ISSN (Print) 0473-453X

Discussion Paper No. 1082

ISSN (Online) 2435-0982

SECTORALINFLATION PERSISTENCE,
MARKET CONCENTRATIONAND
IMPERFECT COMMON KNOWLEDGE
Ryo Kato
Tatsushi Okuda
Takayuki Tsuruga

Revised August 2021
March 2020

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan

Sectoral inflation persistence, market concentration, andimperfect common knowledge∗
Ryo Kato†, Tatsushi Okuda‡, and Takayuki Tsuruga§
August 6, 2021

Abstract
Previous studies have stressed that inflation dynamics exhibit substantial dispersion
across sectors. Using US producer price data, we present evidence that sectoral inflation persistence is negatively correlated with market concentration, which is difficult toreconcile with the prediction of the standard model of monopolistic competition. To
better explain the data, we incorporate imperfect common knowledge into the monopolistic competition model introduced by Melitz and Ottaviano (2008). In the model,
pricing complementarity among firms increases as market concentration decreases. Because higher pricing complementarity generates greater inflation persistence, our model
successfully replicates the observed negative correlation between inflation persistence
and market concentration across sectors.
JEL Classification: E31, D40, L16
Keywords: Information rigidity, Disaggregated prices, Market structure∗

We thank Ryan Chahrour, Jeffrey Fuhrer, Mark Gertler, Yukinobu Kitamura, John McDermott, Benjamin Moll, Satoshi Nakada, Taisuke Nakata, Kiyohiko G. Nishimura, Mototsugu Shintani, Takeki Sunakawa,
Yuki Teranishi, Takashi Ui, Rosen Valchev, and Toshiaki Watanabe for their helpful comments and discussions. Kato and Tsuruga gratefully acknowledge the financial support of the Joint Usage/Research Center at the Institute of Social and Economic Research (ISER) and Kyoto Institute of Economic Research(KIER), Grants-in-Aid for Scientific Research (JP18K01684, JP19K23219, JP20H05631, JP20H05633, andJP21H04397). This paper previously circulated as “Market Concentration and Sectoral Inflation Under
Imperfect Common Knowledge.” The views expressed in this paper are those of the authors and do notnecessarily reflect the official view of the Bank of Japan.
†
Faculty of Economics, Asia University‡
Financial System and Bank Examination Department, Bank of Japan§
Institute of Social and Economic Research, Osaka University; CAMA. Correspondence to: Institute ofSocial and Economic Research, Osaka University 6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan. E-mailaddress: tsuruga@iser.osaka-u.ac.jp1

1

Introduction

Inflation has long been a cornerstone issue in monetary economics. Researchers consider
disaggregated inflation at the industry level (hereafter, sectoral inflation) as a fundamentalbuilding block for aggregate inflation dynamics. Not surprisingly, sectoral inflation dynamics
exhibit substantial heterogeneity across industries. For instance, Leith and Malley (2007) andImbs et al. (2011) estimate sector-level Phillips curves and conclude that sectoral inflation
dynamics are significantly dispersed.1 Boivin et al. (2009, hereafter BGM) and Altissimo
et al. (2009) incorporate sector-specific shocks in their models to replicate the heterogeneous dynamics of sectoral prices and inflation. Further, Bils and Klenow (2004) report anegative correlation between inflation persistence and the degree of price stickiness, while
Clark (2006) reveals little correlation between inflation persistence and price stickiness acrosssectors. Together, these previous studies suggest that substantial heterogeneity in sectoral
inflation dynamics remains an important challenge.
This paper explores factors that give rise to the substantial dispersion in sectoral inflation
persistence. We focus on the relationship between sectoral inflation persistence and market
concentration. Using US producer price index (PPI) data, we reveal that sectoral inflation
persistence, as measured by the first-order autocorrelation, exhibits a negative correlation
with market concentration. Figure 1 provides a scatter plot of the first-order autocorrelation
against the Herfindahl–Hirschman Index (HHI) across 145 US manufacturing sectors. As
shown, while the scatter plot is somewhat noisy, the regression line indeed suggests a negative
correlation.
The negative correlation observed between inflation persistence and market concentrationis robust. The correlation is estimated to be negative in various specifications with alternative
measures of inflation persistence. A negative correlation is also present in nonmanufacturingsectors. Quantitatively, benchmark regressions suggest that an increase in HHI by 2,000
1

See also Byrne et al. (2013) and Luengo-Prado et al. (2018).

2

would reduce the first-order autocorrelation of sectoral inflation by 0.10–0.14.
The canonical models of monopolistic competition under constant elasticity of substitution (CES) preferences fail to reconcile this observed negative correlation. The standard
Dixit–Stiglitz monopolistic competition model implicitly assumes that market concentration
is invariant. Notably, the quadratic preferences proposed by Melitz and Ottaviano (2008,
hereafter MO) allow the degree of market concentration to vary across sectors in the otherwise standard model of monopolistic competition.2 Here, for convenience, we refer to thissetup as the MO model. In the MO model, if the market concentration is high, firms set theirprices with positive markup as in the standard Dixit–Stiglitz model. The MO model canalso represent a perfectly competitive market as a limiting case where market concentration
is at its lowest. Nevertheless, the flexible specification of preferences in the MO model is not
sufficient: inflation persistence has no correlation with market concentration.
We find that incorporating information rigidity into the MO model can account for theobserved negative correlation. Remarkably, our model with a reasonable degree of information
rigidity predicts that the first-order autocorrelation decreases by 0.12 as HHI increases from0 to 2,000, compared with 0.10–0.14 in the data.
In the MO model, the information rigidity can be a powerful mechanism for linking
inflation persistence with market concentration. The information rigidity that we consider isimperfect common knowledge as developed by Woodford (2003) and Morris and Shin (2002).
Among others, Woodford (2003) demonstrates that higher pricing complementarity underimperfect common knowledge generates greater inflation persistence.3 In the MO model,
higher market concentration weakens the pricing complementarity. Thus, our model cansuccessfully generate less inflation persistence in more concentrated markets.4
2

Vives (1990) and Ottaviano et al. (2002) consider similar quadratic utility functions in the context oftrade theory.
3
See also Fukunaga (2007), Nimark (2008), and Angeletos and La’O (2009). Crucini et al. (2015) andCandian (2019) apply a model with imperfect common knowledge to examine the persistence of real exchange
rates.
4
Sbordone (2010) discusses the effects of variable market concentration on pricing complementarities using
Kimball (1995) preferences. In a similar spirit, our model relies on MO preferences.

3

Empirical studies have long discussed the effect of market concentration on pricing. Examples include Carlton (1986), Leith and Malley (2007), and Gopinath and Itskhoki (2011).
These studies examine US data and argue that the sensitivity of individual prices is smaller
in more concentrated markets. Koga et al. (2020) employ survey data on Japanese firms
and explore the sensitivity of individual prices to their competitor’s prices (i.e., the pricing
complementarities). Specifically, Koga et al. (2020) find that pricing complementarity is
weaker in more concentrated markets. This existing evidence supports the key ingredient of
the MO model in generating our main result.
Some theoretical studies have also argued for the importance of market structure on
the pricing of firms. Among others, MO discuss how the toughness of competition affects
markups. While their model structure is well known, our paper focuses on one aspect of
the MO model that they do not highlight. That is, pricing complementarity decreases withmarket concentration. Other theoretical studies also consider the effect of market structure
on firms’ pricing, but the motivations differ from ours. Sbordone (2010) discusses the effects
of increases in traded goods on the slope of the Phillips curve. Afrouzi (2019) argues thatpricing complementarity in an oligopolistic market enhances monetary non-neutrality. Amiti
et al. (2019) develop a model of incomplete pass-through in which endogenous markups varywith firm characteristics in an oligopolistic market.5
Our explanation of the observed negative correlation between inflation persistence andmarket concentration does not necessarily invalidate the importance of sticky prices. Angeletos and La’O (2009) discuss the interaction of sticky prices, imperfect common knowledge,
and pricing complementarity. They also show how pricing complementarities generate inflation persistence under both imperfect common knowledge and sticky prices. Our model
can be cast as a special case in the Angeletos and La’O (2009) framework, whereas we include an additional dimension such that pricing complementarity systematically depends on
5

In the empirical part of their paper, they show that the prices set by large firms are less sensitive tomarginal cost (incomplete pass-through) and more sensitive to the competitor’s prices than those for small
firms.

4

market concentration following MO. In this way, we reaffirm the importance of information
rigidity in accounting for the negative correlation between inflation persistence and market
concentration, which is difficult to explain by sticky prices per se.
The remainder of the paper is organized as follows. Section 2 discusses the empiricalevidence. Section 3 introduces the basic model with comparative statistics and numericalexamples. Section 4 extends the basic model. Section 5 discusses the Calvo sticky pricemodel in the context of our empirical findings. Section 6 concludes.

2

Evidence

2.1

Data

We use US (seasonally adjusted) PPI at the six-digit level of the North American Industry
Classification System (NAICS) code to explore the empirical characteristics of sectoral inflation persistence and market concentration. The PPI is more convenient than the consumer
price index (CPI) because we examine the linkage between sectoral inflation persistence andthe degree of market concentration of sectors. The market concentration data are from the
Economic Census. To inspect the linkage, we match the NAICS codes of the PPI with thoseof the market concentration data.
Two datasets are available for the PPI. The first dataset is from BGM.6 This dataset
includes the PPI prices of 152 manufacturing sectors at monthly frequency.7 The sample
period runs from February 1976 to June 2005. The second dataset covers the more recent
period running from January 2004 to December 2020. We include 270 manufacturing sectors
and 79 nonmanufacturing sectors in this second dataset, which we call the extended dataset.
We use these two datasets separately because the sample periods and the coverage of
6

The
data
are
availableontheAmericanEconomic
Association
website,https://www.aeaweb.org/articles?id=10.1257/aer.99.1.350
7
From 154 sectors in the original BGM dataset, we drop two sectors (311119p and 324393 in the six-digit
NAICS codes) for which market concentration data are not available.

5sectors differ between them. The BGM dataset covers the period from 1976:2 to 2005:6, andthe extended dataset is from 2004:1 to 2020:12. In addition, the coverage of sectors differs
between the two datasets. In particular, only 93 of the 152 manufacturing sectors in the BGM
dataset are included in the 270 manufacturing sectors in the extended dataset. This difference
arises because the highly disaggregated prices in the PPI data are likely to be discontinued
and replaced by new items. Thus, many sectors are unique to one of the two datasets.
Concerning the measurement of market concentration, two indicators are available. The
first is the HHI. The second is the share of the top-four largest firms in the sector, often
referred to as the C4 ratio. Given that the two PPI datasets cover different sample periods,
we use the HHI and the C4 ratio in 2002 for the BGM dataset and those in 2007 for theextended dataset. While the C4 ratio is available for all sectors, the HHI is available only for
manufacturing sectors.
Table 1 suggests that inflation persistence exhibits substantial dispersion across sectors
in both datasets. We define the first-order autocorrelation of the difference in log prices ina particular sector (i.e., sector-level aggregate price) as our measure of sectoral inflation persistence. Here, the first-order autocorrelation is estimated from the first-order autoregressive
process or AR(1). For the within-industry comparison, we focus on manufacturers, i.e., thefirst and second columns in Table 1. The standard deviations of manufacturers are 0.20 and
0.23, both of which are similarly large. The inflation persistence ranges from a maximumof 0.65 (0.71) to a minimum of -0.42 (-0.44) in the BGM dataset (the extended dataset).
Compared with manufacturers, the dispersion observed among nonmanufacturers is similarly
significant as that for manufacturers. The standard deviation for nonmanufacturers is 0.18,
which is close to that for manufacturers.
This considerable dispersion does not appear to be an artifact arising from idiosyncratic
shocks in highly disaggregated data. Each sector (at the NAICS six-digit level) includes,
on average, 375 firms in the BGM dataset and 806 (manufacturing) firms in the extended

6

dataset. This large number of firms in each sector ensures that the influence of idiosyncratic
shocks for individual firms is washed out at the sector level.

2.2

Regressions

We run regressions specifying sectoral inflation persistence ρπ,i as the dependent variable,
where i denotes the sector. The explanatory variable of interest is the indicator of market
concentration. Typically, our estimation equation is
ρπ,i = a + b

HHIi+ controlsi + ui ,
1000

(1)

where b measures how much the first-order autocorrelation ρπ,i changes if the HHI of one
sector is 1,000 points higher.
Table 2 provides the benchmark regression results. Since the HHI is available only in
manufacturers, the results here are for manufacturers. Each panel reports the coefficient on
the HHI in (1) for the cases with and without sector dummies, D31,i and D32,i .8 We report
the results based only on the BGM dataset in the left panel and those based only on theextended dataset in the middle panel. In the right panel, we pool the two datasets and report
the estimated coefficient on HHI along with the coefficient on the dummy variable indicating
the BGM dataset (DBGM,i ).
The estimated coefficients on HHI range between -0.07 and -0.05, depending on thedatasets and the presence of the sector dummies. All the estimated coefficients on HHI are
negative and statistically significant at the conventional significance levels. The coefficient on
DBGM,i is estimated to be positive and is statistically different from zero. This significantly
positive estimate suggests that sectoral inflation tends to be more persistent during the earlier(1976:2–2005:6) than the later (2004:1–2020:12) period.
8

The sector dummies D31,i and D32,i control for some manufacturing sectors at the two-digit level of the
NAICS code. The NAICS codes for manufacturers start with 31, 32, or 33. Of these, D31,i controls for sectors
with a NAICS code starting with 31 (e.g., food and textile industries) whereas D32,i controls for those witha NAICS code starting with 32 (e.g., paper, wood, and chemical industries).

7

Table 3 provides the results where we change the explanatory variable in (1) from HHIi /1000
to C4i . The results are qualitatively similar to those in Table 2. That is, the coefficient on
the market concentration indicator is negative and statistically significant.
In Table 4, we use the data for nonmanufacturers in the extended dataset and compare
the regression results with those for manufacturers. Comparisons of specifications (1) and (2)suggest that the negative correlation between inflation persistence and market concentrationis quantitatively similar to each other. While the coefficient on C4i for manufacturers is 0.16, that for nonmanufacturers is -0.20. We also run a regression pooling both manufacturers
and nonmanufacturers. As shown in specification (3) in the same table, the sector dummy
controlling for manufacturers (D3,i ) is economically and statistically significant. This implies
that while average inflation persistence differs between manufacturers and nonmanufacturers,
market concentration matters for both.
Quantitatively, the estimated coefficients in Table 2 suggest that if the HHI of one sector
is 1,000 points higher, the inflation persistence of that sector is lower by 0.05–0.07. The
estimation results can be reconfirmed from the slope of the regression line in Figure 1, in
which the BGM dataset is used for the regression (i.e., specification (1-1)). A cursory lookat the regression line suggests that inflation persistence decreases from around 0.2 to slightly
below 0.1 against a range of HHI from 0 to over 2,000.

2.3

Robustness

This subsection examines whether the above result is robust to changes in specification.
Table 5 summarizes the results. In the following robustness analysis, we take HHIi /1000 as
an explanatory variable in regressions.9
Measures of inflation persistence The first robustness check replaces the dependent
variable in (1) with alternative measures of inflation persistence. One alternative persistence
9

Table A.1 in the not-for-publication appendix complements the exercises in Table 5 by replacing
HHIi /1000 with C4i .

8measure is the sum of the autoregressive coefficients (SAR) that allows for a general AR process.10 Another alternative measure is the first-order autocorrelation of year-on-year inflation.
We take a year-on-year difference of seasonally unadjusted log prices to check robustness tothe methods of removing the seasonality. Specifications (1) and (2) in Table 5 describe theregression results under these alternative measures of inflation persistence. Our main finding
remains unchanged.

Controlling for heterogeneity in sectors In the benchmark regression, only two sector
dummies D31,i and D32,i deal with heterogeneity in inflation persistence across sectors. We
thus elaborate sector dummies using the more detailed three-digit level. The results arereported in specification (3) in Table 5. The estimated coefficients on HHI remain broadly
unchanged compared with the benchmark regressions in Table 2.

Dropping possible outliers We also run a regression excluding the influence of outliers in
the data. We drop sectors with the top-10% HHI from the sample. As shown in specification(4) in Table 5, this subsample analysis also detects a significant negative correlation similar
to the main results.

3

The basic modelThis section introduces the basic model, which describes a firm’s pricing behavior withina particular sector. Unless otherwise noted, any variable without the index j represents
that variable at the sector level. In contrast, variables indexed by j denote the variables ofindividual firms. In parallel with the previous section, the first-order autocorrelation is thepersistence measure in our model.
10

BGM and Furher (2010) employ the SAR as a measure of inflation persistence. In our analysis, we set the
maximum lag length to 13 as in BGM and select the lag length based on the Bayesian Information Criterion.
The detailed regression results using the SAR can also be found in Section A of the not-for-publication
appendix.

9

In the basic model, all firms are subject to a sector-level common shock to marginal costthat is assumed to be i.i.d. The i.i.d. assumption means that there is no inherited inflation
persistence in the basic model, which helps crystallize the role of an endogenous mechanismthat generates persistent inflation dynamics. Another benefit of inspecting the basic modelis that it provides a simple analytical solution for the sectoral inflation in equilibrium. Laterin Section 4, we extend the model by incorporating both permanent and temporary shocks
and numerically examine whether the property of the basic model remains unchanged.

3.1

Preferences and technology

The consumer maximizes utility by choosing a quantity of consumption of indexed goods,
qt (j) for j ∈ [0, N ] and the numeraire qA . Following MO, we assume that her preferences arequadratic,Zβ
Ut = αqt (j)dj −2
j∈[0,N ]Zγ(qt (j)) dj −2
j∈[0,N ]2

2

Z
qt (j)dj+ qA .

(2)j∈[0,N ]We refer to α(> 0) as the demand shifter because a larger α shifts out the demand for
differentiated varieties.11 As γ(> 0) increases, the total demand for the differentiated goods
compared with the numeraire decreases, amplifying the degree of competition among firms.
The parameter β(≥ 0) indicates the degree of love of variety. If β = 0, the differentiatedvarieties are perfect substitutes and, hence, the consumer is interested only in the total
R
consumption of the differentiated goods ( j∈[0,N ] qt (j)dj). As β increases, the consumer cares
more about the consumption distribution across varieties.
The consumer’s budget constraint is given by,Z
qt (j)pt (j)dj + qA ≤ q A ,(3)j∈[0,N ]where pt (j) is the price of each good and q A is the endowment of the numeraire which is11

We assume that α is constant for simplicity. If a time-varying α = αt follows the same stochastic process
as that of ct , our main results remain unchanged.

10exogenously provided at the same amount in each period. Note that this inequality always
binds because of the monotonicity of the utility function Ut .
As shown in Appendix A, we can derive the following linear demand function:αhpt (j)
pt−+,
1 + βhββ (1 + βh)

qt (j) =(4)

where the average price is given by,
1
pt ≡
N

Z
pt (j)dj,
j∈[0,N ]and h ≡ (γN )−1 ∈ [0, ∞]. If we assume that γ = 10−4 and all firms are identical in size, hcan be interpreted as the HHI.12 We maintain these assumptions for the rest of the analysis.
We treat h as the key indicator representing the degree of market concentration. A notablecharacteristic of (4) is that the sensitivity of the demand for good j with respect to average
price pt varies with h. This variable sensitivity is in sharp contrast to CES preferences, where
demand elasticity is assumed to be constant regardless of market concentration.
We turn to the firms’ decision-making. There is a continuum of firms, indexed by j ∈[0, N ]. Every firm operates under monopolistic competition, and the population of firms
implies a mass of product varieties. Given marginal cost ct , firm j solves the following
maximization problem:
max Ej,t [(pt (j) − ct )qt (j)],{pt (j)}(5)subject to (4). Here ct is common for all firms in the sector. In (5) and hereafter, Ej,t [·] ≡E[·|Ht (j)] denotes the expectations operator of firm j conditional on information set Ht (j)available in period t, which we specify in the next section.
12

If N is interpreted as the number of firms in a sector, the share of each firm is given by (100/N )%.
2
Accordingly, the HHI is calculated as (100/N ) × N = 104 /N .

11

3.2

Shocks and information structure

In the basic model, ct follows a stochastic process given by,
ct = ct−1 + εt ,(6)where εt is drawn from a Gaussian white noise process N (0, σ 2 ).13 Early studies assume thatfirms do not precisely observe their marginal cost. For example, Maćkowiak and Wiederholt(2009) argue that firms pay only limited attention to common shocks. Following Woodford(2003), our model incorporates private signals regarding firms’ costs. Specifically, firms cannot
observe {cs }ts=1 in our model. Instead, firm j receives private signal xt (j) given by,
xt (j) = ct + δt (j),(7)

where δt (j) is drawn from N (0, τ 2 ), and recall that ct is common across firms. With this
information structure, firm j’s information set Ht (j) includes c0 , {xs (j)}ts=0 and all theparameter values, including h. In a special case of perfect information where τ = 0, there
is no noise in the signals and, therefore, Ht (j) includes the entire history of firms’ marginal
costs, {cs }ts=0 together with the parameter values.

3.3

Equilibrium

3.3.1

Equilibrium under perfect information

We first derive the equilibrium prices under perfect information. Plugging (4) into the firm’s
profit function (5) and then taking derivatives with respect to pt (j) leaves the best-response
function for a firm to choose its best price as follows.
pt (j) =
1
Ej,t [ct ]− r α + rEj,t [pt ] +
,
2
2

13

(8)

In our model, we implicitly assume that c0 is sufficiently large. This effectively allows us to ignore thepossibility that ct < 0.

12

where r represents the degree of pricing complementarity, defined as,r(h) ≡

1.
2 (1 + βh)(9)Note that r(h) is decreasing in h. The intuition is that when the market is more concentrated,firms’ products are more differentiated and less substitutable. As substitution becomes moredifficult across products, the price of a firm’s product becomes less sensitive to its competitors’
prices. Thus, pricing complementarity becomes weaker. This dependence of r on h is notpresent in the Dixit–Stiglitz models under CES preferences.
In the case of τ = 0, both Ej,t [pt ] = pt (j) = pt and Ej,t [ct ] = ct hold for any j because
there is no heterogeneity across firms. Therefore, (8) can easily be solved for the equilibrium
individual price pt (j) and the average price pt such that,pt (j) = pt = κα + (1 − κ)ct ,(10)where κ(h) ≡ β/ (2β + h−1 ). Further, we define the quasi-inflation rate as ∆pt ≡ pt − pt−1 .
This is now written as,∆pt = (1 − κ)εt .

(11)Note that sectoral inflation without private signals follows a white noise process.14
3.3.2

Equilibrium under imperfect common knowledge

We turn to the case under imperfect common knowledge, that is, τ > 0 in (7). Here, we
define average and higher-order expectations denoted recursively as,
1
Et [·] ≡ Et [·] ≡
14

1
N

Z
Ej,t [·]dj,j∈[0,N ]Hereafter, we refer to ∆pt simply as inflation, instead of quasi -inflation.

13

and Et

k+1

h k i≡ Et Et [·] for every k ≥ 1. With these notations, the average of pt (j) in (8) over

j is now expressed as,

pt =


1
1 1
1
− r α + rEt [pt ] + Et [ct ],
2
2

(12)

1

where Et [pt ] is no longer equal to pt in general because of the private signals. Plugging pt
1

recursively into Et [pt ] in (12) leaves the equilibrium price expressed as,
pt = κα + (1 − κ)(1 − r)

∞
X

rk Et

k+1

[ct ].

(13)k=0Note that prices adjust more slowly as r increases. Intuitively, when pricing complementarity
is stronger, firms’ pricing places more weight on higher-order expectations of ct . This equationis the same as that presented for general linear-quadratic problems by Morris and Shin (2002).
By simplifying (13) and taking the limit of t to infinity, we now provide the explicit
solution forms for pt in the basic model, namely, a linear combination of the past marginal
costs such that,∞Xµs ct−s .

(14)∆pt = µ∆pt−1 +(1 − κ) (1 − µ) εt ,(15)

pt = κα + (1 − κ) (1 − µ)s=0This equation implies,

where µ is given by,µ≡q(1 + λ) /λ − ((1 + λ) /λ)2 − 42,and λ(r) is increasing in r and is given by,
λ(r) ≡

τ2.
τ 2 + (1 − r) σ 2(16)

In a limiting case where r = 0, 1 − λ coincides with the Kalman gain of steady-state Kalmanfiltering.15 If private signals are imprecise (i.e., τ 2 /σ 2 is large), the Kalman gain is low. A
15

We assume that t is sufficiently large so that λ(r) can be dealt with as a time-invariant parameter.

14lower Kalman gain implies that firms place more weight on the priors in their belief updatingprocess, generating slower dynamics for price and inflation. The same logic applies to a larger. That is, under a larger r, firms place more weight on the priors in their belief-updatingprocess.
In the standard model with perfect common knowledge, δt (j) is common across firms. In
this special case, the equilibrium price shown in (13) is independent of r since Ēkt [·] = Ēt [·]for all k. We can also show that λ in (16) reduces to τ 2 /(τ 2 + σ 2 ). In other words, λ is notaffected by r under perfect common knowledge.

3.4

Comparative statics

The explicit solution forms given by (11) and (15) predict how inflation persistence depends
on market concentration in our model. To express the relationship, we denote the predicted
first-order autocorrelation by ρπ (h), where h represents the HHI as discussed in Section 3.1.
In the case of perfect information, (11) indicates ρπ (h) = 0 for any h because ∆pt is i.i.d.
By contrast, in the case of imperfect common knowledge, (15) implies ρπ (h) = µ(λ) > 0
because ∆pt follows AR(1). Because (16) shows that λ depends on r, µ(λ(r)) also depends
on r. Therefore, applying the implicit function theorem and the chain rule, we have,dρπ (h)dµ dλ dr
=
≤ 0,
dh
dh
|dλ{zdr}|{z}
ICK

(17)

MO

where ICK and M O point to the role of imperfect common knowledge and the prediction of
the MO model, respectively. The derivation of (17) is in Appendix B.
The inequality in (17) indicates that incorporating imperfect common knowledge into the
MO model is critical in explaining the negative correlation between inflation persistence andmarket concentration. The intuition behind (17) can be obtained via two steps. The first stepis to recall that pricing complementarity is higher in less concentrated markets (dr/dh < 0),as shown by (9). This negative correlation between h and r is a unique feature of the MO15

model and is absent in monopolistic competition models under CES preferences.
The second step is to adopt the common mechanism in the models of imperfect common
knowledge, which makes inflation persistence µ(λ(r)) increasing in r. In markets where
pricing complementarity is high, firms place more weight on their priors (dλ/dr ≥ 0) and lesson private signals in the belief-updating process. Under a larger λ, firms revise their prices
more slowly (dµ/dλ ≥ 0). This positive correlation between r and µ is a unique feature ofimperfect common knowledge and is absent in the model with perfect common knowledge.

3.5

Numerical examples

To visualize the results of the basic model, we provide some numerical examples. In our
numerical examples, we set γ = 10−4 so that h can be interpreted as the HHI. For the otherparameters, we assume β = 10−3 and σ = 1.
Figure 2 illustrates how inflation persistence varies against market concentration. The
left panel plots the inflation persistence under perfect common knowledge (i.e., signals xt (j)are common across all j). The right panel shows the inflation persistence under imperfectcommon knowledge (i.e., heterogeneous signals). Each panel presents inflation persistence
under τ = 0, 1.0, 1.5, and 2.0 to examine to what extent imperfect information generates
inflation persistence.
Comparing the two panels highlights the role of imperfect common knowledge. In theleft panel, the lines for inflation persistence shift upward as τ increases but are flat against
h. Thus, the model with perfect common knowledge succeeds in generating persistence but
fails to reconcile the observed negative correlation between inflation persistence and market
concentration. This is because inflation persistence is independent of r under perfect common
knowledge.
In the right panel, however, the lines for inflation persistence not only shift upward as τ
increases but also are downward sloping against h if τ > 0. Thus, the model can explain theobserved negative correlations as well as the intrinsic persistence. This implies that imperfect16

common knowledge is necessary for explaining the observed negative correlation.
While our numerical examples reconfirm the main analytical result of the basic model,
the model does not quantitatively perform well in explaining the magnitude of inflation
persistence. In particular, if τ is positive, inflation persistence in Figure 2 ranges between
0.4 and 0.7, much larger than the median of inflation persistence reported in Table 1. In thenext section, we consider a more general stochastic process for ct to reconcile the data. We
also check the robustness of our main result under the general stochastic process for ct .

4

Extending the basic modelThis section extends the basic model for two purposes. First, we improve the model’s predictions for inflation persistence while confirming the robustness of our results concerning
the negative correlation. For this purpose, we generalize the marginal cost stochastic process.
Second, we derive the macroeconomic implications of our model. For this purpose, we develop
a simple general equilibrium model.

4.1

Generalizing the marginal cost stochastic process

4.1.1

Setup

We allow for (i) temporary deviations of the marginal cost from the stochastic trend ηt and(ii) the persistence of εt parameterized by ρ. Consider the following stochastic process for ct ,ct = c̃t + ηt ,(18)c̃t = c̃t−1 + εt ,(19)

εt = ρεt−1 + et ,(20)

where ηt ∼ N (0, ζ 2 ) and et ∼ N (0, σ 2 ) are Gaussian white noise processes. Recall that εt
was assumed to be i.i.d. and ct = c̃t in the basic model. In the extended model, ct follows an

17autoregressive integrated moving average process, specifically, ARIMA(1,1,2) process, which
generalizes the random walk process assumed for ct in the basic model.
The generalized stochastic process of ct generates much richer inflation dynamics, consistent with the data. The model introduces a temporary shock to ct (i.e., ηt ) and can replicate
a wide range of levels of inflation persistence, including negative values. In fact, Table 1
shows that the minimum value of inflation persistence is negative. Further, our specification
distinguishes between intrinsic and inherited inflation persistence, as emphasized in the literature.16 Note that ρ in (20) represents the degree of inherited inflation persistence because
the persistence of ∆c̃t is ρ from (19) and (20).17
Before we move to the numerical results, we note that the equilibrium price continues to be
expressed as a linear combination of past marginal costs and α owing to the linear-quadratic
nature of the problem. Specifically, it is given by,
pt = κα + (1 − κ)tXφt−s ct−s ,(21)

φt−s ∆ct−s ,

(22)

s=0

and, accordingly, inflation is,∆pt = (1 − κ)

t
X
s=0

where each φt−s is a nonlinear function of h. Note that (21) generalizes (14) in the basicmodel. In what follows, we numerically compute φt−s and examine the inflation dynamics
under various parameter values.18
16

See Fuhrer (2010) for a discussion of intrinsic and inherited inflation persistence.
In addition, the generalized process can generate consistent prediction with the findings by BGM regarding
the contrasting responses of inflation to sector-specific and macroeconomic shocks. If ηt and et in (18)-(20)correspond to the sector-specific shock and the macroeconomic shock identified by BGM, respectively, thesolution for inflation, (22), implies that the inflation dynamics exhibit quicker responses to sector-specificshocks than to macroeconomic shocks.
18
See Appendix C for details of the numerical methods.
17

18

4.1.2

Results

Given plausible dynamics of marginal cost, we numerically check whether the model cangenerate inflation persistence that is consistent with the data. We also confirm the robustnessof the negative correlation between inflation persistence and market concentration when ηt 6=
0 and ρ ∈ [0, 1). We compute inflation persistence under (i) ηt = 0 or ηt 6= 0 and (ii)ρ ∈ {0.00, 0.35, 0.70}. In the computation, τ = 1.5 and ζ = 1 and the other parameters
remain unchanged.
Figure 3 illustrates the result. All lines in the figure are downward sloping against h. Thus,
our results are robust to the general stochastic process of the marginal cost. Next, solid lines
in all panels indicate that the presence of ηt significantly lowers inflation persistence. For
example, see panel (a) under ρ = 0. Inflation persistence ranges between 0.54 and 0.63 whenηt = 0 for all t (see the dashed line). However, when a temporary shock ηt is present, therange is now between -0.08 and 0.03, much closer to the data. Panels (b) and (c) show that,as inherited inflation persistence increases (i.e., ρ increases), both the dashed and solid lines
move upward.
Given the stochastic process of ct , we estimate unknown parameters ρ and τ that areconsistent with the data at the sector level. For estimation, we use ρπ,i and HHIi of thepooled dataset. Define the predicted inflation persistence as ρπ (HHIi , θ), where θ is theparameter vector of θ0 = (ρ, τ ). We then estimate θ by nonlinear least squares (NLS) thatP
solve minθ i [ρπ,i − ρπ (HHIi , θ))]2 . The resulting NLS estimates are θ̂0 = (ρ̂, τ̂ ) = (0.66 ×
10−4 , 2.01). The standard errors are 1.82 × 10−4 for ρ̂ and 0.54 for τ̂ , respectively.19
Figure 4 depicts that the size of the declines in inflation persistence predicted by the modelis close to the data. The solid line, which is the model’s prediction under θ̂0 = (ρ̂, τ̂ ), showsthat inflation persistence decreases by 0.12 from 0.17 to 0.05 as the HHI increases from 0 to
19

We added DBGM,i as an explanatory variable to control for a difference in the mean between the BGM
and extended datasets. Our estimation also imposes the restriction that ρ̂ is strictly positive.

19

2,000.20 The dashed line, which is the regression line based on specification (3-1) in Table
2, indicates that inflation persistence decreases by 0.12 from 0.14 to 0.02, alongside the HHI
increasing from 0 to 2,000. The shaded area indicates the range of one-standard deviationfrom the linear regression line, showing that the model’s prediction is in proximity to theregression result.

4.2

A general equilibrium model

4.2.1

Setup

We next develop a general equilibrium model with multiple sectors in each of which firms
are engaged in the monopolistic competition described in Section 3. The general equilibrium
analysis aims to understand how market concentration affects monetary non-neutrality. It also
derives some implications on the rise of market power and the relevant industrial policies.21
A sketch of our general equilibrium model is as follows.22 The model is a simple closed
economy model with a representative household and multiple sectors. The model has twolevels of aggregation. At the “wholesale level,” aggregation is done by intermediate-good
producers in each sector with MO technology, similar to the MO preferences. Outputs ofRfirm j are aggregated at sector i: qi,t = j∈[0,Ni ] qi,t (j)dj, where qi,t is the sectoral output andRqi,t (j) is outputs of firm j. The sectoral price is given by pi,t = (1/Ni ) j∈[0,Ni ] pi,t (j)dj. In thismodel, Ni is heterogeneous across sectors. As a result, the degrees of market concentration
hi = (γNi )−1 is also heterogeneous. The “retail level” aggregation is done by the final-good
P A (υ−1)/υ υ/(υ−1)
producer using standard CES technology: Qt = [(1/NA )1/υ N], where Qt is
i=1 qi,t
the real GDP and υ is the elasticity of substitution satisfying υ > 1. The price index for final
P A 1−υ 1/(1−υ)
goods is Pt = [(1/NA ) N.
i=1 pi,t ]The representative household maximizes its lifetime utility subject to the budget and cash20

The data presented in Section 2 indicate that the HHIs of 94.2% of sectors are less than 2,000.
For recent discussions on the rise of market power, see Autor et al. (2017), Grullon et al. (2019), and De
Loecker et al. (2020).
22
We leave the details of the model setup to Section B in the not-for-publication appendix.
21

20

in-advance constraints. The period utility is given by ln Qt − Lt where Lt is hours worked.
Turning to firms, we assume that the production function for firm j in sector i is linear inlabor: qi,t (j) = li,t (j), where li,t (j) are the labor inputs of firms. There is no government
expenditure in this model.
The above assumptions lead to an equality between the nominal marginal cost ct and thenominal money supply Mt . Firms solve the same maximization problem as (5), but now themoney supply explicitly affects their prices. If the money supply follows the stochastic process
specified by (18) – (20), we can derive the solution for Pt from ct = Mt . We then compute
output Qt from the cash-in-advance constraint Mt = Pt Qt . In computing the solutions, we
use the parameter values consistent with the NLS estimates in Section 4.1. We calibrate the
distribution of hi to the actual distribution of HHIi in the regressions and refer to this as
the baseline distribution.23
4.2.2

Results

We are now ready to discuss the macroeconomic implications of our model.

Baseline distribution The solid lines in Figure 5 plot the impulse response functions of
aggregate inflation (∆ ln Pt ) and the log deviations of aggregate output from the steady state(ln Qt − ln Qss ) to a 1% increase in money supply under the baseline distribution. In what
follows, we discuss the impulse responses under two counterfactual distributions in comparison
with the baseline distribution. The impulse functions allow us to assess how monetary nonneutrality varies depending on the distribution of hi . In the left panel, the initial responsesof aggregate inflation are standardized at 1.0 to facilitate the comparison across calibrations.

Effects of pro-competition policy The dotted lines in Figure 5 represent the impulse
responses in the case where all sectors are homogeneous and perfectly competitive (i.e., hi = 0
23

We use the distribution of HHIi in the pooled dataset. The impulse responses are not substantiallydifferent, even if we use either the BGM dataset or the extended dataset.

21

for all i). In this case, goods produced by firms are fully substitutable in the MO preferences
and pricing complementarity is at its strongest. Therefore, taking other parameters as given,
the degree of monetary non-neutrality would also be the strongest. Under the counterfactual distribution of hi = 0 for all i, inflation converges more slowly to the steady state thanunder the baseline distribution. Quantitatively, output under the baseline distribution increases by 0.45% on impact while the output response is now 0.64% under the counterfactual
distribution.
An interpretation of this case is that an industrial policy achieves perfect competition inall sectors. Thus, according to our model, the impact of the pro-competition industrial policy
can potentially be large.

Rise of market concentration (or market power) The dashed lines in Figure 5 are theimpulse responses in the case where all observed HHIi are hypothetically doubled. Comparedwith the baseline distribution, output responds less under the increased market concentration(or market power). The response on impact declines from 0.45% in the baseline distribution
to 0.38% in the counterfactual distribution. Thus, the impact of the increased market concentration weakens monetary non-neutrality, but quantitatively, the effects would be marginal.
The result is a straightforward outcome from (9). As (9) shows, the declines in pricing
complementarity become smaller as h increases. Therefore, declines in inflation persistence
are also small when the HHI is high. We can reconfirm this result from Figure 4. The solidline indicates that the slope is steep when h is low but becomes flatter when h is high.

5

An alternative approach: The Calvo model

Apart from our arguments so far, some early studies emphasize the linkage between pricestickiness and market concentration. If (i) the degree of price stickiness varies depending onmarket concentration and (ii) price stickiness increases inflation persistence, we can arrive22

at a reasonable hypothesis that can account for the observed correlation between inflation
persistence and market concentration. Along with this argument, Carlton (1986) reports a
positive correlation between the degree of price rigidity and market concentration. BGMargue that the price adjusts more sluggishly in more concentrated markets.24
In what follows, we explore the prediction of a Calvo model with standard CES preferenceswhere the degree of price stickiness increases as market concentration rises. Let ξ be theprobability of no price changes. In the Calvo model, the (log-linearized) price index is aweighted average of the lagged price and the newly reset prices. Here, we maintain the same
assumption as in our basic model that the (log of) marginal cost follows a random walk.
Then, sectoral inflation follows an AR(1) process as, for instance, shown by Bils and Klenow(2004), such that,∆ ln pt = ξ∆ ln pt−1 + (1 − ξ)εt ,(23)

where ξ now coincides with inflation persistence in the sector. Given the findings by Carlton (1986) and BGM, the Calvo model predicts that inflation persistence would positively
correlate with market concentration. Namely,
dξ
∂ρπ
=
> 0,
∂h
dh
which contradicts the empirical findings presented in Section 2.
The key to reconciling the seemingly contradicting predictions given by the Calvo modeland our model lies in the relationship between the degree of price stickiness and inflation persistence. Bils and Klenow (2004) report empirical findings that point to a negative correlation
between the degree of price stickiness and inflation persistence across sectors.25 Although thedegree of price stickiness could positively correlate with market concentration, greater price24

However, Bils and Klenow (2004) conclude that market concentration does not have robust explanatory
power for the degree of price stickiness as measured by the (in)frequency of price changes.
25
See Figures 2 and 3 in Bils and Klenow (2004). Table 4 of their paper also shows the correlation between(sectoral) inflation persistence and the frequency of price changes.

23

stickiness does not ensure higher inflation persistence. As suggested by studies of information
rigidity models, inflation persistence cannot solely be explained by models of sticky prices.

6

Conclusion

Many previous studies have confirmed the role of information rigidities in firms’ price-setting
decisions. Our contribution to the literature on inflation dynamics is twofold. First, using US
PPI data, we present evidence that sectoral inflation persistence is negatively correlated withmarket concentration. Second, we find that pricing complementarity among monopolisticallycompetitive firms decreases as market concentration increases, given quadratic preferences
over the variety. Because of the varying pricing complementarity, our model with imperfectcommon knowledge can generate the observed negative correlation between inflation persistence and market concentration across sectors.
One caveat in this paper is that our simple general equilibrium model in Section 4 may not
fully quantify the importance of the rise of market power on monetary non-neutrality. Ourmodel predicts that the rise of market power may weaken monetary non-neutrality. However,
the model is small-scale, and its quantitative impact may differ in a full-fledged mediumscale model. The interaction of changes in market power with the imperfect formation in
full-fledged, medium-scale models would be worth exploring for future work.

References[1] Afrouzi, Hassan, 2019, “Strategic Inattention, Inflation Dynamics and the NonNeutrality of Money,” manuscript.
[2] Altissimo, Filippo, Benoit Mojon, and Paolo Zaffaroni, 2009, “Can Aggregation Explain
the Persistence of Inflation?,” Journal of Monetary Economics, 56(2), pp. 231–241.

24

[3] Amiti, Mary, Oleg Itskhoki, and Jozef Konings, 2019, “International Shocks, Variable
Markups and Domestic Prices,” Review of Economic Studies, 86(6), pp. 2356–2402.
[4] Angeletos, George-Marios and Jennifer La’O, 2009, “Incomplete Information, HigherOrder Beliefs and Price Inertia,” Journal of Monetary Economics, 56, pp. S19–S37.
[5] Autor, David, David Dorn, Lawrence F. Katz, Christina Patterson, and John Van.
Reenen, 2017, “Concentrating on the Fall of the Labor Share,” American Economic
Review, 107(5), pp. 180–185.
[6] Bils, Mark and Peter J. Klenow, 2004, “Some Evidence on the Importance of Sticky
Prices,” Journal of Political Economy, 112(5), pp. 947–985.
[7] Boivin, Jean, Marc P. Giannoni, and Ilian Mihov, 2009, “Sticky Prices and Monetary
Policy: Evidence from Disaggregated US Data,” American Economic Review, 99(1), pp.
350–384.
[8] Byrne, Joseph P., Alexandros Kontonikas, and Alberto Montaggnoli, 2013, “International Evidence on the New Keynesian Phillips Curve Using Aggregate and Disaggregate
Data,” Journal of Money, Credit and Banking, 45(5), pp. 913–932.
[9] Candian, Giacomo, 2019, “Information Frictions and Real Exchange Rate Dynamics,”
Journal of International Economics, 116, pp. 189–205.
[10] Carlton, Dennis, 1986, “The Rigidity of Prices,” American Economic Review, 76(4), pp.
637–658.
[11] Clark, Todd E., 2006, “Disaggregate Evidence on the Persistence of Consumer Price
Inflation,” Journal of Applied Econometrics, 21(5), pp. 563–587.
[12] Crucini, Mario J., Mototsugu Shintani, and Takayuki Tsuruga, 2015, “Noisy Information, Distance and Law of One Price Dynamics across US Cities,” Journal of Monetary
Economics, 74, pp. 52–66.
25

[13] De Loecker, Jan, Jan Eeckhout, and Gabriel Unger, 2020, “The Rise of Market Power andthe Macroeconomic Implications,” Quarterly Journal of Economics, 135(2), pp. 561–644.
[14] Fuhrer, Jeffrey C., 2010, “Inflation Persistence,” in Handbook of Monetary Economics
3A, B. M. Friedman and M. Woodford, eds., San Diego CA: North-Holland, pp. 423–486.
[15] Fukunaga, Ichiro, 2007, “Imperfect Common Knowledge, Staggered Price Setting, andthe Effects of Monetary Policy,” Journal of Money, Credit and Banking, 39(7), pp. 35–65.
[16] Gopinath, Gita and Oleg Itskhoki, 2011, “In Search of Real Rigidities,” NBER Macroeconomic Annual 2010, 25, pp. 261–310.
[17] Grullon, Gustavo, Yelena Larkin, and Roni Michaely, 2019, “Are US Industries Becoming
More Concentrated?,” Review of Finance, 23(4), pp. 697–743.
[18] Imbs, Jean, Eric Jondeau, and Florian Pelgrin, 2011, “Sectoral Phillips Curves and theAggregate Phillips Curve,” Journal of Monetary Economics, 58(4), pp. 328–344.
[19] Kimball, Miles S., 1995, “The Quantitative Analytics of the Basic Neomonetarist Model,”Journal of Money, Credit and Banking, 27(4), pp. 1241–1277.
[20] Koga, Maiko, Koichi Yoshino, and Tomoya Sakata, 2020, “Strategic Complementarity
and Asymmetric Price Setting among Firms,” BIS Paper, No. 111.
[21] Leith, Campbell and Jim Malley, 2007, “A Sectoral Analysis of Price-Setting Behavior in
U.S. Manufacturing Industries,” Review of Economics and Statistics, 89(2), pp. 335–342.
[22] Luengo-Prado, Marı́a-José, Nikhil Rao, and Viacheslav Sheremirov, 2018, “Sectoral Inflation and the Phillips Curve: What Has Changed Since the Great Recession?,” Economics
Letters, 172, pp. 63–68.
[23] Maćkowiak, Bartosz and Mirko Wiederholt, 2009, “Optimal Sticky Prices under Rational
Inattention,” American Economic Review, 99(3), pp. 769–803.

26

[24] Melitz, Marc J. and Gianmarco I. P. Ottaviano, 2008, “Market Size, Trade, and Productivity,” Review of Economic Studies, 75(1), pp. 295–316.
[25] Morris, Stephen and Hyun S. Shin, 2002, “Social Value of Public Information,” American
Economic Review, 92(5), pp. 1521–1534.
[26] Nimark, Kristoffer, 2008, “Dynamic Pricing and Imperfect Common Knowledge,” Journal of Monetary Economics, 55(2), pp. 365–382.
[27] Ottaviano, Gianmarco, Takatoshi Tabuchi, and Jacques-François Thisse, 2002, “Agglomeration and Trade Revisited,” International Economic Review, 43(2), pp. 409–436.
[28] Sbordone, Argia M., 2010, “Globalization and Inflation Dynamics: The Impact of Increased Competition,” in International Dimensions of Monetary Policy, J. Galı́ and M.
J. Gertler, eds., Chicago: University of Chicago Press, pp. 547–579.
[29] Vives, Xavier, 1990, “Trade Association Disclosure Rules, Incentives to Share Information, and Welfare,” RAND Journal of Economics, 21, pp. 409–430.
[30] Woodford, Michael, 2003, “Imperfect Common Knowledge and the Effects of Monetary
Policy,” in Knowledge, Information, and Expectations in Modern Macroeconomics: In
Honour of Edmund S. Phelps, P. Aghion, R. Frydman, J. Stiglitz, and M. Woodford,eds., Princeton: Princeton University Press, pp. 25–58.

27

A

Consumer demand under MO preferences

The consumers’ problem is to maximize (2) subject to (3). The first-order condition is
obtained as,Z
pt (j) = α − βqt (j) − γ

qt (j)dj.

(24)

j∈[0,N ]By integrating pt (j) over j ∈ [0, N ] and dividing it by N , we obtain,1
pt =N
Using (24),Z

βpt (j)dj = α −Nj∈[0,N ]ZZ
qt (j)dj − γ
j∈[0,N ]qt (j)dj.

(25)

j∈[0,N ]R

q (j)dj is eliminated from (25), resulting in the demand function given byj∈[0,N ] t

(4).

B

Comparative statics

We first derive the equilibrium price from individual prices. Given (4), the firm’s maximization problem (5) can be rewritten as,
1 
max : − Et (pt (j) − pt )2 r + {pt (j) − (κα + (1 − κ)ct )}2 (1 − r) + Ψ,
pt (j)
β(26)

where Ψ represents all exogenous terms for firm j. Notice that the structure of this optimization problem is exactly the same as those introduced by Morris and Shin (2002). Due tothe linear-quadratic nature of the problem, the optimal price set by firm j has a linear form
given by,pt (j) = κα + (1 − κ) φt (λ) xt (j)0 ,(27)

where φt = [φt φt−1 · · · φ0 ] and xt (j) = [xt (j) xt−1 (j) · · · x0 (j)]. Given the assumptionthat information structure converged in t = 0, we conjecture that φt takes the following form
independent of t such that φt−j = (1 − µ) µj , where µ is the smaller root of λz 2 −(1 + λ) z+λ =28

0. Consequently, firm j’s optimally chosen price should take the form,pt (j) = µpt−1 (j) + (1 − µ) {κα + (1 − κ) xt (j)} .

(28)

This conjecture (28) is substituted back into (26) and it can be confirmed that there exists
a unique µ ∈ [0, 1] that solves the optimization problem. Integrating (28) over j leads to theequilibrium price given by (14).
The comparative statics take three steps. First, we inspect dµ/dλ. Because µ ∈ [0, 1] isthe smaller root of λz 2 − (1 + λ) z + λ = 0, applying the implicit function theorem leaves,
dµ
µ
= 2
dλ
λ

−1


1
−µ
> 0.
(1 − µ) +λ

The second step is to examine dλ/dr, which is clearly positive as shown by (16). In the third
step, dr/dh < 0 is assured by the definition of pricing complementarity given by (9). Finally,
the chain rule combines the three inequalities, which results in (17).

C

Numerical solutions of the extended model

C.1

The solution form

We redefine the vector φt = [φt φt−1 · · · φ0 ] as,"φt = arg

min{φ0 ,φ1 ,...φt }

E r

t−1
X

(
φ2t−s τ 2 + (1 − r) (φt − 1)2 ζ 2 + φ2t τ 2 +

s=0

t−1
X

φ2t−s τ 2 + ζ

s=1+(1 − r)


t
s
X
X
 s=0

u=0

s−u+1

φt−u

1−ρ
1−ρ

s+1

−

1−ρ
1−ρ

!2

)

2

# (29).
σ2The terms in the square bracket on the right-hand side of (29) are obtained by substituting(21) and (27) into (26) under the assumption that ct follows the general process specified in29(18)–(20). It can be confirmed that {φt , φt−1 , · · · , φ0 } satisfies the following t + 1 conditions.
φt = Γ1 + Γ2 (1 + ρ)φt−1 − Γ2 ρφt−2 ,
(
)

s 
1 − ρs+1 X 1 − ρs+2−kφt−s = (1 − Γ2 )−φt+1−k
1−ρ
1
−
ρ
k=1
+Γ2 (1 + ρ)φt−s−1 − Γ2 ρφt−s−2 ,

(30)

(31)

for s ∈ {1, 2, ...t − 2} and,
(

φ1
φ0

)

t−1 
1 − ρt X 1 − ρt+1−k= (1 − Γ2 )−φt+1−k + Γ2 (1 + ρ)φ0 ,
1−ρ
1
−
ρ
k=1

t
t+1
X 1 − ρt+2−k 1−ρ
−=φt+1−k ,
1−ρ
1−ρ
k=1

where,
Γ1 ≡(1 − r) (σ 2 + ζ 2 )rτ 2 + (1 − r)(τ 2 + ζ 2 )
,
Γ
≡
.
2
rτ 2 + (1 − r) (τ 2 + σ 2 + ζ 2 )rτ 2 + (1 − r) (τ 2 + σ 2 + ζ 2 )Then, the solution form is given by,
pt = κα + (1 − κ) φt c0t ,where ct = [ct ct−1 · · · c0 ].

C.2

Approximation

When t is extremely large, it is not feasible to deal with t × t matrices numerically. Therefore,
we further assume that for a large T < t, all elements in [φt−T −1 φt−T −2 ...φ0 ] should be zero.
This additional assumption reduces the dimension of the matrices needed to compute thesolution numerically.
Define a 1 × (T + 1) vector φt,T = [φt φt−1 · · · φt−T ] which satisfies the T + 1 conditions
given by (30) and (31) for s ∈ {1, 2, ...T + 1}. Then, φ0t,T can be obtained by,
φ0t,T = (I − M)−1 G0 ,
30

(32)

where I(T +1)×(T +1) is an identity matrix and M(T +1)×(T +1) and G1×(T +1) are given by,02

Γ2 1−ρ1−ρ
 − (1 − Γ2 ) 1−ρ2
0

1−ρ

2
3
 − (1 − Γ2 ) 1−ρ− (1 − Γ2 ) 1−ρ
1−ρ
1−ρ

M = 
..
...

.

..
..

.
.

T
T
+1
− (1 − Γ2 ) 1−ρ− (1 − Γ2 ) 1−ρ
1−ρ
1−ρ

−Γ2 ρ

0

2

Γ2 1−ρ
1−ρ

−Γ2 ρ
..
.
...
..
.

0
...
..
.
...

3
− (1 − Γ2 ) 1−ρ
1−ρ
T ].

...
...
..

.
...
0
2
− (1 − Γ2 ) 1−ρ
1−ρ

0
..
.






0
,

−Γ2 ρ 1−ρ2 
Γ2 1−ρ 
0

G = [ Γ1 (1 − Γ2 ) (1 + ρ) . . . (1 − Γ2 ) (1 + ... + ρ )C.3

Inflation persistence

The inflation is now approximated by,∆pt ' (1 − κ) φt,T ∆c0t,T ,where ∆ct,T = [∆ct ∆ct−1 · · · ∆ct−T ]. Using this approximated form, inflation persistence iscalculated as follows.
PT22
s=0 bt−s bt−s−1 ζ
s=0 at−s at−s−1 σ +,PT 2 2
PT
2
2+
b
ζ
a
σ
t−s
t−s
s=0
s=0

PT
Corr (∆pt , ∆pt−1 ) =where at−s and bt−s for s ∈ {0, 1, 2, ..T } are characterized by A0 = (I − P)−1 φ0t,T and B0 =Qφ0t,T given by,
A = [ at at−1 . . . at−T ] ,B = [ bt bt−1 . . . bt−T ] , 0 ... ... ... 0  1
0 ... ...
.
.
.
.
 ρ . . . . . . .. 
 −1 1 . . . . . .
. . . . ..  , and Q = 
..
P = 
.
.
.
0
ρ
.


 0 −1 1
 .. . . . . . . ..  .. . . . . . .
.
.
. .
.
.
.
.
.
0 . . . . . . −1
0 ... ... ρ 0

31

0 
.. 
.
.. 

. .

0
1



Figure 1: Scatter plot of inflation persistence and market concentration

First-order autocorrelation

0.6

0.4

0.2

0

-0.2

-0.4
0

500

1000

1500

2000

2500

Herfindahl-Hirschman Index
Notes: The vertical axis is inflation persistence measured by the first-order autocorrelation of the seasonallyadjusted monthly difference of the sectoral (log) prices taken from BGM. The horizontal axis is the Herfindahl–
Hirschman Index.

32

Figure 2: Inflation persistence in the basic model

0.9

0.9

=0
=1
=1.5
=2

0.8

0.7

0.6

First-order autocorrelation

First-order autocorrelation

0.7

0.8

0.5
0.4
0.3
0.2

0.6
0.5
0.4
0.3
0.2

0.1

0.1

0

0

-0.1

-0.1
0

500

1000

1500

2000

2500

0

Herfindahl-Hirschman Index

500

1000

1500

2000

Herfindahl-Hirschman Index

Notes: The vertical axis is inflation persistence measured by the first-order autocorrelation. The horizontal
axis is the Herfindahl–Hirschman Index. In the left panel, signals are common across all j. In the right panel,heterogeneous signals are assumed.

33

2500

Figure 3: Inflation persistence under the generalized stochastic process of marginal cost

t

0.8

0.8

0.8

0.6

0.6

0.6

0.4

0.4

0.4

0.2

0.2

0.2

0

0

0

t

First-order autocorrelation

=0

-0.2

0

1000

0

2000

Herfindahl-Hirschman Index

-0.2

0

1000

2000

Herfindahl-Hirschman Index

-0.2

0

1000

2000

Herfindahl-Hirschman Index

Notes: The marginal cost follows the generalized stochastic process specified by (18) – (20). The vertical axis
is inflation persistence measured by the first-order autocorrelation. The horizontal axis is the Herfindahl–
Hirschman Index.

34

Figure 4: Predicted size of declines in inflation persistence

Notes: The vertical axis is inflation persistence measured by the first-order autocorrelation. The horizontal
axis is the Herfindahl–Hirschman Index. “Data” indicates the regression line of specification (3-1) in Table
2. “Model” presents the prediction of the model with ρ̂ = 0.00 and τ̂ = 2.01. The shaded area indicatesthe range of ±1 standard deviations from the regression line calculated using the residuals of regression in
specification (3-1) of Table 2.

35

Figure 5: Impulse response function of aggregate inflation and aggregate output
1

0.7
Baseline
Perfect competition
Concentrated market

0.9

0.6

0.8
0.5

0.7
0.6

0.4

0.5
0.3

0.4
0.3

0.2

0.2
0.1
0.1
0

0
0

2

4

6

8

10

0

Time

2

4

6

8

10

Time

Notes: Panels (a) and (b) show the impulse response function of aggregate inflation and the log-deviationof aggregate output from its steady state, respectively. In both panels, we calibrate ρ and τ at the NLS
estimates of ρ̂ = 0.00 and τ̂ = 2.01. The solid lines indicate the case of the baseline distribution in which we
calibrate the distribution of hi to the actual distribution of HHIi . The dotted lines correspond to the casewhere all sectors are homogeneous and perfectly competitive (i.e., hi = 0 for all i). The dashed lines show
the case where all observed HHIi are hypothetically doubled.

36

Table 1: Descriptive statistics of sectoral inflation persistence
BGM dataset:

Extended dataset:Sample period

1976:2 – 2005:6

2004:1 – 2020:12

Industries

Manufactureres

Manufactureres

Nonmanufactureres

Average

0.13

0.10

-0.07

Median

0.11

0.06

-0.09

Minimum

-0.42

-0.44

-0.38

Maximum

0.65

0.71

0.50

Standard deviation

0.20

0.23

0.18

Observations

152

270

79

Notes: The inflation persistence is measured by the first-order autocorrelation estimated from an AR(1) model.
Inflation is the monthly difference of seasonally adjusted sectoral log prices (NAICS six-digit classification).

37

Table 2: Benchmark regression results
Dependent variable: ρπ,i

HHIi /1000

BGM dataset:

Extended dataset:

Pooled dataset:

1976:2 – 2005:6

2004:1 – 2020:12

-

(1-1)

(1-2)

(2-1)

(2-2)

(3-1)

(3-2)

-0.071***-0.073***-0.052**-0.057***-0.059***-0.063***(0.024)(0.025)(0.021)(0.021)(0.016)(0.016)

D31,i

D32,i

-0.068*

0.059

0.007

(0.040)(0.041)(0.030)0.036

0.076**

0.061**(0.047)(0.032)(0.026)

DBGM,i

0.041*

0.045**(0.022)(0.022)

Observations

145

145

262

262

407

407

Adjusted-R2

0.042

0.061

0.017

0.034

0.029

0.040

F -statistics

9.208

4.786

6.406

4.632

8.072

5.592

p-value

0.003

0.003

0.012

0.004

0.000

0.000

Notes: Regression results for manufacturers. The dependent variable ρπ,i is the first-order autocorrelationof inflation estimated from an AR(1) model. Inflation is the seasonally adjusted monthly difference of thesectoral log prices and i denotes sectors. HHIi is the Herfindahl–Hirschman Index from the Economic Census.
D31,i is a dummy variable controlling for food and textile industries (NAICS codes starting with 31) while
D32,i is a dummy variable controlling for paper, wood, and chemical industries (NAICS codes starting with32). DBGM,i is a dummy variable that equals one if the data are from the BGM dataset. Constant terms aresuppressed in all specifications. The numbers in parentheses are the heteroskedasticity-consistent standard
errors.
*** p < 0.01,** p < 0.05,* p < 0.1.

38

Table 3: Regression results: C4 ratio
Dependent variable: ρπ,i

C4i

BGM dataset:

Extended dataset:

Pooled dataset:

1976:2 – 2005:6

2004:1 – 2020:12

-

(1-1)

(1-2)

(2-1)

(2-2)

(3-1)

(3-2)

-0.216***-0.199**-0.160**-0.180***-0.179***-0.190***(0.079)(0.084)(0.067)(0.068)(0.052)(0.053)

D31,i

D32,i

-0.070*

0.055

0.003

(0.039)(0.040)(0.029)0.014

0.079**

0.057**(0.046)(0.032)(0.026)

DBGM,i

0.042*

0.046**(0.021)(0.022)

Observations

152

152

270

270

422

422

Adjusted-R2

0.035

0.049

0.015

0.034

0.025

0.034

F -statistics

7.445

4.015

5.711

4.610

6.791

4.882

p-value

0.007

0.009

0.018

0.004

0.001

0.001

Notes: Regression results for manufacturers. C4i is the market share of the top-four largest firms includedin the Economic Census. See the notes for Table 2 for the other details.
*** p < 0.01,** p < 0.05,* p < 0.1.

39

Table 4: Regression results for manufacturers and nonmanufacturers
Dependent variable: ρπ,i
Extended dataset: 2004:1 – 2020:12

C4i

Manufacturers

Nonmanufacturers

Pooled

(1)

(2)

(3)

-0.160**-0.199*-0.169***(0.067)(0.102)(0.056)D3,i0.181***(0.024)

Observations

270

79

349

Adjusted-R2

0.015

0.039

0.111

F -statistics

5.711

3.810

31.867p-value

0.018

0.055

0.000

Notes: Regression results using the extended dataset. C4i is used as the explanatory variable because HHIi
is not available for nonmanufacturers. D3,i is a dummy variable controlling for manufacturing sector (NAICS
codes starting with 3). See the notes for Table 2 for the other details.
*** p < 0.01,** p < 0.05,* p < 0.1.

40

Table 5: Robustness
Dependent variable:

SAR

ρπ,i

ρπ,i

ρπ,i

(year-on-year inflation)
(1)

(2)

(3)

(4)

-0.082***-0.007**-0.046***-0.106***(0.022)(0.004)(0.015)(0.027)-0.054-0.0140.012(0.044)(0.009)(0.033)0.078**

0.008**

0.062**(0.033)(0.004)(0.028)0.176***

0.025***

0.046**

0.051**(0.032)(0.004)(0.020)(0.024)

N

N

Y

N

Observations

407

407

407

366

Adjusted-R2

0.099

0.088

0.225

0.042

F -statistics

11.1149.451

25.8665.064

p-value

0.000

0.000

0.000

0.001

HHIi /1000

D31,i

D32,i

DBGM,iControling for the 3 digit
NAICS industries

Notes: Regression results for manufacturers. The sample includes both the BGM and the extended datasets.
The dependent variable in specification (1) is the sum of the autoregressive coefficients (SAR) estimated froma higher-order autoregressive model. The lag length is chosen by the Bayesian Information Criterion. In
specification (2), the dependent variable is the first-order autocorrelation estimated from an AR(1) model,
but inflation is the year-on-year difference of the seasonally unadjusted sectoral log prices. Specifications (3)and (4) have the same dependent variable as specifications in Tables 2 and 3, but the explanatory variables
are different. In specification (3), the sector dummy is based on the three-digit NAICS code. In specification(4), the top-10% of sectors for HHI are removed from the sample as possible outliers. See the notes for Table
2 for the other details.
*** p < 0.01,** p < 0.05,* p < 0.1.

41