clearing conditions for good j = 1; :::; N are given by

Cj;t = Yj;t =

Z 1

p
1

p

Yj;t (f )p

p

1

for j = 1; :::N:

df

(16)

0

The labor market clearing condition is

Lt =

Z 1

Lt (h)

w 1
w

dh

0

w
w 1

=

N Z 1
X
j=1

16

0

Lj;t (f ) df:(17)

The bond market clearing condition implies Bt = 0 at all dates and states. Finally, theP R1
pro…ts of …rms and transfers from the government are speci…ed as t = N
j=1 0 Dj;t (f )df
and Tt = Mt

Mt 1 , respectively.

An equilibrium of the economy is a collection of allocations and prices, fCj;t ; Yj;t ; Yj;t (f ) ;Lt (h) ; Lj;t (f ); Pj;t ; Pj;t (f ) ; Wt ; Bt g1
t=0 ; for j = 1; :::N; which satisfy the following conditions: (i) the households’allocations and wages solve the utility-maximization problem; (ii)producers’allocations and prices solve the pro…t-maximization problem; (iii) markets for thecomposite goods, composite labor, and bonds all clear; and (iv) monetary policy and pro…ts
are as speci…ed above.
4.1.5

Calibration

We calibrate the parameters based on existing studies. We set the discount factor of households

to 1.04 1=4 and the Frisch labor supply

to unity. The weight for labor disutilityLis calibrated so that the labor services supplied by households are equal to 0.3 in the steadystate. We set the weight for utility from real money balances

m to 0.05.

The elasticity of demand among di¤erentiated labor services

w is 21, which is borrowed

from Christiano, Eichenbaum, and Evans (2005). We follow the literature in setting
11. Regarding the aggregate technology shock, we set

z

p to

= 0:9, consistent with ACEL,

who estimate this parameter over the sample period between 1959:Q2 and 2001:Q4. We
parameterize the Taylor rule (15) as
wages stickinessr = 0:9 and= 1:1. We set the degree of nominal

w to 0:85, according to Barattieri, Basu, Gottschalk (2009), who estimate

the probability of nominal wage adjustment to be approximately 14–16 percent per quarter.11
To calibrate the frequency of price changes 1

j in each sector j, we use the frequencyof regular price changes reported by Nakamura and Steinsson (2008). The number of sectors
N is set to 134 for comparison purpose. Because Nakamura and Steinsson (2008) report themonthly frequency of price changes, we transform them to obtain the quarterly frequency as
follows:

j = (1

f rj )3 .12

11

They also found little heterogeneity in the probability of nominal wage adjustment across industries aswell as across occupations. This fact provides a rationale to not introduce heterogeneity in wage settings
across sectors into the model.
12
In evaluating the model, we calculate the weighted average of the monthly frequency of price changes

17

4.2

Simulation results

4.2.1

The baseline model

In this subsection, we evaluate the baseline model in replicating the features of the disaggregated price responses using the impulse response functions to a positive shock to theaggregate technology ez;t and contractionary shock to the monetary policy rule er;t .
The rectangular markers in Figure 9 show the share of positive price responses to the two
shocks. The share is zero over the entire period after the shocks, because all disaggregatedprices fall immediately after both shocks. In this sense, the disaggregated price responses are
symmetric across shocks. This symmetry comes from (9) and (12). Because both unexpected
technological improvement and contractionary monetary policy shocks lead to a decline
in the nominal marginal cost common to all sectors, there is no reason that some of thedisaggregated prices increase, resulting in a zero share of positive price responses to bothshocks.
The model predicts a negative correlation between the disaggregated price responses andthe frequency of price changes, but the cross-sectional link in the model is quantitatively
much stronger than the data suggest. The rectangular markers in Figure 10 compare thecorrelation coe¢ cients in the model with those in the data. The correlation coe¢ cientsamount to -0.75 at the impact period under both shocks, which suggests a stronger link
than the data (-0.13 for the technology shock and 0.05 for the monetary policy shock). The
correlation coe¢ cient increases to -0.43 for both shocks until six years after the shock andbecomes close to the data (-0.31 for the technology shock and -0.28 for the monetary policy
shock). However, the magnitude is inconsistent with the data because almost all correlationcoe¢ cients are lower than what the 95 percent con…dence intervals suggest. Because thenominal marginal cost is equal in all sectors in the baseline model, all cross-sectional variations in disaggregated price responses originate only from heterogeneity in nominal pricestickiness. As a consequence, the cross-sectional link is very strong.13
based on the ELIs to match the PCE item price indices. Using the weighted average of frequencies mightin‡uence our simulation results for the disaggregated price dynamics. As a sensitivity analysis, we also
simulate a model consisting of a larger number of sectors than the baseline model, where the frequency of
price changes is calibrated at the level of ELIs. The simulation results on the disaggregated price responses
to the two shocks is qualitatively unaltered.
13This result can be obtained under various parameterizations. For example, we simulate the model under18

4.2.2

Cost channel of monetary policy

We now modify the baseline model to account for the asymmetric disaggregated price responses by introducing a cost channel of monetary policy. Some existing studies have argued
that a cost channel explains the temporary increase in the aggregate prices after a monetary
tightening shock.14 A cost channel may be useful in breaking the symmetry in disaggregatedprice responses because marginal cost temporarily increases in response to a contractionary
monetary policy shock but not to a positive technology shock.
Suppose that …rms must borrow the wage bill from …nancial intermediaries in advanceat the interest rate Rt . We replace the nominal marginal cost (9) with

M Ct = Rt

Wt
:
zt

(18)

As a result, the nominal marginal cost depends on the nominal interest rate.15
Figure 9 shows that the cost channel helps generate asymmetric price responses acrossshocks. The share of positive price responses is zero over the entire period after the technology
shock (shown by the triangular markers in the upper panel). In contrast, the share of
positive disaggregated price responses ranges between 23.9 and 66.4 percent over the …rst
year after the contractionary monetary policy shock (shown in the lower panel). Although thetheoretical share is somewhat larger than the empirical share of 45.5–55.2 percent, the costchannel successfully produces asymmetric responses of disaggregated prices across shocks.
To see the intuition behind the asymmetry across shocks, note that the nominal interest
rate responds to the two shocks in opposite directions. For a positive technology growthshock, the improved technology negatively a¤ects nominal marginal cost immediately. This
direct e¤ect on marginal cost tends to dominate the indirect e¤ect of sticky nominal wagea wide range of degrees of nominal wage stickiness w or of values of the persistence of the monetary policy
rule r : Our result is also robust even when the expenditure share of 1=N for each good or elasticity of
substitution among goods p varies across sectors.
14
For instance, Christiano, Eichenbaum and Evans (2005) suggested that when a cost channel exists alongwith sticky wages, the aggregate prices can initially rise in response to a contractionary monetary policy
shock. Empirically, several papers argue for the presence of a cost channel (Barth and Ramey, 2001; Ravenna
and Walsh, 2006; Chowdhury, Ho¤mann, and Schabert, 2006; and Tillmann, 2008). One exception is Rabanal(2007), who estimated a medium-sized New Keynesian model by Bayesian estimation.
15
In addition to the replacement of the marginal cost function, we also modify the households’ budget
constraint (4) because they have deposits with the …nancial intermediaries.

19

increases coming from the increased labor demand. Hence, on average, disaggregated prices
should decrease, implying decreased aggregate in‡ation. Because of the decreased aggregate
in‡ation, the nominal interest rate is lowered according to the monetary policy rule (15),which leads to a further fall in nominal marginal cost and disaggregated prices.
In contrast, the contractionary monetary policy shock directly increases the nominalinterest rate. Thus, it temporarily increases, rather than decreases, the nominal marginal
cost because of (18) together with a gradual decrease in wages. Firms that produce goods
with a low frequency of price changes decrease their current prices because they put a large
weight on the future decreased marginal cost in determining current prices. However, …rms
producing goods with a high frequency of price changes increase their current prices based on
the temporary increase in marginal cost because they have plenty of opportunities to reset
prices in the future. For this reason, about a half of the disaggregated prices show positive
responses to a contractionary monetary policy shock.
A by-product of this mechanism is a positive relationship between the disaggregatedprice responses and the frequency of price changes for the monetary policy shock. Whenthe monetary tightening shock occurs, the responses of frequently adjusted prices should belarger than those of infrequently adjusted prices, implying a positive correlation between
responses and frequencies. Triangular markers in the lower panel of Figure 10 con…rm thisconjecture. Unfortunately, the theoretical correlation coe¢ cients are signi…cantly larger
than the empirical correlation coe¢ cients for the absolute values and lie outside the 95
percent con…dence intervals obtained from the empirical analysis. This result suggests thata further modi…cation is required to account for cross-sectional variations in disaggregatedprice responses.
4.2.3

Heterogeneous real rigidities

To weaken the correlation between price responses and the frequency of price changes, we
introduce heterogeneous real rigidities into the model with the cost channel from the view
point of strategic complementarity. Carvalho (2006) argues that strategic complementarity
matters for aggregate price dynamics using a multi-sector sticky price model. We also adopt
strategic complementarity, but assume that its degree varies across sectors, because such20

real rigidities may work as an additional factor that causes variations in disaggregated priceresponses compared with the case of no real rigidities. In what follows, we consider two
forms of real rigidities: (i) a sector-speci…c …xed factor in the production function and (ii) a
sector-speci…c kinked demand curve.
Sector-speci…c …xed factor of production We …rst discuss the role of a sector-speci…cfactor in the production function. Suppose that the production function for sector j is givenby1

Yj;t (f ) = Zt Lj;t (f ) j Hj
rather than (8), where
j

j

(19)

Fj Zt ;

j is a sector-speci…c parameter for returns to labor satisfying 0 <1. Here Hj denotes a sector-speci…c factor, and Fj is a sector-speci…c …xed cost that

ensures the long-run zero pro…t condition. Chari, Kehoe, and McGrattan (2000) assume
that goods are produced with a …xed factor in addition to labor and capital. Here we follow
Chari, Kehoe and McGrattan (2000) to interpret Hj as an inelastically supplied factor, suchas land. The sector-speci…c factor generates real rigidities because of decreasing-returnsto-scale technology, but the degrees of real rigidities are di¤erent across sectors because ofj.

By normalizing Hj to unity for all sectors for simplicity, we obtain the marginal costfunction

M Cj;t (f ) =1

Lj;t (f )1j

Because

j

Rt

Wt
:
Zt

(20)

j varies across sectors, marginal cost ‡uctuates heterogeneously.

To simulate the model with heterogeneous real rigidity, we assign

j to the production

function in each sector j by targeting the moments from the data. Because there seems
no comprehensive micro evidence onj,

we randomly draw

N
j j=1

from a distribution

and evaluate the model. Here, the moments simulated from a particular set of
depend on the combination of the observedjand the generatedj.

j criticallyHence, we evaluate

the model by taking the average of the theoretical moments obtained from 100 repeated21

simulations. Evaluating the model with the moments averaged over simulations permits us
to see, from the distribution of

j , how much heterogeneity the model requires in the degreeof real rigidities but not to see how the model should assign the value ofj to eachj.

While we leave the detail of computations to the appendix of the paper, the simulation
method is as follows. First, we draw

j from a linear function of

j =

+(1

)xj ; where xj isa random variable that follows a beta distribution with a probability density of fx (xj ;andis a lower bound for

16j.

Second, we choose

x and

x;

x)

x by minimizing the quadratic

form of the distance between the simulated and actual moments from the data. Here ourtarget moments are the averages of correlation coe¢ cients Corrjkj(); f rj over di¤erent= 0; 1; :::; 24 for a positive shock to the technology growth rate and a positive shock to thefederal funds rate. The resulting parameters imply that E( j ) = 0:90; std( j ) = 0:18, andskewness of -1.87. Thus,

j is highly diverse but is highly concentrated on a range of values

near unity. Indeed, approximately 70.5 percent of

j is more than 0.95.

The circular markers in Figure 11 show the simulated share of positive price responses
for each shock. Because of the cost channel of monetary policy, the simulated shares of priceresponses in the lower panel are positive up to one year after the monetary policy shock.
Nevertheless, the heterogeneity in real rigidities improves the …t of the model to the data. In
comparison with the lower panel of Figure 9, the shares for the …rst year after the shock shiftdownward from the range of 23.9–66.4 percent to a range of 9.0–44.8 percent, falling in therange of the data expressed by the white bars. Moreover, the upper panel of Figure 11 showsthat the share of positive price responses is no longer equal to zero for several quarters after
the positive technology shock. The simulated shares of positive price responses are 1.5–4.5
percent for the …rst year after the shock, which remains comparable to the data.
Turning to the correlation coe¢ cients, the model with heterogeneity injfares muchbetter in weakening the correlation between the disaggregated price response and the frequency of price changes than the models previously discussed. The correlation coe¢ cients
depicted by circular markers in Figure 12 are now much closer to the data for both shocks.
Almost all correlation coe¢ cients now lie inside the 95 percent con…dence intervals for each16

We set the lower value of to 1/3 in the simulations. Without this lower bound, the beta distribution
may generate j close to zero and may make the computation impossible because of the in…nitely large
steady-state value of marginal cost (See (20).)22

period and each shock.
To see the role of heterogeneity injin the production function, note that the New

Keynesian Phillips curve at the sector level is given by

j;t =

R
j Dj mcj;t

j Dj qj;t +(21)

Et j;t+1 ;where mcR
j;t denotes the log-deviation of the average sectoral real marginal cost from thesteady state, and qj;t is the log-deviation of the relative price (Pj;t =Pt ) from the steady state.
Furthermore,

j

(1

j )(1

j

)= j and Dj

j =( j + (1j ) p ).

in the nominal price stickiness a¤ects

j;t through the slope parameter

heterogeneity in real rigidities a¤ects

j;t

The heterogeneityj.

In contrast, thethrough the other slope parameter Dj and the

movements in the sector-speci…c real marginal cost mcRj;t .
This heterogeneity in real rigidities results in breaking a strong cross-sectional link between the disaggregated price responses and the frequency of price changes. Therefore,
the portion of variations in disaggregated price responses attributed to heterogeneity in thefrequency of price changes is reduced signi…cantly.
Sector-speci…c kinked demand curve The reason that the model with the above heterogeneous real rigidities can replicate the weak correlation between the disaggregated priceresponses and the frequency of price changes comes from two sources: the sector-speci…cslope coe¢ cient Dj and the sector-speci…c ‡uctuations of real marginal cost mcR
j;t . We argue
that the latter is much more important than the former.
To illustrate this, we next consider the kinked demand curve of Kimball (1995), anothertype of strategic complementarity. As the existing literature emphasizes, the kinked demand
curve is another useful device in generating real rigidities.17 However, an important di¤erence
from the sector-speci…c factor in the production function is that this type of real rigidity can
produce the heterogeneity in real rigidity from the demand function. As a result, the sectorspeci…c kinked demand curve a¤ects the slope parameter in the New Keynesian Phillips
curve but does not a¤ect the marginal cost ‡uctuations.
17

A few examples are: Chari, Kehoe, and McGrattan (2000), Dotsey and King (2005), Eichenbaum andFisher (2007) and Coenen, Levin, and Christo¤el (2007).

23

We assume that the elasticity and curvature of the demand function for di¤erentiatedgoods vary across sectors. Here, we use (18) rather than (20) to isolate the role of thesector-speci…c kinked demand curve. The aggregator of di¤erentiated products in sector j isde…ned as
Z 1

Gj

0

Yj;t (f )Yj;tdf = 1;

where Gj ( ) satis…es G0j ( ) > 0; G00j ( ) < 1, and Gj (1) = 1.18 Here, the standard CES
aggregator (6) corresponds to the special case where Gj (Yj;t (f )=Yj;t ) = (Yj;t (f )=Yj;t )( p 1)= p :Denoting the elasticity of demand for goods Yj (f ) around the steady state byof the sector-speci…c elasticity of demand
Pj (f )=Pjjjj ; the slope

j is given by@ j=@ (Pj (f )=Pj ) Yj (f )=Yj =1

Firms’ pricing behavior is in‡uenced by the parameter

j

G000
j (1)
+
G00j (1)j.

j + 1:For a higher value ofj ; the…rm’s pro…t in sector j is more likely to decline when its price increases. As a result, thepass-through of marginal cost to the price becomes moderate.19
Under the production function of (8) and the sector-speci…c kinked demand curve, the
New Keynesian Phillips curve for sector j is

j;t =

R
j Aj mct

j Aj qj;t +Et j;t+1 ;(22)

where the coe¢ cient Aj is given by

Aj

1

p
p+

j

1

:

Here, mcR
t is the log-deviation of real marginal cost that is common to all sectors. The
coe¢ cient Aj decreases withj , reducing the pass-through from marginal cost to in‡ation.

18

Here we follow Eichenbaum and Fisher’s (2007) speci…cation for the nonconstant elasticity of demand.
Note that the CES aggregator with heterogeneous elasticity of demand alone is incapable of weakening
the correlation. To see this, under linear technology, the elasticity of demand p does not enter the coe¢ cientsfor real marginal cost in the New Keynesian Phillips curve and thus it has no e¤ect on disaggregated in‡ation
dynamics.
19

24

To evaluate the model with the sector-speci…c kinked demand curve, we assign randomly
drawnN
j j=1 to each sector j.

Here, we parameterize

j randomly from the gamma dis-tribution. We again choose parameters of the gamma distribution to match the averagesof correlation coe¢ cients Corrjkj(); f rj over

= 0; 1; :::; 24 for the two shocks. The

resulting gamma distribution has E( j ) = 79 and std( j ) = 333.20
The triangular markers in Figure 11 depict the simulated results under the model withthe sector-speci…c kinked demand curve. The upper panel of the …gure shows that the model
generates no positive price response to a positive shock to the aggregate technology growth
as in the model without heterogeneous real rigidities. For the monetary tightening shockshown in the lower panel, the share of positive price responses for the …rst year after theshocks ranges between 20.8 and 54.5 percent, which is close to the 9.0–44.8 percent range
in the model with the sector-speci…c factor of production. Hence, the model of the sectorspeci…c kinked demand curve is as good as the model with a sector-speci…c …xed factor
of production in accounting for the positive disaggregated price responses to a monetary
tightening shock. In contrast, even though we freely choose the parameters for the gamma
distribution to match the target moments, the e¤ects of introducing the sector-speci…c kinked
demand curve on the weak correlation are rather limited as suggested by Figure 12. The
correlation coe¢ cients depicted by the triangular markers in Figure 12 show that most ofthem lie outside the 95 percent con…dence intervals, regardless of shock k.
To summarize, the sector-speci…c marginal cost responses are more important in generating the weak correlations than the sector-speci…c slope coe¢ cient. Our analysis suggests thatthe di¤erence between these two models with heterogeneous real rigidities stems from themarginal cost responses. Therefore, heterogeneous ‡uctuations in the marginal cost seems
to be a key to understanding heterogeneity in the disaggregated price responses to shocks.
20

The calibrated mean for our heterogeneous j is not directly comparable to the values of homogeneous
used in the one-sector models, but roughly consistent with the values in the previous studies. For example,
while Eichenbaum and Fisher (2007) and Coenen, Levin, and Christo¤el (2007) set homogeneous to 10
and 33, respectively, Kimball (1995) sets to 471 and Chari, Kehoe, and McGrattan (2000) set to 385.

25

5Concluding Remarks

In the light of growing interest in highly disaggregated price dynamics in recent research, thispaper studied the responses of disaggregated prices to aggregate technology shocks comparedwith monetary policy shocks, focusing on the di¤erence across both shocks and sectors.
Using the disaggregated price data in the US, we empirically found two features related
to the disaggregated price responses. First, a substantial number of disaggregated prices
tend to rise temporarily in response to a contractionary monetary policy shock, but such apattern is not observed in response to an aggregate technological improvement. Second, thedisaggregated price responses are weakly correlated with the frequency of price changes.
The standard multi-sector sticky price model fails to replicate these two features. Namely,
the model generates symmetric price responses across shocks and stronger correlation withthe frequency of price changes than the data. We extended the model and found that themodel with a cost channel of monetary policy and heterogeneous real rigidities in the formof a sector-speci…c …xed factor of production can explain these features. The introduction of
the cost channel allows marginal cost to depend on the nominal interest rate that respondsin the opposite direction, resulting in asymmetric responses across shocks. The heterogeneous real rigidity that stems from the sector-speci…c …xed factor of production generates alarge cross-sectional variation in the marginal cost, weakening the correlation between thefrequency of price changes and disaggregated price responses. Thus, we conclude that modeling the marginal cost structure and its ‡uctuations that change across sectors is importantfor understanding disaggregated prices.
Extending the standard multi-sector model to other dimensions is possible and could be
explored to overcome the shortcomings of the standard model. For example, heterogeneity
in …nancial market imperfection and in households’preferences, and heterogeneous beliefs ofeconomic agents about shocks might be a possible explanation. Furthermore, our …nding on
the di¤erences in the disaggregated price responses across shocks may be explained by therational inattention model in a sticky price environment.21 These lines of research would bepromising avenues for future research.
21

For a discussion of the rational inattention model, see Mackowiak and Wiederholt (2009), Paciello (2009b)
and Sims (2003).

26

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30

A

Appendix

A.1

Calibrating heterogeneous real rigidities

In this appendix, we explain how to calibrate

j in the production function andkinked demand curve. Let us consider the case of calibrating

j in thej . Our parameterization takes

several steps similar to the indirect inference. First, we draw a random variable xj 2 [0; 1]
from a beta distribution f (x;j by

j = (1)xj +x;x ) for each sector j. We randomly assignwhere

2 [0; 1] is a lower bound forj.

j to each sector

The lower bound forjis required to avoid having the in…nitely large steady-state marginal cost. The upper boundforj is unity to rule out the possibility of increasing returns to labor.

forj is [

Thus, the support; 1]. Second, we compute the simulated moments that are of interest. Because thesimulated moments depend critically on the particular combination ofdrawn value ofj,j and the randomly

we mitigate the e¤ect on the simulation results by taking S sets ofrandom parameterization. In particular, we compute the vector of the simulated moments
is a vector of calibrated parameters includingjand . Here we set S = 30. Third, we choose the parameters for the beta distributionx

ms ( ;

andx;x ) for s = 1; 2; :::; S, where

x to minimize

mdata

1X
ms ( ;
S s=1
1

x;

!0

x)

mdata

1X
ms ( ;
S s=1
1

x;

!

x)

;

where mdata is the moment from the data.
We pick the two averages of correlation coe¢ cients overas the moment from the data.

Namely, the moment from the data is
1 X Corrj ( Tj ech ( ); j )mdata =;
P
25 =0 Corrj ( M( ); j )j24

where T ech refers to a positive shock to the technology growth rate and M P refers to a
positive shock to the federal funds rate. We set

to a (22) identity matrix.

Next consider the case of the sector-speci…c kinked demand curve. The parameter
generated from the gamma distribution. Let g ( j ; k ;We choose the parameters k and) be the probability density of

to match the simulated moments ms ( ; k ;
31

j is
j.

) in thesame way as the case of the sector-speci…c …xed factor of production. Because the mean andthe standard deviation ofj have a one-to-one relationship to the parameters kreport the mean and the standard deviation in the main text.

32

and

, we

Disaggregate: TFP growth shock

0.5

Aggregate: TFP growth shock

0.5
0

0

-0.5

-0.5

-1
-1
-1.5
-1.5

-2

-2
-2.5

-2.5
0

4

8

12

16

20

-3

24

0

4

8

Quarter
Disaggregate: FF rate shock

1

12

16

20

24

Quarter
Aggregate: FF rate shock

1
0.5

0.5

0
0
-0.5
-0.5
-1
-1
-1.5

-1.5
0

4

8

12

16

20

-2

24

Quarter

0

4

8

12

16

20

24

Quarter

Figure 1: Impulse response functions of disaggregated and aggregate prices to di¤erent shocks
Notes: The upper panels plot the impulse response functions to a one percent increase in the total factor
productivity estimated from the SVAR. The lower panels plot the impulse response function to a one percent
increase in the federal funds rate (expressed as an annual rate). In the left panels, a dotted line denotes themean response of disaggregated prices (cumulative sum of in‡ation), a solid line denotes the median responseof disaggregated prices and a dashed line de…nes the 10th–90th percentile range of responses to express thevariability. A solid line in the right panels is the aggregate price response obtained from the SVAR.

33

Share of positive responses: TFP shock

0.7

Share of pos. responses at 5% sig. level
Share of pos. responses
0.6

0.5

Share

0.4

0.3

0.2

0.1

0

0

4

8

12
Quarters

16

20

24

Share of positive responses: FF rate shock

0.7

Share of pos. responses at 5% sig. level
Share of pos. responses
0.6

0.5

Share

0.4

0.3

0.2

0.1

0

0

4

8

12
Quarters

16

20

24

Figure 2: Shares of positive price responses after aggregate shocks that lower prices

Notes: The upper panel shows the share of positive disaggregated price responses to all price responses to a
one percent increase in aggregate technology growth. The lower panel is the share of positive disaggregatedprice responses to all price responses for a contractionary shock to the federal fund rate. The two shocks are
identi…ed using the SVAR. The height of the shaded bar measures the share of positive price responses thatare signi…cantly positive at the …ve percent signi…cance level.

34

Correlation coefficient

Correlation with Frequency
0
-0.1
-0.2
-0.3
-0.4
0

4

8

12

16

20

24

20

24

Correlation coefficient

Quarters after a technology shock
Correlation with Frequency
0.2
0.1
0
-0.1
-0.2
-0.3
-0.4
0

4

8

12

16

Quarters after a monetary policy shock

Figure 3: Correlation coe¢ cients between disaggregated price responses and the frequency
of price changes

Notes: The circular markers indicate the point estimate of the correlation coe¢ cients between the disaggregated price responses after a shock and the frequency of price changes measured by Nakamura and Steinsson(2008). The length of each bar indicates the 95 percent con…dence intervals of the correlation coe¢ cients.
The upper panel is the case of the responses to a one percent increase in aggregated technology growth,
while the lower panel is the case of the responses to a one percent increase in the federal funds rate.

35

Share of positive responses: TFP shock

0.7

Share of pos. responses at 5% sig. level
Share of pos. responses
0.6

0.5

Share

0.4

0.3

0.2

0.1

0

0

4

8

12
Quarters

16

20

24

Share of positive responses: FF rate shock

0.7

Share of pos. responses at 5% sig. level
Share of pos. responses
0.6

0.5

Share

0.4

0.3

0.2

0.1

0

0

4

8

12
Quarters

16

20

24

Figure 4: Shares of positive price responses to a shock to technology growth identi…ed by
Basu, Fernald, and Kimball (2006) and to a contractionary monetary policy shock identi…edby Romer and Romer (2004)

Notes: The shares are calculated based on the impulse response functions estimated by (2). The shock series
used here are the quarterly version of the puri…ed total factor productivity growth by Basu, Fernald andKimball (2006) over the period 1959:Q3–2008: Q3 and the monetary policy shock measure by Romer andRomer (2004) over the period 1969:Q1–1996:Q4. The height of the shaded bar measures the share of positive
price responses that are signi…cantly positive at the …ve percent signi…cance level.

36Correlation with Frequency

0

Correlation

-0.1
-0.2
-0.3
-0.4
-0.5
-0.6
0

4

8

12

16

20

24

20

24

Quarters after a technology shock
Correlation with Frequency
0.3

Correlation

0.2
0.1
0
-0.1
-0.2
-0.3
-0.4
0

4

8

12

16

Quarters after a monetary policy shock

Figure 5: Correlation coe¢ cients between the disaggregated price responses and the frequency of price changes based on aggregate shocks by Basu, Fernald, and Kimball (2006)
and Romer and Romer (2004)

Notes: The disaggregated price responses are estimated from (2). The upper panel is based on the shockseries identi…ed by Basu, Fernald, and Kimball (2006) and the lower panel is based on the shock series basedon Romer and Romer (2004). See the notes of Figure 3 for further details.

37

Figure 6: Shares of positive price responses to a contractionary shock to the federal fundrate and correlation coe¢ cients between the disaggregated price and the frequency of pricechanges, based on by the factor-augmented VAR in Boivin, Giannoni, and Mihov (2009)

Notes: For both panels, the impulse response functions of the disaggregated prices are obtained from theestimates of Boivin, Giannoni, and Mihov’s (2009) factor-augmented VAR. The shaded bars in the upper
panel represent the share of signi…cantly positive price responses at the …ve percent signi…cance level. The
lower panel is the correlation coe¢ cient (circular markers) between the disaggregated price responses toa monetary policy shock and the frequency of price changes. The length of the bars attached to circularmarkers in the lower panel measures the 95 percent con…dence intervals.

38

Share of positive responses: TFP shock

0.7

Share of pos. responses at 5% sig. level
Share of pos. responses
0.6

0.5

Share

0.4

0.3

0.2

0.1

0

0

5

10
Quarters

15

20

Share of positive responses: FF rate shock

0.7

Share of pos. responses at 5% sig. level
Share of pos. responses
0.6

0.5

Share

0.4

0.3

0.2

0.1

0

0

5

10
Quarters

15

20

Figure 7: Shares of positive price responses to aggregate shocks that lower prices: Based on
the subsample period from 1981:Q1 - 2008:Q3

Notes: The shares are calculated from the impulse response functions estimated over the period 1984:Q1–
2008:Q3. See the notes of Figure 2 for further detail.

39

Correlation coefficient

Correlation with Frequency

0
-0.1
-0.2
-0.3
-0.4
-0.5
-0.6
0

4

8

12

16

20

24

20

24

Correlation coefficient

Quarters after a technology shock
Correlation with Frequency
0.2
0.1
0
-0.1
-0.2
0

4

8

12

16

Quarters after a monetary policy shock

Figure 8: Correlation coe¢ cients between the disaggregated price responses and the frequency of price changes: Post 1984 samples

Notes: The correlation coe¢ cients are calculated based on the impulse response functions estimated using
the subsample periods after 1984:Q1. See the notes of Figure 3 for further detail.

40

Share of positive responses: TFP shock

0.7

0.6

0.5

Share

0.4

0.3

0.2

0.1

0

0

5

10

15

20

Quarters

Share of positive responses: FF rate shock

0.7

0.6

0.5

Share

0.4

0.3

0.2

0.1

0

0

5

10

15

20

Quarters

Figure 9: The simulated shares of positive price responses to all price responses to a one
percent increase in aggregate technology and a contractionary shock in the federal fund rate

Notes: The rectangular markers in both panels indicate the shares of positive disaggregated price responses
to all disaggregated price responses under the baseline model. The triangular markers are the corresponding
share under the model with the cost channel of monetary policy. The …gure repeats the estimation results
shown in Figure 2 for comparisons.

41Correlation coefficient
Correlation coefficient

Correlation with Frequency

1

0.5

0

-0.5

-1

0

5

10

15

20

Quarters after a technology shock
Correlation with Frequency

1

0.5

0

-0.5

-1

0

5

10

15

20

Quarters after a monetary policy shock

Figure 10: The simulated correlation coe¢ cients between the disaggregated price responses
and the frequency of price changes based on the baseline model and the model with a costchannel of monetary policy

Notes: The rectangular markers are the correlation coe¢ cients between the disaggregated price responses
and the frequency of price changes under the baseline model. The triangular markers are the correlationcoe¢ cients under the model with the cost channel of monetary policy. The …gure repeats the estimation
results shown in Figure 3 for comparisons.

42

Share of positive responses: TFP shock

0.7

0.6

0.5

Share

0.4

0.3

0.2

0.1

0

0

5

10

15

20

Quarters

Share of positive responses: FF rate shock

0.7

0.6

0.5

Share

0.4

0.3

0.2

0.1

0

0

5

10

15

20

Quarters

Figure 11: The average of the simulated shares of positive price responses to all disaggregatedprice responses to di¤erent shocks: The model with heterogeneous real rigidity

Notes: The circular markers represent the average of the simulated shares of positive disaggregated priceresponses to all disaggregated price responses to a positive technology shock and a contractionary monetary
policy shock under the sector-speci…c factor of production (19). The triangular markers are the average ofthe simulated shares under the sector-speci…c kinked demand curve. Each simulation is based on a randomly
drawn parameterization ofj or

j : The number of simulations is 100.

43

Correlation coefficient
Correlation coefficient

Correlation with Frequency

1

0.5

0

-0.5

-1

0

5

10

15

20

Quarters after a technology shock
Correlation with Frequency

1

0.5

0

-0.5

-1

0

5

10

15

20

Quarters after a monetary policy shock

Figure 12: The average of the simulated correlation coe¢ cients between the disaggregatedprice responses and the frequency of price changes: The model with the sector-speci…c factor
of production and the model with the sector-speci…c kinked demand curve

Notes: The circular markers show the averages of the simulated correlation coe¢ cients between the disaggregated price and the frequency of price changes based on the model with heterogeneous real rigidities comingfrom the sector-speci…c factor of production. The triangular markers are based on the model with heterogeneous real rigidities coming from the sector-speci…c kinked demand curve. The average of the correlationcoe¢ cients are taken over a randomized parameterization of

j or

j.

Both models incorporate the costchannel of monetary policy. The …gure repeats the estimation results shown in Figure 3 for comparison. The
number of simulations is 100.

44