Research InstituteDiscussion Paper Series№ 03
Title：Sectoral inflation persistence and optimal monetary policy

Ida Daisuke／ 井田

大輔

ida-dai@andrew.ac.jp

Revised version: 2018 年 12 月
（First version）2016 年 7 月

http://www.andrew.ac.jp/soken/

Sectoral inflation persistence and optimal monetary
policyDaisuke Ida∗
Revised version: December 7, 2018
First version: July 5, 2016

Abstract
This study examines the optimal monetary policy in a two-sector New Keynesian model with inflation persistence. In contrast to the results suggested by
one-sector models, the proposed model illustrates that sectoral inflation persistence significantly changes the property of the optimal monetary policy. Sectoral
inflation persistence is likely to generate severe trade-oﬀs between inflation and theoutput gap as well as between inflation in diﬀerent sectors. This paper further considers how sectoral heterogeneities of inflation persistence substantially influence
the gain from commitment.

Keywords: Optimal monetary policy; Sectoral heterogeneities of inflation persistence;
Commitment; Two-sector New Keynesian model;
JEL classification: E52; E58∗

Department of Economics, Momoyama Gakuin University, 1-1, Manabino, Izumi, Osaka 594-1198,

Japan. Tel.: +81-725-54-3131. E-mail: ida-dai@andrew.ac.jp

1

Introduction

Many studies have adopted the New Keynesian (NK) model as a standard framework
for analysing monetary policy (e.g. Clarida et al., 1999; Woodford, 2003; Walsh, 2017;
Gali, 2015). In this purely forward-looking NK framework, however, inflation will jump
immediately in response to economic shocks. Therefore, a purely forward-looking structure might generate imprecise predictions of the New Keynesian Phillips curve (NKPC).
Some studies have incorporated past inflation into the NKPC (Gali and Gertler, 1999;
Christiano et al., 2005) and defended such inclusion by arguing that lagged inflation
improves the NKPC’s fit (Gali and Gertler, 1999; Rudd and Whelan, 2007; Kurmann,2007).1 Moreover, theoretical studies have argued that inflation persistence will significantly aﬀect the behaviour of monetary policy (Amato and Laubach, 2003; Steinsson,
2003; Sheedy, 2009; Paez-Farrell, 2012; Ida, 2013). These theoretical studies imply thatthe central bank faces severe welfare losses when inflation dynamics are mainly depictedby past inflation.
In addition, when examining the eﬀect of aggregate inflation dynamics upon thereal economy, it is important to distinguish between the dynamics of core inflation andthose of flexible-price inflation.2 Aoki (2001) demonstrated that stabilising core inflation,
which excludes food and energy prices, is the optimal monetary policy in a two-sector
NK economy. Airaudo and Zanna (2012) examined whether the central bank should
target headline inflation, which includes both core and flexible inflation, rather thancore inflation alone. They argued that targeting core inflation can result in reaching theunique rational expectations equilibrium, whereas targeting headline inflation will likely
render it indeterminate.

1

Cogley and Sbordone (2008) showed that the NKPC with lagged inflation fails to explain actual

inflation dynamics if the model considers trend inflation.
2

Huang and Liu (2005) showed that in an input-output NK economy, the optimal central bank policy

should stabilise inflation in both final and intermediate goods. Strum (2009) explored the optimaldelegation problem in an input-output economy. Strum (2010) examined the optimal monetary policy
in an input-output economy with inflation persistence.

1

These studies appear to highlight the importance of distinguishing between the dynamics of sticky-price inflation and flexible-price inflation in economies with inflation
persistence. Several studies have considered persistence of sector inflation. Recently,
Imbs et al. (2011) considered a heterogeneous hybrid NKPC by presuming the existenceof sectoral heterogeneities of inflation persistence. Byrne et al. (2013) used a dataset ofaggregated and disaggregated sectoral inflation and showed that a sector hybrid NKPC
has important consequences for aggregate inflation dynamics.
Irrespective of the importance of sectoral inflation persistence, it remains unclear how
sectoral persistence of inflation aﬀects the optimal monetary policy. This paper suggests
how the presence of sectoral heterogeneity of inflation persistence changes the properties of the optimal monetary policy suggested in a one-sector model. In this context,
the following questions arise: how should the central bank conduct monetary policy ineconomies with sectoral inflation persistence, and how large are the gains from commitment policies in such an environment? It appears that no studies have yet answered
these questions.
This study examines theoretically the optimal monetary policy in a two-sector NK
model with inflation persistence. It introduces sectoral heterogeneities of inflation persistence into the model developed by Woodford (2003, 2010). The economy modelledin this paper has two production sectors; the first sector, which models a sticky-pricesector, is labelled S, whereas the second sector, which models the flexible goods sector,
is labelled F. Following Woodford (2010) and Airaudo and Zanna (2012), this paper
assumes that prices in sector F are more flexible than those in sector S. In contrast to
Aoki (2001) and Kosem-ALP (2010), however, this paper allows the case wherein prices
in sector F are not completely flexible. Therefore, sector S’s inflation can be regarded ascore inflation.
This paper presents the theoretical foundation of sectoral inflation persistence ina sticky-price model. It utilises the rule-of-thumb hypothesis assumed by Steinsson(2003) to examine inflation persistence in a two-sector NK model. Under the rule-ofthumb hypothesis, firms use information from past aggregate price levels and inflation2

to set (reset) current prices. According to Steinsson (2003), a generalised rule-of-thumbcontains the past price level, past inflation rate and a lagged output gap.3 This allows aderivation of the generalised NKPC in a two-sector economy with inflation persistence.
Furthermore, this study derives the central bank’s loss function, which corresponds
to the second-order approximation of the household utility function, in a two-sector NK
model with inflation persistence. In this model, inflation persistence in each sector willchange the shape of the central bank’s objective function. Thus, the shape of the loss
function is more complicated than that in a one-sector model. Accordingly, the presenceof sectoral heterogeneities of inflation persistence makes the central bank’s optimisation
problem more complicated.
The theoretical model indicates that inflation persistence in one-sector aﬀects inflation in another sector through a change in the relative price between two sectoral goods,
thereby leading to a large fluctuation in the output gap. The trade-oﬀs following costpush shocks in a sticky-price sector are particularly severe. In contrast, and importantly,
trade-oﬀs following cost-push shocks in a flexible-price sector are attenuated. In addition, the central bank faces a trade-oﬀ between consumer price index (CPI) inflationand the output gap in a one-sector model, whereas this study considers that sectoral
heterogeneities of inflation persistence produce a severe trade-oﬀ not only between inflation and the output gap but also between inflation in diﬀerent sectors in a two-sector
model. These results are in contrast to those obtained for the one-sector model. This
paper, therefore, emphasises that this mechanism can trigger a large fluctuation in the
output gap. Accordingly, the model implies that the central bank needs to consider thesemechanisms of sectoral heterogeneities of inflation persistence when implementing the
optimal monetary policy.
This result raises questions about how sectoral inflation persistence substantially influences the gain from commitment. The main findings of welfare analyses are as follows.
Regardless of the degree of inflation persistence in the flexible-price sector, gain will re-3

Strum (2010) proposed an alternative rule-of-thumb specification that depends on the past pricelevel, past inflation and past real marginal cost.

3

sult as long as core inflation is depicted by a forward-looking component. Therefore, thegain from commitment almost disappears when core inflation is predominately backwardlooking. Accordingly, the gain is small when severe sectoral heterogeneities of inflation
persistence exist in both sectors. When moderate inflation persistence is present in both
sectors gains occur unless prices in both sectors are more flexible. Importantly, the gainfrom commitment almost vanishes when prices in both sectors are stickier.
The findings of this paper suggest that persistence of core inflation plays a significant
role in a two-sector NK model that contains sectoral heterogeneities of inflation persistence. Other findings regarding welfare analyses are summarised as follows. First, when
inflation dynamics are described by the forward-looking component in both sectors, thewelfare gain is large if higher price stickiness counteracts the output gap’s sensitivity to
inflation. As suggested by the standard NK model, this occurs because a central bankthat implements a commitment policy can enhance social welfare by manipulating theprivate sector’s expectations in a sticky-price equilibrium. Second, if moderate inflation
persistence is present in the core inflation sector, larger gains result from commitmentin the case of larger substitutability between two sectoral goods. Third, whether thewelfare gain is aﬀected by a change in the elasticity of substitution between individual
goods depends on the degree of inflation persistence in the sticky-price sector.
This paper is related to Aoki (2001), Woodford (2003, 2010), Steinsson (2003) andKosem-ALP (2010). Kosem-ALP (2010), in particular, also examined optimal monetary
policy with sectoral inflation persistence. His study shares several similarities with thisstudy. However, the model proposed here diﬀers from his model as follows. First, I
explicitly evaluate how the presence of sectoral heterogeneities of inflation persistenceaﬀect the gain from commitment. Second, he derived the sectoral NKPC following
Gali and Gertler (1999)’s specification of the rule-of-thumb pricing rule; in contrast, I
derive it using the specification of Steinsson (2003), who generalised their rule-of-thumbpricing rule. Third, he used a Cobb-Douglas consumption index; this study assumes a
consumption index per Woodford (2003, 2010). Fourth, and associated with the two
previous diﬀerences, this study uses the second-order approximation of the household’s

4utility function to derive the central bank’s generalised loss function.
This study is organised as follows. Section 2 describes a two-sector NK model with
inflation persistence and explains the optimal monetary policy in economies with sectoral
inflation persistence. Section 3 calibrates the deep parameters used in this model. Section
4 presents the results of the study. Section 5 provides a brief conclusion.

2

The modelThis section presents the utilised two-sector NK model with inflation persistence. I
introduce sectoral inflation persistence into the standard two-sector NK model developedby Woodford (2003, 2010). The model economy has two production sectors, denoted S(first sector) and F (second sector). Both sectors face monopolistic competition and set
prices according to Calvo-type (1983) nominal price rigidities.4 As in Woodford (2003,
2010), this paper assumes that nominal price rigidities are more severe in sector S thanin sector F. Thus, sector’s prices are assumed to be more flexible than those in sector S.
Therefore, sector S can be considered as the sticky (core) price sector, whereas sector F
can be regarded as the flexible goods sector.5
Following Steinsson (2003), I introduce a rule-of-thumb pricing rule into the model.
Firms that follow this rule-of-thumb pricing rule set prices based on past inflation, past
price level and past output gap. This pricing rule allows the eﬀect of inflation persistenceon the economy to be examined. More specifically, two types of firms operate in eachsector. One sets prices optimally, whereas the other sets prices following a rule-of-thumbpricing rule.
The infinitely lived household derives utility from consumption and disutility fromsupplying labour to each production sector. Finally, the central bank sets its interest
4

Airaudo and Zanna (2012) assume price adjustment costs for the two sectors to introduce nominal

price rigidities in the economy. They also assume that prices in one sector are more flexible than thosein the other sector.
5

As mentioned earlier, note that unlike Aoki (2001) and Kosem-ALP (2010), this paper does notassume that prices in sector F are completely flexible.

5rate to implement a monetary policy by following the targeting rule.

2.1

The household

The inter-temporal utility of an infinitely lived representative household is
Ut = E0∞∑[∫

β u(Ct ; ξt ) −∫v(hst (j); ξts )dj −

t

Ns

t=0

v(hft (j); ξtf )dj

]
,

(1)

Nf

where Ct denotes aggregate consumption. hst (j) and hft (j) denote the labour supply forgoods j in sectors S and F, respectively. ξt is a preference shock, and ξti denotes theexogenous disturbance for sector i, where i = s, f . The parameter β denotes the discount factor. The utility function u(·) is strictly concave and continuously diﬀerentiable.
The disutility of labour supply v(·) is strictly convex and continuously diﬀerentiable.
Furthermore, the intervals of goods in the two sectors are divided by Ns = [0, ns ] andNf = [nf , 1]. Here, ni (for i = s, f ) denotes the share of each sector in the economy.
Aggregate consumption is defined by the constant elasticity of substitution (CES)
aggregator of the two consumption goods, Cts and Ctf :[]η/(η−1)f (η−1)/η1/η
s (η−1)/η
1/η
,
Ct = (ns φs,t ) (Ct )+ (nf φf,t ) (Ct )(2)where η is the intra-temporal elasticity of substitution between goods S and goods F,and φi,t (for i = s, f ) is the random coeﬃcient. Parameter ni (for i = s, f ) is the shareof goods of each type. Assume that ns + nf = 1. In addition, Cti (for i = s, f ) is givenby the following CES aggregate:
[∫
Cti =

Cti (j)(θt −1)/θt dj]θt /(θt −1)
,

(3)

Ni

Cti (j) is consumption for goods j in sector i, for i = s, f . The parameter θt is the timevarying elasticity of substitution for individual goods, which is common to each sector.
To introduce a cost-push shock in each sector, this study assumes time-varying elasticity
of substitution. As indicated by Steinsson (2003), this assumption enables us to derive6

a cost-push shock in the NKPC.6 Relative demand for goods j of sector i is given by(
Cti (j) =Pti (j)
Pti

)−θt
Cti ,

(4)

for any j ∈ [0, 1] and i = s, f . In addition, the demand for the sectoral composite goodis given by(
Cti = ni φi,t

Pti
Pt

)−η
Ct ,

(5)

for any i = s, f . Furthermore, the price indices in each sector are defined as follows:[∫]1/(1−θt )
1−θt

Pti =

Pt (j)

dj

,

(6)

Ni

for i = s, f . Finally, when aggregate consumption is defined by Equation (2), the CPI isdefined as follows:[
Pt =]1/(1−η)ns φs,t (Pts )1−η + nf φf,t (Ptf )1−η.

(7)

Next, consider the household’s utility maximisation problem. The representative
household maximises Equation (1) subject to the following budget constraint:∫∫ss
Pt Ct + Et Qt,t+1 Bt+1 = Bt +[Wt (j)ht (j) + Γt (j)]dj +[Wt (j)hft (j) + Γft (j)]dj − Tt ,
Ns

Nf

(8)
where Bt denotes the nominal bond, and Tt denotes a lump-sum tax. Wt (j) is thenominal wage and Γit (j) (for i = s, f ) is dividend of firms in each sector.7
The stochastic discount factor Qt,t+1 is defined as
Et Qt,t+1 =1,
1 + it(9)where it is the nominal interest rate.
6

An alternative is to assume that the wage mark-up is time-varying to derive the term for a cost-pushshock in the NKPC. Clarida et al. (2002) discussed this issue in detail.
7

Note that in this paper, I do not assume the labour mobility between two sectors. Therefore, thewage rate is indiﬀerent between two sectors.

7

The following conditions are derived from the first-order conditions of the household’sutility maximisation problem:uc (Ct ; ξt ) = βEt (1 + it )uc (Ct+1 ; ξt+1 )

Pt
,
Pt+1

(10)

and
vh (hit (j); ξti )Wt (j)=,
uc (Ct ; ξt )
Pt

(11)

for i = s, f . Equation (10) represents the Euler equation, which implies that changes in
the real interest rate lead to adjustments in inter-temporal consumption. Equation (11)indicates that the marginal rate of substitution between labour supply and consumption
equals the real wage for each sector.

2.2

The firm

As in the standard NK model, firms in each sector face Calvo-type (1983) nominal
price rigidity. Under the assumption of Calvo pricing, a fraction of firms can set prices
optimally, but the remaining firms cannot set their prices. In a two-sector NK model, theprobability that firms cannot revise prices in a given period depends on the particularsector. As mentioned earlier, this paper assumes that sector F’s prices are more flexible
than those of sector S. In addition, some firms that optimally set their prices use a ruleof-thumb pricing rule. Importantly, in contrast to the standard one-sector NK model,
the fraction of firms that follow a rule-of-thumb depends on each sector. Finally, as in
Woodford (2003), this study assumes a sector-specific disturbance to the preference interms of household labour supply.
The production function for goods j is given by
yi,t (j) = Ai,t f (hit ),(12)

for i = s, f , where yi,t (for i = s, f ) is sector output, Ai,t (for i = s, f ) denotes a
sector-specific productivity disturbance, and f (·) is an increasing and concave function.
Under these settings, the real marginal cost of sector i is given byψti (j) =Wt (j)
1,′−1
Pt Ai,t f (f (yi,t (j)/Ai,t ))8(13)

for i = s, f , where ψti (j) is the real marginal cost for goods j for each sector. Using
the household’s first-order conditions with respect to labour supply for each sector, theabove real marginal cost can be rewritten as follows:ψti (j) =vh (hit (j); ξti )1,′
uc (Ct ; ξt ) Ai,t f (f −1 (yi,t (j)/Ai,t ))(14)

for i = s, f .
Following Calvo (1983), this study assumes price rigidity in each goods sector. Fraction 1 − α of all firms in sector i can adjust prices, whereas the remaining firms (α)
cannot. These firms are uncertain regarding when they can change prices in a future
period. The firm’s optimisation problem is thus given by[( i,o )−θT(( i,o )−θT
)]
∞
∑
Pt
Yi,T
i,o Pt−1
T −t uc (CT ; ξT )Pt
Yi,T − WT f,
Et
(αi β)
uc (Ct ; ξt )
PTi
PTi
Ai,T
T =t

(15)

for i = s, f , where Pti,o denotes the optimal price in sector i in period t. The first-ordercondition of this optimisation problem isEt∞∑

T −t uc (CT ; ξT )(αi β)T =t

uc (Ct ; ξt )((1 − θT )(Pti,o )−1

Pti,o
Pti

)−θT

[(
Yi,T

Pti,o
PTi

)

]
Pti,o i
− µT i ψt,T = 0,PT(16)

for i = s, f . Variable µt denotes the time-varying price mark-up, Yi,t denotes sectori(for i = s, f ) denotes the real marginal cost for the firms that
aggregate output, and ψt,Tlast reset their prices in period t. Here, the real marginal costs for each sector under aflexible-price equilibrium are given by
Pti,o (j)= ψti ,i
PT

(17)

for i = s, f . The price indices for each sector are defined as follows:]1/(1−θt )

[
Pti =

i
)1−θt
(1 − αi )(Pti,o )1−θt + αi (Pt−1

,

(18)

for i = s, f .
Following Steinsson (2003), I presume that among firms that can reset prices in eachsector, 1 − λ number of firms sets prices optimally, whereas the remaining fraction of λ9sets prices according to the following rule-of-thumb:(i,oPti,r = Pt−1

i
Pt−1
i
Pt−2

)(

Yt−1
n
Yt−1

) δi
(19)

,

for i = s, f , where Pti,r (for i = s, f ) denotes the price of firms that employ rule-ofn
thumb pricing; Yt−1 is aggregate output in period t − 1, Yt−1
represents eﬃcient output

in period t − 1, and parameter δi (for i = s, f ) is the weight of the lagged output gap
in the rule-of-thumb pricing rule. I assume that the fraction of firms that sets prices
optimally diﬀers in sector i in order to examine the eﬀect of sectoral inflation persistence
on monetary policy. More concretely, the parameter λs is the fraction of firms that
follow a rule-of-thumb in sector S. Similarly, the parameter λf corresponds to the caseof sector F. Following the empirical fact argued in the introduction, this study assumes
that backward-looking firms exist in the sticky-price sector, whereas sector F’s inflation
is depicted by a purely forward-looking component.8
In this case, the price index for sector i is significantly aﬀected by price setters who
use information in the rule-of-thumb pricing rule. Accordingly, the price index that firms
can change their prices becomes
[
o
Pi,t
=

]1/(1−θt )(1 − λi )(Pti,f l )1−θt + λi (Pti,r )1−θt

,

(20)

for i = s, f , where Pti,f l (for i = s, f ) is the price for firms that optimally set prices.
Using this equation, the price indices defined in Equation (18) now change according toPti =[]1/(1−θt )i,f l 1−θti,r 1−θti1−θt(1 − αi )(1 − λi )(Pt )+ (1 − αi )λi (Pt )+ αi (Pt−1 ).

(21)This equation implies that the presence of price setters who follow a rule-of-thumb rule
becomes the source of endogenous inflation persistence.
I now present a log-linearisation of structural equations around the steady state withzero inflation. Specifically, a log-linearised variable around the steady state is expressedas Ẑt = log(Zt /Z̄), where Z̄ represents a steady state value.

8

As mentioned earlier, this assumption is relaxed in section 4.

10

Following Woodford (2003, 2010), all sector-specific disturbances have common values
in the two sectors: φ̄s = φ̄f and Ās = Āf . In this setup, a log-linearisation of Equation(16) becomes the following sector NKPC in terms of the real marginal cost:n) + ςi (ψ̂ti − p̂i,t ) + ui,t , (22)πi,t = γfi Et πi,t+1 + γbi πi,t−1 − χi1 (Ŷt − Ŷtn ) + χi2 (Ŷt−1 − Ŷt−1

for i = s, f . Each coeﬃcient in Equation (22) is given byαi βλi(1 − αi )αi λi βδi, γbi =
, χi1 =
αi + λi (1 − αi (1 − β))αi + λi (1 − αi (1 − β))αi + λi (1 − αi (1 − β))(1 − αi )λi δi(1 − αi )(1 − αi β)(1 − λi )χi2 =,, ςi =
αi + λi (1 − αi (1 − β))[αi + λi (1 − αi (1 − β))](1 + ω θ̄)γfi =for i = s, f and ω = vyy ȳ/vy . Here, πi,t denotes the sectoral inflation rate for sectori, and ui,t is the sectoral cost-push shock associated with a time-varying price mark-up
and p̂i,t (for i = s, f ) denotes the relative price index for sector i. In contrast to Byrne
et al. (2013), the assumption of a rule-of-thumb used by Steinsson (2003) allows theNKPC in each sector to depend on the lagged output gap. In a two-sector model, thelog-linearised real marginal costs [Equation (14)] are given as follows:ψ̂ti − p̂i,t = (ω + σ −1 )(Ŷt − Ŷtn ) − (1 + ωη)(p̂i,t − p̂ni,t ),(23)

for i = s, f . Here, p̂ni,t is the relative price index for sector i in a flexible-price equilibrium.
The parameter σ is defined as σ = −uyy Ȳ /uy .
Substituting Equation (23) into Equation (22) yields the following NKPC expressedby the output gap:nπi,t = γfi Et πi,t+1 + γbi πi,t−1 + κi (Ŷt − Ŷtn ) + χi2 (Ŷt−1 − Ŷt−1) + γi (p̂R,t − p̂nR,t ) + ui,t .

(24)
for i = s, f , where p̂R,t represents a measure of the relative price of the two sectoralcomposite goods, which is defined as p̂R,t = P̂tf − P̂ts . Here, p̂nR,t is the relative priceunder the eﬃcient level of output. Each coeﬃcient in Equation (24) is defined as follows:κi = ςi (ω + σ −1 ) − χi1 ,11for i = s, f , andγs = nf ςs (1 + ωη) > 0, γf = ns ςf (1 + ωη) < 0.
A one-sector model does not need to consider the eﬀect of relative prices between two sectors on the real economy. However, relative prices play an important role in a two-sector
model (Aoki, 2001; Woodford, 2010). In contrast to the purely forward-looking twosector model, it follows from Equation (24) that the presence of sectoral heterogeneities
of inflation persistence significantly aﬀects the real economy through the relative pricebetween two sectors.
The intuition behind this result is as follows. Consider the case where a cost-pushshock occurs in sector S. This cost-push shock will lead to an increase in core inflation
but a decrease in the output gap. As inflation persistence in sector S becomes more
severe, the central bank faces an even starker trade-oﬀ between core inflation and theoutput gap. The persistence of core inflation negatively propagates to sector F’s inflation
through a change in relative prices between the two sectors. Therefore, the existence ofmore severe inflation persistence in sector S produces a greater trade-oﬀ between core
inflation and sector F’s inflation. This results in huge output gap drops through both
NKPCs. Therefore, in contrast to the situation in a one-sector model, the central bankhere should focus on a trade-oﬀ between not only inflation and the output gap but also
between inflation in each sector.
Log-linearising Equation (10) yields the following dynamic IS equation:
xt = Et xt+1 − σ(R̂t − Et πt+1 − r̂tn ),(25)

where R̂t is the log-linearised gross nominal interest rate, πt is CPI inflation and r̂tn is thenatural rate of interest, which holds the real interest rate in the eﬃcient level of output
and, which is given bynr̂tn = σ −1 [(gt − Ŷtn ) − Et (gt+1 − Ŷt+1)],

where gt is an exogenous disturbance associated with preference shocks.

12

(26)

Finally, relative prices between two sectors evolve according top̃rt = p̃rt−1 + πf,t − πs,t + et ,(27)

where p̃rt is the relative price gap, which is defined as p̂R,t − p̂nR,t , and et denotes theexogenous shock for relative prices.

2.3

Optimal monetary policy

This section describes the optimal monetary policy in a two-sector economy with inflation
persistence. Rotemberg and Woodford (1997) and Woodford (2001, 2003) showed thatthe second-order approximation of the household’s utility function corresponds to thecentral bank’s loss function, which features variables denoting the squared inflation rateand squared output gap.
Both Aoki (2001) and Woodford (2003) derived the central bank’s loss function in a
two-sector economy. In Woodford (2003), the loss function contains both the quadraticterm of the relative price gap and the squared inflation rate in each sector. The presenceof the relative price gap is associated with an ineﬃcient relative level of the two sectors’
goods. The inflation stabilisation is characterised by a weighted average of inflation inboth sticky goods and flexible goods. Unlike in the one-sector model, Woodford (2003,
2010) argues that the central bank stabilises the weighted average of sectoral inflation
because price stickiness generates price dispersions in each sector.
I address that inflation persistence in each sector changes the shape of the central
bank’s objective function in a two-sector economy with inflation persistence. Thus, theshape of the loss function is more complicated in a two-sector model than in a one-sector
model because in the former, the fraction of firms that follow a rule-of-thumb pricing ruledepends on sector i. This study derives the central bank’s loss function approximatedaround a steady state. More specifically, it calculates the second-order approximation of
the household’s utility function as follows:
Ut ≃

∞
∑

β t Lt + t.i.p. + O(||ξ||3 ),

t=0

13

(28)

where t.i.p. includes the terms that are independent of monetary policy, and (||ξ||3 )indicates the terms of third or larger orders. Here, the central bank’s periodic loss
function is given by2222
Lt = ws πs,t
+ wf πf,t+ λx x2t + λr (p̃R,t − p̃nR,t )2 + w1s ∆πs,t+ w1f ∆πf,t+ (w2s + w2f )x2t−1 + w3s ∆πs,t xt−1 + w3f ∆πf,t xt−1 ,

(29)

where
ni κ
ni (1 − αi )2 λi δi i
(1 − αi )(1 − αi β)(ω + σ −1 ) i
ni λi, w2 =, κi =, w1 =,κiαi (1 − λi )αi (1 − λi )αi (1 + ω θ̄)2ni (1 − αi )λi δi
w3i = −,αi (1 − λi )wi =for i = s, f andλx =κ
ns nf (1 + ωη)λx−1 −1, λr =, κ = (ns κ−1
s + nf κf ) .
θ̄
1 + ω θ̄

Variable xt (=Ŷt − Ŷtn ) denotes the output gap. The technical appendix provides a detailed derivation of Equation (29).9 Although Kosem-ALP (2010) also derives the central
bank’s loss function in the presence of sectoral heterogeneities of inflation persistence, heassumed both Gali and Gertler (1999)’s specification of the rule-of-thumb pricing ruleand a Cobb-Douglas consumption index. In addition, under his specification, sector F’s
inflation is specified by flexible prices. In contrast, this paper assumes both Steinsson(2003)’s specification of the generalised rule-of-thumb pricing rule and a CES consumption aggregate. Therefore, compared with the results of Aoki (2001) and Kosem-ALP(2010), the shape of the central bank’s loss function is more complicated in this paper.
Equation (29) includes the weighted average of inflation in sectors S and F. In addition, in contrast with the two-sector forward-looking model, the central bank’s loss
function now includes a quadratic term for the change in each sector’s inflation rate.
Unlike the one-sector NK model with inflation persistence, the central bank should instead stabilise changes in the inflation rate in both sectors. The stabilisation term for achange in inflation for each sector is characterised by the weighted average of inflation
9

See Appendix A for a detailed derivation of the loss function.

14

for goods S and F. Moreover, in contrast with one-sector models, here, the central bankmust also stabilise the quadratic term for the output gap in period t − 1 in both sectors
because firms that adopt rule-of-thumb pricing refer to information from the lagged output gap. In contrast to Steinsson’s (2003) model, the stabilisation term for the lagged
output gap is aﬀected by both price stickiness and degree of inflation persistence in both
sectors. In a benchmark economy, I assume that no inflation persistence exists in sector
F. Therefore, the terms associated with the parameter λf are eliminated in the loss function.10 When inflation persistence is absent in this economy, the loss function reduces tothe one derived by Aoki (2001) and Woodford (2003, 2010).
Accordingly, the presence of sectoral inflation persistence complicates the central
bank’s optimisation problem. Concretely, the central bank minimises Equation (29)subject to the hybrid NKPC in both sectors and the law of motion for relative prices
between the two sectors. In the following central bank’s optimisation problem, this study
focuses on a commitment policy.11 The central bank implements its monetary policy
under the promise that it credibly commits to future monetary policy at the current
period. It regards the commitment policy as an optimal monetary policy from a timelessperspective, as suggested by Woodford (2003).12 Under this commitment policy, thecentral bank can conduct monetary policy by manipulating private sector expectations.
Suppose that ϕ1t and ϕ2t are Lagrange multipliers associated with sector NKPCs
and that ϕ3t is the Lagrange multiplier in terms of the identity of the relative price.
Defining the Lagrangian, the following first order conditions under a commitment policy

10

In section 4, I relax this assumption.

11

In section 4.2, I compare the performance of a commitment solution with that of a discretionary

policy. I explain optimal monetary policy with discretion in section 4.2.
12

Woodford (2003) provides a detailed discussion of a timeless perspective approach.

15

are derived:
πs,t : 2ws πs,t + 2w1s ∆πs,t − 2βw1s Et ∆πs,t+1 + 2w3s (xt−1 − βxt ) − 2β −1 γfs Et ϕ1t+1+ 2βγbs ϕ1t−1 + ϕ1t + ϕ3t = 0,(30)

πf,t : 2ws πf,t + 2w1f ∆πf,t − 2βw1f Et ∆πf,t+1 + 2w3f (xt−1 − βxt ) − 2β −1 γff Et ϕ2t+1+ 2βγbf ϕ2t−1 + ϕ2t − ϕ3t = 0,(31)

xt : 2[λx + β(w2s + w2f )]xt + w3s βEt ∆πs,t+1 + w3f βEt ∆πf,t+1 − 2κs ϕ1t − 2βχs2 Et ϕ1t+1− 2κf ϕ2t − 2βξf 2 Et ϕ2t+1 = 0,(32)

p̃R,t : λr p̃R,t − γs ϕ1t − γf ϕ2t + ϕ3t − βEt ϕ3t+1 = 0.

(33)

Combining these conditions with structural equations, the economic system can be written as follows:
A0 Xt = A1 Xt−1 + B1 Rt + Γt+1 ,

(34)

where
′

Xt = [X1t X2t ]′X1t = [us,t , uf,t , p̃nr,t . πs,t−1 πf,t−1 , xt−1 , p̃r,t , ϕ3t ] ,′X2t = [πs,t , πf,t , xt , ϕ1t , ϕ2t ] ,In addition, A0 , A1 , and B1 are matrices constructed by deep parameters, Rt denotes
a vector of the policy instrument, and Γt denotes a shock vector. This study uses thealgorithm method developed by Soderlind (1999) to numerically explore the optimal planof the above commitment policy and illustrate how inflation persistence will change theproperty of the optimal monetary policy in a two-sector NK model.

3

Calibration

This section describes the deep parameters used in this paper. The discount factor β
is set to 0.99. The relative risk-aversion coeﬃcient for consumption σ −1 is 5.0 basedon the value in Steinsson (2003). Following Steinsson (2003), I also assume that the16parameter ω equals 2.0. Following Woodford (2003), I set the elasticity of substitution
between goods θ to 7.88. In addition, based on the existing literature, for the value ofelasticity of substitution between goods S and F, parameter η is set to 1.5 as a benchmark
calibration. With regard to degree of price stickiness, following Woodford (2010), I set a
Calvo lottery in sector S to 0.75 and to 0.55 in sector F for the benchmark specification.
As mentioned previously, this parameterisation implies that sector F’s prices are more
flexible than those in sector S. This assumption is consistent with Woodford (2010) and
Airaudo and Zanna (2012).13 After a benchmark analysis, I will relax this assumption.14
Following Steinsson (2003), the parameter δs is set to 0.052. This paper further sets
δf to zero.15 Following Woodford (2010), the share of sticky goods in the CES aggregatorof two goods is set to 0.5. Thus, parameter n1 is set to 0.5. Finally, I assume that a
cost-push shock in each sector is temporary, whereas the persistence of a relative priceshock is set to 0.9. The standard deviations of a cost-push shock in each sector and a
relative price shock are both set to 0.01.

4

Results

This section presents the main results. I present the impulse-response analyses and show
how the degree of inflation persistence in sector S changes the property of the optimalmonetary policy. In this impulse-response analysis, as mentioned earlier, I assume inflation persistence in sector S, whereas in sector F, the inflation dynamics are purely
forward-looking.
In the impulse-response analysis, I follow Steinsson (2003) and I select four values forparameter λs . First, I set λs to 0.01 in the case of the forward-looking economy. Then, I
13

In Airaudo and Zanna (2012), NKPCs are derived from the assumption of the quadratic priceadjustment costs. Therefore, they consider that the degree of price adjustment costs are diﬀerent in theeach sector.
14

In section 4, with respect to these parameters, sensitivity analyses are implemented.

15This assumption is natural in the case of a benchmark calibration. I confirm that the results remainunchanged by a change of this parameter.

17set it to 0.2, which is empirically supported by Gali and Gertler (1999). The third valueof 0.7 is empirically supported by Fuhrer and Moore (1995), who argued that backwardlooking inflation is important in the hybrid NKPC. A larger value of the parameter λs
is supported by previous studies for estimation of the NKPC (e.g., Rudd and Whelan,
2007; Kurmann, 2007). Finally, I set λs to 0.90. This value is much larger value, whichcorresponds to the case in which backward-looking components predominately aﬀect
inflation dynamics.

4.1

Impulse response analyses

Figure 1 displays the impulse response to a cost-push shock occurring in sector S. The
cost-push shock increases core inflation and leads to a decline in the output gap. Thus,
the shock generates a trade-oﬀ between inflation and the output gap (Clarida, Gali
and Gertler, 1999; Woodford, 2003). The stronger the persistence of core inflation, thegreater the trade-oﬀ. This result is consistent with the findings of Amato and Laubach(2003) and Steinsson (2003). In a forward-looking economy, the central bank, which
has the ability to manipulate private sector expectations, can alleviate trade-oﬀ between
inflation and the output gap. As inflation persistence becomes more severe, however, thecentral bank needs to accommodate a substantial decline in the output gap instead ofpreventing an increase in inflation.
[Figure 1 around here]
In the two-sector model, a trade-oﬀ exists between inflation in each sector. Unlike
previous studies, this paper considers that the presence of inflation persistence in sector
S generates a worse trade-oﬀ between sectors’ inflation. In this case, a cost-push shock
induces an increase in core inflation although it decreases inflation in sector F. Figure
1 reveals that the stronger the inflation persistence in sector S, the worse the trade-oﬀ
between sectoral inflation. Because sector F’s prices are more flexible than those in sector
S, fluctuations in sector F’s inflation are more volatile than those aﬀecting core inflation.
This result is highly instructive. If the central bank aims to stabilise CPI inflation,
it should focus on stabilising headline inflation when a cost-push shock exists in sector
18

S. However, the central bank might react to movements in each sector’s inflation when acost-push shock occurs in sector S due to concerns of a trade-oﬀ between sectors’ inflationin its objective function. If inflation persistence is more severe in sector S, therefore, thecentral bank’s attempts to stabilise CPI inflation might induce an imprecise reaction of its
policy rate to the shock. This indicates that the central bank’s intention to stabilise CPI
inflation may be guided by a misleading monetary strategy. In other words, the central
bank should be cautious regarding changes in CPI inflation when sectoral heterogeneities
of inflation persistence play a significant role in the economy.
Figure 2 illustrates the impulse response to a cost-push shock in sector S followingchanges in several key parameters. Figure 2(a) depicts the case of an increase in η, whichshows that as the value of λs increases, a larger value of η amplifies the persistent dropin the relative price gap. The drop in output gap is also amplified. On the other hand, alarger value of η alleviates the trade-oﬀ in inflation between sectors. This result is more
predominate in the case of λs = 0.7. In contrast, Figure 2(b) shows that a smaller valueof η worsens the trade-oﬀ between the output gap and relative price gap as inflation
persistence in sector S is more severe.
[Figure 2 around here]
Figure 2(c) presents the case wherein prices are more flexible in sector F. In this
scenario, the output gap and sector S’s inflation remain unaﬀected except for the casewhere a larger value of λs aﬀects sector F’s inflation as well as the relative price gap. This
result continues to hold in Figure 2(d), where a larger value of λf aﬀects only sector F’s
inflation and the relative price gap. Thus, the results in Figure 1 are robust to changes
in any of the key parameters.
Figure 3 shows the impulse response when the cost-push shock occurs in sector F.
Such a shock increases inflation in sector F while leading to a decline in the output
gap, thereby reducing core inflation through the heterogeneous NKPC. Compared withthe case depicted in Figure 1, however, the shock does not introduce severe inflation
persistence in each sector even if inflation persistence exists in sector S. Hence, sector
S’s inflation persistence does not lead to a persistent decline in the output gap. This19

result is not surprising. Given that the shock’s source is associated with a temporary
cost-push shock in sector F, where prices are more flexible, inflation persistence in sector
S does not have a long-lasting eﬀect on the economy. Consequently, changes in inflation
persistence in sector S appear to aﬀect the trade-oﬀ only during the initial period.
[Figure 3 around here]
Importantly, Figure 3 reaﬃrms the observation that sector S’s inflation persistence
induces a worse policy trade-oﬀ between inflation and the output gap, although it dampens the trade-oﬀ in inflation between sectors. Intuitively, because the source of the shockin sector F is characterised by firms’ forward-looking behaviour, the central bank can
impart policy inertia into the economy by using sector F’s forward-looking inflation dynamics. This implies that if the source of the shock is related to sector F, the central
bank can still manipulate sector F’s inflation expectations even when severe inflation
persistence is present in sector S. As a result, in contrast to the case of the cost-pushshock occurring in sector S, a higher severity of inflation persistence in sector S does
not cause persistent fluctuations in the macro-economy. This mechanism is specific to a
two-sector model and is in contrast to a one-sector model.
[Figure 4 around here]
Figure 4 shows the impulse response to a cost-push shock in sector F following changes
in several key parameters. First, Figures 4 (a) and 4(b) show that macro-variables are
unaﬀected by any changes in parameter η. However, Figure 4 (c) indicates that when
there are more flexible prices in sector F will attenuate considerably the trade-oﬀ between
sectors’ inflation. Moreover, a smaller value of αf can ease the trade-oﬀ between the
output gap and relative price gap. On the other hand, moderate inflation persistence
exists in sector F causes the trade-oﬀ between sector F’s inflation and the output gap
to become more severe when sector S’s inflation is mainly depicted by forward-looking
components. Interestingly, such a trade-oﬀ is eased in the case of λs = 0.7. This indicatesthat although a cost-push shock in sector F causes a fluctuations in macro-variables,
inflation persistence in sector S helps alleviate such a large fluctuations.
20

Figure 5 illustrates the impulse response to a relative price shock. Fluctuations inrelative prices generate a policy trade-oﬀ between each sector’s inflation through that
sector’s NKPCs. As shown in Figure 3, the shock causes the core inflation rate to decline,
whereas it increases inflation in sector F.
[Figure 5 around here]
Because sector F’s prices are more flexible than those in sector S, the response of
inflation in sector F would dominate that of core inflation. Thus, the output gap declines
through both NKPCs. Therefore, this relative price shock results in a trade-oﬀ between
not only sector F’s inflation and the output gap but also inflation between sectors. Figure
5 shows that because sector S’s inflation persistence is more severe, such inflation inertia
weakens the trade-oﬀ in inflation between sectors. Instead of attenuating this trade-oﬀ
between sectors’ inflation, however, the central bank must allow a huge decline in the
output gap. Intuitively, severe inflation persistence in sector S forces relative prices tomove slowly, leading to a smaller adjustment in sector inflation. However, such a sizable
decline in the output gap is required in order to accommodate a small adjustment insectoral inflation during the initial period. Consequently, the central bank accommodatea higher output gap in subsequent periods. In other words, in the case of a larger value ofparameter λs , the central bank can succeed in restraining fluctuations in sector inflation
through a change in relative prices between two sectoral goods, although it will therefore
face a large decline in the output gap.
Figure 6 shows the impulse response to a relative price shock following changes toseveral key parameters. According to Figure 6(a), when λs takes a larger value, a larger
value of η serves to amplify the trade-oﬀ between sector F’s inflation and the output gap.
In addition, a trade-oﬀ in inflation between sectors is also more severe. Finally, a larger
value of η dampens a fluctuation in the relative price gap. On the other hand, Figure6(b) demonstrates that a smaller value of η does not aﬀect any macro-variables. Figure6(c) shows that, Figure 5’s results remain unchanged when sector F’s prices are more
flexible. Figure 6(d) illustrates that as in the case shown in Figure 4(b), the presence21

of inflation persistence in sector S helps ease such large fluctuations of macro-variables
generated by a relative price shock.
[Figure 6 around here]
The results of the impulse-response analyses can be summarised as follows. Sectoral
heterogeneities of inflation persistence play a significant role in the two-sector NK model.
Importantly, this paper reveals that inflation persistence in sector S worsens the trade-oﬀ
not only between inflation and the output gap but also between inflation in each sector
when a cost-push shock is generated in sector S. These results also imply that whether or
not CPI inflation has a persistent eﬀect crucially depends on the shock’s origin. This inturn indicates that the central bank should carefully consider the significance of sectoral
inflation persistence if it aims to improve the policy trade-oﬀ between CPI inflation andthe output gap. In contrast to previous studies, this study demonstrates that a two-sector
NK model with inflation persistence has important implications for optimal monetary
policy.

4.2

The gain from commitment

This section examines gains from a commitment policy in a two-sector economy with
inflation persistence. It compares the performance of a commitment policy with that ofa discretionary policy. In contrast to a commitment policy, a discretionary policy enables
a central bank to re-optimise its loss function in every period, given future expectations of
the private sector. In a purely forward-looking model, the performance of a commitment
policy generally dominates that of a discretionary policy, in that the latter cannot use
private sector expectations.16 As Amato and Laubach (2003) and Steinsson (2003) haveshown, severe inflation persistence reduces the gain from commitment by counteracting
the power to manipulate the private sector’s expectations. It remains unclear, however,
16

See, Woodford, (2003), Walsh (2003), McCallum and Nelson (2004), Gali (2008), Dennis (2010) andWalsh (2017) for a detailed discussion of the gains from a commitment policy in a purely forward-looking
NK model.

22

whether gains from commitment are negligible when inflation persistence is severe in a
two-sector model.
This section aims to answer this question. To do so, I calculate the central bank’s
unconditional welfare loss for both commitment and discretion. Gains from commitment
are defined as follows:(
W elf are gain =)Ldisc− 1 × 100,
Lcom

(35)

where Lcom denotes the welfare loss under commitment and Ldisc is the one under discretion. Thus, Equation (35) indicates gains from commitment will occur as long as thisvalue exceeds zero. In this simulation, as in the case of a commitment solution, I solve
the optimal monetary policy with discretion using Soderlind (1999)’s methods.
So far, in section 4.1, this paper assumed that inflation is purely forward-looking insector F. However, sensitivity experiments in impulse response analyses have revealed
that a change in parameter λf will aﬀect macroeconomic dynamics. One important
example is that when inflation persistence is present in sector F, inflation persistence
in sector S is able to counteract fluctuations in macro-variables unless the shock was
generated by a cost-push shock associated with sector S. Thus, the presence of inflation
persistence in both sectors might aﬀect the performance of a commitment policy. Therefore, I examine the case in which inflation is also characterised by inflation persistence insector F. This examination is a quite natural extension simply by assuming that flexible
goods inflation can also become inertial behaviour.
Figure 7 shows the result of this examination. Interestingly, when core inflation isdepicted by a purely forward-looking component, the maximum gain is attained in thecase of λf = 0.99. Indeed, Figure 7 indicates that the maximum gain in that case isapproximately 4%. It seems that this value is smaller than ones reported for a one-sector
model (cf. Steinsson, 2003; Amato and Laubach, 2003). It can also be seen that a larger
value of parameter λs reduces the welfare gain from commitment in the case where sector
F does not contain inflation persistence. Even if inflation persistence is severe in sector F,
no gain will occur as long as severe inflation persistence exists in sector S. This outcome
implies that whether or not the gain from commitment is large crucially depends on the23degree of inflation persistence present in sector S.
[Figure 7 around here]
We can interpret this result as indicating that inflation persistence for core inflation
plays a significant role in two-sector NK models. In fact, Figure 1 indicates that a larger
value of parameter λs produces a persistent eﬀect on macro-variables in the case of acost-push shock for core inflation. Furthermore, even when shocks originate with a sector
F’s cost-push shock or relative price shock, sector S’s inflation persistence creates a nonnegligible eﬀect on endogenous variables (cf., Figures 2 to 6). This result is supported by
the fact that these figures demonstrate that a larger value of λs generates a substantial
decline in both the output gap and relative price gap, whereas it can ease a trade-oﬀ
in inflation between sectors. As a consequence, regardless of the value of λf , almost no
gain occurs when λs takes a larger value. Overall, the results in Figure 7 are consistent
with those of the impulse-response analyses.
[Figure 8 around here]
Next, this paper checks whether welfare gains arise from commitment when both
αs and αf change under several parameterisations of λs and λf .17 Figure 8 reports thewelfare gain when moderate inflation persistence exists in both sectors. This figure showsthat the gain disappears when prices in both sectors are more flexible. Both sectors see
their price dispersions disappear when their prices become more flexible. Therefore, theburdens on price stability are reduced for the central bank in this case. In other words,
the welfare gain from price stability declines. The gain is also negligible when prices are
sticky in both sectors. As in the one-sector model, this occur because in such a case,the central bank cannot fully manipulate the private sector’s expectation. Because theNKPCs for each sector flatten, the gain from commitment disappears. This result is
consistent with those reported by Walsh (2003) and Steinsson (2003).
17

I do not report the case of λs = λf = 0.01. I did check that the result of Figures 8 and 9 remains

unchanged in that case.

24

Interestingly, unlike in the one-sector model, gains arise from commitment when
prices are sticky in either sector S or F. Consider the case of price stickiness in sector
S. Although sector F’s inflation immediately jumps in response to shocks, core inflation
does not. In addition, inflation persistence in sector S forces core inflation’ to make more
persistent movements. These mechanisms, in turn, indicate that a central bank thatstabilises sectoral inflation can smooth inflation dynamics through appropriate management of expectations; therefore, it can achieve the welfare gain from commitment. It
follows from this figure that gains arise for the case of price stickiness in sector F. The
intuition of this result is the same as the interpretation of the case for sector S.
Figure 9 shows the welfare gain from commitment when αs and λs change in thecase where firms in both sectors completely follow a backward-looking pricing rule (i.e.
λs = λf = 0.99). In this case, regardless of the degree of price stickiness in both sectors,
almost no gain from commitment arises. Specifically, the gain from commitment is only
1% when prices are sticky in both sectors. As discussed repeatedly, this occurs because
the central bank loses its ability to manage the private sector’s expectations.
[Figure 9 around here]
The results for the sensitivity analyses when several parameters change are as follows.
Here, I focus on a change in the value of λs based on the results from Figures 7-9. Figure
10 calculates the welfare gain from commitment when parameters αs and λs change. In
general, higher nominal price rigidities cause welfare loss through bigger price dispersions.
In addition, as shown in previous studies, the more severe inflation persistence becomes,
the smaller the gain from a commitment becomes because the central bank cannot fully
manage private sector expectations when inflation persistence is severe.
[Figure 10 around here]
According to Figure 10, when sector F’s inflation dynamics are determined by thepurely forward-looking component, no gain from commitment arises when the degree of
nominal price rigidities is considerably small in sector S. This result is not aﬀected by
the degree of parameter λs . Intuitively, price dispersion in sector S would be eliminated25because sector S’s prices are close to flexible ones, which means that the central bankcan enhance social welfare by putting a smaller weight on inflation stabilisation. Similarly, if prices are completely flexible, almost no gain from commitment would arise. In
other words, this case implies that the diﬀerence between discretion and commitment is
negligible.
In contrast, when the degree of inflation persistence is considerably small in sector
S (i.e. λs = 0.01), the central bank can attain a larger gain from commitment because
prices are stickier. As explained earlier, in a sticky-price equilibrium, the central bankcan ease the policy trade-oﬀ between inflation and output gap by managing the private
sector’s expectations. This result is thus consistent with the argument of the standard
NK model. For instance, the welfare gain from commitment exceeds 10% if most firms
cannot set their prices optimally.
However, the gain from commitment decreases if inflation persistence becomes more
severe in sector S. Because the central bank cannot fully manipulate inflation expectations in sector S, it cannot prevent a huge decline in the output gap. The gain from commitment considerably decreases when inflation dynamics are predominately backwardlooking in sector S (i.e. λs = 0.99). Notably, when price stickiness is extremely high,the central bank still produces a smaller welfare gain of approximately 4% if inflation
persistence is severe. This result contrasts the one obtained by Steinsson (2003).
According to Figure 11, for a smaller value of parameter λs , the welfare gain is smallerwhen parameter η takes a larger value. A larger value of parameter η means higher
substitutability between goods S and F. Thus, there is almost no diﬀerence between thetwo sectors. This means that the central bank does not need to focus on the diﬀerence
in sectoral inflation levels. Hence, higher substitutability leads to no diﬀerence between
commitment and discretion because higher substitutability increases the flexibility of
sector S’s prices. Interestingly, opposite eﬀects emerge if parameter λs takes a larger
value. In that case, higher substitutability between goods S and F produces a larger
welfare gain from commitment. Indeed, in the case of a larger value of parameter η, themaximum gains from commitment are achieved when parameter λs takes a value from26

0.6 to 0.8. For instance, in the case of η = 10, the welfare gain is approximately 8% whenλs = 0.8. Interestingly, this result is also in contrast to the result of Steinsson (2003).
[Figure 11 around here]
The intuition of this result is as follows. Sector F’s inflation jumps immediately after
the shock occurs because this inflation is more flexible and forward-looking. However,core inflation changes sluggishly in response to the shock. The diﬀerence in inflation
between sectors is much smaller when parameter η takes a larger value. As the impulseresponse analyses showed, sector F’s inflation dominates core inflation. In this case, if
the central bank can use the private sector’s expectations through sector F’s inflation,such a policy may induce an imprecise reaction to the shock by the nominal interest rate,
therefore destabilising the economy. If goods S and F are not completely substitutable,
however, the presence of inflation persistence in sector S might prevent inflation in sector
S from jumping immediately in response to the shock. Hence, the central bank canappropriately impart policy inertia into the economy. This is the source of the gain fromcommitment.
Figure 12 demonstrates the welfare gain from a commitment policy when parameters
λs and θ change. Parameter θ represents the elasticity of substitution between individual
goods in each sector. This value aﬀects the structural parameters in NKPCs for eachsector. A larger value of this parameter lowers the sensitivity of both relative prices
between two sectoral goods and the output gap to inflation in each sector. In otherwords, a higher-value parameter θ flattens the NKPC, thereby implying that the central
bank needs to accommodate a large fluctuation in relative prices as well as the output
gap to stabilise sectoral inflation.
[Figure 12 around here]
Furthermore, unless inflation persistence is severe in sector S, a larger value of parameter θ decreases the welfare gain from commitment because the central bank faces asevere trade-oﬀ among relative prices, the output gap and sectoral inflation in the case27

of a flattened NKPC in each sector. In fact, Walsh (2003) showed that discretion performs better than commitment in an extremely flattened NKPC. This paper confirms
these findings of Walsh (2003) for a two-sector NK model with inflation persistence.
Furthermore, it is interesting that in the case of a larger value of the parameter λs , thecentral bank obtains the gain from commitment increases as the value of parameter θ
increases. More concretely, in the case of θ = 14, the maximum gain is around 5% whenλs = 0.8. This might be in contrast to findings of previous studies that used a one-sector
NK model.
Intuitively, even in the two-sector model, a larger value of parameter θ counteracts
the market power of firms in each sector. In addition, inflation responds strongly to a
shock in sector F, where prices are flexible and price setting is forward-looking. Even if
sector S loses market power in its own sector, core inflation might not jump immediately
in response to the shock due to the presence of severe inflation persistence. Therefore,
the presence of sector S’s inflation persistence drives a wedge between inflation in eachsector. This mechanism can create scope for the gain from commitment to emerge.
Hence, a larger value of parameter θ indicates that a central bank aiming to stabilise
sectoral inflation can contain a large fluctuation in both relative prices and output gap.
Therefore, it is possible that commitment leads to a superior performance compared withdiscretion.

5

Conclusions

It has debated whether the presence of inflation persistence will significantly aﬀect thegain from a commitment policy in an NK model. Some have argued that as long as
inflation persistence is not predominately backward-looking, gains will arise from a commitment policy. Recent studies have also addressed the role of sectoral heterogeneities
of inflation persistence. Although the gain from commitment in the presence of inflation
persistence has been reported for a one-sector economy, the gains from commitment in
a two-sector model remained ambiguous.
This paper suggested that inflation persistence in one-sector aﬀects inflation in an28

other sector through a change in the relative price between two sectoral goods, leading toa large fluctuation in the output gap. Trade-oﬀs following cost-push shocks in a stickyprice sector are particularly severe. However, trade-oﬀs following cost-push shocks ina flexible-price sector are attenuated. In addition, the central bank faces a trade-oﬀ
between CPI inflation and the output gap in a one-sector model, whereas this study
addresses the scenario where sectoral heterogeneities of inflation persistence produces asevere trade-oﬀ not only between inflation and the output gap but also between eachsector’s inflation in a two-sector model. These results contrast the ones obtained for the
one-sector model. This paper, therefore, emphasises that the central bank needs to take
into account these mechanisms of sectoral heterogeneities of inflation persistence when
implementing the optimal monetary policy.
This study focused on the mechanisms through which sectoral inflation persistence
substantially influences the gain from commitment. The main findings of welfare analyses
are as follows. Regardless of degree of inflation persistence in the flexible-price sector,
the gain will arise as long as core inflation is depicted by a forward-looking component.
Therefore, the gain from commitment decreases when core inflation is backward-looking.
Naturally, the gain is small when sectoral heterogeneities of inflation persistence areconsiderably more severe in both sectors. When moderate inflation persistence is present
in both sectors, gains occur unless prices are more flexible in both sectors. However, thegain from commitment vanishes again when prices are stickier in both sectors.
This study focused on the property of a commitment policy from a timeless perspective in a standard NK model with sectoral heterogeneities of inflation persistence.
How should the central bank implement its monetary policy if it cannot commit to theimplementation of its future monetary policy? In this case, a possible solution is thatthe government delegates the loss function with policy inertia, which is diﬀerent from asocial loss function, to the central bank (see Jensen, 2002; Walsh, 2003; Bilbiie, 2014).
It will be interesting for future research to examine whether gains from several targeting
regimes is large. In addition, as Cogley and Sbordone (2008) and Ascari and Sbordone(2014) argue for the importance of trend inflation to explain actual inflation dynamics,

29

we will examine the optimal monetary policy in an economy with sectoral trend inflation.
Particularly building upon the findings of this study, it may be interesting to explore
how inclusion of inflation persistence changes the optimal monetary policy in such an
economy.

References
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Sheedy, K. (2009). “Intrinsic inflation persistence.” Journal of Monetary Economics
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32

A

Appendix: Derivation of the central bank’s loss
function (Not for publication)

This appendix provides a detailed derivation of the central bank’s loss function in a
two-sector model. As shown in studies by Woodford (2001, 2003), the loss function corresponds to the second-order approximation of the household utility function. Derivationof the central bank’s loss function in this model is mainly based on Woodford (2003), but
the approximation of a price dispersion term diﬀers. In a two-sector NK model, inflation
persistence in each sector significantly changes the shape of the central bank’s loss function. Compared to one-sector models, the shape of the loss function is more complicatedin a two-sector model because the proportion of firms that follow rule-of-thumb pricing
depends on sector i.
This appendix derives the loss function approximated around a steady state. Before
doing so, I define the relevant notations. First, Z̄ denotes the value of the steady state,
and Ztn is the value of the eﬃcient level. Second, as in the main text, Ẑt = log(Zt /Z̄) is
the deviation of Zt from the steady state.
As in Woodford (2003), I provide the second-order approximation of the household’sutility function, which is given by:∫
Vt = u(Ct ; ξt ) −∫v(hst (j); ξts )dj −Nsv(hft (j); ξtf )dj,

(A.1)

Nf

The second-order approximation of the first term on the right-hand side is derived
by
)
1
−1
2
−1
ũ(Ct ; ξt ) ≃ uc Ȳ Ŷt + (1 − σ )Ŷt + σ gt Ŷt + t.i.p. + O(||ξ||3 ),
2
(

(A.2)

where t.i.p. represents the terms that are independent of monetary policy, O(||ξ||3 ) indicates the terms of third or higher orders, and ũ = u(Y ; ξ).
Next, I derive the second-order approximation of the household’s utility function. To

33

do so, I introduce some useful equations:∑1nj (1 + η −1 φ̂i,t )Ŷi,t + n1 n2 (1 − η −1 )(Ŷs,t − Ŷf,t )2 + t.i.p. + O(||ξ||3 ), (A.3)
Ŷt =2i∈{s,f }1
Ŷi,t = Eji ŷi,t (j) + (1 − θ̄−1 )varji yi,t (j) + O(||ξ||3 ),
2

(A.4)

for i = s, f .18 Also, varji yi,t (j) denotes the variance of yi,t (j) over j.
The second-order approximation of the second and third terms on the right-hand sideis derived as follows:
{
}
∫
1
i
i
i
i 2
i
ṽ(yi,t (j); ξt )dj ≃ ni Ȳ uc (1 − Φ)Ej ŷi,t (j) + (1 + ω)[Ej ŷi,t (j) + varj yi,t (j)] − ωqi,t ŷi,t (j)2Ni+ t.i.p. + O(||ξ||3 ),(A.5)

for i = s, f . Also, qi,t denotes the firm-specific exogenous disturbance required to maintain a constant marginal disutility of the labour supply and ṽ = v(f −1 (y/A); ξ). I used
the following fact held in the steady state:θ̄ũc (Ȳ ; 0)=
≡ 1 − Φ,ṽy (ȳ; 0)θ̄ − 1
where Φ = 1/θ̄.
Summing the second-order approximation of the disutility of labour supply in eachsector and using Equations (A.3 ) and (A.4 ), the following equation is derived:∫∫ssv(ht (j); ξt )dj +v(hft (j); ξtf )dj
Ns
Nf
{
∑
1
ni (ωqi,t + η −1 φ̂i,t )Ŷi,t≃ uc Ȳ (1 − Φ)Ŷt + (1 + ω)Ŷt2 −2i∈{s,f }
}
∑
1
1 −1
−1
2
i
+ n1 n2 (η + ω)(Ŷs,t − Ŷf,t ) + (θ̄ + ω)
ni varj yi,t (j) + t.i.p. + O(||ξ||3 ),22i∈{s,f }(A.6)
Combining Equations (A.2 ) and (A.6 ) yields
{
}
∑
uc Ȳ
−1
∗ 2
−1
2
−1
i
(σ + ω)(xt − x ) + n1 n2 (η + ω)xr,t + (θ̄ + ω)
ni varj yi,t (j)Vt ≃ −
2i∈{s,f }+ t.i.p. + O(||ξ||3 ),
18

(A.7)

See Woodford (2003) for a detailed explanation of these equations.

34

where xr,t = xs,t − xf,t and xi,t = Ŷi,t − Ŷi,tn , for i = s, f . Also,(σ −1 + ω)Ŷtn = σ −1 gt +∑ni qi,t ,i∈{s,f }nn(η −1 + ω)(Ŷs,t− Ŷf,t) = η −1 (φ̂f,t − φ̂s,t ) + ω(qf,t − qs,t ),x∗ =Φσ −1 + ω.

Using the definition of the relative price gap, Equation (A.7 ) can be rewritten as follows:
{
}
∑
uc Ȳ
−1
∗ 2
n 2
−1
i
Vt ≃ −(σ + ω)(xt − x ) + n1 n2 η(1 + ηω)(p̂r,t − p̂r,t ) + (θ̄ + ω)
ni varj yi,t (j)2i∈{s,f }+ t.i.p. + O(||ξ||3 ),(A.8)

Furthermore, from firm j’s demand in each sector, we can obtain the following equations:varji log yi,t (j) = θ̄2 varji log Pi,t (j),for i = s, f . Then, we now defineP̄i,t = Eji log Pi,t (j), ∆ji,t = varji log Pi,t (j).
Based on the derivation of Steinsson (2003), we obtain the following second-order approximation of the sectoral price dispersion:
∞
∑
t=0

[
∞
∑
αi
λi
(1 − αi )λi δi 2
1
t
2
2
β
πi,t
+
∆πi,t
+
x
1 − αi β t=01 − αi(1 − αi )(1 − λi )(1 − λi ) t−1
]
λi δi
−2
∆πi,t xt−1 + t.i.p. + O(||ξ||3 ),(A.9)
1 − λiβ ∆ji,t ≃tfor i = S, F . Substituting Equation (A.9 ) into Equation (A.8 ) yields Equation (29) inthe main text.

35

Output gap

0.02

Relative price gap

0.01
0

0

-0.01

-0.02

-0.02
-0.04
-0.03

purely forward-looking
λ s = 0.2

-0.06
-0.08

λ s = 0.7

-0.04

λ s = 0.9

-0.05

-0.1

-0.06
1

2

3

4

5

6

7

8

9

10

1

2

3

4

5

time

20

×10-3

6

7

8

9

10

8

9

10

time

Sticy price inflation

Flexible price inflation

0.015
0.01

15

0.005
0

10

-0.0055-0.01-0.015

0

-0.02
-5

-0.025
1

2

3

4

5

6

7

8

9

10

time

1

2

3

4

5

6

7

time

Figure 1: Impulse response of optimal monetary policy to a cost-push shock in sector S

36

Output gap(a) η = 5.0

0.05

0.01

10

0
-0.05

η = 1.5, λ s = 0.2

5

η = 1.5, λ s = 0.7

0η = 5.0, λ s = 0.2-0.1
0(b) η = 0.5

×10-3 Sticy price inflation

15

2

4

6

-5

η 8= 5.0, λ10= 0.7sFlexible price inflation

0

0

-0.01

-0.02

-0.02
0

2

4

6

8

10

-0.04
0

2

4

6

8

10

0

0.05

0.02

0.01

0.02

0

0.01

0

0

0

-0.01

-0.02

η = 1.5, λ s = 0.2-0.05

η = 1.5, λ s = 0.7

η = 0.5, λ s = 0.2 -0.01

-0.1
0

2

4

6

η 8= 0.5, λ10
= 0.7
s

-0.02
0

2

4

6

8

10

Relative price gap

0.02

2

4

6

8

10

2

4

6

8

10

2

4

6

8

10

2

4

6

8

10

-0.04
0

2

4

6

8

10

0

-3

(c) α F = 0.25

0.05

15

×10

0.02

10

0

-0.02
-0.02

α F = 0.55,λ s = 0.7 0
α F = 0.25,λ s = 0.2

-0.1
0

2

4

6

0

0

α F = 0.55,λ s = 0.2 5

-0.05

0.02

-5

-0.04

-0.04

α F8= 0.25,10λ s = 0.7 0

2

4

6

8

10

-0.06
0

2

4

6

8

10

0

-3

(d) λf = 0.5

0.05

15

×10

10

0

λ f = 0.01,λ s = 0.2 5-0.05

λ f = 0.01,λ s = 0.7 0
λ f = 0.5, λ s = 0.2

-0.1
0

2

4

6

λ f8= 0.5, λ10s = 0.7

time

-5

0.01

0.02

0

0

-0.01

-0.02

-0.02
0

2

4

6

8

time

10

-0.04
0

2

4

6

time

8

10

0

time

Figure 2: Impulse response of optimal monetary policy to a cost-push shock in sector S:
Sensitivity experiments

37

-3

2

-3

Output gap

×10

2.5

Relative price gap

×10

0
2
-2
-4

1.5

-6

purely forward-looking
λ s = 0.2-81

λ s = 0.7

-10

0.5

λ s = 0.9

-12
0
-14
-16

-0.5
1

2

3

4

5

6

7

8

9

10

1

2

3

4

5

time

6

×10-4

6

7

8

9

10

8

9

10

time

Sticy price inflation

2

4

×10-3

Flexible price inflation

1.5

2
1

0
-2

0.5

-4

0

-6
-0.5

-8
-10

-1
1

2

3

4

5

6

7

8

9

10

time

1

2

3

4

5

6

7

time

Figure 3: Impulse response of optimal monetary policy to a cost-push shock in sector F

38

Figure 4: Impulse response of optimal monetary policy to a cost-push shock in sector F:
Sensitivity experiments

39

4

×10-3

Output gap

5

×10-3

Relative price gap

2
0

0

-2
-4

purely forward-looking
λ s = 0.2-6

λ s = 0.7-8

λ s = 0.9

-5

-10

-10
-12

-15
1

2

3

4

5

6

7

8

9

10

1

2

3

4

5

time

0.5

×10-3

6

7

8

9

10

8

9

10

time

Sticy price inflation

6

0

5

-0.5

4

-1

3

-1.5

2

-2

1

-2.5

0

-3

×10-3

Flexible price inflation

-1
1

2

3

4

5

6

7

8

9

10

time

1

2

3

4

5

6

7

time

Figure 5: Impulse response of optimal monetary policy to a relative price shock40

Figure 6: Impulse response of optimal monetary policy to a relative price shock: Sensitivity experiments

41

Figure 7: Welfare gain when λs and λf change

42Figure 8: Welfare gain when αs and αf change in the case of λs = λf = 0.5

43Figure 9: Welfare gain when αs and αf change in the case of λs = λf = 0.99

44

Figure 10: Welfare gain when λs and αs change

45

Figure 11: Welfare gain when λs and η change

46

Figure 12: Welfare gain when λs and θ change

47